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Quantum Computation Toolbox for Decoherence-Free
Qubits Using Multi-Band Alkali Atoms
Mikhail Mamaev,* Joseph H. Thywissen, and Ana Maria Rey
dipole blockade of atomic ensembles,[6]qubits encoded in vibrational atomic
spin-dependentlattices,[9]states,[7,8][10,11]exchangeor spin-orbit coupleddriving,[12] optical tweezers,[13,14] Rydberg atoms[15–17] and several others.[18]High-ﬁdelity gates have been engineeredusing Rydberg atoms,[19–21] and targeted
phase shifts.[22] Signiﬁcant progress has
also been made toward near-perfectinitialization.[23–25] However, parallel generation of large entangled states remains an
open challenge.
In this work we propose to use fermionicalkali atoms such as 40 K in 3D optical lattices to do initialization, state manipulation, and entanglement generation with logical qubits encoded in states generated by
two atoms in two motional bands. Higher
bands of optical lattices provide usefuladditional degrees of freedom,[26] and have been used for studies of superﬂuidity,[27–29] magnetism,[30,31] interferometry,[32] andinterplay with collisional interactions.[33,34] Our logical qubits
are designed to live in a decoherence-free subspace that resists unwanted eﬀects of stray magnetic ﬁelds, leading to much
longer coherence times[35,36] and accessibility to regimes utilizing stronger magnetic ﬁelds for dynamical control. We arefurther able to take advantage of the tunability of interactions
via Feshbach resonances,[37–39] ﬁeld gradients, and single-site
resolution capabilities oﬀered by state-of-the-art quantum gas
microscopes[40–44] as powerful tools for qubit manipulation.
Our implementation makes use of the intra-band contact
interactions as well as inter-band exchange interactions to implement logic qubit rotations and entanglement generation. We
do not require continuous laser driving, nor any superlattice orspin-dependent lattice conﬁgurations. Through the additionaluse of a ﬁeld gradient, we are able to tune the nearest-neighborinteractions between qubits to generate a desired Hamiltonian.
As a proof-of-principle concept of measurement-based quantum
computation[45,46] we show how to realize an Ising model withsigniﬁcantly stronger spin couplings compared to conventional
superexchange, which we show can create high ﬁdelity 1D cluster
states. One can also tune the system to emulate XXZ or Heisenberg models, which may be used (as well as the Ising) to measurecomplex time-dependent quantities such as out-of-time-orderedcorrelation functions (OTOCs).[47,48] The interactions can be
manipulated via Feshbach resonances, allowing us to turn themoﬀ and on as needed. The single-site resolution also permits
the manipulation of individual atoms before and during such

Protocols for designing and manipulating qubits with ultracold alkali atoms in
3D optical lattices are introduced. These qubits are formed from two-atomspin superposition states that create a decoherence-free subspace immune tostray magnetic ﬁelds, dramatically improving coherence times while still
enjoying the single-site addressability and Feshbach resonance control ofstate-of-the-art alkali atom systems. The protocol requires no continuous
driving or spin-dependent potentials, and instead relies upon the populationof a higher motional band to realize naturally tunable in-site exchange andcross-site superexchange interactions. As a proof-of-principle example of theirutility for entanglement generation for quantum computation, it is shown thatthe cross-site superexchange interactions can be used to engineer 1D cluster
states. Explicit protocols for experimental preparation and manipulation of thequbits are also discussed, as well as methods for measuring more complex
quantities such as out-of-time-ordered correlation functions (OTOCs).

1. Introduction
The ﬁeld of ultracold atomic physics in optical lattices has seen
tremendous recent growth in its experimental implementations.
There are many systems which feature unprecedented levels of
cleanliness, environmental isolation, coherence time, and singlesite addressability. These capabilities, especially the latter, havemotivated recent experiments to apply optical lattice systems to
the more ambitious goals of site-resolved quantum simulation[1]and quantum computation.[2] An ultracold atomic system realizing a quantum computer can oﬀer the aforementioned beneﬁts
of coherence and isolation together with improved scalability, due
to the number of atoms these systems can load simultaneously.
The prospect of quantum computing with ultracold neutral
atoms has been explored in many contexts. These include theuse of collisional gates,[3,4] collective states via cavity QED[5] or
M. Mamaev, Prof. A. M. Rey
JILA, NIST and Department of Physics
Center for Theory of Quantum Matter
University of Colorado
Boulder, CO 80309, USA
E-mail: mikhail.mamaev@colorado.edu
Prof. J. H. Thywissen
Department of Physics and CQIQC
University of Toronto
Ontario M5S 1A7, Canada
The ORCID identiﬁcation number(s) for the author(s) of this article
can be found under https://doi.org/10.1002/qute.201900132

DOI: 10.1002/qute.201900132

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Figure 1. a) Schematic of the optical lattice setup. Atoms can only tunnel along the x direction (eﬀective 1D system). Each site holds two atoms (oneper band), which exhibit intra-band repulsion and inter-band exchange interactions. b) Band schematic for a single lattice site. The atoms are loaded
into the ground motional band g, and the ﬁrst-excited band e with one motional excitation along the x direction. Other singly excited bands along y, z
are not populated, and are nondegenerate with e due to higher lattice depths along y, z. Gray numbers in brackets are the harmonic excitation numbers,and 𝜔𝜈 for 𝜈 ∈ {x, y, z} are the on-site trapping frequencies. c) Eﬀective Bloch sphere for the decoherence-free subspace. Logical 0 (1) states |⇓⟩ (|⇑⟩) are(anti-)symmetric superpositions of two diﬀerent Wannier functions, with the opposite symmetry for the spin wavefunctions due to the overall fermioniĉ U and gradient H
̂ B , and are immune to uniform external magnetic ﬁelds.
nature of the atoms. Both these states are eigenstates of the interaction H

computations, allowing for eﬀective error detection and insighton what the quantum system is doing on a per-site basis.
In Section 2 we give an overview of our model, introduce thedecoherence-free subspace used to deﬁne our logic qubits, andthen derive the superexchange interactions between them. In
Section 3 we detail how these interactions can be tuned throughan external ﬁeld gradient, and show how to use them for cluster state generation and OTOC measurement. In Section 4 we
describe diﬀerent protocols for qubit initialization and readout.

e-band tunneling is roughly 40 times faster due to the more delocalized nature of excited motional states. Since we will be working
in the Mott-insulating regime, lowest-order tunneling dynamics will be second-order superexchange processes proportional totunneling squared (hence ∼ 402 times weaker for ground bandtunneling).
The atoms also have an on-site interaction, whose Hamiltonian is
̂U =
H

∑(
)
Uee n̂ j,e,↑ n̂ j,e,↓ + Ugg n̂ j,g,↑ n̂ j,g,↓j

2. Decoherence-Free Subspace Qubits
2.1. Multi-Band Fermi-Hubbard Model
The system we work with is a 3D optical lattice populated byfermionic atoms, as depicted in Figure 1a. The atoms are prepared in their ground electronic state, and restricted to two populated hyperﬁne states which we denote as 𝜎 ∈ {↑, ↓} acting asa spin-1/2 degree of freedom. We assume the atoms are loaded
into two motional bands e and g, with e an excited band holding
one motional excitation along the x direction in the harmonic approximation (as shown in Figure 1b), and g the lowest band. The
lattice depth is made more shallow along the x direction, conﬁning the system to an eﬀective 1D conﬁguration. We assume
a lattice with L lattice sites along the x direction, each one populated by two atoms (N atoms, N = 2L), one in e and one in g.
The full Hamiltonian describing the system is a two-band FermiHubbard model given by
̂U +H
̂B
̂ =H
̂J +H
H

(1)

where the tunneling Hamiltonian iŝ J = −JH)∑( †ĉj,e,𝜎 ĉj+1,e,𝜎 + h.c.

(2)

j,𝜎

with ĉj,𝜇,𝜎 annihilating an atom on site j in band 𝜇 ∈ {e, g} withspin 𝜎. The lattice indexing is along x only. We have assumed
that only e-band atoms can tunnel with rate J, and neglected anytunneling of g-band atoms. For the parameter regime of interest,

Adv. Quantum Technol. 2020, 1900132+

Ueg ∑ ()
n̂ j,e,↑ n̂ j,g,↓ + n̂ j,e,↓ n̂ j,g,↑
2 j

−

)
Ueg ∑ ( ††ĉj,g,↑ + h.c.
ĉj,e,↑ ĉj,e,↓ ĉj,g,↓
2 j

(3)

†
where n̂ j,𝜇,𝜎 = ĉj,𝜇,𝜎ĉj,𝜇,𝜎 . The ﬁrst line is the interaction energy between atoms in the same band (Uee and Ugg for e and g bands
respectively), while the last two lines are direct and exchange interactions of strength Ueg between atoms in two diﬀerent bands.
The values for these interactions depend on the lattice depths,
but their magnitudes can be globally tuned by using a Feshbachresonance to modify the scattering length. We will operate in thestrongly interacting regime where Uee , Ugg , Ueg ≫ J. Table 1 gives
an overview of sample experimentally realistic parameters for a40
K lattice; these are the parameters used in all plots throughout the paper unless otherwise speciﬁed. See Section A, Supporting Information, for details on the derivation of Equation (1) andits parameters.
In addition to the core lattice dynamics, we also permit an externally imposed linear ﬁeld gradient along the x direction,

̂B = BH2∑()
j × n̂ j,e,↑ − n̂ j,e,↓ + n̂ j,g,↑ − n̂ j,g,↓(4)jwhere B is the diﬀerential energy shift between sites. This shift
could be implemented with a direct magnetic ﬁeld gradient, orwith the synthetic magnetic gradient of a vector light shift that

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Table 1. Sample system parameters for an optical lattice conﬁguration using 40 K atoms. Here Er = h2 ∕(8ma2 ) ≈ h × 4.5 kHz is the recoil energy,
using lattice spacing of a = 527 nm and mass of m = 40 amu. Interactions are computed using a scattering length of as = 120a0 (with a0 the
Bohr radius). The value for a typical diﬀerential shift is chosen to giveB = |U3 | + 10J.
Parameter

SymbolValue

Lattice depth(Vx , Vy , Vz )∕Er(40,60,60)

Excited band tunneling strengthJh × 19 Hz

Excited band repulsion

Ueeh × 3.4 kHz

Inter-band exchange interaction

Ueg

h × 4.5 kHz

Ground band repulsion

Ugg

h × 4.7 kHz

Superexchange energy
denominators

U1 = 2Uee + Ueg

h × 11.3 kHz

U2 = 2Uee − Ueg

h × 2.2 kHz

U3 = 2Uee − 3Ueg−h × 6.8 kHzB

h × 7.0 kHz

Diﬀerential energy shift
between sites
Superexchange spin modelparmameters
Cluster state generation timeJ∥h × 0.47 Hz

J⟂−h × 0.51 Hz

tc

260 mscreates a diﬀerential potential between the two spin states. This
gradient can modify the eﬀective superexchange interactions between adjacent sites, providing a tuning parameter to manipulate
the system dynamics. We do not model the eﬀect of a commonmode (spin-independent) shift between sites; however, such ashift would provide an alternate tuning parameter to control thesuperexchange dynamics, as discussed below.
2.2. Decoherence-Free Subspace
The spin states 𝜎 feel diﬀerent Zeeman shifts from external
magnetic ﬁelds and are therefore vulnerable to uniform magnetic ﬁeld ﬂuctuations which severely reduce coherence times.
To mitigate this eﬀect, we use equal-weight superpositions of thespin states on every lattice site, which will feel no linear shift
from an external magnetic ﬁeld, thereby realizing a decoherencefree subspace.[35] These√superpositions are the singlet and triplet
states, (|↑, ↓⟩ ∓ |↓, ↑⟩)∕ 2 (written in ﬁrst-quantized notation for
two atoms on the same site). Their corresponding spatial wavefunctions must uphold the opposite symmetries to maintainoverall fermionic parity, which can be accomplished by putting
them in respective symmetric and anti-symmetricsuperpositions√of the e, g band states, (|e, g⟩ ± |g, e⟩)∕ 2. The states for a single
lattice site may thus be written as
|⇑⟩ =

(
)
1
(|↑, ↓⟩ + |↓, ↑⟩) |e, g⟩ − |g, e⟩21= √ (|1, 0, 0, 1⟩ + |0, 1, 1, 0⟩)
2
|⇓⟩ =

(
)
1
(|↑, ↓⟩ − |↓, ↑⟩) |e, g⟩ + |g, e⟩21= √ (|1, 0, 0, 1⟩ − |0, 1, 1, 0⟩)2Adv. Quantum Technol. 2020, 1900132

Figure 2. a) Schematic for superexchange in the multi-band system. Statesin the decoherence-free subspace are coupled through a second-order virtual tunneling process by atoms in the e band. The intermediate state will
have two atoms in the same band on the same site, exhibiting Uee repulsion. Provided the energy gaps between the occupied and intermediate
states are large compared to J, this leads to a spin-like interaction in thedecoherence-free subspace. b) Benchmark comparison between the fullFermi–Hubbard model of Equation (1) and the superexchange model of
Equation (8). The time tc is the point where ⟨𝜎̂ x ⟩ vanishes, corresponding
to the time needed to make a cluster state (see Section 3). Note that theoscillation frequency seen is not representative, and is an artifact of ﬁnite
point sampling; the genuine oscillation frequency is set by Ueg .

where in the second lines of each equation we have rewritten thestates in second quantization assuming a Fock basis ordering of|ne,↑ , ne,↓ , ng,↑ , ng,↓ ⟩. These states are also eigenstates of the interaction Hamiltonian,̂ U |⇑⟩ = 0
H(6)

̂ U |⇓⟩ = Ueg |⇓⟩Hand zero-energy eigenstates of the Zeeman Hamiltonian,̂ B |⇓⟩ = 0. For experimentally realistic parameters
̂ B |⇑⟩ = H
H(Table 1), the exchange interaction energy diﬀerence Ueg between
these states can exhibit a > 100-fold reduction in sensitivity to external magnetic ﬁeld ﬂuctuations compared to the shifts that barenuclear-spin states {e, g} would experience.
Altogether, the decoherence-free states {|⇑⟩, |⇓⟩} form a spin1/2 logical qubit subspace on each site of the optical lattice, as
shown in Figure 1c. We can have states along diﬀerent√axesof the associated Bloch sphere such√as |⇒⟩ = (|⇑⟩ + |⇓⟩)∕ 2 =|1, 0, 0, 1⟩ and |⇐⟩ = (|⇑⟩ − |⇓⟩)∕ 2 = |0, 1, 1, 0⟩, which are
maximally entangled between the spin and motional degrees
of freedom. On-site qubit rotations can be made without leaving the subspace as discussed in Section 4. We are also robust
to unwanted band-changing collisions into other singly excited
bands (see Section E, Supporting Information), and to population of higher bands because of the 3D lattice band anharmonicity. Nearest-neighbor tunneling processes can be used to generate superexchange interactions and create entanglement between
the qubits, as we will describe next.
2.3. Superexchange Hamiltonian(5)

The single-site states in our decoherence-free subspace have energies of 0 or Ueg from the exchange interaction. However, if ane-band atom tunnels into an adjacent site, the resulting state will
feel an intra-band Uee interaction, as depicted in Figure 2a (Uggplays no role as g atoms do not tunnel). So long as the energy
diﬀerence between the conﬁgurations before and after tunneling

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is much larger than the tunneling amplitude J, no resonant tunneling will occur. However, the system will exhibit a second-orderinteraction through virtual processes starting and ending in thedecoherence-free subspace, with an oﬀresonant excited state inbetween. This leads to an eﬀective superexchange Hamiltonian̂ ex between neighboring lattice sites[49,50] with matrix elementsH
between states |i⟩, |j⟩ in the decoherence-free subspace given bŷ ex |j⟩ =
⟨i| H

∑

̂ J |k⟩ ⟨k| H
̂ J |j⟩Δijk ⟨i| Hk(1
Δijk =
2

)

(7)

1
1
+
Ei − Ek Ej − Ek̂U +Ĥ B ) |i⟩ = Ei |i⟩,where Ei is the unperturbed state energy (Hand k runs over all excited states outside the decoherence-freesubspace (see Section C, Supporting Information, for details).
̂ B in the dŷU +H
Note that we must also keep the unperturbed H
namics.
After some algebra, we ﬁnd that the superexchange interaction
takes the form of)∑∑( y yxẑ ex = J∥𝜎̂ j 𝜎̂ j+1 + 𝜎̂ jz 𝜎̂ j+1
𝜎̂ jx 𝜎̂ j+1
+ J⟂
H
j

j

)
∑(
∑
z
x
𝜎̂ jx 𝜎̂ j+1
+ Jz
+ Jxz
− 𝜎̂ jz 𝜎̂ j+1
𝜎̂ jz
j

≈

(8)

j

[
]
)
∑
∑ J∥ + J⟂ (y yxz𝜎̂ jx 𝜎̂ j+1+ 𝜎̂ j 𝜎̂ j+1 + J⟂ 𝜎̂ jz 𝜎̂ j+1
𝜎̂ jz
+ Jz
2
j
j

Here, 𝜎̂ j𝜈 are standard Pauli operators acting on the {|⇑⟩ , |⇓⟩}
logic qubits at site j. The coeﬃcients depend on three particular combinations of the repulsive and exchange interactions thatshow up in the energy denominators of Equation (7),Figure 3. Field gradient dependence of the superexchange interaction parameters J∥ and J⟂ , using the parameters in Table 1 (except B). All of theresonance points where B = |U𝛾 | for 𝛾 ∈ {1, 2, 3} are marked with dashedlines.

Note that in going to the second line of Equation (8), we
have used the fact that the external ﬁeld ∼ Jz 𝜎̂ z imposed by thebare exchange interactions [i.e. the (U1 − U2 )∕4 in Jz coming
̂ U ] is much larger than any superexchange processes.
from HAs a result, some of the terms in the ﬁrst two lines of Equation (8) can be neglected. To a good approximation, the system
is projected into a given 𝜎̂ z -eigenvalue manifold, causing theJxz term to be negligible in a rotating-wave approximation.
Furthermore, any 𝜎̂ x 𝜎̂ x or 𝜎̂ y 𝜎̂ y terms will be projected into a(𝜎̂ + 𝜎̂ − + h.c.) form. This allows us to recollect them and write the
Hamiltonian as an eﬀective XXZ-type model with an external
ﬁeld.
This model is valid so long as all of the denominators arelarge compared to the tunneling, that is, |B − |U𝛾 || ≫ J for 𝛾 ∈{1, 2, 3}, to avoid higher-order eﬀects. Figure 2b shows a comparison between the full Fermi–Hubbard dynamics and the superex∑change model, looking at collective observable ⟨𝜎̂ x ⟩ = j ⟨𝜎̂ jx ⟩(Section 4 discusses its measurement).

U1 = 2Uee + Ueg
U2 = 2Uee − Ueg(9)

3. Entanglement Generation

U3 = 2Uee − 3Ueg

3.1. Interaction Form

The corresponding coupling constants are

While the form of the interaction coeﬃcients in our superexchange model is nontrivial, the key features are the non-identicaldenominators. By tuning the parameters such that some of thedenominators become much smaller than the others, we canchoose which interactions get turned on and oﬀ. This permits
the isolation of terms of interest, which can then be employed
for useful entanglement generation.
Figure 3 plots J∥ and J⟂ as a function of ﬁeld gradient B (assuming B > 0). We observe three resonances, where B = |U𝛾 |for some 𝛾 and the corresponding denominator vanishes. While
our second-order perturbative Hamiltonian will not be correct
at those resonance points due to higher-order eﬀects, if we stayclose to them while still obeying |B − |U𝛾 || ≫ J, the superexchange Hamiltonian will remain valid and the near-resonant
terms will dominate the dynamics.
Operating close to the resonance conditions B ≈ |U3 | or
B ≈ |U1 | is of particular utility since in this case, J∥ ≈ −J⟂ andJ2
J∥ = −
2(

U1
B2 − U12+

2U2
B2 − U22+U3)

B2 − U32()
J2 U2 3B2 + U1 U3 (U1 − U2 )2J⟂ = ()()()
B2 − U12 B2 − U22 B2 − U32
2BJ2 U2 (U1 − U2 )Jxz = − ()()
B2 − U12 B2 − U32
(
)
U3
U1
U − U22− 1−
Jz = J222
2
4
B − U1B − U3(10)

Table 1 shows these derived values for experimentally realistic setups.

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to leading order the 𝜎̂ x 𝜎̂ x + 𝜎̂ y 𝜎̂ y terms are cancelled out. We areleft with just an Ising model,lim

B→|U3 |,|U1 |̂ ex = Jzz
H

∑
j

z
𝜎̂ jz 𝜎̂ j+1
+ Jz

∑

𝜎̂ z

(11)

j

where Jzz ≈ J⟂ ≈ −J∥ . Since we cannot be exactly at resonance,
there will be some discrepancy between the two coeﬃcients; asuﬃcient approximation is to take the average, Jzz ≈ (J⟂ − J∥ )∕2.
The nearest-neighbor Ising model has a wide variety of applications for entanglement generation between qubits, and can beused to make cluster states as demonstrated in the next section.
Furthermore, the Ising interaction strength near the resonancescales as ∼ J2 ∕|B − |U𝛾 ||, which is much faster than conventional
superexchange ∼ J2 ∕U𝛾 (the former’s denominator is on the order of ∼ 10J, while the latter is ∼ 100J).
The system’s tunability also permits realization of other spinmodels. If we instead bring the ﬁeld gradient close to the |U2 | resonance, the interaction coeﬃcients are equal in magnitude andsign, which creates a Heisenberg model with a transverse ﬁeld:

̂ ex = J⟂lim H

B→|U2 |∑𝜎⃗j ⋅ 𝜎⃗j+1 + Jz

j

∑

𝜎̂ z

(12)

j

Finally, as seen from Equation (8), we can realize the XXZ modelwith a wide range of coeﬃcients, which has seen recent experimental interest.[51]The homogeneity (how close |J∥ | can be made to |J⟂ |) and positions of the resonance points depend on the interaction strength.
Having higher Uee ∕J, Ueg ∕J leads to better homogeneity, but atthe cost of moving the resonance points outward and thus requiring a larger ﬁeld gradient B∕J. This requirement can be mitigated by reducing the tunneling rate J, implying a compromise
between gradient strength and experimental timescale. Increasing Uee , Ueg without aﬀecting J is done by either increasing transverse lattice depth along the y, z directions or increasing the scattering length as via Feshbach resonance.

A cluster state is deﬁned as
(
)]
[
∏
𝜋
z
z
|𝜓(0)⟩
exp −i 𝜎̂ jz 𝜎̂ j+1− 𝜎̂ jz − 𝜎̂ j+1|𝜓c ⟩ =
4jwhere the exponential operator is a controlled phase gate, applied
across all nearest-neighbor links. The initial-state corresponds toa spin-polarized state along +x,|𝜓(0)⟩ =∏|⇒⟩j

As seen from Equation (11), our system already contains the necessary Ising interaction. The protocol we use is similar to the one
discussed in ref. [12], but without the need for continuous laser
driving and easier to tune with the ﬁeld gradient. We simply prepare the initial state |𝜓(0)⟩ (as described in Section 4), quench theﬁeld gradient or interaction strength (via Feshbach resonance) tosatisfy B ≈ |U3 |, and wait for a time
tc = 𝜋∕(4Jzz ),Jzz ≈ (J⟂ − J∥ )∕2

The Ising model in Equation (11) can be employed to generate
useful entanglement for quantum computation. We can generate a cluster state, which is a multipartite entangled state used formeasurement-based quantum computing.[45] A cluster state is a
resource that can reproduce the results of circuit-based computations without needing explicit entangling gates between individual qubits. All entanglement generation is front-loaded into thecluster state itself. Once this state is prepared, a computation isdone by successive feed-forward measurements. Given the longcoherence times and innate 3D nature of the lattice, we can not
only generate a single cluster state, but an entire array of them
to be used or sorted as needed. While a 1D cluster state alone
is not suﬃcient for universal computation, our protocol can beextended to 2D in a straightforward manner as discussed in Section 5. We focus on 1D as a proof-of-principle demonstration of
the system’s capabilities.

(15)

This will implement the cross-site terms in the controlled phasegate. The only remaining task is to implement the single-particle
terms. Since our Jz is so much larger than the interaction andcommutes with it (to good approximation), we could in principle determine how many full periods of single-particle evolution have occurred during tc , and then compensate by letting the system evolve further such that the total time spent is𝜋∕(2Jz ) mod 𝜋∕Jz . However, a more prudent approach is to use
̂ = e−i𝜋 𝜎̂ x ∕2 halfway through the evolution.
a spin-echo 𝜋 pulse ΠThis will cause any phase accrued from the single-particle terms
to be undone by itself during the second half of the evolution. In
addition, such a pulse can help with unwanted sources of noisethat are not captured by our model. After this evolve-echo-evolve
sequence is done, we let the system evolve further for a time𝜋∕(2Jz ) for the necessary single-particle rotation, and then stop
the dynamics by turning oﬀ the tunneling and interactions (via
Feshbach resonance). The overall protocol is thus,̂Adv. Quantum Technol. 2020, 1900132

(14)

j

̂

̂

̂ −iHex tc ∕2 |𝜓(0)⟩|𝜓c ⟩ = e−iHex 𝜋∕(2Jz ) e−iHex tc ∕2 Πe

3.2. Cluster State Generation

(13)

(16)

Figure 4a shows state ﬁdelity between time-evolution using theecho protocol with our superexchange model of Equation (8) andan ideal Ising model evolution which can generate a perfect cluster state. Inhomogeneity between J∥ and J⟂ and the breakdown of
the rotating-wave approximation (since Jz is not inﬁnite) are themain sources of error, but in general they can be made small.
One of the core advantages to measurement-based quantum
computation is that the quality of a cluster state can be estimatedusing spatially local properties rather than global ﬁdelity. To thisend, we look at multi-body correlators called stabilizer operators,
deﬁned asK̂ j = 𝜎̂ jx∏
i∈{j−1,j+1}

𝜎̂ iz

(17)

i.e. products of an 𝜎̂ x measurement on one site and 𝜎̂ z on all its
neighbouring sites (thus a three-body operator in 1D, 5 in 2D,

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Figure 4. a) Fidelity between the state generated by the spin-echo protocol
of Equation (16), and the state generated by an ideal Ising model with
̂ j for the spinJzz = (J⟂ − J∥ )∕2. System size is L = 8. b) Stabilizer values ⟨K⟩̂echo protocol using the superexchange model Hex (orange line) and full̂ (blue dots), for system size L = 4. The stabilizer
Fermi-Hubbard model H
operators in the corners are deﬁned diﬀerently: The missing 𝜎̂ jz is absentzfrom the correlator, and an overall rotation of e−i𝜋 𝜎̂ ∕4 is applied to thestate before calculating them, since they are missing such a rotation fromthe vacant link on the open end.

and 7 in 3D). An ideal cluster state is an eigenstate of all suchstabilizer operators with eigenvalue 1,
K̂ j |𝜓c ⟩ = + |𝜓c ⟩ ,⟨𝜓c | K̂ j |𝜓c ⟩ = +1

(18)

Note that the sign of the eigenvalue does not matter as long as it is
the same for all sites, since we can ﬂip it by applying a collective𝜎̂ z rotation. The closeness of each stabilizer correlator ⟨K̂ j ⟩ to 1
serves as a local measure of cluster state quality. If we average it∑over a region of the lattice, ⟨K̂ j ⟩avg = j ⟨K̂ j ⟩, we get an estimateof the ﬁdelity for computations done using that region.
Figure 4(b) shows the average value of these cluster correlators
for a small system as our protocol goes on. Note that in our simulations the lattice has boundaries, which means that the cornersites have their respective stabilizer operators deﬁned diﬀerently(see caption). The main deviations come from inhomogeneity between J∥ and J⟂ , as well as leakage of population into the excitedstates due to higher-order processes. These can be remedied bydriving further from resonance (at the cost of longer tc ), and usinglarger interactions to keep the coeﬃcients homogeneous (at thecost of needing a stronger gradient). However, we still ﬁnd highstabilizer values ⟨K̂ j ⟩avg > 0.95 indicating a good cluster state.

3.3. OTOC Measurement
Another powerful feature of the tunable superexchange coeﬃcients (see Figure 3) is their ability to change sign. We can quench
the gradient from just below one of the resonances, say B ≲ |U3 |,to just above, B ≳ |U3 |. This ﬂips the interaction coeﬃcient, allowing for the implementation of a unitary time-reversal. Suchcapabilities are applicable to the study of out-of-time ordered correlation functions (OTOCs), which have garnered much recent
interest due to their applications in quantum chaos, butterﬂy effects and temporal correlation spreading.[48]An OTOC is a two-time, four-operator correlator deﬁned aŝ V⟩,̂ where W,
̂ V̂ are time-independent commutinĝ † (t)V̂ † W(t)⟨Ŵ ̂ −iHt
iHt
̂
operators and W(t) = e We ̂ . The connection to chaos can bê V̂ to be local operators with someunderstood by considering W,
spatial separation. When the system is initialized, they commute

Adv. Quantum Technol. 2020, 1900132

Figure 5. a) Schematic of the OTOC-measuring protocol of Equation (19).
The sign of the Hamiltonian is changed by quenching the ﬁeld gradient
B from below to above a resonance (here the |U3 | resonance), choosing
the J⟂ values to have equal magnitude and opposite sign. 𝜋 pulses aredone in the middle of each evolution to remove unwanted single-particle
rotations. b) Fourier-transformed OTOC components [see Equation (21)],
averaged over all j and computed with the superexchange model. Systemsize is L = 8 (open boundaries). Parameters are from Table 1 with themagnetic ﬁeld quenched from B∕J = 350 to B′ ∕J = 370 about the |U3 |∕J =359 resonance, corresponding to J⟂ ∕J = ±0.025.

and the corresponding OTOC is zero. As correlations spread, thê and thus the OTOC increase. The slope of inspatial extent of W
crease gives information about the propagation of correlations inthe system.
In our case, we can measure an OTOC in a straightforward manner. We time-evolve under the eﬀective superexchange
Hamiltonian to a time t, apply a spin rotation by some angle 𝜃,
quench the ﬁeld gradient to ﬂip the interaction sign, evolve foranother t, and ﬁnally measure some observable (total evolution
time 2t), similar to the sequence used in ref. [52]. If we start in|𝜓(0)⟩, a sample such protocol may be written as
)
(
𝜃 x
̂
̂
⟨Ĉ j (𝜃, t)⟩ = ⟨𝜓(0)| e+iHex t e−i 2 𝜎̂ e−iHex t
(
)
𝜃 x
̂
̂
𝜎̂ jx e+iHex t e−i 2 𝜎̂ e−iHex t |𝜓(0)⟩(19)̂ V̂ |𝜓(0)⟩̂ † (t)V̂ † W(t)= ⟨𝜓(0)| W𝜃 x̂ = e−i 2 𝜎̂ , and V̂ = 𝜎̂ x . In going to the second line,
where Wj
we have used the fact that 𝜎̂ jx |𝜓(0)⟩ = |𝜓(0)⟩. Figure 5a depicts
this protocol.
The sign change can be done about any |U𝛾 | resonance (seeFigure 3). Here we will use the Ising model with |U3 |. Note thateach half of the time-evolution will also include a spin-echo midway to remove the 𝜎̂ z single-particle rotations, since their sign
cannot be fully reversed. The total implementation is thus,
̂

̂

̂

̂

̂

̂ −iHex (B)t∕2
e−iHex t = e−iHex (B)t∕2 Πe′̂′̂ −iHex (B )t∕2
e+iHex t = e−iHex (B )t∕2 Πe(20)

with B, B′ on opposite sides of the chosen resonance, set suchthat the magnitude of Jzz is equal for both. For the parametersof U1 ∕J = 597, U2 ∕J = 119, U3 ∕J = −359 (from Table 1), we canquench the gradient about the |U3 | resonance from B∕J = 350
to B′ ∕J = 370, yielding eﬀective interaction strengths of J⟂ ∕J ≈±0.025.
We can measure diﬀerent OTOCs depending on evolution
time t and rotation angle 𝜃. For 𝜃 = 0, we have trivial unitary reversal, and the corresponding OTOC would be equal to unity. As

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Figure 6. a) Raman scheme for transferring an ↑ atom to the e band. Twolinearly polarized lasers (i) and (ii) couple the states |2 S1∕2 , g; 9∕2, −7∕2⟩
and |2 S1∕2 , e; 9∕2, −7∕2⟩ through an excited state |2 P1∕2 , g; 7∕2, −7∕2⟩(transition wavelength ≈ 770 nm, unpopulated due to large detuning) withsingle-photon Rabi frequencies of Ω1 and Ω2 respectively. Here the statelabel is |Λ, 𝜇; F, mF ⟩, with electronic state Λ, band 𝜇, nuclear-spin F andprojection mF (see Supporting Information for details). A 𝜋 pulse eﬀects
the coherent transfer. b) STIRAP scheme for the same transfer. Instead ofalways-on constant Rabi frequencies for both lasers, we ﬁrst ramp up Ω2
coupling the initially unoccupied states with a Gaussian proﬁle, then ramp
it down while ramping up Ω1 . The state is adiabatically dragged into thedesired conﬁguration. Here Ω0 is the maximum Rabi frequency the lasers
reach, equal for both.

the angle increases, the OTOC exhibits a decaying slope as correlations build up. Another useful piece of information that canbe extracted is the spectral fourier transform of the OTOCs,∑
eim𝜃 ⟨Ĉ j (𝜃, t)⟩(21)⟨Ĉ j (m, t)⟩ =𝜃and n, m ∈ {0, … , L − 1}. The diﬀerent m compowhere 𝜃 = 2𝜋nLnents give information about the system’s connectivity.[52] In a
1D lattice with two neighbours per site, we should see signals
at m = ±2. A 2D lattice would have signals at ±4, ±2, and a 3D
lattice at ±6, ±4, ±2. Figure 5b shows these fourier-transformed
OTOC values for our superexchange model. Note that we do notsee any special peak or trough at t = tc because our system has
open boundaries, and thus the OTOCs will feel edge eﬀects since
we average over every site. However, site-resolved OTOCs can beprobed in this system with a quantum gas microscope.

4. Preparation and Control Tools
4.1. State Preparation
In this section, we detail an explicit experimental protocol for
preparing atoms in the required decoherence-free subspace. The
starting point is a band insulator (two atoms of opposite spin persite) in the lowest band g, which can be prepared by standard cooling techniques.[53–55] We then use a Raman scheme to selectively
drive one of the atoms (the spin-↑ one) into the e band. The result
is a state of the form |1, 0, 0, 1⟩ = |⇒⟩ on every site, which can immediately be used for cluster state generation. One can also rotate
this state into others using the protocols of the next section.
The process is depicted in Figure 6a. We use a linearly polarized laser pair to couple an occupied state in the g band to an
empty target state in the e band via an intermediate excited state
in a higher electronic level. The laser detunings Δ from the excited state are chosen to be equal and large compared to their Rabi
frequencies Ω1 , Ω2 , creating a Raman coupling Ω = Ω1 Ω2 ∕Δ

Adv. Quantum Technol. 2020, 1900132

between the occupied and target states. The other ﬁrst-excited
bands along y or z do not participate because of their energy shift
due to unequal lattice depth. Doing a 𝜋 pulse implements a coherent transfer, populating the e band with one ↑ atom. See Section
B, Supporting Information, for more details.
We note that the speed at which this transfer can be made islimited by the energy gaps to other unwanted bands, which we
do not want to populate. The two-photon Rabi frequency Ω mustbe small compared to all gaps to other excited bands. The lowestof these gaps is an unwanted coupling from ﬁrst-excited bandstate e = (1, 0, 0) to the second-excited state (2,0,0), which is offresonant from the (0, 0, 0) ↔ (1, 0, 0) transition we use by ≈ 5kHz. On the other hand, Ω should also be faster than any intrinsic decoherence processes to ensure that state preparation canbe done before the model breaks down. Finally, the detuning Δ
must be larger than spontaneous decay rate in the intermediate
state. All of the above requirements can be met by ﬁrst making
the detuning Δ suﬃciently large, then providing enough laser
power (increasing Ω1 , Ω2 ) to make Ω satisfy the desired intermediate regime.
If a coherent 𝜋 pulse with the above setup poses experimental challenges, one may instead use a STIRAP protocol.[56,57] The
laser conﬁgurations are the same, but instead we slowly ramp
the laser intensities with oﬀset Gaussian proﬁles as depicted in
Figure 6b. The coupling between the initially unoccupied states[laser (ii)] is counter-intuitively ramped up ﬁrst, and then the
coupling between the initially occupied and excited states [laser(i)] is ramped up while the previous coupling is reduced. The
system follows an adiabatic evolution where the ground-state istransferred into a ﬁnal conﬁguration with the atom in the target
state. Unlike the previous protocol, STIRAP can function evenwith Δ = 0, and enjoys better robustness to laser noise or bandwidth limitations.

4.2. Rotations
Quantum computation requires readily accessible rotations upon
the qubits in the decoherence-free subspace. A 𝜎̂ z -type rotationis straightforward because the exchange interaction creates sucha term implicitly. We tune Ueg to be large enough for fast rotations and far from any resonances, then wait the desired time.
On-site rotations of this type can be implemented with a focusedlaser to change the lattice potential (and thus interactions) that asite feels.
A 𝜎̂ x rotation may be realized with diﬀerential Stark
shifts. The desired rotation operator takes the form of|1, 0, 0, 1⟩ ⟨1, 0, 0, 1| − |0, 1, 1, 0⟩ ⟨0, 1, 1, 0| in the Fock basis,
which can be implemented by energetically shifting one of the{|⇒⟩ , |⇐⟩} states in the decoherence-free subspace (i.e. applyinga 𝜎̂ z rotation in the {|1, 0, 0, 1⟩ , |0, 1, 1, 0⟩} basis). This may bedone with a laser such as (i) in Figure 6a. The detuning Δ isset suﬃciently large compared to the single-photon laser Rabi
frequency Ω1 that no population transfer occurs; the lowest-ordereﬀect is then to shift the state |1, 0, 0, 1⟩ by Ω21 ∕Δ, realizing thedesired rotation up to an identity term in the Hamiltonian. Whilein general one would need to avoid perturbing the e state (separated from g by ∼ 52 kHz gap) via transition (ii) in Figure 6a, thisrequirement may be achieved through symmetry. The laser (ii)1900132 (7 of 10)© 2020 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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matrix element normally vanishes due to the spatial symmetryof the wavefunctions about the lattice site center (see Section
B, Supporting Information). For state preparation, we avoid
this by aiming laser (ii) along the x direction, which introduces
an additional phase to break the symmetry and prevents theintegral from vanishing. For rotations, if we aim laser (i) along y
or z instead, transition (ii) will have a vanishing overlap integral,allowing us to generate the desired rotation. For collective rotations such as the 𝜋 pulse in the cluster state protocol, we can even
use the same beam as the one used in state preparation. We mayneed to have interactions Ueg turned oﬀ to avoid rotating about
multiple axis simultaneously, but this can also be done without
having to ramp Feshbach ﬁelds by adjusting the light shifts.
A 𝜎̂ x rotation may also be realized by using additional external
ingredients. One method is an on-site ﬁeld gradient. A linear
gradient is no longer suﬃcient, because it has no on-site eﬀect
on the decoherence-free subspace (see Section D, Supporting
Information). While a linear gradient was used to tune the crosssite superexchange interactions, there it only shifted the energiesof the states involved: adjacent qubits are tunnel-coupled. For
direct coupling between the decoherence-free states, a quadraticﬁeld variation is required. For example, an eﬀective magnetic
curvature could be realized via state-dependent optical potentials
for the bare spin states 𝜎 using the diﬀerential magnetic moments and vector Stark shifts. One could also apply a magnetic
ﬁeld curvature directly. The Hamiltonian for such a gradient ona site j is,()̂ 𝛿B = 𝛿Ω(x − x0 )2 n̂ j,↑ − n̂ j,↓
H(22)

where the gradient is along the tunneling direction x, focusednear one lattice site and centered about some position x0 . Evaluating the eﬀect of this gradient for a single site (see Section D,
Supporting Information), we arrive at the following Hamiltonianin the basis {|⇑⟩ , |⇓⟩} (on site j):(̂ site =
H

0
ℏ 𝛿Ω
m𝜔x

ℏ 𝛿Ω
m𝜔x

Ueg

)
=−

Ueg
2

𝜎̂ jz +ℏ 𝛿Ω x
𝜎̂
m𝜔x j

(23)

A 𝜎̂ x rotation may be done by either turning the interactions oﬀ
via Feshbach resonance, or ensuring that ℏ 𝛿Ω∕(m𝜔x ) ≫ Ueg . For
most cases, the former will be the better choice as we do not wantarbitrary on-site exchange interactions acting on the qubits unless we are generating a cluster state. One could also overcome
the Ueg interactions with a high 𝛿Ω; while this would need a verystrong ﬁeld gradient, having such a requirement also shows adegree of robustness to unwanted magnetic curvature (see discussion in Section D, Supporting Information).

4.3. Readout
The ﬁnal required piece of the toolbox is to measure the qubits inthe decoherence-free subspace. Measurements of the 𝜎̂ x component can be done by simply reversing the state preparation protocol, then measuring the occupation of the g band. If the qubit
was in |⇒⟩ we would ﬁnd two atoms, and if in |⇐⟩, we would
ﬁnd one, oﬀering a simple metric.

Adv. Quantum Technol. 2020, 1900132

The 𝜎̂ z component is more complex, but can be avoided altogether because we have access to rotations about two axes. We
ﬁrst rotate by 𝜋∕2 about the x axis on the Bloch sphere, then by𝜋∕2 about the z axis, and ﬁnally measure 𝜎̂ x . If we were in |⇑⟩ wewould end up in |⇒⟩ and ﬁnd two atoms, whereas the |⇓⟩ state
would be rotated into |⇐⟩ and yield one.
We note that for the more complex application of doing quantum computation with a cluster state, one requires sequentialsite-resolved measurements that do not cause destructive eﬀects
on adjacent lattice sites. There are several protocols in currentgeneration optical lattice systems which can provide such desired
capabilities. One option is an accordion-type lattice[58] that expands the lattice spacing a without changing the conﬁning laser
wavelength, which can be used after the cluster state is generated to spread the atoms out and permit site-resolved measurement with focused beams. Another option is a local transition ofa single lattice site into a diﬀerent hyperﬁne state, which can beselectively measured.[59] While there can be detrimental eﬀects
to adjacent sites due to insuﬃcient beam focusing, frequency selection on the desired site should allow the damage done to theothers to remain coherent, making it possible to undo the crosstalk with a suitable laser pulse after each measurement.

5. Conclusions
We have proposed protocols for using 3D optical lattices populated by atoms in higher bands to realize qubits for quantum computation. Our design embeds the logical states in a
decoherence-free subspace, which greatly ampliﬁes the systemcoherence time while still having scalability, single-site addressability and tunable interactions via Feshbach resonances andelectromagnetic ﬁeld control. State preparation and on-site rotations can be implemented with Raman transitions and external ﬁeld gradients, allowing for straightforward manipulation of
qubit states. Superexchange interactions between the qubits canalso be tuned in form and amplitude with ﬁeld gradients, allowing (among other things) the generation of an Ising model usefulfor cluster state generation and OTOC measurement.
This system has an abundance of natural extensions that canbe implemented without signiﬁcant modiﬁcations to the protocols. While our design allows the creation of a 1D cluster state,
2D states can be generated by reducing the tunneling along y andinducing superexchange dynamics along that direction as well(still keeping the lattice depth suﬃciently diﬀerent from x so thatunwanted band-changing collisions do not occur). Since the dynamics are controlled by a ﬁeld gradient, we can enable or disable interactions at will, allowing us to build the cluster state up
one dimension at a time as the controlled phase gate operations
commute. Such a state allows for universal computation using
measurement-based protocols. Furthermore, if we seek to study
OTOCs, we can use locally applied ﬁelds to induce local perturbations and measure their propagation directly on a site-to-sitebasis rather than relying upon collective observables.
While our protocols have been described for alkali atoms,
straightforward extensions can also be made to systems usingalkaline earth atoms instead. The higher-band conﬁguration canbe mimicked through the use of diﬀerent electronic states (withthe g, e bands replaced by, e.g., ground and excited clock states,

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while the bare spin 𝜎 remains represented by hyperﬁne states).
The resulting system would still exhibit both inter- and intra-stateinteractions for the two electronic states. Applying an electronicstate dependent lattice intensity can likewise restrict tunnelingto just one of the two electronic states, reproducing our proposed
setup. Alkaline earth atoms can provide additional stability androbustness due to their naturally high coherence times and magnetic ﬁeld insensitivity. Related prospects, including the use of
decoherence-free subspaces, have been explored in refs. [60, 61].
Other possible applications for our proposed setup include thegeneration of other useful entangled states such as spin-squeezed
states,[62] by generating appropriate XXZ-type Hamiltonians[63,64]
or similar gap-protected spin interactions.[65,66] The restrictions
on motion due to the use of higher bands can lead to exotic spatially correlated physics mimicking spin-orbit coupledsystems.[67] As a more ambitious goal, we can also explore thefully resonant regime where B = U𝛾 for some 𝛾 for which nontrivial constrained dynamics can arise on fast timescales.[68] Our
protocols oﬀer a powerful toolbox for quantum computation andsimulation, which is accessible with current state-of-the-art optical lattice experiments, and oﬀers powerful tunability and versatility.

Supporting Information
Supporting Information is available from the Wiley Online Library or fromthe author.

Acknowledgements
This work was supported by the AFOSR grants FA9550-19-1-0275
and FA9550-19-1-7044, the NSF JILA-PFC PHY-1734006 grant, NIST,
and NSERC.

Conﬂict of Interest
The authors declare no conﬂict of interest.

Keywords
cluster states, decoherence-free subspaces, multiband systems, quantum
computation
Received: November 29, 2019
Revised: February 25, 2020
Published online:[1] C. Gross, I. Bloch, Science 2017, 357, 995.
[2] I. Bloch, Nature 2008, 453, 1016.
[3] T. Calarco, E. A. Hinds, D. Jaksch, J. Schmiedmayer, J. I. Cirac, P.
Zoller, Phys. Rev. A 2000, 61, 022304.
[4] O. Mandel, M. Greiner, A. Widera, T. Rom, T. W. Hänsch, I. Bloch,
Nature 2003, 425, 937.
[5] T. Pellizzari, S. A. Gardiner, J. I. Cirac, P. Zoller, Phys. Rev. Lett. 1995,
75, 3788.

Adv. Quantum Technol. 2020, 1900132[6] M. D. Lukin, M. Fleischhauer, R. Cote, L. D. Duan, D. Jaksch, J. I.
Cirac, P. Zoller, Phys. Rev. Lett. 2001, 87, 037901.
[7] K. Eckert, J. Mompart, X. X. Yi, J. Schliemann, D. Bruß, G. Birkl, M.
Lewenstein, Phys. Rev. A 2002, 66, 042317.
[8] E. Charron, M. A. Cirone, A. Negretti, J. Schmiedmayer, T. Calarco,
Phys. Rev. A 2006, 74, 012308.
[9] A. J. Daley, M. M. Boyd, J. Ye, P. Zoller, Phys. Rev. Lett. 2008, 101,
170504.
[10] M. Anderlini, P. J. Lee, B. L. Brown, J. Sebby-Strabley, W. D. Phillips,
J. V. Porto, Nature 2007, 448, 452.
[11] A. M. Kaufman, B. J. Lester, M. Foss-Feig, M. L. Wall, A. M. Rey, C. A.
Regal, Nature 2015, 527, 208.
[12] M. Mamaev, R. Blatt, J. Ye, A. M. Rey, Phys. Rev. Lett. 2019, 122,
160402.
[13] C. Weitenberg, S. Kuhr, K. Mølmer, J. F. Sherson, Phys. Rev. A 2011,
84, 032322.
[14] G. Pagano, F. Scazza, M. Foss-Feig, Adv. Quantum Technol. 2019, 2,
1800067.
[15] M. Saﬀman, T. G. Walker, K. Mølmer, Rev. Mod. Phys. 2010, 82, 2313.
[16] K. Maller, M. T. Lichtman, T. Xia, Y. Sun, M. J. Piotrowicz, A. W. Carr,
L. Isenhower, M. Saﬀman, Phys. Rev. A 2015, 92, 022336.
[17] M. Saﬀman, J. Phys. B: At., Mol. Opt. Phys. 2016, 49, 202001.
[18] A. Negretti, P. Treutlein, T. Calarco, Quantum Inf. Process. 2011, 10,721.
[19] H. Levine, A. Keesling, G. Semeghini, A. Omran, T. T. Wang, S. Ebadi,
H. Bernien, M. Greiner, V. Vuletić, H. Pichler, M. D. Lukin, Phys. Rev.
Lett. 2019, 123, 170503.
[20] I. S. Madjarov, J. P. Covey, A. L. Shaw, J. Choi, A. Kale, A. Cooper,
H. Pichler, V. Schkolnik, J. R. Williams, M. Endres, arXiv:2001.04455,
2020.
[21] Y.-Y. Jau, A. M. Hankin, T. Keating, I. H. Deutsch, G. W. Biedermann,
Nat. Phys. 2016, 12, 71.
[22] Y. Wang, A. Kumar, T.-Y. Wu, D. S. Weiss, Science 2016, 352, 1562.
[23] M. Endres, H. Bernien, A. Keesling, H. Levine, E. R. Anschuetz, A.
Krajenbrink, C. Senko, V. Vuletic, M. Greiner, M. D. Lukin, Science
2016, 354, 1024.
[24] D. Barredo, S. De Léséleuc, V. Lienhard, T. Lahaye, A. Browaeys, Science 2016, 354, 1021.
[25] A. Kumar, T.-Y. Wu, F. Giraldo, D. S. Weiss, Nature 2018, 561, 83.
[26] T. Müller, S. Fölling, A. Widera, I. Bloch, Phys. Rev. Lett. 2007, 99,
200405.
[27] G. Wirth, M. Ölschläger, A. Hemmerich, Nat. Phys. 2011, 7, 147.
[28] L. Niu, S. Jin, X. Chen, X. Li, X. Zhou, Phys. Rev. Lett. 2018, 121,
265301.
[29] P. Soltan-Panahi, D.-S. Lühmann, J. Struck, P. Windpassinger, K. Sengstock, Nat. Phys. 2012, 8, 71.
[30] C. V. Parker, L.-C. Ha, C. Chin, Nat. Phys. 2013, 9, 769.
[31] J. Struck, C. Ölschläger, R. Le Targat, P. Soltan-Panahi, A. Eckardt, M.
Lewenstein, P. Windpassinger, K. Sengstock, Science 2011, 333, 996.
[32] D. Hu, L. Niu, S. Jin, X. Chen, G. Dong, J. Schmiedmayer, X. Zhou,
Commun. Phys. 2018, 1, 29.
[33] W. S. Bakr, P. M. Preiss, M. E. Tai, R. Ma, J. Simon, M. Greiner, Nature
2011, 480, 500.
[34] R. Bücker, J. Grond, S. Manz, T. Berrada, T. Betz, C. Koller, U. Hohenester, T. Schumm, A. Perrin, J. Schmiedmayer, Nat. Phys. 2011, 7,
608.
[35] D. A. Lidar, I. L. Chuang, K. B. Whaley, Phys. Rev. Lett. 1998, 81, 2594.
[36] D. Kielpinski, V. Meyer, M. A. Rowe, C. A. Sackett, W. M. Itano, C.
Monroe, D. J. Wineland, Science 2001, 291, 1013.
[37] E. Tiesinga, A. J. Moerdijk, B. J. Verhaar, H. T. C. Stoof, Phys. Rev. A
1992, 46, R1167.
[38] E. Tiesinga, B. J. Verhaar, H. T. C. Stoof, Phys. Rev. A 1993, 47, 4114.
[39] C. Chin, R. Grimm, P. Julienne, E. Tiesinga, Rev. Mod. Phys. 2010, 82,
1225.

1900132 (9 of 10)© 2020 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

www.advancedsciencenews.com

www.advquantumtech.com[40] M. F. Parsons, F. Huber, A. Mazurenko, C. S. Chiu, W. Setiawan, K.
Wooley-Brown, S. Blatt, M. Greiner, Phys. Rev. Lett. 2015, 114, 213002.
[41] E. Haller, J. Hudson, A. Kelly, D. A. Cotta, B. Peaudecerf, G. D. Bruce,
S. Kuhr, Nat. Phys. 2015, 11, 738.
[42] L. W. Cheuk, M. A. Nichols, M. Okan, T. Gersdorf, V. V. Ramasesh, W.
S. Bakr, T. Lompe, M. W. Zwierlein, Phys. Rev. Lett. 2015, 114, 193001.
[43] A. Omran, M. Boll, T. A. Hilker, K. Kleinlein, G. Salomon, I. Bloch, C.
Gross, Phys. Rev. Lett. 2015, 115, 263001.
[44] G. J. A. Edge, R. Anderson, D. Jervis, D. C. McKay, R. Day, S. Trotzky,
J. H. Thywissen, Phys. Rev. A 2015, 92, 063406.
[45] H. J. Briegel, D. E. Browne, W. Dür, R. Raussendorf, M. Van den Nest,
Nat. Phys. 2009, 5, 19.
[46] R. Raussendorf, D. E. Browne, H. J. Briegel, Phys. Rev. A 2003, 68,022312.
[47] S. H. Shenker, D. Stanford, J. High Energy Phys. 2014, 2014, 67.
[48] J. Maldacena, S. H. Shenker, D. Stanford, J. High Energy Phys. 2016,
2016, 106.
[49] S. Trotzky, P. Cheinet, S. Fölling, M. Feld, U. Schnorrberger, A. M. Rey,
A. Polkovnikov, E. A. Demler, M. D. Lukin, I. Bloch, Science 2008, 319,
295.
[50] S. Bravyi, D. P. DiVincenzo, D. Loss, Ann. Phys. 2011, 326, 2793.
[51] I. Dimitrova, N. Jepsen, A. Buyskikh, A. Venegas-Gomez, J. AmatoGrill, A. Daley, W. Ketterle, arXiv:1908.09870, 2019.
[52] M. Gärttner, J. G. Bohnet, A. Safavi-Naini, M. L. Wall, J. J. Bollinger,
A. M. Rey, Nat. Phys. 2017, 13, 781.
[53] A. Mazurenko, C. S. Chiu, G. Ji, M. F. Parsons, M. Kanász-Nagy, R.

Adv. Quantum Technol. 2020, 1900132

Schmidt, F. Grusdt, E. Demler, D. Greif, M. Greiner, arXiv:1612.08436,
2016.
[54] R. A. Hart, P. M. Duarte, T.-L. Yang, X. Liu, T. Paiva, E. Khatami, R. T.
Scalettar, N. Trivedi, D. A. Huse, R. G. Hulet, Nature 2015, 519, 211.
[55] M. Boll, T. A. Hilker, G. Salomon, A. Omran, J. Nespolo, L. Pollet, I.
Bloch, C. Gross, Science 2016, 353, 1257.
[56] J. Oreg, F. T. Hioe, J. H. Eberly, Phys. Rev. A 1984, 29, 690.
[57] K. Bergmann, H. Theuer, B. W. Shore, Rev. Mod. Phys. 1998, 70, 1003.
[58] S. Al-Assam, R. A. Williams, C. J. Foot, Phys. Rev. A 2010, 82, 021604.
[59] E. M. Seroka, A. Valdés Curiel, D. Trypogeorgos, N. Lundblad, I. B.
Spielman, Opt. Express 2019, 27, 36611.
[60] F. Scazza, C. Hofrichter, M. Höfer, P. De Groot, I. Bloch, S. Fölling,
Nat. Phys. 2014, 10, 779.
[61] G. Cappellini, M. Mancini, G. Pagano, P. Lombardi, L. Livi, M. S. de
Cumis, P. Cancio, M. Pizzocaro, D. Calonico, F. Levi, C. Sias, J. Catani,
M. Inguscio, L. Fallani, Phys. Rev. Lett. 2014, 113, 120402.
[62] J. Ma, X. Wang, Chang-P. Sun, F. Nori, Phys. Rep. 2011, 509, 89.
[63] A. M. Rey, L. Jiang, M. D. Lukin, Phys. Rev. A 2007, 76, 053617.
[64] P. Cappellaro, M. D. Lukin, Phys. Rev. A 2009, 80, 032311.
[65] A. M. Rey, L. Jiang, M. Fleischhauer, E. Demler, M. D. Lukin, Phys.
Rev. A 2008, 77, 052305.
[66] P. He, M. A. Perlin, S. R. Muleady, R. J. Lewis-Swan, R. B. Hutson, J.
Ye, A. M. Rey, arXiv:1904.07866, 2019.
[67] X. Li, W. V. Liu, Rep. Prog. Phys. 2016, 79, 116401.
[68] M. Mamaev, I. Kimchi, M. A. Perlin, R. M. Nandkishore, A. M. Rey,
arXiv:1905.12094, 2019.

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