This paper explores the control of non-Markovian systems via coherent quantum feedback. In the spirit of classical control theory that is widely used in engineering, we acquire completely new insights in the closed-loop design from the frequency domain point of view, which appears missing in quantum control theory. Based on the non-Markovian Langevin equation for the closed-loop quantum dynamics, it is found that, in contrast to the existing time-domain design methods, the frequency domain analysis is more natural on the memory kernel function, which can be reshaped by feedback to suppress the non-Markovian decoherence. For illustration, we consider the case that the coupling strength in the feedback loop is constant. The analysis shows that the coherent feedback shifts the two components in the original noise spectral function towards two opposite directions, thereby makes it possible to suppress the noise near the system's working frequency. When the system to be controlled is Markovian, this simple scheme needs to be replaced by more careful design due to the flatness of the noise spectrum. As an example, the effectiveness of our scheme is demonstrated in photonic crystal systems.
This paper explores the control of non-Markovian systems via coherent quantum feedback. In the spirit of classical control theory that is widely used in engineering, we acquire completely new insights in the closed-loop design from the frequency domain point of view, which appears missing in quantum control theory. Based on the non-Markovian Langevin equation for the closed-loop quantum dynamics, it is found that, in contrast to the existing time-domain design methods, the frequency domain analysis is more natural on the memory kernel function, which can be reshaped by feedback to suppress the non-Markovian decoherence. For illustration, we consider the case that the coupling strength in the feedback loop is constant. The analysis shows that the coherent feedback shifts the two components in the original noise spectral function towards two opposite directions, thereby makes it possible to suppress the noise near the system's working frequency. When the system to be controlled is Markovian, this simple scheme needs to be replaced by more careful design due to the flatness of the noise spectrum. As an example, the effectiveness of our scheme is demonstrated in photonic crystal systems.
In this paper, we present a coherent feedback control scheme for non-Markovian bosonic systems, in which an engineered quantum control field is introduced to couple both the system and the noise bath. The closed-loop dynamics of the system is described by an exact non-Markovian quantum Langevin equation, where the spectral density functions of the noise and the quantum control field, as well as their coupling, are combined into a single memory kernel function. We show that the coupling between the quantum control field with the noise bath can be used as a feedback control to modulate the memory kernel function. As a result, the noise bath can be driven out of resonance with the system and the decoherence can be efficiently suppressed. The effectiveness of the controllability is demonstrated with a photonic circuit in photonic crystals.
Abstract In this paper we present a non-Markovian coherent feedback scheme for decoherence suppression in single quantum dot systems. The feedback loop is closed via a quantum tunnelling junction between the natural source and drain baths of the quantum dot. The exact feedback-controlled non-Markovian Langevin equation is derived for describing the dynamics of the quantum dot. To deal with the nonlinear memory function in the Langevin equation, we analyse the Green’s function-based root locus, from which we show that the decoherence of the quantum dot can be suppressed via increasing the feedback coupling strength. The effectiveness of decoherence suppression induced by non-Markovian coherent feedback is demonstrated by a single quantum dot example bathed with Lorentzian noises.
Phase-space methods are powerful tools for the treatment of Markovian open quantum systems: they map the reduced dynamics of a system S, in interaction with an environment E, exactly onto Langevin equations for c-number stochastic variables, as opposed to Heisenberg-Langevin equations for operators. Langevin equations provide analytical insight in key regimes and excel at handling strong nonlinearities and couplings, where other methods often falter. Extending phase-space methods to non-Markovian dynamics, however, has remained a long-standing challenge. Here we address this gap by applying phase-space representations to the full S+E system; integrating out the environmental degrees of freedom then yields a general Langevin framework for S that incorporates both deterministic and stochastic contributions from E. Normally ordered representations, such as the Glauber-Sudarshan P representation and its positive variant due to Drummond and Gardiner, lead to Langevin equations in which (i) non-Markovian effects emerge exclusively in the deterministic terms, via a memory kernel, and (ii) noise contributions vanish when E is initially in the vacuum state. To demonstrate the power of this framework, we address the paradigmatic problem of delayed coherent feedback, in which the system is driven by its own past state, and study its impact on the laser linewidth: we recover the narrowing observed well above threshold and predict an enhanced narrowing just above it. Crucially, the number of stochastic variables scales linearly with the system size, making the framework suitable for problems ranging from a few degrees of freedom to genuinely many-body systems. This opens the way to the systematic study of non-Markovian driven-dissipative quantum systems using the same analytical and numerical tools that have long made phase-space methods so successful in the Markovian regime.
This paper presents a non-Markovian coherent feedback control scheme for stabilizing a single quantum dot system. The feedback loop is closed via quantum tunneling process between the natural source and drain baths of the quantum dot. The exact feedback-controlled non-Markovian Langevin equation is derived to describe the dynamics of the quantum dot. Since the memory function in the Langevin equation is a nonlinear function of the feedback coupling strength, a method named root locus of the Green's function is presented to analyze the non-Markovian dynamics in the frequency domain. This method graphically reveals the mechanism of decoherence suppression induced by non-Markovian coherent feedback, which is verified by an example of single quantum dot systems.
Abstract The no-knowledge quantum feedback, introduced in Phys. Rev. Lett., 113, 020407 (2014), is a measurement-based feedback protocol for decoherence suppression in a quantum system coupled to noisy environment. By continuously measuring the environmental noise, without directly gathering any information about the system, the decoherence effect can be suppressed by feeding back quantum controls proportional to the measured signal. In the original work, the feedback control was assumed instantaneous, leading to perfect cancellation of noise backaction on the quantum system. However, the instantaneous feedback is difficult to achieve in practice, and close-loop feedback protocols are always associated with finite delayed time. Therefore, in this work, we consider the effects of the delay between the time at which the measurement signal is acquired and the time that such signal is fed back to the system. We investigate the problem with an example of a two-level system (qubit) coupled to a Markovian reservoir, via a Hermitian coupling operator, where a homodyne detection is used to measure the environmental noise. We numerically simulate quantum stochastic trajectories of the qubit and analyse their averaged dynamics. We find that the feedback control with time delay can either enhance or reduce the decoherence effects, depending on whether the delayed time is in-phase or out-of-phase with the unitary dynamics of the qubit system.
In this article, we reconsider a version of quantum trajectory theory based on the stochastic Schrödinger equation with stochastic coefficients, which was mathematically introduced in the 1990s, and we develop it in order to describe the non-Markovian evolution of a quantum system continuously measured and controlled, thanks to a measurement-based feedback. Indeed, realistic descriptions of a feedback loop have to include delay and thus need a non-Markovian theory. The theory allows us to put together non-Markovian evolutions and measurements in continuous time, in agreement with the modern axiomatic formulation of quantum mechanics. To illustrate the possibilities of such a theory, we apply it to a two-level atom stimulated by a laser. We introduce closed loop control too, via the stimulating laser, with the aim of enhancing the 'squeezing' of the emitted light, or other typical quantum properties. Note that here we change the point of view with respect to the usual applications of control theory. In our model, the 'system' is the two-level atom, but we do not want to control its state, to bring the atom to a final target state. Our aim is to control the 'Mandel Q-parameter' and the spectrum of the emitted light; in particular, the spectrum is not a property at a single time, but involves a long interval of times (a Fourier transform of the autocorrelation function of the observed output is needed).