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Model for a Josephson junction array coupled to a resonant cavity J. Kent Harbaugh and D. Stroud Department of Physics, 174 W. 18th Ave., Ohio State University, Columbus, OH 43210 (February 1, 2008) 0 0 below which no emission was observed. The coupling, 0 We describe a simple Hamiltonian for an underdamped and hence N , could be variedby moving the arrayrela- c 2 Josephson array coupled to a single photon mode in a reso- tive to the cavity walls. n nant cavity. Using a Hartree-like mean-field theory, we show Barbaraetal.interpretedtheirresultsbyanalogywith a that, for any given strength of coupling between the photon the Jaynes-Cummings model16,17 of two-level atoms in- J field and the Josephson junctions, there is a transition from teracting with a radiation field in a single-mode reso- 4 incoherence to coherence as a function of N, the number of nant cavity. In this case, each Josephson junction acts Josephson junctionsinthearray. AbovethatvalueofN, the ] energy in the photon field is proportional to N2, suggestive as a two-level atom; the coupling between the “atoms” n is provided by the induced radiation field. A dynamical of coherent emission. These features remain even when the o calculation based on a model similar to that of Jaynes junction parameters have some random variation from junc- c and Cummings has been carried out by Bonifacio and - tion to junction, as expected in a real array. Both of these r features agree with recent experimentsby Barbara et al. collaborators18,19 forJosephsonjunctionarraysina cav- p ity. Theirmodeldoesproducespontaneousemissioninto u the cavity above a threshold junction number, provided s . that the Heisenberg equations are treated in a certain t a semi-classicallimit appropriate to large numbers of pho- m tons in the cavity. I. INTRODUCTION - In this paper, we present a simple model for the on- d set of phase locking and coherent emission by an under- n ResearchershavelongsoughttocauseJosephsonjunc- damped Josephson junction array in a resonant cavity. o tion arraysto radiate coherently.1,2 To achievethis goal, Wealsocalculatethethresholdfortheonsetofphaseco- c [ a standard approach is to inject a d.c. current into an herence,usingaformofmean-fieldtheory. Ourmodelde- overdamped array. If this current is sufficiently large, rives from more conventional models of Josephson junc- 2 it generates an a.c. voltage Vac across the junctions, of tion arrays,but treats the interaction with the radiation v 0 frequency ωJ =2eVdc/¯h, where Vdc is the time-averaged field quantum-mechanically. Within the mean field the- 8 voltageacrossthe junction.3 Eachjunctionthenradiates ory,we findthat for any strengthofthat coupling,there 3 (typicallyatmicrowavefrequencies). Ifthe junctions are existsathresholdnumberofjunctionsNcinalineararray 9 coherently phase-locked, the radiated power P N2, abovewhichthearrayiscoherent. Abovethatthreshold, ∝ 0 where N is the number of phase-locked junctions. This the energyinthe photonfieldisquadratic inthe number 9 N2proportionalityisahallmarkofphasecoherence. But ofjunctions,asfoundexperimentally.14Themodeliseas- 9 manydifficultiesinhibitphasecoherenceinpractice. For ily generalized to two-dimensional arrays. Furthermore, / t example, the junctions always have a disorder-induced as we show, the threshold condition and N2 dependence a m spread in critical currents, which produces a distribu- of the energy in the radiation field, are both preserved tion of Josephson frequencies and makes phase locking eveninthepresenceofthedisorderwhichwillbepresent d- difficult.4–7 Furthermore, in small-capacitance (and un- in any realistic array. Finally, the coupling constant be- n derdamped) Josephson junctions, quantum phase fluc- tween junctions and radiation field can, in principle, be o tuations inhibit phase locking.8–13 Thus, until recently, calculatedexplicitly,giventhegeometryofthearrayand c the most efficient coherent emission was found in two- the resonant cavity. : v dimensional arrays of overdamped Josephson junctions, Theremainderofthispaperisorganizedasfollows. In i where quantum fluctuations are minimal. Sec. II, we describe our model and approximations. Our X Recently, Barbara et al. have reported a remarkable numericalandanalyticalresultsarepresentedinSec.III. ar degree of coherent emission in arrays of underdamped SectionIVpresentsabriefdiscussionandsuggestionsfor junctions.14,15 Their arrays were placed in a microwave future work. cavity, so as to couple each junction to a resonant mode of the cavity. If the mode has a suitable frequency and is coupled strongly enough to the junction, it can be ex- II. MODEL cited by a Josephson current through the junction. The power in this mode then feeds back into the other junctions, causing the arrayto phase-lockand inducing a total power P N2. For a given coupling, Barbara et al. ∝ found that there is a threshold number of junctions N c 1 A. Hamiltonian where ... denotes a quantum-statistical average. We h i willimposeaconstant-voltageboundarycondition,byre- We consider a Josephson junction array containing N quiring that Φ= Nj=1Φj across the linear arrayshould junctions arranged in series, placed in a resonant cavity, take on a specified value. Here Φ represents the to- P arrangedinageometryshownschematicallyinFig.1. It tal, time-averaged voltage across the linear array. It is is assumed that there is a total time-averagedvoltage Φ most convenient to impose the constant-voltage bound- across the chain of junctions; this boundary condition is ary condition by using the method of Lagrange multi- N discussed further below. The Hamiltonian for this array pliers, adding to the Hamiltonian a term µ Φ = j=1 j is taken as the sum of four parts: N µ qn /C ,wheretheconstantµwillbedetermined j=1 j j P later by specifying Φ. H =HJ +HC +Hphot+Hint. (1) PIf we combine all these assumptions, we can finally write an explicit expression for H′, the operator whose Here H = N E cosφ is the Josephson coupling J − j=1 Jj j ground state we seek: energy, where φ is the gauge-invariant phase difference j P acrossthejthjunction,E =h¯I /q,thecriticalcurrent N Jj cj of the jth Josephson junction is Icj, and q = 2e is the H′ =H +µ qnj/Cj | | magnitudeofaCooperpaircharge. HC is the capacitive Xj=1 energy of the array, which we assume can be written in N 1 theformH = N E n2,whereE =q2/(2C ),the =h¯Ω a†a+ + E cosφ +E n2 C j=1 Cj j Cj j 2 − Jj j Cj j capacitance of tPhe jth junction is Cj, and nj is the dif- (cid:18) (cid:19) Xj=1(cid:18) ference in the number of Cooper pairs on the two grains ¯hg connected by the jth junction. The field energy may +µqnj/Cj + ji(a a†)sinφj . (3) √V − be written as H = h¯Ω(a†a + 1/2), where Ω is the (cid:19) phot frequency of the cavity resonant mode (assumed to be the only mode supported by the cavity), and a† and a are the usual photon creation and annihilation opera- B. Mean-Field Approximation tors, satisfying the commutation relations [a,a†] = 1; [a,a] = [a†,a†] = 0. We assume that the number The eigenstates of H′ are many-body wave functions, operator nj and phase φk have commutation relations dependingonthephasevariablesφj andnj,andthepho- [nj,exp( iφk)]= exp( iφj)δjk, which implies that nj ton coordinates a and a†. We will estimate the ground ± ± ± can be represented as i∂/(∂φj). state wave function and energy using a mean-field ap- − The crucialterm in the Hamiltonian for phase locking proximation. To define this approximation, we express is the interaction term Hint. We write this in the form H′ in the form H =(1/c) J Ad3x,whereJistheJosephsoncurrent int density, A is th·e vector potential corresponding to the H′ =Hphase+Hphot+Hint, (4) R electric field of the cavity mode, c is the speed of light, and the integral is carried out over the cavity volume. where Hphase = Nj=1(−EJjcosφj+ECjn2j +µqnj/Cj), pSainscseinJg itshcrooumgphristehde ojufnthcteioJnoss,epwhesomnacyurwrernittes Itchjissinlaφsjt afinelddHapinptr=oxiim(h¯a/tP√ioVn)c(oan−sisats†)of wNj=ri1tignjgs2i2nφj. The mean- N P term as H = E A sinφ . int j=1 Jj j j N ¯h AP= q j+1A ds, (2) Hint ≈i√V ha−a†i gjsinφj j ¯hcZj ·  N Xj=1 wheretheintegralisacrossthejthjunction,i.e.,between +(a a†) g sinφ j j thejthand(j+1)thsuperconductinggrain(seeFig.1).20 − h i j=1 X ThephasefactorA maybeexpressedintermsofthecre- j N ation and annihilation operators for the photon quanta a a† g sinφ . (5) as (in esu) Aj = i ¯hc2/(2Ω)(a a†)αj, where αj is a −h − i jh ji − j=1 suitablecouplingconstantdependingonthepolarization X and electric field opf the cavity mode.21 It is convenient With this approximation, H′ is decomposed into a sum to introduce the notation h¯gj/√V = EJjαj ¯hc2/(2Ω), of one-body terms, each of which depends only on the where V is the cavity volume. photon variables or on the phase variables of one junc- p Finally, we need to discuss suitable boundary condition, plus a constant term. The eigenstates of H′, in tions for this linear array. Let Φj denote the time- this approximation, are of the form Ψ(a,a†, φj ) = averaged voltage across the jth junction. For our as- ψ (a,a†) N ψ (φ ), where ψ and the{ψ ’}s are sumed form of the capacitive energy, Φ = q n /C , phot j=1 j j phot j j h ji j one-body wave functions. Q 2 That part of H′ which depends on photon vari- where we have written n¯ = µ/q. (Note that with our ables may be written H′ = H + i(h¯/√V)(a definitionofµ,itdoesnotha−vethedimensionofenergy.) phot phot − a†) N g sinφ , where sinφ denotes a quantum- Introducing the notation j=1 jh ji h ji mechanical expectation value with respect to ψj(φj). E2 =E2 +E2 , (14) WitPhthedefinitionλ = exp(iφ ) ,H′ takestheform α;j Jj int;j j h j i phot where we define E = 2h¯ηg /(ΩV) and φ = int;j j α;j 1 N ¯hg λ˜ tan−1(Eint;j/EJj), we obtain H′ =h¯Ω a†a+ +i j j(a a†), (6) phot 2 √V − H = E cos(φ φ )+E (n n¯)2 n¯2 . (15) (cid:18) (cid:19) Xj=1 j − α j − α;j Cj j − − The Schrödinger equation (cid:2) (cid:3) where λ˜ =(λ λ∗)/(2i)= sinφ . This is the Hamil- j j − j h ji tonianofadisplaced harmonicoscillator;itsgroundstate H ψ (φ )=E ψ (φ ), (16) j j j j j j energy eigenvalue E is readily found by completing phot;0 where H is given by expression (15), can be trans- the square to obtain j formed into Mathieu’s equation by a suitable change 1 ¯h of variables. Specifically, if we use the representation Hp′hot =h¯Ω(cid:18)b†b+ 2(cid:19)− ΩV η2, (7) nabjle=s ψ−i(∂φ/()∂=φje)x,pa(nidn¯φw)eual(sφo)m,athkeenthEeq.ch(1a6n)getaokfesvatrhie- j j j j j whereb† =a†+i N g λ˜ /(Ω√V),andwehavedefined form j=1 j j ∂2u P N (Ej +n¯2)uj =−Eα;jcos(φj −φα;j)uj −ECj ∂φ2j. (17) η = g λ˜ . (8) j j j Xj=1 Since φj and φj +2π represent the same physical state, thephysicallysignificanteigenstateψ (φ )shouldsatisfy j j (Notethatbandb† havethesamecommutationrelations ψ (φ +2π)=ψ (φ ), or equivalently j j j j as a and a†, i. e., [b,b†]=1.) The resulting ground state energy of H′ is uj(φj +2π)=exp( 2πin¯)uj(φj). (18) phot − Thus, the solutions to Eq. (17) are Mathieu functions 1 ¯h E = ¯hΩ η2. (9) satisfying the boundary condition (18). phot;0 2 − ΩV The total ground-state energy of the coupled system takes the form Similarly in the ground state, since b† =0, h i N a† = i 1 η. (10) Etot = Ej;0+Ephot;0+Ed, (19) h i − Ω√V j=1 X Also, the total energy stored in the photon field is where Ej;0 is the lowest eigenvalue of the Schrödinger equation (16). Note that the E ’s are also functions of j;0 E = h¯Ω a†a+ 1 = ¯hΩ + 1 η2, (11) theλ˜j’s,butonlythroughthevariableη. Edisa“double- phot h 2i 2 ΩV countingcorrection”whichcompensatesforthefactthat theinteractionenergyisincludedinbothE andthe where we use Eq. (10) and the fact that b†b =0 in the phot;0 E ’s;itisgivenbythenegativeoftheexpectationvalue h i j;0 ground state. of the last term on the right-hand side of Eq. (12), i.e., The wavefunction ψ (φ ) is an eigenstate ofthe effec- j j tive single-particle Schrödinger equation Hjψj = Ejψj, ¯h N 2h¯η2 where Ed = i a a† gj sinφj = . (20) − √V h − i h i ΩV j=1 1 X H = E cosφ + (q2n2+2µqn ) j − Jj j 2C j j Hence, the total ground-state energy is j + ¯hgjisinφ a a† , (12) N √V jh − i Etot(η)= Ej;0+Ephot;0+Ed j=1 X and a a† = 2iη/(Ω√V). Using this expression and N comphlet−ing tihe square, we can write = E (η)+ 1¯hΩ+ ¯h η2. (21) j;0 2 ΩV j=1 H =E (n n¯)2 E n¯2 X j Cj j Cj − − The actual ground-state energy is found from this ex- 2h¯g j ηsinφj EJjcosφj, (13) pression by minimizing Etot with respect to the variable − ΩV − η, holding µ (or n¯) fixed. 3 C. Approximate Minimization H = E cos(φ φ )+E [(n n¯)2 n¯2] j α;j α Cj − − − − = E cos(φ φ )+H0. (29) − α;j − α j We begin by considering the case n¯ = 0, for which anapproximateminimization of E (η) can be done an- The coherence threshold occurs in the small-coupling tot alytically as follows. First, one must evaluate the en- regime, Eα;j ECj. The desired ground-state solu- | | ≪ ergies E , which are the ground-state eigenvalues of tion can be obtained as a perturbation expansion about j;0 H ψ (φ ) = E ψ (φ ). For n¯ = 0, E has the approxi- the solutions to the zeroth order Schrödinger equation, j j j j j j j;0 mate value23 Hj0ψj0 = Ej0ψj0. The (unnormalized) solutions to this equation are ψ0 = exp(imφ ), corresponding to eigen- E2 j j Ej;0 ∼−2EαC;jj (22) vn¯alu<es1E/2j0, t=heEgCrjo[u(mnd−stna¯t)e2 −is mn¯2]=, w0.ithThme sinecteognedr-.orFdoerr | | perturbationcorrectionto this energydue to the pertur- for E E , and α;j ≪ Cj bation Hj′ =−Eα;jcos(φ−φα) is Ej;0 ∼−Eα;j (23) 0H′ m 2 ∆E = |h | j| i| , (30) j forE E . Afunctionwhichinterpolatessmoothly E E α;j Cj 0 m ≫ m=±1 − between these limits is X where m denotes the ket corresponding to exp(imφ ). j E E E2 +E2 . (24) After |a ilittle algebra, it is found that ∆E = j;0 ∼ Cj − Cj α;j E2 /[(2E )(1 4n¯2)]. If E E , thenj the q − α;j Cj − | α;j| ≫ Cj Substituting this expression into Eq. (21), we obtain groundstate eigenvalue Ej;0 approaches Eα;j as in the − casen¯ =0. Thegeneralizationofthe formula(25)tothe ¯h N case n¯ =0 is readily shown to be E (η)= η2+ E 6 tot Cj ΩV Xj=1" E (η)= ¯h η2+ N E˜ tot Cj E2 +E2 + 2h¯gj 2η2 . (25) ΩV Xj=1" −s Cj Jj (cid:18) ΩV (cid:19) # E˜2 +E2 + 2h¯gj 2η2 , (31) Setting dEtot/dη=0, we obtain the condition −s Cj Jj (cid:18) ΩV (cid:19) # 2h¯ g2 where E˜Cj = ECj(1 4n¯2). To determine η for a given η =η j . (26) value of n¯, and of th−e E ’s, E ’s, and g ’s, one mini- ΩV Xj EC2j +EJ2j +[2h¯gj/(ΩV)]2η2 mizes this energy with reCspject tJojη, as descjribed above. q In practice, at any value of n¯, and for any given dis- This equation always has the solution η =0. If tribution of the parameters g , C , and E , one can j Jj Jj easily evaluate the energy numerically, using the known 2h¯ N gj2 properties of Mathieu functions, hence obtaining both >1, (27) ΩV E2 +E2 the coherence threshold and the value of the order pa- Xj=1 Cj Jj rameter η. Once η is known, the individual values of the q λ˜ ’scanobtainedbynumericallysolvingtheSchrödinger thenthereisalsoareal,nonzerosolutionforη. Whenever j equation(16),usingtheHamiltonian(12)fortheground- this solution exists, it is a minimum in the energy, and state eigenvalue. Finally, the constant-voltage condition the η = 0 solution is a local maximum. Thus, Eq. (27) N can be imposed by choosing µ so that q n /C represents a threshold for the onset of coherence. j=1 h ji j In the opposite limit, when E E and λ˜ 1, equals the time-averagedvoltage across the array. α;j Cj j P ≫ → N N η = g λ˜ g . (28) III. RESULTS j j j → j=1 j=1 X X Althoughourformalismappliesequallytoorderedand N If we define g¯ = g /N and η¯ = η/N, then we disordered arrays, we will present numerical results for j=1 j see that η¯ rises from zero at a threshold determined ordered arrays only, purely for numerical convenience. P by Eq. (27) and approaches unity when the parameters In the ordered case, the constants g , E , and E are j Cj Jj E are sufficiently large. independent of j. In this ordered case, we denote the α;j | For| n¯ = 0, the threshold can still be approximately parametersg,E =q2/(2C),andE respectively. For a C J found ana6 lytically. Since H′ is periodic in n¯ with period specified value of n¯, we can find the ground-state eigen- unity, one need consider only 1/2 < n¯ 1/2. In this value E numerically by solving Eq. 16, using the well- j;0 − ≤ regime, we write H as knownpropertiesoftheMathieufunctions. We canthen j 4 minimize the total energy E with respect to η. In The voltage Φ across the array is determined by n¯ (or tot the ordered case, as noted, all the λ˜’s are equal, and equivalently µ). In Fig. 3, we plot λ˜ as a function of 0 η = Nλ˜. Furthermore, in this case, n¯ is related to Φ by n¯ for severalarraysizes at fixed coupling constants E , J0 Φ = Nqn¯/C. Hereafter, for given values of g, EC, EJ, EC,andEJ inanorderedarray. Sinceasalreadyshown, and n¯, we define λ˜ as the value of λ˜ which minimizes λ˜ is periodic in n¯ with a period of unity, we plot λ˜ (n¯) 0 0 0 the total energy E . only for a single period, 0 n¯ 1. Fig. 3 shows that, tot In Fig. 2, we plot λ˜ for this ordered array, as a func- for any given N and E , t≤he ca≤lculated λ˜ achieves its 0 J0 0 tion of N, assuming n¯ =0. Two curves are plotted. The maximumvalue when n¯ has a half-integervalue, i.e., the full curve shows λ˜ for the case E = 0, i.e., no direct array is most easily made coherent at such values of n¯. 0 J Josephson coupling. The dashed curve in Fig. 2 shows In particular, an array whose size is slightly below the λ˜ but for a finite direct Josephson coupling. In both thresholdvalueatintegervaluesofn¯ canbe made tobe- 0 cases, there is clearly a threshold array size N , below comecoherent,with anonzeroλ˜ ,whenn¯ isincreased— c 0 which λ˜ = 0. For N > N , we find λ˜ > 0. Since that is, when a suitable voltage is applied. On the other 0 c 0 λ˜ = sinφ (that is, the expectation value of sinφ in hand, for values of N far abovethe threshold, λ˜ is little 0 j 0 j 0 h i this energy-minimizingstate), the Josephsonarrayhas a affected by a change in n¯. netsupercurrentinthisconfiguration. AsN increases,λ˜0 In Fig. 4, we show the quantity hnji as a function of approaches unity, which corresponds to complete phase- n¯, for severalvalues ofN andfixed value of the coupling locking. The value of Nc is larger for finite Josephson constant ratios EJ0/EC and EJ/EC, for a single cycle coupling than for zero direct coupling; thus, it appears, (0 n¯ 1). Thisquantityisrelatedtothevoltagedrop ≤ ≤ paradoxically,thatthe finite directcouplingactuallyim- across one junction, in our model, by Φ/N = q nj /C. h i pedes the transition to coherence. This point will be For sufficiently large arrays, nj n¯ and the voltage h i ∼ discussed further below. drop is nearly linear in n¯ in this mean-field approxima- For EJ =0 and n¯ =0, Nc can easily be found analyt- tion. Forarraysclosertothecoherencethreshold,hnjiis ically from Eq. (27). The threshold is found to satisfy a highly nonlinear function of n¯. However,the deviation from linearity, n n¯, is, once again, a periodic func- j Nc =EC/(2EJ0), (32) tion of n¯ with pheriio−d unity. The discontinuous jumps in n¯ as a function of n represent regions of incoherence where EJ0 = h¯g2/(ΩV). This value agrees quite well (λ˜ = 0), whereas thhejiregions in which n is a smooth with our numerical results (cf. Fig. 2). Note that, for 0 h ji any nonzero value of the coupling EJ0, no matter how function of n¯ are regimes of phase coherence (λ˜0 6=0). small, there always exists a threshold value of N, above In Fig. 5, we again plot λ˜0(N) for two fixed ratios which phase coherence becomes established. EJ/EC, but this time for n¯ = 1/2. From Fig. 3, we ex- If E = 0 and n¯ = 0, N can still be obtained as an pect this choice of n¯ to maximize the tendency to phase J c implicit equation even in the disordered case, in terms coherenceandthusto reducethe thresholdarraysize for of the distribution of the g ’s and E ’s. The result is the onset of phase coherence. Indeed, in the absence of j Cj readily shown to be direct Josephson coupling, this threshold is reduced to belowunity (thatis,λ˜ remainsnonzero,evenatN =1, 0 2h¯ Nc gj2 for our choice of EJ0). In fact, for this value of n¯, only 1= . (33) ΩV E aninfinitesimalcouplingtotheresonantmodeisrequired Cj Xj=1 to induce phase coherenceinthis model. Once again(cf. For a given distribution of the parameter g2/E , there Fig.5),the additionofafinitedirectJosephsoncoupling j Cj actually increases the threshold number for phase coher- will always exist a threshold value of N such that this ence at n¯ =1/2 as it does at n¯ =0. equation is satisfied, no matter how weak the coupling Although we have not carried out a similar series of constants g . Thus, at least in this mean-field approxi- j calculations for a disordered, our analytical results show mation, the disorder has no qualitative effect on the co- that the essential features found in the ordered case will herencetransitiondiscussedhere. Inparticular,the crit- be preserved also in a disordered array. Most impor- ical number N does not necessarily either increase or c tantly,thereremainsacriticaljunctionnumberforphase decrease with increasing disorder; instead, N depends c coherence in a disorderedarray,just as there does in the on the distribution of g , E , and E in the array. j Jj Cj orderedcase. Themostimportantdifferencebetweenthe TheinsettoFig.2showsthetotalenergyinthephoton mode, E = h¯Ω( a†a + 1/2), in the ordered case, two cases is that the individual λ˜j’s will be functions of phot h i j in the disordered case. plotted as a function of N for n¯ =0. From Eq. (11), we findthat E =h¯Ω/2+N2λ˜2E foranorderedarray; phot 0 J0 this is the quantity plotted in the inset. As is evident IV. DISCUSSION from the plot, E varies approximately linearly with phot N2 all the way from the coherence threshold to large values of N, where λ˜ 1. This N2 dependence is a Althoughthepresentworkisonlyamean-fieldapprox- 0 → imation,weexpectthatitwillbequiteaccurateforlarge hallmark of phase coherence. 5 N. The reason is that, in this model, the one photonic ACKNOWLEDGMENTS degree of freedom is coupled to every phase difference, and thus experiences an environment which is very close One of us (DS) thanks the Aspen Center for Physics to the mean, whatever the state of the individual junc- for its hospitality while parts of this paper were being tions. Such small fluctuations are necessary in order for written, and acknowledges valuable conversations with a mean-field approach to be accurate. In fact, a simi- CarlosSa de Melo andAlanDorsey. This workhas been lar approachhas provenverysuccessfulin workon novel supportedbyNSFgrantDMR97-31511andbytheMid- Josephsonarraysinwhicheachwireiscoupledtoalarge west Superconductivity Consortium through DOE grant number of other wires via Josephson tunneling.24,25 DE-FG 02-90-45427. It may appear surprising that a finite direct Joseph- son coupling actually increases the threshold array size for coherence. But in fact this behavior is reasonable. If thereisnodirectcoupling(E =0),thephasedifference J acrosseachjunctionevolvesindependently,exceptforthe globalcoupling to the resonantphoton mode. When the array exceeds its critical size, this coupling produces co- 1See, for example, P. A. A. Booi and S. P. Benz, Appl. herence. If the same arraynowhas afinite E , thereare Phys.Lett. 68, 3799 (1996); S.Han, B. Bi, W. Zhangand J two coupling terms. But these are not simply additive, J. E. Lukens,Appl. Phys.Lett. 64, 1424 (1994). but in fact are π/2 out of phase: the direct coupling fa- 2A. K. Jain, K. K. Likharev, J. E. Lukens, and J. E. Sav- vorsφ =0,whilethe photoniconefavorsφ =π/2. For ageau, Phys.Rep.109, 309 (1984). j j alargeenougharray,thecouplingtothephotonfieldstill 3B. D. Josephson, Phys.Lett. 1, 251 (1962). predominates and produces global phase coherence, but 4K. Wiesenfeld, S. P. Benz, and P. A. A. Booi, J. Appl. this occurs at a higher threshold, at least in our model, Phys.76, 3835 (1994). 5K. Wiesenfeld, P. Colet, and S. H. Strogatz, Phys. Rev. than in the absence of direct coupling. Lett. 76, 404 (1996). A striking feature of our results is the very low coher- 6S. Watanabe, S. H. Strogatz, H. S. J. van der Zant, and ence threshold (N = 1) when n¯ = 1/2. In fact, for any T. P. Orlando, Phys. Rev.Lett. 74, 379 (1995). N and for E = 0, only an infinitesimal coupling to the J 7P. Hadley, M. R. Beasley, and K. Wiesenfeld, Phys. Rev. cavitymodewouldberequiredtoinducephasecoherence B 38, 8712 (1988). at n¯ =1/2. The reason for this low threshold is that, in 8P. W. Anderson, Lectures on the Many-Body Problem, ed. the absence of coupling, junction states with n = 0 h ji E. Caianello (Academic, New York,1964), Vol. II. and hnji = 1 are degenerate. Any coupling is therefore 9B. Abeles, Phys. Rev.B 15, 2828 (1977). sufficientto breakthe degeneracyandproduce phaseco- 10E. Simanek, Solid StateComm. 31, 419 (1979). herence. A related effect has been noted previously in 11S.Doniach, Phys. Rev.B 24, 5063 (1981). studies of more conventional Josephson junction arrays 12L. Jacobs, J. José, and M. A. Novotny, Phys. Rev. Lett. in the presence of an offset voltage.26 53, 2177 (1984). Finally, we comment on what is not included in the 13D.M.WoodandD.Stroud,Phys.Rev.B25,1600(1982). present work. This paper really considers only the mini- 14P.Barbara,A.B.Cawthorne,S.V.Shitov,andC.J.Lobb, mum energy state of the coupled photon/junction array Phys.Rev.Lett. 82, 1963 (1999). systemunder the assumptionthataparticularvoltageis 15A. B. Cawthorne, P. Barbara, S. V. Shitov, C. J. Lobb, applied across the array. It would be of equal or greater K. Wiesenfeld, and gratefully A. Zangwill, Phys. Rev. B interest to consider the dynamical response of such an 60, 7575 (1999). array. Specifically, it would be valuable to develop and 16E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 solveasetofcoupleddynamicalequationswhichincorpo- (1963). rate both the junction and the photonic degrees of free- 17B. W. Shore and P. L. Knight, J. Mod. Optics 40, 1195 dom. Such a set of equations has already been proposed (1993). 18R. Bonifacio, F. Casagrande, and M. Milani, Lett. Nuovo byBonifaciounderaparticularsetofsimplifyingassump- Cimento 34, 520 (1982). tions. A more accurate set of equations is needed, which 19R.Bonifacio,F.Casagrande,andG.Casati,Opt.Commun. would include not only a driving current, but also the 40, 219 (1982). damping arising from both resistive losses in the junctions and losses due to the finite Q of the cavity. In the 20ThisformforHint couldalsohavebeenobtainedbytaking absenceofdamping,suchequationscanbe writtendown theform− jEjcos(φj−Aj)appropriatefortheJoseph- son coupling energy in the presence of a vector potential, from the Heisenberg equations of motion. The inclusion and expandPingit to first order in theAj’s. of damping may be more difficult. We hope to discuss 21The specific form of αj is readily obtained from standard some of these effects in a future publication. expressions for A in terms of a and a†. If we choose a normalization such that V |Ea|2d3x = 1, where V is the volume of the cavity and Ea(x) is the electric field of the normalmode,thenwefinRdthatαj =q/(h¯c)√4π jj+1Ea· R 6 ds. FIG.5. Same as Fig. 2, but for n¯ =1/2. 22This approximation is equivalent to retaining terms only throughfirst orderin fluctuationsaboutthemean.Specif- ically, if 1 and 2 are operators depending respectively O O onthephotonandphasevariables,andifδ j = j j O O −hO i for j = 1,2, then the approximation retains all terms in theproduct 1 2 through first order in theδ j’s. 23ThisvaluefoOllowOsfromthestandardpropertiesOofMathieu functions. See, e.g., Handbook of Mathematical Functions, editedM.AbramowitzandI.A.Stegun(Dover,NewYork, 1965), Eqs. (20.3.1) and (20.3.15). 24V.M. Vinokur,L.B.Ioffe,A.I.Larkin,andM.V.Feigel- man,Zh.Eksp.Teor.Fys.93,343(1987)[Sov.Phys.JETP 66, 198 (1987)]. 25H. R. Shea and M. Tinkham, Phys. Rev. Lett. 79, 2324 (1997). 26See,forexample,C.Bruder,R.FazioandG.Schön,Phys. Rev. B 47, 342 (1993); C. Bruder, R. Fazio, A. Kampf, A. van Otterlo, and G. Schön, Physica Scripta 42, 159 (1992). FIG.1. Schematicofthegeometryusedinourcalculations, consisting of an underdamped array of Josephson junctions coupled to a resonant cavity, and subjected to an applied voltage Φ. The array consists of N +1 grains, represented by the dots, coupled together by Josephson junctions, represented by the crosses. Nearest-neighbor grains j and j +1 are connected by a Josephson junction, and the cavity is assumed to support a single resonant photonic mode. For the specific calculations carried out in this paper, we assume a one-dimensional array as shown, and a specific form for the capacitive energy, as discussed in thetext. FIG. 2. Coherence order parameter λ˜0 which minimizes Etot(λ˜) for a one-dimensional array, plotted as a function of the number of junctions N, for two values of the direct Josephson coupling energy EJ. Other parameters are ¯hg/√V = 0.3EC, ¯hΩ/2 = 2.6EC, and n¯ = 0. The coupling parameter EJ0 = ¯hg2/(ΩV) is given by EJ0 0.017EC. In- ≈ set: totalenergyinthephotonfield,Ephot,plottedasafunc- tionofN2,forthesameparametersandthesametwovalues of EJ. FIG. 3. Energy-minimizing value of the coherence order parameter λ˜0, as a function of the parameter n¯ = µ/q, for several values of the array size N. Other parameters are ¯hg/√V = 0.3EC, ¯hΩ/2 = 2.6EC, EJ = 0, and EJ0 0.017EC. ≈ FIG. 4. The parameter nj = ΦC/(qN), where Φ/N is h i thevoltage dropacross onejunction, plottedas afunctionof the parameter n¯ = µ/q, for several values of the array size N. Other parameters are ¯hg/√V = 0.3EC, ¯hΩ/2 = 2.6EC, EJ =0, and EJ0 0.017EC. ≈ 7 1 2 ... j j+1 ... N N+1 Figure 1. Model for a Josephson jun tion array oupled to a resonant avity. J. Kent Harbaugh and D. Stroud. 0.8 EJ = 0 EJ = 0:4EC 0.7 0.6 0.5 100 ~(cid:21)0 0.4 80 ot 60 h p 0.3 E 40 0.2 20 0 0.1 0 5000 10000 2 N 0 20 30 40 50 60 70 80 90 100 N Figure 2. Model for a Josephson jun tion array oupled to a resonant avity. J. Kent Harbaugh and D. Stroud. 0.8 N = 100 N = 50 N = 30 0.7 N = 20 0.6 0.5 0 ~(cid:21) 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 n(cid:22) Figure 3. Model for a Josephson jun tion array oupled to a resonant avity. J. Kent Harbaugh and D. Stroud.