Spectral degeneracies in Liouvillian generators of dissipative dynamics generically occur as exceptional points, where the corresponding non-Hermitian operator becomes nondiagonalizable. Steady states, i.e., zero modes of Liouvillians, are considered a fundamental exception to this rule since a no-go theorem excludes nondiagonalizable degeneracies there. Here, we demonstrate that the crucial issue of diverging timescales in dissipative state preparation is largely tantamount to an asymptotic approach toward the forbidden scenario of an exceptional steady state in the thermodynamic limit. With case studies ranging from NP-complete satisfiability problems encoded in a quantum master equation to the dissipative preparation of a symmetry protected topological phase, we reveal the close relation between the computational complexity of the problem at hand, and the finite size scaling toward the exceptional steady state, exemplifying both exponential and polynomial scaling. Formally treating the weight W of quantum jumps in the Lindblad master equation as a parameter, we show that exceptional steady states at the physical value W=1 may be understood as a critical point hallmarking the onset of dynamical instability.
Spectral degeneracies in Liouvillian generators of dissipative dynamics generically occur as exceptional points, where the corresponding non-Hermitian operator becomes nondiagonalizable. Steady states, i.e., zero modes of Liouvillians, are considered a fundamental exception to this rule since a no-go theorem excludes nondiagonalizable degeneracies there. Here, we demonstrate that the crucial issue of diverging timescales in dissipative state preparation is largely tantamount to an asymptotic approach toward the forbidden scenario of an exceptional steady state in the thermodynamic limit. With case studies ranging from NP-complete satisfiability problems encoded in a quantum master equation to the dissipative preparation of a symmetry protected topological phase, we reveal the close relation between the computational complexity of the problem at hand, and the finite size scaling toward the exceptional steady state, exemplifying both exponential and polynomial scaling. Formally treating the weight W of quantum jumps in the Lindblad master equation as a parameter, we show that exceptional steady states at the physical value W=1 may be understood as a critical point hallmarking the onset of dynamical instability.
Usually, when investigating exceptional points (EPs) of an open Markovian bosonic system, one deals with spectral degeneracies of a non-Hermitian Hamiltonian (NHH), which can correctly describe the system dynamics only in the semiclassical regime. A recently proposed quantum Liouvillian framework [Minganti et al., Phys. Rev. A 100, 062131 (2019)] enables the complete determination of the dynamical properties of such systems and their EPs (referred to as Liouvillian EPs, or LEPs) in the quantum regime by taking into account the effects of quantum jumps, which are ignored in the NHH formalism. Moreover, the symmetry and eigenfrequency spectrum of the NHH become a part of much larger Liouvillian eigenspace. As such, the EPs of an NHH form a subspace of the LEPs. Here we show that once an NHH of a dissipative linear bosonic system exhibits an EP of a certain finite order n, it immediately implies that the corresponding LEP can become of any higher order $m\ensuremath{\ge}n$ defined in the infinite Hilbert space. These higher-order LEPs can be identified by the coherence and spectral functions at the steady state. The coherence functions can offer a convenient tool to probe extreme system sensitivity to external perturbations in the vicinity of higher-order LEPs. As an example, we study a linear bosonic system of a bimodal cavity with incoherent mode coupling to reveal its higher-order LEPs; particularly, of second and third order via first- and second-order coherence functions, respectively. Accordingly, these LEPs can be additionally revealed by squared and cubic Lorentzian spectral lineshapes in the power and intensity-fluctuation spectra. Moreover, we demonstrate that these EPs can also be associated with spontaneous parity-time (PT) and anti-PT-symmetry breaking in the system studied. These symmetries can be switched in the output fields (the so-called supermodes) of an additional linear coupler with a properly chosen coupling strength. Thus, we show that the initial loss-loss dynamics for the supermodes can be equivalent to the balanced gain-loss evolution.
Abstract Exceptional points are the branch-point singularities of non-Hermitian Hamiltonians and have rich consequences in open-system dynamics. While the exceptional points and their critical phenomena are widely studied in the non-Hermitian settings without quantum jumps, they also emerge in open quantum systems depicted by the Lindblad master equations, wherein they are identified as the degeneracies in the Liouvillian eigenspectrum. These Liouvillian exceptional points often have distinct properties compared to their counterparts in non-Hermitian Hamiltonians, leading to fundamental modifications of the steady states or the steady-state-approaching dynamics. Since the Liouvillian exceptional points widely exist in quantum systems such as the atomic vapors, superconducting qubits, and ultracold ions and atoms, they have received increasing amount of attention of late. Here, we present a brief review on an important aspect of the dynamic consequence of Liouvillian exceptional points, namely the chiral state transfer induced by the parametric encircling the Liouvillian exceptional points. Our review focuses on the theoretical description and experimental observation of the phenomena in atomic systems that are experimentally accessible. We also discuss the ongoing effort to unveil the collective dynamic phenomena close to the Liouvillian exceptional points, as a consequence of the many-body effects therein. Formally, these phenomena are the quantum-many-body counterparts to those in classical open systems with nonlinearity, but hold intriguing new potentials for quantum applications.
In the thermodynamic limit, the steady states of open quantum many-body systems can undergo nonequilibrium phase transitions due to a competition between coherent and driven-dissipative dynamics. Here, we consider Markovian systems and elucidate structures of the Liouville superoperator that generates the time evolution. In many cases of interest, an operator-basis transformation can bring the Liouvillian into a block-triangular form, making it possible to assess its spectrum. The spectral gap sets the asymptotic decay rate. The superoperator structure can be used to bound gaps from below, showing that, in a large class of systems, dissipative phase transitions are actually impossible and that the convergence to steady states follows an exponential temporal decay. Furthermore, when the blocks on the diagonal are Hermitian, the Liouvillian spectra obey Weyl ordering relations. The results apply, for example, to Davies generators and quadratic systems and are also demonstrated for various spin models.
The out of equilibrium dynamics of the Sachdev-Ye-Kitaev model (SYK), comprising N Majoranas with random all-to-all four-body interactions, minimally coupled to a Markovian bath modeled by the Lindblad formalism, displays intriguing nontrivial features. In particular, the decay rate towards the steady state is a non-monotonic function of the bath coupling μ, and an analogue of the Loschmidt echo for dissipative quantum systems undergoes a first order dynamical phase transitions that eventually becomes a crossover for sufficiently large μ. We provide evidence that these features have their origin in the presence of exceptional points in the purely real eigenvalues of the SYK Liouvillian closest to the zero eigenvalue associated with the steady state. An analytic calculation at small N, supported by numerical results for larger N, reveals that the value of μ∼0.1 at which the exceptional point corresponding to the longest living modes occurs is close to a local maximum of the decay rate. This value marks the start of a region of anomalous equilibration where the relaxation rate diminishes as the coupling to the bath becomes stronger. Moreover, the mentioned change from transition to crossover in the Loschmidt echo occurs at a larger μ∼0.3 corresponding with a proliferation of exceptional points in the low energy limit of the Liouvillian spectrum. We expect these features to be generic in the approach to equilibrium in quantum strongly interacting many-body Liouvillians.
The dynamics of an isolated quantum system is coherent and unitary. Weak coupling to the environment leads to decoherence, which is traditionally modeled with a Lindblad equation for the system's density matrix. Starting from a pure state, such a system approaches a steady state (mixed or otherwise) in an underdamped or overdamped manner. This transition occurs at an eigenvalue degeneracy of a Lindblad superoperator, called an exceptional point (EP), where corresponding eigenvectors coalesce. Recent years have seen an explosion of interest in creating exceptional points in a truly quantum domain, driven by the enhanced sensitivity and topological features EPs have shown in their classical realizations. Here, we present Floquet analysis of a prototypical qubit whose drive or dissipator strengths are varied periodically. We consider models with a single dissipator that generate global loss (phase damping) or mode-selective loss (spontaneous emission). In all cases, we find that periodic modulations lead to EP lines at small dissipator strengths and a rich EP structure in the parameter space. Our analytical and numerical results show that extending Lindblad Liouvillians to the Floquet domain is a potentially preferred route to accessing exceptional points in the transient dynamics towards the Lindblad steady state.
We study a model of a quantum collective spin weakly coupled to a spin-polarized Markovian environment and find that the spectrum is divided into two regions that we name normal and exceptional Liouvillian spectral phases. In the thermodynamic limit, the exceptional spectral phase displays the unique property of being made up exclusively of second-order exceptional points. As a consequence, the evolution of any initial density matrix populating this region is slowed down and cannot be described by a linear combination of exponential decays. This phase is separated from the normal one by a critical line in which the density of Liouvillian eigenvalues diverges, a phenomenon analogous to that of excited-state quantum phase transitions observed in some closed quantum systems. In the limit of no bath polarization, this criticality is transferred onto the steady state, implying a dissipative quantum phase transition and the formation of a boundary time crystal.