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Optica Acta International Journal of Optics

Field Fluctuations in a Fabry-Pérot Cavity in Resonance with a Reservoir of Two-level Atoms: II . Non–adiabatic periodic fluctuations

R. K. Colegrave, M. Sebawe Abdalla

Published Jun 1, 1983• 26 citations
Publisher

Abstract

Two kinds of periodic fluctuation in the intensity of the field in a Fabry-Pérot cavity are considered: (a) a strong pulsation with periodically vanishing intensity, identifiable with the interaction of a low-density atomic reservoir and (b) a pulsation of adjustable strength and non-vanishing intensity whose frequency is approximately twice that of a natural mode. Self-consistent values are found in each case for the resonance frequency and for the on-resonance Rabi frequency. Case (b) may be identified with a pump-signal-idler system in which the signal and idler frequencies are approximately equal, and these modes can be damped out if the pulsation is sufficiently strong.

DOI
10.1080/716099648a

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Field Fluctuations in a Fabry-Pérot Cavity in Resonance with a Reservoir of Two-level Atoms: I. Adiabatic fluctuations

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In Chapters 8–13 we always assumed that the nonlinear medium is contained in a unidirectional ring cavity. The case of a Fabry–Perot cavity (Fig. 8.1) is substantially more complex because the interference of the two counterpropagating fields gives rise to a standing-wave pattern that modulates the population difference and the atomic polarization to form gratings with a spatial frequency equal to the inverse of the half-wavelength and its harmonics. Therefore the atomic variables exhibit, in addition to the slow spatial variation which characterizes the envelopes, also a fast spatial variation on the wavelength scale, and the coexistence of these two spatial scales poses a highly nontrivial problem that must be treated with care. In Chapters 9 and 11 we solved exactly the Maxwell–Bloch equations with the ring-cavity boundary conditions for both the active and the passive configuration. Exact solutions of this kind have been obtained also for the equations and boundary conditions of a Fabry–Perot cavity [117–119]. However, for the sake of simplicity we will pass directly to models that are valid in the low-transmission approximation. Again for the sake of simplicity, in this chapter we assume that n B = 1, so that the light velocity in the empty part of the cavity is the same as that in the atomic sample. By following the same line of reasoning as that which we pursued in Chapter 12, in Section 14.1 we introduce a spatial transformation of the electric-field envelopes, which leads to boundary conditions without parameters, while the parameters present in the boundary conditions of the original variables appear in the envelope equations. As in the ring-cavity case, this approach allows one to introduce the cavity modes in a natural way, and to derive a set of modal equations coupled to the equations for the atomic variables. Sets of equations of this kind are common in the laser literature, but are not derived from the true boundary conditions for the cavity.

Cambridge University Press eBooks•Feb 28, 2015•0 citations
Two kinds of periodic fluctuation in the intensity of the field in a Fabry-Pérot cavity are considered: (a) a strong pulsation with periodically vanishing intensity, identifiable with the interaction of a low-density atomic reservoir and (b) a pulsation of adjustable strength and non-vanishing intensity whose frequency is approximately twice that of a natural mode. Self-consistent values are found in each case for the resonance frequency and for the on-resonance Rabi frequency. Case (b) may be identified with a pump-signal-idler system in which the signal and idler frequencies are approximately equal, and these modes can be damped out if the pulsation is sufficiently strong.
A canonical approach using variable mass parameters presented in an earlier paper is applied to the case of an electromagnetic field of slowly varying intensity in a cavity in the presence of a reservoir of resonant two-level atoms. The effect is to replace any cavity frequency y s by a reduced variable frequency z s (t). Calculations have been carried out for the changes in the atomic transition amplitudes that are produced by an externally imposed adiabatic change in the field intensity, or by a small low-frequency periodic fluctuation in the photon population which could be due to the Rabi flopping of the atoms.
We consider the chaotic dynamics of the interaction between an ensemble of two-level atoms in a high-Q Fabry-Perot cavity with a single mode of self-consistent field and with an external amplitude-modulated field. It is shown that in the case of an exact atom-field resonance and at the initial population of the atomic ground state, the Hamiltonian chaos in the system is always transient. The differences in the chaotic dynamics for the Fabry-Perot and ring cavity geometries are considered. The conditions for the experimental observation of the transient Hamiltonian chaos are discussed.
We start from the Maxwell-Bloch equations that govern the dynamics of Fabry-Perot lasers and, considering a doubled cavity, generalize our traveling wave formalism to the case in which the two-level medium does not fill the cavity but occupies only a portion of it. We linearize these equations and focus on the adiabatic elimination of both atomic fluctuations. Next, we restrict our attention to free running lasers under resonant conditions and analyze amplitude and phase instabilities. The examination of the unstable domain leads to the conclusion that the multimode Fabry-Perot instability arises near threshold only when the ratio of the longitudinal to the transverse atomic relaxation rate is substantially smaller than unity. This result agrees with our previous study in the limit of adiabatic elimination of the atomic polarization fluctuations only. We describe the self-pulsing behavior that arises from the multimode instability, and exhibits hysteretic behavior when the pump parameter is swept forward and backward. Finally, we investigate the single-mode instability and show that in the Fabry-Perot case there is no longer the correspondence between single-mode and multimode instabilities that is well known in the case of ring lasers. We confirm that in the case of resonant Fabry-Perot lasers the single-mode amplitude instability arises very far from threshold.
We have investigated nonlinear atom optical effects which arise from atom-atom collisional interactions in a single-mode atomic Fabry-Perot cavity driven by a coherent cw atom laser beam. When the nonlinear interaction energy per single atom is small, the exact numerical solution of the master equation is well reproduced by a mean-field treatment in which quantum fluctuations are included linearizing the stochastic equations of the Positive-P representation. On the other hand, when the damping of the cavity mode is very weak and its wave-function is tightly confined in space, a regime of strong nonlinearity can be achieved. For the specific case of an incident atom laser frequency at resonance with the empty cavity, the numerical calculations predict a sort of atom blockade effect, which is a sort of atom optical analog of the well-known Coulomb blockade effect of electronic transport through microscopic structures: only one atom can occupy the cavity mode at a time and the statistical properties of the transmitted beam, being very similar to the resonance fluorescence from a single two-level system, show definitely nonclassical behaviors such as antibunching. Only at very large incident intensities, more than one atom can be simultaneously forced inside the cavity mode: in this regime, the results of the numerical calculations can be successfully interpreted using a dressed cavity model. From the formal analogy between atomic matter waves and optical light waves in nonlinear media, it follows that the same results hold for photonic systems.
In Chapters 8–13 we always assumed that the nonlinear medium is contained in a unidirectional ring cavity. The case of a Fabry–Perot cavity (Fig. 8.1) is substantially more complex because the interference of the two counterpropagating fields gives rise to a standing-wave pattern that modulates the population difference and the atomic polarization to form gratings with a spatial frequency equal to the inverse of the half-wavelength and its harmonics. Therefore the atomic variables exhibit, in addition to the slow spatial variation which characterizes the envelopes, also a fast spatial variation on the wavelength scale, and the coexistence of these two spatial scales poses a highly nontrivial problem that must be treated with care. In Chapters 9 and 11 we solved exactly the Maxwell–Bloch equations with the ring-cavity boundary conditions for both the active and the passive configuration. Exact solutions of this kind have been obtained also for the equations and boundary conditions of a Fabry–Perot cavity [117–119]. However, for the sake of simplicity we will pass directly to models that are valid in the low-transmission approximation. Again for the sake of simplicity, in this chapter we assume that n B = 1, so that the light velocity in the empty part of the cavity is the same as that in the atomic sample. By following the same line of reasoning as that which we pursued in Chapter 12, in Section 14.1 we introduce a spatial transformation of the electric-field envelopes, which leads to boundary conditions without parameters, while the parameters present in the boundary conditions of the original variables appear in the envelope equations. As in the ring-cavity case, this approach allows one to introduce the cavity modes in a natural way, and to derive a set of modal equations coupled to the equations for the atomic variables. Sets of equations of this kind are common in the laser literature, but are not derived from the true boundary conditions for the cavity.