Spacetime analogue of Bose-Einstein condensates

Transcription

1 Spacetime analogue of Bose-Einstein condensates Bogoliubov-de Gennes formulation Hideki ISHIHARA Osaka City Univ., JAPAN Y.Kurita, M.Kobayashi, T.Morinari, M.Tsubota, and H.I., Phys. Rev. A79, (2009)

2 1. Introduction

3 Analogue spacetime on perfect fluid Curved spacetime (gravitational field) Quantum field Perfect Fluid flow phonon field W. G. Unruh, Phys. Rev. Lett. 46, 1351 (1981).

4 Spacetime analogue of BEC We consider Bose-Einstein Condensation as a play ground. : Quantum fluid Verify effects of quantum field in curved spacetime. Analogy suggests interesting phenomena in condensed matter physics. Backreaction of quantum effects to background.

5 2. Bose-Einstein Condensation (BEC)

6 Many-particle system of Bosons Hamiltonian Trapping potential Particle interaction Decomposition condensation Gross-Pitaevskii Equation fluctuation 400nK, 200nK, 50nK

7 Dynamics of Condensation Gross-Pitaevskii Equation Setting we get where Unruh s idea leads analog model of a quantum field in a curved spacetime

8 Total Hamiltonian Fluctuation decomposition Hamiltonian of fluctuation Bogoliubov-de Gennes (BdG) equations:

9 Bogoliubov quasi-particles BdG equations in a stationary BEC Mode functions: Field expansion: Commutaion relation: BdG Hamiltonian Vacuum:

10 Spectrum of Bogoliubov quasiparticle Dispersion relation E i Excitation spectrum for initial state i Phonon mode at Low energy Analogue massless particles

11 3. Analogy in BEC

12 References: Analogy on BEC L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 85, L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, Phys. Rev. A 63, C. Barcelo, S. Liberati, and M. Visser, Int. J. Mod. Phys. A 18, U. Leonhardt, T. Kiss, and P. Öhberg, J. Opt. B: Quantum Semiclassical Opt. 5, S U. Leonhardt, T. Kiss, and P. Öhberg, Phys. Rev. A 67, S. Giovanazzi, C. Farrell, T. Kiss, and U. Leonhardt, Phys. Rev. A 70, S. Giovanazzi, Phys. Rev. Lett. 94, S. Wüster and C. M. Savage, Phys. Rev. A 76, Y. Kurita and T. Morinari, Phys. Rev. A 76, R. Balbinot, A. Fabbri, S. Fagnocchi, A. Recati, and I. Carusotto, Phys. Rev. A 78, R I. Carusotto, S. Fagnocchi, A. Recati, R. Balbinot, and A. Fabbri, New J. Phys. 10, S. Wüster, Phys. Rev. A 78, R P. Jain, A. S. Bradley, and C. W. Gardiner, Phys. Rev. A 76, H. Nakano, Y. Kurita, K. Ogawa, and C. M. Yoo, Phys. Rev. D 71, S. Basak, e-print arxiv:gr-qc/ F. Federici, C. Cherubini, S. Succi, and M. P. Tosi, Phys. Rev. A 73, C. Barcelo, A. Cano, L. J. Garay, and G. Jannes, Phys. Rev. D 74, C. Barcelo, A. Cano, L. J. Garay, and G. Jannes, Phys. Rev. D 75, H. Takeuchi, M. Tsubota, and G. E. Volovik, J. Low Temp. Phys. 150,

13 References: Analogue universe on BEC C. Barcelo, S. Liberati, and M. Visser, Int. J. Mod. Phys. D 12, C. Barcelo, S. Liberati, and M. Visser, Phys. Rev. A 68, J. E. Lidsey, Class. Quantum Grav. 21, P. Jain, S. Weinfurtner, M. Visser, and C. W. Gardiner, Phys.Rev. A 76, S. Weinfurtner, P. Jain, M. Visser, and C. W. Gardiner, Class.Quantum Grav. 26, M. Uhlmann, Phys. Rev. A 79, P. O. Fedichev and U. R. Fischer, Phys. Rev. Lett. 91, P. O. Fedichev and U. R. Fischer, Phys. Rev. A 69, P. O. Fedichev and U. R. Fischer, Phys. Rev. D 69, U. R. Fischer and R. Schützhold, Phys. Rev. A 70, S. E. C. Weinfurtner, Gen. Relativ. Gravit. 37, M. Uhlmann, Y. Xu, and R. Schutzhold, New J. Phys. 7,

14 BdG analogy BdG equation in BEC set BdG equation becomes equivalently Correspondence

15 Normalization of field Mode expansions Constant C is determined by normalization Then we have

16 Correspondence spacetime BEC

17 4. Particle creation in BEC

18 Analogue universe model of BEC 2 ways for analog universe (1) Varying U 0 by Fishbach resonance: Simple homogeneous universe U 0 is under control from outside (2) Moving BEC, c s, v 0 are time dependent: Inhomogeneous universe model Isolated system

19 Bogoliubov quasi-particles in dynamical BEC Diagonalization of the BdG Hamiltonian By using a new mode functions

20 5.Numerical Simulations

21 Set up of numerical simulation 1-dimensional BEC system for simplicity Stationary state in the harmonic potential with ω i Oscillating BEC in the harmonic potential withω f = 0.7ω i

22 Particle creation

23 Particle spectrum Σ Number m B nm (E) (a) exp[-e (2) j / k B T e ] t = 2.55 / ω i E (2) j / ħ ω 20 i For 87 Rb BEC

24 Summary We consider an analogue universe of BEC We consider oscillating BEC in a trapping potential. Then, the analog universe model is oscillating and inhomogeneous. The BEC with fluctuation system is isolated. Bogoliubov quasiparticle production is calculated by BdG eq. Hamiltonian diagonalization is used for definition of particles. Spectrum of created particles is almost thermal. Temperature for realistic situation is expected as Backreaction to BEC Definition of particles: