We have studied the relaxation and dephasing processes in a superconducting
quantum point contact induced by the interaction with an electromagnetic
environment. Based on a density matrix approach we obtain the rates for the
dissipative dynamics as function of the transmission, the phase difference on
the contact and the external impedance. Our calculation allows to determine the
appropriate range of parameters for the observation of coherent oscillations in
the current through the contact.Comment: 8 pages, 2 figures. To appear in Physical Review 

A. Furusaki

A. Levy Yeyati

A. Martín-Rodero

C. W. Gardiner

C.W.J. Beenakker

D. Averin

D.A. Ivanov

E. Scheer

L.Y. Gorelik

M. A. Despósito

M. Tinkam

M.F. Goffman

N.I. Lundin

T.M. Chang

V. Bouchiat

Y. Nakamura

English

arXiv

Crossref

Controlled dephasing of Andreev states in superconducting quantum point contacts

We have studied the relaxation and dephasing processes in a superconducting quantum point contact induced by the interaction with an electromagnetic environment. Based on a density matrix approach we obtain the rates for the dissipative dynamics as function of the transmission, the phase difference on the contact, and the external impedance. Our calculation allows us to determine the appropriate range of parameters for the observation of coherent oscillations in the current through the contact.Fil: Desposito, Marcelo Arnaldo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; ArgentinaFil: Levy Yeyati, Alfredo. Universidad Autónoma de Madrid; Españ

Desposito, Marcelo Arnaldo

Levy Yeyati, Alfredo

CONICET Digital

RAPID COMMUNICATIONSPHYSICAL REVIEW B, VOLUME 64, 140511~R!Controlled dephasing of Andreev states in superconducting quantum point contactsM. A. Despo´sito1 and A. Levy Yeyati21Departamento de Fı´sica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina2Departamento de Fı´sica de la Materia Condensada C-V. Facultad de Ciencias, Universidad Auto´noma de Madrid,E-28049 Madrid, Spain~Received 28 March 2001; published 21 September 2001!We have studied the relaxation and dephasing processes in a superconducting quantum point contact inducedby the interaction with an electromagnetic environment. Based on a density matrix approach we obtain therates for the dissipative dynamics as function of the transmission, the phase difference on the contact, and theexternal impedance. Our calculation allows us to determine the appropriate range of parameters for the obser-vation of coherent oscillations in the current through the contact.DOI: 10.1103/PhysRevB.64.140511 PACS number~s!: 74.80.Fp, 73.63.2b, 73.63.RtIn a superconducting quantum-point contact ~SQPC! thezero-voltage transport properties are determined by the so-called Andreev states within the superconducting energy gapD , which are given by1E6~f!56DA12t sin2~f/2!, ~1!where f is the phase difference between the superconductingelectrodes and t is the normal transmission coefficient foreach conduction channel. Atomic contacts, produced byscanning tunneling microscope or the mechanically control-lable break-junction ~MCBJ! technique, have provided an al-most ideal realization of an SQPC with a few conductionchannels whose transmissions can be determinedexperimentally.2 Different methods for Andreev level spec-troscopy have been suggested.3The Andreev states in a single channel SQPC constitutean interesting realization of a two-level system with wellcharacterized parameters, which could be proposed as a solidstate realization of a quantum qubit.4 The phase differencethrough the contact can be fixed by the magnetic flux in asuperconducting ring geometry like the one depicted in Fig.1~a!. In addition, by varying the external flux one could pre-pare the system in a given linear combination of the Andreevstates, in close resemblance to the case of a spin-1/2 in amagnetic field. The phase would be, however, affected byfluctuations originated in the coupling of the ring with itselectromagnetic environment, characterized by a finite im-pedance Z(v). These fluctuations provide a source of relax-ation and dephasing for the dynamics of our two-levelsystem.5The aim of this work is to investigate this dissipative dy-namics in order to determine the conditions for the observa-tion of quantum-coherence effects in this system.For a single mode SQPC in the low bias regime one canneglect the continuous part of the quasiparticle spectrum andrestrict the description to the subgap states, introducing thefollowing Hamiltonian,6Hˆ 05DH cosS f2 D sˆ z1r sinS f2 D sˆ xJ , ~2!where sˆ i are the Pauli matrix and r5A12t is the normalreflection amplitude of the contact. The Hamiltonian adopts0163-1829/2001/64~14!/140511~4!/$20.00 64 1405the form ~2! in the basis of ballistic states and becomes di-agonal Hˆ 05E1(f) sˆ z in the basis of the Andreev states. Theballistic and the Andreev states are represented in Fig. 1~b!.In the ring geometry, the phase on the contact will begiven by f5f02df , where f05(2e/\)F ~being F theexternal magnetic flux! and df represents the phase fluctua-tions due to the electromagnetic environment. If the imped-ance viewed by the contact is small compared to the resis-tance quantum RQ5h/4e2 one can expand Eq. ~2! to the firstorder in df to obtain the coupling between the subgap levelsand the environment. Then, the Hamiltonian can be writtenas Hˆ 0(f)5Hˆ 0(f0)1Hˆ c , where the coupling Hˆ c52(\/2e)Iˆdf is proportional to the current operator Iˆ . Per-forming a rotation to the basis of the Andreev states, thecurrent operator is given byIˆ5eD2\E1~f0!HA12t sˆ x2t sinS f02 D cosS f02 Dsˆ zJ . ~3!On the other hand, a generic environment can be repre-sented by a set of L-C circuits, with phases fn , such thatdf5(nfn . The relevant information of the environment iscontained in the phase-phase correlation function7FIG. 1. ~a! Schematic circuit: a superconducting quantum pointcontact ~SQPC! inserted in a superconducting ring threaded by amagnetic flux F . Z(v) denotes the effective impedance seen by thecontact. ~b! Andreev levels with phase dependent energies E6 ~fulllines!. The dotted and dashed lines indicate the ballistic states.©2001 The American Physical Society11-1RAPID COMMUNICATIONSM.A. DESPO´ SITO AND A. LEVY YEYATI PHYSICAL REVIEW B 64 140511~R!C~ t !5(n ,m^fn~ t !fm~0 !&51RQE0‘dvReZt~v!v H cothS b\v2 D cos vt2isin vtJ ,~4!where Zt(v) is the effective impedance of the circuit as seenfrom the contact and b is the inverse temperature.Projection operator techniques allow to derive a time-convolutionless generalized master equation8 ~GME! for thereduced density matrix rˆ of the Andreev levels.9 Up to sec-ond order in the coupling Hamiltonian this equation is givenbyrˆ˙ ~ t !1i\@Hˆ 0 ,rˆ ~ t !#521\2E0tdt$@Iˆ ,Iˆ~2t!rˆ ~ t !#C~t!2@Iˆ ,rˆ ~ t !Iˆ~2t!#C*~t!%, ~5!where Iˆ(t) denotes the current operator ~3! in the interactionrepresentation.One can extract the evolution equations for the matrixelements of the reduced density projecting the GME ~5! inthe $u1&,u2&% basis of the Andreev states. Taking into ac-count that r111r2251, and defining rR and r I as the realand imaginary part of the off-diagonal matrix element r12 ,we finally get the rate equationsr˙ 152~W11W2!r11W12LrR , ~6!r˙ R5Vr I1hr˙ 1 , ~7!r˙ I52~W11W21hL!r I2~V12Vr!rR1hVr~2r121 !1Y , ~8!where r15r11 , h5t sin(f0/2)cos(f0/2)/A12t and V5(E12E2)/\ .The coefficients W6 entering in Eqs. ~6! to ~8! are theupward and downward transition rates which determine thesystem relaxation. The dephasing, i.e., the decay of the offdiagonal elements, is also controlled by the coefficient L ,while Vr plays the role of a level shift as discussed below.The explicit form of all these coefficients depends on theactual impedance Zt(v). Typically, this impedance would bedetermined by the measuring circuit, which could consist ofa SQUID inductively coupled to the ring with the SQPC. Weshall denote by M and L the mutual inductance between thering and the SQUID and the self-inductance of the SQUIDrespectively. The SQUID is also characterized by a finiteresistance R. The charging effects associated with the contactcapacitance can be neglected in the low impedance regimethat we are considering.7 Under these conditions the real partof Zt will be given byReZt~v!5v2M 2RR21v2L2. ~9!14051If the environment has a large cuttoff frequency (g5R/L@D/\), we can make the Markov approximation inthe evolution equations ~6!–~8!. In this case, the involvedcoefficients can be written asW75W02H n~V!11,n~V!,~10!L5W02h2b\V, ~11!Vr5W02H 12cotS 2b\g D2 2g2b\ (k51‘ nk@g22nk2#@V21nk2#J ,~12!Y5W02hVg, ~13!whereW025p4D\ReZt~V!RQ12t@12t sin2~f/2!#3/2~14!is the downward transition rate at zero temperature, n(V) isthe Bose mean occupation number, and nk52pk/b\ are theMatsubara frequencies.The actual value of the coefficients is controlled by theenvironment parameters M, L, and R. The experimental setupcan be designed in order to fix these parameters in the ap-propriate range to observe quantum-coherence effects.Longer decoherence times would be obtained by reducing Mas much as possible. However, SQUID parameters must sat-isfy certain constraints, imposed by the need to avoid hyster-esis and thermal fluctuations.12 These considerations permitto estimate a minimal value of M of the order of 0.5 nH ,with L50.1 nH and R520 V .13 Assuming that the super-conducting material is aluminum (DAl;0.18 meV), one ob-tains values of W02 in the nsec21 range. This rate fixes theorder of magnitude of the coefficients ~10!–~13!.The system of differential equations ~6!–~8! is character-ized by a set of eigenvalues which determine the typicaltimes for the dissipative dynamics. In the limit of small im-pedance these eigenvalues are given by the following simpleexpressions:l152~W11W2! ,~15!l2,35212 ~W11W212hL!6i~V1Vr!.Notice that due to the presence of the environment, theAndreev levels are shifted according to E˜ 6.E66Vr/2.However, for the range of parameters we are considering thisrenormalization is small and can be neglected.Solving Eqs. ~6!–~8! at the lowest order in V21 ~which isequivalent to implement the rotating wave approximation14!one realizes that the eigenvalues ~15! are directly connectedto the relaxation and dephasing rates by the relations GR.2l1 and GD.2Re$l2,3%.1-2RAPID COMMUNICATIONSCONTROLLED DEPHASING OF ANDREEV STATES IN . . . PHYSICAL REVIEW B 64 140511~R!The approximate evolution equations can then be writtenasr15W1GR~12e2GRt!1r1~0 !e2GRt , ~16!rR5e2GDt@rR~0 !cos Vt1r I~0 !sin Vt# , ~17!r I5e2GDt@r I~0 !cos Vt2rR~0 !sin Vt# , ~18!which clearly shows that the diagonal elements of the densitymatrix decay exponentially towards their equilibrium valueson a time scale given by 1/GR while the nondiagonal ele-ments perform oscillations with frequency V which aredamped on a typical time 1/GD .This behavior is reflected in the evolution of the currentmean value which is given by^I~ t !&5I0S 122W1GR D12I0e2GRtS W1GR2r1~0 ! D14eS D\ D2 A12tVe2GDt@rR~0 !cos Vt1r I~0 !sin Vt# , ~19!where I05e(D/\)2tsinf/V is the equilibrium current atzero temperature.According to Eq. ~19! quantum coherence would manifestas damped Rabi oscillations in the mean current through thecontact. These oscillations appear for an initial condition inwhich both the lower and the upper Andreev state are popu-lated. An arbitrary initial condition can be in principlereached from the equilibrium situation by imposing a giventime-dependent evolution to the phase. The amplitude of theoscillations are maximal for f5p where they become trans-mission independent. However, by varying the transmission,the frequency of the oscillations could be tuned in order toreach an experimentally accessible range. For instance, in thecase of aluminum and transmission t50.99 the frequency Vwould be of the order of 331010 Hz.The decay of these oscillations is controlled by thedephasing rate. Figure 2 illustrates the dependence of therelaxation and dephasing rates on the phase f0, the contacttransmission t and the temperature. At zero temperature wehave GR52GD5W02, which means that both rates reach amaximum at f05p regardless of the contact transmission.This maximum decrease with transmission as A12t . At fi-nite but low temperatures we have GR5W02(112e2\V/kT)and GD5W02( 12 12h2kT/\V). As expected, both rates in-14051crease with temperature but the dephasing rate increases lin-early instead of exponentially. The maximum of GR is lo-cated at f05p for all range of temperatures and behave asA12t coth bDA12t . On the other hand, GD develops a dipat f05p and exhibits a double peaked structure locatedf052 arccos6A11@A(12t)/221#/t for high tempera-tures. In this classical limit we get GR5W022kT/\V andGD5GR( 12 1h2).In summary, we have studied the dissipative dynamics ofthe Andreev states in a SQPC coupled to an electromagneticenvironment characterized by a generic frequency-dependentimpedance. Our results show that with the appropriate choiceof the environment parameters one can reach values of thedephasing and relaxation rates within the nsec21 range. ASQPC in a ring geometry thus provides an interesting real-ization of a solid-state two-level system in which quantuminterference effects could be observed.We acknowledge useful discussions with C. Urbina, D.Esteve, and A. Martı´n-Rodero. We are also indebted to F.Vazquez and A. Raguet for their stimulating comments. Thiswork was supported by the Spanish CICyT under ContractNo. PB97-0044.FIG. 2. Relaxation and dephasing rates GR and GD as functionof the phase f for transmission coefficients t50.9, 0.95, and 0.99~from top to bottom! and two different values of the parameter bD .The downward transition rate at zero temperature W02 coincides inthe graphic scale with GR for bD5100.1 A. Furusaki and M. Tsukada, Physica B 165&166, 967 ~1990!;C.W.J. Beenakker, Phys. Rev. Lett. 67, 3836 ~1991!.2 E. Scheer et al., Phys. Rev. Lett. 78, 3535 ~1997!; E. Scheeret al., Nature ~London! 394, 154 ~1998!.3 L.Y. Gorelik et al., Phys. Rev. Lett. 81, 2538 ~1998!; A. LevyYeyati, J.C. Cuevas, and A. Martı´n-Rodero, in Photons andLocal Probes, Vol. 300 of NATO Advanced Study Institute, Ser-ies E: Applied Sciences, edited by O. Marti and R. Mo¨ller ~Klu-wer, Dordrecht, 1995!.4 So far the most advanced results have been obtained on qubitsbased on flux states in small superconducting loops and oncharge states in small superconducting islands. See Y. Nakamura1-3RAPID COMMUNICATIONSM.A. DESPO´ SITO AND A. LEVY YEYATI PHYSICAL REVIEW B 64 140511~R!et al., Physica B 280, 405 ~2000!; and V. Bouchiat et al., J.Supercond. 12, 789 ~1999!.5 Other sources of dephasing for Andreev states like Flicker noisehave been studied recently by N.I. Lundin and Y.M. Galperin,Phys. Rev. B 63, 094505 ~2001!.6 D.A. Ivanov and M.V. Feigel’man, Phys. Rev. B 59, 8444 ~1999!.7 G.L. Ingold and Yu. V. Nazarov, in Single Charge Tunneling, Vol.294 of NATO Advanced Study Institute, Series B: Physics, editedby H. Grabert and M.H. Devoret ~Plenum, New York, 1991!.8 T.M. Chang and J.L. Skinner, Physica A 193, 483 ~1993!.9 Notice that our analysis is restricted to the subgap states neglect-ing coupling with the continuous part of the spectrum. Otherinteractions, such as electron-phonon coupling, would introduce14051a broadening of the subgap states ~Ref. 10!. For aluminumatomic contacts this broadening is, however, small ~Ref. 11! andwould not introduce important corrections to the relaxation anddephasing rates.10 A. Martı´n-Rodero, A. Levy Yeyati, and F.J. Garcı´a-Vidal, Phys.Rev. B 53, 8891 ~1996!; D. Averin and H.T. Imam, Phys. Rev.Lett. 76, 3814 ~1996!.11 M.F. Goffman et al., Phys. Rev. Lett. 85, 170 ~2000!.12 M. Tinkam, Introduction to Superconductivity ~McGraw-Hill,New York, 1996!.13 C. Urbina ~private communication!.14 C. W. Gardiner, Quantum Noise ~Springer-Verlag, Heidelberg,1991!.1-4

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tinaRAPID COMMUNICATIONSPHYSICAL REVIEW B, VOLUME 64, 140511~R!Controlled dephasing of Andreev states in superconducting quantum point contactsM. A. Despósito1 and A. Levy Yeyati21Departamento de Fı´sica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argen2Departamento de Fı´sica de la Materia Condensada C-V. Facultad de Ciencias, Universidad Auto´noma de Madrid,E-28049 Madrid, Spain~Received 28 March 2001; published 21 September 2001!We have studied the relaxation and dephasing processes in a superconducting quantum point contact inducedby the interaction with an electromagnetic environment. Based on a density matrix approach we obtain therates for the dissipative dynamics as function of the transmission, the phase difference on the contact, and theexternal impedance. Our calculation allows us to determine the appropriate range of parameters for the obser-vation of coherent oscillations in the current through the contact.DOI: 10.1103/PhysRevB.64.140511 PACS number~s!: 74.80.Fp, 73.63.2b, 73.63.Rtsgatinrbrol-onine-tuelsonige-eeinbtsimxvey-vacaathptdi-ebea-ed-sis-elstenthepre-isintatheIn a superconducting quantum-point contact~SQPC! thezero-voltage transport properties are determined by thecalled Andreev states within the superconducting energyD, which are given by1E6~f!56DA12t sin2~f/2!, ~1!wheref is the phase difference between the superconducelectrodes andt is the normal transmission coefficient foeach conduction channel. Atomic contacts, producedscanning tunneling microscope or the mechanically contlable break-junction~MCBJ! technique, have provided an amost ideal realization of an SQPC with a few conductichannels whose transmissions can be determexperimentally.2 Different methods for Andreev level spectroscopy have been suggested.3The Andreev states in a single channel SQPC constian interesting realization of a two-level system with wcharacterized parameters, which could be proposed as astate realization of a quantum qubit.4 The phase differencethrough the contact can be fixed by the magnetic flux isuperconducting ring geometry like the one depicted in F1~a!. In addition, by varying the external flux one could prpare the system in a given linear combination of the Andrstates, in close resemblance to the case of a spin-1/2magnetic field. The phase would be, however, affectedfluctuations originated in the coupling of the ring with ielectromagnetic environment, characterized by a finitepedanceZ(v). These fluctuations provide a source of relaation and dephasing for the dynamics of our two-lesystem.5The aim of this work is to investigate this dissipative dnamics in order to determine the conditions for the obsertion of quantum-coherence effects in this system.For a single mode SQPC in the low bias regime oneneglect the continuous part of the quasiparticle spectrumrestrict the description to the subgap states, introducingfollowing Hamiltonian,6Ĥ05DH cosS f2 D ŝz1r sinS f2 D ŝxJ , ~2!where ŝ i are the Pauli matrix andr 5A12t is the normalreflection amplitude of the contact. The Hamiltonian ado0163-1829/2001/64~14!/140511~4!/$20.00 64 1405o-pgyl-dtellida.vay--l-nndesthe form ~2! in the basis of ballistic states and becomesagonalĤ05E1(f) ŝz in the basis of the Andreev states. Thballistic and the Andreev states are represented in Fig. 1~b!In the ring geometry, the phase on the contact willgiven by f5f02df, where f05(2e/\)F ~being F theexternal magnetic flux! anddf represents the phase fluctutions due to the electromagnetic environment. If the impance viewed by the contact is small compared to the retance quantumRQ5h/4e2 one can expand Eq.~2! to the firstorder indf to obtain the coupling between the subgap levand the environment. Then, the Hamiltonian can be writas Ĥ0(f)5Ĥ0(f0)1Ĥc , where the coupling Ĥc52(\/2e) Îdf is proportional to the current operatorÎ . Per-forming a rotation to the basis of the Andreev states,current operator is given byÎ 5eD2\E1~f0!HA12t ŝx2t sinS f02 D cosS f02 D ŝzJ . ~3!On the other hand, a generic environment can be resented by a set of L-C circuits, with phasesfn , such thatdf5(nfn . The relevant information of the environmentcontained in the phase-phase correlation function7FIG. 1. ~a! Schematic circuit: a superconducting quantum pocontact~SQPC! inserted in a superconducting ring threaded bymagnetic fluxF. Z(v) denotes the effective impedance seen bycontact.~b! Andreev levels with phase dependent energiesE6 ~fulllines!. The dotted and dashed lines indicate the ballistic states.©2001 The American Physical Society11-1eneentricthow.theoeeIDitacimrtinheap-cts.gsat-ter-it-cal-plethethisRAPID COMMUNICATIONSM.A. DESPÓSITO AND A. LEVY YEYATI PHYSICAL REVIEW B 64 140511~R!C~ t !5(n,m^fn~ t !fm~0!&51RQE0`dvReZt~v!v H cothS b\v2 D cosvt2 isinvtJ ,~4!whereZt(v) is the effective impedance of the circuit as sefrom the contact andb is the inverse temperature.Projection operator techniques allow to derive a timconvolutionless generalized master equation8 ~GME! for thereduced density matrixr̂ of the Andreev levels.9 Up to sec-ond order in the coupling Hamiltonian this equation is givbyṙ̂~ t !1i\@Ĥ0 ,r̂~ t !#521\2E0tdt$@ Î , Î ~2t!r̂~ t !#C~t!2@ Î ,r̂~ t ! Î ~2t!#C* ~t!%, ~5!whereÎ (t) denotes the current operator~3! in the interactionrepresentation.One can extract the evolution equations for the maelements of the reduced density projecting the GME~5! inthe $u1&,u2&% basis of the Andreev states. Taking into acount thatr111r2251, and definingrR andr I as the realand imaginary part of the off-diagonal matrix elementr12 ,we finally get the rate equationsṙ152~W11W2!r11W12LrR , ~6!ṙR5Vr I1hṙ1 , ~7!ṙ I52~W11W21hL!r I2~V12V r !rR1hV r~2r121!1Y, ~8!where r15r11 , h5t sin(f0/2)cos(f0/2)/A12t and V5(E12E2)/\.The coefficientsW6 entering in Eqs.~6! to ~8! are theupward and downward transition rates which determinesystem relaxation. The dephasing, i.e., the decay of thediagonal elements, is also controlled by the coefficientL,while V r plays the role of a level shift as discussed beloThe explicit form of all these coefficients depends onactual impedanceZt(v). Typically, this impedance would bdetermined by the measuring circuit, which could consista SQUID inductively coupled to the ring with the SQPC. Wshall denote byM andL the mutual inductance between thring and the SQUID and the self-inductance of the SQUrespectively. The SQUID is also characterized by a finresistanceR. The charging effects associated with the contcapacitance can be neglected in the low impedance regthat we are considering.7 Under these conditions the real paof Zt will be given byReZt~v!5v2M2RR21v2L2. ~9!14051-x-effefeteIf the environment has a large cuttoff frequency (g5R/L@D/\), we can make the Markov approximationthe evolution equations~6!–~8!. In this case, the involvedcoefficients can be written asW75W02H n~V!11,n~V!, ~10!L5W02h2b\V, ~11!V r5W02H 12cotS 2b\g D2 2g2b\ (k51` nk@g22nk2#@V21nk2#J ,~12!Y5W02hVg, ~13!whereW025p4D\ReZt~V!RQ12t@12t sin2~f/2!#3/2~14!is the downward transition rate at zero temperature,n(V) isthe Bose mean occupation number, andnk52pk/b\ are theMatsubara frequencies.The actual value of the coefficients is controlled by tenvironment parametersM, L, andR. The experimental setupcan be designed in order to fix these parameters in thepropriate range to observe quantum-coherence effeLonger decoherence times would be obtained by reducinMas much as possible. However, SQUID parameters mustisfy certain constraints, imposed by the need to avoid hysesis and thermal fluctuations.12 These considerations permto estimate a minimal value ofM of the order of 0.5nH,with L50.1 nH andR520 V.13 Assuming that the superconducting material is aluminum (DAl;0.18 meV), one ob-tains values ofW02 in the nsec21 range. This rate fixes theorder of magnitude of the coefficients~10!–~13!.The system of differential equations~6!–~8! is character-ized by a set of eigenvalues which determine the typitimes for the dissipative dynamics. In the limit of small impedance these eigenvalues are given by the following simexpressions:l152~W11W2! ,~15!l2,35212~W11W212hL!6 i ~V1V r !.Notice that due to the presence of the environment,Andreev levels are shifted according toẼ6.E66V r /2.However, for the range of parameters we are consideringrenormalization is small and can be neglected.Solving Eqs.~6!–~8! at the lowest order inV21 ~which isequivalent to implement the rotating wave approximation14!one realizes that the eigenvalues~15! are directly connectedto the relaxation and dephasing rates by the relationsGR.2l1 andGD.2Re$l2,3%.1-2tesies-ntstthipeenth-iotthetwan-s lin-astedofeticdenticetheal-mD.F.hisactRAPID COMMUNICATIONSCONTROLLED DEPHASING OF ANDREEV STATES IN . . . PHYSICAL REVIEW B64 140511~R!The approximate evolution equations can then be writasr15W1GR~12e2GRt!1r1~0!e2GRt , ~16!rR5e2GDt@rR~0!cosVt1r I~0!sinVt#, ~17!r I5e2GDt@r I~0!cosVt2rR~0!sinVt#, ~18!which clearly shows that the diagonal elements of the denmatrix decay exponentially towards their equilibrium valuon a time scale given by 1/GR while the nondiagonal elements perform oscillations with frequencyV which aredamped on a typical time 1/GD .This behavior is reflected in the evolution of the curremean value which is given by^I ~ t !&5I 0S 122W1GR D12I 0e2GRtS W1GR2r1~0! D14eS D\ D2 A12tVe2GDt@rR~0!cosVt1r I~0!sinVt#, ~19!where I 05e(D/\)2tsinf/V is the equilibrium current azero temperature.According to Eq.~19! quantum coherence would manifeas damped Rabi oscillations in the mean current throughcontact. These oscillations appear for an initial conditionwhich both the lower and the upper Andreev state are polated. An arbitrary initial condition can be in principlreached from the equilibrium situation by imposing a givtime-dependent evolution to the phase. The amplitude ofoscillations are maximal forf5p where they become transmission independent. However, by varying the transmissthe frequency of the oscillations could be tuned in orderreach an experimentally accessible range. For instance, incase of aluminum and transmissiont50.99 the frequencyVwould be of the order of 3 1010 Hz.The decay of these oscillations is controlled by tdephasing rate. Figure 2 illustrates the dependence ofrelaxation and dephasing rates on the phasef0, the contacttransmissiont and the temperature. At zero temperaturehaveGR52GD5W02 , which means that both rates reachmaximum atf05p regardless of the contact transmissioThis maximum decrease with transmission asA12t. At fi-nite but low temperatures we haveGR5W02(112e2\V/kT)and GD5W02( 12 12h2kT/\V). As expected, both rates in14051ntytenu-en,ohehee.crease with temperature but the dephasing rate increaseearly instead of exponentially. The maximum ofGR is lo-cated atf05p for all range of temperatures and behaveA12t cothbDA12t. On the other hand,GD develops a dipat f05p and exhibits a double peaked structure locaf052 arccos6A11@A(12t)/221#/t for high tempera-tures. In this classical limit we getGR5W022kT/\V andGD5GR(12 1h2).In summary, we have studied the dissipative dynamicsthe Andreev states in a SQPC coupled to an electromagnenvironment characterized by a generic frequency-depenimpedance. Our results show that with the appropriate choof the environment parameters one can reach values ofdephasing and relaxation rates within thensec21 range. ASQPC in a ring geometry thus provides an interesting reization of a solid-state two-level system in which quantuinterference effects could be observed.We acknowledge useful discussions with C. Urbina,Esteve, and A. Martı´n-Rodero. We are also indebted toVazquez and A. Raguet for their stimulating comments. Twork was supported by the Spanish CICyT under ContrNo. PB97-0044.FIG. 2. Relaxation and dephasing ratesGR andGD as functionof the phasef for transmission coefficientst50.9, 0.95, and 0.99~from top to bottom! and two different values of the parameterbD.The downward transition rate at zero temperatureW02 coincides inthe graphic scale withGR for bD5100.r-ubitsonura1A. Furusaki and M. Tsukada, Physica B165&166, 967 ~1990!;C.W.J. Beenakker, Phys. Rev. Lett.67, 3836~1991!.2E. Scheeret al., Phys. Rev. Lett.78, 3535 ~1997!; E. Scheeret al., Nature~London! 394, 154 ~1998!.3L.Y. Gorelik et al., Phys. Rev. Lett.81, 2538 ~1998!; A. LevyYeyati, J.C. Cuevas, and A. Martı´n-Rodero, inPhotons andLocal Probes, Vol. 300 of NATO Advanced Study Institute, Seies E: Applied Sciences, edited by O. Marti and R. Mo¨ller ~Klu-wer, Dordrecht, 1995!.4So far the most advanced results have been obtained on qbased on flux states in small superconducting loops andcharge states in small superconducting islands. See Y. Nakam1-3oisinleeucnd.,RAPID COMMUNICATIONSM.A. DESPÓSITO AND A. LEVY YEYATI PHYSICAL REVIEW B 64 140511~R!et al., Physica B280, 405 ~2000!; and V. Bouchiatet al., J.Supercond.12, 789 ~1999!.5Other sources of dephasing for Andreev states like Flicker nhave been studied recently by N.I. Lundin and Y.M. GalperPhys. Rev. B63, 094505~2001!.6D.A. Ivanov and M.V. Feigel’man, Phys. Rev. B59, 8444~1999!.7G.L. Ingold and Yu. V. Nazarov, inSingle Charge Tunneling, Vol.294 ofNATO Advanced Study Institute, Series B: Physics,editedby H. Grabert and M.H. Devoret~Plenum, New York, 1991!.8T.M. Chang and J.L. Skinner, Physica A193, 483 ~1993!.9Notice that our analysis is restricted to the subgap states neging coupling with the continuous part of the spectrum. Othinteractions, such as electron-phonon coupling, would introd14051e,ct-rea broadening of the subgap states~Ref. 10!. For aluminumatomic contacts this broadening is, however, small~Ref. 11! andwould not introduce important corrections to the relaxation adephasing rates.10A. Martı́n-Rodero, A. Levy Yeyati, and F.J. Garcı´a-Vidal, Phys.Rev. B 53, 8891 ~1996!; D. Averin and H.T. Imam, Phys. RevLett. 76, 3814~1996!.11M.F. Goffmanet al., Phys. Rev. Lett.85, 170 ~2000!.12M. Tinkam, Introduction to Superconductivity~McGraw-Hill,New York, 1996!.13C. Urbina~private communication!.14C. W. Gardiner,Quantum Noise~Springer-Verlag, Heidelberg1991!.1-4

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Controlled dephasing of Andreev states in superconducting quantum
  point-contacts

Controlled dephasing of Andreev states in superconducting quantum point-contacts

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