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Quantum Optics Journal of the European Optical Society Part B

The theory of the two-photon micromaser

D W G Laughlin, S. Swain

Published Apr 1, 1991• 30 citations
Publisher

Abstract

The authors develop the theory of the degenerate and nondegenerate two-photon micromaser using the full, three-level Hamiltonian under conditions of overall two-photon resonance. Subject to this condition, their results are valid for arbitrary one-photon detunings, but they consider in particular the large detuning and zero detuning limits, where some simplification is possible. For the case of zero detuning, they obtain semianalytic solutions, and find that, depending upon the interaction time, the system lases as either a single-mode or a two-mode system. They have compared their results with the degenerate two-photon micromaser, and find qualitative differences. Their results are valid for arbitrary one-photon detunings, provided the interaction times are not too long. The solution to the Fokker-Planck equation is found using a detailed balance, linearization procedure, and the photon statistics is investigated. Markedly sub-Poissonian behaviour is possible under certain conditions. They find that the behaviour in the zero detuning, nondegenerate situation is quite distinct to that of all other cases.

DOI
10.1088/0954-8998/3/2/001

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Aspects of the degenerate two-photon laser

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This paper presents new results on aspects of the two-photon laser. A semiclassical investigation of threshold behaviour reveals that the atom-field coupling is already so strong at threshold that a non-perturbative treatment of the dynamics is necessary. A master equation is derived on the basis of an effective Hamiltonian generalized in that it includes the effect of an overall two-photon detuning, the effect of ionization of the upper atomic level and the effect of dynamic Stark shifts. From the diagonal elements of the master equation, a Fokker-Planck equation is obtained and shown to yield a general analytical expression for the photon distribution which is found to simplify considerably in special cases. The agreement with the result obtained through numerical solution of the master equation itself is very good. It is further shown that a detuning from two-photon resonance and a significant Stark shift lead to sub-Poissonian statistics. Finally, time-dependent solutions for the photon distribution under an injected coherent state are presented.

Journal of Physics B Atomic Molecular and Optical Physics•Dec 28, 1995•10 citations
The authors develop the theory of the degenerate and nondegenerate two-photon micromaser using the full, three-level Hamiltonian under conditions of overall two-photon resonance. Subject to this condition, their results are valid for arbitrary one-photon detunings, but they consider in particular the large detuning and zero detuning limits, where some simplification is possible. For the case of zero detuning, they obtain semianalytic solutions, and find that, depending upon the interaction time, the system lases as either a single-mode or a two-mode system. They have compared their results with the degenerate two-photon micromaser, and find qualitative differences. Their results are valid for arbitrary one-photon detunings, provided the interaction times are not too long. The solution to the Fokker-Planck equation is found using a detailed balance, linearization procedure, and the photon statistics is investigated. Markedly sub-Poissonian behaviour is possible under certain conditions. They find that the behaviour in the zero detuning, nondegenerate situation is quite distinct to that of all other cases.
Exact results for the photon statistics of a two-photon micromaser are presented, including the effect of finite detuning of the intermediate level. A periodic dependence on interaction times is shown to arise for large detunings, when sufficiently long interaction times are considered. Some typical photon-number distributions are shown, including some exhibiting sub-Poissonian fluctuations and multiple peaks. Previous treatments and the validity of some usual approximations are discussed. By direct comparison with the exact results, it is shown that the effective-Hamiltonian approximation yields incorrect equations for the off-diagonal elements of the field density matrix.
We present the theory of a microscopic maser operating on a nondegenerate two-photon transition between two states of same parity. We start from a three-level Hamiltonian, and discuss both the one-photon resonant cascade and the two-photon regime, when the intermediate state is far off resonance. In the semiclassical limit, we derive rate equations for the field intensities and get the corresponding steady-state values. The vacuum is stable in the two-photon regime but becomes unstable for zero detuning (a clear signature of a one-photon process). Solutions displaying true two-photon operation (intermediate state unpopulated) even with zero detuning are displayed. Using the full quantum density-matrix approach, we derive a master equation for the field and discuss both operating regimes, showing that the spontaneous emission may turn the vacuum unstable in the two-photon case. In the off-resonance two-photon regime, there is a strong correlation between the fluctuations of the intensities of the two modes. We find a squeezing factor of 50% for the fluctuations in the difference of intensities.
In this paper, we study the linewidth of a two-photon micromaser by using an exact Hamiltonian approach. We consider a three-level atomic system interacting with a single-mode radiation field. The intermediate level is considered to be off resonant with the field. In this model, under the large detuning limit, two-photon transitions result. We calculate the linewidth of a two-photon micromaser and present results in the small and large detuning limits. In the small detuning limit, single-photon behavior is evident, whereas in the large detuning limit, true two-photon behavior is exhibited. We compare the results for the linewidth, under a large detuning limit, with the results obtained by using a phenomenological, effective Hamiltonian approach. The absence of the dynamic Stark shift in a phenomenological, effective Hamiltonian leads to qualitatively different results in the two approaches. \textcopyright{} 1996 The American Physical Society.
In this dissertation, we study the quantum theory of two-photon micromaser and laser. To begin with, we study the photon statistics of the two-photon micromaser by considering the interaction of a quantized single-mode field with an effective two-level atom through intermediate states. We present an exact solution for the photon distribution function and study the behaviour of mean photon number and photon number fluctuations. A periodic "collapse" and "revival" type behaviour is predicted. We find that the photon number fluctuations exhibits a periodic sub-Poissonian behaviour with respect to the interaction time. We then discuss the photon statistics of two-photon laser and derive an exact expression for photon distribution function. We find that photon distribution function for two-photon laser is wider than the Poisson distribution function. We also calculate the natural line-width of a two-level two-photon laser by using a Fokker-Planck approach. It is shown that, apart from a constant, the gain contribution to the laser linewidth is equal to the two-photon gain parameter and is independent of the mean number of photons in the field. Finally, we study the two-photon transitions in multi-level atomic system. In this case first we discuss the photon statistics of three-level two-photon micromaser and laser. Then we consider the two-photon transition in four-level atomic system by considering dye laser as an example.
Quantum interference effects are theoretically investigated in the steady-state photon statistics of single-mode Λ- and V-type micromasers with injected three-level atoms in a superposition of their internal states. The two lower (upper) levels are nondegenerate and coherent in a general case. For the Λ-type case, we have discussed how detunings affect the photon statistics of the noninversion micromaser. Under the proper conditions strongly sub-Poissonian field states with a large number of photons can be easily produced in this novel micromaser even if there exists no coherence between the two lower levels. For the V-type case, we have analysed how the photon statistics changes with the increase of the ground-state population. Although the generation of the strongly sub-Poissonian states is always possible in the micromaser under the noninversion case, their photon intensities become very weak compared to the Λ-type case. Our results indicate that it is perfectly possible to experimentally realize a noninversion micromaser with strongly sub-Poissonian statistics in the Λ-type system.
This paper presents new results on aspects of the two-photon laser. A semiclassical investigation of threshold behaviour reveals that the atom-field coupling is already so strong at threshold that a non-perturbative treatment of the dynamics is necessary. A master equation is derived on the basis of an effective Hamiltonian generalized in that it includes the effect of an overall two-photon detuning, the effect of ionization of the upper atomic level and the effect of dynamic Stark shifts. From the diagonal elements of the master equation, a Fokker-Planck equation is obtained and shown to yield a general analytical expression for the photon distribution which is found to simplify considerably in special cases. The agreement with the result obtained through numerical solution of the master equation itself is very good. It is further shown that a detuning from two-photon resonance and a significant Stark shift lead to sub-Poissonian statistics. Finally, time-dependent solutions for the photon distribution under an injected coherent state are presented.