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Physical Review A

Spontaneous emission from a single two-level atom in the presence ofNinitially unexcited identical atoms

F.W. Cummings

Published Mar 1, 1986• 44 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.33.1683

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Abstract One of the greatest successes in quantum theory, and certainly one of the more important parts for application to devices and applications is the prediction of the emission of light through the quantization of an electromagnetic field. Broadly, this is the field of quantum electrodynamics. In this chapter, we develop the Hamiltonian for the classical electromagnetic field. It is seen that the Hamiltonian for each mode (identified by the k-vector and polarization of the field) of the plane wave electromagnetic field is identical to that of the harmonic oscillator. One unit of energy, ℏω, in a mode is a called a photon. The eigenkets for the system are number states (Fock states). We then consider a two-level system described by a Hamiltonian which couples the two-level quantum system to the quantized electromagnetic field. Using the Weisskopf–Wigner formalism developed in Chapter 14, we solve the equations of motion for the time dependent Schrödinger equation assuming the system starts in the excited state with no radiation present in the vacuum field. The results show the creation of one unit of energy in an electromagnetic mode corresponding to the emission of a photon. The excited state probability decays exponentially with the emission of this photon. We consider the important and special case of such a two-level system but in a cavity restricting the radiation field to a single mode. The Jaynes–Cummings Hamiltonian shows that this system, if started in the excited state, Rabi oscillates with no radiation incident on the system.

•Apr 30, 2021•0 citations
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.
The long-time behavior of the number of photons in a mode of the electromagnetic field is given in closed form in terms of elementary functions when there is initially one (two-level) atom excited in the presence of NNN unexcited identical atoms and MMM field modes. The radiation trapping which occurs with only one mode disappears when the number of modes MMM approaches the total number of atoms in the continuum mode approximation.
The problem of the spontaneous emission from a single excited two-level atom in the presence of N unexcited similar atoms is discussed. A formalism is presented for evaluation of the time-development operator (and thus the density matrix) which has certain desirable features for time-dependent problems in the area, e.g. of atom-field interactions, when the off-diagonal elements are large compared to the diagonal elements of the Hamiltonian. By use of this formalism, it is shown that the spontaneous emission from a single atom is often drastically altered in the presence of unexcited atoms with similar resonances. When the number of atoms present becomes very large relative to the number of modes of the electromagnetic field, the excited atom becomes trapped in its excited state.>
The spontaneous emission of a set of NNN identical two-level atoms interacting with a quantized electromagnetic field is studied. The atoms are assumed to be close together compared to the mean wavelength of the emitted radiation, and their initial state is either a Dicke state or a superposition of the ground and excited state for each atom. Approximate expressions are obtained for transition probabilities, expectation values, and correlations as functions of time. The spontaneous emission for a very large separation of the atoms is also studied. In this case the directional properties of the emitted radiation, as well as the time within which the atoms decay to their ground state, depend on the position of each atom.
NNN two-level "atoms" are considered in interaction with a single-mode resonant electromagnetic field. The exact solution is given nonrelativistically for all times for the case of spontaneous emission, when only one atom is initially excited. The solution is given for the general case of the NNN atoms in inequivalent mode positions.
The interaction between electromagnetic radiation and two three-level atoms is investigated. The atom-atom separation is assumed to be smaller than the corresponding mean resonance wavelength. The total radiance rate of the atomic system is calculated as a function of time. Explicit results are given for several different initial states of the atomic system. Some of them exhibit superradiance, and some act as photon-trapping states. Instead of exponential decay, the oscillations appear in the time evolution of the radiation rate due to the correlation between atoms. The decay constants and coupling decay constants show strong effects on the emission rates. Completely differing from two identical double-level atoms, the cooperative effects always exist in the three-level atomic system except for the case in which the coupling decay constants ${\ensuremath{\gamma}}_{12}$ and ${\ensuremath{\gamma}}_{21}$ are zero.
Abstract One of the greatest successes in quantum theory, and certainly one of the more important parts for application to devices and applications is the prediction of the emission of light through the quantization of an electromagnetic field. Broadly, this is the field of quantum electrodynamics. In this chapter, we develop the Hamiltonian for the classical electromagnetic field. It is seen that the Hamiltonian for each mode (identified by the k-vector and polarization of the field) of the plane wave electromagnetic field is identical to that of the harmonic oscillator. One unit of energy, ℏω, in a mode is a called a photon. The eigenkets for the system are number states (Fock states). We then consider a two-level system described by a Hamiltonian which couples the two-level quantum system to the quantized electromagnetic field. Using the Weisskopf–Wigner formalism developed in Chapter 14, we solve the equations of motion for the time dependent Schrödinger equation assuming the system starts in the excited state with no radiation present in the vacuum field. The results show the creation of one unit of energy in an electromagnetic mode corresponding to the emission of a photon. The excited state probability decays exponentially with the emission of this photon. We consider the important and special case of such a two-level system but in a cavity restricting the radiation field to a single mode. The Jaynes–Cummings Hamiltonian shows that this system, if started in the excited state, Rabi oscillates with no radiation incident on the system.