Effects of Resonant Cavity on Macroscopic Quantum Tunneling of Fluxon in Long Josephson Junctions
aa r X i v : . [ c ond - m a t . s up r- c on ] J a n Effects of Resonant Cavity on Macroscopic Quantum Tunnelingof Fluxon in Long Josephson Junctions
Ju H. Kim and Ramesh P. Dhungana ∗ Department of Physics and Astrophysics, University of North Dakota, Grand Forks, ND 58202-7129
We investigate the effects of high- Q c resonant cavity on macroscopic quantum tunneling (MQT)of fluxon both from a metastable state to continuum and from one degenerate ground-state of adouble-well potential to the other. By using a set of two coupled perturbed sine-Gordon equations,we describe the tunneling processes in linear long Josephson junctions (LJJs) and find that MQT inthe resonant cavity increases due to potential renomalization, induced by the interaction betweenthe fluxon and cavity. Enhancement of the MQT rate in the weak-coupling regime is estimated byusing the experimantally accessible range of the model parameters. The tunneling rate from themetastable state is found to increase weakly with increasing junction-cavity interaction strength.However, the energy splitting between the two degenerate ground-states of the double-well potentialincreases significantly with increasing both the interaction strength and frequency of the resonantcavity mode. Finally, we discuss how the resonant cavity may be used to tune the property ofJosephson vortex quantum bits. PACS numbers: 74.50.+r, 74.78.Na, 85.25.Cp
I. INTRODUCTION
Experimentally observed quantum behavior ofJosephson vortices (i.e., fluxons) at ultra-low tempera-tures has opened up a possibility of realizing quantumcomputers based on long Josephson junctions (LJJs).This observation led to much interest on Josephsonvortex quantum bit (qubit) as an alternative to thepreviously proposed superconducting qubits. Similarto other approaches based on Josephson junctions suchas charge, phase, and flux qubits, Josephson vortexqubit (JVQ) is also a promising candidate for quantumcomputation application. Due to its weak interactionwith decoherence sources in the environment at lowtemperatures, the JVQ may have significant advantagesover the other superconducting qubits. For instance, asignificantly longer decoherence time was suggested asone such advantage. The JVQ takes advantage of the coherent superposi-tion of two spatially separated states arising from the lowtemperture property of a trapped fluxon in a double-wellpotential. This property includes (i) energy quantiza-tion and (ii) macroscopic quantum tunneling (MQT). Wenote that, for linear LJJs, the fluxon potential for eithermetastable state or JVQ may be obtained by using Nb-AlO x -Nb junctions and by implanting either one or twomicroresistors in the insulator layer, respectively. For ap-plication of JVQs, tuning both the decoherence time andthe level of entaglement by controlling the qubit prop-erty is essential. However, due to its weak interactionwith external perturbations, an effective tuning mecha-nism for JVQ is less clear. Recent studies on usingmicrowave cavity for both tuning a single phase qubitand inducing interaction between either two charge ortwo phase qubits suggest that resonant cavity may beused for JVQ to serve the same purpose.Earlier studies on the effects of resonant cavity indicate that both electric and magnetic fields ofthe cavity couple to the Josephson junction since thecavity electromagnetic (EM) mode behaves similar to aphonon mode which interacts with the fluxon. The ef-fects of resonant cavity on the fluxon dynamics in LJJstacks have been studied both experimentally and theoretically. These studies show that when thecoupling between LJJ and resonant cavity is spatiallyuniform, no force is exerted on the fluxon by the cavity,but its dynamics may become modified. These studiessuggest that the interaction between LJJ and a resonantEM wave mode of the cavity promotes collective dy-namics of fluxons. The in-phase locking mode of thefluxon dynamics is shown to be enhanced by the cavityEM mode.These studies also suggest that the junction-cavity in-teraction may be used to change the qubit property. Theproperty of JVQ depends on MQT between two spatiallyseparated states of the fluxon. We note that MQT repre-sents quantum particle-like collective exciations. Assemi-classical theories indicate that the MQT rate de-pends on the potential barrier height, the JVQ can betuned by adjusting the potential-well for the fluxon. Thisadjustment can be achieved by potential renormalizationinduced by the junction-cavity interaction since this in-teraction can strongly affect the fluxon tunneling pro-cesses, similar to phonon assisted tunneling in Josephsonjunctions. We note that a two-level atom interactingwith a quantized radiation field, described by the Jaynes-Cummings model, is also similar to the JVQ-cavity sys-tem that we consider in the present work. The potentialrenormalization for fluxon suggests that the resonant cav-ity may be used as a tool for controlling the JVQ prop-erty. As the fluxon tunneling processes may be controlledexternally by tuning either the junction-cavity couplingstrength or the resonant frequency, the effects of the res-onant cavity depend on the nature of the interaction.However, the influence of junction-cavity interaction onthe MQT rate has not been understood clearly.In this paper, we investigate the effects of the junction-cavity both on MQT from metastable state and on theground-state energy splitting in a double-well potential.We note that, to focus on the interaction between LJJand a single resonant cavity mode, we consider only ahigh- Q c cavity. First, we estimate the MQT rate forthe fluxon in a single LJJ and for the phase-locked flux-ons in a coupled LJJ stack by computing the local andnon-local contributions. Then, we estimate the effects ofresonant cavity on the JVQ property by computing theground-state energy splitting. Before proceeding further,we outline the main result. (i) The potential barrier for afluxon in the metastable state is not affected by increasingneither the junction-cavity interaction nor the resonantfrequency of the cavity EM mode . (ii)
The non-local con-tribution to the tunneling rate due to the junction-cavityinteraction is negligible in the weak-coupling regime. (iii)
Due to potential renormalization induced by the junction-cavity interaction, the potenital barrier height for thefluxon trapped in a double-well potential is reduced. Thisreduction leads to increase in the ground-state energysplitting for the JVQ with increasing junction-cavity cou-pling and resonant frequency .The outline of the remainder of the paper is as follows.In Sec. II, we describe the LJJ-cavity system by using aset of two perturbed sine-Gordon equations. In Sec. III,the effects of resonant cavity on the fluxon tunneling ratefrom the metastable state in a LJJ are discussed. In Sec.IV, we discuss MQT of phase-locked fluxons from themetastable state in a vertical stack of two coupled LJJs.In Sec. V, the effects of interaction between LJJ and asingle mode in high- Q c cavity on JVQ are estimated bycomputing the ground-state energy splitting. Finally, wesummarize the result and conclude in Sec. VI. II. COUPLED LONG JOSEPHSON JUNCTIONSIN RESONANT CAVITY
To examine i) one-fluxon tunneling in a single LJJ,ii) phase-locked two-fluxon tunneling in a stack of twocoupled LJJs, and iii) the ground-state energy splittingin JVQ, we start with coupled perturbed sine-Gordonequations for describing two LJJs which interact withresonant cavity ∂ ∂x (cid:18) ϕ − S ϕ (cid:19) − ∂ ϕ ∂t − sin ϕ = F (1) ∂ ∂x (cid:18) ϕ − S ϕ (cid:19) − ∂ ϕ ∂t − sin ϕ = F (2)where x and t are the dimensionless coordinates in unitsof λ J γ − ( S ) and ω − p , respectively. Here γ − ( S ) = √ − S and ω p denotes the plasma frequency. Thedynamic variable ϕ i represents the difference betweenthe phase φ of the superconductor order parameter forthe two superconductor (S) layers i and i − ϕ i = φ i − φ i − ). The strength of magnetic inductioncoupling between two LJJs is denoted by S . Here we set¯ h = k B = c = 1 for convenience. The perturbation term F of for each LJJ which is given by F i = β ∂ϕ i ∂t + f i − g E d q r dt − ǫ i δ ( x − x oi ) sin ϕ i (3)accounts for the contribution from dissipation ( β ), biascurrent ( f = J B /J c ), resonant cavity ( g E ), and microre-sistors ( ǫ = ( J c − J ′ c ) l b /J c λ J ). Here x oi , J B , J c , J ′ c , l b ( ≪ λ J ) and λ J denote the position of microresistors inthe insulator layer of the i -th junction, the bias currentdensity, the critical current density, the modified currentdensity, the length of the LJJ in which J c is modified, andthe Josephson length, respectively. We note that dissi-pation, bias currents, resonant cavity and microresistorson the phase dynamics lead to different effects.We account for the perturbation contribution due toresonant cavity by following Tornes and Stroud and byassuming that the cavity supports a single harmonic os-cillator mode which may be represented by the displace-ment variable q r as d q r dt + ω r Q c dq r dt + ω r q r = g E γ ( S ) M osc Z dx ∂ ∂t (cid:18) ϕ + ϕ (cid:19) . (4)Here ω r , Q c , and M osc are the dimensionless oscillatorfrequency in units of ω p , the cavity quality factor, andthe ”mass” of the oscillator mode, respectively. For sim-plicity, we neglect the second term on the left hand sideof Eq. (4) by assuming that the cavity is non-dissipative(i.e., high- Q c cavity). Also, we assume that the cavityelectric field E is uniform within the junction by consid-ering the spatially uniform junction-cavity coupling g E of g E = − ǫ d e r M osc π E · ˆz , (5)where ǫ d is the dielectric constant. As we will discussbelow, the position independent coupling g E does notchange the fluxon motion directly but yields potentialrenormalization when a microresistor is present.To estimate the effects of interaction between LJJ andresonant cavity analytically, we consider the weak per-turbation F limit. As each perturbation term in Eq. (3)is small and does not change the form of the kink solutionwithin the lowest order approximation, we describe thefluxon motion in terms of the center coordinate q ( t ). Inthe absence of both the perturbation terms ( F = 0) andthe magnetic induction effect ( S = 0), the fluxon solutionto Eq. (1) is given by ϕ i ( x, t ) ≈ − h e γ ( v i )[ x − q i ( t )] i , (6)in the non-relativistic limit (i.e., v ≪ q i ( t ) = v i t denotes the center coordinate for the fluxon, and v is thefluxon speed in units of Swihart velocity. Equation (6)represents propagation of nonlinear wave as a ballisticparticle. The perturbation contributions of F only affectthe dynamics of fluxon expressed in the q coordinate.We now describe the fluxon phase dynamics in the cou-pled LJJ using the center coordinate q i representation.The energy of the fluxon may be seen easily from theEuclidean Lagrangian (i.e., τ = it ), L = L o + L mag + L pert + L osc + L coup . (7)The first three terms for L of Eq. (7) describe the LJJcontributions, while the remaining two terms arise fromthe resonant cavity. First, we discuss the LJJ contribu-tions to Lagrangian L . The unperturbed part of LJJ isdescribed by the Lagrangian L o given by L o = X i Z dx (cid:20)(cid:18) ∂ϕ i ∂τ (cid:19) + (cid:18) ∂ϕ i ∂x (cid:19) +2(1 − cos ϕ i ) (cid:21) . (8)The Lagrangian contribution from the magnetic induc-tion effect, L mag , is given by L mag = S Z dx (cid:18) ∂ϕ ∂x (cid:19) (cid:18) ∂ϕ ∂x (cid:19) . (9)We note that L mag accounts for the interaction energy E int between two LJJs due to the magnetic induction ef-fect. The perturbation contribution to the Lagrangian, L pert = L nd + L d , is expressed as the sum of two terms:i) the non-dissipative ( L nd ) and ii) dissipative ( L d ) part.The non-dissipative contribution comes from the biascurrents and microresistors. The non-dissipative La-grangian L nd is expressed as the sum of the contributionsfrom the bias current ( L bias ) and microresistors ( L pin )(i.e., L nd = L bias + L pin ). The bias current contribution L bias is given by L bias = X i Z dxf i ϕ i , (10)and the inhomogeneity contribution due to microresistors L pin is given by L pin = X i Z dx ǫ i δ ( x − x oi )(1 − cos ϕ i ) . (11)We note that L pin accounts for the fluxon pinning en-ergy E pin . These non-dissipative contributions providethe bare fluxon potential V ( q ). On the other hand, thedissipative Lagrangian L d accounts for the interactionbetween the fluxon and environment. The effects of thiscontribution may be described by following Caldeiraand Leggett and by representing the environment as aheat bath. The heat bath is represented as harmonicoscillators with generalized momenta P i and coordinates Q i . The dissipation Lagrangian L d which accounts forthe coupling between the phase ( ϕ ) and oscillator ( Q i )variables is given by L d = Z dx X i " P i m i + m i ω i (cid:18) Q i − c i ϕm i ω i (cid:19) . (12) Here, the spectral function J β ( ω ), J β ( ω ) = π X i c i m i ω i δ ( ω − ω i ) = βω, (13)is used to reproduce the dissipation effects ( β ) in Eq. (3).The effects of dissipation on a two-state system has beenstudied extensively by using the spin-boson model. Inthe adiabatic approximation, the energy splitting for thetwo-state system is known to be reduced in the dissipa-tive environment. However, this result does not implythat the effects of the interaction between the two-statesystem and a single oscillator, which represents either aphonon or quantized radiation field, on the energy split-ing is similar. In our discussion below, we neglect thedissipation effects by setting β = 0 since these effects aresmall at low temperatures, and we focus on the effectsdue to a resonant cavity.We now discuss the high- Q c resonant cavity contribu-tion to the Lagrangian L of Eq. (7). The resonant cavityis modeled by using Lagrangian for a single harmonic os-cillator which represents a single EM-mode supported bythe cavity. The Lagrangian for this single mode oscillator L osc is written as L osc = M osc (cid:18) dq r dτ (cid:19) + K q r , (14)where K is the ”spring constant” and q r denotes the oscil-lator coordinate. We note that the oscillator frequency ω r in Eq. (4) is given by ω r = ( K/M osc ) / . The ca-pacitive coupling between LJJ and resonant cavity is de-scribed by the Lagrangian L coup as L coup = − g E (cid:18) dq r dτ (cid:19) Z dx X i (cid:18) ∂ϕ i ∂τ (cid:19) . (15)Here we assume that the coordinate q r is spatially homo-geneous and focus on the effects of the uniform E -fieldin the cavity. We note that the interaction between LJJand resonant cavity yields the non-local effects, similarto those from the dissipation term (i.e., β = 0).We estimate MQT of fluxon by using the usual semi-classical approach of starting with the partition func-tion Z for the junction-cavity system Z = Z D [ ϕ ] D [ q r ] exp {− S [ ϕ, q r ] } (16)where S [ ϕ, q r ] = R dτ L is the action and L is the La-grangian of Eq. (7). By noting that shape distortionof the fluxon due to weak perturbation (i.e., small F )is negligible, we may rewrite the partition function Z interms of q ( τ ) and q r ( τ ) as Z = Z D [ q ] Z D [ q r ] e − S [ q,q r ] . (17)Also by noting that the Lagrangian L coup of Eq. (15)which accounts for the interaction between LJJ andresonant cavity is linear in both coordinates q r and ϕ , we separate the partition function Z into the reso-nant cavity and fluxon contribution by expressing Z = Z res Z fluxon . The resonant cavity ( Z res ) and fluxon( Z fluxon ) contribution to Z are given, respectively, as Z res = R D [ q r ( ω n )] exp {− S res [ q r ( ω n )] } and Z fluxon = R D [ q ( τ )] exp {− S eff [ q ( τ )] } . The action for the resonantcavity contribution S res [ q r ] is given by S res [ q r ] = T X ω n M osc (cid:0) ω n + ω r (cid:1) × (cid:20) q r,n + 2 πg E q n ω n M osc ( ω n + ω r ) (cid:21) (cid:20) q r, − n + 2 πg E q − n ω n M osc ( ω n + ω r ) (cid:21) (18)where q r,n = q r ( ω n ), q n = q ( ω n ), ω n = 2 πnT is theMatsubara frequency, and T is the temperature. Theaction for the fluxon contribution S eff [ q ] is given by S eff [ q ] = Z dτ " M e X i =1 ˙ q i + V ( q ) + ¯ g E ω r − S ( X i =1 q i ) − g E − S Z dτ ˙ q ˙ q − ¯ g E − S Z dτ dτ ′ K ( τ − τ ′ ) X i =1 q i ( τ ) X i =1 q i ( τ ′ ) (19)where ˙ q i = dq i /dτ , M e denotes the renormalized fluxonmass M e = M (cid:18) − M g E − S (cid:19) (20)due to the spatially uniform junction-cavity interactionand the M denotes the rest mass of the fluxon. Themass M e accounts for the renormalization effect of bothjunction-cavity and magnetic induction interaction. Thebare potential V ( q ) = V ( q , q ) is given by V ( q ) = − X i =1 (cid:18) πf i q i + 2 ǫ i cosh q i (cid:19) − S ( q − q )sinh( q − q ) . (21)Here, the fluxon potential V ( q ) includes the effects fromthe three contributions: (i) the potential tilting effect ( f ),(ii) the pinning effect ( ǫ ), and (iii) the magnetic inductioneffect ( S ). The third term in [ ] of Eq. (19) accountsfor the potential renormalization due to junction-cavityinteraction. This renormaliztion is similar to that for theelectronic tunneling process with phonon coupling. Inthe discussion below, we refer ¯ g E = 2 π g E /M osc as thestrength of junction-cavity interaction. The cavity kernel K ( τ − τ ′ ) in the third term of Eq. (19) is given by K ( τ ) = ω r ω r / T − ω r | τ | )sinh( ω r / T ) . (22)at non-zero temperature T . This term accounts for thenon-local effect arising from the junction-cavity interac-tion. I S Resonantcavity defectL x S J B J B L y xz y FIG. 1: A LJJ is shown schematically as an insulator ( I )layer is sandwiched between two superconductor ( S ) layers. L x and L y denote the dimensions in x − and y − direction,respectively. J B denotes the bias current density. The filledcircle represents microresistor (i.e., pinning center), and thedashed box represents resonant cavity. After the calcultion , the oscillator coordinate q r in thepartition function Z of Eq. (16) is decoupled from thecenter coordinate q . This separation allows us to inte-grate out the q r -coordinate. Hence, in discussions below,we consider the fluxon contribution Z fluxon to the par-tition function which is described by the action S eff .Using S eff , we discuss how the junction-cavity interac-tion affects both one-fluxon and two-fluxon tunneling inLJJs. III. MACROSCOPIC QUANTUM TUNNELINGIN SINGLE JUNCTION
We now examine the effects of resonant cavity on MQTfrom the metastable state in a single LJJ obtained byimplanting a microresistor in the insulator layer and byapplying the bias current ( J B ) as shown in Fig. 1. Thedimensions of the junction, compared to the Josephsonlength λ J , are chosen so that L x ≫ λ J and L y ≪ λ J .These choices are made to enhance the quantum effectat low temperatures. We describe MQT of the fluxon bystarting with the action S seff [ q ] for the LJJ given by S seff [ q ] = Z dτ (cid:20) M e q + V s ( q ) + ¯ g E ω r q (cid:21) − ¯ g E Z dτ dτ ′ K ( τ − τ ′ ) q ( τ ) q ( τ ′ ) . (23)Here, the action S seff [ q ] is obtained from S eff [ q ] of Eq.(19), by setting S = 0 (i.e., no magnetic induction effect), q = q , and q = 0. Following Caldeira and Leggett, wemay simplify S seff [ q ] by making a usual substitution of w e V o V (x) x x o FIG. 2: The fluxon potential V s due to both the bias cur-rent density and microresistor in a single LJJ is schematicallyillustrated. q ( τ ) q ( τ ′ ) = [ q ( τ ) + q ( τ ′ )] / − [ q ( τ ) − q ( τ ′ )] /
2. Wenote that the first two terms of this substitution cancelthe potential renormalization contribution (i.e., ¯ g E ω r q term) arising from the junction-cavity interaction. Withthis cancellation, the action S seff [ q ] becomes similar tothat for the dissipative system, but the fluxon mass isnow renormalized to M e = M (cid:18) − g E M (cid:19) (24)and β is replaced by the junction-cavity interactionstrength (i.e., β → ¯ g E ). The renormalized mass M e ac-counts for the effects of the uniform E field in the cavity.The bare fluxon potential V s ( q ) is given by V s ( q ) = − πf q − ǫ cosh q . (25)Here the bias current density f = f c − δ f is measuredin terms of the deviation δ f from the critical value f c =4 ǫ/ (3 √ π ). The potential V s ( q ) may be approximatedby a quadratic-cubic potential as shown schematically inFig. 2. The cavity kernel K ( τ − τ ′ ) of Eq. (22) describingthe non-local effect due to the junction-cavity interactionsimplifies to K ( τ − τ ′ ) = ω r e − ω r | τ − τ ′ | (26)in the T = 0 limit.The action S seff [ q ] of Eq. (23) indicates that the res-onant cavity yields i) fluxon mass renormalization andii) non-local effects. The mass renormalization modifiesthe oscillation frequency about the metastable point, asshown in Fig. 2. This change may be easily seen by com-puting the oscillation frequency ω e at the metastatblestate (i.e., local minimum) as ω e = (cid:20) M e d ¯ V s (0) dx (cid:21) / ≈ ω o (cid:18) g E M (cid:19) , (27) FIG. 3: The ratio of the tunneling rates Γ cav (0) / Γ(0) is plot-ted as a function of the junction-cavity coupling strength ¯ g E to illustrate the size of enhancement. where ω o is the oscillation frequency at the metastablepoint in the absence of the resonant cavity. The non-localcontribution due to junction-cavity interaction is similarto that for the dissipative system, but to determine thesize of this contribution more calculation is needed.To estimate the size of these two contributions fromthe junction-cavity interaction, we compute the MQTrate given byΓ cav (0) = A cav (0) e −B cav (0) (28)at T = 0. Here, the prefactor A cav (0) is given by A cav (0) = √ ω e (cid:18) B o,cav π (cid:19) / (29)and the bounce exponent B cav (0) = B o,cav + δ B cav in-cludes both the local contribution B o,cav of B o,cav = Z ∞−∞ dτ (cid:20) M e q + V s ( q ) (cid:21) (30)and the non-local contribution δ B cav of δ B cav = ¯ g E Z ∞−∞ dτ Z ∞−∞ dτ ′ K ( τ − τ ′ ) [ q ( τ ) − q ( τ ′ )] . (31)These two contributions, B o,cav and δ B cav , to B cav (0) areevaluated explicitly to estimate their size.The local contribution B o,cav may be computed easilyby approximating V s ( q ) of Eq. (25) as a usual quadratic-plus-cubic potential of¯ V s ( x ) = V s ( q ) − V s ( q o ) ≈ V o (cid:0) ¯ x − ¯ x (cid:1) (32)where ¯ x = x/x o , x = q − q o , and V o = [8 π δ f / ( √ ǫ )] / is the barrier potential for the fluxon. Here q o is theposition of the metastable point and x o = 9 √ M e ω e / ǫ is the escape point as shown in Fig. 2. The evaluation of B o,cav yields B o,cav = 2 Z x o o dx (cid:2) M e ¯ V s ( x ) (cid:3) / = 36 V o ω e . (33) FIG. 4: The non-local contribution δ B cav to the bounce ex-ponent B cav (0) is plotted as a function of ω r for ¯ g E = 0 . Using this result, we estimate the local contribution toenhancement of the tunneling rate due to the resonantcavity. The ratio of the MQT rates, Γ cav (0) / Γ(0), isgiven by Γ cav (0)Γ(0) ≈ g E M (cid:18) V o ω o (cid:19) , (34)where Γ(0) is the tunneling rate in the absence of the res-onant cavity (i.e., ¯ g E = 0). Equation (34) indicates thatthe tunneling rate increases with increasing junction-cavity interaction strength ¯ g E . In Fig. 3, we plot thenumerically computed ratio Γ cav (0) / Γ(0) as a functionof ¯ g E to illustrate its enhancement in the weak-couplingregime (i.e., ¯ g E ≪ cav (0) / Γ(0) is less than 1%.The non-local contribution δ B cav to B cav (0) of Eq. (28)reduces the tunneling rate Γ cav (0). The size of this re-duction is estimated by evaluating δ B cav of Eq. (31) bywriting δ B cav = ¯ g E ω r x o ω e Z ∞−∞ d ¯ τ d ¯ τ ′ e − ωrωe | ¯ τ − ¯ τ ′ | [¯ x (¯ τ ) − ¯ x (¯ τ ′ )] , (35)where ¯ x ( τ ) = sech ( ω e τ / x ( τ ) is thesolution to the equation of motion for the quadratic-plus-cubic potential in the absence of the non-local effect. Weevaluate Eq. (35) and obtain δ B cav = 2¯ g E √ M e ǫ ! ω r sinh ( πω r /ω e ) . (36)The result for δ B cav indicates that the non-local contri-bution increases almost linearly with ¯ g E in the weak-coupling regime and has a strong dependence on thefrequency ω r of the cavity mode. At low cavity fre-quencies ( ω r ≪ δ B cav ∝ ω r . At high cavity frequencies ( ω r ≫ δ B cav ∝ ω r exp( − πω r /ω e ). I S I Resonantcavity defectsL x SS J B J B1 02 L y xz y FIG. 5: Two LJJs with a vertical column of two microresistorsis shown schematically. L x and L y denote the dimensions in x − and y − direction, respectively. J B denotes the bias currentdensity. The filled circles represent the microresistors. To illustrate the cavity frequency dependence, we plot δ B cav as a function of ω r for ¯ g E = 0.02 (solid line), 0.04(dashed line), and 0.06 (dot-dashed line) in Fig. 4. Thecurves indicate that δ B cav vanishes both in the low andhigh cavity frequency ω r limits. Hence, the non-local ef-fects on the tunneling rate Γ cav (0) is negligible near theselimits. IV. MACROSCOPIC QUANTUM TUNNELINGIN COUPLED JUNCTIONS
In this section, we estimate the effects of resonant cav-ity on the tunneling rate of the phase-locked fluxons fromthe metastable state in two coupled LJJs. Here the flux-ons are trapped by the microresistor on each insulator (I)layer, shown schematically in Fig. 5. Earlier studies in-dicate that uncorrelated one-fluxon tunneling is the dom-inant process in the absence of resonant cavity. However,phase-locking between the fluxons in two LJJs becomesenhanced in the resonant cavity. This enhancement maybe seen more easily from the effective action S eff [ q ] forthe two coupled LJJs of Eqs. (19) and (21) written inthe rotated coordinates ( q + , q − ) as S eff [ q ] = Z dτ (cid:20) M e q + M q − + V ( q ) + ¯ g E ω r − S q (cid:21) − g E − S Z dτ dτ ′ K ( τ − τ ′ ) q + ( τ ) q + ( τ ′ ) , (37)where q ± = ( q ± q ) / √
2. The action S eff [ q ] indicatesthat the potential for the in-phase mode, ( q + , , q − ), is not. Also, the non-local contri-bution appears only for the motion in the q + direction.The bare fluxon potential V ( q ) = V ( q + , q − ) of V ( q + , q − ) = − √ πf q + − √ S q − sinh √ q − − ǫ (cid:16) q + + q − √ (cid:17) + 1cosh (cid:16) q + − q − √ (cid:17) (38)for f = f = f and ǫ = ǫ = ǫ , indicates that theone-dimensional potential along the ( q + ,
0) direction (i.e., V ( q + , V s ( q ) of Eq. (25) under the transforma-tion of 2 f → f , 2 ǫ → ǫ , and q + / √ → q . This sim-ilarity reflects that the phase-locked fluxons moving inthe ( q + ,
0) direction (i.e., q = q ) behave as a singlefluxon. However, the one-dimensional potential for theout-of-phase mode (i.e., V (0 , q − ) or along the (0 , q − ) di-rection) behaves as a potential well near the metastablepoint q o = ( q o + , q o − ), determined from the condition[ ∂V ( q ) /∂q + ] q − = [ ∂V ( q ) /∂q − ] q + = 0.To illustrate these phase-locking modes, we plot thepotential V ( q + , q − ) in Fig. 6 for f = 0 . ǫ = 0 . S = − .
05. Here, the value for ǫ and S are chosenso that when a vertical stack of two interacting JVQsare fabricated using coupled LJJs and microresistors onlyone quantum state is bound on each side of the double-well potential. The metastable point q o is denoted bythe solid circle. The solid lines indicate that the poten-tial is metastble for the in-phase mode (i.e., along the( q + ,
0) direction), but it behaves as a well for the out-of-phase mode (i.e., along the (0 , q − ) direction). Thesecurves show that tunneling of the in-phase mode fromthe metastable state is more favorable than that for theout-of-phase mode.The tunneling rate Γ cav (0) from q o can be estimatedby summing over the contribution from all paths of es-cape, but the dominant contribution comes from themost probable escape path (MPEP) in which S eff isthe minimum. For the physical parameters chosen inFig. 6, the MPEPs correspond to one-fluxon tunneling,indicated by the dashed lines. The MPEPs are deter-mined by the two competing energies: (i) the pinningenergy ( E pin = | E pin | ) and (ii) the magnetic inductioninteraction energy ( E int = | E int | ). When E int ≫ E pin ,the fluxons are not pinned at the microresistor sites butmaintain a large separation distance. However, when E int ≪ E pin , the one-fluxon tunneling processes are fa-vored over the two-fluxon tunneling processes.We now estimate the two-fluxon tunneling rate for thein-phase mode. We simplify the calculation by using thesimialrity between the tunneling of the in-phase modeand the one-fluxon tunneling process discussed in Sec.III. When the bias current f is less than the critical value f c (i.e., f = f c − δ f with 0 < δ f ≪ f c = 4 ǫ/ (3 √ π ), thepotential along the path ( q + ,
0) has the metastable state,as illustrated in Fig. 2. The potential V ( q + ,
0) may be - q + - q - - - H q L FIG. 6: The potential V Q ( q + , q − ) surface is plotted for ǫ =0 .
269 and S = − .
05. The filled circle represents the positionof the metastable state. The dashed and solid lines denotethe most probable escape paths (MPEPs) for one-fluxon andtwo-fluxon tunneling, respectively. approximated as the quadratic-plus-cubic form of V (¯ q + , ≈ V to q − ¯ q ) , (39)where ¯ q + = ( q + − q o + ) /q e + , q e + is the escape point and V to = 2[ d V ( q o + , /dq ] / d V ( q o + , /dq ] denotes thepotential barrier height for two-fluxon tunneling. Wenote that q e + is similar to x o in Fig. 2. Also, similar tothe single LJJ, the semiclassically estimated two-fluxontunneling rate of Γ tcav (0) = A tcav exp[ B tcav (0)] at T = 0depends on both the barrier height and oscillation fre-quency. The factor A tcav (0) and bounce exponent B tcav (0)are calculated in the same way as in Sec. III. The factor A tcav (0) is given by A tcav (0) ≈ √ ω e (cid:18) B to,cav π (cid:19) / . (40)The local and non-local contributions to the bounce ex-ponents B tcav (0) = B to,cav + δ B tcav are given by B to,cav = 2 Z q e + − q o + dq + p M e V ( q + , ≈ V to ω e , (41)and δ B tcav ≈ g E − S √ M e ǫ ! ω r sinh ( πω r /ω e ) , (42)respectively. The result indicates that the two-fluxontunneling rate Γ tcav (0) in the cavity is enhanced from thatΓ t (0) in its absence. Neglecting the non-local contribu-tion, we may write the ratio Γ tcav (0) / Γ t (0) asΓ tcav (0)Γ t (0) ≈ g E M (1 − S ) (cid:18) V to ω o (cid:19) . (43)This enhancement is similar to the tunneling processdiscussed in Sec. III. The estimated value of Γ t (0)for the Nb-Al O x -Nb-Al O x -Nb junction is 8 . × s − . This value is obtained by using the experimentalvalue of J c ∼ × A/m , λ L ∼ λ J ∼ µ m,and ω p ∼ L y ∼ µ m to enhancethe quantum effect and used the experimentally accessi-ble value of ǫ = 0 . S = − .
05 and δ f ∼ × − .On the other hand, the potential V ( q + , q − ) along the( q + ,
0) direction indicates that the two-fluxon tunnelingrate Γ tcav (0) is suppressed from the one-fluxon tunnel-ing rate Γ ocav (0) along either the q + = q − or q + = − q − direction. This reduction in the tunneling rate is givenby Γ tcav (0)Γ ocav (0) ≈ α o s V to V oo e − V to − α oV oo )5 ωo × (cid:20) g E ( V to − α o V oo )5 ω o M (1 − S ) (cid:21) , (44)where α o = { [ d V ( q o + , /dq ] / [ d V ( q o , /dq ] } / is a constant of order unity, V oo =2[ d V ( q o , /dq ] / d V ( q o , /dq ] is the one-fluxontunneling potential barrier height, V ( q,
0) is the fluxonpotential of Eq. (21) along the q + = q − direction,and q o denotes the position of the metastable pointfor one-fluxon tunneling, given by the condition that dV ( q o , /dq = 0. The ratio Γ tcav (0) / Γ ocav (0) ≪ V to ≫ V oo . V. JOSEPHSON VORTEX QUBIT INRESONANT CAVITY
We now examine the effects of high- Q c resonant cav-ity on JVQ. The JVQ may be fabricated by using twoclosely implanted microresistors in the insulator layer ofthe linear LJJ as shown in Fig. 7. As earlier studies indicate, MQT of fluxon between the spatially separatedminima of double-well potential leads to splitting of thedegenerate ground-state energy. In this section, weestimate the effects of junction-cavity interaction on thisenergy splitting.The interaction between the LJJ and resonant cavityyields i) fluxon potential renormalization and ii) non-local contribution to the action. The effects of these con-tributions on the energy splitting may be estimated bystarting with the action S Qeff for the JVQ given by S Qeff [ q ] = Z dτ (cid:20) M e q + V Q ( q ) (cid:21) − g E Z dτ dτ ′ K ( τ − τ ′ ) q ( τ ) q ( τ ′ ) . (45)Without loss of generality, we obtain the potential func-tion V Q ( q ) from the double-well potential V ( q ) of V ( q ) = ¯ g E ω r q − ǫ cosh (cid:0) q − ℓ (cid:1) − ǫ cosh (cid:0) q + ℓ (cid:1) , (46) I S Resonantcavity defectsL x S L y xz y l FIG. 7: A LJJ with two microresistors, representing a Joseph-son vortex qubit, in a resonant cavity is shown schematically.The separation distance between the microresistors is denotedby ℓ . The filled circles and dashed box represent the microre-sistors and resonant cavity, respectively. qV (q) w e V o e = e = e o o d g E2 =0g =0 E2 FIG. 8: A schematic diagram of a double-well potential V Q ( q )due to the two microresistors in the insulator layer of the LJJis shown to illustrate the renormalization of V Q ( q ). The solidand dashed lines represent the potential V Q ( q ) in the absenceand in the presence of the resonant cavity, respectively. where ℓ denotes the separation distance between the twomicroresistors. Here, we have added a constant energy E Q term to V ( q ) (i.e., V Q ( q ) = V ( q ) + E Q ) so that V Q ( q )vanishes at the potenital minima. Here, the potential V Q ( q ) may be characterized by the position of the twominima and the potential barrier height. In the discus-sion below, we do not make the usual substitution of q ( τ ) q ( τ ′ ) = [ q ( τ ) + q ( τ ′ )] / − [ q ( τ ) − q ( τ ′ )] / g E = 0),the double-well structure for V Q ( q ) with the separationdistance ℓ > ℓ o ≈ .
317 is shown schematically in Fig. 8
FIG. 9: The ratio of the potential barrier height V o,cav /V o isplotted as a function of the junction-cavity coupling strength¯ g E for ω r = 0 .
50 (dot-dashed line), 0.70 (dashed line), and0.90 (solid line) to illustrate the suppression in the cavity. as the solid line. The two potential minima are locatedat q = ± q o / q o is determined fromcosh q o = sinh ℓ − ℓ . (47)The energy shift E Q , representing a constant of motion,is given by E Q = − ǫ cosh ℓ cosh ℓ − . (48)Also, the potential barrier height V o between the twominima (i.e., q = ± q o /
2) is given by V o = 2 ǫ (cid:18) cosh ℓ − ℓ (cid:19) . (49)We note that these quantities change in the resonant cav-ity, as shown schematically by the dashed line in Fig. 8.In the resonant cavity (i.e., ¯ g E = 0), on the other hand,the JVQ potential V Q ( q ) acquires an additional ¯ g E ω r q term in Eq. (46). This term arises from the couplingbetween the oscillator coordinate q r and the center co-ordinate q in the coupling Lagrangian L coup of Eq. (15)and accounts for potential renormalization. The mainrenormalization effects are the following: i) the barrierpotential height is reduced, ii) the position of the po-tential minima become closer together, and iii) the os-cillation frequency at the potential minima is modified.These effects become amplified with increasing junction-cavity interaction strength (¯ g E ) and resonant frequency( ω r ).The effects of the junction-cavity interaction on thepotential barrier height V o,cav may be estimated straight-forwardly. In Fig. 9, we plot the numerically computedratio V o,cav /V o as a function of ¯ g E to illustrate the de-pendence on the junction-cavity interaction. The curves FIG. 10: The shift δ o in the position of the potential minima isplotted as a function of the junction-cavity coupling strength¯ g E for ω r = 0 .
50 (dot-dashed line), 0.70 (dashed line), and0.90 (solid line). for ω r = 0 .
50 (dot-dashed line), 0.70 (dashed line) and0.90 (solid line) indicate that the barrier potential heightdecreases with increasing ¯ g E and ω r . Also, the curvesindicate that the ratio decreases linearly in the weak cou-pling regime. To leading order in ¯ g E , the potential barrierheight V o,cav estimated from the renormalized potential V ( q ) of Eq. (46) is given by V o,cav ∼ = V o − ¯ g E ω r q o . (50)This decrease in the potential barrier height leads to theincrease in the ground-state energy splitting.Another important effect of the resonant cavity is theshift δ o in the position of potential minima. As the poten-tial barrier height is reduced, the position of the potentialminima are closer together. The shift δ o from the initialposition of q = ± q o / δ o = ¯ g E q o ω r ǫ cosh ℓ tanh ℓ cosh 2 ℓ − . (51)Here, we obtained δ o by imposing the condition[ dV ( q ) /dq | q =( q o / ± = 0, where ( q o / ± = ± [( q o / − δ o ]denotes the new potential minima. This shift δ o modifiesthe constant of motion E Q . The new value for E Q may beobtained from the condition [ dq ( τ ) /dτ ] ( q o / ± = 0, notingthat the fluxon is initially located at the bottom of eitherside of the double-well potential so that V Q (( q o / ± ) = 0.We plot the numerically computed shift δ o as a func-tion of ¯ g E in Fig. 10 for ω r = 0 .
50 (dot-dashed line),0.70 (dashed line), and 0.90 (solid line) to illustrate theamount of this shift in the weak-coupling regime. Thecurves indicate that δ o increases with ¯ g E and with ω r ,reflecting potential renormalization.The resonent cavity also modifies the oscillation fre-quency ω e at the potential minima. The modified fre-quency ω e is given by ω e ≈ ω o (cid:26) g E M (cid:20) ω r ω o (1 − Υ) (cid:21)(cid:27) , (52)0where ω o is the frequency in the absence of resonant cav-ity and Υ = 6 q o sinh 2 q o tanh ℓ/ǫ (cosh ℓ −
4) sinh ℓ .We now combine these effects together and estimatethe ground-state energy splitting ∆ cav by using theaction S Qeff [ q ] of Eq. (45) and by using the standardmethod of summing over the ”instanton” trajectories. By following Weiss and coworkers, we compute the one-bounce contribution to the partition function Z fluxon ,assuming that the fluxon is initially pinned at one of thepotential minima. We write the partition function as Z fluxon = ∞ X i =0 Z i (53)where Z i denotes the i -bounce contribution. Here thebounce is an instanton-anti-instanton pair. To estimate∆ cav , we compute both the saddle-point ( Z ) and theone-bounce ( Z ) contribution to Z fluxon by noting that Z may be expressed as Z = Z π (cid:18) ∆ cav θ (cid:19) , (54)where θ = 1 /T . For the contribution Z , we assume thatthe fluxon is initially confined at q = ( q o / − and obtain Z = N ∞ Y n =0 λ on ! − / (55)where the eigenvalues λ on are determined from h − M e ∂ τ + V ′′ Q (cid:16) − q o δ (cid:17)i q on ( τ )+4 π ¯ g E Z θ/ − θ/ K ( τ − τ ′ ) q on ( τ ′ ) = λ on q on ( τ ) . (56)Here ∂ τ = ∂ /∂τ , V ′′ Q ( q ) = ∂ V Q ( q ) /∂q , and the cavitykernel K ( τ − τ ′ ) = ( ω r /
2) exp[ − ω r | τ − τ ′ | ] accounts forthe non-local effect.For the one-bounce contribution Z to Z fluxon , we sep-arate the center coordinate q ( τ ) into two parts as q ( τ ) = ¯ q ( τ ) + ∞ X n =0 c n q n ( τ ) , (57)where ¯ q ( τ ) describes a bounce-like trajectory and the re-maining terms describe the arbitrary paths about thisbounce-like trajectory. This separation of q ( τ ) may beused to write the action S Qeff [ q ] as S Qeff [ q ( τ )] = S cavB, (¯ q ( τ )) + ∞ X n =0 λ n c n . (58)Here S cavB, accounts for the one-bounce-like trajectory inthe resonant cavity. We choose q n ( τ ) of Eq. (57) so that the eigenfunctions of the second variational derivative of S Qeff [ q ] at ¯ q and the eigenvalues λ n are determined from h − M e ∂ τ + V ′′ Q (¯ q ) i q n ( τ )+ 4 π ¯ g E Z θ/ − θ/ K ( τ − τ ′ ) q n ( τ ′ ) = λ n q n ( τ ) . (59)We note that the first two eigenvalues, λ and λ , needto be separated from the rest because λ ≤ λ = 0while the other eigenvalues are positive. The one-bouncecontribution ( Z ) may be expressed as Z = N Z ∞ Y n =0 dc n √ π e − ( S cavB, + P ∞ n =0 λ n c n ) , (60)where N is a normalization constant. With the separa-tion of the first two eigenvalues (i.e., λ ≤ λ = 0)from the others, we write the one-bounce contribution tothe partition function as Z ≈ Z θ π "Z θ dτ e − S cavB, ( τ ) ∞ n =0 λ on Q ∞ n =2 λ n (cid:21) / × "Z θ/ − θ/ dτ (cid:18) d ¯ qdτ (cid:19) / "Z θ/ − θ/ dτ ′ (cid:18) d ¯ qdτ ′ (cid:19) / . (61)We now need to evaluate Z of Eq. (61) to estimate∆ cav . Using Eq. (54), we write the ground-state energysplitting ∆ cav as∆ cav = 2 ω e √ π (cid:16) R cav L cav e − S cavB, (cid:17) / (62)where the dimensionless factors R cav and L cav are R cav = 1 M e ω e (cid:18) Q ∞ n =0 λ on Q ∞ n =2 λ n (cid:19) / (63)and L cav = M e "Z dτ (cid:18) d ¯ qdτ (cid:19) / "Z dτ ′ (cid:18) d ¯ qdτ ′ (cid:19) / , (64)respectively. The exponent S cavB, is given by S cavB, = Z θ/ − θ/ dτ " M e (cid:18) dq ( τ ) dτ (cid:19) + V Q ( q ) . (65)This exponent accounts for the contribution from the twotransversal of the potential barrier. We note that the ex-ponent S cavB, of Eq. (65) does not contain the non-localcontribution, as in Eq. (28), because this contribution isalready included in the calculation of Z (see Eq. (60)).We now compute R cav , L cav and S cavB, , separately, to de-termine the ground-state energy splitting ∆ cav . To focuson the effects due to the junction-cavity interaction, we1 FIG. 11: The numerically computed ratio of the dimensionlessfactor R cav /R is plotted as a function of the junction-cavitycoupling strength ¯ g E for ω r = 0.60 (dot-dashed line), 0.75(dashed line), and 0.90 (solid line). present the details of the calculation for R cav and L cav in Appendix A and B, respectively, and discuss the de-pendence of these factors on the junction-cavity couplingstrength ¯ g E .The dimensionless factor R cav in the weak-couplingregime is given by R cav ∼ = 2 + π ¯ g E ω r M ω o X R ( ω r + ω o ) , (66)where X R = ω r + 15 ω r ω o + 12 ω r ω o − ω o . Equation(66) yields the value R cav = 2 in the absence of resonantcavity (i.e., ¯ g E = 0). In Fig. 11, we plot the numericallycomputed ratio R cav /R as a function of ¯ g E for ω r = 0.60(dot-dashed line), 0.75 (dashed line), and 0.90 (solid line)to illustrate enhancement of R due to resonant cavity.The curves indicate that R cav /R increases from 1 almostlinearly with increasing ¯ g E and ω r .For the dimensionless factor L cav , we evaluate the in-tegral of Eq. (64) by expanding the function Q ( τ ) whichaccounts for the non-local contribution to the bounce-liketrajectory as a power series. (See Appendix B.) In theweak-coupling regime (i.e., ¯ g E ≪ L cav ≈ V M [ A o + ¯ g E ( B + B q o + B q o )] , (67)by retaining the leading order contribution (in ¯ g E ). Here V M = q o √ M V o , A o = 1 − q o (2 ǫb / V o ) − q o (4 ǫb / V o ), B = − ( ǫ + 8 b ω r ) / ǫ , B = [ b ǫ + 2(6 b b − ω r + 2 πd ω r ] / V o , and B = [ b ǫ + 20 b b ω r + ǫ ) +3 πd ω r ] / V o . The frequency independent constants b i are b = (cosh ℓ − ( ℓ/ b = (cosh 2 ℓ −
26 cosh ℓ +33) / (cosh ℓ + 1) , and b = (sinh ℓ tanh ℓ ) / (cosh 2 ℓ − L cav in the resonant cavityis larger than L = V M A o in its absence. However, dueto the functional form of L cav , the enhancement of L cav from L deviates from the linear dependence on ¯ g E at a FIG. 12: The numerically computed ratio of L cav /L is plottedas a function of the junction-cavity coupling strength ¯ g E for ω r = 0 .
60 (dot-dashed line), 0.75 (dashed line), and 0.90 (solidline) to illustrate the enhancement. smaller value than that for R cav . To illustrate this de-viation, we numerically compute L cav and plot the ratio L cav /L in Fig. 12 as a function of ¯ g E for ω r = 0 .
60 (dot-dashed line), 0.75 (dashed line), and 0.90 (solid line). Thecurves show nonlinear enhancement of the dimensionlessfactor L cav for much smaller value of ¯ g E than that for R cav shown in Fig. 11.Finally, we estimate the effects of junction-cavity in-teraction on S cavB, . The action S cavB, of Eq. (65) for thebounce-like trajectory is given by S cavB, = 2 Z ( q o / + ( q o / − dq q M e V Q ( q ) . (68)The integral of Eq. (68) is evaluated in the same way asthat for L cav (see Appendix B). Again, we simplify thecalculation by writing V Q as a power series in q and thenexpand p V Q ( q ) in powers of ¯ g E as p V Q ≈ V M √ M e q o (cid:26) − ǫq V o (cid:18) b + 4 b q + · · · (cid:19) − ¯ g E (cid:20)
18 + ω r q o V o − ǫq V o (cid:18) ¯ b + 23 b q (cid:19)(cid:21) (cid:27) , (69)where ¯ b = b + ( ω r /ǫ ). Using this series expansion for p V Q , we evaluate Eq. (68) and obtain S cavB, to the lead-ing order in ¯ g E as S cavB, ≈ V M [ A o + ¯ g E ( B + ¯ B q o + ¯ B q o )] , (70)where ¯ B = B − ( πd ω r / V o ) and ¯ B = B − (3 πd ω r / V o ). The action S cavB, in the presence of cavityis reduced from that in its absence (i.e., S cavB, < S B, ).To illustrate this suppression of the ratio, we plot thenumerically computed ratio S cavB, /S B, as a function of¯ g E for ω r = 0 .
60 (dot-dashed line), 0.75 (dashed line),and 0.90 (solid line) in Fig. 13. The curves indicate that2
FIG. 13: The numerically computed ratio of the action S cavB, /S B, is plotted as a function of ¯ g E for ω r = 0 .
60 (dot-dashed line), 0.75 (dashed line), and 0.90 (solid line) to illus-trate that the one-bounce-like action is reduced. in the one-bounce contribution to the action decreasesalmost linearly with ¯ g E in the weak-coupling region asindicated by Eq. (70). This reduction reflects that thepotential barrier height is reduced (see Fig. 9) and thepotential minima become closer together (see Fig. 10)with increasing junction-cavity interaction strength.We now combine the effects of resonant cavity on R cav , L cav and S B, together and estimate the enhancement ofthe gound-state energy splitting ∆ cav from ∆. Here, ∆denotes the energy splitting in the absence of resonantcavity given by ∆ = 2 A (cid:18) S o π (cid:19) / e − S o , (71)where A = [ Q ∞ n =0 λ on / Q ∞ n =1 λ n ] / and S o denotes theaction integral. In the weak-coupling regime, the ratio∆ cav / ∆ to the leading order in ¯ g E is given by∆ cav ∆ ≈ g E M (cid:26) ω r ω o (cid:20) − Υ + πX R ω r + ω o ) (cid:21) + M A o ( B + B q o + B q o ) (72) − M V M ( B + ¯ B q o + ¯ B q o ) (cid:27) . The result indicates that ∆ cav is enhanced with increas-ing ¯ g E and ω r . To illustrate this enhancement, we nu-merically compute and plot ∆ cav / ∆ as a function of ¯ g E for ω r = 0 .
60 (dot-dashed line), 0.75 (dashed line) and0.90 (solid line) in Fig. 14. The curves show that ∆ cav / ∆increases roughly linearly with ¯ g E from ¯ g E = 0 to 0.02.However, the deviation from this linear behavior becomesnoticeable for ¯ g E ≥ .
02. Also, ∆ cav / ∆ increases signifi-cantly from 1 at ¯ g E = 0 in the weak-coupling regime. Wenote that the corresponding changes in the ratio R cav /R , L cav /L and S cavB, /S B, over the same range of ¯ g E are FIG. 14: The numerically computed ratio of ∆ cav / ∆ is plot-ted as a function of the junction-cavity coupling strength ¯ g E for ω r = 0 .
60 (dot-dashed line), 0.75 (dashed line) and 0.90(solid line) to illustrate the enhancement in resonant cavity. less significant. For instance, ∆ cav / ∆ for ω r = 0 .
90 in-creases from 1.0 to 1.45 for the increase of ¯ g E from 0.0to 0.015. Over the same range of ¯ g E , R cav /R , L cav /L and S cavB, /S B, change from 1.0 to 1.05, from 1.0 to 1.29,and from 1.0 to 0.96, respectively. The notable increasein ∆ cav / ∆ compared to R cav /R , L cav /L and S cavB, /S B, reflects that ∆ is small. Hence, ∆ cav depends sensitivelyon the variation of the exponent S cavB, . VI. SUMMARY AND CONCLUSION
In summary, we investigated the effects of high- Q c res-onant cavity on MQT of fluxon from metastable state ina single LJJ and in a stack of two coupled LJJs. Also, weestimated the ground-state energy splitting for fluxon ina double-well potential. We find that both the tunnelingrate and the ground-state energy splitting are increasedin the resonant cavity. However, the amount of these in-creases is significantly different. For MQT of the fluxon,the tunneling rate increases due to the renormalization offluxon mass, but negligible in the weak-coupling regime.On the other hand, the increase in the ground-state en-ergy splitting is due to potential renormalization, butthis increase can become significant with increasing ¯ g E as shown in Fig. 14. This energy splitting enhancementis consistent with the result of increase in the energy sep-aration due to the interaction between a two-level systemand a quantized radiation field, described by the Jaynes-Cummings (JC) model. Moreover, the consistency between the result of the present work and that of theJC model indicates that the effective Hamiltonian for theJVQ-cavity system may be similar to the JC model.The effects due to i) interaction between the JVQ anda dissipative environment and ii) the losses resulting froma low-Q cavity are neglected in the present work. Thesedissipative effects are expected to be present in real sys-3tems and may be accounted by using an effective spectraldensity which characterizes the form of dissipation. In-clusion of both the dissipative environment and cavitylosses may reduce the size of increase in the ground-stateenergy splitting and may lead to decrease in the energyspliting when the dissipative effects become strong, as in-dicated by the analysis of dissipative two-state systems. However, these dissipation contributions do not reversethe effects due to the potential renormalization com-pletely in weakly dissipative systems.Enhancement of ground-state energy splitting due tothe junction-cavity interaction may have an importantconsequence for the decoherence time of JVQ in the res-onant cavity. Earlier study of the JVQ decoherencetime by Kim, Dhungana and Park indicates that the in-crease in the decoherence time in noisy environment (i.e., T noiseφ ) is correlated with the increasing ground-state en-ergy splitting ∆. This suggests that, as ∆ may be tunedby adjusting the strength of junction-cavity interaction,the resonant cavity may be used to control the prop-erty of JVQ. For instance, the decoherence time T noiseφ may be increased by increasing the strength of interac-tion between fluxon and cavity EM mode. Also, due tothe similarities between a cavity EM mode and an opticalphonon mode, the interaction between fluxon and opticalphonons in the LJJ may affect the decoherence time.Another important property of JVQs is entanglementbetween the qubits. As our result suggests that the deco-herence time for JVQ can be increased by increasing thestrength of junction-cavity interaction, the resonant cav-ity may also be useful for tuning the level of entanglementbetween the JVQs. Our study suggests that the presentapproach for JVQs is similar to the microwave cavity ap-proach used for the other superconductor qubits. Theeffective Hamiltonian for the multiple JVQs in a resonantcavity may resemble the Tavis-Cummings model whichis the extension of the JC model to the case of multiplequbits. This similarity may be exploited by using theresonant cavity to control the level of concurrence forJVQs since the junction-cavity interaction may also pro-mote entanglement. Hence, the effects of resonant cavityon entanglement between the interacting JVQs would bean interesting area for further study.The authors would like to thank W. Schwalm and K.-S. Park for helpful discussions and I. D. O’Bryant forassisting with part of the numerical calculation. APPENDIX A:
CALCULATION OF R cav For convenience, the dimensionless factor R cav of Eq.(63) is estimated in the continuum limit. In this limit,we may write R cav as R cav = exp (cid:26) π Z ∞ M e ω e dλλ [ δ + ( λ ) + δ − ( λ )] (cid:27) , (73) where δ ± ( λ ) denotes the phase shift due to the scatteringpotential U . This phase shift may be expressed as δ ± ( λ ) = cot − (cid:20) U − − g ′ λ (0) ∓ g ′ λ ( τ s ) g ′′ λ (0) ± g ′′ λ ( τ s ) (cid:21) , (74)where τ s = − θ/
2, and g ′ λ ( τ ) and g ′′ λ ( τ ) denote thereal and imaginary part of the Green’s function (i.e., g λ ( τ ) = g ′ λ ( τ ) + ig ′′ λ ( τ )). The phase shift δ ± ( λ ) due tothe scattering from the net potential difference of V ′′ Q (¯ q ) − V ′′ Q (( q o / − ) = − U h δ (cid:16) τ + τ s (cid:17) + δ (cid:16) τ − τ s (cid:17)i (75)consists of two Dirac δ -functions at τ = ± τ s /
2. Thestrength of the scattering potential U is given by U − = g (0) − g ( τ s ) , (76)where g ( τ ) is the Green’s function for the eigenvalue λ = 0. The Green’s function g λ ( τ ) is written as g λ ( τ ) = Z ∞−∞ dω π e iωτ M e [ ω + ζ ( ω ) + ω e ] − λ − iδ . (77)Here the effects of the resonant cavity are accounted forvia M e , ω e and ζ ( ω ). The function ζ ( ω ), obtained fromthe cavity kernel K ( τ ) of Eq. (26), ζ ( ω ) = 4 π ¯ g E M e ω r ω + ω r , (78)reflects that the resonant cavity supports a single-modewith frequency ω r . Using the function ζ ( ω ), we write thereal part of the Green’s function as g ′ λ ( τ ) = g ′ λ, + ( τ ) + g ′ λ, − ( τ ), where g ′ λ, ± ( τ ) = − M e ω λ, ± ± ω r + ω ,λ ω ,λ ! sin ω λ, ± τ , (79) ω λ, ± = ( ω ,λ ± ω ,λ ) / , ω ,λ = [( λ/M e ) − ω e − ω r ] / ω ,λ = { [( λ/M e ) − ω e + ω r ] − (16 πg E /M e ) ω r } / .On the other hand, we write the imaginary part of theGreen’s function as g ′′ λ ( τ ) = g ′′ λ, + ( τ ) + g ′′ λ, − ( τ ), where g ′′ λ, ± ( τ ) = 14 M e ω λ, ± ± Ω + ω ,λ ω ,λ ! cos ω λ, ± τ . (80)We note that the phase shift δ ± ( λ ) has both slowly vary-ing and rapidly oscillating contributions. For an ex-tended bounce (i.e., ω e τ s ≫ R cav of Eq. (73) may be simplified by us-ing the substitution λ = M e ω e (1 + p ), where p is adimensionless momentum variable. With this change ofvariable, we write R cav as R cav = exp (cid:26) π Z ∞ p dp p [ δ + ( p ) + δ − ( p )] (cid:27) . (81)4The factor R cav of Eq. (81) may be further simplifiedby neglecting the rapidly oscillating contributions in thephase shift δ ± ( λ ) of Eq. (74). Neglecting these oscilla-tory contributions, we approximate δ ± ( p ) to a simplerform δ ( p ) and write the factor R cav as R cav = exp (cid:26) π Z ∞ p dp p δ ( p ) (cid:27) . (82)The simplified phase shift δ ( p ) is given by δ ( p ) = cot − " U − − g ′ p (0) g ′′ p (0) , (83)where the scattering potential strength U is given by U − = 14 M e − W q ω , − ω , + 1 + W q | ω , + ω , | , (84)and W = ( ω r + ω , ) /ω , . We note that ω , and ω , are obtained from ω ,λ and ω ,λ of Eq. (79) for the eigen-value λ = 0, respectively. The real and imaginary partof the Green’s function are given, respectively, by g ′ p (0) = 14 M e − W p q ω ,p − ω ,p (85)and g ′′ p (0) = 14 M e W p q ω ,p + ω ,p , (86)where W p = ( ω r + ω ,p ) /ω ,p . We note that ω ,p and ω ,p are obtained from ω ,λ and ω ,λ of Eq. (79), respectively,by setting λ = M e ω e (1 + p ).We now compute R cav to the leading order in ¯ g E toaccount for the effects of resonant cavity in the weakcoupling regime (i.e., ¯ g E ≪ M e = M − g E and express the oscillation frequency ω e as ω e ∼ = ω o (cid:26) g E M (cid:20) ω r ω o (1 − Υ) (cid:21)(cid:27) . (87)Also we rewrite the strength of the potential U as U − ∼ = 12 M ω o − π ¯ g E ω r M ( ω r − ω o ) (cid:18) − X u πω r ω o (cid:19) (88)where X u = M ω o ( ω o − ω r ) − ω r Υ( ω o − ω r ) +2 πω r ( ω r − ω o )]. By combining these expressions to-gether, we rewrite the real and imaginary part of theGreen’s function of Eqs. (85) and (86), respectively, as g ′ p (0) ∼ = − ¯ g E πω r M ( p ω o + ω r ) (89) and g ′′ p (0) = 12 M pω o (cid:20) g E X g M p ω o ( p ω o + ω r ) (cid:21) , (90)where X g = 4 ω r [ πω r (3 p ω o + ω r ) − p ( p ω o + ω r ) ] + M p ω o ( p ω o + ω r ) . Now, we use Eqs. (88) - (90) andrewrite the simplified phase shift δ ( p ) of Eq. (83) as δ ( p ) ∼ = cot − p − ¯ g E πω r X p M p (1 + p ) X ω (91)where X p = − ( ω o ω r + 2 ω o ω r + ω r ) + p (2 ω o + ω o ω r − ω o ω r − ω o ω r − ω o ω r − ω r ) − p (16 ω o ω r + 8 ω o ω r ) − p (8 ω o + 4 ω o ω r ) and X ω = ω o ( ω o + ω r ) ( p ω o + ω r ) .Finally, we substitute δ ( p ) of Eq. (91) into R cav of Eq.(82) and evaluate the integral to obtain R cav ∼ = 2 + π ¯ g E ω r M ω o X R ( ω r + ω o ) , (92)where X R = 5 ω r +15 ω r ω o +12 ω r ω o − ω o . Equation (92)yields R cav = 2 in the absence of the resonant cavity (i.e.,¯ g E = 0) as expected. APPENDIX B:
CALCULATION OF L cav The factor L cav of Eq. (64) may be estimated by deter-mining the bounce-like trajectories q ( τ ). The trajectoriesobey the equation of motion given by − M e d q ( τ ) dτ + dV Q ( q ) dq + 4 π ¯ g E Z ∞−∞ dτ ′ K ( τ − τ ′ ) q ( τ ) = 0 . (93)We rewrite the equation of motion in a convenient formby integrating Eq. (93) by parts and obtain − M e (cid:18) dqdτ (cid:19) + V Q ( q )+4 π ¯ g E Z ∞−∞ dτ ′ K ( τ − τ ′ ) q ( τ ) q ( τ ′ ) = 0 . (94)Using this result, we write the factor L cav as L cav ≈ M e Z dτ (cid:18) dqdτ (cid:19) = Z ( q o / + ( q o / − dq q V Q ( q ) + 2 π ¯ g E ω r qQ ( τ ) (95)where q = q ( τ ) and Q ( τ ) = Z ∞−∞ dτ ′ e − ω r | τ − τ ′ | q ( τ ′ ) . (96)5 q( t ) t q o d + d q o t Q( t ) FIG. 15: Similarity between the function Q ( τ ) of Eq. (96)and the instanton solution q ( τ ) representing the trajectoryof the fluxon from one potential minimum to the other viatunneling is illustrated schematically. Here, the non-local contribution due to resonant cavityis accounted for by Q ( τ ). As discussed in Appendix C,the function Q ( τ ) is similar to q ( τ ). By exploiting thissimilarity, we expand Q ( τ ) in a power series as Q ( τ ) = ∞ X n =0 d n +1 q n +1 ( τ ) , (97)where d n +1 is the expansion coefficients (see AppendixC). The power series expansion for Q ( τ ) allows us toevaluate the factor L cav straightforwardly. By using thispower series expansion, we evaluate the integral of Eq.(95) in the weak-coupling regime (i.e., ¯ g E ≪
1) and ob-tain the factor L cav to the leading order in ¯ g E as L cav ≈ V M [ A o + ¯ g E ( B + B q o + B q o )] , (98)where V M = q o √ M V o , A o = 1 − q o (2 ǫb / V o ) − q o (4 ǫb / V o ), B = − ( ǫ + 8 b ω r ) / ǫ , B = [ b ǫ +2(6 b b − ω r +2 πd ω r ] / V o , and B = [ b ǫ +20 b b ω r + ǫ ) + 3 πd ω r ] / V o . The frequency independent con-stants b i are given by b = (cosh ℓ − ( ℓ/ b = (cosh 2 ℓ −
26 cosh ℓ + 33) / (cosh ℓ + 1) , and b =(sinh ℓ tanh ℓ ) / (cosh 2 ℓ − APPENDIX C:
POWER SERIES EXPANSION OF Q ( τ )The numerically computed function Q ( τ ) of Eq. (96)indicates that Q ( τ ) is similar to the functional form of thebounce-like trajectory q ( τ ). This similarity suggests that Q ( τ ) is a scaled function of q ( τ ) as shown schematicallyin Fig. 15. In this case, we may express the function Q ( τ ) as a power series in q ( τ ) as Q ( τ ) = ∞ X n =0 d n +1 q n +1 ( τ ) , (99)where d n +1 denotes the coefficient for this power seriesexpansion. We compute the coefficients d n +1 by startingwith a series expansion of q ( τ ) in τ as q ( τ ) = ∞ X n =0 a n +1 τ n +1 , (100)noting that the instanton solution q ( τ ) is an odd functionof τ . Here, the coefficient d n +1 is obtained by followingthe five steps as discussed below. First, we write thebounce-like trajectory q in the absence of resonant cavity.This trajectory q may be expressed as q = − b τ + b tanh − ( b tanh q ) , (101)where the constants b = 2 p ǫ/M coth ℓ , b = (cosh 2 q o +cosh ℓ ) / sinh 2 q o , and b = coth q o depend on the param-eters ℓ and ǫ . Second, we expand the right hand side ofEq. (101) as a power series in q as q = − b τ + b b q (cid:18) − − b q + 2 − b + 3 b q + · · · (cid:19) . (102)Here, we find the coefficients a n +1 by substituting theseries expansion for q ( τ ) of Eq. (100) into Eq. (102).The first three coefficients are given by a = b b b − ,a = b b b (1 − b )3( b b − ,a = b b b (1 − b )[ b b (3 − b ) + (2 − b )]15( b b − . Third, we use Eqs. (26) and (100) to evaluate Q ( τ ) ofEq. (96) explicitly as Q ( τ ) = ∞ X n =0 a n +1 Z ∞ dτ ′ e − ω r | τ − τ ′ | τ ′ n +1 . (103)Fourth, we evaluate the integrals of Eq. (103) and write Q ( τ ) in a power series in τ as Q ( τ ) ≈ ω r (cid:20) τ (cid:18) a + 6 a ω r + 120 a ω r (cid:19) + τ (cid:18) a + 20 a ω r (cid:19) + τ a + · · · (cid:21) . (104)Finally, we use the power series expansion for q ( τ ) of Eq.(100) and rewrite Q ( τ ) of Eq. (99) as Q ( τ ) = τ ( d a ) + τ ( d a + d a )+ τ ( d a + 3 d a a + d a ) + · · · (105)6This series expansion allows us to obtain the expansioncoefficients d n +1 by comparing the power series Q ( τ ) ofEqs. (104) and (105). The first three expansion coeffi-cients, d n +1 , are the following: d = 2 ω r (cid:18) ω r a a + · · · (cid:19) ,d = 4 ω r (cid:20)(cid:18) a a − a a (cid:19) + 60 ω r (cid:18) a a − a a a (cid:19) + · · · (cid:21) , d = 12 ω r (cid:20)(cid:18) a a − a a a + 3 a a (cid:19) + · · · (cid:21) . In Sec. 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