Images of the dark soliton in a depleted condensate

The dark soliton created in a Bose–Einstein condensate becomes grey in the course of time evolution because its notch fills up with depleted atoms. This is the result of quantum mechanical calculations which describe the output of many experimental repetitions of creation of the stationary soliton, and its time evolution terminated by a destructive density measurement. However, such a description is not suitable to predict the outcome of a single realization of the experiment where two extreme scenarios and many combinations thereof are possible: one will see either (1) a displaced dark soliton without any atoms in the notch, but with a randomly displaced position, or (2) a grey soliton with a fixed position, but a random number of atoms filling its notch. In either case the average over many realizations will reproduce the mentioned quantum mechanical result. In this paper we use N-particle wavefunctions, which follow from the number-conserving Bogoliubov theory, to settle this issue.

[1]  Interference of Bose condensates. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[2]  S. Burger,et al.  Dark solitons in Bose-Einstein condensates , 1999, QELS 2000.

[3]  A. Leggett,et al.  Measurement theory and interference of spinor Bose-Einstein condensates , 2002 .

[4]  Y. Castin,et al.  Low-temperature Bose-Einstein condensates in time-dependent traps: Beyond the U(1) symmetry-breaking approach , 1998 .

[5]  G. Shlyapnikov,et al.  Stability of standing matter waves in a trap , 1998, cond-mat/9811408.

[6]  J. Dalibard,et al.  Relative phase of two Bose-Einstein condensates , 1997 .

[7]  A. Leggett,et al.  Bose-Einstein condensation in the alkali gases: Some fundamental concepts , 2001 .

[8]  M. Girardeau.  Comment on “Particle-number-conserving Bogoliubov method which demonstrates the validity of the time-dependent Gross-Pitaevskii equation for a highly condensed Bose gas” , 1998 .

[9]  C. W. Gardiner.  Particle-number-conserving Bogoliubov method which demonstrates the validity of the time-dependent Gross-Pitaevskii equation for a highly condensed Bose gas , 1997 .

[10]  Anglin,et al.  Motion of dark solitons in trapped bose-einstein condensates , 2000, Physical review letters.

[11]  Krzysztof Sacha,et al.  Depletion of the dark soliton: The anomalous mode of the Bogoliubov theory , 2002 .

[12]  École d'été de physique théorique,et al.  Coherent atomic matter waves : Ondes de matière cohérentes : : École de Physique des Houches - UJF & INPG - Grenoble, a NATO Advanced Study Institute, Les Houches, Session LXXII, 27 July-27 August 1999 , 2001 .

[13]  Yoo,et al.  Quantum phase of a Bose-Einstein condensate with an arbitrary number of atoms. , 1996, Physical review letters.

[14]  Phillips,et al.  Generating solitons by phase engineering of a bose-einstein condensate , 2000, Science.

[15]  R. Arnowitt,et al.  THEORY OF MANY-BOSON SYSTEMS: PAIR THEORY , 1959 .

[16]  W. Ketterle,et al.  Observation of Interference Between Two Bose Condensates , 1997, Science.

[17]  F. Dalfovo,et al.  Theory of Bose-Einstein condensation in trapped gases , 1998, cond-mat/9806038.

[18]  E. Gross.  Structure of a quantized vortex in boson systems , 1961 .

[19]  J. R. Taylor,et al.  Optical Solitons: Theory and Experiment , 2005 .