Dissipative Quantum Systems Research Articles

A short trajectory is all you need: A transformer-based model for long-time dissipative quantum dynamics.

In this Communication, we demonstrate that a deep artificial neural network based on a transformer architecture with self-attention layers can predict the long-time population dynamics of a quantum system coupled to a dissipative environment provided that the short-time population dynamics of the system is known. The transformer neural network model developed in this work predicts the long-time dynamics of spin-boson model efficiently and very accurately across different regimes, from weak system-bath coupling to strong coupling non-Markovian regimes. Our model is more accurate than classical forecasting models, such as recurrent neural networks, and is comparable to the state-of-the-art models for simulating the dynamics of quantum dissipative systems based on kernel ridge regression.

Kinetics in the Two-Level System with Strong Time-De-pendent Coupling of Its States to the Phonon Bath: Spin-Boson Description

Using the methods of nonequilibrium statistical mechanics, the master equation for the density matrix of an open dissipative quantum system is obtained under conditions, when the coupling between the electronic states of the system and the nuclear displacements in it is controlled by the alternating field. A time-dependent polaron transformation is proposed, which made it possible to solve kinetic equations using an expansion in a parameter characterizing transitions between “phonon-dressed” electronic states of the system. As an example, a mechanism is shown that can control the kinetics in a two-level system by applying a periodic force field to electron-phonon coupling.

Characterisation of gradient flows for a given functional

Let X be a vector field and Y be a co-vector field on a smooth manifold M. Does there exist a smooth Riemannian metric gαβ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g_{\\alpha \\beta }$$\\end{document} on M such that Yβ=gαβXα\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Y_\\beta = g_{\\alpha \\beta } X^\\alpha $$\\end{document}? The main result of this note gives necessary and sufficient conditions for this to be true. As an application of this result we provide a gradient-flow characterisation for dissipative quantum systems. Namely, we show that finite-dimensional ergodic Lindblad equations admit a gradient flow structure for the von Neumann relative entropy if and only if the condition of bkm-detailed balance holds.

Bath-induced interactions and transient dynamics in open quantum systems at strong coupling: Effective Hamiltonian approach.

Understanding the dynamics of dissipative quantum systems, particularly beyond the weak coupling approximation, is central to various quantum applications. While numerically exact methods provide accurate solutions, they often lack the analytical insight provided by theoretical approaches. In this study, we employ the recently developed method dubbed the effective Hamiltonian theory to understand the dynamics of system-bath configurations without resorting to a perturbative description of the system-bath coupling energy. Through a combination of mapping steps and truncation, the effective Hamiltonian theory offers both analytical insights into signatures of strong couplings in open quantum systems and a straightforward path for numerical simulations. To validate the accuracy of the method, we apply it to two canonical models: a single spin immersed in a bosonic bath and two noninteracting spins in a common bath. In both cases, we study the transient regime and the steady state limit at nonzero temperature and spanning system-bath interactions from the weak to the strong regime. By comparing the results of the effective Hamiltonian theory with numerically exact simulations, we show that although the former overlooks non-Markovian features in the transient equilibration dynamics, it correctly captures non-perturbative bath-generated couplings between otherwise non-interacting spins, as observed in their synchronization dynamics and correlations. Altogether, the effective Hamiltonian theory offers a powerful approach for understanding strong coupling dynamics and thermodynamics, capturing the signatures of such interactions in both relaxation dynamics and in the steady state limit.

A Simple Testbed for Stability Analysis of Quantum Dissipative Systems

A Simple Testbed for Stability Analysis of Quantum Dissipative Systems

On the relevance of weak measurements in dissipative quantum systems

We investigate the impact of dissipation, including energy relaxation and decoherence, on weak measurements. While weak measurements have been successful in signal amplification, dissipation can compromise their usefulness. More precisely, we show that in systems with a unique steady state, weak values always converge to an expectation value of the measured observable as dissipation time tends to infinity, in contrast to systems with multiple steady states, where the weak values can remain anomalous, i.e. outside the range of eigenvalues of the observable, even in the limit of an infinite dissipation time. In addition, we propose a method for extracting information about the dissipative dynamics of a system using weak values at short dissipation times. Specifically, we explore the amplification of the dissipation rate in a two-level system and the use of weak values to differentiate between Markovian and non-Markovian dissipative dynamics. We also find that weak measurements operating around a weak atom-cavity coupling can probe the atom dissipation through the weak value of non-Hermitian operators within the rotating-wave approximation of the weak interaction.

Dynamics of a Quantum System Interacting with White Non-Gaussian Baths: Poisson Noise Master Equation.

Quantum systems are unavoidably open to their surrounding degrees of freedom. The theory of open quantum systems is thus crucial to understanding the fluctuations, dissipation, and decoherence of a quantum system of interest. Typically, the bath is modeled as an ensemble of harmonic oscillators, which yields Gaussian statistics of the bath influence on the quantum systems. However, there are also phenomena in which the bath consists of two-state systems, spins, or anharmonic oscillators; therefore, the non-Gaussian properties of the bath become important. Nevertheless, a theoretical framework to describe quantum systems under the influence of such non-Gaussian baths is not well established. Here, we develop a theory to describe quantum dissipative systems affected by Poisson noise properties of the bath, because the Lévi-Itô decomposition theorem asserts that Poisson noise is fundamental in describing arbitrary white noise beyond Gaussian properties. We introduce a quantum bath model that allows for the consistent description of dissipative quantum systems. The obtained master equation reveals non-Gaussian bath effects in the white noise regime, and provides an essential step toward describing open quantum dynamics under the influence of generic baths.

Artificial-Intelligence-Based Surrogate Solution of Dissipative Quantum Dynamics: Physics-Informed Reconstruction of the Universal Propagator.

The accurate (or even approximate) solution of the equations that govern the dynamics of dissipative quantum systems remains a challenging task in quantum science. While several algorithms have been designed to solve those equations with different degrees of flexibility, they rely mainly on highly expensive iterative schemes. Most recently, deep neural networks have been used for quantum dynamics, but current architectures are highly dependent on the physics of the particular system and usually limited to population dynamics. Here we introduce an artificial-intelligence-based surrogate model that solves dissipative quantum dynamics by parametrizing quantum propagators as Fourier neural operators, which we train using both data set and physics-informed loss functions. Compared with conventional algorithms, our quantum neural propagator avoids time-consuming iterations and provides a universal superoperator that can be used to evolve any initial quantum state for arbitrarily long times. To illustrate the wide applicability of the approach, we employ our quantum neural propagator to compute the population dynamics and time-correlation functions of the Fenna-Matthews-Olson complex.

Fluctuating parametric drive of coupled classical oscillators can simulate dissipative qubits

We investigate a system composed of two coupled oscillators subject to stochastic fluctuations in its internal parameters. In particular, we answer the question whether the well-known classical analogy of the quantum dynamics of two-level systems (TLSs), i.e., qubits, provided by two coupled oscillators can be extended to simulate the dynamics of dissipative quantum systems. In the context of nanomechanics, the analogy in the dissipation-free case has already been tested in multiple experimental setups, e.g., doubly clamped or cantilever string resonators and optically levitated particles. A well-known result of this classical analogy is that the relaxation and decoherence times of the analog quantum system must be equal, i.e., T1=T2, in contrast to the general case of quantum TLSs. We show that this fundamentally quantum feature, i.e., T1≠T2, can be implemented as well in the aforementioned classical systems by adding stochastic fluctuations in their internal parameters. Moreover, we show that these stochastic contributions can be engineered in the control apparatus of those systems, discussing, in particular, the application of this theory to levitated nanoparticles and to nanostring resonators. Published by the American Physical Society 2024

Memory-induced weak dissipation in fractional-time-derivative quantum Lindblad-based model

Modeling the interaction between a quantum system and its environment is crucial for practical quantum technologies. The Lindblad master equation is the simplest equation to understand this interaction. In this paper, we have extended the traditional Lindblad equation by fractionalizing its time derivative to account for the memory-induced dissipation. Using this approach, we show weak dissipation of selected quantum systems can be reproduced by this proposed time-fractional Lindblad equation without introducing any specific dissipation terms in the model. By varying the order of the time-fractional Lindblad equation without dissipation terms, we can reproduce the results with good agreements to three tested cases: (a) dissipative Rabi oscillation, (b) dissipative Ising model, and (c) collapse and revival in the Jaynes–Cumming model. We believe this proposed time-fractional Lindblad equation may be a useful modeling tool to characterize weakly dissipative quantum systems in practical quantum technologies, especially if the complex dissipation mechanism is not completely known from the traditional approaches.

Dynamics of a Quantum Dissipative System Coupled with an Oscillator

Dynamics of a Quantum Dissipative System Coupled with an Oscillator

On Krylov complexity in open systems: an approach via bi-Lanczos algorithm

Continuing the previous initiatives [1, 2], we pursue the exploration of operator growth and Krylov complexity in dissipative open quantum systems. In this paper, we resort to the bi-Lanczos algorithm generating two bi-orthogonal Krylov spaces, which individually generate non-orthogonal subspaces. Unlike the previously studied Arnoldi iteration, this algorithm renders the Lindbladian into a purely tridiagonal form, thus opening up a possibility to study a wide class of dissipative integrable and non-integrable systems by computing Krylov complexity at late times. Our study relies on two specific systems, the dissipative transverse-field Ising model (TFIM) and the dissipative interacting XXZ chain. We find that, for the weak coupling, initial Lanczos coefficients can efficiently distinguish integrable and chaotic evolution before the dissipative effect sets in, which results in more fluctuations in higher Lanczos coefficients. This results in the equal saturation of late-time complexity for both integrable and chaotic cases, making the notion of late-time chaos dubious.

Plane wave stability analysis of Hartree and quantum dissipative systems

We investigate the stability of plane wave solutions of equations describing quantum particles interacting with a complex environment. The models take the form of PDE systems with a non local (in space or in space and time) self-consistent potential; such a coupling lead to challenging issues compared to the usual nonlinear Schrödinger equations. The analysis relies on the identification of suitable Hamiltonian structures and Lyapounov functionals. We point out analogies and differences between the original model, involving a coupling with a wave equation, and its asymptotic counterpart obtained in the large wave speed regime. In particular, while the analogies provide interesting intuitions, our analysis shows that it is illusory to obtain results on the former based on a perturbative analysis from the latter.

Parametrically driving a quantum oscillator into exceptionality

The mathematical objects employed in physical theories do not always behave well. Einstein’s theory of space and time allows for spacetime singularities and Van Hove singularities arise in condensed matter physics, while intensity, phase and polarization singularities pervade wave physics. Within dissipative systems governed by matrices, singularities occur at the exceptional points in parameter space whereby some eigenvalues and eigenvectors coalesce simultaneously. However, the nature of exceptional points arising in quantum systems described within an open quantum systems approach has been much less studied. Here we consider a quantum oscillator driven parametrically and subject to loss. This squeezed system exhibits an exceptional point in the dynamical equations describing its first and second moments, which acts as a borderland between two phases with distinctive physical consequences. In particular, we discuss how the populations, correlations, squeezed quadratures and optical spectra crucially depend on being above or below the exceptional point. We also remark upon the presence of a dissipative phase transition at a critical point, which is associated with the closing of the Liouvillian gap. Our results invite the experimental probing of quantum resonators under two-photon driving, and perhaps a reappraisal of exceptional and critical points within dissipative quantum systems more generally.

Resolving mean-field solutions of dissipative phase transitions using permutational symmetry

Phase transitions in dissipative quantum systems have been investigated using various analytical approaches, particularly in the mean-field (MF) limit. However, analytical results often depend on specific methodologies. For instance, the Keldysh formalism shows that the dissipative transverse Ising (DTI) model exhibits a discontinuous transition at the upper critical dimension, dc=3, whereas the fluctuationless MF approach predicts a continuous transition in infinite dimensions (d∞). These two solutions cannot be reconciled, because the MF solutions above dc should be the same. Thus numerical check is required. However, numerical studies in large systems may not be feasible owing to the exponential increase in computational complexity as O(22N) with system size N. Here, we notice that because spins can be regarded as being fully connected at d∞, the spin indices can be permutation invariant, and then the number of quantum states can be considerably contracted as O(N3). The Lindblad equation is transformed into the dynamic equation based on the contracted states. Applying the Runge–Kutta algorithm to the dynamic equation, we obtain all critical exponents including the dynamic exponent as z≈0.5. Moreover, using the property that the DTI model has Z2 symmetry and thus the hyperscaling relation has the form 2β+γ=ν(dc+z), we obtain the relation dc+z=4 in the MF limit. Hence, dc≈3.5 and thus the discontinuous transition in d=3 cannot be treated as an MF solution. We conclude that the permutation invariance at d∞ can be used effectively for checking the validity of an analytic MF solution in quantum phase transitions.

On quantum dissipative systems: ground states and orbital stability

We investigate the existence and stability of ground states for a model coupling the Schrödinger equation to the wave equation in transverse directions. The model is intended to describe complex interactions between quantum particles and their environment. The result can be interpreted as a dissipation statement, induced by the energy exchanges with the environment. The proofs use either concentration-compactness arguments or spectral analysis of the linearized energy. Difficulties arise related to the fact the model does not satisfy scale invariance properties.

Frequency Combs with Parity-Protected Cross-Correlations and Entanglement from Dynamically Modulated Qubit Arrays.

We develop a general theoretical framework to dynamically engineer quantum correlations and entanglement in the frequency-comb emission from an array of superconducting qubits in a waveguide, rigorously accounting for the temporal modulation of the qubit resonance frequencies. We demonstrate that when the resonance frequencies of the two qubits are periodically modulated with a π phase shift, it is possible to realize simultaneous bunching and antibunching in cross-correlations as well as Bell states of the scattered photons from different sidebands. Our approach, based on the dynamical conversion between the quantum excitations with different parity symmetry, is quite universal. It can be used to control multiparticle correlations in generic dynamically modulated dissipative quantum systems.

Adiabatic elimination theory of multi-level system in spin-orbit coupled Bose-Einstein condensate

In quantum optics, adiabatic elimination simplifies multi-level quantum system by eliminating the fast oscillatory degree of freedom and preserving the slow-varying dynamics, thus obtaining an efficient description of the system. Adiabatic elimination has important applications in quantum simulation and quantum precision measurement. For example, spin-orbit coupling has been realized in ultracold atoms by using three-level Raman coupling and adiabatic elimination. In this paper, we investigate the theoretical method and generalize the adiabatic elimination in three-level non-Hermitian systems and multi-level systems on the basis of standard elimination scheme. These can provide theoretical guidance for realizing the interdiscipline of non-Hermitian physics and spin-orbit coupling effects and their potential applications. We mainly discuss the influences of dissipative effect on the population dynamics of the system, the validity and accuracy of the adiabatic elimination theory under different parameters for both non-Hermitian and two types of five-level systems. Specifically, the dynamics satisfying the large detuning condition gives very accurate results for quite a long evolution time with the adiabatic elimination theory, but when the two-photon detuning <i>δ</i> and the Rabi frequency <inline-formula><tex-math id="Z-20231017181120">\begin{document}$\varOmega $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20231052_Z-20231017181120.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20231052_Z-20231017181120.png"/></alternatives></inline-formula> gradually increase, leading to the violation of the large detuning condition <inline-formula><tex-math id="M1">\begin{document}$ \varOmega,\gamma, \delta \ll \Delta$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20231052_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20231052_M1.png"/></alternatives></inline-formula>, the effective two-level model can no longer describe the fast-varying dynamics of the system even in a short evolution time. Thus the choice of system parameters affects the effectiveness of adiabatic elimination of the excited levels. In a non-Hermitian system, the population in the ground state oscillates with gain periodically at the beginning, while that in the ground state oscillates with loss and decreases with time, with the total population decreasing with oscillation. For long-time evolution the gain in the system causes the population to diverge, and the adiabatic elimination of the effective two-energy level system describes this behavior accurately. The effect of the non-Hermitian parameters on the dynamics of the system in the resonance case is manifested in the case that the total population remains conserved, while the total population tends to diverge for finite two-photon detuning. We find that with the increase of detuning, the divergence appears earlier and the total number of particles can be kept constant by choosing the ratio of gain to loss appropriately. This study provides a theoretical basis for state preparation and dynamical manipulation in dissipative multi-energy quantum systems.

Operator growth and Krylov construction in dissipative open quantum systems

Inspired by the universal operator growth hypothesis, we extend the formalism of Krylov construction in dissipative open quantum systems connected to a Markovian bath. Our construction is based upon the modification of the Liouvillian superoperator by the appropriate Lindbladian, thereby following the vectorized Lanczos algorithm and the Arnoldi iteration. This is well justified due to the incorporation of non-Hermitian effects due to the environment. We study the growth of Lanczos coefficients in the transverse field Ising model (integrable and chaotic limits) for boundary amplitude damping and bulk dephasing. Although the direct implementation of the Lanczos algorithm fails to give physically meaningful results, the Arnoldi iteration retains the generic nature of the integrability and chaos as well as the signature of non-Hermiticity through separate sets of coefficients (Arnoldi coefficients) even after including the dissipative environment. Our results suggest that the Arnoldi iteration is meaningful and more appropriate in dealing with open systems.

Model Hamiltonians of open quantum optical systems: Evolvement from hermiticity to adjoint commutativity

In the conventional quantum mechanics of conserved systems, Hamiltonian is assumed to be a Hermitian operator. However, when it comes to quantum systems in presence of dissipation and/or noise, including open quantum optical systems, the strict hermiticity requirement is nor longer necessary. In fact, it can be substantially relaxed: the non-Hermitian part of a Hamiltonian is allowed, in order to account for effects of dissipative environment, whereas its Hermitian part would be describing subsystem’s energy. Within the framework of the standard approach to dissipative phenomena based on a master equation for the reduced density operator, we propose a replacement of the hermiticity condition by a more general condition of commutativity between Hermitian and anti-Hermitian parts of a Hamiltonian. As an example, we consider a dissipative two-mode quantum system coupled to a single-mode electromagnetic wave, where we demonstrate that the adjoint-commutativity condition does simplify the parametric space of the model.