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Physical Review A

Nonlinear atomic Fabry-Perot interferometer: From the mean-field theory to theatom blockadeeffect

Iacopo Carusotto

Published Jan 16, 2001• 38 citations
Publisher

Abstract

We have investigated nonlinear atom optical effects which arise from atom-atom collisional interactions in a single-mode atomic Fabry-Perot cavity driven by a coherent cw atom laser beam. When the nonlinear interaction energy per single atom is small, the exact numerical solution of the master equation is well reproduced by a mean-field treatment in which quantum fluctuations are included linearizing the stochastic equations of the Positive-P representation. On the other hand, when the damping of the cavity mode is very weak and its wave-function is tightly confined in space, a regime of strong nonlinearity can be achieved. For the specific case of an incident atom laser frequency at resonance with the empty cavity, the numerical calculations predict a sort of atom blockade effect, which is a sort of atom optical analog of the well-known Coulomb blockade effect of electronic transport through microscopic structures: only one atom can occupy the cavity mode at a time and the statistical properties of the transmitted beam, being very similar to the resonance fluorescence from a single two-level system, show definitely nonclassical behaviors such as antibunching. Only at very large incident intensities, more than one atom can be simultaneously forced inside the cavity mode: in this regime, the results of the numerical calculations can be successfully interpreted using a dressed cavity model. From the formal analogy between atomic matter waves and optical light waves in nonlinear media, it follows that the same results hold for photonic systems.

DOI
10.1103/physreva.63.023610

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We have investigated nonlinear atom optical effects which arise from atom-atom collisional interactions in a single-mode atomic Fabry-Perot cavity driven by a coherent cw atom laser beam. When the nonlinear interaction energy per single atom is small, the exact numerical solution of the master equation is well reproduced by a mean-field treatment in which quantum fluctuations are included linearizing the stochastic equations of the Positive-P representation. On the other hand, when the damping of the cavity mode is very weak and its wave-function is tightly confined in space, a regime of strong nonlinearity can be achieved. For the specific case of an incident atom laser frequency at resonance with the empty cavity, the numerical calculations predict a sort of atom blockade effect, which is a sort of atom optical analog of the well-known Coulomb blockade effect of electronic transport through microscopic structures: only one atom can occupy the cavity mode at a time and the statistical properties of the transmitted beam, being very similar to the resonance fluorescence from a single two-level system, show definitely nonclassical behaviors such as antibunching. Only at very large incident intensities, more than one atom can be simultaneously forced inside the cavity mode: in this regime, the results of the numerical calculations can be successfully interpreted using a dressed cavity model. From the formal analogy between atomic matter waves and optical light waves in nonlinear media, it follows that the same results hold for photonic systems.
This thesis investigates the properties of an atomic system exhibiting a giant Kerr nonlinearity. The atomic energy level scheme involves four energy levels. A three level A subsystem in the atom exhibits the effect of electromagnetically induced transparency (EIT), reducing the spontaneous emission noise. The fourth level leads to an ac-Stark shift of the ground state, which in turn leads to a giant, noiseless Kerr nonlinearity. Two different environments are explored. First, a system comprising of large number of atoms in an optical cavity is analysed. Detailed aspects of noise reduction in this system are investigated. In particular, strong squeezing in the quadrature in phase with the field driving the cavity mode is found, if the effective coupling of light to the atoms is strong. However, the linewidth of the predicted squeezing is found to be very narrow. This is attributed to a very steep linear susceptibility of the atomic medium. Since the widening of the squeezing window is possible only for weaker effective coupling, in turn reducing the squeezing level, a different environment is proposed. This involves a single four level atom, strongly coupled to the cavity mode. In such a strongly coupled system, the most appropriate approach is found to be that formulated in terms of polaritons – composite excitations of the 'atom-cavity molecule'. Adopting the polariton approach, nonclassical correlations in the field leaving the cavity are investigated. Strong photon antibunching is found and the effect of photon blockade predicted and described. The photon blockade effect can also be found in a system comprised of a two level atom coupled to the cavity mode, if the external driving is tuned to one of the vacuum Rabi resonances. A comparison between the two schemes is performed, and it is found that the four level scheme exhibits much better photon blockade. The reason for this is quantum interference between secondary transitions in the dressed states picture. Destructive interference cancels the transitions that would otherwise introduce a second photon into the system, hence producing a more robust photon blockade. All of these results are valid in the regime where external driving is weak. If the external driving strength is increased, the photon statistics (as measured by the zero-delay second order correlation function) changes from strong antibunching to strong bunching, over a relatively narrow range of driving strengths. The occurrence of this change can again be attributed to quantum interference. It is shown that the interference effect prevents the excitation of the composite system by a second photon, but not excitation by a two-photon transition (following the first excitation). Therefore, the third excitation manifold is excited, which then decays back to the first manifold in a two photon cascade. This two photon cascade is the source of correlated photon pairs causing an increase in the second order correlation function. The dynamics of forward scattering of light is presented, and nonclassical behaviour of the delay dependence of correlation function ('overshoots' and 'undershoots') is discussed. For the analytical treatment of this system, a method based on the polariton approach is devised, which includes the treatment of driving and damping. It is shown that this method is ideally suited to the analysis of strongly coupled systems, where only a few photons contribute to the dynamics.
A two-level system interacting with a cavity field is an important model for investigating the photon blockade (PB) effect. Most work on this topic has been based on the assumption that the atomic transition frequency is resonant with the fundamental mode frequency of the cavity. We relax this constraint and reexamine PB in a more general atom--cavity system with arbitrary atomic and cavity detunings from a driving field. The results show that when the signs of the atomic and cavity detunings are the same, PB occurs only in the strong-coupling regime, but for opposite signs of the atomic and cavity detunings, strong photon antibunching is observed in both the weak- and strong-coupling regimes and a better PB effect is achieved compared with the case when the signs are the same. More interestingly, we find that this PB arises from quantum interference for both weak and strong nonlinearities. These results deepen our understanding of the underlying mechanism of PB and may be help in the construction of single-photon sources with higher purity and better flexibility using atom--cavity systems.
There is increasing interest in the photon statistics and spectroscopic properties of the radiation from atoms confined in an optical cavity. Effects such as the cavity enhancement and inhibition of spontaneous emission, which were first observed using microwave cavities, have recently been measured at optical frequencies.1 Recent measurements of squeezing and the dynamic effects of single-quantum-level splittings also demonstrate the possibilities for quantum-statistical studies of coupled atom-cavity systems at optical frequencies.2 We study a single two-level atom confined inside an optical cavity which is driven by coherent light in resonance with both cavity and atom. For weak driving field intensities we have calculated the spectrum of the light transmitted by the cavity and fluorescence out the sides of the cavity.
The steady-state transmission spectrum of a cavity with two modes of orthogonal polarization strongly coupled to a single atom with multiple Zeeman states is computed. Effects due to cavity birefringence and atomic ac-Stark shifts are included. The transmission spectrum is compared to experimental results for a single Cesium atom trapped via an intracavity FORT in a Fabry-Perot cavity. The excellent agreement of the theory with the data is used to infer the distribution of the position of the trapped atom. The intensity correlation function of this system is also calculated, and found to be strongly antibunched and sub-Poissonian. This effect is explained in terms of photon blockade, based on the structure of the lowest energy eigenvalues. Experimental results confirm the strong nonlinearity at the single-photon level. We present theoretical predictions of the weak field spectra of microtoroid and photonic bandgap cavities strongly coupled to the D2 transition of single Cesium atoms. These calculations include all hyperfine and Zeeman states of the transition and model the cavity as a single-mode, linearly polarized resonator. Finally, we outline a technique for using multiple hyperfine and Zeeman levels of a single atom in a strongly coupled atom-cavity system to generate polarized single photons on demand in a well-defined temporal mode via adiabatic passage. The technique is insensitive to cavity birefringence and only weakly sensitive to atomic position. Variations of this technique for generating entanglement of photon polarization and atomic Zeeman state are also discussed.
We consider a single two-level atom in an optical cavity, in which the system is driven by a coherent driving field. In the bad-cavity limit, when the cavity loss rate is taken to be large compared to the atomic decay rate and atom-field coupling, the cavity field can be adiabatically eliminated from the equations of motion for the atomic operators. The main effect of the cavity in this limit is the enhancement of spontaneous emission into the cavity mode. For this model, the second-order intensity correlation function g(2)() has been shown to exhibit photon antibunching, sub-Poissonian photon number distributions, and other nonclassical effects.1 In this paper, analytic results are presented for g(2)() for both the transmitted and fluorescent light fields when broadband squeezed light is injected into the cavity. Effects of the squeezing on nonclassical behavior is examined.
We explore the photon statistics of light emitted from a system comprising a single four--level atom strongly coupled to a high-finesse optical cavity mode which is driven by a coherent laser field. In the weak driving regime this system is found to exhibit a photon blockade effect. For intermediate driving strengths we find a sudden change in the photon statistics of the light emitted from the cavity. Photon antibunching switches to photon bunching over a very narrow range of intracavity photon number. It is proven that this sudden change in photon statistics occurs due to the existence of robust quantum interference of transitions between the dressed states of the atom-cavity system. Furthermore, it is shown that the strong photon bunching is a nonclassical effect for certain values of driving field strength, violating classical inequalities for field correlations.