non-Markovian Quantum Trajectories Research Articles

Higher-order solutions to non-Markovian quantum dynamics via a hierarchical functional derivative

Solving realistic quantum systems coupled to an environment is a challenging task. Here we develop a hierarchical functional derivative (HFD) approach for efficiently solving the non-Markovian quantum trajectories of an open quantum system embedded in a bosonic bath. An explicit expression for arbitrary order HFD equation is derived systematically. Moreover, it is found that for an analytically solvable model, this hierarchical equation naturally terminates at a given order and thus becomes exactly solvable. This HFD approach provides a systematic method to study the non-Markovian quantum dynamics of an open system coupled to a bosonic environment.

Exact non-Markovian master equations for multiple qubit systems: Quantum-trajectory approach

A wide class of exact master equations for a multiple qubit system can be explicitly constructed by using the corresponding exact non-Markovian quantum state diffusion equations. These exact master equations arise naturally from the quantum decoherence dynamics of qubit system as a quantum memory coupled to a collective colored noisy source. The exact master equations are also important in optimal quantum control, quantum dissipation and quantum thermodynamics. In this paper, we show that the exact non-Markovian master equation for a dissipative N-qubit system can be derived explicitly from the statistical average of the corresponding non-Markovian quantum trajectories. We illustrated our general formulation by an explicit construction of a three-qubit system coupled to a non-Markovian bosonic environment. This multiple qubit master equation offers an accurate time evolution of quantum systems in various domains, and paves a way to investigate the memory effect of an open system in a non-Markovian regime without any approximation.

Approach to solving spin-boson dynamics via non-Markovian quantum trajectories

We develop a systematic and efficient approach for numerically solving the non-Markovian quantum state diffusion equations for open quantum systems coupled to an environment up to arbitrary orders of noises or coupling strengths. As an important application, we consider a real-time simulation of a spin-boson model in a strong coupling regime that is difficult to deal with using conventional methods. We show that the non-Markovian stochastic Schr\"{o}dinger equation can be efficiently implemented as a real--time simulation for this model, so as to give an accurate description of spin-boson dynamics beyond the rotating-wave approximation.

Quantum trajectories: Memory and continuous observation

Starting from a generalization of the quantum trajectory theory (based on the stochastic Schr\"odinger equation - SSE), non-Markovian models of quantum dynamics are derived. In order to describe non-Markovian effects, the approach used in this article is based on the introduction of random coefficients in the usual linear SSE. A major interest is that this allows to develop a consistent theory of quantum measurement in continuous time for these non-Markovian quantum trajectory models. In this context, the notions of instrument, a priori and a posteriori states are rigorously described. The key point is that by starting from a stochastic equation on the Hilbert space of the system, we are able to respect the complete positivity of the mean dynamics for the statistical operator and the requirements of the axioms of quantum measurement theory. The flexibility of the theory is next illustrated by a concrete physical model of a noisy oscillator where non Markovian effects come from random environment, coloured noises, randomness in the stimulating light, delay effects. The statistics of the emitted photons and the heterodyne and homodyne spectra are studied and we show how these quantities are sensible to the non-Markovian features of the system dynamics, so that, in principle, the observation and analysis of the fluorescence light could reveal the presence of non-Markovian effects and allow for a measure of the spectra of the noises affecting the system dynamics.

Non-Markovian quantum trajectories, instruments and time-continuous measurements

The linear and nonlinear non-Markovian quantum state diffusion (NMQSD) equations are well-known tools for the description of certain non-Markovian open quantum systems. In this work, we systematically investigate whether the normalized linear NMQSD or the nonlinear NMQSD solutions can be generated by means of a time-continuous measurement. By considering any conceivable measurement scheme, we derive a necessary criterion for the measurement interpretation of both equations. Concrete examples show that the normalized linear NMQSD solutions are realizable only in the Markovian limit in general. The application of the presented criterion to the nonlinear NMQSD remains an open issue. Furthermore, this criterion is a necessary one not only in the context of the NMQSDs but also for a much broader class of stochastic Schrödinger equations.

Non-Markovian Continuous Quantum Measurement of Retarded Observables

We reconsider the non-Markovian time-continuous measurement of a Heisenberg observable x[over ] and show for the first time that it can be realized by an infinite set of entangled von Neumann detectors. The concept of continuous readout is introduced and used to rederive the non-Markovian stochastic Schrödinger equation. We can prove that, contrary to recent doubts, the resulting non-Markovian quantum trajectories are true single system trajectories and correspond to the continuous measurement of a retarded functional of x[over ].

Quantum trajectories for realistic detection

Quantum trajectories describe the stochastic evolution of an open quantum system conditioned on continuous monitoring of its output, such as by an ideal photodetector. Here we derive (non-Markovian) quantum trajectories for realistic photodetection, including the effects of efficiency, dead time, bandwidth, electronic noise, and dark counts. We apply our theory to a realistic cavity QED scenario and investigate the impact of such detector imperfections on the conditional evolution of the system state. A practical theory of quantum trajectories with realistic detection will be essential for experimental and technological applications of quantum feedback in many areas.

Non-Markovian homodyne-mediated feedback on a two-level atom: a quantum trajectory treatment

Quantum feedback can stabilize a two-level atom against decoherence (spontaneous emission), putting it into an arbitrary (specified) pure state. This requires perfect homodyne detection of the atomic emission, and instantaneous feedback. Inefficient detection was considered previously by two of us. Here we allow for a non-zero delay time τ in the feedback circuit. Because a two-level atom is a non-linear optical system, an analytical solution is not possible. However, quantum trajectories allow a simple numerical simulation of the resulting non-Markovian process. We find the effect of the time delay to be qualitatively similar to that of inefficient detection. The solution of the non-Markovian quantum trajectory will not remain fixed, so that the time-averaged state will be mixed, not pure. In the case where one tries to stabilize the atom in the excited state, an approximate analytical solution to the quantum trajectory is possible. The result, that the purity ( P=2Tr[ ρ 2]−1) of the average state is given by P=1−4 γτ (where γ is the spontaneous emission rate) is found to agree very well with the numerical results.

Continuous measurement and non-Markovian quantum trajectories

Based on a microscopic model of an open system consisting of a boson field coupled linearly to a system, we formulate quantum trajectories as continuous measurements of the output field from the system. We present a formulation for the case when the field quantity that is being monitored effectively commutes with itself at different times. In general, this results in a non-Markovian evolution for the reduced system conditioned on the measurements. Under the finite-memory time approximation this evolution can be numerically simulated. The generalization of the techniques of quantum trajectories to the non-Markovian regime is expected to contribute to our understanding of open system problems beyond the Born-Markov approximation.

A non-Markovian quantum trajectory approach to radiation into structured continuum

We present a non-Markovian quantum trajectory method for treating atoms radiating into a reservoir with a non-flat density of states. The results of an example numerical simulation of the case where the free space modes of the reservoir are altered by the presence of a cavity are presented and compared with those of an extended system approach.

Open System Dynamics with Non-Markovian Quantum Trajectories

A non-Markovian stochastic Schroedinger equation for a quantum system coupled to an environment of harmonic oscillators is presented. Its solutions, when averaged over the noise, reproduce the standard reduced density operator without any approximation. We illustrate the power of this approach with several examples, including exponentially decaying bath correlations and extreme non-Markovian cases, where the `environment' consists of only a single oscillator. The latter case shows the decay and revival of a `Schroedinger cat' state. For strong coupling to a dissipative environment with memory, the asymptotic state can be reached in a finite time. Our description of open systems is compatible with different positions of the `Heisenberg cut' between system and environment.

Non-Markovian quantum trajectories for spectral detection

We present a formulation of non-Markovian quantum trajectories for open systems from a measurement theory perspective. In our treatment there are three distinct ways in which non-Markovian behavior can arise; a mode dependent coupling between bath (reservoir) and system, a dispersive bath, and by spectral detection of the output into the bath. In the first two cases the non-Markovian behavior is intrinsic to the interaction, in the third case the non-Markovian behavior arises from the method of detection. We focus in detail on the trajectories which simulate real-time spectral detection of the light emitted from a localized system. In this case, the non-Markovian behavior arises from the uncertainty in the time of emission of particles that are later detected. The results of computer simulations of the spectral detection of the spontaneous emission from a strongly driven two-level atom are presented.