Liouville Superoperator Research Articles

Arnoldi-Lindblad time evolution: Faster-than-the-clock algorithm for the spectrum of time-independent and Floquet open quantum systems

The characterization of open quantum systems is a central and recurring problem for the development of quantum technologies. For time-independent systems, an (often unique) steady state describes the average physics once all the transient processes have faded out, but interesting quantum properties can emerge at intermediate timescales. Given a Lindblad master equation, these properties are encoded in the spectrum of the Liouvillian whose diagonalization, however, is a challenge even for small-size quantum systems. Here, we propose a new method to efficiently provide the Liouvillian spectral decomposition. We call this method an Arnoldi-Lindblad time evolution, because it exploits the algebraic properties of the Liouvillian superoperator to efficiently construct a basis for the Arnoldi iteration problem. The advantage of our method is double: (i) It provides a faster-than-the-clock method to efficiently obtain the steady state, meaning that it produces the steady state through time evolution shorter than needed for the system to reach stationarity. (ii) It retrieves the low-lying spectral properties of the Liouvillian with a minimal overhead, allowing to determine both which quantum properties emerge and for how long they can be observed in a system. This method is general and model-independent, and lends itself to the study of large systems where the determination of the Liouvillian spectrum can be numerically demanding but the time evolution of the density matrix is still doable. Our results can be extended to time evolution with a time-dependent Liouvillian. In particular, our method works for Floquet (i.e., periodically driven) systems, where it allows not only to construct the Floquet map for the slow-decaying processes, but also to retrieve the stroboscopic steady state and the eigenspectrum of the Floquet map. Although the method can be applied to any Lindbladian evolution (spin, fermions, bosons, …), for the sake of simplicity we demonstrate the efficiency of our method on several examples of coupled bosonic resonators (as a particular example). Our method outperforms other diagonalization techniques and retrieves the Liouvillian low-lying spectrum even for system sizes for which it would be impossible to perform exact diagonalization.

Damping transition in an open generalized Aubry-André-Harper model

We study the damping dynamics of the single-particle correlation for an open system under periodic and aperiodic order, which is dominated by the Lindblad master equation. In the absence of the aperiodic order, the Liouvillian superoperator exhibits the non-Hermitian skin effect, which leads to unidirectional damping dynamics, dubbed as "chiral damping". Due to the non-Hermitian skin effect, the damping dynamics is boundary sensitive: The long-time damping of such open systems is algebraic under periodic boundary conditions but exponential under open boundary conditions. We reveal the phase transition with the inclusion of the hopping amplitude modulation. By using the spectral topology and a finite-size scaling analysis in the commensurate case, we show there exists a phase transition of the skin effect with non-Bloch anti-parity-time symmetry breaking. For the incommensurate case, we find richer phases with the coexistence of the non-Hermitian skin effect and the Anderson localization, which are separated by a generalized mobility edge. We reveal the transition of the damping dynamics as a consequence of the phase transition. Furthermore, we propose a possible scheme with ultracold atoms in a dissipative momentum lattice to realize and detect the damping dynamics.

Generating high-order quantum exceptional points in synthetic dimensions

Recently, there has been intense research in proposing and developing various methods for constructing high-order exceptional points (EPs) in dissipative systems. These EPs can possess a number of intriguing properties related to, e.g., chiral transport and enhanced sensitivity. Previous proposals to realize non-Hermitian Hamiltonians (NHHs) with high-order EPs have been mainly based on either direct construction of spatial networks of coupled modes or utilization of synthetic dimensions, e.g., mapping of spatial lattices to time or photon-number space. Both methods rely on the construction of effective NHHs describing classical or postselected quantum fields, which neglect the effects of quantum jumps and which, thus, suffer from a scalability problem in the quantum regime, when the probability of quantum jumps increases with the number of excitations and dissipation rate. Here, by considering the full quantum dynamics of a quadratic Liouvillian superoperator, we introduce a simple and effective method for engineering NHHs with high-order quantum EPs, derived from evolution matrices of system operator moments. That is, by quantizing higher-order moments of system operators, e.g., of a quadratic two-mode system, the resulting evolution matrices can be interpreted as alternative NHHs describing, e.g., a spatial lattice of coupled resonators, where spatial sites are represented by high-order field moments in the synthetic space of field moments. Notably, such a mapping allows correct reproduction of the results of the Liouvillian dynamics, including quantum jumps. As an example, we consider a $U(1)$-symmetric quadratic Liouvillian describing a bimodal cavity with incoherent mode coupling, which can also possess $\mathrm{anti}\text{\ensuremath{-}}\mathcal{P}\mathcal{T}$ symmetry, whose field moment dynamics can be mapped to an NHH governing a spatial network of coupled resonators with high-order EPs.

Solving the Liouvillian Gap with Artificial Neural Networks.

We propose a machine-learning inspired variational method to obtain the Liouvillian gap, which plays a crucial role in characterizing the relaxation time and dissipative phase transitions of open quantum systems. By using "spin bi-base mapping," we map the density matrix to a pure restricted-Boltzmann-machine (RBM) state and transform the Liouvillian superoperator to a rank-two non-Hermitian operator. The Liouvillian gap can be obtained by a variational real-time evolution algorithm under this non-Hermitian operator. We apply our method to the dissipative Heisenberg model in both one and two dimensions. For the isotropic case, we find that the Liouvillian gap can be analytically obtained and in one dimension even the whole Liouvillian spectrum can be exactly solved using the Bethe ansatz method. By comparing our numerical results with their analytical counterparts, we show that the Liouvillian gap could be accessed by the RBM approach efficiently to a desirable accuracy, regardless of the dimensionality and entanglement properties.

Exact Liouvillian Spectrum of a One-Dimensional Dissipative Hubbard Model.

A one-dimensional dissipative Hubbard model with two-body loss is shown to be exactly solvable. We obtain an exact eigenspectrum of a Liouvillian superoperator by employing a non-Hermitian extension of the Bethe-ansatz method. We find steady states, the Liouvillian gap, and an exceptional point that is accompanied by the divergence of the correlation length. A dissipative version of spin-charge separation induced by the quantum Zeno effect is also demonstrated. Our result presents a new class of exactly solvable Liouvillians of open quantum many-body systems, which can be tested with ultracold atoms subject to inelastic collisions.

Limit cycle phase and Goldstone mode in driven dissipative systems

In this article, we theoretically investigate the first- and second-order quantum dissipative phase transitions of a three-mode cavity with a Hubbard interaction. In both types, there is a mean-field (MF) limit cycle phase where the local U(1) symmetry and the time-translational symmetry of the Liouvillian superoperator are spontaneously broken. In MF, this spontaneous symmetry breaking manifests itself through the appearance of an unconditionally and fully squeezed state at the cavity output, connected to the well-known Goldstone mode. By employing the Wigner function formalism, hence, properly including the quantum noise, we show that away from the thermodynamic limit and within the quantum regime, fluctuations notably limit the coherence time of the Goldstone mode due to the phase diffusion. Our theoretical predictions suggest that interacting multimode photonic systems are rich, versatile test beds for investigating the crossovers between the mean-field picture and quantum phase transitions, a problem that can be investigated in various platforms including superconducting circuits, semiconductor microcavities, atomic Rydberg polaritons, and cuprite excitons.

Bosonic representation of a Lipkin-Meshkov-Glick model with Markovian dissipation

We study the dynamics of a Lipkin-Meshkov-Glick model in the presence of Markovian dissipation, with a focus on late-time dynamics and the approach to thermal equilibrium. Making use of a vectorized bosonic representation of the corresponding Lindblad master equation, we use degenerate perturbation theory in the weak-dissipation limit to analytically obtain the eigenvalues and eigenvectors of the Liouvillian superoperator, which in turn give access to closed-form analytical expressions for the time evolution of the density operator and observables. Our approach is valid for large systems, but takes into account leading-order finite-size corrections to the infinite-system result. As an application, we show that the dissipative Lipkin-Meshkov-Glick model equilibrates by passing through a continuum of thermal states with damped oscillations superimposed, until finally reaching an equilibrium state with a temperature that in general differs from the bath temperature. We discuss limitations of our analytic techniques by comparing to exact numerical results.

Quantum thermodynamics in adiabatic open systems and its trapped-ion experimental realization

Quantum thermodynamics aims at investigating both the emergence and the limits of the laws of thermodynamics from a quantum mechanical microscopic approach. In this scenario, thermodynamic processes with no heat exchange, namely, adiabatic transformations, can be implemented through quantum evolutions in closed systems, even though the notion of a closed system is always an idealization and approximation. Here, we begin by theoretically discussing thermodynamic adiabatic processes in open quantum systems, which evolve non-unitarily under decoherence due to its interaction with its surrounding environment. From a general approach for adiabatic non-unitary evolution, we establish heat and work in terms of the underlying Liouville superoperator governing the quantum dynamics. As a consequence, we derive the conditions that an adiabatic open-system quantum dynamics implies in the absence of heat exchange, providing a connection between quantum and thermal adiabaticity. Moreover, we determine families of decohering systems exhibiting the same maximal heat exchange, which imply in classes of thermodynamic adiabaticity in open systems. We then approach the problem experimentally using a hyperfine energy-level quantum bit of an Ytterbium 171Yb+ trapped ion, which provides a work substance for thermodynamic processes, allowing for the analysis of heat and internal energy throughout a controllable engineered dynamics.

Symmetry and block structure of the Liouvillian superoperator in partial secular approximation

We address the structure of the Liouvillian superoperator for a broad class of bosonic and fermionic Markovian open systems interacting with stationary environments. We show that the accurate application of the partial secular approximation in the derivation of the Bloch-Redfield master equation naturally induces a symmetry on the superoperator level, which may greatly reduce the complexity of the master equation by decomposing the Liouvillian superoperator into independent blocks. Moreover, we prove that, if the steady state of the system is unique, one single block contains all the information about it, and that this imposes a constraint on the possible steady-state coherences of the unique state, ruling out some of them. To provide some examples, we show how the symmetry appears for two coupled spins interacting with separate baths, as well as for two harmonic oscillators immersed in a common environment. In both cases the standard derivation and solution of the master equation is simplified, as well as the search for the steady state. The block-diagonalization does not appear when a local master equation is chosen.

Dissipative time crystal in an asymmetric nonlinear photonic dimer

We investigate the behavior of two coupled non-linear photonic cavities, in presence of inhomogeneous coherent driving and local dissipations. By solving numerically the quantum master equation, either by diagonalizing the Liouvillian superoperator or by using the approximated truncated Wigner approach, we extrapolate the properties of the system in a thermodynamic limit of large photon occupation. When the mean field Gross-Pitaevskii equation predicts a unique parametrically unstable steady-state solution, the open quantum many-body system presents highly non-classical properties and its dynamics exhibits the long lived Josephson-like oscillations typical of dissipative time crystals, as indicated by the presence of purely imaginary eigenvalues in the spectrum of the Liouvillian superoperator in the thermodynamic limit.

Quantum exceptional points of non-Hermitian Hamiltonians and Liouvillians: The effects of quantum jumps

Exceptional points (EPs) correspond to degeneracies of open systems. These are attracting much interest in optics, optoelectronics, plasmonics, and condensed matter physics. In the classical and semiclassical approaches, Hamiltonian EPs (HEPs) are usually defined as degeneracies of non-Hermitian Hamiltonians such that at least two eigenfrequencies are identical and the corresponding eigenstates coalesce. HEPs result from continuous, mostly slow, nonunitary evolution without quantum jumps. Clearly, quantum jumps should be included in a fully quantum approach to make it equivalent to, e.g., the Lindblad master-equation approach. Thus, we suggest to define EPs via degeneracies of a Liouvillian superoperator (including the full Lindbladian term, LEPs), and we clarify the relations between HEPs and LEPs. We prove two main theorems: Theorem 1 proves that, in the quantum limit, LEPs and HEPs must have essentially different properties. Theorem 2 dictates a condition under which, in the ``semiclassical'' limit, LEPs and HEPs recover the same properties. In particular, we show the validity of Theorem 1 studying systems which have (1) an LEP but no HEPs, and (2) both LEPs and HEPs but for shifted parameters. As for Theorem 2, (3) we show that these two types of EPs become essentially equivalent in the semiclassical limit. We introduce a series of mathematical techniques to unveil analogies and differences between the HEPs and LEPs. We analytically compare LEPs and HEPs for some quantum and semiclassical prototype models with loss and gain.

Density matrix superoperator for periodic quantum systems and its application to quantum cascade laser structures

In this work we present a generalization of the Liouvillian superoperator for periodic quantum systems that can be formulated through partitioned Hamiltonians. We formulate a compact algebraic form of the superoperator that allows efficient numerical implementation along with the possibility of further generalization and the inclusion of the system’s boundary effects (i.e. device contacts). We apply this formalism to Quantum Cascade Laser structure where we compare the second nearest and the nearest on approximation, and present the laser dynamics that is independent from the number of states considered.

Spectral theory of Liouvillians for dissipative phase transitions

A state of an open quantum system is described by a density matrix, whose dynamics is governed by a Liouvillian superoperator. Within a general framework, we explore fundamental properties of both first-order dissipative phase transitions and second-order dissipative phase transitions associated with a symmetry breaking. In the critical region, we determine the general form of the steady-state density matrix and of the Liouvillian eigenmatrix whose eigenvalue defines the Liouvillian spectral gap. We illustrate our exact results by studying some paradigmatic quantum optical models exhibiting critical behavior.

Error reduction in quantum annealing using boundary cancellation: Only the end matters

The adiabatic theorem of quantum mechanics states that the error between an instantaneous eigenstate of a time-dependent Hamiltonian and the state given by quantum evolution of duration $\tau$ is upper bounded by $C/\tau$ for some positive constant $C$. It has been known for decades that this error can be reduced to $C_{k}/\tau^{k+1}$ if the Hamiltonian has vanishing derivatives up to order $k$ at the beginning and end of the evolution. Here we extend this result to open systems described by a time-dependent Liouvillian superoperator. We find that the same results holds provided the Liouvillian has vanishing derivatives up to order $k$ only at the end of the evolution. This asymmetry is ascribable to the arrow of time inherent in open system evolution. We further investigate whether it is possible to satisfy the required assumptions by controlling only the system, as required for realistic implementations. Surprisingly, we find the answer to be affirmative. We establish this rigorously in the setting of the Davies-Lindblad adiabatic master equation, and numerically in the setting of two different time-dependent Redfield-type master equations we derive. The results are shown to be stable with respect to imperfections in the preparation. Finally, we prove that the results hold also in a fully Hamiltonian model.

Уравнение Линдблада для квантового диссипативного гармонического осциллятора

Linblad's equation is obtained for a quantum harmonic oscillator with linear in a form convenient for applications. The operator of the equation contains the usual linear Liouville superoperator, which includes the Hamiltonian and the energy operator, and the Linblad quadratic superoperator. Linblad's superoperator consists of the sum of the operators diffusion and diffusion acting in the phase space, and the difference between the dissipation rate operator of the pulse and the coordinate in the phase space. A solution of the system of equations is found for the second moments of the coordinate, the momentum, and their product obtained from the Linblad equation. The equation for the entropy density is derived and it is shown that the entropy density according to the Linblad equation increases.

Critical dynamical properties of a first-order dissipative phase transition

We theoretically investigate the critical properties of a single driven-dissipative nonlinear photon mode. In a well-defined thermodynamical limit of large excitation numbers, the exact quantum solution describes a first-order phase transition in the regime where semiclassical theory predicts optical bistability. We study the behavior of the complex spectral gap associated with the Liouvillian superoperator of the corresponding master equation. We show that in this limit the Liouvillian gap vanishes exponentially and that the bimodality of the photon Wigner function disappears. The connection between the considered thermodynamical limit of large photon numbers for the single-mode cavity and the thermodynamical limit of many cavities for a driven-dissipative Bose-Hubbard system is discussed.

Kato expansion in quantum canonical perturbation theory

This work establishes a connection between canonical perturbation series in quantum mechanics and a Kato expansion for the resolvent of the Liouville superoperator. Our approach leads to an explicit expression for a generator of a block-diagonalizing Dyson’s ordered exponential in arbitrary perturbation order. Unitary intertwining of perturbed and unperturbed averaging superprojectors allows for a description of ambiguities in the generator and block-diagonalized Hamiltonian. We compare the efficiency of the corresponding computational algorithm with the efficiencies of the Van Vleck and Magnus methods for high perturbative orders.

Nonclassical non-Gaussian state of a mechanical resonator via selectively incoherent damping in a three-mode optomechanical system

We theoretically propose a scheme for the generation of a non-classical single-mode motional state of a mechanical resonator (MR) in the three-mode optomechanical systems, in which two optical modes of the cavities are linearly coupled to each other and one mechanical mode of the MR is optomechanically coupled to the two optical modes with the same coupling strength simultaneously. One cavity is driven by a coherent laser light. By properly tuning the frequency of the weak driving field, we obtain engineered Liouvillian superoperator via engineering the selective interaction Hamiltonian confined to the Fock subspaces. In this case, the motional state of the MR can be prepared into a non-Gaussian state, which possesses the sub-Poisson statistics although its Wigner function is positive.

Matrix-product-operator approach to the nonequilibrium steady state of driven-dissipative quantum arrays

We develop a numerical procedure to efficiently model the nonequilibrium steady state of one-dimensional arrays of open quantum systems, based on a matrix-product operator ansatz for the density matrix. The procedure searches for the null eigenvalue of the Liouvillian superoperator by sweeping along the system while carrying out a partial diagonalization of the single-site stationary problem. It bears full analogy to the density-matrix renormalization group approach to the ground state of isolated systems, and its numerical complexity scales as a power law with the bond dimension. The method brings considerable advantage when compared to the integration of the time-dependent problem via Trotter decomposition, as it can address arbitrarily long-ranged couplings. Additionally, it ensures numerical stability in the case of weakly dissipative systems thanks to a slow tuning of the dissipation rates along the sweeps. We have tested the method on a driven-dissipative spin chain, under various assumptions for the Hamiltonian, drive, and dissipation parameters, and compared the results to those obtained both by Trotter dynamics and Monte-Carlo wave function. Accurate and numerically stable convergence was always achieved when applying the method to systems with a gapped Liouvillian and a non-degenerate steady-state.

TurboEELS—A code for the simulation of the electron energy loss and inelastic X-ray scattering spectra using the Liouville–Lanczos approach to time-dependent density-functional perturbation theory

We introduce turboEELS, an implementation of the Liouville–Lanczos approach to linearized time-dependent density-functional theory, designed to simulate electron energy loss and inelastic X-ray scattering spectra in periodic solids. turboEELS is open-source software distributed under the terms of the GPL as a component of Quantum ESPRESSO. As with other components, turboEELS is optimized to run on a variety of different platforms, from laptops to massively parallel architectures, using native mathematical libraries (LAPACK and FFTW) and a hierarchy of custom parallelization layers built on top of MPI. Program summaryProgram title: turboEELSCatalogue identifier: AEXB_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEXB_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public License V 2.0No. of lines in distributed program, including test data, etc.: 1371515No. of bytes in distributed program, including test data, etc.: 37355712Distribution format: tar.gzProgramming language: Fortran 95.Computer: Any computer architecture.Operating system: GNU/Linux, AIX, IRIX, Mac OS X, and other UNIX-like OS’s.Classification: 7.2.External routines: turboEELS is a tightly integrated component of the Quantum ESPRESSO distribution and requires the standard libraries linked by it: BLAS, LAPACK, FFTW, MPI.Nature of problem: Calculation of the electron energy loss and inelastic X-ray scattering spectra of periodic solids.Solution method: The charge-density susceptibility of a periodic system is expressed in terms of the resolvent of its Liouvillian superoperator within time-dependent density functional perturbation theory. It is calculated using non-Hermitian or pseudo-Hermitian variants of the Lanczos recursion scheme, whose implementation does not require the calculation of any virtual states. Pseudopotentials (both norm-conserving and ultrasoft) are used in conjunction with plane-wave basis sets and periodic boundary conditions. Relativistic effects (spin–orbit coupling) can be included in calculations.Restrictions: Linear-response regime. Adiabatic exchange–correlation kernels only. No hybrid functionals. Collinear spin-polarized formalism is not supported, only non-collinear spin-polarized case can be used. Spin–orbit coupling cannot be used with ultrasoft pseudopotentials. No magnetism. No Hubbard U formalism. No PAW pseudopotentials.Unusual features: No virtual orbitals are used, nor even calculated. A single Lanczos recursion gives access to the whole spectrum at fixed transferred momentum.Additional comments: The distribution file of this program can be downloaded from the Quantum ESPRESSO website: http://www.quantum-espresso.org/, and the development version of this program can be downloaded via SVN from the QE-forge website: http://qe-forge.org/gf/project/q-e/.!!!!! The distribution file for this program is over 37 Mbytes and therefore is not delivered directly when download or Email is requested. Instead a html file giving details of how the program can be obtained is sent. !!!!!Running time: From a few minutes for elemental bulk systems with a few atoms in the primitive unit cell on serial machines up to many hours on multiple processors for complex systems (e.g., surfaces with high Miller indices) with dozens or hundreds of atoms.