14 June 1999

Ž .Physics Letters A 256 1999 375–378

Quantum coherence in a mesoscopic Josephson junction

Jian Zou, Bin ShaoDepartment of Applied Physics, Beijing Institute of Technology, Beijing 100081, PR China

Received 8 February 1999; accepted 29 March 1999Communicated by L.J. Sham

Abstract

We consider a mesoscopic Josephson junction in a circular superconducting ring which encloses a magnetostatic flux. Itis shown that the state of the junction can evolve in a quantum superposition of two coherent states. It is also shown that thedestructive interference between these two states suppresses the supercurrent, and the dynamic behavior of the supercurrentshows collapse and revival phenomena. q 1999 Elsevier Science B.V. All rights reserved.

PACS: 74.50.q r; 42.50.Dv; 42.50.Fx

In the last few years the mesoscopic Josephsonjunction has been widely studied, and many new

w xquantum effects have been revealed 1–3 . The es-sential feature of these effects is the phase coherencetunneling of single electron and Cooper pairs, espe-cially, at low temperature. As the coherence lengthof an electron is comparable to the length of amesoscopic Josephson junction, single electron andCooper pairs have a well defined phase all over themesoscopic Josephson junction, and tunneling willshow quantum interference phenomena. In this casethe classical description is no longer sufficient, in-stead it has to be described in fully quantum-mecha-nical way. In order to exploit the fully quantum-mechanical nature of mesoscopic Josephson junction,an ultra-small capacitance weak-link biased with aclassical voltage source, and also coupled to a non-

w xclassical microwave, has been considered 4 , andw xsome interesting results were obtained 4–9 . Re-

cently, a more general Hamiltonian that describesboth inductive and capacitive coupling between

mesoscopic Josephson junctions and nonclassical mi-w xcrowave has been discussed 10,11 .

The linear superposition principle is one of themost fundamental features of quantum-mechanics. Itimplies that probability densities of observable quan-tities, in a superposition state, usually exhibit inter-ference effects instead of simply being added. Thesuperpositions of coherent states were widely studied

w xin recent years in various fields 12–22 , and some-times they can be interpreted as quantum superposi-tions of two macroscopically distinguishable states,so-called Schrodinger cat states. As early as in 1984,¨the possibility of generating Schrodinger cat states in¨superconducting devices was discussed by Leggettw x13 . Recently there have been several proposals forthe generation of the optical superpositions of coher-

w xent states in various nonlinear processes 14–19 ,w xand also a proposal 20 to generate such state in

Josephson junction different from Leggett’s. ThereŽhave been the observation of macroscopic or, strictly

.speaking, mesoscopic Schrodinger cat states of¨

0375-9601r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0375-9601 99 00248-0

( )J. Zou, B. ShaorPhysics Letters A 256 1999 375–378376

w xtrapped ions 21 and of an electromagnetic field in aw xhigh-Q cavity 22 . In this paper we consider a

mesoscopic Josephson junction in a circular super-conducting ring which encloses a magnetostatic flux.We will show that the state of the junction canevolve into a quantum superposition of two coherentstates.

Although the quantum superpositions of coherentstates are formed by the states close to classical ones,the quantum interference between states composingit profoundly affects the behavior of the system. Aconstructive interference between the probability am-plitudes can enhance the probability, while a destruc-tive interference should suppress this probability.Here we show that the destructive interference oc-curs in the dynamic behavior of supercurrent in thejunction, and the supercurrent exhibits collapse andrevival phenomenon, which is well known in the

w xcontext of quantum optics 23,24 .ŽIn this paper a circular superconducting device of

.inductance L with a mesoscopic Josephson junctionwith external inductive coupling through a constantexternal flux F and working at a very low temper-ex

ature T , is considered. The requirement of low tem-perature makes the thermal fluctuations much smallerthan the quantum fluctuations. The junction is

Ždescribed by the following Hamiltonian "sk sB.cs1

22q fyFŽ .exHs qE 1ycosu q , 1Ž . Ž .J2C 2 L

where E is the Josephson coupling constant. u , theJŽphase difference across the junction equivalent to

.f, the total flux through the ring, and us2 ef , andq, the charge tunneling through the junction are apair of quantum mechanical conjugate variables, and

w xgive the commutation relation u ,q s2 ei.Ž .First we expand the 1ycosu , and rewrite Eq.

Ž .1 as

q2 C fFex X2 2Hs q v f y qH , 2Ž .2C 2 L

X2'where vs1r CL q4e E rC, and H contains allJ2 Ž .the higher order terms of f noting that us2 ef

4 6X 2 4H sy E ef q E ef y PPP 3Ž . Ž . Ž .J J3 45

We introduce the creation operator b† and annihila-tion operator b, and the operators q and f can beexpressed as:

1 vC† †fs b qb ,qs i b yb . 4Ž . Ž . Ž .(' 22vC

Ž .Then we can rewrite the Hamiltonian 2 as

Hsv b†bqm bqb† qH X , 5Ž . Ž .'Ž .where msF r L 2vC . We assume that theex

<junction is initially in the vacuum state 0) , and thedriving magnetic field is strong enough, i.e., F isex

large enough. Suppose that the acting time t is shortenough, then within its duration the remaining evolu-tion can be neglected. The effective initial state isjust the coherent state, and can be expressed as

` na2y< a < r2< < <a)sD a 0)s e n) , 6Ž . Ž .Ý 'n!ns0

Ž . Ž † ) .where D a sexp a b ya b is the displacement<operator, and asyimt . A coherent state a) has

a Gaussian spatial distribution in phase space and isa minimum uncertainty state. For the system we

Ž . Ž .discuss, Re a ;-f) , and Im a ;-q) . Af-Ž .ter the action the Hamiltonian 5 becomes

Hsv b†bqH X . 7Ž .

In the case of periodic potentials, perturbative calcu-lations around one of the minima can be unreliablebecause tunneling non-perturbative phenomena be-tween the minima is ignored. In this paper we only

Ž .consider that u is small -u)s2 e-f)<2p ,and omit the higher than fourth-order terms, i.e.,

X Ž .4 Ž .H ,y2r3 ef . Then we rewrite Eq. 7 in therotating-wave approximation as:

2† †HsV b byn b b , 8Ž . Ž .

where

E e4J

ns ,Vsvyn . 9Ž .2vCŽ .

<For the initial state a) considering that asŽ .yimt from Eq. 4 one can obtain

< < < <-a u a)s2 e-a f a)s0. 10Ž .

( )J. Zou, B. ShaorPhysics Letters A 256 1999 375–378 377

Thus as long as the time interval is small, thecondition of -u)<2p is satisfied.

Ž .The Hamiltonian 8 represents the same modelwidely studied in the context of quantum opticsw x14–17 and various nonclassical effects have been

Ž † .2found. In the interaction picture where yn b b isregarded as the interaction part, the state vector< Ž . Ž .C t ) of Hamiltonian 8 evolves according to

` na2 2y< a < r2 in n t< <C t )s e e n) . 11Ž . Ž .Ý 'n!ns0

It is interesting that at a time tspr2n , the state< Ž . w xvector C t ) will be 14,15

1yip r4 ip r4< < <C pr2n )s e ya)qe a) .Ž . Ž .'2

12Ž .

Ž .It can be seen from Eq. 12 that now the state of thesystem is in a quantum superposition of two coherentstates. As long as V)n41, pr2n is small, i.e.,the condition of small time interval is satisfied.

Now we consider the dynamic behavior of thesupercurrent. The supercurrent operator can be ob-

Ž .tained from Eq. 8

'vC†w xIsyE qsy i H ,q s Vyn b qbŽ . Ž .t 2

†2 † 2y2n b bqb b . 13Ž . Ž .

It is easy to obtain the following expressions fromŽ .Eq. 11 :

< <-C t b t C t )Ž . Ž . Ž .

2 2 2 in t< < < <saexp y a y i Vyn tq a e , 14Ž . Ž .

< †2 <-C t b t b t C t )Ž . Ž . Ž . Ž .

2 2 2) y2 in t< < < < < <sa a exp y a q i Vy3n tq a e ,Ž .

15Ž .

†Ž . Ž .where b t and b t are the creation operator andannihilation operator in the interaction picture. From

Ž . Ž .Eqs. 13 – 15 one can easily obtain

n2 2y2 < a < sin n t- I)s I e 1y0 ž /½ V

=

2< <2 a n2< <sin Vyn ty a sin2n t yŽ .

V

=2< <sin Vy3n ty a sin2n t , 16Ž . Ž .5

'< <where I s a V vC . For short times0

n2 2 2y2 < a < n t- I)s I e 1y0 ž /½ V

=

2< <2 a n2< <sin Vyn ty2 a n t yŽ .

V

=2< <sin Vy3n ty2 a n t , 17Ž . Ž .5

so that the collapse time of the supercurrent is'Ž < < .t s1r 2 a n . At tsprn , the supercurrent re-c

vives completely. The collapses and revivals of thesupercurrent are clearly shown in Fig. 1. Revival isdue to the inherently nonlinear nature of the Hamil-

Ž .tonian 8 and indicates the quantum nature of thesystem, i.e., the discreteness of the energy levels. In

w xRefs. 8,9 we have shown that the supercurrent canexhibit collapse and revival phenomenon which ismainly due to the effects of quantum fluctuations ofthe nonclassical electromagnetic fields. Now the col-lapse and revival is mainly due to the destructiveinterference effects.

Ž .It has been shown that in Hamiltonian 8 , quan-tum coherence leads to a recurrence of the initialquantum state and a departure of the quantum dy-

w xnamics from that expected classically 14–17 . The

Fig. 1. Plot of the supercurrent - I)rI against a scaled time0< <n rp for V rn s40, a s2.

( )J. Zou, B. ShaorPhysics Letters A 256 1999 375–378378

coherent state is just like a wave packet in phasespace. The wave packet mimics, for a short period oftime, the classical behavior. As time evolves, thewave packet disperses, but also shows revivals attsprn that are of purely quantum origin. Whentspr2n the state evolves into a superpositions oftwo coherent states. Now the spatial distribution inphase space consists of two coherent wave packetswhich are phase shifted by p with respect to eachother. The destructive interference between the twowave packets results in the suppression of the super-current.

Although the possibility of generating theSchrodinger cat state in a macroscopic Josephson¨

w xjunction has been discussed in Ref. 13 , here weexplicitly derived the quantum superpositions of twocoherent states in a mesoscopic Josephson junction.Schrodinger introduced his famous cat states in order¨to illustrate the fundamental problem of the corre-spondence between the micro and the macro world.Quantum-coherence effects are extremely sensitiveto dissipation, which are usually referred to theinteractions of the system with the environmentsw x25–27 , and dissipation leads to quantum decoher-ence. Decoherence is commonly interpreted as a wayof quantifying the elusive boundary between classi-cal and quantum worlds and almost always precludesthe existence of macroscopic Schrodinger cat states,¨

w xexcept at extremely short time scales 25–28 . Themesoscopic systems have both microscopic andmacroscopic features and the creation of mesoscopicSchrodinger cat states may allow controlled studies¨of quantum coherence. Actually the observations ofSchrodinger cat states experimentally are all in the¨

w xmesoscopic systems 21,22 . It should be noted thatthe quantum superposition of two coherent statesdiscussed here is quite different from the Schrodi-¨

w xnger cat state in Ref. 20 , which consists of asuperposition of Bloch states associated with dcJosephson currents flowing in opposite directionsacross the junction.

In conclusion, we have shown that an initialvacuum state of the junction can evolve into acoherent superposition of two coherent states. Wehave also shown that the destructive interference

occurs in the dynamic behavior of supercurrent, andthe supercurrent exhibits collapse and revival phe-nomenon. However it is noted that the quantumcoherence effects are extremely sensitive to dissipa-tion, which usually are included by coupling the

w xoscillator to a heat bath 26–28 . Dissipation leadsdamping, and will destroy quantum interference ef-fects. But if the dissipation is weak enough, thequantum coherence effects should remain observablein the mesoscopic system, at least in short times aswe have discussed in this paper.

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