In this paper, we give an analytical treatment to study the behavior of the collapse and the revival of the Rabi oscillations in the Jaynes–Cummings model (JCM). The JCM is an exactly soluble quantum mechanical model, which describes the interaction between a two-level atom and a single cavity mode of the electromagnetic field. If we prepare the atom in the ground state and the cavity mode in a coherent state initially, the JCM causes the collapse and the revival of the Rabi oscillations many times in a complicated pattern in its time-evolution. In this phenomenon, the atomic population inversion is described with an intractable infinite series. (When the electromagnetic field is resonant with the atom, the nth term of this infinite series is given by a trigonometric function for $\\sqrt{n} t$ , where t is a variable of the time.) According to Klimov and Chumakov's method, using the Abel–Plana formula, we rewrite this infinite series as a sum of two integrals. We examine the physical meanings of these two integrals and find that the first one represents the initial collapse (the semi-classical limit) and the second one represents the revival (the quantum correction) in the JCM. Furthermore, we evaluate the first- and second-order perturbations for the time-evolution of the JCM with an initial thermal coherent state for the cavity mode at low temperature, and write down their correction terms as sums of integrals by making use of the Abel–Plana formula.In this paper, we give an analytical treatment to study the behavior of the collapse and the revival of the Rabi oscillations in the Jaynes–Cummings model (JCM). The JCM is an exactly soluble quantum mechanical model, which describes the interaction between a two-level atom and a single cavity mode of the electromagnetic field. If we prepare the atom in the ground state and the cavity mode in a coherent state initially, the JCM causes the collapse and the revival of the Rabi oscillations many times in a complicated pattern in its time-evolution. In this phenomenon, the atomic population inversion is described with an intractable infinite series. (When the electromagnetic field is resonant with the atom, the nth term of this infinite series is given by a trigonometric function for $\\sqrt{n} t$, where t is a variable of the time.) According to Klimov and Chumakov's method, using the Abel–Plana formula, we rewrite this infinite series as a sum of two integrals. We examine the physical meanings of these two integrals and find that the first one represents the initial collapse (the semi-classical limit) and the second one represents the revival (the quantum correction) in the JCM. Furthermore, we evaluate the first- and second-order perturbations for the time-evolution of the JCM with an initial thermal coherent state for the cavity mode at low temperature, and write down their correction terms as sums of integrals by making use of the Abel–Plana formula.

Jan 1, 20105 citations

THERMAL EFFECTS IN JAYNES–CUMMINGS MODEL DERIVED WITH LOW-TEMPERATURE EXPANSION

Hiroo Azuma, Masashi Ban

In this paper, we investigate thermal effects of the Jaynes–Cummings model (JCM) at finite temperature with a perturbative approach. We assume a single two-level atom and a single cavity mode to be initially in the thermal equilibrium state and the thermal coherent state, respectively, at a certain finite low temperature. Describing this system with Thermo Field Dynamics formalism, we obtain a low-temperature expansion of the atomic population inversion in a systematic manner. Letting the system evolve in time with the JCM Hamiltonian, we examine thermal effects of the collapse and the revival of the Rabi oscillations by means of the third-order perturbation theory under the low-temperature limit, that is to say, using the low-temperature expansion up to the third-order terms. From an intuitive discussion, we can expect that the period of the revival of the Rabi oscillations becomes longer as the temperature rises. Numerical results obtained with the perturbation theory reproduce well this temperature dependence of the period.In this paper, we investigate thermal effects of the Jaynes–Cummings model (JCM) at finite temperature with a perturbative approach. We assume a single two-level atom and a single cavity mode to be initially in the thermal equilibrium state and the thermal coherent state, respectively, at a certain finite low temperature. Describing this system with Thermo Field Dynamics formalism, we obtain a low-temperature expansion of the atomic population inversion in a systematic manner. Letting the system evolve in time with the JCM Hamiltonian, we examine thermal effects of the collapse and the revival of the Rabi oscillations by means of the third-order perturbation theory under the low-temperature limit, that is to say, using the low-temperature expansion up to the third-order terms. From an intuitive discussion, we can expect that the period of the revival of the Rabi oscillations becomes longer as the temperature rises. Numerical results obtained with the perturbation theory reproduce well this temperature dependence of the period.

arXiv:1103.3612International Journal of Modern Physics CAug 25, 20114 citations

Efeitos dissipativos na inversão atômica segundo o modelo de Jaynes-Cummings

Clarissa do Valle da Silva Lima

We study the Jaynes-Cummings model (JCM) which describes the interaction between a two-level atom and a single-mode resonant or quasi-resonant electromagnetic field in the non-dissipative and issipative cases. In particular, we study the atomic population inversion whose behavior presents some aspects of purely quantum origin: collapse and revivals. This observables expectation value can be btained analytically without use of approximations for the non-dissipative case and using first-order perturbation in the dissipative case. We make use of the sum of Poisson and the steepest descents method to obtain analytical formulas for the collapse and for each revival. In this way, we can obtain relevant information like the occurrence time, the maximum amplitude, the approximate frequency, and width. The dissipative JCM models the action of the environment on the quantum system. We show that the characteristic time of the revival diminishes as the interaction with the environment enlarges. We consider the limit in which it becomes experimentally impossible to distinguish the obtained curve from the classical one.We study the Jaynes-Cummings model (JCM) which describes the interaction between a two-level atom and a single-mode resonant or quasi-resonant electromagnetic field in the non-dissipative and issipative cases. In particular, we study the atomic population inversion whose behavior presents some aspects of purely quantum origin: collapse and revivals. This observables expectation value can be btained analytically without use of approximations for the non-dissipative case and using first-order perturbation in the dissipative case. We make use of the sum of Poisson and the steepest descents method to obtain analytical formulas for the collapse and for each revival. In this way, we can obtain relevant information like the occurrence time, the maximum amplitude, the approximate frequency, and width. The dissipative JCM models the action of the environment on the quantum system. We show that the characteristic time of the revival diminishes as the interaction with the environment enlarges. We consider the limit in which it becomes experimentally impossible to distinguish the obtained curve from the classical one.

Apr 7, 20060 citations

A resummation formula for collapse and revival in the Jaynes–Cummings model

Anatolii Alekseevich Karatsuba, Ekatherina Anatol'evna Karatsuba

We present a new approach to the study of the function of atomic inversion in the model of interaction of a single two-level atom with a single mode of the quantized electromagnetic field in the coherent state in an ideal resonator. The approach suggested is based on an application of certain number-theoretic results to the approximation of the trigonometric sums of a special form, in particular of the functional equation of the Jacobi theta function. New asymptotic formulae for the atomic inversion are found. The asymptotics that we obtain make it possible to predetermine the details of the behavior of the inversion on various time intervals depending on the parameters of the system.We present a new approach to the study of the function of atomic inversion in the model of interaction of a single two-level atom with a single mode of the quantized electromagnetic field in the coherent state in an ideal resonator. The approach suggested is based on an application of certain number-theoretic results to the approximation of the trigonometric sums of a special form, in particular of the functional equation of the Jacobi theta function. New asymptotic formulae for the atomic inversion are found. The asymptotics that we obtain make it possible to predetermine the details of the behavior of the inversion on various time intervals depending on the parameters of the system.

Journal of Physics A Mathematical and TheoreticalApr 27, 200920 citations

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.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 eBooksAug 12, 20200 citations

The revival–collapse phenomenon in the quadrature field components of the two-mode multiphoton Jaynes–Cummings model

Faisal A. A. El-Orany

In this paper we consider a system consisting of a two-level atom in an excited state interacting with two modes of a radiation field prepared initially in l -photon coherent states. This system is described by a two-mode multiphoton (i.e., k 1 , k 2 ) Jaynes–Cummings model (JCM). For this system we investigate the occurrence of the revival–collapse phenomenon (RCP) in the evolution of the single-mode, two-mode, sum and difference quadrature squeezing. We show that there is a class of states for which all these types of squeezing exhibit RCP similar to that involved in the corresponding atomic inversion. Also we show numerically that the single-mode squeezing of the first mode for ( k 1 , k 2 ) = (3,1) provides RCP similar to that of the atomic inversion of the case ( k 1 , k 2 ) = (1,1); however, sum and difference squeezing give partial information on that case. Moreover, we show that single-mode, two-mode and sum squeezing for the case ( k 1 , k 2 ) = (2,2) provides information on the atomic inversion of the single-mode two-photon JCM. We derive the rescaled squeezing factors giving accurate information on the atomic inversion for all cases. The consequences of these results are that the homodyne and heterodyne detectors can be used to detect the RCP for the two-mode JCM.In this paper we consider a system consisting of a two-level atom in an excited state interacting with two modes of a radiation field prepared initially in l -photon coherent states. This system is described by a two-mode multiphoton (i.e., k 1 , k 2 ) Jaynes–Cummings model (JCM). For this system we investigate the occurrence of the revival–collapse phenomenon (RCP) in the evolution of the single-mode, two-mode, sum and difference quadrature squeezing. We show that there is a class of states for which all these types of squeezing exhibit RCP similar to that involved in the corresponding atomic inversion. Also we show numerically that the single-mode squeezing of the first mode for ( k 1 , k 2 ) = (3,1) provides RCP similar to that of the atomic inversion of the case ( k 1 , k 2 ) = (1,1); however, sum and difference squeezing give partial information on that case. Moreover, we show that single-mode, two-mode and sum squeezing for the case ( k 1 , k 2 ) = (2,2) provides information on the atomic inversion of the single-mode two-photon JCM. We derive the rescaled squeezing factors giving accurate information on the atomic inversion for all cases. The consequences of these results are that the homodyne and heterodyne detectors can be used to detect the RCP for the two-mode JCM.

arXiv:0908.1819Journal of Optics B Quantum and Semiclassical OpticsSep 30, 20056 citations

\\sqrt{n} t

, where t is a variable of the time.) According to Klimov and Chumakov's method, using the Abel–Plana formula, we rewrite this infinite series as a sum of two integrals. We examine the physical meanings of these two integrals and find that the first one represents the initial collapse (the semi-classical limit) and the second one represents the revival (the quantum correction) in the JCM. Furthermore, we evaluate the first- and second-order perturbations for the time-evolution of the JCM with an initial thermal coherent state for the cavity mode at low temperature, and write down their correction terms as sums of integrals by making use of the Abel–Plana formula.