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Physica Scripta

Quantum theory of the coherently pumped micromaser

István Németh, János A. Bergou

Published Jul 1, 2009• 0 citations
Publisher

Abstract

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 realized in a micromaser is very close to the ideal case of a single two-level atom interacting with a single quantized mode of a superconducting cavity. This system's ability to coherently transfer quantum states between atoms and photons made it relevant in the context of quantum computation as well. Although considerable work, both theoretical and experimental, has been devoted to this system, with a few exceptions, most cases involved only non-coherent pumping. As a result, the density matrix describing the cavity field remained diagonal, preventing the appearance of coherences. Here, our goal is to provide a summary of the progress we made toward a comprehensive understanding of the features of the coherently pumped micromaser with special respect to coherences, which are central to quantum information processing.

DOI
10.1088/0031-8949/2009/t135/014017

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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 realized in a micromaser is very close to the ideal case of a single two-level atom interacting with a single quantized mode of a superconducting cavity. This system's ability to coherently transfer quantum states between atoms and photons made it relevant in the context of quantum computation as well. Although considerable work, both theoretical and experimental, has been devoted to this system, with a few exceptions, most cases involved only non-coherent pumping. As a result, the density matrix describing the cavity field remained diagonal, preventing the appearance of coherences. Here, our goal is to provide a summary of the progress we made toward a comprehensive understanding of the features of the coherently pumped micromaser with special respect to coherences, which are central to quantum information processing.
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.
The development of a single-atom maser or a micromaser allows a detailed study of the atom–field interaction. The situation realized is very close to the ideal case of a single two-level atom interacting with a single-mode quantized field as treated in Section 6.2. In a micromaser a stream of two-level atoms is injected into a superconducting cavity with a high quality factor. The injection rate can be such that only one atom is present inside the resonator at any time. Due to the high quality factor of the cavity, the radiation decay time is much larger than the characteristic time of the atom–field interaction, which is given by the inverse of the single-photon Rabi frequency. Therefore, a field is built up inside the cavity when the mean time between the atoms injected into the cavity is shorter than the cavity decay time. A micromaser, therefore, allows sustained oscillations with less than one atom on the average in the cavity. The realization of a single-atom maser or a micromaser has been made possible due to the enormous progress in the construction of superconducting cavities together with the laser preparation of highly excited atoms called Rydberg atoms. The quality factor of the superconducting cavities is high enough for periodic energy exchanges between atom and cavity field to be observed. The interesting properties of the Rydberg atoms make them ideal for micromasers. In Rydberg atoms the probability of induced transitions between adjacent states becomes very large and scales as n 4 , where n denotes the principle quantum number.
The microscopically pumped maser, or micromaser is a cavity QED experiment intended to be a physical realisation of the Jaynes-Cummings model of a single two-level atom interacting with a single mode electromagnetic field. This is the simplest model that describes the interaction between light and matter, yet it predicts behaviour unexpected from semiclassical models, such as the revival of Rabi oscillations of an atom interacting with an initially coherent field and non-monotonic linewidth as a function of pumping. The micromaser at the University of Leeds consists of a high quality superconducting microwave cavity designed to be resonant with the transition between two specific Rydberg states of rubidium. These two states behave like an ideal two level atom, and couple strongly to the cavity field due to a large dipole moment. These Rydberg atoms are passed through the cavity in a rarified beam such that in most instances when there is an atom in the cavity, there will only be one, closely approximating the Jaynes-Cummings model. I present experimental work on the build phase of the micromaser. Specifically I routed all of the wiring and microwave lines in the cryostat that contains the micromaser, and designed mounts for various components. I also designed several testing methods for probing high quality microwave cavity resonances and quality factors which are presented. Using the Jaynes-Cummings model as a prototype, I demonstrate how extensions to the model can be used to construct universal quantum logic gates that operate on photonic qubits in a multi-mode cavity. This could be realised in a micromaser with a multi-mode cavity. Conversely, I demonstrate that by using atoms as qubits, detuned cavities can be used to generate entangled resources such as the Greenberger-Horne-Zeilinger state, the W state, and graph states of atoms. I show that single qubit rotations on Rydberg atom qubits have already been experimentally demonstrated so that in combination these entangled resources are useful for quantum metrology, quantum computation and even tests of quantum gravity.
There are only fistful of systems in physics that can be described by an exact theory from first principles and, at the same time, can be investigated under experimental conditions approaching the idealized theoretical ones. The one-atom-maser or micromaser provides such an example, making a detailed study of the fundamental properties of the atom-field interaction possible. The situation realized in a micromaser is very close to the ideal case of a single two-level atom interacting with a single quantized mode of a cavity field. In the micromaser the atoms have a dual purpose of both pumping the field and also probing it via measurements of the outgoing atoms. Any other means of measuring the field inside the resonator has the detrimental effect of lowering its Q-factor and thereby the photon storage time. The long photon storage time of the resonator allows for the decay of the field to be negligible during the passage of an atom and its interaction with the field. The behavior of atoms in a cavity is governed by the oscillatory exchange of energy between the atoms and the field, which is called Rabi oscillation. For cavity fields in the vacuum and few photon number states (Fock states), Rabi oscillations have been measured in the past. To prevent thermal effects from polluting the pure number states these experiments were performed at low temperatures (below 1 K). However, the observed resolution of the Rabi oscillations measurements was quite disappointing (only about 2% of the predicted theoretical value) and precluded more ambitious experiments like atom-atom correlations or time resolved investigations of the maser field dynamics. The observed contrast of Rabi oscillations can be considered as a figure of merit for the experimental resolution of quantum features of a maser field. Therefore, a high resolution in the measurements of Rabi oscillations is crucial. This thesis describes the methods and improvements applied to enhance the resolution of Rabi oscillations by a factor of 8. The following upgrades and developments proved to be crucial in this work: 1) The base temperature of the cryogenic system was lowered to 0.3 K, i.e. one third of the original value. At the same time, the time during which the system remains at the base temperature was improved by almost of an order of magnitude (to more than 12 h). 2) For the promotion of the atoms to the Rydberg regime new three-step diode laser setup was constructed. The excitation efficiency compared to the old dye laser was increased by two orders. Using top-of-fringe stabilization with new spectroscopic methods, specially designed error detection and feedback schemes as well as state-of-the-art PID regulators the continuous locking of lasers is about 8 h (compared to about 20 min with the old dye laser system). 3) The flexibility of using the three step excitation allows to excite different maser states and for the first time to investigate pure maser transition. 4) Various improvements on the atomic oven and the channeltron detection unit led to the reliable and stable production and detection of the atomic beam. Observed distribution of the atomic beam statistics reaches to within 6% the Poisson limit which is expected in the absence of experimental imperfections. 5) New developed techniques in magnetic field compensation allow to perform such measurements with two orders of magnitude better accuracy compared to the previous experiments. To demonstrate the new capabilities of the apparatus and to apply its improved resolution of quantum fields we performed a previously impossible in-depth analysis of mean photon number and the atomic inversion produced by a micromaser for various detunings between the cavity- and atomic resonance frequency.
A distinctive signature of the granulated nature of the electromagnetic field is the revivals known to occur in the dynamics of a two-level atom interacting with a single-field mode (Jaynes-Cummings model). The development of high-Q single-mode cavities combined with progress in Rydberg atom spectroscopy has led to a possibility of verifying these predictions. Revival experiments using micromasers have been performed by Walther et al. Here the cavity mode is first allowed to reach a steady state, and then probe atoms are injected into the resonator. By measuring their state as they exit the cavity, one can infer the state of the field. We show that the micromaser experiences Jaynes-Cummingslike revivals when the pump parameter θ is varied. However, in contrast to the conventional Jaynes-Cummings situation, which considers initially coherent fields, the micromaser photon statistics are usually not Poissonian and depend sensitively on θ. The micromaser revivals cannot be explained by a semiclassical theory or linearized quantum fluctuations about the semiclassical solution. Rather, an exact quantum treatment, considering fully the granularity of the cavity mode, is required.
This chapter discusses the novel aspects of cavity QED that are not only relevant to traditional quantum optics (lasers and masers, coherence theory, non-classical fields, etc.), but simultaneously make contact with nonlinear dynamics and chaos. A micromaser consisting of a single-mode cavity in which two-level atoms are injected at such a low rate that at most one atom at a time is present inside the cavity. In recent work, a complete quantum theory of the micromaser, including resonator losses, is presented. A detailed study of the dynamics of the quantum-mechanical micromaser is called for, and it will hopefully contribute to a better understanding of quantum chaos in dissipative quantum mechanical systems. A semi-classical version of the micromaser is generally chaotic and exhibits a number of coexisting basins of attraction. The micromaser is a truly quantum-mechanical system which is within reach of experimental investigations.