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Physical Review

Induced and Spontaneous Emission in a Coherent Field. II

I. R. Senitzky

Published Jul 15, 1959• 45 citations
Publisher

Abstract

The interaction between the electromagnetic field and a number of identical atomic systems, individually characterized by an electric dipole moment and two energy levels, is analyzed for the case where the atomic systems are inside a lossy cavity and exposed to a coherent driving field, resonance being assumed between atomic system, cavity, and driving field.The problem of introducing loss into a quantum-mechanical formalism is treated first. Formal operator expressions are obtained for the field variables which include the absorption and the fluctuation (both thermal and quantum-mechanical) effects of the loss mechanism. Expectation values are then obtained for the field strength and the field energy which are valid for times short compared to the lifetime of the excited state. It is shown that the spontaneous-emission energy in the field increases initially as the square of the time and approaches a steady-state value after a transient period which is of the order of the cavity relaxation time. The induced emission contains two parts: incoherent emission induced by the thermal field, and coherent emission induced by the driving field. The incoherent induced-emission energy has the same time dependence as the spontaneous-emission energy, and the ratio of the former to the latter is the number of photons in the thermal (inducing) field. The coherent induced-emission energy does not approach a steady-state value, but, after a transient period, increases linearly with the time. The ratio of the coherent induced-emission energy to the spontaneous-emission energy is equal, initially, to the number of photons in the driving-field energy, but becomes $\ensuremath{\beta}t$ times as large after the transient period, where $\ensuremath{\beta}$ is the reciprocal of the cavity relaxation time.The expectation value of the rate of energy emission by the atomic systems is also obtained. It is shown that the ratio of the downward to the upward transition probabilities has the well-known value of $\frac{(n+1)}{n}$, where $n$ is the field energy in units of the photon energy, only in the absence of a coherent field.

DOI
10.1103/physrev.115.227

Related papers

Induced and Spontaneous Emission in a Coherent Field. III

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The problem of induced and spontaneous emission is investigated for an atomic two-level system with incident beams of radiation which are either in a coherent state or in a stationary state (contain a definite number of photons). The treatment is fully quantum-mechanical, and is confined to the case where the frequency spectrum of the incident beam is narrow compared to the natural linewidth of the system. It is shown that, under such conditions, the spontaneous emission for frequencies within the narrow band of the incident radiation is sharply reduced compared to the prediction of the natural lineshape. It is shown that a hole is burned in the natural lineshape within the narrow frequency band, thus effectively quenching the spontaneous emission at some frequency within the band. This effect is shown to occur both for the coherent and stationary beams. Quantities proportional to the induced and spontaneous probability amplitudes and the lifetimes are computed for times comparable to and long compared to the free lifetime of the state. An expression is found for the spectrum of the emergent radiation in terms of these quantities. Its physical meaning is briefly discussed. The density operator of the field for all times is given.

Journal of Mathematical Physics•Feb 1, 1967•18 citations
The interaction between the electromagnetic field and a number of identical atomic systems, individually characterized by an electric dipole moment and two energy levels, is analyzed for the case where the atomic systems are inside a lossy cavity and exposed to a coherent driving field, resonance being assumed between atomic system, cavity, and driving field.The problem of introducing loss into a quantum-mechanical formalism is treated first. Formal operator expressions are obtained for the field variables which include the absorption and the fluctuation (both thermal and quantum-mechanical) effects of the loss mechanism. Expectation values are then obtained for the field strength and the field energy which are valid for times short compared to the lifetime of the excited state. It is shown that the spontaneous-emission energy in the field increases initially as the square of the time and approaches a steady-state value after a transient period which is of the order of the cavity relaxation time. The induced emission contains two parts: incoherent emission induced by the thermal field, and coherent emission induced by the driving field. The incoherent induced-emission energy has the same time dependence as the spontaneous-emission energy, and the ratio of the former to the latter is the number of photons in the thermal (inducing) field. The coherent induced-emission energy does not approach a steady-state value, but, after a transient period, increases linearly with the time. The ratio of the coherent induced-emission energy to the spontaneous-emission energy is equal, initially, to the number of photons in the driving-field energy, but becomes \ensuremathβt\ensuremath{\beta}t\ensuremathβt times as large after the transient period, where \ensuremathβ\ensuremath{\beta}\ensuremathβ is the reciprocal of the cavity relaxation time.The expectation value of the rate of energy emission by the atomic systems is also obtained. It is shown that the ratio of the downward to the upward transition probabilities has the well-known value of (n+1)n\frac{(n+1)}{n}n(n+1)​, where nnn is the field energy in units of the photon energy, only in the absence of a coherent field.
The theory developed in the first two articles of this series, dealing with the interaction between the electromagnetic field in a cavity resonator and a number of two-level molecules, is generalized to include a Gaussian spread in the molecular frequency. The center of the molecular frequency distribution coincides with the cavity resonant frequency. There is a coherent driving field in the cavity at the same frequency, and cavity loss is taken into account.Using the formalism previously developed for a quantum-mechanical field in a lossy cavity, expressions are obtained by means of second-order perturbation theory for the expectation values of the field strength and field energy in the cavity, and of the power loss by the molecules. It is shown that the parts of the field energy resulting from induced and spontaneous emission, respectively, initially increase as the square of the time and approach steady-state values after (different, in general) transient periods, each of which is determined by two time constants: cavity relaxation time and inverse molecular frequency spread.It is also shown that both the induced and spontaneous emission power radiated by the molecules increase initially linearly with the time and approach steady-state values after transient periods. For the induced emission power, the transient period is determined by only one time constant, the inverse molecular frequency spread, while for the spontaneous emission power it is determined both by the inverse molecular frequency spread and the cavity relaxation time. The ratio of induced to spontaneous emission is initially nnn, and approaches a steady-state value $n{[\mathrm{exp}({r}^{2})(1\ensuremath{-}\mathrm{erf}r)]}^{\ensuremath{-}1},$ where nnn is the driving field energy in units of the photon energy, and rrr is the ratio of the cavity resonance width to molecular frequency spread. The seeming inconsistency of this value with the classical value of the ratio of the Einstein coefficients is discussed.
In the previous articles of this series, dealing with the interaction between a number of molecules and the electromagnetic field in a resonant cavity, both the molecules and the field were treated by perturbation theory. The perturbation restriction on the field is removed in the present article, allowing large changes in the field, but the molecules are still assumed to undergo a small change during the time under consideration. The justification for this type of analysis, involving the generalization of the conventional concepts of induced and spontaneous emission, the applicability to a molecular amplifier during the buildup period, and the re-examination of a calculation by Serber and Townes concerning the fundamental limits of molecular amplification, is discussed.Two different molecular distributions are considered. In one (the resonant case) all molecules have the same frequency as the cavity, and in the other (the nonresonant case) there is a uniform frequency distribution. The molecules are assumed to be initially in an emissive state. Several types of driving fields are considered. Expressions are obtained for the field operators by the solution of a Volterra integral equation, and expectation values are obtained for the field strength and field energy.In the resonant case, both the coherent and incoherent fields increase exponentially after a sufficiently long time, no matter how small the initial gain is. Their ratio becomes constant and is equal to the number of photons in the driving field only in the absence of dissipation. An interesting related result is the fact that the signal-to-noise ratio for constant signal input power increases as the cavity dissipation increases. An estimate of the total time for which the theory is valid is obtained from a consideration of the energy emitted by the molecules. Contact is made with perturbation theory for sufficiently small gain and short time.In the nonresonant case the effect of the molecules is shown to be that of a negative dissipation. In contrast to the resonant case, the gain becomes exponential only if the negative dissipation exceeds, in absolute value, the true dissipation. The ratio of induced to spontaneous emission is, in this case also, equal to the number of photons in the driving field only in the absence of disipation. However, the signal-to-noise ratio for constant input power drops with increasing cavity dissipation.
The interaction between a coherently oscillating radiation field, such as is generally encountered in micro-wave experiments, and a number of similar atomic systems coupled to the field through their electric dipole moments is analyzed for the case of resonance between atomic system and field, with both the field and molecules treated quantum-mechanically. Expressions are derived for the expectation value of field strength and the energy, and comparison between the two makes it possible to distinguish between coherent and incoherent parts of the energy, the former having its counterpart in the expectation value of the field strength, while the latter has no such counterpart. Terms corresponding to induced and spontaneous emission are identified, and it is shown that the latter includes both coherent and incoherent components. Special situations related to masers and to the coherence of spontaneous radiation are discussed. The distinction between the behavior of correlated and uncorrelated states is examined, and it is shown that both the coherent emission from uncorrelated states and the incoherent emission from correlated states may be proportional to the square of the number of molecules. The phase of the field is predictable for the coherent emission but unpredictable for the incoherent emission.
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.
The theory of the previous articles of this series is extended to allow analysis of the spectral distribution of induced and spontaneous emission in a cavity and to study the steady-state condition when a molecular beam traverses the cavity.It is shown that the induced emission energy, when averaged over a reasonable time, depends only on the energy spectrum of the driving field. Expressions are obtained for the spectral distribution of induced, spontaneous, and thermally induced emission. These results are then applied to a molecular beam analysis and conditions for a steady state in the cavity are obtained.The amplification by the molecular beam of the driving field (signal) and of the thermal and spontaneous emission energies (noise) is studied. Signal and noise energies are compared and the dependence of the results on both the bandwidth of the signal and the bandwidth of detection is examined.
The problem of induced and spontaneous emission is investigated for an atomic two-level system with incident beams of radiation which are either in a coherent state or in a stationary state (contain a definite number of photons). The treatment is fully quantum-mechanical, and is confined to the case where the frequency spectrum of the incident beam is narrow compared to the natural linewidth of the system. It is shown that, under such conditions, the spontaneous emission for frequencies within the narrow band of the incident radiation is sharply reduced compared to the prediction of the natural lineshape. It is shown that a hole is burned in the natural lineshape within the narrow frequency band, thus effectively quenching the spontaneous emission at some frequency within the band. This effect is shown to occur both for the coherent and stationary beams. Quantities proportional to the induced and spontaneous probability amplitudes and the lifetimes are computed for times comparable to and long compared to the free lifetime of the state. An expression is found for the spectrum of the emergent radiation in terms of these quantities. Its physical meaning is briefly discussed. The density operator of the field for all times is given.