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Journal of Contemporary Physics (Armenian Academy of Sciences)

Jaynes-Cummings Model in Counter Propagating Waves Basis: Large Numbers of Excitations

A. Zh. Muradyan

Published Jun 1, 2023• 0 citations
Publisher

Abstract

Abstract The Jaynes-Cummings (JC) model is presented based on quantized counterpropagating waves, characteristic of a ring resonator. This model, in addition to the Hamiltonian and the operator of the number of excitations, includes an interference operator whose eigenvalues imitate the term cos 2kz of the classical picture of the field. The average number of photons, dispersion, the Mandel parameter Q, and atomic-photon entanglement in states with certain excitations are considered. The collapses and revivals are revealed, which in the standard JC model are present only in states with an indefinite number of excitations. It is shown that photons at the boundary energy levels are antibunched (with Q ≈ –1/2) regardless of the system’s parameters, and photons of other energy levels are grouped to the degree that grows with the total number of photons.

DOI
10.1134/s1068337223020159

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Abstract The Jaynes-Cummings (JC) model is presented based on quantized counterpropagating waves, characteristic of a ring resonator. This model, in addition to the Hamiltonian and the operator of the number of excitations, includes an interference operator whose eigenvalues imitate the term cos 2kz of the classical picture of the field. The average number of photons, dispersion, the Mandel parameter Q, and atomic-photon entanglement in states with certain excitations are considered. The collapses and revivals are revealed, which in the standard JC model are present only in states with an indefinite number of excitations. It is shown that photons at the boundary energy levels are antibunched (with Q ≈ –1/2) regardless of the system’s parameters, and photons of other energy levels are grouped to the degree that grows with the total number of photons.
The Jaynes-Cummings model (JCM), a soluble fully quantum mechanical model of an atom in a field, was first used (in 1963) to examine the classical aspects of spontaneous emission and to reveal the existence of Rabi oscillations in atomic excitation probabilities for fields with sharply defined energy (or photon number). For fields having a statistical distributions of photon numbers the oscillations collapse to an expected steady value. In 1980 it was discovered that with appropriate initial conditions (e.g. a near-classical field), the Rabi oscillations would eventually revive, only to collapse and revive repeatedly in a complicated pattern. The existence of these revivals, present in the analytic solutions of the JCM, provided direct evidence for discreteness of field excitation (photons) and hence for the truly quantum nature of radiation. Subsequent study revealed further non-classical properties of the JCM field, such as a tendency of the photons to antibunch. Within the last two years it has been found that during the quiescent intervals of collapsed Rabi oscillations the atom and field exist in a macroscopic superposition state (a Schrödinger cat). This discovery offers the opportunity to use the JCM to elucidate the basic properties of quantum correlation (entanglement) and to explore still further the relationship between classical and quantum physics. The relative simplicity of the JCM and the ease with which it can be extended through analytic expressions or numerical computation continues to motivate attention, as evidenced by the growing abundance of publications. We here present an overview of the theory of the JCM and some of the many extensions and generalizations that have appeared.
. The Jaynes-Cummings Model (JCM), a soluble fully quantum mechanical model of an atom in a field, was first used (in 1963) to examine the classical aspects of spontaneous emission and to reveal the existence of Rabi oscillations in atomic excitation probability for fields with sharply defined energy (or photon number). For fields having a statistical distribution of photon numbers the oscillations collapse to an expected steady value. In 1980 it was discovered that with appropriate initial conditions (e.g. a near-classical field), the Rabi oscillations would eventually revive – only to collapse and revive repeatedly in a complicated pattern. The existence of these revivals, present in the analytic solutions of the JCM, provided direct evidence for discreteness of field excitation (photons) and hence for the truly quantum nature of radiation. Subsequent study revealed further nonclassical properties of the JCM field, such as a tendency of the photons to antibunch. Within the last two years it has been found that during the quiescent intervals of collapsed Rabi oscillations the atom and field exist in a macroscopic superposition state (a Schrödinger cat). This discovery offers the opportunity to use the JCM to elucidate the basic properties of quantum correlation (entanglement) and to explore still further the relationship between classical and quantum physics.
The Jaynes-Cummings Model (JCM), a soluble fully quantum mechanical model of an atom in a field, was first used (in 1963) to examine the classical aspects of spontaneous emission and to reveal the existence of Rabi oscillations in atomic excitation probability for fields with sharply defined energy (or photon number). For fields having a statistical distributions of photon numbers the oscillations collapse to an expected steady value. In 1980 it was discovered that with appropriate initial conditions (e.g. a near-classical field), the Rabi oscillations would eventually revive -- only to collapse and revive repeatedly in a complicated pattern. The existence of these revivals, present in the analytic solutions of the JCM, provided direct evidence for discreteness of field excitation (photons) and hence for the truly quantum nature of radiation. Subsequent study revealed further nonclassical properties of the JCM field, such as a tendency of the photons to antibunch. Within the last two years it has been found that during the quiescent intervals of collapsed Rabi oscillations the atom and field exist in a macroscopic superposition state (a Schroedinger cat). This discovery offers the opportunity to use the JCM to elucidate the basic properties of quantum correlation (entanglement) and to exploremore » still further the relationship between classical and quantum physics. In tribute to E. D. Jaynes, who first recognized the importance of the JCM for clarifying the differences and similarities between quantum and classical physics, we here present an overview of the theory of the JCM and some of the many remarkable discoveries about it.« less
The eigenstate problem of the Jaynes–Cummings model on the basis of a complete Hamiltonian, including the centre-of-mass kinetic energy operator, is considered. The energy spectrum and wavefunctions in the standing-wave (SW) and counterpropagating-wave (CPW) cases are calculated and compared with each other. It is shown that in the CPW case: (i) the atomic momentum distribution is asymmetric; (ii) the concept of quasimomentum is not applicable and instead the problem concerns the ordinary momentum; (iii) atomic and photonic state distributions are self-consistent; and as a consequence (iv) the mean number of photons in counterpropagating travelling waves and mean atomic momentum are matched. Explicit analytic expressions are proposed to describe the transition from counterpropagating quantized waves to a standing wave. It is also shown that, if the recoil energy is taken into account, the Doppleron resonance is split into two branches, one of which diverges to Bragg-like resonance in high-order range.
In this paper we have investigated the eigen energy spectrum of the k-photon Jaynes-Cummings (JC) model without the rotating-wave approximation (RWA). Our analysis shows that for k > 2 the k-photon JC model without the RWA does not have eigenstates in the Hilbert space spanned by the photon number states, i.e. the model becomes ill-defined. Hence, the k-photon JC model without the RWA is qualitatively different from the RWA counterpart which is valid for all values of k and the coupling parameter.
Squeezed states of the Jaynes-Cummings (JC) model are constructed and the statistical properties of the field oscillator in these states are studied. The presence of interaction between the field oscillator and the atom results in the appearance of slow temporal variations in the average values of the dynamic quantities. Two cases essentially differ: When only one of two modes of the JC model is excited, the variations in time, in particular, of the average value of the coordinate are governed by the mode anharmonicity. In the case when both modes of the JC model with close frequencies are excited, the temporal variations represent collapse and revival of oscillations with the frequency 2kn+12k\sqrt {n + 1} 2kn+1​ , where k is the constant of the atom-field oscillator interaction and n is the number of photons. The constructed squeezed vacuum states are not two-photon states, because the probability of finding an odd number of photons is not equal to zero and varies with time.