Double-periodic Josephson junctions in a quantum dissipative environment

Double-periodic Josephson junctions in a quantum dissipative environment

Tom Morel    Christophe Mora Laboratoire de physique de l’École Normale Supérieure, PSL Research University, CNRS, Université Pierre et Marie Curie-Sorbonne Universités, Université Paris Diderot-Sorbonne Paris Cité, 24 rue Lhomond, 75231 Paris Cedex 05, France
Abstract

Embedded in an ohmic environment, the Josephson current peak can transfer part of its weight to finite voltage and the junction becomes resistive. The dissipative environment can even suppress the superconducting effect of the junction via a quantum phase transition occuring when the ohmic resistance exceeds the quantum resistance . For a topological junction hosting Majorana bound states with a periodicity of the superconducting phase, the phase transition is shifted to . We consider a Josephson junction mixing the and periodicities shunted by a resistor, with a resistance between and . Starting with a quantum circuit model, we derive the non-monotonic temperature dependence of its differential resistance resulting from the competition between the two periodicities; the periodicity dominating at the lowest temperatures. The non-monotonic behaviour is first revealed by straightforward perturbation theory and then substantiated by a fermionization to exactly solvable models when : the model is mapped onto a helical wire coupled to a topological superconductor when the Josephson energy is small and to the Emery-Kivelson line of the two-channel Kondo model in the opposite case.

I Introduction

The tunneling of Cooper pairs in the Josephson effect can be reduced and even suppressed by a shunting resistance . The resistor acts as an ohmic dissipative environment which controls the quantum fluctuations of the superconducting phase in the Josephson junction ingold (). A renormalization group (RG) analysis predicts a quantum phase transition between a superconducting and an insulating state in a single Josephson junction schmid (); shon_zaikin (); fisher_zwerger (); zwerger (); tewari (); herrero (); kimura (); kohler (); werner (); lukyanov (). The location of the quantum phase transition is determined solely by the dimensionless dissipation strength where is a quantum of resistance. When , quantum fluctuations of the phase are suppressed by dissipation and the junction is superconducting. Conversely, for , the dissipation is strong enough to destroy the Josephson current even at zero temperature. Several aspects of this transition have been observed experimentally yagi (); penttila (); kuzmin () in superconducting junctions shunted by metallic resistors.

The model describing the quantum phase transition is well-established and understood. It can be mapped onto the problem of quantum Brownian motion in a periodic potential which has been studied in detail fisher_zwerger (); aslangul (); korshunov (). It is also equivalent to the one-dimensional boundary Sine-Gordon model Zamolodchikov () which describes in particular an impurity in a Luttinger liquid kane_fisher (); giamarchi (); fabrizio (), such as a defect in an interacting nanowire or a point contact in a fractional quantum Hall state saleur-fendley (). More generally, the quantum phase transition and environment fluctuations have a strong impact on the whole current-voltage characteristics of the junction at energies well below the gap ingold (); grabert1999 (); corlevi (); didier ().

The past years have witnessed a tremendous interest for the fractional Josephson effect in junctions hosting Majorana bound states kitaev (); beenakker (). Majorana excitations exhibit a topological protection against small perturbation and, as such, are believed to be building blocks for fault-tolerant quantum computation ioffe (); read (); nayak () via their braiding malciu (). The fractional Josephson effect involves a periodicity of the current as function of the superconducting phase in contrast with the usual periodicity. It has been tested experimentally in semiconducting nanowires and topological junctions via the absence of odd Shapiro steps under radio-frequency irradiation rokhinson (); wiedenmann (); bocquillon (). Physically, the periodicity is in fact associated with coherent single-electron tunneling at zero energy.

Topological junctions most probably combine Josephson energy terms with and periodicity. These multiple periodicities are nevertheless not uncommon since non-sinusoidal Josephson junctions, for instance in atomic point contacts golubev (); janvier (), already involve different harmonics associated with the presence of Andreev levels. At zero energy, an Andreev state produces a -periodic Josephson effect similar to the topological case. The presence of a strong Kondo impurity in the junction has been argued to pin the Andreev level to zero energy zazuno (), thereby achieving a robust fractional Josephson effect. Moreover, there exist other means to realize different periodicities, including hybrid junctions involving superconducting and ferromagnetic layers which have been theoretically predicted to exhibit a controllable Josephson periodicity ouassou (). Another proposal is a specific arrangement of four Josephson junctions with a periodicity called the Josephson rhombus and also appearing in certain Josephson arrays Ulrich_rhombus (); gladchenko (); bell (); doucot (); ioffe ().

In this paper, we study a Josephson junction having the two periodicities and shunted by an ohmic environment. Whereas we use here a full quantum treatment, the classical limit of this model has been investigated in the framework of the resistively capacitively shunted junction (RCSJ) model with the purpose of describing Shapiro steps dominguez (); pico (); hangli (); sau (). For a pristine topological Josephson junction with periodicity, a renormalization group analysis ribeiro () shows that the superconductor-insulator quantum phase transition is just shifted to the critical value , four times smaller than for conventional Josephson junctions. This critical condition also reads , with . It is then easily understood by noting that single-electron tunneling occurs through Majorana bound states in topological Josephson junctions in contrast to Cooper pair tunneling in conventional Josephson junctions.

Figure 1: Renormalization group scaling flows of the dissipative Josephson junction. represents the strength of the potential term ( or ). The full lines represent the -term and the dashed lines the -term. For , is relevant. We therefore expect that the system is in an insulating state for and in a superconducting state for .

In the presence of both periodicities, a competition emerges with the phase diagram shown in Fig. 1. We focus in this work on the values of between and where, (i) the topological Josephson energy , corresponding to single-electron tunneling, is relevant while (ii) the standard Josephson energy , describing Cooper pair tunneling, is irrelevant but commensurate with the topological term. No intermediate fixed point can emerge from this competition since there are only two admissible infrared fixed points for the corresponding conformal field theory CFT_affleck (), representing the superconducting and insulating states. Nevertheless, the two terms can dominate different energy regimes, being always the dominant effect at sufficiently low energy. We study the interplay of the two Josephson terms and the Coulomb interaction at arbitrary temperatures by using a combination of perturbative techniques and mappings to exactly solvable models for . We compute the resistance of the whole system - Josephson junction and ohmic environment - as a function of temperature and exhibit non-monotonic behaviours for different regimes of Josephson and charging energies.

This article is organized as follows: in Sec. II we use a quantum circuit description of the system and show that the dissipative term and charging energy can be absorbed in the Josephson tunneling to recover the usual Sine-Gordon action shon_zaikin (); kane_fisher (). The rest of the paper is devoted to the computation of the zero-bias differential resistance of the circuit at arbitrary temperature. In Sec. III, we identify the different low temperature regimes using renormalization group arguments. We then derive the resistance within linear response theory using perturbation theory and an infinite resummation based on refermionization at , thereby showing the non-monotonic temperature dependence. In Sec. IV, we treat with a tight-binding approach the limit of a deep Josephson periodic potential landscape with the Josephson energy much larger than the charging energy. A Bloch band description is combined with a refermionization procedure valid at to derive a mapping to the Emery-Kivelson model of the two-channel Kondo problem. The topological Josephson energy acts as an effective magnetic field driving the system to a superconducting phase. We obtain an analytical form for the resistance as function of temperature which qualitatively agrees with the shape derived in the opposite regime of small Josephson energies. We conclude in Sec. V.

Ii Circuit theory

ii.1 Model

Instead of starting from an abstract Caldeira-Leggett form, we derive the relevant Hamiltonian from quantum circuit theory michel (); leppakangas (). We consider the quantum device depicted in Fig. 2 composed of three parallel elements: a superconducting junction with a Josephson energy , a second topological junction with a Josephson energy , a capacitance and a resistor . The whole apparatus is biased by a dc-current . The fractional Josephson junction allows for coherent single-electron tunneling i.e a -periodicity of the phase. We neglect in our analysis the quasiparticle excitations above the superconducting gap and we use for simplicity.

....................................................................................................................................................................................................................................................
Figure 2: (a) Schematic representation of a resistively and capacitively shunted Josephson junction combining and -periodic contributions. (b) Sketch of the distributed LC line circuit representing the shunt resistor . The dissipative environment is described as a semi-infinite transmission line with lineic capacitance and lineic inductance . The correspondance between the two representations gives .

In the charge representation, the Hamiltonian of the system is

(1)

The first term is the energy stored in the capacitance where the charge across the Josephson junctions is added to the charge brought by the resistor and the charge integrated from the current source . The third term corresponds to the Josephson tunneling between states with consecutive Cooper pair charge numbers, . The fourth term describes the tunneling of electrons through the topological Majorana fermions, implying that the charge operator takes half-integer values, such that the corresponding phase operator

(2)

with , is defined on a circle of size . An electron is thus seen as half of a Cooper pair. models the resistor in terms of an semi-infinite one-dimensional transmission line, i.e as a collection of harmonic oscillators with lineic inductance and capacitance  thesis (). The Hamiltonian is

(3)

where . The local flux and the local charge are conjugate variables and obey the canonical quantization

(4)

with the additional constraint that and are conjugate operators

(5)

ii.2 Unitary transformation

Before acting on the Hamiltonian (1), we note that the Hamiltonian can be diagonalized by the following mode expansion

(6a)
(6b)

with the dispersion and the velocity , and the boundary conditions

(7)

corresponding to the reflexion of microwaves by the capacitor. The commutation relations Eqs. (4) and (5) are recovered from the canonical quantization

(8)

This is a field theoretical description of a simple RC circuit with the time scale for discharge . Inserting this mode expansion, we find the diagonal form

(9)

In order to disentangle the different variables, it is convenient to apply the time-dependent unitary transformation

(10)

which essentially shifts the charge operator

(11)

and acts as a displacement operator for the propagating modes

(12)

We note however that leaves invariant. The transformed Hamiltonian assumes the simplified form

(13)

where the Cooper and single-electron tunneling terms are dressed by the dissipative phase functions , with , describing the RC environment. Using Eq.(2), the Eq.(13) becomes

(14)

At this point, disappeared and the phase operator commutes with the Hamiltonian . is a constant of motion and it can be absorbed into , i.e. removed from Eq. (14). We emphasize that, although we made no approximation, the charge discreteness and the related phase compactness no longer play a role in Eq. (14).

It is also be possible to formulate the euclidean action corresponding to the Hamiltonian (14), where all modes except are integrated,

(15)

for . This expression recovers the standard action already used by many authors shon_zaikin () for . We have introduced the charging energy and the dimensionless dissipative constant where is the quantum resistance for Cooper pairs.

Hereinafter, we will use equivalently Eq. (14) and Eq. (15) as a starting point to derive the differential resistance of our model.

Iii Differential resistance in the Coulomb blockade regime

iii.1 Linear response theory

The effective resistance of the circuit is defined by the relation where is the bias current and the voltage drop across the junction is

We use linear response theory to compute by treating in Eq. (14) as a perturbation. The details in the imaginary time formalism are provided in appendix A where the expression

(16)

is derived. denotes a Matsubara frequency, the real part, and the inverse temperature. We have also introduced the current-like operator

(17)

where we use the notation . For , we recover as expected.

The computation of the resistance (16) is based on the evaluation of the phase autocorrelation functions , with . This can be done in perturbation theory in , with the expression of the phase at ,

(18)

the thermal occupation

(19)

and the Bose factor . The leading order () is given at zero temperature and in real time by the expression familiar to the theory ingold (); moskova (); golubev (); grabert (); joyez ()

(20)

with the impedance of the RC environment .

Physically, the long-time asymptotics

(21)

measures how fast phase correlations decay in real time. A large corresponds to a slow diffusion indicating a well-defined superconducting phase. The result is that the second term with the time integral in Eq. (16) diverges as at zero temperature indicating a breakdown of perturbation theory and a flow towards zero resistance, i.e. a superconducting state with a Josephson current. In contrast, a small gives a fast phase diffusion resulting in a vanishing second term in Eq. (16) for , i.e. an insulating state with . Just by power counting, the threshold between these two states is found at , as recapitulated in Fig.1.

iii.2 Perturbation theory for the resistance

After setting the basis of the calculation in linear response theory, we compute the resistance at finite temperature from Eq. (16) and perturbatively in . The leading corrections to the fully shunted junction are derived in appendix B and expressed as fisher_zwerger ()

(22)

with the functions

(23)

is the Euler’s constant, the logarithmic derivative of the Gamma function and an hypergeometric function. The result (22) is only valid perturbatively i.e. when the second and third terms are much smaller than . It can be further simplified at low temperatures , leading to

(24)

with

(25a)
(25b)

with is the Gamma function. We recover the results of the renormalisation group analysis that (resp. ) scales down to zero as the temperature is lowered for (resp. ) whereas it becomes increasingly large at low energy in the opposite case (resp. ).

We focus henceforth on the most interesting scenario where is chosen between and , such that is relevant and is irrelevant at low energy. There, a competition emerges between the two temperature corrections of Eq. (24) with opposite limits. Since eventually dominates at sufficiently low energy, the competition is best discussed in the regime . Differentiating with respect to , we find that the resistance reaches a (local) maximum for

(26)

with and within the temperature range of validity of Eq. (24). For the resistance is an increasing function of temperature whereas it decreases for . We note that the temperature correction due to is diverging at zero temperature such that there is a temperature, much lower than , below which the perturbative expansion (24) is insufficient.

For , Eq. (24) is no longer valid; however we can set in Eq. (22) since we assume . The resulting expression for the resistance exhibits a (local) minimum for

(27)

where can be evaluated numerically and is represented in Fig. 3

Figure 3: versus the dimensionless dissipation term . The function evolves between as to zero when . This is in agreement with the result of Fisher and Zwerger fisher_zwerger ().

The distance between the local maximum and the local minimum decreases with the ratio . Quite generally for arbitrary , the resistance can be obtained by a numerical evaluation of the integrals in Eq. (22). We thus observe a critical value of , shown in Fig. 4 as function of , at which the two extrema meet and disappear. Above this critical value, the resistance becomes a monotonic increasing function of the temperature.

Figure 4: Critical ratio , as function of , at which the local maximum and minimum merge. Above, the resistance is a monotonous function of temperature, see also Fig. 5. The curve is close to be linear and well fitted by .

iii.3 Non-perturbative resummation

We mentioned in the preceding section that perturbation theory fails at low temperature since multiplies a relevant operator. The description of the crossover to very low temperatures thus requires a resummation of the whole perturbation series, and such an exact resummation is not available for general when both and are non-zero.For however, a refermionization technique kane_fisher (); mora (); furusaki () has been successfully applied to compute the crossover for the resistance when . We extend it below to non-zero where an exact crossover can also be formulated.

The main idea is to interpret as a bosonized form of a fermion operator . At zero temperature, with a chiral Hamiltonian

(28)

the correlation function is

(29)

At zero temperature and for , the integral (20) gives

(30)

For , the correlator of has the same time dependence as (29). The bosonization formula compatible with (30) and (29) is

(31)

is a local Majorana fermion with and . ensures the anticommutation rules for the fermionic field . With the representation (31), the quadratic part of Eq.(14) can be remplaced by the Hamiltonian (28), and

(32)

A straightforward point-splitting calculation connects to , since due to Fermi statistics. The two operators have scaling dimension when . The precise connection is obtained by identifying the two-point correlators using Eq.(29), with the result

(33)

The refermionized Hamiltonian takes the form

(34)

which is quadratic and exactly solvable. This effective Hamiltonian allows for a complete resummation for energies smaller than . Interestingly, Eq.(34) already appeared in a different context lin (); fidkowski (); zuo (); pikulin (); sela () as it can represent a semi-infinite helical wire - unfolded as a chiral mode on an infinite line - coupled at to a topological superconductor hosting a single Majorana bound state at its edge. plays the role of the tunnel coupling to the Majorana bound state while generates Andreev reflections at the superconductor. In this model, an incoming electron can be reflected as an electron or an hole (and vice versa).

Eq.(34) is easily diagonalized using a mode expansion fidkowski (); sela () for and summarized in Appendix C. Moreover, we can describe the relation between the left-moving electrons/holes and the right-moving electrons/holes with the S-matrix. At the formal level, the second term in the expression (16) of the resistance, involving the correlator , coincides with the differential conductance of the boundary helical model fidkowski (). Hence, the resistance (16) can be expressed using one of the S-matrix components:

(35)

where is the Fermi distribution and is the probability for an incoming electron with energy to be reflected as a hole. The derivation of is reproduced in Appendix C:

(36)

in agreement with Ref. fidkowski, . Inserting Eq.(36) into Eq. (35), the effective resistance reads

(37)

where the dimensionless function is given by

(38)

For , the integration can be performed and we recover the known result shon_zaikin (); weiss_wollensak ()

(39)

At zero temperature, one obtains consistent with a fully coherent Josephson junction. Let us emphasize that the result (37) was obtained assuming and . As a consequence, we have such that second parameter in Eq.(38) is always much smaller than one. At low temperature , we can therefore ignore to obtain the asymptotic expression

(40)

In the opposite limit , we keep and recover the result Eq. (24) of the preceding section for .

We plot in Fig. 5 the interpolation between Eq. (37) and Eq. (22) for different values of (. We thus observe the crossover where the two extrema disappear.

Figure 5: Dimensionless resistance as function of the reduced temperature for and different values of obtained by interpolation between Eq. (37) and Eq. (22). We represent (red line), (blue line), (green line), (black line). The two extrema and merge at , see Fig. 4.

Iv Large Josephson energy

The analysis of Sec. III was restricted to the Coulomb blockade regime where the bare charging energy is the largest energy scale. Coulomb blockade tends to pin the superconductor charge which has the effect of delocalizing the conjugated phase variable. The shunted Josephson junction is then closer to an insulator, with a differential resistance below but in the vicinity of , except at very low temperature where the relevant Josephson energy takes over and reestablishes a dissipationless Josephson tunneling.

The resulting resistance, shown in Fig. 5, exhibits a local minimum at with a distance to the fully shunted junction increasing with , reflecting a partial relocalization of the phase. This scaling suggests that the local minimum keeps decreasing with until it reaches an almost vanishing resistance in a certain temperature range for larger than . In what follows, we consider directly the regime of deep potential wells while is chosen below the plasma frequency .

We first diagonalize the model in the absence of the ohmic environment in Sec. IV.1 and then take in Sec. IV.2 where a mapping to the Emery-Kivelson model can be demonstrated. This gives the exact resistance for and a qualitative picture for between and , extending the analysis of Sec. III.

iv.1 Dissipationless case

In the absence of dissipation, the Hamiltonian (1) simplifies as with the transmon Hamiltonian koch ()

(41)

where is the offset charge of the capacitor. In the phase representation, is acting on -periodic functions, and can be diagonalized exactly using Mathieu functions wilkinson (). For , the energy of the ground state takes the suggestive form

(42)

where koch ()

(43)

The ground state energy has a periodicity of one in as expected from the discreteness of .

Although the above derivation is self-contained, it is instructive to formulate it using a Bloch band description shon_zaikin (); catelani (). The Hamiltonian  (41) with the compact phase in is mathematically equivalent to solving with an extended phase, i.e. between and , with playing the role of the quasimomentum. In this langage, Eq. (42) as function of is a band dispersion. Focussing again on the regime , the wavefunctions of low-energy eigenstates are strongly localized near the minima of the cosine potential and the Wannier function of the lowest band is the ground state of an harmonic oscillator,

(44)

with energy . The small overlap between consecutive Wannier functions induces a nearest-neighbor hopping term . We thus obtain a tight-binding model whose diagonalization reproduces Eq. (42) and is identified as the bandwidth of the lowest band in the cosine potential.

Next, we include and numerically evaluate the spectrum of . The presence of doubles the size of the unit cell folding the spectrum at , corresponding to single-electron tunneling, and opening gaps at the edge of the new Brillouin zone as illustrated in Fig.6. In order to make further analytical progress, we consider such that the different bands of are not mixed, and project the Hamiltonian onto the lowest band. Due to non-zero , the wavefunctions must have a periodicity of in . For a given charge offset , we find two such functions in the lowest band yavilberg (); ginossar ():

(45)

with energies and . These are -periodic functions, even and odd with respect to the parity operator . After projecting the Hamiltonian onto the basis (45), we get

(46)

where () are Pauli matrices operating in parity space and the constant term has been removed. This derivation of the different overlaps in Eq. (46) uses that takes appreciable values only close to . One consequence is that does not enter the diagonal elements - or only with a very small contribution neglected here, whereas there is a perfect overlap along . The overlap along is given by

(47)

and can be also neglected. The and components in Eq. (46) can be seen respectively as a staggered potential and a staggered hopping amplitude in the tight-binding model. We note that applying a non-zero flux between the two Josephson junctions can change the relative values of and . The limiting case and corresponds to the SSH model ssh ().

Figure 6: Lowest bands obtained by a numerical solution of as a function of the charge offset in the absence of dissipation. The numerical solution is found qutip1 (); qutip2 () by performing a truncation in the charge basis (eigenstates of ) and diagonalizing a finite-size version of . In both pictures, and . (a) Spectrum for . (b) Spectrum for . Gaps open at the edges of the reduced Brillouin zone.

iv.2 Effective Hamiltonian with dissipation

The projection to the lowest band of the extended potential can still be applied to the original Hamiltonian (1) provided the ohmic dissipation is not too strong. It amounts to an adiabatic approximation where one replaces the charge offset by in the transmon Hamiltonian  (41). It is justified as long the input current is weak and varies slowly in time, and if  korshunov ().

The projected Hamiltonian is

(48)

where , which we expand to first order in as