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Quantum effects in weakly coupled superfluid liquids
Volume 34A, number 6
QUANTUM
PHYSICS LETTERS
EFFECTS
IN WEAKLY
COUPLED
5April1971
SUPERFLUID
LIQUIDS
L. L E P L A E , F. MANCINI and H. UMEZAWA
Department of Physics, University of Winconsin-Milwaukee, Milwaukee, Wisconsin 53201, USA Received 8 March 1971 The boson method is applied to the study of two superfluid liquids joined together by a weak link. A miccroscopic derivation of experimental results is presented.
Recent e x p e r i m e n t s [1] in s u p e r f l u i d liquid h e l i u m , p e r f o r m e d in analogy with the A n d e r s o n Dayem e x p e r i m e n t [2] in s u p e r c o n d u c t i v i t y , have shown some effects that a r e analogous to the ac Josephson effects in superconductivity. Two baths of s u p e r f l u i d a r e weakly coupled through a s m a l l o r i f i c e ; n e a r the o r i f i c e is placed a quartz c r y s t a l o s c i l l a t o r . A difference, d, in the helium head will produce a difference in the c h e m i c a l potential AI~ = mgd: this is the equivalent to the difference in the c h e m i c a l potential induced by an e x t e r n a l constant voltage in s u p e r c o n d u c t i v i t y . The equivalent of the a l t e r n a t i n g voltage i s played by the q u a r t z c r y s t a l , which acts as an u l t r a s o n i c t r a n s d u c e r . The e x p e r i m e n t s show that when the o s c i l l a t o r i s t u r n e d on the s y s t e m will p r e s e n t d y n a m i c a l stability at v a l u e s d which statisfy the condition
mgd = (nl/n2)2~u ,
(1)
where v is the f r e q u e n c y of the t r a n s d u c e r and n 1 and n 2 a r e i n t e g e r s . In this note we p r e s e n t an application of the boson method [3, 4] to the s y s t e m d e s c r i b e d above and we show how the stability condition (1) can be d e r i v e d in the sphere of our g e n e r a l f o r m u l a t i o n without i n t r o d u c i n g any phenomenological a s s u m p t i o n . In the boson method, which was o r i g i n a l l y d e r i v e d f r o m the m i c r o s c o p i c H a m i l tonian, p e r s i s t e n t c u r r e n t s a r e obtained by m e a n s of the so called "boson t r a n s f o r m a t i o n s " [4]. In o r d e r to d e s c r i b e n o n - s t a t i o n a r y p h e n o m ena we have r e c e n t l y extended [5] the boson method to the case in which the p e r s i s t e n t c u r r e n t is t i m e - d e p e n d e n t . F o r a superfiuid s y s t e m our r e s u l t s show that the boson t r a n s f o r m a t i o n i n d u c e s a ground state d e n s i t y
p ( x , t) = -v2 f d3y c ( x - y ) ] (y, t) ,
(2)
and a ground state c u r r e n t
j (x, t) = 7}2 Vo2fd3y c ( x - y ) F f ( y , t ) .
(3)
Here ~? and vo a r e s o m e c o n s t a n t s ; c ( x - y ) is a c o r r e l a t i o n function, defined in ref. [4], and f , the phase of the condensate wave function, s a t i s fies the equation *
[a 2/at2 _ Vo 2 v2]f
(x, t) : o .
(4)
In the p r e s e n c e of e x t e r n a l fields eq. (4) should be modified and f ( x , t) will statisfy an i n h o m o g e neous equation, where the e x t e r n a l effect acts a s s o u r c e . Since in this p a r t i c u l a r p r o b l e m the e x t e r n a l field is v a r y i n g s i n u s o i d a l l y in the t i m e , we shall look only for solutions of the eq. (4) which a r e periodic in the t i m e and a r e in r e s o n a n c e with the e x t e r n a l effect. To be m o r e p r e c i s e let us s c h e m a t i z e the s y s t e m as two s u p e r f i u i d s connected together by a h o r i z o n t a l junction of negligible t h i c k n e s s , at whose edges t h e r e is a potential difference given by
A I~ = mgd + ( Vo/2yu)cos 2~ut .
(5)
Choosing the origin of the s y s t e m of C a r t e s i a n c o o r d i n a t e s at the c e n t e r of the b a r r i e r and the z - a x i s in the d i r e c t i o n of the g r a v i t a t i o n a l field, we shall look for a solution of eq. (4) statisfying the following boundary conditions: (i) f is a function only of z and t and i s denoted by f(z, t), (ii) the density and the c u r r e n t a r e finite at any t i m e t. Defining ~b(t) = f(0 +, t) - f ( 0 , , t) we s h a l l f u r t h e r r e q u i r e that: • The proof that the boson transformation is an invariant transformation if f i s a solution of eq. (4) can be easily seen by looking at the equation for the boson field B [3, eq. (14)]. 301
Volume 34A. number 6
PHYSICS
LE T T E R S
5 April 1971
(iii) (~(t) i s p e r i o d i c in the t i m e with m o d u l u s 2vl (l = i n t e g e r ) ; i . e . ~(t + T) = ~)(t) + 2vl .
(6) :
F r o m the e q u a t i o n [5]
V/~t = -u ,
(7)
we obtain the b o u n d a r y condition:
(iv)
~/at
= t,~
(8)
.
T h e g e n e t a l s o l u t i o n of eq. (4) s a t i s f y i n g t h e b o u n d a r y c o n d i t i o n s (i) - (iii) is g i v e n by *:
mgd
(9)
f(z,t ) = A (i) t + oo
+ylF~i)sin wherei=
C2~nz
l for z >0andi
A (1) - A (2) = 2 , l / T
T h e r e is an a d d i t i o n a l c o n d i t i o n ~(1) = f~(2) = 0 in o r d e r that the c u r r e n t j be an odd f u n c t i o n of t. By c o m p a r i n g e q s . (10), (11) and (12) we s e e that the v a l u e s of d a r e c o n t r o l l e d by the f r e q u e n c y of the o s c i l l a t o r t h r o u g h t h e r e l a t i o n
•
= (l/p)2~v
.
(15)
T h e e x p r e s s i o n s f o r the d e n s i t y and c u r r e n t d i s t r i b u t i o n s can b e c o m p u t e d by s u b s t i t u t i n g s o l u tion (9) in e q s . (2) and (3). S u m m a r i z i n g we t h u s ring that t h e s y s t e m w i l l p r e s e n t s t a t i o n a r y s t a t e s in r e s o n a n c e with t h e e x t e r n a l f i e l d when and only when eq. (15) i s s a t i s f i e d .
= 2 f o r z < 0 , and
.
(10) References
T h e b o u n d a r y c o n d i t i o n (iv) r e q u i r e s that: A (1) - A (2) = m g d ,
(11)
p v =-T (p = i n t e g e r ) ,
(12)
* Here we disregard a constant t e r m , which does not have any physical relevance. We also do not consider terms linear in z; these kind of t e r m s will produce a constant flow of current in the z-direction. Since we are principally interested in stationary states we do not take these t e r m s into account; however these linear terms will be responsible for the transition b e tween one resonance state to the next one.
302
[1] P . L . Richards and P.W. Anderson, Phys. Rev. Letters 14 (1965) 540; B. M. Khorana and B. S. Chandrasekhar, Phys. Rev. Letters 18 (1967) 230; B. M. Khorana, Phys. Rev. 185 (1969) 299; P. L. Richards, Phys. Rev. A 2 (1970) 1532. [2] P.W. Anderson and A. H. Dayem, Phys. Rev. Letters 13 (1964) 195. [3] L. Leplae and H. Umezawa, J. Math. Phys. i0 (1969) 2038. [4] L. Leplae, F. Maneini and H. Umezawa, Phys. Rev. B 2 (1970) 3594. [5] L. Leplae, F. Maneini and H. Umezawa, to be published.
QUANTUM
PHYSICS LETTERS
EFFECTS
IN WEAKLY
COUPLED
5April1971
SUPERFLUID
LIQUIDS
L. L E P L A E , F. MANCINI and H. UMEZAWA
Department of Physics, University of Winconsin-Milwaukee, Milwaukee, Wisconsin 53201, USA Received 8 March 1971 The boson method is applied to the study of two superfluid liquids joined together by a weak link. A miccroscopic derivation of experimental results is presented.
Recent e x p e r i m e n t s [1] in s u p e r f l u i d liquid h e l i u m , p e r f o r m e d in analogy with the A n d e r s o n Dayem e x p e r i m e n t [2] in s u p e r c o n d u c t i v i t y , have shown some effects that a r e analogous to the ac Josephson effects in superconductivity. Two baths of s u p e r f l u i d a r e weakly coupled through a s m a l l o r i f i c e ; n e a r the o r i f i c e is placed a quartz c r y s t a l o s c i l l a t o r . A difference, d, in the helium head will produce a difference in the c h e m i c a l potential AI~ = mgd: this is the equivalent to the difference in the c h e m i c a l potential induced by an e x t e r n a l constant voltage in s u p e r c o n d u c t i v i t y . The equivalent of the a l t e r n a t i n g voltage i s played by the q u a r t z c r y s t a l , which acts as an u l t r a s o n i c t r a n s d u c e r . The e x p e r i m e n t s show that when the o s c i l l a t o r i s t u r n e d on the s y s t e m will p r e s e n t d y n a m i c a l stability at v a l u e s d which statisfy the condition
mgd = (nl/n2)2~u ,
(1)
where v is the f r e q u e n c y of the t r a n s d u c e r and n 1 and n 2 a r e i n t e g e r s . In this note we p r e s e n t an application of the boson method [3, 4] to the s y s t e m d e s c r i b e d above and we show how the stability condition (1) can be d e r i v e d in the sphere of our g e n e r a l f o r m u l a t i o n without i n t r o d u c i n g any phenomenological a s s u m p t i o n . In the boson method, which was o r i g i n a l l y d e r i v e d f r o m the m i c r o s c o p i c H a m i l tonian, p e r s i s t e n t c u r r e n t s a r e obtained by m e a n s of the so called "boson t r a n s f o r m a t i o n s " [4]. In o r d e r to d e s c r i b e n o n - s t a t i o n a r y p h e n o m ena we have r e c e n t l y extended [5] the boson method to the case in which the p e r s i s t e n t c u r r e n t is t i m e - d e p e n d e n t . F o r a superfiuid s y s t e m our r e s u l t s show that the boson t r a n s f o r m a t i o n i n d u c e s a ground state d e n s i t y
p ( x , t) = -v2 f d3y c ( x - y ) ] (y, t) ,
(2)
and a ground state c u r r e n t
j (x, t) = 7}2 Vo2fd3y c ( x - y ) F f ( y , t ) .
(3)
Here ~? and vo a r e s o m e c o n s t a n t s ; c ( x - y ) is a c o r r e l a t i o n function, defined in ref. [4], and f , the phase of the condensate wave function, s a t i s fies the equation *
[a 2/at2 _ Vo 2 v2]f
(x, t) : o .
(4)
In the p r e s e n c e of e x t e r n a l fields eq. (4) should be modified and f ( x , t) will statisfy an i n h o m o g e neous equation, where the e x t e r n a l effect acts a s s o u r c e . Since in this p a r t i c u l a r p r o b l e m the e x t e r n a l field is v a r y i n g s i n u s o i d a l l y in the t i m e , we shall look only for solutions of the eq. (4) which a r e periodic in the t i m e and a r e in r e s o n a n c e with the e x t e r n a l effect. To be m o r e p r e c i s e let us s c h e m a t i z e the s y s t e m as two s u p e r f i u i d s connected together by a h o r i z o n t a l junction of negligible t h i c k n e s s , at whose edges t h e r e is a potential difference given by
A I~ = mgd + ( Vo/2yu)cos 2~ut .
(5)
Choosing the origin of the s y s t e m of C a r t e s i a n c o o r d i n a t e s at the c e n t e r of the b a r r i e r and the z - a x i s in the d i r e c t i o n of the g r a v i t a t i o n a l field, we shall look for a solution of eq. (4) statisfying the following boundary conditions: (i) f is a function only of z and t and i s denoted by f(z, t), (ii) the density and the c u r r e n t a r e finite at any t i m e t. Defining ~b(t) = f(0 +, t) - f ( 0 , , t) we s h a l l f u r t h e r r e q u i r e that: • The proof that the boson transformation is an invariant transformation if f i s a solution of eq. (4) can be easily seen by looking at the equation for the boson field B [3, eq. (14)]. 301
Volume 34A. number 6
PHYSICS
LE T T E R S
5 April 1971
(iii) (~(t) i s p e r i o d i c in the t i m e with m o d u l u s 2vl (l = i n t e g e r ) ; i . e . ~(t + T) = ~)(t) + 2vl .
(6) :
F r o m the e q u a t i o n [5]
V/~t = -u ,
(7)
we obtain the b o u n d a r y condition:
(iv)
~/at
= t,~
(8)
.
T h e g e n e t a l s o l u t i o n of eq. (4) s a t i s f y i n g t h e b o u n d a r y c o n d i t i o n s (i) - (iii) is g i v e n by *:
mgd
(9)
f(z,t ) = A (i) t + oo
+ylF~i)sin wherei=
C2~nz
l for z >0andi
A (1) - A (2) = 2 , l / T
T h e r e is an a d d i t i o n a l c o n d i t i o n ~(1) = f~(2) = 0 in o r d e r that the c u r r e n t j be an odd f u n c t i o n of t. By c o m p a r i n g e q s . (10), (11) and (12) we s e e that the v a l u e s of d a r e c o n t r o l l e d by the f r e q u e n c y of the o s c i l l a t o r t h r o u g h t h e r e l a t i o n
•
= (l/p)2~v
.
(15)
T h e e x p r e s s i o n s f o r the d e n s i t y and c u r r e n t d i s t r i b u t i o n s can b e c o m p u t e d by s u b s t i t u t i n g s o l u tion (9) in e q s . (2) and (3). S u m m a r i z i n g we t h u s ring that t h e s y s t e m w i l l p r e s e n t s t a t i o n a r y s t a t e s in r e s o n a n c e with t h e e x t e r n a l f i e l d when and only when eq. (15) i s s a t i s f i e d .
= 2 f o r z < 0 , and
.
(10) References
T h e b o u n d a r y c o n d i t i o n (iv) r e q u i r e s that: A (1) - A (2) = m g d ,
(11)
p v =-T (p = i n t e g e r ) ,
(12)
* Here we disregard a constant t e r m , which does not have any physical relevance. We also do not consider terms linear in z; these kind of t e r m s will produce a constant flow of current in the z-direction. Since we are principally interested in stationary states we do not take these t e r m s into account; however these linear terms will be responsible for the transition b e tween one resonance state to the next one.
302
[1] P . L . Richards and P.W. Anderson, Phys. Rev. Letters 14 (1965) 540; B. M. Khorana and B. S. Chandrasekhar, Phys. Rev. Letters 18 (1967) 230; B. M. Khorana, Phys. Rev. 185 (1969) 299; P. L. Richards, Phys. Rev. A 2 (1970) 1532. [2] P.W. Anderson and A. H. Dayem, Phys. Rev. Letters 13 (1964) 195. [3] L. Leplae and H. Umezawa, J. Math. Phys. i0 (1969) 2038. [4] L. Leplae, F. Maneini and H. Umezawa, Phys. Rev. B 2 (1970) 3594. [5] L. Leplae, F. Maneini and H. Umezawa, to be published.