We examine the possibility that pertinent impurities in a condensed matter
system may help in designing quantum devices with enhanced coherent behaviors.
For this purpose, we analyze a field theory model describing Y- shaped
superconducting Josephson networks. We show that a new finite coupling stable
infrared fixed point emerges in its phase diagram; we then explicitly evidence
that, when engineered to operate near by this new fixed point, Y-shaped
networks support two-level quantum systems, for which the entanglement with the
environment is frustrated. We briefly address the potential relevance of this
result for engineering finite-size superconducting devices with enhanced
quantum coherence. Our approach uses boundary conformal field theory since it
naturally allows for a field-theoretical treatment of the phase slips
(instantons), describing the quantum tunneling between degenerate levels.Comment: 11 pages, 5 .eps figures; several changes in the presentation and in
  the figures, upgraded reference

Camalet S

Cardy J

Domenico Giuliano

Kohler H

Oshikawa M

Pasquale Sodano

Tsvelik A M

New Journal of Physics

English

arXiv

Crossref

Frustration of decoherence in Y-shaped superconducting Josephson networks

Giuliano, D.

Sodano, P.

MPG.PuRe

Max Planck Society eDoc Server

Frustration of decoherence in Y -shaped superconducting Josephson networksD. Giuliano1 and P. Sodano21 Dipartimento di Fisica, Universita` della Calabria and I.N.F.N.,Gruppo collegato di Cosenza, Arcavacata di Rende I-87036, Cosenza, Italy2 Max-Planck Institut fu¨r Physik Komplexer Systeme,No¨thnitzer Strasse 38, 01167, Dresden, Germany∗(Dated: October 14, 2007)We show that Y -shaped superconducting Josephson networks allow for engineering two-level quan-tum systems, which may be operated at points where their entanglement with the environment isfrustrated. This happens since Y -shaped networks exhibit a new, finite coupling, infrared fixedpoint in their phase diagram. Our approach uses boundary conformal field theory, which naturallyallows for a full field-theoretical treatment of the phase slips, describing quantum tunneling betweenthe two degenerate levels.PACS numbers: 11.25.Hf, 74.81.Fa, 03.65.YzSuperconducting Josephson devices are promising candi-dates for realizing quantum coherent two-level systems,interesting for their potential relevance to quantum in-formation processing [1]. For the engineering of realisticdevices, one should be able to tame the decoherence, un-avoidably arising from the interaction of the two-levelsystem with both the control circuitry and the plasmonmodes lying outside the subspace spanned by the twolow-lying operating states.Decoherence in a physical system arises since the to-tal state of the two-level system and of its environmentevolves towards an entangled state; thus, it is a quan-tum phenomenon intrinsically different from dissipation,which relies on the transfer of energy from a subsystem toan environment [2]. When a system is such that it is cou-pled to more than one bath, and its entanglement witheach one of the baths is suppressed by the other(s), thedecoherence phenomenon is frustrated [3]. In the follow-ing we evidence that quantum frustration of decoherencemay be induced in a superconducting Josephson device ifits phase diagram admits a finite coupling infrared (IR)fixed point (FFP).Josephson superconducting devices provide remarkablerealizations of quantum systems with impurities, whosephase diagrams, in simple cases, admit only two fixedpoints: an unstable weak coupling fixed point (WFP),and a stable one at strong coupling (SFP), [4]. Neitherone of the above fixed points yields frustration of decoher-ence, since, at weak coupling, there is not even quantumtunneling between the states while, at strong coupling,there is full entanglement between the two degeneratestates and the plasmon modes [4]. In this letter we showthat a FFP emerges in a Y -shaped Josephson junctionnetwork and that its existence is sufficient for inducingquantum frustration of decoherence.∗Permanent address : Dipartimento di Fisica, Universita` di Pe-rugia, and I.N.F.N., Sezione di Perugia, Via A. Pascoli, 06123,Perugia, ItalyFor our analysis it is most convenient to use a descriptionof 1d-superconducting devices in terms of 1+1-boundaryconformal field theories (BCFT)’s, in which the plasmonmodes are described as bosonic conformal fields, whoseboundary conditions are set by the strength of the impu-rity Josephson junction(s) [4–6]. First of all, BCFT ’s en-able to account for the phase slip (instanton) trajectoriesbetween the minima of the boundary potential in terms ofdual vertex operators and, thus, allow for a quantitativeanalysis of the decoherence arising from the interactionof the instantons with the modes of the plasmon field.Furthermore, BCFT ’s provide us with reliable tools notonly for analyzing the phases accessible to junctions ofthree (or more) quantum wires [7] and to models of quan-tum Brownian motion on frustrated planar lattices [8, 9],but also for predicting the response of quantum wires [10]or spin chains [11] to pertinent constrictions. Recently,ultra-cold atoms in Y -shaped potentials have been an-alyzed in [12], evidencing remarkable analogies of thesesystems with the corresponding fermionic systems.A Y - shaped Josephson junction network (see Fig.1) isrealized by weakly connecting three equal superconduct-ing grains forming a ”central region” C, pierced by amagnetic flux Φ¯, with the endpoints of three chains ofJosephson junctions of length L, with lattice step a. The”outer” endpoint of each chain is connected to a bulksuperconductor, at a fixed phase ϕi (i = 1, 2, 3). Thesephases act as control parameters of the device, which maybe regarded as a pertinent planar realization of the tetra-hedral qubit proposed in Ref.[13]. The central triangularregion is regarded as the quantum system, while the threechains provide the environment of plasmon modes. Forthe sake of simplicity, we assume that all the junctionsare identical, except for the ones joining the endpoints ofthe chains to the vertices of the triangle. Defining Ec asthe charging energy of each grain, and EJ as the Joseph-son energy of the junctions (throughout all the paper, weset c = ~ = 1; the electric charge is measured in units ofthe Cooper pair charge e∗), the Hamiltonian describing2the central triangular array C is given byHC =Ec23∑i=1[−i ∂∂φ(i)0−Wg]2−2EJ3∑i=1cos[∆φ(i)0 +ϕ¯3],(1)where ∆φ(i)0 = φ(i)0 − φ(i+1)0 , φ(i)0 is the phase of the su-perconducting order parameter at grain i, ϕ¯ is the fluxΦ¯ in units of the quantum of flux Φ∗0, and Wg is a gatevoltage. If EJ/Ec ¿ 1, W (i)g = N + 12 + h, with integerN and 0 < h < 1/2, the low-energy dynamics is governedonly by the two states with total charge equal to N andto N + 1; the effective Hamitonian may be expressedin terms of spin-1/2 operators and is given by HEffC =−Ech∑3i=1(S(i)0 )z−EJ∑3i=1[ei ϕ¯3 (S(i)0 )+(S(i+1)0 )−+h.c.],with (S(i)0 )± = PCe±iφ(i)0 PC, (S(i)0 )z = PC(−i ∂∂φ(i)0−N − 12 )PC, and PC being an operator projecting ontothe subspace with total charge N or N +1 at each grain.At low energy and long wavelengths, the three chainsmay be described by the Luttinger Liquid Hamiltonian[4]H0 =g4pi3∑i=1∫ L0dx[1v(∂Φi∂t)2+ v(∂Φi∂x)2], (2)where g =√v0v0+4pia∆[1−cos(2akf )] , v =v0√1 + 2 4pia∆[1−cos(2akf )]v0 , with ∆ = Ez − 316 E2JEcand Ez being the interaction energy of charges onneighboring junctions. The Fermi momentum and thebare Fermi velocity are given by kf = arccos(hEcEJ ) andv0 = 2piEJ sin(akf ) [4, 6]. Fixing the phase at the outerboundary of the chains, implies Dirichlet boundaryconditions on Φi(x) at x = L: Φi(L) = ϕi. Furthermore,the charge tunneling between the triangular region Cand the inner boundary of the three chains is describedby a Josephson-like interaction, with nominal strengthλ ¿ EJ . As a result, using Neumann boundaryconditions at the inner boundary, i.e. ∂Φi(0)∂x = 0 ∀i,allows to write the Josephson boundary Hamiltonian asHT = −λ∑3i=1 cos[Φi(0)− φ(i)0 ].A boundary field theory approach allows to trade theinteraction Hamiltonian HEffC + HT with an effectiveboundary Hamiltonian, Hb, involving only Φi(0), andgiven byHb = −2E¯W3∑i=1: cos[~αi · ~χ(0) + χ] : , (3)with χ1(x) = 1√2 [Φ1(x) − Φ2(x)], χ2(x) =1√6[Φ1(x)+Φ2(x)− 2Φ3(x)], ~α1 = (1, 0), ~α2 = (− 12 ,√32 ),~α3 = (− 12 ,−√32 ), χ = atan[3 tan(ϕ¯3 )], andE¯W =(aL) 1g EW , with EW ≈ λ2EJ24(Ec)2h˜2√1 + 2 sin2( ϕ¯3 ).The colons : : denote normal ordering with respect tothe ground state of the plasmon modes, |{0}〉. Hb isthe Bosonic version of the Y-junction Hamiltonian forthree quantum wires [7]. Despite the fact that it doesnot contain the Klein factors, due to the bosonic natureof Cooper pairs, we shall show that there is still a rangeof values of g and χ where the phase diagram exhibitsa FFP. This happens at χ = pi, as for the quantumbrownian motion on a planar frustrated lattice [8], since,in a suitable range of values of g, neither the WFP, orthe SFP, is stable. The second-order Renormalization1/ tan(x)y=v 2y= x/ )( Y2L2EJEJEJΦϕ2ϕ31ϕ λ−FIG. 1: Sketch of the Y -shaped Josephson network; Inset:graphical exact solutions for the energy levels at g = 9/8.Group (RG) equation for the running coupling strengthG = LE¯W near the weakly coupled fixed point is givenby dGd ln( LL0)=(1− 1g)G − 2G2 (L0 is a reference lengthscale), and shows that Hb is a relevant perturbationfor g > 1, while it is irrelevant for g < 1. Since atthe SFP, the fields χj(x) obey Dirichlet boundaryconditions at both boundaries, the strong coupling ef-fective boundary Hamiltonian depends only on the dualfields Θj(x), defined by∂Θj∂x =1v∂χj∂t and1v∂Θj∂t =∂χj∂x ,with a normal mode expansion given by Θj(x, t) =√2g{θ0,j +piPjL vt+ i∑n 6=0 cos[pinxL] αj,nn e−ipinL vt},with [θ0,i, Pj ] = iδi,j and [αi,n, αj,m] = nδn+m,0δi,j .At χ = pi, the eigenvalues p1, p2 of the zero-modeoperators P1, P2 span the honeycomb lattice defined by(p1, p2) =√g2 (n1+√2β1, 1√3 (2n2+n1+2√2β2)), and by(p1, p2) =√g2 (n1− 13+√2β1, 1√3 (2n2+n1−1+2√2β2)),with integer (n1, n2), β1 = (ϕ1 − ϕ2)/(2pi√2),and β2 = (ϕ1 + ϕ2 − 2ϕ3)/(2pi√6). Follow-ing the approach outlined in Ref.[8], the ”dual”boundary Hamiltonian H˜b may be presentedas H˜b = −Y∑3i=1{T−Vi(0) + T+V¯i(0)}, withVi(V¯i) =: exp[−(+)23~αi · ~Θ]:, and ~T being an ef-fective isospin operator, connecting the minima of the3honeycomb lattice of the zero-mode eigenvalues. Y is aneffective coupling strength defined as Y = EJ − E¯W [14].From the O.P.E. of the vertex operators entering H˜b, theRG equation for the running coupling strength y = LYcan be derived as dyd ln(LL0) = (1− 4g9 ) y − 2g3 y3. Thus,one sees that, for χ = pi, and 1 < g < 94 , neither theWFP, or the SFP, are stable. Accordingly, a minimalhypothesis for the phase diagram requires a FFP aty = y∗, with y∗ finite. For instance, for g = 94 − γ, withγ ¿ 1, one obtains y∗ ≈ ( 23) 32 √γ.For y ¿ 1 near the SFP, the low-energy spectrum isgiven by E = piv2L [~p]2 + E′, where ~p = (p1, p2) labelsthe zero-mode contribution, while E′ comes from theplasmon modes. At particular values of β1 and β2, thezero-mode contributions to the total energy coming fromtwo nearest neighboring sites on the honeycomb lattice,may become degenerate with each other: this happens,for instance, if β1 = 1/3√2, β2 = 0. The two degeneratequantum states | ↑〉 and | ↓〉 -labelled by (n1, n2) = (0, 0)on sublattice A and by (n1, n2) = (1, 0) on sublattice B-become degenerate, and are characterized by oppositevalues of the Josephson current flowing across chain-1and chain-2: I1 = −I2 = ±pigve∗3L , while I3 = 0.Quantum tunneling between the two degenerate statesis induced by H˜b, with matrix element −Y . Settingβ2 = 0, and β1 = 1/3√2 + δ/(2pi), with δ/2pi ¿ 1,one easily gets the effective Hamiltonian of a two-levelquantum system asH2 = ²0(δ)I+ ²(δ)σz − Y σx , (4)where ²0(δ) = g2(19 +δ24pi2), ²(δ) = g3δ√2pi, the σa’s arethe Pauli matrices, and δ is a control parameter deter-mined by the phases {ϕi}.To provide an estimate of the entanglement between thetwo-level system described by the Hamiltonian in Eq.(4)and the environment realized by the plasmon modesof the three chains, one should compute χ′′(ω)/ω [3],where χ′′(ω) is the imaginary part of the Fourier trans-form of the “transverse” dynamical spin susceptibilitygiven by χ⊥(ω) = −i∫∞−∞dz2pi{g∗↑,↑(−z)g↓,↓(z + ω) −g∗↓,↓(−z)g↑,↑(z + ω)}, with gσ,σ′(t) = 〈σ|e−i(H0+H˜b)t|σ′〉.Use of the dilute instanton gas approximation, yields anexpression for the Schwinger-Dyson equation for gσ,σ′(ω)given bygσ,σ′(ω) =δσ,σ′ [[g(0)σ¯ ]−1(ω) + Y 2Γσ(ω)] + iY δσ,σ¯′D(ω) , (5)with σ¯ = −σ, g(0)σ (t) = 〈σ|e−iH0t|σ〉, and D(ω) ={[g(0)↑ ]−1(ω) + Y 2Γ↓(ω)}{[g(0)↓ ]−1(ω) + Y 2Γ↑(ω)} + Y 2.The “self-energy” Γσ(ω) is the Fourier transform ofΓ(τ1 − τ2) = 〈{0}| : e±i 23Θ(τ1) :: e∓i 23Θ(τ1) : |{0}〉 =[epiiL vτ1 − epiiL v(τ2+iη)]− 89 g at frequency ω − ²γ , where²↑(↓) = ±²(δ). At low frequencies, one has that Γγ(ω) ≈e89piigΓ[1− 89g](Lpiv) 89 g (ω − ²γ) 89 g−1.When the SFP is stable, that is, for g > 9/4, the bound-ary interaction is irrelevant, and one may neglect cor-rections to the amplitudes of order Y 2, getting χ”(ω)ω ∝[δ(ω + 2²(δ)) − δ(ω − 2²(δ))]/ω. Thus, for g > 9/4,there is no entanglement between the two-level quan-tum system and the plasmon modes. However, one can-not conclude that the decoherence is frustrated, since,in this range of g, there is no tunnel splitting of thetwo degenerate states. At variance, when g < 1, therunning coupling constant y becomes relevant and, bykeeping only the leading contributions in y, one getsχ”(ω)ω ∝ [|2²(δ)+ω|3−169 g−|2²(δ)−ω|3− 169 g]/ω. It shouldbe noticed that, for a finite-size Y -shaped network, theremoval of the degeneracy between the two zero eigen-modes, allows for an extra renormalization of the anoma-lous dimension of the boundary operator, α, [3]. To thezeroth order, one gets α = 49g, while third order contri-butions renormalize α according to dαd ln(LL0) = −αy3. Inthe following, we neglect third order contributions sincethey do not affect the phase diagram of the device. InFig.2, it is presented a plot of χ”(ω)/ω vs. ω near theSFP, the WFP, and the FFP. One sees that, near theWFP, a large part of the spectral weight has moved fromthe side peaks towards ω = 0, thus signaling [3] a strongdecoherence of the two-level system described by Eq.(4).Operating the device near the attractive FFP, not onlyenables one to avoid a strong entanglement with the en-vironment, but also to achieve a good tunnel splittingbetween the two degenerate levels. To verify this, onemay compute χ”(ω)/ω to the leading order in γ, in thelimit for g = 94−γ: as a result one has two peaks centeredaround ±√[²(δ)]2 + (pivL y∗)2, where y∗ is the fixed pointvalue of the running coupling constant. This renormal-ization of the energies of the two-level quantum systemclearly signals a quantum coherent tunneling due to thefrustration of decoherence induced by the weak entan-glement of the system with the environment. However,it should be noticed that the two peaks now have a fi-nite width ∝ pivL (y∗)1+89 g. The results for χ”(ω)/ω arereported in Fig.2.Following the procedure outlined in Ref.[15], for g = 9/8one finds that the exact low-lying energy levels are givenby vL tan(LEv + δ2 )+ Y2E−piv2L− vδ2L= 0. In the inset of Fig.1,it is reported the graphical solution for δ = 0. The twolevels define a two-level quantum system, controllable bytuning δ.For g = 1 + γ′, with γ′ ¿ 1, the existence of a FFPcan be derived also from the RG equation at the WFP,leading to G = G∗ ≈ γ′/2. However, a finite-size deviceoperating near G∗ exhibits a nondegenerate minimum.Only when the FFP is close to the SFP, a finite size, Y -shaped Josephson junction network supports a two-levelquantum system. Furthermore, for a device of finite sizeL, the FFP is stable also against small fluctuations of theflux Φ¯, provided that v/L is sufficiently big. In fact, if,as a result of the fluctuation in Φ¯, the degeneracy point4( ω )"χ / ω0 2∆−2∆ 2ε(δ)−2ε(δ)FFPSFPWFPωFIG. 2: Qualitative behavior of χ”(ω)/ω in the variousregimes.FFPWFPSFP(B)SFP(A)ν>0ν<0FIG. 3: Sketch of the RG flow for χ = pi + ν (ν/pi ¿ 1).χ = pi is displaced by a small amount ν (i.e., χ = pi + ν,with ν/pi ¿ 1), one has that v/L > E¯W sin(ν), whereE¯W sin(ν) is the energy splitting between the minimaof the two triangular sublattices forming the honeycomblattice. A sketch of the RG flow diagram is reported inFig.3. One sees that, for v/L < E¯W sin(ν), the systemflows towards the SFP and that, depending on sgn(ν),the minima of the boundary potential lie on either one ofthe triangular sub-lattices forming the honeycomb latticeNear the stable FFP, the quantum coherence propertiesare universal, since they are independent of the precisevalues of the bare parameters. In fact, for g > 1, inho-mogeneities in the outer chains, induced by differences inthe Josephson energies of the junctions, provide an irrel-evant perturbation, since the pertinent operator scales as(LL0)1−g[4]: thus, inhomogeneities in the device’s fab-rication should not alter the main results of our anal-ysis. Today ’s technology allows to fabricate supercon-ducting devices [16] whose parameters EJ , Ec, Ez, a, L,safely yield values of g ranging from g < 1, to g ∼ 2.The relevant control parameter δ of the effective two-levelsystem supported by the Y - shaped Josephson networkis determined by the phase differences β1, β2, ultimatelydependent on the phase differences between the bulk su-perconductors ending the chains. In a realistic setting, δmay be acted upon if one regards the Y -shaped Joseph-son network as a pertinent planar realization of the su-perconducting tetrahedral qubit, proposed in Ref.[13].We thank I. Affleck, C. Chamon, P. Degiovanni and A.Trombettoni for useful discussions and correspondence.[1] Y. Makhlin, G. Sho¨n, and A. Shnirman, Rev. Mod. Phys.73, (2001), 357.[2] U. Weiss, Quantum Dissiption, Scientific World, Singa-pore.[3] E. Novais, A. H. Castro Neto, L. Borda, I. Affleck, andG. Zarand, Phys. Rev. B 72, (2003), 014417.[4] D. Giuliano and P. Sodano, Nucl. Phys. B 711, (2005),480; Nucl. Phys. B 770, (2007), 332.[5] F. W. J. Hekking and L. I. Glazman, Phys. Rev. B 55,(1997), 6551.[6] L. I. Glazman and A. I. Larkin, Phys. Rev. Lett. 79,3736-3739 (1997).[7] C. Chamon, M. Oshikawa and I. Affleck, Phys. Rev.Lett. 91, (2003), 206403; M. Oshikawa, C. Chamonand I. Affleck, Journal of Statistical Mechanics JS-TAT/2006/P02008.[8] H. Yi and C. L. Kane, Phys. Rev. B 57, R5579 - R5582(1998).[9] I. Affleck, M. Oshikawa and H. Saleur, Nucl. Phys. B594,(2001), 535;[10] C.L. Kane and M. P. Fisher, Phys. Rev. Lett. 68, (1992),1220; Phys. Rev. B46, (1992), 15233;[11] S. Eggert and I. Affleck, Phys. Rev. B46, (1992), 10866;[12] A. Tokuno, M. Oshikawa and E. Demler, ArXiv:cond-mat/0703610;[13] M. V. Feigel’man, L. B. Ioffe, V. B. Geshkenbern, P.Dayal, and G. Blatter, Phys. Rev. Lett 92, 098301(2004); Phys. Rev. B 70, 224524 (2004).[14] A. M. Tsvelik, ”Quantum Field Theory in CondensedMatter Physics”, Cambridge University Press, Cam-bridge, UK.[15] S. Camalet, J. Schriefl, P. Degiovanni, F. Delduc, Euro-physics Letters 68, 37 (2004).[16] D. B. Haviland, K. Andersson, and P. Agren, J. LowTemp. Phys. 124, , 291 (2001).

Frustration of decoherence in $Y$-shaped superconducting Josephson
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Frustration of decoherence in $Y$ -shaped superconducting Josephson networks