Quantum simulation and metrology with multilevelfermions in an optical latticebyMichael A. Perlin
B.S., Oregon State University, 2015
M.S., University of Colorado at Boulder, 2019

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2021
Committee Members:
Ana Maria Rey, Chair
Rahul Nandkishore
Jun YeJames Thompson
Sandeep Sharma

iiPerlin, Michael A. (Ph.D., Physics)
Quantum simulation and metrology with multilevel fermions in an optical lattice
Thesis directed by Prof. Ana Maria Rey

Ultracold atomic systems are exquisite platforms for studying many-body physics and developing quantum technologies. Alkaline-earth(-like) atoms (AEA) in particular offer unique capabilities for pushing the state-of-the-art in quantum simulation and sensing capabilities. These atoms
have an ultranarrow electronic transition that is used as the basis for the world’s best atomic clocks.
Moreover, AEAs can have a rich internal structure owing to nuclear spin degrees of freedom thatare largely untapped as a scientific, metrological, and computational resource. In this thesis, we
explore some of the possibilities and prospects for exploring many-body quantum phenomena andadvancing sensing capabilities with ultracold AEAs.
We begin on the simulation end with a deep dive into the emergence of multi-body interactions between ultracold AEAs. We then present a proposal to harness collisional interactions andinhomogeneities in an AEA-based clock for the preparation of many-body entangled states known
as spin-squeezed states, which allow for a quantum enhancement to clock sensitivity. In order toanalyze this proposal’s prospects and limitations, we develop a numerical technique for simulatingcollective quantum spin systems, which may find external applications for studying operator growthand quantum chaos. Borrowing ideas from the proposal to improve AEA-based clocks, we examine the possibility of spin squeezing using power-law interactions that can be found in a variety
of atomic, molecular, and optical systems. Combining the spirit of our investigation into exotic
interactions with the roadmap of our proposal to improve AEA-based clocks, we then propose away to engineer a multilevel spin model with infinite-range interactions in the nuclear spin degrees
of freedom of AEAs. We study the dynamical phases of this system, characterized by order parameters with a simple physical interpretation, and propose ways to measure these order parameters
using standard techniques.

Dedication

To my friend, partner, and wife, Eva.

Acknowledgements

Graduate school has been an exciting journey. There is no real hope of enumerating thecountless family, friends, colleagues, peers, and mentors that carried me through this journey, buthere I make a partial attempt nonetheless. I would first and foremost like to thank my wife, Eva
Morgun, who has been an invaluable friend and indispensable partner throughout these years. I
would like to thank Eva’s father, Andrey Morgun, for playing the role of both father and mentor
to me after the death of my own father (Alexander Perlin) in 2013. I am also blessed to havethe emotional and psychological support of my mother Natalie Perlin, step-mother Anna Kelbert,
siblings Daniel Perlin, Sophie Perlin, and Boris Perlin, mother-in-law Natalia Shulzhenko, andbrother-in-law Nikolay Morgun.
No less important than a family, particularly for a young adult finding their own path throughlife, is a close network of friends. I would like to thank some my dear friends predating graduate
school: John Shin, Paho Lurie-Gregg, Novela Auprey, Aaron Watt, Austin Valeske, MacKenzie
Lenz, Rodney Snyder, Cameron Oden, Jackie Liu, Rachel Kim, Theresa Russell, Andrew C. Drake,
and Youngjoo Lee. The various happenings of our personal lives, our careers, and a global pandemic
have brought us close or carried us apart at different times, but these wonderful people have hada tremendous positive impact on my personal development and well-being throughout the years,
and I hope to stay in touch with them for the rest of my life.
The last five years brought more people into my life, who have likewise provided me support
and camaraderie throughout our postgraduate careers. From my graduate program in Boulder,these include: “Laser Dan” I. Herman, Eddie G. Nerney, Josh Levin, and Curtis Rau; and from thev
Chicago area: Sam Genardi, Mike Brew, Shivani Baisiwala, Wes Peng, Leonie Schroeder, Ruslan
Shaydulin, and Claire Textor. I have little doubt that these new friends will continue to feature
regularly in my life after graduate school.
Finally, I would like to thank my scientific and career mentors, most notably Ana Maria Rey,
Graeme Smith, and David Roundy, as well as Robert J. Lewis-Swan, Arghavan Safavi-Naini, Martin
Suchara, and James C. Osborn; my colleagues Aki Goban, Peiru He, Chunlei Qu, Diego Barberena,
Mikhail Mamaev, Sean R. Muleady, Asier P. Orioli, Thomas Bilitewski, Bhuvanesh Sundar, Zain
H. Saleem, and Teague Tomesh; and my graduate (qualifying and doctoral) committee members
Rahul Nandkishore, Jun Ye, James K. Thompson, Sandeep Sharma, Oliver DeWolfe, Matt Glaser,and Leo Radzihovsky. A special thanks to the extraordinary assistants and administrators Jeanne
M. Nijhowne, Gwen Dickinson, and Corey Keasling, who are all remarkably helpful, efficient, anda pleasure to work with.
There is much that I wish to say about each person listed above (and more people that arenot!), as well as the role that they played in my life through graduate school. But alas, I am writinga doctoral thesis, not a memoir. Suffice to say that each name carries with it a lengthy story anda wealth of gratitude. Going forward in life, I would be delighted to surround myself with people
even half as wonderful and supportive as those that I have encountered thus far. Onward!

Contents

Chapter
1 Context and outline

1

2 Introduction and background3

2.1

Alkaline-earth(-like) atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

2.2

Individual atoms in a trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.3

Optical lattices and magic wavelengths . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.4

Spin-orbit coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.5

Nuclear spin decoupling and SU(n) symmetry . . . . . . . . . . . . . . . . . . . . . . 13

2.6

Two-body interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.7

Spin operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.8

Collective spins and the Dicke manifold . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.9

Spin squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Effective multi-body SU(n)-symmetric interactions of ultracold fermionic atoms on a 3D
lattice23

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2

Background and overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3

Low-energy effective theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3.1

Diagrammatic representation of effective Hamiltonians . . . . . . . . . . . . . 36

3.3.2

Effective two-body interactions and renormalization . . . . . . . . . . . . . . 38

vii

3.4

3.5

3.3.3

Effective three-body interactions at second order . . . . . . . . . . . . . . . . 40

3.3.4

Effective three-body interactions at third order . . . . . . . . . . . . . . . . . 41

3.3.5

Effective four-body interactions at third order . . . . . . . . . . . . . . . . . . 45

Low-excitation Hamiltonians, eigenstates, and spectra . . . . . . . . . . . . . . . . . 46
3.4.1

Many-body state spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.4.2

Experimental signatures and comparison . . . . . . . . . . . . . . . . . . . . . 50

3.4.3

Orbital-state dynamics of a nuclear spin mixture . . . . . . . . . . . . . . . . 56

Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.A Derivation of the effective Hamiltonian expansion . . . . . . . . . . . . . . . . . . . . 61
3.B Diagram counting and symmetry factors . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.C Effective coupling constants in a lattice . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.D Momentum-dependent s-wave interactions . . . . . . . . . . . . . . . . . . . . . . . . 68
3.E Bounds on theoretical uncertainties from inter-site effects . . . . . . . . . . . . . . . 70
3.F Perturbative parameters for the effective theory . . . . . . . . . . . . . . . . . . . . . 72
3.G Low-excitation M -body Hamiltonian coefficients . . . . . . . . . . . . . . . . . . . . 73
4 Engineering spin squeezing in a 3D optical lattice with interacting spin-orbit-coupled76

fermions
4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.2

Spin squeezing with the Fermi-Hubbard model . . . . . . . . . . . . . . . . . . . . . 80
4.2.1

Model validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.2.2

Two-axis twisting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.3

Experimental implementation and practical considerations . . . . . . . . . . . . . . . 87

4.4

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.A Derivation of the effective one-axis-twisting model . . . . . . . . . . . . . . . . . . . 92
4.B Numerical benchmarking of the OAT model . . . . . . . . . . . . . . . . . . . . . . . 95
4.C Two-axis twisting, decoherence, and the residual axial field. . . . . . . . . . . . . . 98

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4.D Dynamical decoupling in the TAT protocol . . . . . . . . . . . . . . . . . . . . . . . 100
4.E Clock interrogation after squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.F Decoherence in the 3D 87 Sr optical lattice clock . . . . . . . . . . . . . . . . . . . . . 106
4.G Time-series of squeezing via OAT and TAT . . . . . . . . . . . . . . . . . . . . . . . 106
4.H Solving Heisenberg equations of motion for collective spin systems . . . . . . . . . . 109
4.I

Effect of a harmonic confining trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.J

Accounting for p-wave inelastic collisions . . . . . . . . . . . . . . . . . . . . . . . . . 114

5 Short-time expansion of Heisenberg operators in open collective quantum spin systems

118

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.2

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.3

Spin squeezing, benchmarking, and breakdown . . . . . . . . . . . . . . . . . . . . . 126

5.4

Two-time correlation functions and out-of-time-ordered correlators . . . . . . . . . . 131

5.5

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.A Basic spin operator identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.B Commutation relations between powers of transverse spin operators . . . . . . . . . 136
5.C Product of arbitrary ordered collective spin operators. . . . . . . . . . . . . . . . . 141

5.D Sandwich identities for single-spin decoherence calculations . . . . . . . . . . . . . . 141
5.E Uncorrelated, permutationally-symmetric single-spin decoherence . . . . . . . . . . . 143
5.E.1 Decay-type decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.E.2 Dephasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.E.3 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.F Sandwich identities for collective-spin decoherence calculations . . . . . . . . . . . . 146
5.G Collective spin decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.G.1 Decay-type decoherence and dephasing . . . . . . . . . . . . . . . . . . . . . . 147
5.G.2 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.H Emulating particle loss in a spin model . . . . . . . . . . . . . . . . . . . . . . . . . . 149

ix
5.I

Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5.J

Computing correlators with the truncated short-time (TST) expansion . . . . . . . . 153

5.K Analytical results for the one-axis twisting model . . . . . . . . . . . . . . . . . . . . 155
5.L Diagnosing breakdown of the TST expansion . . . . . . . . . . . . . . . . . . . . . . 156
5.M Spin squeezing with strong decoherence . . . . . . . . . . . . . . . . . . . . . . . . . 158
5.N Heisenberg operators in open quantum systems . . . . . . . . . . . . . . . . . . . . . 160
5.O Short-time linear response and two-time correlators . . . . . . . . . . . . . . . . . . . 163
6 Spin squeezing with short-range spin-exchange interactions

164

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

6.2

Background and theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

6.3

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

6.4

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

6.5

Experimental applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

6.A Spectral gap of the long-range XXX model. . . . . . . . . . . . . . . . . . . . . . . 176

6.B Numerical results in one spatial dimension . . . . . . . . . . . . . . . . . . . . . . . . 177
6.C Benchmarking DTWA for the power-law XXZ model . . . . . . . . . . . . . . . . . . 179
6.D Scaling relations for the collective phase in D = 2 spatial dimensions . . . . . . . . . 183
6.E Thermalization and long-range order . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
6.F Sub-unit filling fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
7 Engineering infinite-range SU(n) interactions with spin-orbit-coupled fermions in an opticallattice192

7.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

7.2

From lattice fermions to an SU(n) spin model . . . . . . . . . . . . . . . . . . . . . . 195

7.3

External control fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

7.4

Spin-orbit coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

7.5

Mean-field theory and dynamical phases . . . . . . . . . . . . . . . . . . . . . . . . . 207

x

7.6

7.5.1

Initial spin-polarized state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

7.5.2

Initial kitten states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

Conclusions and future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

7.A Numerical benchmarking of the spin model . . . . . . . . . . . . . . . . . . . . . . . 218
7.B Perturbation theory for SU(n) ferromagnets . . . . . . . . . . . . . . . . . . . . . . . 219
7.B.1 Generating excitation energy eigenstates . . . . . . . . . . . . . . . . . . . . . 222
7.B.2 Recovering spin-wave theory. . . . . . . . . . . . . . . . . . . . . . . . . . . 224

7.C Restricting spin operators to the permutationally symmetric manifold . . . . . . . . 225
7.D Relaxing assumptions of the three-laser drive . . . . . . . . . . . . . . . . . . . . . . 226
7.E Mean-field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
7.F Schwinger boson equations of motion for quadratic spin Hamiltonians . . . . . . . . 228
7.G Lax vector analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
8 Spin qudit tomography and state reconstruction error

232

8.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

8.2

Polarization operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

8.3

Spin tomography as signal recovery on the sphere . . . . . . . . . . . . . . . . . . . . 238

8.4

State reconstruction error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

8.5

Tomography protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

8.A Rotating polarization operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
8.B An improved reconstruction error bound . . . . . . . . . . . . . . . . . . . . . . . . . 252
8.B.1 Errors in the spin-projection basis . . . . . . . . . . . . . . . . . . . . . . . . 253
8.B.2 Errors in the polarization operator basis . . . . . . . . . . . . . . . . . . . . . 254
8.B.3 Revisiting the reconstruction error bound . . . . . . . . . . . . . . . . . . . . 256
8.C Exact reconstruction error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
8.C.1 Comments on a tight reconstruction error bound . . . . . . . . . . . . . . . . 260
8.D Polarization operator product expansion . . . . . . . . . . . . . . . . . . . . . . . . . 261

xi
8.E Optimizing the method of Newton and Young . . . . . . . . . . . . . . . . . . . . . . 262
9 Summary and conclusions

264

References266

TablesTable
3.1

Low-excitation eigenvalues of M -body Hamiltonians HM . Many-body energy eigenstates are labeled by the nuclear spins they occupy (i.e. N with N ≡ |N |) andwhether they are in an orbital ground (0), singly-excited symmetric (+), or singly(M )excited asymmetric (−) state. The corresponding N -body eigenvalues EN X of HM
are given in terms of the coefficients UM X as appearing in (3.47) (first three rows),(M )in addition to the M -body eigenvalues EM X (last three rows). . . . . . . . . . . . . . 48
6.1

2 ∼ 1/N ν ) for the values of J /J and α shown in Figure
Scaling exponents ν (with ξoptz⊥

6.9, in D = 2 spatial dimensions. Though provided here for the sake of practicalinterest and transparency (these are essentially the slopes of the dashed lines Figure6.9), we note that these values are subject to correction in future work, as ruling out
effects such as finite sampling errors may require a more extensive numerical analysis. 186
7.1

Drive Hamiltonians (left column) that can be implemented with different amplitudematching conditions (right three columns), some of which are specified by an arbitrary sign σ ∈ {+1, −1}. The drives shown here are equal to that of Eq. (7.12) up toa possible energy shift of ŝ2x + ŝ2y + ŝ2z = s(s + 1), and come in mutually commuting
pairs: a drive with |Ωm | = 1 and Ωn = 0 for both n 6= m commutes with the drivein which Ωm = 0 and both |Ωn | = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

Figures

Figure2.1

Simplified energy level diagram for the low-lying electronic states of 87 Sr, reproduced from Ref. [14]. Dashed and dotted lines respectively show spin-orbit (LS) andhyperfine (HFI) coupling between bare electronic states. Solid lines show opticaltransitions between electronic eigenstates, with the corresponding wavelengths given
in nanometers. Fractions indicate the net spin of the hyperfine manifold (involving
nuclear spin) within the associated electronic energy level.

2.2. . . . . . . . . . . . . .

5Band structure (orbital energy) of atoms in a periodic lattice potential Vext (z) =
V0 cos2 πzwith depth V0 = 4ER , where the lattice recoil energy ER = kL2 /2m isadefined in terms of the atomic mass m and the wavenumber kL of the lattice light.
Color indicates band index n as specified in the legend. . . . . . . . . . . . . . . . .

2.3

9Band structure identical to that in Figure 2.2, but after relabeling excited-state quasimomenta with ĉ†qz +kz ,e → ĉ†qz ,e and kz = 2π/3. Solid and dashed lines respectively
show kinetic energies for ground- and excited-state atoms.

2.4. . . . . . . . . . . . . . 13

Husimi distributions for N = 40 spins in a spin-polarized state (left), as well as
the Dicke states |−10i (middle) and |0i (right). A darker color at a point v on thesphere indicates a larger overlap with the state |vi maximally polarized along v. For
visibility, the color scale is normalized independently on each sphere.

. . . . . . . . 20

xiv
2.5

Representations of the state |ψ(t)i of N = 40 spins initially polarized along theequator, and subsequently evolved under the OAT Hamiltonian ĤOAT for a time
−2/3 . Figure reproduced fromt up to the optimal OAT squeezing time tOAT
opt ∼ N

Ref. [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1

An example second-order diagram and the corresponding three-body interaction(2)

term in Hint , with n > 0 and ĉµs ≡ ĉ0,µs . Diagrams are read from left to rightto construct a sequence of operators from right to left. Solid (dashed) lines represent field operators acting on the lowest (arbitrary) motional states. For the sake ofpresentation, this diagram has colors associated with nuclear spin and orbital states,
an arrow on each line to emphasize that they are directed left-to-right, and an explicit coupling constant written next to each vertex; we will generally not include
these features, as they are not necessary to uniquely identify the term represented
by a diagram. We will also drop explicit appearances of the ground-state projector
P0 in our expressions, with the understanding that the low-energy effective theory

3.2

implicitly addresses only non-interacting motional ground states. . . . . . . . . . . . 37√
Multiplicities of the magnitudes of reduced Rabi frequencies ωN X ≡ ΩN X /Ω0 N
in a lattice with a uniform mixture of nuclear spins with I = 9/2 and single-site
occupation numbers N which are achievable in current 87 Sr experiments. . . . . . . . 51

xv
3.3

Population (in arbitrary units) of the excited 3 P0 orbital state of 87 Sr atoms in a
uniform mixture of nuclear spins. Atoms are prepared in the ground state of a latticewith depth U = 54ER , where ER ≈ 3.5 × 2π kHz is the lattice photon recoil energyof the atoms, and interrogated by a laser with Rabi frequency ΩI = 50 × 2π Hz for atime t = π/ΩI . (Left) Populations predicted by the low-energy effective theory (withs-wave scattering parameters retrieved from Ref. [53]), averaged over all nuclear spincombinations of N ∈ {1, · · · , 5} atoms per lattice site for a fixed total atom number.
(Right) Experimental measurements of 3 P0 populations retrieved from Ref. [7], withLorentzian fits to each peak as a visual guide. Resonance peaks are identified by the
many-body orbital states which are excited at the peak. . . . . . . . . . . . . . . . . 52

3.4

Multi-body excitation energies of ultracold 87 Sr atoms at various lattice depths. The
top plot in each sub-figure with fixed N, X shows the excitation energies ∆N X ≡EN X − EN,0 measured experimentally in Ref. [7] and those predicted by the lowenergy effective theory at different orders in the coupling constants, when applicable
both with and without four-body contributions. The bottom plot in each sub-figure
experimentshows the relative error ηN X ≡ ∆theory− 1. Error bars represent exN X /∆N X

perimental error or conservatively estimated theoretical uncertainties from nearestneighbor hopping of virtual states in the low-energy effective theory (see Appendix
3.E). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.5
3.6

Summary of the many-body excitation spectra in Figure 3.4, retrieved from Ref. [7].
Net population of the N -body orbital excited states |N ±i after interrogation of an√
initial mixed state ρN,0 = PN,0 / tr PN,0 for a reduced time τN ≡ tΩ0 / N (i.e. with55

real time t) by a laser with bare Rabi frequency Ω0 which is resonant with the N body excitation energy ∆N X . Here PN X , defined in (3.60), is a projector onto thespace of the N -body orbital states |N Xi. . . . . . . . . . . . . . . . . . . . . . . . . 58
3.7

Dependence of the perturbative parameters KGX /∆ on the lattice depth U. . . . . . 73

xvi
4.1

Schematic of the setup for spin squeezing. (a) We consider N fermionic atoms
with two (pseudo-)spin components, represented by red and blue spheres, trappedin the ground band of an optical lattice (shown in 2D for the sake of presentation).
Atoms tunnel to neighboring sites at a rate J and experience on-site interactions withstrength U . An external laser carrying a position dependent phase eikL ·r couples thespin states of the atoms. (b) After a gauge transformation, different spin states
exhibit different dispersion relations with a relative phase φ = kL a, where a is thelattice spacing. The external laser couples spin states with identical quasi-momenta qin the gauge-transformed frame. (c) If interactions are sufficiently weak, all motional
degrees of freedom become frozen in momentum space, with atoms effectively pinned
to fixed quasi-momentum modes q. The dynamics on the frozen q-space lattice canthen be mapped to a spin model in which collisional interactions correspond to a
uniform, all-to-all ferromagnetic Heisenberg Hamiltonian with strength U/L, where
L is the total number of lattice sites. (d) The spin dependence of the dispersion
relation is captured by a mode-dependent axial field Bq that generates inhomogeneous spin precession. This axial field couples exchange-symmetric many-body Dicke
states with total spin S = N/2 to spin-wave states with S = N/2 − 1. The all-to-allinteraction opens an energy gap f U (with f = N/L the filling fraction of spatial
modes) between the Dicke states and the spin-wave states, which forbids populationtransfer between them in the weak-field limit. (e) To generate spin squeezing via
one-axis twisting, we initialize a product state with all spins polarized in −z (i.e. in|↓i), and apply a fast external laser pulse to rotate all spins into x. We then let atoms
freely evolve for a variable time t (with a spin-echo pulse), after which the amountof spin squeezing can be determined experimentally from global spin measurements.
The spin-squeezed state can be used for a follow-up clock interrogation protocol (seeAppendix 4.E). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

xvii
4.2

Benchmarking the spin and one-axis twisting models. Comparisons of maximum squeezing (top panels, a.i and b.i) and optimal squeezing time (lower panels,
a.ii and b.ii) between the Fermi-Hubbard (FH), spin, and one-axis twisting (OAT)
models; obtained numerically via the protocol depicted in Figure 4.1(e) in a 1D lattice with L = 12 sites. Results are shown for half filling with N = 12, f ≡ N/L = 1(left panels, a.i and a.ii) and filling f = 5/6 (right panels, b.i and b.ii) as a functionof U/J and the SOC angle φ. In both cases, the system is initialized in the corresponding ground state. Insets for both f = 1 and f = 5/6 show (in green) regions of
the U -φ plane in which both the optimal squeezing (in dB) and the corresponding
squeezing time of all three models agree to within 20%. At half filling (a.i and a.ii),
mode-changing collisions are suppressed by Pauli blocking, resulting in almost exact
agreement between the FH and spin models; both of these models converge onto the
OAT model in the gap-protected, weak SOC regime of large U/J and small φ. The
spin and OAT models show similar behavior away from half filling (b.i and b.ii),
but the presence of mode-changing collisions results in their disagreement with theFH model as interactions begin to dominate at larger U/J. Even below half filling,
however, the FH exhibits comparable amounts of squeezing to the spin model acrossa broad range of U/J and φ, albeit at earlier times when U/J & 2.

. . . . . . . . . 85

xviii
4.3

Optimal squeezing with one- and two-axis twisting in a 2D section of the 3D 87 Sr
optical lattice clock. (a) The maximum amount of squeezing depends only on theatom number N = `2 , where ` is the number of lattice sites along each axis of thelattice. While the time scales for squeezing generally depend on several experimental
parameters, the time at which maximal squeezing occurs can be minimized at anyegiven lattice depth V0 by choosing SOC angles φ that saturate B/U≈ 0.05, where
e is the variance of the SOC-induced axial field and U is the two-atom on-siteBinteraction energy. Panels (b, c) show these minimal squeezing times as a functionof the depth V0 and linear size ` of the lattice. Lattice depths V0 are normalized
to the atomic lattice recoil energy ER , and the upper axis on panels (b, c) marks
values of U/J at fixed lattice depths. In general, TAT achieves more squeezing thanOAT for any system size, and achieves optimal squeezing faster for N & 400 atoms,
as denoted by a dotted line in panels (b, c).

. . . . . . . . . . . . . . . . . . . . . . 89

xix
4.4

Optimal squeezing with decoherence via one- and two-axis twisting in a 2D
section of the 3D 87 Sr optical lattice clock (OLC). In practice, decoherence due tolight scattering limits the amount of squeezing that is attainable in the the 3D 87 Sr
OLC. Due to growing squeezing times with increasing system size, the maximal
squeezing obtainable via OAT saturates past ` ≈ 30 sites along each axis of thelattice, with N ≈ 103 atoms total. The more favorable size-dependence of TATtime scales, however, allow for continued squeezing gains through ` = 100 (N =104 ). While the OAT results in (a) are exact, the TAT results in (b) reflect only
a lower bound on the maximum squeezing obtainable, albeit one that is likely close(within a few dB) to the actual value. Optimal squeezing times in the presence of
decoherence are generally smaller than the corresponding times shown in Figure 4.3,
as decoherence typically degrades squeezing before it reaches the decoherence-freemaximum. The decoherence considered in this work also limits maximally achievablesqueezing to ∼ 20 dB less than the decoherence-free maxima shown in Figure 4.3.
Sample plots of squeezing over time for particular choices of lattice size (`) and depth(V0 /ER ) are provided in Appendix 4.G. . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.5

Relative error between maximal squeezing (measured in dB) obtained by the OAT[Eqn. (4.4)] and spin [Eqn. (4.3)] models of the main text in a system of 20 particles.
The OAT model correctly captures the maximal squeezing (in dB) of the spin model
to within 3% (marked by the horizontal reference line) within the gap-protectederegime B/U< 0.06. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

xx
4.6

Comparison between the OAT and the spin model in the presence of
decoherence. (a) The difference between the maximal squeezing (measured indB) obtained by the OAT [Eqn. (4.4)] and spin [Eqn. (4.3)] models increases withthe particle number N and the single-particle spontaneous emission rate γ. Thisdisagreement is attributed in part to the fact that spontaneous emission transfers
population of the collective spin state outside of the Dicke manifold, violating anassumption of the OAT model; see panel (b). The rate of population transfer outsideof the Dicke manifold increases with both particle number and spontaneous emission
rate. (Parameters for simulations in this figure: U = 1000 Hz, J = 200 Hz, andφ = π/20).

4.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Optimal squeezing as a function of π-pulses applied prior to the optimal TAT
squeezing time in a CPMG sequence with (a) N = 100 and (b) N = 1000 atoms.
Results are shown for OAT, TAT, and TAT±,z , where TAT±,z denotes squeezing via
D Erms(±,z)(±)
the Hamiltonian ĤTAT ≡ ĤTAT − J0 (β± ) B
Ŝz . Details about experimentalfparameters for these simulations are provided in the text. . . . . . . . . . . . . . . . 101
4.8

Squeezing via OAT and TAT in a 2D section of the 3D 87 Sr optical lattice clock,shown for (a) ` = 40 and (b) ` = 100 sites per axis (with N = `2 atoms total), anda lattice depth of V0 = 4 ER , where ER is the atomic lattice recoil energy. Atoms
are confined along the direction transverse to the 2D layer by a lattice of depth 60
ER . Squeezing over time is shown for OAT (blue) and TAT (green), both with (solidlines) and without (dashed lines) decoherence via uncorrelated decay and dephasingof individual spins at rates of 0.1 sec−1 (see Appendix 4.F). . . . . . . . . . . . . . . 108

xxi
4.9

Dynamics of non-interacting spin-orbit coupled fermions in a 1D lattice with SOC
angle φ = π/50, plus a harmonic trap with Ω/J = 0.01. Starting with a spinpolarized cloud in ↓ ground state, an initial clock laser pulse is applied to rotate
spins into x, and the atoms are allowed to evolve during the dark time. We trackthe dynamics of the ↑ particle density for the cases of (a) N = 20 and (b) N = 60
atoms. Panel (c) shows the time-averaged fluctuations of the ↑ particle density foreach site index j from its initial value following the Ramsey pulse; see Eqn. (4.64).
For N = 60, we have filled all delocalized modes as well as several localized modes,
resulting in a large region of no density fluctuations at the trap center. Panel (d)contains the eigenspectrum for a single internal state in the presence of the trap(with the index n labeling the eigenvalues in order of increasing energy), where thecritical mode nc dividing the spatially delocalized and localized modes is indicatedby a black dash-dotted line. The highest occupied mode in the ↓ ground state for
N = 20 and N = 60 is indicated by the green and red solid lines, respectively.

. . . 110xxii
4.10 Dynamics of interacting spin-orbit coupled fermions in a 1D lattice plus a harmonic
trap for U/J = 1 (a), 2 (b), and 4 (c). For a 1D lattice with 10 sites and an SOC
angle φ = π/50, we apply a π/2 clock laser pulse to the ↓ ground state and let thesystem evolve during the dark time. In (a.i)-(c.i) we show the squeezing dynamics
of the system for both N = 10 (solid lines) and N = 9 (dashed lines) for a varietyof trapping strengths. In (a.ii)-(c.ii), we plot the time-averaged fluctuations in totalP
particle density, δnj (as in Eqn. (4.64) but with n̂j,↑ replaced by α n̂j,α ). In (a.iii)(c.iii), we plot the growth of the doublon population Nd (t) (see Eqn. (4.65)) as afunction of time, noting the absence of squeezing in the presence of a large doublon
population. For the chosen trap strengths, the corresponding values of nc are 28(Ω/J = 0.01), 14 (Ω/J = 0.04), and 6 (Ω/J = 0.2). In panels where the results forthe homogeneous case (orange curves) are not visible, they are nearly identical to theresults for Ω/J = 0.01 (green curves). Here, we utilize periodic boundary conditions
to minimize finite size effects.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.11 p-wave loss rates. Both the averaged p-wave inelastic collision rate γ (orange)
and the ratio of this collision rate to the optimal squeezing rate χopt (blue) aresuppressed as the lattice depth increases. χopt is obtained by choosing SOC anglesee is the variance of the SOC-induced axial field
φ that saturate B/U≈ 0.05, where B
and U is the two-atom on-site interaction energy.

. . . . . . . . . . . . . . . . . . . 115

4.12 Squeezing via OAT in the presence of inelastic collisions. (a) For fixed
particle number N = 100, the optimal squeezing decreases as the inelastic collision
rate increases. Panel (b) shows squeezing over time for γ/χopt = 0.04 (solid lines),
which corresponds to U/J = 6, and compares it with γ = 0 (dashed lines) for different
particle numbers. Inelastic collisions prevent the growth of optimal squeezing withparticle number. For N = 1000, the maximum squeezing saturates to ∼ 10 dB. . . . 116

xxiii
5.1

Spin squeezing of N = 104 spins initially in |Xi under (a) unitary and (b) nonunitary dynamics, computed using exact methods (solid lines), quantum trajectory
simulations (dots), and the TST expansion in Eq. (5.8) with M = 35 (dashed lines).
Solid circles mark the times at which the TST expansion gives an unphysical resultwith ξ 2 < 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.2

Spin squeezing of N = 104 spins initially in |Xi with spontaneous decay, excitation,and dephasing of individual spins at rates γ− = γ+ = γz = 100χ. Computed using
the TST expansion in Eq. (5.8) with M = 35. Solid circles mark the times at which
the TST expansion gives an unphysical result with ξ 2 < 0. . . . . . . . . . . . . . . . 130

5.3

The two-time connected correlator C (t) and squared commutator D (t), respectively
defined in Eqs. (5.13) and (5.14), for N = 104 spins initially in the polarized state⊗N|Xi ∝ |↑i + |↓ievolving under the squeezing Hamiltonians in Eqs. (5.9)–(5.11).
Results are shown for both unitary dynamics (solid lines) and non-unitary dynamics
with γ− = γ+ = γz = γ0 = 100χ (dashed lines), computed using the TST expansion
in Eq. (5.8) with M = 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.4

Spin squeezing throughout decoherence-free evolution of N = 100 spins initiallyin the state |Xi, computed using both exact methods (solid lines) and the TST
expansion in Eq. (5.8) with M = 35 (dashed lines). Solid circles mark the times atwhich the TST expansion gives an unphysical result with ξ 2 < 0. . . . . . . . . . . . 157

5.5

A collective spin correlator in the TAT model with N = 100 spins and no decoherence, computed using the TST expansion with M = 35. The divergence of correlatorsof this sort can be used to diagnose the breakdown of the TST expansion. . . . . . . 157

5.6

Optimal spin squeezing of N = 100 spins undergoing spontaneous decay, excitation,and dephasing at rates γ− = γ+ = γz = γ0 , computed using quantum trajectory
simulations with 103 trajectories per data point. . . . . . . . . . . . . . . . . . . . . 159

xxiv
6.1

Representations of the state ψ(t) of N = 40 spins initially polarized along theequator, and evolved under the OAT Hamiltonian for a time t up to the optimal
2/3 . Darker colors at a point n̂ on the sphere
OAT squeezing time χtOAT
opt ∼ 1/N

correspond to a larger overlap Qψ(t) (n̂) ≡ |hn̂|ψ(t)i|2 , where |n̂i is a state in which
all spins are polarized along n̂. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.2

2 (top), minimal squared magnetization hS 2 i
The optimal squeezing ξopt
min (middle),

and optimal squeezing time topt (bottom) for N = 4096 = 642 = 163 spins in D = 2(left) and D = 3 (right) spatial dimensions. Spins are initially polarized along the2equator and evolved under the XXZ Hamiltonian in Eq. (6.3). Squeezing ξoptis
shown in decibels, and hS 2 imin is normalized to its initial value hS 2 i0 = N2 N2 + 1 .

Dashed grey lines mark α = D, and dotted grey lines track local minima of hS 2 imin ,marking the boundary between regions of collective and Ising-limited squeezing dynamics, respectively denoted “S-collective” and “S-Ising”. Other markers in the
middle panels indicate vales of Jz /J⊥ , α, D that are currently accessible with neutral atoms [211, 212] (cyan line), Rydberg atoms [201, 202, 213] (red dots), polar
molecules [203, 204, 214] (green line), magnetic atoms [205, 206] (pink square), andtrapped ions [191] (blue line). DTWA results are averaged over 500 trajectories.
6.3. . 170

Squeezing ξ 2 and squared magnetization hS 2 i over time for the power-law XXZ modelwith α = 3 on a 2D lattice of 64 × 64 spins. Color indicates the value of Jz /J⊥ , andred lines (at Jz /J⊥ = −2.2) mark the approximate transition between S-collective
and S-Ising phases, when the “collective” squeezing peak at τ ≡ t × |Jz − J⊥ | ∼ 6
drops below the “Ising” peak at τ ∼ 1. For the parameters shown, hS 2 i reaches aminimum at τ ∼ 2, which means that optimal squeezing at τ ∼ 1 is reached before
maximal decay of hS 2 i in the S-Ising phase.

. . . . . . . . . . . . . . . . . . . . . . 172

xxv
6.4

2
Optimal squeezing ξopt
as a function of system size for the power-law XXZ modelwith α = 3 on a 2D lattice of N = L × L spins. Whereas the amount of squeezing
generated in the S-Ising phase is insensitive to system size, squeezing in the Scollective phase grows with system size and as Jz /J⊥ → 1 (from below). Dottedgrey line tracks minima of hS 2 imin as a function of Jz /J⊥ , as in Figure 6.2, marking
the approximate dynamical phase boundary.
6.5. . . . . . . . . . . . . . . . . . . . . . 173

2 (top), minimal squared magnetization hS 2 i
The optimal squeezing ξopt
min (middle),

and optimal squeezing time topt (bottom) for N = 4096 = 642 = 163 spins in
D = 1, 2, 3 spatial dimensions. Spins are initially polarized along the equator andevolved under the XXZ Hamiltonian in Eq. (6.3) of the main text. The results for
D = 2 and 3 shown here are a subset of the results in Figure 6.2, presented in thesame format as that for D = 1 for comparison. . . . . . . . . . . . . . . . . . . . . . 178
6.6

Maximal populations hPn imax of the total spin S = N/2−n manifolds Pn throughout
squeezing dynamics of 7 × 7 spins, initially polarized along the equator and evolved
under the XXZ Hamiltonian in Eq. (6.3) of the main text with a power-law exponent
α = 3. Computed with TS4 simulations and periodic boundary conditions. Shaded
regions indicate hP4 imax > 0.1, where TS4 results cannot be trusted due to thelikeliness of population leakage into truncated states. All states in P1 break translational invariance, so the initial population hP1 i0 = 0 is protected by the absenceof translational symmetry-breaking terms in the Hamiltonian. The population hP3 i,
meanwhile, is small because P3 is only coupled to P2 and P4 by matrix elementsthat are O 1/N smaller than the couplings between P0 ↔ P2 ↔ P4 . . . . . . . . . 180

xxvi
6.7

2 (top) and minimal squared magnetization hS 2 i
Optimal squeezing ξopt
min throughout

squeezing dynamics (bottom) as computed via TS4 and DTWA in the same setting
as Figure 6.6, likewise with shaded regions indicating hP4 imax > 0.1 in the TS4
2 is shown in decibels, and hS 2 isimulations. Here squeezing ξopt
min is normalized to
its initial value hS 2 i0 = N2 N2 + 1 . Dashed and dotted lines respectively mark theexactly solvable limits of uniform (OAT, α = 0) and power-law Ising (Ising, J⊥ = 0)interactions.
6.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

2 (top) and minimal squared magnetization hS 2 i
Optimal squeezing ξopt
min throughout

squeezing dynamics (bottom) on 2D lattices of 3 × 3 (left) and 4 × 4 (right) spins,
as computed by exact methods (solid lines) and DTWA (dots). The color of eachmarker indicates the corresponding value of α, as specified in the legend, and thedashed line marks the OAT limit of α = 0.
6.9

. . . . . . . . . . . . . . . . . . . . . . . 182

2
Dependence of the optimal squeezing parameter ξopt
on system size N within thecollective dynamical phase of the power-law XXZ model in D = 2 spatial dimensions.
Color indicates the value of Jz /J⊥ , sweeping down from +0.5 (dark purple, top) to−1.5 (yellow, bottom) in increments of −0.5. Circles show results computed with
2
DTWA; dashed lines show a fit to ξopt
= a/N ν with free parameters a, ν; and thedotted red line marks the OAT limit for reference. The DTWA results in panel (a)for α = 3 are a subset of those in Figure 6.4 of the main text.

. . . . . . . . . . . . 184

6.10 Dependence of the critical Ising coupling Jzcrit at the collective-to-Ising dynamicalphase boundary on system size N for the power-law XXZ model in D = 2 spatial
dimensions. Circles show results computed with DTWA, and dashed lines show a
fit to Jzcrit /J⊥ = −γ ln N + b with free parameters γ, b. The DTWA results in panel(a) for α = 3 are equivalent to the dashed grey lines in Figure 6.4 of the main text.
DTWA simulations were run with values of Jz /J⊥ that are integer multiples of 0.1,
placing a lower bound on the resolution for Jzcrit /J⊥ .

. . . . . . . . . . . . . . . . . 185

xxvii6.11 Squeezing ξ 2 and squared magnetization hS 2 i as a function of time t for N = 4096 =642 = 163 spins in D = 2 and 3 spatial dimensions. Color indicates the valueof Jz /J⊥ , and the red line highlights behavior at the value of Jz /J⊥ immediately
preceding the transition from the S-collective phase (above the red line) to the SIsing phase (below the red line).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

6.12 Same results as in Figure 6.11, but for values of Jz /J⊥ that cross the dynamicalphase boundary at Jz /J⊥ = 1. The red line highlights behavior at Jz /J⊥ = 1.1,
immediately preceding the transition from the S-Ising phase (Jz > 1) to the Scollective phase (Jz < 1).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

2
6.13 Dependence of the optimal squeezing parameter ξopt
on filling fraction f for the

XXZ model in Eq. (6.3) of the main text with power-law exponent α = 3 in D = 2
spatial dimensions with 50 × 50 lattice sites. Results computed using DTWA, witha random choice of f × 50 × 50 lattice sites to occupy. The shape and color of eachmarker indicates the corresponding value of Jz /J⊥ , as specified in the legend, andthe dotted line marks the OAT limit for reference. . . . . . . . . . . . . . . . . . . . 190
7.1(a) Ultracold atoms on a lattice of L sites tunnel between neighboring lattice sites
at a rate J, and locally repel each other with interaction energy U . (b,c) Whenthe interaction energy U is small compared to the single-particle bandwidth 4J, thefrozen-mode approximation enables the interaction Hamiltonian to be written as aspin model consisting of exchange terms ŝp · ŝq , which swap the states of two spins
pinned to modes p, q. (d) Interactions open an energy gap u = U × N/L between
the manifold of permutationally symmetric states of N spins, and the orthogonal
complement of states that break spin-permutation symmetry. . . . . . . . . . . . . . 198

xxviii
7.2

Whereas the state of a two-level spin (qubit) can be represented by a point on (orinside) the Bloch sphere, the state of an n-level spin is more generally represented bya probability distribution on the Bloch sphere. The distribution shown for n = 10
corresponds to a Haar-random pure state.

7.3. . . . . . . . . . . . . . . . . . . . . . . 199

Sketch of the three-laser drive used to address nuclear spins on a one-dimensionallattice. Two counter-propagating lasers with right-circular polarization and amplitudes Ω± point at an angle θ to the lattice axis. A third, linearly polarized laser
with amplitude Ω0 points in a direction orthogonal to both the lattice and the otherdriving lasers. Absorbing a photon from the laser with amplitude Ωm induces atransition (g, µ) → (e, µ + m) for the (electronic, nuclear spin) state of an atom,
where nuclear spin is quantized along the z axis. . . . . . . . . . . . . . . . . . . . . 200

7.4

Spin-orbit coupling for 2-level (a,c) and 4-level (b,d) spins. Colors indicate different
spin projections µ. In the “lab frame” (a,b), kinetic energy is insensitive to spin, but
a spin transition µ → µ + 1 is accompanied by a momentum kick q → q + φ from thedrive. Changing into the “gauge frame” (c,d), essentially by shifting the momentum
label q for each spin state µ, makes the drive diagonal in the momentum index, but
comes at the cost of making kinetic energy spin-dependent. . . . . . . . . . . . . . . 206

7.5

Time-averaged magnetization σMF and interaction energy hhs̄· s̄iiMF for different spin
dimensions n (indicated in the legend) as determined by mean-field simulations of
N = 100 spins initially in the x-polarized state |Xi for a time T = 105 /u. Insets
show same data after rescaling h → h × (n/2)1/3 , and transforming vertical axes
according to Eq. (7.29). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

7.6

The critical value of hcrit as determined by mean-field simulations of N = 100 spins
−αinitially in the x-polarized state |Xi. A single-parameter fit to hcrit = n/2
finds
α = 0.333(5), and α = 1/3 is consistent with all mean-field results to within an
uncertainty determined by the resolution of h in mean-field simulations. . . . . . . . 211

xxix
7.7

A corollary of Figure 7.5 for the initial state |XXi. The inset for interaction energyhhs̄ · s̄iiMF in Figure 7.5 subtracts off the minimal value of hhs̄ · s̄iiMF and rescales
to lie on the interval [0, 1], as prescribed in Eq. (7.29). Here the subtracting andrescaling is identical, but with a minimal value of 2γ (n) rather than γ (n).

7.8

. . . . . 214

A corollary of Figure 7.5 for the initial state |XXi i. Insets show the same data shifted
and rescaled identically to Figure 7.5. . . . . . . . . . . . . . . . . . . . . . . . . . . 214

7.9

Numerical results (analogous to Figures 7.5, 7.7, and 7.8 of the main text) for thetime-averaged interaction energy and magnetization (both normalized to a maximal
value of 1) in a system of L = 5 lattice sites, for both a Fermi-Hubbard model (dots)
and spin model (lines) with n = 4 internal states per spin. The corresponding initialstate (defined in Section 7.5 of the main text) is indicated in each panel, and observables are averaged over a time tJ = 200. Color indicates the value of U/J, and thefield h corresponds to 2Jφ/u in the case of the Fermi-Hubbard model. Simulations
are performed in real-space, with spin-orbit coupling (SOC) implemented through ahomogeneous drive (with no site or φ dependence) and nearest-neighbor tunnelingterms that contain factors of e±iµφ . Results for the initial kitten state |XXi i are
excluded because they are identical to those of |Xi, and magnetization for the initialstate |XXi is always 0. Note that while panels (a) and (b) are representative of
infinite-time behavior, the inset in panel (c) shows that the Fermi-Hubbard and spinmodels exhibit different behaviors on very long time scales, although good agreement
is restored by rescaling time in the spin model, indicating the likelihood of a need torenormalize spin model parameters. In any case, such time scales are inaccessible incurrent experiments and diverge as N → ∞, so these corrections do not affect themain results of our work.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

7.10 Numerical results identical to Figure 7.9, but with L = 4 lattice sites and n = 6
internal states per spin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

xxx
8.1

Signal recovery on the sphere is the problem of reconstructing an unknown functionf (red distribution) from its values f (v) at specific points v ∈ V (blue dots) on thesphere. For almost all choices of V , reconstruction of f is possible if there are atleast as many points in V as there are degrees of freedom in f . . . . . . . . . . . . . 239

8.2

Serial runtime to compute SV , V , or EV (ρ) with |V | = 2d − 1 randomly chosenmeasurement axes and a randomly chosen qudit state ρ. Each point is an averageover 103 calculations or 5 minutes of runtime, whichever comes first. These results
do not count fixed runtimes to pre-compute quantities that can be recycled for every
new choice of V and ρ. Dashed lines show fits to a runtime t = cdα for the 20 largest
values of d, finding α ≈ 3.8 ± 0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

8.3

Empirical measurement-adjusted error scales β̃(p) with p excess measurement axes,
determined by minimizing over 103 choices of measurement axes V or 5 minutes ofruntime (for each p), whichever comes first. Marker and color indicates the qudit
dimension d. The rapid initial drop in β̃(p) implies that using more measurement
axes can substantially lower the upper bound on reconstruction error provided in
Eq. (8.30), and that these benefits plateau after p ≈ d. Horizontal reference lines
mark the smallest measurement-adjusted error scales minθ βθ /β̃(0) achievable withthe method in Ref. [246], which is parameterized by an arbitrarily chosen angle θ. . . 248

8.4

Quantum error scale θ as a function of the polar angle θ in the tomography methodof Ref. [246] for a few qudit dimensions d. . . . . . . . . . . . . . . . . . . . . . . . . 263

8.5

Optimum angle θopt as a function of qudit dimension d for the tomography method
1of Ref. [246], and a fit to θopt = π2 (1 − xd
) finding x ≈ 1.34. . . . . . . . . . . . . . . 263

Chapter 1

Context and outline

When quantum mechanics was first developed, it was primarily used as abstract microscopic
theory with which to explain the behavior of matter on a larger scale, much the way that statisticalmechanics is used to explain thermodynamic phenomena. In this way, one of the crowning scientificachievements of the 20th century led to technological innovations such as the transistor, the laser,
and the medical magnetic resonance imaging (MRI) scanner. Today, we are in the midst of a newtechnological revolution that aims to manipulate matter on the scale of individual quanta. We arealso just beginning to understand and exploit subtle quantum phenomena such as entanglement tosurpass classical limits on what is considered even possible. This is an exciting time to work on
quantum technologies.
One of the leading platforms for quantum technologies is that of cold atoms in optical lattices.
Cold atoms’ first major claim to fame was an experimental confirmation of a quantum phasetransition between superfluid and (Mott) insulating phases of the bosonic Hubbard model (that is, amodel of particles hopping around on a lattice, with pair-wise local interactions). This achievement
was the first major demonstration of an analog quantum simulator, in which a quantum mechanicalmodel of interest is meaningfully studied “by analogy” in an experimental platform. Tabletop cold
atom experiments also make up the most precise and accurate clocks to date (with ion-based clocks
a close rival), and cold atoms take second place for the most precise scientific instruments ever built(trailing kilometer-scale gravitational wave observatories). In addition to quantum simulation, cold
atoms are thereby a major tool for the science of measurement, known as quantum metrology.

2This thesis explores some of the possibilities and prospects for quantum simulation andmetrology with cold atom systems, more specifically with fermionic atoms that have a multilevelinternal structure. Chapter 2 provides some technical background and a crash course to the mainconcepts and tools that appear throughout this thesis. Chapters 3–8 walk through some highlightsof my graduate work, with each chapter featuring a paper on which (1) I was a primary (i.e. firstor “co-first”) author, and (2) Ana Maria Rey was a principal investigator [1–6]. For the sake ofbrevity and cohesion, I exclude papers from my graduate work in which I played a secondary role[7, 8]; papers that I wrote with colleagues at Argonne National Laboratory and Super.tech [9,10]; and papers from projects that I started before entering graduate school [11–13], although I
consider some of these papers (particularly Refs. [7, 9, 11]) to be equally important parts of mygraduate work. Each of Chapters 3–8 begins with a short prologue that informally introduces andcontextualizes the paper that follows.

Chapter 2

Introduction and background

2.1

Alkaline-earth(-like) atoms
Much of the work in this thesis refers to “fermions” or “spins” of some sort, probably ona lattice, usually interacting. This general language isolates the essential features of the objects
under consideration, but often at the cost of hiding the true star players of our show: alkalineearth(-like) atoms (AEA). The electronic structure of these group-II elements, along with similarlystructured Ytterbrium, allows the breadth of physics explored in this thesis to be observed in asingle experimental platform. Specifically, these atoms have two valence electrons with a 1 S0electronic ground state, and above that a first-excited 3 P0 state. These are often referred to as theclock states due to their central role in an AEA-based atomic clocks. A direct, photon-mediatedtransition between the clock states is doubly forbidden:(i) First, a 1 S0 –3 P0 transition is dipole-forbidden because both of these states have a total
electronic (intrinsic + orbital) angular momentum of 0, as indicated by the “0” in 1 S0 and
3 P . There is no way for a spin-0 system to absorb a spin-1 photon and end up in another0

spin-0 state, as in a direct 1 S0 → 3 P0 transition.
(ii) Second, a 1 S0 –3 P0 transition is spin-forbidden because it couples a state with net intrinsic
spin 0 (1 S0 , with a two-electron spin singlet) to a state with net intrinsic spin 1 (3 P0 , with

4
a spin triplet)a . Photons couple primarily to the orbital angular momentum of electrons in
an atom, and therefore should not change their intrinsic spin directly.
Photon-mediated coupling between the 1 S0 and 3 P0 does occur, of course; we would not be
discussing this transition otherwise. 1 S0 –3 P0 coupling is made possible by the hyperfine (i.e. nuclearspin-mediated) coupling of the “bare” electronic 3 P0 state to “bare” electronic 1 P1 , 3 P1 , and 3 P2states (see Figure 2.1) [14]. As a result of this hyperfine coupling, the electronic eigenstate that welabel “3 P0 ” weakly populates these other states as well. In particular, the bare 1 P1 state couples
directly to the 1 S0 state, and thereby virtually mediates 1 S0 –3 P0 coupling. The result of this long
song and dance, and ultimately the point of the entire discussion about AEAs thus far, is thatthe 3 P0 state of AEAs is metastable and long-lived, with a lifetime of roughly two minutes in 87 Sr[15]. As a corollary, the 1 S0 –3 P0 transition has an exceptionally narrow linewidth (e.g. a ∼ 10−3
Hz linewidth on a ∼ 1015 Hz transition in 87 Sr [15]), making AEAs an excellent tool for precision
metrology and fundamental physics.
From the perspective of metrology, measurements of the 1 S0 –3 P0 transition frequency in
AEAs are used in the most precise and accurate clocks to date, rivaled only by ion-based “quantum
logic” clocks (which, incidentally, address a 1 S0 –3 P0 transition engineered in group-III atoms bystripping them of one valence electron) [16]. Precise clocks are important for setting time andfrequency standards (e.g. defining the second), which as of 2019 define all physical units of measurement in the International System of Units (SI)b . Any wristwatch, bathroom scale, or oventhermometer built today is calibrated to an atomic clock (albeit one based alkaline Caesium atoms
rather than AEAs, for the time being). Moreover, the exquisite precision of atomic clocks admits
applications in navigation (such as an advanced global positioning system, GPS) and geodesy (prea

You can tell that 3 P0 has net intrinsic spin 1 because its intrinsic spin must cancel out with orbital spin (which isspin-1, as indicated by the “P” in 3 P0 ) to result in a total electronic angular momentum of 0.

bMore precisely, the second is the only quantity that must be measured to define SI units. All SI units are thendefined as the second raised to some power and multiplied by prefactors with numerical values that are fixed bydefinition.

7

5
2

11

2
5

13

7

State Mixing
LS
HFI

9

689

2

2

11

461

2

2

2

698
9

2

Figure 2.1: Simplified energy level diagram for the low-lying electronic states of 87 Sr, reproduced
from Ref. [14]. Dashed and dotted lines respectively show spin-orbit (LS) and hyperfine (HFI)
coupling between bare electronic states. Solid lines show optical transitions between electronic
eigenstates, with the corresponding wavelengths given in nanometers. Fractions indicate the net
spin of the hyperfine manifold (involving nuclear spin) within the associated electronic energy level.

cision measurements of the gravitational field, which can be used to determine altitude or geologicalmaterial density) [17]. From the perspective of fundamental physics, the precision of atomic clocks
can be leveraged for applications such as gravitational wave detection, dark matter searches, andsearches for the violation of fundamental symmetries (e.g. charge-parity-time reversal symmetry,
Lorentz invariance, etc.) [18]. These capabilities are quite remarkable, given the fact that AEAbased experiments are roughly the size of a (meter-scale) optical table, in contrast to the stadiumor kilometer-scale observatories and colliders that are usually necessary for fundamental physics
experiments.
Finally, due in part to their metrology-driven technological development, AEA experiments
offer an exquisite platform for studying many-body quantum physics. This scientific venture goes
hand-in-hand with the practical development of better sensors: improved sensitivity yields a more
precise and controllable physics testbed, and a better understanding of relevant physics allows us
to mitigate systematic errors and further improve sensitivity. As we will see in Chapters 4 and 6, a
deep understanding of the physics underpinning AEA-based atomic clocks can be used to engineer

6entangled many-body quantum states that allow us to approach the limits of device performanceallowed by physics.

2.2

Individual atoms in a trap
AEAs are typically cooled and trapped in an electromagnetically induced external potential.

In the absence of interactions, these atoms can be described by the Hamiltonian (~ = 1 throughout
this thesis)
Z
Ĥ0 =

"
3

d x

X

†
ψ̂αµ
(x)

α,µ

#
∇2
α
−
+ Vext (x) + Eα ψ̂αµ (x),
2m

(2.1)

where α and µ respectively index the electronic and nuclear states of an atom; ψ̂αµ is the field
operator for an atom with internal state α, µ; ∇2 = ∂x2 + ∂y2 + ∂z2 is a spatial Laplacian; m is theα is an external potential, which may generally depend on the electronic state
mass of an atom; Vext

α; and Eα is the electronic (internal) energy of state α. The field operator ψ̂αµ can be expandedin the formψ̂αµ (x) =X

wqα (x)ĉqαµ ,

(2.2)

q

where q indexes an orthonormal basis of spatial wavefunctions wqα for atoms in state α, and theoperator ĉqαµ annihilates an atom in mode (q, α, µ). The operator ĉqαµ may generally be bosonicor fermionic depending on the species of the atom, but we will primarily consider fermionic atoms
in this thesis. Substituting this expansion into (2.1) yields
X

Ĥ0 =

α †
Jqq
0 ĉqαµ ĉq 0 αµ +q,q 0 ,α,µwhere N̂α =X

Eα N̂α ,

(2.3)

α

†
q,µ ĉqαµ ĉqαµ counts the number of atoms in state α, and

P

α
Jqq
0 =

Z

α
d3 x wq∗ (x)Hspatial(x)wq0 (x),(2.4)

is a tunneling matrix defined in terms of the Hamiltonian densityαHspatial(x) = −

∇2α
+ Vext
(x).
2m

(2.5)

7
α are essentially the rate
For a fixed electronic state α, the off-diagonal (q 6= q 0 ) components of Jqq
0αat which atoms tunnel between modes q and q 0 . Choosing the eigenfunctions of Hspatial
as the basis
α that is diagonal, in which case J α is simply theof wavefunctions wqα yields a tunneling matrix Jqq
0qq

orbital energy of an α-state atom in mode q (where “orbital” refers to the “orbit” of an α-state atomα ). However, the best choice of basis for wavefunctions will generally
in the external potential Vext

depend on the particular setting and parameter regime under consideration. If atoms are trappedon a lattice, for example (see below), it might be convenient to use a basis of wavefunctions localized
to individual lattice sites, and treat the effect of inter-site tunneling as a dynamical or perturbative
process.

2.3

Optical lattices and magic wavelengths
An important class of traps that will play a central role in this thesis are optical lattices.

An optical lattice is generated by retroreflected or counter-propagating laser beams that form anelectromagnetic standing wave. By off-resonantly coupling different electronic states of atoms in
the optical lattice, the lattice light induces AC Stark shifts that are proportional to the electricfield intensity of the lasers, as well as the electric polarizability of the atoms. In a two-state modelof the AC Stark shift, a laser with frequency ν couples electronic state α with energy Eα to an
excited state β with energy Eβ = Eα + ω. If the detuning ∆ = ω − ν is much larger than the
coupling strength Ω between these statesc , then the coupling can be treated perturbatively, leading
to a second-order shift of the α-state energy by ∼ Ω2 /∆ [19]. In practice, each state α gets coupled
to a large number of excited states, with associated coupling strengths and detunings for every pairof states. Altogether, the electromagnetic potential induced by an optical lattice (standing wave)
oriented along the z axis takes the formαVext(z) = Vα sin2 (kz),cBy a coupling Ω between states α and β, we mean some Hamiltonian of the form Ω |βihα| + |αihβ| .

(2.6)

8
where Vα is the lattice depth for α-state atoms, k is the wavenumeber of the lattice light, and werestrict ourselves to one spatial dimension for simplicity.
αEigenfunctions of the Hamiltonian Hspatialin (2.5) with the sinusoidal potential in (2.6) are

known as Mathieu functions, which can be decomposed into Bloch waves as wqα (z) = eiqz z uαqz n (z)[20]. Here the mode q ∼ (qz , n) is indexed by a quasi-momentum qz with periodicity qz = qz + 2π
and an integer band index n ≥ 0, and the function uαqz n has the same periodicity as the lattice:uαqz n (z) = uαqz n (z + a), where a = π/k is the distance between neighboring lattice sites (i.e. thelattice constant). For theoretical analysis and numerical simulations, it is often convenient toconsider lattice of L sites with periodic boundary conditions, in which case the quasi-momentum
qz = 2π/L × j for some integer j. Efficient numerical methods for computing Mathieu functions
can be found in Ref. [20].
We now (and for the remainder of this thesis) restrict ourselves to the two lowest-lyingelectronic states, i.e. the clock states α ∈ 1 S0 ↔ g, 3 P0 ↔ e . By tuning the lattice light to a socalled “magic wavelength” at which these states have the same electric polarizability, the potential
α can be made independent of the electronic state α, in which case the single-particle Hamiltonian
Vext

becomes [21, 22]
Ĥ0 =

X

Eq ĉ†qαµ ĉqαµ + ω N̂e ,

(2.7)

q,α,µ

where Eq = Jqq is the orbital energy of Bloch mode q (see Figure 2.2)d , and ω = Ee − Eg is the|gi ↔ |ei transition frequency. We assume that atoms are trapped in a magic-wavelength lattice
for the remainder of this thesis.
As a final point, we note that the same light that forms an optical lattice unavoidably inducesdissipation called light scattering in AEA. Although the clock states are themselves (meta)stable,
the lattice light off-resonantly couples the clock states to excited states that can have fast decay
rates. The clock states thereby inherit the loss channels of virtually populated excited states,d

We will often consider atoms that occupy only the lowest Bloch band (n = 0), in which case Eqz = −2J cos (qz a),where J is the tunneling rate between neighboring lattice sites.

9

8
n=0
n=1
n=2

Eq /ER

6
4
∆band
2
0
−π

−π/2

0
qz a

π/2

π

Figure 2.2: Band structure (orbital energy) of atoms in a periodic lattice potential Vext (z) =V0 cos2 πzwith depth V0 = 4ER , where the lattice recoil energy ER = kL2 /2m is defined in terms
aof the atomic mass m and the wavenumber kL of the lattice light. Color indicates band index n as
specified in the legend.

albeit only perturbativelye . For this reason, the coherence time of the clock states of 87 Sr atoms ina magic-wavelength optical lattice is typically ∼ 10 seconds, rather than the ∼ 102 second lifetimeof the 3 P0 state, although it may be possible to mitigate light scattering by the construction of
optical lattices with larger lattice constants [23].

2.4Spin-orbit coupling
As an added ingredient and control knob, we can turn on a linearly-polarized laser withwavenumber k and frequency ν ≈ ω to induce an electric dipole transition between the |gi and|ei states of AEAs. There are several ways to deduce the corresponding drive Hamiltonian governing the AEAs; here we take a semi-classical, phenomenological approach that avoids digressions
into subjects and techniques that are outside the scope of this thesis. We treat driving laser“semi-classically” in the sense that we do not quantize the corresponding electromagnetic field(equivalently, we make the reasonable assumption that the laser light is in a coherent state withe

If a dissipation-free clock state is coupled with amplitude Ω and detuning ∆ to an excited state with dissipation
rate Γ, then virtual occupation of the excited state shifts the clock-state energy by ∼ Ω2 /∆, and causes the clockstate to dissipate at a rate ∼ Γ × Ω2 /∆2 [19].

10a large mean photon number), but we retain the fact that individual photons are spin-1 particles,
which means their absorption or emission by an AEA must accompany an appropriate internal spintransition. In the plane-wave approximation, the “bare” drive Hamiltonian for AEAs is thenbareĤdrive
=

Z

d3 x

X

†
Ωµ ei(k·x−νt) ψ̂e,µ(x)ψ̂g,µ (x) + h.c.,

(2.8)

µ

where the nuclear spin µ ∈ {s, s − 1, · · · , −s} for a spin-s atom (e.g. s = 9/2 for 87 Sr) is quantizedalong an axis in the plane orthogonal to k; Ωµ is a spin-dependent drive amplitude; and “h.c.”denotes the Hermitian conjugate of preceding terms. The drive amplitude Ωµ has its physicalorigins in atom-light coupling to an electric dipole transition, and is therefore proportional to boththe electric field intensity of the driving laser and the differential electric polarizability of the |gi and|ei states. Moreover, the Hamiltonian in (2.8) describes a spin-s atom absorbing a spin-1 photonwith spin projection 0 onto the quantization axis, which implies that Ωµ ∝ hsµ; 1, 0|sµi ∝ µ, whereh`1 m1 ; `2 m2 |`3 m3 i is a Clebsch-Gordan coefficient. We consider a more general case of lasers withany polarization and arbitrary orientation relative to the spin quantization axis in Appendix 7.D
of Chapter 7.
Our first order of business is to eliminate the time dependence from the drive Hamiltonian††in (2.8), which we can do by moving into a rotating frame that takes e−iνt ψ̂e,µ→ ψ̂e,µ. Such atransformation is achieved by subtracting ν off of the energy of |ei-state atoms, in effect replacing
the excitation energy ω in (2.7) by the detuning ∆ = ω − ν f . After making this replacement, thedrive Hamiltonian becomesĤdrive =XΩµ Iqq0 ĉ†q,e,µ ĉq0 ,g,µ + h.c.,(2.9)

q,q 0 ,µf††Mathematically, the transformation e−iνt ψ̂e,µ→ψ̂e,µis implemented by the unitary U
=
 R

3 P
†
exp iνt d x µ ψ̂e,µ ψ̂e,µ . If we rotate states |φi → |φ̃i = U |φi, the accompanying transformation of

the overall Hamiltonian H → H̃ (which can be derived by enforcing the Schrödinger equation H̃ |φ̃i = i∂t |φ̃i) is
given by H̃ = U HU † +i∂t U U † . This transformation removes the time dependence from the bare drive HamiltonianRP †in (2.8), and subtracts ν off the energy of |ei-state atoms by the addition of i∂t U U † = −ν d3 x µ ψ̂e,µψ̂e,µ .

11
where the overlap integral
Z
Iqq0 =

d3 x eik·x wq∗ (x)wq0 (x).

(2.10)

We now assume that our atoms are trapped on a three-dimensional cubic lattice oriented alongthe (x, y, z) axes, and with strong confinement in the x-y plane, such that atoms’ spatial degrees of
freedom are frozen within this plane. This assumption is straightforward to relax when necessary,
but doing so complicates the analysis in this section without added benefit. The effect of thisassumption is essentially to replace the three-dimensional integral in (2.10) by a one-dimensional
integral along the z axis:
Z
Iqq0 →

dz eikz z wq∗ (z)wq0 (z),(2.11)

where kz is the projection of k onto the z axis, and the spatial integral over the x-y plane in (2.10)
can be absorbed into the definition of the drive amplitude Ωµ in (2.9).
The spatial modes q ∼ (qz , n) are indexed by a quasi-momentum qz and band index n (seeSection 2.3). When atoms are ultracold, it may be reasonable to assume that their temperature issmall compared to the energy gap ∆band between the two lowest bands (see Figure 2.2). If atoms
are furthermore initialized in the lowest band n = 0, as long as Ωµ  ∆band (which prevents
drive-induced coupling to higher bands) we can neglect the band index n, unambiguously identify
q ∼ qz , and substitute the Bloch-wave decomposition wq (z) = eiqz uq (z) into the overlap integral in(2.11) to get
Z
I

qq 0

=

0

dz ei(kz +q −q)z u∗q (z)uq0 (z).

(2.12)

We now make the final assumption that kz is commensurate with the lattice, which is to saythat kz is an allowed quasi-momentum along the z axis (i.e. an integer multiple of 2π/La). The
assumptions of periodic boundary conditions and drive commensurability can be more cumbersome
to relax than that of tight transverse confinement, particularly when considering the limit kz a  1(which features in Chapters 4 and 7), or when considering length scales over which the plane-wave

12
approximation for the lattice laser breaks down. We defer a careful scrutiny of these assumptions
and their breakdown to another time (see, for example, Ref. [24] and Appendix 4.I of Chapter 4).
If kz , q, q 0 are all integer multiples of 2π/La, we can split the position z into an integer multipleof a and a remainder, z = aj + r with j = bz/ac and r = z mod a; here j essentially indexes alattice site, and r is a position within that site. The periodicity of uq (z) then implies that theoverlap integral in (2.12) can then be written in the form
Iqq0 =LX

ei(kz +q 0 −q)aj

Z a0

dr ei(kz +q −q)r u∗q (r)uq0 (r).

(2.13)

0j=1

Here (kz +q 0 −q)a is an integer multiple of 2π/L, so the sum over j vanishes unless kz +q 0 −q = 0. We
thus find that the overlap integral Iqq0 = 0 unless q = q 0 + kz , so altogether the drive Hamiltonianin (2.9) becomesĤdrive =XΩqµ ĉ†q+kz ,e,µ ĉq,g,µ + h.c.,(2.14)

q,µwhere Ωqµ = Ωµ × Iq+kz ,q is typically independent of q to a good approximationg .
A few notes concerning the drive Hamiltonian in (2.14). First, the result of the long journeyfrom (2.8) to (2.14) can be summarized by conservation of quasi-momentum: an atom that transitions from |gi to |ei by absorbing a photon must also absorb the quasi-momentum of that photon.
Second, drive Hamiltonian in (2.14) exhibits spin-orbit coupling, which is to say that it changes
electronic (“pseudo-spin”) and spatial (“orbital”) degrees of freedom in a correlated fashion. To
further simplify this drive, we can relabel quasi-momenta for the excited electronic state, takingĉ†q+kz ,e → ĉ†q,e , which takes
Ĥdrive →XΩqµ ĉ†q,e,µ ĉq,g,µ + h.c..

(2.15)

q,µThis relabeling comes at the cost of making kinetic energy electronic-state-dependent (see Figure2.3), but can simplify the theoretical treatment of AEAs, which is most clear when atoms aregThis approximation breaks down in very shallow lattices, in which the inter-site tunneling rate J exceeds the latticerecoil energy ER = kL2 /2m (defined in terms of the atomic mass m and the wavenumber kL of the lattice laser).

13

8
|gi

Eq /ER

6

|ei

4
2
0
−π

−π/2

0
qz a

π/2

π

Figure 2.3: Band structure identical to that in Figure 2.2, but after relabeling excited-state quasimomenta with ĉ†qz +kz ,e → ĉ†qz ,e and kz = 2π/3. Solid and dashed lines respectively show kinetic
energies for ground- and excited-state atoms.

nuclear-spin-polarized and the dependence of Ωqµ on q is negligible, which allows us to write Ĥdrive =PΩ q ĉ†q,e ĉq,g + h.c. (where we suppress a fixed nuclear spin index). We elaborate on this point inour discussion of spin operators in Section 2.7.
On one hand, spin-orbit coupling is a complication that limits the capabilities of atomic clocks
by exacerbating the deleterious effect of inhomogeneities on inter-atomic coherence at low lattice
depths (which are desirable to reduce atomic losses from scattering off of high-intensity latticelight). On the other hand, spin-orbit coupling gives us access to a variety of exotic phenomena,such as topological phases of matter and the possibility to prepare metrologically useful quantum
states [2, 25–27]. Getting a handle on these phenomena is a promising route to push the limits ofcurrent atomic clocks.

2.5

Nuclear spin decoupling and SU(n) symmetry
An important consequence of the fact that the both 1 S0 and 3 P0 states have a total electronic(intrinsic + orbital) angular momentum of 0 is the near perfect decoupling of AEAs’ electronic andnuclear spin degrees of freedom. Mathematically, the (first-quantized) Hilbert space of AEAs
occupying 1 S0 and 3 P0 states neatly factorizes into a tensor product of uncoupled [electronic-

14
state]⊗[nuclear-spin] Hilbert spaces. This decoupling implies that, in the absence of an external
magnetic field, freely evolving atoms that interact only through collisions (which are dominatedby electric forces, and thereby electronic-state parameters) exhibit an SU(n) symmetry, wheren = 2s+1 is the number of nuclear spin levels in a spin-s atom (e.g. n = 10 for 87 Sr with nuclear spins = 9/2) [28]. Specifically, the dynamics of such AEAs are invariant under arbitrary global unitarytransformations of atoms’ nuclear spin states. As a consequence, interaction parameters such ass-wave scattering lengths and p-wave scattering volumes should be (approximately) independent of
spin.
However, the same hyperfine interactions that enable photon-mediated 1 S0 –3 P0 coupling also
violate the SU(n) symmetry that results from the decoupling of atoms’ electronic and nuclear states.
The dominant violations of SU(n) symmetry occur due to hyperfine coupling of the K PJ states(i.e. 1 P1 , 3 P0 , 3 P1 , and 3 P2 ). Violation of SU(n) symmetry in the 1 S0 state is in turn mediatedby virtual occupation of symmetry-breaking K PJ states during collision. By estimating the effectof these virtual occupations on the two-body collisional phase shift Φ, SU(n) symmetry-breaking
corrections δΦ have been estimated to be δΦ ∼ 10−3 for 3 P0 -state atoms, and δΦ ∼ 10−9 for 1 S0[29]h . We will consider such corrections negligible in the remainder of this thesis.

2.6

Two-body interactions
Collisional interactions are primarily governed by electronic-state parameters, which almost

perfectly decouple from atoms’ nuclear spin states. As a result, a pair of atoms occupying nuclear spin states (µ, ν) must occupy the same nuclear spin states after collision, since there is no
mechanism by which to change nuclear spin state occupations. Moreover, interaction energy scales
are typically much smaller than the electronic excitation energy (Uint ∼ 103 Hz vs. ω ∼ 1015 Hz
in 87 Sr), so we can ignore processes in which interactions change the number of |ei-state atoms.
hThe s-wave scattering length a ∼ Φ∆t, where Φ is a collisional phase shift and ∆t is the time that a pair atoms
spend in the short-range part of their intermolecular potential during collision (∆t ≈ 1 picosecond for 87 Sr [29]). A
correction δΦ ∼ η implies that scattering lengths a acquire SU(n) symmetry-breaking corrections δa with δa/a ∼ η.

15
Once both nuclear and electronic state occupation numbers are fixed, there are only four possible
interaction channels: one each for a pair of |gi- or |ei-state atoms that scatter off of each other,
one “direct” channel in which a pair of atoms in distinct electronic states interact without changing
their respective nuclear spin states, and one “exchange” channel in which a pair of atoms in distinct
electronic states swap their nuclear spin states. Schematically, the interaction Hamiltonian musthave the form
g, µ

g, µ

Ĥint ∼

e, µ

+
g, ν

g, ν

g, µ

e, µ

g, µ

+
e, ν

e, ν

g, µ

g, ν

+
e, ν

e, ν

,
e, ν

e, µ

(2.16)
where we indicate excited-state atoms by a thicker line, and use color to indicate nuclear spin. At
low temperatures, the most general low-energy (s-wave) interaction Hamiltonian consistent withthese constraints isij :ZĤint =


X 1
†
†

d3 x
Gαα ρ̂αµ ρ̂αν + G+ ρ̂e,µ ρ̂g,ν + G− ψ̂e,µψ̂g,νψ̂e,ν ψ̂g,µ  ,2µ,νX(2.17)

α∈{g,e}†where GX are two-body coupling constants, ρ̂αµ = ψ̂αµ
ψ̂αµ is an atomic density field operator,

and we suppress the implicit dependence of ρ̂αµ and ψ̂αµ on the position x. Eigenstates of Ĥint
are strictly symmetric or anti-symmetric under the pairwise exchange of atoms’ electronic states.
For example, two-body eigenstates of Ĥint have electronic states of the form |ααi or |gei ± |egi,
with corresponding interaction energies of the form Eαα = Gαα I or Eeg± = (G+ ± G− ) I, where
I is a spatial overlap integral. Matching these interaction energies to s-wave scattering lengths asiTechnically speaking, there are two ways to arrive at the Hamiltonian in (2.17): (i) approximating the intermolecularpotential of two atoms by a delta function (formally an unregularized, energy-independent pseudopotential), or (ii)the phenomenological approach of effective field theory, which considers all possible types of (local) interactions
and keeps only those that are dominant in the low-energy (small momentum) limit.

j

The factor of 1/2 in the first term of (2.17) corrects for the fact that a sum over all µ, ν double-counts the interaction
between a pair of atoms with identical electronic states α ∈ {g, e}.

16
EX = 4πm aX I, we thus find that
Gαα=gg,ee =

4π
aαα ,
m

G± =

4π aeg+ ± aeg−.
m
2

(2.18)

If atoms occupy only the lowest band on a three-dimensional lattice, then we can invoke
the tight-binding approximation and neglect interactions between atoms on different lattice sites,
because these interactions will be proportional to inter-site overlap integrals that are much smaller
than on-site overlap integrals. If further assume that all AEAs occupy the same nuclear spin state(for example, if they are nuclear-spin-polarized with µ = ν = 9/2), then the electronic state ofany pair of interacting atoms must be anti-symmetric under exchange in order to obey fermionicstatistics. Suppressing irrelevant (fixed) nuclear spin indices, the only non-vanishing terms in (2.17)
are thenNSP
Ĥint= Geg− KX †X1ĉj,e ĉj,e ĉ†j,g ĉj,g = Geg− KN̂j (N̂j − 1),2
j(2.19)jwhere Geg− = G+ − G− = 4π
m × aeg− , the overlap integralZK=

d3 x wloc (x)4(2.20)

is defined in terms of the (Wannier) wavefunction wloc of an atom localized to a single lattice sitek[24], ĉjα destroys an atom on lattice site j with electronic state α, and N̂j is the number of atoms
on lattice site j.
Rather than fixing atoms’ nuclear spin states and letting their electronic states evolve, we
can fix all atoms in their electronic ground state |gi and let their nuclear spin states evolve. In thiscase all terms in (2.17) aside from ∼ Ggg vanish, and the interaction Hamiltonian becomesgg
Ĥint= Ggg KX †X1
ĉjµ ĉjµ ĉ†jν ĉjν = Ggg K
N̂j (N̂j − 1),
2

j
µ<ν
k

(2.21)

jIn the plane-wave approximation for the light forming an optical lattice, the overlap integral in (2.20) is independentof the choice of lattice site on which to localize wloc .

17where we have suppressed the irrelevant (fixed) electronic-state index g. In both cases of (2.19) and(2.21), after appropriately fixing nuclear or electronic degrees of freedom the interaction Hamiltonian can be written in terms of number operators N̂j that are insensitive to atoms’ internal states.
These Hamiltonians simply assign an energetic penalty to all pairs of atoms occupying the samelattice site, which are counted by the binomial coefficient 12 N̂j (N̂j − 1) = N̂2j . We investigate
some of the physics accessible with these interactions, which manifest due to an underlying SU(n)symmetry, in Chapters 3 and 7.

2.7

Spin operators
Fermions are notoriously difficult to study with a pen and paper (and even computer). One

simplifying technique for modeling fermionic AEAs is thus to map them onto a more tractablespin model, which may be possible when atoms’ spatial degrees of freedom play no role in theirdynamics. Leaving the details, discussion, and benchmarking of some such mappings to Chapters
4 and 7, the basic idea is to define the spin operators
1ŝx,j ≡ ĉ†je ĉj,g + h.c.,2iŝy,j ≡ − ĉ†je ĉj,g + h.c.,2ŝz,j ≡

1 †ĉj,e ĉj,e − ĉ†j,g ĉj,g ,2(2.22)

where we tentatively assume that atoms are nuclear-spin-polarized for simplicity (allowing us tosuppress the nuclear spin indices), and j indexes a spatial mode (such as a lattice site or quasi
momentum). These spin operators obey standard SU(2) commutation relations: ŝαj , ŝβj = iŝγj
for any (α, β, γ) that is a cyclic permutation of (x, y, z). This is simply a fancy way of saying thatthe operators defined in (2.22) behave, for example, like the standard spin-1/2 operators for the
intrinsic spin of an electron. We can think of each atom as a two-level “spin”, identifying |gi ↔ |↓iwith “spin-down” and |ei ↔ |↑i with “spin-up”. The electronic state of a single atom can thenbe visualized as a point on the Bloch sphere, and exponentiating the spin operators sαj generates
rotations of this state. Specifically, defining the spin vector ŝj = (ŝx,j , ŝy,j , ŝz,j ) and spatial axisv = (vx , vy , vz ) with v · v = 1, the unitary e−iθ v·ŝj rotates the state of spin j about axis v by anangle θ.

18
Summing over the mode index j yields the collective spin operators
Ŝα ≡

X

ŝα,j ,

(2.23)

jwhich obey the same commutation relations as the on-site operators ŝαj . In terms of a collective
Q
spin vector Ŝ = (Ŝx , Ŝy , Ŝz ), the unitary e−iθ v·Ŝ = j e−iθ v·ŝj now rotates all spins identically.
The use of collective spin operators can greatly simplify the theoretical treatment of AEAs. If
atoms’ spatial degrees of freedom are frozen (e.g. due to trapping in a deep lattice), an external
drive detuned by ∆ from atoms’ |gi ↔ |ei transition (as in Section 2.4) induces the HamiltonianĤdrive = ΩŜx + ∆Ŝz ,(2.24)

where we include the energy splitting ∆ between |gi and |ei in the rotating frame of the drive. Thisdriving Hamiltonian simply rotates all spins about the axis (Ω, 0, ∆).
If all atoms are prepared (and kept) in the electronic ground state, it can be similarly useful
to define nuclear spin operators and their collective versions:ŝµνj ≡ ĉ†j,g,µ ĉj,g,ν ,Ŝµν ≡Xŝµν,j .

(2.25)j

The on-site spin operators ŝµνj destroy an atom in state ν and create one in state µ, and can thus
be represented as ŝµνj := |µihν|j . These operators obey the commutation relations of an SU(n)algebra: ŝµνj , ŝρσj = δνρ ŝµσj − δσµ ŝρµj , where δαβ = 1 if α = β and 0 otherwise. As we discuss in
Chapter 7, multi-laser driving schemes addressing the excited state |ei far off resonance can induce
effective Hamiltonians that are expressed neatly in terms of the spin operators ŝµνj and Ŝµν . We
explore some of the physics accessible with nuclear spin driving and SU(n)-symmetric interactionsin Chapter 7.

2.8

Collective spins and the Dicke manifold
Spin systems have a tensor product structure that typically makes them simpler to analyze

than comparably-sized fermion systems. Even so, the Hilbert space of a spin system is still too

19large to simulate more than a handful of spins exactly: the Hilbert space of N two-level spins has
dimension 2N , which is to say that we need 2N complex numbers to represent an arbitrary (pure)state of N two-level spins. However, symmetries can cleave this Hilbert space into independent
subspaces (known as “superselection sectors”) that scale more favorably with system size. One such
symmetry that makes frequent appearances in this thesis and related work is that of permutation
symmetry, in which the dynamics of a spin system are invariant under arbitrary permutations of
its constituent spins.
Regardless of the origin, a spin system that is initially prepared in a PS state (such as a spinpolarized state) and evolves under a PS Hamiltonian remains in the PS manifold. The PS manifoldof N two-level spins is known as the Dicke manifold [30] with spin S = N/2, which is spannedby the set of spin-polarized statesl . However, the set of spin-polarized states is an overcomplete(not to mention non-orthogonal and uncountably large) basis for the PS manifold; a minimal basis
is provided by the Dicke states |mi, which can be classified by their spin projection onto a spinquantization axis: Ŝz |mi = m |mi, with m ∈ N/2, N/2 − 1, · · · , −N/2 . The Dicke state |mi is a
uniform superposition of all states in which exactly N/2 + m ∈ {0, 1, · · · , N } spins point up along
the spin quantization axis:

|mi =

where

N
N/2+m



N
N/2 + m−1/2Xdistinct
permutations
P of |m̃i

P |m̃i ,|m̃i = |↑↑ · · · ↑ ↓↓ · · · ↓i ,| {z } | {z }(2.26)

N/2+m N/2−m
times
times

N!
= (N +m/2)!(N/2−m)!
is a binomial coefficient that is equal to the number of terms inthe sum above. Altogether, the dimension of the PS manifold (namely, N + 1) grows only linearlywith system size N , which makes it amenable to numerical simulations.
The PS manifold is also amenable to visualization and geometric reasoning. A state |ψi within
the PS manifold can be uniquely represented up to global phase by a probability distributionm Qψ
on the sphere, sometimes called the “Husimi distribution” or “Husimi-Q function”, with the valuel

By a spin-polarized state we mean any state in which all spins are polarized along the same axis.

m

For convenience, the “probability distribution” Qψ is normalized to a geometric factor that is determined by N .

20z

Figure 2.4: Husimi distributions for N = 40 spins in a spin-polarized state (left), as well as the
Dicke states |−10i (middle) and |0i (right). A darker color at a point v on the sphere indicatesa larger overlap with the state |vi maximally polarized along v. For visibility, the color scale isnormalized independently on each sphere.

Qψ (v) at a point v = (vx , vy , vz ) on the sphere equal to the overlap of |ψi with a state |vi that is
maximally polarized in the direction of v: Qψ (v) = hv|ψi2n. A spin-polarized state, for example,

corresponds to a Gaussian-like distribution on the sphere, while Dicke states |mi correspond torings at a fixed height along the quantization axis (see Figure 2.4).

2.9

Spin squeezing
A central goal in the field of quantum metrology is the realization of a quantum advantage, inwhich a sensor makes use of uniquely quantum effects such as entanglement to push the state-of-theart in sensing capabilities. Averaging over N independent measurements of a physical parameterθ (for example by using one system to identically measure N times, or by measuring once using√
N independent systems) results in an uncertainty ∆θ ∝ 1/ N . This scaling of uncertainty withmeasurement number N is commonly referred to as the standard quantum limit (SQL), although
the term can be mislading: the SQL is the limit for classical averaging over N independent samplesof θ. As has been known for 15 years now (and suspected for decades prior), the actual limit
imposed by quantum mechanics on measurement precision, known as the Heisenberg limit, is infact ∆θ ∝ 1/N [31]. On a high level, beating the SQL is made possible by quantum correlations
that conspire to reduce the uncertainty in θ: if one system over-estimates the value of θ, anothercorrelated system is (conditionally) more likely to under-estimate it.
n

The Husimi distribution for a mixed state ρ̂ is Qρ̂ (v) = hv|ρ̂|vi.

21
One of the most promising strategies for surpassing classical limits on measurement precision
is known as spin squeezing [32–35]. The paradigmatic setting for spin squeezing is the one-axis
twisting (OAT) model, in which N two-level spins evolve under the Hamiltonian
ĤOAT = χŜz2 ,(2.27)

where χ is the squeezing strength and Ŝz is a collective spin-z operator. Recall that Ŝz generates
P
rotations about the z axis, and has the expansion Ŝz = m m |mihm| into Dicke states |mi thatare labeled by a total projection of spin onto the z axis (namely, m). The OAT HamiltonianP
ĤOAT = m χm Ŝz |mihm| can thus be interpreted as generating rotations about the z axis (byŜz ) at a rate that is proportional to the projection of spin onto the z axis (namely, χm). Living up
to its name, the net effect of ĤOAT is thereby to twist a state about the z axis (see Figure 2.5).
Given an initial state polarized along the equator, represented by a Gaussian-like distribution
on the sphere, the effect of the OAT Hamiltonian is to shear this distribution, resulting in a squeezed
state that has a reduced variance (∆θ)2 along some axis. Here θ is the projection of spin onto an
axis orthogonal to the mean spin direction hŜi = (hŜx i , hŜy i , hŜz i). The reduced variance (∆θ)2
allows for an enhanced sensitivity to rotations of the collective spin state along the squeezed axis.
The amount of squeezing can be quantified by the maximal gain in the resolution ∆θ over that
achieved by a spin-polarized state [34, 35]:
ξ2 ≡(∆θmin )2N= min var(Ŝ · v) ×,2(∆θpolarized )|hŜi|2v⊥hŜi(2.28)

v·v=1

where the minimization is performed over unit vectors v = (vx , vy , vz ) in the plane orthogonal to the2mean spin vector hSi, and var(X̂) ≡ hX̂ 2 i − hX̂i is the variance of X̂. A spin squeezing parameterξ 2 < 1 implies the presence of many-body entanglement that enables a sensitivity to rotations beyond that which is allowed by classical limits on measurement precision (i.e. the standard quantum
limit) [35]. The OAT model can prepare squeezed states with ξ 2 ∼ 1/N 2/3 . Other models, such as
the two-axis twisting model ĤTAT = χ Ŝz2 − Ŝy2 , can prepare squeezed states that saturate theultimate limit on measurement precision that is allowed by quantum mechanics (i.e. the Heisenberg

22
z

t=0

t = 13 tOAT
opt

t = tOAT
opt

Figure 2.5: Representations of the state |ψ(t)i of N = 40 spins initially polarized along the equator,and subsequently evolved under the OAT Hamiltonian ĤOAT for a time t up to the optimal OAT−2/3 . Figure reproduced from Ref. [4].
squeezing time tOAT
opt ∼ Nlimit), namely ξ 2 ∼ 1/N [35]. Note that collective interactions are not necessary for spin squeezing, and that a spin-squeezed state need not lie within the PS manifold. We consider some of theprospects for spin squeezing with power-law interactions in Chapter 6.

Chapter 3

Effective multi-body SU(n)-symmetric interactions of ultracold fermionic atoms
on a 3D lattice

Prologue
One of the exciting aspects of working with cold atomic, molecular, and optical systems is thecapability to study how many-body effects emerge from single- and few-body physics. Understanding the mechanics of this emergence can illuminate universal and non-universal aspects of observed
phenomena, and can help engineer systems of external interest. In this chapter, we take a deep diveinto the theory behind emergent multi-body interactions between ultracold fermions on an opticallattice. These multi-body interactions exhibit an SU(n) symmetry that makes them interestingfrom the standpoint of fundamental physics. The experimental observation of these interactions ismade possible by the exquisite precision of the 3D 87 Sr optical lattice clock, which thereby serves
not only as a cutting-edge timekeeping device, but also as a powerful quantum simulation platform. The bulk of this chapter is taken from Ref. [1], which is a theory paper accompanying theexperimental work of Ref. [7].

Abstract
Rapid advancements in the experimental capabilities with ultracold alkaline-earth-like atoms(AEAs) bring to a surprisingly near term the prospect of performing quantum simulations of spin

24models and lattice field theories exhibiting SU(N ) symmetrya . Motivated in particular by recent experiments preparing high density samples of strongly interacting 87 Sr atoms in a three-dimensional
optical lattice, we develop a low-energy effective theory of fermionic AEAs which exhibits emergent
multi-body SU(N )-symmetric interactions, where N is the number of atomic nuclear spin levels.
Our theory is limited to the experimental regime of (i) a deep lattice, with (ii) at most one atom
occupying each nuclear spin state on any lattice site. The latter restriction is a consequence ofinitial ground-state preparation. We fully characterize the low-lying excitations in our effective
theory, and compare predictions of many-body interaction energies with direct measurements ofmany-body excitation spectra in an optical lattice clock. Our work makes the first step in enabling
a controlled, bottom-up experimental investigation of multi-body SU(N ) physics.

3.1

Introduction
Fermionic alkaline-earth atoms (AEAs), in addition to other atoms such as ytterbium (Yb)sharing similar electronic structure, are currently the building blocks of the most precise atomic
clocks in the world [17, 36, 37]. These atoms have a unique, ultra-narrow optical transition between
metastable 1 S0 and 3 P0 electronic orbital states, i.e. the “clock states”, that allows for coherence
times which can exceed 100 seconds [38, 39]. Furthermore, AEAs can be trapped in fully controllable
optical lattice potentials and interrogated with ultra-stable lasers that can resolve and probe theirrich hyperfine spectra, consisting of N different nuclear spin levels with N as large as 10 in strontium(87 Sr) and 6 in ytterbium (173 Yb).
In 2015 the 87 Sr optical lattice clock (OLC) at JILA, operated in a one-dimensional (1-D)
lattice at microkelvin temperatures, achieved a total fractional uncertainty of 2 × 10−18 [40, 41].
More recently (2017), a new generation of OLCs became operational at JILA, interrogating a Fermi
degenerate gas of 87 Sr atoms in a 3-D lattice at nanokelvin temperatures [42]. All of these atoms’degrees of freedom, including the electronic orbital, nuclear spin, and motional states, can be fullya

To minimize notational collisions and complications, in the bulk of this chapter we denote the number of nuclear
spin states by the capital letter N , rather than the lower-case letter n (as in the rest of this thesis).

25
controlled with high fidelity in a 3-D lattice [43–46]. With frequency measurements reaching the10−19 fractional uncertainty level, the new OLCs are thus enabling an exciting opportunity toprobe, for the first time, quantum dynamics with sub-millihertz spectral resolution [42].
A wonderful consequence of the efforts to build better clocks is the development of highly
controllable quantum simulators of many-body systems in the strongly-interacting regime, whereinter-particle interactions set the largest energy scale relevant for system dynamics [42, 47, 48].
The marriage between precision clock spectroscopy and quantum many-body physics [49–54] hasan enormous potential to enable novel explorations of physics for the same reason that makes
AEAs such remarkable time-keepers. Specifically, due to the lack of electronic orbital angularmomentum in the 1 S0 and 3 P0 states, AEAs exhibit decoupled orbital and nuclear spin degrees of
freedom. For atoms with N nuclear spin levels, this decoupling leads to nuclear-spin-conserving
SU(N )-symmetric interactions governed entirely by orbital-state parameters [49, 53, 55].
The presence of this exotic SU(N ) symmetry in a highly controllable experimental platform
opens the door to experimental studies of e.g. the SU(N ) Heisenberg model, whose phase diagram
is believed to exhibit features such as a chiral spin liquid (CSL) phase with topological order andfractional statistics [56–58]. In addition to illuminating open questions in our understanding of
the fractional quantum Hall effect and unconventional superconductivity [59–61], the CSL can
support non-Abelian excitations which allow for universal topological quantum computation [57,62]. Harnessing the SU(N )-symmetric interactions of AEAs might also enable the simulation ofvarious lattice gauge theories [63, 64], some of which share important qualitative features withquantum chromodynamics such as few-body bound states and confinement [65, 66]. These direct,
quantum simulations have an extraordinary potential to provide novel insights by circumventing
e.g. severe sign problems which plague classical simulations of strongly interacting fermionic systems[63, 67].
In this work, we investigate the first experimental capabilities with ultracold fermionic AEAs
to prepare high-density samples in a 3-D lattice with multiple occupation of individual latticesites [7]. Specifically, we consider ground-state preparation of isolated few-body systems in the26deep-lattice limit, and carry out a bottom-up investigation of emergent multi-body interactions
on multiply-occupied lattice sites. These multi-body interactions appear in a low-energy effective
theory of the atoms, and inherit the SU(N ) symmetry of their bare, pair-wise interactions, thereby
enabling experimental studies of multi-body SU(N ) physics through the exquisite capabilities with
OLCs. Our theory is limited to the experimental regime of at most one atom occupying eachnuclear spin state on any lattice site, which is a consequence of the experimental protocol whichstarts with all atoms in the ground state.
Though effective multi-body interactions have previously been studied in the context of harmonically [68, 69] and lattice-confined [70] neutral bosons prepared in a single hyperfine state,
our work deals for the first time with fermions that have internal degrees of freedom and multiple
collisional parameters. Some past work has detected experimental signatures of multi-body interactions in the form of quantum phase revivals [71]. We instead compare the many-body interactionenergies predicted by our low-energy effective theory to the experimental measurements of thedensity-dependent orbital excitation spectra performed in Ref. [7], similarly to the measurementswith bosons performed in Ref. [72]. To facilitate this comparison of excitation spectra and to characterize the low-lying excitations in our effective theory, we consider a restriction of our theory tostates with at most one orbital excitation per lattice site. In this case, we find that the SU(N ) symmetry of atomic collisions allow the effective multi-body interactions to take a remarkably simple
form.
The remainder of this paper is structured as follows. In Section 3.2 we summarize theexperimental procedures relevant to our work, provide an overview of the one- and two-body physics
of ultracold atoms in a deep lattice, and preview our main technical results. In Section 3.3 wediscuss our method for deriving a low-energy effective theory, provide a perturbative expansion
for the net effective Hamiltonian, and compute all M -body Hamiltonians through third order inthe low-energy effective theory. We then analyze the low-lying excitations of the effective theory in
Section 3.4, comparing spectral predictions with experimental measurements, and study the orbitalstate dynamics of nuclear spin mixtures interrogated via Rabi spectroscopy. Finally, we summarize

27
and conclude our findings in Section 3.5, and provide some discussion of future outlooks.

3.2

Background and overview
The work in this paper is closely tied to the experimental work reported in Ref. [7]; we begin

with a short summary of the relevant experimental procedures therein. The experiment begins
by preparing a degenerate gas of 104 -105 (fermionic) 87 Sr atoms in a uniform mixture of their ten
nuclear spin states and at ∼ 0.1 of their fermi temperature (∼ 10 nanokelvin) [42, 73]. This gas isloaded into a primitive cubic optical lattice at the “magic wavelength” for which both ground (1 S0 )and first-excited (3 P0 ) electronic orbital states of the atoms experience the same lattice potential[22]. Lattice depths along the principal axes of the lattice are roughly equal in magnitude, witha geometric mean that can be varied from 30 to 80 ER , where ER ≈ 3.5 × 2π kHz is the latticephoton recoil energy of the atoms (with the reduced Planck constant ~ = 1 throughout this paper).
These lattice depths are sufficiently large as to neglect tunneling on the time scales relevant to theexperiment. The temperature of the atoms is also low enough to neglect thermal occupation of
motional states outside the ground-state manifold.
Once loaded into an optical lattice, atoms are addressed by an external (“clock”) interrogation
laser with an ultranarrow (26 mHz) linewidth, detuned by ∆ from the single-atom 1 S0 − 3 P0transition frequency ω0 . After a fixed interrogation time, the experiment turns off the interrogation
laser, removes all ground-state (1 S0 ) atoms from the lattice, and uses absorption imaging to countthe remaining excited-state (3 P0 ) atoms. Non-interacting atoms in singly-occupied lattice sites
feature the typical single particle lineshape peaked at ∆ = 0. The lineshapes of multiply-occupied
lattice sites, meanwhile, are shifted by inter-atomic interactions, which results in spectroscopic
peaks (i.e. local maxima in excited-state atom counts) away from ∆ = 0. A sweep across different
detunings ∆ (on the scale of inter-atomic interaction energies) thus constitutes a measurement of
the many-body orbital excitation spectrum. We note that this spectroscopic protocol addresses only
singly-excited orbital states of lattice sites. Doubly-excited states are off resonant due to (i) theinteraction-induced non-linearity (∼kHz) of the orbital excitation energies, and (ii) the ultranarrow28
linewidth (∼mHz) of the interrogation laser.
Although an external trapping potential will generally break discrete translational symmetryof the lattice, any background inhomogeneity can be made negligible by spectroscopically addressing a sufficiently small region of the lattice [7]. Throughout this paper, we work strictly in the
deep-lattice regime with negligible tunneling between lattice sites. We also neglect any lattice inhomogeneities and assume that both atomic orbital states (i.e. 1 S0 and 3 P0 ) experience identicallattice potentials. The single-particle Hamiltonian of the atoms can then be written in the form
H0 =X

En ĉ†inµs ĉinµs ,(3.1)

i,n,µ,swhere ĉinµs is a fermionic operator which annihilates a single atom on lattice site i ∈ Z3 in motional
state n ∈ N30 with nuclear spin µ ∈ {−I, −I + 1, · · · , I} (i.e. projected onto a quantization axis)
and orbital state s ∈ {g, e}; and En is the energy of a single atom in motional state n. In a
harmonic trap approximation we would have En = 3/2 + nx + ny + nz ω for an on-site angular
trap frequency ω, but in general the aharmonicity of the lattice potential will cause a non-negligible
shift in motional state energies.
In the absence of hyperfine coupling, as when addressing the spinless 1 S0 (g) and 3 P0 (e)orbital states of AEAs, interactions between any two atoms are governed by their orbital states
alone, and are therefore characterized by four scattering lengths aX with X ∈ gg, eg− , eg+ , ee ,where the + (−) superscript denotes symmetrization (anti-symmetrization) of a two-body orbital
state under particle exchange. In the low-energy limit, we can write the bare two-body interaction
Hamiltonian in the form [29]
Z
X
XZ
XZ
3
3
†
†
Hint =
Gs d x ρ̂µs ρ̂νs + G+
d x ρ̂µ,e ρ̂ν,g + G−
d3 x ψ̂µ,eψ̂ν,gψ̂ν,e ψ̂µ,g ,
µ<ν
s

µ,ν

(3.2)

µ,ν

where ψ̂µs is a fermionic field operator for atoms with nuclear spin µ and orbital state s; ρ̂µs ≡†ψ̂µsψ̂µs is an atomic density field operator; and the coupling constants GX are defined in terms of

the scattering lengths aY by
Gs=g,e ≡

4π
ass ,
mA

G± ≡

2π
mA


aeg+ ± aeg− ,

(3.3)

29
where mA is the mass of a single atom. Defining for brevity



 Gq
q=r=s=t G+ q 6= r and (q, r) = (s, t)
qr
Gst ≡
,



G− q 6= r and (q, r) = (t, s)






 0
otherwise(3.4)

where q, r, s, t ∈ {g, e} are orbital state indices, we can alternately write the bare two-body interaction Hamiltonian in the more compact form
1 X qrHint =
G
2 q,r,s,t stZ††
d3 x ψ̂µsψ̂νtψ̂νr ψ̂µq .

(3.5)

µ,ν

For nuclear spins µ, ν, the symbol Gqr
st gives the coupling constant between the two-atom states(µ, q) + (ν, r) ↔ (µ, s) + (ν, t).
Note that the Hamiltonian in (3.5) is not the true microscopic interaction Hamiltonian of
AEAs, but rather a generic form for a low-energy effective field theoretic description of two-bodyinteractions [29, 51, 55, 68–70, 74, 75]. There are therefore two important points to keep in mind
concerning our use of (3.5) to describe two-body interactions. First, the use of effective field theory
generically gives rise to divergences that must be dealt with either through regularization, e.g. ofthe zero-range interaction potential implicitly assumed in the expression of (3.5) [76], or throughrenormalization of the coupling constants in the theory. We chose the latter approach, as we will inany case find it convenient to renormalize the coupling constants in the effective theory developed in
Section 3.3. The choice of method to regulate divergences has no effect on the underlying physics.
Second, (3.5) is only the first term in a low-energy expansion of two-body interactions in effective field theory, which generally includes additional terms containing derivatives of field operators.
Derivative terms correspond to the dependence of two-body scattering on the relative momentum kof particles involved, with k → 0 in the zero-energy limit. In the present case of s-wave scattering,
the leading dependence of the two-body interaction Hamiltonian on the relative momentum k can
be captured by use of an energy-dependent pseudo-potential, which amounts to using a k-dependent30
effective scattering length [77]. This effective scattering length can be determined by expanding
the s-wave collisional phase shift in powers of the relative momentum k [76, 78]. Details of thisexpansion will depend on the characteristic length scale of finite-range interactions. In our work,
these corrections to (3.5) will be relevant only for the calculation of two-body interaction energies,
appearing at third order in the coupling constants GX . As we are primarily interested in M -bodyinteractions for M ≥ 3, we defer this calculation to Appendix 3.D. We note that our approach ofusing an unregularized contact potential, renormalizing coupling constants, and separately accounting for momentum-dependent scattering is essentially the same as the approach used for similarcalculations in Refs. [68–70]. While this approach does not provide insight into the microscopic
structure of inter-atomic interactions, it is suitable for the phenomenological description of theseinteractions, and in particular for our eventual development of a low-energy effective theory.
We now expand the field operators ψ̂µs in the Wannier basis for a 3-D lattice, such thatPψ̂µs (x) =i,n φin (x)ĉinµs with spatial wavefunctions φin and fermionic annihilation operatorsĉinµs indexed by lattice sites i and motional states n. Invoking the tight-binding approximation, we
assume that the spatial overlap integral in (3.5) is negligible unless all wavefunctions are localized
at the same lattice site; we discuss the breakdown of this approximation and its consequences for
our low-energy effective theory in Appendix 3.E. The relevant spatial overlap integral is then
k`
Kmn
≡

Z

d3 x φ∗im φ∗in φi` φik ,(3.6)

which for a lattice with discrete translational invariance is independent of the lattice site i. The
two-body interaction Hamiltonian can be written in terms of this overlap integral as
Hint =

1
2

Xi,k,`,m,nq,r,s,t
µ,ν

†
†
k`
Kmn
Gqr
st ĉimµs ĉinνt ĉi`νr ĉikµq ≡1 X k` qr †Kmn Gst ĉmµs ĉ†nνt ĉ`νr ĉkµq ,
2

(3.7)

where for brevity we will henceforth suppress the identical site index (i) on all operators, andimplicitly sum over all free indices in a summand (i.e. indices which do not have a fixed value). Wek` , in which case the
may also at times suppress motional state indices on the overlap integral Kmn

31suppressed indices are implicitly zero (corresponding to a motional ground state); i.e.
`m
Kn`m ≡ Kn,0,m,0Knm ≡ Kn,0,0,0Kmn ≡ Kmn,0,0Kn ≡ Kn,0,0,0K ≡ K0,0.

(3.8)

For simplicity, we will also generally work in a gauge for which all two-body overlap integrals are∗k` = K k`
mnreal, such that Kmn
mn = Kk` . The existence of such a gauge is guaranteed by the analytic

properties of the Wannier wavefunctions φin [79].
Current experiments with 87 Sr can prepare up to five atoms in the same (ground) orbital state
on a single lattice site, and coherently address states with a single orbital excitation per latticesite [7]. At ultracold temperatures well below the non-interacting motional excitation energies∆n ≡ En − E0 for n > 0, atoms only occupy their motional ground state in the lattice. For
this reason, it is common to map the description of these atoms onto a single-band Hubbard
model that captures all dynamics within the subspace of motional ground states of non-interacting
atoms, i.e. with wavefunctions φi,0 . Interactions, however, modify atoms’ motional ground-state
wavefuctions. The true motional ground state of a collection of interacting atoms is then an
admixture of the non-interacting motional eigenstates, and a naive Hubbard model that assumes
atomic wavefunctions φi,0 will fail to reproduce the interacting atoms’ orbital excitation spectrum.
Formally, corrections to the spectrum of interacting atoms can be accounted for by a perturbative
treatment of far-off-resonant terms in the interaction Hamiltonian of (3.7) that create atoms in
excited motional states, e.g. ∼ ĉ†nµs ĉ†0,νt ĉ0,νr ĉ0,µq with n > 0. These corrections can be understood
through interaction-induced virtual occupation of higher bands (i.e. excited motional states), whichbecomes relevant as more atoms occupy the same lattice site, such that their interaction energy
becomes non-negligible compared to the motional excitation energies ∆n .
In order to recast interaction-induced modifications to orbital excitation spectra as corrections
to the simple Hubbard model (i.e. computed using the non-interacting ground-state wavefunctions
φi,0 ), we develop a low-energy effective theory of interacting AEAs in a deep lattice. To simplify
our theory, we assume that any N atoms on a single lattice site occupy distinct nuclear spin states.
This assumption applies for any experimental protocol in which all atoms are initially prepared32
in their orbital and motional ground states (as e.g. in Ref. [7]). In this case, multiple occupationof a single nuclear spin state on any given lattice site is initially forbidden by fermionic statistics.
Subsequent violation of this condition cannot occur in the absence of inter-site effects or hyperfine
coupling between nuclear spin states, as is the case of the experiment in Ref. [7].
Our low-energy effective theory exhibits SU(N )-symmetric multi-body interactions, such thatthe effective interaction Hamiltonian can be written in the form
eff
Hint
=

2I+1
XX

(p)

HM ,

(3.9)

M =2 p≥1(p)where HM is an M -body Hamiltonian of order p in the coupling constants GX , and I is the total
nuclear spin of each atom (e.g. I = 9/2 for 87 Sr). The sum terminates at 2I + 1 because this is
the largest number of atoms which may initially occupy a single lattice site. We explicitly computeP(p)all M -body Hamiltonians HM ≡p HM through order p = 3, yielding effective two-, three-,and four-body interactions. To (i) facilitate a comparison with the experimental measurementsof many-body orbital excitation spectra performed in Ref. [7] and (ii) characterize the low-lying
excitations in our effective theory, we additionally restrict the multi-body Hamiltonians HM tostates with at most one orbital excitation per lattice site. Under this restriction, we find that
the SU(N ) symmetry of atomic collisions allows us to express all multi-body Hamiltonians in thesimple form
X

HM =


(µ ,µ2 )where H2 1µj

=M(µ ,µ2 )

H2 1

M
Y

n̂µα ,g ,(3.10)

α=3

is a two-body Hamiltonian addressing atoms with nuclear spin µ1 , µ2 ; and n̂µs =ĉ†µs ĉµs is a number operator for atoms with nuclear state µ and orbital state s. The sum in(3.10) is performed over all choices of nuclear spins µj with j = 1, 2, · · · , M for which all µj are
distinct, or equivalently all choices of µj for which the set µj contains M elements, for a total of
2I+1
× (M !) nuclear spin combinations. The key feature of the M -body interactions in (3.10) isMthat they ultimately take the same form as two-body interactions, but with the addition of M − 233
spectator atoms. This form is a direct consequence of the SU(N ) symmetry of underlying two-bodyinteractions.

3.3

Low-energy effective theory
The net Hamiltonian H = H0 + Hint for interacting AEAs on a lattice is not diagonalwith respect to single-particle motional state indices (e.g. n ∈ N30 ). The problem of determining
interacting atoms’ orbital excitation spectrum therefore nominally involves all atomic motional
degrees of freedom. At zero temperature, however, each orbital state of a collection of interacting
atoms is associated with a single motional ground state. In order to compute an orbital excitation
spectrum at zero temperature, in principle we need to identify this motional ground state. We canthen ignore all excited motional states, which will be neither thermally occupied nor externallyinterrogated. Such a procedure would drastically reduce the dimensionality of the Hilbert spacenecessary to describe the atoms, thereby greatly simplifying any description of the atoms’ orbital
spectrum and internal (i.e. nuclear and orbital) dynamics. In practice, however, identifying the
motional ground states of interacting atoms and writing down a Hamiltonian restricted to thissubspace is a very difficult process to carry out analytically.
We denote the motional ground-state subspace of the non-interacting Hamiltonian H0 bysingleHground, and the motional ground-state subspace of the interacting Hamiltonian H = H0 + Hint by
multi . That is, all atomic wavefunctions for states within Hsingle are described by φ , while
Hgroundi,0groundmulti are generally unknown, and are in principle
the atomic wavefunctions for states within Hground

determined by minimizing the energy of a state with respect to its motional degrees of freedom.
singlemulti are subspaces of the full Hilbert space H
Both Hgroundand Hgroundfull . When interactions are
singlesufficiently weak compared to the spectral gap ∆ between Hgroundand its orthogonal complement
singleHfull \ Hground, one can identify a particular unitary operator U (acting on the full Hilbert space
multi into Hsingle [80]. This unitary U can be used to construct an effective
Hfull ) which rotates HgroundgroundHamiltonian Heff = U HU † with two key properties: (i) Heff is diagonal in the same (known)
singlebasis as the non-interacting Hamiltonian H0 , and (ii) the spectrum of Heff on Hgroundis identical

34
multi . The use of an effective Hamiltonian H
to that of the interacting Hamiltonian H on Hgroundeffmulti in order to compute the orbital spectrum of H atthus overcomes the need to identify Hgroundzero temperature. This method for constructing an effective theory is commonly known as the
Schrieffer-Wolff transformation, named after the authors of its celebrated application in relating
the Anderson and Kondo models of magnetic impurities in metals [81].
Using the machinery developed in Ref. [80] for performing a rotation between low-energy
subspaces of a perturbed (i.e. interacting) and unperturbed (i.e. non-interacting) Hamiltonian, we
eff in terms two-body interaction
derive an expansion for an effective interaction Hamiltonian Hint

Hamiltonian Hint (see Appendix 3.A). This expansion takes the form
eff
Hint
=

X

(p)

Hint ,(3.11)

p≥1(p)singlewhere Hint is order p in Hint . Letting E0 ≡ Hfull \ Hgrounddenote the orthogonal complement of
singleHground(i.e. E0 is the space of all states with at least one atom in an excited motional state), B0 (E0 )denote an eigenbasis of E0 with respect to the single-particle Hamiltonian H0 , and Eα denote themotional energy (with respect to H0 ) of a state |αi ∈ E0 relative to the corresponding motional
ground-state energy, we define the operator
I≡

X|αi∈B0 (E0 )|αihα|,
Eα(3.12)

which sums over projections onto excited states with corresponding energetic suppression factors.
singleThe operator I together with the projector P0 onto Hgroundallows us concisely write the first few

terms in (3.11) as(1)

Hint = P0 Hint P0 ,(3)

Hint = P0 Hint IHint IHint P0 −(2)

Hint = −P0 Hint IHint P0 ,(3.13)i1h
P0 Hint P0 , P0 Hint I 2 Hint P0 ,2+

(3.14)

where [X, Y ]+ ≡ XY + Y X. Writing down a single-band Hubbard model that simply neglects
excited atomic motional states and uses Hint directly to describe the orbital spectrum of interacting

35
eff at first order. In addition to this firstatoms is thus equivalent to truncating our expansion for Hint

order term, the expansion involves effective corrections to the action of Hint on the non-interacting
singlemotional ground states (i.e. on Hground) in the form of higher-order terms with intermediate or

virtual occupation of excited states, manifest in I.
Substituting the definition of I into (3.13)-(3.14) yields expressions that are highly reminiscent of standard non-degenerate perturbation theory in quantum mechanics, but which nonethelessexhibit crucial differences. The first, and most obvious difference is that these expressions are operator equations, and that the sums over virtual states are performed over a basis for the orthogonal
singlecomplement of the subspace Hground, rather than a basis for the orthogonal complement of a single

state, as in non-degenerate perturbation theory. Second, the non-degeneracy condition in standard
perturbation theory is here elevated to a restriction on the magnitude of the perturbation Hint
singlerelative to the spectral gap ∆ of the non-interacting Hamiltonian H0 between Hgroundand theexcited subspace E0 . Specifically, the validity of (3.13)-(3.14) is conditional only on kHint k ≤ ∆/2,p
where kXk ≡ max|ψi∈H hψ|X † X|ψi is the operator norm, with no restrictions on spectral gaps or
singledegeneracies within Hground[80, 82]. Finally, the effective theory involves no corrections to the non-interacting many-body energy eigenstates; the purpose of constructing the effective Hamiltonian
eff is to reproduce, on the known eigenstates of the non-interacting Hamiltonian H
Heff = H0 + Hint0
singlemulti . “Correctwithin Hground, the spectrum of the interacting Hamiltonian H = H0 + Hint on Hgroundsingleing” the eigenstates of the non-interacting Hamiltonian H0 on Hgroundthus invalidates the effective

theory.
As a last comment, we note that our chosen method for constructing an effective Hamiltonianis distinct from adiabatic elimination methods which are commonly used in the atomic physics andquantum optics communities to develop effective theories for e.g. the low-lying levels of a Lambda
system [83–86]. Unlike the perturbative, but exact Schieffer-Wolff transformation, adiabatic elimination methods use approximations which rely on the fast dynamics of excited states. While
generally reasonable, these approximations must be made carefully to avoid potential problems
with self-consistency (see Section 3 of Ref. [83]), and yield no obvious or straightforward means

36
to compute effective corrections beyond second order in the couplings between low- and highenergy sectors of a Hilbert space [84–86]. While at least one attempt at systematically computing
higher-order corrections in the framework of adiabatic elimination has recently been made [86], theresulting expressions do not lend themselves as nicely to analytical treatment, and were in any casefound by the authors to be equivalent to a Schrieffer-Wolff transformation.

3.3.1

Diagrammatic representation of effective Hamiltonians

The form of the bare two-body interaction Hamiltonian Hint in (3.7) motivates a diagrammatic representation of terms in the effective Hamiltonian Heff in (3.11), similarly to the diagrams
used to represent elements of the scattering matrix in standard quantum field theory. Effective
M -body interaction terms at order p in Hint can be represented by directed graphs containing p
vertices with degree greater than one, which we call internal vervices. Each internal vertex and its
associated edges correspond respectively to a coupling constant and the associated field operators
in Hint . An example 2-vertex diagram representing an effective 3-body interaction term is provided
in Figure 3.1. All diagrams are read from left to right to construct a sequence of operators fromright to left; the internal vertices of a diagram are thus strictly ordered, with the n-th internalvertex from the left corresponding to the n-th interaction Hamiltonian Hint from the right in (3.13)or (3.14). Solid (dashed) lines represent field operators acting on the lowest (arbitrary) motional
states. Spatial overlap factors at each vertex are determined by the motional states of the edges
which connect to (i.e. field operators associated with) that vertex. While it is possible to constructexplicit rules for determining the energetic suppression factors (i.e. from I) of the term represented
by a diagram, these factors are most easily determined by examination of the effective Hamiltoniansin (3.13) and (3.14).
The diagram in Figure 3.1 explicitly labels all edges with indices of the corresponding field
operators, but in general we may suppress these indices, in which case the diagram includes a sum
over the suppressed indices. These sums are performed over all allowed values of the suppressedindices, with the restriction that virtual states (i.e. vertical slices of the diagram between internal

37

µ, g
Gge
ge

µ, g
nν, e

ν, g


= P0

Geg
ge
=

ν, e

ρ, g

ρ, e

1
†
†
Kn Gegge ĉν,g ĉρ,e ĉρ,g ĉnν,e
2



1
En




1
ge †
†
Kn Gge ĉµ,g ĉnν,e ĉν,e ĉµ,g P02

1 Kn2 ge eg
Gge Gge P0 ĉ†µ,g ĉ†ν,g ĉ†ρ,e ĉρ,g ĉν,e ĉµ,g P0
4 En

Figure 3.1: An example second-order diagram and the corresponding three-body interaction term(2)
in Hint , with n > 0 and ĉµs ≡ ĉ0,µs . Diagrams are read from left to right to construct a sequenceof operators from right to left. Solid (dashed) lines represent field operators acting on the lowest(arbitrary) motional states. For the sake of presentation, this diagram has colors associated with
nuclear spin and orbital states, an arrow on each line to emphasize that they are directed left-toright, and an explicit coupling constant written next to each vertex; we will generally not include
these features, as they are not necessary to uniquely identify the term represented by a diagram.
We will also drop explicit appearances of the ground-state projector P0 in our expressions, withthe understanding that the low-energy effective theory implicitly addresses only non-interacting
motional ground states.

38
vertices) represented with dashed lines must have at least one motional excitation. While we include
factors of 1/2 from Hint as expressed in (3.7) in the definition of a diagram, in all but the twobody case these factors of 1/2 will be cancelled out by corresponding symmetry factors, i.e. theappearance of duplicate diagrams which are equal up to a relabeling of indices (see Appendix 3.B).
The explicit signs and factor of 1/2 which appear in the effective Hamiltonians in (3.13) and (3.14)
are not included in the definition of a diagram, and must be kept track of manually.

3.3.2

Effective two-body interactions and renormalization

The effective two-body Hamiltonian in (3.11) has contributions at all orders in the coupling
constants, and can be expanded in the form
H2 =

−

+

+ ··· ≡,

(3.15)

where the blob on the right schematically represents the net effective two-body interaction. On
physical grounds, the net two-body interaction must clearly be finite, but individual sums over excited states in the loop diagrams of (3.15) may generally diverge [68]. These divergences ultimately
appear due to our use of effective field theory to describe inter-atomic interactions in (3.2), (3.5),and (3.7), rather than a detailed microscopic description of two-atom scattering. Divergences ofthis sort are a generic feature of field theories, and can be dealt with using standard techniques suchas renormalization. We therefore renormalize our coupling constants by introducing counter-termse X into the interaction Hamiltonian.
G
The introduction of counter-terms is merely a formal decomposition of the “bare” couplinge X . In performing such a decomposition,
constants Gbare
that are used in (3.15) as Gbare= GX + GXX
e X ≡ Gbare − GX . For
we are free to choose the values of GX , which in turn fixes the values of G
X
convenience, we can choose the values of GX to be those of the net effective coupling constants on
the right-hand side of (3.15). Representing the new coupling constants GX by regular vertices and39e X by a crossed dot (i.e. ⊗), this choice leads to the renormalization condition
the counter-terms G
−

+
O(G)

e
O(G)

+ ··· =
O(G2 ).

(3.16)

O(G)This renormalization condition has the benefit of allowing us to express effective two-body interactions simply in terms of net effective two-body coupling constants, rather than in terms of longsums at all order of the bare coupling constants. By construction, the counter-terms we have introduced exactly cancel all terms beyond leading order in (3.15), which implies that the effective
two-body interaction Hamiltonian is simply
H2 =

1 (1)
= α2
2

X

† †
Gqr
st ĉµs ĉνt ĉνr ĉµq ,(3.17){µ,ν} =2
(1)

where for consistency with existing literature [68] we define α2 ≡ K as the overlap integral between
two atoms occupying non-interacting motional ground states.
Before moving on to consider effective three-body interactions, there are a few comments wemust make concerning renormalization and the result in (3.17). First, the effective two-body interaction H2 in (3.17) takes the same form as the bare two-body interaction Hint in (3.7), but without
excited motional states, and with renormalized coupling constants. Our choice of renormalization
scheme, while convenient for the analytical development of a low-energy effective theory, no longer
allows us to use the coupling constants GX as defined by the free-space scattering lengths aX in(3.3) to compute interaction energies. The renormalization condition in (3.16) explicitly fixes GX
to the net effective coupling constants in any given setting. Instead of using free-space coupling
constants to compute interaction energies in a lattice, we must therefore first compute the effective
coupling constants Glattice(U), which now depend on the lattice depth U, and in turn use theseXeffective coupling constants to compute interaction energies. We discuss the calculation of effective
coupling constants in Appendix 3.C.
e X are second
Second, the renormalization condition in (3.16) implies that the counter terms G
e X ∼ G2 . Although the effective Hamiltonian expansions
order in the coupling constants GX , i.e. GXin (3.13) and (3.14) are organized in powers of the interaction Hamiltonian Hint , the couplings GX

40
are the “small” parameters in which we can formally organize a perturbation theory; more specifically, the formally small quantities organizing our perturbation theory are two-body ground-stateinteraction energies (proportional to the couplings GX ) divided by the spectral gap of the singleatom Hamiltonian H0 (see Appendix 3.F). If M atoms can only couple through terms represented
by a p-vertex diagrams for p ≥ pmin
M , then the leading order contribution to M -body interactions is
minorder pmin
M in the couplings GX . If the same pM -vertex diagrams involve any counter-terms, how-ever, then these diagrams are at least order pmin
M + 1 in the couplings GX . Counter-terms therefore
only appear at next-to-leading order (NLO) in the calculation of effective M -body interactions.
Finally, our result in (3.17) neglects the effect of momentum-dependent two-body scattering.
When the effective range of interactions is comparable to the scattering lengths aX , as is the casefor ultracold 87 Sr, these momentum-dependent effects are third order in the coupling constants

GX . Just as the O G2 counter-terms do not affect M -body interactions until next-to-leadingorder (NLO), the O G3 momentum-dependent terms do not come into play until next-nextleading order (NNLO). Given that we develop our low-energy effective theory through third orderin the coupling constants, these interactions will not appear in any of our three- and four-body
calculations, but they do need to be considered in the calculation of pair-wise interaction energies.
The primary interest of our work, however, concerns effective M -body interactions for M ≥ 3; wetherefore defer the calculation of momentum-dependent two-body interactions to Appendix 3.D.

3.3.3

Effective three-body interactions at second order

Our theory of effective multi-body interactions assumes no non-universal contribution to thethree-body interaction energy, which is to say that we assume the absence of real (as opposed toeffective), bare three-body interactions. Consequently, three-body interactions do not appear until(2)

second order in the coupling constants of the effective theory, in the expansion of Hint in (3.13).
(2)

The virtual state of three-body terms in Hint cannot have two atoms in excited motional states,(2)

as otherwise the second application of Hint in Hint would have to address both of theses atoms
to bring them back down to the ground state, resulting in a two-body process as in the second

41
diagram of (3.15). All second-order three-body terms must therefore have only one excited atomin the virtual state, and take the form
µr0

µr

νs00

nνs0

0

†
†
st †
∝ Kn2 Grsr0 s0 Gs00 t0 ĉµr0 ĉνs00 ĉρt0 ĉρt ĉνs ĉµr .

νs(3.18)

ρt0

ρtUnlike for the two-body diagram in (3.17), the explicit factors of 1/2 which appear in the bare
two-body Hamiltonian Hint in (3.7) are now cancelled out by symmetry factors which account for
duplicate diagrams; this cancellation will generally occur for all connected M -body diagrams with
M > 2 (see Appendix 3.B). The net effective three-body interaction Hamiltonian at second orderis then given by the sum over all diagrams of the form in (3.18), i.e.
(2)

H3

=−

(2)

X

= −α3

0

†
†
st †
Grs
r0 s0 Gs00 t0 ĉµr0 ĉνs00 ĉρt0 ĉρt ĉνs ĉµr ,(3.19){µ,ν,ρ} =3(2)

where α3 ≡
3.3.4

P

(2)
2
n>0 Kn /En , and the preceding minus sign is as prescribed by Hint in (3.13).

Effective three-body interactions at third order(3)

The third-order effective interaction Hamiltonian Hint in (3.14) contains both three- andfour-body terms. To compactly enumerate and evaluate all three-body diagrams at third order, we
introduce an expanded coupling symbol




Gqr

st


Gµq;νr−Gqr
ρs;σt ≡
ts





 0

(µ, ν) = (ρ, σ)(µ, ν) = (σ, ρ)(3.20)

otherwise

for more general (µ, q)+(ν, r) ↔ (ρ, s)+(σ, t) coupling induced by terms proportional to ĉ†ρs ĉ†σt ĉνr ĉµq .
The minus sign in (3.20) accounts for fermionic statistics: if (µ, ν) = (σ, ρ), then we are consideringa term of the form
qr † †
qr † †† †Gµq;νrνs;µt ĉνs ĉµt ĉνr ĉµq = −Gts ĉνs ĉµt ĉνr ĉµq = Gts ĉµt ĉνs ĉνr ĉµq .

(3.21)

42
At the cost of introducing an additional sum over new nuclear spin indices, the expanded coupling
symbol allows us to collect together diagrams which have the same graph topology, but represent
different matrix elements of the effective Hamiltonian due to the exchange of nuclear spins at a(3)

vertex. The third order three-body diagrams in Hint are then
µr00

µr
`µr0

νs

ν 0 s000mνs0

nν 0 s00

ρt

ρ0 t 0µr

0 00(3.22)

µr00

`µr0

nνs00
νs

0

νs ;ρt†††rs∝ K`m Knm K`n Grs
r0 s0 Gν 0 s00 ;ρ0 t0 Gr00 s000 ĉµr00 ĉν 0 s000 ĉρ0 t0 ĉρt ĉνs ĉµr ,

νs000

0 0

00

†
†
rs
s t †
∝ K`m Kn`m Kn Grs
r0 s0 Gr00 s00 Gs000 t0 ĉµr00 ĉνs000 ĉρt0 ĉρt ĉνs ĉµr ,mνs0(3.23)

ρt0

ρt

(3)

and the mirror image of (3.23). As prescribed by Hint in (3.14), these diagrams have an associatedminus sign if they contain only one excited virtual state, and a factor of 1/2 if they contain a virtual
ground state. Remembering that counter-terms are O G2 , there are additionally two third-order(2)

three-body diagrams in Hint , namely
µr0

µr

νs00

nνs0
ρt

νs

e rs0 0 Gs000t 0 ĉ† 0 ĉ† 00 ĉ† 0 ĉρt ĉνs ĉµr
∝ Kn2 Gr s s t µr νs ρt(3.24)

ρt0

e qr is equal to the counter-term associated with Gqr .
and its mirror image, where Gstst
The net contribution to the third-order three-body interaction Hamiltonian from threeparticle-loop diagrams of the form in (3.22) is

−

1
2

−

1
2



(3)
(3)
(3)
= α3,1 − α5 H3,1 ,(3.25)

43
where
X K`m K m K`nn,E`m E`n(3)

α3,1 ≡(3)α5 ≡ K

X K2

n

n>0

`+m>0
`+n>0

En2

,

(3.26)

and(3)

H3,1 ≡

X

0

0 00

νs ρt
†
†
†
rs
Grs
r0 s0 Gν 0 s00 ρ0 t0 Gr00 s000 ĉµr00 ĉν 0 s000 ĉρ0 t0 ĉρt ĉνs ĉµr .

(3.27){µ,ν,ρ} =3
(3)

(3)

Even though this contribution comes from loop diagrams, the factors α3,1 and α5in (3.26) are

finite. At large motional state indices n, atoms become free particles for which n essentiallyindexes discrete momentum states. These atoms thus have an energy which asymptotically scales
as En ∼ n2 ≡ n2x + n2y + n2z . Furthermore, the oscillatory behavior of atomic wavefunctions withincreasing motional state indices `, m implies that the overlap integral K`m becomes sharply peaked(3)at ` ≈ m as ` and m get large. The asymptotic behavior of α3,1 at large `, m, n is therefore(3)
α3,1 ∼

Zd3 ` d 3 m d3 n δ (` − m) δ (` − n) ∼`2 + m2 `2 + n2

Z

d3 `
∼
`4

Z ∞

d`
1
∼
,
2
`min
`min `(3.28)

where in the last integral we changed to spherical coordinates, and `2min is the minimum value of `2
for which (i) the energy E` ∼ `2 , and (ii) the integral expression in (3.28) is a good approximation
to the corresponding sum in (3.26). Note that the introduction of `min amounts to neglecting a(3)

finite number of terms in the sum over `, m, n in (3.26), whose contribution to the value of α3,1 is(3)finite. Convergence of α5

is similarly guaranteed by the fact that the overlap integral Kn does(3)

not asymptotically grow with increasing n, such that α5 asymptotically behaves as
(3)
α5 ∼

Z

d3 n
∼
n4

Z ∞

dn
1
∼
,
2
nmin
nmin n(3.29)

where again nmin is defined similarly to `min .
The sum over loop diagrams in (3.23), meanwhile, contains a divergence that must be cancelled out by the counter-terms in (3.24). To leading order in the coupling constants, the renormalization condition in (3.16) implies that=,

(3.30)

44
which can be expanded to finde rs00 00 =
KG
r s

X K2
0 0
`m rs
Gr0 s0 Grr00ss00 .
E`m
0 0(3.31)`,m,r ,sIn terms of ordinary coupling constants, the counter-term diagram in (3.24) is therefore
µr00

µr

νs000

nνs00

2 K2
K`m
†
†
n
r 0 s0s00 t †
Grs
r0 s0 Gr00 s00 Gs000 t0 ĉµr00 ĉνs000 ĉρt0 ĉρt ĉνs ĉµr .
KE
E
n
`m
0 0

X

=

(3.32)`,m,r ,s

νs

ρt0

ρt

Altogether, the contribution to the third-order three-body interaction Hamiltonian from loop diagrams of the form in (3.23) and counter-term diagrams of the form in (3.24) is

−

−

1
2

−

1
2

=

1 (3) 1 (3)(3)α3,2 − α4,3 − α5
2

2



(3)

H3,2 , (3.33)

where
(3)
α3,2 ≡

X K`m Kn K`m Kn`m
Kn −,E`m EnK(3)

α4,3 ≡ K

X K2
mn
,
2
Emn

(3.34)

m+n>0

`+m>0
n>0

and
(3)

H3,2 ≡

X

0 0

00

†
†
rs
s t †
Grs
r0 s0 Gr00 s00 Gs000 t0 ĉµr00 ĉνs000 ĉρt0 ĉρt ĉνs ĉµr .

(3.35){µ,ν,ρ} =3

An equal contribution comes from the mirror images of these diagrams, such that the net third-orderthree-body interaction Hamiltonian is
(3)

H3





(3)
(3)
(3)
(3)
(3)
(3)
(3)
= α3,1 − α5 H3,1 + 2α3,2 − α4,3 − α5 H3,2 .

(3.36)(3)

Note that the aforementioned divergence and its cancellation are buried in α3,2 . Formally, thisfactor is calculated by imposing an ultraviolet cutoff Λ for the maximum values of motional state
indices `, m, n, and then taking the limit Λ → ∞. This procedure ensures that there are no(3)

divergences in α3,2 .

45
3.3.5

Effective four-body interactions at third order

At third order in the coupling constants, we have four-body terms of the form
ρs0

ρs
µq
mµq 0
nνr0

µq 00

0

0

qs
†
†
†
rt †
∝ Kmn Km Kn Gqr
q 0 r0 Gq 00 s0 Gr00 t0 ĉµq 00 ĉνr00 ĉρs0 ĉσt0 ĉσt ĉρs ĉνr ĉµq(3.37)

νr00

νr

σt0

σt

and its mirror image, as well as
νr0

µq

mµq 0

ρ0 s 0nµ0 q 00

νr

σt0

0

00

µq ρs
q t †
†
†
†
∝ Km Knm Kn Gqr
q 0 r0 Gµ0 q 00 ρ0 s0 Gq 000 t0 ĉµ0 q 000 ĉνr0 ĉρ0 s0 ĉσt0 ĉσt ĉρs ĉνr ĉµq .

(3.38)

ρsµ0 q 000σt

As we are computing the leading-order contribution to effective four-body interactions, there are nocounter-terms contributions. In principle, there is now also the possibility to make the disconnected
diagrams of the form

,

,

and

.

(3.39)(3)

As prescribed by Hint in (3.14), however, the second and third of these diagrams pick up a factorof −1/2, so the sum over disconnected diagrams vanishes.
The contribution to the third-order four-body interaction Hamiltonian from diagrams of theform in (3.37) is
1
−
2



1 (3)
(3)
= α4,1 − α5H4,1 ,
2(3.40)

46
where
(3)

α4,1 ≡X Kmn Km Kn
Emn Enm≥0
n>0,(3.41)

and
(3)

0

0

qs
†
†
†
rt †
Gqr
q 0 r0 Gq 00 s0 Gr00 t0 ĉµq 00 ĉνr00 ĉρs0 ĉσt0 ĉσt ĉρs ĉνr ĉµq .

X

H4,1 ≡(3.42){µ,ν,ρ,σ} =4

An equal contribution comes from the mirror images of these diagrams. The contribution fromdiagrams of the form in (3.38), meanwhile, is

−

1
2

−

1
2



(3)
(3)
= α4,2 − α5 H4,2 ,(3.43)

where
X Km K m Kn
n
,
Em En

(3)

α4,2 ≡(3.44)

m,n>0

and
(3)

H4,2 ≡

X

0

00

µq ρs
q t †
†
†
†
Gqr
q 0 r0 Gµ0 q 00 ρ0 s0 Gq 000 t0 ĉµ0 q 000 ĉνr0 ĉρ0 s0 ĉσt0 ĉσt ĉρs ĉνr ĉµq .

(3.45){µ,ν,ρ,σ} =4

The net third-order four-body interaction Hamiltonian is therefore
(3)

H4

3.4





(3)
(3)
(3)
(3)
(3)
(3)
= 2α4,1 − α5 H4,1 + α4,2 − α5 H4,2 .

(3.46)

Low-excitation Hamiltonians, eigenstates, and spectra
Current experiments with ultracold 87 Sr on a lattice can coherently address ground statesand single orbital excitations of up to five atoms per lattice site [7]. Due to the SU(N ) symmetry ofinter-atomic interactions, manifest in the fact that all coupling constants are independent of nuclearP(p)spin, a restriction of the M -body Hamiltonians HM = p HM to the subspace of experimentally
addressed states takes the form
X

HM =


µj

=M



UM,g n̂µ1 ,g n̂µ2 ,g + UM,+ n̂µ1 ,e n̂µ2 ,g + UM,− ĉ†µ1 ,g ĉ†µ2 ,e ĉµ2 ,g ĉµ1 ,e

M
Y
α=3

n̂µα ,g ,(3.47)

47
where n̂µs ≡ ĉ†µs ĉµs is a number operator, and the coefficients UX can be determined from the(p)coupling constants GY and prefactors αZ of the effective M -body Hamiltonians derived in Section
3.3 (see Appendix 3.G). For a lattice with N ≥ M atoms occupying nuclear spins N = µj for
j = 1, 2, · · · , N , the M -body Hamiltonian HM has a single ground state |N , 0i, and a singly-excitedstate |N , +i which is fully symmetric in the orbital degrees of freedom; these states are


Y
1 X †
|N , 0i ≡ ĉ†µ,g  |vacuumi ,|N , +i ≡ √ĉµ,e ĉµ,g |N , 0i .
N µ∈N
µ∈N

(3.48)As these states are fully symmetric in their orbital degrees of freedom, they are anti-symmetric in
their nuclear spin degrees of freedom, forming an SU(N ) singlet. Furthermore, the symmetric stateis particularly interesting as its orbital degrees of freedom form an N -body entangled W state,
which belongs to a special class of multi-partite entangled states that are robust against disposal orloss of particles. This state thus constitutes an important resource for many quantum information
processing and quantum communication tasks [87].
In addition to the states in (3.48), the multi-body Hamiltonian HM in (3.47) has an (N − 1)fold degenerate excited-state eigenspace which is asymmetric in the orbital degrees of freedom,spanned by the states
1 |N , −, ji ≡ √ ĉ†µ1 ,e ĉµ1 ,g − ĉ†µj ,e ĉµj ,g |N , 0i
2(3.49)

for j = 2, · · · , N . If N > 2, the asymmetric states are not separable in their orbital and nuclear
spin degrees of freedom. An important feature of the excited states in (3.48) and (3.49) is that they
are entirely independent of M , which implies that the effect of multi-body interactions is simply
to modify the many-body atomic energy spectra without affecting the energy eigenstates. The(M )eigenvalues EN X = hN X|HM |N Xi of HM associated with each of the eigenstates in (3.48) and(3.49) are provided in table 3.1, both in terms of the coefficients UM X of HM as expressed in (3.47)(M )and the M -body eigenvalues EM X . Due to the SU(N ) symmetry of the multi-body Hamiltonian(M )HM , the eigenvalues EN X depend on the number of nuclear spins on a lattice site, N , but not on
the actual nuclear spins µ ∈ N which are occupied. The total N -body interaction energies EN X
P
(M )
(M )
are given in terms of the M -body eigenvalues EN X by EN X = M EN X .

48

Table 3.1: Low-excitation eigenvalues of M -body Hamiltonians HM . Many-body energy eigenstates
are labeled by the nuclear spins they occupy (i.e. N with N ≡ |N |) and whether they are in anorbital ground (0), singly-excited symmetric (+), or singly-excited asymmetric (−) state. The(M )corresponding N -body eigenvalues EN X of HM are given in terms of the coefficients UM X as
(M )appearing in (3.47) (first three rows), in addition to the M -body eigenvalues EM X (last threerows).
Eigenstate|N , 0i|N , +i|N , −i|N , 0i|N , +i|N , −i

HM eigenvalue (M ≤ N )

N
M! M
UM,g



N −1
N −1
M ! M UM,g + (M − 1)! M
UM,+ + UM,−
−1



N −1
N −2
M ! NM−1 UM,g + (M − 1)! M−1 UM,+ − (M − 2)! M −2 UM,− (M )NE
 (MM) M,0 (M )N −1
N −1
M,0 + M
M E
−1 EM,+hih (M )
 (M )
 N −1 i (M )
N −1
N −1
N −2
−1
−1
−1 N −2
E
+
M
−
E
+
1
−
M
+
M
EM,−
M,0
M,+
M
M −1
M −2
M −1
M −2

49
3.4.1

Many-body state spectroscopy

Spectroscopic interrogation is a powerful means to probe the internal structure and dynamics
of a system under examination. Consequently, we consider Rabi spectroscopy of the low-lying
energy eigenstates in multiply-occupied lattice sites. If we interrogate a lattice site by a laser
red-detuned by ∆ from the single-atom orbital state excitation energy, we realize the Hamiltonian
HRabi =X

EX PX +XX

∆Tµz + Ωµ Tµx ,(3.50)µ

eff , P is a projector onto thewhere EX is an eigenvalue of the effective interaction Hamiltonian Hint
X

corresponding eigenspace, and
Tµz ≡


1 †
ĉµ,e ĉµ,e − ĉ†µ,g ĉµ,g ,
2

Tµx ≡


1 †
ĉµ,e ĉµ,g + ĉ†µ,g ĉµ,e ,2(3.51)

are single-atom pseudospin operators. The Rabi frequency Ωµ is proportional to the ClebschGordan coefficient hI, µ; 1, 0|I, µi ∝ µ for a photon-induced nuclear-spin-conserving orbital statetransition of an atom with nuclear spin µ. We therefore define the “bare” Rabi frequency Ω0 ≡ Ωµ /µ
to explicitly factor out dependence on nuclear spins µ.
Consider now a single lattice site in the orbital ground state |N , 0i with nuclear spins N ≡µj

for j = 1, 2, · · · , N . If we red-detune the interrogation laser by δ from a many-body orbital

state excitation energy, i.e. set ∆ = ∆N X − δ for ∆N X ≡ EN X − EN,0 and X ∈ {+, −}, then in
the subspace of the target states |N , 0i , |N Xi the Hamiltonian in (3.50) becomeszx
HN X = δSN
X + ΩN X SN X ,(3.52)where
1|N XihN X| − |N , 0ihN , 0| ,
2

1
x
SN
|N XihN , 0| + |N , 0ihN X| ,
X ≡
2
z
SN
X ≡

(3.53)(3.54)are many-body pseudospin operators, and the Rabi frequencies ΩN X are determined by
1 X †
1
1
HRabi |N , 0i = Ω0µĉµ,e ĉµ,g |N , 0i = ΩN ,+ |N , +i + ΩN ,− |N , −i .
222µ

(3.55)

50
While the symmetric excited state |N , +i is given in (3.48), at this point we have not explicitly
solved for the asymmetric excited state |N , −i. The asymmetric state is implicitly defined by(3.55), and lies somewhere in the span of the N − 1 asymmetric states given in (3.49). Determining
the symmetric-state Rabi frequency ΩN ,+ is simply a matter of projecting the expression in (3.55)
onto |N , +i, which yieldsΩN ,+ = hN , +|Ω0X

µĉ†µ,e ĉµ,g |N , 0i = Ω0µwhere µ̄N ≡X µ√√ = Ω0 N µ̄N ,Nµ∈N(3.56)Pµ∈N µ/N is the average nuclear spin µ ∈ N . In order to determine the asymmetric-

state Rabi frequency ΩN ,− , we rearrange (3.55) to findΩN ,− |N , −i = Ω0Xµĉ†µ,e ĉµ,g |N , 0i − ΩN ,+ |N , +i = Ω0µX(µ − µ̄N ) ĉ†µ,e ĉµ,g |N , 0i .

(3.57)µ∈N

Denoting the standard deviation of nuclear spins µ ∈ N by σN , normalization of |N , −i thus
determines the asymmetric-state Rabi frequency
1/2ΩN ,− = Ω0 X(µ − µ̄N )2 √
= Ω 0 N σN ,(3.58)µ∈N

which in turn implies that the asymmetric excited state |N , −i is
1 X|N , −i = √
N µ∈N



µ − µ̄NσNĉ†µ,e ĉµ,g |N , 0i .

(3.59)√
Figure 3.2 shows multiplicities of the magnitudes of reduced Rabi frequencies ωN X ≡ ΩN X /Ω0 N
in a lattice with a uniform mixture of nuclear spins with I = 9/2 for single-site occupation numbers
N which are achievable in current 87 Sr experiments [7]. On average, asymmetric-state Rabi frequencies are greater in magnitude, which becomes more pronounced for larger single-site occupation
numbers.

3.4.2

Experimental signatures and comparison

We now consider samples of 87 Sr atoms in a uniform mixture of nuclear spins µ ∈ {−I, −I + 1, · · · , I}
prepared in motional ground states of a rectangular lattice with depths U = Ux , Uy , Uz =

51

X=−

X=+

N =2

N =3

Multiplicity

9

20

6
10
3
0

0

N =4

N =5
40

Multiplicity

30
20

20
10
0

0
0

1/3

2/3
|ωN ,X /I|

1

0

1/3

2/3

1

|ωN ,X /I|√
Figure 3.2: Multiplicities of the magnitudes of reduced Rabi frequencies ωN X ≡ ΩN X /Ω0 N in a
lattice with a uniform mixture of nuclear spins with I = 9/2 and single-site occupation numbers
N which are achievable in current 87 Sr experiments.

52

2−

2+
4−5−

3+

4+

5

+

2

−

2+

3−
4−
5−

3

3

3

−

1

P0 population

P0 population

1

3+
4+

5+

0

0
−2

0

2
4
6
8
Detuning ∆ (2π kHz)

10

−2

0

2
4
6
8
Detuning ∆ (2π kHz)10

Figure 3.3: Population (in arbitrary units) of the excited 3 P0 orbital state of 87 Sr atoms in a uniform
mixture of nuclear spins. Atoms are prepared in the ground state of a lattice with depth U = 54ER ,where ER ≈ 3.5×2π kHz is the lattice photon recoil energy of the atoms, and interrogated by a laser
with Rabi frequency ΩI = 50 × 2π Hz for a time t = π/ΩI . (Left) Populations predicted by the
low-energy effective theory (with s-wave scattering parameters retrieved from Ref. [53]), averaged
over all nuclear spin combinations of N ∈ {1, · · · , 5} atoms per lattice site for a fixed total atom
number. (Right) Experimental measurements of 3 P0 populations retrieved from Ref. [7], withLorentzian fits to each peak as a visual guide. Resonance peaks are identified by the many-bodyorbital states which are excited at the peak.

53(41, 55, 69) ER , where ER ≈ 3.5 × 2π kHz is the lattice photon recoil energy of the atoms. Suchsamples can be prepared in experiments which can vary the single-site occupation number N , andwhich can control for the total number of atoms that are addressed by an external interrogation
laser. Figure 3.3 shows the population of the excited 3 P0 orbital state when these atoms are interrogated for a time t = π/ΩI by a laser with Rabi frequency ΩI = 50 × 2π Hz (i.e. for individual
atoms with nuclear spin µ = I) and detuning ∆ from the single-atom 1 S0 → 3 P0 orbital excitationenergy. The 3 P0 population peaks when the laser detuning ∆ is equal to the many-body excitationenergy ∆N X ≡ EN X − EN,0 for X ∈ {+, −}, as this is precisely when the on-resonance condition
δ = 0 is satisfied in the many-body Rabi Hamiltonian HN X in (3.52). Due to experimental uncertainties which vary with single-site occupation number N , the heights of experimental peaks in
Figure 3.3 are not well-calibrated between different values of N . Nonetheless, Figure 3.3 exhibits
signatures of larger asymmetric-state Rabi frequencies than symmetric-state ones in the form of
higher asymmetric-state peaks for fixed N .
Identifying peaks in excitation spectra such as in Figure 3.3 constitutes a measurement of
many-body excitation energies, which was performed in Ref. [7] to detect signatures of effective
multi-body interactions. Figure 3.4 shows a comparison between (i) experimental measurementsof the many-body excitation energies ∆N X for all (N, X) ∈ {3, 4, 5} × {+, −} at various mean
lattice depths U, and (ii) the corresponding values of ∆N X predicted by the low-energy effective
theory at different orders in the coupling constants. A known source of error in our effective theory
comes from neglecting the inter-site matrix elements of all Hamiltonians. This error is discussed

in Appendix 3.E, and leads to theoretical uncertainties represented by error bars on the O G3
theory in Figure 3.4. A summary of Figure 3.4 is provided in Figure 3.5. We note that many-bodyinteraction energy shifts are smaller for asymmetric (−) states than symmetric (+) ones due to thecompetition between contributions of opposite sign in the asymmetric case (see rows 2 and 3 oftable 3.1, where as a consequence of positive scattering lengths in the case of 87 Sr, all UM X for fixed
M have the same sign). This competition is a many-body analogue of the two-body case with acompetition between direct and exchange terms in the interaction energies of singly-excited states.

54
O(G2 ) theory

ηN X∆N X (2π kHz)

O(G) theory

N, X = 3, −
6

−0.3

4

0.00

∆N X (2π kHz)
ηN X

40

50

60

70

0.05
0.00
−0.05
80
30

40

N, X = 4, −

50

60

70

80

70

80

50
60
70
Lattice depth U (ER )80

N, X = 4, +
10.0

0.6

7.5

0.4

5.0

0.25
0.00
−0.25

∆N X (2π kHz)

30

Experiment

N, X = 3, +

−0.2

−0.25

ηN X

O(G3 ) theory

30

40

50

60

70

80

N, X = 5, −

1.5

0.1
0.0
−0.1

30

40

50

60

N, X = 5, +
12.5
10.0

1.0

7.5
0.2

0.00
−0.25

0.0
30

40

50
60
70
Lattice depth U (ER )

80

30

40

Figure 3.4: Multi-body excitation energies of ultracold 87 Sr atoms at various lattice depths. The top
plot in each sub-figure with fixed N, X shows the excitation energies ∆N X ≡ EN X − EN,0 measured
experimentally in Ref. [7] and those predicted by the low-energy effective theory at different orders
in the coupling constants, when applicable both with and without four-body contributions. The
experimentbottom plot in each sub-figure shows the relative error ηN X ≡ ∆theory− 1. Error bars
N X /∆N X
represent experimental error or conservatively estimated theoretical uncertainties from nearestneighbor hopping of virtual states in the low-energy effective theory (see Appendix 3.E).

55

2−

3−

1

4−

5−

2+

3+

4+

5+

Lattice depth U (ER )

70
60
50
Experiment
O(G) theory

40

O(G3 ) theory

30
−2

0

2

4
6
Detuning ∆ (2π kHz)

8

10

12

Figure 3.5: Summary of the many-body excitation spectra in Figure 3.4, retrieved from Ref. [7].

56
The results in Figures 3.4 and 3.5 highlight a few important points about ultracold, highdensity 87 Sr experiments and our low-energy effective theory. First, these experiments exhibit clear
signatures of multi-body interactions, as evidenced by a stark disagreement between the observed
many-body excitation energies ∆N X and those that are predicted by the two-body O (G) theory.
Multi-body interactions are thus crucial for understanding these high-density experiments in thecontext of a single-band Hubbard model, which naturally arises in the zero-temperature limit
when all atoms occupy their motional ground state. Second, the inter-atomic interactions in theseexperiments are strong enough to require going beyond leading order for the description of multibody interactions in the low-energy effective theory. The formally small quantities organizing our
perturbation theory are two-body ground-state interaction energies (proportional to the couplings
GX ) divided by the spectral gap of the single-atom Hamiltonian H0 . These reduced (dimensionless)interaction energies vary from ∼ 0.05 − 0.15 in the parameter regimes of the 87 Sr experiments
considered here (see Appendix 3.F). As experiments begin to operate at higher atom densities withamplified interaction effects, reliably predicting interaction energies may require going to yet higher
orders in perturbation theory. Due to a combinatorial explosion of the number of diagrams whichappear at increasing orders in the effective theory, however, we need more systematic methods
to compute effective multi-body Hamiltonians at fourth order. In any case, we are agnostic as towhether such a calculation would provide better agreement between experiment and theory without
first performing a detailed analysis of systematic errors.

3.4.3

Orbital-state dynamics of a nuclear spin mixture

In addition to spectral measurements of many-body interaction energies, we consider the dynamics of multiply-occupied lattice sites during spectroscopic interrogation. While these dynamics
do not provide information about the nature or origin of effective multi-body interactions, they provide tools and intuition for addressing the low-lying orbital excitations which are readily accessible
in an experimental setting. If we initialize all atoms in the N -body ground state with an incoherent

57
mixture of all nuclear spins, then we prepare the mixed state ρN,0 = PN,0 / tr PN,0 , where
PN X ≡X|N XihN X|(3.60)|N |=Nis a projector onto the space of the N -body orbital states |N Xi. Interrogating the atoms for atime t by a laser resonant with the excitation energy ∆N ± then gives us the state(±)ρN (t) =

X


1
x
x
exp −itΩN ± SN
± |N , 0ihN , 0| exp itΩN ± SN ± ,tr PN,0(3.61)|N |=Nx
where the Rabi frequencies ΩN ,+ , ΩN ,− and pseudo-spin operators SN± are respectively given inx(3.56), (3.58), and (3.54). Denoting the eigenstates of SN
± by


1
|N , S± i ≡ √ |N , 0i + |N ±i ,2

1
|N , A± i ≡ √ |N , 0i − |N ±i ,2(3.62)and defining the identity operator projected to the relevant subspace,
1N ± ≡ |N , 0ihN , 0| + |N ±ihN ±| = |N , S± ihN , S± | + |N , A± ihN , A± | ,(3.63)(±)

we can write the state ρN (t) and excited-state projectors PN ± in the form(±)ρN (t) =i
X 1h
1
1N ± + ei2tΩN ± |N , S± ihN , A± | + e−i2tΩN ± |N , A± ihN , S± | ,tr PN,02(3.64)|N |=NandPN ± =X 1

1N ± − |N , S± ihN , A± | − |N , A± ihN , S± | ,2(3.65)|N |=N

from which it follows that the net excited-state population at time t is
h
i 1 1
(±)
hPN ± (t)i ≡ tr ρN (t) PN ± = − hcos (2tΩN ± )i|N |=N ,2 2
where hXi|N |=N ≡(3.66)

P|N |=N X/ tr PN,0 is an average of X over all choices of N distinct nuclear spins.

Figure 3.6 shows the excited-state population hPN ± (t)i for several occupation numbers N .
With the exception of N = 2, the asymmetric-state populations generally have an initial shortperiod of growth before falling back to hPN,− i ≈ 1/2. This behavior can be understood by thefact that for fixed N > 2, any pair of Rabi frequencies ΩN1 ,− , ΩN2 ,− with ΩN1 ,− 6= ΩN2 ,− are

58

X=−

X=+

N =2

1.00

N =3

hPN X i

0.75
0.50
0.25
0.00
0

1

2

3

4

5

0

1

N =4

1.00

2

3

4

5

4

5

N =5

hPN X i

0.75
0.50
0.25
0.00
0

1

2

3
τN /π

4

5

0

1

2

3
τN /π


Figure 3.6: Net population of the N -body orbital excited states |N ±i after interrogation of an√
initial mixed state ρN,0 = PN,0 / tr PN,0 for a reduced time τN ≡ tΩ0 / N (i.e. with real time t) bya laser with bare Rabi frequency Ω0 which is resonant with the N -body excitation energy ∆N X .
Here PN X , defined in (3.60), is a projector onto the space of the N -body orbital states |N Xi.

59
mutually incommensurate, which implies that at times t with min|N |=N 2tΩN ,−& 1 the av-

eraging in (3.66) effectively becomes a pseudo-random sampling average of cos x over values ofx, so hcos 2tΩN ,− i|N |=N ≈ 0. When N = 2, the asymmetric-state Rabi frequencies essentiallytake on the same values as the symmetric-state ones (see Figure 3.2); the behavior of asymmetricstate population dynamics for N = 2 can therefore be understood by the following discussion ofsymmetric-state population dynamics.
To understand the periodic collapse and revival of symmetric-state populations in Figure 3.6,
we observe from (3.56) that the symmetric-state phases in (3.64) and (3.66) take the form
2tΩN ,+ = τN

X

2µ

tΩ0
τN ≡ √ ,
N

with

µ∈N

(3.67)

where for fermionic atoms with half-integer nuclear spin, 2µ is always an odd integer, which implies
that the sum in (3.67) is an integer with the same parity (i.e. even/odd) as the occupation number
N (i.e. the number of elements in N ). At reduced times τN = nπ with integer n, therefore, if theoccupation number N is even then all phases 2tΩN ,+ are integer multiples of 2π, which leads to(±)

a collapse of the excited-state populations as ρN (t)(±)τN =nπ= ρN (0) = ρN,0 . If the occupation

N is odd, meanwhile, then the phases 2tΩN ,+ are all odd (even) integer multiples of π for odd(even) n. This alignment of phases implies a complete population transfer to the excited stateρN,− ≡ PN,− / tr PN,− for odd n, and a collapse back to the orbital ground state ρN,0 for even n,
precisely as observed in Figure 3.6.

3.5

Summary and outlook
Current 3-D optical lattice experiments with fermionic AEAs are capable of operating in

the low-temperature, high-density, strongly-interacting limit where inter-atomic interactions set
the dominant energy scale governing system dynamics. For AEAs with total nuclear spin I and
N = 2I + 1 nuclear spin states, these interactions exhibit an exotic SU(N ) symmetry which is ofgreat interest for near-term quantum simulations of SU(N ) spin models and lattice field theories.
Working in the deep-lattice limit and the experimental regime of at most one atom occupying

60
each nuclear spin state on any lattice site, we have derived a low-energy effective theory of theseatoms. Our theory exhibits emergent multi-body interactions that inherit the SU(N ) symmetryof the bare two-body interactions. Considering a restriction of our theory to the subspace ofat most one orbital excitation per lattice site, we found that the SU(N ) symmetry of all M body Hamiltonians allowed us to express them in a simple form, and to fully characterize theireigenstates and spectra. Capitalizing on the extreme precision of state-of-the-art clock spectroscopy,
we have tested spectral predictions of our theory against direct experimental measurements of the
many-body 87 Sr excitation spectrum. This comparison shows good agreement between theory
and experiment, clearly demonstrating the need to consider multi-body effects for understanding
the low-energy physics of high density AEA samples on a 3-D lattice. Finally, we analyzed the
many-body orbital-state dynamics of multiply-occupied lattice sites prepared in a nuclear spin
mixture and interrogated via Rabi spectroscopy. This analysis is useful for future experimental
probes of many-body state structures, as well as for the preparation of long-lived states with multipartite entanglement (i.e. |N ±i) which may be used as a resource to perform quantum information
processing tasks.
Despite the nominal success of our low-energy effective theory in reproducing experimental
observations, there remains room for improvement in the form of controlled, systematic treatmentof higher-order and tunneling processes. Nonetheless, our work makes a major step towards theexperimental investigation of multi-body SU(N ) physics, providing the necessary framework for
future studies going beyond the deep-lattice limit to realize multi-body super-exchange dynamics
and orbital SU(N ) quantum magnetism with AEAs.

Acknowledgements
We acknowledge helpful discussions with R. B. Hutson, G. E. Marti, S. L. Campbell, J. Ye,
P. Julienne, J. P. D’Incao, C. Kennedy, and L. Radzihovsky; and in particular close correspondence
and technical contributions from A. Goban. This work is supported by the Air Force Office of
Scientific Research (AFOSR) grant FA9550-18-1-0319; the AFOSR Multidisciplinary University

61
Research Initiative (MURI) grant; the Defense Advanced Research Projects Agency (DARPA) andArmy Research Office (ARO) grant W911NF-16-1-0576; the National Science Foundation (NSF)
grant PHY-1820885; JILA-NSF grant PFC-173400; and the National Institute of Standards andTechnology (NIST).

3.A

Derivation of the effective Hamiltonian expansion
Suppose we have a Hamiltonian H0 on a Hilbert space H = G0 ⊕ E0 for a zero-energy man-ifold G0 decoupled from a positive-energy manifold E0 , and that we perturb H0 by an operator V
which weakly couples G0 and E0 . For all |ψg i ∈ G0 and |φe i , |χe i ∈ E0 , we have hψg |H0 |ψg i = 0,hφe |H0 |ψg i = 0, and hφe |V |ψg i  hχe |H0 |χe i. The net Hamiltonian H = H0 + V will naturally
admit a decomposition of the Hilbert space as H = G ⊕ E for a subspace G which is spanned by the
low-energy eigenstates of H and has the same dimension as G0 , i.e. |G| = |G0 |.
We can perform a canonical transformation between G and G0 which yields an effective
Hamiltonian Heff on G0 that reproduces the spectrum of H on G [80]. Given an eigenbasis |α0 ifor H0 on G0 and |αi for H on G, this transformation is implemented by a unitary U for which|α0 i = U |αi and U → 1 as kV k → 0. The effective Hamiltonian is then simply
Heff = U HU † .

(3.68)The prescription in (3.68) for constructing an effective Hamiltonian is commonly known as a
Schieffer-Wolff transformation [81]. Unitaries U which follow this prescription are not unique,and different choices of U amount to different realizations of the Schieffer-Wolff transformation. In
Ref. [80], the authors construct the unique operator S which generates a direct or minimal rotation
Umin = eS between G and G0 , and use this construction to expand (3.68) as a perturbative series in
V . The rotation Umin is minimal in the sense that it minimizes the distance of candiate unitariesq
U from the identity 1 with respect to the Euclidian operator normb kXkE ≡ tr X † X . Thisrotation is determined uniquely by enforcing (i) that the generator S is strictly block-off-diagonalb

The Euclidean operator norm is also known as the L2,2 , Hilbert-Schmidt, or Frobenius norm.

62with respect to G0 and E0 , (ii) that the norm kSkE < π/2, and (iii) that the block-off-diagonal partsof (3.68) are zero.
To summarize the solution in Ref. [80], the effective Hamiltonian Heff induced by a directrotation can be expanded as
Heff =X(p)

Heff ,(3.69)

p≥0(p)where Heff is order p in V . Letting P0 denote the projector onto G0 , Q0 ≡ 1 − P0 denote theprojector onto E0 , and X denote any operator on H, we define the superoperators
DX ≡ P0 XP0 + Q0 XQ0 ,OX ≡ P0 XQ0 + Q0 XP0 ,(3.70)

which select out the diagonal (D) and off-diagonal (O) parts of X with respect to G0 and E0 , andLX ≡X |αihα| OX |βihβ|
α,β

H0 =

where

Eα − EβX

Eα |αihα| .

(3.71)

αThe first few terms of the expansion in (3.69) are then, as derived in Ref. [80],(0)Heff = P0 V P0 ,(1)

(3.72)

1(2)
Heff = − P0 [OV, LV ] P0 ,
2


1 
(3)
Heff = P0 OV, L [DV, LV ] P0 .
2(3.73)

Heff = P0 H0 P0 ,Exploiting the fact that in our case hψ|H0 |ψi = 0 for all |ψi ∈ G0 , we let B0 (E0 ) denote an eigenbasisof H0 for E0 and define the operator
I≡

X|αi∈B0 (E0 )|αihα|,
Eα(3.74)

which sums over projections onto excited states with corresponding energetic suppression factors.
We then expand
LX = O (LX) = Q0 LXP0 + P0 LXQ0 = IXP0 − P0 XI,(3.75)(2)

which simplifies the expression for Heff as
1(2)
Heff = − P0 [OV, IV P0 ] − [OV, P0 V I] P0 = −P0 V IV P0 .
2

(3.76)

63(3)

Working toward a similar expansion for Heff , we compute[DV, LV ] = [DV, IV P0 ] − [DV, P0 V I] = O (V IV ) − IV P0 V P0 − P0 V P0 V I,

(3.77)and in turn
 

1 
(3)
Heff = P0 OV, I [DV, LV ] P0 − OV, P0 [DV, LV ] I P0
2
1
= P0 V I [DV, LV ] + [DV, LV ] IV P0
21
1= P0 V IV IV P0 − P0 V I 2 V P0 V P0 − P0 V P0 V I 2 V P0
22i1h
= P0 V IV IV P0 −P0 V P0 , P0 V I 2 V P0 ,2+

(3.78)(3.79)(3.80)(3.81)

where [X, Y ]+ ≡ XY + Y X. The expressions in (3.72), (3.76), and (3.81) complete the derivation
eff in (3.11) through third order. In
for our expansion of the effective interaction Hamiltonian Hint

the case of ultracold atoms on a lattice, the motional ground-state subspace G0 actually contains
many internal atomic states with different energies. Nonetheless, the total Hilbert space is completely separable into uncoupled subspaces associated with each symmetrized many-body internal
atomic state. One can therefore diagonalize the interaction Hamiltonian with respect to these internal states and derive an effective theory within each of the corresponding subspaces, in eachcase setting the appropriate ground-state energy to zero. This procedure is equivalent to simultaeff for all internal states via the prescriptions weneously calculating the effective Hamiltonian Hint

have provided, but letting Eα denote only the motional excitation energy of states |α0 i ∈ B0 (E0 ).

3.B

Diagram counting and symmetry factors
The fact that we include factors of 1/2 from the bare two-body interaction Hamiltonian Hintin the definition of diagrams implies that p-vertex diagrams acquire a factor of 1/2p . In practice,
however, these factors are exactly cancelled out by corresponding symmetry factors in all M -bodydiagrams with M > 2. As an illustrative example, consider the second-order effective Hamiltonian

64
(2)

Hint in (3.13), which expanded in full reads(2)
Hint = −

X


P0

m+n>0

1
†
†
Kmn Gsts0 t0 ĉρs0 ĉσt0 ĉnσt ĉmρs
2



1
Emn




1
qr †
†
Kmn Gq0 r0 ĉmµq0 ĉnνr0 ĉνr ĉµq P0 . (3.82)
2

The three-body terms in this Hamiltonian have {µ, ν, ρ, σ} = 3 and only one virtually excited
atom. The non-vanishing three-body terms must therefore either have ρ ∈ {µ, ν} and contain afactor of the form ĉ†X ĉ†σt0 ĉσt ĉY , or have σ ∈ {µ, ν} with a factor of the form ĉ†ρs0 ĉ†X ĉY ĉρs , where
the labels X, Y both address whichever nuclear spin (i.e. µ or ν) was excited in the corresponding
term. Diagrammatically, we have terms of the form
µq

Z

X

Y
νr
σt

and
σt0

ρs

Z

µq

ρs0
.

Y
νr

(3.83)

X

Observing that ĉ†ρs0 ĉ†X ĉY ĉρs = ĉ†X ĉ†ρs0 ĉρs ĉY , however, it is clear that both of the terms represented
in (3.83) are equal up to the re-indexing σ, t, t0 ↔ ρ, s, s0 . There is therefore a symmetry factorof 2 associated with the second vertex of the diagrams in (3.83), which cancels out with the explicit
factor of 1/2 at that vertex, i.e. the first factor of 1/2 in (3.82). A symmetry factor of essentially
identical origin appears at every vertex with an “incoming” virtual state, as in e.g. the second andthird vertices of,and(3.84)or the last vertex of.

(3.85)

We can thus account for cancellations of 1/2 at all vertices except those which address two “initial”
ground-state atoms, as in the first vertex of the diagrams in (3.83) and (3.84), or the first two
vertices of the diagram in (3.85). For such vertices, there are two possibilities: either

65(i) both edges leaving the vertex in question (i.e. leaving to the right) terminate at different
vertices, as in the examples above, or(ii) both edges leaving the vertex in question terminate at the same vertex, as in for example
the first vertex of.

(3.86)In the former case, (i), the vertex in question has an associated symmetry factor of 2 to accountfor the possibility of a nuclear spin exchange at that vertex. Considering again our example of the(2)

second-order effective Hamiltonian Hint in (3.82), the non-vanishing three-body terms must either
have m = 0 and contain the factor ĉ†µq0 ĉ†nνr0 ĉνr ĉµq , or have n = 0 with the factor ĉ†mµq0 ĉ†νr0 ĉνr ĉµq ,which diagrammatically translates to
µq

µq 0
nνr0

νr
σt

µq

νr00

µq 00

mµq 0

or
σt0

νr0

νr
σt

.

(3.87)σt0These terms are equal up to the re-indexing ν, r, r0 , r00 , n ↔ µ, q, q 0 , q 00 , m , which implies that
there is a symmetry factor of 2 associated with the first vertex of the diagrams in (3.87). A
symmetry factor of identical origin is associated with the first vertex of the diagrams in (3.84), andthe first two vertices of the diagram in (3.85).
The final case we must consider is (ii), which occurs in the first vertex of (3.86). In this case,the symmetry factor of 2 which appears in case (i) to account for the possibility of a nuclear spin
exchange simply gets “pushed forward” to the vertex at which the two nuclear spins in question

66
part ways, e.g. to account for the two possibilities
µr

`µr0

µr00
nνs00

νs

µr
νs000

νs00
µr000

nµr00

and

mνs0
ρt

`µr0

νs

,

ρt0

ρt

ρt0

(3.88)mνs0which are equal up to the re-indexing ν, s, s0 , s00 , s000 , m ↔ µ, r, r0 , r00 , r000 , ` . The arguments fora symmetry factor in cases (i) and (ii) fail only if the two nuclear spins in question take identicalpaths through the internal vertices of a diagram, such that there is no meaningful sense in which
two diagrams can be said to differ by a nuclear spin exchange, as in (3.87) and (3.88). If two
atoms take identical paths through the internal vertices of a diagram, however, then they have only
participated in a two-body process, as in
,

,

,

··· .

(3.89)

After summing over all free indices, therefore, all two-body diagrams have a remaining factor of
1/2 from the first vertex. In all connected M -body diagrams with M > 2, meanwhile, every factorof 1/2 can be identified one-to-one with a corresponding symmetry factor of 2.

3.C

Effective coupling constants in a latticeDue to our choice of renormalization scheme in Section 3.3.2, the interaction energies pre-scribed by our low-energy effective theory for multiply-occupied lattice sites are not given directlyby the coupling constants GX defined by the free-space scattering lengths aX in (3.3). Instead,
we must first compute effective coupling constants Glattice(U) in a lattice with depth U, and inXturn use the effective coupling constants to compute interaction energies. As the renormalization
procedure GX → Glattice
is identical for all coupling constants, we henceforth drop the subscriptXX ∈ gg, eg− , eg+ , ee on coupling constants GX in the remainder of this Appendix. To further
simplify notation, we will also neglect the explicit dependence of parameters on the lattice depth
U, which we generally keep fixed.

67
Proper calculations of the interaction energy of two ultracold fermions in an optical latticewere performed in Refs. [74] and [88] using a two-channel model of a Feshbach resonance, yielding prescriptions for computing effective coupling constants in a lattice from free-space interaction
parameters. These calculations, however, are both analytically and numerically involved. We therefore instead opt to use a modified version of the considerably simpler single-channel calculation in
Ref. [75] of the interaction energy of two ultracold atoms in a harmonic trap. Our approach isequivalent to the calculation of Hubbard parameters performed in Ref. [89], and has been demonstrated to reproduce correct results in the limit of a deep lattice (compared to the lattice photon
recoil energy) and small positive scattering lengths (compared to the effective harmonic oscillator
length) [88].
The exact result in Eq. 16 of Ref. [75] for the interaction energy of two ultracold atoms in aharmonic oscillator with angular trap frequency ω can be written in the form√

Gfree KHO /ω−1π Γ −GHO KHO /2ω,=Γ −GHO KHO /2ω − 1/2

Z
KHO ≡

4

d3 x φHO,0(3.90)

where GHO is an effective coupling constant in the harmonic trap, φHO
0 (x) is the corresponding
non-interacting ground-state wavefunction, and Γ is the gamma function. The expression in (3.90)
can be solved numerically as is, or expanded about GHO KHO /ω = 0 to get−1
G−1
free = GHO∞
X

cn GHO KHO /ωn,(3.91)π21− ln 2 + (ln 2)2 .
24
2(3.92)

n=0

where the first few coefficients are
c0 = 1,c1 = 1 − ln 2,c2 = −The series in (3.91) can in turn be inverted to solve for GHO with an expansion of the form
GHO = Gfree∞Xc̃n Gfree KHO /ωn,(3.93)

n=0

where if we truncate the series in (3.91) at n = 2, the first few coefficients of (3.93) arec̃0 = 1,c̃1 = 1 − ln 2,c̃2 = −π21
− ln 2 + (ln 2)2 + (1 − ln 2)2 .
24
2(3.94)

68(n+1)

The coefficients c̃n thus found are consistent with the coefficients c2reported in table 1 ofRef. [68], in which the authors compute the first few terms of the two-body Hamiltonian H2 directly
as expressed in (3.15) by using a renormalization scheme which subtracts off divergences term byterm.
All of the above results are exact for two atoms in a harmonic oscillator interacting via s-wave
scattering. In order to adapt these results for a lattice, we expand the lattice potential about alattice site centered at x = (0, 0, 0) as
2

U sin (kL · x) ≈ Uh1/3
kLx kLy kLz
x

i2

1
2 2
= mA ωeff
x ,2qωeff ≡ 2 UkL2 /mA ,(3.95)


where kL = kLx , kLy , kLz is the lattice wavenumber, mA is the atomic mass, and ωeff is an effective
angular harmonic trap frequency. We then use ωeff in place of ω in (3.90), and use an overlap
integral K computed with the ground-state wavefunctions φ0 in a lattice rather than those in aharmonic oscillator. We retrieve free-space s-wave scattering lengths afree for 87 Sr from Ref. [53]
to determine the free-space coupling constants Gfree ≡ 4π/mA afree . This procedure yields aneffective coupling constant Glattice given by
Gfree K/ωeff

−1


√
π Γ −Glattice K/2ωeff,=Γ −Glattice K/2ωeff − 1/2(3.96)

with a solution
Glattice = Gfree∞Xc̃n Gfree K/ωeffn,(3.97)

n=0

where the first few coefficients are provided in (3.94).

3.D

Momentum-dependent s-wave interactions
In addition to the renormalization of coupling constants discussed in Appendix 3.C, comput(2)ing two-body interaction energies EN X at third order in the low-energy effective theory requires
accounting for the contribution of momentum-dependent s-wave interactions. At next-to-leading
order in the relative momentum k between two atoms, the effective momentum-dependent scatter-69
ing length aeff is given in terms of the zero-momentum scattering length a by [76–78]


1
1 1
1
1
= − reff k 2 =1 − reff ak 2 ,
aeff
a 2
a
2

(3.98)

which for reff ak 2  1, implies that11aeff ≈ a 1 + reff ak 2 = a + reff a2 k 2 .
(3.99)
22Here reff is an effective range of O k 2 interactions, determined in atomic units by the scattering
length a and van der Waals C6 coefficient by [78]
1 −2 
2
reff = ξ χ 1 − 2χ + 2χ a,
3

where


Γ 3/4
,
ξ≡
Γ 1/4

χ≡

√

(mA C6 )1/4
2ξ,a(3.100)and Γ is the gamma function. As χ ∼ 1 for 87 Sr, the momentum-dependent correction to theeffective scattering length aeff is O a3 without an additional separation of scales (i.e. which could
have occurred if we had χ  1 or χ  1). The momentum-independent O k 0 contribution to aeffin (3.99) gives rise to the bare two-body interactions in (3.7) by use of an unregularized contact(i.e. δ-function) potential, while the momentum-dependent O k 2 term gives rise to the interaction
Hamiltonian [68, 77]
1
0
Hint
≡
2

X

qr
G0 st

Z

h
i h
i
†
†
d3 x d3 y δ (z) ψ̂µq(x) ψ̂νr(y) k̂z2 ψ̂νt (y) ψ̂µs (x) ,(3.101)wherez ≡ x − y,k̂z2 ≡ −
1 ~ 2~2 ,∇z + ∇
z2(3.102)qr
and the primed couplings G0 qr
st are defined similarly to unprimed couplings Gst in (3.3) and (3.4),

but with scattering lengths a → reff a2 /2 and the effective range reff defined by (3.100) for eachscattering length with an appropriate C6 coefficient. We retrieve C6 coefficients for 87 Sr from the
supplementary material of Ref. [53]. The squared relative momentum operator k̂z2 is represented bysymmetrized left- and right-acting derivative operators in order to preserve manifest Hermiticity of
0 . At third order in the low-energy effective theory developed in Section 3.3, the bare momentumHint

dependent interactions in (3.17) yield only the effective two-body Hamiltonian
1 X 0 qr † †H20 ≡ K 0
G st ĉµs ĉνt ĉνr ĉµq ,2(3.103)

70
where, letting Re [X] denote the real part of X,
1
K ≡
2
0

3.E

Z

3

d x Re



 

2
φ∗0

~ 0 · ∇φ
~ 0 − φ0 ∇~ 2 φ0∇φ

.

(3.104)

Bounds on theoretical uncertainties from inter-site effects
In our overview of the relevant one- and two-particle physics of ultracold atoms on a lattice(Section 3.2), we made two approximations which introduce error into the low-energy effective
theory. Both approximations concern the on-site locality of the single- and two-body Hamiltonians:
we assumed that (i) tunneling between lattice sites and (ii) inter-site interactions are negligible.
These approximations are justified for single-particle motional ground states of atoms in a deep
lattice, but generally break down when considering virtual states occupying highly excited motional
levels, whose spatial wavefunctions can span multiple lattice sites. Nonetheless, we can place upperbounds on the magnitude of inter-site corrections to the effective on-site interaction Hamiltonians bytreating tunneling and inter-site interactions of virtual excited states perturbatively and assuming
no energetic penalty for nearest-neighbor hopping. These bounds can be used to diagnose thebreakdown of the on-site effective theory, and signal when a more careful consideration of inter-site
effects is necessary to make precise predictions about many-body spectra and dynamics.
If we still assume negligible overlap between single-particle ground-state wavefunctions indifferent lattice sites but consider nearest-neighbor wavefunction overlaps of states with motional
excitations, our one-body and bare two-body Hamiltonians become
H0 =X

En ĉ†inµs ĉinµs −X tmn ĉ†jnµs ĉimµs + h.c. ,(3.105)

hi,ji
m,n>0

and
Hint =

1 X k` qr †1 X µq;νr 
Kmn Gst ĉimµs ĉ†inνt ĉi`νr ĉikµq +Gρs;σt Kn ĉ†j,0,ρs ĉ†j,0,σt ĉj,0,νr ĉinµq + h.c.
2
2
hi,ji
n>0

+


1 X µq;νr Gρs;σt Kmn ĉ†j,0,ρs ĉ†j,0,σt ĉinνr ĉimµq + K̃mn ĉ†inρs ĉ†j,0,σt ĉj,0,νr ĉimµq + h.c. , (3.106)
2
hi,ji
m,n>0

71
where tmn is a tunneling rate; Kn , Kmn , K̃mn are inter-site spatial overlap integrals; h.c. denotesa Hermitian conjugate, i.e. (X + h.c.) ≡ X + X † ; and hi, ji denotes the set of all lattice sites itogether with their adjacent sites j. Note that we have neglected terms in (3.106) which involve
more than two field operators addressing states with motional excitations, as these terms will notappear in the leading-order corrections to the effective on-site interaction Hamiltonians. We also
still neglect terms which involve products of atomic wavefunctions for motional ground states indifferent lattice sites.
Diagrammatically representing matrix elements of H0 and Hint which are off-diagonal inlattice site by a dot (i.e. •) and marking lines which represent field operators addressing neighboring
lattice sites by a cross (i.e. + or ×, depending on the line orientation), the dominant terms in theeffective theory which we previously neglected by assuming on-site locality are(2)∼ γ2 G2 ,

(2)

γ2 ≡

X Kmn Kmn,
Emn

(3.107)

n+m>0

(2)

∼ γ3,1 G2 ,(2)

γ3,1 ≡X Kn Kn,(3.108)

X Km tmn Kn,
Em En(3.109)n>0(2)∼ γ3,2 G2 ,(2)

γ3,2 ≡

Enn,m>0

where we have identified, up to an assignment of coupling constants G, the magnitude of allnonzero matrix elements of the diagrams with respect to an eigenbasis of the on-site single-particle
Hamiltonian H0 in (3.1).
The terms in (3.107)-(3.109) can be used to estimate an upper bound on the magnitude of
dominant corrections to the spectrum of the low-energy theory from off-diagonal (i.e. in latticesite) matrix elements of the Hamiltonians in (3.105) and (3.106). Conservatively assuming noenergetic penalty and no Pauli blocking for any inter-site process, the dominant correction δEN
to the interaction energy of a lattice site with N atoms and b neighboring sites (e.g. b = 6 in a

72
primitive cubic lattice) is roughly bounded as
 
n
o
N
(2)
(2)
(2)
|δEN | . b
max γ2 , (N − 1) γ3,1 + γ3,2 G2 ,2where the factor of b accounts for the multiplicity of neighboring sites; the factor of(3.110)N
2accounts

for the number of on-site pairs of atoms which are addressed by the diagrams in (3.107)-(3.109);(2)

and the factor of N − 1 on γ3,X accounts for the number of atoms in a neighboring site which
are addressed by the corresponding processes. These factors count the number of matrix elements(2)in the Hamiltonian with magnitude ∼ γX G2 . The maximization in (3.110) is performed because
the relevant two- and three-body processes are mutually exclusive, requiring a different numberof atoms on neighboring lattice sites. For a conservative bound of |δEN |, the coupling factor G2
in (3.110) can simply be maximized over its allowed values for a given state of atoms on a latticenosite, e.g. G2g for a state with no orbital excitations, or max G2g , G2+ , G2− for a state with one net
orbital excitation (in both cases, assuming no orbital excitations in neighboring sites). In the latter
case, the bound in (3.110) can also be reduced by observing that to conserve energy, it must be the
excited atom which moves to a neighboring site, which reduces the factor of N2 in down to N − 1.
We emphasize that the bound in (3.110) is by no means an exact measure of error, and is merely
intended to provide a conservative range of energies and corresponding time scales for which intersite effects could become relevant despite negligible single-particle ground-state tunneling rates.

3.F

Perturbative parameters for the effective theory
The perturbative effective theory developed in Section 3.3 is organized in powers of thecoupling constants GX . The formally small, dimensionless quantities for this perturbation theory
are the two-body interaction energies KGX divided by the spectral gap ∆ of the non-interacting
Hamiltonian H0 . Here K is a ground-state two-body overlap integral and GX is a coupling constant.
The quantities K, GX , and ∆ all depend on the lattice depth U. Figure 3.7 shows these parametersfor the case of 87 Sr atoms with X ∈ gg, eg− , eg+ , ee at lattice depths U ∈ [30, 80] ER , whereER ≈ 3.5×2π kHz is the lattice photon recoil energy of the atoms. The fact that these perturbative

73
0.175
0.150

KGX /∆

0.125
0.100
0.075
0.050
0.025

gg
eg−

eg+

40

50
60
Lattice depth U (ER )

ee

0.000
30

70

80

Figure 3.7: Dependence of the perturbative parameters KGX /∆ on the lattice depth U.

parameters grow with increasing lattice depth U is a consequence of the fact that the overlap
integral K grows faster with U than the spectral gap ∆. In the case of a harmonic trap withangular frequency ω, for example, by dimensional analysis these parameters would be
KHO GX
GX
=
ω
ω

Z



4 3

1/4
Z
√
GX 
mA ω
2
d3 x |φHO |4 =dx
e−mA ωx /2  ∝ ω,ωπ(3.111)

where we assumed that the coupling constants GX vary weakly with ω. While this result may seemto suggest that the low-energy effective theory should become better at smaller lattice depths,
smaller lattice depths also result in increased theoretical uncertainties from the growing relevanceof the inter-site effects discussed in Appendix 3.E.

Low-excitation M -body Hamiltonian coefficients

3.G

When restricted to the subspace of at most one orbital excitation per lattice site, the M -body
Hamiltonians of the low-energy effective theory developed in Section 3.3 can be written in the form
HM =

X

µj

=M



UM,g n̂µ1 ,g n̂µ2 ,g + UM,+ n̂µ1 ,e n̂µ2 ,g + UM,− ĉ†µ1 ,g ĉ†µ2 ,e ĉµ2 ,g ĉµ1 ,e

M
Y
α=3

n̂µα ,g ,(3.112)

74
P

where the coefficients can be expanded as UM X =(p)p UM X(p)

with terms UM X at order p in the(p)coupling constants GY . The terms UM X can be determined from the M -body p-order Hamiltonians(p)

HM derived in Section 3.3, i.e. in (3.17), (3.19), (3.36), and (3.46). For the effective 2-, 3-, and4-body Hamiltonians through third order in the coupling constants, the coefficients are
1 (1)U2,g = α2 Gg ,
2
(2)

(1)

U2,+ = α2 G+ ,(2)

U3,g = −α3 G2g ,(1)

U2,− = α2 G− ,(3.113)(2)(2)
U3,+ = −α3 G+ G+ + 2Gg ,(3.114)(2)(2)
U3,− = −α3 G− 2G+ + G− + 2Gg ,(3.115)





(3)
(3)
(3)
(3)
(3)
(3)
U3,g = α3,1 − α5 2G3g + 2α3,2 − α4,3 − α5 G3g ,(3.116)




(3)
(3)
(3)
U3,+ = α3,1 − α5G3+ + 4G2+ Gg + G+ G2− + G+ G2g + G3− + G2− Gg



(3)
(3)
(3)
+ 2α3,2 − α4,3 − α5G3+ + G2+ Gg + G+ G2− + G+ G2g + G2− Gg ,(3)
U3,− =(3)(3)
α3,1 − α5





(3)

(3)

G−



3G2+ + 2G+ G− + 8G+ Gg + 3G− Gg + G2g(3)+ 2α3,2 − α4,3 − α5(3.117)
G− 3G2+ + 2G+ G− + 2G+ Gg + G2− + G2g ,(3.118)





(3)
(3)
(3)
(3)
(3)
U4,g = 2α4,1 − α5 G3g + α4,2 − α5 2G3g ,(3.119)  (3)
(3)
(3)
(3)
(3)
U4,+ = 2α4,1 − α5 2G+ Gg G+ + Gg + α4,2 − α5 G+ G2+ + 2G+ Gg + 5G2g ,(3.120)




(3)
(3)
(3)
U4,− = 2α4,1 − α5 2G− Gg 2G+ + G− + Gg




(3)
(3)
+ α4,2 − α5 G− 3G2+ + 3G+ G− + 4G+ Gg + G2− + 2G− Gg + 5G2g ,(3.121)(p)

In terms of the spatial overlap integrals defined in (3.6) and (3.8), the prefactors αX on the(p)coefficients UX are(1)

α2 ≡ K,

(2)

α3 ≡

X K2

n

n>0

En

,

(3)

α5 ≡ K

X K2

n

n>0

En2

,

(3.122)

75(3)

α3,1 ≡

X K`m K m K`nn,E`m E`n

`+m>0
`+n>0

(3)

α4,1 ≡X Kmn Km Knm≥0n>0

Emn En,
X K`m Kn K`m Kn(3)`m
α3,2 ≡Kn −,E`m EnK(3.123)

X Km K m Kn
n
,
Em En

(3.124)

`+m>0
n>0

(3)

α4,2 ≡m,n>0(3)

α4,3 ≡ K

X K2
mn
.
2
Emn

m+n>0

Chapter 4

Engineering spin squeezing in a 3D optical lattice with interacting
spin-orbit-coupled fermions

Prologue
The 3D 87 Sr optical lattice clock (OLC) is one of the most precise and accurate scientificinstruments ever created, rivaled only by other atomic clocks and large-scale gravitational wave
detectors. In principle, this clock (or rather, a series of these clocks) could measure the age ofthe universe (∼ 14 billion years) to within an error measured in milliseconds. However, the 3D
87 Sr OLC still operates in the “single-particle” regime, deriving its precision primarily from (i) the

narrow (millihertz) linewidth of an electronic transition in individual atoms, together with (ii) signal
averaging over a large number of independent atomsa that are addressed simultaneously in the clock.
√Averaging over N atoms improves clock precision by a factor of ∼ 1/ N , which can be traced backto the reconstruction of a classical distribution from N independent samples (one sample per atom).
A major untapped resource for further improving the 3D 87 Sr OLC is the possibility of preparing
entangled clock states in which quantum correlations between atoms conspire to reduce sampling√
error. In theory, this strategy can reduce the “classical” factor of 1/ N to a “quantum” factor to
1/N , which would improve the precision of a 3D clock with 100 × 100 × 100 lattice sites by a factorof 103 (at unit filling).
a

The 3D 87 Sr OLC operates in a Fermi-degenerate Mott insulating regime, so atoms’ external (spatial) degrees
of freedom are highly correlated. However, the clock signal is encoded in atoms’ internal (electronic) degrees of
freedom, which are essentially independent.

77
In this chapter, we present a concrete protocol for preparing spin-squeezed states of many
atoms, whose quantum correlations can be leveraged to improve clock performance. The essential ingredients for the protocol are SU(2)-symmetric inter-atomic interactions, together with asymmetry-breaking inhomogeneous single-particle field. The bulk of this chapter is taken from
Ref. [2]. In addition to myself and Ana Maria Rey, this work featured major contributions from
Peiru He, Sean R. Muleady, Robert J. Lewis-Swan, Ross B. Hudson, and Jun Ye.

Abstract
One of the most important tasks in modern quantum science is to coherently control andentangle many-body systems, and to subsequently use these systems to realize powerful quantum
technologies such as quantum-enhanced sensors. However, many-body entangled states are difficult to prepare and preserve since internal dynamics and external noise rapidly degrade any usefulentanglement. Here, we introduce a protocol that counterintuitively exploits inhomogeneities, atypical source of dephasing in a many-body system, in combination with interactions to generate
metrologically useful and robust many-body entangled states. Motivated by current limitations instate-of-the-art three-dimensional (3D) optical lattice clocks (OLCs) operating at quantum degeneracy, we use local interactions in a Hubbard model with spin-orbit coupling to achieve a spin-locking
effect. In addition to prolonging inter-particle spin coherence, spin-locking transforms the dephasing
effect of spin-orbit coupling into a collective spin-squeezing process that can be further enhanced
by applying a modulated drive. Our protocol is fully compatible with state-of-the-art 3D OLC
interrogation schemes and may be used to improve their sensitivity, which is currently limited by
the intrinsic quantum noise of independent atoms. We demonstrate that even with realistic experimental imperfections, our protocol may generate ∼ 10–14 dB of spin squeezing in ∼ 1 second with∼ 102 –104 atoms. This capability allows OLCs to enter a new era of quantum enhanced sensing
using correlated quantum states of driven non-equilibrium systems.

78

4.1

Introduction
A major frontier of contemporary physics is the understanding of non-equilibrium behaviors

of many-body quantum systems, and the application of these behaviors toward the developmentof novel quantum technologies with untapped capabilities [90]. To this end, ultracold atomic,
molecular, and optical systems are ideal platforms for studying unexplored regimes of many-body
physics due to their clean preparation and readout, high controllability, and long coherence times[91, 92]. The exquisite capabilities of these systems have pushed the frontiers of metrology, quantum
simulation, and quantum information science.
Optical lattice clocks in particular have seen some of the most impressive developments
in recent years, reaching record levels of precision (∼ 3 × 10−19 ) [42, 73] and accuracy (∼ 1 ×
10−18 ) [40, 93]. These advancements required important breakthroughs, including the capability
to cool and trap fermionic alkaline-earth atoms in spin-insensitive potentials [22, 94, 95]; thedevelopment of ultracoherent lasers [96–98] to fully exploit an ultranarrow clock transition [17]; thedetailed characterization of inter-atomic interactions [51–53]; and, more recently, the preparationof a quantum degenerate gas in a three-dimensional (3D) optical lattice [7, 42, 73]. Nonetheless, allimprovements in sensing capabilities to date have been based on single-particle control of internal
atomic degrees of freedom. Such strategies will eventually have diminishing returns due to practicaldifficulties in (i) suppressing decoherence from external (motional) degrees of freedom, and (ii)interrogating more particles without additional systematic errors from interactions [17, 50, 73].
Pushing beyond the current independent-particle paradigm requires leveraging many-bodyquantum correlations. Entangled states such as spin-squeezed states [32, 33, 35] can enhancemeasurement sensitivity, i.e. the uncertainty ∆θ in the estimation of a parameter θ, below the√standard quantum limit ∆θ ∼ 1/ N for N uncorrelated particles [99, 100]. The major challenge forprogress in this direction is that generating entanglement requires interactions, which are generally
undesirable because they degrade atomic coherence, thereby limiting clock performance [17, 47, 48,
50, 54, 101]. In fact, the most precise and accurate optical lattice clocks were designed to operate

79with fermionic atoms in identical nuclear and electronic states to suppress collisional decoherence[42, 47, 102], as identical fermions cannot interact via the otherwise dominant (s-wave) collisions
at ultracold temperatures. However, an initially spin-polarized Fermi gas still exhibits interactionsat later times due to spin-orbit coupling (SOC) that is induced by the laser that drives the clocktransition (i.e. the “clock laser”) [25, 26, 103, 104]. Specifically, the momentum kick imparted
by this laser imprints a position-dependent phase that induces inhomogeneous spin precession andgenerates spin dephasing, thereby making atoms distinguishable and vulnerable to collisions. Whilea deep lattice can suppress SOC, it also intensifies the light scattering which currently limits thecoherence time of the clock [7, 23, 105].
In this work, we describe a scheme that can lead to metrological advances in state-of-theart optical lattice clocks through direct use of quantum entanglement by harnessing the interplay
between nominally undesirable collisions and SOC. This scheme is made possible in the weak SOC
regime by the formation of an interaction-energy gap that suppresses the SOC-induced populationtransfer from the exchange-symmetric Dicke manifold (spanned by spin-polarized, and thus noninteracting states) to the remainder of Hilbert space. Interactions thereby prolong inter-particle
spin coherence through a spin-locking effect, which additionally transforms the dephasing effectof SOC into a collective spin squeezing process. To generate spin squeezing, our protocol only
requires the capability to fix (i) the orientation of the clock laser and (ii) the optical lattice depth.
These controls are straightforward to incorporate into current 3D clock interrogation sequences
without sacrificing atom numbers or coherence times. Additionally, we show that by applying a
modulated drive from the clock laser, one can further prepare states that saturate the Heisenberg
limit ∆θ ∼ 1/N for phase sensitivity [33, 35, 100]. This capability mirrors efforts in other settings,
such as nitrogen-vacancy centers in diamond [106, 107] and trapped ions [108], to enhance quantum
metrology through the use of driven non-equilibrium phenomena.
Despite an abundance of proof-of-principle experiments with entangled states [100, 109], so
far only the remarkable example of LIGO [110, 111] has demonstrated a quantum advantage ina state-of-the-art quantum sensing or measurement system. The new generation of 3D optical

80
lattice systems have fully quantized motional degrees of freedom [42], allowing for precise control of collisional interactions. We demonstrate how these interactions can naturally give rise to
metrologically useful correlated many-body fermionic states, opening a path to not only generateentanglement, but also harness it to achieve a quantum advantage in a world-class sensor. Such anadvance will ultimately deliver gains to real-world applications including timekeeping, navigation,
telecommunication, and our understanding of the fundamental laws of nature [18].

4.2

Spin squeezing with the Fermi-Hubbard model
We consider N fermionic atoms with two spin states (labeled ↑ and ↓) trapped in a 3D

optical lattice. In this discussion, these spin states are associated with the two electronic states of a
nuclear-spin-polarized gas. At sufficiently low temperatures, atoms occupy the lowest Bloch bandof the lattice and interact only through s-wave collisions. A schematic of this system is provided in
Figure 4.1(a), where tight confinement prevents motion along the vertical direction (z), effectively
forming a stack of independent 2D lattices. For simplicity and without loss of generality, however,
we first consider the case when tunneling can only occur along one direction, x, and thus model
the system as living in one dimension.
An external laser with Rabi frequency Ω and wavenumber kL along the tunneling axis resoP
nantly couples atoms’ internal states through the Hamiltonian Ĥlaser /~ = j ΩeikL xj ĉ†j,↑ ĉj,↓ + h.c.,where ĉjα is a fermionic annihilation operator for an atom on site j with internal state α ∈ {↑, ↓}and xj is the position of site j. This laser imprints a position-dependent phase that equates toa momentum kick kL when an atom changes internal states by absorbing or emitting a photon,
thereby generating spin-orbit coupling [26, 103]. After absorbing the position dependence of thelaser Hamiltonian into fermionic operators through the gauge transformation ĉj,↑ → eikL xj ĉj,↑ ,which makes Ĥlaser spatially homogeneous, the atoms are well-described in the tight-binding limit
by the Fermi-Hubbard Hamiltonian [112](φ)

ĤFH /~ = −J

X
j


X
eiφ ĉ†j,↑ ĉj+1,↑ + ĉ†j,↓ ĉj+1,↓ + h.c. + Un̂j,↑ n̂j,↓ ,
j

(4.1)

81

(a)

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(c)

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minhvar(Ŝ✓? )i<latexit sha1_base64="m9jExh1wW7jyZDrO3wVafbz7ARo=">AAACJHicbVDLSgNBEJyN7/iKevSyGAS9hI0IKngQvXhUNEbIxtA76SRDZmeXmV4xLPszXvwVLx6MePDitzh5HDSxYKCoqqanK4ilMOR5X05uZnZufmFxKb+8srq2XtjYvDNRojlWeCQjfR+AQSkUVkiQxPtYI4SBxGrQvRj41UfURkTqlnox1kNoK9ESHMhKjcKpT/hEaShU5ktQbYkj4RF0tud3gNKbrJH61EGC7CH1Y9Rxtu/rYbRRKHolbwh3mpTHpMjGuGoU+n4z4kmIirgEY2plL6Z6CpoEl5jl/cRgDLwLbaxZqiBEU0+HV2burlWabivS9ilyh+rviRRCY3phYJMhUMdMegPxP6+WUOu4ngoVJ4SKjxa1EulS5A4qc5tCIyfZswS4FvavLu+ABk622LwtoTx58jSpHJROSt71YfHsfNzGIttmO2yPldkRO2OX7IpVGGfP7JW9s77z4rw5H87nKJpzxjNb7A+c7x8JbKeg</latexit><latexit sha1_base64="80/dsf+PW6LyEOKiB1zg8vOWhRM=">AAAB8nicbVBNS8NAEN3Ur1q/qh69LBahXkoignorePFYxdhCE8pmO2mXbj7YnYgl9G948aDi1V/jzX/jts1BWx8MPN6bYWZekEqh0ba/rdLK6tr6RnmzsrW9s7tX3T940EmmOLg8kYnqBEyDFDG4KFBCJ1XAokBCOxhdT/32Iygtkvgexyn4ERvEIhScoZE8D+EJgzCvw+mkV63ZDXsGukycgtRIgVav+uX1E55FECOXTOuuY6fo50yh4BImFS/TkDI+YgPoGhqzCLSfz26e0BOj9GmYKFMx0pn6eyJnkdbjKDCdEcOhXvSm4n9eN8Pw0s9FnGYIMZ8vCjNJMaHTAGhfKOAox4YwroS5lfIhU4yjialiQnAWX14m7lnjqmHfntead0UaZXJEjkmdOOSCNMkNaRGXcJKSZ/JK3qzMerHerY95a8kqZg7JH1ifP0U3kWw=</latexit>

initial state
preparation| #, #, · · · , #i
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qx
(e)

fU

q

⇡

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clock laser oﬀt<latexit sha1_base64="7uX8qoGudhkVfhNRdeWycNOvdjg=">AAAB53icbVBNS8NAEJ34WetX1aOXxSJ4KokI6q3oxWMLxhbaUDbbTbt2swm7E6GE/gIvHlS8+pe8+W/ctjlo64OBx3szzMwLUykMuu63s7K6tr6xWdoqb+/s7u1XDg4fTJJpxn2WyES3Q2q4FIr7KFDydqo5jUPJW+Hoduq3nrg2IlH3OE55ENOBEpFgFK3UxF6l6tbcGcgy8QpShQKNXuWr209YFnOFTFJjOp6bYpBTjYJJPil3M8NTykZ0wDuWKhpzE+SzQyfk1Cp9EiXalkIyU39P5DQ2ZhyHtjOmODSL3lT8z+tkGF0FuVBphlyx+aIokwQTMv2a9IXmDOXYEsq0sLcSNqSaMrTZlG0I3uLLy8Q/r13X3OZFtX5TpFGCYziBM/DgEupwBw3wgQGHZ3iFN+fReXHenY9564pTzBzBHzifP05BjMs=</latexit><latexit

spin
squeezed
state

clock
interrogation
protocolFigure 4.1: Schematic of the setup for spin squeezing. (a) We consider N fermionic atoms
with two (pseudo-)spin components, represented by red and blue spheres, trapped in the groundband of an optical lattice (shown in 2D for the sake of presentation). Atoms tunnel to neighboringsites at a rate J and experience on-site interactions with strength U . An external laser carrying a
position dependent phase eikL ·r couples the spin states of the atoms. (b) After a gauge transformation, different spin states exhibit different dispersion relations with a relative phase φ = kL a, wherea is the lattice spacing. The external laser couples spin states with identical quasi-momenta q in
the gauge-transformed frame. (c) If interactions are sufficiently weak, all motional degrees of freedom become frozen in momentum space, with atoms effectively pinned to fixed quasi-momentum
modes q. The dynamics on the frozen q-space lattice can then be mapped to a spin model in whichcollisional interactions correspond to a uniform, all-to-all ferromagnetic Heisenberg Hamiltonian
with strength U/L, where L is the total number of lattice sites. (d) The spin dependence of thedispersion relation is captured by a mode-dependent axial field Bq that generates inhomogeneous
spin precession. This axial field couples exchange-symmetric many-body Dicke states with total
spin S = N/2 to spin-wave states with S = N/2 − 1. The all-to-all interaction opens an energy
gap f U (with f = N/L the filling fraction of spatial modes) between the Dicke states and thespin-wave states, which forbids population transfer between them in the weak-field limit. (e) To
generate spin squeezing via one-axis twisting, we initialize a product state with all spins polarizedin −z (i.e. in |↓i), and apply a fast external laser pulse to rotate all spins into x. We then let atoms
freely evolve for a variable time t (with a spin-echo pulse), after which the amount of spin squeezing
can be determined experimentally from global spin measurements. The spin-squeezed state can beused for a follow-up clock interrogation protocol (see Appendix 4.E).

82
where J is the nearest-neighbor tunneling rate; the SOC angle φ ≡ kL a determines the phase gainedby spin-up atoms upon tunneling from site j to site j + 1 (in the gauge-transformed frame) withlattice spacing a = xj+1 − xj ; U is the on-site interaction energy of two atoms; and n̂jα ≡ ĉ†jα ĉjα isa number operator.
The Fermi-Hubbard Hamiltonian can be re-written in the quasi-momentum basis with anP
nihiliation operators ĉqα ≡ L−1/2 j e−iqxj ĉjα , where q is a quasi-momentum and L is the total
number of lattice sites. In this basis, the single-particle Hamiltonian exhibits shifted dispersion
relations that signify spin-orbit coupling [see Figure 4.1(b)]:(φ)

ĤFH,single /~ = −2JXcos(qa + φ)n̂q,↑ + cos(qa)n̂q,↓ .

(4.2)

q

When U . J, interaction energies are too weak for collisions to change the occupancies of singleparticle quasi-momentum modes. Atoms are then pinned to these modes, which form a lattice inquasi-momentum space [see Figure 4.1(c)] [104]. In this strong-tunneling limit, the Fermi-Hubbard
Hamiltonian [Eqn. (4.2)] can be mapped to a spin-1/2 system with a collective ferromagnetic
Heisenberg interaction and an inhomogeneous axial field, given by [50, 54, 104]Ĥspin /~ = −where Ŝ =X
UŜ · Ŝ −Bq ŝzq ,
L
q

(4.3)

P

ŝj=x,y,z≡ 12q

q ŝq is a collective spin operator; ŝq is a spin-1/2 operator for mode q with components†jj=x,y,z ; the sums over q run overα,β ĉqα σαβ ĉqβ defined in terms of the Pauli matrices σPall occupied quasi-momentum modes; and Bq ≡ −4J sin qa + φ/2 sin φ/2 is the SOC-induced
axial field.
On its own, the collective Heisenberg term (∼ Ŝ · Ŝ) in Eqn. (4.3) opens an energy gap
f U , with f ≡ N/L the filling fraction of spatial modes, between the collective Dicke states
S = N/2, MS

and the remainder of Hilbert space [50, 113–115] with S < N/2. Here S andMS respectively label the eigenvalues of the collective spin operators Ŝ · Ŝ and Ŝz , with eigenvalues
S(S + 1) for non-negative S ∈ N/2, N/2 − 1, · · · and MS ∈ {−S, −S + 1, · · · , S}. The axial field
Bq generally couples states within the Dicke manifold to states outside it. In the weak SOC limit

83(i.e. Bq  f U ), however, the interaction energy gap suppresses population transfer between stateswith different total spin S [see Figure 4.1(d)]. In this regime, the virtual occupation of states outside
the Dicke manifold can be accounted for perturbatively. The symmetries of SOC as expressed in
Eqn. (4.3) dictate that this treatment should yield powers of Ŝz when projected onto the collective
Dicke manifold at higher orders in perturbation theory. At second order in perturbation theory (seeAppendix 4.A), we thus find that SOC effectively yields a one-axis twisting (OAT) model widely
known to generate squeezing dynamics [33, 35]:Ĥeff /~ = −where B ≡P

UŜ · Ŝ − B Ŝz + χŜz2 ,
L

χ≡

e2
B
,
(N − 1)f U

(4.4)


2
e 2 ≡ P Bq − B /N the variance of the axial field. The
B
/N
is
the
mean
and
B
q
q
q

effect of the ∼ Ŝ · Ŝ term is to generate a relative phase between states with different total spin
S and thus has no effect on dynamics restricted to a fixed S. Note also that the collective spinrotation from B Ŝz can be eliminated by going into a rotating frame or by using a spin echo.
The entire protocol for preparing a squeezed state via OAT, sketched out in Figure 4.1(e),reduces to a standard Ramsey protocol with a spin echo: after initially preparing a spin-down(i.e. −ẑ) polarized sample of ultracold atoms populating the lowest Bloch band of a lattice, afast π/2 pulse is applied with the clock laser to rotate all spin vectors into +x̂. The atoms thenfreely evolve for a variable time t (possibly with spin-echo π-pulses), after which the amount of
metrologically useful spin squeezing, measured by the Ramsey squeezing parameter2ξ 2 ≡ min hvar(Ŝθ⊥ )i × N/ hŜi ,
θ

(4.5)

can be determined experimentally from global spin measurements. Here hŜi is the mean collective
spin vector and hvar(Ŝθ⊥ )i is the variance of spin measurements along an axis orthogonal to hŜi,parameterized by the angle θ ∈ [0, 2π).
The above protocol concerns only the preparation of a spin-squeezed state, which would thenbe used as an input state for a follow-up clock interrogation protocol without SOC. While increasing
the lattice depth to turn off SOC during clock interrogation is the simplest approach, this will limit

84
the interrogation time due to light scattering (see discussion below). Alternatively, it is possible
to keep the same lattice depth used for the spin squeezing generation by adding a specific pulse
sequence to suppress SOC. See details in Appendix 4.E.

4.2.1

Model validity

The validity of the OAT model in Eqn. (4.4) relies on two key conditions concerning experimental parameter regimes. First, the on-site interaction energy U should not be much larger inmagnitude than the tunneling rate J (clarified below); otherwise, one cannot assume frozen motional degrees of freedom (i.e. with atoms pinned to fixed quasi-momentum modes) and map theFermi-Hubbard model to a spin model. Second, the SOC-induced fields Bq ∼ J sin φ/2 should
be considerably smaller in magnitude than the interaction energy gap f U , as otherwise one cannot perturbatively transform SOC into OAT. These two conditions can be satisfied by appropriate
choices of U/J and the SOC angle φ, which are respectively controlled by tuning the lattice depth
and changing the angle between the clock laser and the lattice axes [see Figure 4.1(a)].
We demonstrate the importance of these conditions in Figure 4.2, where we show numericalresults from exact simulations of a 1D system with L = 12 sites. Therein, optimal squeezing
2 ), while the time at which
achievable under unitary dynamics is provided in dB, i.e. −10 log10 (ξopt

this squeezing occurs is provided in units of the nearest-neighbor tunneling time 2π/J. At f = 1
atom per lattice site, i.e. half filling of all atomic states in the lowest Bloch band, the spin model[Eqn. (4.3)] agrees almost exactly with the Fermi-Hubbard (FH) model [Eqn. (4.2)] up through(and exceeding) U/J = 8. The agreement at half filling (f = 1) is assisted by Pauli blocking ofmode-changing collisions. Below half filling (f = 5/6), these two (FH and spin) models show good
agreement at U/J . 2, while at U/J & 2 mode-changing collisions start to become relevant andinvalidate the frozen-mode assumption of the spin model. Note that we chose filling f = 5/6 todemonstrate that our protocol should work, albeit sub-optimally, even in this highly hole-doped
case; in practice, optimized experiments are capable of achieving fillings closer to the optimal
f = 1 [116]. Interestingly, even with mode-changing collisions the Fermi-Hubbard model exhibits

85

Figure 4.2: Benchmarking the spin and one-axis twisting models. Comparisons of maximum
squeezing (top panels, a.i and b.i) and optimal squeezing time (lower panels, a.ii and b.ii) between
the Fermi-Hubbard (FH), spin, and one-axis twisting (OAT) models; obtained numerically via theprotocol depicted in Figure 4.1(e) in a 1D lattice with L = 12 sites. Results are shown for halffilling with N = 12, f ≡ N/L = 1 (left panels, a.i and a.ii) and filling f = 5/6 (right panels, b.iand b.ii) as a function of U/J and the SOC angle φ. In both cases, the system is initialized in thecorresponding ground state. Insets for both f = 1 and f = 5/6 show (in green) regions of the U -φplane in which both the optimal squeezing (in dB) and the corresponding squeezing time of all threemodels agree to within 20%. At half filling (a.i and a.ii), mode-changing collisions are suppressed
by Pauli blocking, resulting in almost exact agreement between the FH and spin models; both ofthese models converge onto the OAT model in the gap-protected, weak SOC regime of large U/Jand small φ. The spin and OAT models show similar behavior away from half filling (b.i and b.ii),
but the presence of mode-changing collisions results in their disagreement with the FH model asinteractions begin to dominate at larger U/J. Even below half filling, however, the FH exhibits
comparable amounts of squeezing to the spin model across a broad range of U/J and φ, albeit atearlier times when U/J & 2.

86
comparable amounts of squeezing to the spin model, and achieves this squeezing in less time. Thee ∼ J sin φ/2  f U , and exhibit
spin and OAT models agree in the regime of weak SOC with B
different squeezing behaviors outside this regime as single-particle spin dephasing can no longer betreated as a weak perturbation to the spin-locking interactions.
In realistic implementations, the Gaussian profile of the laser beams always introduces anadditional effective harmonic potential that modifies the translational invariance assumed so far. We
present a detailed discussion of the role of the harmonic trap in Appendix 4.I, where we demonstratethat the addition of harmonic confinement barely modifies the achievable spin squeezing withcurrently accessible trapping frequencies. We find that the existence of single-particle localized
modes in the lattice with harmonic confinement [24, 117] helps to protect spin squeezing and shifts
the optimal parameter window to U/J & 2.

4.2.2

Two-axis twisting2 ∼ N −2/3 with
The above scheme for OAT achieves optimal spin squeezing that scales as ξopt

minimal intervention, i.e. a standard Ramsey protocol. Further improvements upon this scheme
can be made by introducing a time-dependent driving field that transforms the OAT Hamiltonian
into a two-axis twisting (TAT) one. While the OAT model initially generates squeezing faster than
the TAT model, the squeezing generation rate of OAT (measured in dB per second) falls off withtime, while the squeezing generation rate for TAT remains approximately constant until reaching
2 ∼ N −1 [33].
Heisenberg-limited amount of spin squeezing with ξopt

There are two general strategies for converting OAT into TAT: by use of either a pulsed[118] or continuous [119] drive protocol. For simplicity, we consider the latter in this work, although the pulsed protocol could provide additional advantages, as explained at the end of Appendix 4.E. Following the prescription in Ref. [119], we use the clock laser to apply an amplitudemodulated drive Ĥdrive (t)/~ = Ω0 cos(ωt)Ŝx . If the modulation frequency ω satisfies ω  N χ and
J0 2Ω0 /ω = ±1/3, where χ is the OAT squeezing strength in Eqn. (4.4) and J0 is the zero-order
Bessel function of the first kind, then up to (i) an ∼ Ŝ · Ŝ term that contributes only overall phase87
factors, and (ii) an ∼ Ŝz term that can be eliminated with a simple dynamical decoupling pulse(+)

sequence (see Appendix 4.D), the effective Hamiltonian becomes ĤTAT /~ = (χ/3)(Ŝz2 − Ŝx2 ) or(−)

ĤTAT /~ = (χ/3)(Ŝy2 − Ŝx2 ) (see Appendix 4.C), which squeezes an initial state polarized along they or z axis, respectively.

4.3

Experimental implementation and practical considerations
Thus far, we have largely considered the general preparation of spin-squeezed states with the

Fermi-Hubbard model. Here, we discuss the specific implementation of the above protocols in thestate-of-the-art 3D 87 Sr optical lattice clock (OLC). If successful, such an implementation would(to our knowledge) for the first time break through the proof-of-principle stage of spin squeezing
efforts, and achieve a genuine metrological enhancement of a world-class quantum sensor.
As required for our protocol, 3D 87 Sr OLC has demonstrated the capability to load a quantum
degenerate gas into a 3D lattice at the “magic wavelength” (λlattice = 2a ≈ 813 nm) for which both
the ground (1 S0 , ↓) and first excited (3 P0 , ↑) electronic states (i.e. the “clock states”) of the atoms
experience the same optical potential [42]. Furthermore, the 3D 87 Sr OLC currently operates at
sufficiently low temperatures to ensure vanishing population above the lowest Bloch band, suchthat its dynamics are governed by the Fermi-Hubbard Hamiltonian [Eqn. (4.2)] [112].
An external clock laser with wavelength λL ≈ 698 nm resonantly interrogates the 1 S0 and 3 P0states of the atoms and generates spin-orbit coupling (SOC) [103]. While the relative wavelengthsof the lattice and clock lasers do not allow for weak SOC along all three lattice axes, weak SOC
along two axes can be implemented by, for example, (i) fixing a large lattice depth along the z axis,
effectively freezing atomic motion along z, and then (ii) making the clock laser nearly collinearwith the z axis, with only a small projection of its wavenumber kL onto the x-y plane [see Figure
4.1(a)]. The entire 3D OLC then factorizes into an array of independent 2D systems with N = f `2
atoms each, where ` is the number of lattice sites along each axis of the lattice. As in the 1D case,atoms within the 2D system experience all-to-all interactions, as well as spin-orbit coupling along
two directions characterized by SOC angles φx,y = kLx,y a. Generally speaking, higher-dimensional

88systems (e.g. 2D vs. 1D) are more desirable because they allow packing more interacting atoms
into a fixed system volume, thereby increasing the maximally attainable amount of spin squeezing.
Figure 4.3 shows, for both OAT and TAT protocols, the maximally attainable amount ofspin squeezing and the shortest time at which it occurs as a function of the lattice depth V0 andlinear lattice size ` in a single half-filled 2D layer (i.e. f = 1, N = `2 ) of the 3D OLC. Atoms
are confined along the direction transverse to the 2D layer by a lattice of depth 60 ER , where ER
is the atomic lattice recoil energy. The maximally attainable amount of spin squeezing by each
protocol in Figure 4.3 depends only on the atom number N , while the shortest attainable timeeis determined by choosing the largest SOC angles φx = φy ≡ φ which saturate B/U≈ 0.05. Weeimpose this constraint on B/U
to ensure validity of the OAT Hamiltonian perturbatively derived
in Appendix 4.A (see also Appendix 4.B).
Currently, light scattering from the lattice beams induces decoherence of the clock on a timescale of ∼ 10 seconds [7, 23], which is much shorter than the natural 3 P0 lifetime of ∼ 160 seconds(see Appendix 4.F). This limitation imposes significant constraints on achievable spin squeezing, asshown in Figure 4.4 where the maximal squeezing with spin decay in the OAT case was determinedusing exact expressions for spin correlators derived in Ref. [120], while in the TAT case these
correlators were determined by solving Heisenberg equations of motion for collective spin operators[3] (see Appendix 4.H). Due to the fast growth of Heisenberg operators in systems with all-to-allinteractions, the latter method is not always capable of simulating up to the optimal squeezing
time, and thus only provides a lower bound on the maximal squeezing theoretically obtainable via
TAT.
The results in Figure 4.4 show that squeezing via OAT saturates with system size aroundN ≈ 103 (` ≈ 30), while TAT allows for continued squeezing gains through N = 104 (` = 100).
Even with decoherence, our protocol may realistically generate ∼ 10–14 dB of spin squeezing in∼ 1 second with ∼ 102 –104 atoms in a 2D section of the lattice, which is compatible with the atom
numbers and interrogation times of state-of-the-art optical lattice clocks [42, 73]. This amount ofspin squeezing exceeds those reported in the ground-state nuclear spin sublevels of a state-of-the-art

89

U/J
1

2

U/J
3

4

5 6

1

2

3

4

5 6

80

4

OAT
TAT

3
60
2
40

Time (sec)

Linear lattice size (`)

100

1
20

(b) OAT(a)(c) TAT0102030
Squeezing (dB)2

4
6
Lattice depth (V0 /ER )2

4
6
Lattice depth (V0 /ER )Figure 4.3: Optimal squeezing with one- and two-axis twisting in a 2D section of the 3D 87 Sr
optical lattice clock. (a) The maximum amount of squeezing depends only on the atom number
N = `2 , where ` is the number of lattice sites along each axis of the lattice. While the time scales
for squeezing generally depend on several experimental parameters, the time at which maximal
squeezing occurs can be minimized at any given lattice depth V0 by choosing SOC angles φ thatee is the variance of the SOC-induced axial field and U is the two-atomsaturate B/U≈ 0.05, where B
on-site interaction energy. Panels (b, c) show these minimal squeezing times as a function of thedepth V0 and linear size ` of the lattice. Lattice depths V0 are normalized to the atomic latticerecoil energy ER , and the upper axis on panels (b, c) marks values of U/J at fixed lattice depths.
In general, TAT achieves more squeezing than OAT for any system size, and achieves optimalsqueezing faster for N & 400 atoms, as denoted by a dotted line in panels (b, c).

90

U/J
1

2

U/J
3

4

5

6

1

2

3

4

5

6
16

80

14

60
12
40

Squeezing (dB)Linear lattice size (`)10010(a) OAT

20
2

3
4
5
6
Lattice depth (V0 /ER )(b) TAT
7 2

3
4
5
6
Lattice depth (V0 /ER )7

Figure 4.4: Optimal squeezing with decoherence via one- and two-axis twisting in a 2D sectionof the 3D 87 Sr optical lattice clock (OLC). In practice, decoherence due to light scattering limits
the amount of squeezing that is attainable in the the 3D 87 Sr OLC. Due to growing squeezing times
with increasing system size, the maximal squeezing obtainable via OAT saturates past ` ≈ 30 sites
along each axis of the lattice, with N ≈ 103 atoms total. The more favorable size-dependence of
TAT time scales, however, allow for continued squeezing gains through ` = 100 (N = 104 ). While
the OAT results in (a) are exact, the TAT results in (b) reflect only a lower bound on the maximum
squeezing obtainable, albeit one that is likely close (within a few dB) to the actual value. Optimalsqueezing times in the presence of decoherence are generally smaller than the corresponding times
shown in Figure 4.3, as decoherence typically degrades squeezing before it reaches the decoherencefree maximum. The decoherence considered in this work also limits maximally achievable squeezing
to ∼ 20 dB less than the decoherence-free maxima shown in Figure 4.3. Sample plots of squeezingover time for particular choices of lattice size (`) and depth (V0 /ER ) are provided in Appendix 4.G.

91
171 Yb OLC (∼ 6.5 dB) [121]. While the latter protocol might be used to transfer spin squeezing to

the electronic clock state, to date there has been no demonstration of spin squeezing in an opticalclock transition.
In addition to light scattering, p-wave losses from inelastic 3 P0 collisions [50, 53, 122] can alsodegrade the maximum achievable spin squeezing, which becomes more pronounced for shallower
lattices. More details on p-wave losses are discussed in Appendix 4.J, where we show that operating
at lattice depths V0 & 7ER may be necessary to suppress the impact of inelastic collisions on spinsqueezing, at the cost of slightly increasing light scattering.
The sources of decoherence considered above are not fundamental, and can be avoided by,for instance, using two nuclear spin levels as spin-1/2 degrees of freedom that are interrogated by
far-detuned Raman transitions instead of a direct optical transition [123]. The strength of SOC for
Raman transitions is tunable and, moreover, the lifetimes of ground nuclear spin levels are longer
than 100 seconds in the lattice [7]. In this case, our protocol for preparing a squeezed state would
additionally end with a coherent state transfer from nuclear to electronic degrees of freedom toretain metrological utility for the atomic clock. If, for example, the −9/2 and −7/2 nuclear spinstates are used for the preparation of a squeezed state, then the collective-spin entanglement of
atoms can be transferred to electronic states at the end of the spin squeezing protocol with a σ −polarized π pulse. Such a pulse can transfer g, −7/2 to e, −9/2 without affecting g, −9/2 ,where g and e respectively denote the ground and excited (electronic) clock states.

4.4

Conclusions
We have proposed a new protocol to generate spin squeezing in a fermionic 3D optical latticeclock by combining nominally undesirable atomic collisions with spin-orbit coupling. To our knowledge, this is the first proposal to use quantum correlations in a many-body fermionic system topush state-of-the-art quantum sensors beyond the independent-particle regime, thereby achieving a
genuine quantum advantage. Such capability could allow for major improvements in clock sensitivity and bandwidth, enhancing not only traditional timekeeping applications such as measurement

92
standards, navigation (GPS), and telecommunications, but also geodesy and gravitational wave detection, precision tests of fundamental physics, and the search for new physics beyond the standard
model [18].

Acknowledgments
We acknowledge helpful discussions with M. Norcia, C. Sanner, and M. Mamaev. This
work is supported by the Air Force Office of Scientific Research (AFOSR) grant FA9550-18-1-0319;
the AFOSR Multidisciplinary University Research Initiative (MURI) grant; the Defense Advanced
Research Projects Agency (DARPA) and Army Research Office (ARO) grant W911NF-16-1-0576;
the National Science Foundation (NSF) grant PHY-1820885; JILA-NSF grant PFC-1734006; andthe National Institute of Standards and Technology (NIST).

4.A

Derivation of the effective one-axis-twisting model
Suppose we have a Hamiltonian of the form (~ = 1)
H = H0 + V,

(4.6)

with
H0 = −U
S · S,LV =−X

Bn s(n)z + ΩSx ,

(4.7)

n

and we consider N -particle states initially in the ground-state manifold G0 of H0 , which have total
spin S = N/2. If the largest eigenvalue of V is smaller in magnitude than half of the collective
spin gap N U/L = f U , i.e. the energy gap under H0 between G0 and its orthogonal complement E0 ,then we can formally develop a perturbative treatment for the action of V on G0 . Such a treatment
P
(p)
(p)
yields an effective Hamiltonian on G0 of the form Heff =p Heff , where Heff is order p in V .
Letting P0 (Q0 ) be a projector onto G0 (E0 ), we define the super-operators O and L by
OV ≡ P0 V Q0 + Q0 V P0 ,LV ≡X |αihα| OV |βihβ|α,βEα − Eβ

,

(4.8)

93
where H0 =P

α Eα |αihα|.

The first few terms in the expansion of the effective Hamiltonian Heff

are then, as derived in Ref. [80],
(0)

(1)

Heff = P0 H0 P0 ,Heff = P0 V P0 ,

(4.9)

and
1
(2)
Heff = − P0 [OV, LV ]− P0 ,2

(4.10)(0)with [X, Y ]− ≡ XY −Y X. The zero-order effective Hamiltonian Heff = H0 within the ground-state(1)

manifold. To calculate Heff , we note that the ground-state manifold G0 is spanned by the Dicke
statesN/2+m|mi ∝ S+|↓i⊗n ,

S+ ≡

X (n)
s+ ,

(4.11)

nin terms of which we can expand the collective spin-z operator as Sz =P

m m |mihm|.

We canlikewise expand the collective spin-x operator Sx in terms of x-oriented Dicke states |mx i as Sx =P
m m |mx ihmx |. The ground-state projector P0 onto G0 can be expanded in either basis as P0 =PP
m |mihm| =m |mx ihmx |. Defining the mean and residual fields
1 X
Bn ,
N n

B≡

bn ≡ Bn − B,

(4.12)

we can then write
V =−

X


X
bn + B s(n)bn s(n)z + ΩSx = −z − BSz + ΩSx ,n(4.13)nand in turn!
(1)
Heff = P0−

X

bn s(n)z − BSz + ΩSx

P0 = −Xn

bn P0 s(n)z P0 − BSz + ΩSx ,(4.14)nwhere we used the fact that P0 Sj=z,x P0 = Sj within the ground-state manifold. By construction,P
the residual fields are mean-zero, i.e. n bn = 0. Using the particle-exchange symmetry of the
Dicke states, we can therefore expandXnbn P0 s(n)z P0 =

Xn,m,m0bn |mihm| s(n)m0 m0 =
z

X
n

bn

X
m,m0

|mihm| s(1)m0 m0 = 0,z(4.15)

94
which implies(1)

Heff = −BSz + ΩSx .

(4.16)(2)

To calculate the second-order effective Hamiltonian Heff , we let B0 (E0 ) denote an eigenbasis of H0
for the excited subspace E0 , and set the ground-state energy to 0. We then define the operator
X

I≡|αi∈B0 (E0 )|αihα|,
Eα(4.17)

which sums over projections onto excited states with corresponding energetic suppression factors,
in terms of which we can write(2)

Heff = −P0 V IV P0 ,(4.18)

which is simply an operator-level version of the textbook expression for second-order perturbation
theory. The only part of V which is off-diagonal with respect to the ground- and excited-stateP(n)manifolds G0 and E0 is − n bn sz , and the individual spin operators in this remainder can onlychange the total spin S by at most 1. It is therefore sufficient to expand I in a basis for states(n)which span the image of G0 under all sz

within the S = N/2−1 manifold. Such a basis is provided

by the spin-wave states|mki ∝

NX(n)e2πikn/N s+ |m − 1i ,(4.19)

n=1

for k = 1, 2, · · · , N − 1 [47]. Using the fact that all spin-z operators preserve the projection of total
spin onto the z axis, we then have that(2)

Heff = −

1
fU

X

0

(n )
bn bn0 |mihm| s(n)|mihm| ,z |mkihmk| sz(4.20)

m,k,n,n0

where the relevant matrix elements between the Dicke states and the spin-wave states are [47]sDE(N/2)2 − m2
m s(n)
mk = e2πikn/N,(4.21)z
N 2 (N − 1)which implies(2)

Heff = −X
1 X (N/2)2 − m2
2πik n−n0 /N0e|mihm|×bb.
nnf U m N 2 (N − 1)0k,n,n(4.22)

95
Using the fact thatXP

n bn = 0, we can expand0

bn bn0 e2πik n−n /N =k,n,n0X

bn bn0

n,n0

N
−1
X

0



e2πik n−n /N =X

bn bn0

n,n0

k=1

N
−1
X

0



e2πik n−n /N ,(4.23)

k=0

where the sum over k vanishes for n 6= n0 and equals N when n = n0 , soX0e2,
bn bn0 e2πik n−n /N = N 2 B



(4.24)

k,n,n0where
e2 ≡ 1BN

21 X
Bn − B .
N n(4.25)


X N/2 2 − m2
e 2 |mihm| ,
B
(N
−
1)
f
U
m

(4.26)

X

b2n =

n

We therefore have that(2)
Heff = −where the N/2

2term contributes a global energy shift which we can neglect, while the m2 termis proportional to m2 |mihm| = Sz2 . In total, the effective Hamiltonian through second order in
perturbation theory is thus
Heff = −U
S · S − BSz + ΩSx + χSz2 ,L(4.27)

with
χ≡

e2
B
.
(N − 1) f U(4.28)We benchmark the validity of this effective Hamiltonian via exact simulations of the spin [Eqn. (4.3)]
and OAT [Eqn. (4.4)] Hamiltonians in a system of 20 spins, finding that the relative error in maximalesqueezing (in dB) of the OAT model is less than 3% when B/U< 0.06 (see Appendix 4.B).

4.B

Numerical benchmarking of the OAT model
Here we provide additional information about our benchmarking of the one-axis twistingmodel against the spin model. This benchmarking was performed via exact simulations of a 20spin system. Figure 4.5 shows the relative error in maximal squeezing of the OAT model (measured<latexit sha1_base64="TEan9Qw4dF3Rkiparf12osFCqCg=">AAACBHicbVA9SwNBEN2LXzF+RS21WAyBCBIuIqhd0MYygmcCuRD2NpNkyd4Hu3PBcKSx8a/YWKjY+iPs/Ddukis08cHA470ZZuZ5kRQabfvbyiwtr6yuZddzG5tb2zv53b17HcaKg8NDGaqGxzRIEYCDAiU0IgXM9yTUvcH1xK8PQWkRBnc4iqDls14guoIzNFI7f+giPGCiQBphCBSUCtXYPaElt3jczhfssj0FXSSVlBRIilo7/+V2Qh77ECCXTOtmxY6wlTCFgksY59xYQ8T4gPWgaWjAfNCtZPrFmBaN0qHdUJkKkE7V3xMJ87Ue+Z7p9Bn29bw3Ef/zmjF2L1qJCKIYIeCzRd1YUgzpJBLaEQo4ypEhjCthbqW8zxTjaILLmRAq8y8vEue0fFm2b88K1as0jSw5IEekRCrknFTJDakRh3DySJ7JK3mznqwX6936mLVmrHRmn/yB9fkDDeSXzw==</latexit>

relative error (%)

96
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

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B̃/U
<latexit sha1_base64="0Ii4ytsDMttzU3wLyVS5jprlLy8=">AAAB8XicbVBNS8NAEN34WetX1aOXxSJ4qokI6q3Ui8cKxhbSUDabTbt0sxt2J0IJ/RlePKh49d9489+4bXPQ1gcDj/dmmJkXZYIbcN1vZ2V1bX1js7JV3d7Z3duvHRw+GpVrynyqhNLdiBgmuGQ+cBCsm2lG0kiwTjS6nfqdJ6YNV/IBxhkLUzKQPOGUgJWCHnARs6I1Off7tbrbcGfAy8QrSR2VaPdrX71Y0TxlEqggxgSem0FYEA2cCjap9nLDMkJHZMACSyVJmQmL2ckTfGqVGCdK25KAZ+rviYKkxozTyHamBIZm0ZuK/3lBDsl1WHCZ5cAknS9KcoFB4en/OOaaURBjSwjV3N6K6ZBoQsGmVLUheIsvLxP/onHTcO8v681WmUYFHaMTdIY8dIWa6A61kY8oUugZvaI3B5wX5935mLeuOOXMEfoD5/MHdR6Q5w==</latexit>

Figure 4.5: Relative error between maximal squeezing (measured in dB) obtained by the OAT[Eqn. (4.4)] and spin [Eqn. (4.3)] models of the main text in a system of 20 particles. The OAT
model correctly captures the maximal squeezing (in dB) of the spin model to within 3% (markedeby the horizontal reference line) within the gap-protected regime B/U< 0.06.

e . Here squeezing is measured
against the spin model) as a function of the reduced field variance B/U
in decibels (dB) by −10 log10 ξ 2 for the squeezing parameter ξ 2 define in Eqn. (4.5). The relativeeerror in maximal squeezing (in dB) by the OAT model is less than 3% when B/U< 0.06.
In principle, spin-changing decoherence compromises the validity of the OAT model, as its
perturbative derivation in Appendix 4.A relies on spin population remaining primarily within the
Dicke manifold. This assumption breaks down in the presence of, for example, spontaneous emission, which transfers population outside of the Dicke manifold. Nonetheless, we find decent agreement between the OAT and spin models when decoherence is sufficiently weak (see Figure 4.6).

97

(a)

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γ= -

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Figure 4.6: Comparison between the OAT and the spin model in the presence of decoherence. (a) The difference between the maximal squeezing (measured in dB) obtained by
the OAT [Eqn. (4.4)] and spin [Eqn. (4.3)] models increases with the particle number N and thesingle-particle spontaneous emission rate γ. This disagreement is attributed in part to the fact thatspontaneous emission transfers population of the collective spin state outside of the Dicke manifold, violating an assumption of the OAT model; see panel (b). The rate of population transfer
outside of the Dicke manifold increases with both particle number and spontaneous emission rate.
(Parameters for simulations in this figure: U = 1000 Hz, J = 200 Hz, and φ = π/20).

98

4.C

Two-axis twisting, decoherence, and the residual axial field
The protocol we use to transform one-axis twisting (OAT) into two-axis twisting (TAT) is

as previously proposed in Ref. [119]; we provide a summary of this protocol here, in addition tosome brief discussion of its implications for decoherence and the residual ∼ S · S and ∼ Sz termsof our OAT protocol. The TAT protocol begins with the OAT Hamiltonian with a time-dependenttransverse field,
H = χSz2 + Ω (t) Sx ,Ω (t) = βω cos (ωt) ,(4.29)

where β is the modulation index of the driving field and the drive frequency ω  N χ, with N
the total number of spins. Moving into the rotating frame of Ω (t) Sx subtracts this term from the
Hamiltonian, and transforms operators O as
O → U (t)† OU (t) ,(4.30)

where
"

Z t

U (t) ≡ exp −i#dτ Ω (τ ) Sx = exp −iβ sin (ωt) Sx .

(4.31)

0

In particular, the operators S̃± ≡ −Sz ± iSy (i.e. the raising and lowering operators in the x basis)transform simply as
S̃± → U † S̃± U = e±iβ sin(ωt) S̃± .
For any operator O and drive frequency ω  kOk, where kOk ≡ maxψ

(4.32)
phψ|O† O|ψi is the operator

norm of O (i.e. the magnitude of the largest eigenvalue of O), we can generally make a secularapproximation to saye±imβ sin(ωt)

∞
X

O=

Jn (±mβ) einωt O ≈ J0 (±mβ) O = J0 (mβ) O,(4.33)n=−∞where Jn is the n-th order Bessel function of the first kind. Expanding Sz2 =21
S̃+ + S̃− , one
4

can thus work out that the effective Hamiltonian in the rotating frame of the drive is
Heff ≈



 
χ 
1 + J0 (2β) Sz2 + 1 − J0 (2β) Sy2 .
2(4.34)

99
Driving with a modulation index β for which J0 (2β) = ±1/3 then gives us the effective two-axis
twisting Hamiltonians
 χ

χ 2
2Sz + Sy2 'Sz2 − Sx2 ,
3
3
 χ

χ 2
(−)
Heff =
Sz + 2Sy2 'Sy2 − Sx2 ,3
3(+)

Heff =(4.35)(4.36)

where ' denotes equality up to the addition of a term proportional to S 2 = Sz2 + Sx2 + Sy2 , which isirrelevant in the absence of coherent coupling between states with different net spin. In a similar
spirit, one can work out that single-spin operators transverse to the x-axis transform as
s̃± ≡


1
−sz ± isy → U † s̃± U = e±iβ sin(ωt) s̃± ≈ J0 (β) s̃± ,2(4.37)

which implies that shifting into the rotating frame of the time-dependent drive takes
sx → sx ,sy,z → J0 (β) sy,z ,(4.38)

and
s± →



1
1
1 ± J0 (β) s+ +1 ∓ J0 (β) s− .
22

(4.39)(±)

As the TAT Hamiltonians Heff are realized in a rotating frame, to properly account for decoherence
throughout the TAT protocol one must transform jump operators according to Eqns. (4.38)-(4.39).
In practice, our protocols realize the OAT Hamiltonian in Eqn. (4.29) with additional ∼ S · S
and ∼ Sz terms [see Eqn. (4.27)]. The effect of the ∼ S · S term is to generate a relative phasebetween states with different total spin S (with e.g. S = N/2 within the Dicke manifold). In theabsence of coherent coupling between states with different total spin, therefore, the ∼ S · S term
has no effect on system dynamics. The ∼ Sz term, meanwhile, is important; the magnitude of thisterm (as measured by the operator norm) is generally comparable to that of the squeezing termχSz2 . Unlike in the case of OAT, Sz does not commute with the TAT Hamiltonians, so its effects
cannot be eliminated by a single spin-echo π-pulse exp (−iπSx ) half way through the squeezing
protocol. Nonetheless, we find that for N = 102 (103 ) atoms, ∼ 5 (10) π-pulses in a CPMG

100
(Carr-Purcell-Meiboom-Gill) sequence [124, 125] suffice to mitigate the effects of the Sz term in the
TAT protocol (see Appendix 4.D). Phase control over these pulses, specifically choices of whether
to apply exp (±iπSx ) or exp ±iπSy in any given π-pulse, can be used to construct XY-n pulse
sequences [126, 127] that are robust to pulse errors.

4.D

Dynamical decoupling in the TAT protocol
The effective Hamiltonian resulting from a perturbative treatment of SOC is (see Appendix

4.A)
Heff = −U
S · S − BSz + ΩSx + χSz2 ,L

where U is a two-atom on-site interaction strength; L is the number of lattice sites; B ≡(4.40)
Pn Bn /Nis a residual axial field determined by the occupied quai-momentum modes {n} (with {n} = N
atoms total); Ω is the magnitude of a driving field; and χ is an effective OAT squeezing strength.
The effect of the ∼ S · S term is to generate a relative phase between states with different total
spin S (where S = N/2 within the Dicke manifold). In the absence of coherent coupling between
states with different total spin, therefore, the ∼ S · S term has no effect on system dynamics, andwe are safe to neglect it entirely.
In the parameter regimes relevant to our discussions in the main text, the operator normsof B Ŝz and χŜz2 in Eqn. (4.40) will typically be comparable in magnitude. The OAT protocol sets
Ω = 0, and eliminates the effect of B Ŝz with a spin-echo π-pulse exp −iπ Ŝx applied half way(±)

through the squeezing protocol. The TAT protocol, meanwhile, effectively takes χŜz2 +ΩŜx → ĤTAT(as defined in Appendix 4.C) and B Ŝz → J0 (β± ) B Ŝz , where J0 is the zero-order Bessel function of
the first kind and β± is the modulation index of the amplitude-modulated driving field Ω, satisfying
J0 (2β± ) = ±1/3. Unlike in the case of OAT, Ŝz does not commute with the TAT Hamiltonian, so
its effect cannot be eliminated with a spin-echo. Nonetheless, this term can be eliminated with a(±)

dynamical decoupling pulse sequence that periodically inverts the sign of Ŝz while preserving ĤTAT .
Figure 4.7 shows the maximal squeezing generated by N = 102 and 103 atoms via OAT, TAT,

101(a) N = 100(b) N = 1000

14
Squeezing (dB)

22.5
12
20.0
10

17.5

8

OAT
TAT
TAT+,zTAT−,z

15.0

6

12.5
0

5

10
Pulses

15

0

5

10

15

PulsesFigure 4.7: Optimal squeezing as a function of π-pulses applied prior to the optimal TAT
squeezing time in a CPMG sequence with (a) N = 100 and (b) N = 1000 atoms. Results are shown
(±,z)
(±)
for OAT,D TAT,
E and TAT±,z , where TAT±,z denotes squeezing via the Hamiltonian ĤTAT ≡ ĤTAT −J0 (β± ) B
the text.

rmsfŜz . Details about experimental parameters for these simulations are provided in102
and TAT in the presence of the mean field J0 (β± ) B Ŝz as a function of the number of π-pulses
performed prior to the optimal TAT squeezing time. These pulses are applied in a CPMG sequencenτn /2 − πx − τn /2 , where τn /2 denotes Hamiltonian evolution for a time τn /2, πx denotes theapplication of an instantaneous π-pulse exp −iπ Ŝx , and n is the number of pulses, such that thenoptimal TAT squeezing time is tTAT
opt = (τn ) . The label TAT±,z in Figure 4.7 denotes squeezing
D ErmsD Erms(±,z)(±)
through the Hamiltonian ĤTAT ≡ ĤTAT − J0 (β± ) B
Ŝz , where B
is the root-meanffsquare average of B over choices of occupied spacial modes {n} at fixed filling f of all spatial
modes in the lowest Bloch band of a periodic 2D lattice. While the modulation index β+ is
uniquely defined by J0 (2β+ ) = 1/3, there are two choices of β− for which J0 (2β− ) = −1/3; we
use that which minimizes J0 (β− ) . Figure 4.7 assumes an SOC angle φ = π/50 (although resultse
are independent of φ for φ  1), a reduced field variance B/U= 0.05, and a filling f = 5/6. Notethat as the filling f → 1, the residual axial field vanishes (B → 0), so TAT±,z → TAT.

4.E

Clock interrogation after squeezing
The protocols in our work concern the preparation of spin-squeezed states in an opticallattice clock. Here, we discuss the use of these states in a follow-up clock interrogation protocol.
For simplicity, we restrict our discussion to the case of squeezing in 1D, as in Section 4.2 of themain text, with the understanding that a generalization of this discussion to higher dimensions isstraightforward.
A spin-squeezed state is generated by interactions and SOC that are generally undesirable
during the clock interrogation protocol. In the parameter regimes considered in our work, interactions alone have no effect on clock interrogation: absent of coherence between states withdifferent net spin S, collective S · S interactions only generate unobservable global phases withineach fixed-S sector of Hilbert space. Therefore, the remaining task to allow clock interrogation after
spin squeezing is to turn off SOC, which inhomogeneously detunes atomic transition frequencies by
an amount Bq that depends on the quasi-momentum q of an atom. The SOC-induced axial fields

Bq ∼ J sin φ/2 depend on two tunable parameters: the tunneling rate J and the SOC angle φ.

103
The simplest way to turn off SOC is thus to increase the lattice depth prior to clock interrogation,
taking J ∼ Bq → 0. Increasing the lattice depth to turn off SOC is compatible with the current
clock interrogation sequence, but is incompatible with ongoing efforts to mitigate light scatteringfrom the lattice beams (currently the primary source of decoherence in the clock; see Appendix
4.F) by using shallower lattices [23]. We thus devote the rest of this section to discussing strategies
for turning off SOC that are compatible with using the same lattice depth for clock interrogation
as the spin squeezing generation.
If we cannot take the tunneling rate J → 0, our remaining control parameter for turning off
SOC is the SOC angle φ = kL a, where kL is the projection of the clock laser wavenumber onto
the lattice axis and a is the lattice spacing. The squeezing protocol needs a clock laser with asmall but nonzero SOC angle φ  1, while the clock interrogation protocol requires a clock laser
with φ = 0. Simply using one clock laser with φ 6= 0 for the squeezing protocol and another clock
laser with φ = 0 for the clock interrogation protocol, however, does not resolve this discrepancy,
because a state that is squeezed with respect to spin operators that are homogeneous (i.e. of theform Sx , Sy , Sz ) in a particular gauge is not necessarily squeezed with respect to spin operators thatare homogeneous in a different gauge. In this appendix, we will work explicitly in the “lab gauge”
of the clock interrogation protocol, in which the Fermi-Hubbard Hamiltonian is SOC-free and theφ = 0 clock laser is homogeneous. To resolve the fact that our squeezing protocol prepares a statethat is squeezed in the “wrong gauge”, we will construct a simple pulse sequence that transforms
the inhomogeneous spin operators accessible by the φ 6= 0 laser into a homogeneous form in ourlab gauge.
Starting with a spin-down-polarized initial stateY †|−Zi ≡  cj,↓  |vacuumi ,(4.41)

jour OAT protocol prepares the state
E
 (θ)
(0)
(θ)
ξOAT = e−iHFH t e−i π/2 Sx |−Zi ,(4.42)

104(0)

where HFH is the Fermi-Hubbard Hamiltonian in Eqn. (4.2) without SOC; t is some free evolution(θ)

time; and Sx is the “rotated” spin-x-like generator induced by a clock laser with SOC angle φ = θ,namelySx(θ) =

1 X iθj †e cj,↑ cj,↓ + h.c..
2(4.43)

j

Defining on-site spin operators (in the lab gauge)
1 †cj,↑ cj,↑ − c†j,↓ cj,↓
2

1 †
†
s(j)
≡
c
c
+
c
c
j,↓
x
j,↓ j,↑
2 j,↑


i †
†
s(j)
≡
c
c
−
c
c
j,↑
y
j,↑ j,↓ ,
2 j,↓
s(j)
z ≡

(4.44)(4.45)(4.46)

we can identify the rotated collective spin operators
Sx(θ) ≡

XSy(θ) ≡X(j)cos (θj) s(j)
+
sin
(θj)
s
,
x
y

(4.47)

(4.48)j(j)cos (θj) s(j)y − sin (θj) sx

.

j

E
(θ)
The state ξOAT is squeezed with respect to components of the rotated collective spin vector Sθ ≡


E
(θ)
(θ)
(θ)
Sx , Sy , Sz . Therefore, to take advantage of the squeezing in ξOAT , the clock interrogation
protocol effectively needs to rotate this state by some unitary exp (−iη · Sθ ), and then extract
information about the rotation vector η from collective spin observables of the form
D
E
(θ)
(θ)
hOθ iηOAT ≡ ξOAT eiη·Sθ Oθ e−iη·Sθ ξOAT ,(4.49)(θ)(θ) (θ)where Oθ is some product of the rotated collective spin operators in Sθ , e.g. Sx or Sx Sy . In orderto turn off SOC during clock interrogation, however, we are restricted to performing rotations of theform exp (−iη · S0 ) and measuring homogeneous operators O0 . We thus seek a “gauge-switching”
operation Gθ that maps homogeneous operators O0 onto rotated operators Oθ via G†θ O0 Gθ = Oθ .
Equipped with Gθ , we could decompose
D
E D
E
(θ)
(θ)
(θ)
(θ)
hOθ iηOAT = ξOAT G†θ eiη·S0 O0 e−iη·S0 Gθ ξOAT = ξ˜OAT eiη·S0 O0 e−iη·S0 ξ˜OAT(4.50)

105
for a transformed state
E
E
(θ)
(θ)
ξ˜OAT ≡ Gθ ξOAT

(4.51)

that is now squeezed with respect to the homogeneous collective spin operators in S0 , accessible
with the SOC-free (φ = 0) clock laser used during clock interrogation.
(θ)

(θ)

Given the definitions of the rotated spin operators Sx , Sx in Eqns. (4.47), (4.48), a suitable
candidate for the gauge-switching operator Gθ is the site-dependent rotation
Gθ =

Y



exp iθjs(j).
z(4.52)

j

To implement Gθ with “global” (i.e. non-site-selective) experimental controls, we decompose each
local rotation into a product of two reflections:






exp i2αs(j)' exp iπs(j)exp iπs(j)
z
x
α
(j)

(j)

(4.53)(j)where ' denotes equality up to an overall phase, and sα ≡ cos αsx +sin αsy . This decomposition
impliesGθ ' exp iπSx(0) exp iπSx(θ/2) ,(4.54)

which can be implemented using one SOC-free (φ = 0) clock laser, and one clock laser with SOC
angle φ = θ/2. Appending the two π-pulses given by Eqn. (4.54) to our squeezing protocol thusE(θ)
prepares a state ξ˜OAT that is squeezed with respect to the homogeneous collective spin operators
Sx , Sy , Sz accessible to the SOC-free (φ = 0) clock laser used during clock interrogation.
As presented, the combined squeezing and clock interrogation protocols require three clocklasers in total: one without SOC (φ = 0), and one each for SOC angles φ ∈ θ/2, θ . We can use
Gθ , however, to decompose any pulse exp (−iβ · Sθ ) into composite pulses that use only S0 andSθ/2 :
exp (−iβ · Sθ ) = G†θ exp (−iβ · S0 ) Gθ .

(4.55)

106
The state prepared by the OAT squeezing protocol, for example, can be equivalently prepared via
E
 (0)
 (0)
(0)
(0)
(θ/2)
(θ)
ξOAT = e−iHFH t G†θ e−i π/2 Sx Gθ |−Zi ' e−iHFH t e−iπSx ei π/2 Sx |−Zi .

(4.56)

Spin-echo pulses applied throughout OAT can likewise be decomposed according to (4.55), eliminating the need for a clock laser with SOC angle φ = θ. Applying a continuous drive during asqueezing protocol, however, still requires all three clock lasers. If carefully tuning the relativeorientations of three clock lasers proves to be too difficult in practice, converting OAT into TAT
would therefore need to be done with a pulsed drive protocol, as in Ref. [118].

4.F

Decoherence in the 3D 87 Sr optical lattice clock
Currently, light scattering from lattice beams in the 3D 87 Sr optical lattice clock induces

decoherence on a time scale of ∼10 seconds [7, 23]. This decoherence acts identically on all atoms
in an uncorrelated manner, and can be understood by considering the density operator ρ for a single
atom, with effective spin states ↓ and ↑ respectively corresponding to the 1 S0 and 3 P0 electronic
states. Empirically, the effect of decoherence after a time t within the {↓, ↑} subspace of a single
atom is to take ρ → ρ (t) with ρ (0) ≡ ρ andρ↑↑ e−Γ↑↑ t
ρ (t) := ρ∗↑↓ e−Γ↑↓ tρ↑↓ e−Γ↑↓ t

,



ρ↓↓ + 1 − e−Γ↑↑ t ρ↑↑(4.57)

where Γ↑↑ ≈ Γ↑↓ ≈ Γ = 0.1 sec−1 are respectively decay rates for ρ↑↑ and ρ↑↓ . This form of
decoherence can be effectively modeled by decay and dephasing of individual spins (respectively
denoted Γud and Γel in Ref. [120]) at rates Γ. In the language of the section that follows, we would
n o
n o
(j)
(j)
say that this decoherence is captured by the sets of jump operators J− ≡ s− and Jz ≡ sz
with corresponding decoherence rates γ− = γz = Γ.

4.GTime-series of squeezing via OAT and TAT
Figure 4.8 shows an example of squeezing over time via OAT and TAT, both with and without

decoherence via decay and dephasing of individual spins. The OAT model initially generates

107squeezing faster than the TAT model, but the squeezing generation rate of OAT (measured indB per second) falls off with time. The squeezing generation rate for TAT, meanwhile, remains
approximately constant (in the absence of decoherence) until squeezing via TAT surpasses that of
OAT. In the absence of decoherence, OAT achieves a maximal amount of squeezing that scalesas ξ 2 ∼ N −2/3 , while TAT achieves Heisenberg-limited squeezing with ξ 2 ∼ N −1 . Note that our
method for computing squeezing via TAT in the presence of decoherence (described in Appendix
4.H) is not capable of computing squeezing for the full range of times shown in Figure 4.8; thecorresponding time-series data in this figure is therefore shown up to the point at which thismethod breaks down.

108(b) ` = 100

OAT
TAT

30

Squeezing (dB)

Squeezing (dB)(a) ` = 40

20
10
0

OAT
TAT

30
20
10
0

0.0

0.5

1.0
Time (sec)

1.5

2.0

0.0

0.5

1.0
Time (sec)

1.5

2.0

Figure 4.8: Squeezing via OAT and TAT in a 2D section of the 3D 87 Sr optical lattice clock,shown for (a) ` = 40 and (b) ` = 100 sites per axis (with N = `2 atoms total), and a lattice depthof V0 = 4 ER , where ER is the atomic lattice recoil energy. Atoms are confined along the direction
transverse to the 2D layer by a lattice of depth 60 ER . Squeezing over time is shown for OAT (blue)
and TAT (green), both with (solid lines) and without (dashed lines) decoherence via uncorrelated
decay and dephasing of individual spins at rates of 0.1 sec−1 (see Appendix 4.F).

109

4.H

Solving Heisenberg equations of motion for collective spin systems
In order to compute squeezing of a collective spin system, we need to compute expec-

tation values of (homogeneous) collective spin operators. We compute these expectation values using a method recently developed in Ref. [3], and provide a short overview of the methodmmhere. Choosing the basis {Sm } for all collective spin operators, where Sm ≡ S+ + Szmz S− − withm ≡ (m+ , mz , m− ) ∈ N30 , we can expand all collective spin Hamiltonians in the formH=X

hm Sm .

(4.58)

m

The evolution of a general correlator hSn i under a Hamiltonian of the form in Eqn. (4.58) is thengiven by
D
E X
X
X
d
hSn i = ihm [Sm , Sn ]− +γJ D (J ; Sn ) ≡hSm i Tmn ,dtmm(4.59)

J

where [X, Y ]± ≡ XY ± Y X; J is a set of jump operators with corresponding decoherence rate γJ ;the decoherence operator D is defined by
i 
X
1h †
†
D (J ; O) ≡J OJ −J J, O;2+(4.60)J∈J

and Tmn is a matrix element of the time derivative operator T ≡ d/dt. These matrix elements canbe calculated analytically using product and commutation rules for collective spin operators. We
can then expand correlators in a Taylor series about t = 0 to write*+
X tk dk
X tk XhSn i =
S=
Tmn;k hSm it=0 ,nk! dtk
k! m
k≥0

h i
where Tmn;k ≡ T k

mn

t=0

(4.61)

k≥0

are matrix elements of the k-th time derivative. Expectation values ofcollective spin operators can thus be computed via the expansion in Eqn. (4.61), which at shorttimes can be truncated at some finite order beyond which all terms have negligible contribution to
hSn i.

110

N = 20N = 60

(a)

(b)

(c)

(d)

Figure 4.9: Dynamics of non-interacting spin-orbit coupled fermions in a 1D lattice with SOC angle
φ = π/50, plus a harmonic trap with Ω/J = 0.01. Starting with a spin-polarized cloud in ↓ groundstate, an initial clock laser pulse is applied to rotate spins into x, and the atoms are allowed to
evolve during the dark time. We track the dynamics of the ↑ particle density for the cases of (a)N = 20 and (b) N = 60 atoms. Panel (c) shows the time-averaged fluctuations of the ↑ particle
density for each site index j from its initial value following the Ramsey pulse; see Eqn. (4.64). For
N = 60, we have filled all delocalized modes as well as several localized modes, resulting in a large
region of no density fluctuations at the trap center. Panel (d) contains the eigenspectrum for asingle internal state in the presence of the trap (with the index n labeling the eigenvalues in orderof increasing energy), where the critical mode nc dividing the spatially delocalized and localized
modes is indicated by a black dash-dotted line. The highest occupied mode in the ↓ ground state
for N = 20 and N = 60 is indicated by the green and red solid lines, respectively.

111

U/J = 1

U/J = 2

U/J = 4

(a.i)

(b.i)

(c.i)

(a.ii)(b.ii)(c.ii)(a.iii)(b.iii)(c.iii)Figure 4.10: Dynamics of interacting spin-orbit coupled fermions in a 1D lattice plus a harmonic
trap for U/J = 1 (a), 2 (b), and 4 (c). For a 1D lattice with 10 sites and an SOC angle φ = π/50,
we apply a π/2 clock laser pulse to the ↓ ground state and let the system evolve during the darktime. In (a.i)-(c.i) we show the squeezing dynamics of the system for both N = 10 (solid lines) andN = 9 (dashed lines) for a variety of trapping strengths. In (a.ii)-(c.ii), we plot the time-averagedP
fluctuations in total particle density, δnj (as in Eqn. (4.64) but with n̂j,↑ replaced by α n̂j,α ). In(a.iii)-(c.iii), we plot the growth of the doublon population Nd (t) (see Eqn. (4.65)) as a functionof time, noting the absence of squeezing in the presence of a large doublon population. For the
chosen trap strengths, the corresponding values of nc are 28 (Ω/J = 0.01), 14 (Ω/J = 0.04), and 6(Ω/J = 0.2). In panels where the results for the homogeneous case (orange curves) are not visible,
they are nearly identical to the results for Ω/J = 0.01 (green curves). Here, we utilize periodic
boundary conditions to minimize finite size effects.

112

4.I

Effect of a harmonic confining trap
Current 3D optical lattice implementations involve a harmonic confining potential, which sig-

nificantly alters the underlying single-particle eigenstructure and can potentially degrade accessible
squeezing within our protocol. In this appendix, we examine the effect of a harmonic trap on our
protocol and discuss strategies to mitigate undesired effects. We model the trap by the addition of
the termĤΩ = ~ΩX(j − j0 )2 n̂j,α(4.62)j,αto our Fermi-Hubbard model (Eqn. (4.2)), where j0 denotes the trap center and Ω = m(ωa)2 /2~
characterizes the trap strength for atom mass m, trap frequency ω, and lattice spacing a. In current
state-of-the-art 3D 87 Sr OLC implementations, values of ω ≈ 56 × 2π sec−1 can be achieved withineach 2D layer of weak SOC by utilizing in-plane lattice depths of 5ER and a lattice depth of 60ER
in the axial direction, resulting in a value of Ω/J ≈ 0.01. We restrict our discussion to 1D, although
for a separable 3D lattice our arguments should extend in a straightforward manner.
We briefly review the structure of the single-particle eigenstates of the system, before discussing the effects on squeezing. In the quasi-momentum basis, the eigenfunctions ψn,α (q) = hq|n, αi
are given by the π-periodic Mathieu functions, with the corresponding energies described by the
Mathieu characteristic values [24]. In the presence of SOC, using the gauge transformation described in the main text, we obtain the relation
ψn,↑ (q) = ψn,↓ (q − φ/a).

(4.63)

In contrast to the case of a pure harmonic potential, which generically has spatially delocalized
single-particle eigenstates, the addition of a tight-binding lattice causes eigenmodes with quantumpnumber n (index n labels the eigenvalues in order of increasing energy) larger than nc ≈ 2 2J/Ω
to become localized at corresponding lattice sites. Therefore the sites at a distance nc /2 from thetrap center with potential energy 2~J define the boundary between the delocalized modes at thetrap center and the high-energy localized trap edges. Tunneling in the region of localized modes

113
is typically suppressed by large potential energy differences even in the presence of SOC. These
modes are thus largely decoupled and do not contribute to the trap center dynamics. On the other
hand the delocalized modes may be approximated by those of a quantum harmonic oscillator with√effective mass m∗ = ~/(2Ja2 ) and frequency ω ∗ = 4JΩ.
As emphasized in the main text, the key requirements for our protocol are 1) the validityof the spin model, which depends on the pinning of particles in their initial single particle modes,
and 2) the gap protection against SOC dephasing, which arises from collective spin interactions.
Concerning the latter point, it is desirable to maintain a weak trap so as to enable a large numberof delocalized modes in the trap center, which are the only type capable of contributing to thegeneration of squeezing. Though the interactions between these modes are not strictly all-to-all,they remain long-ranged, and can thus still lead to a spin-locking effect and a protective gap [54,115]. For Ω/J = 0.01 we have nc = 28, enabling ∼ 103 contributing modes in each 2D layerof our system. Concerning the validity of the spin model, from a single-particle perspective theeigenmodes of our ↑ states will be initially displaced in quasi-momentum space from equilibriumby φ/a as per Eqn. (4.63), and will generally undergo dipole oscillations and not remain strictly
pinned to their initial modes. However, as long as we ensure the displacement is small enough toguarantee a constant density distribution across the trap center, the spin model will remain valid.
The localized modes at the trap edges can actually help to satisfy this condition, since they canserve as a barrier against motion. This is demonstrated in Figure 4.9 where we show that fillingall delocalized modes guarantees that the trap center maintains a constant density; we characterize
this by the time-averaged fluctuations of the ↑ density at each site j about its initial value following
the Ramsey pulse,sδnj,↑ ≡

1
lim
t→∞ t

Z t


2
dτ hn̂j,↑ (τ )i − hn̂j,↑ (0)i ,(4.64)

0choosing sufficiently large evolution times to ensure convergence.
In the presence of interactions, an additional point of concern is that the interplay between
the trap and interactions may induce resonances that enable the formation of a significant doublon

114
population,
Nd (t) =X

hn̂j,↑ (t)n̂j,↓ (t)i,(4.65)

jwhich in turn may alter the density distribution and invalidate the spin model. Since doublon
formation in the localized edges will not have consequences for our squeezing protocol, we must
only ensure that doublons are not formed in the trap center, which requires U > Ω(nc /2)2 = 2J
[117]. In Figure 4.10, we perform exact simulations to assess the effect of the trap on our system.
Though restricted to small system sizes, the results demonstrate that for U/J . 2, the trap will
always lead to a decrease of squeezing due to the formation of doublons in the trap center, while
for U/J & 2, we are protected from this process even for trap strengths much stronger than theexperimentally relevant ones.

4.J

Accounting for p-wave inelastic collisions
Inelastic 3 P0 (electronic state e or ↑) collisions are detrimental for optical lattice clocks. For

the nuclear-spin-polarized gas discussed in this work, ee losses are only possible via the p-wave
scattering channel since s-wave collisions are suppressed by Fermi statistics. The big advantage
here compared to prior experiments done in a 1D lattice at µK temperature [50] is that in a Fermi
degenerate gas loaded in a 3D optical lattice, p-wave losses are further suppressed by the centrifugal
barrier and Pauli blocking, and only happen through a wave-function overlap between atoms at
different lattice sites. In this appendix, we quantify the effect of p-wave interactions on squeezing.
To account for p-wave losses, we describe the dynamics using a master equation for the system’s
density matrix ρ̂:dρ̂i= − [Ĥeff , ρ̂] + Lρ̂,dt~

(4.66)

where Ĥeff = χŜz2 is the effective one-axis twisting Hamiltonian obtained from the original FermiHubbard Hamiltonian with spin-orbit coupling, and L is the Lindblad superoperator that accounts
for p-wave ee inelastic collisions. This latter term can be written using a pseudo-potential approx-

115
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Figure 4.11: p-wave loss rates. Both the averaged p-wave inelastic collision rate γ (orange) andthe ratio of this collision rate to the optimal squeezing rate χopt (blue) are suppressed as the latticeee isdepth increases. χopt is obtained by choosing SOC angles φ that saturate B/U≈ 0.05, where B
the variance of the SOC-induced axial field and U is the two-atom on-site interaction energy.

imation as [54]:
Lρ̂ =



1 †
Γkk0 Âkk0 ρ̂Â†kk0 −Âkk0 Âkk0 ρ̂ + ρ̂Â†kk0 Âkk0 ,
2
0

X
kk

(4.67)

where the jump operators are Âkk0 = ĉk,↑ ĉk0 ,↑ , and k, k0 sum over all the populated quasimomentum modes. The decay matrix elements Γkk0 are given by:
3π~b3imΓkk0 =mZ
dr W [φk (r), φk0 (r)] ,3(4.68)

where bim = 121a0 [7, 53] is the p-wave inelastic scattering length (with a0 = 5.29 × 10−11 m theBohr radius), φk (r) is the Bloch function with quasi-momentum k, andW φk (r), φk0 (r)
h

i h

i
≡ ∇φ∗k (r) φ∗k0 (r) − φ∗k (r) ∇φ∗k0 (r) · ∇φk (r) φk0 (r) − φk (r) ∇φk0 (r)(4.69)In Figure 4.11, we show the averaged decay rate γ ≡2
kk0 Γkk0 /` , where ` is the number ofP

lattice sites along the x and y axes, as a function of the lattice depth V0 along these axes. Here, we
assume the same lattice depth in the z direction used in the main text, V = 60ER . The decay rate γ
is suppressed exponentially with increasing lattice depth V0 . To quantify the effect of these losses on
the spin squeezing generation process, we follow a similar methodology to the one described in detail116(a) N = 100(b) / = 0.04

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N = 1000

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Figure 4.12: Squeezing via OAT in the presence of inelastic collisions. (a) For fixed particle
number N = 100, the optimal squeezing decreases as the inelastic collision rate increases. Panel(b) shows squeezing over time for γ/χopt = 0.04 (solid lines), which corresponds to U/J = 6, andcompares it with γ = 0 (dashed lines) for different particle numbers. Inelastic collisions preventthe growth of optimal squeezing with particle number. For N = 1000, the maximum squeezing
saturates to ∼ 10 dB.

in Ref. [54]. The basic idea is to take advantage of the so-called Truncated-Wigner Approximation(TWA) [128, 129], which allows us to capture the development of spin squeezing using semi-classicalphase-space methods. In the TWA the quantum dynamics are accounted for by solving mean field
equations of motion supplemented by noise. The mean field equations are derived by assumingNthat the many-body density matrix of the system can be factorized as ρ̂ = i ρ̂(i), where ρ̂(i) is
the reduced density matrix of the particle in quasi-momentum mode qi [see Eqn. (4.57)]. Under
this assumption, the non-linear mean field equations are given byXdρ↑↑ (j)=−Γkj kj 0 ρ↑↑ (j)ρ↑↑ (j 0 ),dt
0dρ↓↓ (j)=0dt

j(4.70)and
X
dρ↑↓ (j)1000= ρ↑↓ (j)iχ(ρ↑↑ (j ) − ρ↓↓ (j )) − Γkj kj 0 ρ↑↑ (j ) ,dt
20(4.71)jwhere ρσσ0 ≡ hρ̂σσ0 i. Since we are interested in the collective behavior, one can define ρTσσ0 =P
j ρσσ 0 (j). For these observables the equations of motion simplify to
dρT↑↑
dt

= −f γ(ρT↑↑ )2 ,

dρT↓↓
dt

=0

(4.72)

117
and
dρT↑↓
dt

= ρT↑↓

1iχ(ρT↑↑ − ρT↓↓ ) − f γρT↑↑2

,(4.73)

where f ≡ N/`2 is the filling fraction.
Under the TWA, one accounts for quantum fluctuations during the dynamics by averaging
over different mean field trajectories generated by sampling over different initial conditions chosen
to reconstruct the Wigner function of the initial coherent spin state at t = 0 [54]. This method has
proven to be successful in simulating quantum spin dynamics. Using this approach, Figure 4.12
shows numerical simulation results of squeezing over time in the presence of inelastic collisions. For
shallow lattices (V0 . 7ER ), the effect of inelastic collisions can limit the spin squeezing to ∼ 10
dB. Thus, in this regime, losses are as relevant as light scattering. The role of inelastic interactions
could be mitigated by either operating at deeper lattices as shown in Figure 4.12, or by using
nuclear spin states to generate the squeezing instead of the clock states directly.

Chapter 5

Short-time expansion of Heisenberg operators in open collective quantum spin
systems

PrologueCollective spin systems, which are invariant under permutations of their constituent spins,
make frequent appearances as idealized models in atomic, molecular, and optical platforms (see,for example, Chapters 4, 6, and 7). The high degree of symmetry in these systems can make them
numerically tractable in the mesoscopic regime of 102 – 105 particles, which is typically inaccessible
for exact simulations of quantum systems. These features make collective spins systems attractivemodels for theoretical and numerical studies of phenomena such as dynamical phase transitions andspin squeezing (i.e. quantum-enhanced metrology). The results shown in Figure 4.3, for example,
were obtained via exact numerical simulations of collective spin systems. However, realistic systems
in the laboratory typically break permutational symmetry, for example through incoherent decay
and dephasing processes that kick the system out of the space of permutationally-symmetric states.
In some cases, such as the Ising model, even these sorts of decoherence processes are analyticallytractable, which enabled the calculation of exact numerical results in Figure 4.4(a). In other cases,
one must resort to suitable approximation schemes or specialized techniques, where available. This
chapter is about one such technique, which was essentially developed to calculate the results in
Figure 4.4(b). The bulk of this chapter is taken from Ref. [3].

119

Abstract
We present an efficient method to compute short-time expectation values in large collective
spin systems with typical forms of Markovian decoherence. Our method is based on a Taylor
expansion of a formal solution to the equations of motion for Heisenberg operators. This expansion
can be truncated at finite order to obtain virtually exact results at short times that are relevant for
metrological applications such as spin squeezing. In order to evaluate the expansion for Heisenberg
operators, we compute the relevant structure constants of a collective spin operator algebra. We
demonstrate the utility of our method by computing spin squeezing, two-time correlation functions,and out-of-time-ordered correlators for 104 spins in strong-decoherence regimes that are otherwise
inaccessible via existing numerical methods. Our method can be straightforwardly generalized
to the case of a collective spin coupled to bosonic modes, relevant for trapped ion and cavity
QED experiments, and may be used to investigate short-time signatures of quantum chaos andinformation scrambling.

5.1

Introduction
Collective spin systems are a versatile resource in quantum science for a range of applications

including quantum-enhanced metrology and quantum simulation. The study of such systems dates
back to the mid-twentieth century with the introduction of the Dicke model [30] that describes atoms
cooperatively interacting with a single mode of a radiation field, and the Lipkin-Meshkov-Glick(LMG) model, a toy model for testing many-body approximation methods in contemporary nuclear
physics [130–132]. On the experimental side, the development of advanced trapping, cooling, andcontrol techniques in atomic, molecular, and optical (AMO) systems have enabled the realization ofcollective spin models in a broad range of platforms, including cold atomic gasses [133, 134], BoseEinstein condensates [135–138], ultracold Fermi gasses [50, 104, 115], trapped ions [139], and opticalcavities [114, 140–148], among others. These implementations compliment innumerable theoreticalstudies in a variety of rich subjects, including quantum phase transitions and criticality [149–152],120
non-equilibrium phenomena [153–160], and precision mentrology [2, 32, 33, 35, 119, 161–167].
One of the primary motivations for studying collective spin systems is their application toquantum-enhanced metrology. Quantum projection noise limits the error ∆φ in the measurement√of a phase angle φ with N independent spins to ∆φ ∼ 1/ N [32, 99]. Collective spin systems
provide a means to break through this limit via the preparation of many-body entangled states suchas spin-cat states [164, 168, 169] and most notably spin-squeezed states [32, 33, 35] that allow formeasurement errors ∆φ ∼ 1/N ε with 1/2 < ε ≤ 1, where ε = 1 saturates the Heisenberg limit [170].
Such entangled states can be prepared either via heralded methods such as quantum nondemolition
measurements [133, 134, 140, 141], or via deterministic methods that require nonlinear dynamics,
typically realized with phonon-mediated [139], photon-mediated [114, 135, 142–148] or collisional[50, 104, 115, 136–138] interactions. Although a truly collective spin model requires uniform, allto-all interactions, as long as measurements do not distinguish between constituent particles, evennon-uniform systems may be effectively described by a uniform model with renormalized parameters[171].
In the absence of decoherence, permutation symmetry and total spin conservation divide thetotal Hilbert space of a collective spin system into superselection sectors that grow only linearlywith system size N , thereby admitting efficient classical simulation of its dynamics. Decoherence
generally violates total spin conservation and requires the use of density operators, increasing thedimension of accessible state space to O N 3 [172, 173]. In this case, exact simulations can be
carried out for N . 100 particles. If decoherence is sufficiently weak, dynamics can be numericallysolvable for N . 105 particles via “quantum trajectory” Monte Carlo methods [174, 175] (alsoknown as “quantum jump” or “Monte Carlo wavefunction” methods) that can reproduce all expectation values of interest. When decoherence is strong, however, these Monte Carlo methods cantake a prohibitively long time to converge, as simulations become dominated by incoherent jumps
that generate large numbers of distinct quantum trajectories that need to be averaged in order toaccurately compute expectation values. Even with strong decoherence, dynamics are sometimes
solvable through the cumulant expansion [176] that neglects all n-body connected correlators for121n > 2. The growth of genuinely multi-body correlations, however, eventually causes the cumulant
expansion to yield incorrect results with no clear signature of failure. In the absence of othermeans to compute correlators, it can therefore be difficult to identify the point at which correlators
computed via the cumulant expansion can no longer be trusted.
In this work, we present an efficient method to compute short-time dynamics of collective
spin systems with typical forms of Markovian decoherence. The only restriction on decoherence(beyond Markovianity) is that, like the coherent collective dynamics, it must act identically on
all constituent particles. Our method is based on a formal solution to the equations of motionfor Heisenberg operators, thereby bearing some resemblance to the Mori formalism [177] and related work [178]. Specifically, we expand a formal solution for a Heisenberg operator into a Taylor
series whose truncation can yield negligible error at sufficiently short times. Evaluating the resulting expansion requires knowing the structure constants of a collective spin operator algebra;
the calculation of these structure constants (in Appendices 5.A–5.C) is one of the main technicalresults of this work, which we hope will empower both analytical and numerical studies of collective
spin systems in the future. We benchmark our method against exact results from both analytical calculations and quantum trajectory Monte Carlo computations of spin squeezing in accessible
parameter regimes, highlighting both advantages and limitations of the short-time expansion. Finally, we showcase applications of our method by computing quantities that are inaccessible toother numerical methods.

5.2

TheoryIn this section we provide the basic theory for our method to compute expectation values ofcollective spin operators, deferring lengthy derivations to the appendices. We consider a system
of N distinct spin-1/2 particles. Defining individual spin-1/2 operators ŝα=x,y,z ≡ σ̂α /2 and ŝ± ≡ŝx ± iŝy = σ̂± with Pauli operators σ̂α , we denote an operator that acts with ŝα on the spin indexed(j)

by j and trivially (i.e. with the identity 1̂) on all other spins by ŝα . We then define the collective
n o
P
(j)
spin operators Ŝα ≡ N
ŝ
for
α
∈
{x,
y,
z,
+,
−}.
Identifying
thesetŜm as a basis for all
α
j=1

122
m

m

collective spin operators, with m ≡ (m+ , mz , m− ) ∈ N30 and Ŝm ≡ Ŝ+ + Ŝzmz Ŝ− − , we can expand
any collective spin operator Ô in the formÔ =X

Om Ŝm(5.1)

m∗ = O ∗ with m∗ ≡with scalar coefficients Om ∈ C. If Ô is self-adjoint, for example, then OmmP(m− , mz , m+ ). The corresponding Heisenberg operator is then Ô (t) = m Om (t) Ŝm +ÊO (t), with

time-dependent coefficients Om (t) for time-independent Schrödinger operators Ŝm , and mean-zero“noise” operators ÊO (t) that result from interactions between the spin system and its environment,
initially ÊO (0) = 0. These noise operators will essentially play no role in the present work, but
are necessary to include for a consistent formalism of Heisenberg operators in an open quantum
system; see Appendix 5.N for further discussion. The expectation values of Heisenberg operators
evolve according toXdhÔ (t)i = hŤ Ô (t)i =hŜm i Tmn On (t)
dt
m,n

(5.2)

with a Heisenberg-picture time derivative operator Ť = d/dt whose matrix elements Tmn ∈ C aredefined byhi
XXŤ Ŝn ≡ i Ĥ, Ŝn +γJ Ď (J ) Ŝn =Ŝm Tmn ,
−

J

(5.3)

m

where [X, Y ]± ≡ XY ± Y X; Ĥ is the collective spin Hamiltonian; J is a set of jump operatorswith a corresponding decoherence rate γJ ; and Ď is a Heisenberg-picture dissipator, or Lindblad
superoperator, defined byĎ (J ) Ô ≡

X
ˆ
J∈J


1 h ˆ† ˆ i†ˆˆJ ÔJ −J J, Ô
.
2
+

(5.4)

Decoherence via uncorrelated decay of individual spins, for example, would be described by the setno(j)
of jump operators J− ≡ ŝ− : j = 1, 2, · · · , N . The commutator in Eq. (5.3) can be computedPby expanding the product Ŝ` Ŝm = n f`mn Ŝn with structure constants f`mn ∈ R that we workout in Appendices 5.A–5.C, and the effects of decoherence from jump operators (i.e. elements of

123J ) of the form ĝ (j) =(j)α gα ŝαP

and Ĝ =P

α Gα Ŝαare worked out in Appendices 5.D–5.G.

We consider these calculations to be some of the main technical contributions of this work, withpotential applications beyond the short-time simulation method presented here. These ingredientsare sufficient to compute matrix elements Tmn of the time derivative operator Ť in Eq. (5.3) inmost cases of practical interest.
We note that particle loss is an important decoherence mechanism in many experimental
realizations of collective spin models [35]. In principle, a spin model has no notion of the particle
annihilation operators that generate particle loss, and therefore cannot capture this effect directly.
Nonetheless, for a system initially composed of N particles, the effect of particle loss can be emulatedPwith O(1/N ) error by the dissipator Ďloss defined by Ďloss Ŝm = −|m|Ŝm , where |m| ≡ α mα (seeAppendix 5.H). Furthermore, the effect particle loss can be accounted for exactly by (i) introducing
an additional index on spin operators, Ŝm → ŜN m , to keep track of different sectors of fixed particle
number within a multi-particle Fock space, and (ii) constructing jump operators that appropriately
couple spin operators within different particle-number sectors. We defer a detailed exact accountingof particle loss to future work.
The time derivative operator Ť will generally couple spin operators Ŝn to spin operators Ŝmwith higher “weight”, i.e. with |m| > |n|. The growth of operator weight signifies the growth of
many-body correlations. Keeping track of this growth eventually becomes intractable, requiring us
to truncate our equations of motion somehow. The simplest truncation strategy would be to takedhÔ (t)i →dtXw(m)<WhŜm iX

Tmn On (t)

(5.5)

n

for some weight measure w, e.g. w (m) = |m|, and a high-weight cutoff W . The truncation in
Eq. (5.5) closes the system of differential equations defined by Eq. (5.2), and allows us to solve itusing standard numerical methods. Some initial conditions for this system of differential equations,
namely expectation values of collective spin operators with respect to spin-polarized (Gaussian)states that are generally simple to prepare experimentally, are provided in Appendix 5.I.
The truncation strategy in Eq. (5.5) has a few limitations: (i) simulating a system of differ-124
ential equations for a large number of operators can be time-consuming, (ii) the weight measure
w may need to be chosen carefully, as the optimal measure is generally system-dependent, and(iii) simulation results can only be trusted up to the time at which the initial values of operators
Ŝm with weight w (m) ≥ W have a non-negligible contribution to expectation values of interest.
The last limitation in particular unavoidably applies in some form to any method tracking only asubset of all relevant operators. We therefore devise an alternate truncation strategy built aroundlimitation (iii).
We can formally expand Heisenberg operators Ô (t) in a Taylor series about the time t = 0
to writehÔ (t)i = hetŤ Ô (0)i =X tk Xk! m,nk≥0

khŜm i Tmn
On (0) ,(5.6)

kwhere the matrix elements Tmnof the k-th time derivative operator Ť k are0Tmn≡ δmn ,1Tmn
≡ Tmn ,

X

k>1
Tmn
≡

Tmpk−1 · · · Tp3 p2 Tp2 p1 Tp1 n ,(5.7)

p1 ,p2 ,··· ,pk−1

with δmn = 1 if m = n and zero otherwise. For sufficiently short times, we can truncate the series
in Eq. (5.6) by takinghÔ (t)i →

M k X
X
t
k=0

k! m,nkhŜm i Tmn
On (0) .

(5.8)

We refer to Eq. (5.8) as the truncated short-time (TST) expansion of Heisenberg operators. Note†that when computing an expectation value hÔ (t)i, the relation Ŝm= Ŝm∗ , which by Hermitian∗ , cuts both the number of initial-time
conjugation of Eq. (5.2) also implies that Tm∗ n∗ = Tmn

expectation values hŜm i and the number of matrix elements Tmn that we may need to explicitly
compute roughly in half.
kUnlike the weight-based truncation in Eq. (5.5), the nonzero matrix elements Tmn
for k =

0, 1, · · · , M in Eq. (5.8) tell us which operators Ŝm are relevant for computing the expectation
value hÔ (t)i to a fixed order M . The TST expansion thereby avoids the introduction of a weightmeasure w that chooses which operators to keep track of, and trades the cost of solving a system of

125
k .
differential equations for the cost of computing expectation values hŜm i and matrix elements Tmn

In all cases considered in this work, we find that the TST expansion is both faster to evaluate andprovides accurate correlators hÔ (t)i until later times t than the weight-based expansion in (5.5)
with weight measure w (m) = |m| and cutoff W ≈ M . We therefore restrict the remainder of our
discussions to the TST expansion in Eq. (5.8), and provide a pedagogical tutorial for computing
correlators using the TST expansion in Appendix 5.J.
Three primary considerations limit the maximum time t to which we can accurately compute a
correlator hŜn (t)i using the TST expansion. First, maintaining accuracy at larger times t requires
going to higher orders M in the TST expansion. An order-M TST expansion of the correlatorhŜn (t)i can involve a significant fraction of operators Ŝm with weight |m| . M , which implies

the need to compute O M 3 initial-time expectation values hŜm i and O M 4 matrix elements
k . In practice, with a straightforward implementation of the TST expansion we find that theseTmnrequirements generally restrict M . 50 – 70 with 8 – 50 gigabytes of random access memory (RAM).
Second, individual terms at high orders of the TST expansion in Eq. (5.8) can grow excessively
large, greatly amplifying any numerical errors and thereby spoiling cancellations that are necessary
to arrive at a physical value of a correlator, i.e. with |hŜn (t)i| . S |n| (where S ≡ N/2). Finally,
the TST expansion is essentially perturbative in the time t, which implies that its validity as aformal expansion eventually breaks down. Precisely characterizing the implications of these lasttwo considerations for the TST expansion requires additional analysis that we defer to future work.
An investigation of connections between the TST expansion and past work related to the Mori
formalism [177, 178], for example, might answer questions about the breakdown and convergenceof the TST expansion. As we show from benchmarks of the TST expansion in Section 5.3, however,(M )a detailed understanding of breakdown is not necessary to diagnose the breakdown time tbreakbeyond which the TST expansion yields inaccurate results. Empirically, we find that going beyondorder M ≈ 35 yields no significant gains in all cases considered in this work.

126

5.3

Spin squeezing, benchmarking, and breakdown
To benchmark our method for computing collective spin correlators, we consider three col-lective spin models known to generate spin-squeezed states: the one-axis twisting (OAT), two-axis
twisting (TAT), and twist-and-turn (TNT) models described by the collective spin Hamiltonians[35]
HOAT = χŜz2 ,χ 2HTAT =
Ŝz − Ŝy2 ,3(5.9)(5.10)HTNT = χŜz2 + ΩŜx ,(5.11)

where we include a factor of 1/3 in the TAT Hamiltonian because it naturally appears in realistic
proposals to experimentally implement TAT [118, 119]. For simplicity, we further fix Ω = χS
(with S ≡ N/2 throughout this work) to the critical value known to maximize the entanglement
generation rate of TNT in the large-N limit [179, 180].
Note that the OAT model is a special case of the zero-field Ising model, whose quantum
dynamics admits an exact analytic solution even in the presence of decoherence [120]. The approximate and numerics-oriented TST expansion is therefore an inappropriate tool for studying the
OAT model, which will merely serve as an exactly solvable benchmark of our methods. Whereverapplicable, we will provide exact results for the OAT model (see Appendix 5.K, as well as the
Supplementary Material of Ref. [139]).
⊗N
The Hamiltonians in Eqs. (5.9)–(5.11) squeeze the initial product state |Xi ∝ |↑i + |↓iwith Ŝx |Xi = S |Xi. Our measure of spin squeezing is the directionally-unbiased Ramsey squeezing
parameter determined by the maximal gain in resolution ∆φ of a phase angle φ over that achievedby any spin-polarized product state (e.g. |Xi) [32, 35],2

ξ ≡(∆φmin )2∆φpolarized

where Ŝ ≡ Ŝx , Ŝy , Ŝz



2 =

N

min
|hŜi|2 v⊥hŜiŜ · v

2 ,(5.12)

v·v=1

is a collective spin operator-valued vector, the minimization is performed

over all unit vectors v orthogonal to the mean spin vector hŜi, and for brevity we have suppressed

127
101

OAT
TAT
TNT

100

Squeezing (ξ 2 )Squeezing (ξ 2 )

101

10−1
10−2
10−3

OAT
TAT
TNT

100
10−1
10−2
10−3

0

5

10
Time (N χt)15(a) Squeezing with unitary dynamics

0

5

10
Time (N χt)15(b) Squeezing with decoherence: γ− = γ+ = γz = χFigure 5.1: Spin squeezing of N = 104 spins initially in |Xi under (a) unitary and (b) non-unitary
dynamics, computed using exact methods (solid lines), quantum trajectory simulations (dots), andthe TST expansion in Eq. (5.8) with M = 35 (dashed lines). Solid circles mark the times at which
the TST expansion gives an unphysical result with ξ 2 < 0.

the explicit time dependence of operators in Eq. (5.12). This squeezing parameter is entirely
determined by one- and two-spin correlators of the form hŜα i and hŜα Ŝβ i. For the unitary dynamics discussed in this work, these correlators are obtainable via exact simulations of quantumdynamics in the (N + 1)-dimensional Dicke manifold of states |S, mi with net spin S and spin
projection m onto the z axis, i.e. with hS, m|Ŝ 2 |S, mi = S (S + 1) and hS, m|Ŝz |S, mi = m form ∈ {−S, −S + 1, · · · , S}. In the presence of single-spin or collective decoherence, meanwhile,these correlators are obtainable with the collective-spin quantum trajectory Monte Carlo method
developed in Ref. [175]. In this work, these exact and quantum trajectory simulations will be used
to benchmark the TST expansion in Eq. (5.8).
Figure 5.1 compares the squeezing parameter ξ 2 for N = 104 spins initially in the state|Xi evolved under the Hamiltonians in Eqs. (5.9)–(5.11), as computed via both benchmarking
simulations and the TST expansion in Eq. (5.8) with M = 35. Squeezing is shown for both unitary
dynamics (Figure 5.1a), as well as non-unitary dynamics in the presence of spontaneous decay,
excitation, and dephasing of individual spins at rates χ (Figure 5.1b), respectively described by then o(j)sets of jump operators Jα ≡ ŝα with corresponding decoherence rates γα = χ for α ∈ {−, +, z}.

128
The results shown in Figure 5.1 were computed in a rotated basis with (ŝz , ŝx ) → (ŝx , −ŝz ) and|Xi → |−Zi ≡ |↓i⊗N , as well as appropriate transformations of the Hamiltonian and jump operators.
The only effects of this rotation on the results presented in Figure 5.1 are to (i) reduce the time ittakes to compute correlators hÔ (t)i with the TST expansion, and (ii) prolong the time for which the
TST expansion of TNT results agree with benchmarking simulations. The speedup in a different
basis occurs because for the initial state |−Zi, all initial-time correlators hŜm i are zero unless
m+ = m− = 0, and all non-zero correlators take O (1) (i.e. constant in N ) time to compute, rather
than O (N ) time (see Appendix 5.I). In total, the use of a rotated basis reduces the computation
time of initial-time correlators from O M 3 N to O (M ). The reason for prolonged agreement of
TNT results in a rotated basis is not entirely understood, and provides a clue into the precise
mechanism by which the TST expansion breaks down (discussed below). We defer a detailed studyof this breakdown to future work.
The main lesson from Figure 5.1 is that the TST expansion yields essentially exact results
right up until a sudden and drastic departure that can be diagnosed by inspection. The breakdownof the TST expansion in Figure 5.1 induces an unphysical squeezing parameter ξ 2 < 0. In general,
however, there is no fundamental relationship between the breakdown of the TST expansion andthe conditions for a physical squeezing parameter ξ 2 . A proper diagnosis of breakdown therefore
requires inspection of the correlators hŜn (t)i used to compute the squeezing parameter ξ 2 , which
upon breakdown will rapidly take unphysical values with |hŜn (t)i| & S |n| (see Appendix 5.L for
an example). The sudden and drastic departure from virtually exact results is consistent with thelimitations of the TST expansion discussed at the end of Section 5.2. Specifically, we identify threepossible mechanisms for breakdown: (i) a rapid growth in the order M necessary for the TST
expansion to converge, (ii) the growth of numerical errors in excessively large terms of the TST
expansion, and (iii) the formal breakdown of the perturbative expansion in the time t. In all of thesecases, a detailed cancellation eventually ceases to occur between large terms at high orders in the
TST expansion. These large terms grow with the time t raised to some large power (as high as M ),and therefore rapidly yield wildly unphysical results. In contrast to other approximate methods

129such as the cumulant expansion [176], the TST expansion can thus diagnose its own breakdown,
which is an important feature when working in parameter regimes that are inaccessible via othermeans to compute correlators. Note that, due to the breakdown mechanisms of the TST expansion,(M )going up through order M = 70 does not significantly increase the breakdown time tbreak in Figure(M )5.1, and in some cases even shortens tbreak .
Although the TST expansion breaks down at short times, it has two key advantages over thequantum trajectory Monte Carlo method to compute correlators in the presence of decoherence.
First, computing spin correlators with the TST expansion is generally faster and requires fewer
computing resources. The TST expansion results in Figure 5.1b, for example, take ∼ 10 seconds
to compute with a single CPU on modern computing hardware. The quantum trajectory Monte
Carlo results in the same figure, meanwhile, take ∼ 104 CPU hours to compute on similar hardware;
the bulk of this time is spent performing sparse matrix-vector multiplication, leaving little room
to further optimize serial runtime. Parallelization can reduce actual runtime of the Monte Carlosimulations to ∼ 10 hours by running all trajectories at once, but at the cost of greatly increasing
computing resource requirements. Though it may be possible to further speed up quantum trajectory Monte Carlo simulations by introducing new truncation schemes, any modifications (i) should
be made carefully to ensure that simulations still yield correct results, and (ii) are unlikely to bridge
the orders of magnitude in computing resource requirements.
The second advantage of the TST expansion is the capability to compute spin correlators
in strong-decoherence regimes of large systems that are entirely inaccessible to other methods. As
an example, Figure 5.2 shows squeezing of N = 104 spins initially in |Xi, undergoing spontaneous
decay, excitation, and dephasing of individual spins at rates γ− = γ+ = γz = 100χ. The systemsize in these simulations is too large for straightforward application of exact methods for open
quantum systems. Quantum trajectory Monte Carlo simulations, meanwhile, take a prohibitively
long time to converge with such strong decoherence due to the multiplicity of quantum trajectories
that require averaging.
The results in Figure 5.2 show that the TNT model can generate more squeezing than the

130

Squeezing (ξ 2 )

100

OAT
TAT
TNT

10−1

0.0

2.5

5.0
7.5
Time (N χt)

10.0

12.5

Figure 5.2: Spin squeezing of N = 104 spins initially in |Xi with spontaneous decay, excitation, anddephasing of individual spins at rates γ− = γ+ = γz = 100χ. Computed using the TST expansion
in Eq. (5.8) with M = 35. Solid circles mark the times at which the TST expansion gives anunphysical result with ξ 2 < 0.

131
OAT or TAT models in the presence of strong decoherence. The better performance of TNT is inpart a consequence of the fact that TNT initially generates squeezing at a faster rate than OAT
or TAT, thereby allowing it to produce more squeezing before the degrading effects of decoherence
kick in. We corroborate this finding with quantum trajectory simulations of a smaller system inAppendix 5.M. Strong-decoherence computations of the sort used for Figure 5.2 put lower bounds
on theoretically achievable spin squeezing via TAT with decoherence in Ref. [2], exemplifying a
concrete and practical application of the TST expansion and the collective-spin structure constants
calculated in this work.

5.4

Two-time correlation functions and out-of-time-ordered correlators
As a final example of collective-spin physics that is numerically accessible via the TST ex-

pansion of Heisenberg operators, we consider the calculation of two-time correlation functions andout-of-time-ordered correlators (OTOCs). In particular, we consider the effect of decoherence on
short-time behavior of the two-time connected correlator
C (t) ≡

1hŜ+ (t) Ŝ− (0)i − hŜ+ (t)i hŜ− (0)i ,Sand the expectation value of a squared commutator,

i† h
i 
1 h
D (t) ≡ 2
Ŝ+ (t) , Ŝ− (0)Ŝ+ (t) , Ŝ− (0)
,
S
−
− nn

(5.13)(5.14)

in the context of the squeezing models in Section 5.3. The subscript on h·inn in Eq. (5.14) stands for“no noise”, and denotes a correlator computed without the noise contributions ÊO (t) to Heisenberg
operators Ô (t). While linear contributions from noise operators as e.g. in Eq. (5.13) always vanish
under Markovian decoherence (see Appendix 5.N), quadratic contributions that would otherwise
appear in Eq. (5.14) generally do not [181]. Determining the effect of these noise terms generally
requires making additional assumptions about the environment, which would be a digression forthe purposes of the present work. We therefore exclude these noise terms in (5.14) in order to keep
our discussion simple and general; see Ref. [181] for more detailed discussions of noise terms andthe quantum regression theorem underlying the calculation of multi-time correlators.

132
In an equilibrium setting, correlation functions similar to that in Eq. (5.13) contain information about the linear response of Heisenberg operators to perturbations of a system; in a
non-equilibrium setting, they contribute to short-time linear response (see Appendix 5.O). Similarcorrelators have made appearances as order parameters for diagnosing time-crystalline phases ofmatter [182]. Squared commutators such as that in Eq. (5.14), meanwhile, are commonly examined
for signatures of quantum chaos and information scrambling [183–185]. In typical scenarios, suchsquared commutators initially vanish by construction through a choice of spatially separated operators. Collective spin systems, however, have no intrinsic notion of locality or spatial separation. In⊗N
our case, therefore, with the choice of initial state |Xi ∝ |↑i + |↓i
we merely have D (0) ∼ 1/N .
Figure 5.3 shows the behavior of C (t) and D (t) for N = 104 spins, initially in the state |Xi,
evolving under the squeezing Hamiltonians in Eqs. (5.9)–(5.11) both with and without spontaneous
decay, excitation, and dephasing of individual spins at rates γ− = γ+ = γz = 100χ. In the case ofunitary evolution under OAT, we find that to an excellent approximation |C (t)| takes the functional
form f (t) = f (0) + aN χt + (bN χt)2 with a ∼ b ∼ 1, and with a virtually perfect fit D (t) =2
D (0) + [N + 1] χt . For unitary evolution under TAT and TNT, we find that to an excellent
approximation both |C (t)| and D (t) take the functional form f (t) = f (0) + a exp (bN χt) − 1
with a ∼ b ∼ 1. As may be expected, the growth of C (t) and D (t) is generally suppressedby decoherence. Figure 5.3 serves as an example for the type of behavior that is accessible atshort times with the TST expansion. These examples are straightforward to extend to equilibrium
settings and spin-boson systems.

133

101

|C(t)|

OAT
TAT
TNT

φ(t)/π

D(t)

101

100

γ0 = 0γ0 = 100χ

100

0.00

10−1

−0.25

10−2

−0.50

OAT
TAT
TNT

102

γ0 = 0γ0 = 100χ
0

0

2

4
Time (N χt)6

24Time (N χt)6(b) Squared commutator D (t).
(a) Two-time correlator C (t) ≡ |C (t)| exp iφ (t) .

Figure 5.3: The two-time connected correlator C (t) and squared commutator D (t), respectively de⊗N
fined in Eqs. (5.13) and (5.14), for N = 104 spins initially in the polarized state |Xi ∝ |↑i + |↓ievolving under the squeezing Hamiltonians in Eqs. (5.9)–(5.11). Results are shown for both unitary
dynamics (solid lines) and non-unitary dynamics with γ− = γ+ = γz = γ0 = 100χ (dashed lines),computed using the TST expansion in Eq. (5.8) with M = 20.

134

5.5

Conclusions
We have presented an efficient method for computing correlators at short times in collective

spin systems. This method is based on truncating a short-time expansion of Heisenberg operators,
and can access correlators on time scales that are relevant to metrological applications such as spinsqueezing. In order to evaluate the truncated short-time (TST) expansion of Heisenberg operators,
we have computed the structure constants of a collective spin operator algebra, which we hope
will empower future analytical and numerical studies of collective spin systems. Even though we
considered only non-equilibrium spin-squeezing processes in this work, our method can be applied
directly in an equilibrium setting, and is straightforward to generalize to systems such as trappedions and optical cavities with collective spin-boson interactions. In such contexts, our method may
be used to benchmark the short-time effects of decoherence, or study the onset of quantum chaos
and information scrambling.

Acknowledgments
We acknowledge helpful discussions with Robert Lewis-Swan, Kris Tucker, and Colin Kennedy;
as well as some technical contributions from Diego Barberena. This work is supported by the Air
Force Office of Scientific Research (AFOSR) grant FA9550-18-1-0319; the Defense Advanced Research Projects Agency (DARPA) and Army Research Office (ARO) grant W911NF-16-1-0576 andW911NF-19-1-0210; the National Science Foundation (NSF) grant PHY-1820885; JILA-NSF grant
PFC-1734006; and the National Institute of Standards and Technology (NIST).

5.A

Basic spin operator identities
The appendices in this work make ubiquitous use of various spin operator identities; wecollect and derive some basic identities here for reference. Note that despite the working definition
P (j)of collective spin operators from Sα = j sα , the identities we will derive involving only collective
spin operators apply just as well to large-spin operators that cannot be expressed as the sum of135
individual spin-1/2 operators. The elementary commutation relations between spin operators are,with µ̄ ≡ −µ ∈ {+1, −1} for brevity,
h
h

(k)
s(j)
z , sµ

i

(k)

i

s(j)
µ , sµ̄−−= δjk µs(j)
µ ,

h
i
h
i
Sz , s(j)= s(j)= µs(j)µz , Sµµ ,−Sz , Sµ − = µSµ ,(5.15)= δjk 2µs(j)
z ,

h

h
i
= s(j),
S= 2µs(j)
µ̄
µ
z ,



Sµ , Sµ̄ − = 2µSz .

(5.16)−(j)Sµ , sµ̄i−−These relations can be used to inductively compute identities involving powers of collective spin
operators. By pushing through one spin operator at a time, we can findm−1 (j)2m(j)(µSz )m s(j)sµ (1 + µSz ) = (µSz )m−2 s(j)µ = (µSz )µ (1 + µSz ) = · · · = sµ (1 + µSz ) , (5.17)andm(j) m−1m−1(j) m−1µs(j)+ s(j)= · · · = Sµm µs(j),z Sµ = Sµ µsz Sµµ Sµ
z + msµ Sµ(j)where we will generally find it nicer to express results in terms of µsz(5.18)(j)and µSz rather than sz

and Sz . Summing over the single-spin index j in both of the cases above gives us the purely
collective-spin versions of these identities:(µSz )m Sµ = Sµ (1 + µSz )m ,µSz Sµm = Sµm (m + µSz ) ,(5.19)

where we can repeat the process of pushing through individual Sz operators ` times to get(µSz )` Sµm = (µSz )`−1 Sµm (m + µSz ) = (µSz )`−2 Sµm (m + µSz )2 = · · · = Sµm (m + µSz )` .

(5.20)

Multiplying (5.20) through by (µν)` (for ν ∈ {+1, −1}) and taking its Hermitian conjugate, we cansay that more generally(νSz )` Sµm = Sµm (µνm + νSz )` ,Sµm (νSz )` = (−µνm + νSz )` Sµm .

(5.21)

Finding commutation relations between powers of transverse spin operators, i.e. Sµ and Sµ̄ , turns
out to be considerably more difficult than the cases we have worked out thus far. We therefore savethis work for Appendix 5.B.

136

5.B

Commutation relations between powers of transverse spin operators
To find commutation relations between powers of transverse collective spin operators, we firstcomputeSµm sµ̄ = Sµm−1 sµ̄ Sµ + Sµm−1 2µs(j)
z

(j)

(j)

(5.22)(j)(5.23)

m−1= Sµm−2 sµ̄ Sµ2 + Sµm−2 2µs(j)2µs(j)z Sµ + Sµz(j)= sµ̄ Sµm +

m−1
X

m−k−1
Sµk 2µs(j).
z Sµ(5.24)

k=0

h
i
(j)
While (5.24) gives us the commutator Sµm , sµ̄, we would like to enforce an ordering on products−of spin operators, which will ensure that we only keep track of operators that are linearly indepen(j)(j)

dent. We choose (for now) to impose an ordering with all sµ̄ operators on the left, and all sz

operators on the right. Such an ordering will prove convenient for the calculations in this sectiona .
This choice of ordering compels us to expand
m−1
X

m−k−1
Sµk 2µs(j)
=
z Sµ

k=0

m−1
X

h
i
m−k−2
m−k−1
(j)
Sµk 2 (m − k − 1) s(j)
S
+
S
2µs
µ
µ
µ
z

(5.25)k=0
m−2= m (m − 1) s(j)+ mSµm−1 2µs(j)
µ Sµ
z ,

(5.26)m−2
Sµm sµ̄ = sµ̄ Sµm + m (m − 1) s(j)+ mSµm−1 2µs(j)
µ Sµ
z ,

(j)

(5.27)Sµm Sµ̄ = Sµ̄ Sµm + mSµm−1 (m − 1 + 2µSz ) .

(5.28)

which implies(j)and in turna(j)

In retrospect, it may have been nicer to push all sµoperators to the right throughout these calculations, due

to the enhanced symmetry that expressions would have with respect to Hermitian conjugation. In any case, we
provide the final result of this section in both ordering conventions, and therefore feel no need to reproduce thesecalculations with a different ordering of spin operators.

137
As the next logical step, we take on the task of computing
Sµm Sµ̄n = Sµm−1 Sµ̄n Sµ + nhSµm−1 Sµ̄n−1 (1 − n + 2µSz )i= Sµ̄n Sµm + n

m−1XSµm−k−1 Sµ̄n−1 (1 − n + 2µSz ) Sµk ,k=0(5.29)
which implies
m−1
h
i
X
Sµm , Sµ̄n = Cmn;µ ≡ nSµm−k−1 Sµ̄n−1 (1 − n + 2µSz ) Sµk .
−(5.30)

k=0

We now need to rearrange the operators in Cmn;µ into a standard order, which means pushing all
Sz operators to the right and, for the purposes of this calculation, all Sµ̄ operators to the left. We
begin by pushing Sµk to the left of Sz , which takes 2µSz → 2µSz + 2k, and then push Sµm−k−1 tothe right of Sµ̄n−1 , giving us
Cmn;µ = n

m−1
X


Sµ̄n−1 Sµm−k−1 + Cm−k−1,n−1;µ Sµk (2k + 1 − n + 2µSz )(5.31)

k=0= Dmn;µ + n

m−2X

Cm−k−1,n−1;µ Sµk (2k + 1 − n + 2µSz ) ,(5.32)

k=0

where we have dropped the last (k = m − 1) term in the remaining sum because Cm−k−1,n−1;µ = 0
if k = m − 1, andDmn;µ ≡ mnSµ̄n−1 Sµm−1 (m − n + 2µSz ) .

(5.33)

To our despair, we have arrived in (5.32) at a recursive formula for Cmn;µ . Furthermore, we havenot even managed to order all spin operators, as Cm−k−1,n−1;µ contains Sz operators that are to
the left of Sµk . To sort all spin operators once and for all, we define
(k)
Cmn;µ
≡ Cm−k,n;µ Sµk ,

(k)
Dmn;µ
≡ Dm−k,n;µ Sµk ,(5.34)

which we can expand as
(k)
Dmn;µ
= (m − k) nSµ̄n−1 Sµm−k−1 (m − k − n + 2µSz ) Sµk= (m − k) nSµ̄n−1 Sµm−1 (k + m − n + 2µSz ) ,(5.35)(5.36)

138
and
(k)
Cmn;µ
= Dm−k,n;µ Sµk + n

m−k−2
X

Cm−k−j−1,n−1;µ Sµj (2j + 1 − n + 2µSz ) Sµk(5.37)

Cm−k−j−1,n−1;µ Sµj+k (2j + 2k + 1 − n + 2µSz )(5.38)

j=0(k)= Dmn;µ
+n

m−k−2
X
j=0

(k)
= Dmn;µ
+n

m−k−2
X

(k+j)

Cm−1,n−1;µ 2 [j + k] + 1 − n + 2µSz

(5.39)

j=0(k)= Dmn;µ
+n

m−2
X

(j)

Cm−1,n−1;µ (2j + 1 − n + 2µSz ) .

(5.40)

j=k

While the resulting expression in (5.40) strongly resembles that in (5.32), there is one crucial difference: all spin operators in (5.40) have been sorted into a standard order. We can now repeatedly(k)substitute Cmn;µ into itself, each time decreasing m and n by 1, until one of m or n reaches zero.
Such repeated substitution yields the expansionmin{m,n}−1(0)
Cmn;µ = Cmn;µ= Dmn;µ +

X

(p)
Emn;µ
,

(5.41)

p=1

where the first two terms in the sum over p are
(1)
Emn;µ
=n

(2)
Emn;µ
=n

m−2
X
k=0
m−2
X
k1 =0

(k)

Dm−1,n−1;µ (2k + 1 − n + 2µSz ) ,(n − 1)

m−3
X(5.42)(k )2
Dm−2,n−2;µ(2k2 + 2 − n + 2µSz ) (2k1 + 1 − n + 2µSz ) ,(5.43)

k2 =k1and more generally for p > 1,
(p)
Emn;µ
=

m−p−1
p
m−2
X m−3
X
X
Y

n!
(kp )
···
Dm−p,n−p;µ2kj + j − n + 2µSz .
(n − p)!
k1 =0 k2 =k1

kp =kp−1(5.44)

j=1

In principle, the expressions in (5.33), (5.36), (5.41), and (5.44) suffice to evaluate the commutatorhi(p)Sµm , Sµ̄n = Cmn;µ , but this result is – put lightly – quite a mess: the expression for Emn;µ in (5.44)−

involves a sum over p mutually dependent intermediate variables, each term of which additionallycontains a product of p factors. We therefore devote the rest of this section to simplifying our resulthifor the commutator Sµm , Sµ̄n .
−139Observing that in (5.44) we always have 0 ≤ k1 ≤ k2 ≤ · · · ≤ kp ≤ m−p−1, we can rearrange
the order of the sums and relabel kp → ` to get
m−p−1
X

n!
(`)
Dm−p,n−p;µ 2` + Fnp;µ(n − p)!

(p)
Emn;µ
=

`=0

X

p−1
Y


2kp−j − j + Fnp;µ ,(5.45)(k,p−1,`) j=1

where for shorthand we defineX

Fnp;µ ≡ p − n + 2µSz ,

X≡

`
`
X
X
k1 =0 k2 =k1(k,q,`)

`
X

···

X.

(5.46)

kq =kq−1

We now further definefnp`;µ (k, q) ≡ (` − k + q) ` + k − q + Fnp;µ ,(5.47)and evaluate sums successively over kp−1 , kp−2 , · · · , k1 , finding
X

p−1
Y


2kp−j − j + Fnp;µ =(k,p−1,`) j=1

p−1
Y

X



2kp−j − j + Fnp;µ fnp`;µ kp−2 , 1(5.48)(k,p−2,`) j=2

1=(r − 1)!

X

p−1
Y

2kp−j − j + Fnp;µ(k,p−r,`) j=r

Y
 r−1

fnp`;µ kp−r , q

q=1(5.49)
p−1=

Y1
fnp`;µ (0, q)(p − 1)!

(5.50)

q=1


 p−1

`+p−1 Y=` − q + Fnp;µ .
p−1(5.51)

q=1(`)

Substitution of this result together with Dm−p,n−p;µ using (5.36) into (5.45) then gives us
(p)
Emn;µ
=

n!
S n−p−1 Sµm−p−1 Gmnp;µ(n − p − 1)! µ̄(5.52)

with
Gmnp;µ ≡

m−p−1
X 
`=0


Y

 p−1

`+p−1
(m − p − `) ` + m − p + Fnp;µ 2` + Fnp;µ` − q + Fnp;µp−1
q=1(5.53)

=

Y
p

m
m − p − q + Fnp;µ .
p+1
q=0(5.54)

140
We can further simplifypYm − p − q + Fnp;µ =

q=0

p
Y

(m − n − q + 2µSz ) =

q=0

p+1
X

p+1−q

(−1)

q=0



p+1
(m − n + 2µSz )q ,q(5.55)wherep

is an unsigned Stirling number of the first kind, and finally
 
 X
p
p
q  
X
X
q
p−q p
q
p−q p
(−1)
(m − n + 2µSz ) =(−1)(m − n)q−` (2µSz )`
q
q
`
q=0
q=0
`=0
  
p
p
X X
p q
=
2`
(−1)p−q(m − n)q−` (µSz )` .
q `
q

`=0

(5.56)(5.57)

q=`Putting everything together, we finally have  
p
m n n−p m−p X p`
(p−1)
Emn;µ = p!
Sµ̄ Sµmn (µSz )` ,p
p

(5.58)

`=0

with
`
p`
mn ≡ 2

p
X
q=`

p−q

(−1)

  
p q(m − n)q−` ,q `(5.59)(0)where in this final form Emn;µ = Dmn;µ , which together with the expansion for Cmn;µ in (5.41)
implies thatmin{m,n}h

Sµm , Sµ̄n

i
−

X

=

p=1

  
p
m n n−p m−p X p`p!
SSµmn (µSz )` ,pp µ̄

(5.60)`=0andmin{m,n}Sµm Sµ̄n =X
p=0  
p
m n n−p m−p X p`
p!
Sµ̄ Sµmn (µSz )` .
p
p

(5.61)`=0

If we wish to order products of collective spin operators with Sz in between Sµ̄ and Sµ , thenmin{m,n}   X
m n n−p (p) m−p
m nSµ Sµ̄ =p!
S
Zmn;µ̄ Sµ ,(5.62)
p
p µ̄
p=0

where
(p)
Zmn;µ̄ ≡

p
X

p`
mn

− [m − p] + µSz

`

`=0

=

p
X

pq
ζmn
(µ̄Sz )q ,(5.63)

q=0

with
pq
ζmn
≡

p
X
`=q

Here

p
s

p`
mn

 
p   
X
`
p s
`
`−q
p q
(−1) (m − p)
= (−1) 2(m + n − 2p)s−q .
q
s
q
s=q

is an unsigned Stirling number of the first kind.

(5.64)

141

5.C

Product of arbitrary ordered collective spin operators
The most general product of collective spin operators that we need to compute ismin{r,`}
pqr
S`mn;µ= Sµp (µSz )q Sµ̄r Sµ` (µSz )m Sµ̄n =X
k=0  
r` p+`−k (k)k!
SZ̃qr`m;µ Sµ̄r+n−k ,kk µ(5.65)

where
(k)

(k)

Z̃qr`m;µ ≡ (` − k + µSz )q Zr`;µ (r − k + µSz )m
 X
 
q
k
m
X
X
q−b q
m−c m
ka
=
ζr`
(` − k)(r − k)(µSz )a+b+c ,
b
c
a=0

b=0

(5.66)(5.67)

c=0(k)

ka as respectively given in (5.63) and (5.64). The (anti-)commutator
is defined in terms of Zr`;µ and ζr`of two ordered products of collective spin operators is then simply
h
i
pqr
`mn
Sµp (µSz )q Sµ̄r , Sµ` (µSz )m Sµ̄n = S`mn;µ± Spqr;µ.
±5.D(5.68)

Sandwich identities for single-spin decoherence calculations
In this section we derive several identities that will be necessary for computing the effectsof single-spin decoherence on ordered products of collective spin operators, i.e. on operators of
the form Sµ` (µSz )m Sµ̄n . These identities all involve sandwiching a collective spin operator between
operators that act on individual spins only, and summing over all individual spin indices. Our
general strategy will be to use commutation relations to push single-spin operators together, andthen evaluate the sum to arrive at an expression involving only collective spin operators.
We first compute sums of single-spin operators sandwiching (µSz )m , when necessary making
use of the identity in (5.17). The unique cases up to Hermitian conjugation are, for S ≡ N/2 and142µ, ν ∈ {+1, −1},
X

m (j)
s(j)
z (µSz ) sz =

j

X

X
j

X
j

m (j)
s(j)
ν (µSz ) sν =

j

X

1X
1
1j (µSz )m = S (µSz )m ,42

(5.69)

1
1(µSz )m νSν = νSν (µν + µSz )m ,2
2(5.70)

j

m (j)
m
s(j)
z (µSz ) sν = (µSz )

j

X

m
(j)
s(j)
z sz (µSz ) =

(j)
s(j)
z sν =m(j)s(j)ν sν (µν + µSz ) = 0,(5.71)

j

X (j)
X (j)
m
m
sν̄ (µSz )m s(j)=sν̄ s(j)νν (µν + µSz ) = (S − νSz ) (µν + µSz ) .
j(5.72)j

We are now equipped to derive similar identities for more general collective spin operators. Making
heavy use of identities (5.18) and (5.27) to push single-spin operators through transverse collectivespin operators, we again work through all combinations that are unique up to Hermitian conjugation, findingXm n (j)`s(j)z Sµ (µSz ) Sµ̄ sz =j1(S − ` − n) Sµ` (µSz )m Sµ̄n + `nSµ`−1 (S + µSz ) (−1 + µSz )m Sµ̄n−1 ,2(5.73)



X
1 `+1
1
m n (j)m n(j) `sz Sµ (µSz ) Sµ̄ sµ = µSµ (1 + µSz ) Sµ̄ − µn S − ` − [n − 1] Sµ` (µSz )m Sµ̄n−122j− µ`n (n − 1) Sµ`−1 (S + µSz ) (−1 + µSz )m Sµ̄n−2 ,
X
j

X

1 `
m n (j)
m n+1
`
s(j)
+ µ`Sµ`−1 (S + µSz ) (−1 + µSz )m Sµ̄n ,z Sµ (µSz ) Sµ̄ sµ̄ = − µSµ (µSz ) Sµ̄
2(5.74)(5.75)m n (j)``+1s(j)(µSz )m Sµ̄n−1 − n (n − 1) Sµ` (S + µSz ) (−1 + µSz )m Sµ̄n−2 , (5.76)µ Sµ (µSz ) Sµ̄ sµ = nSµ

j

X

(j)

m n
m n
`
`
s(j)
µ Sµ (µSz ) Sµ̄ sµ̄ = Sµ (S + µSz ) (−1 + µSz ) Sµ̄ ,(5.77)

j

X (j)
m n
`
`−1
sµ̄ Sµ` (µSz )m Sµ̄n s(j)(µSz )m Sµ̄n−1µ = Sµ (S − ` − n − µSz ) (1 + µSz ) Sµ̄ + `n (2S − ` − n + 2) Sµj+ `n (` − 1) (n − 1) Sµ`−2 (S + µSz ) (−1 + µSz )m Sµ̄n−2 .

(5.78)

143

5.E

Uncorrelated, permutationally-symmetric single-spin decoherence
In this section we work out the effects of permutationally-symmetric decoherence of individual

spins on collective spin operators of the form Sµ` (µSz )m Sµ̄n . For compactness, we define
n

 !
o
X
1 (j) † (j)
(j)
(j) †
(j)
D (g) O ≡ D
g : j = 1, 2, · · · , NO=
g Og −
g g ,O
,2+(5.79)jwhere g is an operator that acts on a single spin, g (j) is an operator that acts with g on spin j andtrivially on all other spins, and N is the total number of spins.

5.E.1

Decay-type decoherence

The effect of decoherence via uncorrelated decay (µ = −1) or excitation (µ = 1) of individual
spins is described by
 X
X  (j)1 h (j) (j) i
µ
(j)
(j)
D sµ O =sµ̄ Osµ −sµ̄ sµ , O=
sµ̄ Os(j)[Sz , O]+ .
µ − SO +
2
2
+


j

(5.80)

j

In order to determine the effect of this decoherence on general collective spin operators, we expand
the anti-commutatorhSz , Sµ` (µSz )m Sµ̄ni+

= Sz Sµ` (µSz )m Sµ̄n + Sµ` (µSz )m Sµ̄n Sz = µSµ` (` + n + 2µSz ) (µSz )m Sµ̄n ,(5.81)

which implies, using (5.77),



 `
1
m n
m n
`
`
D sµ̄ Sµ (µSz ) Sµ̄ = Sµ (S + µSz ) (−1 + µSz ) Sµ̄ − Sµ S + (` + n) + µSz (µSz )m Sµ̄n ,2(5.82)
and, using (5.78),



 `
1
m n
m n
`
`
D sµ Sµ (µSz ) Sµ̄ = Sµ (S − ` − n − µSz ) (1 + µSz ) Sµ̄ − Sµ S − (` + n) − µSz (µSz )m Sµ̄n
2+ `n (2S − ` − n + 2) Sµ`−1 (µSz )m Sµ̄n−1+ `n (` − 1) (n − 1) Sµ`−2 (S + µSz ) (−1 + µSz )m Sµ̄n−2 .

(5.83)

144
0

(j)

Decoherence via jump operators sµ̄ only couples operators Sµ` (µSz )m Sµ̄n to operators Sµ` (µSz )m Sµ̄n(j)with m0 ≤ m. Decoherence via jump operators sµ , meanwhile, makes operators Sµ` (µSz )m Sµ̄n
“grow” in m through the last term in (5.83), although the sum ` + m + n does not grow.

5.E.2

Dephasing

The effect of decoherence via single-spin dephasing is described by
D (sz ) O =

X

1
(j)
s(j)
z Osz −
2

j

h

(j)
s(j)
z sz , O

i 
+

=

X
j

1
(j)
s(j)
z Osz − SO.
2(5.84)

From (5.73), we then have1
D (sz ) Sµ` (µSz )m Sµ̄n = − (` + n) Sµ` (µSz )m Sµ̄n + `nSµ`−1 (S + µSz ) (−1 + µSz )m Sµ̄n−1 . (5.85)
2Decoherence via single-spin dephasing makes operators Sµ` (µSz )m Sµ̄n “grow” in m, although thesum ` + m + n does not grow.

5.E.3

The general case

The most general type of single-spin decoherence is described by

 !
X
1
†
†
D (g) O =g (j) Og (j) −g (j) g (j) , O,g ≡ gz sz + g+ s+ + g− s− .
2+

(5.86)

j

To simplify (5.86), we expandg † Og = |gz |2 sz Osz +X|gµ |2 sµ̄ Osµ + gµ̄∗ gµ sµ Osµ + gz∗ gµ sz Osµ + gµ̄∗ gz sµ Osz ,(5.87)

µ

and


 
1
1X
2
2
∗
∗
g g = |gz | +|gµ | (1 − 2µsz ) + µ gz gµ − gµ̄ gz sµ ,42 µ†(5.88)

which implies
D (g) O =X

|gX |2 D (sX ) O +X∈{z,+,−}

X

(j)
∗
(j)
(j)
∗
(j)
(j)
gµ̄∗ gµ s(j)µ Osµ + gz gµ sz Osµ + gµ̄ gz sµ Osz



µ,j



1X  ∗−µ gz gµ − gµ̄∗ gz Sµ , O + .
4 µ(5.89)

145
In order to compute the effect of this decoherence on general collective spin operators, we expand
the anti-commutatorhSµ , Sµ` (µSz )m Sµ̄n

i
+



= Sµ`+1 (µSz )m + (1 + µSz )m Sµ̄n − nSµ` (n − 1 + 2µSz ) (µSz )m Sµ̄n−1 . (5.90)

Recognizing a resemblance between terms in (5.90) and (5.74), we collect terms to simplify
X
j

i
1 h
m n (j)
m n
`
`
s(j)
S
(µS
)
S
s
−
µ
S
,
S
(µS
)
S
= K`mn;µ + L`mn;µ
z
µ
z
z
µ
µ̄ µ
µ
µ̄
4
+

(5.91)i1 hm n (j)
m n
`
`
s(j)
S
(µS
)
S
s
+
µ
S
,
S
(µS
)
S
= K`mn;µ + M`mn;µ
z
µ
z
µ
µ
µ̄ z
µ
µ̄
4
+

(5.92)and likewise
X
j

with


1
K`mn;µ ≡ µSµ`+1 (1 + µSz )m − (µSz )m Sµ̄n ,(5.93)
4
3
1
L`mn;µ ≡ −µnSµ` S − ` − (n − 1) − µSz (µSz )m Sµ̄n−1 − µ`n (n − 1) Sµ`−1 (S + µSz ) (−1 + µSz )m Sµ̄n−2 ,42(5.94)"
M`mn;µ ≡ µnSµ` (S + µSz ) (−1 + µSz )m −

1
2



#


1
[n − 1] + µSz (µSz )m Sµ̄n−1 .
2(5.95)

Defining for completionP`mn;µ ≡Xm n (j)``+1s(j)(µSz )m Sµ̄n−1 − n (n − 1) Sµ` (S + µSz ) (−1 + µSz )m Sµ̄n−2 ,µ Sµ (µSz ) Sµ̄ sµ = nSµj(5.96)and(g)Q`mn;µ ≡ gµ̄∗ gµ P`mn;µ + gz∗ gµ + gµ̄∗ gz K`mn;µ + gz∗ gµ L`mn;µ + gµ̄∗ gz M`mn;µ ,(5.97)

we finally have
D (g) Sµ` (µSz )m Sµ̄n =

X



h
i†
(g)
(g)
|gX |2 D (sX ) Sµ` (µSz )m Sµ̄n + Q`mn;µ + Qnm`;µ .

(5.98)X∈{z,+,−}

Note that the sum `+m+n for operators Sµ` (µSz )m Sµ̄n does not grow under this type of decoherence.

146

5.F

Sandwich identities for collective-spin decoherence calculations
In analogy with the work in Appendix 5.D, in this section we work out sandwich identities

necessary for collective-spin decoherence calculations. The simplest cases are
Sµ Sµ` (µSz )m Sµ̄n Sµ̄ = Sµ`+1 (µSz )m Sµ̄n+1 ,(5.99)Sµ Sµ` (µSz )m Sµ̄n Sz = µSµ`+1 (n + µSz ) (µSz )m Sµ̄n ,hiSz Sµ` (µSz )m Sµ̄n Sz = Sµ` `n + (` + n) µSz + (µSz )2 (µSz )m Sµ̄n .

(5.100)(5.101)

With a bit more work, we can also findSµ` (µSz )m Sµ̄n Sµ = Sµ`+1 (1 + µSz )m Sµ̄n − nSµ` (n − 1 + 2µSz ) (µSz )m Sµ̄n−1 ,(5.102)

which implies
Sµ Sµ` (µSz )m Sµ̄n Sµ = Sµ`+2 (1 + µSz )m Sµ̄n − nSµ`+1 (n − 1 + 2µSz ) (µSz )m Sµ̄n−1 ,(5.103)Sz Sµ` (µSz )m Sµ̄n Sµ = µSµ`+1 (` + 1 + µSz ) (1 + µSz )m Sµ̄nhi
− µnSµ` ` (n − 1) + (2` + n − 1) µSz + 2 (µSz )2 (µSz )m Sµ̄n−1 .

(5.104)

Finally, we computehi
hi
Sµ̄ Sµ` (µSz )m Sµ̄n Sµ = Sµ` Sµ̄ − `Sµ`−1 (` − 1 + 2µSz ) (µSz )m Sµ Sµ̄n − n (n − 1 + 2µSz ) Sµ̄n−1= Sµ` Sµ̄ (µSz )m Sµ Sµ̄n− Sµ` ` (` + 1) + n (n + 1) + 2 (` + n) µSz (1 + µSz )m Sµ̄nhi
+ `nSµ`−1 (` − 1) (n − 1) + 2 (` + n − 2) µSz + 4 (µSz )2 (µSz )m Sµ̄n−1 ,(5.105)whereSµ̄ (µSz )m Sµ = Sµ̄ Sµ (1 + µSz )m = Sµ Sµ̄ − 2µSz (1 + µSz )m = Sµ (2 + µSz )m Sµ̄ − 2µSz (1 + µSz )m ,(5.106)

147
soSµ̄ Sµ` (µSz )m Sµ̄n Sµ = Sµ`+1 (2 + µSz )m Sµ̄n+1− Sµ` ` (` + 1) + n (n + 1) + 2 (` + n + 1) µSz (1 + µSz )m Sµ̄nhi
+ `nSµ`−1 (` − 1) (n − 1) + 2 (` + n − 2) µSz + 4 (µSz )2 (µSz )m Sµ̄n−1 .
(5.107)

5.G

Collective spin decoherence
In this section we work out the effects of collective decoherence on general collective spin

operators. For shorthand, we definei
1h †D (G) O ≡ D {G} O = G† OG −G G, O ,2+(5.108)where G is a collective spin jump operator.

5.G.1

Decay-type decoherence and dephasing

Making use of the results in Appendix 5.F, we find that the effects of collective decay-type
decoherence on general collective spin operators are given by
D Sµ̄


 `


Sµ (µSz )m Sµ̄n = −Sµ`+1 (1 + µSz )m − (µSz )m Sµ̄n+1
1 + Sµ` ` (` − 1) + n (n − 1) + 2 (` + n) µSz (µSz )m Sµ̄n ,2(5.109)

and
D Sµ


 `


Sµ (µSz )m Sµ̄n = Sµ`+1 (2 + µSz )m − (1 + µSz )m Sµ̄n+1− Sµ` ` (` + 1) + n (n + 1) + 2 (` + n + 1) µSz (1 + µSz )m Sµ̄n
1 + Sµ` ` (` + 1) + n (n + 1) + 2 (` + n + 2) µSz (µSz )m Sµ̄n2hi
+ `nSµ`−1 (` − 1) (n − 1) + 2 (` + n − 2) µSz + 4 (µSz )2 (µSz )m Sµ̄n−1 .
(5.110)Similarly, the effect of collective dephasing is given by
1
D (Sz ) Sµ` (µSz )m Sµ̄n = − (` − n)2 Sµ` (µSz )m Sµ̄n .
2(5.111)

148
5.G.2

The general caseMore generally, we consider jump operators of the formG ≡ Gz Sz + G+ S+ + G− S− ,(5.112)

whose decoherence effects are determined byG† OG = |Gz |2 Sz OSz +X|Gµ |2 Sµ̄ OSµ + G∗µ̄ Gµ Sµ OSµ + G∗z Gµ Sz OSµ + G∗µ̄ Gz Sµ OSz ,µ(5.113)and†

G G = |Gz |2 Sz2 +X

|Gµ | Sµ̄ Sµ + G∗z Gµ Sz Sµ + G∗µ̄ Gz Sµ Sz + G∗µ̄ Gµ Sµ22

,(5.114)µwhich implies
D (G) O =X

|GX |2 D (SX ) O +X

G∗µ̄ Gµ Sµ OSµ + G∗z Gµ Sz OSµ + G∗µ̄ Gz Sµ OSzµX∈{z,+,−}



h
i




1X
−
G∗µ̄ Gµ Sµ2 , O + G∗z Gµ Sz Sµ , O + + G∗µ̄ Gz Sµ Sz , O + .
2 µ+(5.115)In order to compute the effect of this decoherence on general collective spin operators, we expand
the anti-commutatorshSµ2 , Sµ` (µSz )m Sµ̄n

i
+



= Sµ`+2 (2 + µSz )m + (µSz )m Sµ̄n − 2nSµ`+1 (n + 2µSz ) (1 + µSz )m Sµ̄n−1hi+ n (n − 1) Sµ` (n − 1) (n − 2) + 2 (2n − 3) µSz + 4 (µSz )2 (µSz )m Sµ̄n−2 ,(5.116)hSz Sµ , Sµ` (µSz )m Sµ̄n

i
+



= µSµ`+1 (` + 1 + µSz ) (µSz )m + (n + 1 + µSz ) (1 + µSz )m Sµ̄nhi
− µnSµ` n (n − 1) + (3n − 1) µSz + 2 (µSz )2 (µSz )m Sµ̄n−1 ,hSµ Sz , Sµ` (µSz )m Sµ̄ni+(5.117)= µSµ`+1 (` + µSz ) (µSz )m + (n + µSz ) (1 + µSz )m Sµ̄n
− µnSµ`h2(n − 1) + 3 (n − 1) µSz + 2 (µSz )2i(µSz )m Sµ̄n−1 .

(5.118)

Collecting terms and defining
G(±)
z,µ ≡


1 ∗
Gz Gµ ± G∗µ̄ Gz ,2

(5.119)

149
"

#


1
1
(G)
`+1
L̃`mn;µ ≡ µ ` − n +
G(+)
G(−)
(1 + µSz )m Sµ̄nz,µ + ` +z,µ Sµ
2
2
"
#



1
1
`+1
−µ `−n+G(+)
G(−)(µSz )m Sµ̄nz,µ + n +z,µ Sµ
2
2



m
m n
`+1
+ µG(−)Sµ̄ ,z,µ Sµ µSz (1 + µSz ) − (µSz )
"
#



1
1
(G)
(+)
(−)
M̃`mn;µ = −µn (n − 1) ` − n +Gz,µ + ` −Gz,µ Sµ` (µSz )m Sµ̄n−1
2
2
"
#



1
1
(+)
(−)
− 2µn ` − n +Gz,µ + ` + n − 1 Gz,µ Sµ` (µSz )m+1 Sµ̄n−1
22
m+2 n−1`− 2µnG(−)
Sµ̄ ,z,µ Sµ (µSz )1P̃`mn;µ ≡ − Sµ`+2 (2 + µSz )m − 2 (1 + µSz )m + (µSz )m Sµ̄n2+ nSµ`+1 (n + 2µSz ) (1 + µSz )m − (n − 1 + 2µSz ) (µSz )m Sµ̄n−1


2
` 1
− n (n − 1) Sµ(n − 1) (n − 2) + (2n − 3) µSz + 2 (µSz ) (µSz )m Sµ̄n−2 ,
2
(G)

(G)

(G)

Q̃`mn;µ ≡ G∗µ̄ Gµ P̃`mn;µ + L̃`mn;µ + M̃`mn;µ ,(5.120)(5.121)(5.122)(5.123)we then have
D (G) Sµ` (µSz )m Sµ̄n =

X



h
i†
(G)
(G)
|GX |2 D (SX ) Sµ` (µSz )m Sµ̄n + Q̃`mn;µ + Q̃nm`;µ . (5.124)

X∈{z,+,−}

Note that the sum ` + m + n for operators Sµ` (µSz )m Sµ̄n grows by one if Gµ 6= 0 or Gµ̄ 6= 0, anddoes not grow otherwise.

5.H

Emulating particle loss in a spin model
Here we discuss the details of emulating particle loss with O(1/N ) error, where N is theinitial number of particles in a system that we wish to describe with a spin model. Starting withthe full algebra of creation and annihilation operators (whether bosonic or fermionic) in a system,
spin models are typically implemented by identifying a subalgebra of relevant “spin” operatorsthat satisfy appropriate commutation relations. Two-state particles on a lattice, for example, aredescribed by annihilation operators cjs indexed by a lattice site j ∈ Z and an internal state index

150s ∈ {↑, ↓}, enabling the straightforward construction of spin operators
σx(j) ≡ c†j,↑ cj,↓ + h.c., σy(j) ≡ −ic†j,↑ cj,↓ + h.c., σz(j) ≡ c†j,↑ cj,↑ − c†j,↓ cj,↓ , 1(j) ≡ c†j,↑ cj,↑ + c†j,↓ cj,↓ ,(5.125)which satisfy the same commutation relations as the standard Pauli operators. These spin operators
can be more compactly defined in the formXσα(j) ≡c†jr hr|σα |si cjs ,(5.126)r,s∈{↑,↓}

where σα for α ∈ {x, y, z, 1} is a Pauli operator, with σ1 ≡ 1; and hr|σα |si denotes a matrix elementbare ≡ cof σα . This construction exemplifies how the set of jump operators Jloss
js that generateparticle loss cannot be constructed from spin operators, which are generally bilinear in particle
creation or annihilation operators. When working on the level of a spin model, therefore, we canat best only emulate the effect of particle loss by some indirect means.
To understand the effect of particle loss on collective spin operators, we first define a singlenomulti-body spin operator addressing sites j = j1 , j2 , · · · , j|j| ,Y

σα(j)j ,(5.127)

i 
X †
1h †
σjα =
cks σjα cks −c cks , σjα
2 ks+(5.128)

σjα ≡

j∈j

and expand
D



bare
Jloss



k,s

=

i
XX †
Y
XX †
1 X h (k)(j)
cks σα(k)
c
σ
+
c
c
σ
−
1
,
σ
ks
ks
jα
jα
αj
ks
k
2
+
s
s
k∈j

=

XX

j∈j
j6=k

k∈j
/

hq|σαk |ri c†ks c†kq ckr cks

k∈j q,r,sY(5.129)k

σα(j)j − |j|σjα .

(5.130)j∈jj6=k

In order to have an actual spin model, fermionic statistics or energetic considerations must forbid multiple occupation of individual lattice sites. In that case, the on-site four-point productc†ks c†kq ckr cks = 0 vanishes, andbare
D Jlossσjα = −|j|σjα .

(5.131)

151
Up to O(1/N ) corrections, a collective spin operator Sm essentially consists of |m|-body operatorsof the form σjα with |j| = |m|, which implies that the dissipator Dloss defined by Dloss Sm =−|m|Sm describes particle loss with O(1/N ) error. We note that the dissipator Dloss is essentiallyn o(j)the depolarizing channel, i.e. Dloss = D (Jloss ) for Jloss = sαwith α ∈ {x, y, z} and j ∈{1, 2, · · · , N }. A direct implementation of Dloss with Dloss Sm = −|m|Sm , however, is much more
efficient than evaluating the depolarizing channel D (Jloss ) with the ingredients in Appendices 5.D
and 5.E.

5.I

Initial conditions
Here we compute the expectation values of collective spin operators with respect to spin-polarized (also Gaussian, or spin-coherent) states. These states are parameterized by polar andazimuthal angles θ ∈ [0, π), φ ∈ [0, 2π), and lie within the Dicke manifold spanned by statesS+k|ki ∝ S+|↓i⊗N with S ≡ N/2 and Sz |ki = k |ki:

1/2
S 
h
X
 −iφ/2
 iφ/2 i⊗N
S+kS−k −ikφN|θ, φi ≡ cos θ/2 e|↑i + sin θ/2 e|↓i=cos θ/2sin θ/2
e
|ki .
S+k
k=−S

(5.132)
We can likewise expand, within the Dicke manifold,
Sz =

S
X

S−δµ,1k |kihk| ,

Sµ =

k=−S

S−δµ̄,1Xgµ (k) |k + µihk| =k=−S+δµ,−1Xgµ̄ (k) |kihk + µ̄| ,(5.133)k=−S+δµ̄,−1where µ̄ ≡ −µ ∈ {+1, −1} andgµ (k) ≡p(S − µk) (S + µk + 1),(5.134)

152
which implies
S−δµ,1 max{`,n}X

Sµ` (µSzm ) Sµ̄n =(µk)m 

`−1
Y



n−1
Y

gµ (k + µp) 

p=0

k=−S+δµ,−1 max{`,n}gµ (k + µq) |k + µ`ihk + µn|

q=0(5.135)µS−δµ,1 max{`,n}

X

=

(µk)


(S + µk + `)! (S + µk + n)! 1/2|k + µ`ihk + µn|(S + µk)! (S − µk − `)! (S − µk − n)!

m (S − µk)!

µk=−µS−δµ,−1 max{`,n}(5.136)S−max{`,n}

=

X
k=−S


(S + k + `)! (S + k + n)! 1/2kµ (k + `) µ (k + n) .
(S + k)! (S − k − `)! (S − k − n)!
m (S − k)!

(5.137)This expansion allows us to compute the expectation value
S−max{`,n}hθ, φ|Sµ` (µSzm ) Sµ̄n |θ, φi = eiφµ(`−n) N !

X
k=−Sk m (S − k)!fµ`n (k, θ)(S + k)! (S − k − `)! (S − k − n)!

N −max{`,n}X= eiφµ(`−n) (−1)m N !

k=0(5.138)(S − k)m (N − k)!f˜µ`n (k, θ)
k! (N − k − `)! (N − k − n)!

(5.139)wherefµ`n (k, θ) ≡ cos θ/2N +µ(2k+`+n)sin θ/2

N −µ(2k+`+n),(5.140)2N δµ,−1 +µ(2k+`+n)2N δµ,1 −µ(2k+`+n)f˜µ`n (k, θ) ≡ fµ`n (k − S, θ) = cos θ/2
sin θ/2.

(5.141)

Defining the states|+Zi ≡ |0, 0i = |↑i

⊗N

,

⊗N

|−Zi ≡ |π, 0i = |↓i,|Xi ≡ π/2, 0 =some particular expectation values of interest are



N !n!


(S − n)m
(N − n)!
hνZ|Sµ` (µSz )m Sµ̄n |νZi = δ`n ×


m


δn,0 (−S)|↑i + |↓i
√
2

⊗N
,

(5.142)µ = ν,,(5.143)µ 6= ν,andhX|Sµ` (µSz )m Sµ̄n |Xi = (−1)mN!
2NN −max{`,n}X
k=0(S − k)m (N − k)!
.
k! (N − k − `)! (N − k − n)!

(5.144)153

5.J

Computing correlators with the truncated short-time (TST) expansion
Here we provide a pedagogical tutorial for computing correlators using the truncated short-time TST expansion. For concreteness, we nominally consider N spins evolving under the one-axis
twisting (OAT) Hamiltonian
HOAT = χSz2 ,(5.145)

additionally subject to spontaneous single-spin decay at rate γ− , with jump operators J− =
n
o


(j)
` S m S n (t) iss− : j = 1, 2, · · · , N . The equation of motion for a Heisenberg operator S+
z −
h

i 


d D ` m nE
2
` m n
` m n
S+ Sz S− = iχ Sz , S+ Sz S−+ γ− D (J− ) S+ Sz S−,
dt−(5.146)

where we have suppressed the explicit time dependence of operators for brevity. Using the results inhi` SmSnappendices 5.C and 5.E.1 respectively to evaluate the commutator Sz2 , S+and dissipator
z −
−


` S m S n in (5.146), we can expand
D (J− ) S+z −dhS ` S m S n i
dt + z −*`n= iχ h(` − n) S+(` + n + 2Sz ) Szm S−
i+γ−

`
S+

"

# +

`+n
n
(S + Sz ) (−1 + Sz ) − S ++ Sz Szm S−
.
2
m



(5.147)
In practice, we do not want to keep track of such an expansion by hand, especially in the caseof e.g. the two-axis twisting (TAT) and twist-and-turn (TNT) models with more general types ofdecoherence, for which the analogue of (5.147) may take several lines just to write out in full.
mm

Defining the operators Sm ≡ S+ + Szmz S− − with m ≡ (m+ , mz , m− ) for shorthand, we note thatthe vector space spanned by {Sm } is closed under time evolution. We therefore expandXdhSn i = hT Sn i =hSm i Tmn ,dtm(5.148)where T is a superoperator that generates time evolution for Heisenberg operators. In the present
example, the matrix elements Tmn ∈ C of T are defined by (5.147) and (5.148). For any Hamiltonian

154
H with decoherence characterized by sets of jump operators J and decoherence rates γJ , the matrix
elements Tmn are more generally defined by
T Sn = i [H, Sn ]− +X

γJ D (J ) Sn =X

Sm Tmn .

(5.149)m

J

The results in Appendices 5.C, 5.E, and 5.G can be used to write model-agnostic codes that computematrix elements Tmn , taking a particular Hamiltonian H and decoherence processes (J , γJ ) as
inputs.
In order to compute a quantity such as spin squeezing, we need to compute correlators of
the form hSn (t)i, where for clarity we will re-introduce the explicit time dependence of Heisenberg
operators Sn (t). The order-M truncated short-time (TST) expansion takes
hSn (t)i = hetT Sn (0)i =

X tk
k≥0

k!

hT k Sn (0)i =X tk Xk≥0

k! mkhSm (0)i Tmn
→

M k X
X
t
k=0

k! m

k
hSm (0)i Tmn,(5.150)kwhere Tmn
are matrix elements of the k-th time derivative operator T k , given by1 m = n,
X
0
1
k>1
Tmn ≡
, Tmn
≡ Tmn , Tmn≡
Tmpk−1 · · · Tp3 p2 Tp2 p1 Tp1 n . (5.151)

p1 ,p2 ,··· ,pk−10 otherwisekMatrix elements Tmn
and initial-time expectation values hSm (0)i are thus computed as needed

for any particular correlator hSn (t)i of interest, and combined according to (5.150). Note that
initial-time expectation values hSm (0)i are an input to the TST expansion, and need to be computed separately for any initial state of interest; expectation values with respect to spin-polarized(Gaussian) states are provided in Appendix 5.I. In practice, we further collect terms in (5.150) towrite
hSn (t)i →

M
X
k=0

cnk tk ,

cnk ≡

1 X
k
hSm (0)i Tmn,
k! m(5.152)where cnk are time-independent coefficients for the expansion of hSn (t)i. After computing thecoefficients cnk , there is only negligible computational overhead to compute the correlator hSn (t)ifor any time t.

155

5.K

Analytical results for the one-axis twisting model
The one-axis twisting (OAT) Hamiltonian for N spin-1/2 particles takes the form
1 X (j) (k) 1
HOAT = χSz2 = χσz σz + N χ,
2
4

(5.153)j<k(j)where σz

represents a Pauli-z operator acting on spin j. This model is a special case of the zero-

field Ising Hamiltonian previously solved in Ref. [120] via exact, analytical treatment of the quantum
trajectory Monte Carlo method for computing expectation values. The solution therein accounts
for coherent evolution in addition to decoherence via uncorrelated single-spin decay, excitation, anddephasing respectively at rates γ− , γ+ , and γz (denoted by Γud , Γdu , and Γel in Ref. [120]). Letting
S ≡ N/2 and µ, ν ∈ {+1, −1}, we adapt expectation values computed in Ref. [120] for the initial⊗Nstate |Xi ∝ |↑i + |↓iwith Sx |Xi = S |Xi evolving under HOAT , findinghS+ (t)i = Se−κt Φ (χ, t)N −1 ,



µ
1 −κt
h Sµ Sz (t)i = − hSµ (t)i + S S −e Ψ (µχ, t) Φ (χ, t)N −2 ,
2
2





N −2
1 −2κth Sµ Sν (t)i = 1 − δµν S + µ hSz (t)i + S S −eΦ [µ + ν] χ, t,2

(5.154)(5.155)(5.156)

where


λ
Φ (X, t) ≡ e−λt cos (ωX t) +sin (ωX t) ,ωX

Ψ (X, t) ≡ e−λt

∆ + iX
ωX

sin (ωX t) ,(5.157)p
X 2 − λ2 − iX∆.

(5.158)forκ≡

1(γ+ + γ− + γz ) ,2

λ≡1(γ+ + γ− ) ,2∆ ≡ γ+ − γ− ,ωX ≡In order to compute spin squeezing as measured by the Ramsey squeezing parameter ξ 2 definedin (5.12), we additionally need analytical expressions for hSz (t)i and hSz2 (t)i. As these operators(j)commute with both the OAT Hamiltonian and the single-spin operators σz , their evolution isgoverned entirely by decay-type decoherence (see Appendix 5.E.1), which meansdhSz (t)i = S (γ+ − γ− ) − (γ+ + γ− ) hSz (t)i ,
dt


d
1
2
hS (t)i = S (γ+ + γ− ) + 2 S −(γ+ − γ− ) hSz (t)i − 2 (γ+ + γ− ) hSz2 (t)i .
dt z
2(5.159)(5.160)

156
The initial conditions hSz (0)i = 0 and hSz2 (0)i = S/2 then implyhSz (t)i = S

γ+ − γ−γ+ + γ−

1−e−(γ+ +γ− )t,1
hSz2 (t)i = S + S
2




1
hSz (t)i 2
S−
.
2
S

(5.161)

With appropriate assumptions about the relevant sources of decoherence, the expectation values in(5.154)–(5.156) and (5.161) are sufficient to compute the spin squeezing parameter ξ 2 in (5.12) at
any time throughout evolution of the initial state |Xi under HOAT .

5.L

Diagnosing breakdown of the TST expansion
In Figure 5.1 of the main text, the TST expansion provided nearly exact results for squeezing

until a sudden departure that quickly resulted in an unphysical squeezing parameter, ξ 2 < 0. In
general, however, there is no fundamental relationship between the breakdown of the TST expansion
and the conditions for a physical squeezing parameter ξ 2 . A proper diagnosis of breakdown therefore
requires inspection of the correlators hSn (t)i used to compute the squeezing parameter ξ 2 , which
upon breakdown will rapidly take unphysical values with |hSn (t)i| & S |n| . As an example, Figure
5.4 shows the squeezing parameter ξ 2 throughout decoherence-free evolution of N = 100 spins
initially in the state |Xi. In this example, the squeezing computed by the TST expansion for
the TAT model diverges from the exact answer without an immediate and obvious signature ofbreakdown. Nonetheless, breakdown can still be diagnosed by inspection of individual correlators,
2 i as a function of time for N = 100 spins evolving underas shown in Figure 5.5, where we plot Im hS+

the TAT without decoherence. Figure 5.5 shows that breakdown clearly occurs around N χt . 7,
2 i begins to diverge to values & S 2 = N/2 2 = 2500 in magnitude. A joint
when the correlator hS+
inspection of figures 5.4 and 5.5 suffice to trace the anomalous behavior of ξ 2 from N χt ≈ 7 backto N χt ≈ 6, when it first took a sudden turn before becoming unphysical at N χt ≈ 8.

157

OAT
TAT
TNT

Squeezing (ξ 2 )

101

100

10−1

0

2

4
Time (N χt)

68

Figure 5.4: Spin squeezing throughout decoherence-free evolution of N = 100 spins initially in thestate |Xi, computed using both exact methods (solid lines) and the TST expansion in Eq. (5.8)
with M = 35 (dashed lines). Solid circles mark the times at which the TST expansion gives anunphysical result with ξ 2 < 0.

3

×103

2

2i
hS+

1
0
−1
−2
−3

0

2

4
Time (N χt)

68

Figure 5.5: A collective spin correlator in the TAT model with N = 100 spins and no decoherence,computed using the TST expansion with M = 35. The divergence of correlators of this sort can beused to diagnose the breakdown of the TST expansion.

158

5.M

Spin squeezing with strong decoherence
Here we provide supplementary evidence of our finding in Section 5.3 that the TNT model

can produce more squeezing than the OAT or TAT models in the presence of strong decoherence.
2
To this end, Figure 5.6 shows the minimal squeezing parameter ξminachievable with N = 100

spins through the OAT, TAT, and TNT models as a function of the rate γ0 at which individual
spins undergo spontaneous decay, excitation, and dephasing. These results were computed withquantum trajectory simulations, with 103 trajectories per data point. While the OAT and TAT
models produce more squeezing than the TNT model with weak decoherence, this squeezing falls
off faster with an increasing decoherence rate γ0 . The relative robustness of TNT is in part a
consequence of the fact that TNT initially generates squeezing at a faster rate than OAT or TAT,
thereby allowing it to produce more squeezing before the degrading effects of decoherence kick in.

159

2
ξmin

OAT
TAT
TNT

10−1

10−2

10−1

100
γ0 /χ

Figure 5.6: Optimal spin squeezing of N = 100 spins undergoing spontaneous decay, excitation,and dephasing at rates γ− = γ+ = γz = γ0 , computed using quantum trajectory simulations with
103 trajectories per data point.

160

5.N

Heisenberg operators in open quantum systems

Here we explain the origin and character of the mean-zero “noise” operators EO (t) that appearPin the expansion of a Heisenberg operator O (t) = m Om (t) Sm + EO (t) with time-dependent
coefficients Om (t) for time-independent Schrödinger operators Sm . Our discussion should clarify
why noise operators play no role in our calculation of expectation values of the form hO (t)i andhOQ (t)i, despite the fact that noise operators generally do need to be considered in the calculationof more general multi-time correlators in open quantum systems[181].
In any closed quantum system with initial state ρ and propagator U (t), such that the state
at time t is ρ (t) ≡ U (t) ρU † (t), time-dependent Heisenberg operators O (t) are uniquely definedfrom time-independent Schrödinger operators O byhO (t)i ≡ tr ρ (t) O = tr ρO (t) .

(5.162)

Enforcing (5.162) for arbitrary initial states ρ forces O (t) = U † (t) OU (t). In an open quantum
system, however, the definition of a Heisenberg operator is not so straightforward. Open systems
can often be understood as subsystems of a larger closed system. Consider therefore an open system
S with environment E, a joint initial state ρSE , and propagator USE (t). The reduced state ρS (t)of S at time t is
h
i


†
ρS (t) ≡ trE ρSE (t) = trE USE (t) ρSE USE(t) ≡ US (t) ρS ,(5.163)where ρS ≡ ρS (0) is a time-independent state of S in the Heisenberg picture, S denotes the spaceof operators on S, and the quantum channel US (t) has the decomposition[186]US (t) ρS =

X

(j)

(j)†

US (t) ρS US(t)(5.164)j(j)with ordinary operators US (t) on S. We can therefore expand
h
i
h
i




hOS (t)i = tr ρS (t) OS = tr US (t) ρS OS = tr ρS US† (t) OS = tr ρS OS (t) = hOS (t)i ,(5.165)

161
where US† (t) is the adjoint map of US (t) (with respect to a trace inner product between operators
on S), and we define the time-dependent operator
OS (t) ≡ US† (t) OS =

X

(j)†

US

(j)

(t) OS US (t) .

(5.166)j

We thus find that substituting OS (t) in place of OS (t) suffices for the calculation of correlatorshOS (t)i, thereby accounting for the validity of the equation of motion in (5.2). As we show below,
this substitution also suffices for the calculation of two-time correlators of the form hOS QS (t)iwhen the environment E is Markovian.
The problem with defining Heisenberg operators OS (t) by OS (t) only becomes evident when
considering products of Heisenberg operators. One would like for the product of two Heisenberg
operators OS (t) and QS (t) to satisfy OS (t) QS (t) = (OS QS ) (t). This intuition can be formalized
by observing thathOS (t)i = tr ρSE (t) (OS ⊗ 1E ) = tr ρSE (OS ⊗ 1E ) (t) = h(OS ⊗ 1E ) (t)i ,(5.167)

where 1E is the identity operator on E, expectation values of Heisenberg operators on systemA ∈ {S, E, SE} are taken with respect to the state ρA , and†(OS ⊗ 1E ) (t) ≡ USE(t) (OS ⊗ 1E ) USE (t) .

(5.168)By expanding Heisenberg operators similarly to (5.167) and (5.168), we then findhOS (t) QS (t)i = h(OS ⊗ 1E ) (t) (QS ⊗ 1E ) (t)i = h(OS QS ⊗ 1E ) (t)i = h(OS QS ) (t)i .

(5.169)The expression in (5.166), however, makes it clear that generally OS (t) QS (t) 6= OS QS (t). To
correct for this discrepancy, we define
OS (t) ≡ OS (t) + EOS (t)(5.170)in terms of new “noise” operators EOS (t) that are essentially defined to enforce the consistency ofoperator products such as OS (t) QS (t) = (OS QS ) (t). Self-consistency forces noise operators to bemean-zero, as
hEOS (t)i = hOS (t)i − hOS (t)i = 0.

(5.171)

162
Furthermore, if the environment E is Markovian, then noise operators are also uncorrelated withinitial-time observables, i.e. hOS EQS (t)i = 0, which means that noise operators can be neglectedin the calculation of two-time correlators of the form hOS QS (t)i. To see why, we observe that a
Markovian environment is essentially defined to satisfy†
ρSE (t) = USE (t) ρSE USE(t) ≈ ρS (t) ⊗ ρE = US (t) ρS ⊗ ρE ,(5.172)

with ρE a time-independent steady state of the environment. If we enforce (5.172) for all states ρS ,
(1)

(2)

(1)

e.g. the maximally mixed state ρS ∝ 1S and ρS ≡ ρS + OS with OS any traceless operator on(2)

S with operator norm kOS k ≤ 1/ tr 1S (i.e. such that ρS remains positive semi-definite, or a valid
quantum state), then by linearity we find that†
USE (t) (1S ⊗ ρE ) USE(t) ≈ US (t) 1S ⊗ ρE ,†USE (t) (OS ⊗ ρE ) USE(t) ≈ US (t) OS ⊗ ρE ,(5.173)
which implies that the Markov approximation (5.172) holds even if we replace ρS by any operator
on S, and in particular††
USE (t) ρSE (OS ⊗ 1E ) USE(t) = USE (t) (ρS OS ⊗ ρE ) USE(t) ≈ US (t) (ρS OS ) ⊗ ρE .

(5.174)

We can therefore expandhi†hOS QS (t)i = tr ρSE (OS ⊗ 1E ) USE(t) (QS ⊗ 1E ) USE (t)hi†= tr USE (t) ρSE (OS ⊗ 1E ) USE(t) (QS ⊗ 1E ) ,(5.175)(5.176)and invoke the Markov approximation in (5.174) to find thathihOS QS (t)i ≈ tr US (t) (ρS OS ) QS = tr ρS OS US† (t) QS = hOS QS (t)i ,(5.177)

which implies
hOS EQS (t)i = hOS QS (t)i − hOS QS (t)i ≈ 0.

(5.178)Noise operators thus play no role in the calculation of correlators such as C (t) in (5.13). In
contrast, noise operators generally do play a role in the calculation of multi-time correlators of the163
Q (j) form h j OS tj i[181]. Furthermore, these calculations generally require additional assumptions
about the environment. To keep our discussion simple and general, we therefore exclude the effects
of noise terms in Section 5.4.

5.O

Short-time linear response and two-time correlators
Here we discuss the appearance of two-time correlation functions in the short-time linear

response of correlators to perturbations of a Hamiltonian. Consider an initial Hamiltonian H
perturbed by an operator V with kV k  kHk, where kOk denotes the operator norm of O, suchthat the net Hamiltonian is H̃ = H + V . We denote the generator of Heisenberg time evolution
under the perturbed (unperturbed) Hamiltonian by T̃ (T ). These generators are related byT̃ = T + iV

(5.179)where V is a superoperator whose action on operators O is defined by
V O ≡ [V, O]− .

(5.180)Through quadratic order in the time t and linear order in the perturbation V , we can say that
etT̃ ≈i
1 h tT itV i
1 he ,e≈ etT + it etT , V
.
2
2
+
+

(5.181)

Defining perturbed and unperturbed Heisenberg operators Õ (t) and O (t) that respectively satisfyhÕ (t)i = hetT̃ Oi and hO (t)i = hetT Oi, we thus find that for sufficiently small times t and weakperturbations V ,hÕ (t) − O (t)i =



e

tT̃

  1  −eO ≈ it h[V, O]− (t)i + h V, O (t) − i .
2
tT(5.182)

Two-time correlators hV O (t)i and hO (t) V i, in addition to the expectation values h(V O) (t)i andh(OV ) (t)i, thus determine the short-time linear response of correlators hO (t)i to perturbations V
of a Hamiltonian.

Chapter 6

Spin squeezing with short-range spin-exchange interactions

Prologue
Chapter 4 featured a proposal to generate spin-squeezed states using uniform spin-spin interactions with an SU(2) symmetry that is broken down to U(1) by a single-particle field. The same
sort of symmetry breaking can occur by modifying the interactions directly. Instead of adding asingle-body axial field, we can add two-body axial (Ising) interactions, which results in a so-called
XXZ model. A major advantage to this strategy is that it requires little to no novelty on the experimental front: the XXZ model has already been implemented in a variety of atomic, molecular,
and optical platforms. However, most implementations of the XXZ model have interactions thatare not uniform, but rather fall off with distance as a power-law. These observations naturally leadto questions about the spin-squeezing properties of the power-law XXZ model, which is the focusof this chapter. The bulk of this chapter is taken from Ref. [4]. In addition to myself and Ana
Maria Rey, this work featured major contributions from Chunlei Qu.

Abstract
We investigate many-body spin squeezing dynamics in an XXZ model with interactions thatfall off with distance r as 1/rα in D = 2 and 3 spatial dimensions. In stark contrast to the Isingmodel, we find a broad parameter regime where spin squeezing comparable to the infinite-range
α = 0 limit is achievable even when interactions are short-ranged, α > D. A region of “collective”
behavior in which optimal squeezing grows with system size extends all the way to the α → ∞165limit of nearest-neighbor interactions. Our predictions, made using the discrete truncated Wigner
approximation (DTWA), are testable in a variety of experimental cold atomic, molecular, andoptical platforms.

6.1

Introduction
Quantum technologies receive an enormous amount of attention for their potential to pushbeyond classical limits on physically achievable tasks. In order to be useful, however, these technologies must demonstrate a practical advantage over their classical counterparts. While most publicattention has focused on a quantum advantage in the realm of computing, the quantum metrology
community has made tremendous progress in developing strategies and platforms for surpassingclassical limits on measurement precision [31, 109, 187–189]. A key element in these strategies isthe use of entanglement to enhance the capabilities of individual, uncorrelated quantum systems.
Spin squeezing is one of the most promising strategies for using entanglement to achieve a quantum
advantage in practical sensing applications [32, 35].
The paradigmatic setting for spin squeezing is the one-axis twisting (OAT) model [33, 35],
which generates spin-squeezed states by use of uniform, infinite-range Ising interactions that do not
distinguish between the constituent spins. These uniform interactions can be implemented directly
via collisional interactions between delocalized atoms [2, 50, 190], as well as indirectly throughcoupling to collective phonon modes [139, 191, 192] or cavity photons [114, 142, 148, 193, 194].
Despite numerous proof-of-principle demonstrations, however, no spin squeezing experiment todate has achieved a practical metrological advantage, and current platforms relying on infinite-range
interactions face a host of technical and fundamental difficulties that will require new breakthroughs
to overcome.
The Ising model with power-law interactions that fall off with distance r as 1/rα generates
squeezing that scales with system size when α < D in D spatial dimensions [195], which is highly
desirable for metrological applications. Conversely, the power-law Ising model generates only aconstant amount of squeezing that is independent of system size when α > D. In practice, only

166
a limited number of platforms can achieve long-range spin interactions (α < D), making it highly
desirable to shed light on the possibilities for scalable spin squeezing with short-range interactions(α > D), which encompasses e.g. super-exchange, dipolar, Van der Waals, and far-detuned phononmediated interactions.
Motivated by the intuition (echoed in Refs. [2, 113, 196–199]) that adding spin-exchangeinteractions to the Ising model should energetically protect collective behavior reminiscent of the
OAT model, in this work we investigate the spin squeezing properties of the power-law XXZ model,
whose ground-state physics was studied in Ref. [200]. Remarkably, we find a broad range ofparameters for which the power-law XXZ model nearly saturates the amount of squeezing generated
in the infinite-range (α = 0) limit. Even when interactions are short-ranged (α > D), we observe alarge region of collective squeezing behavior in which the amount of achievable spin squeezing grows
with system size. This region extends through to the α → ∞ limit of nearest-neighbor interactions.
Our work opens up a new prospect of spin squeezing in variety of cold atomic, molecular, and optical(AMO) systems, including ultracold neutral atoms [28, 92], Rydberg atoms [201, 202], electric andmagnetic dipolar quantum gasses [203–206], and trapped ions [191, 207].

6.2

Background and theory
We begin with a brief review of spin squeezing and the OAT model, described by the Ising

Hamiltonian
HOAT = χNX

sz,i sz,j = χSz2 ,

(6.1)

i,j=1

where χ is the OAT squeezing strength; the spin-z operator sz,i ≡ σz,i /2 is defined in terms of theP
Pauli-z operator σz,i on spin i; and Sz ≡ N
i=1 sz,i is a collective spin-z operator. Eigenstates ofHOAT can be classified by a (nonnegative) total spin S ∈ N/2, N/2 − 1, · · · , and a projection
mz ∈ {S, S − 1, · · · , −S} of spin onto the z axis. The manifold of all states with maximal total spin
S = N/2 (e.g. spin-polarized states) is known as the Dicke manifold [35]. Equivalently, the Dicke
manifold consists of all permutationally symmetric states that do not distinguish between underlying

167
z

t=0

t = 13 tOAT
opt

t = tOAT
opt

Figure 6.1: Representations of the state ψ(t) of N = 40 spins initially polarized along the equator,and evolved under the OAT Hamiltonian for a time t up to the optimal OAT squeezing time
2/3 . Darker colors at a point n̂ on the sphere correspond to a larger overlap Q
χtOAT
ψ(t) (n̂) ≡opt ∼ 1/N
2|hn̂|ψ(t)i| , where |n̂i is a state in which all spins are polarized along n̂.

spins. States in the Dicke manifold can be represented by distributions on a sphere, whose variancesalong different axes must satisfy an appropriate set of quantum (Heisenberg) uncertainty relations(see Figure 6.1). In the case of a single (two-level) spin, this distribution has a fixed, Gaussian-likeshape that is uniquely characterized by its orientation. Identifying the peak of this distribution
recovers the representation of a qubit state by a point on the Bloch sphere. For N > 1 spins,
meanwhile, this distribution can acquire additional structure with metrological utility.
Given an initial state of N spins polarized along the equator, represented by a Gaussianlike distribution on a sphere, the net effect of the OAT Hamiltonian is to shear this distribution,
resulting in a squeezed state with a reduced variance (∆φ)2 along some axis. This reduced varianceallows for an enhanced measurement sensitivity to rotations of the collective spin state along thesqueezed axis, at the expense of a reduced sensitivity to rotations along an orthogonal axis. Spinsqueezing can be quantified by the maximal gain in angular resolution ∆φ over that achieved by aspin-polarized state [35],ξ2 ≡(∆φmin )2∆φpolarized⊥2 = min var(Sφ ) ×
φN,|hSi|2(6.2)

where S ≡ (Sx , Sy , Sz ) is a vector of collective spin operators; the operator Sφ⊥ ≡ S · n̂⊥
φ is theprojection of S onto an axis n̂⊥φ parameterized by an angle φ in the plane orthogonal to the mean
spin vector hSi; and var(O) ≡ hO2 i − hOi2 denotes the variance of O. A spin squeezing parameterξ 2 < 1 implies the presence of many-body entanglement [208] that enables a sensitivity to rotations
beyond that set by classical limits on measurement precision [31]. The OAT model can prepare

168
squeezed states with ξ 2 ∼ 1/N 2/3 , whereas the fundamental (Heisenberg) limit imposed by quantum
mechanics is ξ 2 ∼ 1/N [31].
To accommodate for the fact that physical interactions are typically local, the OAT Hamiltonian in Eq. (6.1) can be modified by the introduction of coefficients 1/|ri − rj |α in the coupling
between spins i, j at positions ri , rj , resulting in the power-law Ising model. The introductionof non-uniform couplings means that the power-law Ising model breaks permutational symmetry,coupling the Dicke manifold of permutationally symmetric states with total spin S = N/2 to asymmetric states with S < N/2, and thereby invalidating the representation of squeezing dynamics
shown in Figure 6.1. The leakage of population outside the manifold of permutationally symmetricstates can be energetically suppressed by the additional introduction of spin-aligning si · sj interactions, where si ≡ (sx,i , sy,i , sz,i ) is the spin vector for spin i. In total, we thus arrive at an XXZ
model described by the Hamiltonian
HXXZ =X J⊥ si · sj + (Jz − J⊥ )sz,i sz,j|ri − rj |αi6=jWhen interactions are uniform, α = 0, theP

i6=j si · sj

.

(6.3)

∼ S 2 = S (S + 1) term in Eq. (6.3) isa constant of motion within manifolds of definite total spin S, resulting in an OAT model with
χ = Jz − J⊥ .
When Jz − J⊥ = 0, the XXZ model contains only the spin-aligning si · sj terms, and ifinteractions are long-ranged, α ≤ D, then the Dicke manifold is gapped away from all orthogonalstates by a non-vanishing energy difference ∆gap & |J⊥ | (see Appendix 6.A). As a consequence,
for any finite N and α ≤ D there exists a non-vanishing range of coupling strengths Jz ≈ J⊥ forwhich a perturbative treatment of the anisotropic Ising terms in Eq. (6.3) is valid. In this case,the XXZ model becomes precisely the OAT model at first order in perturbation theory, with asqueezing strength χeff = hα (Jz − J⊥ ), where hα is the average of 1/|ri − rj |α over all i 6= j.
If interactions are short-ranged with α > D, then generally ∆gap → 0 as N → ∞, formallyinvalidating perturbation theory for any Jz at sufficiently large N . Nonetheless, the spin-aligningterms of the XXZ model can still enable a non-perturbative emergence of “collective” behavior

169
resembling perturbative, gap-protected OAT. We numerically explore the prospect of spin squeezing
with short-ranged interactions in the following section, finding that squeezing comparable to OAT
may be possible with a wide range of α and Jz , including the α → ∞ limit of nearest-neighborinteractions.

6.3

Results
Whereas the quantum Ising model is exactly solvable [120, 209], the XXZ model in Eq. (6.3)

is not. We therefore investigate the spin squeezing properties of the XXZ model using the discrete
truncated Wigner approximation (DTWA) [129] for N = 4096 = 642 = 163 spins, focusing on thecase of two (D = 2) and three (D = 3) spatial dimensions (see Appendix 6.B for D = 1, where
our main results are less striking but still hold). DTWA has been shown to accurately capture thebehavior of collective spin observables in a variety of settings [129, 210], and we provide additional
benchmarking of DTWA for the XXZ model on lattices of up to 7×7 spins in Appendix 6.C, although
it will ultimately be up to experiments to verify our findings. Our main results are summarized
in Figure 6.2, in which we explore the squeezing behavior of XXZ model in Eq. (6.3) around theisotropic (Heisenberg) point at Jz = J⊥ by varying both Jz /J⊥ and the power-law exponent α.

2 ≡ min ξ 2 (t) = ξ 2 t
Specifically, we examine (i) the optimal squeezing parameter ξoptt
opt , (ii) the
minimal squared magnetization throughout squeezing dynamics, hS 2 imin ≡ mint≤topt hS 2 i (t), and(iii) the optimal squeezing time topt .
First and foremost, Figure 6.2 confirms the theoretical argument that OAT-limited squeezing
should be achievable with any power-law exponent α ≤ D for some non-vanishing range of Ising
couplings, Jz ≈ J⊥ . Moreover, when α ≤ D we observe that this capability persists well beyond
the perturbative window with |Jz − J⊥ |  |J⊥ |, covering all Jz /J⊥ < 1 shown in Figure 6.2 and anincreasing range of Jz /J⊥ > 1 as α → 0. Even more strikingly than the behavior at α ≤ D, Figure
6.2 shows that squeezing well beyond the Ising limit can still achievable for a wide range of Ising
couplings Jz /J⊥ < 1 when interactions are short-ranged, α > D. In a nearest-neighbor XXZ model(α → ∞), the region |Jz | < |J⊥ | corresponds to the equilibrium XY phase, whereas Jz /J⊥ < −1

170

S-I

S-Ising

20
15
10

S-collective

S-collective

5

2−10 log10 ξopt

S-Ising

1.0

∞
6
5
4
3
2
1

α

0.8
0.6
0.4

∞
6
5
4
3
2
1

hS 2 imin / hS 2 i0

S-Ising

D=3

101

α

100
10−1

topt × J⊥

D=2

α

∞
6
5
4
3
2
1

10−2
-2

0
Jz /J⊥2-2

0
Jz /J⊥

2

2
Figure 6.2: The optimal squeezing ξopt(top), minimal squared magnetization hS 2 imin (middle),
and optimal squeezing time topt (bottom) for N = 4096 = 642 = 163 spins in D = 2 (left) and D = 3(right) spatial dimensions. Spins are initially polarized along the equator and evolved under the
2 is shown in decibels, and hS 2 i
XXZ Hamiltonian in Eq. (6.3). Squeezing ξopt
min is normalized to its


N N
2
initial value hS i0 = 2 2 + 1 . Dashed grey lines mark α = D, and dotted grey lines track local

minima of hS 2 imin , marking the boundary between regions of collective and Ising-limited squeezing
dynamics, respectively denoted “S-collective” and “S-Ising”. Other markers in the middle panels
indicate vales of Jz /J⊥ , α, D that are currently accessible with neutral atoms [211, 212] (cyan line),
Rydberg atoms [201, 202, 213] (red dots), polar molecules [203, 204, 214] (green line), magnetic
atoms [205, 206] (pink square), and trapped ions [191] (blue line). DTWA results are averaged over
500 trajectories.

171
and Jz /J⊥ > +1 correspond to the equilibrium Ising ferromagnet and anti-ferromagnet phases(depending on the sign of J⊥ ) [200, 215]. The asymmetry about Jz = J⊥ in Figure 6.2 thus hints
at an interesting connection between equilibrium physics [200] and far-from-equilibrium dynamical
behavior of the XXZ model (discussed further in the next section)a .
Though the attainable amount of squeezing generally decreases with shorter range (increasingα) and stronger anisotropy (decreasing Jz /J⊥ < 1), a region of “collective” squeezing behavior
connected to the OAT limit persists through to the α → ∞ limit of nearest-neighbor interactions.
This region is reminiscent of the 23 D ≤ α < D region of the power-law Ising model (J⊥ = 0), in
which squeezing falls short of the OAT limit, but still grows with system size [195].
In fact, the transition between collective and Ising-limited squeezing regions, which we respectively denote “S-collective” and “S-Ising” (with an “S-” prefix to emphasize the role of squeezing intheir characterization), is marked by a discontinuous change in both the minimal squared magnetization hS 2 imin and the optimal squeezing time topt , signifying the presence of a dynamical phasetransition. The dynamical phases in question can be characterized by the behavior of optimal
2 , which either scales with system size or saturates to a constant value. We discusssqueezing ξopt

and clarify these points below.
The discontinuity in optimal squeezing time topt at the dynamical phase boundary in Figure
6.2 is the result of a competition between local optima in squeezing over time, shown in Figure 6.3.
Large amounts of spin squeezing are generated in the S-collective phase near the isotropic pointat Jz = J⊥ . The amount of squeezing generated by collective dynamics falls off away from theisotropic point, until it finally drops below an “Ising” squeezing peak that is generated at much
short times, resulting in a discontinuous change in the time at which squeezing is optimal. The
discontinuous change in the optimal squeezing time is in turn responsible for the sudden change
in the minimal squared magnetization hS 2 imin , which has less time to decay in the Ising-limited(S-Ising) regime.
a

In fact, when J⊥ < 0 the S-collective region at Jz /J⊥ < 1 is contained within the ground-state XY phase of thepower-law XXZ model [200].

S-collective

S-Ising

0.5−2

hS 2 i / hS 2 i0

−10 log10 ξ 2

24
18
12
6
0

−1

0 2 4 6 8 10
t × |Jz − J⊥ |0.0
0 2 4 6 8 10
t × |Jz − J⊥ |Jz /J⊥

172
1.0

−3

Figure 6.3: Squeezing ξ 2 and squared magnetization hS 2 i over time for the power-law XXZ modelwith α = 3 on a 2D lattice of 64 × 64 spins. Color indicates the value of Jz /J⊥ , and red lines (atJz /J⊥ = −2.2) mark the approximate transition between S-collective and S-Ising phases, when the“collective” squeezing peak at τ ≡ t × |Jz − J⊥ | ∼ 6 drops below the “Ising” peak at τ ∼ 1. For
the parameters shown, hS 2 i reaches a minimum at τ ∼ 2, which means that optimal squeezing atτ ∼ 1 is reached before maximal decay of hS 2 i in the S-Ising phase.

It is no surprise that quantities such as topt and hS 2 imin that are defined via minimization
exhibit discontinuous behavior, and these discontinuities do not by themselves indicate a transition
between different phases of matter. We can formally distinguish the S-collective and S-Ising phasesby examining the nature of squeezing that is generated in these regions. Specifically, the S-Ising
phase generates an amount of squeezing that is insensitive to system size, whereas the S-collective
2
phase generates an amount of squeezing that scales with system size as ξopt
∼ 1/N ν , where theexponent ν generally depends on the values of α and Jz /J⊥ (see Appendix 6.D). Numerically,
we find that the S-collective phase spans all Jz /J⊥ < 1 when α . D, whereas the transition
between S-collective and S-Ising phases occurs at a critical Ising coupling Jzcrit that either diverges
logarithmically with system size (Jzcrit ∼ − log N ) or stays constant when α & D (see Figure 6.4,
where we focus on D = 2 and α = 3 due to its experimental relevance, and Appendix 6.D). We
note that small oscillations in squeezing over time (see Figure 6.3) add minor corrections to the
2 and J crit . These oscillations are responsible for the discontinuous behavior of t
behavior of ξopt
opt
z

and hS 2 imin seen in Figure 6.2 within the S-collective phase.

173

20

L

50
40
20
10

15

S-collective

30

10
S-Ising-2

S-I
0
Jz /J⊥

2−10 log10 ξopt

60

2

2 as a function of system size for the power-law XXZ model with
Figure 6.4: Optimal squeezing ξopt
α = 3 on a 2D lattice of N = L × L spins. Whereas the amount of squeezing generated in the
S-Ising phase is insensitive to system size, squeezing in the S-collective phase grows with systemsize and as Jz /J⊥ → 1 (from below). Dotted grey line tracks minima of hS 2 imin as a function ofJz /J⊥ , as in Figure 6.2, marking the approximate dynamical phase boundary.

174

6.4

Discussion
The mechanism behind the collective dynamics featured by the XXZ model far from theisotropic point at Jz = J⊥ is not obvious, and lies in a parameter regime beyond the reach of exact
treatment with current theoretical capabilities. While an in-depth understanding of collective
dynamics will most likely require experimental investigations in the spirit of quantum simulation,
we discuss possible phenomenological explanations below.
Collective squeezing behavior when α < D is the least surprising, as the XXZ model essentially interpolates between perturbative, gap-protected OAT (near Jz = J⊥ ) and the long-range
power-law Ising model (at Jz → ±∞), both of which generate collective spin squeezing. When
α > D, as long as D > 2 or α < 2D (i.e. all α > 3 when D = 3, and 2 < α < 4 when D = 2)
a generalized version of the Mermin-Wagner theorem [216] allows for the existence of long-rangeorder in the thermodynamic limit, below a critical temperature [197, 198]. Our observations may
therefore be indicative of thermalization to a long-range-ordered steady state in an equilibrium
XY phasea , with significant amounts of collective spin squeezing present in the transient dynamics.
This explanation is supported by the fact that the squared magnetization hS 2 i approaches a nonvanishing steady-state value in Figure 6.3 (see also Appendix 6.E). Nevertheless, the persistenceof long-range order is a necessary but insufficient condition to characterize the types of dynamical phases considered in this work. Instead, these phases are defined operationally by whether
attainable spin squeezing scales with system size, and are thus sensitive to transient effects.
For even shorter range interactions (α ≥ 2D) when D ≤ 2, long-range order is forbidden in
the steady state. Even so, a spin-polarized initial state can still take an appreciable amount of timeto thermalize to a disordered steady state. Squeezing beyond the Ising limit can therefore occur asa transient phenomenon, before long-range order is disrupted (see Appendix 6.E).

175

6.5

Experimental applications
As indicated in Figure 6.2, our results are readily applicable to the generation of spin squeezed

states in a variety of experimental platforms that have been shown to implement the power-law
XXZ model, including neutral atoms (α → ∞) [211, 212], Rydberg atoms (α = 3, 6) [201, 202,213], polar molecules (α = 3) [203, 204, 214], and magnetic atoms (α = 3) [205, 206]. Note that
one may additionally have to consider the effects of a sub-unit filling fraction on the realization of aspin model. In principle, sub-unit filling introduces effective disorder into the XXZ spin couplings[198, 217]. Nonetheless, the precise form of these interactions is not essential to the existence of an
S-collective phase in the XXZ model, as evidenced by the fact that this phase persists through tothe α → ∞ limit of nearest-neighbor interactions (see Appendix 6.F).
Finally, we discuss the application of our results to Ising systems without 3D spin-aligningsi · sj interactions, as in the case of some Rydberg atom (α = 3, 6) [201, 202] and trapped ion(0 ≤ α < 3) [191] experiments. In this case, 2D spin-aligning interactions within the y-z plane canstill be engineered by the application of a strong transverse driving field ΩSx . If the drive strength|Ω|  12 N hα |Jz |, with hα the mean of 1/|ri − rj |α over all i 6= j, then moving into the rotating
frame of the drive and eliminating fast-oscillating terms results in an XX model described by the
HamiltonianHXX =Jz X sy,i sy,j + sz,i sz,j,2
|ri − rj |α

(6.4)

i6=j

which is a special case of the XXZ model in Eq. (6.3), with (J⊥ , Jz ) → (Jz /2, 0). Ising systems witha strong transverse field can thus access a vertical cut along Jz /J⊥ = 0 in Figure 6.2. In a similar
fashion, dynamic Hamiltonian engineering protocols [218, 219] can transform the Ising model into
an XXZ model with any Jz /J⊥ ≥ 0, albeit at the cost of added complexity.

Acknowledgments
We thank Andrew Lucas, Thomas Bilitewski, Sean R. Muleady, and Jeremy T. Young forhelpful technical discussions. This work is supported by the DARPA DRINQs grant, the ARO single

176
investigator award W911NF-19-1-0210, NSF grant PHY-1820885, AFOSR grant FA9550-19-1-0275,
NSF grant PHY-1734006 (JILA-PFC), and by NIST.

6.A

Spectral gap of the long-range XXX model
Here we show that the isotropic (Jz = J⊥ ) XXZ model in Eq. (6.3) of the main text has aspectral gap when α ≤ D, which implies the existence of a finite range of Ising couplings Jz ≈ J⊥
for which the XXZ model formally recovers the OAT model at first order in perturbation theory.
For definiteness, we consider an isotropic XXZ model on a cubic lattice with periodic boundary
conditions in D dimensions. The translational and SU(2) symmetries of the isotropic XXZ modelon such a lattice imply that its lowest-lying excitations can be written as spin waves of the form|mz , ki ∝Xeik·n sz,n |mz i ,

(6.5)

n∈ZD
L

where |mz i is a permutationally-symmetric Dicke state with spin projection mz onto the z axis,n = (n1 , n2 , · · · , nD ) indexes an individual site on the lattice of N = LD spins, and k ∈ ZD
L × 2π/L
is a wavenumber. The energy of the state |mz , ki with respect to the isotropic XXZ Hamiltonian is
Ek = −J⊥

X 1 − cos (k · n)
|n|α

n∈ZD
L

,

(6.6)

|n|6=0where for simplicity we work in units for which the lattice spacing is 1. The energy Ek is minimized(in magnitude) by a wavenumber that underdoes one oscillation across one axis of the lattice,e.g. k = 2π/L, 0, 0, · · · , which implies a spectral gap∆gap = |J⊥ |
X 1 − cos 2πn1 /L
|n|α

n∈ZD
L
|n|6=0.

(6.7)

Letting  ≡ 2/L, we define a rescaled domain S = ZL / ⊂ [−1, 1], and substitute x = n to get∆gap = |J⊥ |α−D

X
x∈SD

|x|≥

D

1 − cos (πx1 )
,
|x|α

(6.8)

177
which in the thermodynamic limit  → 0 is well approximated by an integral that avoids an
infinitesimal region at the origin,∆gap → |J⊥ |α−DZ
ID () ,ID () ≡

dD x

D
TD
1 \T

1 − cos (πx1 )
,
|x|α

(6.9)

where Ta ≡ (−a, a) is a symmetric interval about 0. The integrand of ID () is strictly positive andwell-behaved on the entirety of its domain except for the origin, where depending on the value of
α the integrand may vanish or diverge as |x| → 0. Together, these facts mean that
→0ID () ∼ −γ ,→0∆gap ∼ α−(D+γ) ,(6.10)

for some γ ≥ 0, which implies that ∆gap > 0 when α ≤ D.

6.B

Numerical results in one spatial dimension
Here we provide additional DTWA simulation results for the squeezing behavior of the power-

law XXZ model in D = 1 spatial dimension. Figure 6.5 shows results analogous to those in Figure
6.2 of the main text, for D = 1, 2, 3 spatial dimensions and integer values of the power-law exponent
α (as well as the α → ∞ limit of nearest-neighbor interactions). The existence of an S-collective
dynamical phase persists in one spatial dimension, but for a much narrower range of parametersthan in the case of D = 2 and 3. The achievable squeezing in the S-collective phase also scales lessfavorably with system size in the case of D = 1. Nonetheless, squeezing beyond the Ising limit isstill achievable in D = 1 with e.g. Jz = 0 and α > 1, which is relevant for trapped ion experiments.

178

D=1

∞

D=2

D=3

15

4

10

3
2

5

1
∞

1.0

5

0.8

α

6
4

0.6

3
2
1

0.4

∞

102

5

101

hS 2 imin / hS 2 i0α52−10 log10 ξopt

20

6

4

100

3
2

topt × J⊥

α

6

10−1

1
-2

0
Jz /J⊥2-2

0
Jz /J⊥2-2

0
Jz /J⊥

2

2
Figure 6.5: The optimal squeezing ξopt(top), minimal squared magnetization hS 2 imin (middle),
and optimal squeezing time topt (bottom) for N = 4096 = 642 = 163 spins in D = 1, 2, 3 spatial
dimensions. Spins are initially polarized along the equator and evolved under the XXZ Hamiltonianin Eq. (6.3) of the main text. The results for D = 2 and 3 shown here are a subset of the resultsin Figure 6.2, presented in the same format as that for D = 1 for comparison.

179

6.C

Benchmarking DTWA for the power-law XXZ model
In order to gauge the reliability of DTWA for the XXZ model in this work, we benchmark

against truncated shell (TS4 ) simulations of a 7 × 7 spin lattice whose dynamics are restricted tothe subspace of ∼ N 5 states with definite total spin S ≥ N/2 − 4. These simulations are motivatedby the idea that spin-aligning si · sj interactions energetically suppress the decay of total spin S
from its initial value of N/2 in a spin-polarized state. As long as the total spin decay is small,TS4 simulations should faithfully capture the dynamical behavior of a system. The restriction tosmall total spin decay implies that TS4 simulations are only reliable near the isotropic point of the
XXZ model at Jz = J⊥ , and the O(N 5 ) memory footprint of TS4 means that it can only be used
to simulate moderately-sized systems. Nonetheless, TS4 has the advantage over DTWA of being“self-benchmarking,” in the sense that its breakdown can be diagnosed by a large population of
the S = N/2 − 4 manifold, which indicates further population leakage into truncated states with
S < N/2 − 4 (see Figure 6.6).
We benchmark DTWA simulations against TS4 in Figure 6.7 by comparing two observables
2 ≡ min ξ 2 (t) = ξ 2 (tof interest: (i) the optimal spin squeezing parameter ξoptt
opt ), and the minimalvalue of hS 2 i throughout squeezing dynamics, hS 2 imin ≡ mint≤topt hS 2 i (t). For reference, Figure
2 and hS 2 i
6.7 also shows the values of ξopt
min in the exactly solvable limits of uniform (OAT, α = 0)

and power-law Ising (J⊥ = 0) interactions. For initially spin-polarized states, these limits have
only one relevant energy scale, Jz − J⊥ , so the only effect of changing Jz is to change dynamical
time scales.
The results in Figure 6.7 show that DTWA agrees almost exactly with TS4 in the regimesthat TS4 can be trusted, suggesting that DTWA is a reliable method for studying the spin squeezing
behavior of the XXZ model. Values of squeezing −10 log10 ξ 2 > 0 are highly sensitive to errors incollective spin observables, so when comparing DTWA and TS4 one should expect more pronounced(albeit minor) disagreements in spin squeezing −10 log10 ξ 2 than in squared magnetization hS 2 i.
Also, for clarity we used a simple heuristic to identify regimes of validity for TS4 in Figures 6.6 and180

population

0.8

hP1 imax

0.6

hP2 imax

0.4

hP4 imax

hP3 imax

0.2
0.0
−1

0

1

2

Jz /J⊥

Figure 6.6: Maximal populations hPn imax of the total spin S = N/2 − n manifolds Pn throughout
squeezing dynamics of 7 × 7 spins, initially polarized along the equator and evolved under the XXZ
Hamiltonian in Eq. (6.3) of the main text with a power-law exponent α = 3. Computed with TS4
simulations and periodic boundary conditions. Shaded regions indicate hP4 imax > 0.1, where TS4
results cannot be trusted due to the likeliness of population leakage into truncated states. All statesin P1 break translational invariance, so the initial population hP1 i0 = 0 is protected by the absenceof translational symmetry-breaking terms in the Hamiltonian. The population hP3 i,meanwhile, issmall because P3 is only coupled to P2 and P4 by matrix elements that are O 1/N smaller thanthe couplings between P0 ↔ P2 ↔ P4 .

1.0

9hS 2 imin / hS 2 i0

2−10 log10 ξopt

10
TS4
DTWA
OAT
Ising

8
7

0.9
0.8

6

0.7
−1

0

1
Jz /J⊥

2

−1

0

1

2

Jz /J⊥

2 (top) and minimal squared magnetization hS 2 i
Figure 6.7: Optimal squeezing ξopt
min throughout
squeezing dynamics (bottom) as computed via TS4 and DTWA in the same setting as Figure 6.6,
2
likewise with shaded regions indicating hP4 imax > 0.1 in the TS4 simulations. Here squeezing ξopt

is shown in decibels, and hS 2 imin is normalized to its initial value hS 2 i0 = N2 N2 + 1 . Dashed anddotted lines respectively mark the exactly solvable limits of uniform (OAT, α = 0) and power-law
Ising (Ising, J⊥ = 0) interactions.

181
6.7. This heuristic is not intended to be a precise indicator of quantitative accuracy for TS4 , so it
is no surprise that it does not identify the precise values of Jz at which DTWA and TS4 diverge.
Finally, Figure 6.8 shows comparisons of DTWA with exact simulations in 2D lattices of 3 × 3
2 and minimal squared magnetizationand 4 × 4 spins. Though the optimal squeezing parameter ξopt

hS 2 imin saturate to finite-size values fairly quickly away from the isotropic point at Jz = J⊥ , exact
simulations clearly show a collective region with OAT-limited behavior when Jz ≈ J⊥ . Even on
2 and hS 2 i
small lattices, DTWA does a reasonably good job of estimating ξopt
min . Notably, DTWA

performs better with increasing system size, as can be seen by comparing benchmarks of DTWA
in 3 × 3, 4 × 4, and 7 × 7 systems, shown in Figures 6.7 and 6.8. This finding is consistent with an
ongoing study to benchmark DTWA against state-of-the-art simulations of matrix product states(MPS) using the time-dependent variational principle (TDVP) [220].

182

3×3

7

OAT
1
2

2
−10 log10 ξopt

6

4×4

3
4
5

5
4
3

hS 2 imin / hS 2 i0

1.0

0.9

0.8
−1

0

1
Jz /J⊥

2

−1

0

1

2

Jz /J⊥

2 (top) and minimal squared magnetization hS 2 i
Figure 6.8: Optimal squeezing ξopt
min throughout
squeezing dynamics (bottom) on 2D lattices of 3 × 3 (left) and 4 × 4 (right) spins, as computed byexact methods (solid lines) and DTWA (dots). The color of each marker indicates the corresponding
value of α, as specified in the legend, and the dashed line marks the OAT limit of α = 0.

183

6.D

Scaling relations for the collective phase in D = 2 spatial dimensions
Here we inspect the results in Figure 6.4 of the main text, as well as similar results fordifferent exponents α of the power-law XXZ model, to show that2(i) optimal squeezing scales as ξopt
∼ 1/N ν in the S-collective dynamical phase (Figure 6.9

and Table 6.1), and(ii) the critical Ising coupling Jzcrit at the boundary between S-collective and S-Ising phases
either diverges logarithmically with system size (Jzcrit ∼ − log N ), or remains essentiallyconstant when α & D (Figure 6.10).
2
The exponent ν governing the behavior of ξopt
will generally depend on the values of Jz /J⊥ andα. Similarly, the precise dependence of Jzcrit on N will depend on the value of α. Note that all
DTWA simulations of N -spin systems throughout this work average over 500 × 642 /N trajectories,
i.e. with 500 trajectories (samples of the initial state) for the largest system size, and ∼ 1/N scaling
to account for the fact that DTWA results converge more slowly in smaller systems. We find thatchanging these trajectory numbers does not affect our overall results and conclusions. Nonetheless,precise quantitative predictions, such as the exact value of ν as a function of system size, may bebeyond our current computational capabilities, since they might require a more extensive numericalanalysis to rule out finite sampling errors, or corrections from quantum correlations that are notcaptured by DTWA.

184

15

2
−10 log10 ξopt

2
−10 log10 ξopt

DTWA
fit
OAT

20

20
15
10

10
102

103

102

103

N

N

(a) α = 3(b) α = 42
−10 log10 ξopt

2
−10 log10 ξopt

20

20

15

15

10

10
102

5

103

102

103

N

N

(c) α = 5(d) α = 62
−10 log10 ξopt

20

10

102

103
N

(e) α = ∞2
Figure 6.9: Dependence of the optimal squeezing parameter ξopt
on system size N within thecollective dynamical phase of the power-law XXZ model in D = 2 spatial dimensions. Color
indicates the value of Jz /J⊥ , sweeping down from +0.5 (dark purple, top) to −1.5 (yellow, bottom)
in increments of −0.5. Circles show results computed with DTWA; dashed lines show a fit to
2 = a/N ν with free parameters a, ν; and the dotted red line marks the OAT limit for reference.
ξopt
The DTWA results in panel (a) for α = 3 are a subset of those in Figure 6.4 of the main text.

185

Jzcrit /J⊥−0.4
Jzcrit /J⊥

DTWA
fit

−1.0

−0.6

−1.5
−2.0
10

2

10

−0.8

3

103

N

N

(a) α = 3(b) α = 4

Jzcrit /J⊥−0.2

Jzcrit /J⊥

−0.3

−0.4

−0.5

102

−0.3

102

−0.4

103

102

103

N

N

(c) α = 5(d) α = 6

Jzcrit /J⊥

0.0

−0.1

−0.2

102

103
N

(e) α = ∞Figure 6.10: Dependence of the critical Ising coupling Jzcrit at the collective-to-Ising dynamicalphase boundary on system size N for the power-law XXZ model in D = 2 spatial dimensions.
Circles show results computed with DTWA, and dashed lines show a fit to Jzcrit /J⊥ = −γ ln N + b
with free parameters γ, b. The DTWA results in panel (a) for α = 3 are equivalent to the dashedgrey lines in Figure 6.4 of the main text. DTWA simulations were run with values of Jz /J⊥ thatare integer multiples of 0.1, placing a lower bound on the resolution for Jzcrit /J⊥ .

186

Jz /J⊥

α

3
4
5
6
∞

−1.5
1.0
–
–
–
–

−1.0
0.9
–
–
–
–

−0.5
0.8
0.3
–
–
–

+0.0
0.8
0.3
0.2
0.2
0.1

+0.5
0.7
0.4
0.2
0.1
0.1

2 ∼ 1/N ν ) for the values of J /J and α shown in Figure
Table 6.1: Scaling exponents ν (with ξoptz⊥
6.9, in D = 2 spatial dimensions. Though provided here for the sake of practical interest andtransparency (these are essentially the slopes of the dashed lines Figure 6.9), we note that thesevalues are subject to correction in future work, as ruling out effects such as finite sampling errors
may require a more extensive numerical analysis.

6.E

Thermalization and long-range order
Here we provide time-series DTWA results, similar to those of Figure 6.3 of the main text, toshow that the S-collective phase is compatible with thermalization to a long-range-ordered state of
the power-law XXZ model when D > 2 or α < 2D. To this end, Figures 6.11 and 6.12 show bothsqueezing ξ 2 and the squared magnetization hS 2 i as a function of time for N = 4096 = 642 = 163
spins in D = 2 spatial dimensions with α ∈ {2D − 1, 2D, 2D + 1} = {3, 4, 5}, as well as D = 3
spatial dimensions with α ∈ {2D − 1, 2D, ∞} = {5, 6, ∞}. Figure 6.11 shows simulations withvalues of Jz /J⊥ that sweep from 0 (in the S-collective phase) to −3 (in the S-Ising phase), while
Figure 6.12 shows simulations with values of Jz /J⊥ that sweep from 2 (in the S-Ising phase) to 0 (inthe S-collective phase). As long as D > 2 or α < 2D, the squared magnetization hS 2 i approaches
a nonzero steady-state value when Jz /J⊥ < 1, indicating thermalization to a steady state withlong-range order.

187

α=4

α=5

0.0

−10 log10 ξ 2

α=3
20
15
10
5
0
1.0

−1.0

hS 2 i / hS 2 i0

−1.5

Jz /J⊥

−0.5

−2.0

0.5

−2.5

0.0
0

2

4
6
8
t × |Jz − J⊥ |

10 0

2

4
6
8
t × |Jz − J⊥ |

10 0

2

4
6
8
t × |Jz − J⊥ |

10

−3.0

(a) D = 2 spatial dimensions

α=∞

α=6

−10 log10 ξ 2

α=5

0.0

20
15
10
5
0
1.0

−1.0

hS 2 i / hS 2 i0

−1.5

Jz /J⊥

−0.5

−2.0

0.5

−2.5

0.0
0

2

4
6
8
t × |Jz − J⊥ |

10 0

2

4
6
8
t × |Jz − J⊥ |

10 0

2

4
6
8
t × |Jz − J⊥ |

10

−3.0

(b) D = 3 spatial dimensions

Figure 6.11: Squeezing ξ 2 and squared magnetization hS 2 i as a function of time t for N = 4096 =642 = 163 spins in D = 2 and 3 spatial dimensions. Color indicates the value of Jz /J⊥ , and the
red line highlights behavior at the value of Jz /J⊥ immediately preceding the transition from the
S-collective phase (above the red line) to the S-Ising phase (below the red line).

188

α=4

α=5

2.0

−10 log10 ξ 2

α=3
24
18
12
6
0
1.0

hS 2 i / hS 2 i0

1.0

Jz /J⊥

1.5

0.5

0.5
0.0

0.0
0

2

4
6
8
t × |Jz − J⊥ |

10 0

2

4
6
8
t × |Jz − J⊥ |

10 0

2

4
6
8
t × |Jz − J⊥ |10(a) D = 2 spatial dimensions

α=∞

α=6

−10 log10 ξ 2

α=5

2.0

24
18
12
6
0
1.0

hS 2 i / hS 2 i0

1.0

Jz /J⊥

1.5

0.5

0.5
0.0

0.0
0

2

4
6
8
t × |Jz − J⊥ |

10 0

2

4
6
8
t × |Jz − J⊥ |

10 0

2

4
6
8
t × |Jz − J⊥ |10(b) D = 3 spatial dimensions

Figure 6.12: Same results as in Figure 6.11, but for values of Jz /J⊥ that cross the dynamical phaseboundary at Jz /J⊥ = 1. The red line highlights behavior at Jz /J⊥ = 1.1, immediately preceding
the transition from the S-Ising phase (Jz > 1) to the S-collective phase (Jz < 1).

189

6.F

Sub-unit filling fractions
Though we do not study the effect of variable filling fractions in detail, here we show that

the S-collective phase is stable to filling fractions f < 1. To this end, in Figure 6.13 we show the2
dependence of the optimal squeezing parameter ξopt
on filling fraction f on a 50 × 50 lattice in

D = 2 two spatial dimensions with power-law exponent α = 3 (as in the case of polar molecules, forwhich unit filling is difficult to obtain experimentally). Optimal squeezing generally decreases withfilling fraction, which is in part attributable to a changing particle number. Nonetheless, squeezing
well in excess of the Ising limit is clearly achievable even for small filling fractions, f ∼ 0.1, as longas the XXZ model is tuned sufficiently close to the isotropic point at Jz = J⊥ .
On a high level, decreasing the filling fraction f to a value less than 1 can be seen as a
two-step process: (i) rescaling all distances as r → r/f 1/D , and (ii) adding positional disorder tospin-spin couplings, in effect transforming the XXZ Hamiltonian as
HXXZ =
X Jµ sµ,i sµ,j
X Jµ sµ,i sµ,j
f
α/D
→
×
f
1
+

ij ,
|rij |α|rij |αi6=jµ(6.11)

i6=jµ

where the index µ ∈ {x, y, z} with Jx = Jy = J⊥ , and fij are random variables that vanish (fij → 0)
as f → 1. The factor f α/D merely changes time scales, so any deviation from squeezing behavior
at f = 1 is determined by the random variables fij . The general physics of the XXZ model at unit
filling is maintained as long as these random variables are small enough to preserve the structure(connectivity) of 1/rα couplings. When f gets too small, however, the XXZ model is dominated byrandom variables, and the values of collective observables are essentially governed by the dynamics
of small spin clusters with weak inter-cluster interactions. The question remains: what filling
fraction f is “too small”?
In fact, this sort of physics was studied more closely the prior work of Ref. [198], which
examined the XX model (Jz = 0) with 1/r3 interactions (α = 3) and variable filling fractions thatwere treated as positional disorder. By mapping this system onto one of hard-core bosons, using
both mean field and numerical techniques the authors found that interactions stabilize the 1/r3

190

2
−10 log10 ξopt

20
15
0.9
0
−1
OAT

10
5
0
0.0

0.2

0.4

0.6

0.8

1.0

f

2 on filling fraction f for the XXZ
Figure 6.13: Dependence of the optimal squeezing parameter ξopt
model in Eq. (6.3) of the main text with power-law exponent α = 3 in D = 2 spatial dimensions
with 50 × 50 lattice sites. Results computed using DTWA, with a random choice of f × 50 × 50
lattice sites to occupy. The shape and color of each marker indicates the corresponding value ofJz /J⊥ , as specified in the legend, and the dotted line marks the OAT limit for reference.

191
XX model against disorder, such that a transition from order- and disorder-dominated dynamicalphases occurs at a critical filling fraction of fcrit ≈ 0.15. Our results for the generic power-law XXZ
model are consistent with previous results, although the role of Jz 6= 0 and different α remains
an open question. We suspect, for example, that the resilience to low filling fractions to worsenwith increasing α, and severely so when D ≤ 2 and α ≥ 2D, as a strengthened version of the
Mermin-Wagner theorem [216] only allows for long-range order when D > 2 or α < 2D. Eitherway, we defer a thorough analysis of this question to future work, for now merely highlighting therobustness of our main results to sub-unit filling fractions.

Chapter 7

Engineering infinite-range SU(n) interactions with spin-orbit-coupled fermionsin an optical lattice

PrologueWe began the main technical work of this thesis in Chapter 3 by considering an exotic SU(n)symmetry that naturally occurs between ultracold fermions on a lattice. Subsequent chapters
mainly focused on the special and relatively well-understood case of SU(2)-symmetric interactions, in particular considering their potential applications for quantum-enhanced metrology. The
prospect of similarly exploiting SU(n)-symmetric interactions (with n > 2) to address scientificquestions or develop practical technologies remains largely unexplored. This chapter combines thespirit of Chapter 3 with the roadmap of Chapter 4, exploring some of the physics accessible with
SU(n) interactions between multilevel fermions with spin-orbit coupling.
One of the major difficulties of dealing with SU(n) symmetries is the lack of tools and structure
for taming the vast realm of possibilities with high-dimensional quantum systems. To this end, thischapter brings an optimistic message, showing that SU(n) can share similarities with SU(2) thatenable the use of a common language (e.g. expressions that are agnostic to the value of n), and canlead to qualitatively similar dynamical behaviors. The bulk of this chapter is taken from Ref. [5].
In addition to myself and Ana Maria Rey, this work featured major contributions from Diego
Barberena, Mikhail Mamaev, Bhuvanesh Sundar, and Robert J. Lewis-Swan.

193

Abstract
We study multilevel fermions in an optical lattice described by the Hubbard model with onsite SU(n)-symmetric interactions. We show that in an appropriate parameter regime this system
can be mapped onto a spin model with all-to-all SU(n)-symmetric couplings. Raman pulses that
address internal spin states modify the atomic dispersion relation and induce spin-orbit coupling,which can act as a synthetic inhomogeneous magnetic field that competes with the SU(n) exchangeinteractions. We investigate the mean-field dynamical phase diagram of the resulting model as afunction of n and different initial configurations that are accessible with Raman pulses. Consistent
with previous studies for n = 2, we find that for some initial states the spin model exhibits two
distinct dynamical phases that obey simple scaling relations with n. Moreover, for n > 2 we findthat dynamical behavior can be highly sensitive to initial intra-spin coherences. Our predictions
are readily testable in current experiments with ultracold alkaline-earth(-like) atoms.

7.1

Introduction
SU(n) symmetries play an important role in physics. Underpinning much of high energy

physics, the SU(n) gauge theory known as Yang-Mills theory is central to our understanding of theelectroweak and strong forces. Extensions of Yang-Mills and SU(n) symmetry feature in the mostwell-studied examples of holographic duality [221] and the connection between entanglement andgravity [222] through the anti-de Sitter/conformal field theory (AdS/CFT) correspondence. In a
condensed matter setting, SU(2) appears ubiquitously as a symmetry of the Hubbard model, with
important consequences for the study of quantum magnetism and high temperature superconductivity [59]. The extension of SU(2) Hubbard and spin models to SU(n) has led to predictions of
exotic phases of matter such as valence bond solids [57, 223–225] and chiral spin liquids [56–58,
226], as well as the potential to perform universal topological quantum computation [62, 227] andother phenomena [228, 229]. Furthermore, disordered SU(n) spin models have opened analyticallytractable avenues for studying quantum chaos and information scrambling [230].

194
The tremendous theoretical significance of SU(n) symmetries makes it all the more excitingthat they appear naturally in experimental atomic, molecular, and optical (AMO) platforms withexquisite degrees of microscopic control. This symmetry arises through the independence of atomic
orbital and interaction parameters on the n nuclear spin states of alkaline-earth(-like) atoms, withe.g. n = 10 for 87 Sr [28, 29, 55, 67]. As a result, AMO experiments can directly probe the role ofSU(n) interactions in controllable settings. Recent progress includes studies of the thermodynamic
properties of SU(n) fermionic gases [231–238], SU(n) Hubbard phases and phase transitions [49, 239,
240], single- [241] and two-orbital [51–53, 242] SU(n) magnetism, and multi-body SU(n)-symmetricinteractions [1, 7].
In the spirit of quantum simulation, further investigations in controlled settings will play animportant role in understanding the consequence of SU(n) symmetries for fundamental questions inphysics, as well as their practical use in technological applications. For example, SU(2)-symmetricspin interactions can be harnessed to develop quantum sensors that surpass classical limits on
measurement precision [2, 4]. The prospect of similarly exploiting more general SU(n) symmetries toachieve a technological advantage is still an unexplored avenue of research with untapped potential.
In this work, we consider an experimentally relevant and theoretically tractable regime of theSU(n) Hubbard model, highlighting differences and similarities with the more familiar case of SU(2).
Working at ultracold temperatures and unit spatial filling (one atom per lattice site), we begin bymapping the SU(n) Hubbard model onto a multilevel spin model with all-to-all SU(n)-symmetricinteractions in Section 7.2. In Section 7.3 we consider the use of control fields to address nuclear
spins, finding a simple three-laser driving scheme that allows for the preparation of interesting stateswith nontrivial intra-spin correlations when n > 2. We consider the effect of spin-orbit coupling(SOC) induced by control fields in Section 7.4, finding in particular that the weak-SOC limit
generally gives rise to a (synthetic) inhomogeneous magnetic field, extending previously known
results to n > 2 [2, 25, 26, 103, 104, 123]. Finally, we combine these ingredients to examine
mean-field dynamical behaviors of the SU(n) spin model in Section 7.5, finding that: (i) long-timeaveraged observables obey simple scaling relations with the spin dimension n, exhibiting (for spin-

195
polarized initial states) dynamical ferromagnetic and dynamical paramagnetic phases, as previously
seen for the case of n = 2 [115, 243], and (ii) for n > 2 the long-time dynamics can be highly sensitive
to the intra-spin coherences of the initial state. We conclude and discuss future directions in Section
7.6.

7.2

From lattice fermions to an SU(n) spin model
Here we derive a collective SU(n) spin model for a system of ultracold alkaline-earth(-like)atoms trapped in an optical lattice. Without external driving fields, the evolution of such atoms
in their electronic ground state is governed by the single-body kinetic and two-body interaction
Hamiltonians
Ĥkin = −JXĉ†jµ ĉj 0 µ + h.c.,(7.1)

hj,j 0 i,µĤint =U X †ĉjµ ĉjµ ĉ†jν ĉjν ,
2

(7.2)

j,µ,ν

where hj, j 0 i denotes neighboring lattice sites j and j 0 ; µ, ν ∈ {s, s − 1, · · · , −s} index orthogonal
spin states of a spin-s nucleus, with s = n−1(e.g. s = 92 in the case of 87 Sr with 10 nuclear
2
spin states); ĉjµ is a fermionic annihilation operator, J is a tunneling amplitude (for simplicity
assumed to be the same in all directions); and U is a two-body on-site interaction energy. In thepresent work, we neglect inter-site interactions and interaction-assisted hopping, which is a good
approximation for a sufficiently deep lattice, namely when J . ER , where ER is the atom recoil
energy. For simplicity, we now assume a one-dimensional periodic lattice of L sites, and expand theon-site fermionic operators in terms of operators addressing (quasi-)momentum modes q (in unitsPwith lattice spacing a = 1), ĉjµ = √1L q e−iq·j ĉqµ , finding thatĤkin = −2JX

cos (q) ĉ†qµ ĉqµ ,

(7.3)

q,µ

Ĥint =u2NXĉ†kµ ĉ`µ ĉ†pν ĉqν × δk+p,`+q ,(7.4)

k,`,p,q,µ,ν

where N is the total number of atoms on the lattice, we define u ≡ U × N/L for convenience,δk+p,`+q = 1 if k + p = ` + q and zero otherwise (enforcing conservation of momentum).

196
If the interaction energy U is small compared to the single-particle bandwidth 4J, then themode-changing collisions in Ĥint become off-resonant, motivating the frozen-mode approximation{k, p} = {`, q} (i.e. either k = ` and p = q, or k = q and p = `)a . The terms with k = ` and p = q
u P †are 2Nĉkµ ĉkµ ĉ†pν ĉpν = 12 N u, which is a constant energy shift that we can freely neglect. Defining
the spin operators ŝµνq ≡ ĉ†qµ ĉqν , the remaining terms of the kinetic and interaction HamiltoniansareĤkin = −2JX

cos (q) ŝµµq ,

(7.5)

u X
ŝµνp ŝνµq .
2N p,q,µ,ν

(7.6)

q,µ

Ĥint = −Throughout this work, we will assume that atomic modes are singly-occupied, e.g. due to theinitialization of a spin-polarized state with one atom per lattice site, in which multiple occupationof an atomic mode is forbidden by fermionic statistics (Pauli exclusion). In this case we can simply
treat our system as N distinguishable n-level quantum spins at “lattice sites” p, q. Note that the“kinetic” terms of this spin model (Ĥkin ) are proportional to the identity operator, contributing
an overall shift in energy that we can neglect at this point. Nevertheless, these kinetic terms will
become important in the presence of an external drive, which we discuss in Section 7.4. The validityof approximating the Hubbard model in Eqs. (7.1)–(7.2) by the spin model in Eqs. (7.5)–(7.6) has
been previously benchmarked for SU(2)-symmetric interactions [2, 115], and we provide additional
benchmarking for SU(4) and SU(6) in Appendix 7.A.
To further simplify the interaction Hamiltonian Ĥint and write it in a form reminiscent of
more familiar SU(2) spin models, we now construct the operator-valued spin matrix
ŝq ≡

X

ŝµνq |µihν| ,

(7.7)

µ,ν
a

Note that the frozen-mode approximation neglects correlated momentum-hopping terms of the formĉ†π−p,µ ĉπ−q,µ ĉ†pν ĉqν , which conserve both momentum and energy. We defer a careful treatment of these terms
to future work, noting only that they vanish on the manifold of permutationally symmetric spin states with one
atom per lattice site, and that the frozen-mode approximation is benchmarked in Refs. [2, 115] and Appendix 7.A.

197
and for any pair of such operator-valued matrices Â, B̂, we define the inner productÂ · B̂ ≡X

Â†µν B̂µν .

(7.8)

µ,ν

These definitions allow us to write the spin Hamiltonian in Eq. (7.6) asĤint = −where Ŝ ≡u X
uŝp · ŝq = −Ŝ · Ŝ,
2N p,q
2N

P

(7.9)
~

q ŝq is a collective spin matrix, analogous to the collective spin vector S = (Ŝx , Ŝy , Ŝz )~ ·S~ = Ŝ 2 + Ŝ 2 + Ŝ 2 when n = 2 (here ' denotes equality
in the case of SU(2) [2], with 12 Ŝ · Ŝ ' Sxyz
up to identity terms).
We now discuss the spin Hamiltonian Ĥint in Eq. (7.9). The operator ŝp · ŝq simply swaps thenuclear spin states of two atoms pinned to modes p, q. The term −ŝp · ŝq thereby assigns a definite
energy of −1 (+1) to a pair of spins that are symmetric (anti-symmetric) under exchange. In thissense, ŝp · ŝq is analogous to the enforcement of SU(2) spin alignment by ferromagnetic interactions,which similarly assigns different energies to the anti-symmetric spin-0 singlet |↑↓i − |↓↑i and thesymmetric spin-1 triplets |↑↑i , |↓↓i , |↑↓i + |↓↑i . By summing over all pair-wise exchange termsŝp · ŝq , the interaction Hamiltonian Ĥint energetically enforces a permutational symmetry among
all spins, opening an energy gap u between the manifold of all permutationally symmetric (PS)states and the orthogonal complement of excited (e.g. spin-wave) states that break permutational
symmetry. See Figure 7.1 for a summary of this section thus far.
In the case of SU(2), the PS manifold is precisely the Dicke manifold of collective states|mz i with total spin S = N2 and definite spin projection mz ∈ {S, S − 1, · · · , −S} onto a fixed
quantization axis. Equivalently, Dicke states |mz i = m↑ , m↓ can be labeled by a definite numberof spins m↑ = S + mz (m↓ = S − mz ) pointing up (down) along the spin quantization axis,
with m↑ + m↓ = N . In the general case of SU(n), the PS manifold is similarly spanned byP
states |ms , ms−1 , · · · , m−s i with a definite number mµ of spins in state µ, and µ mµ = N . The
dimension of the PS manifold is equal to the number of ways of assigning N identical spins to n+n−1
distinct internal states, or N n−1∼ N n−1 .

(a)

(b)
U
J

kinetic energy

198

U

4J

momentum(d) asymmetric

(c)
p

q

|ψip

|φiq

ŝp · ŝq
←−−−→

p

q

|φip

|ψiq

u
permutationallysymmetric

Figure 7.1: (a) Ultracold atoms on a lattice of L sites tunnel between neighboring lattice sites
at a rate J, and locally repel each other with interaction energy U . (b,c) When the interactionenergy U is small compared to the single-particle bandwidth 4J, the frozen-mode approximation
enables the interaction Hamiltonian to be written as a spin model consisting of exchange termsŝp · ŝq , which swap the states of two spins pinned to modes p, q. (d) Interactions open an energy
gap u = U × N/L between the manifold of permutationally symmetric states of N spins, and theorthogonal complement of states that break spin-permutation symmetry.

199

→
n=2

n=2

n = 10Figure 7.2: Whereas the state of a two-level spin (qubit) can be represented by a point on (orinside) the Bloch sphere, the state of an n-level spin is more generally represented by a probability
distribution on the Bloch sphere. The distribution shown for n = 10 corresponds to a Haar-random
pure state.

External fields or additional interactions that respect permutational symmetry can induce
nontrivial dynamics within the PS manifold. Moreover, additional terms that explicitly break
permutational symmetry can nevertheless lead to interesting dynamics that can be captured within
the PS manifold perturbatively, as long as the coupling to non-PS states is weak compared u (seeAppendix 7.B) [80]. This perturbative regime is thereby efficiently simulable, as the PS manifold
has dimension ∼ N n−1 (as compared to nN for the entire spin Hilbert space). Simulating dynamics
within the PS manifold requires calculating matrix elements h`|Ô|mi of spin operators Ô withrespect to PS states |`i , |mi; we discuss this calculation in Appendix 7.C.
Finally, we take a moment to discuss individual n-level spins. The state of a two-level spin,or a qubit, is commonly represented by a point on (or within) the Bloch sphere. More generally,
the state |ψi of an n-level spin can be represented by a quasi-probability distribution Qψ on the
Bloch sphere (commonly known as the Husimi-Q function, e.g. in the spin-squeezing community[35]). The value Qψ (v) at a point v on the sphere is equal to the overlap of |ψi with a pure state|vi that is maximally polarized in the direction of v: Qψ (v) ≡ |hv|ψi|2 (see Figure 7.2). In thecase of a mixed state ρ̂, this distribution is defined by Qρ̂ (v) ≡ hv|ρ̂|vi. Closely related sphericalrepresentations of multilevel spin states and operators are discussed in Refs. [244, 245]. In practice,
it is conceptually useful to identify the Hilbert space of a single n-level spin with the Dicke manifoldof n − 1 spin- 12 particles.

200
y

x
z

θ

µ = − 32 − 12
|ei

+ 12

+ 32
Ω+

|gi

Ω+
Ω0

Ω−

|ei
Ω0
|gi

θ

|ei
Ω−
|gi

Figure 7.3: Sketch of the three-laser drive used to address nuclear spins on a one-dimensionallattice. Two counter-propagating lasers with right-circular polarization and amplitudes Ω± pointat an angle θ to the lattice axis. A third, linearly polarized laser with amplitude Ω0 points in a
direction orthogonal to both the lattice and the other driving lasers. Absorbing a photon from thelaser with amplitude Ωm induces a transition (g, µ) → (e, µ + m) for the (electronic, nuclear spin)state of an atom, where nuclear spin is quantized along the z axis.

7.3

External control fields
We now consider the addition of external control fields to address atoms’ internal spin states,

which will determine the observables we can access and initial states we can prepare. Specifically,
we consider off-resonantly addressing an electronic |gi → |ei transition of the atoms, and then perturbatively eliminating electronic |ei excitations to arrive at an effective ground-state Hamiltonian
addressing nuclear spins. For simplicity, we will assume that the total spin s of the ground- andexcited-state (hyperfine) manifolds are the same, as e.g. with the 1 S0 → 3 P0 transition of alkalineearth-like atoms (AEAs). However, the results of this section (namely the general form of effective
nuclear spin Hamiltonians, as well as the corresponding set of accessible observables and initialstates) are the same for transitions that take s → s ± 1, so in practice one is free to address thehyperfine manifolds of the 1 S0 → 3 P1 transition of AEAs.
We consider a specific three-laser driving scheme with a geometry sketched in Figure 7.3.
Here the lattice lies in the y-z plane at an angle θ to the z axis, oriented along ` = (0, sin θ, cos θ).
We set the spin quantization axis along z. The laser setup consists of (i) two counter-propagatingright-circularly polarized lasers with drive amplitudes Ω± and wavevectors κv± , propagating in

201
opposite directions along the z axis, v± = (0, 0, ±1), and (ii) a third laser linearly polarized alongz, with drive amplitude Ω0 and wavevector κv0 , propagating along the x axis v0 = (1, 0, 0). All
driving lasers are detuned by ∆ below an electronic transition. The full Hamiltonian for thisthree-laser drive can be written as
full
Ĥ3LD
=

X



Ωm e−imφj ŝmj ⊗ |eihg|j + h.c. + ∆N̂e ,(7.10)

j,mwhere m ∈ {+1, 0, −1} indexes the laser pointing along vm ; the SOC angle φ ≡ κv+ · ` = κ cos θ (inunits with lattice spacing a = 1); ŝz,j , ŝ+,j ŝ−,j are standard axial, spin-raising, and spin-lowering
operators for the spin at lattice site j; ŝ0,j ≡ ŝz,j for shorthand; |gij and |eij respectively denotePthe ground and excited electronic states of atom j; and N̂e = 1 ⊗ j |eihe|j counts the number of
excited atoms (with 1 the identity operator on all spin degrees of freedom).
In the far-detuned limit |∆|  |Ωm |, a second-order perturbative treatment of electronic
excitations (|ei) yields an effective drive Hamiltonian that only addresses ground-state nuclear spins.
After additionally making the gauge transformation ŝmj → eimφj ŝmj (equivalently ĉ†jµ → eiφµj ĉ†jµ ),the drive Hamiltonian then becomes
Ĥ3LD =

X

singleĤ3LD,j,(7.11)

jsinglesinglewhere Ĥ3LD,jdenotes the action of Ĥ3LD
on spin j:singleĤ3LD= Ω̃+ Ω̃− ŝz + Ω̃0 Ω̃− ŝx + Ω̃0 Ω̃+ (ŝz ŝx + ŝx ŝz ) − Ω̃20 ŝ2z − Ω̃2+ ŝ2x − Ω̃2− ŝ2y ,(7.12)withΩ0Ω̃0 ≡ − √ ,
∆

Ω̃± ≡

Ω+ ± Ω−√,∆(7.13)

where we have made the simplifying assumption that all drive amplitudes are real to arrive at thesingleform of Ĥ3LD
in Eq. (7.12). We relax the assumption of real drive amplitudes in Appendix 7.D.

There are three important observations to make about Eqs. (7.11) and (7.12). First, thefact that Ĥ3LD acts identically on all spins means we can freely replace the site index j with a202

Table 7.1: Drive Hamiltonians (left column) that can be implemented with different amplitudematching conditions (right three columns), some of which are specified by an arbitrary sign σ ∈{+1, −1}. The drives shown here are equal to that of Eq. (7.12) up to a possible energy shift ofŝ2x + ŝ2y + ŝ2z = s(s + 1), and come in mutually commuting pairs: a drive with |Ωm | = 1 and Ωn = 0
for both n 6= m commutes with the drive in which Ωm = 0 and both |Ωn | = 1.
singleĤdrive
−ŝ2z
−ŝ2x
−ŝ2y
σŝz + ŝ2z
σŝx + ŝ2xσ (ŝz ŝx + ŝx ŝz ) + ŝ2y±ŝz ± σŝx + σ(ŝz ŝx + ŝx ŝz )

Ω̃0
1
0
0
0
1
1
1

Ω̃+
0
1
0
1
0
σ
σ

Ω̃−
0
0
1
σ
σ
0
±σ

203
momentum index q (as can be verified by substituting ĉjµ = √1LPke−iq·j ĉ ), which is importantqµ

to ensure that this drive addresses the same spin degrees of freedom as the spin Hamiltonians
previously considered in Section 7.2. Second, each of Ω̃0 , Ω̃+ , Ω̃− can be tuned independently bychanging the amplitudes of the driving lasers; some particular Hamiltonians for specific values ofthese amplitudes are shown in Table 7.1. Third, due to the appearance of mutually commuting
pairs of Hamiltonians in Table 7.1, specifically −ŝ2α and ±ŝα + ŝ2α for α ∈ {z, x}, the three-laser
drive admits pulse sequences that exactly implement arbitrary SU(2) (spatial) rotations of the forme−iχ~n·~s , where χ is a rotation angle, ~n is a rotation axis, and ~s ≡ (ŝx , ŝy , ŝz ). The capability toperform arbitrary spatial rotations, together with the capability to measure the number of atoms
Pwith spin projection µ onto a fixed quantization axis, hŜµµ i (where Ŝµν = j ŝµνj ), implies thePcapability to reconstruct all components of the mean collective spin matrix hŜi = µν hŜµν i |µihν|
via spin qudit tomography [6, 246]. Moreover, we expect that advanced quantum control techniques(similar to those of Refs. [247, 248]) can be used to implement arbitrary SU(n) rotations by designingsuitable time-dependent drive amplitudes.
If the excited-state manifold |ei has total spin s ± 1, the effective ground-state Hamiltonians
in Eq. (7.12) and Table 7.1 remain almost identical, but with some additional n-dependent factorsthat do not affect the general results and discussions above. These results still hold if (for example)
all excited hyperfine manifolds of an electronic 1 S0 → 3 P1 transition (with total spins s + 1, s, s − 1)
are addressed simultaneously. See Appendix 7.D for additional details.
Finally, we comment on the preparation of initial states. Initial states are nominally preparedin the “lab frame”, and must be transformed according to the gauge transformation ĉ†jµ → eiφµj ĉ†jµprior to evolution under the three-laser drive Ĥ3LD in Eq. (7.11), which is expressed in the “gauge
frame”. We assume the capability to prepare an initial state in which all spins are maximallypolarized along the z axis, i.e. |zi⊗N = |si⊗N , which is unaffected by the gauge transformation (upto a global phase). The three-laser then allows us to rotate this state into one that is polarizedalong any spatial axis (in the gauge frame). In addition, when n > 2 the three-laser drive allows
us to prepare product states with nontrivial intra-spin correlations. For example, when n is even204
we can prepare an N -fold product of the “kitten” statee−i π2 ŝy +ŝ2yn even|si ∝ |si + |−si .

(7.14)

This state has a vanishing mean spin vector, hŝx i = hŝy i = hŝz i = 0, but variances hŝ2x i = hŝ2y i = s/2
and hŝ2z i = s2 .

7.4

Spin-orbit coupling
We now consider the effect of spin-orbit coupling (SOC) induced by the control fields in

Section 7.3. Before discussing SOC for n-level fermions, we briefly review the well-studied case of
two-level SOC with a one-dimensional lattice [2, 25, 103, 104]. In this case, SOC is induced by
an external driving field that imprints a phase e−iφj on lattice site j, or equivalently imparts amomentum kick q → q + φ, upon the absorption of a photonb :(φ)

Ĥdrive =ΩX †ĉĉq,↓ + h.c..
2 q q+φ,↑(7.15)

Identifying a numerical spin index µ = + 12 (− 12 ) with the state ↑ (↓), this drive Hamiltonian canbe diagonalized in its momentum index q by the gauge transformation ĉ†qµ → ĉ†q−µφ,µ (equivalentlyĉ†jµ → eiφµj ĉ†jµ ), which takes(φ)

Ĥdrive → Ĥdrive ≡ ΩŜx ,

Ŝx ≡

X

ŝx,q ,

(7.16)q

where ŝx,q = 12 ĉ†q,↑ ĉq,↓ + h.c. for two-level spins.
The two-level SOC drive in Eq. (7.15) has been implemented with an external laser that
couples the two electronic states of nuclear-spin-polarized atoms, with ↓ (↑) indexing the ground(excited) electronic state [2, 25, 26, 103, 104]. In contrast, the drive we considered in Section 7.3
addresses electronic excitations off-resonantly, inducing an effective Hamiltonian in the groundstate hyperfine manifold with spin projections µ ∈ {s, s − 1, · · · , −s} (a similar scheme was used tob(φ)

In order for the drive Hamiltonian Ĥdrive to be well-defined, φ should be commensurate with the lattice, e.g. φ ∈Z × 2π/L on a one-dimensional lattice of L sites.

205
study SOC in a subspace of the ground-state manifold in Ref. [123]). Nonetheless, both the twolevel drive in Eq. (7.15) and the n-level drive in Eq. (7.11) become homogeneous (i.e. independent of
the spatial mode index j or q) and independent of the SOC angle φ after the same spin-symmetricgauge transformationc ĉ†jµ → eiφµj ĉ†jµ .
Of course, spin-orbit coupling cannot be “gauged away” entirely. Making a gauge transformation to simplify the drive comes at the cost of making the kinetic energy in Eq. (7.5) spin-dependent,
taking(φ)

Ĥkin → Ĥkin ≡ −2JX

cos (q + µφ) ŝµµq ,(7.17)

q

as visualized in Figure 7.4. To better interpret this Hamiltonian, we can write it in the form(φ)

Ĥkin = −2J

Xh

i
(φ)
(φ)
cos (q) ŵ+,q − sin (q) ŵ−,q ,(7.18)X

cos (µφ) ŝµµq ,(7.19)sin (µφ) ŝµµq .

(7.20)

q

where
(φ)

ŵ+,q ≡µ(φ)

ŵ−,q ≡

X
µ

(φ)

(φ)

For two-level spins with µ = ± 12 , ŵ+,q is proportional to the identity operator and ŵ−,q =
2 sin φ/2 ŝz,q , so the kinetic Hamiltonian in the gauge frame describes a (synthetic) inhomogeneous magnetic field:
(φ)

Ĥkin

n=2

X
= 4J sin φ/2
sin (q) ŝz,q .

(7.21)q

When n > 2, an inhomogeneous magnetic field is likewise recovered in the weak SOC limit sφ  1,in which case
(φ)

Ĥkin

sφ1

= 2Jφ

X



sin (q) ŝz,q + O (sφ)2 .

(7.22)q

For larger φ, this Hamiltonian acquires terms with higher powers of ŝz,q , up to ŝn−1z,q .
c

The “asymmetric” gauge transformation (ĉ†j,↑ , ĉ†j,↓ ) → (eiφj ĉ†j,↑ , ĉ†j,↓ ), sometimes performed in the two-state SOC
literature, does not generalize as nicely to n > 2.

206

|↑i

(b)
φ

|↓i

Ω
φ

|µi

|↑i

(d)

|µ + 2iΩ|µ + 1iΩφmomentum|↓ilab frame

(c)

n=4

Ω
φ

gauge frame

kinetic energy(a)kinetic energy

n=2

momentum

Figure 7.4: Spin-orbit coupling for 2-level (a,c) and 4-level (b,d) spins. Colors indicate different
spin projections µ. In the “lab frame” (a,b), kinetic energy is insensitive to spin, but a spintransition µ → µ + 1 is accompanied by a momentum kick q → q + φ from the drive. Changing
into the “gauge frame” (c,d), essentially by shifting the momentum label q for each spin state µ,
makes the drive diagonal in the momentum index, but comes at the cost of making kinetic energy
spin-dependent.

207
Finally, the gauge transformation ĉ†qµ → ĉ†q−µφ,µ also transforms the interaction Hamiltonian.
Applying this transformation to Eq. (7.4) and keeping only terms that respect coherences that canbe imposed on initial states by the laser drive in Section 7.3 (applied to an initially spin-downpolarized state) again results in an effective spin model. For sufficiently weak SOC (sφ → 0)
this spin model is still well-approximated by Ĥint in Eqs. (7.6) and (7.9). The validity of thisapproximation has been previously benchmarked for SU(2)-symmetric interactions [2, 115], and we
provide additional benchmarking for SU(4) and SU(6) in Appendix 7.A (which finds that the spin(φ)

model works well even for large φ). To ensure that Ĥkin does not become trivial as φ → 0, we cankeep Jφ/u constant, either by increasing J/U or decreasing N/L. Altogether, the interacting spin
Hamiltonian in the gauge frame becomesĤspin = −XuŜ · Ŝ + 2Jφsin (q) ŝz,q ,2N
q(7.23)

consisting of a spin-locking Ŝ · Ŝ term that energetically favors permutational symmetry, and an
inhomogeneous magnetic field that causes inter-spin dephasing.

7.5

Mean-field theory and dynamical phases
We now study the dynamical behavior of the SOC spin Hamiltonian Hspin in Eq. (7.23),

and henceforth work exclusively in the “gauge frame” of Ĥspin and the three-laser drive Ĥ3LD in
Eq. (7.11). We use a Ramsey-like setup wherein we prepare an initial state with the three-laser
drive (using fast pule sequences), then let the state evolve freely for some time under Ĥspin , andfinally apply again the three-laser drive to map observables of interest onto spin projection measurements (e.g. with spin qudit tomography [6, 246]). At the mean-field (MF) level, the undriven
spin Hamiltonian (neglecting constant energy shifts) becomes
ĤMF = uX− hs̄i · ŝq + h sin (q) ŝz,q .

(7.24)q

where s̄ ≡ N1

P

q ŝq

is the average spin matrix, and h ≡ 2Jφ/u is a dimensionless strength of

the inhomogeneous magnetic field. We assume that all momenta q ∈ ZN × 2π/N are occupied.

208
Fixing the atom number N , the spin Hamiltonian has one free parameter, h, which determines therelative strength of the single-particle and interaction terms. One should therefore expect distinct
dynamical behaviors when h  1, in which case strong spin-locking interactions should give rise
to a long-range ordered phase, as opposed to h  1, in which case long-range order should bedestroyed by the strong inhomogeneous magnetic field [115].
To investigate these behaviors quantitatively, we examine time-averaged observables of theform
1
hhÔiiMF = lim
T →∞ TZ T
0

dt hÔ (t)iMF ,(7.25)

where hÔ (t)iMF is the mean-field value of observable Ô at time t. Specifically, we consider thetime-averaged magnetizationσMF ≡ |hh~σ iiMF |,~ ≡ (Ŝx , Ŝy , Ŝz ) with Ŝα ≡where S

~σ ≡

1
~
× S,
Ns

(7.26)Pq ŝα,q , and the time-averaged (dimensionless) interaction energyhhs̄ · s̄iiMF =1
× hhŜ · ŜiiMF .
N2(7.27)

By design, these non-negative quantities are normalized to lie on the interval [0, 1], independent of
the system size N or spin dimension n. In the remainder of this section we will assume that n iseven, both for the sake of experimental relevance (most relevant atomic nuclei are fermionic) andto avoid complications from parity effects.
Our numerical simulations of mean-field dynamics are performed with a Schwinger boson
decomposition of spin operators: ŝµνq = b̂†µq b̂νq . This decomposition requires no approximations,and reduces the number of variables to keep track of by a factor of ∼ n. See Appendices 7.E and7.F for additional details about our numerical simulations and the Schwinger boson equations of
motion.

209

2

σMF

1.0

4

6

8

10

1
0

0.5

hhs̄ · s̄iiMF

0.0
1.0

100

1
0

0.5

0.0
10−1

100
h

101

Figure 7.5: Time-averaged magnetization σMF and interaction energy hhs̄ · s̄iiMF for different spin
dimensions n (indicated in the legend) as determined by mean-field simulations of N = 100 spins
initially in the x-polarized state |Xi for a time T = 105 /u. Insets show same data after rescalingh → h × (n/2)1/3 , and transforming vertical axes according to Eq. (7.29).

210
7.5.1

Initial spin-polarized state

Figure 7.5 shows the time-averages of the magnetization σMF and interaction energy hhs̄· s̄iiMF
as computed by mean-field simulations of N = 100 spins initially in the x-polarized state |Xi ≡|xi⊗N , whereπ|xi ≡ e−i 2 ŝy |si =
Here

m
k





1 X
2s 1/2|µi .
2s µ s + µ(7.28)

is a binomial coefficient. As expected, the spin model exhibits a mean-field dynamicalphase transition between an ordered phase at small h and a disordered phase at large h. The
ordered phase has a non-zero magnetization σMF and an interaction energy hhs̄ · s̄iiMF that asymptotically approach their maximal values as h → 0. The disordered phase has no (time-averaged)
magnetization, σMF = 0, but the interaction energy hhs̄ · s̄iiMF nonetheless indicates persistent nontrivial inter-spin correlations when n > 2. These nontrivial correlations vanish as h → ∞, in whichcase hhs̄ · s̄iiMF approaches the minimal value allowed by conservation laws (clarified below). By
minimizing the reduced field h for which σMF = 0, we numerically find that the transition between
−αordered and disordered phases occurs at a critical field hcrit = n/2
with α ≈ 1/3 (see Figure7.6). When n = 2, this transition is consistent with the predictions of a Lax vector analysis [115,249–252] that exploits integrability of Ĥspin to determine long-time behavior. However, additional
theoretical tools are necessary to understand this transition when n > 2. We elaborate on thispoint in Appendix 7.G.
As shown in insets of Figure 7.5, mean-field results for different spin dimensions n collapse
1/3
onto each other when normalizing the field h to its critical value, h → h × n/2, and rescaling
σMF →

σMF
,
γ n/2

hhs̄ · s̄iiMF →hhs̄ · s̄iiMF − γ (n),1 − γ (n)(7.29)whereΓ k − 12 k≥2
1
γ (k) ≡ √≈ p.
π Γ (k)π(k − 1)(7.30)The rescaling of magnetization and interaction energy can be understood by considering theirlimiting behavior as h → ∞ or h → 0.

211

hcrit

1.0

mean-field
fit: (n/2)−α

0.8

0.6
2

4

6
n

8

10

Figure 7.6: The critical value of hcrit as determined by mean-field simulations
−α of N = 100 spins
initially in the x-polarized state |Xi. A single-parameter fit to hcrit = n/2
finds α = 0.333(5),and α = 1/3 is consistent with all mean-field results to within an uncertainty determined by the
resolution of h in mean-field simulations.

In the strong-field limit h → ∞, we can ignore interactions and treat spins as though they
simply precess at different rates. The time-averaged transverse magnetization σMF then triviallyvanishes as h → ∞. The interaction energy hs̄ · s̄iMF = hs̄iMF · hs̄iMF + O(1/N ), meanwhile, has
contributions from: (i) the diagonal parts of the mean spin matrix hs̄iMF , which are conserved byinhomogeneous spin precession, and (ii) the off-diagonal parts of hs̄iMF , whose oscillations average
to zero when evaluating the time average in hhs̄·s̄iiMF . Altogether, the interaction energy hhs̄·s̄iiMF in
the strong-field limit is determined by the time-independent diagonal part diag hs̄iMF = diag |xihx|,
namelylim hhs̄ · s̄iiMF = Trhh→∞diag |xihx|

2 i= γ (n) .

(7.31)The same result can be obtained by computing the time-averaged interaction energy of two spinsprecessing at different rates.
In the weak-field limit h → 0, the spin-locking Ŝ · Ŝ interactions of the Hamiltonian Ĥspinenergetically restrict dynamics to the permutationally symmetric (PS) manifold. To first order inh, the effect of the inhomogeneous field can be acquired by projecting it onto the PS manifold,which takes ŝz,q → N1 Ŝz . The first order effect of the inhomogeneous field thus vanishes, asXqsin (q) ŝz,q →Xqsin (q) ×

1
Ŝz = 0.
N

(7.32)

212
At second order in h, the effective Hamiltonian within the PS manifold is related to the varianceof the inhomogeneous field, rather than its (vanishing) average. On a high level, the second-ordereffect of the inhomogeneous field within the PS manifold thus consists of permutation-symmetrized
products of two spin-z operators, ŝz,p ŝz,q (with p, q possibly equal). Altogether, the effective spin
Hamiltonian at second order in h is (see Appendix 7.B)


2
X
h u
eff
Ĥspin=× Ŝz2 − Nŝ2z,q  ,2(N − 1)q

(7.33)

which in the mean-field approximation becomes
X
1
eff
ĤMF
= − h2 uŝ2z,q ,2
q(7.34)

where we have used the fact that the axial magnetizations hŝz,q i = N1 hŜz i within the PS manifold,
and the initial value of hŜz i = 0 is conserved by Ĥspin . The weak-field effective Hamiltonian preserves permutational symmetry, so hhs̄ · s̄iiMF → 1 as h → 0. Moreover, the initial y-magnetizationhŜy i = 0 is conserved by Ĥspin , so the long-time-averaged magnetization σMF is determined by thetime-average of ŝx for a single (any) spin:
1
1
lim σMF = | lim
h→0s T →∞ T

Z T
dτ hx|ŝx (τ ) |xi|,(7.35)

0

where
2

2

ŝx (τ ) = eiτ ŝz ŝx e−iτ ŝz .

(7.36)

We can adapt exact analytical results for the dynamics of an infinite-range Ising model [120]d tofind thathx|ŝx (τ ) |xi = s (cos τ )n−2 ,(7.37)

so for even n
1
lim σMF =
h→0
2π
d

Z 2π

n−2

dτ (cos τ )
0

 
n
=γ
.
2

(7.38)

See Appendix K of Ref. [3] for a simpler adaptation of the analytics in Ref. [120] to the one-axis twisting model
ĤOAT = χŝ2z .

213
When going beyond mean-field theory, inter-spin correlations generated by Ŝz2 in Eq. (7.33) will2

cause hŜx i (and thereby the magnetization h~σ i) to decay as e−O(t /N s) ; the timescale of this decay
diverges as N → ∞. On a lattice of linear size L without periodic boundary conditions, additionalcorrections to the behavior predicted above will appear on O(L/J) timescales.

7.5.2

Initial kitten states

We now consider the same setup as above, but with the initial “kitten” states |XXi ≡ |xxi⊗Nand |XXi i ≡ |xxi i⊗N , where|xi + (−1)s |−xi√|xxi i ≡,2|xi + |−xi√|xxi ≡,
2(7.39)

and |−xi is a state polarized along −x, defined similarly to |xi in Eq. (7.28):−i π2 ŝy|−xi ≡ e
1 X
2s 1/2s+µ|−si = s
(−1)
|µi .
2 µ
s+µ

(7.40)

Similarly to Figure 7.5, Figures 7.7 and 7.8 show the time-averaged magnetization σMF and interaction energy hhs̄ · s̄iiMF throughout mean-field dynamics of the initial states |XXi and |XXi i. Thesefigures exclude the trivial case of spin dimension n = 2, for which |xxi = |−si is an eigenstate ofπĤspin and |xxi i = e−i 2 ŝz |xi ≡ |yi is spin-polarized along the y axis. The first and perhaps mostinteresting observation to make about Figures 7.7 and 7.8 is that they are different, signifying theimportance of intra-spin coherences for the dynamical behavior of multilevel spin models.
Unlike Figure 7.5 (for |Xi), Figure 7.7 (for |XXi) exhibits no sharp transition between distinct
dynamical phases: the time-averaged magnetization σMF = 0 for all values of the field h, and theinteraction energy hhs̄ · s̄iiMF smoothly crosses over from a maximal value of 1 to a minimal valueof 2γ (n). The minimal value of hhs̄ · s̄iiMF approached as h → ∞ can be explained with arguments
identical to those in the paragraph containing Eq. (7.31), which now imply that
lim hhs̄ · s̄iiMF = Tr

h→∞

h

2 i
diag |xxihxx|= 2γ (n) .

(7.41)The vanishing magnetization σMF = 0 in Figure 7.7 is protected by symmetries of Ĥspin and |XXi.
For all initial states that we have considered, the value of hŜz i = 0 is conserved by the spin

214
4

6

σMF

1.0

8

10

1
0

0.5

100

0.0
1.0
hhs̄ · s̄iiMF

1
0

0.5

0.0
10−1

100
h

101

Figure 7.7: A corollary of Figure 7.5 for the initial state |XXi. The inset for interaction energyhhs̄ · s̄iiMF in Figure 7.5 subtracts off the minimal value of hhs̄ · s̄iiMF and rescales to lie on theinterval [0, 1], as prescribed in Eq. (7.29). Here the subtracting and rescaling is identical, but witha minimal value of 2γ (n) rather than γ (n).

4

σMF

1.0

6

8

10

1
0

0.5

hhs̄ · s̄iiMF

0.0
1.0

100

1
0

0.5

0.0
10−1

100
h

101

Figure 7.8: A corollary of Figure 7.5 for the initial state |XXi i. Insets show the same data shifted
and rescaled identically to Figure 7.5.

215
Hamiltonian Ĥspin . Moreover, both the spin Hamiltonian Ĥspin and the state |XXi are invariant(up to global phase) under the action of R̂zπ , where R̂zθ ≡ e−iθŜz , which is to say thatR̂zπ Ĥspin R̂zπ † = ĤspinR̂zπ |XXi ' |XXi ,(7.42)

where ' denotes equality up to an overall phase. This symmetry implies thathŜx i = hR̂zπ † Ŝx R̂zπ i = − hŜx i = 0,(7.43)hŜy i = hR̂zπ † Ŝy R̂zπ i = − hŜy i = 0(7.44)

at all times, so altogether σMF = 0.
Turning now to mean-field results for the initial kitten state |XXi i in Figure 7.8, we remarkthat the magnetization σMF and interaction energy hhs̄ · s̄iiMF behave identically to those for the
initial spin-polarized state |Xi in Figure 7.5. This finding can be understood through the fact that|XXi i ' R̂zπ/2 T̂zπ/2 |Xi ,(7.45)

2

where T̂zθ ≡ e−iθŜz . The operators R̂zθ and T̂zθ are generated by axial fields that respect permutational symmetry, and therefore commute with the spin Hamiltonian Ĥspin , soe−itĤspin |XXi i ' e−itĤspin R̂zπ/2 T̂zπ/2 |Xi(7.46)' R̂zπ/2 T̂zπ/2 e−itĤspin |Xi .

(7.47)

In turn, expanding s̄ · s̄ according to Eq. (7.8) shows thatT̂zθ † R̂zθ † s̄ · s̄ R̂zθ T̂zθ = s̄ · s̄,(7.48)

which implies that the interaction energy hs̄ · s̄i throughout dynamics of the initial kitten state|XXi i is the same as that of the spin-polarized state |Xi.
To make sense of why the magnetization σMF is identical in Figure 7.8 for |XXi i as in Figure
7.5 for |Xi, we follow a four-part argument:(i) The time-averaged magnetization vector hh~σ iiMF can be written as a function of the timeaveraged spin matrix hhs̄iiMF .

216(ii) The spin matrix hhs̄iiMF is only ever nonzero on its diagonal and anti-diagonal, regardlessof the initial state. That is, nonzero components hhs̄µν iiMF of hhs̄iiMF always have µ = ±ν(see discussion below).
(iii) The twist operator T̂zθ acts trivially on the diagonal and anti-diagonal components of s̄,
which together with point (ii) implies that hhT̂zθ † s̄ T̂zθ iiMF = hhs̄iiMF .
(iv) The rotation operator R̂zθ merely rotates the magnetization vector hh~σ iiMF without changing
its magnitude.
Altogether, points (i)–(iv) imply that the magnetization
σMF = |hh~σ iiMF | = |hhT̂zθ † R̂zθ † ~σ R̂zθ T̂zθ iiMF |(7.49)

is the same for the initial state |XXi i as for |Xi.
The only nontrivial step in the above argument is point (ii), which says that hhs̄µν iiMF isguaranteed to be zero unless µ = ±ν. This observation, nominally a numerical result of mean-field
simulations, can be understood as follows. The eigenstates |m, wi of Ĥspin are uniquely identified by definite numbers m = (ms , ms−1 , · · · , m−s ) of atoms occupying each internal spin stateµ ∈ {s, s − 1, · · · , −s}, and an auxiliary index w that encodes how |m, wi transforms under permutations of all spins (see Appendix 7.B)e . The operator s̄µν = N1 Ŝµν with µ 6= ν couples the state|m, wi to states m0 , w0 in which (m0µ , m0ν ) = (mµ + 1, mν − 1). Generically, states |m, wi andm0 , w0 with m 6= m0 will have different energies, so their coherence oscillates and averages to zero
when evaluating time-averaged expectation values.
However, degeneracies yield stationary (time-independent) coherences that survive timeaveraging. In the weak-field limit h → 0, such a degeneracy occurs at the mean-field level between
PS states differing only in the populations mµ , m−µ (with a fixed value of mµ +m−µ ), as the effective
P 2
eff ∝
Hamiltonian becomes ĤMF
µ µ mµ . This symmetry is preserved at all orders in perturbatione

Seen otherwise, since Ŝµµ commutes with Ĥspin , eigenvectors of Ĥspin can be indexed by eigenvalues of Ŝµµ . The
number mµ is then the eigenvalue of |m, wi with respect to Ŝµµ , i.e. Ŝµµ |m, wi = mµ |m, wi, while w encodes allother information required to uniquely specify |m, wi.

217
theoryf , so some coherence between such states is preserved as h → hcrit , although this coherence
decays as perturbative corrections to degenerate eigenstates cause them to leak out of the PS manifold (and thereby have a smaller overlap with the initial state |Xi). Note that beyond-mean-field
effects break the symmetry protecting anti-diagonal components of hhŝiiMF , causing them to decay
on time scales that should diverge as N → ∞.

7.6

Conclusions and future directions
Starting with an SU(n) Hubbard model describing ultracold fermionic alkaline-earth(-like)atoms on an optical lattice, we derived a momentum-space multilevel spin model with all-to-all
SU(n)-symmetric interactions. We then introduced external control fields, finding a simple threelaser drive that homogeneously addresses nuclear spins with a variety of spin Hamiltonians. Taking
a closer look at the effect of the spin-orbit coupling (SOC) induced by the driving lasers, we foundthat maintaining the validity of the spin model requires weak SOC, which in turn gives rise to a(synthetic) inhomogeneous magnetic field. Finally, we examined dynamical behavior of the SU(n)spin model at the mean-field level, finding that long-time observables obey simple scaling relations
with n, and that when n > 2 dynamical behavior can be highly sensitive to intra-spin coherences.
Our work makes important progress in understanding the SU(n) Fermi-Hubbard model inexperimentally relevant parameter regimes, and we expect our findings to be readily testable inexperiments with ultracold atoms. Given the possibility for long-range SU(n) interactions, we hope
our work stimulates further efforts into simulating SY and SYK-like models [230, 253] in cold atomic
platforms. In follow-up work, it would be interesting to study the relationship between initial statesand dynamical phases of our SU(n) spin model more systematically, and to consider the effect ofquantum corrections to mean-field behavior. There is also room to improve on the three-laser drive
introduced in this work, for which it is natural to ask what additional techniques or ingredientsare necessary to implement universal control of individual nuclear spins. Universal control wouldf

Only even powers of the “perturbation”P

q sin (q) ŝz,q can be nonzero within the PS manifold, and even powers of

this perturbation exhibit the same mean-field degeneracy between states differing only in the populations mµ , m−µ .

218allow for an experimental study of n-dependence (including even/odd-n parity effects) in a single
experimental platform, simply by controlling the occupation and coherence of internal spin states.
Finally, one can also study the SU(n) Hubbard model in the super-exchange regime that gives
rise to a real-space (as opposed to momentum-space) spin model, where SOC gives rise to chiralmultilevel spin interactions. Unlike our present work, the super-exchange regime does not require
weak SOC, and therefore has a larger parameter space in which to explore dynamical behavior.

Acknowledgements
We thank Victor Gurarie, Emil Yuzbashyan, Asier P. Orioli, and Jeremy T. Young forhelpful discussions on this work. This work was supported by the AFOSR grants FA9550-18-10319, FA9550-19-1-027, by the DARPA and ARO grant W911NF-16-1-0576, ARO W911NF-19-10210, DOE (QSA), NSF PHY1820885, NSF JILA-PFC PHY-1734006, NSF QLCI-2016244, and by
NIST.

7.A

Numerical benchmarking of the spin model
In this appendix we present numerical evidence to support the validity of the spin models

derived in Sections 7.2 and 7.4. Figures 7.9 and 7.10 show a set of time-averaged observables
computed with numerically exact simulations of a Fermi-Hubbard model and an effective spinmodel, respectively, with n = 4 (Figure 7.9) and n = 6 (Figure 7.10) internal levels per spin.
Details for these simulations are provided in the caption of Figure 7.9. Our main conclusion fromthese figures is that the two models show remarkable agreement for the observables considered in
our work. Note that these results are only intended to benchmark the approximation of a FermiHubbard model by a spin model; these results are not expected to agree with the mean-field theory
in Section 7.5 due to strong finite-size effects.

(a)〈〈S·S〉〉

1.0

| X〉

(b)〈〈S·S〉〉

U/J
0.1
0.25
0.5

1.0

(c)〈〈σ〉〉 |X〉

|XX〉

1.0

0.9

0.8

0.8

0.6

0.7

0.7

0.4

0.6

0.6

0.2

0.9
0.8

0.1

0.5 1

5 10

h

0.1

0.5 1

5 10

h

219
〈σ〉

1

.5
0

0.5 1

5000 10000

5

tJ

10

h

Figure 7.9: Numerical results (analogous to Figures 7.5, 7.7, and 7.8 of the main text) for thetime-averaged interaction energy and magnetization (both normalized to a maximal value of 1) ina system of L = 5 lattice sites, for both a Fermi-Hubbard model (dots) and spin model (lines) withn = 4 internal states per spin. The corresponding initial state (defined in Section 7.5 of the maintext) is indicated in each panel, and observables are averaged over a time tJ = 200. Color indicatesthe value of U/J, and the field h corresponds to 2Jφ/u in the case of the Fermi-Hubbard model.
Simulations are performed in real-space, with spin-orbit coupling (SOC) implemented througha homogeneous drive (with no site or φ dependence) and nearest-neighbor tunneling terms that
contain factors of e±iµφ . Results for the initial kitten state |XXi i are excluded because they are
identical to those of |Xi, and magnetization for the initial state |XXi is always 0. Note that whilepanels (a) and (b) are representative of infinite-time behavior, the inset in panel (c) shows that
the Fermi-Hubbard and spin models exhibit different behaviors on very long time scales, althoughgood agreement is restored by rescaling time in the spin model, indicating the likelihood of a needto renormalize spin model parameters. In any case, such time scales are inaccessible in current
experiments and diverge as N → ∞, so these corrections do not affect the main results of our work.
(a)〈〈S·S〉〉

1.0

| X〉

(b)〈〈S·S〉〉

U/J
0.1
0.25
0.5

1.0

(c)〈〈σ〉〉

|XX〉

1.0

0.9

0.8

0.8

0.6

0.7

0.7

0.4

0.6

0.6

0.2

0.9
0.8

0.1

0.5 1

5 10

h

0.1

0.5 1

5 10

h

0.5 1

5

10

h

Figure 7.10: Numerical results identical to Figure 7.9, but with L = 4 lattice sites and n = 6
internal states per spin.

7.B

Perturbation theory for SU(n) ferromagnets
Here we work out a general perturbation theory for SU(n) ferromagnets with a gapped permu-

tationally symmetric (PS) manifold. We begin with an SU(n)-symmetric interaction Hamiltonian

220
of the form
Ĥ0 =

X

Π̂ij ≡ ŝi · ŝj =gij Π̂ij ,Xŝµνi ŝνµj ,(7.50)µ,ν

i<jwhere gij are (real) scalar coefficients for the permutation operators Π̂ij , and ŝµνi ≡ ĉ†µi ĉνi is atransition operator for spin i. We can then consider the addition of, for example, an inhomogeneous
magnetic field or Ising couplings,
Ĥfield =X

Bi ŝz,i ,ĤIsing =iX

Jij ŝz,i ŝz,j ,(7.51)

i6=j

or more generally an M -body operatorg
Ô(w, X̂) =X

wk X̂k ,(7.52)k∈DN (M )where w is a dimension-M (i.e. M -index) tensor of scalar coefficients wk ≡ wk1 k2 ···kM ; X is an
M -spin operator, e.g. ŝz ⊗ ŝz in the case of Ising interactions with M = 2; k ≡ (k1 , k2 , · · · , kM ) is alist of the individual spins ki ∈ ZN ≡ {1, 2, · · · , N } that the operator X̂k ≡ X̂k1 k2 ···kM acts on; andno:
DN (M ) ≡ k ∈ ZM
allentrieskofkaredistinct,i
N(7.53)

is the strictly “off-diagonal” part of ZM
N , which is necessary to identify for a consistent definitionof X̂k as an M -body operator. In this notation, the magnetic field and Ising Hamiltonians in
Eq. (7.51) respectively become Ô (B, ŝz ) and Ô (J, ŝz ⊗ ŝz ).
If the addition Ô(w, X̂) to the SU(n)-symmetric Hamiltonian Ĥ0 in Eq. (7.50) is sufficiently small, namely with operator norm k∗k Ô(w, X̂) less than half the spectral gap ∆gap ofĤ0 , k∗k Ô(w, X̂) < ∆gap /2, then we can treat the effect of Ô(w, X̂) on the ground-state PS man(1)(2)

ifold E0 perturbatively. The effective Hamiltonians Ĥeff and Ĥeff induced by Ô(w, X̂) on the PS
manifold E0 at leading orders in perturbation theory are [80](1)Ĥeff = P̂0 Ô(w, X̂)P̂0 ,(2)Ĥeff = −

X 1
∆6=0

g

∆

P̂0 Ô(w, X̂)P̂∆ Ô(w, X̂)P̂0 ,(7.54)

At face value, an M -body operator with M > 2 does not typically appear in experiments. Nonetheless, considering
M > 2 illuminates the structure of eigenstates (and eigenvalues) of Ĥ0 , and allows us to go to high orders in
perturbation theory with single- and two-body perturbations.

221
where P̂∆ is a projector onto the eigenspace E∆ of Ĥ0 with interaction energy ∆ above that of(1)

the PS manifold. The first order effective Hamiltonian Ĥeff simply projects Ô(w, X̂) onto the PS
manifold E0 , and takes the form
(1)

Ĥeff = w X,

(7.55)

where the coefficient w is the average of all coefficients wk ; and X is a collective version of X:
1|DN (M )|w≡
with |DN (M )| =

QM −1
j=0

X

X≡

wk ,

k∈DN (M )

X

X̂k ,

(7.56)k∈DN (M )(N − j). In the case of a magnetic field ŝz or Ising interactions ŝz ⊗ ŝz , forexample,
ŝz =

X

ŝ(i)
z = Ŝz ,ŝz ⊗ ŝz =

i

X

(j)
2
ŝ(i)
z ŝz = Ŝz − NXŝ2z,i .

(7.57)

i

i6=j
(2)

The second order effective Hamiltonian Ĥeff in Eq. (7.54) takes more work to simplify due to thepresence of a projector P̂∆ onto the manifold E∆ of states with excitation energy ∆. This projector
essentially picks off the part of Ô(w, X̂) that is strictly off-diagonal with respect to the groundand excited-state manifolds E0 and E∆ . We therefore need to decompose Ô(w, X̂) into componentsthat generate states of definite excitation energy when acting on PS states |ψi ∈ E0 . The SU(n)symmetry of Ĥ0 enables such a decomposition to take the form
Ĥ0 Ô(w, X̂) |ψi =X(E0 + ∆) Ô(w∆ , X̂) |ψi ,

E0 ≡

X

gij ,

(7.58)

i<j∆where E0 is the interaction energy of PS states, and thinking of the tensor w as a |DN (M )|component vector, the tensor w∆ can be found by (i) using the coefficients gij to construct amatrix g (M ) of dimensions |DN (M )| × |DN (M )| ∼ N M × N M , and (ii) projecting w onto theeigenspace of g (M ) with eigenvalue ∆. We construct g (M ) for the single-body (M = 1) case below(in Appendix 7.B.1), and provide explicit forms of g (M ) with arbitrary M .
P∆∆
Equipped with the decomposition Ô(w, X̂) =∆ Ô(w , X̂) with terms Ô(w , X̂) thatgenerate states of definite excitation energy ∆, we can expand(2)Ĥeff = −

X 1
∆6=0

∆

P̂0 Ô(w∆ , X̂)2 P̂0 .

(7.59)

222
If X is a single-body operator, then(2)XĤeff =

∆6=0

w∆ · w∆  2
X − N X2 ,N (N − 1) ∆(7.60)and if furthermore all gij = −U/N , as for Ĥint in Eq. (7.9), then the only relevant excitation energy
is ∆ = U (see Section 7.B.2), andX(wi − w)2 = N w
e2(7.61)

w
e2
X2 − N X2 .
(N − 1) U(7.62)

wU · wU =i

is simply N times the variance w
e2 of w, so(2)Ĥeff =
7.B.1

Generating excitation energy eigenstates

Here we construct the matrix g (M ) that enables decomposing M -body operators Ô(w, X̂)
into terms Ô(w∆ , X̂) that generate states of definite excitation energy ∆ above the PS manifold,as in Eq. (7.58). We work through the calculation of g (1) explicitly, and provide the result forg (M ) from a generalized version of the same calculation. To this end, we consider the action of aP
single-body operator Ô(w, X̂) = i wi X̂i on an arbitrary PS state |ψi ∈ E0 and expand
Ĥ0 Ô(w, X̂) |ψi =1 XX
gij wk Π̂ij X̂k |ψi ,2
i6=j(7.63)kwhere strictly speaking gij has only been defined for i < j, so for completeness we define gji = gij
and gii = 0. The sum in Eq. (7.63) has terms with k ∈ {i, j} and terms with k ∈/ {i, j}. In the caseof k ∈/ {i, j}, the permutation operator Π̂ij commutes with X̂k and annihilates on |ψi, and we canreplace the sumXk∈{i,j}
/

→

X
k

−

X

,

(7.64)k∈{i,j}allowing us to simplify
1X X
1X X
gij wk Π̂ij X̂k |ψi = E0 Ô(w, X̂) |ψi −gij wk X̂k |ψi ,22
i6=j k∈{i,j}/i6=j k∈{i,j}(7.65)

223
where E0 = 12P

i6=j gij is the interaction energy the PS state |ψi ∈ E0 . Switching the order of sums

over i 6= j and k ∈ {i, j} as
X X
i6=j k∈{i,j}

→

X X
k

,

(7.66)

i6=j{i,j}3k

we can simplify
1 X
1X
1X
gij =
gik +
gkj = gk ,
2
2
2
i

i6=j
{i,j}3k

gk ≡

X

j

gik ,

(7.67)iwhich implies that the terms in Eq. (7.63) with k ∈/ {i, j} are
X
1X X
gij wk Π̂ij X̂k |ψi = E0 Ô(w, X̂) |ψi −gk wk X̂k |ψi .
2
i6=j k∈{i,j}/(7.68)k

The terms in Eq. (7.63) with k ∈ {i, j}, meanwhile, areX
1 X
gij wk Π̂ij X̂k |ψi =gij wj X̂i |ψi .
2(7.69)i,ji6=jk∈{i,j}so in total


X X

Ĥ0 Ô(w, X̂) |ψi = E0 Ô(w, X̂) |ψi +gkj wj − gk wk  X̂k |ψi .

(7.70)jk

The action of the single-body perturbation Ô(w, X̂) on a permutationally symmetric state thereforeP
generates an eigenstate of Ĥ0 with interaction energy E0 + ∆ if the vector w = k wk |ki satisfies
the eigenvalue equationg (1) · w = ∆w,
where g ≡g (1) ≡ g − diag ~g ,P

g ≡i,j gij |iihj| is a matrix of all couplings gij ; the vector ~the sum of all columns of g; and the matrix diag ~g ≡(7.71)

Pi,j gij |ii =P

i gi |ii isP

g on the diagonal and zeroes
i gi |iihi| has ~everywhere else.
A similar calculation as above with arbitrary M yields an eigenvalue equation of the formg (M ) · w = ∆w,(7.72)

224
where we treat w as an |DN (M )|-component vector, and g (M ) is a matrix with dimensions |DN (M )|×|DN (M )|. In the case of M = 2, we have

g

(2)

X

=

(k,`)∈DN (2) X|k`i 
i∈ZNi∈{k,`}/gik hi`| + gi` hki| + gk` h`k| − (gk + g` − gk` ) hk`| ,

(7.73)and more generallyg (M ) =Xk∈DN (M )



X X
|ki 
gika hka:i | +a∈ZM i∈ZN
i∈k/X{a,b}∈CM (2)
gka kb hka↔b | − g̃k hk|
,

(7.74)

where ka ∈ k = (k1 , k2 , · · · , kM ); ka:i a list that is equal to k except at the a-th position, where ka
replaced is by i, i.e. ka:i = (· · · , ka−1 , i, ka+1 , · · · ); CL (p) is the set of all subsets (“choices”) of pelements from ZL ; ka↔b is equal to k except at the a-th and b-th positions, at which ka and kb are
switched; and
g̃k ≡

X

gij =

{i,j}∈CN (2)i∈k or j∈k

X
i∈k

gi −

X

gka kb .

(7.75){a,b}∈CM (2)If the tensor w is permutationally symmetric, meaning that wk is invariant under arbitrary permutations of k, then this symmetry is preserved by g (M ) . In this case, we can replace sums
over k ∈ DN (M ) in Eqs. (7.73) and (7.74) by sums over k ∈ CN (M ), and replace vectors|k1 , k2 , · · · , kM i → {k1 , k2 , · · · , kM } , such that e.g. |ka↔b i = |ki. These replacements reduce theQ −1size of g (M ) from |DN (M )|×|DN (M )| to |CN (M )|×|CN (M )|, where |DN (M )| = M
j=0 (N − j) =NN
M! × Mand |CN (M )| = M. Additional symmetries of g and w, such as translational invarianceor lattice symmetries, can be used to further reduce the computational complexity of the eigenvalue
problem in Eq. (7.72).

7.B.2

Recovering spin-wave theory

If the interaction Hamiltonian Ĥ0 is translationally invariant, then the single-body eigenvalue
problem in Eq. (7.71) is solvable analytically. In this case, the couplings gij depend only on the225
separation |i − j|, so eigenvectors of g are plane waves of the formwk ≡X

eid·k |di ,(7.76)d∈ZD
L

where on a D-dimensional periodic lattice of N = LD spins, lattice sites are indexed by vectorsDd ∈ ZD
L , and wavenumbers take on values k ∈ ZL × 2π/L. The eigenvalues of g can be determinedby expandingg · wk =X

gcd eid·k |ci =c,d∈ZDL

X

gc,c+d ei(c+d)·k |ci =c,d∈ZDL

X

g0,d cos (d · k) wk ,(7.77)d∈ZD
L

where the imaginary contributions vanish in the sum over d because g0,d = g0,−d . The remainderPPof Eq. (7.71) that we need to sort out is diag ~g , where all gi = i,j gij = d g0,d are equal, whichP
implies that diag ~g = d g0,d is a scalar. We thus find thatg (1) · wk = ∆k wk ,

∆k ≡

X



g0,d cos (d · k) − 1 ,(7.78)d∈ZD
L

in agreement with standard spin-wave theory. Excitations generated by the action of Ô (wk , X)
on PS states |ψi ∈ E0 are known as spin-waves. If gij = −U/N is constant, then the spin-wave
excitation energies are ∆k = U independent of the wavenumber k.

7.C

Restricting spin operators to the permutationally symmetric manifold
Here we provide the restriction of a general M -body spin operator Ô to the permutationallysymmetric (PS) manifold of N spins (each with n internal states). Denoting the projector onto thePS manifold by P̂0 , our task is essentially to find the coefficients of the expansion
P̂0 ÔM P̂0 =X

ha|ÔM |bi |aihb| ,(7.79)

a,b∈An (N )where An (N ) is the set of all ways to assign N (identical) spins to n (distinct) states, such thatfor any a ∈ An (N ) the state |ai = |a1 , a2 , · · · , an i is labeled by the occupation number aµ of statePµ, with µ aµ = N . Written out explicitly,
P

a
!
X
O
µ
µ
1
|ai = pΠ̂ |ãi ,|ãi ≡|µi⊗aµ ,C (a) ≡ Q.
(7.80)C (a)ν aν !
µ
distinct
permutations
Π̂ of ã

226
Here C (a) is a multinomial coefficient that counts the number of distinct ways to permute thetensor factors of the “standard-ordered” state |ãi, enforcing ha|ai = 1. Using these states, withsome combinatorics we can expandsha|ÔM |bi =Xα,β∈An (M )α≤a,β≤bδa−α,b−β

C (α) C (a − α) C (β) C (b − β)hα|ÔM |βi ,C (a) C (b)(7.81)

where the restriction α ≤ a and the difference a − α are evaluated element-wise, i.e. α ≤ a =⇒αµ ≤ aµ and (a − α)µ = aµ − αµ for all µ; and δcd = 1 if c = d and zero otherwise. We sum
over both α and β above merely to keep the expression symmetric with respect to transposition(a, α) ↔ (b, β); in practice, one can simply sum over α ∈ An (M ) and set β = b − a + α, throwing
out terms with any βµ < 0. Note that, by slight abuse of notation, the operator ÔM on the leftof Eq. (7.81) acts on an arbitrary choice of M spins (out of N ), whereas the operator ÔM on theright of Eq. (7.81) is simply an M -spin operator, with matrix elements hα|ÔM |βi evaluated withrespect to the PS M -spin states |αi , |βi ∈ An (M ).

7.D

Relaxing assumptions of the three-laser drive
In order to arrive at the drive Hamiltonian in Eq. (7.12) of the main text, we made two

simplifying assumptions: (i) that the excited-state hyperfine manifold had the same total spin s
as the ground-state manifold, and (ii) that all drive amplitudes are real (which enforces a phaselocking condition between the driving lasers). To derive an effective drive Hamiltonian for thegeneral case in which the excited-state hyperfine manifold has total spin s + r with r ∈ {+1, 0, −1},
we decompose all lasers into their right- and left-circular polarization components and write thefull drive Hamiltonian in the form
fullĤdrive
=

X



(r)
Ωvσ e−iκv·`j ŝvσj ⊗ |eihg|j + h.c. + ∆N̂e ,(7.82)

j,v,σwhere Ωvσ is the amplitude of σ-polarized light propagating along axis v, with σ = +1 and −1
respectively for right and left circular polarizations; and ŝvσj is a spin-raising/lowering operator

227
for atom j along axis v, defined by appropriately rotating the single-atom spin operatorsrrn(n + 1)(n − 1)2(s + r) + 1 X
(r)
(r)
(r)
ŝ± ≡ −
× T̂± , T̂± ≡ ∓hsµ; 1, ±1|s + r, µ ± 1i |µ ± 1ihµ| .
62` + 1µ(7.83)(r)

Here hj1 m1 ; j2 m2 |j3 m3 i is a Clebsch-Gordan coefficient, and we have normalized T̂± such thathi(r) † (r)tr T̂± T̂± = 1. Still assuming real drive amplitudes, the corresponding effective drive Hamiltonian that replaces Eq. (7.12) in the far-detuned limit |∆|  |Ωvσ | is then

h
i


X
single
(1)
(2)
2 2
2 2
2 2
Ĥ3LD = fr Ω̃+ Ω̃− ŝz + Ω̃0 Ω̃− ŝx + fr Ω̃0 Ω̃+ (ŝz ŝx + ŝx ŝz ) − Ω̃0 ŝz + Ω̃+ ŝx + Ω̃− ŝy − fr(3)Ω̃2m ,
m(7.84)(k)where fr

are scalars that depend on the spin dimension n:
(1)

= 1,

f+1 = −s,

(2)

= 1,

f+1 = −

(3)

= 0,

f0
f0
f0

(1)

(1)

f−1 = s + 1,s,
n+2s(s + 1)2
(3)
f+1 =
,
n+2

(7.85)

s+1,
n−2
s2 (s + 1)
(3)
f−1 =
.
n−2

(2)

(2)

f−1 = −

(7.86)(7.87)

If additionally the drive amplitudes are complex, Ωm → Ωm e−iηm (with real Ωm , ηm ), thensingleĤ3LD= fr(1) Ω̃+ Ω̃− ŝz + Ω̃0X Ω̃+ + σ Ω̃− hi
fr(1) σŝη̃σ ,x + fr(2) ŝz ŝη̃σ ,x + ŝη̃σ ,x ŝz2

σ∈{±1}hiX− fr(2) Ω̃20 ŝ2z + Ω̃+ ŝ2η̃0 ,x + Ω̃− ŝ2η̃0 ,y − fr(3)Ω̃2m , (7.88)mwhere ŝηα ≡ e−iηŝz ŝα eiηŝz is a rotated spin-α operator (e.g. ŝπ/2,x = ŝy ), andη̃± ≡ ± (η± − η0 ) ,η̃0 ≡η+ − η−,2(7.89)

are the relative phases of the drive amplitudes.

7.E

Mean-field theory
Here we describe the mean-field theory used to simulate the spin HamiltonianĤspin = −XuŜ · Ŝ + 2Jφsin (q) ŝz,q2N
q(7.90)

228
in Eq. (7.23) of the main text. We begin by decomposing individual spin operators into Schwinger
bosons as ŝµνq = b̂†µq b̂νq , such that the spin Hamiltonian becomesĤspin → Ĥboson = −Xu X †b̂µp b̂νp b̂†νq b̂µq + 2Jφsin (q) µ b̂†µq b̂µq .
2N p,q,µ,ν
q,µ(7.91)The Heisenberg equations of motion for the Schwinger boson operators are (see Appendix 7.F)i∂t b̂µq = −u X †b̂ b̂µp b̂νq + 2Jφ sin (q) µ b̂µq .
N ν,p νp(7.92)

Our mean-field theory then treats all boson operators in these equations of motion as complex
numbers, b̂µq → hb̂µq iMF , with the initial value hb̂µq (t = 0)iMF equal to the initial amplitude of
N P
spin q in state µ. Specifically, for an N -fold product state of the form |ψi =qµ ψµq |µi we
set hb̂µq (t = 0)iMF = ψµq . For pure initial product states, this mean-field treatment of the boson
operators b̂µq is mathematically equivalent to a mean-field treatment of the spin operators ŝµνq , asin Eq. (7.24), but reduces the number of variables to keep track of by a factor of ∼ n.

7.F

Schwinger boson equations of motion for quadratic spin Hamiltonians
Here we decompose a quadratic spin Hamiltonian into Schwinger bosons, and derive theequations of motion for the resulting boson operators. We begin with a general spin Hamiltonianof the formĤ =Xµ,ν,ρ,σ
j<k

µνj
gρσk
ŝµνj ŝρσk +X

µνj ŝµνj ,(7.93)µ,ν,jµνjwhere µ, ν index orthogonal states of an n-level spin; j, k index one of N spins; gρσkand µνj arescalars; and ŝµνj = |µihν|j is a transition operator for spin j. Strictly speaking, Eq. (7.93) only
µνj
µνk
µνj
µνj
defines the couplings gρσkfor j < k, so we enforce gρσj
= gρσkand gρσj= 0 for completion.

Decomposing spin operators into Schwinger bosons as ŝµνj = b̂†µj b̂νj , where b̂νj a annihilates aboson of type ν on site j, we can write this Hamiltonian asĤ =Xµ,ν,ρ,σ
j<k

µνj †
gρσk
b̂µj b̂νj b̂†ρk b̂σk +

X
µ,ν,j

µνj b̂†µj b̂νj .

(7.94)

229
The Heisenberg equations of motion for the boson operators are then
h
i
i X
h
i
X µνj hi∂t b̂α` = b̂α` , Ĥ =gρσk b̂α` , b̂†µj b̂νj b̂†ρk b̂σk +µνj b̂α` , b̂†µj b̂νjµ,ν,ρ,σ
j<k

=

h
i
h
i
X
µν`
gρσk
b̂α` , b̂†µ` b̂ν` b̂†ρk b̂σk +µν` b̂α` , b̂†µ` b̂ν`

X

X


X


µ,ν

(7.96)µ,ν

µ,ν,ρ,σ,k=(7.95)

µ,ν,j

h
i
µν` †
gρσk
b̂ρk b̂σk + µν`  b̂α` , b̂†µ` b̂ν`

(7.97)

ρ,σ,k

where
h
i
b̂α` , b̂†µ` b̂ν` = δαµ δαν b̂α` + δαµ (1 − δαν ) b̂ν` = δαµ b̂ν` ,(7.98)so
i∂t b̂α` =

X


X


ν

αν` †
gρσk
b̂ρk b̂σk + αν`  b̂ν` .

(7.99)

ρ,σ,kIn the case of uniform SU(n)-symmetric interactions of the form g2 Ŝ · Ŝ and a diagonal external
field, we haveαν`gρσk
= g × δασ δνρ ,αν` = α` × δαν

(7.100)soi∂t b̂α` = g

X †b̂νk b̂αk b̂ν` + α` b̂α` .

(7.101)

ν,k7.G

Lax vector analysis
We start with the spin HamiltonianĤspin = −where Ŝµν =P

q ŝµνq .

X
u X
Ŝµν Ŝνµ + 2Jφsin (q) ŝz,q ,2N µ,ν
q(7.102)The single-body operators that appear in this Hamiltonian have squared

norms


tr ŝ†µνq ŝµνq = 1

and


 X
1
tr ŝ†z,q ŝz,q =µ2 = (n + 1)n(n − 1) ≡ ξ 2 .
12µ(7.103)

230
The Lax formulation (following Refs. [115, 249–252]) requires all single-body operators involved tohave the same normalization, so we substitute ŝz̃,q ≡ ŝz,q /ξ to expand
XĤspin1 X=−
Ŝµν Ŝνµ + ξhsin (q) ŝz̃,q ,u
2N µ,ν
q

h≡

where

2Jφ
.
u

(7.104)The intensive, dimensionless, (n2 − 1)-component Lax vector ~` (z) associated with Ĥspin , which isdefined with an auxiliary complex parameter z, has components`α (z) =1 X ŝα,q+ δα,z̃ ξh,
N q z − sin q(7.105)where α indexes elements of a basis {ŝα } of self-adjoint generators of SU(n), with normalizationP
tr ŝ2α = 1. The squared magnitude ~` (z)2 = α `α (z)2 is a constant of motion (for any z), andits residues provide N mutually commuting quantities whose weighted sum recovers Ĥspin . Whenn = 2, conservation of these residues provides sufficient dynamical constraints to make the spin
system fully integrable. In this case, dynamical behavior is governed by the roots of ~` (z)2 , andthe presence (or absence) of complex roots marks distinct dynamical phases of Ĥspin . However,the size of Hilbert space grows with n, while the number of conserved quantities provided by theLax analysis (namely, N ) does not. When n > 2, there is therefore no guarantee that the rootsof ~` (z)2 will similarly govern dynamical behavior. In fact, a straightforward generalization of theLax analysis to n > 2 makes predictions that are inconsistent with the mean-field results in Figures
7.5–7.8 of the main text. We substantiate this claim with a direct calculation of the roots of ~` (z)2
below.
Within the permutationally symmetric manifold, we can replace ŝα,q → s̄α ≡ N1P

q ŝα,q at

the cost of O(1/N ) errors that vanish as N → ∞, so taking this limit we find`α (z) = I (z) s̄α + δα,z̃ ξh,(7.106)

where
1 X
1
1
I (z) ≡ lim=
N →∞ Nz − sin (q)
2π
q

Z 2π
0

dq
1
=√
2
z − sin (q)
z −1

for

z∈
/ [−1, 1] .

(7.107)

231
The squared magnitude of the Lax vector is therefore~` (z)2 =X`α (z)2 = I (z)2

α

X


2
s̄2α + I (z) s̄z̃ + ξh ,(7.108)α6=z̃where we can define the scalar Q2 ≡2α s̄α to simplify

P


 

~` (z)2 = I (z)2 Q2 − s̄2 + I (z) s̄z̃ + ξh 2 = I (z)2 Q2 + ξ 2 h2 + 2I (z) ξhs̄z̃ .
z̃

(7.109)For initial states with hs̄z i = 0, we thus find that~` (z)2 =Q2z2 − 1+ ξ 2 h2 ,(7.110)which is zero whenhsz =± 1−

Q
ξh

2
.

(7.111)

These roots change character when z = 0, suggesting that the critical field hcrit separating dynamical phases satisfies?

h2crit =Q2,
ξ2(7.112)?

where we use the relation = to indicate that this “prediction” of the Lax analysis is not necessarily
valid for all n. For a permutationally symmetric state, up to vanishing O(1/N ) corrections we can
expand
Q2 =

1
1
n−1
=1− =
,
n
n
n

(7.113)

n−1
12
12
×
= 2
.
n
n(n + 1)(n − 1)n (n + 1)(7.114)Xαs̄2α =

Xs̄µν s̄νµ −µ,ν

which implies that?

h2crit =This Lax analysis correctly predicts that hcrit = 1 when n = 2, but otherwise predicts hcrit ∼ n−3/2 ,which is inconsistent with the finding that hcrit ∼ n−1/3 in the mean-field results of the main text
(see Figure 7.6). We emphasize that this inconsistency is not a failure of the Lax formalism,
but rather an indication that new theoretical tools are necessary to understand multilevel spinmodels.
hStrictly speaking, the zeros in Eq. (7.111) occur at values of z at which I (z) is undefined. We avoid this issue by
analytically continuing I (z)2 to the interval z ∈ (−1, 1).

Chapter 8

Spin qudit tomography and state reconstruction error

Prologue
In Chapter 7, we considered the dynamical behavior of a collective multilevel spin model. In
order to make sense of numerical simulations, we had to make a judicious choice of order parameters
to examine. Having to make such a choice naturally leads to the question: what order parameters
are actually accessible experimentally, and how? Given that the spin model is encoded in nuclear
spin degrees of freedom, we know that experiments should be able to measure the projection ofspin onto a fixed quantization axis with relative ease. Moreover, in Section 7.3 we worked out how
external controls can implement arbitrary spatial rotations of these nuclear spins, which expands theset of accessible observables to include spin projections onto arbitrary spatial axes. The remaining
questions are then (i) whether these spin projection measurements are sufficient to reconstruct any
collective spin observable, (ii) how should this reconstruction be performed, and (iii) how reliable
are these reconstructions (i.e. what are the error bars on the estimates of observables)? These
are the motivating questions of the work in this chapter. Reconstructing all collective single-bodyspin observables is equivalent to performing state tomography on the average single-spin density
matrix; due to its numerous connections with existing literature we use the latter formulation of
the problem in this chapter. The bulk of this chapter is taken from Ref. [6]. In addition to myself
and Ana Maria Rey, this work featured major contributions from Diego Barberena.

233

Abstract
We consider the task of performing quantum state tomography on a d-level spin qudit, using
only measurements of spin projection onto different quantization axes. After introducing a basis ofoperators closely related to the spherical harmonics, which obey the rotational symmetries of spin
qudits, we map our quantum tomography task onto the classical problem of signal recovery on the
sphere. We then provide algorithms with O rd3 serial runtime, parallelizable down to O rd2 , for(i) computing a priori upper bounds on the expected error with which spin projection measurements
along r given axes can reconstruct an unknown qudit state, and (ii) estimating a posteriori thestatistical error in a reconstructed state. Our algorithms motivate a simple randomized tomography
protocol, for which we find that using more measurement axes can yield substantial benefits thatplateau after r ≈ 3d.

8.1

Introduction
Quantum state tomography, the task of reconstructing a quantum state by collecting andprocessing measurement data, is an essential primitive for quantum sensing, quantum simulation,and quantum information processing. The central importance of quantum state tomography has
led to the development of techniques based on least-squares inversion [254], linear regression [255],
maximum likelihood estimation [256, 257], Bayesian inference [258–260], compressed sensing [261,262], and neural networks [263], among others. These techniques are typically developed in a
general, information-theoretic setting, and make minimal assumptions about the physical medium
of a quantum state. As a consequence, even well-established techniques can be ill-suited for physicalplatforms with unique or limited capabilities.
Due to advancements in experimental capabilities to address nuclear spin states (i.e. hyperfine
levels) in ultracold atomic systems [28, 45, 264–267], as well as developments in the control ofultracold molecular systems [203, 268–277], a particular setting of growing interest is the spin qudit,
or a multilevel quantum angular momentum degree of freedom. Spin qudits can provide advantages

234
over their qubit counterparts for quantum sensing [278, 279], enable quantum simulations of SU(d)magnetism [5, 7, 28, 51, 53, 65], and offer unique capabilities for quantum computation and error
correction [280–282]. In all cases, quantum state tomography is necessary to take full advantage ofa spin qudita .
The problem of qudit tomography is not new, with an extensive literature on a variety oftechniques [246, 279, 283–292]. However, most existing protocols either rely on infinite-dimensional
representations of a quantum spin [279, 285, 293], or require the capability to perform essentiallyarbitrary operations on a qudit [286–292], generally resulting in tomographic protocols that canbe highly inefficient or unachievable in practice. The protocols based on infinite-dimensional representations of a quantum spin have the advantage of reconstructing its state from measurementsof spin projection onto different spatial axes, which are generally accessible with any spin qudit.
Nonetheless, these protocols obfuscate the minimal requirements for performing full state tomography, provide no straightforward error bounds or guarantees of accuracy, and (with the notable
exception of Ref. [285]) generally extract only a small fraction of the information contained inmeasurement data.
In this work, we consider the task of performing spin qubit tomography using only measurements of spin projection onto different spatial axes. This sort of task was first considered
in Ref. [246], as well as a few later works [283–285]. Specifically, Ref. [246] provided an explicit
protocol for reconstructing a d-level spin qudit state from measurements of spin projection along
2d − 1 axes, the minimum number necessary for full tomography of an arbitrary (possibly mixed)
qudit state. However, the protocol in Ref. [246] involves a choice of a single (arbitrary) angle θ,and provides no means for comparing different choices of θ, which may result in wildly different
statistical errors (i.e. precision) in a reconstructed state. Other works provide insightful discussions
into the problem of spin qudit tomography, but either (i) require making assumptions about thequdit state in question [285] (making the tomographic protocol only valid for a restricted set ofaNote that the measurement of collective observables for quantum sensing or simulation can be recast as a single-spin
tomography task.

235
possible states), (ii) do not address the question of statistical error [283], or (iii) provide a measureof statistical error that is needlessly conservative and computationally demanding [284]. We addressthese shortcomings in this work, and identify remaining avenues for refining spin qudit tomography
protocols.
In Section 8.2, we introduce a set of qudit operators that are closely related to the sphericalharmonics, and which play a central role in our work. We then map the quantum problem ofspin qudit tomography onto the classical problem of signal recovery on the sphere in Section 8.3,
thereby providing an intuitive perspective on spin qudit tomography. In Section 8.4 we provide a
priori upper bounds and a posteriori estimates of the statistical error in a qudit state reconstructedfrom measurements of spin projection along a given set of r measurement axes. The capability todetermine upper bounds on reconstruction error a priori motivates a simple randomized tomography protocol that we outline in Section 8.5, and for which we numerically find that using moremeasurement axes yields substantial benefits that plateau after r ≈ 3d. To facilitate the use of our
protocols, we make all of our codes publicly available at Ref. [294], which also contains the best
measurement axes we found for d ≤ 30 and r = 3d.

8.2

Polarization operators
We begin by introducing a set of qudit operators that are closely related to the sphericalharmonics (in a sense that will be clarified below), and which play a central role in our work.
Consider a d-state spin qudit with total spin s ≡ d−1
2 . The defining property of a spin qudit,
distinguishing it from other qudits, is that it describes an angular momentum degree of freedom,which has specific implications for how a spin qudit should transform under the group SO(3) ofrotations in 3D space. Due to the central importance of these transformation rules for a spin qudit,
we seek a basis of operators that transform nicely under 3D rotationsb . One such basis is that ofb

Technically speaking, we seek a basis of operators that transform as an irreducible representation of SO(3).

236
the polarization operators [295, 296], defined byrs
2` + 1 XT`m ≡hsµ; `m | sνi |νihµ| ,2s + 1 µ,ν=−s(8.1)

where |µi is an eigenstate of the axial spin projection operator Sz |µi = µ |µi; and hsµ; `m | sνi is
a Clebsh-Gordan coefficient that enforces ` ∈ {0, 1, · · · , d − 1} and m ∈ {−`, −` + 1, · · · , `}, suchthat there are d2 polarization operators in total. For brevity, we will generally treat the value ofd as constant but arbitrary throughout this work, and we will suppress any explicit dependence ofquantities or operators such as T`m on d. The polarization operators are orthonormal with respect
to the trace inner product, and transform nicely under conjugation:(T`m |T`0 m0 ) = δ``0 δmm0 ,†T`m= (−1)m T`,−m ,

(8.2)

P
2
where for any d × d matrix X =µ,ν Xµν |µihν| we define the d -component vector |X) ≡P† Y ; and δ 0 ≡ 1
X
|µνi;
(X|
is
the
conjugate
transpose
of
|X),
such
that
(X|Y
)
=
tr
X
µν
kk
µ,ν
if k = k 0 and 0 otherwise. These properties of the polarization operators allow us to expand any
density operator ρ in the polarization operator basis as
ρ=

d−1 X
`
X

ρ`m T`m ,DE†ρ`m ≡ T`m,`=0 m=−`

ρ(8.3)

where hXiρ ≡ tr (ρX) = (ρ|X), and ρ† = ρ implies that ρ∗`m = (−1)m ρ`,−m . The polarization
operators can be interpreted in terms of an absorption process, whereby T`m |ψi is (up to normalization) the state obtained after a spin-s state |ψi absorbs a particle with total spin ` and spin
projection m onto a fixed quantization axis. Similarly to the complex spherical harmonics Y`m , wewill refer to ` as the degree and m as the order of T`m .
The polarization operators are spherical tensor operators, whose degree is preserved under 3D
rotations generated by the spin operators Sx , Sy , Sz . Moreover, the degree-` polarization operators
T`m transform similarly to spin-` particles and spherical harmonics Y`m under 3D rotations (seeAppendix 8.A). Specifically, for any triplet of angles ω = (α, β, γ), we can define the rotation
operator
R (ω) ≡ e−iαSz e−iβSy e−iγSz ,

(8.4)

237
and expand rotated polarization operators asTω`m ≡ R (ω) T`m R (ω)† =

`
X

`
Dmn
(ω̄)∗ T`n ,

(8.5)

n=−`

where ω̄ = (γ, β, α) is the reversal of ω, and`Dmn(ω̄) ≡ `m R (ω̄) `n(8.6)

are (Wigner) rotation matrix elements. For reasons that will become clear shortly, throughout
this work we will primarily consider rotations of the sphere that take the north pole to a pointv = (α, β) at azimuthal angle α and polar angle β. For ease of notation, we therefore define` (v) ≡ D ` (0, β, α).
R (v) ≡ R (α, β, 0), Tv`m ≡ T(α,β,0),`m , and Dmn
mn

The polarization operators T`m share a connection to the spherical harmonics Y`m that goesbeyond the rules for their transformation under 3D rotations. In fact, the phase-space representation of T`m is proportional to Y`m . The phase-space representation of a spin qudit operator X
assigns, to each point v on the sphere, the complex number
X PS (v) ≡ hsv | X | sv i ,(8.7)

where |sv i ≡ R (v) |si is the state of a spin qudit polarized along v. This representation is faithfulin the sense that X is uniquely determined by the phase-space values X PS (v) at all points v on
the sphere. The transformation rules for polarization operators in Eq. (8.5), together with the factthat hs | T`m | si = 0 unless m = 0, suffice to show thatPST`m(v) = c` Y`m (v) ,(8.8)

where the scalar c` simply enforces (T`m |T`m ) = 1 (see Appendix 8.A). The polarization operators
T`m are thus a quantum analogue of the spherical harmonics Y`m , and play an important role inphase-space formalisms for spin qudits [245].
As a special case, the phase-space representation ρPS of a spin qudit state ρ is commonly
known as its Husimi distribution. Performing tomography on an unknown qudit state ρ is therefore equivalent to reconstructing the unknown distribution ρPS on the sphere. In principle, the238
representation ρPS of a finite-dimensional qudit state ρ can be reconstructed from the valuesρPS (v) = hsv | ρ | sv i at a finite number of points v. In practice, the value hsv | ρ | sv i is determined by measuring spin projection along v, which also provides measurement data on all spin
projections hµv | ρ | µv i with µ ∈ {s, s − 1, · · · , −s} and |µv i ≡ R (v) |µi; one would like to makeuse of this additional data as well. We clarify the connection between the quantum problem ofreconstructing ρ from spin projection measurements and the classical problem of reconstructing
ρPS from its values ρPS (v) in the following section.

8.3

Spin tomography as signal recovery on the sphere
Our goal is to reconstruct an arbitrary state ρ of a spin qudit from measurements of spin pro-jection onto different quantization axes. We are thus nominally restricted to measuring projectors
Πvµ ≡ |µv ihµv |, where |µv i ≡ R (v) |µi is a state with spin projection µ onto the measurement axis
v. For any fixed axis v, the sets {Πvµ } and {Tv`,0 } (i.e. all Tv`m with m = 0) are both complete
bases for the space of operators that are diagonal in the basis {|µv i}. Measuring the projectors{Πvµ } is therefore equivalent to measuring the polarization operators {Tv`,0 }, and provides dataon the expectation values Tv`,0 ρ .
In order to reconstruct an arbitrary density operator ρ from the expectation values Tv`,0 ρ ,we essentially need to find a set of coefficients C`mk (v) that would allow us to recover any matrix
element ρ`m of ρ throughρ∗`m = hT`m iρ =X

C`mk (v) Tvk,0 ρ .

(8.9)

v,k

Expanding the rotated polarization operators Tvk,0 into a sum of un-rotated polarization operators
T`n according to Eq. (8.5), we find that the recovery condition in Eq. (8.9) is satisfied when
T`m =

X

k
C`mk (v) D0,n(v)∗ Tkn .

(8.10)

v,k,n

Orthogonality of the polarization operators then implies the decomposition C`mk (v) = δ`k C`m (v),239

Figure 8.1: Signal recovery on the sphere is the problem of reconstructing an unknown functionf (red distribution) from its values f (v) at specific points v ∈ V (blue dots) on the sphere. For
almost all choices of V , reconstruction of f is possible if there are at least as many points in V as
there are degrees of freedom in f .

and in turnX`C`m (v) D0,n(v)∗ = δmn(8.11)v

for all `.
In fact, the problem of finding suitable axes V and coefficients C`m (v) to satisfy Eq. (8.11)
can be mapped onto the well-studied problem of signal recovery on the sphere (see Figure 8.1)
[297–300]. The signal recovery problem can be stated as follows: given a square-integrable function
f on the sphere, with the spherical harmonic expansionf (v) =Xf`m Y`m (v) ,(8.12)`,mwhere f`m are complex coefficients, find a set of points V = {v} and associated coefficients C̃`m (v)with which we can reconstruct f , or equivalently its coefficients f`m , from knowledge of the function’s value f (v) at all points v ∈ V ; that is
f`m =

X

C̃`m (v) f (v) =v

XC̃`m (v) Ykn (v) fkn .

(8.13)

v,k,nReconstruction of functions with arbitrary coefficients f`m implies that
Xv

C̃`m (v) Ykn (v) = δ`k δmn ,(8.14)

240
which is a stronger version of the condition that we found for the spin qudit tomography problem
in Eq. (8.11). We will refer to Eq. (8.14) as the full recovery problem, and Eq. (8.11) as the reducedq
4π`
recovery problem. Due to the fact that D0,m(v) = 2`+1
Y`m (v), any solution to the full recoveryq∗
problem automatically solves the reduced recovery problem by setting C`m (v) = 2`+1
4π C̃`m (v) .
In principle, this mapping allows us to import a host of existing signal recovery algorithms [297–300] for the task of spin qudit tomography. In practice, spin qudits typically have only a modest
dimension d, which allows for simpler and optimized tomography protocols that are practical despite
worse scaling with d (see Section 8.5). A natural avenue to develop better spin qudit tomography
protocols would therefore be to build on the existing classical signal recovery algorithms, tailoringthem to solve the reduced recovery problem in Eq. (8.11) rather than the full recovery problem in
Eq. (8.14). We leave these developments to future work.
If the function f is band-limited at degree L, which is to say that f`m = 0 for all ` ≥ L,then the full recovery problem in Eq. (8.14) is provably solvable with a suitable choice of |V | = L2
points on the sphere [301, 302]. The existence of these solutions to the full recovery problem inturn implies the existence of d2 measurement axes that suffice to reconstruct arbitrary states ofd-level spin qudit, whose possible states (or rather, phase-space representations) are band-limited
at degree d. Moreover, for any fixed degree `, finding solutions to the reduced recovery problem inP
Eq. (8.11) is equivalent to the recovery of a degree-` function f` = m f`m Y`m , which is provably
possible with |V | = 2` + 1 samples [301]. In the case of spin qudit tomography, the degree ` takes
a maximal value of `max ≡ d − 1, so state recovery requires as many measurement axes as there arepolarization operators with degree `max , namely 2`max + 1 = 2d − 1.

8.4

State reconstruction error
For the practically minded, proving the existence of solutions to a problem is less interesting

than the exposition of a particular solution. On a high level, a spin qudit tomography protocol
consists of (i) selecting a set of measurement axes, (ii) collecting measurement data on spin projection onto these axes, and then (iii) processing the collected data to reconstruct the state of the241spin qudit. Whereas step (ii) can involve a host of platform-dependent technical challenges, in thefollowing sections we discuss the steps to take before and after collecting measurement data.
To this end, we begin by asking a question: what is a “good” choice of measurement axes?
Intuitively, a good choice of axes should minimize the error with which one can reconstruct an
unknown quantum state from associated measurement data. If we can quantify this intuition,then we can optimize over different choices of measurement axes to find a set that (approximately)
minimizes the error in reconstructed states.
A set of measurement axes V = {v} nominally induces a set of projectors {Πvµ } that will bemeasured in an experiment. By a simple change of basis, measuring these projectors is equivalent
to measuring the polarization operators {Tv`,0 }. Flattening each d × d matrix Tv`,0 into the d2 component column vector |Tv`,0 ), we construct the measurement matrix
MV ≡X|v`i (Tv`,0 | .

(8.15)

v,`

Here v and ` label a row of MV , or equivalently label a standard (“one-hot”) basis vector |v`i of
a |V | × d -dimensional vector space, and (Tv`,0 | is the conjugate transpose of |Tv`,0 ). A necessary
and sufficient condition for V to allow for full state tomography is that the measured polarization
operators Tv`,0 , or equivalently the rows of MV , span the entire (d2 -dimensional) space of operators
on a d-level spin qudit. In this case MV must be full rank, with d2 nonzero singular values. IndexingPthese singular values MkV and the corresponding (normalized) left singular vectors xVk ≡ j xVkj |jiby an integer k ∈ {1, 2, · · · , d2 }, we can construct the orthonormal qudit operators
QVk ≡

X

V
qkj

∗

Tj ,

j

V
qkj
≡

xVkj
MkV

,

(8.16)

where for shorthand we use a combined index j = (v, `) to specify both a measurement axis v anda degree `, which identify the polarization operator Tj ≡ Tv`,0 . These operators allow us to expand
any state ρ of a d-level spin qudit in the form
2

ρ=

d
X
k=1

ρVk QVk ,D †EρVk ≡ QVk.
ρ

(8.17)

242
Given empirical estimates T̃j of the expectation values Tj ρ , an empirical estimate ρ̃V of ρ is then
ρ̃V ≡

X

ρ̃Vk QVk ,(8.18)kwhere, using the fact that Tj = Tj† (because they are diagonal polarization operators with degreem = 0),
ρ̃Vk ≡

X

V
qkj
T̃j ≈

X

j

D †E
V
qkj
Tj ρ = QVk= ρVk .
ρj(8.19)The measurement matrix MV allows us to make concrete statements about the statistical error
between the empirical estimate ρ̃V and the true state ρ. Assume, for example, that the estimatesT̃j are equal to Tj ρ up to uncorrelated noise with variance no grater than 2 :hhj j 0 ii ≤ 2 δjj 0 .

T̃j = Tj ρ + j ,(8.20)

Here {j } are independent random variables, and we use the double brackets hh·ii to denote statistical
averaging over experimental trials that estimate Tj ρ . In this case, the mean squared error withwhich ρ̃Vk approximates ρVk isD|ρ̃Vk − ρVk |2

E

=
=

*


ρ̃Vk − ρVk

X

V
qkj

∗

∗ 

ρ̃Vk − ρVkVqkj
0 hhj j 0 ii+


(8.21)(8.22)

j,j 0

≤

X

V 2 2
|qkj
|  =

j


MkV

!2
.

(8.23)

Using the fact that the operators QVk are orthonormal, we can therefore bound the mean squared(Euclidean) distance between ρ̃V and ρ as
DE2
EV (ρ) ≡ kρ̃V − ρk≤ 2 SV2 ,2(8.24)


where kXk2 ≡ (X|X) = tr X † X is the squared (Euclidean, Frobenius, or Hilbert-Schmidt) normof X, and the classical error scale SV is defined by
SV2 ≡

X
k

MkV

−2

= MV−1

2,(8.25)

243
where MV−1 is the left inverse of MV , satisfying MV−1 MV = 1. We refer to the error scale SV
as “classical” because the bound in Eq. (8.24) applies in the presence of classical sources of measurement error. Note that the classical error scale SV diverges if the measurement matrix MV
is singular, which indicates that measuring spin projections along all axes in V does not provide
sufficient information to reconstruct arbitrary quantum states.
Computing the classical error scale SV and estimates ρ̃Vk ≈ ρVk requires building the measurement matrix MV and computing its singular value decomposition. The complexity of this task can
be greatly reduced by the fact that the degree ` of a polarization operator T`m is preserved underrotations, which implies that the unitary
U≡

d−1 X
`
X

|T`m ) h`m| ,(8.26)`=0 m=−`

with vectors |T`m ) in a column indexed by integers (`, m), block-diagonalizes the measurement
matrix into d blocks indexed by the degree `:MV U =d−1
X|`ih`| ⊗ MV ` ,(8.27)`=0where the |V | × (2` + 1)-sized blocks are
MV ` ≡X|vi (Tv`,0 |T`m ) hm| =

v,m

X

`
D0,m
(v) |vihm| .

(8.28)

v,m`Here D0,m(v) is a Wigner rotation matrix element, defined in Eq. (8.6). As the singular values of

MV are invariant under unitary transformations, it follows that
SV2 =

X

SV2 ` ,

SV2 ` ≡ MV−1`

2,(8.29)`

where MV−1` is the left inverse of MV ` . Constructing the block MV ` and computing its singularvalue decomposition takes at most O(|V |d2 ) time. If we assume that |V | ∼ d, then computing theclassical error scale SV takes O(d4 ) serial or O(d3 ) parallel runtime (see Figure 8.2).
The assumption that observables can be estimated up to uncorrelated noise with maximal
variance 2 , summarized by Eq. (8.20), is reasonable when measurement error is dominated by244time (seconds)

101
10−1

SV
V
EV (ρ)

10−3
101
102
qudit dimension dFigure 8.2: Serial runtime to compute SV , V , or EV (ρ) with |V | = 2d − 1 randomly chosen measurement axes and a randomly chosen qudit state ρ. Each point is an average over 103 calculations
or 5 minutes of runtime, whichever comes first. These results do not count fixed runtimes to precompute quantities that can be recycled for every new choice of V and ρ. Dashed lines show fits
to a runtime t = cdα for the 20 largest values of d, finding α ≈ 3.8 ± 0.1.

245classical sources of experimental noise. However, this assumption breaks down when measurement
error is limited by fundamental quantum shot noise (i.e. finite sampling error). We relax theassumption of Eq. (8.20) in Appendix 8.B, where we instead assume that ρ̃V is built from n
independent measurements of spin projection along every axis v ∈ V , with shot noise the dominantsource of error. In this case, the constraints that tr (ρ) = 1 and Πvµ ρ ≥ 0 allow us to bound themean squared distance between ρ̃V and ρ as
EV (ρ)2 <

2V
,
n

2V ≡

X

Γ2` SV2 ` ,(8.30)`

where the quantum error scale V is defined in terms of the spectral range of T`,0 :Γ` ≡maxµ t`µ − minµ t`µ,
2t`µ ≡ µ T`,0 µ .

(8.31)

If d is even or ` is odd, then Γ` = maxµ t`µ . For comparison with the “classical” error boundin Eq. (8.24), we note that 2V < SV2 /2, so the previous bound still holds with the replacement
2 → 1/2n. The factors Γ2` are quick to compute and can be recycled for every new choice of axes
V , so the complexity of computing V is the same as that of SV (see Figure 8.2).
Though straightforward to compute, the bound in Eq. (8.30) is not tight, as it is acquiredby bounding the statistical error v` in the empirical estimate T̃v`,0 of Tv`,0 ρ by hh2v` ii ≤ Γ2` . The
individual bounds on hh2v` ii for each axis v and degree ` are tight, but these bounds cannot all beachieved simultaneously. There is therefore still room for improvement on the bound in Eq. (8.30)
by maximizing EV over the set of all physical qudit states ρ. We discuss this maximization problem
in Appendix 8.C, but leave its full solution to future work. We also note that the reconstruction
error bound in Eq. (8.30) obeys the “standard quantum limit” of ∼ 1/n scaling in the number ofmeasurements. In principle, this scaling can be improved to ∼ 1/n2 by preparing and measuring
entangled copies of many qudits [31].
The error scales SV and V provide pessimistic upper bounds on statistical error, which can becalculated without prior knowledge of the true qudit state ρ. The actual error in the reconstructionρ̃V of a particular state ρ may be considerably smaller, and may depend on ρ itself. Written out

246
in full, the mean squared distance between ρ̃V and ρ is (see Appendix 8.B)
2

EV (ρ) =

X

v



MV−1`

†

MV−1`


w hhv` w` ii.

(8.32)

v,w,`

The covariances hhv` w` ii are generally determined by the sources of measurement error in any
given experiment, but will typically satisfy hhv` w` ii = δvw hh2v` ii because measurements along v
are independent of measurements along w. If measurement error is limited by shot noise, then (seeAppendix 8.C)
SNL 1

EV (ρ)2 =XhχV ` | ρ` i − hρ` | NV ` | ρ` i ,n(8.33)`

SNL

where = indicates equality in the “shot-noise-limited” regime; |ρ` i ≡Pm ρ`m |mi is a vector of

the polarization operator components ρ`m of ρ, defined in Eq. (8.3); and the matrix NV ` and vector|χV ` i are defined below. While the true shot-noise-limited error in ρ̃V cannot be known exactly
without knowing ρ, this error can be estimated a posteriori by EV (ρ) ≈ EV (ρ̃V ). After constructing
an estimate ρ̃V of ρ, the complexity of computing the error EV (ρ̃V ) from Eq. (8.33) is the same asthat of computing SV or V (see Figure 8.2).
We now define NV ` and |χV ` i for the sake of completion, but note that these definitions canbe skipped without consequence for the remaining discussions in this paper. The matrix NV ` isNV ` ≡ MV† ` diag



MV−1`

†

MV−1`


MV ` ,

where diag [X] sets the off-diagonal parts of X to zero. The vector |χV ` i ≡(8.34)Vm χ`m |mi is definedPbyχVLM ≡

X(NV ` |DM |g̃L` ) ,(8.35)†(TL,m+m0 |T`m
T`m0 ) m m0 ,(8.36)

δM,m0 −m mm0 mm0 .

(8.37)`g̃L` ≡

X
m,m0

DM ≡

X
m,m0

Here g̃L` is essentially a matrix of structure constants for the polarization operator algebra (seeAppendix 8.D), and DM simply picks off the M -th diagonal of the matrix it acts on.

247

8.5

Tomography protocol
The ability to certify a statistical error bound on the empirical estimate ρ̃V of an unknown

quantum state ρ motivates the following protocol for spin qudit tomography:(i) Select a random set of measurement axes V by uniformly sampling points on the spherec ,and use any standard minimization algorithm to optimize the 2|V | parameters in V (twoangles for each point v ∈ V ) by minimizing the quantum error scale V in Eq. (8.30).
If |V | is too large for such optimization, you can simply generate many sets of randommeasurement axes, and then choose the set with the smallest quantum error scale V . Notethat computing the error scale V requires, for each ` ∈ {0, 1, · · · , d − 1}, constructing themeasurement matrix MV ` in Eq. (8.28) and computing its singular value decomposition.
Save all measurement matrix data associated with the final measurement axes V for lateruse.
(ii) For each axis v ∈ V , make n measurements of spin projection, and set Π̃vµ ≈ Πvµ ρ tothe fraction of times in which the measurement outcome was µ.
(iii) Use the estimates Π̃vµ of Πvµ ρ to compute estimates of Tv`,0 ρ ,T̃v`,0 ≡Xµ T`,0 µ Π̃vµ ,(8.38)µ

where the matrix elements of T`,0 are provided in Eq. (8.1).
V and the corresponding left singular(iv) Denoting the nonzero singular values of MV ` by M`kP
vectors by xV`k = v xV`kv |vi, compute the operators and coefficients

1 X  V ∗x`kv Tv`,0 ,
V
M`k
v
1 X V
ρ̃V`k ≡ V
x T̃v`,0 ,M`k v `kv

Q`k ≡

c

(8.39)(8.40)

To sample a point (α, β) from the uniform distribution on the sphere (with azimuthal angle α and polar angle β),
you can sample a point (a, b) ∈ [0, 1] × [0, 1] from the uniform distribution on the unit square, and then set α = 2πaand β = arccos (1 − 2b).

248

β̃(p)/β̃(0)1.0d = 10d = 20d = 40d = 80

2

4

0.5

0.0
0

p/d

Figure 8.3: Empirical measurement-adjusted error scales β̃(p) with p excess measurement axes,
determined by minimizing over 103 choices of measurement axes V or 5 minutes of runtime (foreach p), whichever comes first. Marker and color indicates the qudit dimension d. The rapid initial
drop in β̃(p) implies that using more measurement axes can substantially lower the upper bound onreconstruction error provided in Eq. (8.30), and that these benefits plateau after p ≈ d. Horizontal
reference lines mark the smallest measurement-adjusted error scales minθ βθ /β̃(0) achievable withthe method in Ref. [246], which is parameterized by an arbitrarily chosen angle θ.

and combine them into the estimate
ρ̃V =

X

ρ̃V`k Q`k ≈ ρ.

(8.41)`,k

The expected reconstruction error in ρ̃V , or its root-mean-square distance from ρ, is provided
by Eq. (8.32). If measurement error is shot-noise-limited, then the error in ρ̃V is approximately
EV (ρ) ≈ EV (ρ̃V ) and can be computed from Eq. (8.33). If ρ̃V has negative eigenvalues, its distancefrom ρ can be reduced with maximum-likelihood corrections [257], which will additionally guaranteethat ρ̃V satisfies all requirements for being a physical state.
The tomography protocol outlined above leaves open the question of how many measurement
axes to use. Though 2d − 1 measurement axes may be sufficient to perform full state tomography,
this is not necessarily the best choice of |V |. Increasing the number of measurement axes generally
decreases the quantum error scale V , but comes at the cost of having to estimate more observables.
At a fixed total number of measurements, increasing |V | reduces the number of measurements n
devoted to each axis v ∈ V . This trade-off begs the question: how should one choose the numberof measurement axes, |V |?

249
The reconstruction error bound in Eq. (8.30) nominally provides a straightforward answer:
at a fixed total number of measurements, N = n|V |, the number of measurement axes should
be chosen to minimize the (squared) reconstruction error EV (ρ)2 < 2V /n ∝ 2V |V |. We therefore
consider the measurement-adjusted error scale β(p) defined bynoβ(p)2 ≡ min 2V |V | : |V | = 2d − 1 + p ,V(8.42)

where p is the number of “extra” measurement axes exceeding 2d − 1. Though we cannot minimize
over all suitable choices of measurement axes V to compute β(p), we can compute an empiricalupper bound β̃(p) ≥ β(p) by minimizing over a large number of randomly chosen V . Figure 8.3
shows the results of such empirical minimization, where we find that β̃(p) drops substantially withp before plateauing at p ≈ d, after which there are only minor benefits to using more measurement
axes. In the interest of reducing experimental complexity as well as the runtime of our randomized
tomography protocol, which grow linearly in |V |, we therefore conclude that this protocol shouldbe performed with |V | ≈ 3d measurement axes. We provide the best measurement axes that we
found for a randomized tomography protocol with d ≤ 30 and |V | = 3d in Ref. [294].
For reference, Figure 8.3 also shows the smallest measurement-adjusted error scales βθ achievable with the method of Ref. [246], which is comparable to those achieved with our randomized
protocol at |V | ≈ 3d. The method of Ref. [246] requires choosing an angle θ, namely the polar angleof all measurement axes, and provides no prescription for making this choice. We therefore findthe optimal choice of θ by minimizing the error scale βθ over all θ (see Appendix 8.E), and show
min βθ /β̃(0) in Figure 8.3. Empirically, we find that the optimal angle for the method of Ref. [246]1
is θopt ≈ π2 (1 − 1.34d) (see Appendix 8.E), which approaches π/2 as d → ∞. However, the error

scale βπ/2 = ∞, reflecting the fact that full state tomography is impossible with measurement axes
lying in a single plane. The method of Ref. [246] therefore requires extremely careful fine-tuning of
measurement axis orientations for large spin dimensions. For this reason, we expect our randomized
tomography protocol be be more robust to errors in axis orientation. We leave a detailed analysisof robustness to errors in axis orientation and the effect of these errors on state reconstruction to250
future work.
As a final point, we note that any information about an unknown qudit state ρ, obtained fromprior knowledge or preliminary measurement data, can be used to construct tailored or adaptive
measurement protocols [258, 260, 303, 304] that are more efficient in terms of the number ofmeasurements required to estimate ρ to a fixed precision. We leave the development of tailored
and adaptive measurement protocols to future work as well.

Acknowledgements
We thank Anthony M. Polloreno and Jeremy T. Young for helpful feedback. This work is
supported by the DARPA DRINQs grant, the ARO single investigator award W911NF-19-1-0210,
AFOSR grant FA9550-19-1-0275, NSF grant PHY-1820885, NSF grant PHY-1734006 (JILA-PFC),
and by NIST.

8.A

Rotating polarization operators
Denoting the state of a spin-s particle spin spin projection µ onto a quantization axis by |sµi,

we define
s
X

Sz ≡

µ |sµihsµ| ,

µ=−s

S± ≡

s p
X

s (s + 1) − µ (µ ± 1) |s, µ ± 1ihsµ| ,(8.43)µ=−s

as well as
Sx ≡1(S+ + S− ) ,2iSy ≡ − (S+ − S− ) ,2S ≡ Sx , Sy , Sz .

(8.44)The spin vector S generates rotations of a spin-s system in 3D space. Specifically, the operator
e−iθS·n̂ rotates a spin-s system by an angle θ about the unit vector n̂.
Observing that Sz = T1,0 and S± ∝ T1,±1 , we can use the operator product expansion of
the polarization operators (see Appendix 8.D), the properties of Clebsch-Gordan coefficients, theproperties of Wigner 6-j symbols, and a computer algebra system to simplify the commutators[Sz , T`m ] = m T`m ,[S± , T`m ] =p` (` + 1) − m (m ± 1) T`,m±1 ,(8.45)

251
which implies that T`m is a spherical tensor operator, whose degree degree ` is preserved underrotations generated by S. Moreover, by comparing Eqs. (8.43) and (8.45) we see that the polarization operators T`m transform identically to spin-` particles under the (adjoint) action of the spin
operators Sz and S± . For any triplet of angles ω = (α, β, γ), we can therefore define the rotation
operator
R (ω) ≡ e−iαSz e−iβSy e−iγSz ,(8.46)and expand rotated polarization operators asTω`m ≡ R (ω) T`m R (ω)† =

`
X

`
Dmn
(ω̄)∗ T`n ,(8.47)n=−`where ω̄ = (γ, β, α) is the reversal of ω, and∗`Dmn(ω̄) ≡ `m R (ω̄) `n = (T`n |R (ω) ⊗ R (ω)∗ |T`m ) = (T`m |R (−ω̄) ⊗ R (−ω̄)∗ |T`n ) (8.48)

are matrix elements of the rotation operator R (ω) for spin-` particles.
` (v) = D ` (0, β, α)For any angle doublet v = (α, β), we define R (v) ≡ R (α, β, 0) and Dmn
mn

for shorthand. The transformation rules in Eq. (8.47) imply that we can expand the phase-space
representation of T`m as
D
E
PS
`
T`m
(v) ≡ hsv | T`m | sv i = s R (v)† T`m R (v) s = D0,m(v) s T`,0 s ,(8.49)

wherers T`,0 s =2` + 1hss; `, 0 | ssi =2s + 1s

2` + 12s + ` + 1



(2s)!
(2s + `)!




(2s)!
,
(2s − `)!

`and the properties of the rotation matrix elements Dmn
imply thatr
4π`D0,m (v) =Y`m (v) ,2` + 1(8.50)(8.51)

so
s
PS
T`m
(v) =

4π
2s + ` + 1



(2s)!
(2s + `)!



(2s)!
(2s − `)!

Y`m (v) .

In this way, the polarization operators are a quantum analogue of the spherical harmonics.

(8.52)

252

8.B

An improved reconstruction error boundIn Section 8.4 of the main text, we provided a reconstruction error bound using the assumptionof Eq. (8.20), namely that expectation values derived from spin projection measurements can be
estimated up to uncorrelated errors with maximal variance 2 . This assumption is reasonable if
measurement error is dominated by experimental sources of noise, and it yields a simple derivation of
the reconstruction bound in Eq. (8.24). Nonetheless, there are two problems with the assumptionof Eq. (8.20): (i) there is no a priori guarantee for the value of , which must be inferred fromexperimental outcomes, and (ii) the assumption that all measurement errors are uncorrelated isunjustified (and generally false). Here, we relax the assumption of Eq. (8.20) and derive an explicit
error bound in terms of the qudit dimension d and the number of spin projection measurements
made along every measurement axis.
To this end, we fix a particular set of measurement axes V , and consider performing nmeasurements of spin projection along every axis v ∈ V , for a total of N = |V | × n measurements.
Such a procedure is equivalent to making N local measurements of the N -fold product state ρ⊗N .
For convenience, we index the tensor factors of ρ⊗N by the integers (i, j), with i ∈ {1, 2, · · · , |V |}specifying a measurement axis vi ∈ V , and j ∈ {1, 2, · · · , n} specifying the copy of ρ prepared
for the j-th measurement spin projection along a particular axis. We then define the projectors
Πiµ ≡ µvi µvi onto single-qudit states µvi with definite spin projection µ along axis vi ∈ V ,and define Πjiµ to be an N -qudit operator with Πiµ on the (i, j)-th tensor factor and the identity
elsewhere. We denote the experimental outcome of measuring Πiµ in the (i, j)-th copy of ρ byΠ̃jiµ ∈ {0, 1}. In other words, Π̃jiµ is the “single-shot estimate” of Πiµ , with Π̃jiµ = 1 if outcomeµ was observed on the (i, j)-th experimental trial, and Π̃jiµ = 0 otherwise. An empirical estimateof the expectation value Πiµ ρ is provided by the fraction of times that outcome µ was observed
when measuring spin projection along axis vi , that isΠ̃iµ ≡

n

n

j=1

j=1


1X j
1 X  ⊗N j Π̃iµ ≈tr ρ Πiµ = tr ρΠiµ .
nn(8.53)For reasons that will be clarified shortly, it will be useful to think of Π̃iµ as an empirical estimate

253
of Π̄iµ ρ⊗N , wherenΠ̄iµ ≡

1X j
Πiµn(8.54)

j=1

is the average of Πiµ applied to all copies of ρ for which spin projection is measured along the axis
vi . Eq. (8.53) implies thatΠ̃iµ ≈ Π̄iµ ρ⊗N = Πiµ ρ .
8.B.1(8.55)

Errors in the spin-projection basis

Finite sampling error (i.e. shot noise) generally induces statistical error O into the empiricalestimate Õ of an observable O:O ≡ Õ − hOi ,(8.56)

where the single brackets h·i denote an expectation value with respect to the measured quantum
state. On average, this statistical error will be zero, which is to say thathhO ii = hhÕ − hOiii = O − hOi = 0,(8.57)

where the double brackets hh·ii to denote statistical averaging over experimental trials that estimate
hOi. However, the covariance between statistical errors O and Q on the empirical estimates Õ
and Q̃ of observables O and Q is
*
+



D

E
hhO Q ii =Õ − hOi Q̃ − hQi= O − hOi Q − hQi = hOQi − hOi hQi .

(8.58)In the context of spin qudit tomography, we can therefore define the statistical error
iµ ≡ Π̃iµ − Πiµ ρ = Π̃iµ − Π̄iµ ρ⊗N(8.59)

in the empirical estimate of Πiµ ρ , and use Eq. (8.54) to expandhhiµ i0 µ0 ii = Π̄iµ Π̄i0 µ0 ρ⊗N − Π̄iµ ρ⊗N Π̄i0 µ0 ρ⊗N

n 
D
E
D 0 E
1 X D j j0 E= 2
Πiµ Πi0 µ0 ⊗N − Πjiµ ⊗N Πji0 µ0 ⊗N .
n
ρ
ρ
ρ
0
j,j =1(8.60)(8.61)

254

0
If (i, j) 6= i0 , j 0 , then Πjiµ and Πji0 µ0 address different tensor factors of the product state ρ⊗N ,hiso the expectation value of their product factorizes due to the fact that tr (A ⊗ B) A0 ⊗ B 0 =tr AA0 × tr BB 0 . This factorization can also be seen as a consequence of the fact that if0(i, j) 6= i0 , j 0 , then Πjiµ and Πji0 µ0 are “spatially separated” on ρ⊗N , which means that their
expectation values cannot have quantum correlations. The terms in Eq. (8.61) with (i, j) 6= i0 , j 0
therefore vanish, so

n 
D
E
D
E
1 X D j j Ejjhhiµ i0 µ0 ii = δii0 × 2
Πiµ Πiµ0 ⊗N − Πiµ ⊗N Πiµ0 ⊗Nnρρρ(8.62)

j=1i1h
Πiµ Πiµ0 ρ − Πiµ ρ Πiµ0 ρn
1= δii0 × covρ Πiµ , Πiµ0 ,n= δii0 ×(8.63)(8.64)

where covρ (X, Y ) ≡ hXY iρ − hXiρ hY iρ .
8.B.2

Errors in the polarization operator basis

Rather than the statistical errors iµ ≡ Π̃iµ − Πiµ ρ in the estimates Π̃iµ of the projectors
Πiµ , we now consider the statistical errors i` ≡ T̃i` − hTi` iρ in the estimates T̃i` of the polarization
operators Ti` ≡ Tvi `,0 . We can expand the polarization operators Ti` as a sum over projectors Πiµ
as
r
Ti` =

X

t`µ Πiµ ,t`µ ≡ µ T`,0 µ =µand likewise T̃i` ≡2` + 1hsµ; `, 0 | sµi ,d(8.65)Pµ t`µ Π̃iµ . The covariance between errors in the polarization operator basis isthenhhi` i0 `0 ii =Xµ,µ0t`µ t`0 µ0 hhiµ i0 µ0 ii = δii0 ×
1X
1t`µ t`0 µ0 covρ Πiµ , Πiµ0 = δii0 × covρ (Ti` , Ti`0 ) ,
n 0
n
µ,µ

(8.66)
where we used the fact that the covariance covρ (X, Y ) is linear in both X and Y . Due to theappearance of δii0 above and the orthogonality of polarization operators Ti` and Ti0 `0 with degrees

255` 6= `0 , it turns out that only the variances hh2i` ii will ultimately contribute to reconstruction error(see Appendix 8.B.3). We therefore seek to find an upper bound on hh2i` ii.
To this end, we define the probability piµ ≡ Πiµ ρ , collect these probabilities into the classicalPPprobability distribution pi = µ piµ |µi, and define the vector t` ≡ µ t`µ |µi. We then observe
that
2


hh2i` ii =1
× σp2i (t` ) ,nσp2 (X) ≡X

pµ Xµ2 − µXpµ Xµ  ,(8.67)µ

where σp2 (X) is the weighted variance of X. This variance is maximal when p has equal weight on
the largest and smallest values of X, which implies that
σp2 (t` ) ≤ Γ2` ,Γ` ≡maxµ t`µ − minµ t`µ,
2sohh2i` ii ≤1
× Γ2` .
n(8.68)

Note that this bound on hh2i` ii is tight, as equality is achieved by the state
ρ?i =


1
Πiµmax + Πiµmin ,2(8.69)

where µmax (µmin ) is the index that maximizes (minimizes) t`µ .
To find an analytical bound on hh2i` ii that is easier to interpret, we can use normalizationPof the polarization operators, (Ti` |Ti` ) = µ t2`µ = 1, and the fact that all probabilities pµ ≤ 1 tobound
σp2 (t` ) ≤X

pµ t2`µ ≤µXt2`µ = 1,sohh2i` ii <

µ

1
.
n

(8.70)

We can get a tighter bound by considering the fact that t2`µ = t2`,−µ due to the symmetries of theClebsch-Gordan coefficients. It follows that if µmax 6= 0 then
σp2 (t` ) ≤X

pµ t2`µ ≤ t2`µmax =µ µmax 6=0 1 X
1 2
1t`µmax + t2`,−µmax≤
t2 = .
22 µ `µ 2(8.71)

If µmax = 0, then similarlyst2`µmax + 2t2`µmin = t2`µmax + t2`µmin + t2`,−µminµmax =0 X≤µt2`µ = 1, so |t`µmin |µmax =0≤

1 − t2`µmax
2,(8.72)

256
which lets us bounds1Γ` =t`µmax − t`µmin2

 µmax =0 111≤t`µmax + |t`µmin |≤t`µmax +22
2

1 − t2`µmax
2

≡ λ t`µmax .
(8.73)

It is straightforward to show that λ (x) is maximally λ? ≡ maxx λ (x) =Γ2`µmax =0

≤

λ?

2

=

p
3/8, so

3
1
< .
8
2

(8.74)

Altogether, we thus find that in all cases1
σp2 (t` ) ≤ ,2
8.B.3

hh2i` ii ≤

so

1
.
2n

(8.75)

Revisiting the reconstruction error bound

We now revisit the derivation of reconstruction error in Section 8.4 to make use of the bounds
on variances hh2i` ii. To recap, for a set of measurement axes V = {v} and degrees ` ∈ {0, 1, · · · , d−1}
we construct the measurement matrix
MV ≡X|v`i (Tv`,0 | ,(8.76)v,`

which can be block diagonalized as
MV U =

X|`ih`| ⊗ MV ` ,

U≡

`

X

|T`m ) h`m| ,MV ` =

X`Dm,0(v) |vihm| ,(8.77)

m,v`,m` (v) ≡ `m R (v) `n is a (Wigner) rotation matrix element for a spin-` particle.
where Dmn
V and corresponding
The block-diagonal structure of MV allows us to index its singular values M`mP(normalized) left singular vectors xV`m = i xV`mi |vi i by the indices (`, m), where the integer |m| ≤`. These singular vectors and values define the orthonormal operators
QV`m ≡

X

V
q`mi

∗

V
q`mi
≡

Ti` ,

i

xV`mi
,
V
M`m

(8.78)

where i ∈ {1, 2, · · · , |V |} indexes an axis vi ∈ V , with Ti` ≡ Tvi ` . The state ρ can be expanded in
the basis of these operators as
ρ=

XD
`,m

†
QV`m

E
ρ

QV`m ,(8.79)

257
and the estimates T̃i` of hTi` iρ can be used to construct the following estimate ρ̃V of ρ:




E
X X
X X
XD
V
V


ρ̃V ≡
q`mi
T̃i`  QV`m ≈q`mihTi` iρ  QV`m =QV`m QV`m = ρ.
`,m

i

i

`,m

`,m

ρ

(8.80)

Recalling that i` ≡ T̃i` − hTi` iρ , we can use orthonormality of all QV`m to expand the mean squared
distance between ρ̃V and ρ as
D
E
∗
X 
X
1X 2 2
2
VV
V
EV (ρ) ≡ kρ̃V − ρk
=
q`mi
q`mi
|q`mi
|2 hh2i` ii <Γ` SV ` , (8.81)
0 hhi` i0 ` ii =
n
0
2

`,m,i

`,m,i,i`where we used the fact that hh2i` ii ≤ Γ2` /n, and
X
m,i

V
|q`mi
|2 =

X

V
M`m

−2

= MV−1` = SV2 ` .

(8.82)m

Here MV−1` is the left inverse of MV ` . The fact that hh2i` ii < 1/2n also implies that
EV (ρ)2 <S2
1 X 2
SV ` = V .
2n2n(8.83)`

Note that the bound in Eq. (8.81) is not tight, as the individual bounds on the variances hh2i` ii
cannot all be achieved simultaneously. There is therefore still room for improvement on the boundin Eq. (8.30) by maximizing EV over the set of physically achievable qudit states ρ.

8.C

Exact reconstruction error
Here we find exact expressions for reconstruction error, which can be used to estimate theerror in a given reconstruction ρ̃V of an unknown state ρ after performing tomography. To thisend, we start with Eq. (8.81) from Appendix 8.B.3 to write
EV (ρ)2 =

X 

V
q`mi

∗

`,m,i,i0

where q̃`i =Vq`mi
0 hhi` i0 ` ii =1X
|q̃`i |2 covρ (Ti` , Ti` ) ,n|q̃`i |2 =`,iX|q`mi |2 ,(8.84)m∗m (q`mi ) |mi, and we used the fact that hhi` i0 ` ii = δii0 ×covρ (Ti` , Ti` ) /n. IdentifyingP†
the singular value decomposition MV ` = UV ` ΣV ` WV† ` , we then we observe that q̃`i = Σ−1
V ` UV ` |vi i,

which allows us to simplify

D
E 

†
−2 †
−1
−1
|q̃`i | = vi UV ` ΣV ` UV ` vi = vi MV ` MV ` vi .
2(8.85)

258
Using the fact that all Ti` = Ti`† , we can also expand




X
†
∗
`
`
covρ (Ti` , Ti` ) = covρ Ti`† , Ti` =D0,m(vi ) D0,m,0 (vi ) covρ T`m , T`m0

(8.86)

m,m0

which implies that
EV (ρ)2 =



1 X
†
∗
`
2 `
D0,m
.
0 (vi ) |q̃`i | D0,m (vi ) covρ T`m , T`m0
n0

(8.87)`,i,m,mAltogether, this reconstruction error can be expressed more compactly by defining the covariancematrixC` [ρ] ≡X†covρ T`m, T`m0 m m0 ,(8.88)

m,m0

and the noise matrix
NV ` ≡ MV† ` diag



MV−1`

†

MV−1`


MV ` ,(8.89)

where diag [X] sets all off-diagonal entries of X to zero, in terms of which
EV (ρ)2 =1X(NV ` |C` [ρ]) ,n(8.90)`


where (X|Y ) = tr X † Y is a trace inner product.
The result in Eq. (8.90) essentially expresses reconstruction error as a weighted sum of thecovariances covρ (T`m , T`m0 ), where the weights are given by the corresponding matrix elementsof the noise matrix NV ` . This expression is perhaps the most physically meaningful form of thereconstruction error EV (ρ) that we will consider in this work, but in practice it turns out thatEq. (8.90) is inconvenient and inefficient to evaluate for any given state ρ. To find a more practicalexpression of reconstruction error, we use the fact that
D

†
T`m

E
ρ





†
†
†
= (ρ|T`m) = tr ρT`m
= tr T`m
ρ = (T`m |ρ) ,(8.91)

259
to expand the covariance matrix asC` [ρ] =

X

m m0

h

†
†
(ρ|T`mT`m0 ) − (ρ|T`m) (ρ|T`m0 )i(8.92)

m m0

h

†
†
(T`m
0 T`m |ρ) − (T`m |ρ) (T`m0 |ρ)i(8.93)


m m0 I (T`m0 T`m |ρ) − (T`m |ρ) (T`m0 |ρ)(8.94)

m,m0

=

X
m,m0

=

X
m,m0

where we define the inversion operator I ≡mm (−1) |−mihm|.

P

We then expand the productT`m0 T`m as(T`m0 T`m |ρ) =

X

L
g`m
0 m (TL,m0 +m |ρ) ,0

L,m +mLg`m0 m ≡ (TL,m0 +m |T`m0 T`m ) = f`m0 ;`m ,(8.95)

L
0

L,m +mwhere the (real) factors f`m
are provided in Appendix 8.D. Substituting the covariance matrix
0 ;`mback into Eq. (8.90) and replacing (T`m |ρ) → ρ`m , we get




X
X
∗
1
EV (ρ)2 = χV`m ρ`m −m0 INV ` m ρ`m ρ`m0  ,n
0`,m(8.96)

`,m,m

where
χVLM ≡

X

δM,m0 +m m0 INV ` m∗ Lg`m0 m(8.97)`,m,m0=X

0L
δM,−m0 +m m NV ` m0 (−1)m g`,−m0 ,m(8.98)`,m,m0=X(NV ` |DM |IgL` )(8.99)`

can be written in terms of the matrices
gL` ≡

X

L
0
g`m
m ,
0m m

m,m0

DM ≡

X

δM,−m0 +m m0 m m0 m .

(8.100)m,m0Here DM simply picks off the M -th diagonal of the matrix it acts on, such that (NV ` |DM |IgL` ) is
an inner product of the M -th diagonal of IgL` with the (−M )-th diagonal of NV ` . Defining the(2` + 1)-component vectors|ρ` i ≡

Xm

ρ`m |mi ,|χV ` i ≡X`,m

χV`m |mi ,(8.101)

260
we can write the expansion in Eq. (8.96) in the vectorized form
EV (ρ)2 =
1 XhχV ` | ρ` i − hρ` | NV ` | ρ` i .
n(8.102)`

8.C.1

Comments on a tight reconstruction error boundIn principle, maximizing the reconstruction error in Eq. (8.102) over all qudit states ρ would
provide a tight upper bound on reconstruction error for any set of axes V . To simplify this task
somewhat, we first maximize Eq. (8.102) over all ρ with tr (ρ) = 1: this maximum occurs at a“state” σV? whose components are given byσV? ``6=0 1

≡

2

E
1
?
σV,0
≡ √ |0i .
d

NV−1` |χV ` i ,(8.103)The corresponding maximum of EV is given by
EV σV?

2

=



E 1
1X 1D
χV ` NV−1
χ
−
tr
(N
)
,
V`
V`
`
n
4
d

(8.104)`>0

where the tr (NV ` ) terms above come from simplifying the ` = 0 terms of Eq. (8.102) with ρ → σV? .
While EV σV? is a strict upper bound on EV (ρ) over all ρ with tr (ρ) = 1, this bound turns out tobe useless in practice, because σV? will generally be a non-physical “state” with negative eigenvalues.
To find tight bound on EV (ρ) over the space of physical qudit states ρ, we also need to constrainρ to have no negative eigenvalues. Equipped with σV? and EV σV? , we can expand
EV (ρ)2 = EV σV?
where X` ≡

P

2

−

1
ρ − σV?
n2,
V

kXk2V ≡

X

hX` | NV ` | X` i ,(8.105)`m (T`m |X) |mi is a vector of the degree-` components of X

in the polarization

operator basis, and kXkV is a noise-weighted norm of X. Maximizing EV over all qudit states ρthus amounts to finding the closest physical qudit state ρ to σV? , with distance measured by the
metric DV (X, Y ) ≡ kX − Y kV . We leave this minimization problem to future work, and note thatsolving it will likely require making use of the positivity conditions derived in Ref. [295]. A loose
lower bound on ρ − σV?

V

can be found by minimization under the constraint kρk ≤ 1, which may

provide a tighter upper bound on EV (ρ) than that in Eq. (8.30) of the main text.

261

8.D

Polarization operator product expansion
The polarization operators on the d-dimensional Hilbert space of a spin-s system (with s ≡d−12 ) are defined by

r
T`m ≡

s
2` + 1 X
hsµ; `m | sνi |νihµ| ,2s + 1 µ,ν=−s(8.106)

where hsµ; `m | sνi is a Clebsh-Gordan coefficient that enforces ` ∈ {0, 1, · · · , 2s} and m ∈ {−`, −`+1, · · · , `}. We wish to compute the coefficients of the operator product expansion
T`1 m1 T`2 m2 =

X

f`LM
T ,
1 m1 ;`2 m2 LM

f`LM≡ (TLM |T`1 m1 T`2 m2 ) ,1 m1 ;`2 m2(8.107)

L,M

which allow us to simplify the commutators in Eq. (8.45) of Appendix 8.A. Using the symmetry
properties of Clebsch-Gordan coefficients, namelyrh`1 m1 ; `2 m2 | LM i = (−1)`2 +m2

2L + 1hL, −M ; `2 m2 | `1 , −m1 i
2`1 + 1h`1 m1 ; `2 m2 | LM i = (−1)`1 +`2 −L h`1 , −m1 ; `2 , −m2 | L, −M i ,we can find that the polarization operators transform under conjugation asr2` + 1 X†T`m =(−1)m hsν; `, −m | sµi |µihν| = (−1)m T`,−m ,2s + 1 µ,ν

(8.108)(8.109)(8.110)which implies thatsf`LM
= (−1)M
1 m1 ;`2 m2(2L + 1) (2`1 + 1) (2`2 + 1) X
hsν; L, −M | sµi hsρ; `1 m1 | sνi hsµ; `2 m2 | sρi .
(2s + 1) (2s + 1) (2s + 1) µ,ν,ρ
(8.111)

Replacing Clebsch-Gordan coefficients by Wigner 3-j symbols with the identity2`2h`1 m1 ; `2 m2 | LM i = (−1)

L−M

(−1)

√
2L + 1L−M` 2 `1,m2 m1(8.112)

we can use the fact that 2`2 is always even (because `2 is always an integer) to expandf`LM= (−1)M
1 m1 ;`2 m2

p(2L + 1) (2`1 + 1) (2`2 + 1)XsL
3s−µ−ν−ρ
×
(−1)
−µ −M
µ,ν,ρ

s
ν



s
`1 s
−ν m1 ρ




s
`2 s
. (8.113)−ρ m2 µ262This sum can be simplified by the introduction of Wigner 6-j symbols, giving us


`1
`2
L ` 1 `2
−m1 −m2
s s s


p
` ` L
= (−1)2s+L (2`1 + 1) (2`2 + 1) h`1 m1 ; `2 m2 | LM i 1 2
.
s s s

f`LM
= (−1)2s+M
1 m1 ;`2 m2

8.E

p(2L + 1) (2`1 + 1) (2`2 + 1)

LM

(8.114)(8.115)

Optimizing the method of Newton and Young
Ref. [246] constructs an explicit protocol for spin qudit tomography, which involves measuring

spin projection along 2d − 1 axes equally spaced at a polar angle θ. However, this method does notprovide any prescription for choosing θ. Here, we show the importance of making a good choice ofθ, and empirically find the optimal value of θopt that minimizes the corresponding quantum error
scale θ , which controls state reconstruction error. To this end, Figure 8.4 shows the quantum error
scale θ as a function of the polar angle θ in the tomography method of Ref. [246] for a few qudit
dimensions d. While a good choice of θ yields an error scale θ ≈ d (for the dimensions shown), thiserror scale can increase by orders of magnitude for poor choices of θ. In turn, Figure 8.5 shows the1
optimal angle θopt as a function of the qudit dimension d, together with a fit to θopt = π2 (1 − xd)finding x ≈ 1.34.

263

103

d=2

d=3

d=4

d = 10d = 20d = 40

θ /d

102
101
100

103

θ /d

102
101
100
0.00

0.25

0.50

0.75

1.00 0.00

θ
π/2

0.25

0.50

0.75

1.00 0.00

θ
π/2

0.25

0.50

0.75

1.00

θ
π/2

Figure 8.4: Quantum error scale θ as a function of the polar angle θ in the tomography methodof Ref. [246] for a few qudit dimensions d.

data
fit

θ

opt
1 − π/2

10−1

10−2
101
d

102

Figure 8.5: Optimum angle θopt as a function of qudit dimension d for the tomography method of
1
Ref. [246], and a fit to θopt = π2 (1 − xd
) finding x ≈ 1.34.

Chapter 9

Summary and conclusions

We examined some near- and mid-term prospects for quantum simulation and metrology withultracold atomic systems. We began by introducing the essential players and recurring concepts
in our work, including alkaline-earth(-like) atoms AEA and the properties that make them sucha valuable tool for precision science. We discussed the basic physics of individual trapped atoms,
the spin-orbit coupling induced by externally addressing atoms’ internal states, and the SU(n)symmetric interactions that feature prominently in our later work to explore and exploit the manybody physics of AEAs. Considering the limit of frozen spatial degrees of freedom, we discussed
the description of AEAs using both on-site and collective spin operators. Collective spin operators,
in turn, are essential for describing the dynamics and properties of a permutationally symmetricspin system, which we used to introduce spin squeezing as the dominant strategy in this thesis forrealizing a quantum advantage in state-of-the-art quantum sensors.
Equipped with this background, in Chapter 3 we took a deep dive into the emergence ofmulti-body interactions between AEAs. These multi-body interactions inherit the SU(n) symmetryof atoms’ two-body interactions, which simplifies the description of their multi-body excitation
eigenstates. We then presented, in Chapter 4, a proposal to combine spin-orbit coupling and twobody interactions to prepare spin-squeezed states in an AEA-based optical lattice clock. Along theway, we developed (in Chapter 5) a new method to simulate the short-time dynamics of collective
spin systems in strong-decoherence regimes that are inaccessible using other numerical methods.
Borrowing some of the key ideas from Chapter 4, we considered the possibility of spin squeezing in a

265
broader class of spin models (called XXZ models) with interactions that decay as a power-law, whichhave been previously demonstrated in a variety of cold atomic, molecular, and optical platforms.
We then combined elements of Chapters 3 and 4 to study a multilevel spin model with infinite-range
interactions in Chapter 7. Finally, we considered the problem of precisely characterizing collective
observables in the model of Chapter 7, under some realistic restrictions on experimental capabilities.
We mapped this problem onto that of spin qudit tomography, which we studied closely in Chapter
8. Going forward, we hope that the techniques and theory in this thesis prove useful for the pursuitof scientific inquiry and development of quantum technologies in AEA-based platforms.

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