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arXiv:1707.00283 · Journal of Statistical Mechanics Theory and Experiment

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

Najirul Islam, Tanmoy Mondal, Sagar Chakraborty, Shyamal Biswas

Published Nov 1, 2019• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1742-5468/ab4fdc

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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.
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.
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.
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 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.
We consider the temporal evolution of the entropy S of an isolated system from an initial non-equilibrium state 2· of entropy S = S i to the equilibrium state of maximum entropy S = S max > S i. The application of usual density matrix theory of quantum mechanics leads us to the surprising result P = dS/dt = 0. The entropy does not change in time. This result is valid in case of the equilibrium state. However it is wrong for a non-equilibrium state, because the entropy of the isolated system could not spontaneously increase from S i to S max by entropy production P > 0. This contradicts the second law of thermodynamics. We avoid this discrepancy by changing from quantum mechanics to quantum thermodynamics. Now the entropy production P = dS/dt and the time t are the relevant observables. In order to treat the temporal development of S and P, we at first prepare an initial state i of entropy production P i at time t < 0. Subsequently at ϑ t ≥ 0 the initial state i interacts with a continuum of final states f, which transfers the entropy production from i to f. The temporal evolution of the entropy production P(t) of our system is treated by means of the time-dependent perturbation theory. For that reason we split the operator of entropy production P = P 0 + W into two parts. The operator P 0 describes the unperturbed basic vectors of entropy production | i > and | f >, while the perturbation operator W determines the interaction between | i > and | f >. We expand the time-dependent state function < t | ψ > of the isolated system into the complete set of stationary eigenvectors | i > and | f > and calculate the time-dependent expansion coefficients c i(t) and c(f, t) by means of the dynamic equation of quantum thermodynamics − i k (d/dt) < t | ψ > = < t | P | ψ >. Here k means the Boltzmann constant, which is characteristic of quantum thermodynamics and represents the atomic entropy unit. The expansion coefficients are given by a system of coupled differential equations, from which we deduce an integral-differential equation for the coefficient c i(t) of the initial state i. Unfortunately it has been impossible to obtain an analytic solution of the complicated equation. However approximations are available. Accordingly, in the vicinity of the equilibrium state, the probability of still finding the initial state i of entropy production P i at time t is given by ω i(t) = | c i(t) |2 = exp (−Lt). The rate constant L ≥ σ depends on the density ρ(P f) of final states f of entropy production P f near by P i, and on the squared modulus of the perturbation matrix elements W if = < i | W | f >, which determines the interaction between | i > and | f >. The entropy production P(t) = P i ω i(t) = P i exp (−ϑLt) of the isolated system decreases exponentially and irreversibly with proceeding time t. Its rate-determining step is the transfer of entropy production from i to f. The integral over the entropy production of the final states f vanishes. Therefore the continuum states f do not contribute to the total entropy production P(t). The temporal evolution of the entropy S(t) can easily be calculated by integration of P(t). In conclusion, the entropy production of a non-equilibrium state is finite P(t) > 0 and only vanishes in the case of the equilibrium state P(∞) = 0. Thus, quantum thermodynamics agrees with the second law of thermodynamics.