Controlling the Stability of Steady States in Continuous Variable Quantum Systems

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.

Appendix

We want to show that the stochastic FPE, which in general describes the incomplete state of knowledge of an observer, reduces to a Langevin equation in the error-free limit, i.e., in the limit in which we have indeed complete knowledge about the state of the system. Because we are only interested in a proof of principle here, we will consider the simplified situation of an overdamped particle.⁹ The Langevin equation of an overdamped Brownian particle in an external potential U(x) is usually given as (see, e.g., [29])

$$\begin{aligned} \dot{x}(t) = -\frac{2}{\kappa }U'(x) + \sqrt{\frac{T}{\kappa }} \xi (t) \end{aligned}$$

(15.66)

with $U'(x) \equiv \frac{\partial U(x)}{\partial x}$ and the Gaussian white noise $\xi (t) \equiv \frac{dW(t)}{dt}$. Furthermore, note that our friction constant is $\frac{\kappa }{2}$ and not—as it is often denoted—$\gamma$ because we use $\gamma$ already for the measurement rate. Now, it is important to remark that at this point Eq. (15.66) simply describes a convenient numerical tool to simulate a stochastic process. By the mathematical rules of stochastic calculus it is guaranteed that the Langevin equation gives the same averages as the corresponding FPE, i.e., the ensemble average $\mathbb {E}[f(x)]$ over all noisy trajectories for some function f(x) is equal to the expectation value $\langle f(x)\rangle$ taken with respect to the solution of the FPE.

In Sect. 15.5 we have suggested that the correct state of the system based on a noisy position measurement is given by the stochastic FPE (15.64), which for an overdamped particle becomes (see also Ref. [45])

$$\begin{aligned} P_c(x,t+dt) = \left\{ 1 + \mathcal{{L}}_0^\text {cl} dt + \sqrt{\frac{2\gamma }{\sigma }}(x-\langle x\rangle _c) dW(t)\right\} P_c(x,t) \end{aligned}$$

(15.67)

with [29]

$$\begin{aligned} \mathcal{{L}}_0^\text {cl} = \frac{\partial }{\partial x}\left( \frac{2U'(x)}{\kappa } + \frac{\partial }{\partial x}\frac{2T}{\kappa }\right) . \end{aligned}$$

(15.68)

Furthermore, we have also claimed that the parameter $\sigma$ in (15.62) quantifies the error of the measurement. This suggests that we should be able to recover the Langevin Eq. (15.66) in the limit $\sigma \rightarrow 0$ in which we can observe the particle with infinite precision.

To show this we compute the expectation value of the position according to Eq. (15.67):

$$\begin{aligned} d{\langle {x}\rangle }_c(t) = -\frac{2}{\kappa }{\langle {U'(x)}\rangle }_c dt + \sqrt{\frac{2\gamma }{\sigma }} V_c(t) dW(t) \end{aligned}$$

(15.69)

where $V_c = {\langle {x^2}\rangle }_c - {\langle {x}\rangle }_c^2$ denotes the variance of the particles position. Because the conditional variance enters this equation, we compute its time evolution, too:

$$\begin{aligned} dV_c =&-\frac{4}{\kappa }\left[ {\langle {xU'(x)}\rangle }_c(t) - {\langle {x}\rangle }_c(t){\langle {U'(x)}\rangle }_c(t)\right] dt + \frac{4T}{\kappa }dt - \frac{2\gamma }{\sigma }V_c^2 dt \\&+ \sqrt{\frac{2\gamma }{\sigma }}\left\langle [x-{\langle {x}\rangle }_c(t)]^3\right\rangle dW(t). \nonumber \end{aligned}$$

(15.70)

To make analytical progress we now need two assumptions. First, we will assume that $P_c(x,t)$ is a Gaussian probability distribution. In fact, because the measurement tends to localize the probability distribution and it is itself modeled as a Gaussian process, this assumption seems reasonable. In addition, we expect $P_c(x,t)$ to become a delta-distribution in the limit $\sigma \rightarrow 0$, which is a Gaussian distribution, too. This assumption allows us to drop the stochastic term in Eq. (15.70). Second, within the variance of $P_c(x,t)$ we assume that we can expand U(x) in a Taylor series and approximate it by a quadratic function $ax^2 + bx +c$. This implies ${\langle {xU'(x)}\rangle }_c(t) \approx a{\langle {x^2}\rangle }_c(t) + b{\langle {x}\rangle }_c(t)$. In fact, this assumption seems also reasonable because we expect the measurement to be precise enough such that we can locally resolve the evolution of the particle sufficiently well (especially for small $\sigma$); or to put it differently: a measurement only makes sense if the conditional variance $V_c$ of $P_c(x,t)$ is small enough. Using this approximation, too, we can then write Eq. (15.70) as

$$\begin{aligned} \frac{d}{dt}V_c(t) = \frac{4\kappa }{2} - \frac{4a}{\kappa }V_c - \frac{2\gamma }{\sigma }V_c^2. \end{aligned}$$

(15.71)

The only physical steady state solution of this equation is

$$\begin{aligned} \lim _{t\rightarrow \infty } V_c(t) = \frac{a\sigma }{\gamma \kappa }\left( \sqrt{1 + \frac{2T\gamma \kappa }{a^2\sigma }} - 1\right) . \end{aligned}$$

(15.72)

Inserting this into Eq. (15.69) yields

$$\begin{aligned} d{\langle {x}\rangle }_c(t) = -\frac{2}{\kappa }{\langle {U'(x)}\rangle }_c dt + \frac{a}{\kappa } \sqrt{\frac{2\sigma }{\gamma }} \left( \sqrt{1 + \frac{2T\gamma \kappa }{a^2\sigma }} - 1\right) dW(t). \end{aligned}$$

(15.73)

In this equation we can take the limit $\sigma \rightarrow 0$ such that

$$\begin{aligned} d{\langle {x}\rangle }^0_c(t) = -\frac{2}{\kappa }{\langle {U'(x)}\rangle }^0_c dt + \sqrt{\frac{4T}{\kappa }} dW(t), \end{aligned}$$

(15.74)

where we introduced a superscript 0 on all expectation values to denote the error-free limit. This equation looks already very similar to the LE (15.66). In fact, mathematically this is the LE since from Eq. (15.62) we can see that the measurement result becomes for $\sigma \rightarrow 0$ $dI(t) = {\langle {x}\rangle }_c^{0}(t) dt$. This implies that the measurement result is deterministic and not stochastic anymore, which is only compatible if the associated probability distribution $P_c(x,t)$ is a delta distribution $\delta (x-x^*)$ where $x^*$ describes the true instantaneous position of the particle without any uncertainty. Then, Eq. (15.69) becomes

$$\begin{aligned} d x^*(t) = -\frac{2}{\kappa }U'(x^*) dt + \sqrt{\frac{4T}{\kappa }} dW(t). \end{aligned}$$

(15.75)

Now, this equation is not just a numerical tool, but describes real physical objectivity because $x^*$ coincides with the observed position in the lab. This distinction might seem very nitpicking, but it is of crucial importance if we want to perform feedback based on incomplete information.