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arXiv:quant-ph/0209102 · Journal of Physics A Mathematical and General

Time evolution of the Rabi Hamiltonian from the unexcited vacuum

R. F. Bishop, Clive Emary

Published Jul 6, 2001• 25 citations
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Abstract

The Rabi Hamiltonian describes a single mode of electromagnetic radiation interacting with a two-level atom. Using the coupled cluster method, we investigate the time evolution of this system from an initially empty field mode and an unexcited atom. We give results for the atomic inversion and field occupation, and find that the virtual processes cause the field to be squeezed. No anti-bunching occurs.

DOI
10.1088/0305-4470/34/28/302

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The Rabi Hamiltonian describes a single mode of electromagnetic radiation interacting with a two-level atom. Using the coupled cluster method, we investigate the time evolution of this system from an initially empty field mode and an unexcited atom. We give results for the atomic inversion and field occupation, and find that the virtual processes cause the field to be squeezed. No anti-bunching occurs.
The authors used the coupled cluster method (CCM) to investigate the evolution of the Rabi Hamiltonian from an initial state |0,| > composed of an empty field mode and an unexcited atom. Within the rotation-wave approxn. (RWA) the system would remain in this state indefinitely, since it is an eigenstate of the excitation no. N with corresponding eigenvalue zero. However, including of the otherwise neglected \"counter-rotating\" terms means that we allow \"energy non-conserving\" processes to occur, and their presence drives the temporal evolution of the Rabi system. The authors present results for the at. inversion and field occupation as function of time. They also show that no anti-bunching occurs, although the \"virtual\" processes do cause the field to become squeezed. [on SciFinder (R)]
In the preceding chapters concerning the interaction of a radiation field with matter, we assumed the field to be classical. In many situations this assumption is valid. There are, however, many instances where a classical field fails to explain experimentally observed results and a quantized description of the field is required. This is, for example, true of spontaneous emission in an atomic system which was described phenomenologically in Chapter 5. For a rigorous treatment of the atomic level decay in free space, we need to consider the interaction of the atom with the vacuum modes of the universe. Even in the simplest system involving the interaction of a single-mode radiation field with a single two-level atom, the predictions for the dynamics of the atom are quite different in the semiclassical theory and the fully quantum theory. In the absence of the decay process, the semiclassical theory predicts Rabi oscillations for the atomic inversion whereas the quantum theory predicts certain collapse and revival phenomena due to the quantum aspects of the field. These interesting quantum field theoretical predictions have been experimentally verified. In this chapter we discuss the interaction of the quantized radiation field with the two-level atomic system described by a Hamiltonian in the dipole and the rotating-wave approximations. For a single-mode field it reduces to a particularly simple form. This is a very interesting Hamiltonian in quantum optics for several reasons. First, it can be solved exactly for arbitrary coupling constants and exhibits some true quantum dynamical effects such as collapse followed by periodic revivals of the atomic inversion.
The resonant interaction of A initially unexcited two-level atoms with different quantum fields in a high-Q single-mode cavity (Dicke model) is studied analytically in one of the limits in which the eigenvalues spectrum of the model may be assumed as equidistant. The time evolution of atomic inversion of a single two-level atom coupled to squeezed vacuum or thermal cavity fields (Jaynes-Cummings model) is irregular. Here we show that the dynamical response of the inversion of an assemblage of A two-level atoms to such fields may be regular, i.e., may reveal collapses and revivals and that the properties of the model with an initial squeezed vacuum field differ from those for coherent or thermal fields.
The N-photon interaction of two two-level atoms with a single-mode quantized radiation field is studied. The general expression for the state vector of the atom–field system is presented with the probability amplitudes in analytic form. For the initial condition in which the atoms are both at their upper levels and the field is in a squeezed state, we study the time evolution of the atomic population inversions and that of the statistics of the field. We find that the ratio of the two coupling constants, the initial squeezing of the field, and N have significant effects on the features of the atoms and of the field.
The quantum-electrodynamical problem of N identical two-level atoms interacting with M field modes is considered, and an exact solution for the case of spontaneous emission from a single atom is obtained. The behavior of this system is shown to be quite different from the exponential decay of a single excited atom radiating into free space. A ``ringing'' behavior at the enigmatic frequency \ensuremath{\sim} \ensuremath{\surd}N which occurs when only one mode is accessible to the field persists when many modes are present. When a continuum of modes is accessible, the long-time limit shows that the energy of the originally excited atom is equally likely to be shared by the N-1 originally unexcited atoms and the electromagnetic field when M/N\ensuremath{\rightarrow}1. ``Radiation suppression'' occurs whenever N\ensuremath{\gg}1 and M\ensuremath{\ll}N, when the originally excited atom does not emit its energy. When M=N\ensuremath{\gg}1, the single atom decays monotonically to zero (and approximately exponentially) but with a decay rate that is \ensuremath{\surd}N times the Rabi frequency. Such unexpected effects are presently within experimental range.
We study the dynamics of an N-level atom coupled in a lossless cavity to a single-mode near-resonant quantized field. The atomic levels are coupled by the multiphoton transitions and the coupling constants between the field and the atomic levels are supposed to be intensity dependent. We find the exact solution for the state vector describing the dynamics of the atom-plus-field system. As an illustration we use the model for studying (i) the time evolution of the atomic occupation probability with the initially coherent field and (ii) the light squeezing, when the cavity field is initially in the vacuum state and the atom is prepared in the atomic ‘coherent state’ (a superposition of atomic states).