We consider the interference of two overlapping ideal Bose-Einstein condensates. The usual description of this phenomenon involves the introduction of a condensate wave function with a definite phase. We investigate the origin of this phase and the theoretical basis of treating interference. It is possible to construct a phase state for which the particle number is uncertain, but the phase is known. How such a state would be prepared before an experiment is not obvious. We show that a phase can also arise from experiments using condensates with known particle numbers. The analysis of measurements in such states also gives us a prescription for preparing phase states. The connection of this procedure to questions of spontaneously broken gauge symmetry and to hidden variables is discussed.

1.
M. R.
Andrews
,
C. G.
Townsend
,
H.-J.
Miesner
,
D. S.
Durfee
,
D. M.
Kurn
, and
W.
Ketterle
, “
Observation of interference between two Bose condensates
,”
Science
275
,
637
641
(
1997
).
2.
B. P.
Anderson
and
M. A.
Kasevich
, “
Macroscopic quantum interference from atomic tunnel arrays
,”
Science
282
,
1686
1689
(
1998
).
3.
Markus
Greiner
,
Immanuel
Bloch
,
Olaf
Mandel
,
Theador W.
Hänsch
, and
Tilman
Esslinger
, “
Exploring phase coherence in a 2D lattice of Bose-Einstein condensates
,”
Phys. Rev. Lett.
87
,
160405
1
(
2001
).
4.
C.
Orzel
,
A. K.
Tuchman
,
M. L.
Fenselau
,
M.
Yasuda
, and
M. A.
Kasevich
, “
Squeezed states in a Bose-Einstein condensate
,”
Science
291
,
2386
2389
(
2001
).
5.
M. H.
Wheeler
,
K. M.
Mertes
,
J. D.
Erwin
, and
D. S.
Hall
, “
Spontaneous macroscopic spin polarization in independent spinor Bose-Einstein condensates
,”
Phys. Rev. Lett.
93
,
170402
1
(
2004
).
6.
M.
Naraschewski
,
H.
Wallis
,
A.
Schenzle
,
J. I.
Cirac
, and
P.
Zoller
, “
Interference of Bose condensates
,”
Phys. Rev. A
54
,
2185
2196
(
1996
);
[PubMed]
H.
Wallis
,
A.
Röhrl
,
M.
Naraschewski
, and
A.
Schenzle
, “
Phase-space dynamics of Bose condensates: Interference versus interaction
,”
Phys. Rev. A
55
,
2109
2119
(
1997
).
7.
P. W.
Anderson
, “
Measurement in quantum theory and the problem of complex systems
,” in
The Lesson of Quantum Theory
, edited by
J.
de Boer
,
E.
Dal
, and
O.
Ulfbeck
(
Elsevier
, New York,
1986
).
8.
J.
Javanainen
and
Sun
Mi Yoo
, “
Quantum phase of a Bose-Einstein condensate with an arbitrary number of atoms
,”
Phys. Rev. Lett.
76
,
161
164
(
1996
).
9.
T.
Wong
,
M. J.
Collett
, and
D. F.
Walls
, “
Interference of two Bose-Einstein condensates with collisions
,”
Phys. Rev. A
54
,
R3718
R3721
(
1996
).
10.
J. I.
Cirac
,
C. W.
Gardiner
,
M.
Naraschewski
, and
P.
Zoller
, “
Continuous observation of interference fringes from Bose condensates
,”
Phys. Rev. A
54
,
R3714
R3717
(
1996
).
11.
Y.
Castin
and
J.
Dalibard
, “
Relative phase of two Bose-Einstein condensates
,”
Phys. Rev. A
55
,
4330
4337
(
1997
).
12.
P.
Horak
and
S. M.
Barnett
, “
Creation of coherence in Bose-Einstein condensates by atom detection
,”
J. Phys. B
32
,
3421
3436
(
1999
).
13.
K.
Mølmer
, “
Macroscopic quantum-state reduction: Uniting Bose-Einstein condensates by interference measurements
,”
Phys. Rev. A
65
,
R0210607
1
(
2002
).
14.
F.
Laloë
, “
The hidden phase of Fock states: Quantum non-local effects
,”
Eur. Phys. J. D
D 33
,
87
97
(
2005
).
15.
C. J.
Pethick
and
H.
Smith
,
Bose-Einstein Condensates in Dilute Gases
(
Cambridge U. P.
, Cambridge,
2002
), Chap. 13.
16.
P. W.
Anderson
,
Basic Notions in Condensed Matter Physics
(
Benjamin-Cummins
, Menlo Park,
1984
).
17.
P. W.
Anderson
, “
Considerations on the flow of superfluid helium
,”
Rev. Mod. Phys.
38
,
298
310
(
1966
).
18.
A. J.
Leggett
and
F.
Sols
, “
On the concept of spontaneously broken symmetry in condensed matter physics
,”
Found. Phys.
21
,
353
364
(
1991
).
19.
A. J.
Leggett
, “
Broken gauge symmetry in a Bose condensate
,” in
Bose-Einstein Condensation
, edited by
A.
Griffin
,
D. W.
Snoke
, and
S.
Stringari
(
Cambridge U. P.
, Cambridge,
1995
).
20.
A. J.
Leggett
, “
Emergence is in the eye of the beholder
,” review of A Different Universe by R. B. Laughlin in
Phys. Today
58
,
77
78
(
2005
).
21.
J. R.
Johnston
, “
Coherent states in superfluids: The ideal Einstein-Bose gas
,”
Am. J. Phys.
38
,
516
528
(
1970
).
22.
C.
Cohen-Tannoudji
,
B.
Diu
, and
F.
Laloë
,
Quantum Mechanics
(
Wiley
, New York,
1977
), p.
559
.
23.
J. S.
Bell
,
Speakable and Unspeakable in Quantum Mechanics
(
Cambridge U. P.
, Cambridge,
1987
).
24.
F.
Laloe
, “
Do we really understand quantum mechanics
,”
Am. J. Phys.
69
,
655
701
(
2001
).
25.
R. J.
Glauber
, “
Coherent and incoherent states of the radiation field
,”
Phys. Rev.
131
,
2766
2788
(
1963
).
26.
O.
Penrose
and
L.
Onsager
, “
Bose-Einstein condensation and liquid Helium
,”
Phys. Rev.
104
,
576
584
(
1956
).
27.
For example,
S.
Howard
and
S. K.
Roy
, “
Coherent states of a harmonic oscillator
,”
Am. J. Phys.
55
,
1109
1117
(
1987
);
I. H.
Deutsch
, “
A basis-independent approach to quantum optics
,”
Am. J. Phys.
59
,
834
839
(
1991
).
28.
S.
Kohler
and
F.
Sols
, “
Phase-resolution limit in the macroscopic interference between Bose-Einstein condensates
,”
Phys. Rev. A
63
,
053605
1
(
2001
).
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