Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Rabi model result for Einstein's $B$ coefficient: role of fundamental processes in nonequilibrium statistical mechanics

Shyamal Biswas, Tanmoy Mondal

Published Jul 2, 2017• 0 citations
OpenAlex

Abstract

Starting from the Rabi Hamiltonian, which is useful in arriving at non-perturbative results within the rotating wave approximation, we have found Einstein's $B$ coefficient to be time-dependent: $B(t)\propto|J_0(\omega_\gamma t)|$ for a two-level system (atom or molecule) in thermal radiation field. Here $\omega_\gamma$ is the corresponding Rabi flopping (angular) frequency and $J_0$ is the zeroth order Bessel function of the first kind. The resulting oscillations in the $B$ coefficient---even for very small $\omega_\gamma$---drives the system away from thermodynamic equilibrium at any finite temperature contrary to Einstein's assumption. The time-dependent generalized $B$ coefficient facilitates a path to go beyond Pauli's formalism of non-equilibrium statistical mechanics involving the quantum statistical Boltzmann (master) equation. In this context, we have obtained entropy production of the two-level system by revising Einstein's rate equations, while considering the $A$ coefficient to be the original time-independent one and the $B$ coefficient to be time-dependent.

Related papers

Re-examining Einstein’s B coefficient and rate equations with the Rabi model

Najirul Islam, Tanmoy Mondal, Sagar Chakraborty +1 more

Abstract Starting from the Rabi Hamiltonian, which is useful in arriving at non-perturbative results within the rotating wave approximation, we have found Einstein’s B coefficient to be time-dependent: for a two-level system (atom or molecule) in thermal radiation field. Here is the corresponding Rabi flopping (angular) frequency and J 0 is the zeroth order Bessel function of the first kind. The resulting oscillations in the B coefficient—even for very small —drives the system away from thermodynamic equilibrium at any finite temperature contrary to Einstein’s assumption. The time-dependent generalized B coefficient facilitates a path to go beyond Pauli’s formalism of non-equilibrium statistical mechanics involving the quantum statistical Boltzmann (master) equation. In this context, we have obtained entropy production of the two-level system by revising Einstein’s rate equations, while considering the A coefficient to be the original time-independent one and the B coefficient to be time-dependent.

arXiv:1707.00283•Journal of Statistical Mechanics Theory and Experiment•Nov 1, 2019•1 citation

Application of Einstein’s Methods in a Quantum Theory of Radiation

Richard Joseph Oldani

Einstein showed in his seminal paper on radiation that molecules with a quantum-theoretical distribution of states in thermal equilibrium are in dynamical equilibrium with the Planck radiation. The method he used assigns coordinates fixed with respect to molecules to derive the A and B coefficients, and fixed relative to laboratory coordinates to specify their thermal motion. The resulting dynamical equilibrium between quantum mechanical and classically defined statistics is critically dependent upon considerations of momentum exchange. When Einstein’s methods relating classical and quantum mechanical statistical laws are applied to the level of the single quantum oscillator they show that matrix mechanics describes the external appearances of an atom as determined by photon-electron interactions in laboratory coordinates, and wave mechanics describes an atom’s internal structure according to the Schrödinger wave equation. Non-commutation is due to the irreversibility of momentum exchange when transforming between atomic and laboratory coordinates. This allows the “rotation” of the wave function to be interpreted as the changing phase of an electromagnetic wave. In order to describe the momentum exchange of a quantum oscillator the Hamiltonian model of atomic structure is replaced by a Lagrangian model that is formulated with equal contributions from electron, photon, and nucleus. The fields of the particles superpose linearly, but otherwise their physical integrity is maintained throughout. The failure of past and present theoretical models to include momentum is attributed to the overwhelming requirement of human visual systems for an explicit stimulus.

IntechOpen eBooks•May 14, 2021•6 citations

Generalization of the Einstein coefficients and rate equations under the quantum Rabi oscillation

Najirul Islam, Shyamal Biswas

Abstract We have generalized Einstein coefficients and rate equations from quantum field theoretic point of view by bringing the fundamental processes and the quantum Rabi oscillation in a single footing for the light–matter interactions for nonzero Rabi frequency. We have analytically obtained multimode Jaynes–Cummings model results for the quantum Rabi oscillation of a two-level system in a lossy resonant cavity containing (i) thermal photons and (ii) injected photons of a coherent field. We have renormalized the coupling constant for the light–matter interactions for these cases. Our results match well with the seminal experimental data obtained in this regard by Brune et al (1996 Phys. Rev. Lett. 76 1800). We also have studied the population dynamics in this regard by applying the generalized Einstein rate equations.

arXiv:2210.04032•Journal of Physics A Mathematical and Theoretical•Mar 8, 2021•1 citation

Limitations of the Rabi Model for Rydberg Transitions

P. L. Knight, Simon J. D. Phoenix

A two-level system driven by a classical single frequency field in rotating-wave approximation is a basic model for resonance in quantum optics. The probability of making a transition depends sinusoidally on time with a characteristic frequency called the Rabi frequency. The Jaynes-Cummings model (JCM) consists of a two-level atomic system driven by a single quantized electromagnetic field mode. The field entropy in the case of an atom interacting with an initially coherent field is expected to reflect the collapse and revival of Rabi oscillations. The photon statistics of the JCM are examined and the entropy concepts are used to describe the apparently disordered collapses. A density matrix method is used to describe the influence of photon statistics in the JCM. The quantum generalization defines entropy through where the trace is taken over a convenient basis set. The thermal field JCM shows enhanced photon variances as expected for an initially Bose-Einstein field with large fluctuations.

CRC Press eBooks•Aug 12, 2020•0 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.

Journal of Physics Conference Series•Feb 1, 2009•2 citations

Periodic thermodynamics of a two spin Rabi model

Heinz-Jürgen Schmidt

Abstract We consider two s = 1/2 spins with Heisenberg coupling and a monochromatic, circularly polarized magnetic field acting only onto one of the two spins. This system turns out to be analytically solvable. Also the statistical distribution of the work performed by the driving forces during one period can be obtained in closed form and the Jarzynski equation can be checked. The mean value of this work, viewed as a function of the physical parameters, exhibits features that can be related to some kind of Rabi oscillation. Moreover, when coupled to a heat bath the two spin system will approach a non-equilibrium steady state (NESS) that can be calculated in the golden rule approximation. The occupation probabilities of the NESS are shown not to be of Boltzmann type, with the exception of a single phase with infinite quasitemperature. The parameter space of the two spin Rabi model can be decomposed into eight phase domains such that the NESS probabilities possess discontinuous derivatives at the phase boundaries. The latter property is shown to hold also for more general periodically driven N -level systems.

arXiv:1911.10488•Journal of Statistical Mechanics Theory and Experiment•Apr 1, 2020•6 citations
Rabi model result for Einstein's BBB coefficient: role of fundamental processes in nonequilibrium statistical mechanics
Starting from the Rabi Hamiltonian, which is useful in arriving at non-perturbative results within the rotating wave approximation, we have found Einstein's BBB coefficient to be time-dependent: B(t)∝∣J0(ωγt)∣B(t)\propto|J_0(\omega_\gamma t)|B(t)∝∣J0​(ωγ​t)∣ for a two-level system (atom or molecule) in thermal radiation field. Here ωγ\omega_\gammaωγ​ is the corresponding Rabi flopping (angular) frequency and J0J_0J0​ is the zeroth order Bessel function of the first kind. The resulting oscillations in the BBB coefficient---even for very small ωγ\omega_\gammaωγ​---drives the system away from thermodynamic equilibrium at any finite temperature contrary to Einstein's assumption. The time-dependent generalized BBB coefficient facilitates a path to go beyond Pauli's formalism of non-equilibrium statistical mechanics involving the quantum statistical Boltzmann (master) equation. In this context, we have obtained entropy production of the two-level system by revising Einstein's rate equations, while considering the AAA coefficient to be the original time-independent one and the BBB coefficient to be time-dependent.
Abstract Starting from the Rabi Hamiltonian, which is useful in arriving at non-perturbative results within the rotating wave approximation, we have found Einstein’s B coefficient to be time-dependent: for a two-level system (atom or molecule) in thermal radiation field. Here is the corresponding Rabi flopping (angular) frequency and J 0 is the zeroth order Bessel function of the first kind. The resulting oscillations in the B coefficient—even for very small —drives the system away from thermodynamic equilibrium at any finite temperature contrary to Einstein’s assumption. The time-dependent generalized B coefficient facilitates a path to go beyond Pauli’s formalism of non-equilibrium statistical mechanics involving the quantum statistical Boltzmann (master) equation. In this context, we have obtained entropy production of the two-level system by revising Einstein’s rate equations, while considering the A coefficient to be the original time-independent one and the B coefficient to be time-dependent.
Einstein showed in his seminal paper on radiation that molecules with a quantum-theoretical distribution of states in thermal equilibrium are in dynamical equilibrium with the Planck radiation. The method he used assigns coordinates fixed with respect to molecules to derive the A and B coefficients, and fixed relative to laboratory coordinates to specify their thermal motion. The resulting dynamical equilibrium between quantum mechanical and classically defined statistics is critically dependent upon considerations of momentum exchange. When Einstein’s methods relating classical and quantum mechanical statistical laws are applied to the level of the single quantum oscillator they show that matrix mechanics describes the external appearances of an atom as determined by photon-electron interactions in laboratory coordinates, and wave mechanics describes an atom’s internal structure according to the Schrödinger wave equation. Non-commutation is due to the irreversibility of momentum exchange when transforming between atomic and laboratory coordinates. This allows the “rotation” of the wave function to be interpreted as the changing phase of an electromagnetic wave. In order to describe the momentum exchange of a quantum oscillator the Hamiltonian model of atomic structure is replaced by a Lagrangian model that is formulated with equal contributions from electron, photon, and nucleus. The fields of the particles superpose linearly, but otherwise their physical integrity is maintained throughout. The failure of past and present theoretical models to include momentum is attributed to the overwhelming requirement of human visual systems for an explicit stimulus.
Abstract We have generalized Einstein coefficients and rate equations from quantum field theoretic point of view by bringing the fundamental processes and the quantum Rabi oscillation in a single footing for the light–matter interactions for nonzero Rabi frequency. We have analytically obtained multimode Jaynes–Cummings model results for the quantum Rabi oscillation of a two-level system in a lossy resonant cavity containing (i) thermal photons and (ii) injected photons of a coherent field. We have renormalized the coupling constant for the light–matter interactions for these cases. Our results match well with the seminal experimental data obtained in this regard by Brune et al (1996 Phys. Rev. Lett. 76 1800). We also have studied the population dynamics in this regard by applying the generalized Einstein rate equations.
A two-level system driven by a classical single frequency field in rotating-wave approximation is a basic model for resonance in quantum optics. The probability of making a transition depends sinusoidally on time with a characteristic frequency called the Rabi frequency. The Jaynes-Cummings model (JCM) consists of a two-level atomic system driven by a single quantized electromagnetic field mode. The field entropy in the case of an atom interacting with an initially coherent field is expected to reflect the collapse and revival of Rabi oscillations. The photon statistics of the JCM are examined and the entropy concepts are used to describe the apparently disordered collapses. A density matrix method is used to describe the influence of photon statistics in the JCM. The quantum generalization defines entropy through where the trace is taken over a convenient basis set. The thermal field JCM shows enhanced photon variances as expected for an initially Bose-Einstein field with large fluctuations.
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.
Abstract We consider two s = 1/2 spins with Heisenberg coupling and a monochromatic, circularly polarized magnetic field acting only onto one of the two spins. This system turns out to be analytically solvable. Also the statistical distribution of the work performed by the driving forces during one period can be obtained in closed form and the Jarzynski equation can be checked. The mean value of this work, viewed as a function of the physical parameters, exhibits features that can be related to some kind of Rabi oscillation. Moreover, when coupled to a heat bath the two spin system will approach a non-equilibrium steady state (NESS) that can be calculated in the golden rule approximation. The occupation probabilities of the NESS are shown not to be of Boltzmann type, with the exception of a single phase with infinite quasitemperature. The parameter space of the two spin Rabi model can be decomposed into eight phase domains such that the NESS probabilities possess discontinuous derivatives at the phase boundaries. The latter property is shown to hold also for more general periodically driven N -level systems.