Continuous quantum measurement of cold alkali-atom spins

He gave me the opportunity to experience what will be one of the best scientific and personal periods of my life, and I will always be grateful. Thanks to JM Geremia for his contributions to much of the experimental and theoretical work described in this thesis.

Getting to the Point

1.5) If we now perform a measurement of the collective spin in the z direction, ˆFz, we get a. An illustration of the Faraday rotation measurement we use in the laboratory is shown in figure 1.2.

Figure 1.1: (A) Graphical representation of the spin-polarized atomic sample as a classical magnetization vector with transverse quantum uncertainty

Personal Lab History

Our own attempt to characterize the effect of the tensor terms ultimately resulted in reference [29]. Without the nonlinear (quadratic) terms of the full Hamiltonian, these practical tasks would be impossible.

Chronological History of Concepts

A few years later, spin squeezing was reported using QND measurements of a hot thermal ensemble of atoms [37]. Over the past few years, continuous QND measurements of cold atomic ensembles have been reported in our own work and also in others, including the Jessen and Takahashi groups.

Thesis Organization

This Hamiltonian is then used to derive an unconditional master equation for the collective spin state of the atomic cloud investigated in Chapter 7. Semiclassical experimental results supporting the theoretical analysis of the interaction Hamiltonian are presented in Chapter 13.

Introduction

Review of Literature

Stochastic processes and differential equations are ubiquitous in the context of noisy measurement and are treated mathematically in. These references are useful for understanding the open system trajectory formalism, but many do not discuss conditional measurement and control as much as more recent works.

Measurement Operator Formalism

The 'classical world' consists of the detector (with efficiency η) and the observer who estimates the spin state and controls the system via a controller. Now we describe the process of inferring the state of the system after the system has interacted with the environment and measured the environment.

Figure 2.1: A general quantum measurement and feedback control schematic as described in the text.

The Continuous Limit

M η(y(t)dt − hFˆzi(t)dt) is a Wiener boost and dW(t)/dt is a Gaussian white noise associated with the local homodyne oscillator shock noise. If M 6= 0, but the quantum efficiency of the detector η = 0, then we get the unconditional evolution of the state due to the measurement.

Modeling the Interaction

This form of SME is quite general and represents the canonical example of a continuous measurement. Another approach is to assume that the collective spin state is in a Gaussian state and only work with a few moments of the total distribution.

QND Measurement

This is the basis for the classical model of the quantum measurement that was used in chapter 10. However, there are several places where the QND characteristic of the measurement is practically compromised.

Quantum Parameter Estimation

In particular, we investigate the long-term measurement limit in Chapter 9, which produces the results of a discrete QND projective measurement. Following this paper, we adapted the techniques for measuring a magnetic field to our system in a set of theoretical papers [18, 20] that are summarized in chapter 10.

Quantum Feedback Control

The use of feedback is useful in the quantum domain for many of the same reasons as it is used in the classical world. The example of magnetometry is representative of the larger drive to analyze robustness and risk sensitivity in the context of quantum estimation and control.

Experimental Scenario

In this chapter we introduce the notation used in this thesis to represent the optical and atomic states and operators common to our experiment. Next, we introduce the atomic spin states, starting with a description of the individual cesium spin and moving on to the definition of spin-squeezed states and other collective states.

Optical States

We now discuss ways of representing the polarization of an optical beam propagating along the z-direction. For the same mode, we define a coherent state [68] in terms of Fock states as

Atomic Spin-States

The single atomic Dicke states corresponding to the sublevels of the f = 4 ground state of cesium are depicted visually in Figure 3.2. For much of the work described in this thesis, where the spin squeezing is relatively small, is Gaussian.

Figure 3.1: Cesium D 2 transition levels as described in the text. The probe and pump laser configurations are also shown.

Introduction

This chapter contains a very abbreviated summary of [17], which deals with the efficient computation of entanglement measures for symmetric spin states. For the sake of brevity, I have omitted most of the details and relevant references in order to examine a few key ideas.

Computing Entanglement

Computing Entanglement in the Symmetric Subspace

Therefore, the entanglement of states in SN will generally be more limited than that in the tensor product space. The unobtainable bound log2(k+ 1) is the entropy that can be achieved by a non-symmetric product of the two symmetric subsystems, {A, B}.

Entanglement of Reference States

The fact that an overall symmetric state could have an entanglement that scaled with this upper bound was not known before this paper and is a testament to the power of the numerical tools developed here. A plot of the equally distributed entanglement of formation (and entropy), EF(ρ,{N/2, N/2}), for a system of N spin-1/2 particles evolved under a counter-rotating spin -pressure Hamiltonian.

Figure 4.2: (A) Plot of the even split entropy of entanglement, E(ρ, {N/2, N/2}), for rep- rep-resentative states as a function of the number of particles, N (which is also equal to the entanglement of formation and distillation)

Spin-Squeezing and the Dynamics of Entanglement

One purpose of this chapter is to provide a more extensive treatment of the adiabatic elimination procedure than is usually provided in the literature. The resulting master equation consists of the polarizability Hamiltonian (including all hyperfine levels) and also spontaneous emission terms, corresponding to weighted jumps between ground state sublevels.

Figure 4.4: Spin-squeezing evolution for a system of N = 50 spin-1/2 particles evolving by the countertwisting Hamiltonian as measured by the squeezing parameter, ξ 2

Master Equation Derivation

In fact, in the end we are only concerned with the simplification of equation (5.29) and do not have to worry about the full inversion of the superoperator. This can be seen using both the cyclic property of the trace and the jump operator identity of equation (5.12).

Applications of the Master Equation

In the next chapter we start from this point to analyze the evolution of the optical polarization due to the atomic spin state. We then consider the evolution of the spin state due to the tensor term in special, experimentally relevant configurations.

Figure 5.1: (A) Schematic showing the relative orientation of the spin, magnetic field, and optical polarization

Deriving the Irreducible Representation

Using the Wigner-Eckart theorem, the angular dependence of the matrix element, hf0, m0|d|f, miˆ can be incorporated into the product of a Clebsch-Gordan coefficient and a reduced matrix element. 6.17). Furthermore, if we use the definition of ˆTm(j) and fill in the Clebsch-Gordan coefficients explicitly, we get:

Interpreting the Irreducible Hamiltonian

The symmetry of the tensor term can be seen by noting the identity ˆfxfˆy + ˆfyfˆx = ˆfx20 −fˆy20. For a linearly polarized input beam, the tensor term leads to an elliptically polarized scattered probe field [32, 26].

Coefficient Spectra

It is also important that this frequency occurs close to the frequency of the forbidden f = 4 to f0 = 2 transition. Again, the polarization dependence of the light shift can be removed by using a frequency close to the forbidden transition (f = 3 to f0 = 5).

Figure 6.1: Plotted, in order, are ˜ α 0,f=3 (A), ˜ α 1,f=3 (B), and ˜ α 2,f=3 (C). The green curve is a magnified version (×20) of the red curve to better see the zero-crossings

Semiclassical Evolution of the Probe State

In Chapter 8, we then reconsider the full analysis including the atomic quantum noise (related to spin-squeezing) for a particular alignment of the collective spin state. In this semiclassical approximation, we have completely neglected any evolution of the atomic state due to the probe beam.

Figure 6.5: Definition of the spherical coordinate angles used to describe the orientation of the collective atomic magnetization vector, F, relative to the fixed laboratory cartesian coordinate system

Tensor Suppression at the Magic Angle

This time-averaged Hamiltonian clearly vanishes at the so-called magic angle θ = arctan degrees with respect to the x-axis, or 54.7 degrees with respect to the field along the y-axis [25].

Tensor Suppression with Parallel Polarizations

Tensor-Driven Oscillations

Indeed, this is how we fine-tune this alignment in the laboratory, and experimental results along these lines are presented in Section 13.1.3. For example, one can determine the degree of optical pumping via the tensor dynamics as discussed in the next section.

Tensor Pumping Tomography

For example, one can determine the rate of optical pumping via tensor dynamics as discussed in the next section. influenced by the pumping efficiency, we have. with others zero again. In Chapter 5, we used a classical description of light to derive a governing equation of an atom with spontaneous emission.

Figure 6.6: Measurement of pumping efficiency as described in the text. On the left, the upper plot corresponds to dragging the spins in the xz-plane and measuring h S ˆ y0 i

Collective Master Equation Derivation

Henceforth we assume that the initial optical state is the same at each time step and simplify this equation to derive the unconditional master equation describing the atomic spins. This is a well-known unconditional master equation that unconditionally contradicts the initial coherent spin condition along x.

Moment Evolution

Consider a tuning close to resonance (c1/c2 = 11) and with an optical polarization along the y-axis (hSˆxi = +hSˆ0i). Consider again a tuning close to resonance (c1/c2= 11) but with an optical polarization along the x-axis (hSˆxi=−hSˆ0i).

Unconditional Squeezing via Collective Tensor Terms

For case 3 (with conditional squeezing), the evolution of the spin-squeezing parameter is given by. Here, the evolution of the spin-squeezing parameter in the initial time step is given by d(ξ2).

Polarimetry

By comparing the signal-to-noise ratios for the two analyzes above, we find that the signal-to-noise polarimetry is a factor of √. If compressed local oscillators were used, the SN R of the polarimetry measurement could be recovered.

Conditional Dynamics

So na¨ıvt one wants to be close to the 4-5' transition, where the negative effect of the 4-4' transition on the measuring strength can be neglected, which gives a higher expected ratio between times and thus more spin-squeezing. In the next chapter, we investigate numerically and analytically the behavior of the ideal SME of equation (8.31).

Figure 8.1: Comparisons of the of the spontaneous emission timescales (perpendicular po- po-larizations τ sc,⊥ , parallel polarizations τ sc,k ) and the spin-squeezing timescale (τ ss ) as a function of detuning from the f = 4 to f 0 = 5 transition.

Abstract

This chapter is adapted directly from [21] and somewhat repeats the previous chapters in order to stand alone. This work and the review [23] also discuss the constructive derivation of the relevant control laws rather than the intuitive process used in this chapter.

Introduction

This is just one example of the growing convergence of quantum measurements with classical estimation and control theories [56, 75]. In this chapter we focus on the long term of the QND measurement and feedback process.

Figure 9.1: The results of a single numerical simulation [36] of the SSE, equation (9.5), with M = 1, η = 1, and N = 10 spins initially aligned along the x-axis

Representations of the Conditional Evolution

See [53, 52] for an introduction to stochastic differential equations (SDE).) Photodetection sensitivity to. In short times, the development of the variance (shown on the logarithm) is deterministic and given by h∆ ˆFz2is(t).

Measurement of Evolution without Feedback

The projective character of the measurement is made clear by the evolution of 100 trajectories of hFˆzi(t). The distribution of the final results is given by the first histogram of Figure 9.1A. C) The evolution of the 100 trajectories all starting in an x-polarized CSS.

Closed-Loop Evolution

Those trajectories that do not land directly on the goal (about 10 out of 100) are recycled and spin back into the attractive region of the target state. The obvious solution to the above problem is to try a controller that ensures that the target state is the only fixed point of the SME/SSE.

Figure 9.3: One hundred closed-loop moment trajectories [36] of the SSE with feedback law b(t) = λh F ˆ x Fˆ z + ˆF z F ˆ x i(t)/2 and λ = 10 chosen from numerical considerations

Solution of the SME without a Field

Also, while the symmetry of eb1(t) will allow it to lock on both sides of the Bloch sphere, b2(t) will lock on only one side of the sphere. About 90% of trajectories are transported directly to the target state, but the remainder are "lost" on the first pass.

Moment Evolution via Cumulants

Now we are interested in finding the evolution of the cumulants that best describes the distribution [68]. Ultimately we arrive at what we were looking for: the evolution of the cumulants in terms of each other.

Few Level Dynamics

Note that the second-order cumulant equation (dκ2) contains a determinant Riccati term (−κ22dt), but also an initially small stochastic term (κ3dW). In other words, this is the fastest one of the eigenstates can be prepared.

Conclusion

Here we discuss the measurement of magnetic fields with our apparatus in terms of quantum parameter estimation. This chapter is adapted directly from the latter paper, which is entitled Robust quantum parameter estimation: Coherent magnetometry with feedback.

Abstract

This work is significantly expanded in a more complete paper [20] where open and closed loop configurations are discussed in detail. The results of these papers were discussed in the context of traditional magnetometry in [79].

Introduction

This theory is applied in Section 10.5, where we simultaneously derive mutually dependent magnetometry and spin-squeezing limits in the ideal case where the observer is certain of the spin number. Finally, in Section 10.6 we show that the estimate can be made robust to the uncertainty about the total number of spins by using precise feedback control.

Quantum Parameter Estimation

The correct description of the system would then be a density matrix ˆρθ(t), depending on the measurement registration(t). M η[y(t)dt− hFˆzib(t)dt]≡dW¯(t) is a Wiener boost (Gaussian white noise with variance dt) based on the optimality of the filter.

Optimal Estimation and Control

Equation (10.20) is the matrix form of the standard integrating factor solution for time-dependent scalar ordinary differential equations [52]. 10.21) When all parameters are known (and F0 =F), this description of the whole state is unnecessary because σbE(t) =σbR(t).

Optimal Performance: F Known

Inputs to the problem include the field fluctuation strength σbF, equation (10.7) and the measurement sensitivity σM, equation (10.8). This solution is simply the constant field solution (σbF = 0 andγb = 0) that uniformly saturates at the steady-state value of equation (10.24) at time t2.

Figure 10.2: The Riccati equation solution gives the ideal field estimation performance.

Robust Performance: F Unknown

Yet all the transfer functions generated from any value of λ will limit to the same performance at long times. In this case, we will not know whether to attribute the magnitude of the measured slope to the magnitude of F or to the magnitude of b.