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Physical Review A

Bounds on sub-Poissonian field statistics in the Jaynes-Cummings model

Mark S. Hillery

Published May 1, 1987• 29 citations
Publisher

Abstract

The Jaynes-Cummings model describes the interaction between a two-level atom and a single mode of the radiation field. This system has a conservation law which restricts its dynamics. On the basis of this conservation law alone one can make useful statements about the photon statistics of the fields which arise in the model. Two cases are considered. In the first the field is initially in a coherent state. Here it is found that the field statistics become less sub-Poissonian as the amplitude of the initial state increases. The case of an initial thermal state is also considered. In that situation, if the average number of photons present in the initial state is greater than 2.55, then the field statistics can never become sub-Poissonian.

DOI
10.1103/physreva.35.4186

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The Jaynes-Cummings model describes the interaction between a two-level atom and a single mode of the radiation field. This system has a conservation law which restricts its dynamics. On the basis of this conservation law alone one can make useful statements about the photon statistics of the fields which arise in the model. Two cases are considered. In the first the field is initially in a coherent state. Here it is found that the field statistics become less sub-Poissonian as the amplitude of the initial state increases. The case of an initial thermal state is also considered. In that situation, if the average number of photons present in the initial state is greater than 2.55, then the field statistics can never become sub-Poissonian.
There is a conservation law present in the Jaynes-Cummings model which restricts its dynamics. This means that from a given initial state only a subset of all possible states of the system can be reached. This allows one to place bounds on the expectation values of observables of the system. Using this procedure bounds are found on the extent to which the field statistics can become sub-Poissonian for an initial coherent state. Mandel’s Q parameter, which measures the extent to which the statistics are sub-Poissonian, is shown to be a decreasing function of the initial field amplitude. For an initial thermal field it is found that there is a temperature (which depends on the field frequency) above which the field never becomes sub-Poissonian. These methods can be extended to multiatom systems, and results for such systems are presented.
We study N two-level atoms coupled to a single mode of the electromagnetic field. This system obeys a conservation law that restricts the set of states which can be reached by time evolution from a given initial state. This fact allows one to calculate time-independent bounds for certain observables, in particular, the uncertainty in photon number, \ensuremath{\Delta}n. These bounds can be used to determine how sub- or super-Poissonian the photon statistics can become. This is done for an initial coherent state and an initial thermal state. For the coherent state the number of atoms must be greater than the square root of the initial number of photons for highly sub-Poissonian fields to be possible. On the other hand, for the thermal state N must be greater than the initial photon number for this to occur. In the case of the thermal state, we also find that the field will never become sub-Poissonian if the ratio of the initial photon number to N is greater than a quantity of order 1.