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 (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.

arXiv:2406.01511Jun 3, 20240 citations

Langevin Theory of Non-Markovian Quantum Dynamics: Application to Delayed Coherent Feedback and the Laser Linewidth

Marc Cuenca-Laràs, Ming Li, Carlos Navarrete-Benlloch +1 more

Phase-space methods are powerful tools for the treatment of Markovian open quantum systems: they map the reduced dynamics of a system S, in interaction with an environment E, exactly onto Langevin equations for c-number stochastic variables, as opposed to Heisenberg-Langevin equations for operators. Langevin equations provide analytical insight in key regimes and excel at handling strong nonlinearities and couplings, where other methods often falter. Extending phase-space methods to non-Markovian dynamics, however, has remained a long-standing challenge. Here we address this gap by applying phase-space representations to the full S+E system; integrating out the environmental degrees of freedom then yields a general Langevin framework for S that incorporates both deterministic and stochastic contributions from E. Normally ordered representations, such as the Glauber-Sudarshan P representation and its positive variant due to Drummond and Gardiner, lead to Langevin equations in which (i) non-Markovian effects emerge exclusively in the deterministic terms, via a memory kernel, and (ii) noise contributions vanish when E is initially in the vacuum state. To demonstrate the power of this framework, we address the paradigmatic problem of delayed coherent feedback, in which the system is driven by its own past state, and study its impact on the laser linewidth: we recover the narrowing observed well above threshold and predict an enhanced narrowing just above it. Crucially, the number of stochastic variables scales linearly with the system size, making the framework suitable for problems ranging from a few degrees of freedom to genuinely many-body systems. This opens the way to the systematic study of non-Markovian driven-dissipative quantum systems using the same analytical and numerical tools that have long made phase-space methods so successful in the Markovian regime.Phase-space methods are powerful tools for the treatment of Markovian open quantum systems: they map the reduced dynamics of a system S, in interaction with an environment E, exactly onto Langevin equations for c-number stochastic variables, as opposed to Heisenberg-Langevin equations for operators. Langevin equations provide analytical insight in key regimes and excel at handling strong nonlinearities and couplings, where other methods often falter. Extending phase-space methods to non-Markovian dynamics, however, has remained a long-standing challenge. Here we address this gap by applying phase-space representations to the full S+E system; integrating out the environmental degrees of freedom then yields a general Langevin framework for S that incorporates both deterministic and stochastic contributions from E. Normally ordered representations, such as the Glauber-Sudarshan P representation and its positive variant due to Drummond and Gardiner, lead to Langevin equations in which (i) non-Markovian effects emerge exclusively in the deterministic terms, via a memory kernel, and (ii) noise contributions vanish when E is initially in the vacuum state. To demonstrate the power of this framework, we address the paradigmatic problem of delayed coherent feedback, in which the system is driven by its own past state, and study its impact on the laser linewidth: we recover the narrowing observed well above threshold and predict an enhanced narrowing just above it. Crucially, the number of stochastic variables scales linearly with the system size, making the framework suitable for problems ranging from a few degrees of freedom to genuinely many-body systems. This opens the way to the systematic study of non-Markovian driven-dissipative quantum systems using the same analytical and numerical tools that have long made phase-space methods so successful in the Markovian regime.

quant-pharXiv:2608.28506Aug 31, 20260 citations

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

Samuel Böhringer, Fabian Kienle, Richard Lopp

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 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.

Physical Review ResearchApr 14, 20253 citations

Quantum Dynamics of Interacting oscillators in a thermal medium: A novel scheme

Ronak Moradi, Fardin Kheirandish

We develop an exact analytical framework for studying the nonequilibrium dynamics of interacting quantum harmonic oscillators coupled to a modelled thermal reservoir and driven by external classical fields. The total Hamiltonian is diagonalized via successive Bogoliubov transformations, enabling a nonperturbative treatment of dissipation and driving. For two coupled oscillators with an external source applied to the first, we derive explicit energy expressions and analyze their dependence on coupling and bath parameters. The phase-space structure is characterized through Husimi $Q$ -functions for initial separable coherent and number states. We obtain the reduced density matrix elements in the number-state basis and demonstrate that, in the absence of driving and at zero temperature, the reduced density matrix of the main system satisfies the Lindblad master equation. A generating function for the Husimi function is introduced to facilitate computation of higher-order correlations. The analysis is then generalized to $n$ interacting oscillators, with analytical energy expressions provided and the case $n=3$ examined in detail. Our results establish a versatile and exact framework for driven-dissipative quantum systems, with potential applications in quantum thermodynamics, open quantum systems, and many-body physics.We develop an exact analytical framework for studying the nonequilibrium dynamics of interacting quantum harmonic oscillators coupled to a modelled thermal reservoir and driven by external classical fields. The total Hamiltonian is diagonalized via successive Bogoliubov transformations, enabling a nonperturbative treatment of dissipation and driving. For two coupled oscillators with an external source applied to the first, we derive explicit energy expressions and analyze their dependence on coupling and bath parameters. The phase-space structure is characterized through Husimi $Q$-functions for initial separable coherent and number states. We obtain the reduced density matrix elements in the number-state basis and demonstrate that, in the absence of driving and at zero temperature, the reduced density matrix of the main system satisfies the Lindblad master equation. A generating function for the Husimi function is introduced to facilitate computation of higher-order correlations. The analysis is then generalized to $n$ interacting oscillators, with analytical energy expressions provided and the case $n=3$ examined in detail. Our results establish a versatile and exact framework for driven-dissipative quantum systems, with potential applications in quantum thermodynamics, open quantum systems, and many-body physics.

quant-pharXiv:2605.24478May 26, 20260 citations

Perturbative dissipation dynamics of a weakly driven Jaynes-Cummings system

S. M. Yu, Gao, Y. B., Hou Ian

We generalize a microscopic master equation method to study the dissipation dynamics of Jaynes-Cummings two-level system with a weak external driving. Using perturbative analysis to extend the damping bases theory, we derive the corrected Rabi oscillation and vaccum Rabi splitting analytically. The evolution of the decoherence factor of the weakly driven system reveals that the off-diagonal density matrix elements are oscillating at a frequency dependent on the driving strength and the initial population inversion. For highly-inverted systems at the weak-driving limit, this frequency reduces to twice the value for the non-driven system, showing the dissipation dynamics unable to be discovered using more conventional approaches.We generalize a microscopic master equation method to study the dissipation dynamics of Jaynes-Cummings two-level system with a weak external driving. Using perturbative analysis to extend the damping bases theory, we derive the corrected Rabi oscillation and vaccum Rabi splitting analytically. The evolution of the decoherence factor of the weakly driven system reveals that the off-diagonal density matrix elements are oscillating at a frequency dependent on the driving strength and the initial population inversion. For highly-inverted systems at the weak-driving limit, this frequency reduces to twice the value for the non-driven system, showing the dissipation dynamics unable to be discovered using more conventional approaches.

arXiv:1612.01248Dec 5, 20160 citations

Theory of damped Rabi oscillations at finite temperatures

Shigemasa Matsuo, Toshiyuki Fujii, Norihito Kosugi +1 more

Rabi oscillations in the presence of dissipation are theoretically studied at finite temperatures. The system that we consider is a quantum two-level system with both internal relaxations and external decays under an alternating field. The environment modeled by a set consisting of an infinite number of harmonic oscillators is employed in order to introduce temperature into the system. The equations of motion for the density matrix (Liouville equation) of such a quantum two-level system coupled to a Caldeira-Leggett-type environment are obtained with the Born-Markov approximation. Internal relaxations of the system are incorporated into the Liouville equations using the Redfield theory, resulting in Liouville equations with temperature-dependent relaxation rates. The exact solutions are obtained by using the Torrey method based on the Laplace transform. The solution for damped Rabi oscillations can be divided into elementary relaxation processes such as a simple decay and a damped oscillation. Since the coefficients of the solution involve physical relaxation rates corresponding to the processes, all the relaxation rates such as the dephasing rate are determined by simple calculations of the coefficients in these different relaxation processes.Rabi oscillations in the presence of dissipation are theoretically studied at finite temperatures. The system that we consider is a quantum two-level system with both internal relaxations and external decays under an alternating field. The environment modeled by a set consisting of an infinite number of harmonic oscillators is employed in order to introduce temperature into the system. The equations of motion for the density matrix (Liouville equation) of such a quantum two-level system coupled to a Caldeira-Leggett-type environment are obtained with the Born-Markov approximation. Internal relaxations of the system are incorporated into the Liouville equations using the Redfield theory, resulting in Liouville equations with temperature-dependent relaxation rates. The exact solutions are obtained by using the Torrey method based on the Laplace transform. The solution for damped Rabi oscillations can be divided into elementary relaxation processes such as a simple decay and a damped oscillation. Since the coefficients of the solution involve physical relaxation rates corresponding to the processes, all the relaxation rates such as the dephasing rate are determined by simple calculations of the coefficients in these different relaxation processes.

Journal of Physics Conference SeriesFeb 1, 20092 citations

Q

-functions for initial separable coherent and number states. We obtain the reduced density matrix elements in the number-state basis and demonstrate that, in the absence of driving and at zero temperature, the reduced density matrix of the main system satisfies the Lindblad master equation. A generating function for the Husimi function is introduced to facilitate computation of higher-order correlations. The analysis is then generalized to

n

interacting oscillators, with analytical energy expressions provided and the case

n=3

examined in detail. Our results establish a versatile and exact framework for driven-dissipative quantum systems, with potential applications in quantum thermodynamics, open quantum systems, and many-body physics.