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Physical Review A

Effects of damping on the quantum limits to optical phase shifts in Kerr nonlinear media

Ariel Liebman, Gerard J Milburn

Published May 1, 1993• 6 citations
Publisher

Abstract

It was recently shown [Phys. Rev. A 45, 1919 (1992)] that the phase shift induced by a control beam on a signal in cross Kerr modulation is limited by the quantum nature of the control. We show that dissipation rapidly restores the classical phase shift.
DOI
10.1103/physreva.47.4528

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We consider two copropagating fields in a nonlinear Kerr medium, each with a particular phase and intensity. The Kerr medium possesses an intensity-dependent refractive index and the phase shift of each field thus depends on the intensities of the fields. Classically it is possible to induce an arbitrary phase shift of one field (the signal field) by either increasing the intensity of the other field (the control field) or by increasing the interaction legnth. We show that if the intensity of the control field is low, the phase shift on the signal is limited by the discrete nature of the photon-number distribution in the control field and cannot be increased simply by increasing the interaction length. In general the maximum phase shift of the signal field is \ensuremath{\varphi} if the control field possesses \ensuremath{\varphi} photons. This limit arises as a consequence of quantum recurrence effects.
A cross-Kerr interaction produces a phase shift on two modes of light proportional to the number of photons in both modes and is sometimes called cross-phase modulation. Cross-Kerr nonlinearities have many applications in classical and quantum nonlinear optics, including the possibility of a deterministic and all-optical controlled-phase gate. We calculate the one- and two-photon SSS matrices for fields propagating in a medium where the cross-Kerr interaction is spatially distributed at discrete interaction sites comprised of atoms. For the interactions considered, we analyze the cases where the photons copropagate and counterpropagate through the medium and give a physical interpretation to the differences between the two cases. Finally, we obtain the SSS matrix in the limit of infinitely long chains, showing that it corresponds to a perfect controlled-phase operation.
Quantum theory of self-phase and cross-phase modulation of ultrashort light pulses in the Kerr medium is developed, taking into account the response time of an electronic nonlinearity. The corresponding algebra of time-dependent Bose operators is elaborated. It is established that the spectral region of the pulse where the quadrature fluctuation level is lower than the shot-noise one depends on the value of the nonlinear phase shift, the intensity of another pulse and the relaxation time of the nonlinearity. It is shown that the frequency of the pulse spectrum at which the suppression of fluctuations is maximum can be controlled by adjusting the other pulse intensity.
By mapping photons into the sample of cuprous oxide with Rydberg excitons, it is possible to obtain a significant optical phase shift due to third-order cross-Kerr nonlinearities realized under the conditions of electromagnetically induced transparency. The optimum conditions for observation of the phase shift over π in Rydberg excitons media are examined. A discussion of the application of the cross-phase modulations in the field of all-optical quantum information processing in solid-state systems is presented.