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Journal of Optics B Quantum and Semiclassical Optics

Micromasers: a quantum jump approach

E. S. Guerra, Carlos R. de Carvalho

Published Jul 30, 2001• 3 citations
Publisher

Abstract

We have developed a quantum jump approach for the one photon micromaser with a Poissonian atomic pumping beam. In our approach we can describe the dynamics of the system, completely unravelling the master equation in terms of stochastic wavefunctions. We have applied the approach to the most general case in which the atoms enter the cavity in an arbitrary coherent superposition of the atomic states with different initial conditions for the cavity field. Hence, we are able to study the system dynamics when the atoms enter the cavity in the excited state and interacting initially with the field in a pure state as well as when the atoms enter the cavity in a coherent superposition with the field initially in a statistical mixture leading to the production of a cotangent state.

DOI
10.1088/1464-4266/3/4/310

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We have developed a quantum jump approach for the one photon micromaser with a Poissonian atomic pumping beam. In our approach we can describe the dynamics of the system, completely unravelling the master equation in terms of stochastic wavefunctions. We have applied the approach to the most general case in which the atoms enter the cavity in an arbitrary coherent superposition of the atomic states with different initial conditions for the cavity field. Hence, we are able to study the system dynamics when the atoms enter the cavity in the excited state and interacting initially with the field in a pure state as well as when the atoms enter the cavity in a coherent superposition with the field initially in a statistical mixture leading to the production of a cotangent state.
We investigate the quantum dynamics of a two-photon micromaser pumped by atoms injected in the superposition state of the upper and intermediate levels. We simulate a master equation governing the system by the Monte Carlo wavefunction approach and analyse the steady-state behaviour as a function of the atomic transit time. The atomic coherence can effectively enhance the intensity and sub-Poissonian of the cavity field as compared with the atomic mixture. It is also discovered that the phase of the cavity field can be shifted by adjusting the detuning between the atom and field. This result shows that it is possible to manipulate the phase of the cavity field by detuning, due to atomic coherence.
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.
In this paper results are reported for the application of the quantum trajectory method to analysing the dynamics of the micromaser. The master equation for the cavity field is derived by a method which highlights the close relationship between the quantum trajectory method and the usual model for the one-atom micromaser. Possible quantum trajectory unravellings of the micromaser dynamics are discussed, and a simple number-state unravelling is described and used to generate simulations from which the cavity-field intensity correlation function is calculated. Attention is principally confined to the case of a Poissonian atomic beam, though a method of dealing with the non-Markovian master equation obtained for non-Poissonian beams is also examined. A formal theory of atomic detection analogous to the Glauber - Kelly - Kleiner theory of photodetection is proposed, based on a quantum-field description of the atomic beam. This theory is used to establish the connection between the atomic detection record obtained by continuous monitoring of the atoms leaving the cavity and the quantum trajectories followed by the cavity field. Results obtained by other treatments of atomic detection are regained. A classical correlation function obtained from the detection record of ground-state atoms emerging from a micromaser cavity is then shown, by use of a quantum trajectory analysis, to be identical to the cavity field intensity correlation function, apart from shot-noise-type contributions, provided the cavity reservoir is at zero temperature.
A general theory of the micromaser is described. The model is based on treating the input atomic beam as a two-component quantum field so that the two-level atoms in the beam are ``quanta'' of this field. This approach makes it possible to formulate a general quantum Langevin description of the dynamics with the input atomic field as a source of quantum noise. The passage to a master-equation description is then effected by use of an adjoint-operator method, and by introducing a general class of statistical states for the atomic beam known as generalized shot noise. The result is a (non-Markovian) master equation for the field inside the cavity which is valid for a broad range of statistical properties of the input atomic beam. The approximate steady-state solutions to this master equation for the photon statistics of the cavity field for sub-Poissonian (antibunched) atomic beams found by other researchers are regained. The theory is then extended to treat super-Poissonian (bunched) atomic beams. An exact result is found in the limit of a strongly bunched beam in which the cavity-field state is shown to converge to a mixture of a thermal-field state and the state produced by a random beam of twice the intensity. A physical explanation of this result is also presented.
We present a quantum theory of the micromaser which allows for arbitrary (sub- as well as super-Poissonian) fluctuations of the pumping beam. It handles the interaction with the active atoms (gain) and cavity decay (loss) on an equal footing. In conventional reservoir theory the rate of change of the cavity field is a sum of the changes due to separate interactions with the individual reservoirs, i.e., the interactions are uncorrelated. In our approach, corrections to reservoir theory arise. They contain the commutator of the gain and loss operators. The magnitude of these additional terms is characterized by the quantity p/Nex{\mathit{N}}_{\mathrm{ex}}Nex​ where p is a parameter describing pump-beam fluctuations. The parameter is, in fact, the negative of the Mandel Q parameter of the pump beam so that p=1 corresponds to regular pumping, p=0 to Poissonian one and p0 to super-Poissonian pump beam fluctuations. Nex{\mathit{N}}_{\mathrm{ex}}Nex​ is the number of excited atoms passing through the cavity during the lifetime of the intracavity field. Thus we recover the conventional reservoir limit if p=0 and/or Nex{\mathit{N}}_{\mathrm{ex}}Nex​ is large. In all other cases the interactions with the gain and loss reservoirs are correlated. We exploit some of the consequences of the additional terms, presenting analytical as well as numerical results for the steady-state photon statistics (mean photon number and photon-number fluctuations, in particular) without resorting to the p expansion.
A detailed theoretical and experimental study of the atom-field interaction starting from first principles was made possible by the realization of the single-atom micromaser. The situation is very close to the ideal case of a single two-level atom interacting with a single quantized mode of a superconducting cavity. In spite of the considerable amount of work devoted to the study of the micromaser, it still remains to be established how the system behaves under coherent pumping. We present a comprehensive study of the coherently pumped micromaser; we develop an analytic method to obtain the steady-state solution of the master equation governing the time evolution of the cavity field. We illustrate this method with a simple example and present results for the photon distribution and purity of the steady state of the coherently pumped micromaser.