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The effect of environmental coupling on tunneling of quasiparticles in Josephson junctions

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2013 Supercond. Sci. Technol. 26 125013

(http://iopscience.iop.org/0953-2048/26/12/125013)

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Supercond. Sci. Technol. 26 (2013) 125013 (7pp) doi:10.1088/0953-2048/26/12/125013

The effect of environmental coupling ontunneling of quasiparticles in Josephsonjunctions

Mohammad H Ansari1, Frank K Wilhelm1,2, Urbasi Sinha1,3 andAninda Sinha4

1 Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo,200 University Avenue West, Waterloo, ON, N2L 3G1, Canada2 Theoretical Physics, Saarland University, D-66123 Saarbrucken, Germany3 Raman Research Institute, Sadashivanagar, Bangalore 560080, India4 Centre for High Energy Physics, Indian Institute of Science, Bangalore, India

E-mail: fwm@lusi.uni-sb.de

Received 13 June 2013, in final form 21 August 2013Published 4 November 2013Online at stacks.iop.org/SUST/26/125013

AbstractWe study quasiparticle tunneling in Josephson tunnel junctions embedded in anelectromagnetic environment. We identify tunneling processes that transfer electrical chargeand couple to the environment in a way similar to that of normal electrons, and processes thatmix electrons and holes and are thus creating charge superpositions. The latter are sensitive tothe phase difference between the superconductors and are thus limited by phase diffusion evenat zero temperature. We show that the environmental coupling is suppressed in manyenvironments, thus leading to lower quasiparticle decay rates and better superconductor qubitcoherence than previously expected. Our approach is nonperturbative in the environmentalcoupling strength.

(Some figures may appear in colour only in the online journal)

1. Introduction

The physics of micro- and nanoscale Josephson junctions is aparadigmatic application of macroscopic dissipative quantummechanics of open systems [1–3]. This, on the one hand,makes them ideal test-beds for that theory, with largelytunable parameters [4, 5]. On the other hand, Josephsonjunctions have an ample range of applications in sensing [6],amplification [7], metrology [8] and quantum informationprocessing [9–12]. For these applications, it is imperativeto thoroughly understand the dissipative quantum physicsof Josephson junctions in order to achieve optimal deviceperformance.

For a long time, work on this topic has focusedon the transport properties of Josephson junctions, theircurrent–voltage and noise characteristics [13]. The ‘P(E)’-theory of treating environmental fluctuations [3, 14] isestablished as a powerful tool in the case of small junctions

when (Josephson or quasiparticle) tunneling can be treatedperturbatively.

For applications in quantum computing, another questionin this framework has emerged [15, 16]: rather than computingthe current–voltage characteristic, the total quasiparticletransition rate is the quantity of interest. This is importantbecause quasiparticle transitions in any direction are highlydetrimental to qubit coherence [17]. In [17, 15, 16, 18, 19]this problem is studied in an approach that is perturbativein the coupling to the environment as measured by theenvironmental impedance in units of the quantum resistanceRQ = h2/e' 25.8 k�. In this paper we are going to generalizethat study to arbitrary system–environment interactionstrength. Some of the processes mix the electron-like andhole-like branches of the quasiparticle spectrum and dependon the phase difference between the superconductors. Theseare generally phase-dependent but highly sensitive to the

10953-2048/13/125013+07$33.00 c© 2013 IOP Publishing Ltd Printed in the UK & the USA

Supercond. Sci. Technol. 26 (2013) 125013 M H Ansari et al

environment. These observations lead to slower relaxationthan expected in previous work.

The paper is organized as follows. We introducethe mathematical formulation of the tunneling model insection 2.1. We will compute the tunneling rate in lowest orderof the tunnel coupling but to all orders in the coupling of thequasiparticles to the environment in section 2.2, which canbe a general linear impedance. We specialize to it being anundamped harmonic oscillator in section 3.1, which directlyrelates to [15, 16, 18, 19] but shows that the phase-dependentcomponent is usually reduced due to dressing by zero-pointfluctuations. We discuss an overdamped environment that mayoccur in other applications in section 3.2.

2. Mathematical formulation

2.1. Model

We start from the Hamiltonian

H = HBCS,1 + HBCS,2 + HT + Henv + Henv−c. (1)

Here, we have the BCS Hamiltonians [20, 21] in meanfield form describing the two electrodes

HBCS,i =∑k,σ

(ξk − µi) c†k,σ,ick,σ,i

+ 1i

∑k

ck,↑,ick,↓,i +1∗i

∑i

c†k,↑,ic

†k,↓,i (2)

where ξk =h2k2

2m is the kinetic energy of the electron and theµiare the chemical potentials in superconductors i = 1, 2, whichare µ1 = EF and µ2 = EF+eV where V is the applied voltage.We keep1i = |1| eiφi/2 complex in order to allow for a phasedifference. The tunneling Hamiltonian is

HT =∑

kl

(Tklc

†k,σ,1cl,σ,2 + T∗klc

†k,σ,2cl,σ,1

). (3)

Before discussing the electromagnetic environment, wediagonalize the BCS Hamiltonians, equation (2), through theBogoliubov transformation

ck↑ = u∗k γk↑ + vkγ†−k↓,

c†−k↓ = −v∗k γk↑ + ukγ

†−k↓.

Here, the BCS coherence factors are defined

uk,i =

√12

(1+

ξk,i − µi

Ek,i

)eiφi/2 (4)

vk,i =

√12

(1−

ξk,i − µi

Ek,i

)e−iφi/2 (5)

where we introduced the quasiparticle energy Ek,i =√(ξk,i − µi)2 + |1|2. This allows us to rewrite the tunneling

Hamiltonian, equation (3), in the following form:

HT =∑

k,σ=↑,↓

(Tkl|uk1ul2|eiφ/2

− T∗−l−k|vk1vl2|e−iφ/2

)×

(γ

†kσ1γkσ2 + γ

†−kσ2γ−lσ1

)+ HT2 (6)

where we introduced the phase difference φ = φ1 − φ2.The term HT2 contains operators that change the number ofquasiparticles by two, i.e., contain terms of the structure γ γand γ †γ † hence changing the total number of quasiparticlesin the setup, which do not contribute to the quasiparticlerate—these terms contribute to the Josephson and Andreevprocesses. We now want to evaluate the Fermi golden rule ratefor a transition that transfers a quasiparticle from electrode 1to electrode 2. The relevant matrix element is

〈(N1 − 1)kσ , (N2 + 1)lσ |HT|N1kσ ,N2lσ 〉 = Tkleiφ/2|uk1ul2|

− T∗−k−le

−iφ/2|vk1vl2| .

We now use that for a nonmagnetic barrier, Tkl = T∗−k−l

can be chosen real. In the Fermi golden rule transition rate, weneed the absolute square

|〈(N1 − 1)kσ , (N2 + 1)lσ |HT|N1kσ ,N2lσ 〉|2

= T2kl(|u|

2+ |v|2 − 2|uv| cosφ)

where u(E,E′) = |u1(E)u2(E′)| and v = |v1(E)v2(E′)| withu1/2(E) and v1/2(E) given by the BCS coherence formulas,equations (4) and (5). Thus, the transition probability containsan interference term that is sensitive to the phase across thejunction. This term describes a process that, even though ittransfers a genuine quasiparticle from electrode 1 to electrode2, actually consists of a superposition of electron and holetransfer, i.e., it is not diagonal in charge space, see thediagrammatic representation in figure 2. We will see later onhow this is important in the sensitivity to the environment.This term bears analogy to the famous cosφ term in theJosephson effect [22–25].

2.2. Tunneling rate

In order to capture the influence of the environment, weapply the ideas of P(E)-theory [3]. There, the environmentalHamiltonian is described by an oscillator bath Hbath =∑

nωna†nan and couples the oscillators linearly to our

quasiparticle system Henv−c = N1eδV , where we assumethat only the first reservoir fluctuates—this corresponds toa specific choice of gauge. The total number and voltageoperators are

N1 =∑k,σ

c†kσ1ckσ1, δV =

∑λi

(ai + a†

i

). (7)

We can now work out the total tunneling rate by summingover momentum states at constant energy from the initial state|i〉 = |R〉|E〉 to the final state |f 〉 = |R′〉|E′〉 and consideringthe matrix element of the environmental Hamiltonian〈E|Tkqγ

1†qσ γ

2kσ |E

′〉 thermal distribution of quasiparticles. The

transition rate can be written formally as

E01(V) =1

e2RN

∫∞

−∞

dE dE′ D1(E)D2(E′)

× f1(E)[1− f2(E′)]Ptot(E,E′) (8)

where Di is the reduced density of states and fi is thecorresponding distribution function on the lead i. We consider

2

Supercond. Sci. Technol. 26 (2013) 125013 M H Ansari et al

here that the nonequilibrium quasiparticle tunneling keeps thesystem in thermal and chemical-potential equilibrium. Theinfluence of the external circuitry on quasiparticle tunneling isencoded in the dressed vertex Ptot(ε), the probability densityfor exchanging energy ε with the environment. From theFermi golden rule the tunneling transition rate can be writtenin more detail as

E01(V) =1

e2RN

∫dE dE′D1(E)D2(E

′)f1(E)(1− f2(E′))

×

∑R,R′|〈R′|HT|R〉|

2Pβ(R)

× δ(E + eV − E′ + ER − ER′),

where Pβ(R) = 〈R|ρβ |R〉 for ρβ = e−βHenv/∑

e−βHenv andthe arrow on 0 indicates the direction of tunneling from lead1 to 2. RN is the tunnel resistance of the junction in the normalstate5. Considering that the phase has fluctuations around theclassical value φ = ϕ + δφ and substituting all of these onecan get the transition rate

E01(V) =4

e2RN

∫dE dE′

∫dt

2π he

ih (E+eV−E′)

× D1(E)D2(E′)f1(E)(1− f2(E

′))

×

((u2+ v2

)e〈δφ(t)δφ(0)〉

4 − 2uv cosϕe−〈δφ(t)δφ(0)〉

4

)× e−

〈δφ(0)δφ(0)〉4 (9)

where u(E,E′) = |u1(E)u2(E′)| and v = |v1(E)v2(E′)| withu1/2(E) and v1/2(E) given by the BCS formulas, equations (4)and (5).

Note that the phase fluctuation defined here by theeffective voltage across the junction

δφ(t)

2=

e

h

∫ t

0dt′ δV(t′) (10)

is the conjugate to the Cooper pair charge. The conventionused here is consistent with assuming the resistance quantumto be RK/4 = h/4e2.

The vertex probability is defined as the probabilitydensity for exchanging energy ε between the system and theenvironment and is defined formally as

Ptot(E,E′) =∫∞

−∞

dt

2π heJ(t)+ i

h (E+eV−E′)t. (11)

By comparing equations (8) and (9) one can correctlyexpect the definition eJ(t)

= (u2+ v2)eJs(t) − 2uv cosϕeJa(t),

where eJs(t) = 〈eiδφ(t)/2e−iδφ(0)/2〉 = e〈(δφ(t)−δφ(0))δφ(0)〉/4 and

eJa(t) = 〈eiδφ(t)/2eiδφ(0)/2〉 = e−〈(δφ(t)+δφ(0))δφ(0)〉/4. From

these equalities one can simplify the definitions to

5 By expanding the Dirac delta as a temporal integral δ(x) =

(2π h)−1 ∫∞−∞

dt e−ih xt and writing the operator 〈R′|e−ER′ tHTeERt

|R〉 =

〈R′|HT(t)|R〉 the time evolution of the Hamiltonian makes the rate becomeproportional to 〈〈HT(t)HT(0)〉〉, where 〈〈· · ·〉〉 =

∑R,R′ 〈R

′| · · · ρβ |R〉.

Substituting the tunneling Hamiltonian into this formula one gets〈〈HT(t)HT(0)〉〉 = 〈〈(ueiφ(t)/2

− ve−iφ(t)/2)(ue−iφ(0)/2− veiφ(0)/2)〉〉.

Reordering using properties of Gaussian states leads to the followingexpressions: 〈e±iφ(t)e∓iφ(0)

〉 ∼ e〈(φ(t)−φ(0))φ(0)〉 and 〈e±iφ(t)e±iφ(0)〉 ∼

e−〈(φ(t)+φ(0))φ(0)〉.

Js(t) = 〈(δφ(t) − δφ(0))δφ(0)〉/4 and Ja(t) = −〈(δφ(t) +δφ(0))δφ(0)〉/4. In this notation, one needs to pay attention tothe observation that whereas the Js/a(t) are purely propertiesof the environment, the combined quantity exp(−J(t))inevitably contains coherence factors of the superconductor.Concurrently, we decompose Ptot into two contributions

Ps/a(E) =∫

dt

2π heiEteJs/a(t). (12)

Note that in the definitions we considered that the meanvalue of the phase is set by the external bias and theeffect of the environment is summarized into the presenceof phase fluctuations about this mean value. The symmetriccombination Js(t) captures all-electron and all-hole processesand the corresponding vertex Ps(E) coincides with theall-electron P(E) known from [3].

However, Ja(t) = 〈(δφ(t) + δφ(0))δφ(0)〉/4 = Js(t) +〈δφ2(0)〉/2 captures processes that mix electrons and holes.Although Js(0) = 0 the asymmetric correlation functiondoes not vanish at t = 0 as it becomes Ja(0) = 〈δφ2(0)〉/2which captures zero-point fluctuation. Ps(E) is a normalizedprobability as

∫Ps(E) dE = exp(−Js(0)) = 1; however,

Pa(E) is not normalized,∫

Pa(E) dE = exp(−Ja(0)) =exp〈δφ2(0)/2〉, as it describes electron–hole coherence forwhich no conservation law should be expected. We canunderstand this as follows. Even though electron–hole mixingprocesses are diagonal in the Fock space of quasiparticles,they are off-diagonal in charge space. The electromagneticnoise couples to these electrical charges. The environmentalmodes dress the charge in an attempt to measure thecharge and localize it. This has an analogy: creation ofa superposition of charge states occurs, following thecharge–flux uncertainty relation, whenever the phase acrossa Josephson junction is becoming localized, i.e., when aJosephson junction is behaving classically. Thus, on the onehand, the scaling factor exp〈δφ2(0)/2〉 directly measures thedegree of charge localization. On the other hand, researchaiming at maximum supercurrent (hence maximally localizedphase and maximally extended charge) in small Josephsonjunctions in a dissipative environment finds exp(−〈φ(0)2〉/2)to be the relevant reduction factor [26, 27].

In order to compute Js/a we introduce an oscillator bathmodel and match its properties to the fluctuation-dissipationtheorem as it describes Johnson-Nyquist noise. Applying thestandard procedures used in P(E)-theory, we find

S(t) = 〈δφ(t)δφ(0)〉 = 2∫∞

−∞

dωω

Re Zeff(ω)

RK

e−iωt

1− e−βhω.

(13)

Notably, this integral seems to diverge at its infrared end,where it formally appears to be'

∫ dωω

Zeff(0) at T = 0 and∝T∫ dωω2 Re Zeff(0) at T > 0. In the combination 4Js(t) = S(t)−

S(0) this divergence is removed, but the same argument doesnot apply to 4Ja(t) = S(t) + S(0). Thus, Zeff(0) 6= 0 enforcesPa = 0, meaning that all electron–hole mixed processes woulddisappear. We will see later that this does not occur in standardphysical environments.

3

Supercond. Sci. Technol. 26 (2013) 125013 M H Ansari et al

The final expression for the probability density P(E,E′)is thus

Ptot(E,E′) =∫

dt

2π he−i(E+eV−E′)te−

S(0)4

×

[(u2+ v2)e

S(t)4 − 2uv cosϕe−

S(t)4

]. (14)

3. Models for the environment

In order to evaluate the typical Ps/a(E) and the re-sulting tunneling rates, we need to project likely mod-els of the electromagnetic environment providing theimpedance Zeff. Following the resistively and capacitativelyshunted junction (RCSJ) model and its microscopic analogs[28, 29, 2, 1], we consider the quasiparticle channel to beput in parallel to the supercurrent, the Josephson channel, thejunction capacitance, and an external impedance Z(ω), seefigure 1. We model the linear impedance of the Josephsonchannel by its Josephson inductance LJ = 80/(2π Ic). Thetotal effective impedance of this parallel setup leads to

Re Zeff =ω2 Re(Z)

C2|Z|21

(ω2 − ω2p)

2 + ωω3P

Im Z|Z|2+

ω2

C2|Z|2

(15)

where ωp = (CLJ)−1/2 is the junction’s plasma frequency.

In the case of a simple resistor or lossless transmission line,Z(ω) = R and we find

Re[Zeff] =ω2

RC2

1

(ω2 − ω2p)

2 + ω2/(RC)2. (16)

Here, we can observe that at low frequencies, Re Zeff ∝ ω2.

Physically, the reason for this is that the noise from theresistor gets shunted through the Josephson junction at lowfrequencies and does not affect the quasiparticle channel.Thus, although Pa will be normalized to a value smaller thanunity, the scenario of it scaling all the way to zero does notoccur. This is consistent with the fact that phase qubits doshow a zero-voltage current. The same general conclusionholds for other physical environments tested.

3.1. Infinite-quality environmental mode

Given the high quality of qubit junctions, it is appropriate tostart from the limit of an infinite-quality factor, R→∞. Inthat case, the effective impedance reduces to

Zeff =π

2C

[δ(ω − ωp

)+ δ

(ω + ωp

)]. (17)

We introduce the dimensionless parameter waveimpedance ρc = 4πZ0/RK, where Z0 =

√LJ/C, which

emerges from equation (17) and ωp = 1/√

LJC and LJ =

80/2π Ic. The index c indicates that in this definition theCooper pair quantum of resistance RK/4 was considered,differently from [3]. The Cooper pair vacuum fluctuation iscontrolled by its corresponding wave impedance S(0) = ρc.The total environmental transition probability is defined as

P(E) =∞∑

k=−∞

pk(ρc, ωP,T)δ(ω − kωp), (18)

Figure 1. Quasiparticle processes labeled by their dispersionconnected to an external circuit. By linearization, the Josephsoncircuit is replaced by an effective impedance.

which has the form of a series of sidebands, correspondingto the emission/absorption of k photons to/from the plasmamode. The weight in the low temperature limit of kBT � hωpis Poissonian, pk(ρ, ωp) = e−ρc/4(ρc/4)k/k!. This coincideswith that of [16] in the limit of ρc→ 0. The total quasiparticletunneling rate hence reads

E01 =

∞∑n=−∞

01n

=4

RTe2

∞∑n=−∞

∫∞

1

dE D1(E)D2(E + nhω)f1(E)

× (1− f2(E + nhω))[(u2(E,E + nhω)

+ v2(E,E + nhω))eS(t)

4 − 2u(E,E + nhω)

× v(E,E + nhω) cosϕe−S(t)

4

]e−

ρc4 pn(ρ, ω,T).

For large capacitance junctions one can Taylor ex-pand in small phase fluctuations (u2

+ v2) exp[S(t)/4] −2uv cosϕ exp[−S(t)/4] ∼ (u− v)2 + (u+ v)2S(t)/4. Consid-ering that the phase fluctuation is small and stable, i.e. S(t) ≈S(0) = ρc, the final result is consistent with the results takenfrom [15, 24] except that ours shows that there is a prefactorexp(−〈δφ(0)2〉/4) in the rate originating from zero-pointfluctuations that can reduce the quasiparticle environmentallymediated transition rate.

In order to finally compute the rate, we use equations (4)and (5). One can show that

(u1u2)2+ (v1v2)

2=

12

(1+

ξ1ξ2

E1E2

),

u1u2v1v2 =1112

4E1E2.

(19)

Substituting the density of states D(E) = E/ξ =E/√

E2 −12 and the coherence factors in the transition rateand ignoring the pure thermal transition we arrive at the

4

Supercond. Sci. Technol. 26 (2013) 125013 M H Ansari et al

Figure 2. Diagrammatic representation of the different contributions to the quasiparticle tunneling matrix elements. Left: processes closedin quasiparticle and charge space; right: a quasiparticle process that creates charge superpositions. The typical state |n,m〉 represents nparticles in lead 1 and m particles in lead 2.

complete formula for nonequilibrium quasiparticle tunneling

E01 =

∞∑n=−∞

e−ρc2

(ρc4

)nn!

×

(0bare

2+

2

e2RN

∫∞

1

dE f1(E)(1− f2(E + nhω))

×E (E + nhω) eS(t)/4

−1112 cosϕe−S(t)/4√(E2 −12

1

) ((E + nhω)2 −12

2

))

(20)

where 0bare = (4/RNe2)∫∞

1dE f1(E)(1 − f2(E)) and cor-

responds to the normal electron tunneling rate under abias voltage, and the second term in equation (20) is adressed term rooted from the quasiparticle tunneling. Wesee several nonperturbative features in this expression. Oneis, of course, the occurrence of higher-order sidebandscorresponding to exchange of multiple photons with theenvironment. The other one is that, as a consequence ofnormalization of the total probability, even the single-photonpeak obtains a nonpeturbative weight factor. In the limit oflarge superconducting gap 1 � hω and a large capacitancejunction, one particle exchange at low-lying quasiparticlelevels indicates that the total one particle rate becomes

01 = e−ρc2ρc

40bare +

2

e2RNe−

ρc2

(ρc

4

)2

×

∫∞

1

dE f1(E)(1− f2(E + hω))

×E (E + hω)+1112√(

E2 −121

) ((E + hω)2 −12

2

) , (21)

where the second term in the parenthesis of equation (21) is infact the dressed tunneling rate 0dr,1. Thus, we directly see thereduction of the phase-sensitive tunneling term by zero-pointfluctuations described above made quantitative.

This rate depends on the details of the energy distributionfunction. One of us [30] derived the explicit temperaturedependence of this rate out of equilibrium. For a junctionwith small macroscopic phase in thermal equilibrium atlow temperatures T � 1, by substituting the Fermi–Diracdistribution function, the rate is

01 =1

2e2RNρc exp

(Eif

2kBT−

1

kBT−ρc

2

)K0

(Eif

2kBT

)(22)

where Eif is the parity transition energy in the qubit fromodd to even states and K0 is a Bessel function. This resultis different from that of [24, see equation (35)] by the

dressing factor exp(−√

Ec/2EJ). More general formulations

for arbitrary phase ϕ are worked out in [30].For phase qubits, whose impedance is engineered to be

Z0 ' 50 �, this correction does not qualitatively changethe physics; however, it improves the quality of fits tothe data. In other types of qubits with higher impedancethese corrections will be crucial. A qubit relaxation/excitationexperiment involving nonequilibrium quasiparticles [15] willattempt to relax the qubit energy splitting which is 'ωp intoquasiparticles; hence, only the main band is relevant. For aquantitative estimate we can identify

ρc =

√2Ec

EJ(23)

where the charging energy is defined for electrons, i.e. Ec =

e2/2C.Note that the ρ2

c in the second term of equation (21)provides the coefficient 2Ec/EJ for the 0dr,1, which makesthis term equivalent to equation (2) in [15]. For a transmonEJ/Ec ≈ 30 and for a flux qubit ≈50, making this correctionlarge enough to be visible in a reduction of the quasiparticlerate. More explicitly, for a transmon we get a difference ofaround 7% from the perturbative approximation (1−e−ρc/2 ≈

ρc/2) and for a flux qubit we get a deviation of around5%. For a traditional charge qubit, ρc is large, leading to adifferent regime where linearization of phase fluctuations isimpossible. From [31], EJ/Ec ≈ 0.35, and with this caveatwe can estimate a large deviation of around 40% from theperturbative approximation.

3.2. Overdamped environmental mode

We now study the opposite case, an environmental modethat is overdamped by an external impedance. Overdampingmeans that the width of the resonance in Zeff, equation (16),is larger than its frequency, i.e. ωpRC � 1 or equivalentlyR � Z0. In that case, to lowest order in the small parameterr = R

Z0, the poles of Zeff, equation (16), are at ω = ±iγ and at

ω = ±iγ r2, where γ = 1/RC. Note that unlike other work foroverdamped oscillators [32] we keep the next-to-leading orderwhich keeps the second set of poles at zero, rendering thequasiparticle current nonzero in view of equation (14). Thiswill have an important consequence later on. We can now, inthe overdamped case, rewrite equation (16) as

Zeff ' Rγ 2 1

1− r4

[1

ω2 + γ 2 −r4

ω2 + γ 2r4

]. (24)

5

Supercond. Sci. Technol. 26 (2013) 125013 M H Ansari et al

This can be read as the difference between two Ohmicenvironments with a Drude cutoff [32]. This allows us to relyon known results [3, 14, 32, 5] for computing J(t). Here,we focus on the zero-temperature regime. We find for thezero-point term S(0) = − 2R

RKlog r.

Now, interestingly, this will be a large positive term forr → 0 which can be achieved by going to small ωp, i.e. theuv-term in equation (14) will be suppressed by a factor r4R/RK

to a very small value.It is known [14] that for Zd =

R01+ω2/ω2

0we have

JS(t) =R0ω0

RK

[e−ω0τE1(−ω0τ)− eω0τE1(ω0τ)

](25)

and that for long times, ω0τ → ∞, this reduces toJS(t) = −

2R0RK

(logωct + γe + iπ2

), where γe ' 0.5772 is the

Euler–Mascheroni constant. For short times, ω0τ � 1, wefind J(t) = iπαω0π t. We can now find P(E) in three regimes.For E� γ, γ r2, both terms in equation (24) should be treatedin the long-time limit. In lowest order in r we find

J(t) = −4R

RKlog r, (26)

i.e., no time-dependence, in agreement with the fact that theenvironment is super-Ohmic at these low frequencies, seeequation (24). This leads to Ps(E) = rρcδ(E).

In the intermediate regime, γ r2� E � γ , we can

combine the short- and long-time limits as

J(t) = −R0

RK

[log γ t + γe + i

π

2− iπγ r2t

]. (27)

This leads to an Ohmic P(E) that is energetically shiftedby δE = R0

RKπγ r2

=R2

LRK. This results in

P(E) =e−2γeRK/R

0(2RK/R)

1E

(πRK

R

E

Ec

)2RK/R

×

(1+

(2RK

R− 1

)δE

E

). (28)

Finally, at large energies, and hence entirely short timesand large energies, E � γ , we can approximate

J(t) 'iπC

t, (29)

hence leading to a simple capacitive contribution from

the junction’s charging energy and P(E,E′) = e−S(0)

4 ((u2+

v2)eS(t)

4 − 2uve−S(t)

4 )δ(E′ − E − e2/2C).

4. Conclusion

In conclusion, we have developed the theory of quasiparticletunneling for superconducting tunnel junctions in anarbitrary linear dissipative environment. We worked out theunperturbed tunneling rate of nonequilibrium quasiparticlesin the junction. In the perturbation regime P(E) governsthe processes of an electron or a hole tunneling, which arerepresented by the diagonal transition matrix elements incharge space. We showed that processes that create charge

superpositions are additionally suppressed by zero-pointfluctuations of the phase. For the case of low damping,quasiparticle tunneling exchanges energy with the plasmamode in the form of a sequence of sidebands.

Acknowledgments

This work was supported by NSERC through the discoverygrants program, Office of the Director of National Intel-ligence (ODNI), Intelligence Advanced Research ProjectsActivity (IARPA). AS gratefully acknowledges support fromPerimeter Institute for Theoretical Physics where most of hiscontribution was worked out. Discussions with J M Martinis,A J Leggett, and G Catelani are gratefully acknowledged.

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