Stochastic Schroedinger Equations Research Articles

Time Resolved Raman Scattering of Molecules: A Quantum Mechanics Approach with Stochastic Schroedinger Equation.

Raman scattering is a very powerful tool employed to characterize molecular systems. Here we propose a novel theoretical strategy to calculate the Raman cross-section in time domain, by computing the cumulative Raman signal emitted during the molecular evolution in time. Our model is based on a numerical propagation of the vibronic wave function under the effect of a light pulse of arbitrary shape. This approach can therefore tackle a variety of experimental setups. Both resonance and nonresonance Raman scattering can be retrieved, and also the time-dependent fluorescence emission is computed. The model has been applied to porphyrin considering both resonance and nonresonance conditions and varying the incident field duration. Moreover the effect of the vibrational relaxation, which should be taken into account when its time scale is similar to that of the Raman emission, has been included through the stochastic Schroedinger equation approach.

Application of stochastic calculus for certain classes of quantum models

In this work, stochastic calculus is applied for describing open quantum systems. The suggested approach is based on a modified stochastic Schroedinger equation. We apply the approach to investigate a qubit relaxation in Markovian and non-Markovian regimes.

Simulation of non-Markovian dynamics of dipole-dipole interacting atoms

We investigate non-markovian dynamics of two interacting two-level systems by numerical solution of the stochastic Schroedinger equation (SSE). It is shown that non- markovian noises have significant influence on dynamics of atomic levels populations.

Quantum Stochastic Equations for an Opto-Mechanical Oscillator with Radiation Pressure Interaction and Non-Markovian Effects

The quantum stochastic Schroedinger equation or Hudson-Parthasareathy (HP) equation is a powerful tool to construct unitary dilations of quantum dynamical semigroups and to develop the theory of measurements in continuous time via the construction of output fields. An important feature of such an equation is that it allows to treat not only absorption and emission of quanta, but also scattering processes, which however had very few applications in physical modelling. Moreover, recent developments have shown that also some non-Markovian dynamics can be generated by suitable choices of the state of the quantum noises involved in the HP-equation. This paper is devoted to an application involving these two features, non-Markovianity and scattering process. We consider a micro-mirror mounted on a vibrating structure and reflecting a laser beam, a process giving rise to a radiation-pressure force on the mirror. We show that this process needs the scattering part of the HP-equation to be described. On the other side, non-Markovianity is introduced by the dissipation due to the interaction with some thermal environment which we represent by a phonon field, with a nearly arbitrary excitation spectrum, and by the introduction of phase noise in the laser beam. Finally, we study the full power spectrum of the reflected light and we show how the laser beam can be used as a temperature probe.

Stochastic Schroedinger equation from optimal observable evolution

In this article, we consider a set of trial wave-functions denoted by | Q〉 and an associated set of operators A α which generate transformations connecting those trial states. Using variational principles, we show that we can always obtain a quantum Monte-Carlo method where the quantum evolution of a system is replaced by jumps between density matrices of the form D = | Q a 〉〈 Q b |, and where the average evolutions of the moments of A α up to a given order k, i.e., 〈 A α 1 〉 , 〈 A α 1 A α 2 〉 , … , 〈 A α 1 ⋯ A α k 〉 , are constrained to follow the exact Ehrenfest evolution at each time step along each stochastic trajectory. Then, a set of more and more elaborated stochastic approximations of a quantum problem is obtained which approach the exact solution when more and more constraints are imposed, i.e., when k increases. The Monte-Carlo process might even become exact if the Hamiltonian H applied on the trial state can be written as a polynomial of A α . The formalism makes a natural connection between quantum jumps in Hilbert space and phase-space dynamics. We show that the derivation of stochastic Schroedinger equations can be greatly simplified by taking advantage of the existence of this hierarchy of approximations and its connection to the Ehrenfest theorem. Several examples are illustrated: the free wave-packet expansion, the Kerr oscillator, a generalized version of the Kerr oscillator, as well as interacting bosons or fermions.

Stochastic mean-field dynamics for fermions in the weak-coupling limit

Assuming that the effect of the residual interaction beyond mean-field is weak and has a short memory time, two approximate treatments of correlation in fermionic systems by means of Markovian quantum jump are presented. A simplified scenario for the introduction of fluctuations beyond mean-field is first presented. In this theory, part of the quantum correlations between the residual interaction and the one-body density matrix are neglected and jumps occur between many-body densities formed of pairs of states $D=| \Phi_a > < \Phi_b |/< \Phi_b . |\Phi_a >$ where $| \Phi_a >$ and $| \Phi_b >$ are antisymmetrized products of single-particle states. The underlying Stochastic Mean-Field (SMF) theory is discussed and applied to the monopole vibration of a spherical $^{40}$Ca nucleus under the influence of a statistical ensemble of two-body contact interaction. This framework is however too simplistic to account for both fluctuation and dissipation. In the second part of this work, an alternative quantum jump method is obtained without making the approximation on quantum correlations. Restricting to two particles-two holes residual interaction, the evolution of the one-body density matrix of a correlated system is transformed into a Lindblad equation. The associated dissipative dynamics can be simulated by quantum jumps between densities written as $D = | \Phi >< \Phi |$ where $| \Phi >$ is a normalized Slater determinant. The associated stochastic Schroedinger equation for single-particle wave-functions is given.

Optimizing stochastic trajectories in exact quantum-jump approaches of interacting systems

The quantum-jump approach, where pairs of state vectors follow the stochastic Schroedinger equation (SSE) in order to treat the exact quantum dynamics of two interacting systems, is described. In this work the nonuniqueness of such stochastic Schroedinger equations is investigated to propose strategies to optimize the stochastic paths and reduce statistical fluctuations. In the proposed method, called the ``adaptative noise method,'' a specific SSE is obtained for which the noise depends explicitly on both the initial state and on the properties of the interaction Hamiltonian. It is also shown that this method can be further improved by introduction of a mean-field dynamics. The different optimization procedures are illustrated quantitatively in the case of interacting spins. A significant reduction of the statistical fluctuations is obtained. Consequently a much smaller number of trajectories is needed to accurately reproduce the exact dynamics as compared to the SSE without optimization.

States of quantum systems and their liftings

Let H1, H2 be complex Hilbert spaces, H be their Hilbert tensor product and let tr2 be the operator of taking partial trace, with respect to the space H2, of trace class operators in H. The operation tr2 maps states in H (=positive trace class operators in H with trace equal to 1) into states in H1. In this paper we give the full description of mappings that are linear right inverse to tr2. More precisely, we prove that any affine mapping F(W) of the convex set of states in H1 into the states in H that is right inverse to tr2 is given by W⟼W⊗D for some state D in H2. In addition we investigate a representation of the quantum mechanical state space by probability measures on the set of pure states and a representation—used in the theory of stochastic Schrödinger equations—by probability measures on the Hilbert space. We prove that there are no affine mappings from the state space of quantum mechanics into these spaces of probability measures.

Quantum Trajectories for Brownian Motion

We present the stochastic Schroedinger equation for the dynamics of a quantum particle coupled to a high temperature environment and apply it the dynamics of a driven, damped, nonlinear quantum oscillator. Apart from an initial slip on the environmental memory time scale, in the mean, our result recovers the solution of the known non-Lindblad quantum Brownian motion master equation. A remarkable feature of our approach is its localization property: individual quantum trajectories remain localized wave packets for all times, even for the classically chaotic system considered here, the localization being stronger the smaller $\hbar$.

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.