The recently proposed Langevin formulation of quantum dynamics yields the propagator at imaginary time as a noise average involving the solutions to a certain Langevin equation with a Gaussian white noise. The main advantage of this approach is to permit a stochastic treatment of quantum systems even when no solutions to the Schrödinger equation are known. We show how the Langevin formulation works in the space of a (nonrelativistic) particle in a uniform magnetic field with arbitary time dependence. We are able to evaluate the quantum-mechanical propagator exactly. As far as we know, this is a new result of considerable interest for atomic and molecular physics.
The recently proposed Langevin formulation of quantum dynamics yields the propagator at imaginary time as a noise average involving the solutions to a certain Langevin equation with a Gaussian white noise. The main advantage of this approach is to permit a stochastic treatment of quantum systems even when no solutions to the Schrödinger equation are known. We show how the Langevin formulation works in the space of a (nonrelativistic) particle in a uniform magnetic field with arbitary time dependence. We are able to evaluate the quantum-mechanical propagator exactly. As far as we know, this is a new result of considerable interest for atomic and molecular physics.
We first point out that nonrelativistic quantum mechanics formulated at imaginary time s = it can formally be recognized as the Fokker-Planck description of a classical Brownian motion, which occurs (in general) in a medium where particles are absorbed and emitted. We next offer a new formulation of quantum dynamics, which is basically the Langevin treatment of that Brownian motion. More explicitly, we derive a noise average representation of the transition probability W ( X", s"|X', s' ) in terms of the solutions to a Langevin equation with a Gaussian white noise. Upon analytic continuation back to real time W ( X", s"|X', s' ) becomes the propagator of the original Schrödinger equation, namely we have ⟨ X", t"|X', t' ⟩ = W ( X", it"|X', it' ). The main advantage of this approach is to permit a systematic application to quantum-dynamical problems of the mathematical techniques developed in the theory of classical stochastic processes, even when vector potentials and/or time-dependent potentials are present.
Recently, a novel quantization technique has been put forward, which yields the imaginary-time quantum-mechanical propagator as a noise average involving the solutions of a Langevin equation in configuration space with a Gaussian white noise. Contrary to previous attempts, this strategy does not require any knowledge about the ground-state quantum dynamics. Our aim is to show that this approach can be cast in a different form, which exhibits a manifest connection with classical mechanics (at imaginary time). The advantages of this strategy are pointed out and illustrated by taking the harmonic oscillator as an example.
Abstract We study the dynamics of a two-level quantum system interacting with a single frequency electromagnetic field and a stochastic magnetic field, with and without making the rotating wave approximation (RWA). The transformation to the rotating frame does not commute with the stochastic Hamiltonian if the stochastic field has nonvanishing components in the transverse direction, hence, applying the RWA requires transformation of the stochastic terms in the Hamiltonian. For Gaussian white noise, the master equation is derived from the stochastic Schrödinger–Langevin equations, with and without the RWA. With the RWA, the master equation for the density matrix has Lindblad terms with coefficients that are time-dependent (i.e., the master equation is time-local). An approximate analytic expression for the density matrix is obtained with the RWA. For Ornstein–Uhlenbeck noise, as well as other types of colored noise, in contradistinction to the Gaussian white noise case, the non-commutation of the RWA transformation and the noise Hamiltonian can significantly affect the RWA dynamics when , where ω is the electromagnetic field frequency and is the stochastic magnetic field correlation time.
The dynamics of a Brownian particle in a constant magnetic field and time‐dependent electric field is studied in the limit of white noise, using a Langevin approach for the classical problem and the path‐integral Feynman‐Vernon and Caldeira‐Leggett framework for the quantum problem. A first goal of this study is to use the two‐dimensional problem of a Brownian particle in a magnetic field to show that a proper re‐formulation of the oscillator model of quantum Brownian motion allows one to recover the classical limit of the dynamics correctly. Furthermore, the probability distribution in configuration space of an initial pure state represented by an asymmetrical Gaussian wave function is worked out and its general time evolution is decomposed in the superposition of basic processes: (a) the classical motion of the center of mass, (b) a rotation around the mean position, and (c) spreading processes along the principal axes.
We study the stochastic dynamics of a two-level quantum system interacting with a stochastic magnetic field, and a single frequency electromagnetic field, with and without making the rotating wave approximation (RWA). The transformation to the rotating frame does not commute with the stochastic Hamiltonian if the stochastic field has nonvanishing components in the transverse direction. Hence, making the RWA modifies the stochastic terms in the Hamiltonian. Modification of the decay terms is also required in a master equation approach (i.e., the Liouville--von Neumann density matrix equation) for describing the dynamics. For isotropic Gaussian white noise, the RWA dynamics remains Markovian, although the Lindblad terms in the master equation for the density matrix become time-dependent when the non-commutation of the RWA transformation and the noise Hamiltonian is properly accounted for. We also treat Ornstein--Uhlenbeck noise, and find, in contra-distinction to the white noise case, a significant difference in the dynamics calculated with the RWA when the non-commutation of the RWA transformation and the noise Hamiltonian is taken into account. These findings are applicable to the modeling of any open quantum system coupled to an electromagentic field.
The stochastic representation of quantum mechanics is used to give a purely stochastic treatment of the quantum Langevin equation. A general coupling involving positions and velocities of the particle and harmonic modes, with arbitrary dependence on the particle's position, is dealt with. Summing over the modes in the path integral and using stochastic integrals, the author obtains explicitly the reduced measure, that is, the stochastic process representing the time evolution of the small system in the presence of noise. Contact is made with the operatorial representation, and a relation is established between the correlations of the noise in both representations. A general condition on the coupling constants is given for recovering time locality in the continuum limit, and the effects of renormalisation are re-analysed within this purely stochastic approach.