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Physical Review Research

Evolution of momentum-dependent observables under stochastic phase noise in Rabi oscillations

Samuel Böhringer, Fabian Kienle, Richard Lopp

Published Apr 14, 2025• 3 citations
Publisher

Abstract

We derive the equations of motion of arbitrary momentum-dependent observables for a fully first-quantized, driven two-level atom in a linear potential under the presence of a stochastic laser phase. In particular, we show that such momentum-dependent observables evolve under stochastic differential equations (SDEs) in the form of geometrical Brownian motion and find that these SDEs reduce to ordinary differential equations (ODEs) for the expectation values under white laser phase noise. As an example, we solve these ODEs numerically and determine the effect of laser phase noise on Rabi oscillations including quantized center-of-mass degrees of freedom of the atom. Moreover, the SDE formalism we derive holds for any phase noise source and can also be applied to extended noise models such as color noise.

DOI
10.1103/physrevresearch.7.023048

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We derive the equations of motion of arbitrary momentum-dependent observables for a fully first-quantized, driven two-level atom in a linear potential under the presence of a stochastic laser phase. In particular, we show that such momentum-dependent observables evolve under stochastic differential equations (SDEs) in the form of geometrical Brownian motion and find that these SDEs reduce to ordinary differential equations (ODEs) for the expectation values under white laser phase noise. As an example, we solve these ODEs numerically and determine the effect of laser phase noise on Rabi oscillations including quantized center-of-mass degrees of freedom of the atom. Moreover, the SDE formalism we derive holds for any phase noise source and can also be applied to extended noise models such as color noise.
We show how a laser driven two-level system including quantized external degrees of freedom for each state can be decoupled into a set of oscillator equations acting only on the external degrees of freedom with operator valued damping representing the detuning. We give a way of characterizing the solvability of this family of problems by appealing to a classical oscillator with time-dependent damping. As a consequence of this classification we (a) obtain analytic and representation-free expressions for Rabi oscillations including external degrees of freedom with and without an external linear potential, (b) show that whenever the detuning operator can be diagonalized (analytically or numerically) the problem decomposes into a set of classical equations and (c) we can use the oscillator equations as a perturbative basis to describe Rabi oscillations in weak but otherwise arbitrary external potentials. Moreover, chirping of the driving fields phase emerges naturally as a means of compensating the Ehrenfest/mean-value part of the detuning operator's dynamics while the presence of driving phase noise leads to a stochastic evolution equation of Langevin type. Lastly, our approach is representation free with respect to the external degrees of freedom and as consequence a suitable representation or basis expansion can be chosen a posteriori depending on the desired application at hand.
We show how a laser driven two-level system including quantized external degrees of freedom for each state can be decoupled into a set of oscillator equations acting only on the external degrees of freedom with operator valued damping representing the detuning. We give a way of characterizing the solvability of this family of problems by appealing to a classical oscillator with time-dependent damping. As a consequence of this classification we (i) obtain analytic and representation-free expressions for Rabi oscillations including external degrees of freedom with and without an external linear potential, (ii) show that whenever the detuning operator can be diagonalized (analytically or numerically) the problem decomposes into a set of classical equations, and (iii) we can use the oscillator equations as a perturbative basis to describe Rabi oscillations in weak but otherwise arbitrary external potentials. Moreover, chirping of the driving fields phase emerges naturally as a means of compensating the Ehrenfest/mean-value part of the detuning operator's dynamics while in the presence of driving phase noise leads to a stochastic evolution equation of Langevin type. Lastly, our approach is representation free with respect to the external degrees of freedom and as consequence a suitable representation or basis expansion can be chosen a posteriori depending on the desired application at hand. Published by the American Physical Society 2024
We propose a stochastic Schrödinger equation in which the momentum is coupled to a white Gaussian noise. In the stochastic representation, the kinetic energy representing the self-interaction of momentum is reduced to a linear term of momentum. As such, the quantum evolution operator factorizes into two contributions due to the momentum and the potential, respectively. The exact quantum propagator thereby becomes an expectation of the stochastic one in which the amplitude results from the potential with a fluctuating position: The particle moves with a constant velocity, subjected to a complex Brownian motion. We demonstrate that the stochastic Schrödinger equation can be feasibly used to derive the quantum propagators for the linear potential and the harmonic oscillator system. Novel semiclassical and other approximations may be developed from the new representation of quantum mechanics.
We study the effect of noise on the quantum nonlinear motion of a two-level atom in a laser standing wave. This system has recently been shown to exhibit dynamic localization in the atomic momentum. Noise is introduced through the phase of the standing wave, and has one of two forms: Weiner noise or Ornstein - Uhlenbeck noise. Using the theory of stochastic Hamiltonians we derive a master equation to describe the quantum mechanical evolution of the momentum statistics. We show that the Weiner noise process is then equivalent to a continuous measurement of the momentum of the atom, and the Ornstein - Uhlenbeck process describes a measurement of the position of the atom up to the period of the optical potential. We study the effect of noise when the classical dynamics shows global chaos.
We show that well above threshold the use of linearized Ito quantum stochastic differential equations allows a systematic description of all quantum stochastic properties of laser light including the phase properties. This is accomplished by the use of mixed field-atom operators which are phase independent and around whose finite mean values we may linearize. As a consequence we are able to derive an equation for an operator describing the phase properties of the laser light. We illustrate our method by applying it to three different models for sub-shot-noise lasers.
We discuss strong laser-atom interactions that are subjected to jump-type (random telegraph) random-phase noise. Physically, the jumps may arise from laser fluctuations, from collisions of various kinds, or from other external forces. Our discussion is carried out in two stages. First, direct and partially heuristic calculations determine the laser spectrum and also give a third-order differential equation for the average inversion of a two-level atom on resonance. At this stage a number of general features of the interaction are able to be studied easily. The optical analog of motional narrowing, for example, is clearly predicted. Second, we show that the theory of generalized Poisson processes allows laser-atom interactions in the presence of random telegraph noise of all kinds (not only phase noise) to be treated systematically, by means of a master equation first used in the context of quantum optics by Burshtein. We use the Burshtein equation to obtain an exact expression for the two-level atom's steady-state resonance fluorescence spectrum, when the exciting laser exhibits phase telegraph noise. Some comparisons are made with results obtained from other noise models. Detailed treatments of the effects of other than phase noise are given in other papers.