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Physical Review A

Quantum squeezing and correlation of self-induced transparency solitons

Ray‐Kuang Lee, Yinchieh Lai

Published Sep 25, 2009• 12 citations
Publisher

Abstract

A general quantum theory of self-induced transparency (SIT) solitons is developed with nonlinear quantum effects of atoms taken into account. The quantization starts from the coarse-grain-averaged light-atom interaction Hamiltonian by which the quantum effects of ensemble atoms are modeled. The calculation of quantum properties is performed by the backpropagation method, which takes into account the contribution of the field continuum parts and the atomic fluctuations. The detectable squeezing ratio with a general homodyne local oscillator can be simulated for practical experimental realizations. Schemes for generating quantum correlation through SIT soliton interaction are also proposed and analyzed.

DOI
10.1103/physreva.80.033839

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A general quantum theory of self-induced transparency (SIT) solitons is developed with nonlinear quantum effects of atoms taken into account. The quantization starts from the coarse-grain-averaged light-atom interaction Hamiltonian by which the quantum effects of ensemble atoms are modeled. The calculation of quantum properties is performed by the backpropagation method, which takes into account the contribution of the field continuum parts and the atomic fluctuations. The detectable squeezing ratio with a general homodyne local oscillator can be simulated for practical experimental realizations. Schemes for generating quantum correlation through SIT soliton interaction are also proposed and analyzed.
We investigate the quantum squeezing of slow-light solitons generated in a $\mathrm{\ensuremath{\Lambda}}$-shaped three-level atomic system working under condition of electromagnetically induced transparency (EIT). Starting from the Heisenberg-Langevin and Maxwell equations governing the quantum dynamics of atoms and probe laser field, we derive a quantum nonlinear Schr\"odinger equation controlling the evolution of the probe-field envelope. By using a direct perturbation approach to diagonalize the effective Hamiltonian (where the atomic variables have been eliminated), we carry out a detailed calculation on the quantum fluctuations of a slow-light soliton, expanded as a superposition of the complete and orthonormalized set of eigenfunctions obtained by solving the Bogoliubov-de Gennes (BdG) equations describing the quantum fluctuations. We show that due to the giant Kerr nonlinearity contributed from the EIT effect, significant quantum squeezing of the slow-light soliton can be realized within a short propagation distance. The results reported here are helpful for understanding the quantum property of slow-light solitons and for realizing light squeezing via EIT in cold atomic gases experimentally.
Linearization of the self-induced transparency equations reduces the problem to a single linear integro-differential equation for the perturbation field of a soliton. The adjoint system is introduced to establish orthogonality relations that lead to the relations between the perturbed field and perturbed scattering data. The commutation relations of the perturbation field provide the quantization rules for the perturbed ``scattering data.'' The quantized solitons are described by operators for photon number, phase, momentum, and position. The evolution and detection of these fluctuations are studied. The contribution of the continuum part can be suppressed by suitable detection schemes.
The optical solitons that play the role of information carriers, are moving, requiring careful experimental timing to achieve the desired logic operation. In contrast, self-induced transparency (SIT) is a strong resonant nonlinear effect, supporting the propagation of optical solitons and are usually associated with a two-level system. By the appropriate application of light fields, the optical soliton, resonant with the SIT transition, can generate an excited region with ground state coherence in the medium, allowing logic operations to be performed on this coherence.
Making use of c-number stochastic theory and soliton perturbation theory, we study the quantum fluctuations of a self-induced transparency (SIT) soliton propagating through a lossless two-level medium. Considering the fluctuations as small corrections to the classical soliton, we are able to construct and solve four stochastic equations that govern the evolution of four soliton parameters: photon number (intensity), phase, timing, and momentum (frequency). We find excellent agreement between our stochastic theory of SIT solitons and the second-quantized theory of Lai Haus [Phys. Rev. A42, 2925 (1990)].
Solitons are known to occur in the context of atom-light interaction via the well-known semiclassical phenomenon of self-induced transparency (SIT). Separately, in the regime where both light and atoms are fully treated quantum mechanically, quantum few-photon bound states are known to be a ubiquitous phenomenon that arises in different systems such as atoms coupled to chiral or bidirectional waveguides, and in Rydberg atomic media. In the specific case of two-level atoms coupled to a chiral waveguide, a recent analysis based on Bethe ansatz has established that SIT emerges from the quantum realm as a superposition of quantum many-photon bound states. Beyond this case, however, the nature of any connection between the full quantum many-body regime and semiclassical behavior has not been established. Here, we employ a general spin-model formulation of quantum atom-light interfaces to numerically investigate this problem, taking advantage of the fact that this approach readily allows for powerful many-body simulations based on matrix product states (MPS). We first analytically derive the two-photon bound state dispersion relation for a variety of atom-light interfaces and then proceed to numerically investigate the multiexcitation bound states dynamics. Interestingly, for all the specific systems studied, we find that the large-photon number limit always coincides with the soliton phenomenon of self-induced transparency or immediate generalizations thereof.