Measurement Operator Formalism

Here we describe a simple measurement formalism adapted heavily from [57] and [63] in order to introduce basic concepts of quantum measurement. The measurement operator formalism discussed here is more suited to discrete time measurement events, rather than the experimental measurement scheme of this thesis, but the general concepts essentially transfer. An example using this formalism is presented in appendix B, where we analyze an atom-cavity system with direct photon detection at the output of both sides of the cavity.

Typically, an experimentalist does not directly measure the system of interest. The system, through a chain of interactions with larger and larger systems, imprints information about itself on an apparatus of a scale similar to our own. In some cases, this mapping may be trivial. In others, the connection may not be so obvious. To deduce this mapping, we must model the measurement apparatus to a sufficient level of accuracy. The more of the apparatus we describe quantum mechanically the better our predictions will be. Clearly, the choice of this “Heisenberg cut along the Von Neumann chain” must be pragmatically chosen. In practice, we essentially always make this cut at the optical photodiode interface where the probe beam is converted into a classical current.

Here we formally describe the measurement apparatus as the environment (or bath) and

consider our measurements as being projections of the environment state. These do not, in general, correspond to projections of the system state. Thus describing the environment quantum mechanically will fundamentally alter our predictions from a description where we chose measurement to correspond with direct projections on the system.

The basic measurement scheme is displayed in figure 2.1. The world is divided into several parts, including the system of interest (S) and the environment (E). In the instance of our experiment, we imagine the system to be the spin-state of a collection of atoms, and the environment to be the optical state of a laser beam by which we indirectly measure the spin-state. The “classical world” consists of the detector (with efficiency η) and the observer who estimates the spin-state and actuates the system via a controller. There also exist unobserved environment modes through which the observer can lose information (e.g., spontaneous emission). We have also considered the possibility of an unknown Hamiltonian (with parameterλ) acting on the system, under which case the observer performs quantum parameter estimation to update a classical distributionP_λ(t). Due to physical interactions, we graphically indicate in figure 2.1 the presence of correlations between the system and the outgoing environment modes. There may also be correlations within the system due to either direct interaction, or via the observer conditioning on the environment detection (e.g., spin-squeezing).

The total state (system plus environment) is represented as ˆρT, while the subcomponents are represented by tracing out the other so that the environment state is ˆρ_E = Tr_S( ˆρ_T) and the system state is ˆρ = ˆρS = TrE( ˆρT). In a single timestep, the joint state evolves according to the Hamiltonian ˆH_T = ˆH_S + ˆH_E + ˆH_SE where ˆH_S evolves the state, ˆH_E evolves the environment, and ˆH_SE lets the system interact with the environment, allowing a measurement of the environment to give information about the system. In the time dt, the joint state then evolves via the propagator ˆU(dt) = exp[−iHˆ_Tdt].

2.3.1 Indirect Measurement

Now we describe the process of inferring the state of the system after allowing the system to interact with the environment and measuring the environment. The logic of the reduction is as follows (following section 2.2 of [57]). We describe both the system and environment as pure states for simplicity. The environment is prepared in the state |ψ_Ei. The total state, which we assume to be initially separable, evolves into the generally entangled form

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

|ψ_T(t +dt)i = ˆU(dt)|ψ_Ei|ψ_S(t)i. Through measurement, the environment is randomly projected into a particular state α via the projector ˆP_α = 1_S ⊗ |ψ_E,αihψ_E,α|. So the resulting unnormalized total state is

|ψ_T,α(t+dt)i= ˆP_α|ψ_T(t+dt)i=|ψ_E,αi|ψ_S,α(t+dt)i (2.1) where

|ψ_S,α(t+dt)i = Ωˆ_α(dt)|ψ_S(t)i (2.2) Ωˆ_α(dt) = hψ_E,α|Uˆ(dt)|ψ_Ei. (2.3)

Notice that the measurement operator ˆΩ_α(dt), which acts on only the system subspace, is not necessarily a projector. Also note that this reduction is only useful if the environment has certain dynamical properties. If we are to use the same measurement operators for every timestep, the environment must quickly return to its prepared state (equilibrium approxima- tion) and never return the measured information to the system (Markovian approximation).

Of course, we may explicitly return information to the system via Hamiltonian feedback, but here we are considering the open-loop dynamics alone.

2.3.2 Dynamics of Continuous Measurement

Now we assume the existence of measurement operators ˆΩα(dt) based on an adequate model of the system-environment interaction and work out the dynamics. The continuous mea- surement operators are usually specified as follows, with the {Lˆi} being the so-calledjump operators

Ωˆ_i(dt) = √

dtLˆ_i (2.4)

Ωˆ₀(dt) = 1−iHˆ_effdt= 1−i( ˆH_S−iK)dtˆ (2.5)

withi= 1,2,3, ...and ˆK = ¹₂P

i>0Lˆ^†_iLˆi from the Kraus normalization condition.

2.3.2.1 Conditional Evolution

Let us define an effect ˆΥ_i to be associated with a measurement operator through ˆΥ_i = ˆΩ^†_iΩˆ_i. In the simulation of conditional evolution, we imagine flipping a coin weighted according to P_i = Tr( ˆΥ_iρ) to get a random resultˆ i on every timestep dt. In the experiment, we let nature flip the coin for us, and we measure for time period dt to get a result i. Note that one possible measurement result is thenull measurement ofi= 0. In any case, given result i, the evolution rule is to apply the appropriate jump operator by

ˆ

ρ(t+dt) = Ωˆ_i(dt) ˆρ(t) ˆΩ^†_i(dt) Pi

. (2.6)

The random state evolution described by this formalism is referred to as a quantum tra- jectory. Given a model of the environment and the interaction this conditional rule for updating the quantum state is called many things, but here we refer to it as the stochas- tic master equation (SME), the stochasticity coming from the inherent randomness of the detection in the environment. Note that if the detection of the environment is perfectly efficient, then the state changes but remains pure because we have not lost any information.

It is also important to point out that this equation is a experimental filter meant to be used in the lab: the inputs are the initial state (of both the system and environment), the model of the interaction,and the measurement record, and the output is the updated system state. The filtering procedure is shown schematically in figure 2.1. Here the dependence on the measurement record is somewhat implicit, but later it will be made more clear. This

point is worth making because some physicists use trajectories as merely a computational tool to evolve the unconditional master equation, discussed below, and do not associate the trajectories with any particular measurement. (The advantage is that with the pure state trajectories one works with the state vector, which is smaller than the state matrix used in the unconditional master equation evolution.)

Also note that the update rule above is very discrete in nature, as we imagine applying the rule every dt. This is fine if the number of detection events expected in the time dt is very small (see appendix B), however often we use another high power beam as a local oscillator in our measurement scheme. In this case, the number of detection events in anydt is much larger and we use a different description as discussed below, incorporating Wiener white noise increments dW to represent the randomness of the measurement [52]. This limit is mathematically distinct, but the underlying physical intuition remains the same as above.

2.3.2.2 Unconditional Evolution

In the case where the measurement record is completely ignored, but the system is still interacting with the environment, our best estimate of the system state must be given by the average of all possibilities

ˆ

ρ(t+dt) =X

i

Ωˆ_i(dt) ˆρ(t) ˆΩ^†_i(dt) (2.7)

or

dˆρ

dt = ˆLρˆ (2.8)

where ˆL is the Liouvillian (i.e., Lindbladian). This is the unconditional master equation.

Note that, by definition, the average of the conditional trajectories should reproduce this unconditional behavior.

2.3.3 Transforming the Measurement

Formally, one can unitarily rearrange the measurement operators without changing the unconditional evolution, thus creating what is called a different unravelling of the master

equation. Thus the transformation

Ωˆ⁰_i =X

j

Uˆi,jΩˆj (2.9)

results in no change of the unconditional evolution ˆ

ρ(t+dt) =X

j

Ωˆ_j(dt) ˆρ(t) ˆΩ^†_j(dt) =X

i

Ωˆ⁰_i(dt) ˆρ(t) ˆΩ^0†_i (dt) (2.10)

where P

rUˆ_r,sUˆ_r,q^∗ =δ_s,q. Whether or not a particular unravelling is physically realizable is another question. For the detection of a laser beam, different unravellings correspond to different means of optical detection. Direct detection results in conditional evolution with jump-like behavior. But by adding a local oscillator to the output field one can perform a homodyne measurement, which in the large oscillator limit results in diffusive motion of the conditional state. The switch from direct detection to homodyne (or heterodyne) corresponds to a unitary rearrangement of the direct detection measurement operators.

This is depicted by the unitary rotation element in figure 2.1. In the case of polarimetry as discussed in section 8.1 different unravellings can be achieved through different settings of the wave-plates prior to the polarimeter. In appendix B, a more explicit example is given.

The concept of unravelling is a very intuitive and important: in essence, what we know about a system depends on how we look at it. Depending on our measurement and control objective, the choice of unravelling will be critical and should be optimized [70].