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Physical Review A

Effective cross-Kerr Hamiltonian for a nonresonant four-level atom

Gary F. Sinclair, Natalia Korolkova

Published Mar 25, 2008• 16 citations
Publisher

Abstract

We derive an effective cross-Kerr Hamiltonian for the four-level atom interacting with three electromagnetic fields in the $N$ configuration. When the atom has relaxed into the ground state, a cross-Kerr nonlinearity arises between two weak probe fields. As a development on earlier work, we derive the form of the atom-field interaction for all detunings and include the spontaneous decay of the upper atomic levels. In general, the atom will also display a linear and self-Kerr response, but if certain resonance conditions are satisfied then only the cross-Kerr interaction will remain. We consider the application of our theory to cold four-level rubidium atoms. The electrical susceptibilities of the probe transitions are explored, and it is shown that a large, pure cross-Kerr nonlinearity can be generated with vanishing absorption of both probe fields.

DOI
10.1103/physreva.77.033843

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A four-level inverted-Y scheme with an electromagnetically induced transparency (EIT) core is investigated for the enhancement of cross-Kerr effect in rubidium atoms. When detunings of the coupling and control fields are appropriately set, an enhanced EIT window is observed, and the induced phase shift of the probe field due to cross-phase modulation (XPM) is obtained by measuring the dispersive property of the probe transition. The maximal XPM phase shift is about 12° under the current experimental conditions. The experimental measurements agree well with the theoretical calculations. The enhanced XPM phase shift in such an atomic system has applications in quantum nonlinear optics and quantum information science.

Journal of the Optical Society of America B•Sep 16, 2010•43 citations
We derive an effective cross-Kerr Hamiltonian for the four-level atom interacting with three electromagnetic fields in the NNN configuration. When the atom has relaxed into the ground state, a cross-Kerr nonlinearity arises between two weak probe fields. As a development on earlier work, we derive the form of the atom-field interaction for all detunings and include the spontaneous decay of the upper atomic levels. In general, the atom will also display a linear and self-Kerr response, but if certain resonance conditions are satisfied then only the cross-Kerr interaction will remain. We consider the application of our theory to cold four-level rubidium atoms. The electrical susceptibilities of the probe transitions are explored, and it is shown that a large, pure cross-Kerr nonlinearity can be generated with vanishing absorption of both probe fields.
We derive a cross-Kerr type effective Hamiltonian for the four-level atom interacting with three electromagnetic fields in the N-configuration. When the atom has relaxed into the ground state a cross-Kerr nonlinearity arises between two weak probe fields. As a development on earlier work we show in general that the atom will also display a linear and self-Kerr response. However, if certain resonance conditions are satisfied then the linear and self-Kerr interactions will vanish. The electrical susceptibilities of the probe transitions are also explored and it is shown that a large, pure cross-Kerr nonlinearity can be generated with vanishing absorption of both probe fields.
A theoretical investigation is put forward for obtaining effective cross-Kerr nonlinearity of the four-level atom interacting with three electromagnetic fields.Using density-matrix equations and perturbation iterative method,the variation dependence of the third-order susceptibility on the detunings of the applied electromagnetic fields is derived.The results show that a large cross-Kerr nonlinearity can be generated when the detunings of the two weak probe fields are Raman resonant,and the corresponding linear susceptibility and self-Kerr interactions vanish.The realization of the effective Kerr nonlinearity with vanishing absorption is possible in a Raman resonant four-level system.
Abstract We theoretically investigate the self-nonlinear and cross-nonlinear responses of the Λ-type, V-type, and ladder-type three-level atomic systems. The analytical solutions of the cross-nonlinearity are obtained by using an iterative method. These analytical expressions reveal that the atomic decay rates and two-photon resonance are closely related to the cross-nonlinear response. For the Λ-type system, the relaxation rate between the ground-state sublevels determines the amplitude and linewidth of the cross-nonlinearity. For the ladder-type system, the lifetime of the upper excited state governs the amplitude and linewidth of the cross-nonlinearity. In both systems, the amplitude of the cross-nonlinearity is significantly enhanced, and the linewidth is greatly compressed simultaneously. In contrast, for the V-type three-level system, the amplitude and linewidth of the cross-nonlinearity are related to the spontaneous emission rate of the excited states, where neither the amplitude enhancement nor the linewidth compression is observed. We conclude that the cross-nonlinear response can reach or even exceed the linear response in the Λ-type system when the relaxation rate between the ground states is sufficiently small, or in the ladder-type system when the spontaneous emission rate of the upper excited state is sufficiently small. Compared with the crossnonlinearity, the self-nonlinearity is related to the Rabi frequency and detuning of the coupling field. Its amplitude decreases with the increase of the coupling Rabi frequency, while it can be increased to a certain extent with the increase of the coupling detuning. Our results provide a theoretical basis for the optimization and application of strong nonlinear effects in three-level atomic systems.
We derive the form of the cross-Kerr interaction in a four-level atomic system in an N configuration. We use time-independent perturbation theory to calculate the eigenenergies and eigenstates of the Schr\"odinger equation for the system. The system is considered as a perturbation of a Raman-resonant three-level $\ensuremath{\lambda}$ scheme for which exact solutions are known. We show that within the strong control field limit the cross-Kerr interaction can arise between two weak probe fields. The strength of this nonlinear coupling is several orders of magnitude larger than that achievable using optical fibres.
A four-level inverted-Y scheme with an electromagnetically induced transparency (EIT) core is investigated for the enhancement of cross-Kerr effect in rubidium atoms. When detunings of the coupling and control fields are appropriately set, an enhanced EIT window is observed, and the induced phase shift of the probe field due to cross-phase modulation (XPM) is obtained by measuring the dispersive property of the probe transition. The maximal XPM phase shift is about 12° under the current experimental conditions. The experimental measurements agree well with the theoretical calculations. The enhanced XPM phase shift in such an atomic system has applications in quantum nonlinear optics and quantum information science.