Spin squeezing in cavity-quantum electrodynamics

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Spin squeezing for metrology

This thesis aims to apply spin squeezing to an atomic clock at a metrologically relevant level. This chapter gives a brief introduction to both aspects: atomic clocks and their stabilities on one hand, and spin squeezing of neutral atoms in cavity-QED systems on the other. The experimental setting of the thesis is introduced in a metrological context, and we will discuss the prospects of this new apparatus in terms of achievable squeezing.

 Atomic clocks and clock stability

Generally, the extreme sensitivity of atomic sensors is commonly achieved in interferome-ters: the measured quantity is mapped to a phase shift between two modes (two atomic states) and is read out using interferometric techniques (Fig. 1.1). For example, certain types of atomic clocks are interferometers with two internal states (|↑i and |↓i) whose tran-sition frequency serves as a frequency standard. The interferometer phase shift is set by the frequency of the electromagnetic field that composes the interferometer, realising an ultra sensitive frequency measurement. Another prime example is atomic inertial sensors measuring gravity (gravimeter), AC acceleration (accelerometer), or rotations (gyroscope). They are matter-wave interferometers: the atomic wave packets are spatially separated (two modes of momentum states) and then recombined such that the phase difference between the states in different trajectories is sensitive to inertial forces.
In practice, the most common working scheme of atomic clocks is a servo loop to lock a local oscillator (LO) – a continuously running clock signal – on the atomic transition frequency. The frequency of the LO is repeatedly compared with the atomic transition by “interrogating” an atomic sample and is subsequently corrected. The spectroscopic mea-surement of the local oscillator can be performed in various ways, all based on the coherent evolution of the clock states under a driving field, which will be explained in this section.
While this interrogation scheme is known as “passive”, an “active” scheme also exists such that a clock signal is directly generated from the atomic transition through a resonant cavity. The most important example would be the active hydrogen maser [57], which plays an important role in timekeeping and dissemination of time reference.
In this section, I will be focusing on atomic clocks based on two-mode interferometer. After some basic formalism in the language of pseudo-spin (spin-1/2 or N spin-1/2), we discuss the figures of merit of clocks in a context relevant to our experiment – trapped-atom clocks, among which are some of the state-of-the-art apparatuses.
The Ramsey interferometer offers superior sensitivity compared to the Rabi spectroscopy. The pulses can be performed in short time, while the quality factor of the clock is only limited by the free evolution time, during which the atoms are free from the interrogation field.
It is appreciably at the heart of the ingenious design of atomic fountain clocks, where cold atoms are launched upwards in a parabolic flight and traverse a microwave cavity twice, implementing the two π/2 pulses of the Ramsey interferometer. The Ramsey free evolution time is then determined by the height of the trajectory, limited by the size of the apparatus.
Many atomic inertial sensors are also based on the atomic fountain configuration. Simi-larly, the sensitivity to inertial forces scales with the interferometer time which is determined by the size of the trajectories [59]. To achieve higher sensitivities, one has to enlarge the apparatus, but also to fight against the free expansion of the cloud, requiring colder samples. An extreme example is the pico-Kelvin temperature required in a 10 m tall atomic fountain interferometer [60].

Trapped-atom clocks and TACC

With trapped atoms, limitations of the free-falling and expanding atoms can be alleviated. Long Ramsey evolution times and at the same time a compact setup make trapped-atom clocks extremely attractive. Similarly, trapped-atom inertial sensors, or with atoms in a “waveguide” for guided matter-wave interferometry, are promising for various applica-tions [61].
However, trapping the atoms might seem contradictory to metrology that measures an unperturbed transition by definition. A trap is an inhomogeneous energy potential, which is in general different for the two clock states. The trap can then induce shift on the clock transition frequency that compromises the clock accuracy and stability. The inhomogeneous shift also deteriorates the coherence in the spectroscopic measurement. Luckily, “magic” traps are found in various systems, in which the two clock states are perturbed equally such that the clock transition is unaffected. As a prime example, the discovery and realisation of “magic wavelengths” for various narrow-line optical transitions lead to the rapid development of optical lattice clocks, reducing the uncertainty by two orders of magnitude in merely a decade [62, 63].
Atomic clocks suffer not only from single-atom uncertainties, such as the perturbations from the trap, but also from the ensemble uncertainties from collisional interactions among atoms, leading to density dependent frequency shift. The latter has been a major issue of trapped-atom clocks in which the atomic density is generally high.
In fermionic optical lattice clocks, strong interactions contrarily suppress collisional shift [64], leading to the recent development of 3D lattice clocks with degenerate Fermi gases [65]. But this is not the case for microwave clocks with Rb or Cs atoms, which are currently closer to broader applications. Then the QPN is harder to mitigate as increasing the atom number has a price to pay. In this context, spin-squeezing beyond the SQL is particularly relevant for trapped-atom clocks.
We will from now on focus on our experiment – trapped-atom clock on a chip (TACC) at SYRTE. It is a microwave clock using magnetically trapped, ultra-cold 87Rb atoms. TACC has been a unique realisation of metrology-grade trapped-atom clock based on atom chip technologies.

Atom chips

Since their invention eighteen years ago, atom chips [66] have enabled experiments from fast preparation of BECs to the zero-g experiments in a drop-tower [67]. Atom chips continue to attract tremendous efforts with the prospects not only as compact and robust platforms for atomic sensors and clocks, but also allowing to trap molecules, charged particles and Rydberg atoms for the study of quantum technologies and fundamental physics. A recent review can be found in [68].
Like conventional magnetic traps, atom chips create magnetic field minima (static or dynamic) that can trap atoms with positive magnetic moment (low-field seekers). Various types of traps can be generated by a combination of bias fields and electric currents running on the chip in various geometries. Here, I shall only mention two common types of static traps concerned in this thesis. Quadrupole traps Commonly generated simply by a pair of anti-Helmholtz coils, quadrupole traps can provide tight confinement, due to non-vanishing field gradient at the trap centre. However, zero-field at the trap centre leads to spin-flip losses (Majorana losses) [69], more significant for the colder atoms.
For atom chips, the building block of micro-traps is the 2D quadrupole trap created by a single wire and a homogeneous bias field perpendicular to it [66]. Confinement in the third dimension while maintaining a finite field gradient at the trap centre can be achieved by bending the wire in a U-shape. This is widely used for the magneto-optical trap (MOT) in atom chip experiments. A pedagogical demonstration can be found in [70].
Ioffe-Pritchard traps The Ioffe-Pritchard traps alleviate the Majorana losses by having a finite B-field at the minimum (field gradient also vanishes). The trap potential near the trap centre then approximates a harmonic potential. Ioffe-Pritchard traps are realised on atom chips by a Z-shape wire, or more versatilely by dimple traps – two wires cross perpendicularly. The latter is commonplace for ultra-cold atoms and is predominantly used in TACC experiment.

A pseudo-magic trap for TACC

In the 87 Rb ground state manifold, states |1, −1i, |2, 1i and |2, 2i are magnetically trappable. Luckily, The energy difference between |1, −1i and |2, 1i – a function of magnetic field – exhibits a minimum at a “magic field” Bm ‘ 3.229 G, where the transition frequency is first-order insensitive to the magnetic field. |1, −1i → |↓i and |2, 1i → |↑i are then used as the clock states, with the Zeeman shift near Bm given by ΔνB = b(B − Bm)2 ≡ bΔB2 (1.10) where b ‘ 431.356 Hz/G2 [71].
We will consider an atomic cloud trapped in a harmonic trap. We shall see that a pseudo-magic trap can be found, in which the inhomogeneity in the transition frequency can be largely cancelled, resulting in very long coherence times.
Zeeman shift In a harmonic trap with trapping frequencies ωx, ωy and ωz, the Zeeman shift (Eq. 1.10) is position dependent (r = (x, y, z)) [47]: ΔνB(r) = bm2 ω2×2 + ω2 y2 + z2 − 2gz + ΔB µB 2 (1.11) where m is the atomic mass , g the gravitational acceleration, and µB the Bohr magneton. The gravity shifts the cloud away from the trap centre, the so-called “gravitational sag”. It has a considerable influence which will become clear later. The curvature of the shift at the trap centre can have different signs depending on the bias field ΔB.
Spatial inhomogeneity compensation While the collisional shift is stronger towards the cloud centre (a positive curvature), the Zeeman shift can be tuned (ΔB < 0) to have the opposite curvature that almost cancels the inhomogeneity close to the trap centre. The imperfections are twofold: different forms of the magnetic potential (quadratic) and the den-sity (Gaussian); and the fact that gravitational sag displaces the centre of the density profile away from that of the magnetic potential (Fig. 1.2(c)). Nevertheless, this mutual compen-sation between the two inhomogeneity sources allows a coherence time of a few seconds in TACC [47].
Two-photon transition To drive the clock transition, two photons are needed via an intermediate state |1, 0i or |2, 0i. In the experiment we use a microwave (MW) field ∼ 6.834 GHz and a radio-frequency (RF) field ∼ 1.7 MHz with Rabi frequencies Ωmw and Ωrf respectively, with a detuning Δi/(2π) ∼ 500 kHz to the intermediate |2, 0i. The two-photon Rabi

Clock stability

In this subsection, after a general introduction to the metrics of evaluating a clock, we will briefly discuss some common noise sources of our clock. A detailed analysis of the TACC-2 stability is given in Sec. 3.1.

Figures of merit

In general, locking the LO to the atomic transition realises a clock signal νclk(t) = νat0(1 + + y(t)) (1.16) where νat0 denotes the unperturbed atomic frequency. We will use frequencies in Hz for convenience. We identify two terms that are commonly used for evaluating clocks.
Accuracy denotes a systematic shift from the unperturbed atomic frequency. It de-pends on the particular realisation and its uncertainty quantifies the accuracy of the clock. It is less of a concern for secondary standards for applications, as long as the systematic inaccuracies can be calibrated with the primary standards. However, the uncertainty of the systematic error cannot be distinguished from the random error of a measurement, therefore also contributes to the clock instability. The clock accuracy is not studied in this thesis.
Stability y(t) is the fractional frequency fluctuation and quantifies the stability of the clock. More specifically, one also distinguishes short-term (seconds) and long-term (hours, days) stability, depending on the application. The noise mechanisms and limits are also very different. In this thesis, we focus on the applications of quantum technologies, which target on improving the short-term stability.
Allan variance The Allan variance is the standard way to characterise the clock stability. It resolves the problem that the standard variance is not well defined at low frequencies if the clock signal has flicker noise or drift. Specifically, the Allan variance (AVAR) is defined as [74]: σy2(τ) = 1 1) M−1 (¯yi+1 − y¯i)2 (1.17) 2(M X − i=1
It is the expectation value of the two-sample variance, where y¯i(τ)’s are contiguous samples
averaged over time τ:1 1 Z iτ y¯i(τ) = y(t)dt (1.18) (i−1)τ τ which reveals the noise at different time scales.
If we consider the noise spectrum (a function of the Fourier frequency), the Allan variance can be understood as a bandpass filter near the frequency 1/(2τ). The scaling of σy2(τ ) as a function of τ then reflects different noise sources in the power spectrum of the clock signal [75]. For example, white frequency noise, which usually dominates in timescales between seconds to hours, scales as τ− 1. The flicker frequency noise appears flat in σy2 versus τ, and random walk frequency noise diverges as ∼ τ.

Noise in P↑ measurement

Uncertainties in determining P↑ originate from different mechanisms:
1 In practice, for limited clock measurement samples, one uses the overlapping Allan variance or Total variance [74].
Quantum projection noise As we have already discussed in the introduction, projec-tive measurements of uncorrelated atoms lead to the binomial distribution of the outcome.
Operating at mid-fringe with P = 1 , the variance of N from measuring N atoms reads: ↑ 2 ↑ √ Var(N↑) = P↑ (1 − P↑)N = N/4. Hence the noise in P↑ due to QPN reads σP↑ = 1/(2 N). One can only fight with it by increasing the atom number in an unfavourable scaling N−1/2. Moreover, density-related frequency shift and technical difficulties also limit this approach, as we will see in particular for TACC (Sec. 3.1).
Detection noise The error in counting atoms in the two states also leads to an uncertainty in P↑. It is usually of technical origin and is subject to the experimental methods. It may also boil down to some form of shot noise: for example, the photon shot noise of the imaging beam or the fluorescence. In TACC, the primary detection is absorption imaging. The detection noise is nearly limited by the photon shot noise.
Other technical noise In a Ramsey sequence, we assume two π/2 pulses. However, noise in the pulse area of the π/2 pulses due to e.g. power fluctuations of the LO field directly leads to error in the final P↑.2 Apart from technical improvements, slow variations can be rejected by e.g. probing alternately on both sides of the Ramsey fringe and only extracting the frequency from the differential signal.

Local oscillator noise

Most naively, as a phase modulo π is measured in a Ramsey scheme, the phase is subject to an ambiguity when its deviation might exceed π. Moreover, a large frequency deviation reduces the sensitivity of the Ramsey spectroscopy (cf. Eq. 1.9, with P↑ different from 1/2). In other words, the clock has a fairly narrow bandwidth in correcting the LO frequency.
More profoundly, as we can see from Eq. 1.20, how precisely the LO frequency is measured is also determined by the sensitivity function. The clock is blind to the LO noise outside the Ramsey time. With the T R only a small fraction of the cycle time Tc, the clock resembles a discrete data acquisition that periodically samples the LO frequency and its fluctuations, suffering from aliasing such that high frequency LO noise (multiples of the sampling frequency 1/Tc) can further degrade the clock stability.
This is known as the Dick effect, which has been one of the most important noise sources for optical clocks today. The Cs fountain clock at SYRTE is supported by a cryogenic sapphire oscillator, which is not accessible for broader metrological applications. In fact, the Dick effect can be the most prominent limit for compact devices where only a quartz crystal is affordable.
This also motivates new techniques such as non-destructive measurements to use the same atoms in multiple clock measurements [56]. In the case where the excess LO noise limits the Ramsey time, multiple short Ramsey sequences sharing a single phase of atom preparation effectively improve the clock duty cycle (TR/Tc), alleviating the Dick effect.
In TACC, the Dick effect is a major contribution to clock instability. Although recycling atoms with non-destructive measurements has not been studied in this thesis, it is one of the two major objectives of TACC-2 and experiments will be carried out in the near future.

Collisional shift

As introduced in Eq. 1.13, the collisional shift is one of the most important systematic effects in atomic fountains and has a major contribution to the uncertainty [78, 79]. The situation is much severer for trapped-atom clocks in which the density can be 4 orders of magnitude higher.
Correction to the systematic error can be applied shot-to-shot based on the measurement of the total atom number which determines the atomic density. However, this correction is compromised by the detection noise in the atom number. Furthermore, in the presence of atom loss, only the final atom number is known. The statistical nature of atom loss imposes an uncertainty in the average atom number during the interrogation time, hence in the collisional shift. In fact, as we will see, for TACC-2 the uncertainty of this correction can be an important source of clock instability, if a large part of the atoms is lost.

Other instabilities in TACC

There are other technical fluctuations which deteriorate the clock stability in TACC. Here I only take an overview and a detailed analysis will be given in Sec. 3.1.
Magnetic field fluctuations As I mentioned, the pseudo-magic trap for TACC requires a bias field slightly lower than the magic field. In this case, however, the Zeeman shift is more sensitive to magnetic field variations. Further is the bias field away from the magic field, the bigger the contribution from the magnetic field fluctuation. As we will see, this is one of the major noise sources in TACC.
Atom temperature and its fluctuations Both the Zeeman shift and the density shift depend on the atom temperature, but an optimum field exists at which the total shift is first-order insensitive to temperature fluctuations. However, this optimum field differs from the magic field hence the insensitivity to the temperature fluctuation and that to the magnetic field fluctuation can not reconcile. The overall optimum bias field inevitably suffers from noise both in magnetic field and in atom temperature.
Motivations for trapped-atom clocks As we know, the SQL is not a remote theoretical limit. The state-of-the-art atomic fountains have long reached a short-term stability limited by the QPN [23]. Here I would like to emphasise that spin squeezing surpassing the SQL is particularly relevant for trapped-atom clocks:
• For trapped-atom clocks in which the collisional shift can impose a large uncertainty due to high density, the number of atoms is often limited. Employing spin-squeezed states can mitigate the severe SQL.
• In recently developed 3D lattice clocks in which collisional interactions are suppressed by keeping a single atom per lattice site [65], increasing atom number becomes techni-cally difficult and the QPN limit is expected to be approached soon.
• More generally, squeezed states can benefit compact applications where the number of atoms is limited, either technically or fundamentally.

Overview of spin-squeezing generation

With all these prospects, there have been tremendous experimental efforts over a decade to demonstrate and study spin squeezing in atomic systems. Here I give a brief summary of the most studied methods for squeezing generation.
As a particular type of entanglement, spin squeezing correlates the local spin observables of the atoms. Let us distinguish two categories of entanglement creation, namely by dynamics due to interactions between atoms or by a partial projection of the collective state.

Squeezing by inter-atomic interactions

Described in terms of collective spin operators, entanglement requires non-linearity. A bench-mark model is the so-called one-axis twisting (OAT) Hamiltonian, with the simplest non-linearity: ˆ = ˆ2 (1.28) HOAT ~χSz
The collective operator ˆ2 actually means that each atom interacts with all others. But it Sz also gives a very intuitive picture: the precession rate is itself proportional to Sz, distorting the Bloch sphere (Fig. 1.3(b)). The initially isotropic noise distribution is twisted under the dynamics. It can be shown that the process almost preserves the minimum uncertainty area and exhibits squeezing along a certain axis [28]. The dynamics reaches a maximum squeezing parameter around tmax ∼ χ−1N−2/3 and later loses the squeezing as the state wraps around the Bloch sphere, but the entanglement keeps increasing, and the assessment of which requires more complex measures like the nonlinear squeezing [82]. The maximum squeezing at tmax scales with ξ2 ∼ N−2/3. Collisional interactions in BECs As widely explored, collisional interactions in a BEC lead to an OAT Hamiltonian, in which χ depends on the scattering lengths and the overlap of the wavefunctions of the condensate modes. By controlling the scattering lengths via Feshbach resonances [33, 36] or modifying the wavefunction overlap via state-dependent potentials [34, 35], substantial spin squeezing has been achieved. However, strong interactions in a BEC generally limit their application for clocks.
Cavity feedback As will be detailed below, light-mediated interactions between atoms in an optical cavity can also produce an effective OAT Hamiltonian [43, 44]. The χ term now depends on the dispersive coupling between atoms and cavity photons, and can be very large in a strong-coupling cavity-QED system. However, there is usually excessive noise enhancement in the anti-squeezed quadrature that far exceeds the squeezing due to cavity decay. This so-called non-unitary anti-squeezing compromises possible metrological gain in clock applications [83], but recent progress has approached near unitary squeezing [84, 45].
Beyond OAT A few other schemes show metrological advantages compared to the OAT, including the “twist-and-turn” [85] and two-axis-counter-twisting (TACT) [28]. The for-mer has been demonstrated in a BEC [86], while the TACT, despite extensive theoretical studies, remains experimentally challenging. Ideally, the TACT, described by a Hamiltonian ˆ ˆ2 ˆ2 2 ∼ 1/N, HTACT = ~α(Sθ −Sθ+π/2) can achieve maximum squeezing at the Heisenberg limit ξ and it squeezes faster compared to OAT (ξ2 ∼ N−2/3). However, the benefits can be lost in the presence of decoherence [87].

Table of contents :

Introduction 
1 Spin squeezing for metrology 
1.1 Atomic clocks and clock stability
1.1.1 Basic notions
1.1.2 Trapped-atom clocks and TACC
1.1.3 Clock stability
1.2 Concepts of spin squeezing
1.2.1 Surpassing SQL with spin-squeezed state
1.2.2 Overview of spin-squeezing generation
1.3 Spin squeezing in cavity-quantum electrodynamics
1.3.1 Real-world cavities
1.3.2 Cavity-QED in the dispersive regime
1.3.3 Squeezing by QND measurement
1.3.4 Squeezing by cavity feedback
2 Experimental methods 
2.1 Setup
2.1.1 Atom chip assembly
2.1.2 Vacuum system
2.1.3 Optical system
2.1.4 Magnetic fields
2.2 Cold atom preparation and interrogation
2.2.1 Laser cooling and optical pumping
2.2.2 On-chip transport of atoms
2.2.3 Evaporative cooling
2.2.4 The clock trap
2.2.5 Interrogation photons
2.2.6 Interrogation pulse tuning
2.2.7 Absorption imaging
2.3 Cavity probing and stabilisation
2.3.1 Cavity parameters
2.3.2 Laser scheme and setup
2.3.3 PDH lock with minimum intra-cavity power
2.3.4 Digital filter cancelling mechanical resonances
2.3.5 Feed-forward targeting the thermal drift
2.3.6 Locking “without” light
3 A highly stable cavity-QED platform 
3.1 Clock stability analysis
3.1.1 Detection noise
3.1.2 Dick effect
3.1.3 Atom number fluctuation
3.1.4 Magnetic and temperature fluctuations
3.1.5 Ramsey Contrast
3.1.6 Preliminary stability results
3.1.7 Prospects with spin-squeezed states
3.2 Characterisation of the atom-cavity coupling
3.2.1 Vacuum-Rabi splitting
3.2.2 Cavity shift in the dispersive regime
3.2.3 Atom-cavity alignment
3.2.4 Intra-cavity optical lattice
4 Spin squeezing by measurement 
4.1 Inhomogeneous coupling and decoherence
4.1.1 Phase shift by cavity probe
4.1.2 Monte-Carlo simulations
4.1.3 Contrast and phase measurements
4.2 Composite measurements
4.2.1 Spin echo
4.2.2 Composite pulse
4.2.3 Coherence measurements
4.3 Conditional spin squeezing
4.3.1 Measurement uncertainty
4.3.2 Spin noise estimation
4.4 Squeezing by cavity feedback
4.5 Outlook
4.5.1 Squeezing lifetime
4.5.2 Alternative inhomogeneous-light-shift compensation
5 Quantum amplification by ISRE 
5.1 Identical spin rotation effect (ISRE)
5.1.1 Basic principles
5.1.2 An intuitive picture with atoms in two energy classes
5.1.3 Experimental signatures
5.1.4 Mean-field kinetic equation
5.1.5 Observation of ISRE via motional energy
5.2 Interplay between ISRE and cavity measurements
5.2.1 Origin: inhomogeneous coupling
5.2.2 Cavity shift in a continuous probing
5.2.3 ISRE triggered by a probe pulse
5.2.4 Dynamics in motional energy sensed by cavity shift
5.3 Amplification of quantum fluctuations
5.3.1 Experimental observations
5.3.2 Simple model with two energy classes
5.3.3 Simulation using classical spins
5.3.4 Circumventing the amplification
5.3.5 Future work
Conclusion and outlook 
A PDH error signal
B Alternative light shift compensation
C Résumé en français
List of Figures
Bibliography 

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