Quantum fluctuations of voltage in superconducting nanowires

At low temperatures non-equilibrium voltage fluctuations can be generated in current-biased superconducting nanowires due to proliferation of quantum phase slips (QPS) or, equivalently, due to quantum tunneling of magnetic flux quanta across the wire. In this paper we review and further extend recen...

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Date:	2017
Main Authors:	Semenov, Andrew G., Zaikin, Andrei D.
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Subjects:	Сверхпроводящие и мезоскопические структуры. К 70-летию со дня рождения А.Н. Омельянчука
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Cite this:	Quantum fluctuations of voltage in superconducting nanowires / Andrew G. Semenov Andrei D. Zaikin // Физика низких температур. — 2017. — Т. 43, № 7. — С. 1011-1022. — Бібліогр.: 29 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-1295352025-02-09T10:12:18Z Quantum fluctuations of voltage in superconducting nanowires Semenov, Andrew G. Zaikin, Andrei D. Сверхпроводящие и мезоскопические структуры. К 70-летию со дня рождения А.Н. Омельянчука At low temperatures non-equilibrium voltage fluctuations can be generated in current-biased superconducting nanowires due to proliferation of quantum phase slips (QPS) or, equivalently, due to quantum tunneling of magnetic flux quanta across the wire. In this paper we review and further extend recent theoretical results related to this phenomenon. Employing the phase-charge duality arguments combined with Keldysh path integral technique we analyze such fluctuations within the two-point and four-point measurement schemes demonstrating that voltage noise detected in such nanowires in general depends on the particular measurement setup. In the low frequency limit we evaluate all cumulants of the voltage operator which turn out to obey Poisson statistics and exhibit a power law dependence on the external bias. We also specifically address a non-trivial frequency dependence of quantum shot noise power spectrum S Ω for both longer and shorter superconducting nanowires. In particular, we demonstrate that S Ω decreases with increasing frequency Ω and vanishes beyond a threshold value of Ω at T → 0. Furthermore, we predict that S Ω may depend non-monotonously on temperature due to quantum coherent nature of QPS noise. The results of our theoretical analysis can be directly tested in future experiments with superconducting nanowires. This work was supported in part by RFBR grant No. 15-02-08273. 2017 Article Quantum fluctuations of voltage in superconducting nanowires / Andrew G. Semenov Andrei D. Zaikin // Физика низких температур. — 2017. — Т. 43, № 7. — С. 1011-1022. — Бібліогр.: 29 назв. — англ. 0132-6414 PACS: 73.23.Ra, 74.25.F–, 74.40.–n https://nasplib.isofts.kiev.ua/handle/123456789/129535 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Сверхпроводящие и мезоскопические структуры. К 70-летию со дня рождения А.Н. Омельянчука Сверхпроводящие и мезоскопические структуры. К 70-летию со дня рождения А.Н. Омельянчука
spellingShingle	Сверхпроводящие и мезоскопические структуры. К 70-летию со дня рождения А.Н. Омельянчука Сверхпроводящие и мезоскопические структуры. К 70-летию со дня рождения А.Н. Омельянчука Semenov, Andrew G. Zaikin, Andrei D. Quantum fluctuations of voltage in superconducting nanowires Физика низких температур
description	At low temperatures non-equilibrium voltage fluctuations can be generated in current-biased superconducting nanowires due to proliferation of quantum phase slips (QPS) or, equivalently, due to quantum tunneling of magnetic flux quanta across the wire. In this paper we review and further extend recent theoretical results related to this phenomenon. Employing the phase-charge duality arguments combined with Keldysh path integral technique we analyze such fluctuations within the two-point and four-point measurement schemes demonstrating that voltage noise detected in such nanowires in general depends on the particular measurement setup. In the low frequency limit we evaluate all cumulants of the voltage operator which turn out to obey Poisson statistics and exhibit a power law dependence on the external bias. We also specifically address a non-trivial frequency dependence of quantum shot noise power spectrum S Ω for both longer and shorter superconducting nanowires. In particular, we demonstrate that S Ω decreases with increasing frequency Ω and vanishes beyond a threshold value of Ω at T → 0. Furthermore, we predict that S Ω may depend non-monotonously on temperature due to quantum coherent nature of QPS noise. The results of our theoretical analysis can be directly tested in future experiments with superconducting nanowires.
format	Article
author	Semenov, Andrew G. Zaikin, Andrei D.
author_facet	Semenov, Andrew G. Zaikin, Andrei D.
author_sort	Semenov, Andrew G.
title	Quantum fluctuations of voltage in superconducting nanowires
title_short	Quantum fluctuations of voltage in superconducting nanowires
title_full	Quantum fluctuations of voltage in superconducting nanowires
title_fullStr	Quantum fluctuations of voltage in superconducting nanowires
title_full_unstemmed	Quantum fluctuations of voltage in superconducting nanowires
title_sort	quantum fluctuations of voltage in superconducting nanowires
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2017
topic_facet	Сверхпроводящие и мезоскопические структуры. К 70-летию со дня рождения А.Н. Омельянчука
url	https://nasplib.isofts.kiev.ua/handle/123456789/129535
citation_txt	Quantum fluctuations of voltage in superconducting nanowires / Andrew G. Semenov Andrei D. Zaikin // Физика низких температур. — 2017. — Т. 43, № 7. — С. 1011-1022. — Бібліогр.: 29 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT semenovandrewg quantumfluctuationsofvoltageinsuperconductingnanowires AT zaikinandreid quantumfluctuationsofvoltageinsuperconductingnanowires
first_indexed	2025-11-25T18:38:38Z
last_indexed	2025-11-25T18:38:38Z
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fulltext	Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 7, pp. 1011–1022 Quantum fluctuations of voltage in superconducting nanowires Andrew G. Semenov1,3 and Andrei D. Zaikin2,1 1I.E. Tamm Department of Theoretical Physics, P.N. Lebedev Physical Institute, Moscow 119991, Russia 2Institute of Nanotechnology, Karlsruhe Institute of Technology (KIT), Karlsruhe 76021, Germany E-mail: andrei.zaikin@kit.edu 3National Research University Higher School of Economics, Moscow 101000, Russia Received February 2, 2017, published online May 25, 2017 At low temperatures non-equilibrium voltage fluctuations can be generated in current-biased superconducting nanowires due to proliferation of quantum phase slips (QPS) or, equivalently, due to quantum tunneling of mag- netic flux quanta across the wire. In this paper we review and further extend recent theoretical results related to this phenomenon. Employing the phase-charge duality arguments combined with Keldysh path integral tech- nique we analyze such fluctuations within the two-point and four-point measurement schemes demonstrating that voltage noise detected in such nanowires in general depends on the particular measurement setup. In the low fre- quency limit we evaluate all cumulants of the voltage operator which turn out to obey Poisson statistics and ex- hibit a power law dependence on the external bias. We also specifically address a non-trivial frequency depend- ence of quantum shot noise power spectrum SΩ for both longer and shorter superconducting nanowires. In particular, we demonstrate that SΩ decreases with increasing frequency Ω and vanishes beyond a threshold value of Ω at T → 0. Furthermore, we predict that SΩ may depend non-monotonously on temperature due to quantum coherent nature of QPS noise. The results of our theoretical analysis can be directly tested in future experiments with superconducting nanowires. PACS: 73.23.Ra Persistent currents; 74.25.F– Transport properties; 74.40.–n Fluctuation phenomena. Keywords: quantum phase slips and shot noise 1. Introduction Perhaps the most fundamental property of any bulk su- perconducting material is its ability to conduct electric current without any resistance, i.e., a non-dissipative cur- rent below some critical value can pass through such ma- terials. It is clear that in this case neither non-zero aver- age voltage nor voltage fluctuations across the superconductor can be expected. While this simple physi- cal picture holds for sufficiently large superconducting samples (usually well described by means of the standard mean field theory approach), it may change drastically as soon as superconductor dimensions become sufficiently small. In this case thermal and/or quantum fluctuations start playing an important role and the system properties may qualitatively differ from those of bulk superconduct- ing structures. For instance, in the case of ultrathin super- conducting wires such fluctuations are responsible for temporal local suppression of the superconducting order parameter = \| \| eiϕ∆ ∆ inside the wire and, hence, for the phase slippage process. This process gives rise to interest- ing physical phenomena which cannot be captured with the aid of the mean field theory. In the low temperature limit thermal fluctuations are unimportant and the system behavior is essentially deter- mined by quantum phase slips (QPS) [1–4]. Each QPS event implies the net phase jump by = 2δϕ ± π accompa- nied by a voltage pulse = /2V eδ ϕ as well as tunneling of one magnetic flux quantum 0 / = \| ( ) \|e V t dtΦ ≡ π δ∫ across the wire normally to its axis (here and below we set = 1 ). Formally different QPS events can be considered as logarithmically interacting quantum particles [5] forming a 2D gas in space-time characterized by an effective fugacity proportional to the QPS tunneling amplitude per unit wire length [6] 0( / ) exp ( ), 1.QPS g ag aξ ξγ ∆ ξ −  (1) © Andrew G. Semenov and Andrei D. Zaikin, 2017 mailto:andrei.zaikin@kit.edu Andrew G. Semenov and Andrei D. Zaikin Here 0∆ is the mean field superconducting order parame- ter, 2= 2 /( ) 1Ng s eξ πσ ξ >> is the dimensionless normal state conductance of the wire segment of length equal to the coherence length ξ , s and Nσ are respectively the wire cross section and its Drude conductance. At 0T → long superconducting wires exhibit a quan- tum phase transition [5] controlled by the dimensionless parameter sλ ∝ which we will specify later. In ultrathin wires with < 2λ superconductivity is fully suppressed by quantum fluctuations, and such wires may even go insulat- ing at 0T → . In somewhat thicker wires with > 2λ quan- tum fluctuations are not so efficient, the wire resistance R decreases with T and one gets [5] 2 2 3 0 2 2 3 0 , ,ˆ = , . QPS QPS T T Id VR dI I T I λ− λ− γ >> Φ〈 〉 ∝  γ << Φ (2) Here and below V̂〈 〉 is the expectation value of the voltage operator across the wire. According to Eq. (2) the wire non- linear resistance does not vanish down to lowest temperatures, as it was later confirmed in a number of experiments [7–10]. Can one also expect to observe non-vanishing voltage fluctuations in superconducting nanowires? The presence of QPS-induced equilibrium voltage fluctuations in such nanowires can be predicted already making use of the re- sult (2) combined with the fluctuation–dissipation theorem (FDT). The issue of non-equilibrium voltage fluctuations (e.g., shot noise) is somewhat more complicated. At this stage it is worth to remind the reader two key pre-requisites of shot noise: (i) the presence of discrete charge carriers (e.g., electrons) in the system and (ii) scattering of such carriers at disorder. Although discrete charge carriers — Cooper pairs — are certainly present in superconducting nanowires, they form a superconducting condensate flow- ing along the wire without any scattering. For this reason the possibility for shot noise to occur in superconducting nanowires need to be investigated in more details. In this paper we will review and extend our recent theo- retical analysis of QPS-induced voltage fluctuations in ultrathin superconducting wires [11–13]. In particular, we will proceed beyond FDT and demonstrate that quantum phase slips can generate not only equilibrium but also non- equilibrium voltage fluctuations in ultrathin superconduct- ing wires. Such fluctuations are caused by quantum tunnel- ing of magnetic flux quanta 0Φ and — as we will show — obey Poisson statistics. In what follows we will mainly focus our attention to QPS-induced shot noise of the volt- age in both long and short nanowires within different measurement schemes and identify highly non-trivial de- pendencies of the noise power spectrum on temperature, frequency and external current. The structure of the paper is as follows. In Sec. 2 we define the two models to be analyzed here and present a simple operator derivation of the dual Hamiltonian for a superconducting nanowires in the presence of quantum phase slips. In Sec. 3 we outline our real time Keldysh tech- nique based approach that generally allows us to evaluate all cumulants of the voltage operator perturbatively in the QPS amplitude (1). General expressions for the voltage correlators are derived in Sec. 4. In Sec. 5 we illustrate a direct relation between our real time technique and the quasiequilibrium imaginary time (the so-called Im F) approach. Our general results for voltage fluctuations (in particular for shot noise) are further analyzed in Sec. 6 in a number of important limits. In Sec. 7 we consider a four-point measurement setup and com- pare our results derived in this case with those for the two- point measurement setup discussed in previous sections. The paper is concluded by a brief summary in Sec. 8. 2. Basic models and phase-charge duality In this paper we will consider two somewhat different setups which allow to experimentally study voltage fluctu- ations in superconducting nanowires. The first setup is displayed in Fig. 1. This system consists of an ultrathin superconducting wire of length L and cross section s. A capacitance C and a shunt resistor sR are switched in par- allel to this wire. The whole system is biased by an exter- nal current = /x xI V R . The right wire end ( =x L) is grounded as shown in the figure (here and below x is the coordinate along the wire ranging from 0 to L). The volt- age ( )V t at its left end = 0x fluctuates and such fluctua- tions can be measured by a detector. Another possible setup is shown in Fig. 2. It consists of a superconducting nanowire attached to a current source I and two voltage probes located in the points 1x and 2x . The wire contains a thinner segment of length L where quan- tum phase slips can occur with the amplitude (1). Both structures displayed in Figs. 1 and 2 can be treated within the same formalism which we are going to outline below. The system shown in Fig. 2 will be addressed be- low in Sec. 7 of this paper. Here we stick to the system of Fig. 1. An effective Hamiltonian for this system can be expressed in the form ch dis wire ˆ ˆ ˆ ˆ= /2 .H H H I e H+ − ϕ + (3) The first three terms in the right-hand side of Eq. (3) define respectively the charging energy [14], 2 ch 1ˆ = , 2 ( /2 ) H i Q C e  ∂ − + ∂ ϕ  (4) Fig. 1. (Color online) The first setup under consideration. The figure also illustrates creation of two plasmons by a QPS. 1012 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 7 Quantum fluctuations of voltage in superconducting nanowires the Caldeira–Leggett contribution of the shunt resistor sR [14] and the energy tilt produced by an external current I . The variable ( ) (0, )t tϕ ≡ ϕ denotes the phase of the super- conducting order parameter field ( , )x t∆ at = 0x . Here we also define ( , ) 0L tϕ ≡ . The last term wireĤ in Eq. (3) accounts for the super- conducting wire. This part of the effective Hamiltonian can be expressed in terms of both the modulus \| ( , ) \|x t∆ and the phase ( , )x tϕ of the order parameter field [5,6,15]. Here, however, it will be convenient for us to proceed differently and to employ the duality arguments. The duality between the phase and the charge variables was discussed in details in the case of ultrasmall Josephson junctions [14,16–18]. Later the same duality arguments were extended to short [19] and long [20–22] superconducting wires. Below we will illustrate the formal path integral re- sults [22] by means of a simple operator analysis. In the absence of quantum phase slips an effective low energy Hamiltonian for a superconducting nanowire can be written in the form 22 eff kin0 ˆ ˆ ( )( ) 1ˆ = , 2 2 2 L x w xQ xH dx C e  ∂ ϕ  +      ∫  (5) where wC and kin 0= 1/( )N sπσ ∆ are respectively the ge- ometric wire capacitance (per length) and the kinetic wire inductance (times length), ˆ ( )Q x and ˆ ( )xϕ are canonically conjugate local charge and phase operators obeying the commutation relations ˆ ˆ[ ( ), ( )] = 2 ( ),Q x x ie x x′ ′ϕ − δ − (6) As the contribution of the external current source I is already accounted for in Eq. (3), for the sake of our deriva- tion and without loss of generality we can now assume that the superconducting wire is isolated from any external cir- cuit. Then the current at its end points = 0x and =x L vanishes and, hence, we can define the boundary condi- tions for the phase in the form ˆ ˆ(0) = ( ) = 0.x x L∂ ϕ ∂ ϕ (7) Employing the Fourier series expansion, we get 0 =1 2ˆ ˆ ˆ( ) = cos( / ),n n x nx L L ∞ ϕ ϕ + ϕ π∑ (8) 0 =1 ˆ 2ˆ ˆ( ) = cos( / ),n n Q Q x Q nx L L L ∞ + π∑ (9) where 0 0 ˆ ˆˆ ˆ[ , ] = 2 , [ , ] = 2 .m n mnQ ie Q ieϕ − ϕ − δ (10) Let us now perform the dual transformation. For this purpose we introduce the following (dual) operators ˆ ˆ( ) = ( )/2xx x eΦ ∂ ϕ (11) and 0 ( )ˆ ˆˆ ( ) = ( ) ( ), L L x L xx dx Q x dx Q x e eL π π −′ ′ ′ ′χ − +∫ ∫ (12) which can also be expressed as 2 2 3 =1 ˆ ˆ( ) = sin( / ), 2 n n x n nx L e L ∞π Φ − ϕ π∑ (13) 2 =1 ˆ2ˆ ( ) = sin( / ).n n QLx nx L ne ∞ χ π∑ (14) These new canonically conjugate flux and charge operators obey the commutation relations 0ˆ ˆ[ ( ), ( )] = ( )x x i x x′ ′Φ χ − Φ δ − (15) and obvious boundary conditions ˆ ˆ ˆ ˆ(0) = ( ) = 0, (0) = ( ) = 0.L LΦ Φ χ χ (16) Substituting the relations 0 ˆ ˆˆˆ ˆ( ) = 2 ( ), ( ) = ( )x x Q ex e x Q x x L ∂ ϕ Φ + ∂ χ π (17) into Eq. (5), we obtain 2 0 eff ˆ ˆ ˆ= , 2 TL w Q H H LC + (18) where 22 2 kin 00 ˆ ˆ( )ˆ = 2 2 L x TL w H dx C  ∂ χΦ +  Φ  ∫  (19) is the Hamiltonian for a transmission line formed by a su- perconducting wire. The physical meaning of the operator ˆ ( , )x tχ is trans- parent: It is simply proportional to the operator for the charge that has passed through the point x up to the time moment t . Hence, the local current and the local charge density operators are defined respectively as 0 0 ˆ ˆ ˆ ˆ( , ) = ( , )/ , ( , ) = ( , )/ .t xI x t x t x t x t∂ χ Φ ρ −∂ χ Φ (20) The charge Q in Eq. (4) equals to 0( ) = (0, )/Q t tχ Φ . The above consideration does not yet account for quan- tum phase slips. In order to specify the QPS contribution to Fig. 2. (Color online) The second setup to be analyzed in Sec. 7. Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 7 1013 Andrew G. Semenov and Andrei D. Zaikin the wire Hamiltonian let us first define the phase field con- figurations as ˆ ( ) \| ( ) = ( ) \| ( )x x x xϕ ϕ 〉 ϕ ϕ 〉 (21) and bear in mind that the phase of the superconducting order parameter is a compact variable. Accordingly, the field configurations ( )xϕ and ( ) 2xϕ + π correspond to the same quantum state of our system. Furthermore, in the absence of QPS, i.e., provided the absolute value of the order parameter does not fluctuate 0\| ( , ) \|=x t∆ ∆ , also the states ϕ and 1( ) = ( ) 2 ( )x x x xϕ ϕ + πθ − (where 10 < <x L and ( )xθ is the standard Heaviside step function equal to 0 for 0x ≤ and to 1 for > 0x ) are physically indistinguisha- ble. For instance, the supercurrent operator proportional to the combination 2 0 ˆ ˆexp( ( )) exp ( ( ))xi x i x∆ − ϕ ∂ ϕ remains the same in both cases. Let us now make the step function continuous by effec- tively smearing it at the scale of the superconducting coher- ence length ξ , i.e., we substitute ( ) ( )x xξθ → θ . The corre- sponding field configuration 1( ) = ( ) 2 ( ),x x x xξ ξϕ ϕ + πθ − on one hand, remains very close to ( )xϕ and, on the other hand, is already physically distinguishable from the latter. The QPS process can be viewed as quantum tunneling be- tween these two close but physically different phase con- figurations. What remains is to make use of the fact that the wire Hamiltonian does not depend on the operator 0ϕ̂ , implying that any shift by a constant phase does not change the state of our system. Hence, without loss of generality we can set 0ˆ \| = 0ϕ ψ〉 for any system state ψ . This condition applies for the evolution controlled by the Hamiltonian (5) and it is also maintained in the presence of quantum phase slips. With this in mind we conclude that the QPS process corre- sponds to quantum tunneling of the phase between the states ( )xϕ and 1 1 0 ( ) = ( ) 2 ( ) 2 ( ). L x x x x dx x xξ ξ′ϕ ϕ + πθ − − π θ −∫ (22) In the operator language this tunneling process can be denoted as 1 ˆ ( ) \| ( ) = \| ( )U x x xξ ′ϕ 〉 ϕ 〉 , where the expression for 1 ˆ ( )U xξ just follows from the commutation relations and reads 1 0 1 0 ˆ ˆˆ ( ) = exp ( ( ) / ) ( ) . LiU x dx Q x Q L x x eξ ξ  π − θ −     ∫ (23) As a result, the part of the Hamiltonian which explicitly accounts for the QPS process takes the form 1 1 0 ˆ = ( ) L QPS QPSH dx x− γ ×∫ 0 1 0 ˆ ˆcos ( ( ) / ) ( ) , L dx Q x Q L x x e ξ  π × − θ −     ∫ (24) where 1( )QPS xγ is the QPS amplitude at the wire point 1=x x . Setting now 0ξ → and making use of the second Eq. (17), in the case of a uniform wire with ( ) =QPS QPSxγ γ we obtain 0 ˆ ˆ= cos . L QPS QPSH dx−γ χ∫ (25) This result completes our derivation of the dual representa- tion for the Hamiltonian of a superconducting nanowire. Note that the first term in the right-hand side of Eq. (18) describes an extra contribution to the system charging en- ergy (4), i.e. this term can simply be eliminated by absorb- ing the total wire capacitance wLC into C as wC LC C+ → . The dual Hamiltonian of the wire in Eq. (3) is then defined by an effective sine-Gordon model wire ˆ ˆ ˆ= .TL QPSH H H+ (26) 3. Keldysh perturbation theory and Green functions Let us now investigate fluctuations of the voltage ( )V t in the presence of quantum phase slips. In order to proceed we will employ the dual Hamiltonian (3) derived in the previ- ous section and make use of the Keldysh path integral tech- nique. As usually, our variables of interest are defined on the forward and backward time branches of the Keldysh contour, i.e. we now have , ( )F B tϕ and , ( , )F B x tχ . We also routinely introduce “classical” and “quantum” variables, respectively ( ) = ( ( ) ( ))/2F Bt t t+ϕ ϕ + ϕ and ( ) = ( ) ( )F Bt t t−ϕ ϕ −ϕ (the same recipe holds for the χ-fields). Employing the Josephson relation between the voltage and the phase one can formally express the expectation value of the voltage operator across the the superconduct- ing wire in the form 1 1 0 1( ) = ( )e 2 iSQPSV t t e +〈 〉 ϕ , (27) where 0 = 2 sin( )sin( / 2) L QPS QPSS dt dx + −− γ χ χ∫ ∫ (28) and [ , ]2 2 00... = ( ) ( , )(...)eiSt x t ϕ χ〈 〉 ϕ χ∫   (29) implies averaging with the Keldysh effective action 0S corresponding to the Hamiltonian 0 ˆ ˆ ˆ= QPSH H H− . Analogously, for any higher order correlator of voltages we have 1 2 1( ) ( )... ( ) = (2 ) n nV t V t V t e 〈 〉 × 1 2 0 ( ) ( )... ( )e , iSQPS nt t t+ + +× ϕ ϕ ϕ   (30) 1014 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 7 Quantum fluctuations of voltage in superconducting nanowires At this stage let us emphasize that Eq. (30) defines the symmetrized voltage correlators. E.g., for = 2n we have 1 2 1 2 2 1 1 ˆ ˆ ˆ ˆ( ) ( ) = ( ) ( ) ( ) ( ) , 2 V t V t V t V t V t V t〈 〉 〈 + 〉 (31) while for = 3n one finds [23] 1 2 3 1 2 3 1 ˆ ˆ ˆ( ) ( ) ( ) = { ( )( ( ) ( )) 8 V t V t V t V t V t V t〈 〉 〈 〉 + 2 3 1 2 1 3 ˆ ˆ ˆ ˆ ˆ ˆ( ( ) ( )) ( ) ( )( ( ) ( ))V t V t V t V t V t V t+ 〈 〉 +〈 〉 +  1 3 2 3 1 2 ˆ ˆ ˆ ˆ ˆ ˆ( ( ) ( )) ( ) ( )( ( ) ( ))V t V t V t V t V t V t+ 〈 〉 +〈 〉 +  1 2 3 1 2 3 ˆ ˆ ˆ ˆ ˆ ˆ( ( ) ( )) ( ) ( ) ( ) ( )V t V t V t V t V t V t+ 〈 〉 +〈 〉 +  1 2 3 ˆ ˆ ˆ( ) ( ) ( ) },V t V t V t+ 〈 〉 (32) where  and  are, respectively, the forward and back- ward time ordering operators. A formally exact expression (30) can be evaluated perturbatively in the tunneling amplitude QPSγ (1). In the zero order in QPSγ the problem is described by the quad- ratic (in both ϕ and χ) action 0S . In that case it is neces- sary to employ the averages 0 0 0( ) = ( ) = ( , ) = 0,t t x t+ − −〈ϕ 〉 〈ϕ 〉 〈χ 〉 0 0( , ) = ,x t It+〈χ 〉 Φ (33) as well as the following Green functions 0 0 0( , ) = ( ) ( ) ( ) ( ) ,K abG X X i a X b X i a X b X+ + + +′ ′ ′− 〈 〉 + 〈 〉 〈 〉 0( , ) = ( ) ( ) ,R abG X X i a X b X+ −′ ′− 〈 〉 (34) where ( )a X and ( )b X stand for one of the fields ( )tϕ and ( , )x tχ . As these fields are real, the Green functions satisfy the condition ( ) = ( )A R ab baG Gω −ω . Then the Keldysh function KG takes the form ( )1( ) = coth ( ) ( ) . 2 2 K R R ab ab baG G G T ω ω ω − −ω    (35) Expanding Eq. (30) up to the second order in QPSγ and performing all necessary averages, one can express the results in terms of the Green functions (34). These results can be represented graphically in the form of the so-called candy diagrams [11]. These diagrams for the first and the second moments of the voltage operator are displayed in Fig. 3. They involve four different propagators ( ,R KGχχ and ,R KGϕχ ) and plenty of vertices originating from Taylor ex- pansion of the cosine terms. Summing up all the diagrams in the same order in QPSγ one arrives at the final expres- sion containing the exponents of the Green functions. What remains is to evaluate all the above Green func- tions for the system depicted in Fig. 1. This task can be carried out in a straightforward manner. E.g., for the func- tion RGϕϕ we obtain [11] 2 2 tot 1( ) = , coth 2 4 R C G i L E e R ϕϕ ω ω ω ωλ ω + −  π  v (36) where 2= 2 /CE e C , kin= 1/ wCv is the plasmon veloci- ty [24] and the parameter λ already introduced above is defined as = /(2 )Q wR Zλ with 2= /(2 )QR eπ being the “superconducting” quantum resistance unit and kin= /w wZ C being the wire impedance. We also de- fined tot = /( )x s x sR R R R R+ . The corresponding expressions for RGχϕ and RGχχ read [11] ( ; ) = ( ; )R RG x G xχϕ ϕχω − ω = 2 2 tot ( )2 cos = , sin cos 2 4C L xi i L L E e R ω − λ      ω ω ω ωλ ω   + −       π     v v v (37) Fig. 3. Candy-like diagrams which determine both average volt- age V〈 〉 (upper diagram) and voltage–voltage corrrelator VV〈 〉 (six remaining diagrams) in the second order in QPSγ . The fields +ϕ , +χ and −χ in the propagators (34) are denoted respectively by wavy, solid and dashed lines. Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 7 1015 Andrew G. Semenov and Andrei D. Zaikin 1 ( ) ( )( , ; ) = 4 cos cos ( ) cos cos ( ) sin R L x x L x xG x x x x x x Lχχ ′ ′ ω − ω ω − ω        ′ ′ ′ω πλ θ − + θ − +        ω          ω     v v v v v 2 2 2 tot ( ) ( )4 cos cos . sin sin cos 2 4C L x L x L i L L E e R ′ω − ω −   λ        +   ω ω ω ω ωλ ω      + −        π        v v v v v (38) ________________________________________________ In order to simplify the above expressions let us make use of the momentum conservation for plasmons propagating along the wire. Such plasmons can only be created in pairs with the total zero momentum. Excitations moving towards the grounded end of the superconducting nanowire eventu- ally vanish there with no chance to reappear again while plasmons propagating in the opposite direction produce voltage fluctuations measured by a detector. Then in the long wire limit the general expressions for RGϕχ and RGχχ reduce to more simple ones 2 e( ; ) , ( 0) 2 xi R C G x ii E ω ϕχ λ ω −  ω λ ω+ + π   v (39) \| \| 2( , ; ) e . 0 x xiR iG x x i ′ω − χχ π λ′ ω − ω+  v (40) In Eqs. (39) and (40) we also set ,x sR R →∞. 4. I–V curve and voltage noise: general results Making use of the above results it is now straightforward to derive general expressions for the voltage correlators (30). Here we restrict our analysis to the first two moments of the voltage operator. For the expectation value of this operator we obtain 2 00 0 = ( ; )lim4 L L QPS Ri V dx dx G x e ϕχ ω→ γ  ′〈 〉 ω ω ×   ∫ ∫ ( ), 0 , 0( ) ( ) ,x x x xI I′ ′× −Φ − Φ  (41) where , , ,( ) = ( ) ( )x x x x x xP P′ ′ ′ω ω + ω and ( , ; ,0) , 0 ( ) = e e ,i t i x x t x xP dt ∞ ′ω ′ ω ∫  (42) 1( , ; ,0) = ( , ; ,0) ( , ; , ) 2 K Kx x t G x x t G x x t tχχ χχ′ ′ − − 1 1( , ; 0,0) ( , ; ,0). 2 2 K RG x x G x x tχχ χχ′ ′ ′− + With the aid of the identity 0 ( ; ) = 2lim RG x iω→ ϕχω ω π Eq. (41) can be expressed in the form ( )0= ( ) ( ) ,QPS QPSV I I〈 〉 Φ Γ −Γ − (43) where we defined 2 , 0 0 0 ( ) = ( ). 2 L L QPS QPS x xI dx dx I′ γ ′Γ Φ∫ ∫  (44) Turning to voltage fluctuations we identify three differ- ent contributions to the noise power spectrum (0)= e ( ) (0) = .i t r aS dt V t V S S SΩ Ω Ω ΩΩ〈 〉 + +∫ (45) The first of these contributions (0)SΩ is unrelated to QPS. It just defines equilibrium voltage noise for a transmission line and reads ( ) 2 (0) 2 coth 2= ( ) ( ) . 16 R R i TS G G e ϕϕ ϕϕΩ Ω Ω     Ω − −Ω (46) The other two terms are due to QPS effects. The term rSΩ is also proportional to coth ( /2 )TΩ and depends on the prod- ucts of two retarded (advanced) Green functions: 2 2 2 0 0 coth 2= Re ( ; ) 8 L LQPS r RTS dx dx G x e Ω ϕχ Ω γ Ω     ′ Ω ×∫ ∫ , ,( ( ) ( ; ) (0) ( ; )) .R R x x x xG x G x′ ′ϕχ ϕχ ′× Ω Ω − Ω   (47) Here we also denoted , , 0 , 0( ) = ( ) ( )x x x x x xP I P I′ ′ ′Ω − Ω+Φ − Ω−Φ + , 0 , 0( ) ( ).x x x xP I P I′ ′+ −Ω +Φ + −Ω−Φ (48) The remaining term aSΩ, in contrast, contains the product of one retarded and one advanced Green functions. We get 2 2 2 0 0 = ( ; ) ( ; ) 16 L L QPSa R RS dx dx G x G x e Ω ϕχ ϕχ γ Ω ′ ′Ω −Ω ×∫ ∫ ( ), 0 , 0( ) ( ) ,x x x xI I′ ′± ±   × Ω ±Φ − −Ω Φ     ∑    (49) 1016 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 7 Quantum fluctuations of voltage in superconducting nanowires where 0= coth coth . 2 2 I T T± Ω ±Φ Ω   −        (50) Eqs. (45)–(50) together with the expressions for the Green functions (36)–(38) fully determine the voltage noise power spectrum of a superconducting nanowire in the perturbative in QPS regime. 5. Relation to ImF-method Comparing Eq. (43) for the average voltage with the corresponding result [5] we can identify the quantity ( )QPS IΓ (44) as a quantum decay rate of the current state due to QPS. In [5] this rate was derived with the aid of the so-called ImF -method [25]. It is of interest to establish a direct relation between the latter approach and the real time Keldysh technique employed here. Let us introduce the generalized Green function ( , ; )x xχ ′ σ which depends on the complex time σ and satisfies the condition ( , ; 0) = ( , ; ,0)x x t i x x tχ ′ ′−  at > 0.t This function reads ( , ; ) = coth( ( )) 2 iTx x dt T tχ ′ σ π −σ ×∫ ( )e ( , ; ) ( , ; ) 2 i t R Rd G x x G x x− ω χχ χχ ω ′ ′× ω − −ω π∫ (51) The function (51) is analytic, periodic in the imaginary time, ( , ; ) = ( , ; / ),x x x x i Tχ χ′ ′σ σ −  (52) and has branch cuts at Im ( ) = /N Tσ for all integer N . On the imaginary axis the function χ matches with the Matsubara Green function ( , ; ) = ( , ; ).Mx x i iG x xχ χχ′ ′− τ τ (53) The quantum decay rate Γ of a metastable state can be evaluated by means of the well known formula [25] = 2Im ,FΓ (54) where F is the system free energy. In order to establish the QPS contribution to Γ it is necessary to identify the corre- sponding correction to the free energy Fδ . In the leading order in QPSγ one can consider just one QPS–antiQPS pair [5] which yields the following contribution 2 1/ pair 0 0 0 = e , 2 L L T SQPSF dx dx d −γ ′δ τ∫ ∫ ∫ (55) where pair 0= ( , ; ,0)S I x x′−Φ τ + τ (56) and τ is the imaginary time interval between QPS and anti- QPS events. The term ( ; ; ,0)x x′ τ accounts for the inter- action between these events which occur respectively at the points x and x′. Expressing this interaction term via the Matsubara Green function, we find ( ; ; ,0) = ( , ; )Mx x G x xχχ′ ′τ τ − 1 1( , ; 0) ( , ; 0). 2 2 M MG x x G x xχχ χχ ′ ′− − (57) An attentive reader may have already noticed that the integral over τ in Eq. (55) formally diverges at low tem- peratures. As a consequence, the free energy acquires an imaginary part ImF derived with the aid of a proper ana- lytic continuation of Fδ . Evaluating the integral (55) by the steepest descent method we routinely determine a sta- tionary point sτ from the stationary condition for the action 0 = ( , ; ).M sI G x xτ χχ ′Φ ∂ τ (58) A closer inspection allows to conclude that this stationary point delivers a maximum to the action rather than a mini- mum, thus indicating an instability with respect to quantum decay to lower energy states. In this case the correct recipe is to deform the integration contour along the steepest de- scent path. This procedure is illustrated in Fig. 4. The ini- tial integration contour goes vertically from 0 to i− β. This contour can be deformed and directed along the real time axis after passing through the point sτ . Then we obtain 2 ( ; ; ,0)0 0 0 0 = e 2 L L sQPS I x xF dx dx d τ ′Φ τ− τγ ′δ τ +∫ ∫ ∫  2 ( ) ( ; ; ,0)0 0 0 0 e . 2 L L QPS I i x x is sdx dx id ′Φ τ + τ − τ + τγ ′+ τ∫ ∫ ∫  (59) The imaginary part of this expression reads 2 ( ) ( ; ; ,0)0 0 0 0 2Im = e 2 L L QPS I i x x is sF dx dx d ′Φ τ + τ − τ + τγ ′ τ +∫ ∫ ∫  2 ( ) ( ; ; ,0)0 0 0 0 e . 2 L L QPS I i x x is sdx dx d ′Φ τ − τ − τ − τγ ′+ τ∫ ∫ ∫  (60) Expressing Eq. (60) as a single integral along the contour passing through the point sτ in the direction perpendicular to real τ axis, we get Fig. 4. (Color online) Integration contour. Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 7 1017 Andrew G. Semenov and Andrei D. Zaikin 2 ( ) ( ; ; ,0)0 0 0 2Im = e . 2 L L QPS I i x x is sF dx dx d ′Φ τ + τ − τ + τγ ′ τ∫ ∫ ∫  (61) Combined with Eqs. (57) and (53), this expression can be cast to the form 2 ( )0 0 0 2Im = e 2 L L QPS I itsF dx dx dt Φ τ +γ ′ ×∫ ∫ ∫ ( , ; ,0) ( , ;0,0) ( , ;0,0) 2 2e . i ii x x t i x x x xs′ ′ ′− τ − −χ χ χ×    (62) Then making use of the relation ( , ; 0) ( , ;0,0) ( , ;0,0) 2 2, ( ) = e e , i ii x x t i x x x xi t x x dt ∞ ′ ′ ′− − −χ χ χω ′ −∞ ω ∫     we arrive at the final result 2 , 0 0 0 2Im = ( ). 2 L L QPS x xF dx dx I′ γ ′ Φ∫ ∫  (63) This expression together with Eq. (54) confirms that Eq. (44) indeed determines the QPS-mediated decay rate of the cur- rent states in a superconducting nanowire, thus proving the equivalence of the ImF -approach [5] and the real time Keldysh technique combined with duality arguments elab- orated here. 6. I–V curve and voltage fluctuations Now we turn to concrete results. As a first step, let us reconstruct the results [5] for the average voltage .V〈 〉 Making use of Eqs. (43), (44) together with the relation (42) and the expressions for the Green functions (36)–(40) and keeping in mind the detailed balance condition , ,( ) = exp ( )x x x xT′ ′ ω ω −ω      (64) we obtain 2 0 2 0 0= sinh , 2 2 2 QPSL I I V T Φ γ Φ Φ   〈 〉 ς         v (65) where we introduced 1 0 2 2 2 2( ) = (2 ) , ( ) i i T TTλ λ− λ ω λ ω   Γ − Γ +   π π   ς ω τ π Γ λ (66) 0 01/τ ∆ is the QPS core size in time and ( )xΓ is the Gamma-function. Further assuming that 0 0xτ v (where 0x ξ is the QPS core size in space) we observe that the result (65) fully matches with that derived in [5] by means of a different technique. Now let us analyze the general expressions for the volt- age noise (45)–(50). At zero bias 0I → the term aSΩ van- ishes, and the equilibrium noise spectrum (0)= rS S SΩ ΩΩ + can be obtained directly from FDT, see also [22]. At non- zero bias values the QPS noise becomes non-equilibrium. In the limit 0Ω→ the terms (0)SΩ and rSΩ vanish and the voltage noise 0 0S SΩ→ ≡ is determined solely by aSΩ. Then from Eq. (49) we get ( )2 0 0= ( ) ( )QPS QPSS I IΦ Γ +Γ − = 0 0= coth , 2 I V T Φ Φ 〈 〉    (67) where V〈 〉 is specified in Eq. (65). Combining the result (67) with Eqs. (65), (66) we obtain 2 2 0 0 2 2 0 , , , . T T I S I T I λ− λ−  >> Φ∝  << Φ (68) At higher temperatures 0T I>> Φ (although 0T << ∆ ) Eq. (68) accounts for equilibrium voltage noise 0 = 2S TR of a linear Ohmic resistor 2 3= /R V I T λ−〈 〉 ∝ [5]. In the low temperature limit 0T I<< Φ it describes QPS-induced shot noise 0 0=S VΦ 〈 〉 obeying Poisson statistics with an effective “charge” equal to the flux quantum 0Φ . The above analysis allows to answer the question about the physical origin of shot noise in superconducting nano- wires. We conclude that voltage shot noise is produced by coherent tunneling of magnetic flux quanta 0Φ across the wire. In the dual picture employed here such flux quanta can be viewed as charged quantum particles passing through and being scattered at an effective “tunnel barrier”. We also note that the result analogous to Eq. (67) was previously derived for thermally activated phase slips (TAPS) [26]. It is instructive to mention that our analysis also allows to recover higher correlators of the voltage operator (30). Let us define the voltage cumulants 1 1 0 =0 1= ( ) log exp ( )lim t n n n z t z i iz dt V t t→∞      − ∂      ∫ . (69) Within the accuracy of our perturbation theory the terms 2 k QPS∝ γ with <k n generated in the right-hand side of Eq. (69) can be ignored. Then n coincides with the Fouri- er transformed correlators (30), i.e., 2 0= S etc. Proceed- ing perturbatively in QPSγ and employing Eqs. (65), (66), at 0T → we obtain [12] 2 2 2 0 01 2 2 0 02 2 2= = \| \| . 2 ( ) n QPSn n L V I λ − λ− λ− π γ τ Φ Φ 〈 〉 Φ Γ λ  v (70) The above results allow to fully describe statistics of QPS- induced voltage fluctuations in superconducting nanowires in the low frequency limit. 1018 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 7 Quantum fluctuations of voltage in superconducting nanowires Another interesting situation is that of sufficiently high frequencies and/or long wires 0/L << Ω << ∆v . In this limit we find (0) 2 2 2 coth 2= . 8 ( /2 ) ( / )C TS e EΩ Ω Ω  λ   π Ω + λ π (71) It is easy to observe that this contribution does not depend on the wire length L . At low T and 2/ = /2CE e CΩ λ  we have (0) 1/SΩ ∝ Ω, i.e. the wire can generate 1/f voltage noise. Let us now evaluate the QPS terms rSΩ and aSΩ. In doing so, it is straightforward to demonstrate that the latter term scales linearly with the wire length L whereas the former shows weaker dependence on L . Hence, the term rSΩ can simply be dropped in the long wire limit. For the remaining QPS term aSΩ we get 2 2 0 0 2= 2 24 QPSa L I I S e Ω λ γ  Φ Φ    ς −Ω − ς +Ω ×          v ( ) 0 0 2 2 sinh 2 2 . ( /2 ) ( / ) sinh 2C I I T E T Φ Φ   ς       × Ω Ω + λ π     (72) At 0T → Eq. (72) yields 1 1 0 0 0 ( 2 / ) , < /2, 0, > /2. a I I IS I λ− λ− Ω  − Ω Φ Ω Φ∝  Ω Φ (73) In order to interpret this threshold behavior it is necessary to bear in mind that at = 0T each QPS event can excite 2N plasmons ( = 1, 2 ...N ) with total energy 0=E IΦ and total zero momentum. The left and the right moving plasmons (each group carrying total energy /2E ) eventual- ly reach respectively the left and the right wire ends. One group gets dissipated at the grounded end of the wire while another one causes voltage fluctuations with frequency Ω measured by a detector. Clearly, at = 0T this process is only possible at < /2EΩ in the agreement with Eq. (73). The result (72) is also illustrated in Fig. 5. At sufficient- ly small Ω one observes a non-monotonous dependence of SΩ on T . This behavior is a direct consequence of quantum coherent nature of QPS noise. We also emphasize that at non-zero T Eq. (72) does not coincide with the zero fre- quency result (67) even in the limit 0Ω→ . This difference has to do with the order of limits: Before taking the zero frequency limit in Eq. (72) one should formally set L →∞ . Then one finds 2 2 0 0 0 0 0( ) = sinh . 2 2 2 a QPS I I I S I LT TΩ→ Φ Φ Φ     ′− γ Φ ς ς            v (74) Comparing Eqs. (74) and Eq. (67) (in the latter equation the limit 0Ω→ was taken prior to sending the wire length L to infinity) one observes the identity 0 0( , ) ( , ) = 2 ( , ),aS I T S I T TR I TΩ→− (75) implying that both expressions (74) and (67) coincide only at = 0T , while at any non-zero T the noise power 0 ( , )S I T (67) exceeds one in Eq. (74) and grows monotonously with temperature. The above analysis is merely applicable to sufficiently long wires in which case the main dissipation mechanism is due Mooij–Schön plasmons [24] propagating along the wire and carrying energy out of the system. One can also consider the limit of shorter wires where such plasmons are irrelevant and other dissipation mechanisms come into play. In such wires one typically has /L T<< v , i.e., the system spatial dimension is much shorter than that in time direction. Under such conditions it is convenient to split our analysis into two parts and consider the effect of high frequency modes (short scales) and low frequency ones sep- arately. This procedure was already described elsewhere [22,27] and is known as the so-called two stage scaling. High frequency modes can be accounted for by means of the well known Berezinskii–Kosterlitz–Thouless (BKT) renormalization group (RG) approach. The corresponding RG equations read 2 2 2= (2 ) , = 32 ( ), ln ln d d K d d ζ λ −λ ζ − π ζ λ λ Λ Λ (76) where 2= QPSζ γ Λ is the dimensionless coupling parame- ter, Λ is the renormalization scale and ( )K λ is some nonuniversal function (which depends on the renormaliza- tion scheme) equal to one at the quantum BKT phase tran- sition point = 2λ which separates a superconducting (or- dered) phase > 2λ with bound QPS–antiQPS pairs and a disordered phase < 2λ with unbound QPS [5]. Fig. 5. (Color online) The frequency dependence of the QPS noise spectrum SΩ (72) at = 2.7λ , large CE and different T in the long wire limit. The inset shows SΩ as a function of T . Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 7 1019 Andrew G. Semenov and Andrei D. Zaikin As usually, we start renormalization at the shortest scale 2 2 2= /cΛ ξ ξ + ∆ v and proceed to bigger scales. As above, for simplicity we set cξ ξ . Within the first order perturbation theory in ζ one can ignore weak renormaliza- tion of the parameter λ. With this accuracy the solution of Eqs. (76) takes the form ( ) = ( / )QPS QPS λγ Λ γ ξ Λ . Termi- nating this RG procedure at the maximum scale LΛ  we arrive at the renormalized QPS amplitude for our system = ( / ) .QPS QPS L λγ γ ξ (77) This equation demonstrates that interaction-induced renor- malization of the QPS amplitude is usually quite important. This effect can be disregarded only for very small values of 1/ ln( / )Lλ << ξ which is not the case here. At all time scales exceeding /L v the system behaves as effectively zero-dimensional one characterized by the QPS amplitude (77). Repeating the whole analysis of voltage fluctuations we again arrive at the general results for the voltage-voltage correlator in the form (45)–(50), with all the Green functions being independent of the spatial coor- dinates. This general result can be rewritten as 2 2 2 (0) 2 coth 2= ( ) ( ) 16 QPS R R L TS S G G e Ω ϕχ ϕχΩ Ω γ Ω    − Ω Ω ×  0 0 0 0( ( ) ( ) ( ) ( )P I P I P I P I× Ω+Φ + Ω−Φ − −Ω−Φ − −Ω+Φ − 0 0 0 0( ) ( ) ( ) ( ))P I P I P I P I− Φ − −Φ + Φ + −Φ + 2 0 2 coth coth 2 2 ( ) ( ) 16 R R I T T G G e ϕχ ϕχ  Ω+Φ Ω   Ω −       + Ω −Ω × 0 0 0( ( ) ( ) ( )P I P I P I× Ω+Φ + Ω+Φ − −Ω−Φ − 0( )) ( ),P I− −Ω−Φ + Ω→ −Ω (78) where ( ) (0) ( ) 2 0 ( ) = e e , iK K RiG t G G ti tP dt ∞ − +χχ χχ χχωω ∫ (79) and the Green functions are equal to 2( ) = , 2(1 ) R RC G i i L ϕϕ π ω λ µω − ωτ − v (80) 4( ) = , 2( 0) (1 ) R RC iG L i i i L ϕχ π λ ω − λ ω+ µω − ωτ −    v v (81) 4 (1 ) ( ) = . 2( 0) (1 ) R RC RC i i G L i i i L χχ π µλ − ωτ ω λ ω+ µω − ωτ −    v v (82) Here =RC sR Cτ is the RC-time and = /Q sR Rµ is the shunt dimensionless conductance. One can further simplify the above expression provided all relevant energy scales, such as Ω , 0IΦ and T remain smaller than both 1/ RCτ and /( )Lλ µv . In that case ( ) 2RG iϕχω ω ≈ π is approximately constant and 2 2 2 0 0 0 1 coth ( ( ) 4 2QPS I S L P I TΩ Ω+Φ ≈ Φ γ Ω+Φ +     0 0 0( ) ( ) ( ))P I P I P I+ Ω+Φ − −Ω−Φ − −Ω−Φ + 2 2 2 0 0 0 1 coth ( ( ) 4 2QPS I L P I T Ω−Φ + Φ γ Ω−Φ +     0 0 0( ) ( ) ( )).P I P I P I+ Ω−Φ − −Ω+Φ − −Ω+Φ (83) The frequency dependence of QPS-induced shot noise power spectrum in the short wire limit is illustrated in Fig. 6. 7. Comparison with four-probe measurement scheme Let us now consider another system configuration dis- played in Fig. 2. We will again stick to the wire Hamiltoni- an wireĤ in its dual representation defined by Eqs. (26), (19) and (25), where, as before, canonically conjugate flux and charge operators ˆ ( )xΦ and ˆ ( )xχ obey the commutation relation (15). The effect of an external current bias is now accounted for within Eq. (19) by means of the shift of the flux operator kinˆ ˆ( ) ( )x x IΦ →Φ + . The phase difference Fig. 6. The frequency dependence of the shot noise spectrum =0IS SΩ Ω− at = 1.025µ , 0 = 0.3RCIΦ τ , 0 2 = 3.33v I L λ Φ µ , and different T in the short wire limit. 1020 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 7 Quantum fluctuations of voltage in superconducting nanowires between the two wire points 1x and 2x can then be de- fined as 1 1 2 2 ˆˆ ˆ( ) ( ) = 2 ( ) x x x x e dx xϕ −ϕ Φ∫ (84) Employing the Josephson relation one easily recovers the operator for the voltage difference between the points 1x and 2x in the form: ( )1 2 0 1ˆ ˆ ˆ= ( ) ( ) . w V x x C ∇χ −∇χ Φ (85) The above expressions allow to directly evaluate volt- age correlators perturbatively in QPSγ . In the case of the four-point measurement scheme of Fig. 2 the calculation is similar to one already carried out above for the two-point measurements. Therefore we can directly proceed to our final results. Evaluating the first moment of the voltage operator V̂〈 〉 we again reproduce Eqs. (43), (65). For the voltage noise power spectrum SΩ we now obtain (0)= e ( ) (0) = ,QPSi tS dt V t V S SΩ Ω Ω Ω〈 〉 +∫ (86) where the term (0)SΩ describes equilibrium voltage noise in the absence of QPS (71) and [13] ( 2 2 02 2 coth 2= ( ) ( ) 24 QPS QPS k k w e dkTS P I CΩ Ω γ     Ω Φ − ππ ∫  )0 0 0( ) ( ) ( )k k kP I P I P I− −Φ + −Φ − Φ + ( 2 2 02 2 coth 2 ( ) ( ) ( ) 24 QPS k k k w e dkT P I C − Ω γ    + Ω Ω −Ω−Φ − ππ ∫   )0 0 0( ) ( ) ( )k k kP I P I P I− Ω+Φ + −Ω+Φ − −Ω−Φ + 2 2 2 2 /coth coth 2 2 ( ) 24 QPS k w I ee T T dk C  Ω + π Ω    γ −        + Ω × ππ ∫  ( 0 0( ) ( ) ( )k k kP I P I−× −Ω Ω+Φ + Ω+Φ − )0 0( ) ( ) ( )k kP I P I− −Ω−Φ − −Ω−Φ + Ω→ −Ω (87) is the voltage noise power spectrum generated by quantum phase slips. Eq. (87) contains the function ( , ) (0,0) ( , ) 2 0 ( ) = e e e iK K RiG x t iG G x tikx i t kP dx dt ∞ − +χχ χχ χχωω ∫ ∫ (88) and geometric form-factors ( )k Ω and ( )k Ω which ex- plicitly depend on 1x and 2x . E.g., setting 1 = /2x L and 2 = /2x L− we obtain 2 sin 2sin 2 2( ) = (4 ) e i L k kL kL k k Ω           Ω πλ +     v v v (2 ) (2 )sin sin 2 2 , 2 2 k L k L k k Ω+ Ω−             + + Ω+ Ω−    v v v v v v (89) ( ) ( )sin sin 2 2( ) = 4 e . i L k k L k L k k Ω  Ω+ Ω−             Ω πλ + Ω+ Ω−       v v v v v v v 2 (90) These form-factors oscillate as functions of Ω due to the interference effect at the boundaries of a thinner wire seg- ment. Such oscillations make the result for the shot noise in general substantially different as compared to that eval- uated for the setup of Fig. 1. For /LΩ >> v one has 2 2 (4 )( ) e ( ), i L k L k Ω πλ Ω ≈ π δ v v (91) 2 2 (4 )( ) ( ) e 2 i L k k L k Ω − πλ  π Ω Ω Ω ≈ δ + +       v vv , 2 L kπ Ω  + δ −    v (92) 2 2 (4 )( ) ( ) 2k k L k− πλ  π Ω Ω −Ω ≈ δ + +       vv . 2 L kπ Ω  + δ −    v (93) Neglecting the contributions (91) and (92) containing fast oscillating factors e i LΩ v and combining the remaining term (93) with Eq. (87), we obtain = /2,QPS aS SΩΩ (94) where aSΩ is defined in Eq. (72). This result implies that shot noise measured by each of our two detectors in the configuration of Fig. 2 is 4 times smaller than that detected with the aid of the setup of Fig. 1. The result (94) is also illustrated in Fig. 7. At 0T → Eq. (94) obviously yields the same threshold behavior (73). The physical reasons for this behavior are the same as before, one should just bear in mind that in the long wire limit and for non-zero Ω the two groups of plasmons — left moving and right moving ones — each carrying the energy /2E become totally uncorrelated im- plying that at = 0T voltage noise can only be detected at < /2EΩ in the agreement with Eq. (73). Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 7 1021 Andrew G. Semenov and Andrei D. Zaikin 8. Concluding remarks In this paper we combined the phase-charge duality ar- guments with Keldysh path integral technique and demon- strated that quantum phase slips may cause voltage fluctua- tions in superconducting nanowires. In the presence of a current bias I quantum tunneling of the magnetic flux quanta 0Φ across the wire yields. Poissonian statistics of such fluctuations. In both limits of longer and shorter nan- owires shot noise exhibits a non-trivial power law depend- ence of its spectrum SΩ on temperature T , external bias I and frequency Ω . We also demonstrated that in the zero temperature limit SΩ decreases with increasing frequency and vanishes beyond a threshold value of Ω . At low enough frequencies SΩ may depend non-monotonously on temperature due to quantum coherent nature of QPS noise. It is important to emphasize that the perturbative in QPSγ approach employed here is fully justified in the so- called “superconducting” phase, i.e. for longer wires with > 2λ [5] and for shorter wires at <s QR R [14]. In the “non-superconducting” regime, i.e. for wires either with < 2λ or with >s QR R the QPS amplitude gets effectively renormalized to higher values and, hence, the perturbation theory eventually turns obsolete. Nevertheless, even in this case our predictions may still remain applicable at high enough temperature, frequency and/or current values. In the opposite low energy limit long wires with < 2λ show an insulating behavior, as follows from the exact solution of the corresponding sine-Gordon model [28]. The same is true also for shorter wires at low energies and sR →∞. This behavior suggests that also voltage fluctuations be- come large in this limit. Finally, we would like to point out that voltage fluctua- tions detected in superconducting nanowires may in general depend on the particular measurement setup. This dependence can be important and should be observed while performing noise measurements in such nanowires. In addition, the results of our theoretical analysis need to be taken into account while optimizing the operation of QPS qubits [29]. This work was supported in part by RFBR grant No. 15-02-08273. 1. K.Yu. Arutyunov, D.S. Golubev, and A.D. Zaikin, Phys. Rep. 464, 1 (2008). 2. 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Controzzi, F.H.L. Essler, and A.M. Tsvelik, Phys. Rev. Lett. 86, 680 (2001) and References therein. 29. J.E. Mooij and C.J.P.M. Harmans, New J. Phys. 7, 219 (2005). Fig. 7. (Color online) The dependence of QPS noise power SΩ (72) on frequency Ω and temperature T at = 2.3λ . 1022 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 7 1. Introduction 2. Basic models and phase-charge duality 3. Keldysh perturbation theory and Green functions 4. I–V curve and voltage noise: general results 5. Relation to ImF-method 6. I–V curve and voltage fluctuations 7. Comparison with four-probe measurement scheme 8. Concluding remarks

Quantum fluctuations of voltage in superconducting nanowires

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