Quantum Stochastic Dynamics Research Articles

Dynamics of a small quantum system open to a bath with thermostat.

We investigate dynamics of a small quantum system open to a bath with thermostat. We introduce another bath, called a superbath, weakly coupled with the bath to provide it with a thermostat, which has either the Lindblad or Redfield type. We treat the interaction between the system and bath via a rigorous perturbation theory. Due to the thermostat, the bath behaves dissipative and stochastic, for which the usual Born-Markov assumption is not needed. We consider a specific example of a harmonic oscillator system of interest and a photonic bath in a large container, and a superbath of the Caldeira-Legget oscillators distributed on the inner surface of the container. After taking the trace over the superbath states, we use the P representation for the total harmonic system of the system and bath. We derive the reduced time-evolution equationfor the system by explicitly finding the correlation between the system and bath beyond the product state that was not obtainable in the previous theory for the system and bath isolated from environment, and marginalizing bath degrees of freedom. Remarkably, the associated dynamic equationfor the system density matrix is of the same form as the Redfield master equationwith different coefficients depending on thermostat used. We find the steady state does not depend on the thermostat but the time-dependent state does, which agrees with common expectation. We expect to apply our theory to general systems. Unlike the usual quantum master equations, our reduced dynamics allows investigation for time-dependent protocols, and nonequilibrium quantum stochastic dynamics will be investigated in the future.

Quantum velocity limits for multiple observables: Conservation laws, correlations, and macroscopic systems

How multiple observables mutually influence their dynamics has been a crucial issue in statistical mechanics. We here introduce a new concept, “quantum velocity limits,” to establish a quantitative and rigorous theory for nonequilibrium quantum dynamics for multiple observables. Quantum velocity limits are universal inequalities for a vector that describes velocities of multiple observables. They elucidate that the speed of an observable of our interest can be tighter bounded when we have knowledge of other observables, such as experimentally accessible ones or conserved quantities, compared with conventional speed limits for a single observable. Moreover, quantum velocity limits are conceptually distinct from the conventional speed limits because we need to introduce the velocity vector and solve an optimization problem for multiple variables to obtain them. We first derive an information-theoretical velocity limit in terms of the generalized correlation matrix of the observables and the quantum Fisher information. The velocity limit has various novel consequences: (i) conservation law in the system, a fundamental ingredient of quantum dynamics, can improve the velocity limits through the correlation between the observables and conserved quantities; (ii) speed of an observable can be bounded by a nontrivial lower bound from the information on another observable, while most of the previous speed limits provide only upper bounds; (iii) there exists a notable nonequilibrium tradeoff relation, stating that speeds of uncorrelated observables, e.g., anticommuting observables, cannot be simultaneously large; (iv) velocity limits for local observables in locally interacting many-body systems are described by the fluctuation of a local Hamiltonian, which is convergent even in the thermodynamic limit, with a nontrivial finite-size correction. Moreover, we discover another distinct velocity limit for multiple observables on the basis of the local conservation law of probability current, which becomes advantageous for macroscopic transitions of multiple quantities. Our newly found velocity limits ubiquitously apply not only to unitary quantum dynamics but to classical and quantum stochastic dynamics, offering a key step towards universal theory of far-from-equilibrium dynamics for multiple observables. Published by the American Physical Society 2024

State Estimation and Control for Stochastic Quantum Dynamics With Homodyne Measurement: Stabilizing Qubits Under Uncertainty

State Estimation and Control for Stochastic Quantum Dynamics With Homodyne Measurement: Stabilizing Qubits Under Uncertainty

Causal potency of consciousness in the physical world

The evolution of the human mind through natural selection mandates that our conscious experiences are causally potent in order to leave a tangible impact on the surrounding physical world. Any attempt to construct a functional theory of the conscious mind within the framework of classical physics, however, inevitably leads to causally impotent conscious experiences in direct contradiction to evolution theory. Here, we derive several rigorous theorems that identify the origin of the latter impasse in the mathematical properties of ordinary differential equations employed in combination with the alleged functional production of the mind by the brain. Then, we demonstrate that a mind–brain theory consistent with causally potent conscious experiences is provided by modern quantum physics, in which the unobservable conscious mind is reductively identified with the quantum state of the brain and the observable brain is constructed by the physical measurement of quantum brain observables. The resulting quantum stochastic dynamics obtained from sequential quantum measurements of the brain is governed by stochastic differential equations, which permit genuine free will exercised through sequential conscious choices of future courses of action. Thus, quantum reductionism provides a solid theoretical foundation for the causal potency of consciousness, free will and cultural transmission.

Theory and computation of Markovian quantum antenna systems

We describe a general framework for the modeling and analysis of Markovian quantum antenna systems viewed as a special problem in quantum stochastic dynamics. The quantum radiator, which can be used to spatially direct radiated quantum states in future quantum communication systems, is modeled as an open quantum system capable of controlling the composition of its radiation modes through external source manipulations while in continuous interaction with the surrounding thermal and random environments. Our analysis and computational methods are based on deploying the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) master equation to account for environment-induced jump processes such as quantum dissipation and decoherence. We construct a definition of the quantum antenna directivity inspired by the corresponding formulas in classical antenna theory and present several versions of the quantum directivity formula. As a computational example, we study the flow of the density operator of a coupled two-level quantum dot (qubit) array, excited by classical external signals with variable amplitude and phase, which is evolved in time using the quantum master equation. It is shown that by manipulating the amplitude and phase excitations of individual quantum dots, one may significantly enhance the directive radiation properties of the Markovian quantum antenna system.

Entangled multiplets and spreading of quantum correlations in a continuously monitored tight-binding chain

We analyze the dynamics of entanglement in a paradigmatic noninteracting system subject to continuous monitoring of the local excitation densities. Recently, it was conjectured that the evolution of quantum correlations in such system is described by a semi-classical theory, based on entangled pairs of ballistically propagating quasiparticles and inspired by the hydrodynamic approach to unitary (integrable) quantum systems. Here, however, we show that this conjecture does not fully capture the complex behavior of quantum correlations emerging from the interplay between coherent dynamics and continuous monitoring. We unveil the existence of multipartite quantum correlations which are inconsistent with an entangled-pair structure and which, within a quasiparticle picture, would require the presence of larger multiplets. We also observe that quantum information is highly delocalized, as it is shared in a collective {\it nonredundant} way among adjacent regions of the many-body system. Our results shed new light onto the behavior of correlations in quantum stochastic dynamics and further show that these may be enhanced by a (weak) continuous monitoring process.

Large Deviations at Level 2.5 for Markovian Open Quantum Systems: Quantum Jumps and Quantum State Diffusion

We consider quantum stochastic processes and discuss a level 2.5 large deviation formalism providing an explicit and complete characterisation of fluctuations of time-averaged quantities, in the large-time limit. We analyse two classes of quantum stochastic dynamics, within this framework. The first class consists of the quantum jump trajectories related to photon detection; the second is quantum state diffusion related to homodyne detection. For both processes, we present the level 2.5 functional starting from the corresponding quantum stochastic Schrödinger equation and we discuss connections of these functionals to optimal control theory.

Control of stochastic quantum dynamics by differentiable programming

Control of the stochastic dynamics of a quantum system is indispensable in fields such as quantum information processing and metrology. However, there is no general ready-made approach to the design of efficient control strategies. Here, we propose a framework for the automated design of control schemes based on differentiable programming. We apply this approach to the state preparation and stabilization of a qubit subjected to homodyne detection. To this end, we formulate the control task as an optimization problem where the loss function quantifies the distance from the target state, and we employ neural networks (NNs) as controllers. The system’s time evolution is governed by a stochastic differential equation (SDE). To implement efficient training, we backpropagate the gradient information from the loss function through the SDE solver using adjoint sensitivity methods. As a first example, we feed the quantum state to the controller and focus on different methods of obtaining gradients. As a second example, we directly feed the homodyne detection signal to the controller. The instantaneous value of the homodyne current contains only very limited information on the actual state of the system, masked by unavoidable photon-number fluctuations. Despite the resulting poor signal-to-noise ratio, we can train our controller to prepare and stabilize the qubit to a target state with a mean fidelity of around 85%. We also compare the solutions found by the NN to a hand-crafted control strategy.

Solution to the Quantum Symmetric Simple Exclusion Process: The Continuous Case

The quantum symmetric simple exclusion process (Q-SSEP) is a model for quantum stochastic dynamics of fermions hopping along the edges of a graph with Brownian noisy amplitudes and driven out-of-equilibrium by injection-extraction processes at a few vertices. We present a solution for the invariant probability measure of the one dimensional Q-SSEP in the infinite size limit by constructing the steady correlation functions of the system density matrix and quantum expectation values. These correlation functions code for a rich structure of fluctuating quantum correlations and coherences. Although our construction does not rely on the standard techniques from the theory of integrable systems, it is based on a remarkable interplay between the permutation groups and polynomials. We incidentally point out a possible combinatorial interpretation of the Q-SSEP correlation functions via a surprising connexion with geometric combinatorics and the associahedron polytopes.

How to win friends and influence functionals: deducing stochasticity from deterministic dynamics

The longstanding question of how stochastic behaviour arises from deterministic Hamiltonian dynamics is of great importance, and any truly holistic theory must be capable of describing this transition. In this review, we introduce the influence functional formalism in both the quantum and classical regimes. Using this technique, we demonstrate how irreversible behaviour arises generically from the reduced microscopic dynamics of a system-environment amalgam. The influence functional is then used to rigorously derive stochastic equations of motion from a microscopic Hamiltonian. In this method stochastic terms are not identified heuristically, but instead arise from an exact mapping only available in the path-integral formalism. The interpretability of the individual stochastic trajectories arising from the mapping is also discussed. As a consequence of these results, we are also able to show that the proper classical limit of stochastic quantum dynamics corresponds non-trivially to a generalised Langevin equation derived with the classical influence functional. This provides a further unifying link between open quantum systems and their classical equivalent, highlighting the utility of influence functionals and their potential as a tool in both fundamental and applied research.

Quantum jump Monte Carlo approach simplified: Abelian symmetries

We consider Markovian dynamics of a finitely dimensional open quantum system featuring a weak unitary symmetry, i.e., when the action of a unitary symmetry on the space of density matrices commutes with the master operator governing the dynamics. We show how to encode the weak symmetry in quantum stochastic dynamics of the system by constructing a weakly symmetric representation of the master operator: a symmetric Hamiltonian, and jump operators connecting only the symmetry eigenspaces with a fixed eigenvalue ratio. In turn, this representation simplifies both the construction of the master operator as well as quantum jump Monte Carlo simulations, where, for a symmetric initial state, stochastic trajectories of the system state are supported within a single symmetry eigenspace at a time, which is changed only by the action of an asymmetric jump operator. Our results generalize directly to the case of multiple Abelian weak symmetries.

Stochastic dissipative quantum spin chains (I) : Quantum fluctuating discrete hydrodynamics

Motivated by the search for a quantum analogue of the macroscopic fluctuation theory, we study quantum spin chains dissipatively coupled to quantum noise. The dynamical processes are encoded in quantum stochastic differential equations. They induce dissipative friction on the spin chain currents. We show that, as the friction becomes stronger, the noise induced dissipative effects localize the spin chain states on a slow mode manifold, and we determine the effective stochastic quantum dynamics of these slow modes. We illustrate this approach by studying the quantum stochastic Heisenberg spin chain.

Quantum Noise from Reduced Dynamics

We consider the description of quantum noise within the framework of the standard Copenhagen interpretation of quantum mechanics applied to a composite system environment setting. Averaging over the environmental degrees of freedom leads to a stochastic quantum dynamics, described by equations complying with the constraints arising from the statistical structure of quantum mechanics. Simple examples are considered in the framework of open system dynamics described within a master equation approach, pointing in particular to the appearance of the phenomenon of decoherence and to the relevance of quantum correlation functions of the environment in the determination of the action of quantum noise.

Quantum stochastic dynamics in multi-photon optics

Multi-photon models are theoretically and experimentally important because in them quantum properly phenomena are verified; as well as squeezed light and quantum entanglement also play a relevant role in quantum information and quantum communication (see Refs. 18–20).In this paper we study a generic model of a multi-photon system with an arbitrary number of pumping and subharmonics fields. This model includes measurement on the system, as could be direct or homodyne detection and we demonstrate the existence of dynamics in the context of Continuous Measurement Theory of Open Quantum Systems (see Refs. 1–11) using Quantum Stochastic Differential Equations with unbounded coefficients (see Refs. 10–15).

Quantum Stochastic Dynamics in the Presence of a Time-Periodic Rapidly Oscillating Potential: Nonadiabatic Escape Rate

Escape from a metastable state in the presence of a high-frequency field (where the driving becomes nonadiabatic) underlies a broad range of phenomena of physics and chemistry, and thus its understanding is of paramount importance. We study the problem of intermediate-to-high-damping escape from a metastable state of a dissipative system driven by a rapidly oscillating field, one of the most important classes of nonequilibrium systems, in a broad range of field driving frequencies (ω) and amplitudes (a). We construct a Langevin equation using quantum gauge transformation in the light of Floquet theorem and exploiting a systematic perturbative expansion in powers of 1/ω using "Kapitza-Landau time window". The quantum dynamics in a high-frequency field are found to be described by an effective time-independent potential. The temperature dependence of escape rate and the change of its form with varying parameters of the field have been analyzed. It may decrease upon increasing the temperature which is contingent on the effects of intricate interplay between external modulation and dissipation. The crossover temperature between tunnelling and thermal hopping increases with an increase in external modulation so that quantum effects in the escape are relevant at higher temperatures. These observations are uncommon and counterintuitive and, therefore, of considerable interest. Our results might be valuable for the exploration of the dynamics of cold atoms in electromagnetic fields.

Stabilization of Stochastic Quantum Dynamics via Open- and Closed-Loop Control

In this paper we investigate parametrization-free solutions of the problem of quantum pure state preparation and subspace stabilization by means of Hamiltonian control, continuous measurement and quantum feedback, in the presence of a Markovian environment. In particular, we show that whenever suitable dissipative effects are induced either by the unmonitored environment or by non Hermitian measurements, there is no need for feedback control to accomplish the task. Constructive necessary and sufficient conditions on the form of the open-loop controller can be provided in this case. When open-loop control is not sufficient, filtering-based feedback control laws steering the evolution towards a target pure state are provided, which generalize those available in the literature.

Quantum Lévy–Itô algebras and non-commutative stochastic analysis

We review the basic concepts of quantum probability and revise the classical and quantum stochastic (QS) calculus using the universal Itô B*-algebra approach. A non-commutative generalization of Lévy process is defined in Fock–Guichardet space in terms of the underlying B*-Itô modular algebra. The main notions and results of classical and QS analysis are reformulated in terms of the non-commutative but associative stochastic covariation of quantum integrals defined as sesquilinear forms adapted to a quantum Lévy process in Fock space. The general semi-Markov QS dynamics is defined in terms of a hemigroup of semi-morphic transformations as adjoints to a semiunitary hemigroup resolving a QS Schrödinger equation in Fock space. The corresponding semi-Markov hemigroup of completely positive contractions resolving a quantum master equation with generalized Lindblad generator as a non-commutative analogue of Feller–Kolmogorov equation is derived by taking vacuum conditional expectation in Fock space.

Quantum stochasticity and neuronal computations

AbstractThe nervous system probably cannot display macroscopic quantum (i.e. classically impossible) behaviours such as quantum entanglement, superposition or tunnelling (Koch and Hepp, Nature 440:611, 2006). However, in contrast to this quantum ‘mysticism’ there is an alternative way in which quantum events might influence the brain activity. The nervous system is a nonlinear system with many feedback loops at every level of its structural hierarchy. A conventional wisdom is that in macroscopic objects the quantum fluctuations are self-averaging and thus not important. Nevertheless this intuition might be misleading in the case of nonlinear complex systems. Because of a high sensitivity to initial conditions, in chaotic systems the microscopic fluctuations may be amplified upward and thereby affect the system's output. In this way stochastic quantum dynamics might sometimes alter the outcome of neuronal computations, not by generating classically impossible solutions, but by influencing the selection of many possible solutions (Satinover, Quantum Brain, Wiley & Sons, 2001). I am going to discuss recent theoretical proposals and experimental findings in quantum mechanics, complexity theory and computational neuroscience suggesting that biological evolution is able to take advantage of quantum-computational speed-up. I predict that the future research on quantum complex systems will provide us with novel interesting insights that might be relevant also for neurobiology and neurophilosophy.

Dynamical programming of continuously observed quantum systems

We develop dynamical programming methods for the purpose of optimal control of quantum states with convex constraints and concave cost and bequest functions of the quantum state. We consider both open loop and feedback control schemes, which correspond, respectively, to deterministic and stochastic master equation dynamics. For the quantum feedback control scheme with continuous nondemolition observations, we exploit the separation theorem of filtering and control aspects for quantum stochastic dynamics to derive a generalized Hamilton-Jacobi-Bellman equation. If the control is restricted to only Hamiltonian terms this is equivalent to a Hamilton-Jacobi equation with an extra linear dissipative term. In this work, we consider, in particular, the case when control is restricted only to observation. A controlled qubit is considered as an example throughout the development of the formalism. Finally, we discuss optimum observation strategies to obtain a pure state from a mixed state of a quantum two-level system.

The quantum rate of escape from a metastable state non-linearly coupled to a heat bathdriven by external colored noise**This paper is dedicated to Professor S P Bhattacharyya on the occasion of his sixtiethbirthday.

We have studied the quantum stochastic dynamics of a system whose interaction with areservoir is non-linear in system coordinates. In addition, the bath particles are driven byexternal noise with finite correlation. The effects of the space dependent frictionand correlation of the external noise on the rate of decay of a particle from ametastable well have been examined. At a critical value of the correlation, thevariation of the quantum decay rate versus the external noise strength exhibits aresonance.