Jul 26, 2005 - A microscopic theory of a nondissipative drag in a two-component superfluid Bose gas is developed. The expression for the drag current in the ...
PHYSICAL REVIEW A 72, 013616 共2005兲
Nondissipative drag of superflow in a two-component Bose gas D. V. Fil1,2 and S. I. Shevchenko3
1
Institute for Single Crystals, National Academy of Sciences of Ukraine, Lenin av. 60, Kharkov 61001, Ukraine 2 Ukrainian State Academy of Railway Transport, Feyerbakh Sq. 7, 61050 Kharkov, Ukraine 3 B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, Lenin av. 47, Kharkov 61103, Ukraine 共Received 11 January 2005; published 26 July 2005兲 A microscopic theory of a nondissipative drag in a two-component superfluid Bose gas is developed. The expression for the drag current in the system with the components of different atomic masses, densities, and scattering lengths is derived. It is shown that the drag current is proportional to the square root of the gas parameter. The temperature dependence of the drag current is studied and it is shown that at temperature of order or smaller than the interaction energy the temperature reduction of the drag current is rather small. A possible way of measuring the drag factor is proposed. A toroidal system with the drag component confined in two half-ring wells separated by two Josephson barriers is considered. Under certain condition such a system can be treated as a Bose-Einstein counterpart of the Josephson charge qubit in an external magnetic field. It is shown that the measurement of the difference of number of atoms in two wells under a controlled evolution of the state of the qubit allows one to determine the drag factor. DOI: 10.1103/PhysRevA.72.013616
PACS number共s兲: 03.75.Kk, 03.75.Lm, 03.67.Lx
I. INTRODUCTION
Macroscopic quantum coherence manifests itself in many specific phenomena. One of them is a nondissipative drag that takes place in superfluids and superconductors. The nondissipative drag, also known as the Andreev-Bashkin effect, was considered, for the first time, in Ref. 关1兴, where a three velocity hydrodynamic model for 3He- 4He superfluid mixtures was developed. It was shown that superfluid behavior of such systems can be described under accounting the “drag” term in the free energy. This term is proportional to the scalar product of the superfluid velocities of two superfluid components. A similar situation may take place in mixtures of superfluids of Sz = + 1 and Sz = −1 pairs in liquid 3He in the A-phase 关2兴. Among other objects, where the nondissipative drag may be important, are neutron stars, where the mixture of neutron and proton Cooper pair Bose condensates is believed to realize 关3,4兴. The possibility of realization of the nondissipative drag in superconductors was considered in 关5兴. The nondissipative drag in bilayer Bose systems was treated microscopically in 关6,7兴 for a special case of two equivalent layers of charged bosons. The case of a bilayer system of neutral bosons was studied in 关8兴 in the limit of small interlayer interaction. The most promising systems where the nondissipative drag can be observed experimentally are two-component alkali metal vapors. In such systems the interaction between atoms of different species is of the same order as the interaction between atoms of the same specie and the effect is expected to be larger than in bilayers. In Bose mixtures the components are characterized by different densities, different masses of atoms, and different interaction parameters. In this paper we consider such a general case and obtain an analytical expression for the drag current for zero and finite temperatures. In the system under consideration the drag force influences the dynamics of atoms in the drag component in the 1050-2947/2005/72共1兲/013616共9兲/$23.00
same manner as the vector potential of the electromagnetic field influences the dynamics of electrons in superconductors. In particular, in neutral superfluids with Josephson links the drag effect may induce the gradient of the phase of the order parameter in the bulk and, as a consequence, control the phase difference between weakly coupled parts of the system. Therefore, one can expect that the effect reveals itself in a modification of Josephson oscillations between weakly coupled Bose gases. In this paper we discuss possible ways for the observation of such a modification. We consider the Bose gas confined in a toroidal trap with two Josephson links. In the Fock regime 关9兴 the low energy dynamics of the system can be described by the qubit model of general form 共the model, where all three components of the pseudomagnetic field can be controlled independently兲. The parameters of the qubit Hamiltonian depend on the drag factor. The measurement of the state of the qubit under controlled evolution allows one to observe the effect caused by the nondissipative drag and determine the drag factor. In this paper we consider two particular schemes of the measurement. In the first scheme one should determine the time required to transform a reproducible initial state to a given final state. In the second scheme the geometrical 共Berry兲 phase should be detected. In Sec. II the microscopic theory of the nondissipative drag in two-component Bose gases is developed. In Sec. III a model of the Bose-Einstein qubit subjected by the drag force is formulated and the schemes of measurement of the drag factor are proposed. Conclusions are given in Sec. IV. II. NONDISSIPATIVE DRAG IN A TWO-COMPONENT BOSE SYSTEM: MICROSCOPIC DERIVATION
Let us consider a uniform two-component atomic Bose gas in a Bose-Einstein condensed state. We will study the most general situation where the densities of atoms in each component are different from one another 共n1 ⫽ n2兲, the at-
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PHYSICAL REVIEW A 72, 013616 共2005兲
D. V. FIL AND S. I. SHEVCHENKO
oms of each components have different masses 共m1 ⫽ m2兲, and the interaction between atoms is described by three different scattering lengths 共a11 ⫽ a22 ⫽ a12兲. The Hamiltonian of the system can be presented in the form H=
1
兺 共Ei − iNi兲 + 2 兺 i=1,2
int
i,i⬘=1,2
H2 =
冕 冉兺 再 冋 i
បvi · 兵nˆi共r兲 ˆ i共r兲 + 关 ˆ i共r兲兴nˆi共r兲其 2
+
iប2 兵关nˆi共r兲兴 · ˆ i共r兲 − 关 ˆ i共r兲兴 · nˆi共r兲其 2mi
+
␥ii 关nˆi共r兲兴2 + ␥12nˆ1共r兲nˆ2共r兲 . 2
where Ei =
冕
ប2 ˆ †共r兲兴 · ⌿ ˆ 共r兲 d 3r 关⌿ i i 2mi
共2兲
is the kinetic energy, int
Eii⬘ =
冕
ˆ †共r兲⌿ ˆ † 共r兲␥ ⌿ ˆ ˆ d3r⌿ ii⬘ i⬘共r兲⌿i共r兲 i i⬘
共3兲
is the energy of interaction, ␥ii = 4ប2aii / mi and ␥12 = 2ប2共m1 + m2兲a12 / 共m1m2兲 are the interaction parameters, and i are the chemical potentials. For the further analysis it is convenient to use the density and phase operator approach 共see, for instance, 关10,11兴兲. The approach is based on the following representation for the Bose field operators: ˆ 共r兲 = exp关i 共r兲 + iˆ 共r兲兴冑n + nˆ 共r兲, ⌿ i i i i i
共4兲
ˆ †共r兲 = 冑n + nˆ 共r兲exp关− i 共r兲 − iˆ 共r兲兴, ⌿ i i i i i
共5兲
where nˆi and ˆ i are the density and phase fluctuation operators, i共r兲 are the c-number terms of the phase operators, which are connected with the superfluid velocities by the relation vi = ប i / mi. In what follows we specify the case of the superfluid velocities independent of r. Substituting Eqs. 共4兲 and 共5兲 into Eq. 共1兲 and expanding it in series in powers of nˆi and ˆ i we present the Hamiltonian of the system in the following form 共6兲
H = H0 + H2 + ¯ . In Eq. 共6兲 the term H0 = V
冉兺 冋 i=1,2
册
1 ␥ii miniv2i + n2i − ini + ␥12n1n2 2 2
冊
共i = 1,2兲.
冎
冊
共9兲
The quadratic part of the Hamiltonian determines the spectra of the elementary excitations. Hereafter we will neglect the higher order terms in the Hamiltonian 共6兲. These terms describe the scattering of the quasiparticles and they can be omitted if the temperature is much smaller than the temperature of Bose-Einstein condensation. Let us rewrite the quadratic part of the Hamiltonian in terms of the operators of creation and annihilation of the elementary excitations. As the first step, we use the substitution nˆi共r兲 =
ˆ i共r兲 =
冑
1 2i
ni 兺 eik·r V k
冑
冑
1 兺 eik·r n iV k
⑀ik 关bi共k兲 + b†i 共− k兲兴, Eik
冑
共10兲
Eik 关b 共k兲 − b†i 共− k兲兴, 共11兲 ⑀ik i
where operators b†i , bi satisfy the Bose commutation relations. Here ⑀ik = ប2k2 / 2mi is the spectrum of free atoms, and Elk = 冑⑀ik共⑀ik + 2␥iini兲
共12兲
is the spectrum of the elementary excitations at ␥12 = 0 and vi = 0. The substitution 共10兲 and 共11兲 reduces the Hamiltonian 共9兲 to the form quadratic in b†i and bi operators:
冋 冉
H2 = 兺 Ei共k兲 b†i 共k兲bi共k兲 + ik
冊 册
1 1 − ⑀ik 2 2
+ 兺 gk关b†1共k兲b2共k兲 + b1共k兲b2共− k兲 + H.c.兴. 共13兲
共7兲
k
does not contain the operator part. Here V is the volume of the system. The minimization conditions for the Hamiltonian H0 yield the equations 1 miv2i + ␥iini + ␥12n3−i − i = 0 2
册
+
共1兲
Eii⬘ ,
ប2 关nˆi共r兲兴2 + ni关 ˆ i共r兲兴2 2mi 4ni
dr
共8兲
Under the conditions 共8兲 the terms, linear in the density fluctuation operators, vanish in the Hamiltonian. Taking into account the · 关ni i共r兲兴 = 0, we find that the terms, linear in the phase fluctuation operators, vanish in the Hamiltonian as well. The part of the Hamiltonian quadratic in ˆ i and nˆi operators reads as
Here Ei共k兲 = Eik + បk · vi
共14兲
and gk = ␥12
冑
⑀1k⑀2kn1n2 . E1kE2k
共15兲
The Hamiltonian 共15兲 contains nondiagonal in Bose creation and annihilation operator terms and it can be diagonalized using the standard procedure of u-v transformation 关13兴. The result is
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H2 = 兺 k
冋兺 冉 =␣,
E共k兲 † 共k兲共k兲 +
册
冊
1 1 − 兺 ⑀ik , 2 2 i=1,2 共16兲
where † 共k兲 and 共k兲 are the operators of creation and annihilation of elementary excitations. The energies E共k兲 satisfy the equation
冉
A − EI
B
B
A + EI
E1共k兲
0
gk
0
0
E1共− k兲
0
gk
gk
0
E2共k兲
0
0
gk
0
E2共− k兲
det
冊
As required in the procedure 关13兴, we take positive valued solutions of Eq. 共17兲. The energies 共22兲 should be real valued quantities. This requirement yields the common condition for 2 the stability of the two-component system: ␥12 艋 ␥11␥22. If this condition were not fulfilled, spatial separation of two components 共at positive ␥12兲 or a collapse 共at negative ␥12兲 would take place. At nonzero superfluid velocities we present the solutions of Eq. 共17兲 as series in vi:
共17兲
= 0,
where
A=
冢
B=
冢
0
0
0 gk
0
0 gk 0
0 gk 0
0
0
0
gk 0
冣
冣
冉
2 E2 − E2k 1 + បk · v2 1 − 21k 2 E␣k − E2 k
共18兲
,
+
共19兲
,
1 F , ji = V vi
共20兲
where F is the free energy of the system. Here the quantity ji is defined as the density of the mass current. The free energy of the system, described by the Hamiltonian 共6兲, is given by the formula
+ T兺 k
冋兺
=␣,
E共k兲 −
冋 冉
兺 ln =␣,
兺 ⑀ik
i=1,2
1 − exp −
T
冊册
冉
冑
E␣k共E␣2 k − E2 k兲3
冉
−
冉
ប2共k · v1 − k · v2兲2 ,
冊
2 2␥12 n1n2⑀1k⑀2k共E␣2 k + 3E2 k兲
Ek共E␣2 k − E2 k兲3
共23兲
冊
ប2共k · v1 − k · v2兲2 . 共24兲
Note that at v1 = v2 = v the spectra 共23兲 and 共24兲 are reduced to common expressions for the energies of quasiparticles in a moving condensate: E␣共兲共k兲 = E␣共兲k + បk · v. Using Eqs. 共21兲, 共23兲, and 共24兲 we obtain the following expression for the free energy: F = F0 +
V 关共1 − n1兲v21 + 共2 − n2兲v22 − dr共v1 − v2兲2兴, 2 共25兲
共21兲
.
where F0 does not depend on vi. In Eq. 共25兲 i = mini are the mass densities, the quantities
The second term in Eq. 共21兲 is the energy of the zero-point fluctuations and the third term is the standard temperature dependent part of the free energy for the gas of noninteracting elementary excitations. We specify the case of small superfluid velocities 共much smaller than the critical ones兲. In this case the currents can be approximated by the expressions linear in vi. To obtain these expressions we will find the free energy as series in vi, neglecting the terms higher than quadratic. At v1 = v2 = 0 the equation 共17兲 is easily solved and the spectra are found to be 2 2 E1k + E2k E␣共兲k = ± 2
2 n1n2⑀1k⑀2k共3E␣2 k + E2 k兲 2␥12
2 E2 − E2k 1 + បk · v2 1 + 21k 2 E␣k − E2 k
册
E共k兲
冊
冊
2 E2 − E2k 1 E共k兲 = Ek + បk · v1 1 − 21k 2 E␣k − E2 k
and I is the identity matrix. The densities of superfluid currents in two components can be obtained from the relation
1 F = H0 + 兺 2 k
冉
2 E2 − E2k 1 E␣共k兲 = E␣k + បk · v1 1 + 21k 2 E␣k − E2 k
2 2 2 共E1k − E2k 兲 2 + 4␥12 n1n2⑀1k⑀2k 4
冊
1/2
. 共22兲
冋
n1 = −
dN m1 dN 兺 ⑀1k dE␣␣kk + dEkk 3V k
+
2 2 E1k − E2k dN␣k dNk − 2 2 E␣k − Ek dE␣k dEk
n2 = −
dN␣k dNk m2 ⑀2k + 兺 3V k dE␣k dEk
−
2 2 E1k − E2k dN␣k dNk − E␣2 k − E2 k dE␣k dEk
冉
冊册
冋
冉
,
冊册
共26兲
共27兲
describe the thermal reduction of the superfluid densities, and the quantity
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D. V. FIL AND S. I. SHEVCHENKO
dr =
冋
The integral in Eq. 共31兲 can be evaluated analytically. To present the answer in a compact form it is convenient to introduce the dimensionless parameters
2 3/2 4 冑m1m2 兺 ␥12n1n2共⑀1k⑀2k兲 1 + N␣k + N3k E ␣kE k 3V 共E␣k + Ek兲 k
−
冉
N ␣k − N k 2E␣kEk dN␣k dNk + 2 2 2 3 + 共E␣k − Ek兲 共E␣k − Ek兲 dE␣k dEk
冊册
,
=
共28兲 which we call the “drag density,” yields the value of redistribution of the superfluid densities between the components. In Eqs. 共26兲–共28兲 N␣共兲k = 关exp共E␣共兲k / T兲 − 1兴−1 is the Bose distribution function. Using Eqs. 共20兲 and 共25兲 we arrive at the following expressions for the supercurrents:
2 a12 and = a11a22
冑
n1a11 + n2a22
冑
n2a22 n1a11
共0 艋 艋 1 and 艌 2兲. Using these notations we have 3 3 dr = 冑12冑n1a11 n2a22 4
冑 F共, 兲,
共32兲
where j1 = 共1 − n1 − dr兲v1 + drv2 ,
共29兲 F共, 兲 =
j2 = 共2 − n2 − dr兲v2 + drv1 .
共30兲
One can see that at nonzero dr the current of one component contains the term proportional to the superfluid velocity of the other component. It means that there is a transfer of motion between the components. In particular, at v1 = 0 the current in the component 1 共j1 = drv2兲 is purely the drag 2 current. Since dr is the function of ␥12 关see Eqs. 共28兲 and 共22兲兴 the drag current does not depend on the sign of the interaction between the components. Equation 共28兲 is the main result of the paper. This equation yields the value of the drag for the general case of the two-component Bose system with components of different densities, different masses of atoms, different interaction parameters, and for zero as well as for nonzero temperatures. Moreover, this equation is valid not only for the point interaction between the atoms, but for any central force interaction. In the latter case the interaction parameters ␥ik in Eq. 共28兲 and in the spectra 共22兲 and 共12兲 should be replaced with the Fourier components of the corresponding interaction potentials. To estimate the absolute value of the drag we, for simplicity, specify the case m1 = m2 = m, that is realized when two components are two hyperfine states of the same atoms. At T = 0 Eq. 共28兲 is reduced to
dr =
4m 3
冕
⬁
0
d⑀
2 n1n2共⑀兲⑀1/2 ␥12
冑共⑀ + w1兲共⑀ + w2兲共冑⑀ + w1 + 冑⑀ + w2兲3 , 共31兲
where
共⑀兲 =
m3/2
冑2 2 ប 3
冑⑀
is the density of states for free atoms, and 2 w1共2兲 = ␥11n1 + ␥22n2 ± 冑共␥11n1 − ␥22n2兲2 + 4␥12 n 1n 2 .
共 + 3冑1 − 兲冑
256
45冑2 共冑 + 冑2 − 4 + 4 + 冑 − 冑2 − 4 + 4兲3
.
共33兲 Direct evaluation of Eq. 共33兲 shows that at allowed and 共0 艋 艋 1 and 艌 2兲 the factor F共 , 兲 is almost the constant 共the range of variation of F is 关0.7–0.8兴兲 and one can neglect the dependence of F on the parameter of the system. At a11n1 = a22n2 we obtain from Eq. 共32兲 the following approximate relation: a2 a2 1 1 3 3 dr ⬇ 1 12 冑n1a11 = 2 12 冑n2a22 . 2 a11a22 2 a11a22
共34兲
If the density of one component is much larger than that of the other and a11 ⬃ a22, the “drag density” is approximated as
dr ⬇ 0.81
dr ⬇ 0.82
2 a12 2 a22
2 a12 2 a11
3 冑n2a22 at n1 Ⰶ n2 ,
3 冑n1a11 at n2 Ⰶ n1 .
共35兲
One can see that the “drag density” is proportional to the square root of the gas parameter. It means that the drag effect is larger in “less ideal” Bose gases. The temperature dependence of the “drag density” at small T can be evaluated analytically from Eq. 共28兲 using the linear approximation for the spectra of the excitations. It yields dr共T兲 = dr共0兲共1 − ␣TT4 / T40兲, where T0 = 冑␥11n1␥22n2 and the factor ␣T is positive. Numerical evaluation of the sum over k in Eq. 共28兲 shows that the analytical approximation is valid only at T Ⰶ T0. At T ⲏ T0 the “drag density” decreases much slower under increase of the temperature. As an example, the dependence of dr共T兲 at n1 = n2 = n, ␥11 = ␥22 = ␥, and = 0.5 is shown in Fig. 1. Now let us discuss how the drag effect can reveal itself in a real physical situation. If one deals with the stationary superflow one implies that it is the circulating superflow, e.g., the tangential superflow in a hole cylinder. In such a case the
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FIG. 1. temperature.
Dependence
of
the
“drag
density”
on
the
superfluid velocities satisfy the Onsager-Feynman quantization condition
冖
vi · dl =
2បNi , mi
共36兲
where the vorticity parameters Ni are integer. Then, the drag effect can be understood as the appearance of the circulating current in the drive component 共e.g., specie 1兲, when the circulation of the superfluid velocity of the drive component 共e.g., specie 2兲 is fixed 共N2 = const兲. The current of the specie 1 关Eq. 共29兲兴 depends on the superfluid velocities of both species and if the superfluid velocity of the drag component is directed antiparallel to the superfluid velocity of the drive component the current of the drag component might vanish. But since the velocities are quantized it may happen only under certain special conditions 共see below兲. The superfluid velocity of the drag component is determined by that at fixed N2 the free energy 共25兲 has a minimum with respect to discrete values of v1 = បN1 / 共m1R兲 关where the R is the radius of the contour in Eq. 共36兲兴. Depending on the value of the parameter ␣ = 关dr / 共2 − n2 − dr兲兴共m1 / m2兲N2 several possibilities can be realized. At 兩␣兩 ⬍ 1 / 2 the minimum of the energy 共25兲 corresponds to N1 = 0 共and v1 = 0兲. In this case the current of the drag component is directed along the drive current and it is proportional to the drag density. At 兩␣兩 = p 共p is natural兲 the value N1 = −p minimizes the energy. In this case two terms in Eq. 共29兲 compensate each other and the current in the drag component vanishes. At half-integer ␣ the degenerate situation takes place: two states 共with codirected currents, and counterdirected currents兲 have the same energy. At p − 1 / 2 ⬍ 兩␣兩 ⬍ p the state with counterdirected currents gains the energy and at p ⬍ 兩␣兩 ⬍ p + 1 / 2 the codirected currents are energetically preferable. In the latter two cases the nonzero vorticity of the drag component 共N1 ⫽ 0兲 is also induced. This behavior is analogous to the behavior of a superconducting ring in a magnetic field. We note that since dr Ⰶ 2, the most realistic case is 兩␣兩 ⬍ 1 / 2 when the simple picture of the transfer of part of the motion from the drive to the drag component takes place. In this study we have concentrated on the analytical derivation of the drag effect in the uniform Bose gases. The consideration of the nonuniform case requires the solution of the eigenvalue problem for the elementary excitations in the
two-component Bose gas in the external potential. But even for the simplest case of a spherically symmetric trap this problem can be solved analytically only in the longwavelength limit and the Thomas-Fermi approximation 关12兴 共the spectrum of elementary excitations in one-component Bose gases was obtained analytically for a number of potentials but also in the same limit 关11,14–16兴兲. Since the main contribution to the drag density comes from the excitations with the wave vectors of order of the healing length 关see Eq. 共28兲兴, the rigorous analysis of the drag effect can be done only numerically. Nevertheless, in the Tomas-Fermi situation the drag effect can be evaluated basing on the following arguments. When the linear size of the Bose cloud is much larger than the healing length, the spectrum of the excitations at the wave vectors of order or higher than the inverse healing length is well-described by the quasiuniform approximation. Therefore the drag effect can be described by the same equations, as in the uniform case with the only modification that the quantities n1 and n2, and, correspondingly, i, ni, dr, and ji in Eqs. 共26兲–共30兲 are understood as functions of coordinates. At an arbitrary symmetry of the trap potential the superfluid velocity of the drag component cannot be equal to zero in each point. Indeed, in the general case of space dependent i, ni, and dr the velocity field v2共r兲 cannot satisfy two independent continuity conditions · 关共2 − n2 − dr兲v2兴 = 0 and · 共drv2兲=0. To analyze this case one should find the velocity fields v1共r兲 and v2共r兲 that satisfy the continuity conditions and the quantization conditions. To illustrate this point let us consider a simple example of a trap having the shape of a hollow cylinder with the densities that depend only on the polar angle . We will seek the velocity fields that do not have radial components. Then, Eqs. 共29兲 and 共30兲, written for the tangential components of the currents and the velocities, can be presented in the matrix form
冉冊 冉
冊
j1 v1共r, 兲 , = Rˆ j2 v2共r, 兲
where
冉
s1共兲 − dr共兲 dr共兲 Rˆ = s2共兲 − dr共兲 dr共兲
共37兲
冊
共38兲
with si共兲 = i共兲 − ni共兲. Due to the continuity conditions the currents j1 and j2 in Eq. 共37兲 do not depend on . According to Eq. 共37兲 the velocities v1共r , 兲 and v2共r , 兲 are connected with the currents by the equation
冉
v1共r, 兲
冊 冉 冊
j1共r兲 = Rˆ−1 . j2共r兲 v2共r, 兲
共39兲
Integrating Eq. 共39兲 over and taking into account the quantization conditions 共36兲 we obtain the equation for the currents
冉 冊
冉 冊
j1共r兲 2ប N1/m1 = , Tˆ j2共r兲 r N2/m2 where
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D. V. FIL AND S. I. SHEVCHENKO
Tˆ =
冢
冕 冕
2
d
0
2
−
s2 − dr s1s2 − dr共s1 + s2兲
d
0
dr s1s2 − dr共s1 + s2兲
If a given vorticity of the drive component N2 is not very large the minimum of energy is reached at N1 = 0. In the latter case the solution of Eq. 共40兲 in the leading order in dr yields the following expression for the current of the drag component:
j1共r兲 ⬇
2បN2 m 2r
冕
冕
2
0
2
0
d
dr共兲 s1共兲s2共兲
1 d s1共兲
冕
2
0
1 d s2共兲
.
共42兲
One can see that if at some the density s1 has a sharp minimum the first factor in the denominator in Eq. 共42兲 becomes large. On the other hand, the integral in the numerator is not very sensitive to lowering of s1 关see Eqs. 共35兲兴. Thus in a system with a “bottle neck” in the drag component the drag current decreases strongly and the main consequence of the drag effect is the emergence of the gradient of the phase of the order parameter of the drag component. A similar situation takes place in a system with a weak link. The latter case is analyzed in the next section. In the uniform case Eq. 共42兲 is reduced to j1 = drv2. To complete the discussion we emphasize that the crossed term 共drv1 · v2兲 in the free energy 共25兲 关and, consequently, the drag terms in the currents 共29兲 and 共30兲兴 comes only from the second and third terms in Eq. 共21兲. Consequently, the drag effect considered in this paper is solely by the excitations. At the mean field level of approximation 共which can be also formulated in terms of the Gross-Pitaevsky equation兲 the effect does not appear, while the coupling between the components is also present at that level of approximation. We would note that at the mean field level the drag effect of another type may emerge. That effect takes place in the case when one of the species is subjected by an asymmetric rotating external potential 共see, for instance, 关17兴, where such an effect has been studied with reference to the system of two coupled traps兲.
冕 冕
2
−
d
0
2
d
0
dr s1s2 − dr共s1 + s2兲
s1 − dr s1s2 − dr共s1 + s2兲
冣
.
共41兲
fined in a toroidal trap and the Bose cloud of the component 1 共the drag component兲 is situated inside and overlaps with the Bose clouds of the component 2 共the drive component兲. Such a situation can be realized if 兩␥12兩 ⬍ min共␥11 , ␥22兲. Deforming the confining potential one can cut the drag component into two clouds of a half-torus shape 共separated by two Josephson links兲 leaving the Bose cloud of the drive component uncut 共Fig. 2兲. In what follows we use the following notations: Rt is the large radius of the toroidal trap, and rt1 and rt2 are the small radii of the toroidal Bose clouds of the drag and the drive components, correspondingly. Rotating this trap one can excite a superflow in the drive component. After the rotation is switched off there will is a circulating superflow in the drive component and no superflow in the drag component 共at negligible small Josephson coupling兲. The superfluid velocity of the drive component is v2 =
N 2ប . m 2R t
共43兲
In Eq. 共43兲, we imply that Rt Ⰷ rt1 , rt2 and neglect, for simplicity, the effect caused by a dependence of rt2 on the polar angle. Since j1 = 0, the phase gradient 1 should be nonzero to compensate for the drag effect. In the polar coordinates the component of the phase gradient is given by the relation 共 1兲 = −
N2 f dr = − f dr共 2兲 , Rt
共44兲
where
III. MODEL OF BOSE-EINSTEIN QUBIT WITH EXTERNAL DRAG FORCE
It is known that Bose systems in the Bose-Einstein condensed state may demonstrate Josephson phenomenon 关9兴. In this paper we consider the external Josephson effect that takes place in two-well Bose systems. It was shown in 关18兴 that in such systems one can realize the situation, when two states, that differ in the expectation value of the relative number operator, can be used as qubit states. To include the drag force into the play we consider the following geometry. Let our two-component system be con-
FIG. 2. Schematic shapes of Bose clouds for the drag 共top figure兲 and drive 共bottom figure兲 components. The drag component is situated inside and overlaps with the drive component.
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PHYSICAL REVIEW A 72, 013616 共2005兲
NONDISSIPATIVE DRAG OF SUPERFLOW IN A TWO-…
f dr =
m1 dr . m2 s1 − dr
共45兲
The quantity f dr yields the ratio between the phase gradients in the drag and the drive components in the situation when the drag component is in the open circuit 共i.e., the current cannot flow in the circuit兲. We call this quantity the drag factor. We imply that rt1 and rt2 are much larger than the healing lengths that allows one to describe the drag effect in quasiuniform approximation. For definiteness, we specify the case of 1 Ⰶ 2 and 2 ⬇ const in the overlapping region. In this case one can neglect the space dependence the drag factor 关see Eqs. 共35兲兴. At nonzero Josephson coupling the current j1 can be nonzero, but it cannot exceed the maximum Josephson current jm. Relation 共44兲 remains approximately correct at nonzero Josephson coupling if an inequality jm Ⰶ ប1 / 共m1Rt兲 is satisfied. Here we specify just such a case. It is important to emphasize that we consider the situation, when there is only the external Josephson effect between two half-torus traps, and there is no internal Josephson effect between the drag and the drive species. The drag force can be considered as an effective vector potential Adr = −បf dr 2 共in units of e = c = 1兲 that corresponds to an effective magnetic flux ⌽dr = −2បf drN2. Thus our Bose system is similar to the Cooper pair box system that implements the Josephson charge qubit with the Josephson coupling controlled by an external magnetic flux 关19兴. To extend this analogy we formulate the model of the BoseEinstein qubit subjected by the drag force. In what follows we use the approach of Ref. 关18兴. In the two mode approximation the Bose field operators for the drag component can be presented in the form: ˆ 共r,t兲 = ⌿ 1
ˆ †共r,t兲 = ⌿ 1
a†l 共t兲⌿*l 共r − rl兲,
共46兲
l=L,R † and aL共R兲 are the operators of creation and anniwhere aL共R兲 hilation of bosons in the condensates confined in the left共right兲 half-torus, and ⌿L, ⌿R are two almost orthogonal local mode functions
冕
d
3
r⌿*l 共r兲⌿l⬘共r兲
⬇ ␦ll⬘,
兺
J=
冋
冕 冋 d 3r
册
ប2 2 + Vtr + ␥12⌿*2⌿2 ⌿l , 2m
␥11 2
冕
d3r兩⌿l兩4 ,
共49兲
册
ប2 ⌿L* · ⌿R + Vtr⌿L* ⌿R . 2m
⌿L共R兲共r兲 = 兩⌿L共R兲共r兲兩exp关− iN2 f drL共R兲共r兲兴,
共51兲
共52兲
where rA and rB are the radius-vectors of Josephson links. Substituting Eq. 共51兲 into Eq. 共50兲, using Eq. 共52兲, and taking into account that the functions ⌿L and ⌿R overlap in a small vicinity of A and B links, we obtain the following expression for the Josephson coupling parameter:
冉 冊
J = 共JA + JB兲cos
冉 冊
⌽dr ⌽dr + i共JA − JB兲sin , 共53兲 ⌽0 ⌽0
where ⌽0 = 2ប is the “flux quantum” and
冕
VA共B兲
共47兲
共50兲
where L , R are the polar angles counted from the centers of L and R half-torus, correspondingly 共see Fig. 3兲. The angles L共R兲共r兲, defined as shown in Fig. 3, satisfy the relation
l,l⬘ = L,R
共Kla†l al + la†l a†l alal兲 + 共JaL† aR + J*aR† aL兲,
共48兲
The functions ⌿L and ⌿R contain the phase factors eiL共r兲 and eiR共r兲, where the phases satisfy Eq. 共44兲. Taking these factors into account, one can choose the following basis for the one mode functions
d 3r
冋
ប2 兩⌿L兩· 兩⌿R兩 + Vtr兩⌿L兩兩⌿R兩兴. 2m 共54兲
Here VA and VB are the areas of overlapping of two one mode functions at links A and B, correspondingly. Considering the Hilbert space in which the total number operator Nˆ = aL† aL + aR† aR
l=L,R
with
冕
d3r⌿*l −
l =
JA共B兲 ⬇
that describe the condensate in the left and right traps 关20兴. Substituting Eq. 共46兲 into Hamiltonian 共1兲, we obtain the following expression for the parts of the Hamiltonian that depend on the operators a†l and al: Ha =
Kl =
R共rA兲 − L共rA兲 = L共rB兲 − R共rB兲 = ,
兺 al共t兲⌿l共r − rl兲, l=L,R 兺
FIG. 3. Left 共L兲 and right 共R兲 half-torus of the drag component, separated by Josephson links A and B.
共55兲
is a conservative quantity 共Nˆ = N兲 we present the Hamiltonian 共47兲 in the following form:
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PHYSICAL REVIEW A 72, 013616 共2005兲
D. V. FIL AND S. I. SHEVCHENKO
Ha = Ec共nˆRL − ng兲2 + 共JaL† aR + H.c.兲 + const,
共56兲
where nˆRL =
aR† aR − aL† aL 2
共57兲
is the number difference operator, Ec = R + L
共58兲
is the interaction energy, and the quantity ng =
1 关KL − KR + 共N − 1兲共L − R兲兴 2Ec
共59兲
describes an asymmetry of L and R half-tore. In what follows we imply that the system is in the Fock regime 关9兴 共兩J兩N Ⰶ Ec兲 and use the number representation 兩nRL典 ⬅ 兩nR,nL典 ⬅
冏
冔
N N + nRL, − nRL . 2 2
In this representation the first term in Eq. 共56兲 is diagonal. The second term in Eq. 共56兲 can be considered as a small nondiagonal correction. But if ng is biased near one of the degeneracy points ndeg =
再
M+ M
1 2
for even N for odd N
冎
共60兲
共where M is an integer and 兩M兩 ⬍ N / 2兲, the second term in Eq. 共56兲 results in a strong mixing of two lowest states 共兩 ↑ 典 = 兩ndeg + 1 / 2典 and 兩 ↓ 典 = 兩ndeg − 1 / 2典兲 and the low energy dynamics of the system can be described by a pseudospin Hamiltonian Hef f = −
⍀x ⍀y ⍀z ˆ x − ˆ y − ˆ z , 2 2 2
where ˆ i are the Pauli operators, and
共61兲
冉 冊 冉 冊
2 cos ⍀x = − 共JA + JB兲冑共N + 1兲2 − 4ndeg
⌽dr , ⌽0
2 sin ⍀y = − 共JA − JB兲冑共N + 1兲2 − 4ndeg
⌽dr , ⌽0
⍀z = 2Ec共ng − ndeg兲
is relaxed to the state 兩in典 = 兩 ↑ 典. This state can be used as the reproducible initial state. The quantity that should be measured is the expectation value of the number difference operator. In the initial state the expectation value of this operator is nRL = 1 / 2. When the system is switched suddenly to the degeneracy point ng = 0 and the Josephson couplings are switched on for some time the initial state evolves to another state with another nRL. If one sets JA = JB = J the result of evolution 共兩 f 典 = U兩in典 is described by the unitary operator U 1共 兲 =
cos共␣1兲
− i sin共␣1兲
− i sin共␣1兲
cos共␣1兲
冊
where ␣1 = 共J / ប兲共N + 1兲cos共⌽dr / ⌽0兲. One can see that at the time of evolution = 1 = / 共4兩␣1兩兲 the expectation value of the number difference operation will be equal to zero. For the case JA = J and JB = 0 the operator of evolution reads as U 2共 兲 =
冉
cos共␣2兲
− ie−i⌽dr/⌽0 sin共␣2兲
− iei⌽dr/⌽0 sin共␣2兲
cos共␣2兲
冊
with ␣2 = 共J / 2ប兲共N + 1兲. Respectively, the expectation value nRL will be equal to zero at = 2 = / 共4␣2兲. The ratio 2 / 1 = 兩cos共⌽dr / ⌽0兲兩 / 2 depends only on ⌽dr and the quantity ⌽dr can be extracted from the measurements of 1 and 2. It is important to note that to provide this scheme one should control only the ratio of JA and JB, but not their absolute values. Another possibility can be based on detection of the Berry phase 关21兴. Equation 共61兲 contains all three components of the field ⍀ and they can be controlled independently. The general scheme of detection of the Berry phase in such a situation was proposed 关22兴. A concrete realization of this scheme in the Josephson charge qubit was described in 关23兴. Here we extend the ideas of 关22,23兴 to the case of the “dragged” Bose-Einstein qubit. We start from the same initial state and switch to JA = JB = J and ng = 0. The initial state 兩↑典 can be presented as the superposition of two instantaneous eigenstates 兩ea典 = 共兩 ↑ 典 + 兩 ↓ 典兲 / 冑2 and 兩eb典 = 共兩 ↑ 典 − 兩 ↓ 典兲 / 冑2:
共62兲
are the components of the pseudomagnetic filed. In experiments one can control the parameters ng, JA, and JB independently and, consequently, the pseudomagnetic field ⍀共t兲 can be switched arbitrarily. It means that Eq. 共61兲 represents the standard Hamiltonian of the qubit system. The parameters of the qubit 共61兲 depend on the “drag flux” ⌽dr. Therefore one can determine its value from the measurement of the state of the system after a controlled evolution of a certain reproducible initial state. Let us consider two possibilities. For definiteness, we specify the case of odd N and the degeneracy point ndeg = 0. If the Josephson coupling is switched off and ng is switched on to some positive value 共much less than unity兲 the system
冉
兩in典 =
1
冑2 共兩ea典 + 兩eb典兲.
共63兲
An adiabatic cyclic evolution of the parameters of the Hamiltonian 共61兲 results in the appearance of the Berry phase in the 兩ea典 and 兩eb典 eigenstates if the vector ⍀ subtends a nonzero solid angle at the origin. Let us consider the following four stage cyclic adiabatic evolution starting from the point JA = JB = J and ng = 0: 1 — JB is switched off; 2 — JA is switched off and simultaneously ng is switched to ng1 ⬎ 0; 3 — ng is returned to the same degeneracy point 共ng = 0兲 and JB is switched to JB = J; and 4 — JA is switched to JA = J 共all switches should be done slowly: ប兩d⍀ / dt兩 Ⰶ ⍀2兲. After such an evolution the system arrives at the state
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PHYSICAL REVIEW A 72, 013616 共2005兲
NONDISSIPATIVE DRAG OF SUPERFLOW IN A TWO-…
兩 m典 =
1
冑2
共ei␦a+i␥兩ea典 + ei␦b−i␥兩eb典兲,
共64兲
where ␥ = ⌽dr / ⌽0 is the Berry phase 共equal to half of the solid angle subtended by ⍀兲 and ␦a, ␦b are the dynamical phases. Elimination of the dynamical phases can performed by swapping the eigenstates 共-transformation兲 and repeating the same cycle of evolution in a reverse direction 共see 关22兴兲. The -transformation can be done by fast switching off the Josephson coupling and switching on ng = ng2 ⬎ 0 during the time interval t = ប / 共2Ecng2兲. After the -transformation the state becomes 兩 m典 = −
i
冑2 共e
i␦a+i␥
兩eb典 + ei␦b−i␥兩ea典兲.
共65兲
After the cyclic evolution in the reverse direction we arrive at the state 兩 f 典 = −
i
冑2 e
i共␦a+␦b兲
共e2i␥兩eb典 + e−2i␥兩ea典兲.
共66兲
One can see that the expectation value of the number difference operator in the final state 共66兲 nRL = cos共4␥兲 / 2 = cos共4⌽dr / ⌽0兲 / 2 depends only on ⌽dr and the measurement of this difference allows one to determine the value of the “drag flux.” Thus the measurements of a relative number of atoms in left and right condensates under controlled evolution of the state of the system allows one to observe the nondissipative drag and determine the drag factor 共if the vorticity of the drive component is known兲.
关1兴 A. F. Andreev and E. P. Bashkin, Zh. Eksp. Teor. Fiz. 69, 319 共1975兲 关Sov. Phys. JETP 42, 164 共1975兲兴. 关2兴 A. J. Leggett, Rev. Mod. Phys. 47, 331 共1975兲. 关3兴 M. A. Alpar, S. A. Langer, and J. A. Sauls, Astrophys. J. 282, 533 共1984兲. 关4兴 E. Babaev, Phys. Rev. D 70, 043001 共2004兲. 关5兴 J. M. Duan and S. Yip, Phys. Rev. Lett. 70, 3647 共1993兲. 关6兴 B. Tanatar and A. K. Das, Phys. Rev. B 54, 13827 共1996兲. 关7兴 S. V. Terentjev and S. I. Shevchenko, Fiz. Nizk. Temp. 25, 664 共1999兲 关Low Temp. Phys. 25, 493 共1999兲兴. 关8兴 D. V. Fil and S. I. Shevchenko, Fiz. Nizk. Temp. 30, 1028 共2004兲 关Low Temp. Phys. 30, 770 共2004兲兴. 关9兴 A. J. Leggett, Rev. Mod. Phys. 73, 307 共2001兲. 关10兴 S. I. Shevchenko, Fiz. Nizk. Temp. 18, 328 共1992兲 关Sov. J. Low Temp. Phys. 18, 223 共1992兲兴. 关11兴 D. V. Fil and S. I. Shevchenko, Phys. Rev. A 64, 013607 共2001兲. 关12兴 A. S. Parkins and D. F. Walls, Phys. Rep. 303, 1 共1998兲. 关13兴 N. N. Bogolyubov and N. N. Bogolyubov, Jr., Introduction to
IV. CONCLUSIONS
We have investigated the nondissipative drag effect in three-dimensional weakly interacting two-component superfluid Bose gases. The expression for the drag current is derived microscopically for the general case of two species of different densities, different masses, and different interaction parameters. It is shown that the drag current is proportional to the square root of the gas parameter. The drag effect is maximal at zero temperatures and it decreases when the temperature increases, but at temperatures of order of the interaction energy the drag current remains of the same order as at zero temperature. We have considered the toroidal double-well geometry, where the nondissipative drag influences significantly the Josephson coupling between the wells. In the system considered the drag force can be interpreted as an effective vector potential applied to the drag component. The effective vector potential is equal to Adr = −បf dr drv 共in units of e = c = 1兲, where drv is the phase of the drive component, and f dr is the drag factor. In the toroidal geometry the effective vector potential can be associated with an effective flux of external field ⌽dr = 2បf drNv, where Nv is the vorticity of the drive component. In the Fock regime the system can be considered as a Bose-Einstein counterpart of the Josephson charge qubit in an external magnetic field. The measurement of the state of such a qubit allows one to observe the drag effect and determine the drag factor. ACKNOWLEDGMENT
This work was supported by the INTAS Grant No. 012344.
关14兴 关15兴 关16兴 关17兴 关18兴 关19兴 关20兴 关21兴 关22兴 关23兴
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