Bogoliubov dispersion relation and the possibility of superfluidity for weakly interacting photons in a two-dimensional photon fluid

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 60, Issue 5, 1999, Pages 4114-4121

  Chiao, Raymond Y ^a   Boyce, Jack ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0011289825     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.60.4114     Document Type: Article

References (20)

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10. Since we have assumed a zero-temperature Bose gas, following Bogoliubov we start this calculation with the ground state of the system in the macroscopically occupied zero-momentum Fock or number state (Formula presented) However, it is also possible to derive the same Bogoliubov dispersion relation starting from a system in a coherent state (Formula presented) where (Formula presented) We thank Professor David Thouless for sharing with us his unpublished notes concerning this last point.
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14. It may be asked how this photon gas problem in two dimensions is different from the Planck blackbody problem in three dimensions. The first part of the answer is that the Fabry-Perot resonator makes the problem effectively two-dimensional, since the resonator is excited by a nearly monochromatic laser beam with a narrow linewidth, which selects out only a single longitudinal mode of the Fabry-Perot to be excited. Thus the z degree of freedom for the photons inside the resonator is eliminated from the problem. The second part of the answer is that in contrast to the Planck problem, where the chemical potential of the photon vanishes, here there is a nonvanishing chemical potential, Eq. (20), of the photon in the photon fluid, which arises from the repulsive pairwise interactions between photons in the Bose condensate inside the Fabry-Perot resonator. For the same reasons, a two-dimensional phase transition of the Kosterlitz-Thouless type should be possible in the photon fluid.
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16. The classical dispersion relation, Eq. (40), is also identical to the one that was derived in an early paper on stimulated light-by-light scattering [R. Y. Chiao, P. L. Kelley, and E. Garmire, Phys. Rev. Lett. 17, 1158 (1966)], apart from a sign change for the Kerr nonlinear coefficient (Formula presented) from the self-focusing to the self-defocusing sign. The transverse spatial modulational instability of a paraxial traveling-wave configuration in a nonlinear cell, which was predicted in this paper for the self-focusing sign, upon reversal of the sign to the self-defocusing sign, turns into a kind of transverse spatial modulational stability, whose dispersion relation is identical in form to the Bogoliubov one. (This early paper should be consulted if one wants to generalize this dispersion relation to the case of a noninstantaneous response of the Kerr nonlinearity due to a finite relaxation time of the medium.) However, there is a fundamental difference between the case of light inside a nonlinear cavity and the case of light in a paraxial traveling-wave configuration inside a nonlinear cell. In a nonlinear cavity, the field envelope (Formula presented) evolves in time t, and genuine dynamics occurs inside the cavity. Hence genuine sound waves result in the (Formula presented) limit from the classical nonlinear optical dispersion relation, Eq. (40), which would require second quantization, since we know that phonons indeed exist inside the cavity as a result of the quantum dispersion relation, Eq. (30). However, in a paraxial traveling-wave configuration, the field envelope (Formula presented) evolves in the spatial variable z, and not in the time variable t. Hence the “sound waves” inside this cell are frozen, nonpropagating spatial patterns.
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