Control of Self-Organizing Nonlinear Systems pp 289-313 | Cite as

# Controlling the Stability of Steady States in Continuous Variable Quantum Systems

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## Abstract

For the paradigmatic case of the damped quantum harmonic oscillator we present two measurement-based feedback schemes to control the stability of its fixed point. The first scheme feeds back a Pyragas-like time-delayed reference signal and the second uses a predetermined instead of time-delayed reference signal. We show that both schemes can reverse the effect of the damping by turning the stable fixed point into an unstable one. Finally, by taking the classical limit \(\hbar \rightarrow 0\) we explicitly distinguish between inherent quantum effects and effects, which would be also present in a classical noisy feedback loop. In particular, we point out that the correct description of a classical particle conditioned on a noisy measurement record is given by a non-linear *stochastic* Fokker-Planck equation and *not* a Langevin equation, which has observable consequences on average as soon as feedback is considered.

## Keywords

Harmonic Oscillator Master Equation Classical Limit Wigner Function Feedback Scheme## 15.1 Introduction

Continuous variable quantum systems are quantum systems whose algebra is described by two operators \(\hat{x}\) and \(\hat{p}\) (usually called position and momentum), which obey the commutation relation \([\hat{x},\hat{p}] = i\hbar \). Such systems constitute an important class of quantum systems. They do not only describe the quantum mechanical analogue of the motion of classical heavy particles in an external potential, but they also arise, e.g., in the quantization of the electromagnetic field. Understanding them is important, e.g., in quantum optics [1], for purposes of quantum information processing [2, 3], or in the growing field of optomechanics [4]. Furthermore, due to the pioneering work of Wigner and Weyl, such systems have a well-defined classical limit and can be used to understand the transition from the quantum to the classical world [5].

To each quantum system there is an operator associated, called the Hamiltonian \(\hat{H}\), which describes its energy and determines the dynamics of the system if it is isolated. However, in reality each system is an *open* system, i.e., it interacts with a large environment (we call it the bath) or other degrees of freedom (e.g., external fields). Since the bath is so large that we cannot describe it in detail, it induces effects like damping, dissipation or friction, which will eventually bring the system to a steady state. Classically as well as quantum mechanically it is often important to be able to counteract such irreversible behaviour, for instance, by applying a suitably designed feedback loop.

In this contribution we will apply two measurement based control schemes to a simple quantum system, the damped harmonic oscillator (HO), by correctly taking into account measurement and feedback noise at the quantum level (Sects. 15.3 and 15.4). These schemes will reverse the effect of dissipation and—to the best of our knowledge—have not been considered in this form elsewhere. However, we will see that our treatment is conceptually very close to a classical noisy feedback loop. With this contribution we thus also hope to provide a bridge between quantum and classical feedback control. For pedagogical reasons we will therefore first present the necessary technical ingredients (continuous quantum measurement theory, quantum feedback theory and the phase space formulation of quantum mechanics) in Sect. 15.2. Due to a limited amount of space we cannot derive them here, but we will try to make them as plausible as possible. Section 15.5 is then devoted to a thorough discussion of the classical limit of our results showing which effects are truly quantum and which can be also expected in a classical feedback loop. In the last section we will give an outlook about possible applications and extensions of our feedback loop.

## 15.2 Preliminary

### 15.2.1 The Damped Quantum Harmonic Oscillator

We will focus only on the damped HO in this paper, but we will discuss extensions and applications of our scheme to other systems in Sect. 15.6. Using a canonical transformation we can rescale position and momentum such that the Hamiltonian of the HO reads \(\hat{H} = \omega (\hat{p}^2 + \hat{x}^2)/2\) with \([\hat{x},\hat{p}] = i\hbar \). Introducing linear combinations of position and momentum, called the annihilation operator \(\hat{a} \equiv (\hat{x}+i\hat{p})/\sqrt{2\hbar }\) and its hermitian conjugate, the creation operator \(\hat{a}^\dagger \), we can express the Hamiltonian as \(\hat{H} = \hbar \omega (\hat{a}^\dagger \hat{a} + 1/2)\). Note that we explicitly keep Planck’s contant \(\hbar \) to take the classical limit (\(\hbar \rightarrow 0\)) later on.

*T*, it is possible to derive a so-called master equation (ME) for the time evolution of the density matrix [1, 6, 13]:

*free*time evolution of the HO

*without*any measurement or feedback performed on it [6]. Furthermore, it will turn out to be convenient to introduce a superoperator notation for the commutator and anti-commutator:

^{1}

### 15.2.2 Continuous Quantum Measurements

In introductory courses on quantum mechanics (QM) one only learns about projective measurements, which yield the maximum information but are also maximally invasive in the sense that they project the total state \(\hat{\rho }\) onto a single eigenstate. QM, however, also allows for much more general measurement procedures [6]. For our purposes, so-called continuous quantum measurements are most suited. They arise by considering very weak (i.e., less invasive) measurements, which are repeatedly performed on the system. In the limit where the time between two measurements goes to zero and the measurement becomes infinitely weak, we end up with a continuous quantum measurement scheme. For a quick introduction see Ref. [16]. Using their notation, one needs to replace \(k \mapsto \gamma /(4\hbar )\) to obtain our results.

*instantaneously vanish*.

The ME (15.6) is, however, only half of the story because it tells us only about the average time evolution of the system, i.e., about the whole ensemble \(\hat{\rho }\) averaged over all possible measurement records. The distinguishing feature of closed-loop control (as compared to open-loop control) is, however, that we want to influence the system based on a *single* (and not ensemble) measurement record. We denote the density matrix conditioned on a certain measurement record by \(\hat{\rho }_c\) and call it the conditional density matrix. Its classical counterpart would be simply a conditional probability distribution.

*I*(

*t*) associated to the continuous position measurement scheme above can be shown to obey the stochastic process [6, 16]

*dW*(

*t*) is the Wiener increment. According to the standard rules of stochastic calculus, it obeys the relations [6, 16]

*I*(

*t*).

^{2}We remark that the SME for \(\hat{\rho }_c(t)\) is nonlinear in \(\hat{\rho }_c(t)\), due to the fact that this in an equation of motion for a

*conditional*density matrix.

*dt*. Furthermore, we remark that a solution of a SME is called a

*quantum trajectory*in the literature [6, 13, 22].

### 15.2.3 Direct Quantum Feedback

*after*the measurement as it must due to causality. Now, expanding \(e^{\mathcal{{F}} dt}\) to first order in

*dt*with

*dI*(

*t*) from Eq. (15.8) (note that this requires to expand the exponential function up to

*second*order due to the contribution from \(dW(t)^2 = dt\)) and using the rules of stochastic calculus gives the effective SME under feedback control:

Before we give a short review about the last technically ingredient we need, which is rather unrelated to the previous content, we give a short summary. We have introduced the ME (15.1) for a HO of frequency \(\omega \), which is damped at a rate \(\kappa \) due to the interaction with a heat bath at inverse temperature \(\beta \). We then started to continuously monitor the system at a rate \(\gamma \) with a detector of efficiency \(\eta \). This procedure gave rise to a SME (15.13) conditioned on the measurement record (15.8). Finally, we applied feedback control by instantaneously changing the system Hamiltonian using the operator \(\hat{z}\), which resulted in the effective ME (15.17).

### 15.2.4 Quantum Mechanics in Phase Space

The phase space formulation of QM is an equivalent formulation of QM, in which one tries to treat position and momentum on an equal footing (in contrast, in the Schrödinger formulation one has to work either in the position or (“exclusive or”) momentum representation). By its design, phase space QM is very close to the classical phase space formulation of Hamiltonian mechanics and it is a versatile tool for a number of problems. For a more thorough introduction the reader is referred to Refs. [1, 5, 22, 26, 27, 28].

*F*(

*x*,

*p*) in phase space can be computed via

*F*(

*x*,

*p*) via the Wigner-Weyl transform [5, 22, 28]. Roughly speaking this transformation symmetrizes all operator valued expressions. For instance, if \(F(x,p) = xp\), then \(\hat{f}(\hat{x},\hat{p}) = (\hat{x}\hat{p} + \hat{p}\hat{x})/2\).

*x*,

*p*) as

*D*the diffusion matrix. It is then straightforward to confirm that the ME (15.1) corresponds to a FPE with

Finally, we point out that the transition from quantum to classical physics is mathematically accomplished by the limit \(\hbar \rightarrow 0\) [5]. Physically, of course, we do not have \(\hbar = 0\) but the classical action of the particles motion becomes large compared to \(\hbar \). We will discuss the classical limit of our equations in detail in Sect. 15.5.

## 15.3 Feedback Scheme I

^{3}In our case we have to use the nosiy signal (15.8), i.e., we perform feedback based on

*invasive*feedback scheme, because the feedback-generated force does not vanish even if our goal to reverse the effect of the damping was achieved. We emphasize that such feedback schemes are widely used in classical control theory to influence the behaviour of, e.g., chaotic systems or complex networks [34, 35] and quite recently, there has been also a considerable interest to explore its quantum implications [36, 37, 38, 39, 40, 41, 42, 43]. However, except of the feedback scheme in Ref. [41], the feedback schemes above were designed as

*all-optical*or

*coherent*control schemes, in which the system is not subjected to an explicit measurement, but the environment is suitable engineered such that it acts back on the system in a very specific way. We will compare our scheme (which is based on explicit measurements) with these schemes towards the end of this section.

*time-delayed noise*enters the equation of motion for \(\hat{\rho }_c(t)\). Because we do not know what \(\mathbb {E}[\hat{\rho }_c(t) dW_\tau (t)]\) is in general, there is a priori no ME for the average time evolution of \(\hat{\rho }(t)\). Approximating \(\mathbb {E}[\hat{\rho }_c(t) dW_\tau (t)] \approx 0\) yields nonsense (the resulting ME would not even be linear in \(\hat{\rho }\)). This is, however, not a quantum feature and is equally true for classical feedback control based on a noisy, time-delayed measurement record (also see Sect. 15.5).

Due to the fact that there is no average ME, we are in principle doomed to simulated the SME (15.26) and average afterwards. However, as it turns out Eq. (15.26) can be transformed into a stochastic FPE whose solution is expected to be a Gaussian probability distribution. We will then see that the covariances indeed evolve *deterministicly*. Furthermore, it is possible to analytically deduce the equation of motion for the mean values on average. Within the Gaussian approximation we then have full knowledge about the evolution of the system.

*on average*. Unfortunately the treatment of delay differential equations is very complicated and our goal is not to study these equations in detail now. However, the reasoning why we can turn a stable fixed point into an unstable one goes like this: for \(k=0\) we clearly have a stable fixed point but for \(k\gg \kappa \) we might neglect the term \(-\frac{\kappa }{2}{\langle {x}\rangle }\) for a moment. If we choose \(\tau = \pi /\omega \) (corresponding to half of a period of the undamped HO), we see that the “feedback force” \(k({\langle {x}\rangle } - {\langle {x}\rangle }_{\tau })\) is always positive if \({\langle {x}\rangle } > 0\) and negative if \({\langle {x}\rangle } < 0\) (we assume \(k>0\)). Hence, by looking at the differential equation it follows that the feedback term generates a drift “outwards”, i.e., away from the fixed point (0, 0), which at some point also cannot be compensated anymore by the friction of the momentum \(-\frac{\kappa }{2}{\langle {p}\rangle }\). From the numerics, see Fig. 15.2, we infer that the critical feedback strength, which turns the stable fixed point into an unstable one is \(k\ge \frac{\kappa }{2}\), also see Ref. [30] for a more detailed discussion of the domain of control.

^{4}

*dW*involve third order cumulants, which would in turn require to deduce equations for them as well. However, if we assume that the state of our system is Gaussian, these terms vanish due to the fact that third order cumulants of a Gaussian are zero. In fact, the assumption of a Gaussian state seems reasonable

^{5}: first of all, if the system is already Gaussian, it will also remain Gaussian for all times because then the Eqs. (15.31) and (15.32) as well as Eqs. (15.35)–(15.37) form a closed set. Second, even if we start with a non-Gaussian distribution, the state is expected to rapidly evolve to a Gaussian due to the continuous position measurement and the environmentally induced decoherence and dissipation [44]. Then, the time evolution of the conditional covariances becomes indeed deterministic, i.e., the covariances (but not the means) behave identically in each single realization of the experiment:

Because the time evolution of the conditional covariances is the same for the second feedback scheme, we will discuss them in more detail in Sect. 15.4. Here, we just want to emphasize that we cannot simply average the conditional covariances to obtain the unconditional ones, i.e., \(\mathbb {E}[V_{x,c}] \ne V_x \equiv \int dx dp x^2 W(x,p) - [\int dx dp x W(x,p)]^2\) in general. In fact, the conditional and unconditional covariances can behave very differently, see Sect. 15.4.

Finally, let us say a few words about our feedback scheme in comparison with the coherent control schemes in Refs. [36, 37, 38, 39, 40, 42, 43], which are designed for quantum optical systems and use an external mirror to induce an intrinsic time-delay in the system dynamics. Clearly, the advantage of the coherent control schemes is that they do not introduce additional noise because they avoid any explicit measurement. On the other hand, in our feedback loop we have the freedom to choose the feedback strength *k* at our will, which allows us to truely reverse the effect of dissipation. In fact, due to simple arguments of energy conservation, the coherent control schemes can only fully reverse the effect of dissipation if the external mirrors are *perfect*. Otherwise the overall system and controller is still loosing energy at a finite rate such that the system ends up in the same steady state as without feedback. Thus, as long as the coherent control loop does not have access to any external sources of energy, it is only able to counteract dissipation on a transient time-scale except one allows for perfect mirrors, which in turn would make it unnessary to introduce any feedback loop at all in our situation. It should be noted, however, that for transient time-scales coherent feedback might have strong advantages or it might be the case that one is not primarily interested in the prevention of dissipation (in fact, in Ref. [40] they use the control loop to *speed up* dissipation). The question whether one scheme is superior to the other is thus, in general, undecidable and needs a thorough case to case analysis.

## 15.4 Feedback Scheme II

*synchronize*the motion of the HO with the external reference signal \(x^*(t)\). Choosing \(\hat{z} = k\hat{p}\) and using Eq. (15.15) yields the SME

*dW*(

*t*) due to Eq. (15.12). We thus have a fully Markovian feedback scheme here.

^{6}However, it is important to emphasize that we do not have an open-loop control scheme here although it looks like it at the average level of the means. The asymptotic solution of Eqs. (15.49) and (15.50) is given by

*with*access to the measurement result would associate to the state of the system) evolve as for the first scheme according to

*without*access to the measurement results would associate to the state of the system) obey

*x*-direction. This is the effect of the continuous measurement performed on the system, which tends to localize the state. However, if we average over (or, equivalently, ignore) the measurement results, this effect is missing. Furthermore, note that Eqs. (15.53)–(15.55) do not contain the parameter

*k*, which quantifies how strongly we feed back the signal.

^{7}Especially, at zero temperature (\(n_B = 0\)), we have a minimum uncertainty wave packet satisfying the lower bound of the Heisenberg uncertainty relation, \(V_xV_p = \hbar ^2/4\). Now, for \(\kappa = 0\), we can expand the conditional covariances in powers of \(\gamma \):

*no*conditional dynamics. This becomes also clear by looking at Eq. (15.56), in which the last term diverges in this limit because we would feed back an infinitely noisy signal.

Thus, an observer with access to the measurement record would associate very different covariances to the system in comparison to an observer without that knowledge. However, a detailed discussion of the time evolution of the covariances is beyond the scope of the present paper. Instead, we find it more interesting to discuss the relationship between the present quantum feedback scheme and its classical counterpart, to which we turn now.

## 15.5 Classical Limit

*x*. We explicitly wish, however, to model a

*noisy*classical measurement. We thus additionally demand that \(\eta \rightarrow 0\). More specifically, we set \(\eta \equiv \hbar /\sigma \) with \(\sigma \) finite such that Eq. (15.8) becomes

*W*(

*x*,

*p*) becomes an ordinary probability distribution in the classical limit, we denoted it by

*P*(

*x*,

*p*). As expected, we see that Eq. (15.63) corresponds to a FPE for a Brownian particle in a harmonic potential where position and momentum are both damped (usually one considers only the momentum to be damped [29]). This peculiarity is a consequence of an approximation made in deriving the ME (15.1), which is known as the secular or rotating-wave approximation. Nevertheless, one easily confirms that the canonical equilibrium state \(P_\text {eq} \sim \exp [-\beta \omega (p^2+x^2)/2]\) is a steady state of this FPE as it must be.

*stochastic*FPE, which is nonlinear in \(P_c\). It describes how our state of knowledge changes if we take into account the measurement record (15.62). However, averaging Eq. (15.64) over all measurement results yields Eq. (15.63), which reflects the fact that a classical measurement does not perturb the system.

^{8}This is in contrast to the quantum case where the average evolution is still influenced by \(\mathcal{{L}}_\text {meas}\), see Eq. (15.6). Hence, the term \(\mathcal{{L}}_\text {meas}\) in Eq. (15.6) is

*purely*of quantum orgin and it describes the effect of decoherence on a quantum state under the influence of a measurement. This effect is absent in a classical world. Exactly the same equation and the same conclusions were already derived by Milburn following a different route [45].

The impact of these conclusions is, however, much more severe if one additionally considers feedback. As we will now show, applying feedback based on the use of the stochastic FPE (15.64), *does* indeed yield observable consequences even *on average*. Please note that trying to model the present situation by a classical Langevin equation is nonsensical. If we would use a Langevin equation to describe our state of knowlegde about the system, we would implicitly ascribe an objective reality to the fact that there *is* a definite position \(x_0\) and momentum \(p_0\) of the particle corresponding to a probability distribution \(\delta (x-x_0)\delta (p-p_0)\). This is, however, *not true* from the point of view of the observer who has to apply feedback control based on incomplete information (i.e., the noisy measurement record). Results we would obtain from a Langevin equation treatment can only be recovered in the limit of an error-free measurement, i.e., for \(\sigma \rightarrow 0\), as we will demonstrate in Appendix 15.6.

*x*direction simply due to the fact that the observer applies a slightly wrong feedback control compared to the “perfect” situation without measurement errors.

We thus conclude this section by noting that the treatment of continuous noisy classical measurements faces similar challenges as in the quantum setting. On average the measurement itself does not influence the classical dynamics, but we see that we obtain new terms even on average if we use this measurement to perform feedback control. Most importantly, because a feedback loop has to be implemented by the observer who has access to the measurement record, it is in general not possible to model this situation with a Langevin equation. Furthermore, we remark that the situation is expected to be even more complicated for time-delayed feedback, where no average description is a priori possible.

## 15.6 Summary and Outlook

Because we discussed the meaning of our results already during the main text in detail, we will only give a short summary together with a discussion on possible extensions and applications.

We have used two simple feedback schemes, which are known to change the stability of a steady state obtained from linearizing a dynamical system around that fixed point in the classical case. For the simple situation of a damped quantum HO we have seen that *on average* we obtain the same dynamics for the mean values as expected from a classical treatment and thus, classical control strategies might turn out to be very useful in the quantum realm, too.

However, the fact that a classical control scheme works so well in the quantum regime depends on two crucial assumptions. First of all, we have used a linear system (the HO). Having a non-linear system Hamiltonian (e.g., a Hamiltonian with a quartic potential \(\sim \hat{x}^4\)) would complicate the treatment because already the equations for the mean values would contain higher order moments, as e.g. \({\langle {x^3}\rangle }\) in case of the quartic oscillator. Simply factorizing them as \({\langle {x^3}\rangle } \approx {\langle {x}\rangle }^3\) would imply that we are already using a classical approximation. However, as in the classical treatment, where the equations of motion are obtained from linearizing a (potentially non-linear) dynamical system around the fixed point, it might also be possible in the quantum regime to neglect non-linear terms in the vicinity of the fixed point. Whether or not this is possible crucially depends on the localization of the state in phase space, i.e., on its covariances. Here, continuous quantum measurements can actually turn out to be helpful because they tend to localize the wavefunction and counteract a possible spreading of the state.

The second important assumption we used was that we restricted ourselves to continuous variable quantum systems. The reason why we obtained simple equations of motion is related to the commutation relation \([\hat{x},\hat{p}] = i\hbar \), which we implicitly used to obtain the evolution equation for the Wigner function. Formally, phase space methods are also possible for other quantum systems, but the maps are much more complicated [27]. For such systems the methods presented here might be useful under certain special assumptions, but in general one should expect them to fail.

## Footnotes

- 1.
The situation of an unstable fixed point would be modeled by exchanging the operators \(\hat{a}\) and \(\hat{a}^\dagger \) in the dissipators. This would correspond to a negative \(\kappa \) in the equation for the mean position and momentum. The feedback schemes presented here also work in that case.

- 2.
We explicitly adopt a Bayesian probability theory point of view in which probabilities (or more generally the density matrix \(\hat{\rho }\)) describe only (missing) human information. Especially, different observers (with possibly different access to measurement records) would associate

*different*states \(\hat{\rho }\) to the*same*system. - 3.
In fact, in Ref. [30] they did not only feed back the results from a position measurement, but also from a momentum measurement. The simultaneous weak measurement of position and momentum can be also incorporated into our framework [17, 31, 32], but this would merely add additional terms without changing the overall message.

- 4.
Pay attention to the fact that we are using an Itô stochastic differential equation where the ordinary chain rule of differentiation does not apply. Instead, we have for instance for the stochastic change of the position variance \(dV_{x,c} = d\langle x^2\rangle _c - 2{\langle {x}\rangle }_c d{\langle {x}\rangle }_c - (d{\langle {x}\rangle }_c)^2\).

- 5.
- 6.
Indeed, if we would choose the feedback operator \(\hat{z} = k\hat{x}\), the resulting differential equations for \(\langle x\rangle \) and \(\langle p\rangle \) would exactly resemble the differential equation of a classical harmonic oscillator with sinusoidal driving force.

- 7.
- 8.
This is true at least in our context. In principle, it is of course possible to construct classical measurements, which perturb the system, too [6].

- 9.
The complete description of an underdamped particle (i.e., a particle descibed by its position

*x**and*momentum*p*), which is based on a continuous measurement of its position*x*alone, Eq. (15.62), faces the additional challenge that we have to first estimate the momentum*p*based on the noisy measurement results.

## Notes

### Acknowledgments

PS wishes to thank Philipp Hövel, Lina Jaurigue and Wassilij Kopylov for helpful discussions about time-delayed feedback control. Financial support by the DFG (SCHA 1646/3-1, SFB 910, and GRK 1558) is gratefully acknowledged.

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