Time-Delayed Quantum Feedback Control

Abstract

A theory of time-delayed coherent quantum feedback is developed. More specifically, we consider a quantum system coupled to a bosonic reservoir creating a unidirectional feedback loop. It is shown that the dynamics can be mapped onto a fictitious series of cascaded quantum systems, where the system is driven by past versions of itself. The derivation of this model relies on a tensor network representation of the system-reservoir time-propagator. For concreteness, this general theory is applied to a driven two-level atom scattering into a coherent feedback loop. We demonstrate how delay effects can qualitatively change the dynamics of the atom, and how quantum control can be implemented in the presence of time-delays. Introduction.-Delayed autonomous feedback, where a signal is directly fed back to a system after a controllable time-delay, is an important control tool for classical systems [1][2][3]. It is highly attractive as a tool for stabilizing non-equilibrium states of fast dynamical systems, where avoiding any time-costly signal-processing is crucial. Such stabilization is of great experimental and technological relevance [4][5][6]. In particular, delayed autonomous feedback has been used to stabilize the high frequency dynamics of optical systems and high speed electrical circuits [7, 8].Autonomous feedback is also receiving substantial and growing interest for controlling quantum systems [9][10][11][12][13][14][15][16]. Because of the relatively short coherence time and fast dynamics of quantum systems, very fast feedback control possible with autonomous feedback is highly desirable. In addition, any measurement of the feedback signal will necessarily destroy its quantum character, making a fully quantum mechanical feedback loop that preserves coherence attractive from a fundamental point of view. Compelling evidence that this type of coherent feedback can outperform any measurement-based counterpart for important quantum information processing tasks has been given [17,18].A natural way of implementing coherent feedback control loops is by coupling remote quantum systems via waveguides [19][20][21][22]. Time-delays are unavoidable in practice in such setups and are likely to become important if current experiments are scaled up to larger and more complex networks [23][24][25]. Despite of this, relatively little theoretical research has been done on delay effects for coherent quantum feedback. A major obstacle is the lack of tractable and general theoretical models for treating the highly non-Markovian dynamics induced by this type of feedback. The theoretical difficulty lies in the quantum correlations between the control target system and the in-loop quantum field: The field cannot simply be traced out, and one has to deal with a highly entangled quantum state over a continuum of degrees of freedom.Previous investigations have typically been limited to

Collision models in quantum optics

“…One can show that this is equivalent to allowing S to interact with each discrete input mode twice (the time interval between the two interactions representing the delay time), a dynamics that was tackled in Ref. [16] through a nice diagrammatic method.…”

“…The last property alongside their simple and intrinsically discrete nature make CMs advantageous case studies to investigate major open problems in quantum non-Markovianity once the basic model outlined above is modi ed so as to introduce a memory mechanism. Among the ways to endow a CM with memory are: adding ancillaancilla collisions [4][5][6][7][8][9][10], embedding S into a larger system [11][12][13][14][15], allowing S to collide with each ancilla more than once [16,17], assuming a correlated initial bath state instead of a product one [18][19][20][21][22][23][24][25] or initial system-bath correlations [26][27][28]. Typical tasks that can be accomplished through NM CMs constructed in one of these ways are: deriving well-de ned (i.e., unconditionally completely positive) NM MEs [4,5,[37][38][39], gaining quantitative information about the role of system-bath and/or intra-bath correlations in making a dynamics NM [6,10,[19][20][21][22], simulating highly NM dynamics or indivisible channels [7,18,24].…”

Intensified antibunching via feedback-induced quantum interference

“…To solve this problem methods have been proposed employing the Liouville space [26], Matrix product state (MPS) evolution [27], and the Heisenberg picture [21].…”

Composite quantum collision models

“…Remarkably, (i) in particular can suggest schemes to perform experimental simulations of non-Markovian dynamics [17] or provide valuable theoretical tools in the analysis of very large Hilbert space problems [18] and time-delayed quantum feedback [19]. Concerning (ii), in particular, in 2013 two of us [12] showed that one can use a collision model to work out a non-Markovian master equation that is both capable to interpolate between a Markovian and a strongly non-Markovian regime and, additionally, is ensured to be completely positive and trace preserving, namely two requirements that are in general hard to meet at the same time.…”