Quantum-Enhanced Measurements with Atoms in Cavities:
Superradiance and Spin Squeezingby
Kevin C. Cox
B.S., College of William and Mary, 2011
M.S., University of Colorado, 2014

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2016This thesis entitled:
Quantum-Enhanced Measurements with Atoms in Cavities: Superradiance and Spin Squeezing
written by Kevin C. Cox
has been approved for the Department of Physics

Prof. James K. Thompson

Prof. Jun Ye

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
Cox, Kevin C. (Ph.D., Department of Physics)
Quantum-Enhanced Measurements with Atoms in Cavities: Superradiance and Spin Squeezing
Thesis directed by Prof. James K. Thompson

Advances in engineering quantum systems are expected to lead to a new generation of quantum technology with fundamentally new capabilities and no classical analogue. Specifically, in thenear future, quantum entanglement may become useful for enhancing state-of-the-art atomic clocks
and sensors. I have performed experiments using laser-cooled rubidium atoms trapped in a highfinesse optical cavity to explore quantum and collective enhancements to precision measurements.
In this thesis, I will present a recent experiment to create record amounts of entanglementenhancement, or spin squeezing, in a proof-of-principle atomic sensor using entanglement-generatingcollective measurements. We have demonstrated up to a factor of 60 in directly observed spinsqueezing beyond the standard quantum limit for an unentangled quantum sensor and have demonstrated squeezing with real-time feedback to create deterministic entangled states. Second, I will
present a new method that has generated over a factor of 10 in homogeneous entanglement that
could be resolvable in free-space quantum sensors such as matter-wave interferometers and discussa new method to reduce errors in manipulating collective spin states using reversible dephasing.
These experiments and methods are directly applicable to some of the world’s best optical latticeclocks such as those housed here at JILA and NIST.
In addition, I have studied and demonstrated a proof-of-principle superradiant laser that relies
on collectively enhanced laser emission. These lasers have the potential to realize state-of-the-art
frequency purity useful for optical atomic clocks and long baseline interferometry. I will discuss
an experiment that demonstrates injection locking of a superradiant laser for the first time as well
as explores the collective synchronization behaviors in the system. This study of synchronization
informs research on current and future narrow linewidth superradiant lasers and may also provide
a platform for future studies of quantum phase transitions in open quantum systems.

Dedication

To Caitie, love beyond measure.

v

Acknowledgements

I have had a truly wonderful experience working in JILA. Working in the Thompson group has
been exciting, challenging, nurturing, friendly, professional, and fun. Thank you James for being
patient and kind in addition to being a world class physicist. It makes all the difference. Thankyou to all the great labmates I have had over the years: Zilong, Justin, Josh, Matt, Graham, andBaochen. There are few places where one can find such excellent people to work with. Thank you
Hans Green, Terry Brown and the other shop workers who are always willing to help students. I
believe you are a big part of what makes JILA work.
Thank you to Mom and Dad for all the love and support over the last five years.
Once again, specific thanks to James Thompson, Justin Bohnet, Josh Weiner, Matt Norcia, Graham Greve, and Baochen Wu, who have all been primary contributors to the work herein.

viContents

Chapter
1 Motivation and Introduction

1

1.1

Quantum capability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Quantum certainty and uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.3

Brief overview of this work

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.3.1

Entanglement-enhanced sensors . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.3.2

Superradiant Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6Format and goals of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.4

2 Experimental System Overview
2.1

Laser cooling and trapping

9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.1.1

Magneto-optical trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.1.2

Vapor pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.3

Polarization gradient cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.4

Optical lattice and cavity stabilization . . . . . . . . . . . . . . . . . . . . . . 11

2.2

Vacuum system and optical cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3

State preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Superradiant Injection Locking

18

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2

Experiment: superradiant injection locking . . . . . . . . . . . . . . . . . . . . . . . 19

vii
3.3

Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4

Van Der Pol description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.5

Optical Bloch equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.6

Derivation of small-signal gain

3.7

Bloch vector interpretation of phase diagram . . . . . . . . . . . . . . . . . . . . . . 32

3.8

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.7.1

Attractive regime: mapping to 2D Van der Pol oscillator . . . . . . . . . . . . 34

3.7.2

Repulsive regime: 3D dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Review: Spin Squeezing and Joint Measurements with Atoms in Cavities

38

4.1

Quantum sensors and the Bloch sphere . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2

Squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3

Joint measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.4

Quantifying entanglement through spin squeezing . . . . . . . . . . . . . . . . . . . . 44

4.5

Joint measurements of atoms in a cavity mode . . . . . . . . . . . . . . . . . . . . . 46

4.6

Signal to noise when probing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.7

Contrast loss from free-space scattering and optimal squeezing . . . . . . . . . . . . 49

5 Improved Bloch Vector Rotations with Reversible Dephasing

51

5.1

Experiment: dephased rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.2

Reduction of applied frequency and amplitude errors . . . . . . . . . . . . . . . . . . 61

5.3

Dephased rotation of quantum noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6 3rd Generation Spin Squeezing and Deterministic Squeezing with Feedback

67

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.2

Spin squeezing with real-time feedback . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.3

17.7 dB of conditional squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.4

Experimental details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

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6.4.1

Atomic state preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.4.2

Science cavity and lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.4.3

Relative frequency noise between cavity and probe . . . . . . . . . . . . . . . 79

6.4.4

Decoherence from the cavity probe . . . . . . . . . . . . . . . . . . . . . . . . 85

6.4.5

Atomic probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.5

Background contrast correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.6

Antisqueezing and area of the noise distribution . . . . . . . . . . . . . . . . . . . . . 96

6.7

Feedback implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.7.1

Experimental details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.7.2

Limitations to squeezing with feedback . . . . . . . . . . . . . . . . . . . . . . 99

7 Spatially Homogeneous Spin Squeezing for Atom Interferometry

101

7.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

7.2

Homogeneous squeezing in an effective optical dipole trap . . . . . . . . . . . . . . . 102

7.3

Coupling oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.4

Outlook for atom interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

7.5

Squeezing limits from time averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.5.2

Noise reduction limits with inhomogeneous coupling . . . . . . . . . . . . . . 113

7.5.3

Dephasing from inhomogeneous back-action . . . . . . . . . . . . . . . . . . . 121

7.5.4

Dephasing from imperfect time averaging . . . . . . . . . . . . . . . . . . . . 125

8 Conclusion and Future Outlook

127

8.1

Future outlook: Spin-squeezed optical lattice clocks . . . . . . . . . . . . . . . . . . . 128

8.2

Future outlook: squeezed atom interferometer . . . . . . . . . . . . . . . . . . . . . . 128

8.3

Future outlook: superradiant cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

8.4

Future outlook: Heisenberg limited states . . . . . . . . . . . . . . . . . . . . . . . . 130

8.5

Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

ix
Bibliography

132

Appendix
A Homodyne and Heterodyne Detection

145

B Feedback Loops

148

B.1 Feedback model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
B.2 Loop filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
B.3 Example: beatnote lock between 2 lasers . . . . . . . . . . . . . . . . . . . . . . . . . 150

xTablesTable
2.1

Relevant cavity parameters at the atomic and cavity probe laser wavelength λ =780 nm and at the lattice laser wavelength λ = 823 nm. The cavity’s mirror transmission coefficients, T1 on the probed end (1) and T2 on the closed end (2), areexpressed in terms of coupling rates κ1,2 = T1,2 × (free spectral range). . . . . . . . . 14

6.1

Relevant cavity parameters at the atomic and cavity probe laser wavelength λ =780 nm and at the lattice laser wavelength λ = 823 nm. The symmetric, standing wave cavity’s mirror transmission coefficients, T1 on the probed end (1) andT2 on the closed end (2), are expressed in terms of coupling rates κ1,2 = T1,2 ×(free spectral range). The atomic decay linewidth of |ei is Γ = 2π × 6.07 MHz. The
dressed cavity linewidths κ0 include broadening of the cavity resonance at ωc0 due tospontaneous scattering from the atoms. . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.2

Quantum efficiency summary table. Quantum efficiency losses come from sources ofsignal loss and added noise floors. Qturnon comes from finite laser turn-on times andringing-cancelling “kicks” (see Sec. 6.4.5.5) during which the probe is on but we do(0)

not collect information. The total quantum efficiency Q1 = 0.37(5) is the productof all the measured contributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
B.1 Ranges for setting for the JILA TJ011-A3 high speed loop filter. . . . . . . . . . . . 150

xi
B.2 Example settings for a beatnote lock. The additional parameters DC gain and leadspan are continuously adjusted using potentiometers. For this setup they were tuned
near their maximum values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

xii

FiguresFigure1.1

Simple experimental diagram. 105 to 106 87 Rb atoms (red) are trapped in a 1D
optical lattice, forming pancake-shaped layers of atoms. The atoms are cooled to 10
to 20 µK. An atomic transition is tuned near resonance with the mode of an opticalcavity (blue). Collective atomic excitations and cavity excitations (green photons)
are exchanged at rate Ω. By probing the optical cavity with a laser (green) we canperform precise measurements of the quantum noise of the atoms and project theensemble into entangled states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2

4

History of observed squeezing. Results from the Thompson lab are shown in brown[39, 40, 42, 31]. Experiments using collective measurements are shown as solid circles[68, 4, 142, 180, 164]. Experiments using other schemes with neutral atoms (nonlinear Hamiltonians from collisions or cavity feedback) are shown as open circles[62, 101, 95, 114, 28]. Experiments with ions are shown as open squares [111, 113,
92, 93, 24]. The largest observed squeezing in an optical field is shown as a cross[163].

2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6Level diagram for the D2 transition in 87 Rb. The F = 1 and F = 2 ground state
manifolds are metastable and the excited state lifetime is Γ = 2π × 6.06 MHz.

. . . 10

xiii
2.2

Experimental diagram for science cavity frequency stabilization. The main opticalcavity is stabilized to an optical lattice using Pound-Drever-Hall (PDH) spectroscopy
and feedback to a piezo. A lattice laser sideband created with a tunable EOM islocked to the transfer cavity that bridges the gap between the 795 nm and 823 nm
laser frequencies. The transfer cavity is locked using PDH spectroscopy and feedback
to a piezo to a 795 nm reference laser. The reference laser is stabilized to an atomic
transition in rubidium using FM saturated spectroscopy (FMS).

2.3. . . . . . . . . . . 12

Optical cavity CAD diagram. A Macor “hat” (red) is used to join the Zerodur cavity
spacer (light purple) and a piezo (white). The piezo and mirror (blue) were glued
together beforehand using a plastic (Delrin) alignment jig. Optical cavity parameters
are shown in Table 2.1.

2.4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Mirror heater calibration. The mirror surface temperature is plotted versus appliedcurrent in a vacuum test under the same configuration as the final science cavity.

2.5

. 15

Optical pumping scheme to stretched state. a) Two optical pumping beams copropagate at an angle from the vertical axis with polarization that decomposes~ b) Eninto σ+ and π along the quantization axis set by the magnetic field B.
ergy level diagram for optical pumping. Pumping channels are shown by red andblue lines for the F=1 and F=2 pumps respectively. Atoms accumulate in |↓i =|52 S1/2 , F = 2, MF = 1i.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

xiv
3.1

Experimental setup and level diagram. (a) Atoms interact with both the externallyapplied drive (grey) and the intra-cavity field generated by their collective emission(blue and red). The superradiant laser primarily responds at two frequencies, thedrive frequency ωd and a self-lasing frequency ω` . (b) The characteristic frequencies
are displayed in a level diagram, and all lie within one cavity mode of width κ.
The Raman laser system is approximated as a 2-level laser incoherently repumped
through intermediate optically excited states (not shown) at rate W . W is also
the primary source of broadening of the lasing transition (shown as broadening of|↓i). In this work, the ratio of atomic and optical linewidths is W/κ ≈ 5 × 10−2 to
5 × 10−3  1, placing the system deep into the bad-cavity or superradiant regime.
The state |↑i is a dressed state consisting of a ground hyperfine state of Rb coupled
non-resonantly to an optically excited state as described in [23, 18, 171, 21]. The
applied drive couples |↓i and |↑i with an on-resonance Rabi frequency Ωd . . . . . . . 20

3.2

The predicted phase diagram for the driven superradiant laser is shown in a plane
defined by the applied drive strength Ωd and the drive detuning δd , normalized to
the repumping rate W , which is fixed to N Cγ/2 here. The regions are first divided
by the number of distinct emission frequencies (1 or 2). Region (2) is further divided
by the frequency shift of the self-lasing component at ω` , which can be attracted(2A) or repelled (2R) from the applied drive frequency ωd . When Ωd < 0.2 × W , thelaser synchronizes by smoothly coalescing in frequency with the drive (dashed line).
For larger drives, the self-lasing component remains distinct and is quenched. The
two trajectories (black arrows) refer to the two parameter trajectories explored by
the data in Fig. 3.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

xv
3.3

Experimental observation of coalescing attractive (a) and repulsive (b) synchronizations. (Left) 2D spectrograms are taken with fixed drive strength Ωd as the detuningof the drive δd is varied along the representative vertical trajectories in Fig. 3.2.
Darker colors indicate higher power in a frequency bin (i.e. PSD). The red line indicates the expected self-lasing trajectory in the absence of an applied drive. (Right)
Two panels show the frequency shift δ` = ω` − ω`0 between the lasing frequency
and the lasing frequency when no drive is present. In each region, we qualitatively
identify attraction and repulsion by the sign of δ` and label and color each region similarly to the phase diagram in Fig. 3.2. The behaviors follow the prediction for theirrespective trajectories across the phase diagram, which are represented by verticallines in Fig. 3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4

Synchronization and gain saturation. (a) Measured gain and phase response of thesuperradiant laser at the drive frequency ωd . At large drive detunings, the response
displays linear small-signal gain (red fit to perturbative model overlayed). The gain
saturates (grey shaded region) at small detunings as the laser approaches the synchronization transition. (b) The same gain and phase response are represented in aphasor picture. Points of small signal gain lie along the straight line, and the regionof saturation is approximately described by a curve of maximum stimulated electricfield (inner curve). (c) The stimulated output powers Ps and P` , at the self-lasing
frequency ω` (red) and at the drive frequency ωd (blue) respectively, are displayed
as the laser is driven across a repulsive synchronization transition at approximatelyΩd /2π ≈ 40 kHz. Theoretical predictions (solid lines) show good agreement withthe data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

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3.5

Quantitative small-signal gain measurements. (a) We measure the total transmitted
power at the drive frequency ωd and define power gain G as the measured transmitted
power normalized to the drive power transmitted through the cavity on resonancewith no atoms present. For these measurements δd is scanned over a frequency range
greater than the cavity linewidth κ. The data is fit to the model in Eq. 3.7 (redline). Note that here, the cavity resonance marked by the vertical solid line is a few
MHz higher than the atomic resonance. (b) From fits to data such as (a), we plot the2variation of the fitted gain coefficient α (see text) versus W . The prediction α = γ⊥

is shown in grey. The width of the grey band corresponds to the uncertainty in an
independant calibration of the repumping rate W . . . . . . . . . . . . . . . . . . . . 31
3.6

Bloch vector interpretation of phase diagram. (a) The types of behavior for the driven
superradiant laser can be characterized by a phase diagram. The characteristic rates
that determine the lasing behavior are drive Rabi frequency Ωd , detuning δd , andrepumping rate W . The regions correspond to the number of distinct emission
frequencies (1 or 2) and the frequency shift (attraction or repulsion) of the carrier(A and R respectively). The behavior of the synchronization (a), attraction (b),and repulsion (c) configurations are shown in a Bloch sphere picture. In the frame~ d,of the atomic transition frequency ωa , the drive is represented by a rotation Ωwith an orientation which rotates along the dashed green trajectory at frequency δd .
In the unsynchronized case this modulates the Bloch vector (red vector), causing
drift toward or away from the drive, with the average precession of the Bloch vector
indicated in each case via the large blue and red arrows. In the synchronized case,the Bloch vector follows the drive all the way around the sphere. . . . . . . . . . . . 33

4.1

A 2-level atom’s quantum state is represented by a Bloch vector ~j that lives on thesurface of a Bloch sphere. The state can be parameterized by vector components jx ,jy , and jz or angles θ and φ.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

xvii
4.2

A collective Bloch sphere. The individual projection of each single-atom Bloch vector
leads to an uncertainty in the angular resolution of the Bloch vector of ∆θ = ∆φ =√1/ N for unentangled atoms, called the standard quantum limit. . . . . . . . . . . 40

4.3

A squeezed collective Bloch vector. Entanglement can reduce the quantum fluctuation in an ensemble of atoms or spins, leading to a sharper atomic clock hand.
The blue probability is squeezed since it has an angular resolution narrower than√
the standard quantum limit of ∆θSQL = 1/ N . The red distribution represents a
coherent spin state made up of identically prepared unentangled atoms. . . . . . . . 42

5.1

The reduced sensitivity of a dephased rotation to a small rotation error is graphically represented on collective Bloch spheres. A single representative Bloch vector
prepared in the x̂ − ẑ plane, along with the quantum uncertainty in its position, isshown (red arrow and noise distribution). Each sphere also has a series of coloredlines denoting the tips of Bloch vectors that are at a constant Jz in the initial configuration. The original rings of constant Jz are shown in parts (b-e) as thin black lines
for reference. A small rotation Rŷ (π/16) representing an error is applied without (b)and with (c,d,e) dephasing. By reversibly dephasing the Bloch vector to Cd = 0.14,
the impact of the rotation is greatly reduced. Rotation errors that would otherwise
dominate can be supressed well below the fundamental quantum noise. . . . . . . . . 53

5.2(a) The standing wave intensity of each beam is shown inside the cavity (blue mirrors). The atoms are trapped at antinodes of the 823 nm optical lattice (blue). The
probe laser at 780 nm (red dashed) and dephasing beam at 795 nm (green dotted)
cause dephasing due to their inhomogeneous light shifts. We detect the phase of theprobe light to infer N↑ . (b,c) The reduction in transverse coherence after dephasing Cd (red and green squares) and rephasing CF (black circles) is measured versus
the average number of transmitted photons from the probe beam (b) and dephasing
beam (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

xviii
5.3(a) The rms noise in the measured spin projection JzF , ∆JzF , after applying an
integer number of π-pulses is displayed for three different amounts of applied dephasing, quantified by Md . The contribution to ∆JzF due to finite measurement
resolution (i.e. ∆JzF at 0 π-pulses) is subtracted out. For Md = 0, a linear fit
extracts the rotation-added noise per pulse (red line). Predictions (green and blackbands) using Cd from Fig. 5.2 reasonably explain the reduction in rotation-added
noise with increased Md . All shaded regions represent 68% confidence intervals. (b)Dephased rotations are applied in a sequence designed to resolve spin populations
below QPN. N↑ is measured before and after a π-pulse with outcomes labeled N↑,p ,N↓,p , N↑,f and N↓,f to determine the spin noise reduction R. Both the 780 nm probe
and 795 nm dephasing beams are applied during each measurement of N↑ and N↓ .
(c) R is measured as a function of probe strength Mp for Md = 6.1(3) × 106 (bluedata and fit) and Md = 0 (red data and fit). All quantities are displayed in units ofdB relative to QPN, dBQPN . The fit to the measurement background Rbck is shownin black. (d) The rotation-added noise Rrot , shown in the inset, can be inferred fromthe data of part (c). Rrot with no dephasing is shown as a dashed line. Dephasing
can reduce Rrot by more than 21 dB.

. . . . . . . . . . . . . . . . . . . . . . . . . . 60

xix
5.4

Reduction in sensitivity to amplitude and frequency rotation errors. (a) The ensemble is subjected to a rotation with arbitrary amplitude ψ and equitorial rotation axisα̂ and zero detuning (measurement sequence shown below graph). The resulting Jzf(blue points) are plotted versus amplitude of the rotation ψ for three different valuesof dephasing, characterized by Md . At each amplitude the average magnitude of Jzf(red points) is compared to a prediction using the measured transverse coherence
from Fig. 2 (black line). (b) The ensemble is subjected to a rotation with aribitrary
detuning δ ∗ and azimuthal rotation axis α̂. The amplitude is constrained so that atzero detuning, the rotation is a π-pulse. Jzf (blue points) are plotted as a functionof δ ∗ , again for three different values of dephasing, and the average magnitude of Jzf(red points) is in agreement to a prediction from the measured transverse coherence(black line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.1(a) A coherent spin state’s spin-projection noise (pink distribution) is projected onto
a squeezed state by a measurement of Jz . The quantum state randomly collapses
within the original distribution, creating a conditionally squeezed state. The premeasurement’s outcome is then used to rotate the spin state’s polar angle to a desired target spin projection (black solid line) Jz = Jztar , creating a deterministicallysqueezed state. (b) The relevant 87 Rb energy levels (black) and cavity resonancefrequency ωc (blue). (c) Simplified experimental diagram. The cavity is probed inreflection. Homodyne detection of the probe is sampled by a microcontroller thatthen applies microwaves at 6.8 GHz to achieve the desired feedback rotation θf b tocreate the deterministically squeezed state in (a). See Section 6.4 for experimental
details.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

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6.2

(a) Measured cavity resonance frequency for a single trial versus time, subtracting
a constant 12 MHz frequency offset. (b) The time windows in which the probe isturned on (green) and the populations determined from each window. The fixedmicrowave rotations are shown in black with the feedback rotation shown in orange.
(c) The pre-measurements Jzp (left) and final measurements Jzf (right) of Jz areplotted versus trial number and accumulated into histograms. Five different Jz
states are targeted (five distinct colors on right) and reached with noise below QPN.
The maximum deterministic squeezing is S = −7.4(6) dB relative to the SQL. (d)Feedback reduces the noise distribution of the final measurement relative to the initialquantum noise in the pre-measurement. (e) If no feedback is applied the final andpre-measurement are strongly correlated (black), allowing for conditional squeezing(S = -10.3(6) dB) by using the differential quantity Jzf − Jzp (gold). The increasein noise from feedback is discussed in the Supplementary Material. . . . . . . . . . . 73

6.3(a) Experimental sequence for conditional spin squeezing, with labeling mirroringthat of Fig. 6.2a. (b) Squared contrast C 2 (blue), spin noise R (red), and spinsqueezing S (black) are plotted versus the average number of incident photons Mi ina single measurement window. The solid lines are fits, the blue band is the predicted
loss of contrast from free-space scattering, and the grey band indicates the totalsqueezing error bar. (c) The experimental sequence used to observe the back-action
spin-projection. (d) The measured spin noise R is plotted versus ψ with fit (purple).
(e) The reconstructed conditional probability distribution of the quantum state (red)on a Bloch sphere with Bloch (black) vector. The distribution is magnified with a 1:1
aspect ratio and plotted with the equivalent coherent spin state (blue) in the lower
panel. (f) Thermal radial motion of the atoms causes the spin noise R to oscillate at
twice the radial trap frequency as the time separation T between the pre- and final
measurements is increased.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

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6.4

Experimental frequency diagram. Relevant frequencies described in the text are
shown along with the locking scheme of the atomic (blue) and cavity (red) probe
lasers. The two longitudinal resonances of the cavity that these two lasers probe are
separated by 122 GHz and shown on the upper graph. The unshifted n+15th cavity
mode at ωc is detuned δc blue from the atomic resonance ωa . The presence of atoms in|↑i typically shift this cavity mode by approximately 70 MHz, to ωc0 . The homodyne
local oscillator beam is shown in purple (dashed), and feedback stabilization steps
are shown as gold arrows with descriptions. . . . . . . . . . . . . . . . . . . . . . . . 80

6.5

Optical block diagram. The resonance frequency of the optical cavity ωc0 is detectedusing homodyne detection of the atomic probe laser (red). Homodyne detection is
performed on an fs = 81.1 MHz sideband on the atomic probe laser. This sideband
can be applied at half power by the “kick” switch to provide an extra impulsive
kick to the atoms in order to cancel optomechanical ringing (described in Section
6.4.5.5). The carrier of the atomic probe laser is detected in heterodyne (RF port) toprovide a path length reference (see Fig. 6.6) for stabilizing the homodyne detection
phase. The cavity probe laser (blue) is P.D.H. locked, via the Lcav loop filter, toanother longitudinal mode of the optical cavity, unshifted by atoms, and provides
stabilization of the atomic probe laser’s frequency to the cavity frequency. The
atomic probe and cavity probe are separated optically via polarization. Real-time
feedback is applied using an Arduino microcontroller that controls the sign andduration of 6.8 GHz µ-wave pulses. More details are given in Fig. 6.6. . . . . . . . . 82

xxii
6.6

Electronic block diagram. The homodyne detection phase is stabilized by detecting
the carrier of the atomic probe beam with the signal appearing at 81.1 MHz at the
RF port. The phase of this signal is locked to a DDS frequency reference by applying
feedback through the Lstab loop filter to a VCO controlling the homodyne AOM. The
homodyne difference signal (DIFF) is used to stabilize the atomic probe laser to theatom-shifted cavity mode at ωc0 . The signal is high-pass filtered at 1 Hz to remove
slowly drifting DC offsets and then passed through a variable gain amplifier (usedto maintain constant loop gain as Mi is varied) before entering the loop filter LSHA .
The output of LSHA is used to control a VCO which provides a phase reference to aphase lock between the atomic probe laser and the cavity probe laser using loop filter
La . The cavity frequency ωc0 is detected by sampling the output of LSHA . When theatomic probe is off, a sample and hold circuit is used to hold the output of the loop
filter. A separate synthesizer (DDS) can be used to perform sweeps of the atomic
probe. Real-time feedback is applied by the Arduino based on the sampled output
of LSHA . The Arduino can control the sign of the feedback by switching (sign)
between two 6.8 GHz sources that are 180◦ out of phase.

6.7. . . . . . . . . . . . . . . 83(a) The measured power spectral density of instantaneous frequency fluctuationsSν (f ) between the atomic-probe and an empty cavity mode. The frequency stabilization described in the text reduces the noise by close to a factor of 50 over a broad
range relative to the Sν one expects for the linewidth of our free-running 200 kHz
FWHM external cavity diode lasers. For this data, the atomic and cavity-probes
were set to a high enough power that increasing either did not decrease Sν (f ), sothat we are sensitive only to technical noise floors. Also, heterodyne path-lengthstabilization had not yet been implemented, and this is largely responsible for therise below 2 kHz. (b) The integrated noise in the difference of two frequency measurement windows, plotted as a function of window length Tm , with a fixed t = 0 µs
window separation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

xxiii
6.8(a) The 2-window noise variance (∆fd )2 is plotted versus the detected power ofthe cavity-probe beam for Tm = 40 µs and t = 0. Above 1 µW, the noise variancesaturates to 27 dB below the quantum projection noise level. (b) The measured lightshift of the 87 Rb clock transition is plotted versus the detected power of the cavityprobe beam. The shift is approximately 660 Hz per µW. The non-zero light shift at
Pc = 0 is due to an additional constant light shift from the 823 nm optical lattice.
Due to differing Clebsch-Gordan coefficients, the shift of the |↓i to |↑i transition
frequency ω↑↓ is approximately 13 times larger, but still causes very little coherence
loss for the preliminary squeezing results presented here. . . . . . . . . . . . . . . . . 88

6.9

Probe induced oscillations partially cancelled by a staggered turn-on sequence. The
oscillations are fully present with no kick (red, 43 traces averaged) during a 40
µs measurement, but greatly reduced by a half-power 2.5 µs kick (blue, 30 traces
averaged). The 2.5 µs kick length corresponds to a quarter of the axial trap oscillation
period. There is an 80 MHz offset subtracted from the vertical axis. . . . . . . . . . 95

6.10 (a) The antisqueezing A is plotted versus Mi (black circles). The linear contribution
to the rise in A, A1 , is shown in blue and the quadratic contribution A2 in red. The
squeezing (gold diamonds and fit) is plotted on the right axis. (b) The area of thenoise distribution is calculated from the data in part (a) and plotted in purple. The
measured effective quantum efficiency Q̃1 is plotted in gold with an error bar shown(0)

as a gold band. At low Mi , Q̃1 is consistent with the prediction (green dash) Q1

from Table 6.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

xxiv
7.1

(a) Optical lattice sidebands separated by one free spectral range (FSR) are injected
into the cavity to create an axially homogeneous “dipole” trap. Dipole trap intensity(blue) and its envelope (red) plotted inside of the optical cavity, with exaggerated
wavelength λl ×103 . (b) The envelope of the residual lattice potential Vres (z) normalized to the peak lattice potential depth V0 is plotted near the cavity center, optimized
for a minimum at z = 0 (gold, β = 1.20) and for the minimal fraction of trapped
atoms determined experimentally (red, β = 1.32). (c) Fraction of atoms remaining
in the cavity mode (blue points) vs. fall time, fit to a model (red dash) described in
the text. Fluorescence images show the falling atom cloud at various times (inset).

7.2

104

(a) Projection noise scaling versus total atom number N , measured in the lattice(red points) including a theoretical prediction (red line) and in the dipole trap (bluepoints) including a fit to infer a coupling fraction ζ (blue line, with 68% confidence interval bands). Sequences are inset. Dashed boxes represent Bloch vector
rotations through a given angle using resonant microwaves. Solid boxes represent
cavity frequency measurements. (b) Quantum noise reduction in the dipole trap with
6.3(3)×105 atoms. A histogram of Jzf − Jzp (black data points) shows a standard
deviation 13.9(6) dB below projection noise ∆Jz,QP N = 397 atoms (gold line andshaded distribution). The measurement sequence is inset. . . . . . . . . . . . . . . . 106

7.3(a) Power spectra showing coupling oscillations for fall times of 1 ms (blue), 7.5 ms(red) and 15 ms (green) with their respective fits. (inset) Center frequency f0 of thefitted Boltzmann distribution for various fall times (points) compared to a freefall
prediction line of f0 = 2at/λp (line), see text for definitions. (b) Power spectrum
showing coupling oscillations at the trap frequency when atoms are trapped in the
optical lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

xxv
7.4

Sensitivity of atoms at different velocities to photon shot noise. A moving atom
couples to the probe mode with a transfer function (blue) with sensitivity at DC
and at a frequency fi corresponding to its velocity, shown for an atom with velocity
5 cm/s and 15 cm/s. Stationary atoms only couple at DC (red). The distribution ofoscillation frequencies is given by the Boltzmann distribution P (fi ) (black). Atoms
at different frequencies sample photon shot noise (PSN, purple dash) at different
frequencies leading to dephasing that can limit squeezing with time averaging. . . . . 124

8.1a) Momentum transfer diagram for 2-photon Bragg pulses. Two laser beams (red),
detuned by the frequency corresponding to 2 quanta of photon momentum change areused to make transitions along the kinetic energy curve of the atom (blue) b) Timing
diagram for atom interferometry in a cavity. Atoms’ internal states are manipulated
by microwave pulses (black and grey) and state-dependent momentum transfers areachieved using 2-photon Bragg pulses (red and grey). The spin-momentum entanglement is used to separate the wave function into different spatial paths leading
to a differential phase φ, that is subsequently mapped onto the final wave function
and read out by a final Ramsey π/2 pulse. Squeezing is generated by the collective
premeasurement of Jz shown with a blue and grey pulse. . . . . . . . . . . . . . . . . 129

A.1 Homodyne detection. A signal field Es (t) (blue) and strong local oscillator field
ELO (t) (red) are overlapped on a 50/50 beamsplitter and measured on a detector
with sensitivity S [Amps/Watt]. The signals are then subtracted and sent througha transimpedance amplifier with gain Rt [Ohms]. The output voltage, Vout can beused either as a measurement of the signal amplitude or signal phase, depending on
the phase φ between the signal and LO. . . . . . . . . . . . . . . . . . . . . . . . . . 146

xxvi
B.1 Simple system with feedback. In order to stabilize a noisy input voltage Vin∗ (t), one
can compare the voltage to a set point, Vset and apply feedback by multiplying thedifference by large gain L∗ (ω).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

B.2 Transfer function parameters of JILA high speed loop filters. The gain profile L∗ (ω)
is designed to have high gain at low frequencies and then can be optimized to achieve
the highest feedback bandwidth on the system without losing phase margin.

. . . . 151

B.3 Power spectrum of interference between two beatnote-locked lasers as normallyviewed on a spectrum analyzer and taking 100 averages. The majority of the laser
power resides in a phased-locked carrier that appears as a delta function in the
power spectrum. The unity gain bandwidth is indicated by the shoulders of the
distribitution, measured with markers to be 1 MHz away from the carrier.

. . . . . 152

Chapter 1

Motivation and Introduction

1.1

Quantum capability
We already live in a world full of quantum technology. The development of quantum physics

has lead to a rich and detailed understanding of atoms, solids, light, electricity, and magnetism:
quantum building blocks that are essential for many modern devices. Computers, lasers, andadvanced materials, to name a few, would not exist without modern quantum understanding.
On the other hand, even without modern quantum physics, the majority of modern technologies could, in principle, be realized through strictly “classical” means. For example, consider
a transistor, the basic element of modern computers. Transistors are quantum devices made ofjoined semiconducting materials that display non-linear conduction versus applied voltage due tothe structure of the available quantum energy levels in the solid. Despite this, transistors can beviewed simply as switches or amplifiers. Their relevant logical behavior can be achieved using components that are adequately explained without quantum mechanics. A fully “classical” computer,for example, that can calculate difficult polynomials was suggested by Charles Babbage in the
1800’s and has since been built with Lego children’s toys [37], a sufficiently classical infrastructure.
Quantum technologies with no classical analogue are difficult to find. This fact leads to an
important broad question that motivates research in every sub-field of quantum science: What are
the intrinsic capabilities of quantum systems? Researchers in quantum computing, for example,
study which types of problems quantum computers can solve efficiently [126]. A large body ofresearch is also aimed at understanding errors that occur in quantum objects or information and2
how to control and correct them [153]. The Nobel prize was even awarded this year for the studyof exotic phases exhibited by quantum matter, how they behave, and how they could be useful oneday.
The question of quantum capability also forms a central theme for this thesis. Specifically, I
will explore how quantum physics both limits and improves precision measurements.

1.2

Quantum certainty and uncertainty
Perhaps the most powerful, yet often under appreciated feature of quantum systems is theircertainty or “identicality”. Two 87 Rb atoms (the atoms used in this thesis) are identical. One canbuild two atomic clocks on opposite sides of the world and be certain that, given an ideal apparatus,
they would tick at exactly the same rate. Atoms are so identical that when one considers a quantum
wave function describing their behavior, the wave function fundamentally must have very specificsymmetry when the particles are exchanged with one another.
Quantum identicality is the primary power of atomic clocks and quantum sensors that performthe most precise absolute measurements in the world [115, 66, 162]. I would argue that such devices
are indeed a truly useful quantum technology with no classical analogue because their unmatchedmeasurement precision is a direct outcome of quantum certainty. Similarly, in Section 1.3.2 andChapter 3 of this thesis I will describe experiments to demonstrate a proof-of-principle superradiantlaser that relies on the quantum identicality of the atomic gain medium to achieve a high degree of
frequency purity. Superradiant lasers may one day be able to surpass the frequency purity of thecurrent state-of-the-art lasers based on manmade reference cavities.
Quantum identicality comes with a price. For any quantum sensor, random collapse of theatom’s, ion’s, or photon’s wave-function leads to uncertainty in the final readout of that sensor. Thismeans that many identical quantum particles must be used to average down quantum uncertainty.
Quantum noise is the “yang” to the identicality “yin”.
Quantum sensors based on 2-level qubits work much like flipping coins. By flipping N coins,
one can measure the error E in the coins’ probability E = PH − PT where PH is the probability

3
for the coin to land on heads and PT is the probability for the coin to land on tails. I assume thatthe coins are nearly fair (E ≈ 0), but small biases could be detected around E = 0. Flipping asingle coin would be a poor measurement of E. But by flipping many coins, an estimate of E, that
I label Ẽ, can be obtained using the equation Ẽ = (NH − NT )/N where NH is the number of coinsthat landed in heads and NT is the number that landed on tails. The error in this estimate can√be shown to be, ∆Ẽ = 1/ N where ∆X refers to the standard deviation in the quantity X. This
coin-flip noise is a classical analogue of the quantum noise that plagues quantum sensors.
For real atoms or qubits with two quantum states labeled |↑i and |↓i, quantum collapse
into |↑i or |↓i leads to uncertainty in measuring the quantum phases θ and φ in the superposition|ψi = cos(θ/2) |↑i + sin(θ/2)eiφ |↓i . A quantum sensor operates by biasing the probability in theangles θ or φ based on a measured quantity such as time, gravity, or electric or magnetic fields.
For N unentangled atoms identically prepared in state |ψi, the random collapse of each atom
causes an uncertainty in an estimate of θ and φ known as the standard quantum limit (SQL)√∆θSQL = ∆φSQL = 1/ N .
What would be required to to surpass the SQL in an atomic sensor? To gain insight forthis process, the coin analogy still carries some insight. First, when flipping N coins with E ≈ 0,
the randomness of each coin cannot be avoided. Second, one cannot predetermine the outcomeof any individual coin, because if you could, there would be no way to be sensitive to changes in
E. From these two assumptions, the only way to improve the measurement of E is to engineer
a non-local connection between the coins leading to correlation. If the true value that we wish
to estimate is E = 0, then when one coin noisily lands on heads, another coin must, through theconnection between the coins, know to land on tails. 1 The ability for quantum systems to display
such nonlocal connection was pointed out in the famous EPR paradox [46]. Today such quantum
connections, which are a manifestations of a non-factorable wavefunction, are generally referred toas entanglement. Most of this thesis deals with creating large amounts of quantum entanglement
1

A reduction in ∆Ẽ could also be explained by the coins communicating a “hidden variable”, but in quantum
mechanics, theories of this type can be ruled out by Bell tests [63, 65, 51, 143]

4
to surpass the SQL in atomic sensors.

1.3

Brief overview of this work
A simplified diagram of our experimental system is shown in Fig. 1.1. We trap and coola cloud of 87 Rb atoms (red) to a temperature of 10 µK inside an optical cavity (blue) using the
now-standard techniques of laser cooling and trapping. The cavity resonance frequency is tuned
near an atomic transitions in 87 Rb, between the qubit state |↑i and an optically excited state |ei.
In this configuration, the optical cavity has a strong rate of coupling to atoms in the excited state|ei, so that the ensemble and cavity exchange excitations (between atomic excitations and cavity
photons) at a rate of up to Ω = 2π × 6 × 108 Hz where Ω is often called the collective vacuum Rabi
frequency. In this configuration, the cavity resonance frequency is highly sensitive to the qubit
state of the atoms. Atoms in |↑i affect the cavity strongly, and atoms in |↓i do not. By measuringchanges in the resonant frequency of the cavity we can infer the total number of atoms in |↑i, andtherefore the superposition angle θ, very precisely.

Figure 1.1: Simple experimental diagram. 105 to 106 87 Rb atoms (red) are trapped in a 1D opticallattice, forming pancake-shaped layers of atoms. The atoms are cooled to 10 to 20 µK. An atomic
transition is tuned near resonance with the mode of an optical cavity (blue). Collective atomic
excitations and cavity excitations (green photons) are exchanged at rate Ω. By probing the opticalcavity with a laser (green) we can perform precise measurements of the quantum noise of the atoms
and project the ensemble into entangled states.

5
1.3.1

Entanglement-enhanced sensors

The majority of this thesis describes experiments to create entanglement enhancements in theatomic ensemble useful for precision measurements. By making a measurement of cavity resonancefrequency we measure θ below the SQL while maintaining the quantum superposition of the atoms.
By doing this, we essentially use the measured value of quantum noise in θ to subtract from
subsequent measurements, allowing differential sensing in an atomic clock or sensor below the
SQL. Equivalently, we can say that the first measurement projects the atoms into an entangledstate that is characterized by improved resolution in the quantum angle θ. We have shown directly
observed improvements in the variance in θ of up to a factor of 60 times lower than the SQL.
Although the concept of entanglement enhancements in sensors has been around for over 15
years (spin squeezing was first demonstrated in ions before the year 2000 [137, 159]), significant
entanglement enhancements have been previously elusive. Figure 1.2 shows a historical perspectiveof directly observed entanglement enhancements, defined by S ≡ (∆θ/∆θSQL )2 , usually called
spin squeezing. For this data, noise subtraction was not allowed in order to highlight the actual
observed sensitivity improvement relative to the SQL. The value S = 1 indicates no entanglement.
Along with another very recent experiment [68], the enhancement observed in this thesis S ≡(∆θ/∆θSQL )2 = 59(8) is by far the largest entanglement enhancement observed to date in any
system. This amount of squeezing, if implemented in a quantum-noise limited atomic clock or othersensor, would nominally mean that a particular quantum limited resolution could be achieved witha factor of 60 fewer atoms, or in a factor of 60 less time. Such a large enhancement establishes
entanglement as a practical way to improve atomic clocks and other sensors. In addition to thegeneration of large amounts of spin squeezing, I will present, for the first time, a demonstrationof spin squeezing with real-time feedback to a deterministic entangled state, where the result of
the squeezing measurement is no longer necessary to observe entanglement. Using this real-time
feedback we created one of the largest amounts of deterministic squeezing ever observed, comparable
to the amounts generated by the best naturally deterministic schemes such as 1-axis twisting using

6
cavity feedback or collisions in an ultra-cold gas [95, 62].
Last, I will present another experiment to observe over a factor of 10 in squeezing that additionally has a homogeneous character. This squeezing, while still one of the largest enhancements
ever observed, could also be used in free space measurements such as matter wave interferometers
used for tests of gravity [135], Lorentz invariance [141], inertial sensing [9], probes of new physics[131, 61], and potentially as gravitational wave detectors [54].

Figure 1.2: History of observed squeezing. Results from the Thompson lab are shown in brown[39, 40, 42, 31]. Experiments using collective measurements are shown as solid circles [68, 4, 142,
180, 164]. Experiments using other schemes with neutral atoms (non-linear Hamiltonians fromcollisions or cavity feedback) are shown as open circles [62, 101, 95, 114, 28]. Experiments withions are shown as open squares [111, 113, 92, 93, 24]. The largest observed squeezing in an opticalfield is shown as a cross [163].

1.3.2

Superradiant Lasers

As well as creating a proof-of-principle entanglement-enhanced sensor, I have also worked
on experiments to demonstrate cold-atom based superradiant lasers. Lasers are the best rulers forboth space and time. Lasers have not only become ubiquitous in everyday life but are also used for7some of the most precise measurements in the world, including gravitational wave interferometers[1] and optical atomic clocks [115].
The most frequency-precise laser beams are formed by actively stabilizing the frequency ofa laser to an optical reference cavity [77]. The primary limit to these classical, manmade lengthreferences has been thermal, Brownian motion of particles on the mirror surface that create fluctuations in the cavity’s length and therefore impose fluctuations on the laser’s frequency. Engineering
more stable reference cavities remains a major effort to improve atomic clock performance [77].
As previously mentioned in Section 1.2, the motivation behind cold-atom superradiant lasers
is to replace these manmade length/frequency references with nature’s identical quantum frequency
references, atoms. This can be accomplished by building a laser where the gain medium is alaser-cooled ensemble of atoms with a narrow optical transition such as strontium or ytterbium[108, 122, 120]. Such a laser would operate in the bad-cavity, superradiant regime where the atomic
coherence times are much longer than the cavity decay time [83]. In this regime, atoms form thephase flywheel for the laser and emit collectively into the cavity mode, leading to a superradiantenhancement in the emitted power.
Superradiant lasers were suggested as a powerful type of frequency reference by Meiser et
al. in Ref. [108] and a proof of principle Raman laser was first demonstrated in the Thompson
lab soon before I arrived as a first year graduate student [23]. We have continued to study theproperties and relevant physics behind superradiant lasing over the subsequent years [21, 171, 22],and due to the success of this work, a new Strontium superradiance project has begun, aimed atrealizing a narrow optical superradiant laser [122, 120].
In Chapter 3, I will describe an experiment observing the synchronization physics of a superradiant laser by injection locking the laser to an external drive and discussing synchronization
behavior of the superradiant ensemble. I derive a phase diagram for the superradiant laser anddemonstrate similarities and differences between the superradiant oscillator and standard classicalmodels of synchronization. By studying superradiant synchronization, we develop a framework
for understanding the sensitivity of future narrow-linewidth superradiant lasers to external fields.

8Additionally, it is common in atomic systems to have multiple possible lasing transitions. By
studying the interaction of these different lasing transitions, this work has informed and lead torecent experiments to interfere multiple lasing transitions to observe the fundamental linewidth of
the superradiant laser [122]. Lastly, the synchronization threshold is an example of a second orderphase transition in an open quantum system, and may be useful in the future for studies of phasetransitions in open quantum systems with similarities to recent experiments on the Dicke model[10, 6].

1.4

Format and goals of this thesis
Chapter 2 will be devoted to an overview of various experimental techniques used in this work,

and Chapter 3 will discuss experiments related to superradiant synchronization. I will give a basic
review of the relevant physics for describing and creating entanglement enhanced, spin squeezed
states in Chapter 4, and I will review an experiment to use reversible dephasing to improve coherent
rotations of the atomic state in Chapter 5. Then, I will focus on the current largest entanglement
enhancement observed in our lab, based on the third generation squeezing experiment in Chapter
6 as well as present a demonstration of squeezing with real-time feedback to create deterministic
entangled states. In Chapter 7, I will discuss a recent experiment that used a time averaging
scheme to observe a large entanglement enhancement in a homogeneous configuration that would
be appropriate for matter wave interferometers.
All of the experiments in this thesis have been (or are in the process of being) published injournal articles. Since much effort was given to clearly describe the experiments in those documents,
the bulk of each chapter will rely heavily on these previous writings. Apart from this introduction,
I have spent the majority of my original writing focus on experimental details that have not beenpreviously published. These can be found in Chapters 2, 4, and the appendices. I hope that inaddition to giving a complete collection of my thesis work and compiling many relevant results,
some of these additional details will be useful as a detailed resource for readers wishing to utilize
our techniques in the future.

Chapter 2

Experimental System Overview

Many of the experimental details for this work can be found in previous writings [32, 38,
39, 170, 20, 30]. However, I will provide a brief overview of the experimental apparatus here.
In addition to the general information in this chapter, additional experimental details that are
more relevant or specific to a particular experiment can be found in the chapter describing thatexperiment. Additional information about homodyne and heterodyne detection as well as the useof feedback loops in our lab can be found in the Appendices.

2.1

Laser cooling and trapping
The first step in the experiment is to laser cool and trap a large ensemble of up to 106 87 Rb

atoms inside the TEM00 mode of an optical cavity. The relevant energy level structure is shown in
Figure 2.1. Cooling and trapping is performed in two stages. First the atoms are cooled an trappedin a magneto-optical trap (MOT). Second, the atoms are transferred from the MOT into an opticallattice formed by a resonant cavity TEM00 mode and cooled to a final temperature of 10 to 20 µK.

2.1.1

Magneto-optical trap

MOTs are a very robust technology that have become a standard tool in atomic physics [110].
Our MOT is described in detail in refs [20, 30], and relies on 6 laser beams that provide Doppler
cooling in every direction as well as repumping from the F=1 ground states where the cooling is
not active. A quadrupole magnetic field with gradient of approximately 10 G/cm works in concert

10

Figure 2.1: Level diagram for the D2 transition in 87 Rb. The F = 1 and F = 2 ground state
manifolds are metastable and the excited state lifetime is Γ = 2π × 6.06 MHz.

with these Doppler cooling beams to form a trapping potential. In our experiment, each MOT laser
beam has a characteristic diameter of approximately 1 cm and a power of 10 mW. The MOT beams
are red-detuned by around 3 MHz from the F = 2 to F 0 = 3 transition with circular polarization.
The repumper laser operates on resonance with D2, F = 1 → F 0 = 2 transition with a total powerof 6 mW. For simplicity, we overlap the MOT and repumper beams before splitting into the 6orthogonal beams although this is not critical. Atoms in the MOT are cooled near the Doppler
temperature, set by the linewidth Γ = 2π × 6.06 MHz of the optical transition, of TD = ~Γ/(2kb )where kb is Boltzmann’s constant. For Rubidium this is 145 µK.

2.1.2

Vapor pressure

The MOT is loaded directly from a background vapor. The background Rb pressure isobserved to be the limiting contribution to the vacuum pressure and is measured using an ion
gauge to be approximately 4 × 10− 9 Torr. As a useful note, the background vapor pressure canalso be approximately measured from the MOT loading time [5], that is of order a second for oursystem. The vapor pressure is created by heating a Rb ampule to between 50 and 100 degrees
C. Recently, we observed that the small aperture in the ampule that allows Rb to escape can get

11plugged with an unknown layer, presumed to be rubidium oxide. When this happens, Rb will not
be sufficiently released into the main chamber. To avoid this, care should be taken with theseobjects not to expose the Rb to any air.

2.1.3

Polarization gradient cooling

Sub-doppler cooling occurs very naturally in laser cooling schemes with Rb. In fact, the effect
was a mystery for several years leading to a Nobel prize award for the observation and theoreticalexplanation of sub-Doppler cooling mechanisms [43]. In our experiment, we use the same circularpolarized MOT cooling beams to achieve the polarization gradient cooling (PGC). This coolingworks best with beams detuned from the optical transition. PGC cooling also requires a zeromagnetic field so that Zeeman sublevels are degenerate. This is achieved with bias coils in all three
dimensions that can zero the earth’s magnetic fields as well as any other biases in the system. In
the PGC cooling step, the MOT quadrupole magnetic field is shut off, and atoms are PGC cooled
and loaded into the optical trap for around 20 to 50 ms before they have a chance to fall away fromthe cavity region. After this step, their temperature is measured to be 10 to 20 µK.

2.1.4

Optical lattice and cavity stabilization

The final optical trap of the atoms is created by a cavity standing wave at 823 nm, that werefer to as the optical lattice. This lattice forms an attractive potential for the atoms due to theoptical dipole force [110]. The lattice typically has 0.3 W of circulating power and forms a trapwith a depth of 115 µK, mode waist of 71 µK and Rayleigh length of 2.05(5) cm. The radial andaxial trap frequencies are 1 kHz and 180 kHz respectively. Atoms load the central 1 to 2 mm of
the Lattice, and once they are loaded, the ensemble has a rms thermal displacement in the radial
direction of approximately 7 µm and a thermal rms displacement in the axial direction of 40 nm
at each lattice cite. We load between 105 and 106 atoms, which means there are 100 to 1000 atoms
per site.
Fig. 2.2 shows a simplified experimental diagram of the science cavity frequency stabilization

12

Figure 2.2: Experimental diagram for science cavity frequency stabilization. The main opticalcavity is stabilized to an optical lattice using Pound-Drever-Hall (PDH) spectroscopy and feedback
to a piezo. A lattice laser sideband created with a tunable EOM is locked to the transfer cavity
that bridges the gap between the 795 nm and 823 nm laser frequencies. The transfer cavity islocked using PDH spectroscopy and feedback to a piezo to a 795 nm reference laser. The reference
laser is stabilized to an atomic transition in rubidium using FM saturated spectroscopy (FMS).

chain. First a 795 nm reference laser (red) is stabilized to a Rb vapor cell using FM saturated
spectroscopy [59] and feedback to the laser’s current and piezo. An optical transfer cavity is thenstabilized to this laser via the Pound-Drever-Hall (PDH) stabilization method [16, 45] and feedback
to a cavity piezo. This cavity serves as an effective “gear” to transfer stabilization to the 823 nm
optical lattice. Specifically, phase modulation sidebands are applied to the lattice laser and one ofthose sidebands is stabilized to the transfer cavity using PDH spectroscopy and feedback. Afterthese three steps, the optical lattice frequency is set. Then, the science cavity is PDH-locked to thelattice laser giving absolute frequency stability to the optical cavity mode. Feedback to the cavity
piezos occurs with a bandwidth of approximately 1 kHz limited by resonances in the piezos. As a

13note, since the transfer cavity length is not known precisely enough to give an absolute frequency
conversion between the 795 nm reference laser and the 823 nm lattice laser, we usually probe thescience cavity with a “probe” laser beam with well defined frequency relative to the Rb transition,and then empirically adjust the lattice laser frequency by changing the EOM sideband frequency
in Fig. 2.2.

2.2

Vacuum system and optical cavity
The central element to the experiment is the atom cavity system under ultra-high vacuum. A

SolidWorks CAD drawing of the cavity is shown below in Fig. 2.3. The cavity was constructed bygluing piezos (Piezomechanik HPCh 150/12-6/2) and mirrors (custom from Advanced Thin Films)
to a ZeroDur spacer machined in the JILA machine shop. In order to assemble the system, themirrors and piezos were first glued together on a clean-room workbench using a Delrin plastic jig
that controlled the placement. Meanwhile, an additional “hat” constructed of Macor was used tojoin the piezos and the cavity spacer, and was secured with additional drops of Torrseal. Vacuumcompatible Kapton-dipped wires were glued to the piezo electrodes with low-outgassing silver epoxy(Epoxy Technology H21D). Additionally, before gluing the mirror-piezo combination to the spacer,
the mirror was wrapped with 5 turns of 140 µm diameter Kapton wire purchased from Kurt J.
Lesker and secured with thermal epoxy (Epoxy Technology H77), to allow for heating the mirror to
expel Rb that can stick to the surface. A plot of the mirror temperature, as measured in vacuum,
versus current through the heater wires is shown below in Fig. 2.4.
The measured cavity parameters are shown below in Table 2.1.
One important note is that the cavity linewidth is measured to be 3.15 MHz versus an
expected linewidth from construction of approximately 2.8 MHz. During curing, we observedthat some epoxy outgassed onto the mirror surface and initially raised the cavity linewidth toapproximately 6 MHz. We were able to clean the mirrors and achieve the observed linewidth using
the plasma cleaner in the JILA clean room.
The optical cavity is mounted to the vacuum chamber using a vibration isolation mount that

14

Cavity Parameters (probe λ = 780 nm)
2Single-atom cooperativity C = 4g
κΓ

0.044(6)Input coupling κ12π × 2.60(5) MHz
Output coupling κ22π × 0.17(1) MHz
Internal losses κL
2π × 0.38(8) MHz
Linewidth κ
2π × 3.15(10) MHz
Q.E. due to internal losses κ1 /κ
0.83(3)Finesse2532(80)
Free spectral range
8.105(2) GHz
Frequency difference TEM00 -TEM10
2.290(5) GHz
TEM00 waist size w070(1) µm
Cavity length
1.849(1) cm
Mirror radius of curvature
4.999(5) cm
Cavity Parameters (lattice λ = 823 nm)
Input coupling κ12π × 4.40(10) MHz
Output coupling κ22π × 0.23(1) MHz
Linewidth
2π × 5.8(6) MHz
Finesse1400(150)
Trap depth
115 µK
Power Buildup (Pcirc /Pinc )800(130)
TEM00 waist size w0
71(1) µm
Table 2.1: Relevant cavity parameters at the atomic and cavity probe laser wavelength λ = 780 nm
and at the lattice laser wavelength λ = 823 nm. The cavity’s mirror transmission coefficients,
T1 on the probed end (1) and T2 on the closed end (2), are expressed in terms of coupling rates
κ1,2 = T1,2 × (free spectral range).

15

Figure 2.3: Optical cavity CAD diagram. A Macor “hat” (red) is used to join the Zerodur cavity spacer (light purple) and a piezo (white). The piezo and mirror (blue) were glued together
beforehand using a plastic (Delrin) alignment jig. Optical cavity parameters are shown in Table
2.1.

Figure 2.4: Mirror heater calibration. The mirror surface temperature is plotted versus appliedcurrent in a vacuum test under the same configuration as the final science cavity.

was adapted from an old version that is described in [30, 20]. However, the previous version wasobserved to oscillate at 17 Hz leading to oscillations in the transmission power of an optimallyaligned, on-resonance laser of up to 20%. In the new version, we simply inserted a clamp to
the Viton isolation system to increase the rigidity of the system and rely on the higher quality
vibration isolation of the JILA X-wing. Oscillations in the probe laser’s coupling to the cavity due

16
to vibrations have not been observed in the new system.
The cavity system is mounted inside of a large vacuum chamber that extends down througha hole in the optical table. The system is pumped with an ion pump and also has a Titaniumsublimation pump that can be engaged if desired. The system reaches a base pressure of approximately 1 × 10−9 Torr that may be limited epoxy outgassing and the small conductance of the upper
part of the vacuum chamber. However, this base pressure should only limit the vacuum lifetimeof atoms to nearly a second or greater [5], and has allowed plenty of time to perform current spinsqueezing and superradiance experiments.

2.3

State preparation
For most of the experiments described in this thesis we use the qubit states |↑i = |52 S1/2 , F = 2, MF = 2i

and |↓i = |52 S1/2 , F = 2, MF = 1i, the stretched hyperfine ground states.

The advantage ofthese states is that the |↑i state can be probed on an optical cycling transition from |↑i to|ei = |52 P3/2 , F = 3, MF = 3i. In order to prepare the atoms in the |↓i state, we use the optical pumping scheme shown in Fig. 2.5. An F = 1 beam is tuned between F = 1 and the excitedF 0 = 0 state, leaving a dark state in |↓i. An F = 2 beam is tuned from F = 2 to excited F 0 = 2
to clear out all the F = 2 levels. The quantization axis is defined by the magnetic field, which isaligned with the cavity axis as shown in Fig. 2.5(a). The polarization is set to a combination ofσ+ and π-polarization along the z-axis. In all, this allows atoms to accumulate in |↓i as shown in
Fig. 2.5(b) with approximately 95% of the atoms in the correct state. A small number of atoms inother F=1 sub-states do not significantly affect experiments, but atoms remaining in |↑i can cause
serious errors. For this reason, after optical pumping, we often apply a strong final F=2 depumping
pulse which ensures that less than 1% of atoms remain in the F=2 manifold. After this step, thevarious experimental sequences of chapters 3-7 can be performed at will.

17

Figure 2.5: Optical pumping scheme to stretched state. a) Two optical pumping beams copropagate at an angle from the vertical axis with polarization that decomposes into σ+ and π~ b) Energy level diagram for optical pumpalong the quantization axis set by the magnetic field B.
ing. Pumping channels are shown by red and blue lines for the F=1 and F=2 pumps respectively.
Atoms accumulate in |↓i = |52 S1/2 , F = 2, MF = 1i.

Chapter 3

Superradiant Injection Locking

3.1

Introduction
In a superradiant (or “bad-cavity”) laser, the atomic coherence decays much more slowly

than the optical cavity field. As such, the atomic coherence primarily stores the laser’s phase
information and is initially established via spontaneous synchronization of the individual atomic
dipoles (as in Fig. 3.1(a)). Unlike in conventional good-cavity lasers, coherence has been shown topersist with less than one, and even zero, intracavity photons [23, 21, 171]. This bad-cavity regimeof laser physics has generated recent interest because it offers a promising route for overcoming
fundamental thermal mirror noise in order to realize laser linewidths of one milliHertz or less [109].
More broadly, cold atom-cavity systems are extremely well-controlled experiments useful forobserving many-body phenomena with the cavity mode providing strong long-range interactions
between the atoms. For example, the spontaneous spatial ordering [73, 14, 148, 55] and realization of
the Dicke model [11] in cold atom-cavity systems are examples of nonequilibrium phase transitions
and provide insights into our fundamental understanding of phase transitions in condensed matterphysics [158]. Further, atomic ensembles coupled to many cavity modes may allow the creationof exotic phases of matter with emergent crystallization and frustration [52, 152], and could serve
as a model system for associative memories [53]. Superradiant lasers have been identified as aninteresting system in which to study the problem of synchronization of quantum oscillators [104,
167].

19

3.2

Experiment: superradiant injection locking
In this chapter, I will discuss an experiment to study synchronization of a superradiant laser

to an externally applied optical field that is injected into the lasing cavity mode (Fig. 3.1(a)). The
synchronization is analogous to injection locking in a good-cavity laser, but in this superradiantsystem phase locking is manifested as collective synchronization of an ensemble of cold atoms to theapplied drive. For a weakly injected optical driving field, the system can be approximately mapped
to a driven Van der Pol self-oscillator, a canonical system in synchronization physics [125, 124].
We directly observe the two synchronization behaviors predicted for such a system. However,when the applied drive’s poer and/or detuning from the self-lasing freqeuency are large, we observe
two additional effects which are not explained by either the Van der Pol model or traditional
injection locking theory [145]. First, the stimulated emission component at the self-lasing frequency
is repulsed from, rather than attracted to, the drive frequency. Second, as the drive strength isincreased the stimulated output power at the self-oscillation frequency actually decreases. These two
effects arise from the full three-dimensional description of the atomic spin or Bloch vector dynamics,
in comparison to a two-dimensional description of a Van der Pol oscillator. The new, third degreeof freedom corresponds to the atomic inversion, which is no longer approximately constant at large
detunings or drive strengths. In each regime we show good quantitative understanding of oursystem, providing a solid foundation for future work in fundamental physics using superradiantlasers.
Complex injection locking behaviors beyond the simple Van der Pol description, including
instability, chaos, and repulsion, have been theoretically studied and observed in lasers which
operate in the crossover regime of laser physics where the cavity decay rate and one or more atomic
decay rates are similar [90, 112, 147]. Frequency repulsion is predicted in these lasers from acoupling between the injection locking dynamics and relaxation oscillations [123]. In a distributed
feedback (DFB) laser, a one-sided frequency repulsion was seen to arise from tuning of the cavity
frequency [99]. In our system, frequency repulsion and three-dimensional dynamics are a direct

20

(a)

applied drive(b)heterodyne
detectionWlaser-cooled Rb

W

Figure 3.1: Experimental setup and level diagram. (a) Atoms interact with both the externallyapplied drive (grey) and the intra-cavity field generated by their collective emission (blue and red).
The superradiant laser primarily responds at two frequencies, the drive frequency ωd and a selflasing frequency ω` . (b) The characteristic frequencies are displayed in a level diagram, and all lie
within one cavity mode of width κ. The Raman laser system is approximated as a 2-level laser
incoherently repumped through intermediate optically excited states (not shown) at rate W . W isalso the primary source of broadening of the lasing transition (shown as broadening of |↓i). In thiswork, the ratio of atomic and optical linewidths is W/κ ≈ 5 × 10−2 to 5 × 10−3  1, placing thesystem deep into the bad-cavity or superradiant regime. The state |↑i is a dressed state consistingof a ground hyperfine state of Rb coupled non-resonantly to an optically excited state as describedin [23, 18, 171, 21]. The applied drive couples |↓i and |↑i with an on-resonance Rabi frequency Ωd .

21consequence of injection locking deep in the bad-cavity regime, a regime largely inaccessible byprevious work [25].
The apparatus for the superradiant laser and principles behind its basic operation havebeen described in detail in previous work [23, 18, 171, 21]. The atomic gain medium consists ofN ≈ 1.1 × 106 87 Rb atoms cooled to 10 to 20 µK and trapped in a 1D optical lattice inside of an
optical cavity with power decay linewidth κ = 2π × 11.8 MHz. The atoms are tightly confined to λ (i.e. the Lamb-Dicke regime) along the cavity axis, but only weakly confined perpendicular
to the cavity axis.
A dressing laser is applied transverse to the cavity to induce spontaneous Raman transitions
between two hyperfine gound states |↑i ≡ |5S1/2 , F = 2, mF = 0i to |↓i ≡ |5S1/2 , F = 1, mF = 0i,with typical single-atom free-space Raman transition rates γ = 2π × 100 Hz to 2π × 300 Hz.
The cavity frequency ωc is tuned to be on or near resonance with the spontaneously emitted
light’s frequency ωa . The effective two-photon coupling to the cavity is characterized by the rms
value of the Jaynes-Cummings coupling constant g2 [108] and single atom cooperativity parameter
4g 2

C = κγ2 = 5 × 10−3 . The collective (or superradiant) emission rate for a single atom scales as
N Cγ.
To maintain population inversion and steady-state emission, additional lasers are applied toincoherently repump atoms through optically excited states from |↓i back to |↑i. The characteristic
repumping rate from |↓i (including Rayleigh scattering) is W ≈ 2π × 60 kHz to 2π × 500 kHz.
The repumping process is the primary contribution to the atomic transverse decoherence rateγ⊥ ≈ W/2 + ΓD . We measure a small additional contribution to the transverse broadening ΓDwhich is primarily due to doppler broadening of the two-photon transition from |↑i to |↓i resulting
from the weak transverse confinement of the atoms.
A conceptually simplified experimental diagram for this work is shown in Fig. 3.1. The key
distinct feature in this work is the application of an additional coherent drive to the superradiantlaser’s cavity mode (Fig. 3.1). The drive couples the upper |↑i and lower |↓i lasing states witha single-atom Rabi frequency Ωd . The drive frequency ωd is detuned from the effective atomic

22transition frequency ωa by δd ≡ ωd − ωa (Fig. 3.1(b)). The behavior of the system depends on therelative magnitudes of drive strength Ωd , detuning δd , and characteristic rates of the superradiantlaser: the repumping rate W and characteristic collective emission rate into the cavity N Cγ givenby the collective cooperativity N C and the single-atom decay rate γ from |↑i to |↓i.
When no drive is applied the laser emits at frequency ω`0 near, but not necessarily identicalto ωa . When the drive is applied, the lasing frequency ω` is shifted by the atoms’ interaction withthe drive by δ` ≡ ω` − ω`0 . Additionally, the laser can emit at the drive frequency ωd . We detect
the light emitted from the cavity using heterodyne detection. This gives complete information
about the emission spectrum and allows us to measure ω` , ωd and the phases and amplitudes ofthe electric fields emitted from the cavity at these frequencies. Other frequency components in thelaser emission are expected and observed at sums and differences of ω` and ωd . These additional
components can become rather large near synchronization (see Ref. [125]).

3.3

Phase diagram
The predicted behavior of the emitted field of the laser is summarized by the theoreticalphase diagram in Fig. 3.2. The phase diagram is calculated by numerically integrating optical
Bloch equations based on a simplified 2-level model for the superradiant laser (see [22] and Section
3.5). For simplification, the repumping rate W is set to a value that optimizes the output powerof the laser Wopt = 21 N Cγ [109], so that the characteristic rates governing the phase diagram are
the two ratios Ωd /W and δd /W .
The primary feature of the phase diagram is the synchronization or non-synchronization of
the superradiant emission to the drive. In the unsynchronized phase of region (2), the atomic
dipoles are not perfectly synchronized to the drive and the spectrum of light contains two distinct
frequency components at the drive frequency ωd and the self-lasing frequency ω` . In contrast in
region (1), the atomic dipoles become synchronized to the drive and all light emission occurs at ωd .
For the optimum repumping W = Wopt here, the synchronization transition occurs roughly whenΩd = δd .

23

2R
1.0

2A

1.0

Drive Detuning,

1.0

0.5

0

1

0.5

0.5
0

0.5

2A

0.5

0

2R
0.0

1.0

0.5

0.5

Drive Strength,
Figure 3.2: The predicted phase diagram for the driven superradiantlaser is shown in a plane0defined by the applied drive strength Ωd and the drive detuning δd , normalized to the repumping
rate W , which is fixed to N Cγ/2 here. The regions are first divided by the number of distinct
emission frequencies (1 or 2). Region (2) is further divided by the frequency shift of the self-lasing
component at ω` , which can be attracted (2A) or repelled (2R) from
0.5 the applied drive frequency
ωd . When Ωd < 0.2 × W , the laser synchronizes by smoothly coalescing in frequency with thedrive (dashed line). For larger drives, the self-lasing component remains distinct and is quenched.
The two trajectories (black arrows) refer to the two parameter trajectories explored by the data in
Fig. 3.3.

24

(a)

(b)

Figure 3.3: Experimental observation of coalescing attractive (a) and repulsive (b) synchronizations.
(Left) 2D spectrograms are taken with fixed drive strength Ωd as the detuning of the drive δd isvaried along the representative vertical trajectories in Fig. 3.2. Darker colors indicate higher power
in a frequency bin (i.e. PSD). The red line indicates the expected self-lasing trajectory in theabsence of an applied drive. (Right) Two panels show the frequency shift δ` = ω` − ω`0 between thelasing frequency and the lasing frequency when no drive is present. In each region, we qualitatively
identify attraction and repulsion by the sign of δ` and label and color each region similarly to thephase diagram in Fig. 3.2. The behaviors follow the prediction for their respective trajectories
across the phase diagram, which are represented by vertical lines in Fig. 3.2.

The unsynchronized region (2) is broken into two subregions, delineated by the self-lasing’s
attraction toward or repulsion from the drive frequency. The region of attraction ( δδd` > 0) is labeled(2A) in the phase diagram. The region of repulsion ( δδd` < 0) is labeled (2R). As δd or Ωd are tuned,the behavior of the approach to synchronization depends on whether one enters region (1) fromregion (2A) or (2R). For attractive synchronization (from (2A) to (1)), as the Ωd = δd boundary

25
is crossed, ω` is pulled toward the drive. For drive strengths Ωd < 0.2 × W (dashed line in Fig3.2), the self-lasing component synchronizes by smoothly coalescing with the drive at ωd . WhenΩd > 0.2 × W , the self-lasing is driven to zero before coalescence can occur. In the repulsive
synchronization (from (2R) to (1) in the phase diagram), as one approaches synchronization ω` isrepelled in frequency from ωd and the self-lasing component is driven to zero so that the superradiantensemble is emitting power only at the drive frequency ωd .
In the limit Ωd , |δd |  W , the laser inversion Jz is approximately fixed, and the threedimensional Bloch vector (Jx , Jy , Jz ) describing the atomic ensemble can be reduced to an effective
two dimensional object described by the transverse coherence J− = Jx − iJy . In this case, theequation describing the time evolution of J− is closely equivalent to that of a driven Van der
Pol oscillator, for which attractive synchronization ((2A) to (1)) with and without coalescence(characterized by saddle-node and Hopf bifurcations respectively) have been well studied [125, 124].
We outline this mapping explicitly in Section 3.7.
However, when either Ωd or |δd | & W , the inversion Jz can no longer be approximated as
fixed and the dynamic response of the full three-dimensional Bloch vector must be considered. The
response of the extra degree of freedom Jz leads to the repulsive behavior in region (2R) which canbe interpreted as an AC Stark shift (Section 3.7).
Experimental examples of the two synchronization transitions are shown in Fig. 3.3, withapproximate trajectories in the phase diagram represented by black arrows in Fig. 3.2. Fig. 3.3(a)demonstrates attractive, coalescing synchronization and Fig. 3.3(b) represents repulsive synchronization. The left plots are two-dimensional power spectra of the laser emission. Each horizontal
slice corresponds to a single power spectrum of laser emission where color represents the opticalpower in each frequency bin. The nominal detuning δd is changed between experimental trials andplotted on the left axis. On the horizontal axis, the drive freqency ωd is set to zero so that in theabsence of an applied drive (Ωd = 0), the emission frequency ω` would follow the diagonal red line.
At sufficiently small detunings synchronization occurs and only power at ωd is observed. A faintspectral component in Fig. 3.3 appears on the opposite side of zero from the self-lasing component.

26
We believe this small feature is an artifact of nonlinearities in the detection system.
The spectrograms illustrate the qualitative differences between the two types of synchronization. In Fig. 3.3(a), the self-lasing frequency ω` is attracted toward and joins ωd as |δd | becomes
small. In Fig. 3.3(b), the two emission components remain distinct until the ω` component is extinguished. To more clearly illustrate the attraction and repulsion, the measured quantity δ` = ω` −ω`0
is plotted on the right. The plots are overlaid with color and labeled to help identify the repulsion(red) and attraction regions (blue) matching the phase diagram of Fig. 3.2. For the trajectory at
small Ωd the laser goes from synchronized (1), to attraction (2A), to repulsion (2R) as predicted.
For the trajectory at larger ωd there is a single transition from (1)- to (2R)-like responses. Bothbehaviors qualitatively agree with the prediction from the phase diagram for a 2-level laser, shownby the trajectories in Fig. 3.2.
The qualitative behavior of the data exhibits asymmetry with respect to δd = 0, whereas
the theoretical phase diagram of Fig. 3.2 is symmetric. This is because the data did not strictly
follow the vertical trajectories shown in Fig. 3.2. As the detuning δd was changed, the fractionalamount of the fixed incident drive power coupled into the cavity changed, following the Lorentziancavity resonance profile. The symmetry of this effect about δd = 0 is broken by the fact thatδc ≡ ωc − ωa 6= 0 (where ωc is the cavity resonance frequency), leading to the observed asymmetry
in the data of Fig. 3.3.

3.4

Van Der Pol description
We now turn to the development of a perturbative description of the system far from synchro-nization and the break down of this description as the system approaches and ultimately crosses
the synchronization threshold.
Deep into region (2) of the phase diagram, the superradiant laser’s response to the drive issmall and can be understood as a small modulation of the initially undriven Bloch vector describing
the atomic coherence. The modulated Bloch vector then radiates an additional field into the cavity
at the drive frequency ωd , producing gain. In Fig. 3.4(a) we measure this power gain G and phase

27

(b)
Imaginary Quadrature,
Im[Ed/Ed0]

Power Gain, G φ (deg)

(a)
45
0
-45
12

(c)

8
4
0
-2

2
1
0
-1
-2
0

-1
0
1
2
Drive Detuning, δd/2π (MHz)1
2
3
4Real Quadrature,
Re[Ed/Ed0]

P
Ps

1.4
Power (nW)

5

1.2
1.0
0.8
0.6
0.4
0.2
0.0
0

20
40
60
3
Drive Strength, Ωd/2π (kHz)x10

80

Figure 3.4: Synchronization and gain saturation. (a) Measured gain and phase response of thesuperradiant laser at the drive frequency ωd . At large drive detunings, the response displays linear
small-signal gain (red fit to perturbative model overlayed). The gain saturates (grey shaded region)
at small detunings as the laser approaches the synchronization transition. (b) The same gain andphase response are represented in a phasor picture. Points of small signal gain lie along the straight
line, and the region of saturation is approximately described by a curve of maximum stimulated
electric field (inner curve). (c) The stimulated output powers Ps and P` , at the self-lasing frequencyω` (red) and at the drive frequency ωd (blue) respectively, are displayed as the laser is driven acrossa repulsive synchronization transition at approximately Ωd /2π ≈ 40 kHz. Theoretical predictions(solid lines) show good agreement with the data.

response φ of the laser at the drive frequency ωd versus the drive detuning δd . This corresponds to
da vertical trajectory on the phase diagram where, in this dataset, ΩW ≈ 0.04. A fit to a perturbativemodel based on the optical Bloch equations in Section 3.6 is shown in red.
At small detunings (grey region), the gain and phase response begin to saturate, and deviate
from the predicted small-signal values. This is roughly the point when the laser begins to synchro-28
nize to the drive and the emission at frequencies other than ωd begin to disappear. In Fig. 3.4(a)dwe have chosen a specific drive strength ( ΩW ≈ 0.04) to illustrate the transition from linear gain tosaturation. If a smaller(larger) drive strength is chosen, we observe that, as expected, saturation
occurs at a smaller(larger) detuning.
In Fig. 3.4(b), the equivalent complex electric field response Ed at the drive frequency isshown in a phasor diagram with each point corresponding to the measured field at a given detuningin Fig. 3.4(a). The drive response when no atoms are present Ed0 is normalized to be real and oflength 1. In the perturbative limit, the additional stimulated field follows a straight line. The lineof small-signal gain is tilted due to an additional phase shift of the stimulated field arising fromnonzero detuning of the drive from the optical cavity ωd −ωc 6= 0. At saturation, the stimulated fielddeviates from the straight line and qualitatively follows a contour of constant stimulated electricfield (solid semi-circle).
In Fig. 3.4(c) we show an example of how optical power is “stolen” from the self-lasing
frequency ω` and transferred to the drive frequency. Here the drive strength Ωd is increased at a
fixed detuning δd /W = 2.2, and the vertical axis shows the self-lasing power P` and the stimulated
drive power Ps ≡ Pd − Pd0 . Pd0 is the detected power at the drive frequency in the absence of
any atomic response, scaling as Pd0 ∝ Ω2d , such that Ps represents the extra stimulated power at
ωd . This dataset corresponds to tuning the system along a horizontal line in Fig. 3.2 that lies
outside the plotted range and such that the system crosses from the repulsive region (2R) to thesynchronized region (1).
Numerical solutions (solid lines in Figure 3.4(c)) of the optical Bloch equations (Section 3.5l)
give reasonable agreement with the data. The theoretical model includes approximate corrections
for an additional cavity tuning effect [21], and the absolute vertical scale of the theory has beenscaled so that the P` agrees with the data at Ωd = 0.
The synchronization point in this data is represented by the sharp point when P` hits zero,with a discontinuous first derivative in P` and Ps . At the synchronization point, Ps is approximately
the original output power of the laser when Ωd is zero. At large drive strengths in Fig. 3.4(c), Ωd29
becomes much larger than W and the total output power of the laser decreases due to repumpinginduced dephasing of the rapid Rabi oscillations caused by the drive. This reduction in stimulated
output power is another unique aspect to injection locking in the bad-cavity laser.

3.5

Optical Bloch equations
The average behavior of the superradiant laser with an applied drive can be understood withslight modifications to the optical Bloch equations presented in [22]. The optical Bloch equations
describe the time evolution of expectation values of the cavity annihilation operator â and thecollective atomic operators, Jˆz and Jˆ− , defined as,
Jˆz =
Jˆ− =

N
X
|↑i i h↑i | − |↓i i h↓i |
i=1
N
X

2
|↓i i h↑i | .

(3.1)

(3.2)

i=1

Jˆz and Jˆ− represent the atomic inversion and transverse coherence respectively. The optical Blochequations govern the time evolution of the expectation values of these operators, Jz = hJˆz i, J− =hJˆ− i, and E = hâi. The atomic response can be visualized as a three-dimensional Bloch vector
with x and y projections of the vector given by J− = (Jx − iJy )/2. E is a complex representationof the optical cavity electric field such that |E|2 is the average number of photons inside the cavity.
The nonlinear equations are closed by approximating that the expectation values of products ofoperators can be factorized into products of expectation values. Assuming uniform coupling to thecavity mode, the coupled equations for a 2-level system with an applied drive can be written asĖ = −κκ+ iδc E − ig2 J− + Edi eiδd t
2
2

(3.3)

J˙− = −γ⊥ J− + i2g2 Jz E

(3.4)

N
∗
J˙z = −W Jz + W + ig2 (J− E ∗ − J−
E).
2

(3.5)

The equations are written in a frame rotating at the atom’s natural transition frequency ωa .
Edi is proportional to the amplitude of the electric field of the applied drive incident on the optical

30
cavity. δc = ωc − ωa is the detuning of the cavity from the natural atomic transition frequency ωa .
The rest of the equation parameters are defined in previous text. The Rabi frequency of the drive
is related to these parameters by Ωd =2g2 Edi1+δ̃c2A, where quantities Ã ≡ κ/2. The measured intracavity

Efield when no atoms are present is Ed0 = 1+idδ̃i , where δd,c is the detuning of the driving electricd,cfield from the cavity resonance, δd,c = ωd − ωc .
We have numerically solved these equations to derive the phase diagram of Fig. 2 and tocreate theoretical curves for Fig. 4. The two-level model does a reasonable job of predicting the
behavior of synchronization versus Ωd and δd . However, due to the true multi-level structure of
the atom, these equations cannot be used to predict the total output power of the system. Whenthese equations are not adequate, multi-level optical Bloch equations from Ref. [22] can be used.
Additionally, to approximately account for dispersive shifts of the optical cavity in our Raman
laser, the cavity detuning δc is made a function of Jz to generate the theory for Fig. 4(c). Details
can be found in Ref. [22].

3.6

Derivation of small-signal gain
The small-signal regions of gain and phase response, as described in the main text, follow

a simple form that can be derived from the optical Bloch equations. We first assume κ is larger
than all other characteristic frequencies in the system so that E adiabatically follows J− . We
then assume that J− primarily responds at two frequencies, ωa and ωd , and make the ansatz
J− = J−a + J−d eiδd t . Lastly, we assume that in this perturbative limit Jz is unaffected by the weak2δ̃c )drive, and retains its steady state value with no drive, Jzss = W (1+2Cγ . With these approximations,

we find the small-signal complex field response at the drive frequency. The total detected field at
the drive frequency, Ed = Ed0 + Es , has two contributions. One contribution (Ed0 ) comes from thedrive alone, and the other (Es ) from atomic stimulation. The small-signal form of Es is given byEs =−iEd0 γ⊥ (1 + δ̃c2 ).
δd (1 + iδ̃c )

(3.6)

31
In the limit δ̃c = 0, Es can be written in the form Es =√√−i αEd0, where α characterizes theδd2.
stimulated field gain. From Eq. 3.6 one expects α = γ⊥

Eq. 3.6 explains several features of the measured small-signal gain shown in Fig. 4 of themain text. For δc = 0, the stimulated field is in the orthogonal quadrature to the driving field.
However, when δc 6= 0 as is the case in the data of Fig. 4(b), the response of the cavity to thedriving atomic dipole causes the stimulated field to be partially rotated into the same quadratureas the driving field. This rotation breaks the symmetry about δd = 0 for both the measured phaseand power gain shown in Fig. 4(a).

(a)

(b)

Figure 3.5: Quantitative small-signal gain measurements. (a) We measure the total transmitted
power at the drive frequency ωd and define power gain G as the measured transmitted power
normalized to the drive power transmitted through the cavity on resonance with no atoms present.
For these measurements δd is scanned over a frequency range greater than the cavity linewidth κ.
The data is fit to the model in Eq. 3.7 (red line). Note that here, the cavity resonance marked by
the vertical solid line is a few MHz higher than the atomic resonance. (b) From fits to data such as2(a), we plot the variation of the fitted gain coefficient α (see text) versus W . The prediction α = γ⊥
is shown in grey. The width of the grey band corresponds to the uncertainty in an independantcalibration of the repumping rate W .

To quantify the small-signal gain experimentally, we measure the total output power of the32laser at the drive frequency for a large range of drive detunings δd . We fit the total output power
to the gain model shown in Eq. 3.7. Equation 3.7 is the magnitude squared of Eq. 3.6 with theEd0 dependence on δ̃d,c written explicitly.
()√
G0α(δ̃c − δ̃0 ) α[1 + (δ̃c − δ̃0 )2 ]G(δd , δ̃d,c ) =1−2+2(δd − δ0 )(δd − δ0 )2
1 + δ̃d,c(3.7)

For a single scan of δd , we allow fitting of the parameters G0 , α, δc , κ, and δ0 . The fit model
constrains the transmitted power to be G0 when the drive is on resonance with the cavity in theabsence of an atomic response. The δ0 coefficient allows for an arbitrary offset of the atomic
transition frequency ωa from zero. Figure 3.1(a) displays an example of this measured total output
power as a function of δd with the fit overlaid in red. In Fig. 3.1(a), the data has been rescaled suchthat the fitted coefficient G0 = 1. After this rescaling, the data represents the total power emitted
at the drive frequency, normalized to the power transmitted through the cavity when the drive ison resonance with the cavity and no atoms are present. Also, the frequency axis of the data has
been adjusted such that δ0 = 0.
We follow this procedure, measuring the output power and fitting to Eq. 3.7, for many
repumping rates W . We plot the fitted gain coefficients α versus W in Fig. 3.1(b). The prediction
2 = (W/2 + Γ )2 is overlaid in grey. Uncertainty in the prediction (width of the greythat α = γ⊥Dband) is due to uncertainty in the experimental calibration of W . The prediction shows reasonableagreement with the theory over a significant range of W .

3.7

Bloch vector interpretation of phase diagram
The predicted phase diagram is shown in Fig. S2. The behavior in each of the 3 regions

can be visualized by the behavior of the Bloch vector in each regime. In a frame rotating at theatom’s natural transition frequency ωa , the applied drive can be represented by a rotation of the~ d = Ωd (x̂ cos(δd t) + ŷ sin(δd t)). |Ωd | is the angular frequency of the rotation, andBloch vector, ΩΩ̂d is the axis about which the Bloch vector rotates. The azimuthal phase of the applied rotation
axis precesses at frequency δd . When Ωd  |δd |, the drive primarily acts to slightly modulate the

1.0
1.0

33
0.5
0.5

(a)

2R

1.0
1.0
0
0

z

2A

1.0
1.0

Drive Detuning,

0.5
0.5

1

y

1.0

Region 1:
Synchronization

0.5
0.5
0.5

0.5

0
0

0

0

0.5

1.0

0.5
0.5

0.0

0.5

0.5

z

0

Drive Strength,
(c)

(b)

0.5

6
4
2
0246

0

0.5

y
0.5

Region 2A:
Attraction

Region 2R:
Repulsion

Figure 3.6: Bloch vector interpretation of phase diagram. (a) The types of behavior for thedriven superradiant laser can be characterized by a phase diagram. The characteristic rates that
determine the lasing behavior are drive Rabi frequency Ωd , detuning δd , and repumping rate W . The
regions correspond to the number of distinct emission frequencies (1 or 2) and the frequency shift(attraction or repulsion) of the carrier (A and R respectively). The behavior of the synchronization(a), attraction (b), and repulsion (c) configurations are shown in a Bloch sphere picture. In the~ d , with anframe of the atomic transition frequency ωa , the drive is represented by a rotation Ωorientation which rotates along the dashed green trajectory at frequency δd . In the unsynchronizedcase this modulates the Bloch vector (red vector), causing drift toward or away from the drive,with the average precession of the Bloch vector indicated in each case via the large blue and red
arrows. In the synchronized case, the Bloch vector follows the drive all the way around the sphere.

34
orientation of the Bloch vector (both Fig. 3.6(b) and (c)). However, when Ωd > |δd |, the applied
modulation is so large that the Bloch vector can actually follow the drive all the way around thesphere. This is the synchronized region (1) in Fig. 3.6(a). Near the synchronization transition, therepulsive (2R) versus attractive (2A) behavior is determined by the size of the repumping rate W
compared to δd and Ωd .

3.7.1

Attractive regime: mapping to 2D Van der Pol oscillator

In the case |δd |, Ωd . W (i.e. 2A) the drive does not significantly perturb the laser fromits steady-state inversion because any change in Jz caused by the applied field is quicky healed by
the repumping process [18]. When Jz is not modified, the azimuthal phase φ is partially or fully
dragged in the same direction as the rotating axis Ω̂d (Fig. 3.6(a)). The lasing can be viewed as
being captured by the applied drive.
Furthermore, in this regime of weak drive, |δd |, Ωd . W , the system can be mapped onto a
Van der Pol self-oscillator model with a nonlinear driving term,̇− = −iδ− + λ− (1 − |− |2 ) + Ω(1 − β|− |2 )(3.8)

with complex amplitude  and characteristic rates λ, δ, Ω, and β. Equation 3.8 has an equilibrium
amplitude |− |2 = 1. The nonlinearity of the applied drive is governed by the parameter β. For
β = 0 this model has been well studied [125, 124].
To explicitly show the mapping of the optical Bloch equations (Eq. 3.3) onto this form, wefirst eliminate the cavity field E by assuming operation in the deep bad-cavity limit, and assume
that Jz is not perturbed by the drive. Setting J˙z equal to zero gives the nominal steady state value,
Jzss =NCγ−|J− |2 .
2W

One can then insert Jzss into the J− equation in Eq. 3.3, which leads to



(Cγ)2
N Cγ W2˙
J− = −iδd J− + J−
−
−
|J− |
2
2
W


Ωd N
Cγ
2
+
−
|J− | .
2
2
W

(3.9)

(3.10)

35
Eq. 10 is identical in form to Eq. 3.8. For the case of optimal repumping, we set W = Wopt = N 2Cγ ,and normalize J− to its steady state value defining − = J− /J−ss , where J−ss = √N8 , giving

N CγΩd̇− = −iδd − +(1 − |− |2 ) + √42|− |21−.
2(3.11)ΩdThis equation is of the same form as Eq. 3.8 with δ = δd , Ω = √, β = 12 , and λ = N 4Cγ . We can2|− |2Ωddefine an effective drive strength Ω0d = √1−. For a weak drive, the laser remains close to22

its steady state. One finds Ω0d = √18 Ωd , and the system can be thought of as behaving similarly tothe standard driven Van der Pol oscillator of Ref. [125, 124] with a constant driving term.

3.7.2

Repulsive regime: 3D dynamics

When the drive is applied with a large detuning |δd | & W (i.e. (2R)), the repumping at rate
W cannot heal the changes in the inversion Jz caused by the applied drive [18]. Jz can no longer
be considered static and thus introduces a third degree of freedom (in addition to Jx and Jy ) in thesystem. In this regime, the Van der Pol model breaks down, and small oscillations in Jz must be
taken into account. The unhealed modulations of Jz at frequency δd coherently interact with theapplied rotation due to the drive to cause the Bloch vector to on average aquire a small precession
in the opposite sense to the precession of the drive rotation axis Ω̂d (shown in Fig. 3.6(c)). The
precession rate is second order in Ωd and can be identified as an AC Stark shift that leads to theobserved frequency repulsion in region (2R). To emphasize, this shift does not appear in region (2A)
because there the repumping process acts to smooth out the modulations in Jz that are essential
for creating the AC Stark shift.
The AC Stark shift can be derived to leading order by perturbatively allowing for small
oscillations in Jz . In this way we can mathematically show the additional repulsive behavior not
evident in the 2-dimensional model of Eq. 3.8. The optical Bloch equation for Jz allowing formodulation, and written in the frame of the drive frequency ωd is,NJ˙z = −W Jz + W − Cγ|J− |2 − Ωd Re(J− )2(3.12)

36
We treat this equation perturbatively by assuming that J− primarily oscillates at the self-lasing
frequency. From this we can derive a leading order repulsion term giving a new self-lasing frequency,

δ 0 = δd +

Ω2d δd,2(δd2 + W 2 )(3.13)

which arises from oscillations in Jz coupling into an average frequency repulsion of the Bloch vector.
This repulsive physics is not present in the driven Van der Pol oscillator and arises from the Blochvector occupying a higher, 3-dimensional parameter space.

3.8

Outlook
We have observed for the first time two different types of synchronization transitions of asuperradiant laser to an external drive, one attractive and one repulsive in nature. The synchronization transition is analogous to a ferromagnet in the presence of an applied magnetic field, thedrive breaking a continuous symmetry of the laser with respect to phase [44]. However, the laser
steady state is far from thermodynamic equilibrium, making our well-controlled cold atom-cavity
system an interesting avenue for continued study of nonequilibrium phase transitions with modern
approaches [158].
It is often useful to apply an external drive to a superradiant laser. Such drives have been
used, for instance, to probe the frequency of the optical cavity in Raman-laser systems such as
ours (as was done in ref. [18]) or perhaps in future narrow linewidth superradiant lasers to reduce
errors, inaccuracies, and technical noise due to cavity frequency pulling. This work establishes
understanding for how such a technical probe will affect the system. Furthermore, the phase
response within the saturation region of Fig. 3.4(a) could be used as an error signal for a form ofactive spectroscopy of the gain medium, in some sense, the inverse approach to that of Ref. [106],
although the fundamental signal-to-noise of such an approach is an open question.
In the future, this work will guide the interpretation of other proposed experiments in cold
atom-cavity systems. For instance, multiple superradiant sub-ensembles each with an independent37
transition frequency ωa can be engineered to interact with each other through one or multiple
cavity modes [52, 152, 53, 178, 104, 167]. Furthermore, while injection locking is well-described bya mean field description, and therefore can be considered classical behavior, recent theoretical works
propose systems of multiple superradiant ensembles where quantum noise becomes observably large
and may serve to drive the phase transitions and affect the average behavior [178, 167, 104].

Chapter 4

Review: Spin Squeezing and Joint Measurements with Atoms in Cavities

4.1

Quantum sensors and the Bloch sphere
Many quantum sensors use 2-level atoms or qubits as the sensing device. Such sensors areused for state-of-the-art measurements of time and frequency [115, 66], electric and magnetic fields[85, 2], fundamental constants [135, 175, 36], tests for new physics [61, 7], biological systems [81],and more. An arbitrary quantum state of a 2-level atom used for these sensors can be written as,
 
 
θ
θ
iφ
|ψi = cos|↑i + e sin
|↓i ,
2
2

(4.1)

where |↑i ≡ |ei is the excited state and |↓i ≡ |gi is the ground state. The superposition isparametrized by the two angles θ and φ which tell the population and phase of the superposition
respectively. The 2-level system is equivalent to a quantum spin-1/2 particle that, when measured,
must collapse into either spin “up” or spin “down”. It is convenient to visualize the 2-level system’squantum state as a pseudo-spin vector, or Bloch vector [48], labeled ~j that lives on the surface
of a Bloch sphere. The Bloch vector is defined as ~j = jx x̂ + jy ŷ + jz ẑ with the α ∈ {x, y, z}
component equal to a first order expectation value jα ≡ hĵα i of the spin operator ĵα , that are
simply proportional to the Pauli spin operators, ĵα = σ̂α /2. By these definitions, the Bloch sphere isnormalized so that the z projection of the Bloch vector, jz = (P↑ −P↓ )/2, is equal to the difference inprobabilities of the atom existing in the two states |↑i and |↓i. Similarly, the transverse componentsof the Bloch vector give the magnitude and phase of the coherence between the two states. An
example Bloch vector is plotted below in Fig. 4.1.

39

Figure 4.1: A 2-level atom’s quantum state is represented by a Bloch vector ~j that lives on thesurface of a Bloch sphere. The state can be parameterized by vector components jx , jy , and jz orangles θ and φ.

In all of our experiments, we deal with a large ensemble of 104 to 106 atoms. So we mostcommonly think in terms of a collective Bloch vector composed of many individual Bloch vectors
arranged tip to tail. Going forward, unless otherwise states, all Bloch vectors will be assumed tobe collective.
The collective Bloch vector is denoted with a capitol J~ in terms of collective operators Jˆi =PN

i=1 ĵi .

An example collective Bloch vector, composed of many individual ones, is shown in Fig.

4.2. The Bloch vector shown represents a coherent spin state (CSS) where every atom is identically
prepared in an arbitrary superposition state described by the angles φ and θ. However, due to therandom projection of each atom when measured, uncertainty arises in the two angles, that is oftenplotted as a probability distribution of the Bloch vector on the sphere (red distribution). For a CSS
on the equator the uncertainties in θ and φ are equal, and analogous to the coin flip uncertainty√
discussed in Chapter 1, ∆φ = ∆θ = 1/ N . This uncertainty is called the standard quantum limit(SQL) for unentangled atoms [173, 74].

40

Figure 4.2: A collective Bloch sphere. The individual projection of each single-atom Bloch√vector
leads to an uncertainty in the angular resolution of the Bloch vector of ∆θ = ∆φ = 1/ N forunentangled atoms, called the standard quantum limit.

The angular uncertainty in the Bloch vector can also be interpreted in terms of variancesin each component Jα calculated using second order expectation values of the spin operators. For
2example, (∆Jz )2 = hJˆz2 i − hJˆz i . For a coherent spin state on the equator of the Bloch sphere,√
a common example, (∆Jz )2 = N /2. This spin noise is referred to as quantum projection noise(QPN).
In addition to representing a collective spin state and its uncertainty, Bloch vectors haveimmense intuitive power for other reasons as well. First, the Bloch sphere can be used to visualize
coherent rotations of the atom’s quantum state. For any 2-level system or spin-1/2, coherent state
rotations can be accomplished by applying an AC field at or near the frequency difference ωabetween |↑i and |↓i. Applying a resonant coupling, for example, will cause Rabi flopping of the
spin. However, general applied pulses can be used to achieve arbitrary rotations of the Bloch vector.
For an applied field with Rabi frequency Ω and detuning δ, the Bloch vector is rotated through an√angle γ = Ω2 + δ 2 × t about an axis with polar angle θ = π/2 + arctan (δ/Ω). The azimuthal

41
angle of the rotation axis is set by the phase of the applied field.
Second, and perhaps most importantly, the equatorial face of the Bloch sphere gives a visualization of the “face” of any atomic clock or quantum sensor. A CSS oriented on the equator will
undergo phase oscillation at the difference frequency ω between the |↑i and |↓i states. An atomic
sensor operates by measuring the difference in φ and the phase of the local oscillator φLO . This
phase difference is converted into a population difference using a π/2 rotation of the Bloch sphere,
and the populations of the two state are measured. This sequence is called Ramsey spectroscopy
and is ubiquitous among all types of quantum sensors.

4.2

Squeezing
The goal of this work is to create entanglement-enhanced quantum clock hands that have an

enhanced resolution in φ or θ below the SQL, as shown in Fig. 4.3. In particular, the entanglement
enhancement is measured using the Wineland parameter for spin-squeezing S = (∆θ/∆θSQL )2 (orsimilar for φ). One can show that any state with S < 1 must be in a nonfactorable, entangled
quantum state [149]. Whichever quadrature is squeezed, θ or φ, the anti-squeezing quadrature will
always have an increased uncertainty given by the Heisenberg uncertainty relation ∆φ∆θ ≥ 1/N .
Over the past 10 years or so, researchers have demonstrated many ways to create spinsqueezed states of neutral atoms. Many schemes use unitary interactions between qubits [136,93, 113, 118, 102, 132, 62, 114, 28, 95]. However, we and others create entanglement using a joint,
or collective measurement (sometimes called quantum-nondemolition) [88, 4, 140, 169, 96, 31, 19,
12, 142, 119, 133, 134].

4.3

Joint measurements
To understand how entanglement can be generated with joint measurements, consider a CSS

of 4 atoms,|ψCSS i =

N
=4
Y

1
√ (|↑i i + |↓i i).
2
i=1

(4.2)

42

Figure 4.3: A squeezed collective Bloch vector. Entanglement can reduce the quantum fluctuationin an ensemble of atoms or spins, leading to a sharper atomic clock hand. The blue probability issqueezedsince it has an angular resolution narrower than the standard quantum limit of ∆θSQL =√1/ N . The red distribution represents a coherent spin state made up of identically prepared
unentangled atoms.

The number of atoms in |↑i is uncertain in this state. In principle the state could have anywherefrom zero to four atoms in |↑i. However, one could simply perform a measurement of N↑ , thenumber of atoms in |↑i, in a collective way without gaining any information about which atom isin |↑i, and if the outcome is N↑ = 2, for example, then the resulting state would be,1|ψi = √ (|↑↑↓↓i + |↑↓↑↓i + |↑↓↓↑i + |↓↑↑↓i + |↓↑↓↑i + |↓↓↑↑i)
6

(4.3)

that is, in some sense, a maximally squeezed state of the 4 atoms. In fact, unless the outcome of thecollective or joint measurement had been |↑↑↑↑i or |↓↓↓↓i the resulting state would be entangled
no matter the outcome of the measurement. For a very large number of atoms, such as the 105 to
106 in our experiment, the outcome of a measurement can be very well approximated by a normal,−θ 2√2(∆θSQL )2
1√
where ∆θSQL = 1/ N is very
Gaussian probability distribution P (θ) =e22π(∆θSQL )small. This means that a collective measurement will, essentially, always give an entangled state43
near the equator.
The state shown in equation 4.3 is a Dicke state of 4 atoms, which would be drawn as aring around the equator of the Bloch sphere. In our experiments, our entanglement generating
measurements are not precise enough to create such states, but rather we collapse into states thatare somewhere in between Eq. 4.2 and 4.3. Such states are a bit difficult to write down, but very
easy to plot on the Bloch sphere. These states are the classic “squeezed” states that have reduced
uncertainty in the θ direction on the Bloch sphere and an increased uncertainty in the φ direction.
Of course, since quantum sensors measure the quantum phase φ, our squeezed states will need to
be rotated 90◦ before use in atomic clocks and most other precision measurements.
One mathematical formalism that can be used to write down and analyze squeezed states
is a quantum Baysian formalism [50]. The Baysian formalism simply implements the rule that aquantum state must be consistent with any measurement outcome. In this formalism, one defines
a projection operator for a measurement that we write as P̂B . For this work, we can assume thatthe measurement is a collective measurement that has an outcome of JzM and a measurement
uncertainty of ∆JzM . The collapse operator is then,22

1 (Jˆz −JzM )P̂B = e 2∆JzM
A

(4.4)

where A is a normalization constant that renormalizes the wave function after the collapse. In thecase that the measurement resolution ∆JzM is much smaller than the original projection noise, thismeasurement simply collapses the Bloch sphere into a state that is centered around the measurement
outcome Jz = JzM with an uncertainty equal to the measurement uncertainty ∆Jz = ∆JzM . In
order to analytically derive the wave function of a squeezed state, one can simply apply the operatorof Equation 4.4 to a coherent spin state written in the Dicke basis,
ψCSS (θ = π/2, φ = 0) = 2

−N/2

N  
X
N
i=0

i

|N↑ = N − iiN .

(4.5)

Since the state collapses around the measurement outcome JzM , this outcome must berecorded and subtracted from any final measurement for the entanglement-enhancement to be44
realized. For this reason, this type of squeezing is often called “conditional”, since the final measurement is certain conditioned on the premeasurement. However, conditional squeezing is still
deterministic in the sense that squeezing is created on every trial. In addition, in chapter ?? wediscuss an experiment to use real-time feedback after the measurement in order to create squeezing
with a specific targeted population value Jz .

4.4

Quantifying entanglement through spin squeezing
As I have already discussed, spin squeezing is an entanglement metric that quantifies the

reduction in a group of atoms’ quantum noise. This reduction in spin noise means there must
be quantum correlation between the atoms. In fact, the amount of squeezing and the quantum
spin correlation can be directly related to each other by writing squeezing in terms of second ordermoments of the spin operators. Assume a state at the equator of the Bloch sphere is squeezed inthe θ direction so that the variance in Jz is reduced ∆Jz < ∆Jz,SQL . I can re-write the spin noiseusing expectation values of the second order moment, assuming Jz = 0,(∆Jz )2 = hJˆz2 i
=

(4.6)

X

hĵzi ĵzj i

(4.7)

X

(4.8)

all atoms i,j

=

X

2
hĵzi
i+

all i

=

hĵzi ĵzj i

all i6=jXN+hĵzi ĵzj i .
4

(4.9)

all i6=j

The expectation value can be written in terms of second order operators of a single atom (line 4.8
first term) and joint operators between each atom (line 4.8 term two). The single atom operators
are what lead to the original projection noise (∆Jz )2 = N/4 as shown in line 4.9. Any reduction inthis quantum noise must come from quantum correlation between the atoms given by, for example,
a nonzero expectation value hĵzi ĵzj i for atoms i and j.
In an ideal system, it is reasonable to assume that the correlations would be symmetric among

45
the atoms so that line 4.9 could then be written without the sum, for arbitrary i and j,(∆Jz )2 =N+ N 2 hJˆzi Jˆzj i .
4(4.10)

Additionally, since we assume the state is fully coherent and lies on the equator of the Bloch sphere,
we can write the squeezing fully in terms of the spin noise reduction or squeezing, and cancel all
the factor of two by writing the quantity in terms of the simple Pauli spin operators,(∆Jz )2(∆Jz,SQL )2(4.11)= 1 + N hσ̂zi σ̂zj i .

(4.12)S=

Therefore, in this case, the squeezing is directly expressed in terms of the total atom number andthe atom-atom quantum correlation. However, for large atom number, the maximum correlationof a single pair will be whatever is required to fully cancel the original spin noise so that S goes toapproximately zero, that ishσ̂zi σ̂zj imax = −1/N

(4.13)

However, even though the quantum correlation between two atoms pulled out of the squeezed
ensemble may be weak, the total quantum correlation in the system is very high. This is because
there are N 2 links between atoms that can be correlated at order 1/N . The total correlation is thenthe product of N 2 and 1/N , namely, N . Therefore, a maximally squeezed state has the same total
quantum correlation as N perfectly entangled pairs. In fact, for our best squeezed states wich haveP
S ≈ 1/60, the total quantum correlation is approximately, all i6=j hσ̂zi σ̂zj i = 390, 000, equivalent
to the quantum correlation of 390,000 maximally entangled pairs! This quantity of entanglement isenormous, and created by a single measurement operation. Could there be some way to use joint
measurements as an entanglement resource to create, distill, and then harvest the entanglement forquantum information processing or other quantum tasks?
As another note, one other popular entanglement metric for non-classical states has been theso-called “entanglement depth” [150]. This entanglement metric proves that entanglement exists

46
and gives a minimum provable number of particles that must be participating in an entangled
group. However, entanglement depth does not quantify the strength of the quantum correlation/entanglement in the system, and can even be maximal in the limit of infinitesimal quantum
correlation. Entanglement depth might be more accurately labeled entanglement breadth. For thisreason, spin squeezing may provide a more useful metric for thinking of entanglement as a resourcefor quantum correlations.

4.5

Joint measurements of atoms in a cavity mode
So far the joint measurements have been described in an abstract way that is implementation

independent. I will now turn to an overview of how we implement joint measurements by placing
the atoms inside an optical cavity mode.
The |↑i and |↓i states are formed in rubidium by the F = 2 and F = 1 hyperfine groundstates, where F is the quantum number labeling total angular moment of the atom including
electron spin, orbital angular momentum, and nuclear spin. For all of the work in this thesis we
use the stretched mF = 1 and mF = 2 states. The optical cavity is then tuned near resonanceto the |↑i → |ei transition, where |ei is an optically excited state on either the D2 (780 nm) or
D1 (795 nm) transition. In this situation, photons in the cavity can be exchanged with atomic
excitations, if the atoms begin in |↑i. We work in a regime where the number of photons in thecavity is much smaller than the number of atoms, so the excitation fraction is low. In this regime,
the atom-cavity system can be approximated (known the Holstein-Primakoff approximation) as
two coupled harmonic oscillators. This approximation and formalism is described in detail in Ref.
[32], but I will briefly overview the physics here.
Under the Holstein-Primakoff approximation, the atom-cavity system can be approximatedby the following Hamiltonian, written for a single excitation, neglecting cavity losses and other|↑i|1idecoherence mechanisms, and in matrix form in the |ei|0i basis, where |1i represents a photon

47in the cavity and |0i represents zero photons in the cavity.


Ω/2 
.  δ
H = ~
.
Ω/2
0(4.14)

The cavity mode is detuned by δ ≡ ωc − ωa away from the atomic transition ωa (which is set
to zero energy in this Hamiltonian). The collective coupling between the atoms and the cavitypmode is defined by Ω, called the vacuum Rabi splitting, where Ω = N↑ 2g and 2g, called the
Jaynes-Cummings coupling parameter is the single photon Rabi frequency of atoms in the cavity.
The Eigenstates of this Hamiltonian are characterized by new symmetric and antisymmetricnormal modes of the coupled oscillators. The new normal modes of the system are, relative to thenominal atomic transition ωa
ω± =

δ±

√

δ 2 + Ω2.
2

(4.15)

Most importantly, the size of the vacuum Rabi splitting (the splitting between the two modes when
δ = 0), is dependent on the number of atoms in |↑i. This collectively-enhanced mode splitting is
the phenomenon that allows us to make a joint measurement of N↑ . In general, even if the cavity
is detuned off of resonance (taken to be blue detuning here), the atoms cause a shift ∆ω to thecavity resonance equal to
∆ω =

−δ +

√

δ 2 + Ω2
2

(4.16)

that depends linearly on atom number in the dispersive regime, where δ 2  Ω2 .
In all, by measuring ∆ω, and knowing our detuning δ, we perform a nondestructive, joint
measurement of the number of atoms in |↑i. This allows us to project the atoms into a squeezed
state. A more complete description of the atom cavity system, the Jaynes Cummings Hamiltonian,etc. can be found in [32]. In the next two sections, I will describe the signal to noise of this probing
as well as the fundamental limitation to this scheme from free space scattering.

48

4.6

Signal to noise when probing
In order to probe the cavity shift ∆ω, we send probe photons with frequency ωp at the cavity

near the atom-shifted cavity resonance frequency ωc0 . Small changes in the cavity frequency aroundωp will then be imposed as a phase shift on either the transmitted or reflected electric fields. This
phase shift can either be measured in a homodyne or a heterodyne detection scheme by interfering
the transmitted and reflected light against a reference beam. In the most recent third generation
experiments, we have used homodyne phase measurements in reflection, that are advantageousover any heterodyne measurements which are limited to a 50% fundamental quantum efficiency
due to gaining both phase and amplitude information from the reflected light (see Section A).
However, importantly for this chapter, vacuum fluctuations of the transmitted or reflected fieldlead to noise variance in the estimated phase φH of of the homodyne fringe. The homodyne phasecan be written as the amplitude of the q-quadrature of the reflected electric field to the nominallymaximal i-quadrature, φH = q(δ)/i(δ ≈ 0) where δ = ωp − ωc0 is the detuning of the probe fromthe atom-shifted cavity. The noise in the homodyne phase signal, when sitting at a zero of theq-quadrature, is then(∆φH )2 =

∆q
1
=
imax
qp 4Md

(4.17)

where Md is the total number of detected photons as inferred from the voltage on the photodiodeand the photodiode sensitivity S. qp is a quantum efficiency after the detection plane (that is,in this case, taken to be directly in front of the photodetector) that includes detector quantum
efficiencies and effective quantum efficiency loss from added technical noise sources in and after thephotodetector. The homodyne phase noise leads to noise in the measured cavity frequency ω̃c of,(∆ω̃c )2 = α2 /(qp 4Md )(4.18)

δ≈0 is the phase shift of the relfected probe when probing the cavity near resowhere α−1 = 1i dqdδ |nance. α depends on the input and output cavity linewidths κ1 and κ2 and whether one probes inreflection or transmission. The general phase response of the cavity can be found in Ref. [32]. For

49
probing in reflection in the dispersive regime, the result is,
α−1 =

4κ1
κ(κ − 2κ1 )(4.19)

where κ is the full cavity linewidth. For a perfect single-ended cavity with κ1 = κ, the result
simplifies to α = −4/κ.
Now that we have calculated the photon shot noise in the measurement of the cavity resonance, this noise must be compared to the fluctuations in the cavity resonance induced by fluctuations in the number of atoms in |↑i due to quantum projection noise. As already mentioned,√√∆Jz,QP N = N /2. This leads to equivalent fluctuations of N↑ , ∆N↑,QP N = N /2 and fluctuationsin the outcome of a single measurement of the cavity resonance frequency of,gΩ∆ωc,QP N = √ √.
2 2 Ω2 + δ 2(4.20)

Given the quantum projection noise and the noise in probing, we can note that the amount ofspin noise reduction R when limited by the fundamental photon shot noise is, RP SN ≡ (∆ω̃c /∆ωc,QP N )2 .
In the dispersive regime, one finds, for two equal measurements,
RP SN =

κ2 δ 2κ2 δ 2=.
q p Ω 2 g 2 Md
4qp g 4 Md N

(4.21). Probing with more photons improves this photon shot noise-limited resolution of the cavity,
although in practical systems there will be technical noise floors that limit the decrease in R.

4.7

Contrast loss from free-space scattering and optimal squeezing
However, competing with the photon shot noise concern, which demands more probe photons,

is contrast loss from free space scattering that worsens as more photons are used to probe the cavity
frequency. To lowest order, due to low transverse optical depth, every photon that is emitted intoa free space mode reveals whether a single atom is in |↑i or down and therefore causes that atom’s
wave-function to collapse. This causes a reduction in the transverse projection of the Bloch vector,
measured by contrast C ≡ (Jx2 +Jy2 )/(N/2)2 . Specifically, C = e−Ms /N where Ms is the total number

50
of free-space scattered photons. The Wineland squeezing parameter can then be equivalently rewritten in terms of R and C as S ≡ ∆θ/∆θSQL = R/C 2 . This leads to an optimal squeezing, inthe far detuned limit of, Sopt = e/2qN C, again for two equal probing windows, [32] where q is thetotal quantum efficiency for detecting the probe photons.
For our recent spin-squeezing experiments, this fundamental free-space scattering has been
relevant limit to spin squeezing, along with technical noise floors from laser frequency noise andoptomechanics (See chapter 6). In addition, spin flips induced by state-changing Raman transitions
can also add significant limitations to spin noise reduction and squeezing [32], but these have beensignificantly reduced by working on optical cycling transitions.

Chapter 5

Improved Bloch Vector Rotations with Reversible Dephasing

In the summer of 2013 we performed an experiment to directly observe over a factor often in entanglement enhancement of an atomic clock hand, the largest amount of spin squeezing
ever observed at the time [19]. The details of this experiment have been discussed in detail inprevious theses [20, 170]. One outcome of this experiment was the realization of the difficultyof performing precise rotations of the atomic Bloch vector necessary for Ransey clock sequences,differential measurements of the Bloch vector, or any quantum information protocol. This difficultyis exacerbated when working on a magnetic field sensitive transition. For this reason, we conducted
an experiment to demonstrate a new method to achieve cancellation of certain classes of Blochvector rotation errors in a collective system using reversible dephasing [41]. In this chapter, I will
describe this work.

5.1

Experiment: dephased rotations
Decoherence destroys entanglement, degrades precision measurement signals, and limits awide range of coherent processes from lasing to operating quantum gates [138, 89]. Therefore,
most technologies relying on real or synthetic atoms try to minimize decoherence resulting fromloss, relaxation, and inhomogeneous broadening. Recently, however, specifically engineered forms of
decoherence have been used to enhance certain processes. Dissipative decoherence, for example, canremove information from a system leading to stabilization of polar molecules from lossy collisions[179] or generation of entanglement [49, 87, 130]. Also, non-dissipative, reversible, decoherence in

52
the form of inhomogeneous broadening can be used to stabilize coherent operations allowing, forexample, storage of non-classical light signals [67], or as we show in this Letter, insensitivity toerrors in collective quantum state rotations.
Precision measurements using one or many atoms require precise rotations of the atoms’
quantum state. These rotations, achieved by applying a coherent field at or near the atomic transition frequency, are used to excite an atomic transition [127, 17], map the evolution of a quantum
phase into a measurable quantity [129, 66], or simply transfer state populations for precision readout [31]. Imperfections in these rotations lead to classical uncertainty in the atoms’ quantum state,
which can dominate fundamental quantum uncertainty and limit precision measurements.
In this Chapter, I will present an approach to suppress rotation errors using reversible inhomogenous broadening, an alternative to the composite coupling pulses that are often used to correctstate rotation errors [160, 172, 78, 98, 161, 151, 128, 157]. I first theoretically show how collective
rotations of many qubits can be performed with greatly reduced errors if controlled inhomogeneous
broadening of the transition is applied prior to the desired rotation. I also show that collective
coherence is restored by reversal of the inhomogeneous broadening after the rotation.
Next I apply dephased rotations in a specific experiment, demonstrating a maximum suppression of technical noise of greater than 21 dB when rotating the internal states of laser-cooled
and trapped 87 Rb atoms. Dephased rotations aid the generation and observation of entangled,
spin-squeezed states with a directly observed enhancement in quantum phase estimation 9.5(5) dB
below the standard quantum limit for an unentangled ensemble, one of the largest such enhancements in atomic systems reported to date [19, 103, 62, 113, 95]. In the absence of any reversible
inhomogeneous broadening, either incidental or deliberate, we estimate that little to no squeezing
would have been observed in this experiment due to imperfections in the required quantum state
rotations 1 .
1

A different measurement sequence [19] that avoids any rotations achieved a comparable amount of directly
observed spin squeezing in the same system without relying on dephased rotations, but only by sacrificing a factor of
two in fundamental measurement resolution. This additional factor of two was not realized in this work due to theprobe laser’s frequency noise coupling more strongly into the measurment sequence of Fig. 5.3

53

(a)

dephase(c)rotate(d)rotate(b)rotate(e)rephaseFigure 5.1: The reduced sensitivity of a dephased rotation to a small rotation error is graphically
represented on collective Bloch spheres. A single representative Bloch vector prepared in thex̂ − ẑ plane, along with the quantum uncertainty in its position, is shown (red arrow and noise
distribution). Each sphere also has a series of colored lines denoting the tips of Bloch vectors thatare at a constant Jz in the initial configuration. The original rings of constant Jz are shown in parts(b-e) as thin black lines for reference. A small rotation Rŷ (π/16) representing an error is applied
without (b) and with (c,d,e) dephasing. By reversibly dephasing the Bloch vector to Cd = 0.14,
the impact of the rotation is greatly reduced. Rotation errors that would otherwise dominate canbe supressed well below the fundamental quantum noise.

54
Dephased rotations are a general concept and could be applied to a variety of applications,having several advantages over traditional composite rotation sequences. Composite sequences rely
on cancellation between the errors of each individual rotation. However, cancellation fails if theerrors fluctuate on time scales comparable to the time required for the composite pulse sequence.
Furthermore, increasing the rate of rotations to enhance the correlation in errors may actually be
detrimental depending on the form of the noise spectrum [33]. Lastly, composite pulses requireprecise control over the phase of the coupling field, and the most effective composite sequences
require many pulses, increasing the time required for a measurement sequence. The approachpresented here to a large degree avoids these requirements. We note that intense efforts to apply
composite pulses to reduce rotation-added noise were largely unsuccessful in our experiment.
We describe our system of N 2-level atoms as spin-1/2 particles using a collective Blochvector J = Jx x̂ + Jy ŷ + Jz ẑ = ΣN
i=1 Ji , where the ith Bloch vector Ji = hĴi i is the expectation valueof the quantum spin projection operator for the ith atom. The ẑ projection of the collective Blochvector Jz ≡ J · ẑ = (N↑ − N↓ ) /2, is directly determined by measuring the number of atoms in spin
up N↑ and down N↓ . Precision measurements with 2-level systems are fundamentally limited byquantum uncertainty in the angles describing the orientation of the Bloch vector. This quantum
uncertainty appears as quantum projection noise (QPN) in the measurement of the spin projection
Jz . For unentangled atoms, the rms fluctuation for a coherent spin state (CSS) with J = N/2 x̂ is√∆Jz,QP N = N /2. The projection noise limits the estimate of the Bloch vector’s polar angle to√an rms uncertainty of ∆θSQL = 1/ N , the so-called standard quantum limit (SQL). Due to thisscaling, states with large N are desirable for precise phase estimation, but in these states, classicalrotation errors become more challenging to reduce below the smaller SQL.
The rotation of the ith Bloch vector through angle ψi about an axis n̂ is defined by therotation matrix Rn̂ (ψi ). If the rotation is uniform (ψi = ψ for all i) then the result is a rigid
rotation in which the length of the Bloch vector is conserved. The errors we wish to suppress
are those generated by uniform rotation errors associated with the coupling field, in particular,
an arbitrary erroneous rotation through a small angle φ described by Rn̂ (φ). The suppression of

55
the rotation errors will be achieved by introducing a brief, controlled inhomogeneous broadeningof the energy difference between |↑i and |↓i before and after the imperfect rotation. The timeintegrated effect of the broadening on the ith vector is characterized by the non-uniform rotationRẑ (ψi ). The amount of dephasing is quantified by the fractional reduction in the collective Blochqvector’s transverse projection J⊥ ≡ Jx2 + Jy2 . Specifically, we define the transverse coherence
Cd = J⊥d /J⊥0 , where the subscript d refers to J⊥ after dephasing and 0 refers to J⊥ prior todephasing. In the present work, the inhomogeneous broadening will be achieved through lightshifts, but could also be realized through magnetic fields or electric fields. Whatever method isused, the key is that the dephasing must be reversible: at a later time the opposite rotation can berealized Rẑ (−ψi ) to fully or partially undo the dephasing. Here the dephasing will be undone byusing a π-pulse (e.g. Rŷ (π)) followed by identical inhomogeneous broadening.
To theoretically show that dephased collective spin vectors are protected from small rotation
errors, we analyze the rotation error of a nominal π-pulse, with fractional amplitude error  anddetuning error δ of the applied coupling field from the atomic transition, that is preceded andfollowed by dephasing steps. The final Bloch vector after such a sequence is
JF = ΣN
i=1 Rẑ (ψi )Rγ̂ (β)Rẑ (ψi )Ji0 ,(5.1)

where the subscript F indicates a quantity after all rotations.
The effective rotation angle is a function of both  and δ and can be written β = πp(1 + )2 + δ ∗2

where δ ∗ = δ/Ω and Ω is the on resonance Rabi frequency of the applied rotation. In the rotating
frame of the applied field, the rotation axis depends on the detuning error, γ̂ ∝ Ωα̂ + δẑ. For an
arbitrary initial Bloch vector, the rotation axis α̂ = ŷ can be chosen without loss of generality.
As an example, we assume that the inhomogeneous phase rotation angles ψi are drawn froma Gaussian distribution with mean of zero and rms value σ. The reduction in transverse coherence2

due to the applied inhomogeneous broadening in this case is Cd = e−σ /2 . The complete sequenceof applied broadening and imperfect rotations can then be averaged over all atoms to compute thefinal Bloch vector JF with solution,

56









2
 Jx0 (1 − η ) + Cd πJz0

 JxF 









 JyF  ≈ −  Jy0 (1 − η 2 ) − Cd 2δ ∗ Jz0









2
∗
JzF
Jz0 (1 − 2η ) − Cd (2δ Jy0 + πJx0 ).

(5.2)

where η 2 ≡ π 2 2 /4 + δ ∗2 . We have assumed here that π, δ ∗ , and Cd  1, and neglected all termsof third order in products of these quantities.
The key result is that all rotation errors that are first order in π and δ ∗ are reduced by afactor Cd . The cost of this error suppression is shortening of the Bloch vector, but only at second
order in the rotation error η. The final transverse Bloch vector component CF = J⊥F /J⊥0 is
reduced as CF ≈ 1 − η 2 , and the ẑ projection of the Bloch vector is reduced to JzF /Jz0 ≈ 1 − 2η 2 .
Fig. 1 graphically demonstrates the reduced sensitivity of an arbitrary CSS to a rotation
about an axis on the equator. The rings of constant color indicate the location of the tips of
the Bloch vectors with equal Jz at the beginning of a rotation sequence (top left). Subsequent
steps indicate how these points are mapped to new positions due to rotations and dephasing, with
the initial ring locations shown in black for reference. The figure depicts the effect of an errorπ = π/16, δ ∗ = 0 rotation about the y-axis with and without dephasing to Cd = 0.14, a reasonable
experimental value. Without dephasing, the rotation error can cause angular deflections greaterthan the representative quantum noise distribution (shown for N = 120 for visual clarity). Withdephasing, the rotation error is greatly reduced, causing negligible error compared to the quantum
noise.
The dephased rotation scheme exhibits an additional useful attribute for suppressing rotation
errors in the generation and manipulation of spin-squeezed ensembles. Dephased rotations cansignificantly reduce the amount of anti-squeezing projected into the low noise squeezed quadrature
by a rotation error. We show this theoretically in the Supplementary Material.
We apply the proposed scheme to collective measurements of N = 2.1 × 105 to N = 5 ×
105 87 Rb atoms laser-cooled and trapped inside an optical cavity of finesse F = 660 (see ref. [19]for experimental details). The atoms are tightly confined by a 1D optical lattice formed by exciting

57

(a)

87

Rb atom

Intensity

Detect
Probe

(b)

(c)

Figure 5.2: (a) The standing wave intensity of each beam is shown inside the cavity (blue mirrors).
The atoms are trapped at antinodes of the 823 nm optical lattice (blue). The probe laser at
780 nm (red dashed) and dephasing beam at 795 nm (green dotted) cause dephasing due to theirinhomogeneous light shifts. We detect the phase of the probe light to infer N↑ . (b,c) The reduction
in transverse coherence after dephasing Cd (red and green squares) and rephasing CF (black circles)is measured versus the average number of transmitted photons from the probe beam (b) anddephasing beam (c).

58
a longitudinal TEM00 mode of the cavity with wavelength λl = 823 nm. The atoms fill latticesites long the central 2 mm of the cavity. The spin system is defined by hyperfine ground states|↑i = |F = 2, mf = 2i and |↓i = |F = 1, mf = 1i. Coherent rotations between these states are
performed by applying microwaves at the transition frequency 6.83 GHz. N↑ can be inferred bymeasuring the dispersive frequency shift of another TEM00 cavity mode tuned ≈ 200 MHz fromresonance with the optical transition between |↑i and an excited state |ei = |F 0 = 3, mf = 3i on
the 780 nm D2 line [19].
The probe light at λp = 780 nm that is used to measure the cavity frequency shift and inferN↑ also creates an inhomogeneous light shift that dephases the atoms. Since the standing waves ofthe lattice and probe are incommensurate (λp 6= λl ), the atoms at different lattice sites experience
different light shifts from the probe, leading to dephasing (shown in Fig. 5.2).
We can also apply an additional dephasing laser tuned to resonance with yet another TEM00
longitudinal mode of the cavity. This dephasing beam is detuned ≈ 50 GHz from the 795 nm D1
optical transition and allows us to modify the amount of dephasing without modifying the signal tonoise of the atom number probe or causing additional unwanted free-space scattering. The 795 nm
beam also serves to dephase the sub-class of atoms at lattice sites that are at anti-nodes of theprobe mode (see Fig. 5.2(a)). Because the atoms are tightly confined with respect to the cavity
axis, the same light shifts can be applied at a later time. after a π-pulse, to reverse the applied
phase shifts.
We can measure Cd due to dephasing from the probe and dephasing lasers by first preparing
a coherent spin state along x̂. We then apply either the probe or dephasing laser for a varying
amount of time, after which we apply the rotation Rα̂ (π/2) about a random axis α̂ lying in thex̂-ŷ plane. Lastly, we measure the number of atoms N↑ . When averaged over all rotation axes,
the standard deviation of N↑ is proportional to Cd . In Fig. 5.2(b) and (c), Cd and CF are plottedversus the average number of probe Mp and dephasing Md photons transmitted through the cavity.
For small Md the transverse coherence only shows second order reduction (1 − CF ∝ Md2 )), due tothe large detuning of the dephasing beam from the optical transition. In contrast, for small Mp

59
the transverse coherence loss is linear (1 − CF ∝ Mp ) due to the higher probability of single-atomwave function collapse from free-space scattering of probe photons.
In Fig. 5.3(a) we demonstrate reduced sensitivity to rotation noise arising from environmental noise sources using our dephased rotation scheme. Data showing reduced sensitivity tointentionally applied rotation errors can be found in the Supplementary Material. In our experiment, undesirable environmental rotation noise arises primarily from microwave amplitude noise
and frequency fluctuations in the magnetic field-sensitive hyperfine transition. To demonstrate areduction in sensitivity to environmental noise sources, JzF is measured after a large even numberof π-pulses. With increased dephasing, the rotation-added noise can be reduced below QPN evenafter eight π-pulses.
We now show how dephased rotations can be used in experiments to generate entangled,
spin squeezed states by making precise collective measurements of the spin projection Jz . Theseexperiments are treated in detail in a related work [19]. Here we primarily emphasize the roledephased rotations can play, enabling large reductions in technical rotation noise and allowing
resolution of the spin projection far below the quantum projection noise level. In our experiment,
dephased rotations are highly advantageous to composite pulse sequences because they do not
require any control of the applied rotation axis and do not increase the duration of the measurement
sequence, which would increase sensitivity to low frequency noise.
To verify that the noise in Jz is below QPN, two consecutive measurements of Jz , labeledJz,p and Jz,f must be correlated below ∆Jz,QP N . The degree of spin noise compared to quan2
tum projection noise is characterized by the spin noise reduction R = [∆(Jz,f − Jz,p )]2 /∆Jz,QP
N,

where ∆(Jz,f − Jz,p ) is the standard deviation in the differential quantity Jz,f − Jz,p . The measurement sequence for R is shown in Fig. 5.3(b). The spin noise reduction has two contributions
R = Rbck + Rrot . One, Rbck , we attribute to measurement imprecision of the experiment along withmeasurement back-action. The other, Rrot , is rotation-added noise from the two π-pulses in themeasurement sequence. We estimate Rbck (black line in Fig. 5.3(c)) by performing the measurement sequence of 5.3(b) without the π-pulses. The experiment is then repeated with the π-pulses

60

(a)

(b)

Rotate:{(c){

Probe/
Dephase:

time

(d)

21 dB

Figure 5.3: (a) The rms noise in the measured spin projection JzF , ∆JzF , after applying an integer
number of π-pulses is displayed for three different amounts of applied dephasing, quantified by
Md . The contribution to ∆JzF due to finite measurement resolution (i.e. ∆JzF at 0 π-pulses) is
subtracted out. For Md = 0, a linear fit extracts the rotation-added noise per pulse (red line).
Predictions (green and black bands) using Cd from Fig. 5.2 reasonably explain the reduction inrotation-added noise with increased Md . All shaded regions represent 68% confidence intervals.
(b) Dephased rotations are applied in a sequence designed to resolve spin populations below QPN.
N↑ is measured before and after a π-pulse with outcomes labeled N↑,p , N↓,p , N↑,f and N↓,f todetermine the spin noise reduction R. Both the 780 nm probe and 795 nm dephasing beams areapplied during each measurement of N↑ and N↓ . (c) R is measured as a function of probe strength
Mp for Md = 6.1(3) × 106 (blue data and fit) and Md = 0 (red data and fit). All quantities are
displayed in units of dB relative to QPN, dBQPN . The fit to the measurement background Rbck isshown in black. (d) The rotation-added noise Rrot , shown in the inset, can be inferred from thedata of part (c). Rrot with no dephasing is shown as a dashed line. Dephasing can reduce Rrot by
more than 21 dB.

61
included, and any increase in R is assigned as rotation-added noise Rrot .
In Fig. 5.3(c), the measured spin noise reduction and measurement background are shown
versus Mp (transmitted probe photons in a single measurement window). With the probe beamalone (i.e. Md = 0), the spin noise reduction R (red data and fit) lies well above the measurement
background Rbck (black line). However, when the additional dephasing is applied with strength
Md = 6.1(3) × 106 , the observed R (blue points and fit) is improved to values very close to themeasurement background.
Fig. 5.3(d) (inset) displays the inferred rotation-added noise Rrot with and without theadditional dephasing applied (blue and red lines respectively). The combined dephasing of the probe
and dephasing beams allows a reduction of the rotation-added noise of greater than approximately
21 dB compared to the original rotation noise with no dephasing (i.e. Rrot ≈ 0 dBQPN when Mp = 0
and Md = 0) enabling up to R = 13(1) dB of spin noise reduction below the QPN at 2.1 × 105
atoms.
The rephasing nearly completely restores coherence, as demonstrated in Fig. 5.2. As a result,the state generated after the premeasurement can be viewed as a determinstically generated spinsqueezed state (i.e. no post-selection), conditioned on knowledge of the measurement outcome Jz,pon a given trial. After accounting for both the degree of spin-noise reduction R and the loss ofcoherence CF , the optimum measurement sequence with dephasing provides a directly observed
enhanced phase resolution 9.5(5) dB below the SQL. In contrast, without any reversible dephasing,rotation-added noise would have precluded the observation of any enhancement beyond the SQL.

5.2

Reduction of applied frequency and amplitude errors
The main text shows both experimentally and theoretically that dephasing reduces an en-

semble’s sensitivity to collective rotation errors arising from imperfections in the coupling field used
for state manipulation. To gain intuition, consider the case when the initial Bloch vector lies on
the equator of the Bloch sphere (Jz = 0). In this case, the length of the dephased Bloch vector Jd
is reduced, by definition of Cd , to Jd = Cd J0 . Since the length of the Bloch vector is reduced by

62
Cd , the possible change in the Bloch vector’s Jz projection due to a rotation must also be reduced
by Cd .
Fig. 5.4 demonstrates this reduced sensitivity to intentionally applied rotations representing
amplitude and frequency errors in the coupling field. Measurement sequences are shown at thebottom of Fig. 5.4. A π/2-pulse initializes the Bloch vector at the equator (black pulse). Dephasing
is applied with a strength characterized by the average number of photons transmitted throughthe cavity Md (yellow pulse). A rotation representing an error (either amplitude (a) or detuning(b)) is applied, after which N↑ is measured by measuring the shift of the optical cavity resonancefrequency[19, 32]. From the measured N↑ , we infer the z-projection of the final Bloch vector Jzf .
For part (a), the applied rotation is Rα̂ (ψ), where ψ is the arbitrary rotation amplitude,and α̂ is a random rotation axis lying in the x̂ − ŷ plane of the Bloch sphere. Jzf (blue points)
is plotted as a function of the applied amplitude ψ for three different values of dephasing. The
randomization of α̂ causes large scatter of Jzf over positive and negative values. To compare withan expectation, we plot the average magnitude of the measured Jzf in red, and a prediction basedon the independently measured transverse coherence Cd from Fig. 2 of the main text is shown as a
black line. The envelope of the data decays linearly with Cd in reasonable agreement Eq. 2 in themain text and our intuitive expectation.
To demonstrate the reduction in sensitivity to rotations for which the coupling field is detuned
from the atomic resonance frequency, we apply a nominal π-rotation with variable detuning. The√
applied rotation is Rγ̂ (π 1 + δ ∗2 ) where γ̂ ∝ Ωα̂ + δẑ, and the azimuthal axis α̂ is randomizedδ
between each trial. δ ∗ = κ/2
is the detuning of the coupling field from atomic resonance in cavity

half-widths. In Fig. 5.4(b), Jzf (blue points) is plotted versus δ ∗ for three different values of Md ,and the average magnitude of Jzf (red points) are compared to a prediction (black line). Just as
with the amplitude errors, the magnitude of the deflections of Jzf scale linearly with Cd in good
agreement with the prediction.

63

(a)

(b)

MeasureMeasure

Time

Time

Figure 5.4: Reduction in sensitivity to amplitude and frequency rotation errors. (a) The ensemble issubjected to a rotation with arbitrary amplitude ψ and equitorial rotation axis α̂ and zero detuning(measurement sequence shown below graph). The resulting Jzf (blue points) are plotted versus
amplitude of the rotation ψ for three different values of dephasing, characterized by Md . At each
amplitude the average magnitude of Jzf (red points) is compared to a prediction using the measured
transverse coherence from Fig. 2 (black line). (b) The ensemble is subjected to a rotation witharibitrary detuning δ ∗ and azimuthal rotation axis α̂. The amplitude is constrained so that at zerodetuning, the rotation is a π-pulse. Jzf (blue points) are plotted as a function of δ ∗ , again for threedifferent values of dephasing, and the average magnitude of Jzf (red points) is in agreement to a
prediction from the measured transverse coherence (black line).

64

5.3

Dephased rotation of quantum noise
When dealing with spin squeezed states, classical rotation errors can rotate the anti-squeezed

quadrature of a system into the measurement basis. This leads to additional noise above the noisein classical rotation errors primarily considered in the main text. In this section, we treat thisproblem theoretically with a fully quantum mechanical description of the spin state, instead oftreating each spin as a classical vector as in the main text. We show that dephasing protects
squeezed noise distributions from rotation of the anti-squeezed spin projection into the originallysqueezed quadrature.
We assume for simplicity that there is an arbitrary initial state oriented along ŷ with asqueezed noise distribution in Jˆz (and anti-squeezed in Jˆx ). The state is subjected to a small
rotation amplitude error of size π around the ŷ-axis which rotates the anti-squeezed spin projection
into the z quadrature. The initial squeezed noise distribution is characterized by the second order
expectation valuehJˆz2 i0 ≡ hψ0 | Jˆz2 |ψ0 i ,(5.3)

where |ψ0 i describes the initial state. We wish to evaluate the noise distribution of Jˆz for the final
state,hJˆz2 if ≡ hψf | Jˆz2 |ψf i ,(5.4)

where the final state |ψf i can be written in terms of |ψ0 i using the dephasing operator
D̂ ≡

N
Y

R̂ẑ (θi )

(5.5)

i

and rotation operator (around ŷ) R̂ŷ (π), |ψf i = D̂† R̂ŷ (π)D̂ |ψ0 i. For now, we do not need to
specify a specific form of dephasing (i.e. the inhomogeneous rotations θi ). Using the dephasing androtation operators, Eq. 5.4 becomes65hJˆz2 if = hψ0 | D† R̂ŷ (π)† D̂Jˆz2 D̂† R̂ŷ (π)D̂ |ψ0 i .

(5.6)

The rotation operators can be written in terms of single atom spin projection operators as, tosecond order in the small parameter π,R̂ŷ (π) ≈

N
Y
(π)2
(Ik + iπJˆŷ,k −
Ik ),
2

(5.7)

k

where Ik is the identity operator for the kth atom, and the collective spin projection operator JˆŷP
can be written as a sum over all atoms’ individual spin operators Jˆŷ = Ni Ĵi · ŷ. Using thesedefinitions and single atom commutation relations, we simplify Eq. 5.6 keeping to second order in,hJˆz2 if = hJˆz2 i0 + (π)2 hJˆx2 id − (π)2 hJˆz2 i0 + 2πhJˆz Jˆx id .

(5.8)

The expectation values in the second and final terms (with subscript d) are calculated with respect
to the dephased state, |ψd i = D̂ |ψ0 i. This equation is particularly useful because it gives thequantum noise rotation in terms of measureable quantities for an arbitrary form of the dephasing.
For squeezed states oriented along ŷ with symmetry around x̂ and ẑ (generated, for example,by 2-axis twisting or quantum non-demolition measurement [32, 79]) the final term in Eq. 5.8 iszero, givinghJˆz2 if = hJˆz2 i0 + (π)2 hJˆx2 id − (π)2 hJˆz2 i0 .

(5.9)

This result shows that the back-action quadrature is introduced at order 2 through hJˆx2 id insteadof hJˆx2 i0 . In the limit of random Gaussian dephasing (as considered in the main text), hJˆx2 id canbe written,2hJˆx2 id = Cd2 hJˆx2 i0 + (1 − Cd2 )∆JQP
N.

(5.10)

66
In the limit of small and moderate dephasing, the standard deviation of the back-action quadraturethat is rotated into ẑ is reduced linearly with Cd (first term). However, for complete randomdephasing, the back-action can only be reduced to the quantum projection noise level for a CSS,√∆JQP N = N /2 as seen by the second term. Applying greater dephasing provides marginal returns
when the two terms in Eq. 5.10 become equal. This occurs at a value of Cd which we label Cd0 ,Cd02 =

1.
2
2
ˆ
1 + hJx i0 /∆JQP
N(5.11)

2
For a spin squeezed state with a large back-action quadrature, hJˆx2 i0  ∆JQP
N , more dephasing

is required to reduce hJˆx2 id , the dephased back-action projection, to near the QPN level.

Chapter 6

3rd Generation Spin Squeezing and Deterministic Squeezing with Feedback

6.1

Introduction
After achieving a factor of 10 in squeezing, the obvious question arose: how could we reach afactor of 100? Working on the cycling transition had effectively removed the limit to squeezing duepto Raman spin flips, S = 8p/(qN C) where p is the probability for a free space scattered photon
to cause a Raman spin flip, q is the total quantum efficiency for detecting cavity photons, and N C
is the collective cooperativity [32]. The new observed squeezing was instead limited by decoherence
from free space scattering, laser frequency noise, and noise induced from atomic motion due tooptomechanical forces [19]. These issues prompted a third spin squeezing attempt which achieveda factor of 59 in spin squeezing and was realized through significant engineering improvements anda rebuilt experiment in the basement of the JILA X-wing.
One of the primary improvements was a significant increase in the quantum efficiency q for
detecting probe light. This lead to a reduction in optomechanics, since more spin resolution could
be gained with lower probing power, and an improvement in the fundamental limit due to freespace scattering S = 1/(qN C). In the 2nd generation experiment, we only reached a quantum
efficiency of q ≈ 5%, limited by a 50% lossy optical cavity, use of heterodyne detection, various
small losses due to detecting both sides of the symmetric optical cavity, path efficiency, quantum
efficiency of silicon photodetectors, and technical noise in the AD8015 transimpedance amplifiers
used for heterodyne detection [19].
By building a new, lower loss, single-ended optical cavity and moving to homodyne detection

68with a custom detector sensitive at both DC (for homodyne) and AC (for path length stabilization),
we achieved a quantum efficiency in a single measurement of nearly 40%, roughly 8 times better
than in the second generation. This new system and quantum efficiencies are described in detail in
Sect. 6.4.
In generation 2, the frequency stabilization of the probing laser was achieved by narrowing
Photodigm DBR lasers with a long external optical cavity [100]. Details of this technique can befound in Ref. [170]. Using this technique, the optimum frequency resolution of the optical cavity
which was achieved was approximately 20 kHz rms, or 17 dB below the typical projection noise
driven fluctuation of the cavity freqeuency of 140 kHz [19]. In order to improve this floor to achieve
close to 20 dB of directly observed squeezing, we implemented a relative stabilization of the atomic
probe laser to the cavity using an additional “cavity probe” laser resonant with another TEM00
optical cavity mode 122 GHz away from the D2 atomic transition. This system achieved frequency
resolution of the cavity of approximately 5 kHz and is explained in detail in Sect 6.4.
In all, these and other engineering improvements lead to a directly observed squeezing of
17.7(6) dB below the standard quantum limit using 4 × 105 atoms. This result, along with thesimilar result reported by [68] is the largest entanglement enhancement of any kind in atomic or
optical systems.
Further, with this new spin squeezing apparatus, we demonstrated real-time feedback to asqueezed state to create so-called “deterministic” squeezing, where the result of the pre-measurement
is no longer necessary to obtain enhanced resolution. Using this real-time feedback we created oneof the largest amounts of deterministic squeezing ever observed, comparable to the amounts generated by the best naturally deterministic schemes such as 1-axis twisting using cavity feedback or
collisions in an ultra-cold gas [95, 62].
In this Chapter, in Sections 6.2 and 6.3, I will first present the primary results from thethird generation of spin squeezing experiment. Then, I will elaborate on the apparatus used forthese measurements in Sect 6.4 and report additional relevant information regarding backgrounddecoherence, back-action measurements, and the feedback scheme in Sections 6.5 6.6 and 6.7 re-69
spectively.

6.2

Spin squeezing with real-time feedback
This section features two main results. First, following Fig. 6.1(a), we use the outcome of acollective, or joint, measurement to actively steer the collective spin-projection of an ensemble of 5×
104 laser-cooled and trapped 87 Rb atoms to a target entangled quantum state. Real-time feedback
allows generation of the target state with enhanced angular resolution S −1 ≡ (∆θSQL /∆θ)2 =5.5(8), or 7.4(6) dB below the SQL, with no background subtractions. Second, we perform a direct
subtraction of quantum noise without feedback and directly observe a conditionally enhanced phaseresolution S −1 = 59(8) or equivalently 17.7(6) dB below the SQL, along with [68], the largest phaseestimation enhancement from entanglement to date in any system.
Entanglement is often created and manipulated via unitary interactions between qubits [136,93, 113, 118, 102, 132, 62, 114, 28, 95]. However, the joint measurements on two or more qubits
used here (sometimes referred to as quantum non-demolition measurements) have shown promisefor creating entanglement, particularly among large numbers of qubits [88, 4, 140, 169, 96, 31, 19,
12, 142, 119, 133, 134]. By adding real-time feedback guided by the outcome of joint measurements,
one can access a more diverse range of quantum technologies including Heisenberg-limited atomic
sensors [26], reduction of mean field shifts in atom interferometers [47, 75], quantum teleportation[144, 86], and error correction [116, 165]. Quantum noise suppression with real-time feedback has
been considered theoretically [155, 154] and demonstrated in a previous experiment [72] but without
the critical enhancement in phase resolution that signifies entanglement.
We visualize a collection of N spin-1/2 atoms as a single collective Bloch vector J =Jx x̂ + Jy ŷ + Jz ẑ given by first order expectation values Jα ≡ hJˆα i of collective spin projection
operators with α = {x, y, z}. The quantum projection noise (QPN) and resulting SQL can be
intuitively visualized by a quasi-probability distribution perpendicular to the classical Bloch vector (Fig. 6.1(a)). The distribution’s rms fluctuations along a given spin-projection direction areq
2given by ∆Jα ≡ hJˆα2 i − hJˆα i . In this chapter ∆ will refer to the standard deviation of a given

70

Figure 6.1: (a) A coherent spin state’s spin-projection noise (pink distribution) is projected onto asqueezed state by a measurement of Jz . The quantum state randomly collapses within the original
distribution, creating a conditionally squeezed state. The pre-measurement’s outcome is then used
to rotate the spin state’s polar angle to a desired target spin projection (black solid line) Jz = Jztar ,creating a deterministically squeezed state. (b) The relevant 87 Rb energy levels (black) and cavity
resonance frequency ωc (blue). (c) Simplified experimental diagram. The cavity is probed inreflection. Homodyne detection of the probe is sampled by a microcontroller that then applies
microwaves at 6.8 GHz to achieve the desired feedback rotation θf b to create the deterministicallysqueezed state in (a). See Section 6.4 for experimental details.

71
quantity. For a coherent spin state oriented at the equator of the Bloch sphere, the spin projection
Jz and spin population N↓ both fluctuate from one trial to the next with a standard deviation√∆Jz,QP N = ∆N↓QP N ≡ N /2.
We calculate the enhancement in phase resolution S ≡ (∆θ/∆θSQL )2 = R/C 2 [174], whereR ≡ (∆Jz /∆Jz,QP N )2 is the observed spin projection noise relative to the projection noise level,ˆ /N is the fractional atomic coherence remaining (or “contrast”) after a joint
and C ≡ 2 h|J|i
measurement. An additional 0.2 dB correction is applied to S for a 4% background loss of contrast(see Supplementary Material). Observing S −1 > 1 serves as a witness for entanglement between
atoms [149] and the magnitude usefully quantifies the degree of entanglement [174, 79].
A joint measurement of the population of atoms N↑ is engineered by measuring the frequency
shift of a TEM00 cavity mode. The cavity is tuned δc = 2π × 400 MHz to the blue of the 87 Rb|↑i ≡ |52 S1/2 , F = 2, MF = 2i to |ei ≡ |52 P3/2 , F = 3, MF = 3i optical atomic transition as shownin Fig. 6.1(b). The second state forming the pseudo-spin system is |↓i ≡ |52 S1/2 , F = 1, MF = 1i.
The cavity has finesse 2532(80) and power decay linewidth κ = 2π × 3.15(10) MHz. The atoms arelaser-cooled to 10 µK and trapped tightly on axis in an intracavity 1D optical lattice (Fig. 6.1(c)).
Spatially inhomogeneous coupling of atoms to the cavity mode is handled as in [31, 32, 19, 69].
Atoms in |↑i strongly phase shift the intracavity probe light, causing the empty cavity resonancefrequency ωc to shift to ωc0 . A measurement of the shift ωc0 − ωc using homodyne detection ofprobe light reflected from the cavity can then be used to infer the population N↑ . To measure thepopulation N↓ , a π-pulse microwave coupling can then be applied to swap the populations between|↑i and |↓i, and a measurement of the new population in |↑i can be made with the measurement
outcome now labeled N↓ .
The experimental sequence is shown in Fig. 6.2 (a) and (b). All atoms are prepared in |↓i,then a microwave π/2-pulse is applied to place each atom in an equal superposition of spin states,
equivalent to preparing the Bloch vector along ŷ. We make a measurement of the spin projection Jz
with measurement outcome labeled Jzp = (N↑p − N↓p )/2. Each population measurement outcomeN↑p and N↓p is obtained by averaging the cavity-probe signal over a 40 µs window. In each run

72of the experiment, a microcontroller calculates Jzp and applies feedback to steer the state towarda targeted value of spin projection Jztar . The feedback is accomplished by applying microwaves
to rotate the Bloch vector through polar angle θf b ≈ 2 × (Jztar − Jzp )/(N C). After the feedback,a final measurement of the spin projection Jz is made with measurement outcome labeled Jzf =(N↑f − N↓f )/2. Feedback toward Jzf = 0 is evident in the time trace (Fig. 6.2 (a)), since the final
two cavity frequency measurement windows that provide N↑f and N↓f are more nearly equal than
was the case for the two pre-measurement windows.
The microcontroller sets the sign of the rotation θf b by digitally toggling between two microwave sources that are 180◦ out of phase. The magnitude of the rotation |θf b | is controlled byvarying the duration tf b for which the microwaves are applied, with a discrete timing resolution of
approximately 12 ns. The input technical noise floor, timing jitter, and timing resolution of the
microcontroller are all sufficient to allow up to 20 dB of squeezing.
The outcomes Jzp and Jzf are plotted versus trial number and collated into histograms in
Fig. 6.2(c). Projection noise for this data (independently confirmed by measuring the scaling of∆Jz with N ) is ∆Jz,QP N = 218(10), consistent with the measured ∆Jzp = 235(24). The data on
the right shows the final measurement outcomes Jzf after applying feedback for five different target
states Jztar . By implementing the feedback, each target state was reached with noise below theoriginal projection noise.
To observe deterministic squeezing or phase resolution enhancement, the atomic coherence
that remains after the pre-measurement and feedback must be evaluated. The contrast is determined in a separate set of experiments by using microwave rotations after the feedback step torotate the Bloch vector to determine its total length. Accounting for the loss of coherence, we
directly observe up to S −1 = 5.5(8) (7.4(6) dB) of deterministic squeezing via pre-measurement
and feedback.

73

Figure 6.2: (a) Measured cavity resonance frequency for a single trial versus time, subtracting a
constant 12 MHz frequency offset. (b) The time windows in which the probe is turned on (green)
and the populations determined from each window. The fixed microwave rotations are shown inblack with the feedback rotation shown in orange. (c) The pre-measurements Jzp (left) and final
measurements Jzf (right) of Jz are plotted versus trial number and accumulated into histograms.
Five different Jz states are targeted (five distinct colors on right) and reached with noise below
QPN. The maximum deterministic squeezing is S = −7.4(6) dB relative to the SQL. (d) Feedback
reduces the noise distribution of the final measurement relative to the initial quantum noise in
the pre-measurement. (e) If no feedback is applied the final and pre-measurement are strongly
correlated (black), allowing for conditional squeezing (S = -10.3(6) dB) by using the differential
quantity Jzf − Jzp (gold). The increase in noise from feedback is discussed in the Supplementary
Material.

74

Figure 6.3: (a) Experimental sequence for conditional spin squeezing, with labeling mirroring thatof Fig. 6.2a. (b) Squared contrast C 2 (blue), spin noise R (red), and spin squeezing S (black)are plotted versus the average number of incident photons Mi in a single measurement window.
The solid lines are fits, the blue band is the predicted loss of contrast from free-space scattering,and the grey band indicates the total squeezing error bar. (c) The experimental sequence used toobserve the back-action spin-projection. (d) The measured spin noise R is plotted versus ψ withfit (purple). (e) The reconstructed conditional probability distribution of the quantum state (red)on a Bloch sphere with Bloch (black) vector. The distribution is magnified with a 1:1 aspect ratio
and plotted with the equivalent coherent spin state (blue) in the lower panel. (f) Thermal radial
motion of the atoms causes the spin noise R to oscillate at twice the radial trap frequency as thetime separation T between the pre- and final measurements is increased.

75

6.3

17.7 dB of conditional squeezing
For many applications, the feedback is not necessary. Instead of applying feedback, one

can cancel the quantum noise by directly subtracting the pre-measurement Jzp from the final
measurement Jzf , a technique known as conditional squeezing [88, 4, 140, 169, 96, 31, 19, 12, 142].
In Fig. 6.2(d) and (e), we compare conditional and deterministic spin noise reductions taken underidentical settings. Jzf is plotted versus Jzp and the results are collated into histograms on eachaxis. With feedback (red), Jzf is driven to zero with resolution below ∆Jz,QP N , regardless of Jzp .
Without feedback (black), Jzp and Jzf are correlated, and the quantum noise can be conditionallysubtracted from the final measurement by taking the difference Jzf − Jzp (gold).
The deterministic squeezing with feedback is primarily limited by errors in the π-pulses due
to microwave amplitude and frequency noise. However, by increasing the number of atoms to
N = 4 × 105 , we improve the amount of conditional spin squeezing to S −1 = 59(8) or 17.7(6) dB.
The experimental measurement sequence is the same, but to avoid added noise from the π-pulses,
we only consider the reduction in the noise of the difference of two population measurements of thesame spin state R = (∆(N↓f − N↓p ))/∆N↓QP N )2 ( Fig. 6.3(a)). The information gained from thefirst measurement N↑p is not used here, but its presence serves to spin-echo away probe-induced
inhomogeneous light shifts at the end of the pre-measurement pair N↑p and N↓p . Because theBloch vector lies at the equator, small angular displacements of the polar angle could be sensed
from changes in a single spin state’s population alone.
In Fig. 6.3(b), we show the noise reduction R versus the average number of photons Mi
incident upon the cavity during a single probe measurement window. Again, this is the directly
observed noise reduction with no background subtractions or removal of noise of the final measurement applied. The maximum quantum noise reduction is R−1 = 92(9), or 19.6(4) dB below
QPN and is limited by both a technical noise floor 25 dB below QPN and optomechanical effects
induced by the probe light being turned on and off, an effect that increases with Mi . Also apparent
in Fig. 6.3(b), the atomic coherence or contrast (blue) after the pre-measurement decreases with

76
increasing Mi due primarily to undesired free space scattering causing collapse of individual atoms’
wave functions into spin up (blue prediction band). The background contrast CBG is obtained
from a measurement with Mi = 0 in the two pre-measurement windows. The black data and fit inFig. 6.3(b) display the squeezing obtained by combining the reduction in noise with the reduction
in contrast.
We also examine the back-action or anti-squeezed spin-projection. The experimental sequenceis shown in Fig. 6.3(c) and is distinguished by the replacement of the rotation θf b with a microwave
rotation about an axis parallel to the Bloch vector through a fixed angle ψ. Figure 6.3(d) shows
the increase in spin noise R moving from the 17 dB squeezed (at ψ = 0) to anti-squeezed (atψ = ±90◦ ) projections. Using an inverse Radon transform, we construct a visualization of theequivalent squeezed state, shown in Fig. 6.3(e). The original coherent state noise is shown in blue.
The state has ∆Jz ∆Jx /(∆Jz,QP N )2 = 6.1 > 1 and is no longer a minimum uncertainty stateowing to finite quantum efficiency for detecting the probe light. From the increase in area and its
scaling with Mi we can infer the quantum efficiency of a joint measurement of a single populationis Q̃1 = 38(14)%, in good agreement with an independent prediction of 37(5)% from measuring
path efficiencies, cavity loss, detector efficiencies, technical noise floors, and laser turn-on times (seeSect. 6.4). Here, the total quantum efficiency of the full measurement sequence (N↑p , N↓p , N↓f ) is
effectively 4 times lower than Q̃1 due to the additional noise in the final measurement N↓f and thepresently unused pre-measurement N↑p .
In Fig. 6.3(f), we evaluate how well the conditional noise reduction can be maintained
over a variable evolution time T . This is an important consideration for implementing conditionalsqueezing in atomic sensors. The contribution to R from technical noise sources is partially removedby performing the measurement sequence of Fig. 6.3(a) with no atoms present and subtracting themeasured noise variance from the noise variances obtained with atoms present. The spin noise R
is seen to oscillate at twice the radial frequency of the trapping potential due to thermal radialatomic motion that causes an oscillation in each atom’s coupling to the cavity mode. The additionalmonotonic increase in R is not currently understood. A 3D optical lattice or a smaller atomic

77
temperature to lattice depth ratio can be used to reduce the noise oscillations in the future.
The improved squeezing relative to previous work [19, 41] was achieved by increasing the
net quantum efficiency for probe detection from 5% to 37% (by constructing a single-ended cavity,
reducing losses on cavity mirrors, and using homodyne detection), increasing the cavity finesse
by 3.5, and implementing a two-probe laser technique that reduced the relative frequency noise
between the probe laser and the empty cavity from 16 to 25 dB relative to projection noise [38].
See Sect. 6.4.
It is physically reasonable to expect that the majority of the atoms participate in a single multipartite entangled state. The entanglement depth, or we believe more appropriately “entanglement
breadth” ζ quantifies the minimum number of atoms that provably particpate in a multipartite entangled state, no matter how weakly [150, 107]. We find the largest breadth ζ = 400(120) atoms
at squeezing S −1 =15 dB, but at the largest squeezing we find ζ=170(30) atoms.
Applying real-time feedback based on the outcome of joint measurements may allow for new
applications in both quantum information technology and precision measurement. For instance, theutility of highly spin-squeezed states suffers from the fact that the state lives on a sphere, causing theback-action spin projection to couple into the measured spin projection Jz if the state is rotated
too far from the equator. In clock applications, this results in needing to reduce the Ramsey
phase evolution time such that the net enhancement in clock precision is far from approaching
the Heisenberg limit [3]. It was recently proposed that joint measurement and feedback similar
to that used here would allow one to actively measure and steer the back-action noise out of
the measured spin projection and would thus allow enhancements in precision approaching theHeisenberg limit [26]. With improved atom-cavity coupling (e.g higher finesse, smaller mode waistsize), even greater amounts of squeezing than that reported here can be achieved in principle [32].
However, it will be critical to consider current limiting effects such as optomechanical ringing andtime-varying couplings between measurements due to atomic motion in order to achieve significant
improvements. Having now shown that large enhancements in phase resolution using entanglement
are achievable in real systems that are compatible with state-of-the-art precison measurements,

78
next steps may include application to matterwave interferometers [47] and optical lattice clocks[119].

6.4

Experimental details
Several technological improvements from previous experiments were key to achieving theresults of this chapter. Full experimental diagrams of the optics and electronics are shown in Fig.
6.5 and Fig. 6.6 respectively in an attempt to highlight differences from previous work, and ageneral procedure is described.

6.4.1

Atomic state preparation

The atoms are loaded from a magneto-optical trap (MOT) whose loading time sets theexperimental repetition rate of 1 second. Polarization gradient cooling is then used to cool theatoms to 10 µK and to load them into 823 nm optical lattice sites spanning approximately 1 mmalong the cavity axis. A bias magnetic field along the cavity axis of 1.1 Gauss sets the quantization
axis. Optical pumping beams are used to polarize the atoms with > 90% probability into the|↓i = |52 S1/2 , F = 1, MF = 1i ground state. After optical pumping, the MOT beams are applied
once more to clear any remaining atoms from the F = 2 manifold. Atoms not in |↓i are notrotated by the microwaves into the measured |↑i state due to the Zeeman splitting between states.
As a result, they do not contribute to the experiment. Lastly, to account for inhomogeneous
coupling of the atomic ensemble to the probe mode, the reported atom numbers N↑ , coupling g,
and cooperativity parameter C, in both the main text and here, are effective values as described inRefs. [31, 32, 19, 69]. Neglecting small corrections for radial inhomogeneity, the total atom numberN↑,tot in the lattice in state |↑i, the single-atom Rabi frequency 2g0 at an antinode of the probe
mode, and the cooperativity parameter C0 at an antinode of the probe mode are related to theeffective quantities by N↑ ≈ 23 N↑,tot , g 2 ≈ 43 g02 , and C ≈ 34 C0 , respectively.

79
6.4.2

Science cavity and lattice

The optical cavity parameters are given in Table 6.1. Compared to previous work [19, 31], thecavity finesse has been increased by a factor of 3.5, or equivalently the cavity power decay linewidthκ is smaller by the same factor. The cavity is now primarily transmissive at a single end, i.e. theinput mirror’s transmission coupling rate κ1 is much greater than the output mirror’s transmission
coupling rate κ2 . As a result, measurement of the probe light in reflection from the cavity captures
nearly all of the information transmitted out of the cavity mode, with effective quantum efficiency
now of κ1 /κ = 0.83(3) compared to the previous effective quantum efficiency of 0.23 in reflection
alone. This eliminates the need for a second detection system for the transmitted cavity light.
The cavity’s frequency is actively stabilized to the frequency of the 823 nm optical latticelaser. This is achieved by a Pound-Drever-Hall (PDH) frequency servo that feeds back to piezos tocontrol the cavity length. The bandwidth of the servo is about 1.5 kHz. The lattice laser is frequency
stabilized to an independent transfer cavity using PDH detection with servo bandwidth of 1 MHz.
The lattice laser’s frequency is tuned relative to the transfer cavity by using a high frequency phasemodulator to place 5 to 8 GHz sidebands on the lattice laser light probing the cavity. The opticalfrequency of a first order sideband is locked to the transfer cavity, such that tuning the modulation
frequency then allows the lattice laser’s frequency to be tuned continuously over several GHz. The
microwave voltage-controlled oscillator (VCO) that provides the modulation is phase-locked to a
DDS that is controlled by the data acquisition computer for ease of tuning. Finally, the frequencyof the transfer cavity is stabilized with 1.5 kHz bandwidth by PDH probing of another longitudinalmode using a 795 nm laser that is stabilized using Doppler-free FM spectroscopy to the D1 transitionin 87 Rb. 795 nm light is used simply for historical reasons.

6.4.3

Relative frequency noise between cavity and probe

Relative frequency noise between the atomic probe laser (200 kHz FWHM nominal linewidth
ECDL laser) and the empty cavity hinders our ability to determine the atomic-induced shift ωc0 −ωc80Figure 6.4: Experimental frequency diagram. Relevant frequencies described in the text are shown
along with the locking scheme of the atomic (blue) and cavity (red) probe lasers. The two longitudinal resonances of the cavity that these two lasers probe are separated by 122 GHz and shown on
the upper graph. The unshifted n+15th cavity mode at ωc is detuned δc blue from the atomic resonance ωa . The presence of atoms in |↑i typically shift this cavity mode by approximately 70 MHz,to ωc0 . The homodyne local oscillator beam is shown in purple (dashed), and feedback stabilization
steps are shown as gold arrows with descriptions.

81

Cavity Parameters (probe λ = 780 nm)
2Single-atom cooperativity C = 4g
κΓ

0.044(6)Single-atom vacuum Rabi splitting g
2π × 0.44(3) MHz
Input coupling κ12π × 2.60(5) MHz
Output coupling κ22π × 0.17(1) MHz
Internal losses κL
2π × 0.38(8) MHz
Linewidth κ
2π × 3.15(10) MHz
Dressed-cavity linewidth∗ κ0
2π × 3.6(1) MHz
Dressed-cavity linewidth† κ0
2π × 3.2(1) MHz
Q.E. due to internal losses κ1 /κ
0.83(3)Finesse2532(80)
Free spectral range
8.105(5) GHz
Frequency difference TEM00 -TEM10
2.290(5) GHz
TEM00 waist size w070(1) µm
Cavity length
1.849(1) cm
Mirror radius of curvature
4.999(5) cm
Cavity Parameters (lattice λ = 823 nm)
Input coupling κ12π × 4.40(10) MHz
Output coupling κ22π × 0.23(1) MHz
Linewidth
2π × 5.8(6) MHz
Finesse1400(150)
Trap depth
115 µK
Circulating power Pcirc
0.30(3) W
Power Buildup (Pcirc /Pinc )800(130)
Axial trap frequency
181(20) kHz
Radial trap frequency
900(50) Hz
TEM00 waist size w0
71(1) µm
Table 6.1: Relevant cavity parameters at the atomic and cavity probe laser wavelength λ = 780 nm
and at the lattice laser wavelength λ = 823 nm. The symmetric, standing wave cavity’s mirror
transmission coefficients, T1 on the probed end (1) and T2 on the closed end (2), are expressed interms of coupling rates κ1,2 = T1,2 × (free spectral range). The atomic decay linewidth of |ei isΓ = 2π × 6.07 MHz. The dressed cavity linewidths κ0 include broadening of the cavity resonanceat ωc0 due to spontaneous scattering from the atoms.

82

AtomicProbe
Laser

Homodyne
AOM
Lcav

From
Fig. S3µFull Power
Half Power

fs=81.1MHz,φs

kick

EOM

rot

6.8 GHzOff

λ/2

9.7 MHz P.D.H.
Demod.
I

ArduinoL

Cavity
Probe
Laser
λ/2

Isolator
99:1 (R:T)

Rb

R

EOM

9.7 MHz

λ/4

λ/2

PBS

Homodyne
Detector

50/50 BS-

RF

To
Fig. S3

DC

Figure 6.5: Optical block diagram. The resonance frequency of the optical cavity ωc0 is detectedusing homodyne detection of the atomic probe laser (red). Homodyne detection is performed on an
fs = 81.1 MHz sideband on the atomic probe laser. This sideband can be applied at half power by
the “kick” switch to provide an extra impulsive kick to the atoms in order to cancel optomechanicalringing (described in Section 6.4.5.5). The carrier of the atomic probe laser is detected in heterodyne(RF port) to provide a path length reference (see Fig. 6.6) for stabilizing the homodyne detection
phase. The cavity probe laser (blue) is P.D.H. locked, via the Lcav loop filter, to another longitudinalmode of the optical cavity, unshifted by atoms, and provides stabilization of the atomic probe laser’s
frequency to the cavity frequency. The atomic probe and cavity probe are separated optically via
polarization. Real-time feedback is applied using an Arduino microcontroller that controls the signand duration of 6.8 GHz µ-wave pulses. More details are given in Fig. 6.6.

83

½fs=40.55MHz,φs

REF

RF

From
Fig. S2

To Fig. S2 µ

DC

-

τ=1s

-

Homodyne
AOM

VCO

Lstab

PFD

81.1 MHz

Feedback µwave pulse
for targeting Jz states

Variable Gainsign

Homodyne
Detector

Path length stabilization

6.8GHz: ϕ ϕ+πSampleHold

SHA

Remove DC offset

CavityProbe
Laser

Sample/Hold

13.6 GHz

EOM

-

SHA

LSHA

Hold probe
frequency whenprobe is off

Phase-lock atomic probe
to cavity probe

Atomic
Probe
Laser

La

PFD

REF

fca

DAQ

Arduino

VCO

I

R

L

90-100 MHz

sweep

DDS

Freq. Offset
70 MHzFigure 6.6: Electronic block diagram. The homodyne detection phase is stabilized by detecting
the carrier of the atomic probe beam with the signal appearing at 81.1 MHz at the RF port. The
phase of this signal is locked to a DDS frequency reference by applying feedback through the Lstab
loop filter to a VCO controlling the homodyne AOM. The homodyne difference signal (DIFF) is
used to stabilize the atomic probe laser to the atom-shifted cavity mode at ωc0 . The signal ishigh-pass filtered at 1 Hz to remove slowly drifting DC offsets and then passed through a variable
gain amplifier (used to maintain constant loop gain as Mi is varied) before entering the loop filter
LSHA . The output of LSHA is used to control a VCO which provides a phase reference to a phaselock between the atomic probe laser and the cavity probe laser using loop filter La . The cavity
frequency ωc0 is detected by sampling the output of LSHA . When the atomic probe is off, a sample
and hold circuit is used to hold the output of the loop filter. A separate synthesizer (DDS) can beused to perform sweeps of the atomic probe. Real-time feedback is applied by the Arduino basedon the sampled output of LSHA . The Arduino can control the sign of the feedback by switching(sign) between two 6.8 GHz sources that are 180◦ out of phase.

84
and hence the atomic populations that constitute our joint measurement of spin projection Jz . To
remove this, an improved stabilization scheme of the atomic probe laser has been implemented
similar to that described in [38], described below and shown in Fig. 6.4.
A second 200 kHz FWHM laser at 780 nm, called the cavity probe laser, is PDH locked with
servo bandwidth 800 kHz to a longitudinal mode of the science cavity that is 122 GHz (15 free
spectral ranges) away from resonance with the atomic transition |↑i to |ei. The large detuningmeans that even at much higher circulating powers inside the cavity, the cavity probe produces
sufficiently small atomic dephasing and spontaneous emission. Frequency noise on the science cavity
is thereby imposed on the cavity probe laser for spectral noise at frequencies below the unity gain
frequency of the servo. Conversely, the original frequency noise of the cavity probe laser is also
reduced relative to that of the empty cavity.
The cavity probe light is circularly polarized σ − , opposite to that of the primary or atomic
probe which is σ + polarized. This allows the reflected cavity probe light to be polarization separated
from the atomic probe after probing the cavity. An avalanche photodiode (Hamamatsu S2381, gain
≈ 150) is used to detect the PDH signal generated by typically 50 nW of total optical probe
power. To maximize the signal-to-noise for a given amount of circulating cavity probe power in thecavity, the PDH signal is derived by phase modulating the cavity probe light at frequency 9.7 MHz
 (κ/2)/2π so that the phase modulation sidebands do not enter the cavity.
To link the cavity probe laser to the atomic probe laser, approximately 2 mW of the cavity
probe laser’s light is phase modulated at fm = 13.6 GHz. The modulation frequency is derived
from a low-phase noise microwave source [34]. The atomic probe laser is then phase-locked to a
9th order sideband at a total offset of 9 × fm = 122.4 GHz from the cavity probe frequency using
loop filter Lcav . The heterodyne signal between the sideband and the atomic probe appears in
the rf spectrum at 500 to 700 MHz. This heterodyne beat note is phase-locked to an rf voltagecontrolled oscillator (VCO) with center frequency fca , and with servo bandwidth of 2 MHz. Bytuning the VCO frequency fca , we can thereby tune the atomic probe laser relative to the cavity,
while frequency noise on the atomic probe is now common-mode with the cavity.

85
A representative power spectral density of instantaneous frequency noise between the atomic
probe laser and the cavity, Sν (f ) is shown in Fig. 6.7a. This data was taken using a heterodyne
detection setup which existed before implementing the final homodyne detection scheme describedin this chapter [38]. However, for this data, the power in the atomic probe laser beam was turned
up so that technical noise due to laser freqeuency fluctionations dominated the fundamental photon
shot noise and any technican noise associated with the detection scheme. The data in Fig. 6.7
faithfully characterizes the quality of frequency stabilization between the atomic probe and theoptical cavity for all of the generation three spin squeezing experiments.
In the central flat region of Fig. 6.7(a), Sν ≈ 1.5 × 103 Hz2 /Hz. This corresponds to theinstantaneous frequency noise of a laser with Lorentzian FWHM ∆ν = π ×Sν = 5 kHz, significantly
smaller than the initial laser linewidth of 200 kHz.
The roll off at high frequency results from 300 kHz anti-aliasing low pass filters after the
IQ demodulation. The rise at low frequencies is largely due to uncontrolled relative path lengthchanges between the atomic-probe LO and atomic-probe paths. We have found it unnecessary tostabilize this path length phase for the preliminary results presented here because it only required
200 µs to measure the differential quantity fcf − fcp .
We are interested in characterizing the noise in the difference between the final and premesaurement. Each measurement is the average of the measured frequency in a window of length
Twin and the two measurement windows have a time gap Tdiff between them. The variance is obR∞
tained by integrating (∆fd )2 = 0 Sν (f ) T (f ) df . The transfer function is T (f ) = 4 sin2 (πf (Twin +Tdiff )) sin2 (πf Twin )/(πf Twin )2 . Figure 6.7b shows the measured noise variance ∆fd as a function of
the measurement window length Twin with t = 0 µs. For this data, the minimum is ∆fd = 6 kHz
at Twin = 100 µs.

6.4.4

Decoherence from the cavity probe

The measurements of Fig. 6.7 were made at very high probe powers with no atoms in thecavity. As a result, the photon shot noise and technical noise sources of the detectors were negligible

86

Figure 6.7: (a) The measured power spectral density of instantaneous frequency fluctuations Sν (f )between the atomic-probe and an empty cavity mode. The frequency stabilization described in
the text reduces the noise by close to a factor of 50 over a broad range relative to the Sν one
expects for the linewidth of our free-running 200 kHz FWHM external cavity diode lasers. For thisdata, the atomic and cavity-probes were set to a high enough power that increasing either did notdecrease Sν (f ), so that we are sensitive only to technical noise floors. Also, heterodyne path-lengthstabilization had not yet been implemented, and this is largely responsible for the rise below 2 kHz.
(b) The integrated noise in the difference of two frequency measurement windows, plotted as afunction of window length Tm , with a fixed t = 0 µs window separation.

87
compared to contributions from other technical noise sources. However, as the power in the cavityprobe is increased, the amount of squeezing may become limited by additional scattering of lightfrom the atoms, potentially leading to single-particle wavefunction collapse (loss of signal) and
Raman transitions to other ground hyperfine states (a source of additional noise.) Inhomogeneous
differential light shifts of the spin transition frequency can also lead to dephasing that can be
spin-echoed away, but perhaps imperfectly.
Figure 6.8a shows the noise variance (∆fd )2 between two measurements of the cavity resonance frequency versus the cavity-probe power incident on the cavity Pc . Here the atomic-probe
power is increased such that its photon shot noise contribution is negligible. The right hand axis
translates the noise variance into an equivalent uncertainty relative to a quantum projection noiselevel ∆fP JN = 97 kHz. For powers above 1 µW, the technical noise floor saturates to 27 dB below
the projection noise level. For lower powers, the noise variance scales as (∆fd )2 ∝ 1/Pc2 , indicatingthat lower powers Pc might be utilized with improved photodetection.
The cavity-probe induces a differential light shift of the transition frequency ωhf between
down and up. We measure the differential shift of the clock transition frequency |F = 1, mF = 0i
to |F = 2, mF = 0i versus Pc in Fig. 6.8b. Appropriately rescaling for transition strengths anddetunings gives an average shift of ω↑↓ by 8.5 kHz per µW. The shift is highly inhomogeneous andleads to dephasing. However, spin-echo measurements have shown that the atomic contrast (i.e.,length of the Bloch vector) is negligibly affected at Pc = 0.5 µW with 40 µs measurement windows.
Thus, the atomic state is not significantly decohered by the cavity stabilization presented here.

6.4.5

Atomic probe

The atomic probe is used to determine the shifted cavity mode frequency ωc0 − ωc . The cavity
probe is σ + polarized to take full advantage of the cycling transition for strong light-atom coupling
as well as to avoid Raman transitions to other ground states caused by spontaneous emission [32].
The circular polarization also facilitates easy separation of the cavity probe light reflected from thecavity for sending to a homodyne detector.

88

Figure 6.8: (a) The 2-window noise variance (∆fd )2 is plotted versus the detected power of thecavity-probe beam for Tm = 40 µs and t = 0. Above 1 µW, the noise variance saturates to 27 dB
below the quantum projection noise level. (b) The measured light shift of the 87 Rb clock transitionis plotted versus the detected power of the cavity-probe beam. The shift is approximately 660 Hz
per µW. The non-zero light shift at Pc = 0 is due to an additional constant light shift from the823 nm optical lattice. Due to differing Clebsch-Gordan coefficients, the shift of the |↓i to |↑itransition frequency ω↑↓ is approximately 13 times larger, but still causes very little coherence loss
for the preliminary squeezing results presented here.

89
Near resonance, the reflected qr quadrature response of the field is directly related to the√incident field ii ∝ Mi and the detuning δp between the probe light and the cavity resonance by
[32]:




4δp  κ1  
qr

=

ii
κ
κ 


(κ/κ0 )2

2 
√


1+

(6.1)

N↑ 2g
2δc0

where δc0 = ωc0 −ωa ; ωa is the optical atomic transition frequency, and the dressed cavity linewidth is
2
2
p
p
κ0 = (κ + ΓN↑ g/δc0 )/(1 +N↑ g/δc0 ). Homodyne measurements of qr allow us to determine
the detuning of the probe from ωc0 .

6.4.5.1

Homodyne phase stabilization

Homodyne detection requires stabilization of the relative path length between the homodyne
reference path and the probe path, as well as removing other sources of relative phase noise. To
achieve this, the homodyne reference light derived from the same laser is shifted up in frequency byan acousto-optic modulator (AOM) driven by a VCO with nominal frequency fh = 81.1 MHz. The
atomic probe light is weakly phase modulated at fixed frequency fs = 81.1 MHz. The lower sideband
is tuned close to resonance with ωc0 while the much stronger carrier component is 81.1 MHz off
resonance from the cavity. The strong carrier component primarily reflects off of the cavity without
creating any additional light circulating inside of the cavity–important for avoiding dephasing andspontaneous emission from this frequency component.
The carrier component acts as a phase reference for stabilizing the homodyne detection phase,and it appears on the homodyne detector as a signal at fs . The signal is separately amplified fromthe DC signal by AC coupling the homodyne detector’s signal to a high frequency transimpedance
amplifier AD8015 (RF port). The RF port of the homodyne detector is sensitive to frequencies
above 5 MHz while the DC difference and sum ports (not shown) of the detector have a bandwidthF
1.5 MHz. The two detector ports are balanced well enough that PPDIF< 7×10−3 , where PDIF F andSU M

PSU M are the difference and sum of the two powers detected on the two photodiodes comprising the

90
homodyne detector. An additional electronic relative gain adjustment between the two photodiodes
allows cancellation of power noise on the homodyne reference by typically < 3 × 10−4 .
The carrier/homodyne reference beat note is then phase-locked to a stable DDS reference
frequency at fs . The phase of this reference frequency φs sets the quadrature of detection in
homodyne and is under the control of the data acquisition computer. The phase lock is implemented
with 50 kHz bandwidth and is achieved by feedback to the VCO that drives the homodyne frequency
shifting AOM at fh . This feedback loop works to continuously readjust the homodyne reference’s
phase to compensate for relative path length noise and other relative phase noise so that the qquadrature of the light reflected from the cavity appears at the difference port of the homodyne
detector. The rms noise in this phase lock is low enough to resolve the cavity frequency with
precision at least 28 dB below quantum projection noise.

6.4.5.2

Locking of atomic probe to cavity

We actively feedback to lock the atomic probe’s sideband to ωc0 . This improves the dynamic
range of the detection system, removes sensitivity to scale-factor noise, creates more consistent
optomechanical effects, and removes nonlinearities associated with the dispersive error signal. The
error signal is the detected q quadrature of the atomic probe’s lower sideband as measured inhomodyne at the difference port. The signal is a dispersive feature with a zero crossing appearing
as the atomic probe laser’s frequency is swept through resonance with ωc (or ωc0 ) [32] .
During each measurement window of ωc0 the atomic probe’s lower sideband is turned on for
approximately 40 µs, and the DC homodyne signal is used to actively lock the sideband’s frequency
to ωc0 . This is achieved by feedback to the VCO that provides the frequency reference fca to which
the cavity/atomic probe beat note is phase-locked. The phase-locking is achieved by adjusting theatomic probe laser’s frequency via the loop filter LSHA . The characteristic settling time of theservo is 1 µs for a unity gain frequency of 160 kHz. In order to record ωc0 , the output of the LSHA
loop filter that sets the VCO control voltage is directly sampled at 2.5 MHz by the data acquisition
computer (DAQ).

91
Since the atomic probe lower sideband is turned off between measurements, the atomic probe
laser’s frequency must be held fixed using the sample-and-hold circuit as shown in Fig. 6.6. Whenthe atomic probe sideband is turned on, the circuit samples the loop filter voltage provided to the
VCO that provides fca . When the sideband is turned off, the circuit holds the output voltage ofthe loop filter so that fca is held at its previous value.
Trial-to-trial fluctuations in atom number are significantly larger than fluctuations due toprojection noise. This increases the range over which the probe laser must slew its frequency toalign the lower sideband with ωc0 during the first N↑p premeasurement. To reduce this initial offset,
a “pre-centering” measurement is performed 1.5 ms before each experimental squeezing trial: aπ/2 microwave pulse rotates the atoms to a superposition of |↑i and |↓i and the lower sideband iscentered by the feedback loop at ωc0 . The atomic probe frequency is then held, and the probe lightis switched off. The atoms are then optically pumped back to |↓i for the actual spin squeezing
measurements described in the main text.
We often wish to scan the power in the atomic probe lower sideband (quantified by the
number of incident probe photons in a single measurement window Mi ) to look at variation inmeasurement noise. This is accomplished by changing the rf power supplied to the EOM at fs tomodify the phase modulation index. For reference, a typical sideband/carrier ratio for Mi = 36500
incident photons is 0.004. Thus, the carrier power and hence the open loop gain of the path length
phase stabilization for homodyne detection is relatively unaffected as we vary Mi .
In contrast, the open loop amplitude gain of the lower sideband to cavity lock scales as

√

Mi .

To compensate, a variable gain amplifier (VGA; Analog Devices AD8337) is inserted after thehomodyne detector. When the data acquisition computer changes the rf power that sets Mi , it also
simultaneously scales the VGA’s gain to keep the net loop gain fixed. DC offsets in the homodyne
difference port are problematic when the gain is scaled and are therefore removed using a low passfilter (τ = 1 s) and differential amplifier that essentially make a low bandwidth measurement of
the DC offset that is then subtracted from the fast 40 µs measurement windows.
With this approach, we achieve a very large dynamic range from Mi = 150 to Mi = 3 × 105 .

92
When Mi . 100 in a 40 µs window, the average number of detected photons within the servo’s
time scale of 1 µs approaches unity. The photon shot-noise then imposes rms fluctuations on theatomic probe’s frequency that are comparable to the cavity half-linewidth, leading to a reduction
in fundamental signal to noise for estimating ωc0 .
Lastly, for diagnostic reasons, it is often useful to do broad sweeps of the lower sideband’s
frequency across the cavity resonance frequency. To accomplish this, the atomic probe laser’s beatnote with the cavity probe laser can be phase-locked to a direct digital synthesizer (DDS) sourcethat provides the reference frequency fca in place of the usual VCO. The DDS frequency can bephase-coherently swept at programmable rate and range, accomplishing the desired sweep of theatomic probe frequency.

6.4.5.3

Calibration of incident photon number Mi

The number of incident photons on the cavity Mi is determined from the homodyne signal
and measured quantum efficiencies. The locking scheme used for homodyne detection allows precise
control of the relative phase between the homodyne reference beam and the atomic probe sideband
by tuning the phase φs . Experimentally, when the atomic probe sideband is off resonance fromthe cavity and one scans φs over 2π, a sinusoidal interference fringe is observed in the homodyne
difference port. The size of this fringe and the independently measured total power in the homodyne
reference beam (130 µW typical) are used to determine the rate Ri of incident photons in the atomic
probe lower sideband, coupled to the cavity, that would have been required to produce the observed
fringe. The number of incident photons is Mi = Ri × 40 µs.
Physically, this means Mi can be understood as the average number of photons in the atomic
probe lower sideband crossing an imaginary plane directly in front of the cavity input mirror,
counting only those that are spatially mode matched to the cavity TEM00 mode, and integrated
into a 40 µs window. The uncertainty in the absolute calibration Mi is approximately 25% due touncertainty in the spatial mode-matching of the incident atomic probe beam and the homodyne
reference beam. This uncertainty leads to uncertainty in the prediction of contrast lost in Fig.

93
3(b) of the main text, but does not lead to any uncertainty in the amount of squeezing or the(0)

experimental quantum efficiency Q1 to be discussed in Section 6.4.5.4.
6.4.5.4

Quantum efficiency

To determine the probe detuning δp , we estimate the ratio qr /ii from the detected fields
qd /id . Vacuum or photon shot noise that appears in the detection of the qd quadrature limits theresolution on our ability to determine ωc0 . We express the noise in the ratio as(∆qd )2
1=+ (∆q)2 ,
(0)
i2d
4Mi Q

(6.2)

1

(0)
where the one-window quantum efficiency Q1 includes fundamental losses of signal to noise re-sulting from both photon losses and technical noise floors shown in Table 6.2.
The additional term (∆q)2 = f + rMin represents noise contributions from the technical noise
floor f associated with residual frequency noise on the atomic probe laser relative to the cavity
mode frequency, and noise from optomechanical ringing r, which we model with an arbitrary nthorder polynomial scaling with n 6= −1 . These noise sources have different scalings with Mi than
the fundamental quantum noise (first term).
We define a new effective quantum efficiency Q1 which includes the effects of the technicalnoise floor and optomechanics and write the noise in homodyne detection as(∆qd )2
1=
2
4Mi Q1
id(6.3)

where Q1 is given by
(0)

Q1 =

Q1

(0)

1 + 4Mi Q1 (f + rMin )

.

(6.4)

This effective quantum efficiency provides a useful figure of merit for the experiment and can becompared to measurements of the increase in area of the Bloch vector’s noise distribution (discussed
in Section 6.6).

Source
Path efficiency, Qpath
Cavity-mode/homodyne overlap, Qo
QE of cavity (κ1 /κ), Qcav
Technical noise from detector, Qelec
Detector QE, QP D
Probe turn-on time, Qturnon(0)
Total, Q1

Q
0.75(3)0.95(3)0.83(3)0.86(1)0.86(2)0.86(1)0.37(5)94

Table 6.2: Quantum efficiency summary table. Quantum efficiency losses come from sources ofsignal loss and added noise floors. Qturnon comes from finite laser turn-on times and ringingcancelling “kicks” (see Sec. 6.4.5.5) during which the probe is on but we do not collect information.
(0)
The total quantum efficiency Q1 = 0.37(5) is the product of all the measured contributions.

6.4.5.5

Limits to noise reduction, optomechanics

The primary limitation to noise reduction R is currently set by optomechanical effects fromthe probing light. Due to the incommensurate probing and trapping potentials, when the probe lightis turned on, the atoms are given an impulse that drives axial oscillations in the trap. Additionally,
the minimum of the trapping potential moves in space. This ringing effect is shown in Fig. 6.9(red) over the 40 µs probing period.
To partially cancel the optomechanical ringing, we employed a 2.5 µs half-power turn-on
sequence of the probe laser. The initial half power turn-on induces ringing, while the second,
full-power turn-on (applied one quarter of an axial oscillation period later) coherently zeroes theinitial axial ringing such that the atoms come to rest at the new trap minimum. As shown in
Fig. 6.9, this technique significantly reduced the amount of ringing but only somewhat improved
the optimal squeezing in Fig. 3 by an estimated 0.6 dB. Mitigation of optomechanical effects will
present a challenge for future experiments aimed at generating even more spin squeezing. Tighter
trapping or homogeneous coupling of atoms to the atomic probe could be avenues toward reducing
optomechanical effects.

(kHz)

95

Time (µs)
Figure 6.9: Probe induced oscillations partially cancelled by a staggered turn-on sequence. The
oscillations are fully present with no kick (red, 43 traces averaged) during a 40 µs measurement,
but greatly reduced by a half-power 2.5 µs kick (blue, 30 traces averaged). The 2.5 µs kick length
corresponds to a quarter of the axial trap oscillation period. There is an 80 MHz offset subtracted
from the vertical axis.

96

6.5

Background contrast correction
The spin squeezing values presented in the main text were calculated using the slightly

modified relationship S = RCBG /C 2 described in Ref. [173]. Here CBG = 0.96 is the backgroundcontrast as determined from measurements of the contrast at Mi = 0 probe photons. For clarity ofpresentation we approximated CBG = 1 for the expressions in the main text. Using the more exact
formula represents a small 0.2 dB improvement in the reported squeezing compared to what would
be calculated with S = R/C 2 .

6.6

Antisqueezing and area of the noise distribution
The noise in the backaction (or antisqueezed) quadrature of the squeezed state was measured

using the sequence in Fig. 3(c) of the main text, with the variable rotation ψ inserted, and measurement outcomes here labeled Jzp and Jzf (ψ) for the first and second measurements respectively.
This measurement sequence was used to make the visualization of the squeezed state shown in Fig.
3(e) of the main text. In order to construct a meaningful probability distribution describing ourstate, we constructed the normalized probability distribution P (Jzf (ψ) − cos ψJzp ) for obtaining
a differential measurement outcome Jzf (ψ) − cos ψJzp . The weighting of the premeasurement bycos ψ ensures that we only condition the final measurement on the premeasurement to the degree
that the two spin projections overlap. We performed an inverse Radon transform [94] on the measured P (Jzf (ψ) − cos ψJzp ), yielding the conditional probability distribution shown in Fig. 3(e) of
the main text.
We now consider the magnitude of the noise in the backaction quadrature versus the numberof probe photons Mi . We generalize the spin noise reduction to now be a function of ψ as R(ψ) =22∆ (Jzf (ψ) − cos ψJzp )2 /∆Jz,QP
N . The antisqueezing is defined as A ≡ R(π/2)CBG /C , in direct

analogy to the Wineland squeezing parameter, S = R(0)CBG /C 2 . The antisqueezing parametercan be interpreted as the noise variance in the azimuthal phase of the Bloch vector relative to thestandard quantum limit A ≈ (∆φ/∆φSQL )2 , up to the small correction for the background contrast

97
CBG = 0.96.
The antisqueezing A is plotted in Fig. 6.10 versus Mi . The data is fit to a model that includes
three contributions A = A0 + A1 Mi + A2 Mi2 . The quantum backaction should rise linearly with Mi
and is therefore parameterized by A1 . The contribution of this term to the total backaction is shownby the blue shaded region. Classical intensity noise on the probe laser power circulating inside thecavity (for example) would contribute backaction noise scaling as Mi2 . The classical backaction istherefore parameterized by A2 , with this classical contribution to the total backaction shown by
the red shaded region. Lastly, the constant term A0 is attributed primarily to the projection noise
as well as noise in the rotations.
Squeezing data S was taken at the same experimental settings (gold points and line in2
Fig. 6.10). This allows us to infer the angular area of the quantum noise distribution, ∆φ∆θ/∆θSQL=q
2 , shown in purple in Fig. 6.10(b).
SA1 /CBG

The increase of the area of the noise distribution can also be used as an alternate, global
measurement of the quantum efficiency Q1 in Section 6.4.5.4. Specifically, the total quantum
efficiency of the entire measurement sequence is proportional to the square of the increase in the
2 ). The factor C 2
angular area of the noise distribution and can be written Q̃1 = 4/(A1 SC 2 /CBG

comes from the angular momentum uncertainty relation and accounts for the fact that the SQL
increases as the Bloch vector shrinks. As mentioned in the main text, the factor of four arises due
to finite measurement strength and an unused premeasurement. Q̃1 as measured by the area of
the noise distribution is plotted in Fig. 6.10(b) in gold. The gold shaded region represents theuncertainty in the extrapolation of Q̃1 due to uncertainty in the fit of the antisqueezing data of(0)

Fig. 6.10(a). At low photon number, Q̃1 agrees with the predicted value of Q1

from Table 6.2.

At higher photon number, Q̃1 begins to rise due to the effective quantum efficiency losses from thetechnical noise floor and optomechanics discussed in Section 6.4.5.4.

98

Figure 6.10: (a) The antisqueezing A is plotted versus Mi (black circles). The linear contribution
to the rise in A, A1 , is shown in blue and the quadratic contribution A2 in red. The squeezing (golddiamonds and fit) is plotted on the right axis. (b) The area of the noise distribution is calculated
from the data in part (a) and plotted in purple. The measured effective quantum efficiency Q̃1is plotted in gold with an error bar shown as a gold band. At low Mi , Q̃1 is consistent with the(0)prediction (green dash) Q1 from Table 6.2.

99

6.7

Feedback implementation

6.7.1

Experimental details

Real-time feedback to steer the atomic spin projection to a target spin projection Jz is implemented with an Arduino Due microcontroller with an internal clock of 84 MHz. The microcontroller
is programmed to sample the loop filter LSHA output voltage (directly related to N↑ ) during theN↑p and N↓p measurement windows, and the sampling rate allows for averaging 18 points in each
40 µs window. The microcontroller then calculates Jzp from the difference of the two measurement windows and applies the feedback microwave rotation to the atoms through the formulaθf b ≈ 2 × (Jztar − Jzp )/N where Jztar is the target value for Jz . Fluctuations in N are small enoughthat it can be taken as a constant. The microcontroller controls the microwave rotation angle byvarying the duration for which the microwaves are applied using a high speed microwave switch(Minicircuits ZASW-2-50-DR+) (labelled rot in Fig. 6.6) with single clock cycle (12 ns) resolution. The sign of the rotation is controlled by another digital output of the Arduino that toggles
a switch denoted sign between two microwave sources that are 180◦ apart in phase, as shown in
Fig. 6.6. Microwave rotations to accomplish π/2 and π pulses can also be applied independentlyof the Arduino using digital outputs from the data acquisition computer (not shown), though withless timing resolution.

6.7.2

Limitations to squeezing with feedback

As mentioned in the main text, the primary limitation to deterministic squeezing is noise
imposed from microwave rotations. We estimate these noise sources by performing two additionalvariations of the measurement sequence of Fig. 2 of the main text, removing either the feedback
rotation θF B or all rotations.
At the optimal deterministic spin squeezing with feedback, we achieve R−1 = 9.5(4) dB. To
estimate the noise added from feedback, we measure conditional spin noise Jzf − Jzp in a sequencewith no feedback and find R−1 = 12.4(7) dB. Feedback leads to approximately 2.9 dB of added

100
noise. Next, we perform the sequence with no microwave rotations of any kind, effectively measuring
the same spin population N↑ four times. In this sequence we attain R−1 = 14.0(5) dB, 1.6 dB lessthan the sequence with rotations but no feedback. This measurement suggests a rotation noise floor
due to microwave amplitude and frequency noise that is approximately 17.5 dB below projection
noise. Further, we suspect that rotation noise is also a primary contribution to the additional noisefrom adding feedback, since certain rotation errors which cancel after two π pulses will no longer
cancel when feedback is applied. Improving the precision of microwave rotations remains a major
obstacle in working with atomic spin states with extreme phase resolution.

Chapter 7

Spatially Homogeneous Spin Squeezing for Atom Interferometry

7.1

Introduction
Spin-squeezed states could be used to improve a wide range of quantum sensors, with today’s

best atomic clocks [115, 66, 76] being particularly promising candidates [121]. In this chapter
I focus on preparing spin-squeezed states appropriate for matter-wave atom interferometry withapplications including inertial sensing [9], measurements of gravity and freefall [135, 141], and eventhe search for certain proposed types of dark matter and dark energy [131, 61].
A major challenge arises for cavity-based atom interferometry and other applications involving release of spin-squeezed atoms into free space. The problem is that the probe mode used to perform the collective measurement is a standing wave, but the atoms are trapped in a 1-dimensionallattice defined by a standing wave cavity mode with a significantly different wavelength. Some
atoms will sit in lattice sites positioned near nodes and some near anti-nodes of the entanglementgenerating probe light. As a result, the atoms will contribute to the collective measurement withdifferent strengths. In this common case, the large degree of squeezing exists only for this specific coupling configuration and would be largely lost after releasing the atoms into the arm of
an interferometer, since their final coupling to the cavity mode or other readout detector will bedifferent from the original configuration [70]. In contrast, we wish to create spatially homogeneous
entanglement, quantified by the amount of observed phase resolution beyond the SQL that one canachieve when every atom couples equally to the final measurement apparatus.

102

7.2

Homogeneous squeezing in an effective optical dipole trap
In this section, I demonstrate a method to create homogeneous spin-squeezed states in astanding wave optical cavity by allowing the atoms to traverse many wavelengths of the standing
wave probe during each collective measurement. Atoms experience a time-averaged coupling to thecavity so that every atom is measured with the same strength, ensuring homogeneous entanglement.
We do this by creating an optical trap with a uniform axial potential, which we refer to as an effective“dipole trap” as opposed to the standing-wave “lattice”. The dipole trap maintains transverse
confinement of the atoms while allowing free movement subject to gravity along the vertical cavity
axis. We demonstrate 11(1) dB of directly observed squeezing via collective measurements in the
dipole trap, the most directly observed spin squeezing ever achieved apart from Refs. [39, 68], anduse fluorescence images and noise scalings to show that the generated squeezing is homogeneously
shared among the atoms to a large degree, in principle allowing significant amounts of squeezing forfree space or guided matter-wave interferometry. We also discuss the limits placed on entanglement
generation with time-averaged measurements.
Homogeneous squeezing can also be obtained using a travelling wave “ring” cavity [13], butbirefringence must be controlled to maintain the efficacy of utilizing cycling transitions [32]. Another appealing approach is to introduce a commensurate lattice [68, 91]. This approach requires
special mirror coatings and frequency doubling equipment and doesn’t permit guided movement for
atom interferometry within the cavity mode. Homogeneous entangled states can also be obtained
without using a cavity [56, 4, 12, 114, 62], but free space experiments have not yet achieved thelarge amounts of squeezing observed using optical cavities.
In this work, we use the pseudo-spin states defined by the ground hyperfine states of 87 Rb,
with |↓i ≡ |52 S1/2 , F = 1, mF = 1i and |↑i ≡ |52 S1/2 , F = 2, mF = 2i split by 6.8 GHz. As in Refs.
~ with[39, 32], we describe the total pseudo-spin state of N atoms by a collective Bloch vector J,
spin projections Jx , Jy , and Jz . The spin projection on a single trial Jz = N↑ − N2 is determined bymaking a collective measurement of the total number of atoms in the upper spin state N↑ . For an103unentangled, coherent spin state (CSS), quantum projection noise (QPN) leads to fluctuations in√
Jz of size ∆Jz,QP N = N /2. In this work, ∆X will refer to the standard deviation of a quantity
X as measured over repeated trials of the experiment.
The collective measurement is performed using the experimental apparatus and techniques
described in Ref. [39]. In brief, we trap 87 Rb atoms in the central 2 mm of a 2 cm optical cavity
with finesse F = 2532(80). A cavity mode is tuned δc = 2π × 400 MHz to the blue of the |↑i to|ei ≡ |52 P3/2 , F = 3, mF = 3i transition. The cavity resonance frequency ω is shifted by an amount
depending on the number of atoms in |↑i due to the dispersive interaction between the atoms andcavity. The cavity’s resonance frequency is measured by probing the cavity in reflection for 40 µs.
The probing is collective because it is not possible to tell from the single probe mode precisely
which atoms are in |↑i.
In a single trial, we apply resonant microwaves to prepare each atom in an equal superposition√(|↑i + |↓i)/ 2. We then perform two consecutive measurements of the projection Jz , with the two
measurement outcomes labeled Jzp and Jzf , with subscripts denoting pre and final measurement.
The quantum projection noise is common to the two measurements and is removed when we takethe difference between the pre and final measurements, yet the atoms nearly completely retain
coherence of the quantum phase between |↑i and |↓i. This allows one to sense a quantum phasethat evolves between the final and premeasurements below the SQL.
The atoms are initially cooled to approximately 10 µK and trapped in a far off resonance red
detuned optical lattice at λl = 823 nm (with corresponding wave vector k0 = 2π/λl ). We thenconvert this standing-wave lattice into an effective dipole trap. This is achieved by simultaneously
driving multiple TEM00 longitudinal modes of the cavity near 823 nm. Adjacent longitudinal modes
have opposite symmetry with respect to the cavity center. To lowest order, near the center of thecavity, one mode creates a cos2 (k0 z) standing-wave intensity profile while the next mode creates asin2 (k0 z) intensity profile such that the sum of the two standing waves cos2 (k0 z) + sin2 (k0 z) = 1
creates a net uniform intensity profile along the cavity axis as shown in Fig. 7.1(a).
To drive adjacent longitudinal modes, we phase modulate the lattice light at the cavity

104

Figure 7.1: (a) Optical lattice sidebands separated by one free spectral range (FSR) are injected
into the cavity to create an axially homogeneous “dipole” trap. Dipole trap intensity (blue) andits envelope (red) plotted inside of the optical cavity, with exaggerated wavelength λl × 103 . (b)The envelope of the residual lattice potential Vres (z) normalized to the peak lattice potential depth
V0 is plotted near the cavity center, optimized for a minimum at z = 0 (gold, β = 1.20) and forthe minimal fraction of trapped atoms determined experimentally (red, β = 1.32). (c) Fraction of
atoms remaining in the cavity mode (blue points) vs. fall time, fit to a model (red dash) describedin the text. Fluorescence images show the falling atom cloud at various times (inset).

free spectral range (FSR), FSR = 2π × 8.1050(5) GHz, using a fiber-coupled phase modulator.

105
The resulting axial component of the potential at distance z from the cavity center can be written
2 (β) sin2 ((k + δk )z)+J 2 (β) sin2 ((k + δk )z)+. . . ], where J (β)
V (z) = V0 [J02 (β) cos2 (k0 z)+J−1
0
−1
0
1
n
1

is the nth Bessel function and β is the modulation index. δkn = nFSR/c is the additional wave
vector for the sidebands offset by n cavity free spectral ranges, with speed of light c. Interferenceterms between sidebands are neglected since they oscillate at 8 GHz.
Figure 7.1(b) shows the depth of the residual standing-wave lattice potential in the dipole
trap Vres (z) as a function of distance from the center of the cavity for two different values of β.
We find β ≈ 1.32 (overdriving the dipole trap) to be the optimum value for freeing atoms to move.
This is due to a wider minimum of Vres (z) which overlaps the atomic spatial distribution as wellas the fact that overdriving causes the lattice potential wells to be converted into small potential
peaks, giving atoms additional potential energy.
When an atom begins to fall in the dipole trap, the increase in the residual lattice depth isnot sufficient to stop the atom from continuing to fall; rather, we expect the atom to be guided by
the optical dipole trap until it collides with the lower mirror. In Fig. 7.1(c), we measure the numberof atoms in the cavity as a function of freefall time, tdrop , by continuously monitoring the dispersive
shift of the cavity resonance frequency. The data is renormalized to account for background atom
loss and is reasonably described by a fit (purple line) which assumes atoms are guided by the net
transverse intensity profile of the dipole trap until they are lost when they collide with the lower
mirror. For comparison, ballistic expansion out of the cavity mode would occur in only 2 ms were
we to simply turn off the optical lattice. The free fall and guiding are corroborated by fluorescencemeasurements such as shown in Fig. 7.1(c) inset for various tdrop . Figure 7.1(c) and fluorescence
images indicate that at long times only 5(1)% of the atoms remain trapped in a residual lattice. The
majority of the atoms move along the cavity axis, the key for obtaining time-averaged homogeneity
in the coupling of the atoms to the standing-wave probe mode.
For a fixed total atom number, we expect the projection noise induced fluctuations in thecavity resonance frequency ∆ωQPN to be smaller in the dipole trap than in the lattice. While
the total dispersive shift is the same in both cases, in the lattice the dominant contribution is

106

Figure 7.2: (a) Projection noise scaling versus total atom number N , measured in the lattice (redpoints) including a theoretical prediction (red line) and in the dipole trap (blue points) including
a fit to infer a coupling fraction ζ (blue line, with 68% confidence interval bands). Sequences
are inset. Dashed boxes represent Bloch vector rotations through a given angle using resonant
microwaves. Solid boxes represent cavity frequency measurements. (b) Quantum noise reductionin the dipole trap with 6.3(3)×105 atoms. A histogram of Jzf − Jzp (black data points) shows astandard deviation 13.9(6) dB below projection noise ∆Jz,QP N = 397 atoms (gold line and shaded
distribution). The measurement sequence is inset.

107from the subset of atoms situated near antinodes of the probe. These atoms have a JaynesCummings coupling parameter gi near the maximum value g0 = 2π × 0.519(5) MHz and provide
stronger than average fluctuations. In the ideal time-averaged situation, on the other hand, the√full ensemble only couples with the rms coupling strength grms = g0 / 2, actually leading toweaker cavity frequency fluctuations. To quantify the level of homogeneous coupling, we definea model where fractionally, ζ of the atoms release into the dipole trap and are assumed to haveperfectly homogeneous coupling. 1 − ζ of the atoms remain fixed in position and maintain their
original coupling. In this model, the projection noise induced fluctuations in the cavity resonancepp2
frequency can be written ∆ωQPN = grms
N (3 − ζ)/ 8(g02 N + δc2 ) (See Section 7.5).
We observe this change in the projection noise scaling between the lattice and dipole trap by
performing the measurement sequences of Fig. 7.2(a) in the lattice (red, superscript L) and in the
dipole trap (blue, superscript D) versus the total atom number in the cavity N . The ω↑ and ω↓
windows represent the outcome of a measurement of the cavity resonance frequency, sensitive to N↑
or N↓ respectively, and we plot the observed projection noise fluctuations ∆ωQPN,meas = ∆(ω↑ −ω↓ )in either the lattice or the dipole trap. A small amount of technical noise that does not have theproper scaling with atom number has been subtracted out of this data. The lattice data is used
as a calibration of g0 with the theoretical scaling plotted in red. The dipole trap data is fit to themodel 2 × ∆ωQPN (since the measurement sequence includes two anti-correlated windows, ω↑ andω↓ ) with ζ as a free parameter. We fit ζ = 1.0(2), consistent with our expectation of 95% from thedata in Fig. 7.1(c).
D and ω D we can
By consecutively performing a pre and final measurement ω↓D , labeled ω↓p↓fshow a large degree of spin noise reduction below QPN and correspondingly demonstrate thecreation of entangled, spin-squeezed states in the dipole trap. We measure spin squeezing using
the Wineland criterion for phase enhancement relative to the SQL, (∆θ/∆θSQL )2 ≡ S = R/C 2[174, 39]. The observed spin noise reduction normalized to the quantum projection noise levelis R = (∆(Jzf − Jzp )/∆Jz,QP N )2 < 1. Squeezing or enhanced phase resolution also requires
the additional demonstration of retained coherence, or Bloch vector length, often referred to as

108~“contrast”, C ≡ 2|J|/N.
The measurement sequence is shown in the inset of Fig. 7.2(b) and is the same as that ofRef. [39]. We use tdrop = 13 ms, which accelerates the atoms enough to average over approximately
13 cycles of the probe standing wave during the 40 µs measurement window. Figure 7.2(b) showsD − ω D in the dipole trap with total atom number N = 630(30) ×
noise in measurements of ω↓f↓p

103 atoms. Experimental parameters grms , δc , and N are used to scale between cavity frequencyp
2 / 4g 2 N + δ 2 . The data is collated into a histogrammeasurements and Jz , ∂ω/∂Jz = grmscrms ↑on the left, showing a standard deviation 13.9(6) dB less than the projection noise level shown inyellow. The remaining contrast after the premeasurement was independently measured, C =0.70(5).
Together with the noise reduction, this yields a directly observed phase resolution, or spin squeezing,of S = 1/13(3) or −11(1) dB below the SQL

7.3

Coupling oscillations
When the cavity frequency is measured in a 40 µs window using the dipole trap, oscillationsin the signal are observed, indicating the atomic motion over the probe standing wave. Specifically,
we measure the number of atoms N↑ that are coupled to the cavity as a function of time by applying
a scale factor to convert cavity frequency to atom number. We refer to this rescaled time signal as
PN
2 . We observe noise in the atom’s coupling in the frequency domain, which
N (t) = i ↑ gi2 (t)/grms
can be used to infer the distribution of atoms’ coupling oscillation frequencies. Most of the coupling
oscillations average away, since the oscillation of each atom occurs with a random phase. However,
the residual uncancelled coupling oscillations are observed in N (t) such that the squared Fouriertransform of the time signal, |Ñ (f )2 | has units of Atoms/Hz and is closely related to the atomic
velocity distribution.
Figure 7.3(a) shows |Ñ (f )2 |, recorded using 2 ms of data and taking the average power
spectrum of time traces from approximately 65 trials. The data was taken after 1 ms (blue), 7.5 ms(red), and 15 ms (green) of freefall time after release into the dipole trap. Each power spectrum is fit
to an appropriately folded 1D Boltzmann distribution that accounts for the inability to distinguish

109

Figure 7.3: (a) Power spectra showing coupling oscillations for fall times of 1 ms (blue), 7.5 ms (red)and 15 ms (green) with their respective fits. (inset) Center frequency f0 of the fitted Boltzmann
distribution for various fall times (points) compared to a freefall prediction line of f0 = 2at/λp (line),
see text for definitions. (b) Power spectrum showing coupling oscillations at the trap frequency
when atoms are trapped in the optical lattice.

110
between upwards and downwards velocities. The fit center frequency f0 is plotted as a functionof the freefall time t in the inset of Fig. 7.3(a). The result is consistent, particularly at longtimes, with the simple prediction, f0 = a t/(λp /2), where a = 9.81 m/s2 is the acceleration due togravity. The widths of the distributions are consistent with Boltzmann distributions giving final
axial temperatures of 25 µK. To contrast, Fig. 7.3(b) shows |Ñ (f )2 | for atoms in the lattice. Insteadof a large thermal distribution, a narrow distribution is observed at the lattice trap frequency, about
200 kHz.

7.4

Outlook for atom interferometry
In summary, we infer that we have created a spatially homogeneous squeezed state from the

combined observations of Figs. 1-3. First, we observe release of 95(1)% of the atoms (Fig. 7.1)
at a sufficient velocity (Fig. 7.3) to ensure, on average, 13 averaging cycles of the probe standing
wave during a 40 µs collective measurement. In Fig. 7.2(a) we also confirm the transformation
to homogenous coupling by the change in scaling of projection noise fluctuations of the cavity.
The demonstration of 11(1) dB of observed squeezing in the homogeneous configuration proves ourability to create a large amount of entanglement in this highly time-averaged scheme.
We investigate the limits to squeezing using our time-averaged scheme in the Section 7.5.
Since the ac signals in Fig. 7.3 yield additional information about the spin state of each velocity
component of the atomic ensemble, time averaging will fundamentally limiting the squeezing to
1order S ∝ qN
, where q is the total quantum efficiency of the experiment. For q ∼ 1, this is close

to the Heisenberg limit. The more relevant limitation for our system is imperfect averaging of theprobe standing wave. For 40 µs measurements after a 13 ms drop time and for a 25 µK ensemble,
we estimate that the observed noise reduction should be limited to 15 dB below QPN, which we
believe to be a primary limitation to our observed spin noise reduction of 13.9(6) dB. In the future,
this limit could be improved using longer measurement windows to average over more cycles of theprobe standing wave during each measurement.
To realize a free space matter-wave interferometer, atoms could be prepared in the cavity for

111
the entanglement generating premeasurement, then released into free space for an interferometry
sequence. The final measurement could be performed with fluorescence detection. The 5% of atoms
with non-uniform coupling during the premeasurement would lead to an additional noise floor, notobserved in this work, of 13 dB below the SQL. Additionally, inhomogeneity from radial motion
will lead to another additional noise floor of approximately 10 dB below the SQL [39]. Notably,
this radial motion would also equally affect systems using ring cavities, commensurate lattices, orother axial averaging techniques.
Another possibility is to perform guided interferometry inside the cavity mode. Here the
pre and final measurements would both be performed with collective cavity measurements. In thiscase, the noise from the 5% of atoms remaining trapped and radial motion largely cancels at shorttimes. The 11(1) dB of squeezing observed in this work would in principle fully translate to thistype of interferometer. In addition to the possibility of using entangled states, performing the final
readout via a cavity measurement may allow for reduced technical noise, higher bandwidth, cleaner
optical modes, and power buildup for Raman transitions [60].
Similarly, higher order transverse modes, atom-chip technologies [168, 176], or tailored potentials [58, 97] might be combined with the cavity measurement technique presented here to create
new varieties of matter-wave Sagnac interferometers and other inertial sensors. The real-timeobservation of mechanical motion also opens the path to stochastic cooling schemes based on measurement and feedback [166] with applications to more complex systems such as molecules, which
can be challenging to laser cool using conventional Doppler cooling methods.

7.5

Squeezing limits from time averaging

7.5.1

Introduction

As shown in the Main Text, time averaging the probe standing wave during collective cavity
measurements can be used to create homogeneous squeezing within an atomic ensemble in a cavity.
However, the technique introduces limitations to spin squeezing in addition to the fundamental

112
limits calculated in Ref. [32] set by finite collective cooperativity N C, finite quantum efficiency
for detecting the probe light q, and the probability of the probe light inducing a spin flip p.
hipSpecifically, the best achievable spin squeezing is S = Max e/(qN C), 8p/(qN C) where the firstlimit (e/(qN C)) is due to wave-function collapse from free space scattering and the second limitp( 8p/(qN C)) is due to additional noise from Raman spin flips. Here, we calculate the additional
limits to S imposed by time averaging the spatially inhomogeneous coupling.
First, we establish the language for describing the limits on spin-noise reduction R when
the coupling coefficient of each atom differs between the pre and final measurement windows, withresults closely matching those of Hu et al. [70]. We apply this formalism to determine the best
achievable spin-noise reduction, and therefore best spin squeezing for two concrete scenarios relevant
to matter-wave interferometry with squeezed states. First, we consider the scenario in which we
perform the premeasurement of the spin state with the atoms trapped in an incommensurate
1D intracavity lattice and then perform the final measurement with uniform coupling. Next, weconsider the case of incomplete cycle averaging over the standing wave probe mode if the atoms
are allowed to move along the cavity axis during the pre and final measurements.
Next, we calculate two sources of inhomogeneous broadening, or dephasing, that arise fromtime averaging. This dephasing leads to a loss of contrast or signal. The first source is fundamental
quantum back-action that arises from the atoms sampling the photon shot noise of the intracavity
probe light at different times, or equivalently, spectral frequencies. The second source arises fromimperfect cycle averaging of the probe coupling. We show that these two limiting effects do not significantly impact current experiments, and can in principle allow spatially homogeneous squeezingnear the Heisenberg limit.

113
7.5.2
7.5.2.1

Noise reduction limits with inhomogeneous coupling
Noise between the final and premeasurement

To find the optimum spin-noise reduction with inhomogeneous coupling, we begin with an
expression for a measurement of atoms in a cavity where we assume each atom i couples to theoptical cavity with a factor ηmi (t) in the pre (m = p) or final (m = f ) measurement. Concretely, we
2 (t)/δ when we are in the dispersive regime of cavity measurements, where
can write ηmi (t) = gmic√δc , the optical cavity detuning from the atomic transition, is much greater than Ω = 2grms N , thecollectively enhanced coupling rate between the atoms and the cavity, or vacuum Rabi splitting [32].
2 (t) is the ith atom’s time dependent Jaynes-Cummings coupling parameter. We consider the
gmi

average outcome of the measurement by taking the time average of ηmi (t), denoted by dropping2
the time dependence (t). As a reminder, grms
is the average of gi2 over the atomic ensemble.

Note however, that by changing the units and scaling of ηmi , the expressions we derive can begeneralized for any type of population measurement such as fluorescence detection. The operator
ω̂mi that measures the time-averaged cavity frequency shift from the ith atom is then0
ω̂mi = σ̂z,i+ γ ηmi .

(7.1)

0 = (1 − γ)( |↑ i h↑ | − |↓ i h↓ | ) is the Pauli spin operator σ̂
Here, σ̂z,iiiiiz,i for the ith atom, rescaled by(1 − γ). The constant 0 ≤ γ ≤ 1/2 is used to account for the fact that the measurement may besensitive to some linear combination of state populations N↑ and N↓ , given by (N↑ −(1−2γ)N↓ ). We
include this factor since a number of spin squeezing experiments have used an optical cavity detuned
halfway between the |↑i → |ei and |↓i → |ei transitions [68, 140, 4]. In such cases, the dispersive
cavity shift is sensitive only to population differences between |↑i and |↓i and can be modeled with
γ = 0. However, in this work we set the cavity resonance frequency near the |↑i → |ei transition(modeled with γ = 1/2), so there is no sensitivity to N↓ . This scheme is advantageous for probing
on a cycling transition p = 0, as mentioned in the introduction [39, 32, 19]. Additionally, even when
the cavity only couples to atoms in |↑i, a π-pulse can be used to swap the state populations and114
measure N↑ and N↓ sequentially, again achieving a differential measurement with γ = 0. However,this method is only valid if the atoms’ couplings to the measurement do not change between theN↑ and N↓ measurement.
We wish to cancel the quantum projection noise in the final measurement of all N atoms
PPNω̂f = Ni=1 ω̂f i using a premeasurement ω+ =i=1 ω̂pi of the noise value. However, if the ith atom’s
couplings ηf i and ηpi differ, its projection noise cannot be exactly cancelled. In order to optimize
the cancellation of the noise, we construct a weighted difference ω̂diff ≡ ω̂f − W ω+ , with weightfactor W . It is important to note that, since our entanglement-generating collective measurementsof ω̂f and ω+ must not reveal single-particle information, we can only use a single weight factor forthe entire ensemble. Thus, if the change between ηf i and ηpi is inhomogeneous, that is, different
for each atom i, there is no value of W that can be chosen to achieve full cancellation of the noisein the final measurement.
To derive the best achievable squeezing limit with inhomogeneous coupling, we will calculate
the noise in the weighted difference ω̂diff and find the optimum value for the weight factor W .
First, using Eq. 7.1, we calculate the variance (∆ωdiff,i )2 in ω̂i,diff for a single atom. Next, we will
independently sum the noise contributions from each atom to calculate the total variance (∆ωdiff )2 .
We neglect all sources of noise in the system except the quantum projection noise and fluctuationsin the couplings ηmi . Other realistic noise sources such as photon shot noise and laser frequency
noise are neglected in order to calculate the squeezing limit just from inhomogeneous coupling.
Importantly, the optimum weight factor W for cancelling quantum noise and coupling noise islikely to not be the optimum for cancelling the other technical noise sources such as laser frequency
noise. In fact, in our experiments optimum squeezing was always observed with W = 1.
The noise variance in ω̂i,diff is calculated using(∆ωdiff,i )2 ≡ h(ω̂f i − W ω̂pi )2 i − h(ω̂f i − W ω̂pi )i2 ,(7.2)

where we denote the average over many independent experimental trials with the h...i notation. This
trial average will simultaneously evaluate the quantum fluctuations of the spin projection operator

115
0 as well as possible fluctuations in the time-averaged couplings η . In calculating averages, we
σ̂z,i
mi

will reasonably assume that fluctuations in the couplings are uncorrelated with fluctuations in thespin projection operator, leading to0(∆ωdiff,i )2 = h(σ̂z,i+ γ)2 ih(ηf i − W ηpi )2 i0
− hσ̂z,i+ γi2 hηf i − W ηpi i2 .

(7.3)
(7.4)

We now limit the discussion to the relevant case of atoms in an equal superposition of |↑i and |↓i
0 i = 0. However, we leave the expectation value hσ̂ 02 i unevaluated. This helps tosuch that hσ̂z,iz,iilluminate different contributions to (∆ωdiff,i )2 and also allows for the possibility of atoms being indifferent entangled or mixed states.
ih202− 2W ηf i ηpi i(∆ωdiff,i )2 = hσ̂z,ii hηf2i + W 2 ηpihi+ γ 2 (∆ηf i )2 + W 2 (∆ηpi )2 − 2W Cov(ηf i , ηpi )

(7.5)
(7.6)

where Cov(X, Y ) = hXY i − hXihY i is the covariance between X and Y. As a reminder, thecovariance quantifies the correlation of the fluctuations of X and Y . If X and Y are uncorrelated,then Cov(X, Y ) = 0. If X and Y are perfectly correlated, then Cov(X, Y ) = ∆X∆Y where ∆X
is the standard deviation in the quantity X. There are two contributions to (∆ωdiff,i )2 . The first(line 7.5) is the term arising from uncancelled quantum projection noise and will be nonzero if
the couplings, W ηpi and ηf i , are not equal to one another. This term captures the fundamental
problem of inhomogeneous coupling to the collective measurement.
The second contribution proportional to γ (line 7.6) results from trial-to-trial noise in the
couplings ηf i and ηpi and is fully classical, since it has no contribution from projection noise, that
02 i. The physical origin of this term can be thought of as trial-to-trial noise in the scale factor
is, hσ̂z,irelating an observed cavity frequency shift to an estimated population of atoms in spin up or down.
When γ = 0, the cavity frequency shifts are proportional to N↑ − N↓ which is on average zero forthe case considered here. As a result, any noise in the scale factor contributes no additional noise.

116
However, if γ = 1/2, then the cavity frequency shifts are proportional to N↑ which for the caseconsidered here is approximately N/2. In this case, classical scale factor noise can easily contributeat or above the quantum noise level.
As an example, the second term (line 7.6) is important when the atoms are trapped in theintracavity lattice for the premeasurement and then released to into an optical dipole trap for the
final measurement. An atom in the optical lattice sees fluctuations with a range of 100% fromtrial to trial in the standing wave coupling ηpi . This leads to classical noise in the premeasurementthat is of the same order as quantum projection noise. The quantum (line 7.5) and classical (line7.6) contribution to (∆ωdiff,i )2 are approximately equally. The classical noise could, in principle,
be removed if we knew every atom’s couplings ηf i and ηpi on each trial, but we do not. Instead,
we can only measure the average couplings hηf i i and hηpi i by linking a cavity shift, averaged overmany trials, to a change in the state population N↑ .
The second relevant example that is sensitive to both quantum and classical noise in ω̂diff isthat of our time-averaged scheme. Atoms will have different initial positions and velocities fromtrial to trial, leading to classical noise between ηf i and ηpi . We theoretically predict the contribution
to our observed squeezing from this effect in section 7.5.2.4.
The total noise variance (∆ωdiff )2 is found by summing the noise contribution from each
atom, assuming that they are uncorrelated:2(∆ωdiff ) =NX(∆ωdiff,i )2 .

(7.7)

i=1
2 i) contributing to this sum can in principle be different, but
Every second order moment (e.g. hηmi

we make the often-valid assumption that the particle labels for every atom are interchangeable. In
this case, the trial average h...i can equivalently be viewed as an average over all of the atoms in
the ensemble on a single trial. In either picture, each term in Eq. 7.7 is equal so that(∆ωdiff )2 = N (∆ωdiff,i )2 .

(7.8)

117
Since we have assumed that expectation values are the same for all atoms i, we will drop the indexes
02 i ≡ hσ̂ 02 i.
in following expressions such that hηmi ηm0 i i ≡ hηm ηm0 i, hηmi i ≡ hηm i, and hσ̂z,iz

7.5.2.2

Quantum projection noise

To calculate the quantum projection noise (QPN) limit to the noise difference (∆ωdiff )2 , wecan consider the case in Eq. 7.5-7.6 when W = 0 and only take the resulting noise from the quantum
term of line 7.5,(∆ωQPN )2 = N hσ̂z02 ihηf2 i.

(7.9)

Equation 7.9 can be used to derive the prediction (given in the Main Text) for projection noise
fluctuations in the effective dipole trap when fractionally (1 − ζ) of the atoms remained trappedin the residual lattice. We assume that the atoms are identically prepared in a pure state with
02 i = 1/4. We have set γ = 1/2, to model ouran equal superposition of |↑i and |↓i, so that hσ̂z,imeasurements that are sensitive to N↑ . For the ζ atoms in the dipole trap, hηf2 i = g04 /(4δc2 ) where
g0 is the Jaynes-Cummings coupling parameter at an anti-node of the cavity. For the 1 − ζ atoms
in the residual lattice, hηf2 i = 3g04 /(8δc2 ). Simply adding the projection noise variance of all of theatoms leads to the observed decrease of the projection noise versus ζ,
N g04 1
3(∆ωQPN ) (ζ) =ζ + (1 − ζ) .
4δc2 482(7.10)The equation for the QPN level given in the Main Text is more general in that it does not assume√
the dispersive limit δc  2 N grms as we have done here.

7.5.2.3

Optimum spin-noise reduction

The observable spin-noise reduction relative to the quantum noise in the final measurement
is given by R = (∆ωdiff )2 /(∆ωQPN )2 . Here we consider atoms independently prepared in a pure

118
state with equal superposition of |↑i and |↓i, such that hσ̂z02 i = 1/(1 − γ)2 . We then find,R=+hηf2 + W 2 ηp2 − 2W ηf ηp i
hηf2 ii
γ 02 h
2
2
2
(∆η
)
+
W
(∆η
)
−
2W
Cov(η
,
η
)
p
p
f
f
hηf2 i(7.11)(7.12)

where we define γ 0 = γ/(1 − γ). As a reminder, γ = γ 0 = 0 represents a measurement of N↑ − N↓and γ = 1/2, γ 0 = 1 represents a measurement of only N↑ . Similar to Eq. 7.5-7.6, this result
possesses a quantum (line 7.11) and classical (line 7.12) contribution. The classical term is givenby the variances, or fluctuations of measurement strengths while the quantum term is given by thesquared magnitude of the pre and final measurement coupling strengths. Additionally, if γ 0 = 0
the classical term vanishes, since the measurement signal will be centered around zero. If γ 0 = 1,
the two components can be of the same size. This shows that engineering measurements sensitive
to N↑ − N↓ can be advantageous to measuring N↑ .
The optimum weight factor that minimizes R is
Wopt =γ 02 Cov(ηf , ηp ) + hηf ηp i,
hηp2 i + γ 02 (∆ηp )2(7.13)

with optimized noise reduction Ropt ,γ 02 (∆ηf )2
hηf2 i(7.14)

2hηf ηp i + γ 02 Cov(ηf , ηp ).
−hηf2 ihηp2 i + γ 02 hηf2 i(∆ηp )2(7.15)

Ropt = 1 +This result is the best noise reduction possible when the final and premeasurement have different
coupling strengths. Line 7.14 gives the projection noise and classical noise in the final measurement.
Line 7.15 represents the optimum cancellation of the final measurement’s noise provided by the
optimally weighted premeasurement. In the case that only population differences (N↑ − N↓ ) are
mesured, such that γ 0 = 0, Ropt simplifies to
Ropt = 1 −hηf ηp i2,
hηf2 ihηp2 i

the ratio of the second order moments of the coupling strengths.

(7.16)

119
7.5.2.4

Fundamental limits for specific cases

We now apply the previous results to two important cases. First, we show that spin-squeezed
states created in an optical lattice with incommensurate coupling to the probe standing wave do
not lead to significant noise reduction below the SQL after they are launched into a homogeneousenvironment such as free space or our time-averaged optical dipole trap. Second, we derive thelimit to spin-noise reduction in our time-averaged scheme due to imperfect time averaging of theprobe standing wave.
For applications such as atom interferometry, it is interesting to consider the case in which
the atoms uniformly couple to the final measurement (homogeneous coupling), but during thepremeasurement the atoms are held in an optical lattice that has an incommensurate wavelengthwith the probe standing wave (inhomogeneous coupling.)
We take the final coupling to be the same for all atoms ηf = g02 /2δc . This represents the casewhen the final measurement is performed with the atoms moving along the cavity axis such thatthey perfectly time average away spatially inhomogeneous coupling to the probe standing wave.
This will also capture the physics of similar such measurement scenarios including ring cavities,
commensurate lattice/probe standing waves and spatially homogeneous fluorescence detection.
In our experiment, the lattice λl = 823 nm and probe λp = 780 nm wavelengths are higly
incommensurate. The two standing wave antinodes go from fully aligned to misaligned to fully
realigned in roughly 7.5 µm, a much shorter length than the characteristic 1 mm range of latticesites into which atoms are loaded. To describe this scenario, we take the premeasurement couplingof the ithatom to depend on its fixed position xi on a single trial as ηpi = (g02 /δc ) sin2 (Φi ), where
the phase of the coupling is Φi = 2πxi /λp . We further assume that the atoms independently andrandomly load into different lattice sites from one trial to the next such that the ithatom uniformly
samples coupling phases Φi = 0 to 2π from trial to trial. However, we re-emphasize that to modela premeasurement performed with the atoms trapped in the lattice, on a single trial the couplingphase of atom i does not vary in time. One then finds second order moments hηp2 i = 3g04 /8δc2 and120
4 /δ 2 , and variances (∆η )2 = hη 2 i − hη i2 = g 4 /8δ 2 , (∆η )2 = 0 andhηf2 i = hηp ηf i = g04 /4δc2 = grms
p
p
f
c
p
c
0

Cov(ηf , ηp ) = 0.
The best achievable spin-noise reduction for this case is Ropt = 1/3 (1/2) or -4.8 dB (-3 dB)
when we consider the two different cases, γ 0 = 0 (1). This shows that squeezed states createdby a premeasurement in an incommensurate standing wave lattice [39] cannot provide significant
entanglement enhancement for free space sensors. Specifically, this degree of spin-noise reduction
is far worse than the R ≈ −18 dB of spin-noise reduction achieved in experiments in which theatoms are trapped for both the pre and final measurement [68, 39].
This optimal spin-noise reduction is achieved using the optimum weight factor Wopt =2/3 (1/2). For comparison, if the relative weight factor is not optimized, but simply set to W = 1,
one finds that Ropt = 1/2 (1) or -3 dB (0 dB).
For the case γ 0 = 0, it is interesting to note that in previous squeezing work with incommensurate standing waves [140, 95, 31, 19, 39], an effective atom number Neff was defined in relationshipto the total atom number N as Neff = 2N/3 = Wopt N . The above results provide a nice physicalinterpretation of this effective atom number. The premeasurement can in principle perfectly measure the spin noise of Neff of the total atoms N , here 2/3 of all the atoms. The spin noise of the
Neff atoms can then be perfectly cancelled from the final measurement of N atoms. The remaining
spin noise of the “unmeasured” atoms N − Neff , here 1/3 of the total atoms, cannot be canceled at
all such that the best achievable spin-noise reduction is Ropt = (N − Neff )/N = 1/3.
As another example, consider the case in the Section 7.2, where both the pre and final
measurements are performed by time averaging the probe coupling as the ithatom moves along thecavity axis at velocity vi . The atom moves from anti-node to anti-node of the probe at frequency
fi = 2vi /λp , a frequency we call the coupling oscillation frequency. The time dependent coupling
can then be written as ηmi (t) = (g02 /δc ) sin2 (πfi t + φmi ), where φmi sets the coupling at t = 0.
We assume both the pre and final measurements last for a time Twin , and that the pre and final
measurements start at t = 0 and Tdiff (Tdiff ≥ Twin ) respectively. We take the projection noiselevel to be that for a perfectly time-averaged scenario in which each atom moves exactly an integer

121
number of cycles of the standing wave: (∆ωQPN )2 = N g04 /16δc2 . Under the same assumption, we
set the weight factor W = 1. The spin-noise reduction, averaging over the normalized thermalvelocity distribution of atoms P (fi ), isZ ∞P (fi )

R=−∞

4 sin2 (πfi Tdiff ) sin2 (πfi Twin )dfi .
2fi2 π 2 Twin

(7.17)

To gain some insight, consider an example when the atomic distribution P (fi ) is Gaussian withmean f0 and standard deviation ∆f . In the limit that Tdiff , Twin  1/∆f and f0  ∆f , the terms
sin2 (πfi Tdiff ) and sin2 (πfi Twin ) in Eq. 7.17 will oscillate rapidly with fi and so can be replacedin the integrand by their average 1/2. The resulting spin-noise reduction is then R = 1/(πNosc )2
where Nosc = f0 Twin is the number of cycles averaged by an atom at coupling oscillation frequency
f0 .
We estimate the maximum possible spin-noise reduction expected for the conditions used forspin squeezing in the Section 7.2, for which the above simplifying approximations are not valid.
To do this, we keep the full expression in Eq. 7.17, set Tdiff = Twin = 40 µs, and use the directly
measured distribution of coupling oscillation frequencies shown in Fig 3(a) of the main text toobtain an experimentally measured probability distribution P (fi ). We find a limit from imperfect
averaging R ≈ −15 dB. We believe this is one of the primary limits to the observed spin-noise
reduction R = −13.9(6) dB. However, we expect that this limit can be improved to beyond 20 dB
by allowing the atoms to fall for longer or by using longer measurement windows Twin , changes thatare difficult to implement with current technical constraints of the experiment but that could bestraightforward to implement in the future.

7.5.3

Dephasing from inhomogeneous back-action

In Section 7.2, the atoms are allowed to move along the cavity axis in order to achieve time
averaging of the inhomogeneous coupling to the standing wave probe mode. As atoms traverse
the probe standing wave, they produce fast oscillations in the measured cavity frequency shift in
addition to the desired average pre and final measurement signals that are used to resolve phases

122
below the standard quantum limit. This power spectrum of the oscillations in the cavity shift isshown in Fig. 7.3 with the spread in frequency components reflecting the thermal spread in atomic
velocities. These oscillating signals yield information about the spin state of atoms moving at aparticular velocity and therefore must cause some degree of additional quantum collapse or backaction, which may limit the amount of squeezing. A full treatment of this effect is very difficult
since it likely moves the total wave function away from the restricted fully symmetric Hilbert spaceof N dimensions and toward the full 2N dimensional Hilbert space. However, we attempt to derive
an estimate for the scale of the deleterious back-action using a classical back-action model driven
by quantum noise on the optical probing field. We estimate that the time-averaging scheme, at
2 = 1/N 2 rad2 .
worst, only provides an additional squeezing limit near the Heisenberg limit ∆θHThis is far from being a relevant limit for the best current experiments that achieve enhanced phaseresolutions of approximately ∆θ2 ≈ 104 /N 2 rad2 with approximately N = 106 atoms [68, 39].
One explanation of the quantum back-action is that the probe light causes a differential AC
Stark shift between the spin states such that |↑i + |↓i → |↑i + eıψi |↓i, equivalent to the ith atom’s
Bloch vector changing its azimuthal angle by ψi . Photon shot noise (PSN) of the probing beamcauses a noisy, unknown contribution to the phase shift with rms fluctuation ∆ψ that is equivalent
to the observed quantum back-action in the azimuthal angle. The PSN level can be plotted (dotted
purple line in Fig. 7.4) in frequency space as a power spectral density of photon number fluctuationsin the cavity SM = 4Mc /κ [105], valid for frequencies much less than the cavity linewidth κ, where
Mc is the average number of photons in the cavity mode and κ is the cavity linewidth. Stationary
atoms sample this PSN in a frequency window centered at zero frequency, with characteristic
bandwidth 1/Twin (the exact sensitivity function shown in red in Fig. 7.4) However, if an atom
moves along the cavity axis, its coupling to the probe field during the premeasurement oscillates as
2 (t) = g 2 sin2 (πf t+φ ), and it will sample the PSN with a modified transfer function, sensitive at
gpi
i
pi
0
2 (t)) as well as a component oscillating at f (shown
DC (due to the time-averaged component of gpi
i

for two different velocities or fi in blue in Fig 7.4). In the time domain picture, this is equivalent to
the intra-cavity photon number fluctuating on a time scale 1/κ. Atoms at different velocities and123
initial positions sample these fluctuations differently. As a reminder, we only consider the backaction of the premeasurement (p) because loss of coherence during or after the final measurement
does not affect the desired sensitivity for detecting a phase that is applied between the pre andfinal measurement.
Given a thermal velocity distribution of atoms, the distribution of coupling oscillation frequencies fi will be given by a Gaussian probability distribution P (fi ) with standard deviationq
2π × ∆f = 4π kb Twin /(mλ2p ) where kb is Boltzmann’s constant, and m is the mass of 87 Rb.The
number of sub-ensembles that receive uncorrelated back-action will be of order Ne ∼ ∆f × Twin . If
2Ne  1, the total Bloch vector length will be reduced, leading to a loss of contrast C = e−∆ψ /2 .
We can write the rms phase shift about the mean due to back-action ∆ψ as∆ψ = αpMr ,(7.18)

where Mr is the total number of probe photons reflected from the cavity input mirror during
a measurement, assuming a loss-less, single-ended cavity. On resonance, the intra-cavity photon
2 /(κδ ) characterizes
number Mc can be related to Mr by Mr = Mc (Twin κ)/4. The constant α = 4grmsc

the average azimuthal phase shift to an atom’s Bloch vector per reflected photon.
Using the Heisenberg uncertainty relation for a collective spin state, ∆θ∆ψ ≥ 1/N , the PSN
limited spin-noise reduction, ignoring all other noise sources, can be shown to be writtenR=

1
α2 Mr qN.

(7.19)In real experiments, the squeezing is fundamentally limited by contrast loss from free space scattering or diffusion of the Bloch vector due to Raman transitions [32]. However, here we neglect theselimits to squeezing and instead focus on the squeezing limit solely due to the contrast loss C frominhomogeneous back-action due to time averaging. In this case, the total squeezing as a functionof Mr becomes2

eα Mr
S = R/C = 2.
α Mr qN2(7.20)

124

Figure 7.4: Sensitivity of atoms at different velocities to photon shot noise. A moving atom couples
to the probe mode with a transfer function (blue) with sensitivity at DC and at a frequency ficorresponding to its velocity, shown for an atom with velocity 5 cm/s and 15 cm/s. Stationary
atoms only couple at DC (red). The distribution of oscillation frequencies is given by the Boltzmann
distribution P (fi ) (black). Atoms at different frequencies sample photon shot noise (PSN, purple
dash) at different frequencies leading to dephasing that can limit squeezing with time averaging.

125
which has an optimum value
Sopt =

e.
Nq(7.21)

Sopt represents an estimate of the quantum limit to squeezing with our time averaging scheme.
However, Equations 20 and 21 show that, for q near 1, the squeezing is only affected by time
averaging near the Heisenberg limit.
The physical mechanism and scaling of the squeezing limit (Eq. 21) is quite similar to regularquantum back-action, where every atom receives an identical random phase with an rms magnitudeof ψrms due to photon shot noise in the measurement. In this standard case, since the phase shift
is the same for all atoms, the Bloch vector retains its rms extent J 2 ≡ hJˆx2 i + hJˆy2 i ≈ N 2 /4. The
difference in the time-averaged case is that each subensemble receives a random phase causing adecrease in the collective Bloch vector extent J 2 < N 2 /4. In this case, the intrinsic phase resolutionof the ensemble is lowered, but only when one approaches the Heisenberg limit.

7.5.4

Dephasing from imperfect time averaging

In addition to quantum back-action driven dephasing, there can also be classical dephasing:each atom receives a different average AC Stark shift due to imperfect cycle averaging of the probe
standing wave. In realistic experiments, this classical dephasing will usually be much larger than
the quantum dephasing described in the previous section.
The phase shift on a single atom can be written, for a single premeasurement window,
Z Twinψi =0

2 (t)Mgpicdt
δc(7.22)

where Mc is the average intracavity photon number, taken to be constant for this calculation.
To estimate the dephasing due to classical imperfections in the time averaging, we calculate thestandard deviation in ψi , ∆ψ, over the atomic distribution . The coupling for the ith atom is
2 (t) = g 2 sin2 (πf t + φ ), with a normalized probability distribution for f and φ , denoted
gpi
i
pi
i
pi
0

126
P (fi , φpi ).
Z ∞

Z 2π

2

dφpi

∆ψ =

−∞
Z ∞

0
Z 2π

(7.23)

2

dfi P (fi , φpi )ψidφpi
0dfi P (fi , φpi )ψi2 −(7.24)−∞

Assuming the phase φpi of the coupling oscillations is random for each atom, the result can be
simplified to∆ψ 2 =Z ∞dfi P (fi )
−∞

1
2



g02 Mc2πfi δc2

sin2 (πfi Twin ),(7.25)

2

reducing the contrast after a single premeasurement by e−∆ψ /2 . Similar to the uncancelled noise
reduction of Eq. 7.17, this result has the interpretation of being due to the phase shift from thefinal, uncancelled non-integer fraction of an atom’s coupling oscillation. However, the result will
be slightly modified by the use of a spin-echo pulse for the premeasurement, as in Fig. 7.2 (b), butsince the phase of each atom’s coupling oscillation changes for each window based on it’s velocity,
significant spin-echo cancellation of this dephasing is not expected. For our system we estimatethat classical dephasing from imperfect time averaging leads to a small contrast loss of less than
1 dB at the optimal squeezing and could be improved to arbitrary levels with better averaging byincreasing the number of periods of oscillation in a measurement window while holding the total
number of incident photons in the window fixed.

Chapter 8

Conclusion and Future Outlook

Over the course of my graduate studies, my labmates and I have performed a number ofexperiments to explore how quantum atom-cavity systems can be used to improve precision measurements. I hope that this collection of work will continue to advance the field of quantum scienceand precision measurement by leading to practical entanglement-enhanced sensors, providing newtechniques for building high quality atom-cavity systems, understanding how to generate and understand large amounts of entanglement, and providing new techniques to control precise atomic
states.
In general, as atomic sensors become more accurate and precise, we will be able to moresensitively probe the most fundamental principles of the universe, perhaps leading to physics beyond the standard model [61, 156, 7, 175]. Additionally, as atomic experiments become easier toconstruct and operate, commercialized and miniaturized systems will make their way into industrial applications [80]. Lastly, the ability to control and manipulate quantum systems including
entangled states is critically important to the quickly growing field of quantum information science.
As we continue to explore and improve the capabilities of atom-cavity systems, I expect advancesin quantum communication and quantum simulation to occur as well as advances in precision
measurement.

128

8.1

Future outlook: Spin-squeezed optical lattice clocks
In the near future it may be possible to translate the spin squeezing techniques of this thesis

to state-of-the-art optical lattice clocks [115, 66, 121]. Many optical lattice clocks already employ
optical cavities to create 1-D optical lattices, and have convenient, dipole-allowed, nearly cyclingtransitions on which to perform low noise collective probing [121]. An initial proof-of-concept experiment has already been demonstrated in the new strontium experiment in the Thompson lab[121]. Since today’s best optical lattice clocks are already significantly limited by quantum noise,and laser-frequency noise from the Dick effect continues to be improved through lower noise lasers[77], and continuous probing schemes [139], spin squeezing has the potential to lead to significant
improvements. Increasing atom number to improve quantum noise, on the other hand, may actually be more difficult than generating entanglement since higher atom density leads to unwanted
collisional shifts in the clock [71]. However, our technique for generating squeezing fundamentally
improves with increasing atom number, so the future of atomic clocks may include both 3D opticallattices for high atom number (small SQL) and very large entanglement enhancements.

8.2

Future outlook: squeezed atom interferometer
Another short term outlook for our work is to demonstrate a proof-of-principle atom interfer-ometer with resolution below the SQL. The work of chapter 7 is a clear first step towards this goal.
In particular we could demonstrate a guided interferometer operating inside the optical cavity modethat would not need to perform fluorescence detection, that can be difficult for large ensembles ofatoms.
Such an atom interferometer could use Bragg pulses through the cavity in order to drive
spin-dependent momentum transfer to the atoms. An experimental diagram that shows the level
diagram for the Bragg pulses as well as an experimental sequence is shown below in Fig. 8.1.
In a similar experiment [60], state changing pulses and momentum transfer were accomplishedusing Raman light pulses. This was possible to do through the cavity since the cavity free spectral

129

Figure 8.1: a) Momentum transfer diagram for 2-photon Bragg pulses. Two laser beams (red),
detuned by the frequency corresponding to 2 quanta of photon momentum change are used tomake transitions along the kinetic energy curve of the atom (blue) b) Timing diagram for atominterferometry in a cavity. Atoms’ internal states are manipulated by microwave pulses (black andgrey) and state-dependent momentum transfers are achieved using 2-photon Bragg pulses (red andgrey). The spin-momentum entanglement is used to separate the wave function into different spatial
paths leading to a differential phase φ, that is subsequently mapped onto the final wave function
and read out by a final Ramsey π/2 pulse. Squeezing is generated by the collective premeasurementof Jz shown with a blue and grey pulse.

range was tuned equal to the hyperfine splitting in the atom. However, in our system, this may
not be the case, so I have proposed operating the interferometer with separate microwave spinchanging pulses and Bragg momentum transfers, that can easily be injected into the cavity since
the frequency difference between the momentum states is only in the kHz range. An experimental
diagram showing a possible atom interferometry sequence and the energy/momentum diagram forthe 2-photon Bragg pulses is shown in Fig. 8.1. The interferometer would operate in the z-direction
parallel to the cavity axis. Since precise rotations of the Bloch vector have remained challenging
in this experiment, operating at lower atom number and higher cavity finesse may be the easiest
route to demonstrate a first squeezed interferometer.

130
As a first proof-of-principle experiment, the evolution time in such a sequence could be verysmall and one could simply show the ability to generate and read-out momentum-spin entanglement
below the standard quantum limit. However, in the future this type of interferometer could be usefulin a wide range of atom interferometer applications [9, 135, 141, 131, 61].

8.3

Future outlook: superradiant cooling
Another future outlook for the experiment that is currently being investigated is a collec-

tive superradiant cooling mechanism. This phenomenon was recently theoretically introduced inRef. [177]. While it is certainly unclear whether superradiant cooling will provide a practical or
technological advancement to the field of cold atom research, it is clear that this is one of the only
collective (could not occur with a single-atom) cooling mechanisms to date and for this reason isof significant scientific value to study.

8.4

Future outlook: Heisenberg limited states
Speculating further for the future, given the recent advances in atom-cavity systems and joint

measurement, it is reasonable to believe that we may be able to create maximally-entangled stateswith Heisenberg-limited phase resolution. Such states could be generated, for example, in a cavity
with a single atom cooperativity C much greater than 1. These cavities can be constructed using
higher finesse mirrors and/or smaller mode waists. Given our increased ability to engineer highquantum efficiency joint measurements and low technical-noise probing of the optical cavity, thismay be possible. However, as the phase resolution on the Bloch sphere continues to improve, state
manipulation will continue to be a challenge. For this reason, it may be advisable to first attempt
creating maximally entangled states using a smaller atom number, perhaps closer to 103 than 106 .

8.5

Final remarks
Entanglement-enhanced sensors may be among the first practical technologies with no clas-

sical analogue. In fact, squeezed states are already being added to a few precision measurements

131[57, 121]. As previously mentioned, entangled states are a reasonable avenue toward improving
optical lattice atomic clocks and atom interferometers. Additionally, gravitational wave interferometers [57] have demonstrated the use of squeezed light in long baseline interferometry and is
expected to implement squeezed light to improve sensitivity of gravitational wave detectors [64].
In addition to using atom-cavity sensors to create spin squeezed states, atom-cavity systems
are being explored for other entanglement-enhanced technologies as well. Atoms in cavities canbe used to build quantum repeaters to build up quantum networks [35, 29, 146, 15]. They arealso being explored for quantum simulation and exploration of highly complex many body physics[117, 82]. They are being used to control optomechanical systems [27], and they are allowing thecreation of more exotic entangled states with negative Wigner functions and complex structure[107, 8]
But what comes next? Although it is impossible to definitively predict the future impact ofentanglement-enhanced technology, allow me to spend the last paragraph of my thesis in wild speculation. Perhaps one day top down approaches to entanglement generation such as spin squeezing
could be used as entanglement resources. This entanglement could be created, manipulated at will,
and then shared among large quantum networks. Perhaps large ensembles of atoms in a cavity
could be used for analog quantum algorithms, where the probability distribution of the collective
spin is manipulated in specific ways to perform algorithms that are difficult for classical computers.
Lastly, as has already been proposed, perhaps entangled atomic clocks could be connected in a
network [84] to achieve enhanced global timekeeping and synchronization.
I believe the outlook for atom-cavity systems and other quantum technologies is promising,and the most impactful uses of these quantum platforms are yet to come.

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Appendix A

Homodyne and Heterodyne Detection

For all of the experiments in this thesis, we wish to make phase or frequency measurementsof an optical field used to probe the optical cavity or emitted from the atoms inside the opticalcavity. To do this, we overlap the emitted field with another local oscillator (LO) beam used asa phase reference, and perform optical interferometry. There are generally two possible ways todo this. The first is for the measured field and LO to have the same frequency and measure thedirect DC interference. This is known as homodyne detection. The second option is to detune themeasured beam from the LO and observe oscillations in the interference between them, allowing
a simultaneous measurement of both the laser’s amplitudes and relative phases. This is known asheterodyne detection.
We have used both heterodyne and homodyne detection to measure cavity fields for various
experiments. Heterodyne is often considered easier since it does not require as strong phase stability
between the probe and LO, there is an infinite dynamic range for phase excursions, and the signals
lie away from DC where technical 1/f noise sources can be troublesome. On the other hand,heterodyne detection also leads to a sub-optimal measurement of the phase of the emitted lightsince half the time the heterodyne effectively measures the phase quadrature and half the time theamplitude of the emitted light. This leads to an effective loss of 50% of the emitted information.
In recent experiments (Chapters 6,7), a homodyne measurement was used to perform an
entanglement-generating joint measurement of atoms in a cavity as well as to calibrate the powerin the probing beam. For this reason, and since the factors of two that determine the size of these146interference signals have caused difficulty in the past, I will derive the size of homodyne signals
here.

Figure A.1: Homodyne detection. A signal field Es (t) (blue) and strong local oscillator field
ELO (t) (red) are overlapped on a 50/50 beamsplitter and measured on a detector with sensitivity
S [Amps/Watt]. The signals are then subtracted and sent through a transimpedance amplifier
with gain Rt [Ohms]. The output voltage, Vout can be used either as a measurement of the signal
amplitude or signal phase, depending on the phase φ between the signal and LO.

As shown in Fig. A.1, the signal and LO beam are overlapped on a 50/50 beamsplitter anddetected on a balanced homodyne detector. The signal beam and LO beam can be written, for thisanalysis, as classical electric fields with peak amplitudes Es and ELO .
Es (t) = Es sin(ωt + φ)
ELO (t) = ELO sin(ωt)

(A.1)
(A.2)

The photocurrents I1 (t) and I2 (t) that would be generated on an infinite bandwidth detector
on each arm are given by a constant 2 which converts squared electric fields into photo-current.
2  2 22
Es sin (ωt + φ) + ELO
sin2 (ωt) + 2ELO Es sin(ωt + φ) sin(ωt)
22  2 2
2
I2 (t) =Es sin (ωt + φ) + ELO
sin2 (ωt) − 2ELO Es sin(ωt + φ) sin(ωt)
2
I1 (t) =

(A.3)
(A.4)

147
The final terms in I1 (t) and I2 (t) must differ by a sign due to energy conservation requirements inthe beamsplitter.
The products of oscillating signals in I1 (t) and I2 (t) can be rewritten in terms of sums anddifferences.

2  2 2
2
E sin (ωt + φ) + ELO
sin2 (ωt) + ELO Es cos(φ) − ELO Es cos(2ω + φ)
2 s
2  2 2
2
I2 (t) =Es sin (ωt + φ) + ELO
sin2 (ωt) − ELO Es cos(φ) + ELO Es cos(2ω + φ) .
2
I1 (t) =

(A.5)
(A.6)

Since photodetectors do not have the bandwidth to directly measure the optical frequency of thebeam ω, the measured difference of the two photocurrents I diff also includes a time average ofthe oscillations, signified as an over bar. The cos terms which oscillate rapidly at freqeuency ωwill quickly average to zero, and the sin2 terms will simple become DC offsets which cancel in the
difference. The result of the time average is,I diff = (I1 − I2 ) = 2 Es ELO cos φ.

(A.7)

The only term that remains from I1 (t) and I2 (t) is the DC interference between the signal and
LO. Additionally, the conversion constant  can be re-written in terms of the sensitivity of thephotodiode when the electric fields are re-written in terms of the average optical power in thesignal and LO beams Ps and PLO and the photodetector’s sensitivity S in units of Amps per
Watt. This relation can be easily derived by considering the effect of a single beam (here labeled“a”) incident on a single photodetector using I a = SPa = 2 Ea2 /2. The final homodyne signal,in terms of measured quantities Vout , Rt , S, and optical powers, is found by substituting Es andELO by their average powers Ps and PLO as measured on a DC photodiode and multiplying by thetransimpedance gain Rt .
Vout = Rt I diff = 2Rt S

p
Ps PLO cos(φ).

(A.8)

When performing heterodyne detection, one finds the exact same solution where φ now becomes afunction of time φ → φ(t).

Appendix B

Feedback Loops

For the optical and electrical elements of our experiment to work in concert requires a numberof feedback loops used for laser frequency, amplitude, and temperature stabilization, as well as
dynamic control and synchronization of the optical and rf signals required in an experimental
sequence. Inserting PID (Proportional-Integral-Differential) feedback loops has become a general
purpose tool in the lab. In this section, I will review a simplified feedback theory and useful
information about the standard loop filter circuits most commonly used for feedback in our lab.

B.1

Feedback model

Figure B.1: Simple system with feedback. In order to stabilize a noisy input voltage Vin∗ (t), one
can compare the voltage to a set point, Vset and apply feedback by multiplying the difference bylarge gain L∗ (ω).

A simple steady-state model of feedback can elucidate the most important results for design-

149
ing feedback loops. Assume there is an AC signal Vin∗ (t) represented in the complex plane that wewish to stabilize. I assume that this signal, presumed to be a voltage, is a constant oscillation witha complex amplitude Vin∗ , Vin∗ (t) = Vin∗ eiωt . In order to stabilize Vin∗ (t) to a constant real value Vset ,we apply feedback through a loop filter with a complex, frequency dependent gain labeled, L∗ (ω).
∗ can be found by solving the recursive equation for the
The steady state output voltage Vout

feedback loop,∗∗
Vout= Vin∗ (t) + (Vout− Vset ) × L∗ (ω).

(B.1)

The equation has a solution,
∗
Vout
=

Vin∗ (t) − Vset L∗ (ω)
1 − L∗ (ω)

(B.2)

∗ → V
If L∗ (ω) is very large, Vout
set as desired; the feedback successfully suppresses the noise at

frequency ω. However, in reality, the system will be subjected to a continuum of noise at all
frequencies ω. For the feedback to work properly, therefore, the system must be well behaved at all
ω. In particular, the most important point is at the unity gain frequency ω1 such that |L∗ (ω1 )| ≡ 1.
The feedback will actually amplify noise near frequencies ω1 unless L∗ (ω1 ) is negative or imaginary,that is, unless Re[L∗ (ω1 )] < 0. Therefore, the optimization of a feedback loop is this: maximize thegain L and feedback bandwdith ω1 while still making sure that the phase shifts around ω1 (called
loss of phase margin) are sufficiently small to avoid positive feedback at the unity gain frequency.
Positive feedback begins to occur when the loss of phase margin surpasses 90◦ . Often, phase losses
slightly larger than 90◦ are allowable, but lead to a small amount of noise gain (and therefore avisible noise bump in the spectrum) near the unity gain frequency.

B.2

Loop filters
Loop filters are built to allow a large amount of tunability in the amplitude and bandwidth ofL∗ (ω). PID loop filters have a constant, integral L∗ (ω) ∝ 1/ω, and differential gain term L∗ (ω) ∝ ωthat allow one to craft the most optimal feedback. Importantly, integrators give a large gain boost

TJ011-A3 Parameter Ranges
Proportional Gain
-50 to 20 dB
PI1
15 kHz to 7 MHz
PI2
20 kHz to 10 MHz
D
20 kHz to 10 MHz
Lead Span×3.2 to ×10 (trimpot)
DC Gain
20 to 1000 (trimpot)

150

Table B.1: Ranges for setting for the JILA TJ011-A3 high speed loop filter.

at low frequencies but also cause a 90◦ phase lag. Differentiators cause a 90◦ phase advance andare usually used to correct for some loss of phase margin at high frequency. One of the mostcommon problems for a feedback loop is when the feedback crosses unity gain with two integrators,
and therefore a 180◦ phase shift, causing the loop to go into positive feedback and oscillate out ofcontrol.
For reference, the high speed JILA-built loop filters most commonly used in the Thompson
lab have transfer functions decided by various frequency corners that are represented below in Fig.
B.2. Corner freqeuncies are set by the PI1, PI2, DC gain, D, and Lead span settings. In the
TJ011-A3 (most recent) version of the JILA high speed loop filters, the loop filter settings canbe adjusted by the amount shown in Table B.1. The DC gain limitation on the integrators canbe engaged (using a jumper) on one or two of the integrators. importantly, since real systems oflasers and electronics also have their own transfer functions, in order to predict the performance ofa feedback loop one must multiply L∗ (ω) times the gain of the rest of the system.

B.3

Example: beatnote lock between 2 lasers
As a specific example of a feedback loop in our lab, consider the most common application:phase locking two lasers together with an offset frequency. This is performed using a Hittite
HMC440 phase freqeuncy detector (PFD) that is sensitive to both frequency and phase errors
between the two lasers. When viewed as freqeuncy feedback, the phase detection can be viewed
as an additional integrator in the system that integrates frequency errors into phase errros. For
this reason, often only one integrator is necessary to achieve a high quality phase lock between

151

Figure B.2: Transfer function parameters of JILA high speed loop filters. The gain profile L∗ (ω)
is designed to have high gain at low frequencies and then can be optimized to achieve the highest
feedback bandwidth on the system without losing phase margin.

the two lasers. The feedback settings for an example laser beatnote phase lock are shown in Table
B.2. The resulting power spectrum of interference between the two laser, as normally viewed ona spectrum analyzer in the lab, is shown in Fig. B.3. The feedback is seen to effectively remove
power from the background Lorentzian power-spectrum between the two lasers and concentrate themajority of the power within a phase-locked carrier that appears as a delta function on the centerof the spectrum analyzer. The approximate unity gain frequency of 1 MHz is seen as the locationof the “shoulders” of the phase lock. The fact that these shoulders do not peak strongly indicatessufficient phase margin at the unity gain freqeuncy.
As an additional note, one observed difficulty in setting up phase locks using the HMC440
PFDs is the need for high signal to noise in the phase frequency detector. Empirically we observethat greater than 40 dB of signal to noise between the carrier and the noise floor (measured in a
3MHz spectrum anaylzer bandwidth), is usually needed for high quality phase lock.

152

Example Beatnote Lock Settings
G1
0 dB
G2
−10 dB
PI1
150 kHzPI2
Off
D
100 kHz
Table B.2: Example settings for a beatnote lock. The additional parameters DC gain and leadspan are continuously adjusted using potentiometers. For this setup they were tuned near theirmaximum values.

Figure B.3: Power spectrum of interference between two beatnote-locked lasers as normally viewed
on a spectrum analyzer and taking 100 averages. The majority of the laser power resides in aphased-locked carrier that appears as a delta function in the power spectrum. The unity gain
bandwidth is indicated by the shoulders of the distribitution, measured with markers to be 1 MHz
away from the carrier.