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Journal of Modern Optics

Exact trapping state dynamics for the 85Rb atom micromaser at very high Q and/or very low T

Amitabh Joshi

Published May 1, 1996• 1 citation
Publisher

Abstract

Matrix continued fraction methods pioneered previously [1–3] are used to solve essentially exactly the dynamics of the one 85Rb atom micromaser. Ranges of cavity Q values, of average atomic input rates R = T p −1 and of atomic interaction times t int are studied with the principal aim of minimizing the cavity field variance v. As expected for T p ≳ 200t int the system does evolve to a steady state with v < 1 and sub-Poissonian photon statistics. But even Q = 5 × 1010 and temperature T = 0·070 K are apparently insufficient conditions for avoiding evolution to a steady state with v < 0 when t int = 1·54/(44 kHz) unless T p is also substantially reduced. Only in this way is actual trapping state dynamics obtained for this value of t int, while no example is found where the system evolves to v = 0 and a single Fock state is produced. The field variance v considered alone is not a good diagnostic for observing such trapping state dynamics and more information will be needed in order to describe adequately the cavity field.

DOI
10.1080/095003496155689

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Matrix continued fraction methods pioneered previously [1–3] are used to solve essentially exactly the dynamics of the one 85Rb atom micromaser. Ranges of cavity Q values, of average atomic input rates R = T p −1 and of atomic interaction times t int are studied with the principal aim of minimizing the cavity field variance v. As expected for T p ≳ 200t int the system does evolve to a steady state with v < 1 and sub-Poissonian photon statistics. But even Q = 5 × 1010 and temperature T = 0·070 K are apparently insufficient conditions for avoiding evolution to a steady state with v < 0 when t int = 1·54/(44 kHz) unless T p is also substantially reduced. Only in this way is actual trapping state dynamics obtained for this value of t int, while no example is found where the system evolves to v = 0 and a single Fock state is produced. The field variance v considered alone is not a good diagnostic for observing such trapping state dynamics and more information will be needed in order to describe adequately the cavity field.
Matrix continued fraction methods pioneered previously [1–3] are used to solve essentially exactly the dynamics of the one 85Rb atom micromaser. Ranges of cavity Q values, of average atomic input rates R = T p −1 and of atomic interaction times t int are studied with the principal aim of minimizing the cavity field variance v. As expected for T p ≳ 200t int the system does evolve to a steady state with v < 1 and sub-Poissonian photon statistics. But even Q = 5 × 1010 and temperature T = 0·070 K are apparently insufficient conditions for avoiding evolution to a steady state with v < 0 when t int = 1·54/(44 kHz) unless T p is also substantially reduced. Only in this way is actual trapping state dynamics obtained for this value of t int, while no example is found where the system evolves to v = 0 and a single Fock state is produced. The field variance v considered alone is not a good diagnostic for observing such trapping state dynamics and more information will be needed in order to describe adequately the cavity field.
We have used matrix continued fraction methods pioneered previously, to solve essentially exactly for the dynamics of the one 85Rb atom micromaser. A range of cavity Q values and of average atomic input rates R equals Tp-1 and of atomic interaction times tint is studied. For Tp approximately greater than 200tint the system does evolve to a steady state. And even Q equals 5 multiplied by 1010 and temperature T equals 0.070 degrees K are apparently insufficient to avoid evolution to a steady state with finite variance of the cavity field when tint equals 1.54/44 kHz unless Tp is also substantially reduced. Only in this way are actual trapping state dynamics obtained. The field variance v is not a good diagnostic for observing such trapping state dynamics.
We report a numerical simulation of the action of the one 85 Rb atom micromaser which shows that the micromaser field evolves towards a steady state equilibrium when the condition that Tp t int satisfies. In this steady state regime the cavity field reaches a steady state after a sufficient number of atoms have passed through the cavity. The cavity temperature T and the cavity quality factor Q can produce a quantitive change when one of them or the two are changed but they are not responsible for the qualitative changes in the micromaser field. Evidently the only responsible parameter for these qualitative changes is the repetition time T p namely if T p t int then the micromaser field evolution is towards a steady state equilibrium. In contrast when T p reduced to the order of t int then the micromaser field evolution is controlled by the trapping state dynamics – no steady state is apparent in this case. The first going-up trapping state at n=3 plays a significant role in the early dynamics of the micromaser field. Moreover the state of the field ultimately evolves towards a mixed state rather than a pure state.
Trapping states in the one-atom maser or micromaser are predicted by maser theory and are a direct consequence of the quantization of the electromagnetic field. They have been found in our experiment when the inversion of atoms leaving the cavity is observed while the interaction time is scanned. They correspond to quantum states of the atom-cavity system and under certain conditions represent Fock states. The trapping states occur when the atom-field dynamics complete a full evolution (or multiples thereof). Owing to feedback through the maser dynamics, a stabilization of the photon number in the steady state field of the maser cavity is achieved.
An exact method for the simulation of micromaser action is presented. The equation of motion governing the atom–field interaction is solved by inclusion of the effect of cavity dissipation at finite temperature. When the cavity is without an atom, the cavity–reservoir equation as it appears in quantum optics is solved analytically, which is valid for any photon-distribution function required by the initial conditions. The solutions can be fitted into any statistical input of atoms with the condition that, at most, one atom passes through the cavity at any time, and the field as well as the atomic statistics can be obtained. Numerical results are discussed for a Poissonian arrival of atoms at the cavity, and it is observed that the fluctuations in the photon-distribution function, caused solely by the randomness of the pump mechanism, may make the attainment of a Fock state of the cavity field a difficult task experimentally.
Using a Monte Carlo wavefunction approach we investigate the dynamics of a micromaser driven by a resonant coherent field. At steady state, for increasing interaction times, the system exhibits driven Rabi oscillations, followed by collapse as the range of micromaser trapping states is approached. The system operates in regimes ranging from a strong to a weak amplifier. In the strong-amplifier regime the cavity mode shows a preferred phase and can exhibit quadrature squeezing and sub-Poissonian photon statistics. In the weak-amplifier regime the cavity mode has no preferred phase, is super-Poissonian and is influenced by trapping effects; no revival of Rabi oscillations occurs. The main predictions can be compared with experimental measurements on the populations of atoms leaving the cavity.