Session D01: Open Quantum Systems I

Discovering feedback strategies for open quantum systems via deep reinforcement learning

Recent rapid advances in deep neural networks are helping to revolutionize science and technology. In this talk, I will describe how neural networks can discover from scratch feedback strategies to help control open quantum systems, by exploiting the toolbox of reinforcement learning. A few-qubit quantum memory can be protected against decoherence via quantum error correction strategies that have been autonomously constructed in this way. Additional illustrative examples from our recent research include reinforcement learning applied to systems from the domain of cavity quantum electrodynamics and to arrays of coupled modes.
  

Integrability Meets Dissipation: Critical Dynamics of Open Driven Systems

Driven quantum systems coupled to an environment are generally non-integrable and typically exhibit relaxational dynamics. We investigate the paradigmatic open Dicke model which describes collective light-matter interactions subject to dissipation. In a certain limit (at large detuning), this model is governed by an effective driven-dissipative Ising model in a transverse field, which is integrable in the absence of dissipation. In the limit of weak dissipation though, integrability is only weakly broken. We show that, in this regime, the system undergoes a dynamical crossover from relaxational dynamics to underdamped critical dynamics, each described by a distinct dynamical exponent. We identify these critical behaviors with the infinite-range classical (stochastic) and quantum (unitary) Ising models at finite temperature, respectively. These results are obtained through a non-equilibrium quantum-to-classical mapping in addition to an efficient numerical analysis that exploits the permutation symmetry.   

Dissipative generation of highly entangled states of light and matter

We investigate the full quantum evolution of ultracold interacting bosonic atoms confined to a chain geometry and coupled to the field of an optical cavity. Extending the time-dependent matrix product state techniques to capture the global coupling to the cavity mode and the open nature of the cavity, we fully include the light-atom entanglement. We examine the long time behavior of the system beyond the mean-field elimination of the cavity field. We show that in the self-organized phase the steady state consists in a mixture of the mean-field predicted density wave states and coherent states with lower photon number, with a large entanglement between the atomic and photonic degrees of freedom. In the regime of large dissipation strengths we develop a variant of the many-body adiabatic elimination technique and obtain a highly entangled steady state with a fully mixed atomic sector. We observe numerically the crossover from the density wave state towards the fully mixed state.   

Boundary dissipation in Ising CFT

Boundary conformal field theory (CFT) is one of few exact analytical tools for the study of quantum non-equilibrium systems, yet so far has been primarily restricted to closed Hamiltonian systems. We study one of the simplest CFTs, namely thequantum critical Ising model with boundary dephasing along the direction of the most relevant boundary operator. Naïve scaling arguments suggest that such dissipation should be marginal. Numerically for large systems, we find that boundary spin in the presence of dissipation, with or without boundary field, depolarizes and reaches a nonequilibrium steady state (NESS). In the presence of RG-relevant boundary term, scaling functions appear unmodified, suggesting that the dissipation is marginally irrelevant in that case. In the absence of boundary field, we study modifications of the Zeno effect by Keldysh RG and numerical calculations of the two-time correlation function of the boundary spin. Finally, we comment on potential application to other CFTs   

On the nature of the non-equilibrium phase transition in the non-Markovian driven Dicke model

The Dicke model exhibits a phase transition to a superradiant phase with a macroscopic population of photons and is realized in multiple settings in open quantum systems. In this work, we study a variant of the Dicke model where the cavity mode is lossy due to the coupling to a Markovian environment while the atomic mode is coupled to a colored bath. We find a simple effective theory for this model allowing us to derive analytical expressions for various critical exponents, including those, such as the dynamical critical exponent, that have not been previously considered. We find excellent agreement with previous numerical results when the non-Markovian bath is at zero temperature; however, contrary to these studies, our low-frequency approach reveals that the same exponents govern the critical behavior when the colored bath is at finite temperature unless the chemical potential is zero. Furthermore, we show that the superradiant phase transition is classical in nature, while it is genuinely non-equilibrium. Finally, we consider finite-size effects at the phase transition and identify the finite-size scaling exponents, unlocking a rich behavior in both statics and dynamics of the photonic and atomic observables.   

Anomalous exceptional point and non-Markovian Purcell effect at threshold in 1-D open quantum systems

We show that when a quantum emitter is coupled near threshold to a generic 1-D continuum with a van Hove singularity in the density of states, a characteristic spectral configuration appears involving a bound state, a resonance state and an anti-resonance state, as well as several exceptional points (EPs). At one EP appearing below the threshold, the resonance and anti-resonance states coalesce while the bound state instead experiences an avoided crossing. Meanwhile, if one considers the limit in which the coupling g vanishes, all three states converge on the continuum threshold itself. For small g values the eigenvalue and norm of each of these states can be expanded in a Puiseux expansion in terms of powers of g^2/3, which suggests a third order EP occurs at the threshold. However, in the actual g ––> 0 limit, only two discrete states in fact coalesce as the system can be reduced to a 2x2 Jordan block; the third state instead merges with the continuum. We further demonstrate the influence of the EP on non-Markovian dynamics characterizing the relaxation process of the quantum emitter in the vicinity of the threshold.   

Non-Markovian collective emission from macroscopically separated emitters

We study the collective radiative decay of a system of two two-level emitters coupled to a onedimensional waveguide in a regime where their separation is comparable to the coherence length of a spontaneously emitted photon. The electromagnetic field propagating in the cavity-like geometry formed by the emitters exerts a retarded backaction on the system leading to strongly non-Markovian dynamics. The collective spontaneous emission rate of the emitters exhibits an enhancement or inhibition beyond the usual Dicke super- and sub-radiance due to a self-consistent coherent timedelayed feedback.   

Analysis of matter-wave emission dynamics and polariton formation in a quantum-emitter array coupled to a band structure

Recent progress on a spontaneous emitter for atomic matter waves [1] has enabled studies of exotic emission phenomena in bandgap materials. Here we present analytic solutions for the single excitation dynamics in a 1D sinusoidal lattice potential and a finite or infinite array of emitters. The calculated phenomenology ranges from non-Markovian decay into bound states for the case of one emitter, to the emergence of an additional band structure of polaritons for the case of infinite emitters.

¹L. Krinner, M. Stewart, A. Pazmiño, J. Kwon, D. Schneble, Nature 559, 598 (2018).   

Generalized Theory of Pseudomodes for Exact Descriptions of Non-Markovian Quantum Processes

In this talk we develop a general approach to analyzing the non-Markovian behavior of an open quantum system in a setting where the interaction is modelled by a generalized class of spectral density function. By introducing an auxiliary model in which the environment is replaced by a set of discrete bosonic modes exhibiting Markovian dissipative interactions, we prove the validity of the conditions allowing for a non-Markovian open system dynamics to be amended to a Markovian description, where the dynamics in the latter is governed by an exact master equation of Lindblad form. Initially we apply our result to obtain a generalization of the pseudomode method [1] in cases where the spectral density function has a Lorentzian structure. For many other types of spectral density function, we extend this result to show that an open system dynamics may be modelled physically using discrete modes which admit a non-Hermitian coupling to the system, and for such cases determine the equivalent master equation to no longer be of Lindblad form. For applications involving two discrete modes, we demonstrate how to convert between pathological and Lindblad forms of the master equation via [1].

[1] B. M. Garraway, Nonperturbative decay of an atomic system in a cavity, Phys. Rev. A 55, 2290 (1997).   

Stabilization of Fractional Quantum Hall States of Light: Effects of Fractionalization

Recently, the possibility of realizing strongly-correlated states of matter with photons has started to become an experimental reality. In combination with artificial gauge fields this paves the way to simulation of fractional quantum Hall states with light. A major hindrance to such simulations is the inevitable dissipation of photons into the environment which creates holes in the correlated state. The leakage of light can be counteracted by a stabilization scheme in which lost particles are irreversibly refilled. We investigate the efficiency of such a scheme for the preparation of a photonic Laughlin state at half-filling. We explore the limitations imposed by the ability of holes in the Laughlin state to fractionalize into several spatially separated quasiholes. Quasiholes correspond to the absence of a fraction of a particle and thus cannot be refilled efficiently. We find that the fractionalization drastically restricts the steady-state fidelity of the Laughlin state and leads to a long relaxation time in the system.   

Transport through a quantum critical system: A thermodynamically consistent approach

Quantum phase transitions are striking phenomena of many-body systems at low temperatures. The current experimental feasibilities enable us to bring such critical systems out of equilibrium in a controlled manner. Due to the vanishing energy gap above the ground state [1], appropriate methods have to be developed to study the dynamics and thermodynamical applications. We show for a class of critical systems connected to several non-Markovian heat baths that by combining the reaction coordinate mapping [2] and a polaron technique [3] it is possible to find analytic expressions of the reduced system dynamics in the vicinity of quantum critical points, which are consistent with the laws of thermodynamics. As an example we consider the Lipkin-Meshkov-Glick model in a transport setup, where the underlying phase transition manifests itself in the heat transfer statistics.
[1] S. Sachdev, Quantum Phase Transitions (Cambridge Univ. Press, Cambridge, 1999).
[2] P. Strasberg et. al, New J. Phys. 18, 073007 (2016).
[3] P. Kirton and J. Keeling, Phys. Rev. Lett. 111, 100404 (2013).   

Deconstructing Effective Non-Hermitian Dynamics in Quadratic Bosonic Hamiltonians

Unlike their fermionic counterparts, the dynamics of Hermitian, bosonic, quadratic Hamiltonians are governed by a generally non-Hermitian Bogoliubov de-Gennes Hamiltonian. This effective non-Hermiticity gives rise to two distinct dynamical phases: one with bounded evolution of observables in time, and one without. We elucidate the physical manifestations of the transitions between these two dynamical phases. We show how a generalized notion of PT symmetry may be used to classify the mechanisms by which this transition can occur. By combining this understanding with tools from Krein stability theory, we derive an indicator of dynamical phase boundaries inspired by the notion of phase rigidity in non-Hermitian quantum systems. As an example, we fully characterize the dynamical phase diagram of a bosonic analogue to the Kitaev-Majorana chain under a wide class of boundary conditions, and further establish a connection between phase-dependent transport properties and the onset of instability. We discuss potential applications of our techniques to quadratic Lindblad dynamics.   

Non-Hermitian Linear Response Theory

Linear response theory lies at the heart of quantum many-body physics because it builds up connections between the dynamical response to an external probe and correlation functions at equilibrium. Here we consider the dynamical response of a Hermitian system to a non-Hermitian probe, and we develop a non-Hermitian linear response theory that can also relate this dynamical response to equilibrium properties. As an application of our theory, we consider the real-time dynamics of momentum distribution induced by one-body and two-body dissipations. We find that, for many cases, the dynamics of momentum occupation and the width of momentum distribution obey the same universal function, governed by the single-particle spectral function. We also find that, for critical state with no well-defined quasi-particles, the dynamics are slower than normal state and our theory provides a model independent way to extract the critical exponent. We apply our results to analyze recent experiment on the Bose-Hubbard model and find surprising good agreement between theory and experiment. We also propose to further verify our theory by carrying out a similar experiment on a one-dimensional Luttinger liquid.