Emergent phases and novel critical behavior in a non-Markovian open quantum system

Emergent phases and novel critical behavior in a non-Markovian open quantum system
H. F. H. Cheung, Y. S. Patil and M. Vengalattore
Physical Review A 97, 052116 (2018)
arXiv:1707.02622 (2017)

Abstract

Open quantum systems exhibit a range of novel out-of-equilibrium behavior due to the interplay between coherent quantum dynamics and dissipation. Of particular interest in these systems are driven, dissipative transitions, the emergence of dynamical phases with novel broken symmetries, and critical behavior that lies beyond the conventional paradigm of Landau-Ginzburg phenomenology. Here, we consider a parametrically driven two-mode system in the presence of non-Markovian system-reservoir interactions. We show that the non-Markovian dynamics modifies the phase diagram of this system, resulting in the emergence of a broken symmetry phase in a universality class that has no counterpart in the corresponding Markovian system. This emergent phase is accompanied by enhanced two-mode entanglement that remains robust at finite temperatures. Such reservoir-engineered dynamical phases can potentially shed light on universal aspects of dynamical phase transitions in a wide range of nonequilibrium systems, and aid in the development of techniques for the robust generation of entanglement and quantum correlations at finite temperatures with potential applications to quantum control, state preparation, and metrology.

Bibtex

@article{cheungEmergentPhase2018,
author = {{Cheung}, H.~F.~H. and {Patil}, Y.~S. and {Vengalattore}, M.},
title = "{Emergent phases and critical behavior in a non-Markovian open quantum system}",
journal = {\pra},
keywords = {Quantum Physics, Condensed Matter - Quantum Gases}, year = 2018,
month = may,
volume = {97},
number = {5},
eid = {052116},
pages = {052116},
doi = {10.1103/PhysRevA.97.052116},
archivePrefix = {arXiv},
eprint = {1707.02622},
primaryClass = {quant-ph},
adsurl = {https://ui.adsabs.harvard.edu/abs/2018PhRvA..97e2116C},
adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

Figure 1

(a) Schematic of the two-mode system. (b) The phase diagram as a function of the drive strength \(\mu \equiv F_P/F_{cr}\) and the normalized reservoir decay rate \((\gamma_0 \tau_r)^{-1}\). The color scale indicates the least negative real part of the eigenvalues of the susceptibility matrix (see text). Critical points and phase boundaries (dashed lines) correspond to the vanishing of this real part, i.e. a divergent relaxation time.

Figure 2

The transition between the \(U(1)\) and the \(U(1) \times \mathbb{Z}_2\) phase versus the normalized reservoir decay rate, \((\gamma_0 \tau_r)^{-1}\) The critical point occurs at \((\gamma_0 \tau_r)^{-1} = \frac{1}{2}\), corresponding to a divergent variance, \(\mathrm{Var}(\dot{\phi})\), of the instantaneous frequency of the signal and idler modes. Below this critical point, these modes no longer self-oscillate at their nominal resonances but shift to \(\omega_{i} \rightarrow \omega_i \pm \Delta, \omega_{s} \rightarrow \omega_s \mp \Delta\), corresponding to a breaking of the \(\mathbb{Z}_2\) symmetry (see inset, bottom). In contrast to the spontaneously chosen but constant phase difference between the two modes in the \(U(1)\) phase, the phases of these modes now oscillate (see inset, top) at a frequency \(\Delta\) that continuously grows from zero below the critical point.

Figure 3

The behavior of the low lying eigenspectrum corresponding to the disordered phase (orange), \(U(1)\) phase (green) and \(U(1) \times \mathbb{Z}_2\) phase (blue) with increasing reservoir coherence time \(\tau_r\) showing the relative positions of the exceptional point (blue circle) and the critical point (green circle) {\em vs} the drive strength. The imaginary part of the eigenvalue is represented as the width of the eigenmode. The exceptional point corresponds to a coalescence of the eigenvalues and eigenmodes and a vanishing imaginary part. The critical point occurs when the disordered phase becomes unstable (\(\mathrm{Re}[\lambda] >0\)) and gives way to the broken symmetry phases. 
(a) In the Markovian regime, i.e. \((\gamma_0 \tau_r)^{-1} \gg \frac{1}{2}\), the exceptional point occurs before the critical point governing the transition to the \(U(1)\) phase. The eigenvalues are purely real in the vicinity of the critical point. 
(b) At \((\gamma_0 \tau_r)^{-1} = \frac{1}{2}\), the exceptional point and the critical point coincide, i.e. the real and imaginary parts of the eigenvalues vanish simultaneously at the critical point, indicating the emergence of the \(U(1) \times \mathbb{Z}_2\) phase. 
(c) Deep in the non-Markovian regime, i.e. \((\gamma_0 \tau_r)^{-1} \ll \frac{1}{2}\), the critical point occurs before the exceptional point and the transition to the \(U(1) \times \mathbb{Z}_2\) phase occurs when the eigenvalues are purely imaginary. The displayed eigenspectra correspond to (a) \((\gamma_0 \tau_r)^{-1} = 1.25\), (b) \((\gamma_0 \tau_r)^{-1} = 0.50\), and (c) \((\gamma_0 \tau_r)^{-1} = 0.15\). 

Figure 4

(Top) The logarithmic negativity \(\mathcal{E}_N\) as a measure of the bipartite entanglement \cite{peres1996, plenio2005} between the signal and idler modes {\em vs} the drive strength \(\mu\) and the normalized reservoir decay rate \((\gamma_0 \tau_r)^{-1}\). (Bottom) In the \(U(1) \times \mathbb{Z}_2\) phase, the entanglement between the two modes extends well beyond the quantum regime and can be observed even for large thermal occupancy of the two modes. The logarithmic negativity is shown for increasing thermal occupancy \(\bar{n}_{th}\) {\em vs} drive strength for \((\gamma_0 \tau_r)^{-1}= \frac{1}{5}\). For comparison, the logarithmic negativity for a Markovian system with \(\bar{n}_{th} = 5\) is shown in dashed lines.