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Spontaneous symmetry breaking in Bose–Einstein condensates
22 May 1998
Chemical Physics Letters 288 Ž1998. 248–252
Spontaneous symmetry breaking in Bose–Einstein condensates Roger A. Hegstrom Department of Chemistry, Wake Forest UniÕersity, Winston-Salem, NC 27109, USA Received 26 February 1998
Abstract The structure of the many-particle wavefunction for a pair of ideal gas Bose–Einstein condensates a, b in the number eigenstate < Na Nb : is analyzed. It is found that the most probable many-particle position or momentum measurement outcomes break the configurational phase symmetry of the state. Analytical expressions for the particle distribution and current density for a single experimental run are derived and found to display interference. Spontaneous symmetry breaking is thus predicted and explained here simply and directly as a highly probable measurement outcome for a state with a definite number of particles. q 1998 Elsevier Science B.V. All rights reserved.
The realization of Bose–Einstein ŽBE. condensation in dilute, ultracold gases and the recent observation 1 of interference between pairs of such condensates has attracted widespread interest in this new form of matter. The relative simplicity of these many-boson systems inspires the hope of obtaining a more complete understanding of important concepts, such as condensate phase and spontaneous symmetry breaking, which have been widely used in describing more complicated systems as well w1–3x. The work presented in this Letter may be considered, in a sense, an explanation of the numerical results of Javanainen and Yoo ŽJY. w3x, who have, by means of a computer simulation, calculated the particle density for a pair of BE condensates and found interference fringes without ever having assumed a phase relationship between the pair. The JY results are highly significant because all previous treatments of BE condensation and related phenomena Že.g. superfluidity, superconductivity. inserted these phase relationships ‘by hand’, invoking the idea of spontaneous breaking of gauge symmetry as an assumed justification w1–3x. Like JY, the present work assumes that the exact quantum state of a condensate pair a, b is, in the language of field theory, a number eigenstate or Fock state
Ž 1.
where Na and Nb are the numbers of bosons occupying the one-particle states a and b. A final result for the particle distribution similar to Žbut more general than. that of JY, including the prediction of spontaneous symmetry breaking, is obtained here. However, whereas in JY a photon detection model of quantum measurement and stochastic computer simulation of an experiment to measure the positions of the particles was 1 Ref. w1x contains an extensive list of references to earlier experimental and theoretical work. For a more recent theoretical treatment, also see Ref. w2x.
0009-2614r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 2 7 0 - X
R.A. Hegstromr Chemical Physics Letters 288 (1998) 248–252
249
used, in the present work an analytical expression for the particle distribution is calculated directly from the many-particle wavefunction, providing a deeper insight into the nature of the spontaneous symmetry breaking process. An interesting new result is the derivation of an approximate, effective N-particle wave function which displays the effects of the broken symmetry, including a definite relative phase. Spontaneous symmetry breaking is thus found to follow rather directly and relatively simply from the fundamental probabilistic postulates of quantum mechanics without the need for specific measurement models or computer simulations. The system consists of N s Na q Nb spinless noninteracting identical bosons in the state given in Eq. Ž1.. In a slight generalization of the treatment of JY w3x, the single particle states, hereafter called condensate orbitals and denoted a and b for the respective condensates, are taken to be normalized wave packets in three dimensions with their centers located at x a and x b , respectively. The N-particle wavefunction which corresponds to Eq. Ž1. may be written
C Ž1 . . . N . s
(W
N
N!
Ý P aŽ 1. aŽ 2. . . . aŽ Na . b Ž Na q 1. . . . b Ž N .
Ž 2.
P
where the summation is taken over all permutations P of the particle coordinates 1 . . . N, and where WN is the total number of distinct terms, equal to N! WN s . Ž 3. Na !Nb ! It is assumed that the condensate orbitals are spatially sufficiently small and well separated initially so that they can be taken to be orthogonal. Then, since their unitary time evolution is generated by the same Hamiltonian operator, they are orthogonal at all times and the wavefunction C Ž1 . . . N . is normalized at all times. The condensate orbitals may be written quite generally as a Ž j . s Fa Ž j . exp i fa Ž j . 4 b Ž j . s Fb Ž j . exp i f b Ž j . 4
Ž 4a. Ž 4b.
where the moduli Fa , Fb and phases fa , f b are real functions of the spatial coordinates or momenta of the particles, and j s 1 . . . N. At this point it is important to note that the wavefunction C given in Eq. Ž2. possesses a symmetry which will be called configurational phase symmetry or configurational phase invariance. Defining the phase transformation
fa Ž j . ™ fa Ž j . q ua fbŽ j. ™fbŽ j. qu b ,
Ž 5a . Ž 5b .
where the angles ua , u b may depend upon the condensate orbital but not upon the particle coordinates, it is easy to see that the N-particle wavefunction in Eq. Ž2. transforms as
C ™ C exp i Ž Na ua q Nb u b . 4
Ž 6.
so that the probability density P Ž 1 . . . N . s
Ž 7.
is invariant. As a necessary and sufficient condition for possessing this symmetry, a state C must be a single configuration, e.g. a Na b N b in Eq. Ž2.. Precisely this kind of state symmetry is broken in the analysis given below. As will now be shown, the symmetry breaking may be considered to arise concomitantly with the measurement of the positions or the momenta of the particles, at a particular time t, according to the standard interpretation of quantum mechanics. The probability density in Eq. Ž7. gives the probability of an N-particle individual measurement outcome event EŽ1 . . . N .. The most probable events are those for which the terms in
R.A. Hegstromr Chemical Physics Letters 288 (1998) 248–252
250
Eq. Ž2. add coherently. This is seen to be the case when the phase differences faŽ j . y f b Ž j . are concentrated near the values 2 n j p q 2 u where n j is an integer and where u is an angle which is arbitrary but fixed for a given N-particle measurement. Each of these highly probable events breaks the configurational phase symmetry. For a large number of particles, the effect is striking since an event of this kind is of order WN ( 2 N times more likely than an event corresponding to random phases. The effect is seen to be closely analogous to the coherent scattering of waves from a collection of regularly spaced identical scatterers, which concentrates very high intensities at certain angles in contrast to the much weaker intensities found when the spacing is irregular. A detailed demonstration including a derivation of the effective broken symmetry wave function will now be given. For the sake of simplicity, the condensate orbitals are taken to be identical except for the spatial locations x a and x b of their centers, and the wave function C is expressed as a function of the momenta of the particles, so that we may write without essential loss of generality that FaŽp j . s Fb Žp j . ' F Žp j . in Eqs. Ž4a. and Ž4b.. The phases are then given, to within a common term, by pjPxa fa Ž p j . s y Ž 8a . " pjPxb fbŽpj . sy Ž 8b . " which follows from translational symmetry. Further simplification is achieved by choosing the origin of the spatial coordinates to be at the midpoint between the condensate centers so that x a s x abr2 and x b s yx abr2, where x ab ' x a y x b , and by taking Na s Nb s Nr2. The wave function C then may be expressed
C Ž p1 . . . p N . s
(W
N
N!
F Ž p 1 . . . . F Ž p N . Ý P exp i Ž f 1 q f 2 q . . . qf y f N 2
N 2
q1 y . . . yf N
.4
Ž 9.
P
where
f j ' fa Ž p j . s yf b Ž p j . s y
p j P x ab
. Ž 10 . 2" From Eq. Ž9. it follows that an N-particle measurement event for which the phases f j are concentrated near the values n j p q u , where the n j are integers and u is a fixed arbitrary angle, and for which N is large, has an exceedingly high probability compared to an event with randomly distributed phases — when a measurement of the momenta Žor positions, see below. is performed, an event of this kind, with a definite u value, is almost certain to be the outcome. Given the wave function in Eq. Ž9. and the constraint on the phases given in the above paragraph, an effective, approximate N-particle wavefunction and its associated particle density, both corresponding to one of the highly probable single measurement events, can be derived. First Eq. Ž9. is expanded in a power series about f j s n j p q u to obtain, after some straightforward calculation,
(
C Ž p 1 . . . p N . s WN F Ž p 1 . . . . F Ž p N . Ž y1 .
½
= 1y
N
n 1 q . . . qn N
N
Ý ž f ynj pyuyf / 2 Ž N y 1 . js1 j
2
qO
4
½Žf yn pyu . 5 j
j
where f is the average deviation of the phases from n j p q u 1 N f' Ý Žf ynj pyu . , N js1 j
5
Ž 11 .
Ž 12 .
which, without loss of generality, is set equal to zero. Eq. Ž11. is exact to order Ž f j y n j p y u . 3 for any number of particles. In the limit N ™ ` it can be written exact to order Ž f j y n j p y u . 3 as
(
C Ž p 1 . . . p N . s WN F Ž p 1 . . . . F Ž p N . cos Ž f 1 y u . cos Ž f 2 y u . . . . cos Ž f N y u . .
Ž 13 .
R.A. Hegstromr Chemical Physics Letters 288 (1998) 248–252
251
Then, using Euler’s relations for the cosines and the definitions of the condensate orbitals given in Eqs. Ž4a. and Ž4b., Eq. Ž13. is expressed as
C Ž p1 . . . p N . s
WN
1r2
ž /
c Ž p1 . c Ž p 2 . . . . c Ž p N . ,
2N
Ž 14 .
where
cs
aeyi u q be i u
Ž 15 .
'2
is a normalized one-particle wavefunction. Finally, renormalizing the many-particle wavefunction C, Eq. Ž14. is written in the abbreviated form
C Ž 1 . . . N . s c Ž 1. c Ž 2. . . . c Ž N . .
Ž 16 .
In passing, one may notice a formal correspondence to the theory of molecular electronic structure, with the condensate orbitals a, b corresponding to atomic orbitals and the linear combination of condensate orbitals Ž15. corresponding to an LCAO molecular orbital. Also, whereas the exact N-particle wavefunction Ž2. is similar to a valence–bond wavefunction, the approximate N-particle wavefunction Ž16. is similar to a many-particle molecular orbital wavefunction. The corresponding particle density is now obtained from Eq. Ž16. quite simply as
rsN
Ž 17 .
This particle density exhibits interference between the condensates a and b, as evident from the expression for c given in Eq. Ž15.. By including the time-dependent factors exp Ž yi pj2 t . r Ž 2 m" . in Eq. Ž9. Žheretofore ignored since they cancel out for all probability density calculations in momentum space. and performing Fourier transforms, it is easy to show that Eqs. Ž15. – Ž17. remain valid to the same degree of approximation when the momenta are replaced by the spatial coordinates of the particles. In that case, the condensate orbitals appearing in Eq. Ž15. are nontrivial functions of the time aŽx, t ., bŽx, t . Žthey describe expanding wave packets for free particles.. The angle u is fixed for a given N-particle measurement and varies randomly for repeated state preparations and the corresponding N-particle measurements. The results given in Eqs. Ž15. and Ž17. are in good agreement with the experimental results w1x and are also in essential agreement with previous theoretical results obtained using considerably more complicated methods w1–3x. The results may be generalized further to arbitrary large Na and Nb ; the only essential difference is that Eq. Ž15. is replaced by
ž
cs
(
Na N
aeyi u q
(
Nb N
be i u .
/
Ž 18 .
If the same procedure as detailed in the previous paragraphs is used to calculate the particle current density, the corresponding result for large N is N" J Ž x, t . s Re yi c ) Ž x, t . =c Ž x, t . 4 Ž 19 . m where m is the particle mass. Eqs. Ž4a., Ž4b., Ž15. and Ž19. may then be combined to obtain N" Js F 2 =fa q Fb2 =f b q Ž Fa=Fb y Fb=Fa . sin Ž f b y fa q 2 u . q Fa Fb=Ž fa q f b . 2m a =cos Ž f b y fa q 2 u . 4
Ž 20 .
which contains a Josephson-like current term. Since this current depends upon the relative phase, it may be amenable to experimental manipulation by subjection of the two condensates to unequal potentials. ŽThe spatial
252
R.A. Hegstromr Chemical Physics Letters 288 (1998) 248–252
coordinate and momentum distributions may be similarly manipulated.. Using data from Ref. w1x in Eq. Ž20., a rough order-of-magnitude of the maximum current can be estimated as I f Nsy1 . In summary, it has been shown from standard quantum mechanics and the structure of the many-particle wavefunction that, for a pair of BE condensates a, b described by a single configuration a Na b N b , the most probable many-particle measurement outcomes break the configurational phase symmetry of the state with respect to the condensate orbitals a, b. Each of these highly probable outcomes can be described in terms of an approximate effective many-particle wave function which corresponds to a single configuration c NaqN b that breaks the configurational phase symmetry with respect to a and b, introduces a relative phase u , and exhibits configurational phase symmetry with respect to the ‘dicondensate’ orbital c which is delocalized over both condensates Žsee Eqs. Ž15. and Ž16... The particle number is definite and constant at all times. Since it provides a perspective different from and it is simpler than all previous treatments w1–3x, the approach used here also should provide new insight into the nature of other symmetry-breaking phenomena in many-boson systems, such as the interference between two ‘independent’ lasers and aspects of superfluidity and superconductivity. In this connection, it should be remembered that the two subsystems a, b in the composite state given in Eq. Ž1. are not really independent, as seen clearly in Eq. Ž2., because the particles in a are correlated with those in b by means of exchange symmetry.
Acknowledgements I am grateful to my friend and colleague D. Kondepudi for his support and his helpful suggestions during the course of this work.
References w1x M.R. Andrews, C.G. Townsend, H.-J. Miesner, D.S. Durfee, D.M. Kurn, W. Ketterle, Science 275 Ž1997. 637. w2x Y. Castin, J. Dalibard, Phys. Rev. A, 55 Ž1997. 4330. w3x J. Javanainen, S.M. Yoo, Phys. Rev. Lett. 76 Ž1996. 161.
Chemical Physics Letters 288 Ž1998. 248–252
Spontaneous symmetry breaking in Bose–Einstein condensates Roger A. Hegstrom Department of Chemistry, Wake Forest UniÕersity, Winston-Salem, NC 27109, USA Received 26 February 1998
Abstract The structure of the many-particle wavefunction for a pair of ideal gas Bose–Einstein condensates a, b in the number eigenstate < Na Nb : is analyzed. It is found that the most probable many-particle position or momentum measurement outcomes break the configurational phase symmetry of the state. Analytical expressions for the particle distribution and current density for a single experimental run are derived and found to display interference. Spontaneous symmetry breaking is thus predicted and explained here simply and directly as a highly probable measurement outcome for a state with a definite number of particles. q 1998 Elsevier Science B.V. All rights reserved.
The realization of Bose–Einstein ŽBE. condensation in dilute, ultracold gases and the recent observation 1 of interference between pairs of such condensates has attracted widespread interest in this new form of matter. The relative simplicity of these many-boson systems inspires the hope of obtaining a more complete understanding of important concepts, such as condensate phase and spontaneous symmetry breaking, which have been widely used in describing more complicated systems as well w1–3x. The work presented in this Letter may be considered, in a sense, an explanation of the numerical results of Javanainen and Yoo ŽJY. w3x, who have, by means of a computer simulation, calculated the particle density for a pair of BE condensates and found interference fringes without ever having assumed a phase relationship between the pair. The JY results are highly significant because all previous treatments of BE condensation and related phenomena Že.g. superfluidity, superconductivity. inserted these phase relationships ‘by hand’, invoking the idea of spontaneous breaking of gauge symmetry as an assumed justification w1–3x. Like JY, the present work assumes that the exact quantum state of a condensate pair a, b is, in the language of field theory, a number eigenstate or Fock state
Ž 1.
where Na and Nb are the numbers of bosons occupying the one-particle states a and b. A final result for the particle distribution similar to Žbut more general than. that of JY, including the prediction of spontaneous symmetry breaking, is obtained here. However, whereas in JY a photon detection model of quantum measurement and stochastic computer simulation of an experiment to measure the positions of the particles was 1 Ref. w1x contains an extensive list of references to earlier experimental and theoretical work. For a more recent theoretical treatment, also see Ref. w2x.
0009-2614r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 2 7 0 - X
R.A. Hegstromr Chemical Physics Letters 288 (1998) 248–252
249
used, in the present work an analytical expression for the particle distribution is calculated directly from the many-particle wavefunction, providing a deeper insight into the nature of the spontaneous symmetry breaking process. An interesting new result is the derivation of an approximate, effective N-particle wave function which displays the effects of the broken symmetry, including a definite relative phase. Spontaneous symmetry breaking is thus found to follow rather directly and relatively simply from the fundamental probabilistic postulates of quantum mechanics without the need for specific measurement models or computer simulations. The system consists of N s Na q Nb spinless noninteracting identical bosons in the state given in Eq. Ž1.. In a slight generalization of the treatment of JY w3x, the single particle states, hereafter called condensate orbitals and denoted a and b for the respective condensates, are taken to be normalized wave packets in three dimensions with their centers located at x a and x b , respectively. The N-particle wavefunction which corresponds to Eq. Ž1. may be written
C Ž1 . . . N . s
(W
N
N!
Ý P aŽ 1. aŽ 2. . . . aŽ Na . b Ž Na q 1. . . . b Ž N .
Ž 2.
P
where the summation is taken over all permutations P of the particle coordinates 1 . . . N, and where WN is the total number of distinct terms, equal to N! WN s . Ž 3. Na !Nb ! It is assumed that the condensate orbitals are spatially sufficiently small and well separated initially so that they can be taken to be orthogonal. Then, since their unitary time evolution is generated by the same Hamiltonian operator, they are orthogonal at all times and the wavefunction C Ž1 . . . N . is normalized at all times. The condensate orbitals may be written quite generally as a Ž j . s Fa Ž j . exp i fa Ž j . 4 b Ž j . s Fb Ž j . exp i f b Ž j . 4
Ž 4a. Ž 4b.
where the moduli Fa , Fb and phases fa , f b are real functions of the spatial coordinates or momenta of the particles, and j s 1 . . . N. At this point it is important to note that the wavefunction C given in Eq. Ž2. possesses a symmetry which will be called configurational phase symmetry or configurational phase invariance. Defining the phase transformation
fa Ž j . ™ fa Ž j . q ua fbŽ j. ™fbŽ j. qu b ,
Ž 5a . Ž 5b .
where the angles ua , u b may depend upon the condensate orbital but not upon the particle coordinates, it is easy to see that the N-particle wavefunction in Eq. Ž2. transforms as
C ™ C exp i Ž Na ua q Nb u b . 4
Ž 6.
so that the probability density P Ž 1 . . . N . s
Ž 7.
is invariant. As a necessary and sufficient condition for possessing this symmetry, a state C must be a single configuration, e.g. a Na b N b in Eq. Ž2.. Precisely this kind of state symmetry is broken in the analysis given below. As will now be shown, the symmetry breaking may be considered to arise concomitantly with the measurement of the positions or the momenta of the particles, at a particular time t, according to the standard interpretation of quantum mechanics. The probability density in Eq. Ž7. gives the probability of an N-particle individual measurement outcome event EŽ1 . . . N .. The most probable events are those for which the terms in
R.A. Hegstromr Chemical Physics Letters 288 (1998) 248–252
250
Eq. Ž2. add coherently. This is seen to be the case when the phase differences faŽ j . y f b Ž j . are concentrated near the values 2 n j p q 2 u where n j is an integer and where u is an angle which is arbitrary but fixed for a given N-particle measurement. Each of these highly probable events breaks the configurational phase symmetry. For a large number of particles, the effect is striking since an event of this kind is of order WN ( 2 N times more likely than an event corresponding to random phases. The effect is seen to be closely analogous to the coherent scattering of waves from a collection of regularly spaced identical scatterers, which concentrates very high intensities at certain angles in contrast to the much weaker intensities found when the spacing is irregular. A detailed demonstration including a derivation of the effective broken symmetry wave function will now be given. For the sake of simplicity, the condensate orbitals are taken to be identical except for the spatial locations x a and x b of their centers, and the wave function C is expressed as a function of the momenta of the particles, so that we may write without essential loss of generality that FaŽp j . s Fb Žp j . ' F Žp j . in Eqs. Ž4a. and Ž4b.. The phases are then given, to within a common term, by pjPxa fa Ž p j . s y Ž 8a . " pjPxb fbŽpj . sy Ž 8b . " which follows from translational symmetry. Further simplification is achieved by choosing the origin of the spatial coordinates to be at the midpoint between the condensate centers so that x a s x abr2 and x b s yx abr2, where x ab ' x a y x b , and by taking Na s Nb s Nr2. The wave function C then may be expressed
C Ž p1 . . . p N . s
(W
N
N!
F Ž p 1 . . . . F Ž p N . Ý P exp i Ž f 1 q f 2 q . . . qf y f N 2
N 2
q1 y . . . yf N
.4
Ž 9.
P
where
f j ' fa Ž p j . s yf b Ž p j . s y
p j P x ab
. Ž 10 . 2" From Eq. Ž9. it follows that an N-particle measurement event for which the phases f j are concentrated near the values n j p q u , where the n j are integers and u is a fixed arbitrary angle, and for which N is large, has an exceedingly high probability compared to an event with randomly distributed phases — when a measurement of the momenta Žor positions, see below. is performed, an event of this kind, with a definite u value, is almost certain to be the outcome. Given the wave function in Eq. Ž9. and the constraint on the phases given in the above paragraph, an effective, approximate N-particle wavefunction and its associated particle density, both corresponding to one of the highly probable single measurement events, can be derived. First Eq. Ž9. is expanded in a power series about f j s n j p q u to obtain, after some straightforward calculation,
(
C Ž p 1 . . . p N . s WN F Ž p 1 . . . . F Ž p N . Ž y1 .
½
= 1y
N
n 1 q . . . qn N
N
Ý ž f ynj pyuyf / 2 Ž N y 1 . js1 j
2
qO
4
½Žf yn pyu . 5 j
j
where f is the average deviation of the phases from n j p q u 1 N f' Ý Žf ynj pyu . , N js1 j
5
Ž 11 .
Ž 12 .
which, without loss of generality, is set equal to zero. Eq. Ž11. is exact to order Ž f j y n j p y u . 3 for any number of particles. In the limit N ™ ` it can be written exact to order Ž f j y n j p y u . 3 as
(
C Ž p 1 . . . p N . s WN F Ž p 1 . . . . F Ž p N . cos Ž f 1 y u . cos Ž f 2 y u . . . . cos Ž f N y u . .
Ž 13 .
R.A. Hegstromr Chemical Physics Letters 288 (1998) 248–252
251
Then, using Euler’s relations for the cosines and the definitions of the condensate orbitals given in Eqs. Ž4a. and Ž4b., Eq. Ž13. is expressed as
C Ž p1 . . . p N . s
WN
1r2
ž /
c Ž p1 . c Ž p 2 . . . . c Ž p N . ,
2N
Ž 14 .
where
cs
aeyi u q be i u
Ž 15 .
'2
is a normalized one-particle wavefunction. Finally, renormalizing the many-particle wavefunction C, Eq. Ž14. is written in the abbreviated form
C Ž 1 . . . N . s c Ž 1. c Ž 2. . . . c Ž N . .
Ž 16 .
In passing, one may notice a formal correspondence to the theory of molecular electronic structure, with the condensate orbitals a, b corresponding to atomic orbitals and the linear combination of condensate orbitals Ž15. corresponding to an LCAO molecular orbital. Also, whereas the exact N-particle wavefunction Ž2. is similar to a valence–bond wavefunction, the approximate N-particle wavefunction Ž16. is similar to a many-particle molecular orbital wavefunction. The corresponding particle density is now obtained from Eq. Ž16. quite simply as
rsN
Ž 17 .
This particle density exhibits interference between the condensates a and b, as evident from the expression for c given in Eq. Ž15.. By including the time-dependent factors exp Ž yi pj2 t . r Ž 2 m" . in Eq. Ž9. Žheretofore ignored since they cancel out for all probability density calculations in momentum space. and performing Fourier transforms, it is easy to show that Eqs. Ž15. – Ž17. remain valid to the same degree of approximation when the momenta are replaced by the spatial coordinates of the particles. In that case, the condensate orbitals appearing in Eq. Ž15. are nontrivial functions of the time aŽx, t ., bŽx, t . Žthey describe expanding wave packets for free particles.. The angle u is fixed for a given N-particle measurement and varies randomly for repeated state preparations and the corresponding N-particle measurements. The results given in Eqs. Ž15. and Ž17. are in good agreement with the experimental results w1x and are also in essential agreement with previous theoretical results obtained using considerably more complicated methods w1–3x. The results may be generalized further to arbitrary large Na and Nb ; the only essential difference is that Eq. Ž15. is replaced by
ž
cs
(
Na N
aeyi u q
(
Nb N
be i u .
/
Ž 18 .
If the same procedure as detailed in the previous paragraphs is used to calculate the particle current density, the corresponding result for large N is N" J Ž x, t . s Re yi c ) Ž x, t . =c Ž x, t . 4 Ž 19 . m where m is the particle mass. Eqs. Ž4a., Ž4b., Ž15. and Ž19. may then be combined to obtain N" Js F 2 =fa q Fb2 =f b q Ž Fa=Fb y Fb=Fa . sin Ž f b y fa q 2 u . q Fa Fb=Ž fa q f b . 2m a =cos Ž f b y fa q 2 u . 4
Ž 20 .
which contains a Josephson-like current term. Since this current depends upon the relative phase, it may be amenable to experimental manipulation by subjection of the two condensates to unequal potentials. ŽThe spatial
252
R.A. Hegstromr Chemical Physics Letters 288 (1998) 248–252
coordinate and momentum distributions may be similarly manipulated.. Using data from Ref. w1x in Eq. Ž20., a rough order-of-magnitude of the maximum current can be estimated as I f Nsy1 . In summary, it has been shown from standard quantum mechanics and the structure of the many-particle wavefunction that, for a pair of BE condensates a, b described by a single configuration a Na b N b , the most probable many-particle measurement outcomes break the configurational phase symmetry of the state with respect to the condensate orbitals a, b. Each of these highly probable outcomes can be described in terms of an approximate effective many-particle wave function which corresponds to a single configuration c NaqN b that breaks the configurational phase symmetry with respect to a and b, introduces a relative phase u , and exhibits configurational phase symmetry with respect to the ‘dicondensate’ orbital c which is delocalized over both condensates Žsee Eqs. Ž15. and Ž16... The particle number is definite and constant at all times. Since it provides a perspective different from and it is simpler than all previous treatments w1–3x, the approach used here also should provide new insight into the nature of other symmetry-breaking phenomena in many-boson systems, such as the interference between two ‘independent’ lasers and aspects of superfluidity and superconductivity. In this connection, it should be remembered that the two subsystems a, b in the composite state given in Eq. Ž1. are not really independent, as seen clearly in Eq. Ž2., because the particles in a are correlated with those in b by means of exchange symmetry.
Acknowledgements I am grateful to my friend and colleague D. Kondepudi for his support and his helpful suggestions during the course of this work.
References w1x M.R. Andrews, C.G. Townsend, H.-J. Miesner, D.S. Durfee, D.M. Kurn, W. Ketterle, Science 275 Ž1997. 637. w2x Y. Castin, J. Dalibard, Phys. Rev. A, 55 Ž1997. 4330. w3x J. Javanainen, S.M. Yoo, Phys. Rev. Lett. 76 Ž1996. 161.