Approximate stabilization of an infinite dimensional quantum stochastic system
Abstract
We propose a feedback scheme for preparation of photon number states in a microwave cavity. Quantum NonDemolition (QND) measurements of the cavity field and a control signal consisting of a microwave pulse injected into the cavity are used to drive the system towards a desired target photon number state. Unlike previous work, we do not use the Galerkin approximation of truncating the infinitedimensional system Hilbert space into a finitedimensional subspace. We use an (unbounded) strict Lyapunov function and prove that a feedback scheme that minimizes the expectation value of the Lyapunov function at each time step stabilizes the system at the desired photon number state with (a prespecified) arbitrarily high probability. Simulations of this scheme demonstrate that we improve the performance of the controller by reducing “leakage” to high photon numbers.
I Introduction
Quantum NonDemolition (QND) measurements have been used to detect and/or produce highly nonclassical states of light in trapped superconducting cavities [1, 2, 3] (see [4, Ch. 5] for a description of such quantum electrodynamical systems and [5] for detailed physical models with QND measures of light using atoms). In this paper we examine the feedback stabilization of such experimental setups near a prespecified target photon number state. Such photon number states, with a precisely defined number of photons, are highly nonclassical and have potential applications in quantum information and computation.
The state of the cavity may be described on a Fock space , which is a particular type of Hilbert space that is used to describe the dynamics of a quantum harmonic oscillator (see e.g. [4, Sec 3.1]). The cannonical orthonormal basis for this Hilbert space consists of the set of Fock states . Physically, the state corresponds to a cavity state with precisely photons. In this paper we study the possibility of driving the state of the system to some prespecified target state . The feedback scheme uses the so called measurement back action and a control signal, which is a coherent light pulse injected into the cavity, to stabilize the system at the target state with high probability.
Such feedback schemes for this experimental setup were examined previously in [6, 7]. The overall control structure used in [6] is a quantum adaptation of the observer/controller structure widely used for classical systems (see, e.g. [8, Ch. 4]). The observer part consists of a discretetime quantum filter, based on the observed detector clicks, to estimate the quantumstate of the cavity field. This estimated state is then used in a statefeedback based on Lyapunov design, the controller part.
As the Hilbert space is infinite dimensional it is difficult to design feedback controllers to drive the system towards a target state (because closed and bounded subsets of are not compact). In [6], the controller was designed by approximating the underlying Hilbert space with a finitedimensional Galerkin approximation . Here, is the linear subspace of spanned by the basis vectors and , our target sate. Physically this assumption leads to an artificial bound on the maximum number of photons that may be inside the cavity. In this paper we wish to design a controller for the full Hilbert space without using the finite dimensional approximation. The need to consider the full Hilbert space is motivated by simulations (see Section IV) which indicate that using the controller designed on a finite dimensional approximation results in “leakage” to higher photon numbers with some finite probability.
Controlling infinite dimensional quantum systems have previously been examined in the deterministic setting without measurements. Various approaches have been used to overcome the noncompactness of closed and bounded sets. One approach consists of proving approximate convergence results which show convergence to a neighborhood of the target state [9, 10]. Alternatively, one examines weak convergence for example, in [11]. Other approaches such as using strict Lyapunov functions or strong convergence under restrictions on possible trajectories to compact sets have also been used in the context of infinite dimensional statespace for example in [12, 13].
The situation in our paper is different in the sense that the system under consideration is inherently stochastic due to quantum measurements. The system may be described using a discrete time Markov process on the set of unit vectors in the system Hilbert space as explained in Section II. We use a strict Lyapunov function that restricts the system trajectories with high probability to compact sets as explained in Section III. We use the properties of weakconvergence of measures to show approximate convergence (i.e. with probability of convergence approaching one) of the discrete time Markov process towards the target state.
We use a similar overall feedback scheme that is used in [6]. The entire feedback system is split into an observer part, a quantum filter, and a controller part based on a Lyapunov function. The quantum filter used to estimate the state is identical to the one used in [6] and we do not discuss the filter further in this paper. However we do not use the Galerkin approximation to design the controller. We show in Theorem III.2 that given any , we can drive our system to the target state with probability greater than . Simulations (see Section IV) indicate that this controller provides improved performance with lower probability of having trajectory escaping towards infinite photon numbers. The precise choice of Lyapunov function is motivated by [14] that uses a similar form of the Lyapunov function in a finite dimensional setting.
Ia Outline
The remainder of the paper is organised as follows: in the following Section we describe the experimental setup and the Markovian jump dynamics of the system state. In Section III we state the main result of our paper including an outline of the proof of Theorem III.2. We then present our simulation results in Section IV and then our conclusions in the final Section.
Ii System description
The system, illustrated in Figure 1, consists of 1) a high microwave cavity , 2) an atom source that produces Rydberg atoms, 3) two lowQ Ramsey cavities and , 4) an atom detector and 5) a microwave source . The system may be modeled by a discretetime Markov process, which takes into account the backaction of the measurement process (see e.g. [4, Ch. 4] and [6]).
Rydberg Atoms are sent from , interact with the cavity , entangling the state of the atom with that of the cavity and are then detected in . Each timestep, indexed by the integer , corresponds to atom number crossing the cavity and interacting with the cavity. The state of the cavity in time step is described by a unit vector for . Here, is the set of possible cavity states. The change of the cavity state at timestep to the state at timestep consists of two parts corresponding to the projective measurement of the cavity state, by detecting the state of the Rydberg atom in detector and also due to an appropriate coherent pulse (the control) injected into C.
Let and be the photon annihilation and creation operators where and is the Hermition conjugate of . Also, let be the diagonal number operator satisfying . Let be the displacement operator which is a unitary operator that corresponds to the input of a coherent control field of amplitude that is injected into the cavity. The amplitude of the coherent field is the control that is used to manipulate the system. Let and be the measurement operators, where and are experimental parameters. Physically, the measurement operator , correspond to the state of the detected atom in either the ground state or the excited state .
We model these dynamics by a Markov process
(1)  
(2) 
Here and the control .
Remark II.1
The time evolution from the step to , consists of two types of evolutions: a projective measurement by the operators and a coherent injection involving operator . For the sake of simplicity, we will use the notation of to illustrate this intermediate step.
Iii Global (approximate) feedback stabilization
We wish to use the control to drive the system into a prespecified target state with high probability. That is, we wish to show that the sequence converges to the set of probability measures where for all , is big.
In order to achieve this we use a Lyapunov function (5) and at each time step we choose the feedback control to minimize the Lyapunov function. Before discussing the choice of the Lyapunov function in Subsection IIIB we recall some facts concerning the convergence of probability measures
Iiia Convergence of probability measures
We refer the interested reader to [15, 16] for results pertaining to convergence of probability measures. We denote by the set of all continuous bounded functions on .
Definition III.1
We say that a sequence of probability measure converges (weak) to a probability measure if for all
and we write
It can be shown that if then for all open sets ,
(3) 
A set of probability measures is said to be tight [16, p. 9] if for all there exists a compact set such that for all , .
Theorem III.1 (Prohorov’s theorem)
Any tight sequence of probability measures has a (weak) converging subsequence.
We also recall Doob’s inequality. Let be a Markov process on some state space X. Suppose that there is a nonnegative function satisfying , then Doob’s inequality states
(4) 
IiiB Lyapunov function and control signal
We now introduce our Lyapunov function and explain the intuition behind this peculiar form of this function. The function, is defined
(5)  
Here
is a small positive number and
(6) 
We set to be the set of all where the above Lyapunov function is finite. We note that coherent states, which are states that are of relevance in practical experiments are in .
We choose a feedback that minimizes the expectation value of the Lyapunov function in every timestep . Indeed, applying the result of the ’th measurement, we know the state and we choose as follows
(7) 
for some positive constant .
Remark III.1
The Lyapunov function is chosen to be this specific form to serve three purposes 

We choose the sequence as . This guarantees that if we choose to minimize the expectation value of the Lyapunov function then the trajectories of the Markov process are restricted to a compact set in with probability arbitrarily close to 1. This implies that the limit set of the process is nonempty (see Step 2 in the Proof of Theorem III.2).

The term is chosen such that the Lyapunov function is a strict Lyapunov functions for the Fock states. This implies that the support of the limit set only contains Fock states (see Step 3 in the Proof of Theorem III.2).

The relative magnitudes of the coefficients have been chosen such that is a strict global minimum of . Moreover given any we can choose such that for all , and for all in a neighborhood of , does not have a local minimum at . This implies that if is in this neighborhood of then we can choose an to decrease the Lyapunov function and move away from by some finite distance with probability 1 (see Steps 4 and 5 in the Proof of Theorem III.2).
IiiC Main Result
We make the following assumption.

The eigenvalues of and are nondegenerate. This is equivalent to the assumption that is not a rational number.
The quantum filter uses the statistics of the measurement of whether the atom is in the ground or excited state to estimate the cavity’s state. Therefore if one of the eigenvalues of (or ) is degenerate then the measurement statistics will be the same for more than one photon number state. Therefore it is not possible to control the system effectively in this case (However, as explained in Remark III.2 below, we may weaken this assumption slightly).
The following Theorem is our main result.
Theorem III.2
If we assume to be true then given any and , there exist constants and such that for all satisfying , converges to a limit set . Moreover for all , only if is one of the Fock states and
The proof is split into 5 steps:

is a supermartingale that is bounded from below.

The sequence of measures is tight and therefore has a converging subsequence. Hence the set is nonempty.

If then the support set of only consists of Fock states.

Let be given. Then for all , and may be chosen small enough such that for small enough and all in the neighborhood
(8) of , satisfying , we have for the polynomial approximation
and for some constant . The term only depends on and not on and the term is independent of both and .

Because is negative, we can choose and small enough such that the probability of convergence to the Fock states for may be made arbitrarily small. Therefore
may be made arbitrarily big.
Below we sketch the proofs of each of the above steps. The interested reader is referred to [17] for further details on the proof which are beyond the scope of a short note. {proof}[Proof of step 1] We can write
where,
(9)  
It is obvious that and after simple but tedious manipulations, we get
(10) 
Therefore, is a supermartingale. {proof}[Proof of step 2] Let be given. Because is a supermartingale, Doob’s inequality (11) gives us
(11) 
If we set,
then for all , . Because, the sequence as , the set can be shown to be precompact in . We can now apply Prohorov’s Theorem III.1 to show that has a converging subsequence. Therefore the limit set is nonempty. {proof}[Proof of step 3] Suppose some subsequence of converges to . From step 1 we have as and because and are both nonnegative we have
But, from (10) and the boundedness of and , we know that is a continuous function on . Therefore from Definition III.1 of (weak) convergence of measures we get
(12) 
But implies . The CauchySchwartz inequality gives
with equality if and only if and are colinear. Therefore implies (by Assumption) that is a Fock state. Hence from (12) we can conclude that the support set of only consists of the set of Fock states. {proof}[Proof of step 4]
Set
It can be shown [17] that is an analytic function of if satisfies . Moreover, for all satisfying we have the second order polynomial approximation
for all . In particular the term only depends on and is independent of . Here is the derivative of w.r.t. evaluated at .
If we let and recall that then after some manipulations, we get
If and we have
and for we get
For any Fock state with , , where is the Kroneckerdelta function and we have
Because the terms and are bounded by the norm in , it can be shown that for small enough we have in the neighborhood of , where is given as in Equation (8).
But,
Hence, given any , step 4 above is true with . {proof}[Proof of step 5] Let be given. We show that . From step 3 we know that the support of only consists of Fock states. Therefore using (3), we only need to show that there exists an open neighborhood of such that for big enough the .
We construct the set using two disjoint parts and . We first show that there exists a big enough and a neighborhood of such that for all . We then construct a neighborhood of such that for large enough.
Construction of
Because there exists an large enough such that for all , . We can choose a small enough neighborhood of such that for all in this neighborhood,
Because , Doob’s inequality implies the probability of is less than . Therefore,
(13) 
Construction of
From Doob’s inequality, we have
(14) 
for all . Therefore we can complete the proof if we show that for small enough
where
In Step 4 we set and and let be small enough so that is as given in step 4. Then, because , we can choose and small enough so that there exists a constant such that for all , , for some . Because and are bounded operators and and , can be chosen small enough such that if then with probability .
We claim that . To see this note that because for all , with probability 1 and this implies
So if then the Markov process is outside the set within a finite number of steps less than with probability . So if does not approach zero, then the Markov process must enter the set from outside the set infinitely many times. But by Doob’s inequality (11) the probability of this happening once is less than . Therefore the probability of this happening infinitely many times is zero. Thus . This combined with (13) and (14) gives, .
Therefore,
Remark III.2
In step 2 we show that the only vectors in the support of are those corresponding to eigenvector of . We then used assumption to claim that the only eigenvectors of are the Fock states. We can however weaken this assumption to the following: eigenvalues corresponding to eigenvectors , are nondegenerate. This is because, we can show that if some eigenvector is in the span of the set then using the same argument as that used for , we can show that the probability of is small. This is significant for cases where is a more complicated nonlinear function of , as is the case in a practical system.
Iv Simulations
To illustrate Theorem III.2, we performed closedloop simulations of the controller designed using the finitedimensional approximation [6] and the one in Theorem III.2. Both simulations were performed on a system truncated to 21 photons. However the quantum filter (and therefore the controller) was truncated to 10 photons.
The initial state was chosen to be the coherent state having an average of photons:
The measurement operators are , . We take and to ensure the Lyapunov function is strictly concave near the Fock states , . To compute the feedback law given by the minimisation (7), we approximate, for each step , by the polynomial of degree two with the same first and second order derivatives at . Figure 3 shows good convergence properties of such feedback strategy with an average asymptotic value of close to . The remaining trajectories that do not converge to can be interpreted as the in theorem III.2.
Figure 3 is devoted to similar simulations but with the feedback law of [6, 7] based on a finite dimensional model:
(15) 
The average asymptotic value of is then around with this ”finite dimensional” feedback. Around of the trajectories do not converge towards and escape towards high photon numbers. Figure 4 shows a typical example of such a trajectory which converges towards photon number 15 and 20.
V Conclusion
In this paper we examine the stabilization of a quantum optical cavity at a prespecified photon number state . In contrast with previous work, we designed a Lyapunov function on the entire infinite dimensional Hilbert space instead of using a truncation approximation. The Lyapunov function was chosen so that it is a strict Lyapunov function for the target state and the feedback consisted of a control that minimizes the expectation value of the Lyapunov function at each timestep. Simulations indicate that this feedback controller performs better than the one designed using the finite dimensional approximation.
Vi Acknowledgments
The authors thank M. Brune, I. Dotsenko, S. Haroche and J.M. Raimond for enlightening discussions and advices.
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