Stochastic Schr Research Articles

Coherent qubit measurement in cavity-transmon quantum systems

A measurement of the time between quantum jumps implies the capability to measure the next jump. During the time between jumps the quantum system is not evolving in a closed or unitary manner. While the wave function maintains phase coherence it evolves according to a non-Hermitian effective Hamiltonian. So under null measurement the timing of the next quantum jump can change by very many orders of magnitude when compared to rates obtained by multiplying lifetimes with occupation probabilities obtained via unitary transformation. The theory developed in 1987 for atomic fluorescence is here extended to transitions in transmon qubits. These systems differ from atoms in that they are read out with a harmonic cavity whose resonance is determined by the state of the qubit. We extend our analysis of atomic fluorescence to this infinite level system by treating the cavity as a quantum system. We find that next photon statistics is highly nonexponential and when implemented will enable faster readout, such as on timescales shorter than the decay time of the cavity. Commonly used heterodyne measurements are applied on timescales longer than the cavity lifetime. The overlap between the next photon theory and the theory of heterodyne measurement which are described according to the stochastic Schr\"odinger equation is elucidated. In the limit of large dispersion the intrinsic error for next jump detection, at short time, tends to zero. Whereas for short-time dyne detection the error remains finite for all values of dispersion.

Quantum spin helices more stable than the ground state: Onset of helical protection

Topological magnetic structures are promising candidates for resilient information storage. An elementary example is spin helices in one-dimensional easy-plane quantum magnets. To quantify their stability, we numerically implement the stochastic Schr\"odinger equation and time-dependent perturbation theory for spin chains with fluctuating local magnetic fields. We find two classes of quantum spin helices that can reach and even exceed ground-state stability: spin-current-maximizing helices and, for fine-tuned boundary conditions, the recently discovered ``phantom helices.'' Beyond that, we show that the helicity itself (left or right rotating) is even more stable. We explain these findings by separated helical sectors and connect them to topological sectors in continuous spin systems. The resulting helical protection mechanism is a promising phenomenon for stabilizing helical quantum structures, e.g., in ultracold atoms and solid-state systems. We also identify a third type of phantom helix in the system.

Synchronization of persistent oscillations in spin systems with nonlocal dissipation

We explore the synchronization phenomenon in the quantum few-body system of spins with the nonlocal dissipation. Without the external driving, we find that the system can exhibit stable oscillatory behaviors in the long-time dynamics accompanied by the appearance of the purely imaginary eigenvalues of the Liouvillian. Moreover, the oscillations of the next-nearest-neighboring spins are completely synchronized, which is revealed by the quantum trajectory analysis within the stochastic Schr\"odinger equation. The possibility of the appearance of the long-time oscillations in an infinite-size lattice by means of cluster mean-field approximation is also discussed.

Dissipation-driven formation of entangled dark states in strongly coupled inhomogeneous many-qubit systems in solid-state nanocavities

We study quantum dynamics of many-qubit systems strongly coupled to a quantized electromagnetic cavity field in the presence of decoherence and dissipation for both quantum emitters and cavity photons, taking into account the varying coupling strength of different qubits to the cavity field and the spread of their transition frequencies. Compact analytic solutions for time-dependent quantum state amplitudes and observables are derived for a broad class of open quantum systems in Lindblad approximation with the use of the stochastic Schr\"odinger equation approach. We show that depending on the initial quantum state preparation, an ensemble of qubits can evolve into a rich variety of many-qubit entangled states with destructive or constructive interference between the qubits. In particular, when only a small fraction of qubits are initially excited, the dissipation in a cavity will inevitably drive the system into robust dark states that are completely decoupled from the cavity and live much longer than the decay time of the cavity field. We also determine the conditions under which coherent coupling to the quantized cavity field overcomes the dephasing caused by a spread of transition frequencies in multiqubit systems and leads to the formation of a decoupled dark state.

Open quantum dynamics for plant motions

Stochastic Schrödinger equations that govern the dynamics of open quantum systems are given by the equations for signal processing. In particular, the Brownian motion that drives the wave function of the system does not represent noise, but provides purely the arrival of new information. Thus the wave function is guided by the optimal signal detection about the conditions of the environments under noisy observations. This behaviour is similar to biological systems that detect environmental cues, process this information, and adapt to them optimally by minimising uncertainties about the conditions of their environments. It is postulated that information-processing capability is a fundamental law of nature, and hence that models describing open quantum systems can equally be applied to biological systems to model their dynamics. For illustration, simple stochastic models are considered to capture heliotropic and gravitropic motions of plants. The advantage of such dynamical models is that they allow for the quantification of information processed by the plants. By considering the consequence of information erasure, it is argued that biological systems can process environmental signals relatively close to the Landauer limit of computation, and that loss of information must lie at the heart of ageing in biological systems.

Invariant measures of stochastic Schrödinger delay lattice systems

In this paper, we investigate stochastic Schrödinger lattice systems with time delay, whose drift and diffusion coefficientsare locally Lipschitz continuous. Firstly, some uniform estimates of solutions areestablished which include higher-order moment estimates and uniform tail-ends estimates. Then the tightness of a family ofprobability distributions of solutions in $C([-\\rho,0];l^2)$ is proved by the Arzel${\\rm~\\grave{a}}$-Ascoli theorem and the technique of diadicdivision. Finally, the existence of invariant measures for the Markov semigroup of the system is provedby the Krylov-Bogolyubov method.

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.

Stochastic Schr\xf6dinger equation derivation of non-Markovian two-time correlation functions

We derive the evolution equations for two-time correlation functions of a generalized non-Markovian open quantum system based on a modified stochastic Schrödinger equation approach. We find that the two-time reduced propagator, an object that used to be characterized by two independent stochastic processes in the Hilbert space of the system, can be simplified and obtained by taking ensemble average over one single noise. This discovery can save the cost of computation, and accelerate the converging process when taking the average over noisy trajectories. As a result, our method can be widely applied to many open quantum models, especially large-scale systems and extend the quantum regression theory to the non-Markovian case. In the short-time simulations, it is observed a significant difference between Markovian and non-Markovian cases, which can be applied to realize the environmental spectrum detection and enhance the measurement sensitivity in varying open quantum systems.

Almost conservation laws for stochastic nonlinear Schr\xf6dinger equations

In this paper, we present a globalization argument for stochastic nonlinear dispersive PDEs with additive noises by adapting the I-method (= the method of almost conservation laws) to the stochastic setting. As a model example, we consider the defocusing stochastic cubic nonlinear Schrödinger equation (SNLS) on {mathbb {R}}^3 with additive stochastic forcing, white in time and correlated in space, such that the noise lies below the energy space. By combining the I-method with Ito’s lemma and a stopping time argument, we construct global-in-time dynamics for SNLS below the energy space.

Continuous quantum measurement for general Gaussian unravelings can exist

Quantum measurements and the associated state changes are properly described in the language of instruments. We investigate the properties of a time continuous family of instruments associated with the recently introduced family of general Gaussian non-Markovian stochastic Schr\"odinger equations. In this Letter we find that when the covariance matrix for the Gaussian noise satisfies a particular $\delta$-function constraint, the measurement interpretation is possible for a class of models with self-adjoint coupling operator. This class contains, for example the spin-boson and quantum Brownian motion models with colored bath correlation functions. Remarkably, due to quantum memory effects the reduced state, in general, does not have a closed form master equation while the unraveling has a time continuous measurement interpretation.

Unraveling the non-Hermitian skin effect in dissipative systems

The non-Hermitian skin effect, i.e. eigenstate condensation at the edges in lattices with open boundaries, is an exotic manifestation of non-Hermitian systems. In Bloch theory, an effective non-Hermitian Hamiltonian is generally used to describe dissipation, which however is not norm-preserving and neglects quantum jumps. Here it is shown that in a self-consistent description of the dissipative dynamics in a one-band lattice, based on the stochastic Schr\"odinger equation or Lindblad master equation with a collective jump operator, the skin effect and its dynamical features are washed out. Nevertheless, both short- and long-time relaxation dynamics provide a hidden signature of the skin effect found in the semiclassical limit. In particular, relaxation toward a maximally mixed state with the largest von Neumann entropy in a lattice with open boundaries is a manifestation of the semiclassical skin effect.

A numerical approach for the stochastic control of a two-level quantum system

The aim of this paper is to study the problem of controlling the stochastic evolution of a two-level quantum system in the presence of two randomly fluctuating electromagnetic fields, given by a Wiener process. The system is modeled by the stochastic Schr¨odinger equation dependent on time. We set up the quantum optimal control problem by choosing a cost functional type Bolza. By applying the Pontryagin Maximum Stochastic Principle to an extended Hamiltonian, we express the stochastic optimal controls in terms of the co-state of the system. To solve numerically the resulting stochastic differential equations we propose an iterative algorithm using the Euler-Maruyama method. Finally, we obtain the optimal trajectories on the Bloch sphere.

Variational Quantum Simulation of General Processes

Variational quantum algorithms have been proposed to solve static and dynamic problems of closed many-body quantum systems. Here we investigate variational quantum simulation of three general types of tasks-generalized time evolution with a non-Hermitian Hamiltonian, linear algebra problems, and open quantum system dynamics. The algorithm for generalized time evolution provides a unified framework for variational quantum simulation. In particular, we show its application in solving linear systems of equations and matrix-vector multiplications by converting these algebraic problems into generalized time evolution. Meanwhile, assuming a tensor product structure of the matrices, we also propose another variational approach for these two tasks by combining variational real and imaginary time evolution. Finally, we introduce variational quantum simulation for open system dynamics. We variationally implement the stochastic Schrödinger equation, which consists of dissipative evolution and stochastic jump processes. We numerically test the algorithm with a 6-qubit 2D transverse field Ising model under dissipation.

An open quantum system theory for polarizable continuum models.

The problem of a solute described by Quantum Chemistry within a solvent represented as a polarizable continuum model (PCM) is here reformulated in terms of the open quantum systems (OQS) theory. Using its stochastic Schrödinger equation formulation, we are able to provide a more comprehensive picture of the electronic energies and the coupling between solute and solvent electronic dynamics. In particular, the OQS-PCM proves to be a unifying theoretical framework naturally including polarization and dispersion interactions, the effect of solvent fluctuations, and the non-Markovian solvent response. As such, the OQS-PCM describes the interplay between the solute and the solvent typical electronic dynamical times and yields both the standard PCM and the so-called Born-Oppenheimer solvation regime, where the solvent electronic response is considered faster than any electronic dynamics taking place in the solute. In analyzing the OQS-PCM, we obtained an expression for the solute-solvent dispersion (van der Waals) interactions, which is very transparent in terms of a physical interpretation based on fluctuations and response functions. Finally, we present various numerical tests that support the theoretical findings.

Exponential integrators for stochastic Schrödinger equations

We present a class of exponential integrators to compute solutions of the stochastic Schrödinger equations arising from the modeling of open quantum systems. To be able to implement the methods within the same framework as the deterministic counterpart, we express the solution using Kunita's representation. With appropriate truncations, the solution operator can be written as matrix exponentials, which can be efficiently implemented by the Krylov subspace projection. The accuracy is examined in terms of the strong convergence by comparing trajectories, and in terms of the weak convergence by comparing the density-matrix operators. We show that the local accuracy can be further improved by introducing third-order commutators in the exponential. The effectiveness of the proposed methods is tested using the example from Di Ventra et al. [J. Phys.: Condens. Matter 16, 8025 (2004)JCOMEL0953-898410.1088/0953-8984/16/45/024].

On the stochastic nonlinear Schr\xf6dinger equations at critical regularities

We consider the Cauchy problem for the defocusing stochastic nonlinear Schrödinger equations (SNLS) with an additive noise in the mass-critical and energy-critical settings. By adapting the probabilistic perturbation argument employed in the context of the random data Cauchy theory by Bényi et al. (Trans Am Math Soc Ser B 2:1–50, 2015) to the current stochastic PDE setting, we present a concise argument to establish global well-posedness of the mass-critical and energy-critical SNLS.

Weak convergence rates for stochastic evolution equations and applications to nonlinear stochastic wave, HJMM, stochastic Schrödinger and linearized stochastic Korteweg–de Vries equations

We establish weak convergence rates for noise discretizations of a wide class of stochastic evolution equations with non-regularizing semigroups and additive or multiplicative noise. This class covers the nonlinear stochastic wave, HJMM, stochastic Schr\"odinger and linearized stochastic Korteweg-de Vries equation. For several important equations, including the stochastic wave equation, previous methods give only suboptimal rates, whereas our rates are essentially sharp.

A white noise approach to stochastic partial differential equations driven by the fractional L\xe9vy noise

In this paper, based on the white noise theory for d-parameter Lévy random fields given by (Holden et al. in Stochastic Partial Differential Equations: A modeling, white noise functional approach, 2010), we develop a white noise frame for anisotropic fractional Lévy random fields to solve the stochastic Poisson equation and the stochastic Schrödinger equation driven by the d-parameter fractional Lévy noise. The solutions for the two kinds of equations are all strong solutions given explicitly in the Lévy–Hida stochastic distribution space.

Martingale solutions for the stochastic nonlinear Schr\xf6dinger equation in the energy space

We consider a stochastic nonlinear Schrödinger equation with multiplicative noise in an abstract framework that covers subcritical focusing and defocusing Stochastic NLSE in H^1 on compact manifolds and bounded domains. We construct a martingale solution using a modified Faedo–Galerkin-method based on the Littlewood–Paley-decomposition. For the 2d manifolds with bounded geometry, we use the Strichartz estimates to show the pathwise uniqueness of solutions.

Stochastic unravelings of non-Markovian completely positive and trace-preserving maps

We consider open quantum systems with factorized initial states, providing the structure of the reduced system dynamics, in terms of environment cumulants. We show that such completely positive (CP) and trace preserving (TP) maps can be unraveled by linear stochastic Schr\"odinger equations (SSEs) characterized by sets of colored stochastic processes (with n-th order cumulants). We obtain both the conditions such that the SSEs provide CPTP dynamics, and those for unraveling an open system dynamics. We then focus on Gaussian non-Markovian unravellings, whose known structure displays a functional derivative. We provide a novel description that replaces the functional derivative with a recursive operatorial structure. Moreover, for the family of quadratic bosonic Hamiltonians, we are able to provide an explicit operatorial dependence for the unravelling.