Makoto Mine

Condition for emergence of complex eigenvalues in the Bogoliubov-de Gennes equations

Y. Nakamura, ∗ M. Mine, † M. Okumura, 4, ‡ and Y. Yamanaka § Department of Materials Science and Engineering, Waseda University, Tokyo 169-8555, Japan Department of Physics, Waseda University, Tokyo 169-8555, Japan CCSE, Japan Atomic Energy Agency, 6-9-3 Higashi-Ueno, Taito-ku, Tokyo 110-0015, Japan CREST (JST), 4-1-8 Honcho, Kawaguchi-shi, Saitama 332-0012, Japan Department of Electronic and Photonic Systems, Waseda University, Tokyo 169-8555, Japan (Dated: April 4, 2008)

Quantum field theoretical description of unstable behavior of trapped Bose-Einstein condensates with complex eigenvalues of Bogoliubov-de Gennes equations

Abstract The Bogoliubov–de Gennes equations are used for a number of theoretical works on the trapped Bose–Einstein condensates. These equations are known to give the energies of the quasi-particles when all the eigenvalues are real. We consider the case in which these equations have complex eigenvalues. We give the complete set including those modes whose eigenvalues are complex. The quantum fields which represent neutral atoms are expanded in terms of the complete set. It is shown that the state space is an indefinite metric one and that the free Hamiltonian is not diagonalizable in the conventional bosonic representation. We introduce a criterion to select quantum states describing the metastablity of the condensate, called the physical state conditions. In order to study the instability, we formulate the linear response of the density against the time-dependent external perturbation within the regime of Kubo’s linear response theory. Some states, satisfying all the physical state conditions, give the blow-up and damping behavior of the density distributions corresponding to the complex eigenmodes. It is qualitatively consistent with the result of the recent analyses using the time-dependent Gross–Pitaevskii equation.

Relation between generalized Bogoliubov and Bogoliubov–de Gennes approaches including Nambu–Goldstone mode

The two approaches of consistent quantum field theory for systems of the trapped Bose–Einstein condensates are known, one is the Bogoliubov–de Gennes approach and the other is the generalized Bogoliubov approach. In this paper, we investigate the relation between the two approaches and show that they are formally equivalent to each other. To do this one must carefully treat the Nambu–Goldstone mode which plays a crucial role in the condensation. It is emphasized that the choice of vacuum is physically relevant.

Decay of resonance structure and trapping effect in potential scattering problem of self-focusing wave packet

Potential scattering problems governed by the time-dependent Gross–Pitaevskii equation are investigated numerically for various values of coupling constants. The initial condition is assumed to have the Gaussian-type envelope, which differs from the soliton solution. The potential is chosen to be a box or well type. We estimate the dependences of reflectance and transmittance on the width of the potential and compare these results with those given by the stationary Schrodinger equation. We attribute the behaviors of these quantities to the limitation on the width of the nonlinear wave packet. The coupling constant and the width of the potential play an important role in the distribution of the waves appearing in the final state of scattering.

Effect of the Zero-Mode on the Response of a Trapped Bose-Condensed Gas

The dynamical response of a trapped Bose-Einstein condensate (BEC) is formulated consistently with quantum field theory and is numerically evaluated. We regard the BEC as a manifestation of the breaking of the global phase symmetry. Then, the Goldstone theorem implies the existence of a zero energy excitation mode (the zero-mode). We calculate the effect of the zero-mode on the response frequency and show that the contribution of the zero-mode to the first excitation mode is not so important in the parameter set realized in the existing experiment. This is the reason that experimental results can be described using the Bogoliubov prescription, although it breaks the consistency of the description in quantum field theory.

Interference pattern formation between bound solitons and radiation in momentum space: Possible detection of radiation from bound solitons with bose-einstein condensate of neutral atoms

We propose an indirect method for observing radiation from an incomplete soliton with a sufficiently large amplitude. We show that the radiation causes a notched structure on the envelope of the wave packet in the momentum space. The origin of this structure is the interference between the main body of oscillating solitons and the small radiation in the momentum space. We numerically integrate the nonlinear Schrodinger equation and perform Fourier transformation to confirm that the predicted structure really appears. We also show a simple model which reproduces the qualitative result. The experimental detection of the notched structure with the Bose–Einstein condensation of neutral atoms is discussed and suitable parameters for this detection experiment are shown.

Quantum field theoretical analysis on unstable behavior of Bose–Einstein condensates in optical lattices

K. Kobayashi, ∗ M. Mine, † M. Okumura, 4, ‡ and Y. Yamanaka § Department of Materials Science and Engineering, Waseda University, Tokyo 169-8555, Japan Department of Physics, Waseda University, Tokyo 169-8555, Japan CCSE, Japan Atomic Energy Agency, 6-9-3 Higashi-Ueno, Taito-ku, Tokyo 110-0015, Japan CREST(JST), 4-1-8 Honcho, Kawaguti-shi, Saitama 332-0012, Japan Department of Electronic and Photonic Systems, Waseda University, Tokyo 169-8555, Japan (Dated: February 1, 2008)

Condition for the existence of complex modes in a trapped Bose-Einstein condensate with a highly quantized vortex

Abstract We consider a trapped Bose–Einstein condensate with a highly quantized vortex. Pu et al. found numerically the parameter region in which complex eigenvalues arise. Recently, the splitting of a highly quantized vortex into two singly quantized vortices is observed in the experiment. We derive analytically the condition for the existence of complex eigenvalues by using the small coupling constant expansion and the two-mode approximation. We check that our results agree with those by Pu et al.

Quantum field theoretical description of unstable behavior of a Bose-Einstein condensate with a highly quantized vortex in a harmonic potential

The Bogoliubov-de Gennes equations are used for a number of theoretical works to describe quantum and thermal fluctuations of trapped Bose-Einstein condensates. We consider the case in which the condensate has a highly quantized vortex. It is known that these equations have complex eigenvalues in this case. We give the complete set including a pair of complex modes whose eigenvalues are complex conjugates to each other. The expansion of the quantum fields which represent neutral atoms in terms of the complete set brings the operators associated with the complex modes, which are simply neither bosonic nor fermionic ones. The eigenstate of the Hamiltonian is given. Introducing the notion of the physical states, we discuss the instability of the condensates in the context of Kubo’s linear response theory.

Effect of zero mode on the response of trapped bose-condensed atoms

The response of the trapped Bose-Einstein condensate (BEC) is investigated. We regard the BEC as a manifestation of the spontaneous breakdown of the global phase symmetry. Then, the Goldstone theorem leads to the existence of the zero energy excitation mode (zeromode). We calculate the effect of the zero-mode to the response frequency and show that the contribution of the zero-mode to the first excitation mode becomes dominant as the temperature and/or the coupling constant are increased.