Entropy Removal and Coherence with Lasers
byJarrod Reilly

A thesis submitted to the Faculty of the
University of Colorado in partial fulfillmentof the requirements for the degree of
Bachelor of Arts
Department of Physics
2020

This thesis entitled:
Entropy Removal and Coherence with Lasers
written by Jarrod Reilly
has been approved for the Department of Physics

Murray Holland

Prof. Jun Ye

Prof. Magdalena Czubak

Date

The final copy of this thesis has been examined by the signatories, and we find that both thecontent and the form meet acceptable presentation standards of scholarly work in the above
mentioned discipline.

iii
Reilly, Jarrod (B.A., Physics)
Entropy Removal and Coherence with Lasers
Thesis directed by Prof. Murray Holland

The creation of low entropy states is crucial to modern atomic and molecular physics asparticle cooling has opened the way for the creation of ultracold superfluid, quantum simulators, andincreasing the precision of atomic clocks. However, it is often unclear how the particle ensemble’s
entropy is redistributed to other subsystems during the cooling process. Furthermore, experimental
control of quantum systems during its interaction with laser fields is often impaired by incoherent,
spontaneous processes that may transfer particles to unwanted states.
With these points in mind, my honors thesis focuses on protocols that create low entropy
states as well as schemes that increase coherent control in quantum evolution. First, we present
a method for creating high impulse laser slowing protocols that mitigate the effects of incoherent
jumps by evolving the system with an adiabatic shortcut. Once the particles are slowed, they areoften cooled using laser fields. This is the process under investigation next as we demonstratethat the laser fields themselves can absorb atomic entropy during their coherent dynamics. We
then dedicate our study to the steady-state superradiance model in the weak pumping regime.
Here, we investigate the properties of the subradiant state and the appearance of an “enhancement
threshold” where the subradiant state may be extracted for high atom numbers in an experimental
setting. These states are insensitive to spontaneous emission and can therefore protect the evolutionof a system during a quantum metrological process on atomic platforms.

Dedication

To Sandra Bichefsky.

vContents

Chapter
1 Introduction and Motivation

1

2 Atom-Laser Interaction Background3

2.1

Adiabatic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.2

Two-Level Atom Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.3

Rapid Adiabatic Passage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.4

Adiabatic Shortcuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4.1

Berry’s Transitionless Quantum Driving Algorithm . . . . . . . . . . . . . . . 13

2.4.2

Inverse Engineering Based on Lewis-Riesenfeld Invariants . . . . . . . . . . . 17

3 Speeding Up Particle Slowing Using Shortcuts to Adiabaticity
3.1

3.2

20

Application to Particle Slowing in Two-Level Atoms . . . . . . . . . . . . . . . . . . 23
3.1.1

Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.2

Bloch Sphere Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.3

Quantum Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1.4

Slowing Dynamics in Classical Phase Space . . . . . . . . . . . . . . . . . . . 35

3.1.5

Slowing Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1.6

Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Application to Particle Slowing in Three-Level Atoms . . . . . . . . . . . . . . . . . 47
3.2.1

Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

vi

3.3

3.2.2

Slowing Dynamics in Classical Phase Space . . . . . . . . . . . . . . . . . . . 57

3.2.3

Slowing Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4 Entropy Removal by a Laser Field
4.1

64

Preliminary Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.1.1

Quantum Statistical Mechanics Background . . . . . . . . . . . . . . . . . . . 65

4.1.2

Interactions Between Atoms and Quantized Fields . . . . . . . . . . . . . . . 67

4.2

Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.3

Gedankan Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.4

Displacement of Laser Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.4.1

Block Matrix Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.4.2

Fidelity Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.4.3

Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.5

Bayesian Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.6State Purification and Mutual Information . . . . . . . . . . . . . . . . . . . . . . . . 81

4.7

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5 Subradiant State Selection by Photon Detection
5.1

5.2

5.3

86

Dicke Basis Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.1.1

Addition of Angular Momenta . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.1.2

Superradiance and Subradiance . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.1.3

Permutation Invariant Simulations . . . . . . . . . . . . . . . . . . . . . . . . 91

Enhancement and Selection of J = 0 State . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2.1

Superradiant Lasing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.2.2

Subradiant State Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.2.3

Detection Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

vii
6 Conclusion

110

Bibliography

111

Appendix
A Two-Level Hamiltonian Derivation

121

A.1 Atom-Field Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
A.2 Dipole Operator in the Energy Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
A.3 Rotating-Wave Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
A.4 Rabi Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
A.5 Time-Dependent Detuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
B Interaction Picture126

B.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
B.2 Laser Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
B.3 Particle Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
C Open Quantum Systems

131

C.1 Quantum Master Equation in Lindblad Form . . . . . . . . . . . . . . . . . . . . . . 131
C.2 Monte Carlo Wave Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

viii

TablesTable
3.1

Comparison of slowing dynamics between the shortcut slowing scheme based on
Lewis-Riesenfeld invariants and radiation pressure slowing. If Ω sin β  πΓ ρee ,the shortcut slowing scheme results in larger forces, shorter slowing times, shorter
slowing distances, and fewer scattered photons than RP. . . . . . . . . . . . . . . . . 35

3.2

Fundamental properties for various atomic and molecular candidates for shortcut
slowing on a narrow transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3

Slowing results for various atomic and molecular candidates using a 0.1 W/cm2 laser
intensity and starting from a momentum of 200~k. The Rabi frequency, slowing time,and slowing distance follow from Eqs. (3.49), (3.40), and (3.43) respectively, with
β = 0.85 π/2. The simulated capture fraction, defined as the fraction of particles with
momentum |p| ≤ 3~k after the slowing process, was calculated over 1,000 trajectories
for Ca, Sr, and YO, and 10,000 trajectories for Yb and BaH. . . . . . . . . . . . . . 40

5.1

Source: Ref. [154]. Table of quantum jumps resulting from the jump operator associated for the different dissipative processes we consider. . . . . . . . . . . . . . . . . 97

5.2

Source: Ref. [154]. Table of the transition probabilities in a short time interval δtassociated with the quantum jumps involved in the superradiant lasing model. Here,dJN is the degeneracy of the specific J manifold, given by Eq. (5.18). . . . . . . . . . 102

ix
B.1 Source: Ref. [121]. Table of the three main pictures used to determine the time
evolution of a quantum system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

xFiguresFigure2.1Frequency diagram of a two-level particle interacting with a classical field. The
particle has two internal states |ei and |gi representing an excited state and longlived ground state, respectively, separated by the transition frequency ωa . The laser
has time-dependent frequency ω(t) that is detuned from resonance by δ(t) = ω(t) − ωa .

2.2

6

Eigenfrequency diagram for a linear sweep δ(t) = αt and constant Rabi frequencyΩ over resonance. The bare states (|gi and |ei, dashed line) exchange identities inthe dressed eigenstate (|+i , |−i, solid lines with parabolic shape) picture. Driving
the system in the lower eigenstate is shown by the path of arrows. The eigenstates
are split by the generalized Rabi frequency Ω̃(t) which becomes Ω at resonance.
Therefore, eigenstate mixing decreases as Ω increases. . . . . . . . . . . . . . . . . .

2.3

9Stimulated evolution of the excited state population Pe for a linear sweep with constant Rabi frequency. Here, the final excited state populations for the 3 sweep rates
are 0.9996 (κ = 5, black line), 0.7921 (κ = 1, red line), and 0.5441 (κ = 0.5,blue line).
Numerical calculations employed the fouth-order Runge-Kutta integration method. . 11

2.4

Path of the Bloch vector (thick curves) in the laser’s rotating frame for regularadiabatic passage (blue, solid curve), SHAPE with a Lorentzian profile (red, dashedcurve), and SHAPE with a Gaussian profile (yellow, dotted curve) with parametersµ = 0.8761 and τ = Ω1 . Here, κ = 1 and thus, the adiabatic condition is not satisfied. 16

xi
3.1

One-dimensional schematic of a generalized laser cooling experiment. (a) A thermal particle beam exits a particle source and enters a slowing region where a large
classical force is applied opposite to the particles motion (orange arrow). Once theparticles have sufficiently small kinetic energy, they are are then trapped and cooledby causing a large range of velocity classes to be transferred towards zero momentum within a decreased region of physical space (green arrows) and, as a result, lower
entropy. (b) Schematic of the normalized velocity distributions P (v) of the particle
beam before (solid) and after (dashed) the slowing process. We characterize theinitial distribution by a Gaussian function with a standard deviation σv and mean
velocity v̄. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2

Particles exit an atom or molecule source and are collimated into the slowing region.
The spatial setup of the sequentially-pulsed counterpropagating lasers, which havetime-dependent frequencies ω1 (t) and ω2 (t), and a sample particle with velocity v inthe laboratory frame are displayed in the slowing region. The circular inset shows
the two-level internal structure of each particle. . . . . . . . . . . . . . . . . . . . . . 24

3.3

Left: Frequency diagram of an isolated subset of states in the lab frame. The
laser frequencies ωi (p̄, t) [see Eq. (3.1)] are dynamically updated according to thesolution derived from the Lewis-Riesenfeld invariant shortcut method to promote
quick, coherent transfer from (a) |g, p̄i to |e, p̄ − ~ki, followed by (b) |e, p̄ − ~ki to|g, p̄ − 2~ki. Right: Experimental parameters and particle dynamics over a |p̄i →|p̄ − 2~ki sequence of period 2T . (c) Square-pulse Rabi frequency profiles Ω1 (t) (red)and Ω2 (t) (blue) with amplitude Ω [see Eq. (3.28)]. (d) Lewis-Riesenfeld detuning
profile δ(t) [see Eq. (3.29)] with cutoff frequency ±δcut [see Eq. (3.48)] for eachlaser. (e) Ideal excited state fraction Pe dynamics. (f) Ideal average momentum hp̂i
dynamics. Parameters are: T = 0.032/ωr , Ω = 100ωr , δcut = 250ωr , Γ = 0, p̄ =100~k, and β = 0.85 π/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

xii
3.4

Bloch sphere trajectories (thick curves) between the initial state |Gi and final state|Ei parameterized by the auxiliary angle β [see Eq. (3.14)] in the free-energy interaction picture [see Eq. (3.30)]. We set the Rabi frequency Ω to be equal for all
trajectories, and the slowing periods T are given by Eq. (3.28). The cutoff detunings
1 π
1πfor β = 40
2 and β = 2 2 are δcut = 318Ω and δcut = 225Ω, respectively [see Eq. (3.48)]. 32

3.5

Initial (blue) and final (orange) momentum distributions P (p) of a system subject tothe shortcut slowing protocol. The system is initialized in the internal ground state|gi and with a momentum distribution of either the 100~k eigenstate (top row) or a
Gaussian state with average momentum hp̂i = 100~k and width σp = 10~k (bottomrow). The excited state linewidth is Γ = 0 in (a) and (c), and Γ = ωr in (b) and(d). Further details are discussed in text. Other parameters are: Ω = 100ωr , T =0.032/ωr , ttot = 1.6/ωr , and δcut = 244ωr , β = 0.85 π/2, and p̄0 = 100~k. Subplots(b) and (d) are averaged over 1,000 trajectories. . . . . . . . . . . . . . . . . . . . . . 39

3.6

Robustness comparison of π-pulse slowing (purple, plus), SWAP slowing (green,cross), and the shortcut scheme (blue, circle with cross) over a p̄ → p̄ − 2~k transfer
process. Ideally, the resulting impulse ∆p satisfies ∆p/2~k = −1. The evolution ispurely coherent (Γ = 0), the momentum of the particle (which is addressed by thelaser frequencies) is p̄ = 2~k, and the Rabi frequency is Ω = 10ωr for all processes.
(a) Impulse ∆p experienced by the particle, in units of the ideal impulse magnitude
2~k, as a function of the error in the Rabi frequency amplitude  [see Eq. (3.53)]. The
shortcut scheme is the most robust protocol when  > 0. (b) ∆p/2~k as a functionof the relative momentum of the particle δp with respect to the momentum p̄ = 2~kused in the laser frequencies [see Eqs. (3.1) and (3.54)]. For this set of parameters,
the shortcut and π-pulse scheme are generally more robust than SWAP slowing. πpulse parameters are: T = 0.314/ωr and β = π/2. SWAP slowing parameters are:T = 1/ωr and ∆s = 50ωr . Shortcut parameters are: T = 0.44/ωr , δcut = 230ωr , and
β = 0.5 π/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

xiii
3.7

Impulse ∆p experienced by the particle, in units of the ideal impulse magnitude 2~k,
as a function of the laser pulse overlap fraction f . The impulse is calculated underboth coherent (Γ = 0) and dissipative (Γ = ωr ) dynamics. Other parameters are:T = 0.44/ωr , δcut = 230ωr , Ω = 10ωr , p̄ = 2~k, and β = 0.5 π/2. All points areaveraged over 1,000 trajectories.

3.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Particles exit an atom or molecule source and enter a slowing region. The spatial
setup of the counterpropagating lasers, which have time-dependent frequencies ω1− (t)and ω1+ (t) and polarizations σ − and σ + for cycle 1, and a sample particle withvelocity v in the laboratory frame are displayed in the slowing region. The circularinset shows the three-level internal structure of each particle. . . . . . . . . . . . . . 48

3.9

The two-cycle process that makes up the slowing scheme with a large detuningfrom the excited state ∆. The internal states |ei, |g1 i, and |g2 i have magnetic
quantum numbers 0, 1, and −1, respectively. (a) Cycle 1: The counterpropagating(copropagating) laser has polarization σ − (σ + ) to drive the |g1 i → |ei (|ei → |g2 i)transition during the two-photon process |g1 i → |g2 i for an impulse of ∆p = −2~k.
(b) The laser polarizations are flipped in order to now drive the |g1 i → |g2 i transition
with another impulse of ∆p = −2~k. . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.10 Sanpshots of the momentum distribution P (p) of the system at different times
throughout the slowing process. The system is initialized (purple line) in a Gaussianstate with average momentum hp̂i = 400~k and width σp = 50~k. Snapshots of the
momentum distribution at 50 (green distribution), 100 (blue distribution), 150 (golddistribution), and 200 sweeps (red distribution) are shown. The other parameters
are: Γ = ωr , ∆ = 5000ωr , β = − π4 , δcut ≈ 89ωr , τ = 0.025/ωr , and T = 0.1/ωr ,which leads to a maximum Rabi frequency of Ωmax ≈ 1004ωr . Results are averaged
over 500 trajectories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

xiv
3.11 The detuning profile (dashed green curve) and the absolute value of the Raman
Rabi frequency profile (blue curve) over a single sweep. Also displayed is the cutoff
scheme for the detuning profile at cutoff time tcut (purple curve). The parameters
are: ∆ = 5000ωr , β = − π4 , δcut ≈ 89ωr , τ = 0.025/ωr , and T = 0.1/ωr . . . . . . . . . 61
4.1

Schematic diagram of laser cooling gedanken experiment. (a) Energy diagram ofthe particle’s internal state structure which is a Λ-configuration. It is coupled tothe optical cavity with strength g on the bright transition (|ei ↔ |bi) and has alinewidth Γ on the dark transition (|ei → |di). (b) A schematic of the experimental
setup. The quantum gas (orange circle) is placed in a lossless (κ = 0) optical cavity(dark blue mirrors) that contains a laser field (light blue). The particle can undergo
spontaneous emission into free space with rate Γ . . . . . . . . . . . . . . . . . . . . . 71

4.2

Fidelity F Eq. (4.50) between the initial and final laser states ρ̂F (0) and ρ̂F (∞)(solid blue curve) as a function of the initial average intracavity photon number n̄0 .
The dashed red curve displays F in the infinite cavity photon limit n̄0 → ∞. The
fidelity F is equal to unity only when n̄0 = 0, demonstrating that the laser state ischanged upon interaction with the particle. The parameters are g = Γ = 1. . . . . . 79

4.3

Standard deviation σN (x) of the Gaussian fit to the posterior probability distribution P (x|n) as a function of the initial bright fraction x after N = 25 (blue circles),
50 (orange targets), and 100 (gold stars) iterations. Inset: An example of a simulated P (x|n) (blue, dotted) and associated Gaussian fit σ1 00(x) (black, solid). Both
distributions are maximized near the actual bright fraction x = 0.1 (red, dot-dashed). 82

4.4

Entropy of various subspaces and quantum mutual information (dashed gold curve)
between the field and non-interacting atoms I(R : F ) = S(ρ̂R ) + S(ρ̂F ) − S(ρ̂RF ).
We display the entropy of the atomic (blue curve), field (green curve), and atomauxiliary atom (red curve) subspaces, as well as the total entropy (purple curve) intime. The parameters are g = Γ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

xv
5.1

Source: Ref. [111]. Two-atom metrology plotting the optimal sensitivity δω as afunction of τ Γ . This displays the enhancement of sensitivity for long time scales
while using the anti-symmetric Ramsey sequence. The atoms are separated by adistance r = 0.3λ, with wavelength λ. . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.2

Source: Ref. [63]. Comparision of the general characteristics of ordinary fluorescence(a) and superradiant fluorescence (b) for decay time τSp = 1/Γ . . . . . . . . . . . . . 92

5.3

Source: Ref. [90]. Schematic diagram of the superradiant lasing model. N two-levelatoms (black dots with internal structure given in the inset) with individual decayrate Γ are collected at the anti-nodes of a single-mode cavity field. The cavity has
single atom cooperativity parameter C, decay rate κ, and coupling constant g. The
atoms are incoherently repumped at a rate w. The output of the cavity field ismonitored by two detectors D1 and D2 with a variable time delay τ between them. . 95

5.4

Source: Ref. [91, 150]. Schematic diagram for implementing the incoherent pumping
scheme with a three-level atom at an effective rate w = |Ωp |2 /γp (brown arrow).
The atoms are coherently pumped (black arrow) from the ground state |gi to theauxiliary state |ai at a rate Ωp . This then decays to the excited state |ei at a rate
γp (yellow arrow). This can then decay back to the ground state at an individual
decay rate Γ (blue arrow). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.5

Source: Ref. [154]. Dicke ladder diagrams with N = 4 for (a) individual decay,(b) collective decay through the cavity, (c) individual pumping, and (d) individualdephasing. The thickness of the arrows indicates the probabilities of the different
jumps (see, for example, Table 5.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.6

Source: Ref. [90]. Second-order intensity correlation function g (2) (0) as a function of
repump rate for N = 10. Results were calculated using Eq. (5.28) and the dissipative
picture master equation Eq. (5.24). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

xvi
5.7

Example of the time evolution of the density matrix in the superradiant lasing modelwith N = 2 and repump rate w = 0.1Γc . The red areas indicate highly populated
states. (a) The system may, for example, begin in the excited state ρ̂(0) = |1, 1ih1, 1|.
(b) No matter the initial state, the system evolves under Eq. (5.24) to a uniquesteady-state ρ̂ss . Also displayed is the block diagonal form of the density matrix inthe Dicke basis where the different blocks represent different J manifolds. . . . . . . 101

5.8

The population of the J = 0 state as a function of the atom number. The values of thesuperradiant lasing scheme (black dots) are given by the steady-state of Eq. (5.24),
while the infinite temperature state values (red dots) are calculated by Eq. (5.19).
The repump rate for each N is given by w = Γc /N . . . . . . . . . . . . . . . . . . . . 103

5.9

Contour plot of the steady-state subradiant state population as a function of atom
number N and repump rate w. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.10 Verification of an “enhancement threshold” near w = Γc /N , as displayed by the significant change of behavior in steady-state near this region. Each subfigure contains
plots for N values in the set {2, 4, 6, . . . , 18, 20, 30, 50, 70}. (a) Steady-state population in the J = 0, subradiant state. (b) Expectation value hJˆ+ Jˆ− i in steady-statenear the threshold, which is related to hâ† âi. (c) Sum of the steady-state probabilities in the states with the lowest M value of each Dicke block. (d) Sum of thesteady-state probabilities in the |0, 0i and |1, −1i states. . . . . . . . . . . . . . . . . 105
5.11 Density matrix after the detection phase for N = 2, Ω = Γc , and w = 0. The
system is initialed in the steady-state of the superradiant lasing model from Fig. 5.7
and evolved under the detection phase master equation Eq. (5.32) for a total time
tf = 10/Γc . The red areas indicate highly populated states. . . . . . . . . . . . . . . 107
C.1 Algorithm for implementing the Monte Carlo wave function formalism into our simulations of open quantum systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Chapter 1

Introduction and Motivation

The creation of ultra-low temperature particle ensembles through cooling techniques has
been at the forefront of contemporary atomic and molecular physics since the 1970’s. In such
experiments, cooling occurs when phase space is compressed as the entropy of the ensemble istransferred to other subsytems of the universe. This entropy exchange is governed by the second
law of thermodynamics, a law so fundamental in our current understanding of physics that Albert
Einstein believed [69]:
It is the only physical theory of universal content concerning which [he is] convincedthat within the framework of the applicability of its basic concepts, it will never be
overthrow.
However, the fact that only the total entropy of the universe must increase often makes it unclear
how the entropy is redistributed during the cooling process. It is often assumed that this removal
relies of the irreversibly of incoherent processes. Yet, the spontaneity of incoherent quantum jumps
may lead to poor control in experimental settings, as well as the loss of particles to undesired states.
Similar undesirable dynamics occurs in many techniques in the ever growing field of quantum
metrology. Here, incoherent processes can severely impact the precision of measurements andtherefore mitigate the enhancement achieved by the exploitation of quantum effects.
It is thus desirable to increase coherent control of the system under study to reduce thereliance and occurrence of spontaneous processes. In doing so, one may ask the question if thesystem preforming the coherent dynamics, typically a laser field, may also absorb some of the2initial entropy in the particle ensemble during the cooling process. Furthermore, increased control
over the system dynamics may open the pathway for metrological techniques that can sustain
many-body systems for long periods of operation. In this spirit, we identify three major themesthat we address throughout the duration of this thesis to advance the cooling of quantum states:(1) Presenting novel techniques that minimize spontaneous events and instead rely to a greater
extent on coherent dynamics,(2) The creation of low entropy states through an ensemble’s interaction with laser fields,(3) Studying the transfer of entropy from an atomic gas to the laser fields during laser cooling.
In this thesis, we present our contributions to the study of coherent control in creating andmaintaining quantum systems. We begin with an introduction to the interaction of a particle witha classical laser field in Chapter 2. In this, we will discuss the field of adiabatic shortcuts, where
coherent dynamics can be achieved in prodigious evolution times. In Chapter 3, we exploit theadvantages of adiabatic shortcuts to create a novel particle slowing protocol. This falls into thefirst theme as open dynamics with scattered photon counts on the order of unity may be achieved.
We then turn our attention to the second and third themes in Chapter 4 as we directly study theexchange of entropy during a laser cooling gedanken experiment. Here, we find that the laser field
can indeed remove entropy from the atomic gas during the cooling process. We present anotherproposal for creating low entropy quantum states in Chapter 5 using the model for steady-statesuperradiance. This again falls under the second theme. While a low-entropic system is the result of
our investigation, this chapter is primarily motivated by the first theme as the procedure enhances
population in subradiant states that are insensitive to spontaneous emission. These states are thenextracted and may be used for quantum metrological purposes to mitigate the effects of incoherent
processes. We conclude in Chapter 6 with a brief summary of our work.

Chapter 2

Atom-Laser Interaction BackgroundI begin with one of the crucial problems in theoretical quantum optics, describing the interaction of a two-level particle with a classical field. The applicability of the model extends muchfurther than what we cover here, as it is also useful to describe phenomena ranging from qubits to
Bragg diffraction to neutrino oscillations [70, 59, 54]. I narrow my focus in this thesis to its application to particle-laser interactions that results in population transfer from one internal electronic
state to another. In particular, I first look at systems that have a decreased reliance on incoherent,
irreversible spontaneous emission compared to traditional schemes that are ubiquitously employedthroughout the state of the art. Therefore, the focus of this background chapter is the coherent
dynamics resulting for a particle’s interaction with a classical laser field which drives stimulated
absorption and stimulated emission between internal states. I switch my analysis in Chapter 4 toa particle’s interaction with a laser when the latter is also treated quantum mechanically. I follow
this by considering an ensemble of N two-level particles in Chapter 5, where collective incoherent
processes can emerge by way of symmetry properties and interatomic coherence [46, 133, 63].
First demonstrated in Einstein’s seminal paper on the quantum theory of radiation [48],stimulated emission offers many advantages over the randomness of spontaneous emission in thefield of laser slowing and cooling, as well as elsewhere. As we shall see in Chapter 3, time-orderingof stimulated emission and absorption events driven by counterpropagating lasers has the potential
to generate very large forces without reliance on the relaxation of the particle to return to theground state manifold, leading to a nearly coherent slowing scheme that has a small associated4
slowing distance. We study coherent dynamics in a somewhat deeper manner in Chapter 4, where
we analyze entropy production in the laser fields during laser cooling. Before this, however, I offer
background information on the coherent dynamics of a two-level particle interacting with a classicalfield.
I begin this chapter by presenting an introduction to the adiabatic approximation. This isfollowed by studying the approximation’s application to a two-level particle dressed by a laser. We
then show how these processes can be sped up using adiabatic shortcuts: processes that achieve
the same dynamics as ideal, adiabatic dynamics, but in much less time. I present both the counterdiabatic driving and Lewis-Riesenfeld invaniant (LRI) shortcuts, the latter of which is used in
the context of laser slowing in Chapter 3. This chapter exclusively studies coherent dynamics.

2.1

Adiabatic Approximation
In this section, I offer a brief description of the adiabatic approximation, following Sakurai’s

explanation in [121, Chapter 5, § 6]. This is the the foundation of adiabatic passage between
internal states, as demonstrated in the next section. Assuming no degeneracy, the instantaneousenergy eigenvalue equation,Ĥ(t) |n(t)i = En (t) |n(t)i ,(2.1)

implies that a general solution to the time-dependent Schrödinger equation,i~∂t |Ψ (t)i = Ĥ(t) |Ψ (t)i ,(2.2)

can be written in the energy basis as|Ψ (t)i =X|ψn (t)i =nwhere θn (t) = − ~1Rt00
t0 En (t )dt .

X

cn (t)eiθn (t) |n(t)i ,

(2.3)

n

Plugging this into the Schrödinger equation and taking the innerproduct with hm(t)|, we findċm (t) = −Xneiθn (t)−iθm (t) hm(t)|∂|n(t)i .
∂t

(2.4)

5
By differentiating the instantaneous energy eigenvalue equation Eq. (2.1), we arrive atċm (t) = −cm (t) hm(t)|∂t m(t)i −X

cn (t)ei(θn (t)−θm (t))n6=mhm(t)| ∂t Ĥ |n(t)i,
En − Em(2.5)

where the dot denotes a derivative with respect to time. It is clear that, for time-dependent
Hamiltonians, the second term of Eq. (2.5) will cause mixing between the |m(t)i and |n(t)i states
when m 6= n. By assuming
hm(t)| ∂t Ĥ |n(t)i hm(t)|∂t m(t)i ,(En − Em )(2.6)

the adiabatic approximation removes this eigenstate mixing:ċm (t) ≈ −cm hm(t)|∂t m(t)i .

(2.7)

Furthermore, it can be shown that the nth state vector takes the form
Rt

0

0

0

|ψn (t)i = e− 0 hn(t )|∂t n(t )idt eiθn (t) |n(t)i .

(2.8)

Thus, the adiabatic approximation Eq. (2.6) implies that a system can be driven in its instantaneous
eigenstates, up to a phase, under the condition that the Hamiltonian changes gradually.
The adiabatic approximation has led to revolutionary discoveries such as Berry’s phase [18]and the Aharonov-Bohm effect [5] which have changed our understanding of the quantum phase. I
focus on its application to the time-dependent two-level particle Hamiltonian to derive a condition
that defines a regime where eigenstate mixing is limited.

2.2

Two-Level Atom Hamiltonian
We now model a particle as a two-level system consisting of a single internal ground state

and a single internal excited state separated by atomic frequency ωa , as depicted in Fig. 2.1. Anexample of this system is the 1 S0 → 3 P1 transition in 88 Sr. We investigate the Hamiltonian of a
two-level particle driven by a classical field that is varying in time. The Hamiltonian, after therotating-wave and dipole approximations, is given by
Ĥ =

R

~ωa
~Ω(t) i t ω(t0 )dt0(|eihe| − |gihg|) +|gihe| e t0
+ H.c. ,
2
2

(2.9)

6

Figure 2.1: Frequency diagram of a two-level particle interacting with a classical field. The particle
has two internal states |ei and |gi representing an excited state and long-lived ground state, respectively, separated by the transition frequency ωa . The laser has time-dependent frequency ω(t)that is detuned from resonance by δ(t) = ω(t) − ωa .

7where ω(t) is the laser’s frequency, ~ is Planck’s reduced constant with units [~] = J · s, H.c. standsfor Hermitian conjugate, and I have defined E = 0 to be halfway between the electronic states.
√
The laser’s Rabi frequency Ω(t) is related to its intensity Ω ∝ I [see Eq.(3.49)] and given by
Eq. (A.20). The first two outer products in Eq. (2.9) represent the bare energies of the internal
states. The terms proportional to the Rabi frequency represent stimulated emission |gihe| andstimulated absorption |eihg| of a photon due to the particle’s interaction with the laser field. The
detailed derivation of Eq. (2.9) is provided in Appendix A.
We next move into a rotating frame by defining an interaction picture:
Ĥ0 =

~ω(t)
(|eihe| − |gihg|) .
2(2.10)

This gives, as shown in Appendix B.2,
~δ(t)
~Ω(t)
˜
Ĥ =
(|gihg| − |eihe|) +(|gihe| + |eihg|) ,2
2(2.11)

where we have defined the time-dependent laser detuning δ(t) = ω(t) − ωa . Eq. (2.11) can be
written in matrix form as~  −δ(t) Ω(t) ˜Ĥ = 
,2Ω(t) δ(t)where |gi =01

and |ei =10

(2.12)

are the ground and excited states, respectively.

Due to the interaction with the classical field, the bare states |gi and |ei are no longer
the eigenvectors of the system’s Hamiltonian. We now find the new eigenstates that diagonalize
Eq. (2.11). The instantaneous energy eigenvalues are proportional to the generalized Rabi frequency√Ω̃ = δ 2 + Ω 2 :E± (t) = ±~Ω̃(t).
2(2.13)

We note that
2

2



2

Ω̃ (t) = δ (t) + Ω (t) −→ 1 =

δ(t)
Ω̃(t)

2


+

Ω(t)
Ω̃(t)

2
,

(2.14)

which motivates the definition of the Stückelberg angle,χ(t) ≡ arccos

δ(t)
Ω̃(t)


,

(2.15)

8
in deriving the system’s instantaneous eigenvectors, named the dressed states [130]:




χ(t)
χ(t)
|+(t)i = cos|ei + sin
|gi ,
2
2




χ(t)
χ(t)
|ei − cos|gi .
|−(t)i = sin22(2.16)

We will see that the instantaneous eigenvectors of the system will give a good amount of insighton effectively transferring population between two electronic quantum states.

2.3

Rapid Adiabatic Passage
The goal of this section is to achieve population transfer between internal states via theparticle’s interaction with a classical field. The most basic way to achieve such a transfer between
two quantum states is to apply a resonant (δ(t) = 0) light pulse with an “area” of π, 1
Z tfΩ(t0 )dt0 = π,(2.17)

t0

with initial and final times t0 and tf . While this Rabi-flopping “π-pulse” method, in theory,completely transfers population, it is not robust to small errors in Ω, the frequency ω of thelight source, or coupling to other states outside of the two-level manifold. These are importantconsiderations when the goal of the process is to transfer a wide distribution of momentum states,
as is the goal consider in the next chapter.
To overcome these problems, one can instead chirp the laser frequency from well below
resonance to well above resonance. We see from Eq. (2.16) that transfer from one bare state to theother can be accomplished by driving the particle in one of its dressed states, which Eq. (2.6) states
is possible if the time-dependent perturbation applied to the system varies slowly, and changing
χ(t) by π. The transfer between bare states becomes apparent in the avoided crossing depicted inFig. 2.2, where a frequency gap in the eigenvalue spectrum causes an exchange of the uncoupledbare eigenstates. This process, when satisfying the adiabatic approximation, is called adiabatic
passage.
1
This is seen from Eq. (2.19), with δ(t) = 0, by decoupling through differentiation and noting the normalization
condition when solving the acquired second-order differential equations.

9

Figure 2.2: Eigenfrequency diagram for a linear sweep δ(t) = αt and constant Rabi frequency Ω overresonance. The bare states (|gi and |ei, dashed line) exchange identities in the dressed eigenstate(|+i , |−i, solid lines with parabolic shape) picture. Driving the system in the lower eigenstate isshown by the path of arrows. The eigenstates are split by the generalized Rabi frequency Ω̃(t)which becomes Ω at resonance. Therefore, eigenstate mixing decreases as Ω increases.

10
One can then ask how much population is lost if the frequency is swept too quickly so thatEq. (2.6) is not satisfied. This problem can be solved analytically, as first shown by Landau andZener [153], and is often called the Landau-Zener crossing problem. Let us define the atomic statevector as |ψ(t)i = cg (t) |gi + ce (t) |ei, where we note the normalization condition for the complex
P2
amplitudes of the state vector:
i |ci | = 1. The time-dependent Schrödinger equation Eq. (2.2)
using the Hamiltonian given in Eq. (2.11) then reads

i~

∂cg
∂ce
|gi +
|ei
∂t
∂t


=

~δ(t)
~Ω(t)
(cg |gi − ce |ei) +(ce |gi + cg |ei) .
22(2.18)

Projecting with hg| and he|, the equations of motion of the bare state amplitudes are
iΩ(t)
iδ(t)
cg −
ce ,
2
2
iδ(t)
iΩ(t)
ċe =
ce −
cg ,
2
2

ċg = −

(2.19)

which we can now solve for the atomic evolution by decoupling by differentiation. The resulting
second-order equations take the form of Weber’s equation,
1 z 2 t2
∂t2 + ν + −24


f = 0,(2.20)and can be solved analytically (see [130, Chapter 5, § 3]). The resulting Landau-Zener equation
gives the probability of ending the sweep in the excited state, Pe (t → ∞) = |he|ψ(t → ∞)i|2 , for agiven sweep slope. Assuming a real and constant Rabi frequency, as well as the particle beginning
the sweep in the ground state, it is given byPe (t → ∞) = |ce |2 = 1 − exp −πΩ 2 

2 δ̇(t) πκ 
= 1 − exp −.
2(2.21)

Here, we have defined the adiabaticity parameter κ, and we see that the adiabatic approximation
for transfer in the dressed states of the system is given by κ  1. Eq. (2.21) demonstrates that in
the regime of very large laser detuning slope, δ̇(t)  Ω 2 , the excited state population decreases
substantially as a result of tunneling across the energy gap in Fig. 2.2. This causes the particle toend the sweep in the opposite dressed state, and thus the same bare state, in which it began thesweep as Eq. (2.6) is not satisfied. This is illustrated in Fig. 2.3 for a linear detuning sweep.

11

Figure 2.3: Stimulated evolution of the excited state population Pe for a linear sweep with constantRabi frequency. Here, the final excited state populations for the 3 sweep rates are 0.9996 (κ = 5,
black line), 0.7921 (κ = 1, red line), and 0.5441 (κ = 0.5,blue line). Numerical calculations
employed the fouth-order Runge-Kutta integration method.

12
We have thus found a solution to stimulate absorption and emission with a classical laser
field that is robust to many of the problems encountered when preforming π-pulses. We have so far
omitted the incoherent effects of spontaneous emission that will cause the excited state to decay
down to the ground state at the natural decay rate Γ of the atomic state. This omission is justified
in a process called rapid adiabatic passage by requiring that the chirp must be fast compared tothe decay rate of |ei. Additionally, the field must be strong enough so that the dressed states arewell resolved in the presence of the homogeneous broadening of the energy levels.
The velocity-independent feature of rapid adiabtic passage motivated the development ofsawtooth-wave adiabatic passage (SWAP) cooling [108, 14, 62, 13, 127, 104], which coherently
transfers a particle between quantum states via periodic Landau-Zener chirps in order to reduce themomentum of the system. However, the requirement to satisfy Eq. (2.6) may cause long evolution
times and the process can be sensitive to the laser’s phase. Furthermore, in the neighborhoodof the atomic resonance, there are significant oscillations in the populations that result from theprecession of the Bloch vector as it travels along the Bloch sphere [52, 147], which can complicate
the amount of population transfer. As I shall show, an alternative scheme for coherent transfer by
use of adiabatic shortcuts ameliorates the issues that have been described for π-pulses and adiabatic
passage.

2.4

Adiabatic Shortcuts
Over the past decade, there has been a rapid development and growth of interest in short-cuts to adiabaticity. These are processes with the same outcome as ideal, infinitely long adiabatic
processes, but in a much quicker time. A variety of techniques have been proposed and experimentally demonstrated: transitionless quantum driving [45, 19, 31, 16], inverse engineering based on
Lewis-Riesenfeld invariants [80, 32, 31, 78, 33, 65], optimal control bang-bang type [146, 27, 12, 38],
fast-forward techniques for Schrödinger [87, 88, 142] and Dirac dynamics [44], “environment” assisted methods [89], fast quasiadiabatic dynamics (FAQUAD) [85], and using the properties of Liealgebras [123, 86, 141, 112]. So far, these adiabatic shortcut techniques have been employed in fric-13
tionless cooling in harmonic traps [31], in Penning traps [38], in cavity quantum electrodynamics[33], in suppressing pair production [44], in atom interferometry [47], in implementing the AllenEberly scheme [31], in STIRAP demonstrations [31, 89, 81, 103], in quantum simulators [12, 11],in quantum computing [135], and even in quantum gaming [125]. A review of the growing shortcut
research field is presented in [64]. In this section, we focus on shortcut techniques that employ
inverse engineering. In Chapter 3, we demonstrate how coherent particle slowing processes can besped up using adiabatic shortcuts.

2.4.1

Berry’s Transitionless Quantum Driving Algorithm

The idea of counteradiabatic driving was first developed in the early 2000’s [45], but grew inpopularity significantly after it was rediscovered by Michael Berry in [19] and when it was applied
to two- and three-level atoms in [30]. Berry called his procedure the transitionless quantum driving(TQD) algorithm and it is based on the idea of inverse engineering. Rather than finding the stateof the system for a known time-dependent Hamiltonian, inverse engineering prescribes that thedriving Hamiltonian is determined from the desired state evolution.
Our desired evolution is obtained, in ansatz to Eq. (2.8), by defining the unitary timeevolution operator |Ψ (t)i = Û (t) |Ψ (0)i as [19]Û (t) =

X

Rt

0

0

0

e− 0 hn(t )|∂t n(t )idt eiθn (t) |n(t)ihn(0)| .

(2.22)

n

It can be shown [93, Chapter 8, § 8] that the time-evolution operator satisfiesi~∂ Û (t)= Ĥ(t)Û (t),∂t(2.23)and the instantaneous Hamiltonian can therefore be written as
Ĥ(t) = i~

∂ Û †Û .
∂t(2.24)

Plugging in Eq. (2.22), we find that the driving Hamiltonian can be written as
Ĥ(t) = Ĥ0 + Ĥ1 = Ĥ0 + i~

X X |m(t)ihm(t)| ∂t Ĥ0 |n(t)ihn(t)|m6=nEn − Em,(2.25)

14
where Ĥ0 =P

n En (t) |n(t)ihn(t)| is the original Hamiltonian in its instantaneous eigenbasis. Notethe similarity of the counteradiabatic term Ĥ1 to Eq. (2.5). Performing the unitary transformationon the initial state ket |ψ(0)i, we find the state evolution is|ψ(t)i = Û (t) |ψ(0)i =

X

Rt

0

0

0

cn (0)e− 0 hn(t )|∂t n(t )idt eiθn (t) |n(t)i .

(2.26)

nThus, the system will now be driven exactly in the instantaneous eigenstates with no state mixing,
as the adiabatic approximation becomes exact [see Eq. (2.8)]. In other words, we have found a
nearby Hamiltonian with the property that the transition amplitudes between any eigenstates ofthe original Hamiltonian are exactly zero for any speed.
The simplest counterdiabatic Hamiltonian is given byĤ1 = i~X|∂t nihn| .

(2.27)n

For the Landau-Zener scheme with a linear detuning, δ(t) = αt + t0 , and constant Rabi frequencyΩ(t) = Ω, this gives [64]

Ĥ1 =

~

2

0

−iΩ 0 (t)iΩ 0 (t)

0



,

(2.28)

where Ω 0 (t) = δ̇(t)Ω/Ω̃ 2 (t). Noting that δ̇(t) is a constant, we see that Ω 0 (t) has a Lorentzian
profile. Taking the same physical interpretation of the counterdiabatic term presented in [30], we
see that Ĥ1 amounts to adding a second laser 90◦ out of phase from the first (note i = exp[iπ/2])with a Lorentzian Rabi frequency profile. This extra laser will exactly cancel out any eigenstate
mixing that would take place.
While this is the simplest counteradiabatic Hamiltonian, an infinate number of driving Hamiltonians are possible (see [16], for example). A nearby solution is given by a Gaussian Rabi frequency
profile on the auxiliary laser,t2Ω (t) = µΩ exp
,
2τ 2
0



(2.29)

where I have set t0 = −tf , δ(0) = 0, defined the dimensionless parameter µ, and defined theparameter τ with dimensions of time. To visualize the speed up of the adiabatic process, I take

15
advantage of the pseudo-spin 21 nature of the system to examine the dynamics on the Bloch spherein the so-called Feynman-Vernon-Hellwarth representation [52]. Assuming a pure state, the density
operator ρ̂ is given by ρee ρge ρ̂(t) = |ψ(t)ihψ(t)| = ,ρeg ρgg(2.30)

where ρij = hi| ρ̂ |ji = ci c∗j . Note the normalization condition now reads Tr[ρ̂] = 1, where Tr[·] isthe trace. We can map this to a 3D representation via the expectation values of the three Paulimatrices,hσ̂x i = hσ̂ − i + hσ̂ + i = ρeg + ρge ,hσ̂y i = ihσ̂ − i − ihσ̂ + i = iρeg − iρge ,(2.31)hσ̂z i = hσ̂ + σ̂i− − hσ̂ − σ̂ + i = ρee − ρgg ,where σ̂ − = |gihe| (σ̂ + = |eihg|) is the lowering (raising) operator and we have used the cyclic
property of the trace in the expectation value equation [121, Chapter 3, §4]h ihÂi = Tr Âρ̂ .

(2.32)

hσi = hσ̂x iı̂ + hσ̂y i̂ + hσ̂z ik̂,

(2.33)The Bloch vector

is plotted in Fig. 2.4 for the adiabatic process, as well as for shortcuts to adiabatic passage (SHAPE)
with the Lorentzian and Gaussian schemes. We can see from Eq. (2.31) that the south pole of the
Bloch sphere corresponds to an unexcited particle, while the north pole is a completely excited
particle. Fig. 2.4 demonstrates that, for a particular value of κ, the adiabatic process has significant
loss [Pe (t → ∞) ≈ 0.8], but both shortcut schemes reach the north pole along two different, but
nearby, paths on the Bloch sphere.
While the transitionless quantum driving algorithm drives the system perfectly in the dressedstates of the system, the question that arises is about a trade-off between speed and energetic
resources for applying the shortcut. In other words, at what value of κ does the amount of auxiliary
laser power needed to drive the system via TQD become so great that simply increasing the original

16

jei

<z i
h^

Adiabatic
LorentzianGaussian

jgi
h^
<y i

h^
<x i

Figure 2.4: Path of the Bloch vector (thick curves) in the laser’s rotating frame for regular adiabatic
passage (blue, solid curve), SHAPE with a Lorentzian profile (red, dashed curve), and SHAPE witha Gaussian profile (yellow, dotted curve) with parameters µ = 0.8761 and τ = Ω1 . Here, κ = 1 andthus, the adiabatic condition is not satisfied.

17
laser power by that amount would satisfy that new adiabatic condition? The concept of an energetic
cost for implementing adiabatic shortcuts and the intrinsic relationship between this cost andshortcut speed has been rigorously studied for Berry’s TQD algorithm [26], various other shortcuts[1], and in its applications to quantum computing [41]. The study of multiple shortcuts in [1]suggests that counteradiabatic driving schemes (such as TQD) are, in general, more energeticallycostly than other shortcut protocols such as the bang-bang shortcut and inverse engineering basedon Lewis-Riesenfeld invariant theory. This motivates the use of the latter as the shortcut protocol
to speed up particle slowing. I will now briefly introduce the Lewis-Riesenfeld invariant shortcut
protocol, followed by its application to particle slowing in Chapter 3. More on this concept of theenergetic cost to implement a shortcut scheme will be discussed in Section 3.3.

2.4.2

Inverse Engineering Based on Lewis-Riesenfeld Invariants

The theory of Lewis-Riesenfeld invariants (LRI) was developed in the late 1960’s to derive
the solutions of the Schrödinger equation for explicitly time-dependent Hamiltonians [80]. Lewis
and Riesenfeld derived a simple relationship between the eigenstates of the Hamiltonian and a
dynamical invariant of the system. They then demonstrated how to track the evolution of a statein the time-dependent harmonic oscillator potential and to describe a charged particle in a timedependent electromagnetic field using this protocol.
ˆ
A dynamical invariant I(t)is a Hermitian operator with a time-independent expectation
value, i.e.,ˆ = hΨ (t)| I(t)ˆ |Ψ (t)i = const,
hIi(2.34)

where |Ψ (t)i is the state vector evolved by the Hamiltonian Ĥ(t). This satisfiesi~ˆ∂ I(t)
ˆ− [Ĥ(t), I(t)]=0∂t(2.35)

in the Schrödinger picture [80], where [A, B] = AB − BA is the commutator. The states |ψn (t)i,defined by a gauge transformation|ψn (t)i = eiαn (t) |φn (t)i

(2.36)

18
ˆ |φn (t)i = λn |φn (t)i are each a solution to the time-dependent Schrödingerof the eigenbasis I(t)equation, and it can be shown that λ̇n = 0. In Eq. (2.36), the “Lewis-Riesenfeld phases” αn (t) aredefined as1αn (t) =~Z t

φn (t0 ) i~t0∂
− Ĥ(t0 ) φn (t0 ) dt0 .
∂t0(2.37)

It follows that a general solution |Ψ (t)i to the Schrödinger equation can be decomposed as|Ψ (t)i =X

cn |ψn (t)i =X

cn eiαn (t) |φn (t)i ,(2.38)

nn

where cn are time-independent amplitudes. Thus, the unitary time-evolution operator can be
written asÛ (t) =X

eiαn (t) |φn (t)ihφn (t0 )| ,(2.39)nand the evolution of the system can now be tracked in the invariant basis.
Lewis and Riesenfeld originally used this definition of the time evolution operator with thegoal of finding the unknown time-evolution of the state given some known time-dependent Hamiltonian. However, interest in Lewis-Riesenfeld invariants grew immensely decades later when theconcept of inverse engineering was applied [31]. Here, the desired time-evolution of the system isgiven, and one uses Eq. (2.39) along with Eq. (2.24) to explicitly solve for the driving Hamiltonian,Ĥ(t) = −~X

α̇n |φn (t)ihφn (t)| + i~nX|∂t φn (t)ihφn (t)| .

(2.40)nEquating the Hamiltonian in the invariant basis to the original Hamiltonian creates a map between
the physical parameters and the auxiliary parameters that define the invariant operator. We use
this procedure in the next chapter to derive laser detuning and Rabi frequency profiles that drive
ˆ area system perfectly from its initial state to a final target state. In particular, if Ĥ(t) and I(t)designed to commute at t0 and tf , i.e.,hi hiˆ 0 ) = Ĥ(tf ), I(tˆ f ) = 0,
Ĥ(t0 ), I(t(2.41)

then the final state |Ψ (tf )i will maintain the initial populations for each eigenstate [32, 64] as
the operators will be simultaneously diagonalizable. A way of visualizing this is a similar avoided

19crossing picture as in Fig. 2.2, but now in the invariant basis. Here, the eigenvalues are constantλ̇n = 0 which results in a constant energy splitting so that tunneling between eigenstates does notoccur. We will therefore recover the results of an adiabatic process without the requirement of slow
time-evolution. I note in passing the possible connection between inverse engineering using Berry’s
algorithm and Lewis-Riesenfeld invariants [32, 64]. In the next chapter, I apply the adiabatic
shortcut to a velocity-selective particle slowing scheme, affording us with the robustness of anadiabatic slowing scheme, but with a smaller slowing time and associated slowing distance. This is
the first step in creating low entropy quantum states by way of laser cooling.

Chapter 3

Speeding Up Particle Slowing Using Shortcuts to Adiabaticity

For decades, laser cooling atoms and molecules to near absolute zero has been at the forefront of research into the interactions of light and matter [35, 114, 37, 96]. Ultracold ensembles
provide testbeds for exploring fundamental physics [120], can create low-temperature superfluids
such as Bose-Einstein and fermionic condensates [8, 117], and can be used as platforms for quantum simulators [73]. A typical design of a laser cooling scheme for a thermal beam of particles
emerging from a particle source, such as an oven or supersonic nozzle, is depicted schematically in
Fig. 3.1(a). Since atom and molecule sources produce particle beams with a high average velocity,
it is common to first preform a precursor slowing stage that generates a large classical force against
the particle’s motion [see Fig. 3.1(a)] to remove a substantial fraction of the particle’s kinetic energy [see Fig. 3.1(b)]. An assortment of particle slowing methods have been developed for thisprecursor stage, including Stark and Zeeman decelerators [105, 67, 6, 113], centrifuge decelerators[34], electrostatic trapping methods [21], frequency-chirped laser slowing [144], white light slowing[39, 66], and angled slowing [84], among others. Once slowed, the particles may possess sufficiently
low kinetic energy that they can be efficiently loaded into a finite-depth electromagnetic trap [96],such as a magneto-optical trap (MOT), and then be cooled using light [132, 145, 13], cooled throughevaporation that redistributes energy through two-body collisions [134], or sympathetically cooledwith another species [49]. These basic steps have been the key to opening up the world of quantum gases to many modern applications of atomic and molecular physics in quantum science andengineering.

21

Figure 3.1: One-dimensional schematic of a generalized laser cooling experiment. (a) A thermal
particle beam exits a particle source and enters a slowing region where a large classical force isapplied opposite to the particles motion (orange arrow). Once the particles have sufficiently small
kinetic energy, they are are then trapped and cooled by causing a large range of velocity classes tobe transferred towards zero momentum within a decreased region of physical space (green arrows)
and, as a result, lower entropy. (b) Schematic of the normalized velocity distributions P (v) of theparticle beam before (solid) and after (dashed) the slowing process. We characterize the initial
distribution by a Gaussian function with a standard deviation σv and mean velocity v̄.

22
The ability to produce a large number of ultracold particles is made difficult by practicalshortcomings of slowing methods, such as a large slowing distance that requires significant physicalspace, or substantial spread in the final velocities of the particles. Furthermore, the main hindrance
to slowing particles that lack closed cycling transitions is that there may be leakage of population
to dark electronic states that are not coupled with the fields that perform the laser slowing andcooling, resulting in the loss of the particle from the system. Even if this does not occur, spontaneousemission of many photons creates momentum diffusion due to the random emission direction, andthis results in heating and a finite limit on the achievable temperatures. These issues may be
alleviated through tailored coherent dynamics in a timescale fast compared to the natural decayof the excited state, as discussed with rapid adiabatic passage in Section 2.3. This makes methods
that increase slowing forces and minimize the number of scattering events though mostly coherent
dynamics, such as SWAP cooling, the Allen-Eberly scheme [7], stimulated Raman adiabatic passage(STIRAP), the adiabatic passage force, and the bichromatic force [148, 128, 28, 100, 129, 95] enticing
candidates to consider for particle slowing. However, one concern is that in order to satisfy the
intrinsic adiabaticity condition, the time evolution should typically be slow and this could result ina long stopping time and associated large stopping distance.
To overcome these issues, we present a method to develop fast, simple, and robust particle
slowing schemes that employ the Lewis-Riesenfeld invariant shortcut method. The slowing protocol
involves driving sped-up transitions using counterpropagating, pulsed lasers with intensity anddetuning profiles prescribed by the LRI shortcut. By applying this protocol many times, theparticle can in principle be subject to the impulse of many photon momenta without emitting
spontaneous photons. In Section 3.1, we drive a two-level system from its ground state to its
excited state and back by time-ordered, single-photon transitions. We apply the protocol to aslightly more complicated model in Section 3.2, where slowing occurs by way of two-photon Raman
transitions between internal ground states while maintaining a small excited state population. We
then turn our attention to the next state of the cooling process (right-hand side of Fig. 3.1(a)) in
Chapter 4 to create a quantum gas with low entropy.

23

3.1

Application to Particle Slowing in Two-Level Atoms
I now present a novel method for slowing particles via time-ordered laser sweeps sped up by

adiabatic shortcuts that address a narrow electronic transition between two internal states. This
work has the potential to generate large classical forces without a reliance on spontaneously emitted
photons and culminated in a publication in Physical Review A [15]. We show that the applicationof Lewis-Riesenfeld invariants will afford our slowing scheme the advantages of adiabatic slowing
schemes similar to SWAP cooling, such as robustness against the precise strength and detuning of
the laser pulses, but with a much smaller slowing time and associated slowing distance. Furthermore, we demonstrate that in the limit that the excited state linewidth is much smaller than thelaser’s Rabi frequency, the shortcut slowing scheme has many advantages over traditional, radiation pressure based slowing schemes such as Zeeman slowers. I begin by introducing a theoreticalmodel for the slowing scheme, presenting an analytical study of the slowing scheme in classicalphase space, and studying the evolution of a momentum eigenstate and a Gaussian momentum
distribution under repeated absorption and emission cycles. I conclude this section by comparing
the robustness of the adiabatic shortcut to SWAP slowing and a π-pulse technique.

3.1.1

Model

We consider the experimental setup depicted in Fig. 3.2. A thermal beam of particles emerge
from a particle source and enter a slowing region where a narrow transition is addressed by counterpropagating lasers. Each particle is modeled by a two-level atom, as shown in the inset. The
excited state |ei can decay to the ground state |gi at a rate given by the natural linewidth Γ .
The particles exit the source with a high average velocity and velocity large spread, as depicted inFig. 3.1(b). The goal is to translate an appreciable fraction of the initial momentum distribution
from p ≈ p̄ to p ≈ 0, where p is the particle’s momentum and p̄ is the mean particle momentum.
We track motion along one dimension, for which the particle has position and momentum operatorsẑ and p̂. We choose to represent the slowing dynamics of the system in the momentum basis. To

24

Figure 3.2: Particles exit an atom or molecule source and are collimated into the slowing region.
The spatial setup of the sequentially-pulsed counterpropagating lasers, which have time-dependent
frequencies ω1 (t) and ω2 (t), and a sample particle with velocity v in the laboratory frame aredisplayed in the slowing region. The circular inset shows the two-level internal structure of eachparticle.

quickly reduce the kinetic energy of an appreciable fraction of the momentum distribution, thelasers are pulsed in a time-ordered fashion to stimulate emission and absorption of photons thatcause momentum impulses in the direction opposite to the particle’s motion.

3.1.1.1

Slowing Mechanism

The ideal coherent dynamics are presented in Fig. 3.3, where it is assumed that the particle
begins in |gi. The counterpropagating laser (laser 1) is switched on first and its frequency ω1 isswept over resonance. This stimulates absorption of a photon in the direction opposite to its motion,and the atom transitions to |ei. In a timescale where it is assumed that the particle does not decay
back to the ground state, the first laser is switched off and the copropagating laser (laser 2), withfrequency ω2 , is turned on which causes stimulated emission back to the ground state in the directionof the laser. Therefore, by conservation of momentum, the particle experiences an impulse of 2~kwithout the adverse affects of momentum diffusion that is associated with spontaneous emission.
Repeating this cycle many times, we remove a plethora of momentum from a selected portion of
the particles’ velocity distribution through predominately coherent dynamics.

25

Figure 3.3: Left: Frequency diagram of an isolated subset of states in the lab frame. The laser
frequencies ωi (p̄, t) [see Eq. (3.1)] are dynamically updated according to the solution derived fromthe Lewis-Riesenfeld invariant shortcut method to promote quick, coherent transfer from (a) |g, p̄i
to |e, p̄ − ~ki, followed by (b) |e, p̄ − ~ki to |g, p̄ − 2~ki. Right: Experimental parameters andparticle dynamics over a |p̄i → |p̄ − 2~ki sequence of period 2T . (c) Square-pulse Rabi frequency
profiles Ω1 (t) (red) and Ω2 (t) (blue) with amplitude Ω [see Eq. (3.28)]. (d) Lewis-Riesenfeld
detuning profile δ(t) [see Eq. (3.29)] with cutoff frequency ±δcut [see Eq. (3.48)] for each laser. (e)Ideal excited state fraction Pe dynamics. (f) Ideal average momentum hp̂i dynamics. Parameters
are: T = 0.032/ωr , Ω = 100ωr , δcut = 250ωr , Γ = 0, p̄ = 100~k, and β = 0.85 π/2.

26
We choose a square-wave profile for each laser’s Rabi frequency Ωi (t), whose peak value isdetermined by the LRI shortcut protocol. This makes the scheme robust to aligning the centersof periodic Rabi frequency and detuning profiles in time. The two lasers’ Rabi frequencies Ω1 andΩ2 have the same maximum amplitude Ω, but are, initially, completely offset [see Fig. 3.3(c)] to
avoid multi-photon processes that may hinder the desired system dynamics. Our scheme is velocity
selective because of the velocity-dependent Doppler shift ±kv of the two laser fields. Thus, weparameterize the instantaneous frequencies of each laser, ω1 (t) and ω2 (t), to account for the mean
particle velocity v̄ = p̄/m,
ω1 (p̄, t) = ωa − δ(t) − kv̄ + ωr
(3.1)
ω2 (p̄, t) = ωa + δ(t) + kv̄ − 3ωr ,where ωr ≡ ~k 2 /2m is the recoil frequency and m is the particle’s mass. The photon wavenumber
k is approximated to be constant throughout the slowing process, although the frequecy is varying. Each laser is on resonance with the associated transition displayed in Fig. 3.3(a-b) when the
detuning δ(t) is zero, where we have introduced the notation |i, pi = |ii ⊗ |pi since the internal andmomentum Hilbert spaces are simultaneously diagonalizable. Here, |ii , i ∈ {g, e} is in the internal
Hilbert space and p̂ |pi = p |pi is a momentum eigenstate with eigenvalue p. The explicit form of
δ(t) is derived from the LRI shortcut method, and is the aim of this subsection. Similar results to
those presented here may also be obtained by flipping the sign of δ(t) in Eq. (3.1).

3.1.1.2

System Dynamics

We apply the Hamiltonian to a system with momentum p̄ = mv̄. It takes the formĤ(p̄, t) = Ĥself + Ĥint (p̄, t),(3.2)

where the particle’s free evolution Hamiltonian is given byĤself (p̄, t) =

p̂2
~ωa z
+
σ̂ ,
2m
2

(3.3)

where the first term represents the particle’s kinetic energy and σ̂ z ≡ |eihe| − |gihg| is the usual
Pauli spin matrix. Under the dipole and rotating wave approximations (see Appendix A), the27particle-field interaction Hamiltonian in the Schrödinger picture is given byĤint (p̄, t) =i
~ −hσ̂ Ω1 (t)ei[kẑ+η1 (p̄,t)] + Ω2 (t)e−i[kẑ−η2 (p̄,t)] + H.c.,
2

(3.4)

where σ̂ − ≡ |gihe| is the lowering operator, exp[−ikẑ] |pi = |p − ~ki is the momentum-shift operator [130, Chapter 6], and the accumulated phase of each laser field is given byZ t
ηi (p̄, t) ≡ωi (p̄, t0 )dt0 .

(3.5)

t0

The relevant physics of the laser-field interaction can be captured in the first transfer whenlaser 2 is off (Ω2 = 0), where the interaction Hamiltonian becomesĤint (p̄, t) =~Ω1 (t)  − i[kẑ+η1 (p̄,t)]σ̂ e+ H.c. .
2(3.6)

We move into the interaction picture defined byĤ0 (t) =p̂2~+ [ωa − δ(t)] σ̂ z ,2m 2

resulting in the Hamiltonian (see Appendix B.3)

 




p̂
~δ(t) z ~Ω1 (t)−σ̂ +σ̂ exp i kẑ + k− v̄ t + ωr t + H.c. .
Ĥ1 (p̄, t) =
2
2
m

(3.7)

(3.8)

The Hamiltonian associated with laser 2 being on Ĥ2 (p̄, t) is found by the substitutions 1 → 2,
δ(t) → −δ(t), k → −k, and ωr → −3ωr into Eq. (3.8). After application of Ĥ1 (p̄, t) followed byĤ2 (p̄, t), the particle ideally has decreased its momentum by 2~k, and thus the addressed momentum
state p̄ is updated to p̄ − 2~k in the laser detunings [Eq. (3.1)] and the process is repeated. This
repeated cycle takes place until the particle ideally reaches p̄ = 0.
We are now tasked with the duty of finding the appropriate forms of the detuning profile δ(t)using the LRI shortcut protocol. To simplify the analytical complexity of this, we limit our scope
to a small, isolated subset W (p̄) of the full composite Hilbert space during transfer 1,
W (p̄) = {|g, p̄i , |e, p̄ − ~ki} ≡ {|Gi , |Ei}.

(3.9)

In this subspace, the interaction Hamiltonian Ĥ1 (t) is given by
(W )

Ĥ1

(t) =

~δ(t) z~Ω1 (t) x
σ̂W +
σ̂W ,
2
2

(3.10)

28z ≡ |EihE| − |GihG| and σ̂ x ≡where we have introduced the W subsystem Pauli matrices σ̂W
W|GihE| + |EihG|. In parallel to Eqs. (2.13) and (2.16), the eigenvectors are given by




χ(t)
χ(t)
|+(t)i = cos|Ei + sin
|Gi ,
2
2




χ(t)
χ(t)
|−(t)i = sin|Ei − cos
|Gi ,
2
2

(3.11)

with associated eigenvalues
E± (t) = ±~Ω̃1 (t),2and we have introduced the generalized Rabi frequency Ω̃1 ≡(3.12)pδ 2 + Ω12 and mixing angle cos χ ≡δ/Ω̃1 . Similar to [32], we cast the eigenvectors of our invariant operator in a similar form to the(W )eigenvectors of Ĥ1. The subset
W 0 (p̄) = {|e, p̄ − ~ki , |g, p̄ − 2~ki}(3.13)

is then used to calculate the corresponding quantities for transfer 2 which follows in a straightforward manner.

3.1.1.3

Shortcut Application

We are now in a position to design an invariant operator in order to end up with the desired
final populations. The construction of the invariant operator and its associated eigenvalue equation
can be a difficult process, and various methods have been introduced to overcome this [80, 55, 74,43]. However, since we have the form of the Hamiltonian and its eigenvectors, we need only toˆ and |φn (t)i in the same functional forms. Thus, in an identical manner to [32],parameterize I(t)we use the inverse engineering approach to parameterize δ(t) and Ω1 (t) as to begin and end thesweep with the desired populations. Note that this derivation holds for time-dependent Ω(t) unless
noted otherwise.
ˆ in parallel to Eq. (3.11):
We parameterize the eigenvectors of an invariant operator I(t)γ eiβ |Ei + sin
|Gi ,
2
2
γ 
γ 
|φ− (t)i = sin|Ei − cose−iβ |Gi ,2
2|φ+ (t)i = cosγ (3.14)

29
where γ = γ(t) and β = const are auxiliary angles. The unitary phase eiβ was introduced as anadditional degree of freedom to define the shortcut solution, as we shall see in subsection 3.1.2.
It follows that the invariant with eigenvalues λ± = ±~Ω 0 /2 can be written in the basis of the
Hamiltonian as


ˆ =
I(t)

~Ω 0
2

cos γ



sin γ e−iβ
sin γ eiβ ,− cos γ(3.15)

ˆ with units of energy. Substituting
where Ω 0 is an arbitrary constant frequency in order to keep I(t)Eqs. (3.10) and (3.14) into Eq. (2.37), we calculate the Lewis-Riesenfeld phases for transfer 1 as
1α± (t) = ∓2

Z t(δ cos γ + Ω1 sin γ cos β) dt0 .

(3.16)0

Substituting these phases into Eq. (2.40), we find that the Hamiltonian is parameterized by


iβ
Be 
~ A
(W )
Ĥ1 (t) = (3.17)
,
2
−iβ
Be
−A
where
A = δ(t) cos2 γ + Ω1 (t) cos γ sin γ cos β,(3.18)

B = δ(t) cos γ sin γ + Ω1 (t) sin2 γ cos β − iγ̇.

(3.19)

Equating the two forms of the Hamiltonian [Eqs. (3.10) and (3.17)], we arrive at the auxiliary
equationsγ̇ = Ω1 (t) sin β,(3.20)

δ = Ω1 (t) cot γ cos β,(3.21)

from which we determine the experimental parameters δ(t) and Ω1 (t), subject to the boundary
conditions on the auxiliary variables γ and β required for state transfer from state |Gi to |Ei,
which we now determine.
As seen from Eq. (3.14), γ must satisfyγ(t0 ) = πn,γ(tf ) = γ(t0 ) + π(2m + 1),(3.22)

30
to induced state transfer, where n and m are integers. For simplicity and without loss of generality,
we choose n = m = 0 so thatγ(t0 ) = 0,γ(tf ) = π(3.23)and causes the state |φ− (t)i to align with |Gi and |Ei at initial and final times t0 and tf , respectively.
(W )Since the commutator between Ĥ1
h

(W )

Ĥ1

ˆ is
(t) and I(t)i ~2 Ω 0 zˆ(t), I(t)=−iΩ1 sin γ sin β σ̂W2iβ+ (δ sin γe(3.24)+−− Ω1 cos γ)σ̂W+ (Ω1 cos γ − δ sin γe−iβ )σ̂W



,

−
+ †
= (σ̂W) ≡ |Gi hE|, we should also impose
where σ̂WΩ1 (t∗ ) sin γ(t∗ ) sin β = 0,(3.25)

β = qπ(3.26)(W )for t∗ = t0 and t∗ = tf and integer q to align the eigenbases of Ĥ1

ˆ at the beginning(t) and I(t)and end of the shortcut process [see Eq. (2.41)]. The condition (3.25) is automatically satisfiedby (3.23), whereas the condition (3.26) is not necessary to enforce in the case of complete statetransfer because it only affects non-physical, global phases.
We have now parameterized the Hamiltonian in terms of the auxiliary angles and foundboundary conditions for these angles for the interaction with laser 1. The corresponding quantities
for the subsequent interaction with laser 2 are derived in a similar manner. Note that the particular
choice of invariant eigenvector does not affect the resulting form of δ(t) and Ω1 (t). As we shall see
in Section 3.2, the general case of Ω1 (t) 6= const results in the task of choosing, from an infinite
set of Hamiltonians, a particular form of γ(t) and β(t) that satisfy the boundary conditions. We
now use Eqs. (3.20), (3.21), and (3.23) to determine the detuning profile associated with a constantRabi frequency over a single sweep Ω1 (t) = Ω = const.
We first define the initial and final times t0 = 0 and tf = T . From the auxilary equation
Eq (3.20) and the boundary condition γ(0) = 0, we see
γ(t) = (Ω sin β)t.

(3.27)

31
To satisfy the other γ boundary condition γ(T ) = π, we require thatπ = ΩT sin β,(3.28)

which we combine with Eqs. (3.21) and (3.27) to arrive at
π cot βδ(t) =
cot
T



πt
T


.

(3.29)

This is the main result of this subsection, and a plot of δ(t) for a particular choice of β and T isgiven in Fig. 3.3(d). We now have a particular Hamiltonian that will drive the |g, p̄i → |e, p̄ − ~kitransition with dynamics that do not have to be adiabatic. The process is repeated to derive thesame profiles for laser 2 to drive the transition |e, p̄ − ~ki → |g, p̄ − 2~ki. Ideal dynamics of thiscycle is shown in Fig. 3.3(e-f). The method may be optimized further with respect to different
cost functions, but we do not explore such solutions here (e.g., see [131]). We now, similar tosubsection 2.4.1, examine the trajectories of the Bloch vector for different shortcut solutions.

3.1.2

Bloch Sphere Trajectories

As stated with Eq. (3.14), different choices of the auxiliary angle β can result in distinct
dynamics as the particle is transferred from |Gi and |Ei. This can be seen by investigating the
Bloch sphere trajectories. As seen from Eqs. (3.10), (3.29), and Ω1 (t) = Ω = const, the trajectories
under Ĥ1 (t) are purely longitudinal and parameterized by β. For a better comparison, we insteadtransform into a time-dependent interaction picture defined by the free evolution Hamiltonian
Eq. (3.3). This results in the interaction Hamiltonian in the W subspace,
(W )

ĤI

(t) =


~Ω  iθ(t)e|EihG| + H.c. ,2(3.30)

where the accumulated phase of the detuning is defined as
Z tf
θ(t) ≡
t0

with δ(t) given by Eq. (3.29).

δ(t0 )dt0 ,(3.31)

32

Figure 3.4: Bloch sphere trajectories (thick curves) between the initial state |Gi and final state |Eiparameterized by the auxiliary angle β [see Eq. (3.14)] in the free-energy interaction picture [seeEq. (3.30)]. We set the Rabi frequency Ω to be equal for all trajectories, and the slowing periods
1 π
1π
T are given by Eq. (3.28). The cutoff detunings for β = 40
2 and β = 2 2 are δcut = 318Ω andδcut = 225Ω, respectively [see Eq. (3.48)].

33
We plot the Bloch vector trajectories for three choices of β with a fixed Rabi frequency Ω in
Fig. 3.4. The slowing periods T vary from trajectory to trajectory since it is completely determinedby β from Eq. (3.28) when Ω is a constant. A choice of β = (r + 21 )π for integer r minimizes theslowing period. Since this results in δ(t) = 0, this corresponds to a simple, resonant π-pulse (seeSection 2.3). The choice of r = 0 results in the longitudinal Bloch sphere trajectory (yellow, dottedcurve) in Fig. 3.4. However, this choice of β can lead to many experimental challenges, such as
sensitivity to errors in Ω. This point will be investigated further in subsection 3.1.6. The choice of
β = sπ for integer s results in an adiabatic process as T tends towards infinity. We approximate
1 π
s = 0 to have a finite slowing time by using β = 40
2 , which results in the highly processing

curve (blue, solid curve) in Fig. 3.4. This results in a long slowing distance, increased probabilityof spontaneous emission, and a large sensitivity to laser phase. We therefore use a choice fromthe range 0  β < π/2 that results in an intermediate trajectory for our slowing example insubsection 3.1.5. The middle trajectory (red, dash-dotted curve) in Fig. 3.4 results from the choice
β = π/4.

3.1.3

Quantum Master Equation

So far, we have only considered the system’s coherent, reversible dynamics. We now include
the incoherent, irreversible dynamics due to spontaneous emission, incoherent pumping, and dephasing, among others, by considering an open quantum system. Here, the full system-environment
density matrix ρ̂SB (t) that undergoes unitary dynamics is assumed, by the Born-Markov approximation, to factorize ρ̂SB (t) = ρ̂S (t) ⊗ ρ̂B , where ρ̂B is a stationary-state of the reservoir. We thenpreform a partial trace over the degrees of freedom of the reservoir to arrive at the reduced density
matrixρ̂(t) ≡ ρ̂S (t) = TrB [ρ̂SB (t)].

(3.32)

We can now derive an equation of motion for the state of the system alone that undergoes dynamics
that need not be unitary. See Appendix C for a comprehensive study on open quantum systems.
Typically, the system’s incoherent dynamics are incorporated by evolving the reduced density

34
matrix via the quantum master equation in Lindblad form [82] (see Appendix C.1):
i
∂ ρ̂
1 h
Ĥ, ρ̂ + L̂(ρ̂).
=∂ti~(3.33)The Lindblad superoperator L̂(ρ̂) depicts the irrevesible dynamics and takes the general formL̂(ρ̂) =X

D̂[Ĉj ]ρ̂ =

j

X
j

1
Ĉj ρ̂Ĉj† − {Ĉj† Ĉj , ρ̂},2

(3.34)

where the summation is over all of the system’s jump operators Ĉj and {A, B} = AB + BA is theanti-commutator. Meanwhile, the first term on the right hand side of Eq. (3.33) represents theunitary dynamics of the system, and reduces to the Schrödinger equation when L̂ = 0. The righthand side of the master equation can be written compactly as the Liouvillian superoperator,
X
iD̂[Ĉj ]ρ̂.
L̂[ρ̂] = − [Ĥ, ρ̂] +~(3.35)

j

For the slowing schemes considered in this chapter, the only incoherent process is spontaneousemission from the excited state to the ground state manifold at a rate Γ . Since we use a discreet
momentum grid that is parameterized by integer multiples of photon momenta, we approximate thecontinuous dipole radiation pattern to only have three discreet recoil probabilities that correspondto whole integers of ~k. For the two-level system, this can be described by three jump operatorsfor each momentum state,
r
3Γ −
(p)
Ĉ0 =
σ̂ ⊗ |pihp| ,
5

(p)
Ĉ− =

r

Γ −
σ̂ ⊗ |p − ~kihp| ,5

Thus, the system’s Lindblad superoperator L̂(ρ̂) =

P P
p

(p)
Ĉ+ =

r

Γ −
σ̂ ⊗ |p + ~kihp| .
5(3.36)(p)j D̂[Ĉj ]ρ̂ takes the form [102]


i
Γ
2
L̂(ρ̂) = −σ̂ + σ̂ − ρ̂ + ρ̂σ̂ + σ̂ − −3 σ̂ − ρ̂σ̂ + + eikẑ σ̂ − ρ̂σ̂ + e−ikẑ + e−ikẑ σ̂ − ρ̂σ̂ + eikẑ ,2
5(3.37)

which implies possible impulses of −~k, 0, and ~k along the slowing axis with associated probabilities of 15 , 53 , and 51 , respectively. I will study numerical simulations of Eq (3.33) for our slowing
system in Section 3.1.5. Beforehand, however, I preform an analytical study of the system’s slowing
dynamics in classical phase space to determine the usefulness of our slowing protocol and compare
to results from methods that rely on radiation pressure.

35

slowing method

force F

slowing time ∆tslowing distance ∆xscattering rate Rs

number of scattered photons N

Lewis-RiesenfeldΩ~k sin βππζ0Ω sin βωr ζ02λ2Ω sin β

Γ
2

Γ π
ζ0
Ω 2 sin β

Radiation pressureΓ ~kρeeζ0Γ ρeeωr ζ02λ2πρee ΓΓ ρee

ζ0Table 3.1: Comparison of slowing dynamics between the shortcut slowing scheme based on LewisRiesenfeld invariants and radiation pressure slowing. If Ω sin β  πΓ ρee , the shortcut slowing
scheme results in larger forces, shorter slowing times, shorter slowing distances, and fewer scatteredphotons than RP.

3.1.4

Slowing Dynamics in Classical Phase Space

We now determine whether the LRI shortcut method can achieve a practical device that iscompetitive with slowing methods that are regularly employed. To do this, we compare different
quantities associated with the slowing process in LRI slowing to commonly-implemented processes
that rely on radiation pressure (RP) such as Zeeman slowers or Stark decelerators. We calculatequantities such as the classical force associated with each sweep, total slowing distance, and numberof scattered photon below, and present the results in Table 3.1.
We first note the classical force F = ∆p/∆t exerted on the particle during each sweep isgiven byFLRI =~kΩ~k sin β=,Tπ(3.38)

where we have used Eq. (3.28). In contrast, the classical force from RP is given by
FRP = Γ ~kρee ≤

Γ ~k
,
2

(3.39)

where ρee is the excited state fraction and equality is reached with infinite laser power [96].
Assuming the particle begins the slowing process in the eigenstate |g, p̄0 i, it receives animpulse of ~k against its motion for a total time
∆tLRI =

T p̄0
πζ0
=
,
~k
Ω sin β(3.40)

36
until it reaches zero momentum, where ζ = p/~k is the momentum number and ζ0 = p̄0 /~k is theinitial momentum number.
We can now calculate the distance that a particle that undergoes the slowing process travels
in physical space in order to reach zero momentum. We approximate p(t) during the LRI scheme
to be linear,


~kt
t
p(t) ≈ p̄0 −,= ~k ζ0 −TT(3.41)

so that we may write the slowing distance as∆xLRI =pavg
ωr T ζ02∆tLRI =λ,m
2π(3.42)

where pavg ≡ p̄0 /2 is the time-averaged momentum and λ is the wavelength of the electronictransition, which is typically O(10−7 m). We rewrite this using Eq. (3.28) in terms of the Rabi
frequency:∆xLRI =

ωr ζ02λ.
2Ω sin β(3.43)

Using a similar method, we obtain the related expression for the RP slowing distance,∆xRP =

ωr ζ02λ.
2πρee Γ(3.44)

Comparing the LRI and RP results, we find that the shortcut slowing scheme is able to exert higher
forces, and therefore slow particles in a shorter distance, when Ω sin β  πΓ ρee , which reduces toΩ  Γ when sin β is chosen to be on order of unity.
So far, we have not incorporated the effects due to dissipation as a result of spontaneousemission in the analysis of the LRI scheme. Since the mechanism to reduce momentum in thisscheme is purely coherent, the ideal case is when there is no scattered photons during the process.
This can be nearly accomplished by using ultra-narrow linewidth transitions or by applying theentire slowing protocol in a time that is shorter that the lifetime of the excited state. The lattermethod typically requires an extremely large laser power, but also results in an extremely small
slowing distance. Furthermore, we will show in Section 3.2 that application of the shortcut to a
two-photon Raman transition between internal ground states can greatly reduce the scattering rate

37
through an engineered excited state linewidth [see Eq. (3.70)]. Nevertheless, the scattering rate Rsof the LRI slowing technique is approximately
s
RLRI
≈

Γ
2

(3.45)

since the particles are in the excited state for roughly half the slowing time. This is the saturationof the inequalitys
RRP≈ Γ ρee ≤Γ2(3.46)

for the RP scattering rate [96]. However, the expected number of scattered photons, N = Rs ∆t
for each technique is
NLRI ≈Γ π
ζ0 ,Ω 2 sin β

NRP ≈ ζ0 .

(3.47)

This again implies that the LRI slowing scheme is advantageous to RP schemes when Ω  Γ , withsin β on order of unity, as it generates much fewer scattered photons.

3.1.5

Slowing Example

By repeatedly driving the transition |g, p̄i → |e, p̄ − ~ki followed by |e, p̄ − ~ki → |g, p̄ − 2~ki
according to our shortcut solution, we now show that we are able to slow a significant number ofparticles when the full composite Hilbert space Hi ⊗ Hp is taken into account. After each pair oftransitions, we update the momentum p̄ to p̄ − 2~k in the laser frequencies Eq. (3.1), as we assume
that the initial population in |g, p̄i has moved to |g, p̄ − 2~ki. Note that this procedure results inthe slowing of a pulse of particles, as opposed to a steady-state ensemble. We assume that thejump in laser frequency at the end of each ramp is perfectly diabatic. Since there is very little
population transfer when δ(t)  Ω, we do not track the detuning profile’s asymptotic behavior, as
this can cause numerical errors. We thus take δ(t) to be equal to a constant ±δcut in these regions,defined byδ(t < t0 + tcut ) = −δ(t > tf − tcut ) = δcut

(3.48)

for some cutoff time tcut , as shown in Fig. 3.3(d). We evolve the quantum master equation Eq. (3.33)
using the method of quantum Monte Carlo wave functions [102], which is presented in Appendix C.2.

38
Species

Transition

Wavelength (nm)Linewidth (kHz)Recoil frequency (kHz)Saturation intensity (W/cm2 )

40 Ca

1S

0 →

3P

1

657

0.4

11.5

1.8 × 10−7

88 Sr

1S

0 →

3P

1

689

7.5

4.8

3.0 × 10−6

690

5.9

4.0

2.3 × 10−6

556

180

3.7

1.4 × 10−4

1061

1200

1.3

1.3 × 10−4

YO
174 YbBaH

X 2 Σ + → A02 ∆3/2
1S

0

→ 3P

1

X 2 Σ + → A2 Π

Table 3.2: Fundamental properties for various atomic and molecular candidates for shortcut slowing
on a narrow transition.

Fig. 3.5 presents the momentum distribution of an ensemble of particles before (blue line) andafter (orange distribution) application of our slowing scheme over 100 sweeps of each laser in thecase of purely coherent (left column) and dissipative (right column) dynamics. In the dissipative
case, we chose Γ = ωr as an order-of-magnitude estimate for a narrow-linewidth transition thatroughly corresponds to the linewidth for the 1 S0 → 3 P1 transition in 88 Sr (see Table 3.2). We choseto initialize the momentum distribution as either a momentum eigenstate at 100~k (top row) or a
Gaussian profile with an average momentum hp̂i = 100~k and a standard deviation of σp,0 = 10~k(bottom row), which typically corresponds to an initial average particle speed on the order of 1
m/s. While actual physical systems may have much higher initial particle beam speeds, this choicesufficiently demonstrates the slowing effects of our protocol. We have chosen to use the average
momentum of the Gaussian as the initial momentum used in the laser frequencies, p̄0 = hp̂i, because
it results in the largest fraction of slowed particles. For experimental accessibility and robustness,
we chose β = 0.85 π/2 and Ω = 100Γ , which results in a shortcut time T = 0.032/Γ typically on
the order of microseconds, much shorter than the associated timescale of radiation pressure slowing
on the transition. This choice of Ω also sets the expected number of scattered photons N to be on
the order of unity [see Eq. (3.47)].
As seen in Fig. 3.5, a substantial fraction of the distribution is slowed to near zero momentum.
We first discuss the results in the case of purely coherent dynamics (left column). If the system
begins in the eigenstate |g, 100~ki [Fig. 3.5(a)], about 99.3% of the population ends in the zero

39

Figure 3.5: Initial (blue) and final (orange) momentum distributions P (p) of a system subject tothe shortcut slowing protocol. The system is initialized in the internal ground state |gi and with amomentum distribution of either the 100~k eigenstate (top row) or a Gaussian state with average
momentum hp̂i = 100~k and width σp = 10~k (bottom row). The excited state linewidth is Γ = 0
in (a) and (c), and Γ = ωr in (b) and (d). Further details are discussed in text. Other parameters
are: Ω = 100ωr , T = 0.032/ωr , ttot = 1.6/ωr , and δcut = 244ωr , β = 0.85 π/2, and p̄0 = 100~k.
Subplots (b) and (d) are averaged over 1,000 trajectories.

40

Species

Rabi frequency (MHz)Initial speed at 200~k (m/s)

Slowing time (µs)

Slowing distance (µm)

Capture fraction

40 Ca

0.2

3.0

494

746

5.6 × 10−1

88 Sr

1.0

1.3

106

70

1.5 × 10−1

YO

0.9

1.1

119

66

2.1 × 10−1

174 Yb

3.4

0.82

30

12

6.8 × 10−6

BaH

23.6

0.54

4

1

1.6 × 10−5

Table 3.3: Slowing results for various atomic and molecular candidates using a 0.1 W/cm2 laser
intensity and starting from a momentum of 200~k. The Rabi frequency, slowing time, and slowing
distance follow from Eqs. (3.49), (3.40), and (3.43) respectively, with β = 0.85 π/2. The simulated
capture fraction, defined as the fraction of particles with momentum |p| ≤ 3~k after the slowing
process, was calculated over 1,000 trajectories for Ca, Sr, and YO, and 10,000 trajectories for Yb
and BaH.

41
momentum eigenstate. (We attribute the lack of 100% transfer to our use of a cutoff frequency
δcut = 244ωr .) If the system begins in a Gaussian distribution [Fig. 3.5(c)], about 35% of the
population ends with momentum |p| ≤ 10~k, which corresponds to about half of the population
within one standard deviation of the average momentum in the initial distribution.
In the case of dissipative dynamics, 29% of the population ends with momentum |p| ≤ ~k if
the system is initialized in the eigenstate |g, 100~ki [Fig. 3.5(b)], and 13% of the population ends
with momentum |p| ≤ 10~k if the system is initialized in the Gaussian state [Fig. 3.5(d)]. In bothcases, there was an average of 1.6 scattered photons per particle, which agrees with the predictedscattering rate given in Table 3.1. This is a two order-of-magnitude improvement over the numberof scattered photons that occur during radiation pressure. These results demonstrate that our
protocol can potentially slow a significant fraction of particles to zero momentum in a very shortdistance and with a virtually negligible reliance on spontaneous emission.
As a more realistic demonstration, we present several atomic and molecular species that arereasonable candidates [110, 108, 39, 152, 137] for our slowing protocol in Table 3.2. We simulate theslowing of each species with an initial momentum of 200~k using a 0.1 W/cm2 laser intensity, andprovide the expected slowing times ∆t, slowing distances ∆x, and capture fractions C in Table 3.3.
The saturation intensity Isat , laser intensity I, and Rabi frequency Ω can be calculated according
to [96]

2

Ω
Γ

2
=

I
Isat

,

Isat ≡πhcΓ
,
3λ3(3.49)

where h = 2π~ is Planck’s constant and c is the speed of light. The calculated slowing times anddistances become smaller as the transition dipole matrix element increases, as can be seen for the
species with larger linewidths. However, the required temporal control of the Rabi frequency anddetuning profiles in these cases may be an experimental challenge. One solution is to simply reduce
the laser power, but this can quickly move the parameters away from the regime Ω  Γ , which
increases the chance of spontaneous emission and hence reduces the capture fraction C.
Due to our method’s sensitivity to spontaneous emission, a rough estimate for the capture

42
fraction C is given by the fraction of particles that do not emit a single spontaneous photon
throughout the slowing process. Using Eq. (3.47), this is approximatelyΓ πC ≈ exp (−NLRI ) = exp −ζ0 .
Ω 2 sin β(3.50)

This formula is on the same order as the simulated capture fractions given in Table 3.3 for Ca,
Sr, and YO, but it underestimates the results for Yb and BaH, suggesting that particles canstill be slowed after emitting spontaneous photons. It should be noted that Eq. (3.50) is not a
fundamental limit, as it can be improved with a more sophisticated implementation of the shortcut
process, such as introducing occasional waiting periods so that the particles return to the internal
ground state, allowing the laser pulses to overlap in time (as studied in Fig. 3.7), or repeating theslowing protocol over a range of momentum states. For example, we were able to increase thesimulated 174 Yb capture fraction by over an order of magnitude to C = 1.7 × 10−4 by introducing
a laser pulse overlap fraction f = 0.2 as discussed in the next subsection. Furthermore, it should
be noted that atomic and molecular sources can often produce particle fluxes on the order 1012 1014 particles/(cm · s) [97], so an O(10−4 ) capture fraction may still be valuable.

3.1.6

Robustness

We now study the robustness of the shortcut slowing scheme to various systematic errorsthat may arise in an experimental setting. Specifically, we modify the Rabi frequency amplitude,
then separately consider the result of slowing a particle with a momentum p that is not equal tothe momentum accounted for in the laser frequencies p̄ (Fig. 3.6). We also consider the scheme’srobustness to the temporal overlap f of the square pulses (Fig. 3.7), which may minimize scattering
events since it potentially reduces the amount of time the particle remains in the internal excitedstate. For simplicity, we focus on a single |gi → |ei → |gi process and calculate the resulting
impulse ∆p experienced by the particle. Moreover, we employ a phase relation between the laser
pulses such that the Rabi frequency of each pulse is purely real. The correct detuning profiles
are employed throughout this section. We set the Rabi frequency to Ω = 10 ωr across all slowing

43
methods, but necessarily allow for different slowing times.
Using the auxiliary variable β = 0.5 π/2, we compare the robustness of our scheme to both
π-pulse and SWAP slowing. As discussed in subsection 3.1.2, the π-pulse solution is a special caseof our slowing method with the choice β = π/2, which results in a fixed detuning [see Eq. (3.29)].
SWAP slowing is simulated by using a sawtooth-wave detuning profile with full range ∆s and period
T for each single-photon transfer. Unlike SWAP cooling, the lasers are sequentially pulsed as inFig. 3.3(a) and 3.3(b), and the laser detunings are centered on the momentum p̄.
As discussed in Section 2.3, the population of the (initially unoccupied) excited state Pe , if
the laser is linearly chirped from a detuning of minus infinity to positive infinity, is
π Ω2
Pe = 1 − exp −
2 α



,

(3.51)

where α is the frequency chirping rate in rad/s2 . Therefore, to obtain a transfer probability of atleast 95%, we chose to set Ω 2 /α = 2. Next, we increased the chance of population transfer by
setting the shortcut period to be several times larger than the approximate time τj required totransfer population between quantum states in the adiabatic regime [147]:
τj =

2Ω
0.4
=
α
ωr

⇒

T =

1
> τj .
ωr(3.52)

These choices constrained the sweep range to be ∆s = 50 ωr . Note that the resulting SWAP
slowing period T = 1/ωr is significantly longer than the π-pulse (T ≈ 0.3/ωr ) and shortcut slowing(T ≈ 0.4/ωr ) periods [see Eq. (3.28)].
Fig. 3.6(a) presents the resulting particle impulse, which reaches a maximum of ∆p = −2~k,as a function of the error in the Rabi frequency amplitude, which is characterized by the small
parameter :Ω → Ω(1 + ).

(3.53)

While the shortcut scheme applies a similar impulse ∆p compared to the π-pulse method for  < 0,
it is the most robust method for  > 0. The SWAP slowing result, while being the most robust
method for  < 0, can change significantly with small changes to the parameters. This instability is

44

Figure 3.6: Robustness comparison of π-pulse slowing (purple, plus), SWAP slowing (green, cross),and the shortcut scheme (blue, circle with cross) over a p̄ → p̄ − 2~k transfer process. Ideally,
the resulting impulse ∆p satisfies ∆p/2~k = −1. The evolution is purely coherent (Γ = 0), themomentum of the particle (which is addressed by the laser frequencies) is p̄ = 2~k, and the Rabi
frequency is Ω = 10ωr for all processes. (a) Impulse ∆p experienced by the particle, in units of
the ideal impulse magnitude 2~k, as a function of the error in the Rabi frequency amplitude  [seeEq. (3.53)]. The shortcut scheme is the most robust protocol when  > 0. (b) ∆p/2~k as a functionof the relative momentum of the particle δp with respect to the momentum p̄ = 2~k used in thelaser frequencies [see Eqs. (3.1) and (3.54)]. For this set of parameters, the shortcut and π-pulse
scheme are generally more robust than SWAP slowing. π-pulse parameters are: T = 0.314/ωr andβ = π/2. SWAP slowing parameters are: T = 1/ωr and ∆s = 50ωr . Shortcut parameters are:T = 0.44/ωr , δcut = 230ωr , and β = 0.5 π/2.

45a result of high-frequency population oscillations [147], as can be seen in Fig. 2.3. Moreover, when
dissipation is included, the much longer SWAP slowing period increases the chance of spontaneousemission, which can disrupt the slowing process. These results demonstrate the utility of theshortcut scheme, as it is robust to small errors  and takes much less time than SWAP slowing.
Note that intensity modulators typically have errors less than approximately 5%.
Fig. 3.6(b) presents the effects of applying each slowing method to a particle with a momentum p which is not equal to the momentum p̄ accounted for in the laser frequencies [see Eq. (3.1)].
Such an error occurs when slowing a cloud of particles with a distribution of momenta. We parameterize this difference in momentum with the variable δp as:
p = p̄ + δp.

(3.54)In this case, we find that the shortcut scheme is most robust for small δp, but SWAP slowing
becomes more robust for |δp| > 3~k. This change in the trend for SWAP slowing near the particularvalues δp = ±3~k is due to population oscillations, and the results can again change significantlywith small changes in the parameters. In the limit of adiabatic dynamics and time-resolved transfers(which necessarily takes a long time), SWAP slowing is generally most robust to this error since
the rate of change of the detuning profile in SWAP slowing is constant, thereby removing the needto align the center of the laser detuning profile with the Doppler shift of the particle. It should be
noted that it is not necessarily a major problem that the shortcut scheme is less robust to large
deviations of δp from zero because we scan through values of p/~k in integer steps by design. Whatis more important is that the shortcut scheme is more robust when |δp| < ~k than SWAP slowing,as particles will not necessarily have integer momentum.
Fig. 3.7 presents the effects of allowing the laser pulses to overlap in time. We define theoverlap fraction of the square pulses f such that f = 0 when the pulses are completely time-resolved
but occur sequentially with no delay, and f = 1 when the pulses occur at the same time for theentire pulse duration. We find, in the case of purely coherent dynamics (Γ = 0), that a pulse
overlap fraction as large as f = 0.2 does not change the impulse experienced by the particle. In46

Figure 3.7: Impulse ∆p experienced by the particle, in units of the ideal impulse magnitude 2~k,
as a function of the laser pulse overlap fraction f . The impulse is calculated under both coherent(Γ = 0) and dissipative (Γ = ωr ) dynamics. Other parameters are: T = 0.44/ωr , δcut = 230ωr , Ω =10ωr , p̄ = 2~k, and β = 0.5 π/2. All points are averaged over 1,000 trajectories.

47
the case of dissipative dynamics, we find that there is an optimal f that maximizes the impulse∆p. This optimal f , which depends on system specifics, must be small enough to allow for full
population transfer to |ei before the transfer back to |gi, but large enough to minimize the time
the particle spends in the excited state, thereby minimizing the chance of spontaneous emission.
It should be noted that a pulse overlap introduces the possibility of multiphoton, or Doppleron,resonances [101] which may interfere with the single-photon slowing dynamics.
When slowing a particle with a large initial momentum p  ~k to rest, we emphasize thatthe slowing efficiency is significantly affected by even a small deviation from the ideal impulse∆p = −2~k since the error compounds exponentially with the number of transfer processes, andthe particle is generally not transferred back to |gi for the next pulse sequence. It may be possible
to further enhance the robustness of the slowing protocol to the rapidly changing detuning profileat the beginning and end of the transfers by using a Rabi frequency profile that satisfies Ω(t0 ) =Ω(tf ) → 0, such as a Gaussian or a sinusoidal function [124, 100, 129]. However, in order to satisfy
the auxiliary equations and boundary conditions, the peak Rabi frequency may need to be larger
than what we considered here.
We have now demonstrated how adiabatic shortcuts may be used to exert immense forcesin order to remove a substantial fraction of a particle’s kinetic energy. However, the scattering
rate Eq. (3.45) for this time-ordered protocol is high as the particle spends half of the slowing
protocol in its excited state. This problem can somewhat be overcame using the overlap strategy
just discussed. We now turn to another slowing scheme that utilizes the LRI shortcut methods
with the focus of limiting the excited state population throughout the slowing process by way of
two-photon transitions.

3.2

Application to Particle Slowing in Three-Level Atoms
In this section, we consider a similar experimental setup as Section 3.1. However, we now

model each particle as a three-level, Λ-system with an excited state |ei and two internal groundstates, |g1 i and |g2 i with different magnetic quantum numbers mF , as depicted in Fig. 3.8. We

48

Figure 3.8: Particles exit an atom or molecule source and enter a slowing region. The spatial
setup of the counterpropagating lasers, which have time-dependent frequencies ω1− (t) and ω1+ (t)and polarizations σ − and σ + for cycle 1, and a sample particle with velocity v in the laboratory
frame are displayed in the slowing region. The circular inset shows the three-level internal structureof each particle.

49

Figure 3.9: The two-cycle process that makes up the slowing scheme with a large detuning fromthe excited state ∆. The internal states |ei, |g1 i, and |g2 i have magnetic quantum numbers 0, 1,
and −1, respectively. (a) Cycle 1: The counterpropagating (copropagating) laser has polarization
σ − (σ + ) to drive the |g1 i → |ei (|ei → |g2 i) transition during the two-photon process |g1 i → |g2 i
for an impulse of ∆p = −2~k. (b) The laser polarizations are flipped in order to now drive the|g1 i → |g2 i transition with another impulse of ∆p = −2~k.

speed up a two-photon Raman transition between the internal ground states that results in an
impulse of ∆p = −2~k. Here, the magnetic quantum number of |ei, |g1 i, and |g2 i are 0, 1, and−1, respectively, which could represent, for example, hyperfine split states. Moreover, the lasers
are no longer time-ordered to drive the transition, nor are they linearly polarized in order to avoid
multi-photon physics at Doppleron resonances that may interfere with the slowing dynamics. Here,multi-photon physics refers to transitions between motional states within the individual groundstate manifolds (i.e., |gi , p1 i → |gi , p2 i) which is now forbidden due to conservation of angularmomentum [42], as the circularly polarized photons have angular momentum of ±~ [20]. Multiphoton transitions between the two ground state manifolds can occur when the counterpropagating
lasers have opposite polarizations. Assuming the particle begins in |g1 i, the counterpropagating(copropagating) has circular polarization σ − (σ + ) for the |g1 i → |g2 i transition during transfer 1,
which is switched to σ + (σ − ) during transfer 2 from |g2 i back to |g1 i, as shown in Fig. 3.9. This will
ensure that each two-photon transition causes a 2~k momentum impulse opposite the direction of
the particle’s motion during both cycles, and the two cycles are repeated until the particle reaches
zero momentum.

50
The main advantage of the use of multi-photon transitions to reduce the particle’s momentumis that the particle remains, for the most part, in the ground state manifol which virtually eliminates
the incoherent scattering of photons during the slowing process. This assumes that the laser
fields are far-detuned from atomic resonance, so that single-photon transitions become negligible.
Another obvious advantage of the two-photon transition is that it results in a change of momentumof 2~k after each sweep, rather than just ~k in the single-photon case considered in Section 3.1
and radiation pressure methods. I begin this section by presenting the Λ-configuration’s model,
preform an adiabatic elimination of the excited state, and applying the LRI adiabatic shortcut tothe acquired effective Hamiltonian. I follow this with an examination of the slowing dynamics inclassical phase space. I conclude by studying an application of the LRI Raman transition slowing
scheme to a Gaussian momentum distribution.

3.2.1

Model

We again track the dynamics in momentum space, where we assume that the motion is onedimensional with position and momentum operators ẑ and p̂. The thermal beam of particles exit
the source with an initial average velocity v̄0 and enter a slowing region where they undergo a seriesof Raman transitions driven by counterpropagating lasers. The ideal slowing process is shown in
Fig. 3.9 and is described by the following: A particle begins in |g1 i and in the central momentum
state |ζ0 i, where p̂ |ζ0 i = ζ0 ~k |ζ0 i = p̄0 |ζ0 i is a momentum eigenstate. Far-detuned, counterpropagating lasers with opposite circular polarizations drive the |g1 , ζ0 i → |g2 , ζ0 − 2i transition
with little excited state population throughout the process. Here, the counterpropagating (copropagating) laser has polatization σ − (σ + ) to ensure that it causes stimulated absorption from thecounterpropagating beam and stimulated emission into the copropagating beam. Once the population is entirely in |g2 i, the lasers’ polarizations are exchanged and drive the |g2 , ζ0 − 2i → |g1 , ζ0 − 4itransition. The two cycles are repeated down the dispersion curve until it reaches zero momentum.

51
3.2.1.1

System Dynamics

The two internal ground states are assumed to be energetically degenerate, Eg1 = Eg2 , andthus both separated from the excited state by a frequency ωa . We assume that the decay channels
from the excited state to the two ground states in the Λ-transition have equal branching ratios,
which is modeled by having the same linewidth Γ . Furthermore, we assume that the lasers areintensity locked,Ω1 (t) = Ω2 (t) ≡ Ω(t),(3.55)

for the ease of both analytical calculations and of experimental implementation. Note that thisdifferentiates our mechanism from stimulated Raman adiabatic passage (STIRAP) and its adiabatic
shortcuts, such as the application of Berry’s TQD in [30].
During cycle 1, the laser frequencies, addressing the Doppler shifts of the momentum stateζ, are given byδ(t)+ ωr (1 − 2ζ),2δ(t)
ω1+ (ζ, t) = ωa − ∆ −+ ωr (2ζ − 3),2
ω1− (ζ, t) = ωa − ∆ +(3.56)

where ω1− and ω1+ are the counterpropagating and copropagating laser frequencies, respectively,∆ = const is both lasers’ large detuning from the respective single-photon resonance, and δ(t)  ∆is a small detuning from ∆ that will be determined via the LRI shortcut method. This is followedby cycle 2, where the laser polarizations are switched and momentum number is updated from ζ
to ζ − 2:δ(t)+ ωr (−3 − 2ζ),2δ(t)
ω2− (ζ − 2, t) = ωa − ∆ −+ ωr (2ζ − 7),2
ω2+ (ζ − 2, t) = ωa − ∆ +(3.57)

such that ω2+ and ω2− are the counterpropagating and copropagating laser frequencies, respectively.
Defining the energy to be zero at the ground states, Eg1 = Eg2 = 0, the Hamiltonian of thecomposite Hilbert space is given byĤ1 (ζ, t) =−+~Ω(t) p̂2
+ ~ωa |eihe| +|g1 ihe| ei[kẑ+η1 (ζ,t)] + |g2 ihe| e−i[kẑ−η1 (ζ,t)] + H.c. ,2m2(3.58)

52
where the accumulated phase of each laser is
ηi± (ζ, t) ≡

Z t
t0


ωi± ζ, t0 dt0 .

(3.59)

We now transform into the interaction picture defined by −~[δ(t) + 4ωr ]p̂2
ω1 (ζ, t) + ω1+ (ζ, t)
Ĥ0 (ζ, t) =+~+ ωr |eihe| +(|g2 ihg2 | − |g1 ihg1 |) ,2m
2
2

(3.60)

which results in an interaction Hamiltonian of~[δ(t) + 4ωr ]˜Ĥ1 (ζ, t) =~∆ |eihe| +(|g1 ihg1 | − |g2 ihg2 |)
2
p̂t
−
θ
~Ω(t) |g1 ihe| ei(k[ẑ+ m ]+η1 (ζ,t)−ωa t+t∆− 2 −2ωr t)+
2(3.61)

p̂t+
θ
+ |g2 ihe| e−i(k[ẑ+ m ]−η1 (ζ,t)+ωa t−t∆− 2 −2ωr t) + H.c. .
Here, we have defined ωa − ∆ = ω1− (ζ, t) + ω1+ (ζ, t) /2 + ωr andθ1 (ζ, t) ≡ η1− (ζ, t) − η1+ (ζ, t) = θ(t) − 4ωr (ζ − 1) t,
where θ(t) is the accumulated phase of of the small δ(t) detunings. We note thatp̂
ωr + 2ζ̂ωr + ωr (1 − 2ζ) − 2ωr = 2ωr (ζ̂ − ζ) = k− v̄ ,mp̂
ωr − 2ζ̂ωr + ωr (2ζ − 3) + 2ωr = −2ωr (ζ̂ − ζ) = −k− v̄ ,
m(3.62)(3.63)

with ζ̂ = p̂/~k and v̄ = ζ~k/m. This allows us to rewrite Eq. (3.61) as~[δ(t) + 4ωr ]˜Ĥ1 (ζ, t) =~∆ |eihe| +(|g1 ihg1 | − |g2 ihg2 |)2~Ω(t) +|g1 ihe| eikẑ eiϕ̂(ζ,t) + |g2 ihe| e−ikẑ e−iϕ̂(ζ,t) + H.c. ,2(3.64)

where we defined ϕ̂(ζ, t) ≡ k (p̂/m − v̄) t. This is the Hamiltonian we use for cycle 1 in our numericalsimulations presented Section 3.2.3. By moving into a similar interaction picture as Eq. (3.60), butwith an opposite sign on the final term, we arrive at the interaction Hamiltonian for cycle 2,~[δ(t) + 4ωr ]˜Ĥ2 (ζ, t) =~∆ |eihe| −(|g2 ihg2 | − |g1 ihg1 |)2~Ω(t) +|g1 ihe| e−ikẑ e−iϕ̂(ζ,t) + |g2 ihe| eikẑ eiϕ̂(ζ,t) + H.c. .
2(3.65)

53
3.2.1.2

Adiabatic Elimination of Excited State

Focusing on cycle 1, we now, once again, isolate our analysis to a subset W1 (ζ) of the fullcomposite Hilbert space:
W1 (ζ) = {|g1 , ζi , |g2 , ζ − 2i , |e, ζ − 1i} ≡ {|1i , |2i , |3i}.

(3.66)In this subsystem, the cycle 1 Hamiltonian Eq. (3.64) is given by
(W )

Ĥ1

~ [δ(t) + 4ωr ](ζ, t) =~∆ |3ih3| +(|1ih1| − |2ih2|)
2
h
i
h
i

~Ω(t) |1ih3| exp iϕζζ−1 (t) + |2ih3| exp −iϕζζ−1 (t) + H.c. ,+2(3.67)

where ϕζl (t) ≡ k (l~k/m − v̄) t for momentum number l. The time-dependent Schrödinger equation
with this Hamiltonian and atomic state vector |ψ(t)i = c1 (t) |1i + c2 (t) |2i + c3 (t) |3i results in thefollowing equations of motion:ihi [δ + 4ωr ]iΩ
c1 −c3 exp iϕζζ−1 ,22ihi [δ + 4ωr ]
iΩ
ċ2 =
c2 −
c3 exp −iϕζζ−1 ,
2
2
i
h
i
h
iΩ 
ċ3 = −i∆c3 −c1 exp −iϕζζ−1 + c2 exp iϕζζ−1 .
2ċ1 = −

(3.68)In the limit ∆  Ω(t), δ(t), the form of our Hamiltonian places the fast time-dependence on the
excited state |3i, which allows us to make the approximation that the excited state amplitude
damps to equilibrium instantaneously ċ3 ≈ 0 [130, Chapter 6]. Thus, we find from Eq. (3.68) thatthe excited state population can be determined fromc3 ≈ −

h
i
h
i
Ω 
c1 exp −iϕζζ−1 + c2 exp iϕζζ−1 ,2∆(3.69)

which predicts the scattering rate
Rs = Γtot ρee = 2Γ |c3 |2 Γ,
2

(3.70)

to be very small since |c3 |2  1 in the limit Ω  ∆. Here, we have defined the Λ-configuration’sP
total decay rate Γtot given by the sum of each individual channel’s rate Γtot = i Γi = 2Γ . Notethat adiabatic elimination of |3i is used only to simplify the mathematical complexity of applying the

54
adiabatic shortcut, and that the excited state manifold will be included in the numerical simulations
in subsection 3.2.3.
We now plug Eq. (3.69) into the first two lines of Eq. (3.68) to find an effective Hamiltonian˜ĤR,1 describing the effective two-level evolution of the Raman system formed by |1i and |2i withthe result~ [δ(t) + 4ωr ]~ΩR (t)˜ĤR,1 (ζ, t) =(|1ih1| − |2ih2|) +(|1ih1| + |2ih2|)
2
2
h
i

~ΩR (t) +|1ih2| exp 2iϕζζ−1 (t) + H.c. .
2(3.71)Here,ΩR (t) ≡ −Ω 2 (t)2∆(3.72)

is the two-photon Rabi frequency, or Raman Rabi frequency, and ~ΩR (t)/2 is the AC Stark shift.
Lastly, we shift the definition of zero energy by the AC Stark shift and move into another rotating
frame by use of the interaction picture~ΩR (t)˜Ĥ0 (ζ, t) =(|1ih1| + |2ih2|) + 2~ωr (|1ih1| − |2ih2|) ,2(3.73)

resulting in the Raman interaction Hamiltonian for cycle 1:
ĤR,1 (ζ, t) =

i

h 
~δ(t)
~ΩR (t) (|1ih1| − |2ih2|) +|1ih2| exp 2i ϕζζ−1 (t) + 2ωr t + H.c. .
22(3.74)

Noting that ϕζζ−1 (t) = k(ζ~k/m− v̄)t−2ωr t, we can write the final form of the Raman Hamiltonian,ĤR,1 (ζ, t) =~ΩR (t)~δ(t)(|1ih1| − |2ih2|) +(|1ih2| + H.c.) ,22(3.75)

which can be written in matrix notation as
ĤR,1 (ζ, t) =~  −δ(t) ΩR (t) .
2ΩR (t) δ(t)(3.76)

A similar process can be done from cycle 2. We have found that the effective two-level dynamics
of the subsystem again takes a similar form to Eq. (2.16), and thus the application of the LRI
shortcut protocol will be similar to that given in subsection 3.1.1.3. Here, however, we pursue
shortcut solutions with a Gaussian Rabi frequency profile.

55
3.2.1.3

Shortcut Application and Analytical Solution

We again parameterize the invariant eigenvectors in parallel to the eigenvectos of Eq. (3.75),
γ eiβ |1i + sin
|2i ,
2
2
γ 
γ 
|φ− (t)i = sin|1i − cose−iβ |2i ,22|φ+ (t)i = cosγ (3.77)

with auxiliary angles γ = γ(t) and β = β(t). This leads to the same auxiliary equations as in
Eqs. (3.20) and (3.21) when we assume β = const:γ̇ = ΩR (t) sin β,(3.78)

δ(t) = ΩR (t) cot γ cos β,(3.79)

We impose the boundary condition defined in Eq. (3.23), as well as those in Eqs. (3.25) and (3.26),
so that we drive the system from |1i at t0 into |2i at tf via the upper invariant eigenstate |φ+ (t)i.
We are now faced with the task of choosing, from an infinite set of Hamiltonians, a particularform of γ and β that satisfy these boundary conditions. This particular solution is often found by
trying a quadratic ansatz for γ and β [32]. However, the resulting laser frequency and intensity
profiles may be difficult to implement experimentally, which led to the use of a constant Rabi
frequency profile solution in subsection 3.1.1.3. This method also has drawbacks such as a therequirement to maintain a large laser intensity throughout the entire slowing process. This problem
is compounded when driving Raman transition as, in the limit ∆  Ω(t), the maximum singlephoton Rabi frequency will be large compared to the Raman Rabi frequency that is found fromthe shortcut method Ω(t)  ΩR (t) [see Eq. (3.72)]. I now present an analytical solution that can
produce experimentally implementable solutions, such as Gaussian Rabi frequency profiles, that
may be advantageous when considering these issues.
Let us define a time-dependent function f (t) such thatΩR (t) = Ω0 f (t),(3.80)

with a frequency constant Ω0 . Plugging this into Eq. (3.78) and integrating, we find (noting

56
β = const),
Z tγ(t) = Ω0 sin β

f (t0 )dt0 ,(3.81)

t0

to which we apply the boundary condition Eq. (3.23) to arrive at
Z tfΩ0 sin β

f (t0 )dt0 = π.

(3.82)

t0

The second auxiliary equation Eq. (3.79) then reads
Z t



0

f (t )dt

δ(t) = Ω0 f (t) cos β cot Ω0 sin β0
.

(3.83)

t0

Suppose, for example, we choose ΩR (t) to have a Gaussian waveform. We also assume
the single-photon Rabi frequencies to be real Ω(t) = Ω(t)∗ . Therefore, we impose Ω0 ≤ 0 [seeEq. (3.72)] and22

f (t) = e−t /τ ,(3.84)

where τ > 0 is a time constant and t0 = −tf . This choice gives
γ(t) =

√

  
 
Ω0 τ
t
t0
π sin β
erf
− erf
,
2
τ
τ

(3.85)

which, in conjunction with Eq. (3.82), allows us to derive the minimum Raman Rabi frequencyΩ0 = ΩR (0) =√ −1Tπ τ sin β erf< 0,2τ

(3.86)

for total shortcut time T = 2tf . Here, we have used the error function erf. The condition ΩR (t) ≤ 0
imposes β ≤ 0 and Eq. (3.86) shows that the choice of a particular β directly determines therequired maximum of the absolute value of the Raman Rabi frequency. This should be minimized
for experimental simplicity. A choice that minimizes the laser intensity is given by β = (r + 12 )π,but again results in a constant detuning (i.e., a Raman π-pulse). Thus, we examine solution in theregime −π/2  β < 0. We also note that Eq. (3.83) becomes−t2 /τ 2

δ(t) = Ω0 cos βe

cot


√

  
 
Ω0 τ
t
t0
π sin β
erf
− erf
.
2
τ
τ

(3.87)

We again impose a cutoff time tcut to avoid tracking the asymptotic behavior of the detuning [seeEq. (3.48)].

57
3.2.2

Slowing Dynamics in Classical Phase Space

We now examine the slowing dynamics of the Raman slowing scheme in classical phase spacein order to interpret its usefulness. This analysis will be similar to subsection 3.1.4, but now with
an impulse of ∆p = 2~k per sweep as well as factors resulting from the time-dependence of the
Rabi frequency. Eq. (3.82) can be written, for the Gaussian solution, as

 r
πG
G
√
Ω0 sin β erf=.
(3.88)2T
2 2√Here, we have defined G ≡ 2T /τ and D ≡ Ωmax /∆, where Ωmax is the maximum single-photon
Rabi frequency,Ωmax = Ω(0) =p−2∆Ω0 .

(3.89)The variable G measures the total number of standard deviations of the Rabi frequency profile thatis used during each sweep, which must be large to approximately satisfy Eqs. (3.25) and (3.26).
For the adiabatic elimination of the excited state to remain valid, the condition D  1 must be
satisfied.
We use Eq. (3.88) to calculate the classical force exerted of the particle during each sweep,
r


2~k
2 2~kΩ0G√
FLRI ==
sin β erf.
(3.90)T
π G
2 2
We assume that D = const which implies, from Eq. (3.89), that the force scales with Ωmax . We
now use the force to derive the associated slowing distance of the Raman slowing protocol.
The particle begins the process in the state |g1 , ζ0 i and we again approximate the momentum
to be linear2tp(t) ≈ ~k ζ0 −T,

(3.91)

so that we may write∆xLRI =pavg
T ωr ζ02
∆t =
λ,
m
4π

(3.92)

where we have used pavg = ζ0 ~k/2 = ζ0 mωr /k and ∆t = ζ0 T /2. Using Eq. (3.88), we write this interms of the Raman Rabi frequency as
r


−1
1 Gωr ζ02 λG
ζ 2 ωr λ√∆xLRI =
sin β erf
∝ 0.
2π 4Ω0|Ω0 |2 2(3.93)

58
We can compare this quantity to the corresponding expression for radiation pressure [see Eq. (3.44)]
by computing the ratio∆xLRIρee Γtot T
ρee Γ
=
∝
,
∆xRP
2
|Ω0 |

(3.94)

where we have used the Λ-configuration’s total decay rate Γtot = 2Γ in our ∆xRP calculation for asensible comparison between schemes. We once again find that our scheme significantly reduces theslowing distance when |Ω0 |  Γ and D  1 or, more specifically, when ∆  Ωmax  Γ . Since theexcited state population is minimal in the Raman slowing scheme, its number of scattered photon
is on order of unity or less, which is a massive improvement to the approximately ζ0 scatteredphotons for RP.

3.2.3

Slowing Example

By repeatedly driving the two-photon transition |gi , ζi → |gj , ζ − 2i according to our Raman
shortcut solution, we now show that we are able to generate an impulse of 2N ~k against theparticle’s motion in N sweeps with an engineered excited state linewidth. Compared to slowing
processes that rely on radiation pressure such as the Zeeman slower, our procedure significantly
reduces the number of scattered photons which can be crucial for slowing atoms and moleculesthat lack closed cycling transitions. To exemplify our slowing scheme, we simulate the results ofapplying the analytic solution developed in subsection 3.2.1.3 to a specific momentum family W1 (ζ)
when we track a large subset of the composite Hilbert space. After each cycle, we flip the laser
polarizations and update the momentum number we are tracking ζ → ζ − 2 as we assume the initial
population in the first ground state has fully transferred to the other ground state.
The incoherent dynamics of our system are governed by the master equation Eq. (3.33) with,in a similar fashion as Eq. (3.36), six decay channels for each momentum state:
r
r
r
3Γ
Γ
Γ
(ζ)
(ζ)
(ζ)
Ĉ1,0 =
|g1 ihe| ⊗ |ζihζ| , Ĉ1,− =|g1 ihe| ⊗ |ζ − 1ihζ| , Ĉ1,+ =|g1 ihe| ⊗ |ζ + 1ihζ| ,
5
5
5
r
r
r
3Γ
Γ
Γ
(ζ)
(ζ)
(ζ)
Ĉ2,0 =
|g2 ihe| ⊗ |ζihζ| , Ĉ2,− =|g2 ihe| ⊗ |ζ − 1ihζ| , Ĉ2,+ =|g2 ihe| ⊗ |ζ + 1ihζ| .
55
5(3.95)

59
We again simulate the incoherent dynamics by way of Monte Carlo wave functions (see Appendix C.2) over many trajectories to ease the numerical complexity.
An example of our Raman LRI slowing scheme over 200 sweeps is shown in Fig. 3.10 anddiscussed below. The momentum distribution of the ensemble of particles is initialized as a Gaussian
profile (purple line) with a standard deviation σp,0 = 50~k centered at p̄0 = 400~k. We choose thecenter of the Gaussian distribution to be our central momentum state to account for the Doppler
shifts, ζ0 = 400/~k. We first apply Ĥ1 (ζ, t) to the full, composite Hilbert space, followed by theapplication of Ĥ2 (ζ − 2, t), and then Ĥ1 (ζ − 4, t) and so on until the slowed momentum packet
reaches zero average momentum. Fig. 3.10 presents snapshots of the momentum distribution after
50 (green distribution), 100 (blue distribution), 150 (yellow distribution), and 200 (red distribution)
cycles of our slowing scheme. The applied laser detuning and Raman Rabi frequency profiles are
determined through the adiabatic shortcut by Eqs. (3.84), (3.86), and (3.87), which we plot for ourchosen parameters in Fig 3.11. We note that our choice of β = −π/4 satisfies the inequality givenin the text before Eq. (3.87).
We find that about 4% of the initial momentum distribution has been slowed to aroundzero momentum at the conclusion of the slowing process, as shown by the red spike near 0 in
Fig. 3.10. There is only an average of 0.18 scattered photons per particle throughout the slowing
scheme, which is in stark contrast to the O(ζ0 ) scattered photons in radiation pressure schemes,
as well as an order of magnitude improvement over the single-photon slowing scheme presented insubsection 3.1.5. This comes at the cost of a larger single-photon Rabi frequency which may bedifficult to create experimentally, although some possible solutions to this problem are discussed
in Section 3.3. Furthermore, we note that our choice of shortcut time, T = 0.1/ωr , is typicallymuch shorter than the associated timescale of radiation pressure, and thus slowing occurs in a much
shorter distance (see subsection 3.2.2).
We can calculate the 1D temperature Tf of the final, slowed momentum packet,
2
σp,f1
hp̂2 ikB Tf =
≈
,
2
2m
2m

(3.96)

60

Figure 3.10: Sanpshots of the momentum distribution P (p) of the system at different times throughout the slowing process. The system is initialized (purple line) in a Gaussian state with average
momentum hp̂i = 400~k and width σp = 50~k. Snapshots of the momentum distribution at 50(green distribution), 100 (blue distribution), 150 (gold distribution), and 200 sweeps (red distribution) are shown. The other parameters are: Γ = ωr , ∆ = 5000ωr , β = − π4 , δcut ≈ 89ωr ,τ = 0.025/ωr , and T = 0.1/ωr , which leads to a maximum Rabi frequency of Ωmax ≈ 1004ωr .
Results are averaged over 500 trajectories.

61

Figure 3.11: The detuning profile (dashed green curve) and the absolute value of the Raman Rabi
frequency profile (blue curve) over a single sweep. Also displayed is the cutoff scheme for the
detuning profile at cutoff time tcut (purple curve). The parameters are: ∆ = 5000ωr , β = − π4 ,δcut ≈ 89ωr , τ = 0.025/ωr , and T = 0.1/ωr .

62
where kB is Boltzmann’s constant. Here, σp,f is the variance in p of the final momentum packet
which we calculate using a Gaussian fit of the final momentum distribution in the range −15 ≤ ζ ≤15. We find that, for the example given in Fig. 3.10, the final distribution has a temperature ofTf ≈ 5Tr , where kB Tr = ~ωr is the recoil temperature. We note that the slowed packet of particles
will spread out in position space during the slowing process, so phase space compression, and thus
cooling, does not occur. However, this final momentum packet is now in a suitable state to be
trapped and cooled. We turn our attention to laser cooling in the next chapter.

3.3

OutlookBy periodically modifying the frequency and intensity profiles of counterpropagating lasers

based on the LRI shortcut protocol, we have created a novel slowing scheme that has the potential
to generate a large classical force against the motion of a thermal beam of particles. The coherent
nature of our scheme allows for a very low scattered photon number and can greatly reduce theslowing distance compared to radiation pressure techniques for narrow-line transitions or systems
that lack a closed cycling transition. We also demonstrated that our slowing blueprint has advantages over adiabatic and purely resonant schemes. We presented an operating regime for our
slowing schemes, which we showed to be when the maximum Rabi frequency is much larger thanthe linewidth. The use of a ring-cavity may be used to achieve such high laser intensities [61],which is a particularly important consideration in implementing the Raman slowing scheme.
A possible practical implementation of our scheme is to directly apply it to a particle beam
exiting a supersonic nozzle or buffer gas cell. Our scheme may also be utilized as a second slowing
stage after particles exiting an effusive oven have been slowed in a precursor stage to the order of
10 m/s and the initial spread in velocity has been greatly reduced. Additionally, it may be possible
to implement our protocol as a steady-state slowing procedure by compensating for the changing
particle velocity with a magnetic field gradient instead of the time-dependent laser frequencies, ina similar approach to a Stark or Zeeman decelerator. This has the potential to greatly increase theslowing blueprint’s capture range and is an enticing candidate for future work.

63
A natural next step would be to optimize our shortcut solutions with respect to different
criteria. Time-optimization of different adiabatic shortcuts has been studied previously for two-levelsystems [118, 119], STIRAP [103], and frictionless cooling in harmonic traps [131]. Additionally,
quantum optical control based protocols could be implemented to derive further solutions [58].
Another optimization criterion comes from the fact that adiabatic shortcuts cannot be implemented
without an energetic cost, an intrinsic relationship that has been rigorously studied for various
shortcuts [26, 1, 41], as noted in subsection 2.4.1. Thus, minimizing the energetic cost for a fixed
sweep period is, in effect, finding the most efficient shortcut.
Furthermore, it is impossible to implement the LRI scheme to the entire momentum Hilbert
space by hand. An intriguing possible solution to this problem is to employ advanced optimization
techniques, such as quantum optimal control and reinforcement learning, to find a solution thatmaximizes the slowing scheme’s capture range or minimizes the energetic cost of implementing theshortcut for a given sweep period.
There are similarities between our results and others [124, 83, 100, 129] in the sense that
two-level inversion is achieved in the diabatic limit. A more thorough investigation could further
our understanding of the connection between these solutions and adiabatic shortcuts.
Having successfully slowed a substantial fraction of the momentum distribution to near zero
momentum with a low velocity spread, the final packet of particles is now in a position to beloaded into an electromagnetic trap. Here, the particles are then trapped and cooled to both a lowtemperature and low entropy state. Such is the study of the next chapter.

Chapter 4

Entropy Removal by a Laser Field

As mentioned in Section 3.2.3, the slowing schemes developed in the previous chapter does
not lead to entropy removal. To see this, we can quantify the entropy S of a system by identifying
the volume it occupies in a phase space consisting of pairs of conjugate variables. The slowing
schemes considered in Chapter 3 only produce a translation in momentum space, and therefore do
not lead to phase space compression. However, the final slowed state is in an ideal position to be
trapped and cooled in which the entropy of the particle cloud can be reduced.
We now expand on the idea of phase space compression. While the total phase space volume
must be conserved, the volume can be redistributed so that a subspace can experience phase spacecompression [50]. In the language of open quantum systems (see Appendix C), the system subspacedeposits its entropy into the degrees of freedom of an external reservoir so that it may reduce its
entropy without violating the second law of thermodynamics [50, 115].
Our particular interest is entropy transfer in the process of laser cooling and trapping where
the atomic gas redistributes the photons of a coherent laser field while spontaneous emission into
the background radiation field is possible. The compression of the atomic phase space, as thecloud’s spread in both momentum and position is reduced, decreases both the entropy and thetemperature of the atomic subspace. It is often assumed that the only subspace that is suitable toabsorb the entropy lost by the atomic subsystem is the radiation field via the spontaneous emission
process. As Wolfgang Ketterle stated in his Nobel lecture [72]:Real cooling needs an open system which allows entropy to be removed from the65system – in laser cooling, in the form of scattered photons.
This commonly accepted explanation lies in the fact that the reservoir can absorb a substantial
amount of entropy due to the large number of possible configurations of the outgoing light, suchas various frequencies, polarizations, and direction [96, Chapter 5, §1]. Furthermore, it is often
assumed that the state of the laser field is not significantly perturbed, and therefore can notremove entropy from the gas. However, there are some studies that predict entropy removal fromthe gas via interaction with the laser field [76, 94, 40, 29, 99].
To investigate this point, we propose a simple gedanken experiment that probes the change
in the state of a laser field that interacts with a system with non-zero initial entropy. We begin
this chapter with a condensed background on quantum statistical mechanics and Fock space, preliminaries necessary for understanding the usefulness of our approach. We then further motivate
the possibility of entropy transfer to the laser fields in Section 4.2, and we propose a gedankanexperiment to test our rationale. In Section 4.3, we present the model of our gedankan experiment,
followed by an analysis of the time evolution of the cavity field. Our main results are presented in
Sections 4.5 and 4.6 where we provide both an experimental and a theoretical approach to quantify
the entropy exchange between the laser and atomic cloud.

4.1

Preliminary Information

4.1.1

Quantum Statistical Mechanics Background

In Section 2.4.1, we introduced the density matrix ρ̂ to map a pseudo-spin 12 system to thethree dimensional Bloch vector representation. We saw further usefulness of the density matrix in
Section 3.1.3 as the dynamics of the system subspace is governed by the quantum master equation
Eq. (3.33) under the Born-Markov approximation. Let us now generalize our conversation ofdensity matrices as they are critical for representing mixed ensembles. Here, all of the membersof the ensemble can not be characterized by a single ket which undergoes Schrödinger dynamics.
Throughout the last chapter, this statistical mixture arose because the randomness of the system’s

66
incoherent dynamics. This can be thought of in a quantum trajectories picture as the jump operator
associated with spontaneous emission being applied to particles at different times, resulting indifferent final states.
We can write the density matrix for an arbitrary mixed ensemble asρ̂(t) =

Xwi ψ (i)EDψ (i) ,

(4.1)

iwhere ψ (i) is a ket that characterizes a fraction of the ensemble’s population with weight wi . The
P
normalization condition is now i wi = 1. For a pure state, we have wj = 1, wi6=j = 0 and thus,ρ̂ = |ψj ihψj |. To measure the overlap between two systems, we define the fidelity
q

p p 2


F ρ̂, σ̂ = Trρ̂σ̂ ρ̂ ,(4.2)

which, using the cyclic property of the trace, reduces toF ρ̂, |ψj ihψj | = hψj | ρ̂ |ψj i ,(4.3)

when σ̂ is a pure state.
We now define the von Neumann entropy (see [121, Chapter 3, §4])
S(ρ̂) = − Tr[ρ̂ ln ρ̂],

(4.4)

which is written in the diagonal basis of ρ̂ asS(ρ̂) = −X

ρii ln ρii ≥ 0,

(4.5)

i

where ρii = hi| ρ̂ |ii, we have set Boltzmann’s constant kB = 1, and we have used 0 ≤ ρii ≤ 1. Thisis, as traditionally stated, a measure of disorder of which the second law of thermodynamics states
will never decrease in the universe. For a pure ensemble, we either have wi6=j = 0 → ρii = 0 or
wj = 1 → ln ρjj = 0 and thus, the entropy of a pure ensemble is zero:
S(|ψj ihψj |) = 0.

(4.6)

We can further define the mutual information between two subsystems A and B as
I(A : B) = S(ρ̂A ) + S(ρ̂B ) − S(ρ̂AB ),

(4.7)

67
where one assumes the two systems are uncorrelated and measures the error of that assumption.
This therefore expresses how much information is common to both subsystems, which is containedin the total correlations between A and B [107]. This is an important measure when differentiating
between entropy that is created during the interaction from entropy that is transferred between
subsystems. We use mutual information in Section 4.3 to look at correlations between atoms anda quantized field.

4.1.2

Interactions Between Atoms and Quantized Fields

In Chapters 2 and 3, we treated the laser field as a purely classical electric field [see Eq. (A.1)].
However, the situation often arises when an electromagnetic field must also be treated quantum
mechanically – that is, as a collection of identical particles of light called photons. Such a situation
occurs when considering a quantum gases inside of a cavity: two mirrors that form an opticalresonator that allows for stable, contained, standing-wave modes of oscillation.

4.1.2.1

Fock SpaceWe spare the details of the quantization of the electromagnetic field (see, for example, [130,
Chapter 8, §3] or [121, Chapter 7, §6]) and present the important points here. The energy eigenstates
take the form of the number basis |ni, often referred to as Fock space, so that n̂ |ni = n |ni forn = 0, 1, 2, . . ., and the number operator n̂. This gives the number of quanta in the field (photons).
For a single cavity mode, the system’s Hamiltonian takes the form [98, Chapter 13, §1]1ĤF = ~ωc n̂ +≈ ~ωc n̂,2

(4.8)

where ωc is the cavity mode frequency and we have, in the last step, neglected the zero-point energy~ωc /2. Since the energy eigenvalues are quantized by integer multiples of ~ωc , it is convenient tointroduce ladder operators that raise and lower the energy eigenstates and thus, adds or removes
a quanta of energy to or from the field. These take the form of the non-Hermitian creation â† and68
annihilation â operators that raise and lower the eigenstate:â† |ni =√n + 1 |n + 1i ,â |ni =√n |n − 1i .

(4.9)

The number operator can be written in terms of ladder operators as n̂ = â† â, so that the field
Hamiltonian becomesĤF = ~ωc â† â.

(4.10)The eigenstates of the annihilation operatorâ |αi = α |αi ,(4.11)are called the coherent states:1|αi = e− 2 |α|

2

∞
X
αn
√ |ni .
n!
n=0

(4.12)

This is the quantum state that shares the most similarities with a classical coherent laser field [4]and has the property hn̂i = |α|2 .

4.1.2.2

Jaynes-Cummings Hamiltonian

Now having a quantum description of the electromagnetic field, we move to how atomic
systems interact with this quantized field. We consider a two-level atom with ground state |gi andexcited state |ei, separated by atomic frequency ωa , and follow the presentation of [130, Chapter
10]. The free evolution Hamiltonian is thenĤself = ĤA + ĤF = ~ωa |eihe| + ~ωc â† â,(4.13)

where A and F represent the atomic and field Hilbert spaces, respectively. Similar to Appendix A,
we write the atom-field interaction in dipole form of Eq. (A.4):
Ĥint = −dˆ · Ê,(4.14)

where the dipole operator dˆ takes the form [see Eq. (A.12)]dˆ = hg| dˆ|ei (|gihe| + |eihg|) = hg| dˆ|ei σ̂ + σ̂ † .

(4.15)

69
The interaction between the atoms and the cavity is given by (see [130, Chapter 8, §3] for theelectric field operator Ê derivation)rĤint =


h
i
~ωc 
σ̂ + σ̂ † hg| dˆ|ei · f (r)â + f ∗ (r)â† ,20(4.16)

where 0 is the permittivity of free space and f (r) is a mode function that satisfies the Helmholtzequation ∇2 + k 2 f (r) = 0.
We now define the cavity coupling constant1r2ωchg| dˆ|ei · f (r),g(r) = −~0(4.17)

so that, at a given point, the interaction Hamiltonian becomesĤint =~g 
σ̂ + σ̂ † â + â† .
2(4.18)We use the rotating-wave approximation (see Appendix A.3) to write this asĤint =

 ~g ~g σ̂ ⊗ â† + σ̂ † ⊗ â =σ̂â† + σ̂ † â ,2
2

(4.19)

where we have dropped the tensor products for compactness. As should be expected, the first termin the parenthesis describes the creation of a cavity photon in conjunction with the lowering of theatomic state |ei → |gi while the second term represent the absorption of a photon â that excites
the atom |gi → |ei. The total Hamiltonian, called the Jaynes-Cummings model [71], is given by
ĤJC = ~ωa |eihe| + ~ωc â† â +4.2

~g  †
σ̂â + σ̂ † â .
2

(4.20)

Motivation
With this rudimentary knowledge presented, we can now move on to the idea of entropy

removal in laser cooling. As discussed in this chapter’s introduction, it is often assumed that thelaser fields do not remove entropy from an atomic gas during laser cooling. A typical argumentthat attempts to denounce the possibility of entropy removal by the laser field is the fact that a
1Some literature [98, 130] include a factor of 1/2 in this definition, i.e., this is the definition of 2g.

70
Hamiltonian process can not change the entropy of a quantum system. This can be seen using thedefinition of von Neumann entropy Eq. (4.4), the cyclic property of the trace, and the unitary time
evolution, under the Hamiltonian Ĥ, of the density matrix (see Eq. B.5):
ρ̂(t) = e−iĤ(t−t0 )/~ ρ̂(t0 )eiĤ(t−t0 )/~ ,(4.21)

so that S(t) = const. Thus, it is asserted that an interaction that is described in dipole form,
ĤAF = dˆA · Ê F ,(4.22)

can not change the entropy of the system, where A and F represent the atomic and field Hilbert
spaces, respectively.
While the entropy SAF of the total AF system is indeed a constant, this argument does notconsider the entanglement entropy between the two subspaces that can be seen using ρ̂A = TrF [ρ̂AF ]or ρ̂F = TrA [ρ̂AF ] in Eq. (4.4) to calculate SA (t) or SF (t). Since, in laser cooling, one is only
concerned about reducing the entropy of the atomic subsystem, we see that the entropy can bechanged by coherent interaction with a laser. This analysis also explains the mechanism for entropy
exchange via spontaneous emission if we replace the laser field’s Hilbert space with that of the
background radiation bath B. These concepts will be expanded upon in Sections 4.4 and 4.6.
We now present a gedanken experiment that highlights the essential physics of entropy removal in laser cooling, but with a much simpler model. The system that we consider is depicted
in Fig. 4.1 and discussed below. A quantum gas consisting of three level atoms is placed inside
a lossless optical cavity. We model the initial state of the cavity field as a coherent state |αi [seeEq. (4.12)], as it is the quantum state that has the most in common with a classical laser field.
Furthermore, a coherent state is pure (ρ̂F (0) = |αihα|) and the laser field therefore begins the interaction with zero entropy SF (0) = 0. We represent the quantum gas as a single, motionless particle
that exists at an anti-node of the optical cavity. While the lack of particle motion distances ourinvestigation from a direct study of laser cooling and trapping, we can still investigate the entropy
exchange process by instead studying the evolution of the entropy contained in the particle’s internal degrees of freedom. The particle’s internal structure is a Λ-configuration with two ground71

Figure 4.1: Schematic diagram of laser cooling gedanken experiment. (a) Energy diagram of theparticle’s internal state structure which is a Λ-configuration. It is coupled to the optical cavity
with strength g on the bright transition (|ei ↔ |bi) and has a linewidth Γ on the dark transition(|ei → |di). (b) A schematic of the experimental setup. The quantum gas (orange circle) is placedin a lossless (κ = 0) optical cavity (dark blue mirrors) that contains a laser field (light blue). The
particle can undergo spontaneous emission into free space with rate Γ .

72states |bi and |di and an excited state |ei, as shown in Fig. 4.1(a). The |bi ↔ |ei (bright) transition
is resonant with the optical cavity, which models the coherent interaction between the laser andthe atoms. Meanwhile, the dark state |di does not interact with the laser field and is instead only
mediated by the spontaneous emission process, which models the incoherent interaction between
the background radiation field and the gas. We assume all scattered photons leave the cavity, as
shown in Fig. 4.1(b). Our goal now is to understand the evolution of the entropy contained in
the particle and laser field subspaces, A and F , and to determine whether or not the F subspaceabsorbs entropy from the A subspace.
Of course, the quantum state of the laser field must change if it is to absorb entropy fromthe gas. While a coherent state is unperturbed by the removal of a photon [see Eq. (4.11)], its
evolution upon gaining a photon is non-trivial (action of â† ) [3]. Therefore, one should expect a
coherent state to change upon interaction with near-resonant atoms and it is interesting to quantify
the change in the laser field state after interaction with the gas. We will investigate this further in
Section 4.4 using the model developed in the next section.

4.3

Gedankan Experiment
We now present the model of our gedanken experiment. The full system consists of threesubspaces: a three-level particle A and a single mode optical cavity F that is loseless (κ = 0)
except for spontaneous emission into free space into a background radiation field B. The total
Hamiltonian is given by
ĤAF B = Ĥ0 + Ĥint ,(4.23)

where the free energy isĤ0 = ĤA + ĤF + ĤB= ~ωeb |ei he| − ~ωbd |di hd| + ~ωc1â â +
2
†





1
†
+~
ωn ĉn ĉn +.
2nX(4.24)

Here, ~ωij = Ei − Ej is the energy difference between states |ii and |ji of the particle, and ĉn
are the annihilation operators for the background radiation field, which is modeled as an infinite

73
bandwidth bosonic bath. The Hamiltonian describing the interaction between the particle and lightfields Ĥint can be written in Jaynes-Cummings form as
Ĥint =ĤAF + ĤBF
 ~
~g 
=
|bi he| â† + H.c. +22(4.25)!
|di he|X

λn ĉ†n + H.c.

,nwhere g and λn are the coupling strengths of the particle to the cavity and bath modes [seeEq. (4.17)].
We, under the Born-Markov approximation (see Appendix C.1), trace out the backgroundradiation field and study the dynamics of the reduced density matrix in the atom-field subspaceρ̂AF . Furthermore, we assume that the branching ratio of the |ei → |di transition is much larger
than |ei → |gi transition so that Γd  Γb , where Γi is the respective |ei → |ii transition’s linewidth.
Moving into an interaction picture defined by Ĥ0 and assuming the cavity is on resonance with thebright transition ωc = ωeb , we arrive at the system’s quantum master equation (see subsection 3.1.3)ih√i∂ ρ̂AF1 h=
Ĥ, ρ̂AF + D̂
Γ |dihe| ρ̂AF ,∂ti~(4.26)

in which the coherent particle-laser interaction is determined by the Hamiltonian
Ĥ =


~g 
|bi he| â† + H.c. ,2(4.27)

and we have set the spontaneous emission rate Γ ≡ Γd . In the sections that follow, we consider
both analytical and numerical solutions to Eq. (4.26) with the atom initialized in a mixed state in
the ground state manifold:ρ̂A (0) = x |bi hb| + (1 − x) |di hd| .

(4.28)We use the QuantumOptics package in the Julia programming language for all numerical calculations [77]. Since |di is a dark state that does not undergo stimulated absorption throughout theprocess, any population that is in |di will remain in that state. Therefore, as t → ∞, one would
expect the atomic ensemble ρ̂A to end the process in a near pure state ρ̂A (∞) ≈ |dihd|, limited onlyby the probability of having 0 photons in the cavity, P0 = exp[−hn̂i]. However, the dynamics ofthe field ρ̂F (t) is not obvious.

74

4.4

Displacement of Laser Field
We first demonstrate a change of the laser state upon interaction with the particle, which

we define as the fidelity F between the initial laser state ρ̂F (0) and the laser state after the AF
system has reached equilibrium: ρ̂F (∞). Here, the laser field state ρ̂F (t) is defined as the reduced
density matrix on the F subspace ρ̂F (t). Since the laser state is initialized as a coherent state |αi,the fidelity F is simply [see Eq. (4.3)]
F ρ̂F (∞), ρ̂F (0) = hα| ρ̂F (∞) |αi .

4.4.1(4.29)

Block Matrix Formalism

In order to derive an analytic form of the final state of the cavity, we now present a method
for separating the density matrix into Jordan blocks to allow us to solve for the steady-state using
the master equation. Our initial stateρ̂(0) = [x |bihb| + (1 − x) |dihd|] ⊗ |αihα|
!
!
0
∞
∞
X
X
αn
(α∗ )n− 21 |α|20− 12 |α|2√√ hb, n|
=x e
b, n
e
0!
n!
n
0
n=0
n =0
!
!
0
∞
∞
X
X
αn
(α∗ )n− 12 |α|20− 12 |α|2√√ hd, n| ,+ (1 − x) e
d, n
e
n!
n0 !
n=0
n0 =0

(4.30)

explicitly assumes there are no coherences between the three internal states. This motivates us
solving the dynamics of the block matrices:ρ̂b = hb| ρ̂ |bi |bihb| + hb|ρ̂ |ei |bihe| + he| ρ̂ |bi |eihb| + he| ρ̂ |ei |eihe| ,(4.31)
ρ̂d = hd| ρ̂ |di |dihd| .
Taking a time derivative and using the master equation Eq. (4.26) in the form [see Eq. (C.28)]

∂ ρ̂
1 
†
=
Ĥeff ρ̂ − ρ̂Ĥeff+ L̂jump (ρ̂) ,∂ti~(4.32)

we findρ̂˙ d = Γ he| ρ̂ |ei |dihd| ,i
1 h†ρ̂˙ b =Ĥeff ρ̂b − ρ̂b Ĥeff,i~(4.33)

75
where we have used hi| L̂jump |ji = 0 for i, j ∈ {b, e} and defined [see Eqs. (C.26) and (C.27)]
Ĥeff = Ĥ −i~Γ|eihe| ,2

L̂jump (ρ̂) = Γ |dihe| ρ̂ |eihd| .

(4.34)

We first look at the subset {|e, n − 1i , |b, ni} and diagonalize the non-Hermitian matrix√~  −iΓ g n (4.35)
Ĥn = 
,
√
2
g n
0
(n)

(n)

for n > 0. The eigenvalues are given by λ± = −i~Λ± , with
r
i
Γ2
Γ
(n)
ng 2 −,
Λ± ≡ ±
42
4(4.36)(n)and the kernel of the matrix Ĥn − λ± Î is spanned by


√
(n)
g n
− 2 
~  −iΛ+
V̂n =
,

√
cn
(n)
g n
−iΛ
+
2

(4.37)

P(n) 2with normalization factor: c2n = −i~Λ++ ~2 g 2 n/4. We now define V̂ = n V̂n ⊗ Iˆn with nth
projection operator Iˆn so that the bright manifold matrix can be written as ρ̂b = V̂ ρ̂V V̂ † .
R∞
We can use Laplace’s equation, L[Â](s) = 0 dte−st Â, to solve for the transformed density
matrix (see [130, Chapter 5, §5]):h isL[ρ̂] = L ρ̂˙ (s) + ρ̂(0),(4.38)

which we then can solve for steady-state:˙ρ̂(t → ∞) = lim L[ρ̂](s)+ ρ̂(0).
s→0(4.39)

Projecting with hd| and |di and using the master equation Eq. (4.26), hd| ρ̂b |di = 0, and he| ρ̂d |ei =0, we findhd| ρ̂d (∞) |di = Γ lim he| L[ρ̂b ](s) |ei + hd| ρ̂d (0) |di .
s→0(4.40)

To solve for the first term on the right-hand side, we findhi
X
L[ρ̂V ](s) = L V̂ −1 ρ̂b V̂ −† (s) =R̂n0 ,n ⊗ Iˆn0 ,n ,n,n0 ≥1(4.41)

76
where Iˆn0 ,n is a matrix with 1 at position n0 , n and zeros everywhere else, and


√
√
(n)∗
g 2 nn0~2

R̂n0 ,n =∗cn0 c 
n

(n0 )
(n)∗
4(s+Λ+ +Λ+ )√ (n0 )
ig nΛ+(n0 )(n)∗2(s+Λ− +Λ+ )−ig n0 Λ+
(n0 )

(n)∗

2(s+Λ+ +Λ−
(n0 )

)

(n)

Λ+ Λ+
(n0 )

(n)∗

(s+Λ− +Λ−

 −|α|2 αn0 (α∗ )n
e
√ √ .

n0 ! n!

(4.42))

We next calculateXL[ρ̂b ](s) =V̂n0 R̂n0 n V̂n† ⊗ Iˆn0 n ,(4.43)

n,n0 ≥1and project with he| and |ei to find the first term on the right hand side of Eq. (4.40). Rememberingthat the excited state in our basis in Eq. (4.35) is |e, n − 1i, Eq. (4.40) becomes√X
nn0
hd| ρ̂d (∞) |di =xn0 − 1 n − 1 n0 α hα|ni
2
g
n+n0
0 )2
+
(n
−
n
n,n0 ≥1 2
4Γ 2+ (1 − x)Xn0(4.44)

0n n α hα|ni ,n,n0 ≥0where we have used hb, n0 | ρ̂b (0) |b, ni = x hn0 |αi hα|ni and hd, n0 | ρ̂d (0) |d, ni = (1 − x) hn0 |αi hα|ni.
Projecting hb| and |bi on Eq. (4.39) and noting that the steady-state bright state population resultsfrom the cavity being in the vacuum state |0i, we havehb| ρ̂b (∞) |bi = x |0ih0| h0|αi hα|0i .

(4.45)

Now that we have traced over the atomic states, we find the steady-state cavity density matrix


√
0
n
∗
n
0
X
2
2 α (α )nnρ̂F (∞) =x e−|α| |0ih0| +n0 − 1 n − 1 e−|α| √ √ 
2
g2
n+n0
0
n0 ! n!
n,n0 ≥1 2 + 4Γ 2 (n − n )(4.46)n0∗ nX
2 α (α )+ (1 − x)n0 n e−|α| √ √ .
n0 ! n!
n,n0 ≥0
4.4.2

Fidelity Calculation

We now introduce the critical photon number
 
1 Γ 2
,
m0 ≡
2 g

(4.47)

which can be understood as the minimum Fock state that significantly participates in the dynamics [75]. We can therefore rewrite the final cavity field Eq. (4.46) as


0
n
∗
n
X
2
2 α (α )ρ̂F (∞) = (1 − x)ρ̂F (0) + x e−|α| |0ih0| + |α|2Kn,n0 n0 n e−|α| √ √  ,n0 ! n!
n,n0 ≥0(4.48)

77
where ρ̂F (0) = |αihα| and
Kn,n0 ≡

n + n0 (n − n0 )2
1+
+
2
8m0

−1
.

(4.49)

Defining the initial intracavity photon number n̄0 ≡ hâ† â(0)i = |α|2 , we find the fidelity between
the initial and final states of the cavity Eq. (4.29):
(n+n0 )
X


n̄
 .
F ρ̂F (∞), ρ̂F (0) = 1 − x 1 − e−2n̄0 1 + n̄0
Kn,n0 0 0n!
n!
0(4.50)n,n ≥0The classical limit results from a field that contains many photons n̄0 → ∞. We now consider two
limiting cases that exemplify the displacement of the laser field.
We first consider the limit of zero linewidth m0 → 0 in which Kn,n0 → δn,n0 /(1 + n):
X 1  n̄n 20 .
(4.51)
Fm0 →0 = 1 − x 1 − e−2n̄0 1 + n̄0
1 + n n!
n≥0

We note that the the modified Bessel functions of the first kind are [10, Chapter 11, §5]
Iv (x) =∞X
s=0

 x (2s+v)
1,s!(s + v)! 2(4.52)

such thatI1 (2n̄0 ) =

∞
X
n=0

∞

X n̄0
1
(2n+1)n̄0=n!(n + 1)!
n+1
n=0



n̄n0
n!

2
.

(4.53)We therefore find
Fm0 →0 = 1 − x 1 − e−2n̄0 [1 + I1 (2n̄0 )] ,(4.54)

which we take in the classical limit
1
lim Fm0 →0 = 1 − x 1 − √≈ 1 − x,n̄0 →∞
2 πn̄0(4.55)e2n̄0
lim I1 (2n̄0 ) = √.
n̄0 →∞
2 πn̄0(4.56)

where we have used

Interestingly, we find that Fm0 →0 ≈ 1 − x in the limit of zero linewidth. The fact that Fm0 →0 6= 1 inthis case provides direct evidence that the laser state can be substantially changed upon interaction
with a quantum gas.

78
We now look at the opposite limit, that of infinite linewidth m0 → ∞. We have Kn,n0 →0−1 and thus,(1 + n+n2 )
Fm0 →∞ = 1 − x 1 − e

−2n̄0



1 − e2n̄0 (1 − 2n̄0 )
1+
2n̄0


.

(4.57)In the classical limit, we findlim Fm0 →∞ = 1 −n̄0 →∞

x
≈ 1.
2n̄0

(4.58)

This demonstrates the importance of allowing the atom and field to interact for a substantial period
before decaying to the dark state. The classical limit fidelities are thus:






1
m0 = 0 1 − x 1 − 2√πn̄0 ,lim F ρ̂F (∞), ρ̂F (0) =n̄0 →∞
1 − x ,m0 → ∞(4.59)

2n̄0

We interpret the deviation from F = 1 as result of entanglement generation between the atoms
and field through the coherent interaction Ĥ [see Eq. (4.27)].

4.4.3

Numerical Results

Having demonstrated in the two extreme regimes that the cavity state can change, we now
turn to numerical simulations to check our analytic conclusions. We exemplify the change in thefield in the intermediate regime by simulating the master equation Eq. (4.26) with g = Γ = 1(and thus m0 = 1/2). In Fig. 4.2, we plot the fidelity (blue curve) between the initial and final
states of the field as a function of the initial intracavity photon number n̄0 . We see that the fidelity
only equals unity in the limit n̄0 → 0, which is a result of the atomic gas and field not interacting
when no photons are in the cavity. In contrast, the fidelity falls below unity for n̄0 > 0 which
demonstrates that the laser state has changed as a result of its interaction with the atom. The
fidelity asymptotes to some value less than unity in the classical limit n̄0 → ∞ (dashed red line),demonstrating that the laser field is changed even when treated as a classical electric field. We
verify the value of limn̄0 →∞ F using the analytic calculations from subsection 4.4.2.

79

F [ρ̂F (∞), ρ̂F (0)]

1
0.95
0.9
0.85
0.8

0

5

10
n̄0

15

20

Figure 4.2: Fidelity F Eq. (4.50) between the initial and final laser states ρ̂F (0) and ρ̂F (∞) (solid
blue curve) as a function of the initial average intracavity photon number n̄0 . The dashed red
curve displays F in the infinite cavity photon limit n̄0 → ∞. The fidelity F is equal to unity only
when n̄0 = 0, demonstrating that the laser state is changed upon interaction with the particle. The
parameters are g = Γ = 1.

80

4.5

Bayesian Analysis
Now that we have shown that the laser field is displaced upon its interaction with an atomic

mixture, we demonstrate that information about the atom can be imprinted on the laser field. To
this end, we first determine the dependence of the equilibrium laser state on the initial bright state
fraction x. We will then examine the information transfer from a quantum information perceptivein the next section.
We first reduce the number of cavity photons by shifting the laser state toward the vacuum
state by application of the displacement operator D̂ of the initial coherent state [4, Chapter 1, §4]hiD̂(α) ≡ exp αâ† − α∗ â ,|αi = D̂(α) |0i ,(4.60)

so thatσ̂F (x) = D̂† (α)ρ̂F (∞)D̂(α) = D̂(−α)ρ̂F (∞)D̂(α).

(4.61)The dependence on the initial bright fraction x is obvious from Eq. (4.48). While Eq. (4.61) may
be analytically tractable for m0 = 0 and n̄0 → ∞, we do not pursue such an analytical expression
here.
The probability distribution for x, P (x|n), can then be generated through Bayesian inferenceby sampling the resulting photon number distribution P (n|x) = hn| σ̂F |ni [68]. The procedure
adopted is as follows. Bayes theorem states
P (x|n) =1
P (n|x)P (x),P (n)(4.62)

where P (x) is the prior knowledge about the initial bright fraction and 1/P (n) is a normalization
factor. We assume that the prior is initially flat. For a given x, we first calculate the photon
number distribution P (n|x) and use the distribution to simulate a photon detection event. We
use this to update our prior P (x) and apply Eq. (4.62) to arrive at a new posterior probability
distribution P (x|n). To quantify how well the posterior predicts the initial bright fraction x, we
fit P (x|n) to a normalized Gaussian distribution and calculate its standard deviation σN (x) as a
metric of coincidence, as exemplified in the inset of Fig. 4.3. We then repeat the process N times

81
for each x ∈ {0, 0.01, 0.02, . . . , 1}. Furthermore, we repeat each run of x and average the results tominimize the inherent statistical unpredictability of each run.
The results of our Bayesian analysis are displayed in Fig. 4.3 and discussed below. We plot
the standard deviation σN (x) of the Gaussian fit of P (x|n) for three different numbers of iterations,
N = 25 (blue circles), N = 50 (orange crosses), and N = 100 (gold stars), with m0 = 1/2 andn̄0 = 5. We see that the curves for σN (x) decreases as N increases for every x. This indicates that
the Gaussian fits are converging on the initial bright population as we generate more data fromphotodetection events. Furthermore, the figure demonstrates that the minimum of σN (x) occurs
at a value x 6= 0, 1. Noting that the two extreme values of x are when the atom is initially pure(|bihb| and |dihd|), this behavior indicts that atomic states with initial entropy are easier to predict
from cavity field measurements. In addition, the minimum occurs at values with a large initialdark state population, rather than the state x = 0.5 which has the maximum initial entropy. Thissuggests that the initial atomic state is easier to predict when there is less atom-field interaction
but the atom has initial entropy.

4.6

State Purification and Mutual Information
A further complication of the thought experiment is that it may be unclear whether any of

the final field entropy is directly transferred from the atom or if this is new entropy generated byentanglement between the atom and the field during the interaction. To investigate this question
further, we now look at the gedanken experiment from a quantum information prospective. This willallow us to separate atomic entropy that must be removed during the process from the generated
entanglement entropy (GEE).
We first purify the atomic state by way of its Schmidt decomposition. That is, we view theatomic ensemble as a marginal of a pure state, ρ̂A = TrR [ s(AR) s(AR) ], where
E Xp
E
E
(A)
(R)
s(AR) =
βj ψj
⊗ φj
.

(4.63)j(A)

Here, ρ̂A ψj

E(A)= βj ψj

E(R)

are the atomic eigenvectors and φj

E∈ HR for some auxiliary

82

Figure 4.3: Standard deviation σN (x) of the Gaussian fit to the posterior probability distribution
P (x|n) as a function of the initial bright fraction x after N = 25 (blue circles), 50 (orange targets),and 100 (gold stars) iterations. Inset: An example of a simulated P (x|n) (blue, dotted) andassociated Gaussian fit σ1 00(x) (black, solid). Both distributions are maximized near the actual
bright fraction x = 0.1 (red, dot-dashed).

83
Hilbert space HR . It can be shown that the marginals of s(AR) s(AR) have the same eigenvalues[107]. This motivates the choice of ρ̂R (0) = TrA [ s(AR) (0) s(AR) (0) ] to be an identically preparedensemble to ρ̂A (0), but with atoms that are not interacting with the field. Thus, the total system’s
density matrix becomes ρ̂ = ρ̂ARF , but only the identity operator of the HR basis appears in the√√system’s Hamiltonian and jump operators in Eq. (4.26): Ĥ → Ĥ⊗ÎR and Γ |dihe| → Γ |dihe|⊗ÎR .
Since the auxiliary state in HR purifies the initial state of the atom, it contains information
about ρ̂A (0). We can therefore quantify the amount of information that is common to the cavity
and auxiliary atom subsystems using quantum mutual information Eq. (4.7):
I(R : F ) = S(ρ̂R ) + S(ρ̂F ) − S(ρ̂RF ).

(4.64)We characterize the information flow between the real atoms and the cavity by determining if there
are any correlations between the field and the non-interacting atom, I(R : F ).
In Fig. 4.4, we track the von Neumann entropy of the various subsystems of the interaction,
as well as the mutual information between the field and auxiliary atom. We start the atom in the
maximum entropy state x = 0.5 and the field in a coherent state with n̄0 = 5. Furthermore, wechoose m0 = 1/2. It is clear that the field entropy (green line) increases throughout the process.
This demonstrates that the field is not only displaced from its initial state during the interaction,
but that the field ends the process in some mixed state rather than a new coherent state. We
differentiate the entropy that is transferred from the atom during the interaction from the GEE
by studying the mutual information between the field and auxiliary atoms Eq. (4.64). Fig. 4.4
demonstrates the the mutual information (dashed gold curve) increases during the interaction.
This exemplifies, from a quantum information perspective, that part of the initial entropy in theatomic state has been transferred to the field as a result of the coherent dynamics of the system.

4.7

Outlook
Our results conclude that the laser field can also remove entropy from the gas, and thatthe laser field is consequently altered in a detectable manner. We do emphasize, however, that

84

Figure 4.4: Entropy of various subspaces and quantum mutual information (dashed gold curve)
between the field and non-interacting atoms I(R : F ) = S(ρ̂R ) + S(ρ̂F ) − S(ρ̂RF ). We display
the entropy of the atomic (blue curve), field (green curve), and atom-auxiliary atom (red curve)
subspaces, as well as the total entropy (purple curve) in time. The parameters are g = Γ = 1.

85
our conclusion requires the existence of spontaneous emission, so we do not claim the existence ofsteady-state gas-laser entropy exchange in its absence. Furthermore, comparing the initial atomic
entropy and the final mutual information in Fig. 4.4 suggests that the field absorbs only a fractionof the initial atomic entropy. This implies, for our specific simulation parameters, that the majorityof the initial atomic entropy has been transferred to the background radiation field.
It would be interesting to investigate if a regime exists where I(R : F ) increases significantly.
It is possible that this may be done by deriving an analytic solution for the mutual information ina similar manner as Section 4.4. Another possible discussion point may be the the use of quantum
discord [51] as a further measure of quantum correlations. This may allow us to connect our research
to Liouville’s theorem in a more appropriate way, by the use of Wigner trajectories [122].
Additional future work will explore the possibility of deriving analytic solutions using theblock matrix formalism. As mentioned in Section 4.5, the displaced equilibrium laser field Eq. (4.61)
might be analytically tractable for m0 = 0 and n̄0 → ∞. This calculation could inform how theminimum in Fig. 4.3 tends towards as N → ∞. A non-zero value would be quite interesting as
it would demonstrate that states with initial entropy are easier to predict in the limit of infinite
iterations of the experiment.

Chapter 5

Subradiant State Selection by Photon Detection

In the last chapter, we demonstrated that an atomic system tends towards zero entropy as
it becomes a pure state during the laser cooling process, subject to the condition of the second
law of thermodynamics. This extends to more general systems where similar entropy reduction
processes can occur when a subsytem is coupled to an environmental bath [2]. In this chapter, we
present a protocol for creating low entropy many-body states that results from the steady-statesuperradiance model in the weak pumping regime [91, 90]. Here, we first drive the system to havea high probability of occupancy of its subradiant state, and then implement a detection phase to
separate this state from the mixture. In doing so, we are reducing the system’s entropy in a similarmanner to the energy redistribution that occurs during evaporative cooling by way of two-bodycollisions. This cooling analogue is explored in Section 5.3.
While entropy removal is a consequence of our subradiant state selection protocol, its mostuseful application lies in the ever growing field of quantum metrology where quantum effects are
exploited to enhance the precision of measurements beyond the capability of purely classical approaches [57]. This is because the subradiant state has the property that the total spontaneousemission scattering rate is significantly reduced compared to the independent decay rate as a resultof interatomic interference. This occurs even in the presence of a significant amount of excited state
population. The weakening of the system to the background radiation field allows for protocols
that are robust to dissipative effects. This is an important consideration when the evolution times
in metrological processes approach the lifetime of the excited state and thus, spontaneous emission

87

5.1: Source: Ref. [111]. Two-atom metrology plotting the optimal sensitivity δω as a
FIG. 2Figure(colorTwo-atom(a) time
Normalized
mufunctionof τ Γonline).
. This displays theenhancement metrology.
of sensitivity for long
scales while usingtheanti-symmetricRamseysequence. The atomsfrequency
are separated byshift
a distancetual decayrate anddipole-dipole
forr =a 0.3λ,pairwith
ofwavelength
atoms
λ.
as a function of r=. For positive (negative) , the asymmetric stateis subradiant (superradiant) (indicated by dashed/non-dashed recritical factor.
To exemplify
the usefulnessof subradiantin quantum
metrology,
gions).becomes(b) aLevel
schemein thedressedbasis states
showing
the
two
A .
independent
with Ramsey
modifiedSthatandwe present a decay
result from channels
the the anti-symmetrictechnique rates(ART) [111]demonstrates(c) Optimal
sensitivity
as a functionof .
atom
the significant
sensitivity improvement
for timescalesτ  Γ The
. Fig. 5.1
shows separation
the improvementisofr= ¼population
0:3 corresponding
to 
end
0:41
and  [116] sequence,
0:29.butTheinversion measurements
at theof a Ramsey-likewithminithe freemum evolution
for thenow
asymmetricaddressing
is reachedaround’2=Ato. a
occurring with thesystem in its anti-symmetricstate(blue curve)
as opposed

mag
N
more
in va
be g
lattic
quic
To i
the t
geom
for a
inde
for t
eluc
indic
emit
ij
onal

symmetric state (red curve). The basis of this improvement lies in the fact that decoherent spontaneous emission can cause a detrimental effect on population inversion, especially as the scattered123601-3

photon number becomes proportional to the atom number [79, 53]. This can lead, for example, toimproved measurements of transition frequencies [149, 22, 136].
Having motivated the usefulness of systems that are weakly coupled to the background radiation field in quantum metrology, we now turn our attention to the problem of effectively creating
the subradiant state. While this is often an untrackable problem [111], we demonstrate a procedurethat first enhances the subradiant state population and then selects this subradiant state out of amixed ensemble by way of a photodetection scheme. We begin this chapter with some backgroundinformation by introducing the concept of angular momentum in quantum mechanics and apply it
to the problem of super- and subradiance. We then introduce a many-body model that leads to88steady-state superradiance in the strong pumping limit, but a collective dark state with high overlap with the subradiant state in the weak pumping regime. We next propose a protocol to selectthe subradiant state from the statistical mixture. We then conclude the chapter by connecting ourscheme to the idea of entropy removal by making an analogue to evaporative cooling.

5.1

Dicke Basis Background

5.1.1

Addition of Angular Momenta

I now provide a brief introduction to the addition of angular momentum in quantum mechanics. I again follow a similar format to Sakurai’s presentation and refer the reader to [121,Chapter 3, §1,5,8] for a detailed derivation of the equations presented. This is useful in describing
a collection of N two-level atoms because of the psuedo-spin 21 nature of each atom, as discussed
in subsection 2.4.1.

5.1.1.1

Eigenbasis of Angular-Momentum Operators

We begin with the quantum operator D̂(R) that induces, in the appropriate basis, the transformation|ψ(n̂, θ)i = D̂[R(n̂, θ)] |ψi ,(5.1)

on a state |ψi into a rotated system defined by a three-dimensional orthogonal matrix R characterized by a unit vector n̂ and angle θ. For infinitesimal rotations dθ, this is given by the unitary
operatorD̂[R(n̂, dθ)] = 1 − i

Jˆ · n̂
~

!
dθ,

(5.2)

where Jˆ ≡ Jˆx + Jˆy + Jˆz is the angular-momentum operator. The SO(3) rotation group is nonAbelian as the individual generators Jˆi about the three axes do not commute,hiJˆi , Jˆj = i~εijk Jˆk ,

(5.3)

89
with Levi-Civita symbol εijk . Since the generators are not simultaneously diagonalizeable, we
h 2 i2introduce a new operator Jˆ ≡ Jˆx Jˆx + Jˆy Jˆy + Jˆz Jˆz with the property Jˆ , Jˆi = 0 so that1
2Jˆ |a, bi = a |a, bi ,Jˆz |a, bi = b |a, bi .

(5.4)

We determine the eigenvalues a and b using the non-Hermitian ladder operators Jˆ± ≡ Jˆx ± iJˆy ,such thatJˆz Jˆ± |a, bi = (b ± ~)Jˆ± |a, bi ,Jˆ± |a, bi = c± |a, b ± ~i ,(5.5)

with normalization constants c± . We see that b will be quantized by integer multiples of ~ and that
the Jˆz eigenstate is raised (lowered) by Jˆ+ (Jˆ− ).
By imposing boundary conditions on b, bmin ≤ b ≤ bmax , it can be shown thatJˆz |j, mi = m~ |j, mi ,Jˆ |j, mi = j(j + 1)~2 |j, mi ,(5.6)
p
Jˆ± |j, mi = ~ (j ∓ m)(j ± m + 1) |j, m ± 1i ,with the allowed values ofm = −j , −j + 1 , . . . , j − 1 , j ,
|
{z
}

(5.7)

2j+1 states

where we have switched to the notation j ≡ bmax /~ and m ≡ b/~.

5.1.1.2

Coupled Basis

Let us now consider two angular-momentum operators Jˆ1 and Jˆ2 in different subspaces, suchithat Jˆ1,i , Jˆ2,j = 0, ∀ i, j. We now consider the total angular-momentum operatorhJˆ = Jˆ1 ⊗ Î2 + Î1 ⊗ Jˆ2 = Jˆ1 + Jˆ2 ,(5.8)

h 2 2ih 2iwhich satisfies Jˆ , Jˆ1 = 0 and Eq. (5.3) for the total generators, but Jˆ , Jˆ1,i 6= 0. There
are similar equations for the Jˆ2 generators. We chose to represent the base kets as eigenstates of222Jˆ1 , Jˆ2 , Jˆ , and Jˆz :2Jˆ1 |j1 , j2 ; J, M i = j1 (j1 + 1) ~2 |j1 , j2 ; J, M i ,2Jˆ2 |j1 , j2 ; J, M i = j2 (j2 + 1) ~2 |j1 , j2 ; J, M i ,2Jˆ |j1 , j2 ; J, M i = J (J + 1) ~2 |j1 , j2 ; J, M i ,Jˆz |j1 , j2 ; J, M i = M ~ |j1 , j2 ; J, M i ,
(5.9)

1

We make the arbitrary choice of working in the Jˆz basis by convention.

90
with M = m1 + m2 and |j1 − j2 | ≤ J ≤ j1 + j2 . We usually denote these states simply as |J, M iand assume the j1 and j2 values are understood.
A simple example is the singlet-triplet representation of two spin- 21 systems. Here, j1 = j2 =1/2 and thus, |j1 − j2 | = 0 ≤ J ≤ j1 + j2 = 1. This leads to four eigenstates |0, 0i , |1, −1i , |1, 0i ,and |1, 1i, which we write out explicitly, with + = 1/2 and − = −1/2, as
1|1, −1i = |m1 = −, m2 = −i , |1, 0i = √ [|m1 = +, m2 = −i + |m1 = −, m2 = +i] ,2
1|1, 1i = |m1 = +, m2 = +i , |0, 0i = √ [|m1 = +, m2 = −i − |m1 = −, m2 = +i] .
2
|
{z
}

(5.10)

Spin Singlet

The states with J = 1 are symmetric and are called the spin triplet states. The J = 0 state isanti-symmetric and corresponds to the spin singlet state. To add N spins, one first adds Jˆ1 andJˆ2 , followed by the addition of Jˆ12 and Jˆ3 , and so on.

5.1.2

Superradiance and Subradiance

Let us now consider an ensemble of N two-level atoms which are modeled as spin- 21 systems.
We study atomic states with permutational invariance such that they are indiscernible with respect
to photon emission or absorption processes [63, 150]. The symmetric states, called the Dicke
PN
states [46], correspond to the largest J value: J =i=1 ji = N/2. The simplest examples ofthis are the excited |N/2, N/2i and ground state |N/2, −N/2i which corresponds to the N -atom
permutation invariant states,NN
J = ,M =22

≡ | e, e, . . . , e i,
| {z }NN
J = ,M = −22

≡ |g, g, . . . , gi ,(5.11)

N

where, for example, |e, e, . . . , ei =NN

i=1 |eii . The other Dicke states are given by the “symmetriza-tion operator” Ŝ [63]:|JM i = Ŝ[| e, e, . . . , e,
| {z }
J+M

g, g, . . . , g i].
| {z }(5.12)J−M

A straightforward application of this is for N = 2, where the Dicke states are given by the J = 1
spin triplet states in Eq. (5.10).

91
These symmetric states are the superradiant states. To see this, let us define the single-atomspin-flip operators in the nth atom’s subspace,
!
n−1
O
(n)
σ̂− ≡
Îi ⊗ |gin he|n ⊗
i=1

N
O

!
Îi

h
i
(n) †
,
= |gin he|n = σ̂+(5.13)

i=n+1

to introduce the collective operators,
Jˆ± =

N
X

(n)

σ̂± .

(5.14)

n=1

Assuming every excited state has linewidth Γ , the rate of photon emission from N atoms iss
RN= Γ hJˆ+ Jˆ− i = Γ (J + M ) (J − M + 1) ,(5.15)

which, for J = N/2, demonstrates that the photon emission rate greatly increases, with a maximum
rate proportional to N 2 , until it reaches the ground state [63], as depicted in Fig. 5.2. This
demonstrates the collective enhancement of spontaneous emission by way of interatomic coherence.

As demonstrated by the spin singlet state in Eq. (5.10), lower values of J can be obtained
through superpositions of atomic states. The subradiant state J = 0 is called the subradiant state
as it only has one M value, i.e.,
Jˆ± |0, 0i = 0,

(5.16)

s = 0 in spite of nearly half the atoms being in the excited state [90]. This illustrates
and thus, RN

the suppression of spontaneous emission due to interatomic decoherence, and the |0, 0i dark stateis the target state in the next section.

5.1.3

Permutation Invariant Simulations

The irreducible Hilbert space of the N permutation invariant two-level atoms obviously has a
2N dimensionaity [9]. While numerical calculations can now be preformed in this uncoupled basis,the exponential scaling of the Hilbert space can make simulations quickly intractable for all but
the smallest N values [151, 126]. This is compounded when studying open quantum systems as the
Liouvillian space scales as 4N . Therefore, we instead take advantage of the permuational, SU(4)rogressively the basic physical concepts relevant to the superradiance phenomenon.

ntroduction. Motivation of this essay

92

n “ordinary” fluorescence experiments, a gas of atoms (or molecules) initially prepared in thel of an electronic transition decays by spontaneous emission of light towards lower states ac
a process in which the atoms (or molecules) of the sample can be considered as inte
ependently from each other with the radiation field. The emission, obeying an exponential law
aracteristic time r~,equal to the reciprocal of the radiative decay rate F of the initially excite
the radiation pattern of the atomic sample is essentially isotropic (see fig. la). These featu

D~ectortime.a.
Detector

0.7

b.

time

Comparison between the general characteristics of ordinary fluorescence and superradiance experiments. (a) Ordinary spontaneou

5.2: Source:
Ref. [63].
Comparision ofbetween
the general thecharacteristics
ordinary fluorescenceof ordinary
gure isotropic
2.1: Figure
Source:Ref. [42].
Comparison
generalofcharacteristics
ntiallywithan exponentiallydecaying
intensity (time constant;~,).(b) Superradiance is anisotropic with an emission occurring(a) and superradiant fluorescence (b) for decay time τSp = 1/Γ .
scencesuperradiance experiment. (a) Ordinary spontaneous emission is essentially isotofdurationand~~rw/N.
ith exponential decaying intensity (time constant τsp ). (b) Superradiance is anisotropic wit
mission occurring in a short burst of duration ∼ τsp /N .

uration of the order of τsp /N .

Since Dicke’s original work [28], superradiance has been extensively studied and demonsta wide variety of physical systems [42]. More recently, it was demonstrated in quantum dots

V centers [62], and atoms trapped along a photonic crystal waveguide [37]. In spite o93symmetry that the system possesses to map its generators (the Gell-Mann matrices) to the coupledbasis [151, 126]. Here, we note that the density matrix takes block diagonal form as no coherences
are created when we project ρ̂ into the Dicke basis [126]:
ρ̂ =

X
π

X

P̂π ρ̂P̂π† =ρJ,J 0 ,M,M 0 J, M J 0 , M 0 δJ,J 0 ,(5.17)

J,J 0 ,M,M 0

where P̂π is the permutation operator for permutation π, δi,j is the Kronecker delta, and ρJ,J 0 ,M,M 0 =hJ, M | ρ̂ |J 0 , M 0 i. There is adJN =N !(2J + 1)
 N

N
2 +J +1 !

2 −J



!

(5.18)

degeneracy of each |J, M i Dicke state. We note that the state in the limit of infinite temperature
T → ∞ is given byρ̂T →∞ =

X dJ

N

J,M

2N

|J, M ihJ, M | ,(5.19)

as every state is equally occupied with a dJN degeneracy and normalization factor of 2−N . Each
“Dicke block” has (2J + 1) values of M [see Eq. (5.7)] which therefore allows us to work with a
PN/2
reduced density matrix with only J=0 (2J + 1)2 = (1/6)(N + 1)(N + 2)(N + 3) = O(N 3 ) totalelements. This work has been collectivized into QuTip’s permutational invariant quantum solverin Python [126] which is the core of my simulations.

5.2

Enhancement and Selection of J = 0 State
With these preliminaries complete, we now turn our attention to a specific model that op-

erates in the bad-cavity limit. It has been shown that this system can achieve steady-state superradiance [92, 91, 90, 150], as well as lasing [23, 109], for intermediate repump rates. We instead
focus, in this section, on the weak pumping limit in which the J = 0 state population is greatly
enhanced. We then demonstrate a photon detection scheme that isolates the subradiant state fromthe steady-state of this previous model.

94
5.2.1

Superradiant Lasing Model

We now consider the simple steady-state superradiance model proposed [92] and developed [91, 90] to achieve lasing by Dominic Meiser, Jun Ye, and Murray Holland. The model
was further expended by Minghui Xu [150, 151], David Tieri [139, 140], James Thompson [23], andMatt Norcia [109]. A schematic of the model is depicted in Fig. 5.3, and I will refer to the system
as the superradiant lasing model.

5.2.1.1

System Dynamics

An ensemble of N atoms are, ideally, collected at the anti-nodes of a cavity mode so that they
have an identical cavity-coupling constant g [see Eq. (4.17)]. Each atom has excited and groundstates |ei and |gi, separated by the atomic frequency ωa , and an excited state linewidth Γ . The
many-body Hamiltonian is given by~g  † ˆĤ = ~ωa Jˆz + ~ωc â† â +â J− + âJˆ+ ,2(5.20)

with cavity frequency ωc and collective z-operator
1
Jˆz =
2

N
X
i=1

N

σ̂z(i) =1X|eii he|i − |gii hg|i .
2(5.21)

i=1

The cavity field decays at a rate κ and its output can be monitored by two detectors D1 and D2 witha variable delay time τ between them. When the decay from the atomic ensemble is collectively
enhanced through a cavity, we can assume that the spontaneous decay into free space Γ can beneglected. This comes from the approximation N C  1, where C = g 2 /(κΓ ) ∝ 1/N0 is the single
atom cooperativity parameter with critical atom number N0 .
We also include an incoherent process that represents noncollective pumping at rate w with(n)jump operator σ̂+ for each atom. Here, the repumping can be implemented by coherently driving,at a rate Ωp , a transition between |gi and an auxiliary state |ai. This state then decays, at a rate
γp , to the |ei state, as shown in Fig. 5.4. Note that, in the limit of large γp , this leads to an effective
pumping rate w = |Ωp |2 /γp [150]. We choose to ignore the effects of incoherent dephasing of the

95

Figure 5.3: Source: Ref. [90]. Schematic diagram of the superradiant lasing model. N two-levelatoms (black dots with internal structure given in the inset) with individual decay rate Γ arecollected at the anti-nodes of a single-mode cavity field. The cavity has single atom cooperativity
parameter C, decay rate κ, and coupling constant g. The atoms are incoherently repumped at a
rate w. The output of the cavity field is monitored by two detectors D1 and D2 with a variable
time delay τ between them.

Figure 5.4: Source: Ref. [91, 150]. Schematic diagram for implementing the incoherent pumping
scheme with a three-level atom at an effective rate w = |Ωp |2 /γp (brown arrow). The atoms arecoherently pumped (black arrow) from the ground state |gi to the auxiliary state |ai at a rate Ωp .
This then decays to the excited state |ei at a rate γp (yellow arrow). This can then decay back to
the ground state at an individual decay rate Γ (blue arrow).

96
(n)

atomic dipole (with jump operator σ̂z ) by assuming its characteristic time T2 is very long. Thus,
the system’s master equation Eq. (3.33) is written in Lindblad form as
N
i
X
∂ ρ̂
1 h
(i)
Ĥ, ρ̂ + κD̂[â]ρ̂ +=wD̂[σ̂+ ]ρ̂,∂ti~

(5.22)

i=1

with Lindblad superoperators in the form of Eq. (3.34).
As it turn out, the problem can be simplified further by exploiting the fact that the cavity
field decays much faster than the atomic coherence [90], and therefore is suitable for adiabatic
elimination [24, 36]. The validity of the assumption that the intensity decay rate is much larger
than atomic rates takes the formκ  N CΓ.

(5.23)

This is the so-called bad-cavity limit. Elimination of the light field yields [90], in the so-called“dissipation picture”,
N

X
∂ ρ̂˜
(i)
= Γc D̂[Jˆ− ]ρ̂˜ +wD̂[σ̂+ ]ρ̂˜
∂t
i=1
!


X (i) (i) 1 (i) (i)
1
˜ +w
˜
˜
= Γc Jˆ− ρ̂˜Jˆ+ − {Jˆ+ Jˆ− , ρ̂}σ̂+ ρ̂σ̂− − {σ̂− σ̂+ , ρ̂} ,2
2(5.24)iwhere the collective decay rate of the atoms through the cavity is given byΓc = CΓ =g2.
κ(5.25)We are now in a position to only study the dynamics of the atomic evolution as the cavity has
effectively become a decay channel for the atoms with jump operator Jˆ− . Based on [154], I include
a Dicke-state diagram (see Fig. 5.5) and a table (see Table. 5.1) listing the different decay processes
that have been discussed. We can now consider numerical results of Eq. (5.24).

5.2.1.2

Second-Order Correlation Function

To motivate the usefulness of the superradiant lasing model, we will discuss fluctuations of thelight intensity in a similar manner to [90]. By using a Hanbury Brown-Twiss photodetector setup(see right side of Fig. 5.3), one can measure a joint probability P2 (t, t + τ ) of detecting a photon in97

Figure 5.5: Source: Ref. [154]. Dicke ladder diagrams with N = 4 for (a) individual decay,(b) collective decay through the cavity, (c) individual pumping, and (d) individual dephasing.
The thickness of the arrows indicates the probabilities of the different jumps (see, for example,
Table 5.2).

Dissipation

Rate

Jump Operator

Jump(J, M ) → (J, M − 1)

Individual decay

Γ

(n)

σ̂−

(J, M ) → (J − 1, M − 1)
(J, M ) → (J + 1, M − 1)

Collective decay

Γc

P (n)
Jˆ− = n σ̂−(J, M ) → (J, M − 1)
(J, M ) → (J, M + 1)

Individual pumping

w

(n)

σ̂+

(J, M ) → (J − 1, M + 1)
(J, M ) → (J + 1, M + 1)
(J, M ) → (J, M )Individual dephasing

1
T2

1 (n)
2 σ̂z

(J, M ) → (J − 1, M )(J, M ) → (J + 1, M )Table 5.1: Source: Ref. [154]. Table of quantum jumps resulting from the jump operator associatedfor the different dissipative processes we consider.

98
two separate time intervals P1 (t). We express this as the second-order correlation function, givenby [60]g (2) (t, τ ) =P2 (t, t + τ )∆t∆τhâ† (t)â† (t + τ )â(t + τ )â(t)i,= †P1 (t)∆tP1 (t + τ )∆τhâ (t)â(t)ihâ† (t + τ )â(t + τ )i

(5.26)

which is a measure of detecting a photon both in a time interval ∆t at time t and ∆τ at time t + τ .
In steady-state, this does not depend on t [90], so we simply write g (2) (τ ). In Fig. 5.6, we plot thesecond-order correlation function as a function of repump rate at zero time delay, which is relatedˆ
to fluctuations of the photon flux I:hiˆ 2 = hIi
ˆ 2 g (2) (0) − 1 + BhIi
ˆ
∆I 2 = hIˆ2 i − hIi(5.27)

for detection bandwidth B. Here, we are able to writeg (2) (0) =hJˆ+ Jˆ+ Jˆ− Jˆ− i,hJˆ+ Jˆ− i2(5.28)

as, in the bad-cavity limit Eq. (5.23), the cavity field is slaved to the atomic dipoles [90]:
â ∼
=

g ˆ
J− .
iκ

(5.29)

When g (2) (0) = 1 (blue dotted line), the field is in a coherent state and photons have aˆ which is often a sign of lasing. A large second-order correlation function
mean rate of arrival of hIi
g (2) (0) > 1 has super-Poissonian properties, which indicates that photons will arrive in bunches.
For a thermal state, which occurs in the limit of very strong pumping w  Γc , this behavior reaches
a maximum of g (2) (0) = 2 (dotted orange line). In contrast, the second-order correlation function
can reach higher values in the weak pumping regime w ≤ Γc . As discussed in the next subsection,
this is because the system has evolved into near subradiant, collective dark states [17, 106] in which
the emission of photons is highly suppressed.

99

3.5
3.0

g(2)(0)

2.5
2.0
1.5
1.0
0.5

10 1

100

101

w/ c

102

Figure 5.6: Source: Ref. [90]. Second-order intensity correlation function g (2) (0) as a function of
repump rate for N = 10. Results were calculated using Eq. (5.28) and the dissipative picture
master equation Eq. (5.24).

100
5.2.2
5.2.2.1

Subradiant State Enhancement
N = 2 Case

We first simulate the case of two atoms (N = 2) so that we may use the singlet-triplet spin
language. Throughout this subsection, our simulations initialize the state along the diagonal in
the J = N/2 manifold, which we demonstrate in Fig. 5.7(a) by initializing it in the excited state|1, 1i. No matter the choice, the system reaches a unique steady-state, which we investigate in the
weak pumping regime. Fig. 5.7(b) demonstrates the final state ρ̂ss of the system for the choice
w = 0.1Γc . Here, almost all of the population ends in the |1, −1i and |0, 0i states, both of whichhave final populations of 0.464. This is a slight enhancement over the 2−2 = 0.25 population in the
J = 0 state for the infinite temperature state ρ̂T →∞ [see Eq. (5.19)]. We study this enhancement
further in the next subsubsection.
First, we can now discuss the significance of a large g (2) (0) function in the weak pumping
regime. Let us take the limit that w/Γc → 0 [138, 91, 90]. Here, the population is pumped entirely
into the |1, −1i and |0, 0i states. Fig. 5.5 demonstrates that this state may only be pumped into
the |1, 1i state (as collective decay dominates the |1, −1i ↔ |1, 0i transition when w/Γc ≈ 0). From
|1, 1i, the atoms collectively decay back to the |1, −1i state by way of a quick photon cascade ona time scale O(Γc−1 ). This in itself has potential quantum metrological applications as one would
expect this collective dark state is robust to perturbations of the system. In particular, when thesystem is perturbed, such as a single atom dephasing event, there is an avalanche of photons until
the system finds its way back to the collective dark state, suggesting that there can be a large
output signal from a small perturbation. However, we do not pursue this potential project here.

5.2.2.2

Many-body Case

Having now demonstrated the subradiant state enhancement in the weak pumping limit with
two atoms, let us explore how the subradiant population changes as a function of atom number.
We choose to set the repump rate to be w = Γc /N . As demonstrated in Fig. 5.8, the steady-

101

1

0

1

0
1

0

1

0

1

0

1

0

Figure 5.7: Example of the time evolution of the density matrix in the superradiant lasing modelwith N = 2 and repump rate w = 0.1Γc . The red areas indicate highly populated states. (a) The
system may, for example, begin in the excited state ρ̂(0) = |1, 1ih1, 1|. (b) No matter the initialstate, the system evolves under Eq. (5.24) to a unique steady-state ρ̂ss . Also displayed is the block
diagonal form of the density matrix in the Dicke basis where the different blocks represent different
J manifolds.

102

Dissipation

Jump

Jump Probability

Collective decay(J, M ) → (J, M − 1)Γc (J + M )(J − M + 1)dJN δt(J, M ) → (J, M + 1))(J+M +1) Jw (2+N )(J−MdN δt
4J(J+1)(J, M ) → (J − 1, M + 1))(J−M −1) JdN δtw (N +2J+2)(J−M
4J(J+1)(J, M ) → (J + 1, M + 1)

+1)(J+M +2) JdN δtw (N −2J)(J+M
4(J+1)(2J+1)

Individual pumping

Table 5.2: Source: Ref. [154]. Table of the transition probabilities in a short time interval δtassociated with the quantum jumps involved in the superradiant lasing model. Here, dJN is thedegeneracy of the specific J manifold, given by Eq. (5.18).

state value of the J = 0 state population in the infinite temperature case (red dots) scales like apolynomial of N . Meanwhile, the subradiant population in the superradiant lasing scheme (blackdots) approximately saturates at around N = 10. This leads to an order of magnitude improvement
using the superradiant lasing model for the case of N = 20.
We preform a more thorough investigation into the system’s parameter space in the contour
plot presented in Fig. 5.9. This confirms the manifestation of an apparent “enhancement threshold”
in the weak pumping regime at which the steady-state J = 0 state population significantly falls off.
Fig. 5.9 also suggests that the threshold value is approximately linear with N and that, for a given
N , the subradiant state overlap does not vary much with N below this threshold.
Expanding on the idea of a threshold approximately linear with N , we look at the transition
rates for each jump associated with collective decay and individual pumping. Table 5.2 displays
the jump probabilities of each jump in a short time interval δt for a |J, M i state with degeneracydJN given by Eq. (5.18). Interestingly, the transition rates for the individual repumping jumps
increase with N while the collective decay probability does not. This suggests that the enhancement
threshold value occurs at
w ≈ Γc /N.

(5.30)

This motivated our choice of repump rate in Fig. 5.8 as w/(Γc /N ) seems to be the natural scalingJ = 0| |J = 0

103

10 1

0, 0| ss|0, 0
0, 0| T |0, 0
10 2
0

10

20

N

30

40

50

Figure 5.8: The population of the J = 0 state as a function of the atom number. The values ofthe superradiant lasing scheme (black dots) are given by the steady-state of Eq. (5.24), while theinfinite temperature state values (red dots) are calculated by Eq. (5.19). The repump rate for each
N is given by w = Γc /N .

100

10 1

0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00

J = 0| ss|J = 0

w/ c

101

2.5

5.0

7.5

10.0 12.5 15.0 17.5 20.0

N

Figure 5.9: Contour plot of the steady-state subradiant state population as a function of atom
number N and repump rate w.

104
for the repump strength in the subradiant regime, similar to w/(N Γc ) in the superradiant regime.
We test this conclusion by plotting the steady-state J = 0 population as a function of w/(Γc /N ) in
Fig. 5.10(a). It is clear that the steady-state subradiant state population begins to sharply decreasefor every N value at the enhancement threshold predicted in Eq. (5.30), while the value for a given
N remains approximately constant below the threshold. Moreover, Fig. 5.10(c) shows that the
population below threshold ends up almost entirely in the lowest M value of each Dicke block. We
investigate this point further in Fig. 5.10(d). Here, we demonstrate that the population not only
evolves to the lowest M value in each block, but that the population head towards lower J values
as well, ending up almost entirely in the |0, 0i and |1, −1i states. This is because, for M < 0 in
the weak pumping limit, the repumping predominantly drives J → J − 1 transitions rather thanJ → J + 1 transitions [91], as is demonstrated by the darker arrows towards lower J values inFig. 5.5(c).
We also show in Fig. 5.10(b) that the expectation value hJˆ+ Jˆ− i becomes O(1) at the enhancement threshold. This offers significant insight into the dynamics of the system, as this is related tohâ† âi [see Eq. (5.29)]. As displayed in Fig. 5.5(c), the repump couples the states with the lowest M
values in each Dicke block to higher M values in the same J manifold. However, these states decay
too fast in the weak pumping limit as Γc dominates over w. When the repump rate becomes large
enough to put at least one atom in the excited state in the timescale 1/Γc , one excitation in thefield becomes possible. Since the probability of finding an atom in the excited state is ∼ w/Γc , we
see that the total excited state population for N atoms becomes N w/Γc . Therefore, the system will
begin to leave the collective dark state when N w/Γc ≈ 1, which is consistent with our enhancement
threshold.
The upside of the enhancement of the subradiant state in the weak pumping limit is best
understood in the language of Monte Carlo wave functions [see Appendix C.2]. In the infinite
temperature case and for N = 50, only one out of 1/(d050 2−50 ) ≈ 231 trajectories will result in the
J = 0 state at the end of a stochastic shot. On the contrary, one out of every five trajectories
end in the subradiant state in the the superradiant lasing model which is now an experimentally

105

Figure 5.10: Verification of an “enhancement threshold” near w = Γc /N , as displayed by thesignificant change of behavior in steady-state near this region. Each subfigure contains plots for N
values in the set {2, 4, 6, . . . , 18, 20, 30, 50, 70}. (a) Steady-state population in the J = 0, subradiantstate. (b) Expectation value hJˆ+ Jˆ− i in steady-state near the threshold, which is related to hâ† âi.
(c) Sum of the steady-state probabilities in the states with the lowest M value of each Dicke block.
(d) Sum of the steady-state probabilities in the |0, 0i and |1, −1i states.

106
viable proposal. Thus, this enhancement is necessary for the subradiant state selection by photon
detection protocol that we present in the next subsection.

5.2.3

Detection Phase

We can now present a novel protocol that can extract the subradiant state from the atomic
mixture. For simplicity, we consider the N = 2 case, but the conclusions drawn extends to higher
atom numbers. To differentiate between the J = 0 and J = 1 manifolds, we introduce a coherent
driveĤd = ~Ω Jˆx =~Ω  ˆ
J+ + Jˆ− .
2(5.31)

Therefore, the subradiant state will remain unchanged [see Eq. (5.16)] by the system’s coherent
dynamics, in the dissipation picture. We, once again, turn our attention to Fig. 5.5 to determine
the necessary incoherent dynamics. Since we want a scheme that does not affect the subradiantstate population, we only allow for collective emission through the cavity, and therefore turn therepumping laser off (Ωp = 0). The detection-phase master equation, in the dissipation picture, is
thus


i
∂ ρ̂˜
1 h
1 ˆ ˆ ˜˜˜ˆˆ=Ĥd , ρ̂ + Γc J− ρ̂J+ − {J+ J− , ρ̂} .
∂ti~
2(5.32)

Here, ρ̂˜ is initialized as the steady-state density matrix ρ̂ss of the superradiant lasing scheme.
Numerical results for this interaction are presented in Fig. 5.11 with total time tf = 10/Γc , N = 2,
and Ω = Γc . It is clear that the J = 0 state population has remained constant during the process,while the populations in the J > 0 manifolds are displaced from their steady-state values in the
superradiant lasing model ρ̂ss .
Let us again frame our discussion in a quantum trajectories picture. If a single emitted photon
is detected, the prescribed system dynamics indicates that the state was in a J > 0 manifold andthat run is removed. However, in the limit t → ∞, a system that does not emit a photon isnecessarily in the subradient state. Since the J = 0 state population does not change during theinteraction, the absence of any photodetection events should occur ntrj h0, 0| ρ̂ss |0, 0i times in ntrj

107

Figure 5.11: Density matrix after the detection phase for N = 2, Ω = Γc , and w = 0. The system
is initialed in the steady-state of the superradiant lasing model from Fig. 5.7 and evolved under
the detection phase master equation Eq. (5.32) for a total time tf = 10/Γc . The red areas indicate
highly populated states.

108
trajectories. This demonstrates why we first drive the system in the weak pumping limit with thesuperradiant lasing model to enhance the initial J = 0 value for our proposed detection scheme.

5.3

Outlook
We have therefore shown that by first enhancing the subradiant state population in the weak

pumping regime, a photodetection test can be used to isolate this J = 0 state in an experimentally viable number of iterations. We can now relate our proposed scheme to the idea of entropy
reduction as we are driving the system toward a pure state, up to the subradiant state degeneracy. Furthermore, we can create an analogue to evaporative cooling as the incoherent pumping in
the initial phase plays the role of atomic collisions, leading to compression towards lower J and
M values. Meanwhile, the detection phase is analogous to lowering the trap depth to allow hot
atoms to escape, as trajectories with one or more photodetection events are disregarded and only
trajectories in the J = 0 state are keep.
While these preliminary results suggest the J = 0 state may be isolated in an experimental
setting, what we have presented is just the tip of the iceberg. There are two possible avenues topursue with this project. One route is to present our work as the first detailed exploration of thesubradiant regime. We could, for instance, investigate how the slopes in Fig. 5.10 vary as we change
N . We are also studying an analytical solution for the steady-state J = 0 population in the limitw/(Γc /N ) → 0. The other approach is to present our work as an experimental proposal. In thiscase, future work will be dedicated mostly to the detection phase where we would optimize thetime for each trajectory. Furthermore, a proposal to test experimentally if the acquired trajectoryis truly in J = 0 would most likely be needed.
For both approaches, it is thought-provoking to try and enhance the subradiant state further
by including more incoherent processes. For example, keeping the incoherent pump on during
the detection phase would allow population to exit the J = 0 manifold. However, the fact thattransition rates towards lower J values are dominant in the weak pumping regime suggests thatcompression towards J = 0 might allow for a even larger subradiant state population. We can then109
look for entanglement [143] in the system as well as applications of the subradiant state in quantum
metrology [56].

110

Chapter 6

Conclusion

In Chapter 1, we listed three areas of study for the dynamics of cooling particle ensembles.
As a precursor to laser cooling, we presented a novel scheme in Chapter 3 to create nearly-coherent
laser slowing protocols utilizing adiabatic shortcuts. We were therefore able to create slowing
techniques with scattered photon numbers on the order of unity for a large amount of momentum
removal. We demonstrated in Chapter 4 that in the laser cooling stage, contrary to popular belief,
the laser fields are able to remove part of the entropy initially in the atomic system. In doing so,
we directly demonstrated the creation of a low entropy, near-pure atomic state during the coolingprocess. We created another low entropy state in Chapter 5, this time by driving the system in
the weak pumping limit of the steady-state superradiance model. Furthermore, we demonstrated
that the population of the subradiant state in greatly enhanced and may be separated from theremainder of the ensemble in an experimentally viable number of stochastic shots. We hope thatthe creation of many-spin dark states will allow for an improvement on quantum control during
quantum metrological investigations.

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Appendix A

Two-Level Hamiltonian Derivation

This appendix is dedicated to the derivation of the Hamiltonian of a two-level atom in aclassical electric field Eq. (2.9). It follows a similar format as [130, Chapter 5].

A.1

Atom-Field Interaction
If we assume a monochromatic field, with angular frequency ω, to model a laser, we see thatthe elctric field operator can be written as
Ê(t) = ε̂E0 cos (ωt) ,(A.1)

where ε̂ is the unit polarization vector of the field. Note we have made the diople approximation,
which assumes the wavelength of the field is much longer than the particle’s size so that we may
neglect the spatial dependence of the field. This is appropriate for most optical transitions thatare O(10−7 m) while the inter-atomic distances are on the scale of ångströms (Å = 10−10 m). It isconvenient to decompose the field into oppositely rotating components
Ê(t) = ε̂E0 −iωt(+)(−)
e+ eiωt = Ê (t) + Ê (t).
2

(A.2)

We then makes the two-level atom approximation where we consider ground |gi and excited state|ei, and assume that transitions to all other levels are negligible. If the energy difference between
the states is ∆E = Ee − Eg , the de Broglie relation gives us the tranisition frequency ∆E = ~ωa .
With the definition of zero energy to be halfway between the states, the atomic free-evolution

122
Hamiltonian is
ĤA =

~ωa
(|eihe| − |gihg|) .
2

(A.3)

The atom-field interaction Hamiltonian, under the dipole approximation, is [98, Chapter 3]
Ĥint = −dˆ · Ê,(A.4)

where dˆ is the atomic dipole operator.

A.2

Dipole Operator in the Energy Basis
Let us consider an interaction with a single electron, such that the dipole operator can be

written in terms of the electron’s position r̂ and charge q = −e asdˆ ≡ qr̂ = −er̂.

(A.5)

We then us a parity argument to get the dipole operator in a more convenient form.
We define the unitary parity operator asΠ̂ r̂ Π̂ † = −r̂,(A.6){Π̂, r̂} = 0 → ha| {Π̂, r̂} |bi = 0.

(A.7)

(πa + πb ) ha| r̂ |bi = 0,

(A.8)

so that Π̂ r̂ = −r̂ Π̂ . We can thus write

Using the energy basis, we see

for a, b ∈ {g, e} and parity eigenvalues πa and πb . Since the parity operation has the involution
property1 , these eigenvector must correspond to ±1 which corresponds to even and odd parity,
respectively. This suggests that either (πa + πb ) = 0 or ha| r̂ |bi = 0. For the diagonal elements, thelatter must be true as the eigenvalues are equal:hg| dˆ|gi = he| dˆ|ei = 0.
1Π̂ = Π̂ −1 → Π̂ 2 = Î

(A.9)

123
However, this need not be the case for the off diagonal terms provided that the states have opposite
parity and thus,hg| dˆ|ei = he| dˆ|gi∗ .

(A.10)

This expresses that the dipole operator may couple the ground and excited states, but it does notproduce any first-order energy shifts of the individual states. Applying the identity operator onˆ we findboth sides of d,dˆ = hg| dˆ|ei |gihe| + he| dˆ|gi |eihg| .

(A.11)Let us suppose, by choice of phase, that the dipole matrix elements are real, which allows us towrite the dipole operator asdˆ = hg| dˆ|ei (|gihe| + |eihg|) ,(A.12)

Ĥint = hg| dˆ|ei · Ê (|gihe| + |eihg|) = hg| dˆ|ei · Ê σ̂ − + σ̂ + ,(A.13)and thus,with lowering operator σ̂ − = |gihe|.

A.3

Rotating-Wave Approximation
We now can decompose the dipole operator into positive- and negative-rotating parts as the

expectation value of σ̂ − has time dependence of e−iωa t [130, Chapter 5]. We find

(+)
(−)
dˆ = hg| dˆ|ei σ̂ − + σ̂ + = dˆ + dˆ ,(+)

with dˆ(A.14)(−)∼ σ̂ − and dˆ∼ σ̂ + . Therefore, the atom-field interaction Hamiltonian becomes (+)
  (+)
(−)(−)
Ĥint = − dˆ + dˆ
· Ê
+ Ê
(+)
(−)
(+)
(−)
(+)
(−)
(−)
(+)
= −dˆ · Ê− dˆ · Ê− dˆ · Ê− dˆ · Êhi= − e−i(ω+ωa )t + ei(ω+ωa )t + e−i(ω−ωa )t + ei(ω−ωa )t .

(A.15)

Here, we find that the leading two terms oscillate rapidly, while the cross terms evolve slowly. This
prompts us to make the rotating-wave approximation (RWA):|ω − ωa |  ω + ωa .

(A.16)

124
This allows us to replace the fast oscillating terms by their average value. We are thus able to focus
on the slow dynamics of the system:
(+)
(−)
(−)
(+)
Ĥint = −dˆ · Ê− dˆ · Ê .

(A.17)As pointed out in [130], the RWA approximation and the two-level approximation are at the samelevel of accuracy as they both amount to disregarding interaction that are far off resonance. If thedetuning is large enough that the counter-rotating terms are not negligible, than the coupling toother energy levels is not negligible either.

A.4

Rabi Frequency
We can now write Eq. (A.17) out explicitly,


(−)
(+)
Ĥint = − hg| ε̂ · dˆ|ei E0 σ̂ − eiωt + E0 σ̂ + e−iωt ,(A.18)

which becomesĤint =

~Ω − iωtσ̂ e + σ̂ + e−iωt .
2(A.19)

Here, we have defined the Rabi frequency as
(+)

Ω=−
(+)

where we have assumed E0

2 hg| ε̂ · dˆ|ei E0~=−

hg| ε̂ · dˆ|ei E0,~(A.20)

is real. In the case of a linearly polarized field2 , this simplifies toΩ=−hg| dz |ei E0.
~(A.21)We see that the Rabi frequency characterizes the strength of the atom-field coupling.

A.5

Time-Dependent Detuning
We now generalize to a chirped field in which the frequency ω is now the rate at which thephase accumulates, e−iη(t) with ω = dη(t)dt . This leads to the replacements
 Z t

 Z t

iωt
0
0
−iωt
0
0
e → exp i
ω(t )dt ,e→ exp −iω(t )dt ,t0
2

We make the arbitrary choice of ε̂ = ẑ.

t0(A.22)

125
where ω(t0 ) is the laser’s instantaneous frequency. Further assuming a time-dependent Rabi frequency gives the final, desired form of the Hamiltonian [Eq. (2.9)]:Ĥ = ĤA + Ĥint =R~ωa~Ω(t) i t ω(t0 )dt0(|eihe| − |gihg|) +|gihe| e t0
+ H.c. .
2
2(A.23)

Appendix B

Interaction PictureThis appendix provides some basic background of the quantum interaction picture. I thenprovide two specific examples that occur commonly throughout the thesis. I refer the reader to [121,Chapter 5, §5] for a more detailed discussion on the interaction picture.

B.1

BackgroundWe consider a time-dependent Hamiltonian, which we can split into two separate part,Ĥtot = Ĥ0 + Ĥ

(B.1)

Let us denote a state vector in the Schrödinger picture as |iS . We now define a state vector |iI in
the so-called interaction picture, or Dirac picture, as
iĤ0 t|ψ(t)iI ≡ e ~ |ψ(t)iS ,

(B.2)

˜
so that an observable in the interaction picture Â is given byiĤ0 tiĤ0 t˜Â ≡ e ~ Âe− ~ ,(B.3)

where Â is an observable in the Schrödinger picture. Note that this is different than the third
picture in quantum mechanics, the Heisenberg picture, where one has time-independent state ket
and time-dependent observable defined by|ψiH ≡ eiĤtot t~|ψ(t)iS ,ÂH ≡ eiĤtot t~Âe−iĤtot t
~

,

(B.4)

127
Schrödinger pictureInteraction picture

Heisenberg pictureState ket

Evolution determined by Ĥtot

˜
Evolution determined by Ĥ

No change

Observable

No change

Evolution determined by Ĥ0

Evolution determined by Ĥtot

Table B.1: Source: Ref. [121]. Table of the three main pictures used to determine the time evolutionof a quantum system.

that satisfies the Heisenberg equation of motion:
i
dÂH
1 h
ÂH , Ĥtot .
=
dt
i~

(B.5)

Using Eqs. (B.2) and (B.3) along with the Schödinger equation Eq. (2.2), one finds theequations of motion in the interaction picture [121, Chapter 5, §5]:
iĤ0 t∂~i~∂t |ψ(t)iI = i~e|ψ(t)iS∂t
iĤ0 tiĤ0 t= −Ĥ0 e ~ |ψ(t)iS + e ~ Ĥ0 + Ĥ |ψ(t)iS=eiĤ0 t~Ĥe−iĤ0 t~eiĤ0 t~(B.6)|ψ(t)iS˜= Ĥ |ψ(t)iI ,which is a Schrödinger-like equation. Furthermore, using Eq. (B.3) it can be shown that the
Heisenberg-like equation,
˜
i
dÂ
1 h˜
=
Â, Ĥ0 ,
dt
i~

(B.7)

is satisfied with the evolution determined by Ĥ0 . The interaction picture is therefore a sort of intermediate picture that connects the Schrödinger and Heisenberg pictures, as displayed in Table B.1.

B.2

Laser Frame
Let us now consider an illustrative example. We begin with Eq. (2.9),
Ĥtot =

~ωa~Ω(t)  − i Rtt ω(t0 )dt0(|eihe| − |gihg|) +σ̂ e 0+ H.c. ,
2
2

(B.8)

128
and move into the frame rotating with the laser frequency,
Ĥ0 =

~ω(t)
(|eihe| − |gihg|) .
2

(B.9)

Here, we write −iĤ0 t
iĤ0 t˜Ĥ = e ~ Ĥtot − Ĥ0 e ~
=−

~Ω(t)
~δ(t)
(|eihe| − |gihg|) +22



iĤ0 te ~ σ̂ − e−iĤ0 t~eRi tt ω(t0 )dt00


+ H.c. ,(B.10)

where we have defined the detuning δ(t) = ω(t) − ωa .
We now calculate the lowering operator transformed into the laser frame,iĤ t

˜ − = e ~0 σ̂ − eσ̂−iĤ0 t~.

(B.11)Taking a time derivative, denoted by a dot, we seeiĤ t˜˙ − = iĤ0 σ̂
˜ − + e ~0 σ̂ −
σ̂
~

iĤ0
−
~

!
e

−iĤ0 t~,(B.12)

which we can, noting that Ĥ0 commutes with itself, write asii −iĤ0 t hiĤ0 tii  ˜− ˜− 
ih
˙−
−
−
˜
˜
Ĥ0 σ̂ − σ̂ Ĥ0 =
Ĥ0 , σ̂ =σ̂ =e ~ Ĥ0 , σ̂ e ~~~~(B.13)

Plugging in for Ĥ0 and σ̂ − , we findiĤ t−iĤ t˜˙ − = iω(t) e ~0 ([|eihe| , |gihe|] − [|gihg| , |gihe|]) e ~ 0σ̂2−iĤ0 tiω(t) iĤ0 t
˜−.
e ~ (−2 |gihe|) e ~ = −iω(t)σ̂=2This differential equation is solved by separation of variables,Z ˜− Z t˜−dσ̂dσ̂
= −iω(t)dt −→=−iω(t)dt0 ,
−
−
˜
˜
σ̂
σ̂
t0

(B.14)(B.15)

which yields
˜− =
ln σ̂

Z t

Rt

0

0

˜ = σ̂ − e−i t0 ω(t )dt ,−iω(t )dt −→ σ̂00(B.16)t0˜ − (t = 0) = σ̂ − . The same process gives its Hermitian conjugate,
where we have used σ̂
Rt

0

0

˜ + = σ̂ + ei t0 ω(t )dt .
σ̂(B.17)We plug these back into Eq. (B.10) and see that the laser phases now cancel out and thus, we arrive
at Eq. (2.11):~δ(t)~Ω(t) −˜Ĥ =(|gihg| − |eihe|) +σ̂ + σ̂ + ,2
2(B.18)

129

B.3

Particle Frame
Here, we present a general approach to move into a particle’s frame, i.e.,
Ĥ0 =

p̂2
.
2m

(B.19)

We consider how the momentum-shift operator transforms into this frame:


eikẑ

iĤ0 tI= e ~ eikẑ e−iĤ0 t~,(B.20)

which we expand via Taylor series1 as
"∞
#
∞


X (ikẑ)n −iĤ0 t X
iĤ0 t(ik)n iĤ0 t n −iĤ0 t
ikẑ
~
~
e
=
e ~ ẑ e ~ ,
e
=e
n!
n!
I
n=0

(B.21)

n=0

h
i
h
i
and noting exp iĤ0 t/~ exp −iĤ0 t/~ = Î, we can write


e

ikẑ


I

=

∞
X
(ik)n

e
n! |

n=0

iĤ0 t~ẑe−iĤ0 t~eiĤ0 t~−iĤ0 t
~

ẑe{z

...e

iĤ0 t
~

ẑe


n
∞
X
iĤ0 t−iĤ0 t
(ik)n
~
~
e
ẑe
.
}=
n!

−iĤ0 t~n=0n terms(B.22)
We can expand the term in the parenthesis using the Baker-Hausdorff lemma (see [121,Chapter 2, §3]):hi (iλ)2 h hiieiB̂λ Âe−iB̂λ =Â + iλ B̂, Â +B̂, B̂, Â
2!
n h
h
h
h
iii i
(iλ)
+ ··· +
B̂, B̂, B̂, . . . B̂, Â
... + ··· ,n!

(B.23)

with real parameter λ, so thatiĤ0 te ~ ẑe−iĤ0 t~= ẑ +

i
h
ii
it h
t2 h
Ĥ0 , ẑ − 2 Ĥ0 , Ĥ0 , ẑ + · · · .
~2~(B.24)

Focusing on the commutator, we expandhi

1  2
i~ ∂i~p̂
Ĥ0 , ẑ = −ẑ, p̂ = −p̂2 = −,
2m
2m ∂ p̂m(B.25)

where we have used [121, Chapter 2, §2][x̂i , f (pˆi )] = i~
1

ex =

P∞

n=0 x

n

/n!

∂f
.
∂ p̂i

(B.26)

130
We see from Eq. (B.25) thathhii
Ĥ0 , Ĥ0 , ẑ = 0,

(B.27)and therefore
iĤ0 te ~ ẑe−iĤ0 t~= ẑ +

p̂t.
m(B.28)

Plugging this into Eq. (B.24), we find the momentum-shift operator in the atomic frame:


e

ikẑ






 
∞
X
p̂t n
(ik)n
p̂t
ẑ +
.
=
= exp ik ẑ +
n!
m
m
I
n=0

(B.29)

Appendix C

Open Quantum Systems

This appendix is dedicated to study of a system’s interaction with its environment, which welabel as an external reservoir. Here, we are describing a coupling between a small system and alarge “bath” system, such as a continuum of modes of the electromagnetic field. Typically, however,
one is only concerned with the time evolution of the small system and only wishes to describe theeffects of the bath on the system. This results, under the Weisskopf-Wigner approximation (see [98,Chapter 15]), in an irreversible decay of the populations of non-stable states. Therefore, information
contained in the system can be lost to its environment which motivates the labeling of an “opensystem.” We first present the derivation of the Lindblad form of the quantum master equation.
Then, we introduce the “quantum jumps” picture which is both a convenient numerical tool, as
well as a picture that provides considerable insight into the way a system behaves during a single
experiment.

C.1

Quantum Master Equation in Lindblad Form
This section uses results from in [130, Chapter 4, §5] and follows its presentation. The goal

is find the equations of motion of the system alone, after taking a partial trace over the reservoir
states:ρ̂(t) ≡ ρ̂S (t) = TrB [ρ̂SB (t)].

(C.1)

132
We first use the density matrix of a mixed ensemble,
ρ̂(t) =

X

wj ψ (j)EDψ (j) ,

(C.2)

jalong with the Schrödinger equation Eq. (2.2) to derive the unitary time evolution of the density
matrix in the Schrödinger equation:
E D
i
E D
∂ ρ̂ X hψ (j) + ψ (j) ∂t ψ (j)=wj ∂t ψ (j)
i~
∂t
j

=

X

(C.3)


wj Ĥ ψ

(j)

ED

ψ

(j)

− ψ

(j)

ED

ψ

(j)

i
Ĥ = Ĥ, ρ̂ ,hjwhich is reminiscent of the Heisenberg equation of motion Eq. (B.5), but with the opposite sign.
For a typical quantum system, the density matrix that undergoes the unitary, reversible dynamics
is the composite density matrix of both the system and the environment,i∂ ρ̂SB
1 h=Ĥtot , ρ̂SB ,
∂t
i~

(C.4)

where S denotes the system, B denotes the bath, and Ĥtot is the total Hamiltonian:Ĥtot = ĤS + ĤB + ĤSB .

(C.5)

To rotate out the fast dynamics of the uncoupled subsystems, we move into an interaction
picture defined byĤ0 = ĤS + ĤB ,

(C.6)

i
∂ ρ̂˜SB
1 h˜=
ĤSB , ρ̂˜SB .
∂t
i~

(C.7)

so thatFormally integrating for a time ∆t,1ρ̂˜SB (t + ∆t) = ρ̂˜SB (t) +i~

Z t+∆t hi˜ĤSB (t0 ), ρ̂˜SB (t0 ) dt0 ,

(C.8)

t

and then using this equation in an iterative manner, we find∆ρ̂˜SB (t) = ρ̂˜SB (t + ∆t) − ρ̂˜SB (t)Z
Z t+∆t
Z t0
i
h
h
ii
1 t+∆t 0 h ˜
1
˜
˜
0 ˜
0
0
dt ĤSB (t ), ρ̂SB (t ) − 2dt=dt00 ĤSB (t0 ), ĤSB (t00 ), ρ̂˜SB (t00 ) .
i~ t
~ t
t
(C.9)

133
We now take a partial trace over the reservoir degrees of freedom to derive the time evolutionof the reduced density matrix,hi˜ ≡ ∆ρ̂˜S (t) = TrB ∆ρ̂˜SB (t) .
∆ρ̂(t)(C.10)

Furthermore, we make the Born approximation,˜ ⊗ ρ̂˜B ,ρ̂˜SB (t) ≈ ρ̂(t)(C.11)

where we have assumed a stationary state of the reservoir ρ̂˜B (t) = ρ̂˜B , so that
h
i
h
i
˜
˜
TrB ĤSB (t0 ), ρ̂˜B = 0 −→ TrB ĤSB (t0 ), ρ̂˜SB (t) = 0.

(C.12)

This approximation is a statement that the reservoir is large and that its coupling to the system is
weak [130, Chapter 4, §5], and Eq. (C.9) reduces to˜ ≈−1∆ρ̂(t)~2

Z t+∆t
dt
t

0

Z t0

h
h
ii
˜
˜
dt00 TrB ĤSB (t0 ), ĤSB (t00 ), ρ̂˜SB (t00 ) .

(C.13)t˜ and not its past
We now assume that the evolution of the density operator only depends on ρ̂(t)history. This is the Markov approximation,
˜ 00 ) = ρ̂(t),˜ρ̂(t(C.14)and the Born-Markov approximation can now be written as˜ ⊗ ρ̂˜B .
ρ̂˜SB (t00 ) ≈ ρ̂(t)(C.15)

One further approximation is that the interaction Hamiltonian can be written as a product ofsystem and reservoir operators,
X˜ ˜˜ĤSB = ~Ĉj R̂j ,(C.16)j˜˜where Ĉj and R̂j are operators in the system and reservoir subspaces, respectively, that need not˜
be Hermitian. When the operators on the right-hand side are non-Hermitian, such that ĤSB =˜˜
˜ ˜Ĉ R̂† + Ĉ † R̂, there can be exchange of quantized energy between the system and reservoir. This
thus represents dissipation from a system to a reservoir, such as spontaneous emission.

134
Using these approximations and changing variables, τ ≡ t0 − t00 , Eq. (C.13) becomesZ ∞ Z t+∆t h
i
i h
˜ ˜
˜
˜
˜ − Ĉ˜j (t0 − τ )ρ̂(t)˜ Ĉ˜i (t0 ) G R̂
˜ ≈−R̂∆ρ̂(t)dτ
dt0 Ĉi (t0 )Ĉj (t0 − τ )ρ̂(t)i j (τ )0t(C.17)
i

h
i h
˜ ˜
˜ Ĉ˜j (t0 − τ )Ĉ˜i (t0 ) − Ĉ˜i (t0 )ρ̂(t)˜ Ĉ˜j (t0 − τ ) G R̂
+ ρ̂(t)j R̂i (−τ ) ,where the reservoir correlation function is
h
i
h
i
˜ ˜
˜
˜
˜
˜
˜
˜
G R̂i R̂j (τ ) = TrB R̂i (t0 )R̂j (t0 − τ )ρ̂˜B = hR̂i (t0 )R̂j (t0 − τ )iB = hR̂i (τ )R̂j (0)iB ,(C.18)˜
and we have used Eq. (2.32). Assuming that ĤSB is Hermitian, we take the limit ∆t → 0 so that
this reduces to [121, Chapter 4, §5]
i Xh
iZ ∞
1 h
∂ ρ̂(t)˜˜††ĤS , ρ̂(t) +Ĉj ρ̂(t)Ĉj − Ĉj Ĉj ρ̂(t)=dτ e−iωj τ hR̂j† (τ )R̂j (0)iB
∂t
i~
0
j
h
iZ ∞

˜
˜
+ Ĉj ρ̂(t)Ĉj† − ρ̂(t)Ĉj† Ĉjdτ eiωj τ hR̂j† (0)R̂j (τ )iB(C.19)

0where we have transformed out of the interaction picture and assumed−iĤS tiĤS t˜Ĉj (t) = e ~ Ĉj e ~ = Ĉj eiωj t .

(C.20)

Defining
ωj+ =

Z ∞
0

˜
˜
dτ e−iωj τ hR̂j† (τ )R̂j (0)iB ,(C.21)

we now separate the density matrix time evolution into real and imaginary parts:

i X
h i
h ih
i
X
∂ ρ̂(t)
1 h
1 †
†
+
=
ĤS , ρ̂(t) +2 Re ωjĈj ρ̂(t)Ĉj − {Ĉj Ĉj , ρ̂(t)} − i
Im ωj+ Ĉj† Ĉj , ρ̂(t) .
∂t
i~
2
j

j

(C.22)The last term is an effective Hamiltonian term generated by the reservoir interaction, called thegeneralized Lamb shift, so that the Hamiltonian becomesĤ = ĤS + ~Xh i
Im ωj+ Ĉj† Ĉj .

(C.23)j

We now associate the non-commutator term with the irreversible dynamics of the system by defining
the Lindblad superoperator [82]:L̂(ρ̂) =XjΓj D̂[Ĉj ]ρ̂ =

X
j



1 †
†
Γj Ĉj ρ̂Ĉj − {Ĉj Ĉj , ρ̂} .
2(C.24)

135
The Ĉj are called the system’s jump operators and the decay rate of the j th jump operator is given
h i
by Γj ≡ 2 Re ωj+ . These effective decay channels, in general, cause non-unitary dynamics of thesystem, and we therefore have captured the system’s coupling to the reservoir in an effective masterequation of an open quantum system:
i
∂ ρ̂
1 h
Ĥ, ρ̂ + L̂(ρ̂).
=
∂t
i~

C.2

(C.25)

Monte Carlo Wave Functions
This section is dedicated to the method of Monte Carlo wave functions to study open quantum

systems. This involves unraveling the master equation into trajectories based on, for example,
stochastic simulation of spontaneously emitted photons. I follow the presentation in [98, Chapter
15, §4]. As hinted at in Chapters 4 and 5, there is a useful connection between the quantum jumps
picture presented here and the quantum theory of measurement.
We rewrite the quantum master equation Eq. (C.25) in terms of a non-Hermitian effective
Hamiltonian,Ĥeff ≡ Ĥ −i~ X †Ĉk Ĉk ,2(C.26)Ĉk ρ̂Ĉk† ,(C.27)kand a “jump” superoperator,
L̂jump (ρ̂) =Xkso that

∂ ρ̂
1 
†
=
Ĥeff ρ̂ − ρ̂Ĥeff+ L̂jump (ρ̂) .
∂ti~(C.28)

We now reintroduce a state vector description of the system often referred to as the quantum jumps
picture. However, the spontaneity of the dissipative effects of the system resulting from Eq. (C.27)
cause the evolution to no longer be Schödinger-like, nor deterministic. Instead, the method ofMonte Carlo wave functions evolves pure states of the system under Schrödinger-like evolution
governed by Eq. (C.26) with random quantum jumps, and then takes a statistical average overthem in the end. The quantum jumps picture is therefore intrinsically stochastic.

136
Plugging the general form of a mixed state’s density matrix,ρ̂ =X

wj |ψj ihψj | ,(C.29)

jinto the master equation in the form of Eq. (C.28) results in
"
#
 X
X
1 
†
wj (∂t |ψj i) hψj | + |ψj i (∂t hψj |) =Ĥeff |ψj ihψj | − |ψj ihψj | Ĥeff+
Ĉk |ψj ihψj | Ĉk† .
i~jk(C.30)We restict our discussion to a single state ket |ψi in the mixture and note that the first term on
the right-hand side is a the Schrödinger-like evolution with a non-Hermitian Hamiltonian:i~∂t |ψi = Ĥeff |ψi .

(C.31)

However, the second term results in a discontinuous evolution in which the state is incoherently
projected onto another state:|ψi → |Ψk i = Ĉk |ψi ,(C.32)

which motivates the labeling of a “jump” superoperator in the effective Liouvillian.
We now evolve Eq. (C.31) for a short time δt,Eψ̃(t + δt) =iĤeff1−~!
|ψ(t)i ,(C.33)

such that the state ket no longer remains normalized:!
!
†ED
iĤeffiĤeff1−|ψ(t)i = 1 − δP,ψ̃(t + δt) ψ̃(t + δt) = hψ(t)| 1 +~~(C.34)

where δP is the “missing norm,”
δP =

X
X
i
†
δt hψ(t)| Ĥeff − Ĥeff|ψ(t)i = δthψ(t)| Ĉk† Ĉk |ψ(t)i ≡
δPk .
~
k

(C.35)k

The full master equation Eq. (C.28) preserves the norm of the density matrix, so the missing norm
must be a result from the quantum jumps portion L̂jump that projects the system into state |Ψk iwith probability δPk .

137
We now must determine whether of not a jump occurred in time δt. This is achieved numerically by choosing a random number 0 ≤ r ≤ 1 and comparing its value to δP . If r > δP , we saythat no jump occurred, normalize the state vector|ψ(t + δt)i =Eψ̃(t + δt)
E ,ψ̃(t + δt)(C.36)and now proceed with the next integration step. However, if r ≤ δP , we say a jump occurred andproject the state vector onto the new normalized state|Ψk i|ψ(t + δt)i ==||Ψk i|

s

δt
|Ψk i
δPk

(C.37)

with probability δPk /δP . This state is then the initial condition for the next integration step.
This procedure is then repeated for many integration steps which yields a possible time
evolution of the initial state, as shown in Fig. C.1. This is what is meant by a quantum trajectory.
Each run results in a different result, so the trajectories are averaged over at the end of thesimulation. In the limit that the number of trajectories ntrj becomes large ntrj → ∞, we recover
the results that would occur under the evolution of the density matrix. The advantage of the Monte
Carlo wave function approach is in simulations that in involve a large number of states N , as thestate vector scale as N , while the density matrix scaling is N 2 .

138

Initiate Wavefunction
Select Random Number
Propagate until either its time
to store the wavefuntion or aquantum jump is detected?

Store time

Store Observables

Quantum jump
Apply the jump operator andselect new random number

FinishFigure C.1: Algorithm for implementing the Monte Carlo wave function formalism into our simulations of open quantum systems.