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arXiv:quant-ph/0104128 · Physical Review A

Conditional evolution in single-atom cavity QED

Andrei N. Soklakov, Rüdiger Schack

Published Dec 5, 2001• 2 citations
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Abstract

We consider a typical setup of cavity QED consisting of a two-level atom interacting strongly with a single resonant electromagnetic-field mode inside a cavity. The cavity is resonantly driven and the output undergoes continuous homodyne measurements. We derive an explicit expression for the state of the system conditional on a discrete photocount record. This expression takes a particularly simple form if the system is initially in the steady state. As a byproduct, we derive a formula for the steady state that had been conjectured before in the strong driving limit.

DOI
10.1103/physreva.65.013804

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We consider a typical setup of cavity QED consisting of a two-level atom interacting strongly with a single resonant electromagnetic-field mode inside a cavity. The cavity is resonantly driven and the output undergoes continuous homodyne measurements. We derive an explicit expression for the state of the system conditional on a discrete photocount record. This expression takes a particularly simple form if the system is initially in the steady state. As a byproduct, we derive a formula for the steady state that had been conjectured before in the strong driving limit.
A common experimental setup in cavity quantum electrodynamics consists of a single two-level atom interacting with a single mode of the electromagnetic field inside an optical cavity. The cavity is externally driven and the output is continuously monitored via homodyne measurements. We derive formulas for the optimal rates at which these measurements provide information about (i) the quantum state of the system composed of atom and electromagnetic field, and (ii) the coupling strength between atom and field. We find that the two information rates are anticorrelated.
A single two-level emitter coupled to a single mode of an electromagnetic resonator constitutes the most fundamental instance of light--matter interaction. Realizations of this type of system are investigated in the field of cavity quantum electrodynamics (cavity QED). Particularly important are schemes in which the coupling is strong enough to exceed the spontaneous emission rate as well as cavity damping. In this case, the coherent exchange of excitation between source and photon is the dominating process, providing the unique possibility to deterministically control the quantum evolution of the system. Great progress has recently been accomplished, technologically as well as conceptually, and the focus section in this issue of Journal of Optics B: Quantum and Semiclassical Optics features five articles highlighting important recent developments in the field. Strong coupling between single atoms and the electromagnetic cavity field have been achieved in two distinct regimes, the optical and the microwave region. The latter is represented by the micromaser, a high-Q microwave cavity pumped by a beam of highly excited Rydberg atoms. It has proved to be an almost ideal system for the generation of non-classical field states. A constraint in these experiments is the fact that the field inside the cavity can only be examined indirectly, by analysing the occupation of atoms leaving the cavity. While only two atomic states were previously identified for this purpose, V A Reshetov shows in his article (pages 119--126) that more information on the field may be extracted by taking advantage of the Zeeman substructure of the atomic levels. In the optical domain of cavity QED, by contrast, the state of the resonator field may be probed directly by measuring the light leaking from the cavity using photodetectors. In order to minimize damping, high-quality cavities must be employed. In most optical experiments these are Fabry--Perot resonators with ultra-high reflectivity mirrors, with losses as low as a few parts per million being state of the art. At the same time, the coupling of atoms and photons is enhanced by reducing the length of the cavity, so that strong coupling conditions are reached. A very attractive application of this type of cavity QED setup is the possibility of exerting control over the quantum dynamics of the system. An impressive example is the implementation of quantum feedback, reported in the paper by W P Smith and L A Orozco (pages 127--134). Using information gained from detecting a single photon escaping from the cavity, active feedback is applied to halt the dynamical evolution of the system. This is the first time active feedback has been used to modify the quantum evolution of a system at the single-photon level. The authors discuss how detunings between the subsystems affect the quantum feedback, an issue particularly relevant for ensembles of atoms with a spread in detuning. Perfect quantum feedback in this case requires the use of additional parameters to control the system. The majority of optical cavity QED experiments performed so far have used as a medium atoms traversing the cavity on random trajectories, such as, for example, thermal atomic beams. In this case, the observation of quantum effects is degraded in two ways. The Doppler effect and the finite transit time lead to line broadening, while the spatial structure of the cavity mode results in atoms in different locations experiencing different coupling with the photonic field. J E Reiner et al (pages 135--142) investigate the impact of these effects on quantum fluctuations in their cavity QED system. In order to overcome line broadening, they have developed a continuous cold atomic beam, extracted from a magneto-optical trap. An even better control of broadening effects and fluctuating coupling strength is provided by trapping the atoms inside the cavity and cooling them to temperatures at which their position spread is smaller than the optical wavelength. Much recent activity has been devoted to developing such an ultimate cavity QED system. A Vukics et al (pages 143--153) present a theoretical proposal for dynamically trapping and cooling atoms inside a cavity. By driving the atoms with a standing wave injected from the side of the cavity and by weakly exciting the cavity field itself, they predict three-dimensional trapping of the atom for periods around 1 second, an extremely long time on the typical scale of cavity QED dynamics. The essential component of their scheme is the interference between the two driving fields, determining the temperature of the particle and the stable trapping positions. In the quest for better confinement of radiation, lately, an appealing alternative to the Fabry--Perot cavity has been investigated -- the optical microsphere resonator. A sphere with a diameter of a few tens of microns, it supports so-called whispering gallery modes near its circumference, for which quality factors up to 10 10 have been measured. In order to exploit the correspondingly low field damping rates for cavity QED, a method must be found to couple atomic or solid state emitters to these modes. S Göotzinger et al (pages 154--158) report the coupling of single nanocrystals on the surface of a microsphere through the evanescent field of the mode. Selective excitation of a single nano-emitter was accomplished with a scanning confocal microscope. The system is a promising candidate for cavity QED with single-quantum emitters. The ability to fully control the quantum dynamics of atoms and light is beginning to spawn early applications. The most notable example is quantum information processing. This requires the coherent manipulation of single-quantum systems, which is ideally provided by cavity QED interaction in the strong-coupling limit. The development of practical devices will be among the principal future tasks for research in the field. This will include physical systems, which have not been covered in this issue, such as semiconductors, quantum dots as emitters, or photonic crystals as resonator structures. Cavity QED is therefore guaranteed to remain a vigorous field in the years to come.
We propose experimental schemes for realizing frequency up- and down-conversion in two-mode cavity QED by considering the atom-cavity interaction in the presence of a strong driving classical field. In contrast to the recent paper based on dispersive atom-cavity interaction [Serra et al., Phys. Rev. A 71, 045802 (2005)], our scheme is based on resonant interaction of the cavity modes with a single driven three-level atom, so that the quantum dynamics operates at a high speed, which is important in view of decoherence. It is shown that, with the help of a strong driving classical field, frequency up- and down-conversion operations can be realized by initially preparing the atom in a certain state.
As atom-cavity systems are becoming more sophisticated, the limitations of the Jaynes-Cummings model are becoming more apparent. In this paper, we therefore take a more dynamical approach to the modelling of atom-cavity systems and do not reduce the electromagnetic field inside the resonator to a single mode. Our approach shows that the decay rate [Formula: see text] of an emitter inside a subwavelength cavity with metallic mirrors can be much larger than its free space decay rate [Formula: see text] due to constructive interference effects of the emitted light. In general, however, we find that [Formula: see text] to a very good approximation which might explain why atom-cavity experiments with planar mirrors have not been able to operate in the so-called strong coupling regime.
Summary form only given. The second order intensity correlation function can provide evidence of the non-classical nature of an electric field, as well as information about the dynamics which produce the field. In practice, this is a conditional measurement of the probability to detect a photon given that another photon was detected in coincidence or some time before. We have measured the correlation function of the photons escaping from a strongly coupled cavity QED system consisting of a high finesse interferometer traversed by a beam of optically pumped Rb atoms. In this regime, the rate of reversible exchange of energy between the atoms and the cavity is larger than the cavity decay and spontaneous emission rates, and the saturation photon number is less than one. The correlations exhibit the non-classical features of antibunching and sub-Poissonian statistics, and an oscillation related to the exchange of energy taking place in the coupled system of atoms and cavity field. We view the initial photon detection as causing a projection of the system into a well defined initial state from which it coherently evolves back to steady state. A full characterization of this conditional evolution requires the measurement of the electric field as it evolves from the initial state.
Summary form only given. We have developed a correlation technique that permits us to follow the evolution of the mean amplitude of the conditional electromagnetic field emitted from an optical system. The detection of a photon is correlated with the photocurrent record from a balanced homodyne detector. The measured signal in the homodyne detector is proportional to a correlation between particle wave aspects of light. One application of this technique is the measurement of quadrature amplitude fluctuations which violate classical bounds by orders of magnitude.We use this wave-particle correlator to study the time evolution of the electromagnetic field emitted from a cavity QED system. We operate in the optical regime of cavity QED where a beam of optically pumped two-level Rb atoms traverse the TEM/sub 00/ mode of a high finesse Fabry-Perot cavity.