Diagnostics of macroscopic quantum states of Bose–Einstein condensate in double-well potential by nonstationary Josephson effect
We propose a method of diagnostics of a degenerate ground state of Bose condensate in a double- well potential. The method is based on the study of the one-particle coherent tunneling under switching of the time-dependent weak Josephson coupling between the wells. We obtain a simple expression th...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1218012017-06-19T03:03:07Z Diagnostics of macroscopic quantum states of Bose–Einstein condensate in double-well potential by nonstationary Josephson effect Vol, E.D. Квантовые жидкости и квантовые кpисталлы We propose a method of diagnostics of a degenerate ground state of Bose condensate in a double- well potential. The method is based on the study of the one-particle coherent tunneling under switching of the time-dependent weak Josephson coupling between the wells. We obtain a simple expression that allows one to determine the phase of the condensate and the total number of the particles in the condensate from the relative number of particles in the two wells Δn = n₁ - n₂ measured before the Josephson coupling is switched on and after it is switched off. The specifics of the application of the method in the cases of the external and the internal Josephson effect are discussed. 2005 Article Diagnostics of macroscopic quantum states of Bose–Einstein condensate in double-well potential by nonstationary Josephson effect / E.D. Vol // Физика низких температур. — 2005. — Т. 31, № 2. — С. 131-133. — Бібліогр.: 7 назв. — англ. 0132-6414 PACS: 03.75.Fi, 05.30.Jp http://dspace.nbuv.gov.ua/handle/123456789/121801 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Квантовые жидкости и квантовые кpисталлы Квантовые жидкости и квантовые кpисталлы |
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Квантовые жидкости и квантовые кpисталлы Квантовые жидкости и квантовые кpисталлы Vol, E.D. Diagnostics of macroscopic quantum states of Bose–Einstein condensate in double-well potential by nonstationary Josephson effect Физика низких температур |
| description |
We propose a method of diagnostics of a degenerate ground state of Bose condensate in a double-
well potential. The method is based on the study of the one-particle coherent tunneling under
switching of the time-dependent weak Josephson coupling between the wells. We obtain a simple
expression that allows one to determine the phase of the condensate and the total number of the
particles in the condensate from the relative number of particles in the two wells Δn = n₁ -
n₂ measured
before the Josephson coupling is switched on and after it is switched off. The specifics of the
application of the method in the cases of the external and the internal Josephson effect are discussed. |
| format |
Article |
| author |
Vol, E.D. |
| author_facet |
Vol, E.D. |
| author_sort |
Vol, E.D. |
| title |
Diagnostics of macroscopic quantum states of Bose–Einstein condensate in double-well potential by nonstationary Josephson effect |
| title_short |
Diagnostics of macroscopic quantum states of Bose–Einstein condensate in double-well potential by nonstationary Josephson effect |
| title_full |
Diagnostics of macroscopic quantum states of Bose–Einstein condensate in double-well potential by nonstationary Josephson effect |
| title_fullStr |
Diagnostics of macroscopic quantum states of Bose–Einstein condensate in double-well potential by nonstationary Josephson effect |
| title_full_unstemmed |
Diagnostics of macroscopic quantum states of Bose–Einstein condensate in double-well potential by nonstationary Josephson effect |
| title_sort |
diagnostics of macroscopic quantum states of bose–einstein condensate in double-well potential by nonstationary josephson effect |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2005 |
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Квантовые жидкости и квантовые кpисталлы |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/121801 |
| citation_txt |
Diagnostics of macroscopic quantum states
of Bose–Einstein condensate in double-well potential
by nonstationary Josephson effect / E.D. Vol // Физика низких температур. — 2005. — Т. 31, № 2. — С. 131-133. — Бібліогр.: 7 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT voled diagnosticsofmacroscopicquantumstatesofboseeinsteincondensateindoublewellpotentialbynonstationaryjosephsoneffect |
| first_indexed |
2025-07-08T20:32:39Z |
| last_indexed |
2025-07-08T20:32:39Z |
| _version_ |
1837112241457987584 |
| fulltext |
Fizika Nizkikh Temperatur, 2005, v. 31, No. 2, p. 131–133
Diagnostics of macroscopic quantum states
of Bose–Einstein condensate in double-well potential
by nonstationary Josephson effect
E.D. Vol
B. Verkin Institute for Low Temperature Physics and Engineering
of the National Akademy of Sciences of Ukraine, 47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: vol@ilt.kharkov.ua
Received June 10, 2004, revised July 15, 2004
We propose a method of diagnostics of a degenerate ground state of Bose condensate in a dou-
ble-well potential. The method is based on the study of the one-particle coherent tunneling under
switching of the time-dependent weak Josephson coupling between the wells. We obtain a simple
expression that allows one to determine the phase of the condensate and the total number of the
particles in the condensate from the relative number of particles in the two wells �n n n� �1 2 mea-
sured before the Josephson coupling is switched on and after it is switched off. The specifics of the
application of the method in the cases of the external and the internal Josephson effect are dis-
cussed.
PACS: 03.75.Fi, 05.30.Jp
Beginning from its first observation [1] the
Bose–Einstein condensation (BEC) of atoms in alkali
metal vapors remains a source of new possibilities for
the study of macroscopic quantum phenomena. One of
these phenomena is the coherent tunneling of atoms
between two coupled Bose condensates (BC) [2],
which is analogous to the Josephson effect in super-
conductors. It is known [3] that for the stationary
case, when the total number of atoms in the trap
N n n� �1 2 is conserved and the trap is symmetric re-
lative to the two BC, the average relative number of
atoms n n n n1 2 1 2� � � � �� �| � � | is equal to zero in the
ground state and in any excited state. Therefore, one
can expect that in such a situation the study of
nonstationary coherent tunneling (which is realized
when one or several parameters of the system depends
on time) is more informative for the diagnostics of the
macroscopic wave functions of the condensates than is
the study of the stationary case. In the nonstationary
case the average value of the relative number of atoms
n n1 2� measured at a time t0 is generally nonzero and
depends on the history of the systems at all t t� 0. In
this paper we show that nonstationary Josephson ef-
fect can be used for the diagnostics of a macroscopic
state of BC and the total number of the atoms in the
condensate.
We consider a simple model of coherent tunneling
between two BC, described in [4] (see also references
therein). The model is based on the two-mode approx-
imation, which implies that each of N bosons can be in
one of two states, and the dynamical coupling be-
tween these states allows the bosons to jump from one
state to the other. Such a model is applicable for a de-
scription of the external as well as the internal
Josephson effect in Bose systems. The external Jo-
sephson effect [5] can be realized if the Bose gas is
confined in a double-well trap and the tunneling be-
tween two wells is small. In this case two modes corre-
spond to self-consistent ground states in each well.
The internal Josephson effect [6] can be realized in a
Bose gas with two macroscopically occupied hyperfine
states (e.g., the | ,F mF� � � �1 1 and | ,F mF� � �2 1
states of 87Rb atoms). The dynamical coupling be-
tween the two states is settled by a resonant laser field
applied to the system. At the beginning we specify the
simplest case of the external Josephson effect at T � 0.
The Hamiltonian of the symmetric two-mode model
has the form
H H H t
K
n ns� � � �0 1 1 2
2
8
( ) ( � � )
� ��E t
a aJ ( )
( )
2 1 2 h. c , (1)
© E.D. Vol, 2005
where ai
� (ai ) are the creation (annihilation) opera-
tors for the well i, and �n a ai i i� � are the number ope-
rators. The parameter K is determined by the interac-
tion between the atoms in the well. Here we consider
the case K
0, corresponding to a repulsive interac-
tion. The value of EJ is determined by the overlap-
ping of the wave functions of two modes and it can be
controlled by a variation of the height and (or) one
can width of the barrier. For the external Josephson
effect one can without loss of generality choose the
Josephson coupling E tJ ( ) to be real.
Let us consider the situation when the dynamical
coupling between two condensates is switched on at
t ti� � 0 and switched off at t tf� . At t � 0 and t tf
the coupling parameter E tJ ( ) � 0 and the occupation
numbers operators �n1 and �n2 as well as the relative
number operator � �n n1 2� commute with the Hamil-
tonian and do not depend on time. During the time
when the coupling is switched on, the operator � �n n1 2�
is changed. Suppouse that at t � 0 the wave function of
the two-mode Bose condensate is �( )0 and that at t tf�
it becomes �( )tf . The task we consider is how to find
the characteristics of the function �( )0 from the mea-
surements of the mean relative number � � �� �| � � |n n1 2 .
Let us specify the case of an odd total number of parti-
cles (the case of an even N is discussed below). At
N M� �2 1 and Ej � 0 the ground state of the Ha-
miltonian (1) is doubly degenerate. The minimum of
the energy, equal to K/8, is reached for the orthogo-
nal states | | ,g M M1 1� � � � and | | ,g M M2 1� � � � as
well as for an arbitrary superposition of these states
| | |g a g b g� � � � �1 2 (| | | |a b2 2 1� � ). The state at t � 0 can
be parameterized as | ( ) cos ( )| ,� 0 2 1� � � � ��/ M M
� � ��sin ( ) | ,�/ M Mi2 1e . At �
� 0, this is the entan-
gled state. The angle � is connected with the initial
relative number by the relation
� � �n n n( ) ( )| � � | ( ) cos0 0 01 2 � � � � �. (2)
Since this value does not depend on � the phase can-
not be extracted from the result of measurements of
the initial relative number. But the phase � is also an
essential characteristic of the macroscopic state of the
Bose condensate. In particular, the interference pat-
tern emerging under an overlapping of two such sys-
tems (two BC in degenerate states with internal
phases �1 and �2) is determined by the relative phase
� � � � �1 2. In this case we keep in mind that total
phases of both condensates are fixed and equal to
each other. We will show that the value of � can be
determined from the measurements of the final rela-
tive number � � �n t n n tf f f � � �( )| � � | ( )1 2 . To do this
the amplitude of the Josephson coupling should
be taken so small that the strong inequality
NE /KJ
max �� 1 is satisfied. Then at 0 � �t tf the sys-
tem remains in the Fock regime. In this regime the
dynamics of the system is realized mainly on the
states for which | ( )| � � | ( ) |� � � �� �t n n t1 2 1. Therefore to
find the evolution of the function � one can use the
basis (| g1�, | g2 �). Note that the regime considered is
the same as required for a realization of the Bose
qubit [7].
It is more convenient to use the unitary trans-
formed basis of symmetric | (| | )s g g /� � �� �1 2 2 and
antisymmetric | (| | )a g g /� � �� �1 2 2 states. In this ba-
sis the wave function of the BC reads as
�( ) ( )| ( )|t s t s a t a� � � �. Using the nonstationary
Schrödinger equation i H� �� �� one finds that the
functions s t( ) and a t( ) satisfy the equations
i s
K
s
E t M
sJ
��
( )( )
� �
�
8
1
2
, (3)
i a
K
a
E t M
aJ
� �
( )( )
� �
�
8
1
2
. (4)
Integrating equations (3), (4), we find the mean
value of the relative number at the time tf
� �n t s t a t s af f f
i( ) ( ) ( ) ( ) ( )* *� � � � ��c. c. e c. c.0 0 2
� � �cos cos ( ) sin sin ( )sin� �2 2� � , (5)
where � � � � ��( )( ) ( )1 2 1
0
/ M E t dtJ
tf
� .
Equation (5) determines the relation between the
measured quantity �n tf( ) and the parameter �. One
can see that for an entangled initial state the relative
number in a final state depends on the phase � and this
phase can be found from the measurement of �n tf( ).
Thus, if one has a system in a reproducible (but un-
known) initial state | ( )� 0 � the parameters � and � that
describe this state can be found from two sets of mea-
surements of the relative number at t � 0 and t tf�
(under the assumption that the total number of parti-
cles N in the condensate is known). If the total num-
ber of particles is unknown an additional set of mea-
surements is required: measurement of the final
relative occupation number for another value of tf .
Using the results of three sets of measurements one
can determine the initial state and find the total num-
ber of particles in the condensate.
It is necessary to point out an essential restriction for
the maximum value of t tf i� . In deriving (5) we did
not take into account that the coupling between two
condensates induces small (of order of E M /KJ ( )� 1 )
but nonzero occupation of the excited states | ( )ek
1 � �
� � � � �| ,M k M k1 and | | ,( )e M k M kk
2 1� � � � � �
(with k
0). Due to such processes the phases of s(t)
and a t( ) are shifted from the values given by the solu-
132 Fizika Nizkikh Temperatur, 2005, v. 31, No. 2
E.D. Vol
tion of Eqs. (3), (4). If such a shift is of the order of
unity the relation (5) is not valid any more. Ne-
vertheless, one can show that for t tf i� ��
�� ��K E MJ/ [ ( )]max 1 2 the phase shifts are very
small and Eq. (5) is applicable. The fulfillment of the
mentioned restriction on the value of t tf i� is re-
quired for the use of the diagnostic methods proposed.
Let us now discuss the case of BC with an even
number of atoms. In the symmetric double-well trap
the ground state of the condensate with N M� 2 is
| | ,g M M� � �. This state is nondegenerate and �n( )0 �
� ��n tf( ) 0. If initially the system in an excited state
| | , | ,e a M M b M M� � � � � � � � �1 1 1 1 then �n( )0 �
� �2 2 2(| | | | )a b can be nonzero, but for t tf i� ��
�� ��K E MJ/ [ ( )]max 1 2 the difference �n tf( ) �
��n( )0 is of the order of ME KJ
max / �� 1. Such be-
havior differs from the case of odd N, where the
change of �n can be order of unity. This feature can be
used for determining the parity of the number of at-
oms in the BC. We point out again that this conclu-
sion is for a confining potential symmetric relative the
two BC.
If the confining potential is asymmetric the Hamil-
tonian (1) is modified to (see e.g. [4])
H H H ta
1 0 1� � ( )
� � � � ��K
n n n n
E t
a aJ
8 21 2
2
1 2 1 2( � � ) ( � � )
( )
( )�� h.c .
(6)
One can see that if the potential bias �� � K/4, the
ground state of the system with an even number of at-
oms N M� 2 is doubly degenerate at EJ � 0, and its
wave function can be presented in the form | g� �
� � � � � �a M M b M M| , | ,1 1 . This situation is analo-
gous to the symmetric case with odd N. The only dif-
ference is that the values of �n( )0 and �n tf( ) given
above are counted from �n � 1. Thus, under assump-
tion that one can control the value of �� with the ac-
curacy | | max�� � ��K/ MEJ4 the method of diagnos-
tics of the ground state wave function and the total
number of atoms suggested is applicable for BC with
even N.
Hitherto we have discussed the case of the external
Josephson effect. The case of the internal Josephson
effect is also described by the Eq. (6) (in the rotating
frame of reference) [4]. In this case the expression for
the chemical potential �� reads as
�� �
�� � � �
4 2
11 22
N
m
a a
� ~( ), (7)
where � is the detuning of the laser field from the re-
sonant frequency, a11 and a22, the s-wave scattering
amplitudes of macroscopically occupied internal
states |1� and |2�, respectively, m is the mass of the at-
oms, and ~� is the renormalized constant K. In a situa-
tion where the value of the detuning can be varied
smoothly, one can achieve the regime of the degene-
rate ground state of the Hamiltonian Ha
0 both for
even (� �� e) and for odd (� �� î) numbers of atoms
N. In such a regime one can apply the method of di-
agnostics of the initial state of BC proposed in this
paper. We note that for the case of the internal
Josephson effect the value of �n tf( ) is just propor-
tional to the expectation value of M tz f( ) – the pro-
jection of the magnetic momentum of the BC on the
axis chosen. Therefore, using Eq. (5) one can deter-
mine the phase � and the total number of atoms in the
condensate from the measurement of M tz f( ).
I would like to acknowledge R.I. Shekhter, S.I.
Shevchenko, and D.V. Fil for discussion of the results
presented in this article. This work is supported by
INTAS Grant No 01-2344.
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Diagnostics of macroscopic quantum states of Bose–Einstein condensate
Fizika Nizkikh Temperatur, 2005, v. 31, No. 2 133
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