Flux noise in a superconducting transmission line
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A ug Flux noise in a superconducting transmission line
F. T. Vasko ∗ QK Applications, San Francisco, CA 94033, USA (Dated: September 26, 2018)We study a superconducting transmission line (TL) formed by distributed LC oscillators andexcited by external magnetic fluxes which are aroused from random magnetization (A) placed insubstrate or (B) distributed at interfaces of a two-wire TL. Low-frequency dynamics of a randommagnetic field is described based on the diffusion Langevin equation with a short-range source causedby (a) random amplitude or (b) gradient of magnetization. For a TL modeled as a two-port networkwith open and shorted ends, the effective magnetic flux at the open end has non-local dependencyon noise distribution along the TL. The flux-flux correlation function is evaluated and analyzed forthe regimes (Aa), (Ab). (Ba), and (Bb). Essential frequency dispersion takes place around theinverse diffusion time of random flux along the TL. Typically, noise effect increases with size fasterthan the area of TL. The flux-flux correlator can be verified both via the population relaxation rateof the qubit, which is formed by the Josephson junction shunted by the TL with flux noises, andvia random voltage at the open end of the TL.
I. INTRODUCTION
Recent progress in the implementation and the bench-marking of the quantum-information-processing proto-cols is based on different types of superconducting fluxqubits connected through TLs, which provide inter-qubitlinks, control of qubit’s states, and readout, see [1–4] andthe references therein. These elements form an essen-tial part of quantum hardware and their effect on thefidelity of different computational protocols should beelucidated. Although the dynamic properties of multi-qubit devices have been analyzed with the use of thelumped-element approach, [5] the effects of environmenton quantum hardware has not been investigated com-pletely. Superconducting circuits have been studied ex-tensively for sensor applications at temperatures around1 K, [6] where noise arises due to two-level defects. Foroperating temperatures of quantum hardware, near 10 − K, microscopic mechanisms of interaction with flux noisesare under investigation now, see a detailed discussion[3] and recent experimental data [7, 8] with referencestherein. In particular, mechanisms of low-frequency fluxnoise in TL have not been analyzed and their effects onquantum hardware, including spectral and size depen-dencies, have not been characterized. Thus, it is timelynow to study flux noises caused by random magnetiza-tion around TLs and to consider physical effects causedby these noises.In this paper, we consider a two-wire TL (Fig. 1a)formed by distributed LC oscillators under the influenceof different types of low-frequency flux noises. Based onthe standard electrodynamic approach, [9, 10] we con-sider the TL as a two-port network and evaluate a ran-dom e.m.f. due to fluctuations of magnetic flux acrossthe area of the TL. Within the low-frequency approxi-mation, when the characteristic frequency of TL ω LC is ∗ Electronic address: [email protected]
FIG. 1: (a) Geometry of a two-wire TL with random exter-nal flux (the green arrows) penetrated through the interval( x, x + dx ). (b) TL placed on substrate with random mag-netization (the red arrows); a is the distance between thewires of radius r w . (c) Y0Z cross section of a two-wire TLwith fluctuations of tangential magnetization along the wires(the red arrows). Gray curves sketch the bipolar coordinatesystem ( τ, σ ). (d) Circuit model for a TL of length ℓ underlow-frequency noise. Part of the TL between x and x + dx with inductance l TL dx , capacitance c TL dx , and random dropof potential ∆ v xt is shown in the left cell. Shunting of the TLvia two-pole scheme of admittance y ω and via a Josephsonjunction is shown at the left and the right, respectively. greater than ω , this evaluation gives an effective non-local flux through a TL of length ℓ as ∆Φ xt | x = ℓ/ x = − ℓ/ + ℓ ( d ∆Φ xt /dx ) (cid:12)(cid:12)(cid:12) x = ℓ/ x = − ℓ/ with random flux along the TL,∆Φ xt , and its derivative taken at the ends of TL, x = ± ℓ/
2, see Eq. (12) below. We consider the case of classical external noises which are aroused from randommagnetization (A) placed in substrate (Fig. 1b) or (B)distributed along interfaces of the superconducting wireswhich form the TL (Fig. 1c).The low-frequency dynamics of a random magneticfield is governed by the Langevin equation with short-range sources caused by (a) random amplitude or (b)gradient of magnetization. We employ the hydrody-namic approach with an isotropic diffusion coefficient anda relaxation rate which are introduced phenomenologi-cally. A general out-of-equilibrium regime of fluctuationsis considered here, without any restrictions imposed bythe fluctuation-dissipation theorem. The models of noisesuggested here are consistent with recent experimentaldata reporting effects of random magnetization in a sub-strate and along interface regions, see [11] and [12, 13]respectively, as well as with an essential stray populationof qubit (hot qubits) at low temperatures, reported onin Refs. [13–15]. Beside of this, a typical flux qubit isformed by the Josephson junction’s loop shunted by longTL (not an effective LC circuit as supposed in simplifiedmodels), and a coupling of this qubit to low-frequencynoise is described by the approach suggested here. De-pendencies of noise on ω and ℓ can be verified via dis-sipative dynamics of this qubit or via measurements ofa random voltage appearing at the open end of the TLif the other end is shortened. These regimes of mea-surements are illustrated schematically in Fig. 1d andthey are described by the flux-flux correlation functionof noises penetrating through the TL’s area.The obtained results demonstrate that spectral andsize dependencies of the correlators are rather differentfor the cases (Aa), (Ab), (Ba), and (Bb). Spectral disper-sion takes place if ω is comparable to the inverse diffusiontime along TL, τ − D . For different cases, the correlator isdecreased ∝ ( ωτ D ) − , ∝ ( ωτ D ) − . , or ∝ ln(1 /ωτ D ) athigh frequencies ( ωτ D ≫
1) and it is increased ∝ ( ωτ D ) − or saturated in the low-frequency region ( ωτ D ≪ to a verificationof noise’s effect on quantum hardware and a determina-tion of phenomenological parameters describing flux fluc-tuations. Characterization of noises in multi-qubit clus-ters, which include qubits, the inter-qubit connections,and the qubit’s control lines, opens the way to improv-ing the fidelity of the hardware for quantum-informationprocessing.The rest of the paper is organized as follows. Theeffects of low-frequency noise on TLs are described in Sec.II. The flux-flux correlation functions for the mechanismsof noise under consideration are analyzed in Sec. III.Concluding remarks with a summary of spectral and sizedependencies, a list of assumptions, and an outlook aregiven in the last section. The dissipative dynamics of aflux qubit shunted by a TL and the integrals used in Sect.IIIB are considered in Appendixes A and B, respectively. II. EFFECT OF NOISE ON A SHUNTED TL
Based on the lumped-element approach, we analyzehere the effective circuit for TL under random magneticflux arising from the above-listed mechanisms. For a TLof length ℓ ( | x | < ℓ/ V xt and I xt , are connected by thesystem of equations [5, 9] ∂V xt ∂x + l T L ∂I xt ∂t = ∂ ∆ v xt ∂x ,∂I xt ∂x + c T L ∂V xt ∂t = 0 . (1)Here l T L = L/ℓ and c T L = C/ℓ are the inductanceand the capacitance per unit length, which are writtenthrough the total inductance and capacitance, L and C . Performing an integration of the Maxwell equationcurl E = . . . over inter-wire area S = a × dx (see Fig. 1a)we obtain the random e.m.f. in Eq. (1), ∂ ∆ v xt /∂x , asfollows (cid:18) ∂ ∆ v xt ∂x (cid:19) dx = − c ∂∂t Z ( S ) d S · h r t , (2)where h r t is the external magnetic field penetratingthrough two-wire TL; see the standard electrodynamicderivation of the telegraph equations. [9, 10] The bound-ary conditions at x = ± ℓ/ V x = ± ℓ/ ,t ≡ V ± ,t and I x = ± ℓ/ ,t ≡ I ± ,t . After the Fourier transform inthe t -domain and substitution of I xω through dV xω /dx in Eqs. (1) one obtains the second-order telegraph equa-tion (cid:18) d dx + k ω (cid:19) V xω = d ∆ v xω dx ≡ w xω , (3)where k ω = ω √ l T L c T L = ω/ ( ω LC ℓ ) is the wave vectorat frequency ω for TL with the characteristic frequency ω LC = 1 / √ LC .From the upper line of Eq. (1) we obtain I xω throughvoltages V xω and ∆ v xω as I xω = iωl T L (cid:18) dV xω dx − d ∆ v xω dx (cid:19) . (4)The voltage and current distributions along the TL aregiven by V xω = v s sin k ω x + v c cos k ω x − ∆ V xω , (5) I xω = ik ω ωl T L ( v s cos k ω x − v c sin k ω x ) − ∆ I xω , where we separate the noise-induced contributions, ∆ V xω and ∆ I xω , which are written in the form∆ V xω = sin k ω x Z ℓ/ x dx ′ k ω cos k ω x ′ w x ′ ω + cos k ω x Z x − ℓ/ dx ′ k ω sin k ω x ′ w x ′ ω , (6)∆ I xω = iωl T L cos k ω x Z ℓ/ x dx ′ cos k ω x ′ w x ′ ω − sin k ω x Z x − ℓ/ dx ′ sin k ω x ′ w x ′ ω + d ∆ v xω dx ! . The constants v s,c in Eqs. (5) are determined throughedge voltages V ± ,ω and random contributions ∆ V ± ℓ/ ,ω according to (cid:12)(cid:12)(cid:12)(cid:12) v s v c (cid:12)(cid:12)(cid:12)(cid:12) = (sin k ω ℓ ) − (cid:12)(cid:12)(cid:12)(cid:12) cos( k ω ℓ/ V − sin( k ω ℓ/ V + (cid:12)(cid:12)(cid:12)(cid:12) , (7)where we introduce V ± ≡ V + ,ω + ∆ V ℓ/ ,ω ± V − ,ω ± ∆ V − ℓ/ ,ω . After substitution of v s,c into Eqs. (6) oneconnects the edge currents I ± ,ω and voltages V ± ,ω : (cid:12)(cid:12)(cid:12)(cid:12) I + ,ω I − ,ω (cid:12)(cid:12)(cid:12)(cid:12) = ˆ Y ω (cid:12)(cid:12)(cid:12)(cid:12) V + ,ω + ∆ V ℓ/ ,ω V − ,ω + ∆ V − ℓ/ ,ω (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) ∆ I ℓ/ ,ω ∆ I − ℓ/ ,ω (cid:12)(cid:12)(cid:12)(cid:12) , ˆ Y ω = ik ω ωl T L sin k ω ℓ (cid:12)(cid:12)(cid:12)(cid:12) cos k ω ℓ − − cos k ω ℓ (cid:12)(cid:12)(cid:12)(cid:12) . (8)Here ˆ Y ω is the admittance matrix [17] and the noise con-tributions, ∆ V ± ℓ/ ,ω and ∆ I ± ℓ/ ,ω , are given by integralsover the TL determined by Eqs. (6).Below we consider a TL which is shunted by a loss-less two-pole circuit with the imaginary admittance y ω connected at x = − ℓ/
2, when I − ,ω = y ω V − ,ω . Usingthis relation and Eqs. (8) we write the current-voltagerelation at the other end of the TL, x = ℓ/
2, as I + ,ω = Y ω V + ,ω − δ J ω , Y ω = Y + Y Y y ω − Y . (9)Here we introduce the effective admittance of shuntedTL, Y ω , and the noise-induced random current δ J ω at ℓ/ x = ± ℓ/ δ J ω = ∆ I ℓ/ ,ω − Y ∆ V ℓ/ ,ω − Y ∆ V − ℓ/ ,ω (10) − Y y ω − Y (cid:0) ∆ I − ℓ/ ,ω − Y ∆ V ℓ/ ,ω − Y ∆ V − ℓ/ ,ω (cid:1) . For the case of the open edge condition at x = ℓ/ I + ,ω = 0, Eq. (9) gives the random voltage as δV ω = δ J ω / Y ω and the voltage-voltage correlation func-tion takes the form: h δV δV i ω = h δ J δ J i ω / |Y ω | , (11)where h . . . i means averaging over a random e.m.f. Thespectral dependencies of h δV δV i ω are determined byboth the correlations of the random contributions at ± ℓ/ Y ω and y ω .Thus, a random e.m.f. along a TL shunted at − ℓ/ δ J ω at ℓ/ x = − ℓ/
2, where the edge condition is V − ,ω → | y ω | → ∞ , so Y ω ≃ Y and δ J ω is determinedby the upper line of Eq. (10).In the low-frequency region, k ω ℓ = | ω | /ω LC ≪
1, weuse the series expansion cot z ≈ /z − z/ − z / − . . . in the diagonal elements of ˆ Y ω given by Eq. (8) andthe effective admittance determined by Eq. (9) takes theform Y ω ≈ i (1 /ωL − ωC/ xω = ∆ v xω /iω and performing integrations byparts in ∆ V ℓ/ ,ω and ∆ I ± ℓ/ ,ω according to Eqs. (6), onetransforms Eq. (10) into δ Φ t ≡ − Lδ J t ≈ ∆Φ xω | x = ℓ/ x = − ℓ/ + ℓ d ∆Φ xω d x (cid:12)(cid:12)(cid:12) x = ℓ/ x = − ℓ/ , (12)where δ Φ t is the effective flux at x = ℓ/ ∝ ( ω/ω LC ) contributions are omitted. As a result, thevoltage-voltage correlator (11) at the open end of the TLis written through the flux-flux correlator, h δV δV i ω = ω h δ Φ δ Φ i ω , so the fluctuations of voltage are suppressedif ω → III. FLUX-FLUX CORRELATION FUNCTIONS
Random voltage at the open end of the TL and dissi-pative dynamics of a qubit shunted by the TL are deter-mined by the correlation function h δ Φ t +∆ t δ Φ t i . In orderto calculate this correlator using Eqs. (2) and (12), weconsider the Fourier transform of the flux distribution∆Φ xt , which is determined from the relation (cid:18) d ∆Φ xω dx (cid:19) dx = − c Z ( S ) d S · h r ω . (13)Next, we describe the transverse to the TL componentof a low-frequency magnetic field, h ⊥ r t , with the Langevinequation by taking into account diffusion and magnetiza-tion damping. Within the isotropic approximation, theLangevin equation is written using the scalar diffusioncoefficient and relaxation rate, D and ν , in the form (cid:18) ∂∂t − D ∇ r + ν (cid:19) h ⊥ r t = ζ ⊥ r t . (14)Random source ζ ⊥ r t is determined by the short-range cor-relation function in the spatiotemporal domain, (cid:10) ζ ⊥ r t ζ ⊥ r ′ t ′ (cid:11) = (cid:2) W + W ( ∇ r · ∇ r ′ ) (cid:3) δ ( r − r ′ ) δ ( t − t ′ ) , (15)where W and W determine level of fluctuations causedby random amplitude and gradient of magnetization, re-spectively. [18] Below, we analyze the models of noise inlower half-space (A) or at interfaces (B), when ∇ r is 3D-or 2D-gradient, and we restrict ourselves to the case of along and wide TL, ℓ ≫ a ≫ r w . We consider the region ω ≥ ν because h δ Φ δ Φ i ω is ω -independent if ω ≪ ν . A. Noise excited in a substrate
For the case of substrate with a random magnetic fieldshown in Fig. 1b, Eqs. (12) and (13) suggest that effec-tive flux δ Φ t is obtained through the transverse compo-nent of field h ⊥ r t = ( h r t · e z ) at interface z = 0 accordingto δ Φ t = − a/ Z − a/ dyc ℓh ⊥ r t (cid:12)(cid:12) x = ℓ/ x = − ℓ/ + ℓ/ Z − ℓ/ dxh ⊥ r t z =0 . (16)The random field h ⊥ r t is governed by Eq. (14) with thezero-flow boundary condition ( ∂h ⊥ r t /∂z ) z =0 = 0. Thesymmetry with respect to the z = 0 solution of Eq. (14)is written with the bulk Green’s function G ∆ r ∆ t and aspatiotemporal correlator of random fields is given by (cid:10) h ⊥ r t h ⊥ r ′ t ′ (cid:11) z,z ′ =0 = Z Z ( z ,z ′ < d r d r ′ Z Z dt dt ′ × G r − r t − t G r ′ − r ′ t ′ − t ′ D ζ ⊥ r t ζ ⊥ r ′ t ′ E , (17)where r = ( r k , z ). Using Eq. (15), performing integrationby parts over r , r ′ , and introducing ∆ t = t − t ′ oneobtains (cid:10) h ⊥ r t h ⊥ r ′ t ′ (cid:11) z,z ′ =0 = Z ( z < d r Z dt h W G r − r t − t G r ′ − r , − ∆ t − t + W ( ∇ r G r − r , ∆ t − t ) · ( ∇ r G r ′ − r , − ∆ t − t i . (18)After substitution of the Fourier transform of the Green’sfunction G k t = θ ( t ) exp( − Dk t ) / (2 π ) and integrationsover time t and in-plane coordinates ( r ) k , we re-writethis correlator in the form [here ∆ r k = r k − r ′k and k =( k k , k ⊥ )] (cid:10) h ⊥ r t h ⊥ r ′ t ′ (cid:11) z,z ′ =0 = e − ν | ∆ t | D Z −∞ dz Z dk ⊥ Z dk ′⊥ e i ( k ⊥ + k ′⊥ ) z (19) × Z d k k (2 π ) e i k k · ∆ r k − D ( k || + k ⊥ ) | ∆ t | k k + k ⊥ + k ′⊥ h W + W ( k k − k ⊥ k ′⊥ ) i . Furthermore, we integrate over k ′⊥ and the transversecoordinate z , so the correlator of the random fields istransformed into (cid:10) h ⊥ r t h ⊥ r ′ t ′ (cid:11) z,z ′ =0 = πe − ν | ∆ t | D Z d k (2 π ) W + W k k × exp( i k k · ∆ r k − Dk | ∆ t | ) . (20)The flux-flux correlator h δ Φ δ Φ i ω in Eqs. (11) and (A6)is obtained after the Fourier transform over ∆ t , whichgives 2( Dk + ν ) / [( Dk + ν ) + ω ], and integrations overthe X0Y-plane, according to Eq. (16). The integrationover the area of the TL gives the form-factor Ψ ( X, Y ) = (sin X sin Y ) [1 + (2 X ) ] / ( XY ) and the correlator takesthe final form: h δ Φ δ Φ i ω = π ( aℓ ) Dc Z d k (2 π ) Ψ (cid:18) k x ℓ , k y a (cid:19) × ( Dk + ν )( W + W k ) k [( Dk + ν ) + ω ] . (21)Using the dimensionless wave vector u = k √ aℓ/ ψ u = R d Ω u Ψ( u x , u y /p ) / π which is dependent on the aspectratio p = ℓ/a , one arrives at the integral over u : h δ Φ δ Φ i ω = u max Z duψ u ( u + ντ D )( A + Bu )( u + ντ D ) + ( ωτ D ) . (22)Here, we introduce the diffusion time along the TL τ D = ℓ / D , the coefficients A = W ℓ ( aℓ/Dc ) and B = W ℓ (2 a/Dc ) , which determine strength of noises,and a cut-off factor u max which is due to the inapplica-bility of Eq. (14) for short scales.The averaged form-factor ψ u is shown in Fig. 2a to-gether with the hyperbolic asymptotes ψ u ≈ . p/u which are valid at u > e u ∼ p . Separating contribu-tions (a) and (b) as the sum h δ Φ δ Φ i ω = K ω + K ω andintegrating over the tail region ( e u, u max ), we transformEq. (22) into (cid:12)(cid:12)(cid:12)(cid:12) K ω /AK ω /B (cid:12)(cid:12)(cid:12)(cid:12) = e u Z du ψ u ( u + ντ D )( u + ντ D ) + ( ωτ D ) (cid:12)(cid:12)(cid:12)(cid:12) u (cid:12)(cid:12)(cid:12)(cid:12) +0 . p (cid:12)(cid:12)(cid:12)(cid:12) (cid:2) π/ − arctan (cid:0)e u /ωτ D (cid:1)(cid:3) /ωτ D ln( u ) − ln p ( ωτ D ) + e u (cid:12)(cid:12)(cid:12)(cid:12) , (23)where K ω is logarithmically divergent if u max → ∞ whilearctan (cid:0) u /ωτ D (cid:1) → π/ K ω . The dependencies of K ω /A versus ωτ D are shown in Fig. 2b [case (Aa)] fordifferent aspect ratios and relaxation rates. K ω /A is con-stant if ω < ν and, for the wide intermediate spectral re-gion ν < ω < τ − D , it is approximated by 0 . / √ ωτ D )(another possible fitting is K ω ∝ ω − a with a ∼ . ωτ D ≥ e u , the correlator decreases to 0 . p/ωτ D ; we donot consider the region ωτ D ∼ u max where K ω ∝ ω − .If ωτ D ≤ e u or ℓ ≤ √ e uℓ ω ( ℓ ω = 2 p D/ω is the dif-fusion length during times ∼ ω − ), there is no depen-dency on aspect ratio and K ω ∝ ( aℓ ) ln( ℓ ω /ℓ ). By con-trast, for the 1/f spectral region the integral contribu-tion in Eq. (23) is weak [ ∝ ( ωτ D ) ], K ω /A is not de-pendent on ν , and the size dependence appears to beweaker, K ω ≈ . Ap/ωτ D ∝ aℓ . For the case (Ab), K ω /B is logarithmically divergent, if u → ∞ , andit increases with p . Spectral dependencies vary slowly,from constant at ωτ D ≤ ∝ p ln( u max / √ ωτ D ) at u max ≫ √ ωτ D ≫ ντ D = 0 .
05; these results remainweakly dependent on relaxation rate even for ντ D ≥ K ω is determined by the factor FIG. 2: (a) Averaged form-factor ψ u versus the dimen-sionless wave vector u = kℓ/ p =5 (red),10 (blue), and 20 (green). Dashed curves show asymptoticbehaviors determined by 0 . p/u . (b) Flux-flux correlationfunction K ω /A versus the dimensionless frequency ωτ D forthe case (Aa) with ντ D =0.01 (the black dotted curves) and0.1 (the gray curves). The red dashed lines show asymptotesfor ωτ D ≫
1. (c) Function K ω /B [case (Ab)] versus ωτ D for ντ D =0.05, p =5, 10, or 20 (marked), and u max =350 (black)or 500 (gray) plotted in semi-log scale. The red logarithmicasymptotes correspond ( u max , p ) =(350,5) and (500,20). Bp ln( u max / e u ), if ωτ D ≤ e u , and the ln-factor should bereplaced by ln( u max ℓ ω /ℓ ) if ωτ D ≥ e u . Neglecting weakln-variations we obtain K ω ∝ aℓ , i. e. h δ Φ δ Φ i ω in-creases with TL’s sizes faster than the area of the TL, aℓ . B. Noise excited at interfaces of wires
We turn now to case (B) when random magnetizationis localized at interfaces of superconducting wires. Themagnetic field around the two-wire TL is governed by thequasistatic equation ∆ h r t = 0. Below, we analyze thisequation in the bipolar coordinate system, r = ( x, τ, σ ),with −∞ < τ < ∞ and 0 < σ < π which are variedover the Y0Z plane as shown in Fig. 1c. If a ≫ r w ,the outside-of-wire region corresponds | τ | < ln( a/r w ) ≡ τ w and the line connecting wires is along σ →
0. Theboundary conditions at the left ( − ) and the right (+)wire’s interfaces S ± , where τ → ± τ w and 0 < σ < π , arewritten for tangential and normal components of the fieldas h ( τ ) r t | S ± = e h ( ± ) xσt and h ( σ ) r t | S ± = 0, respectively. Here e h ( ± ) xσt represents random magnetic fields at the interfacesand there are no fields normal to the superconductingwires. At | r | → ∞ , we use the zero boundary condition.After the Fourier transform ( k is now the wave vectoralong 0X), one obtains the quasi-static equation for thenon-zero τ -component of random magnetic field (cid:20) ∂ ∂τ + ∂ ∂σ − ( ka/ (cosh τ − cos σ ) (cid:21) h ( τ ) kτσt = 0 (24)with the periodic boundary condition over σ and h ( τ ) k, ± τ w σt = e h ( ± ) kσt . The random flux δ Φ t is determinedthrough h ( τ ) xτσt from Eq. (13) in a form similar to Eq.(16): δ Φ t = − τ w Z − τ w dτ g ττ c h ⊥ xτσt (cid:12)(cid:12) x = ℓ/ x = − ℓ/ + ℓ/ Z − ℓ/ dxh ⊥ xτσt σ =0 , (25)where g ττ = a/ (cosh τ +1) is the square root of the metrictensor. [19]In analogy to Eq. (14), the Fourier transform over σ and t for the tangential component of the field at in-terfaces of the m -th wire, e h ( m ) knω (with n = 0 , ± , . . . and m = ± for a wire placed at around τ = ± τ w ), is governedby the 2D diffusion equation (cid:20) iω + ν m − D m (cid:18) d dx − n r w (cid:19)(cid:21) e h ( m ) xnω = ζ ( m ) xnω , (26)where D m and ν m are the diffusion coefficient and relax-ation rate of the m -th wire and | x | < ℓ/
2. The boundaryconditions at the ends of the TL are ( e h ( m ) xnω /dx ) x = ± ℓ/ =0. If the scale of correlations in D ζ ( m ) xω ζ ( m ′ ) x ′ ω ′ E is sig-nificantly greater than r w , no contributions other than e h ( m ) xn =0 ω ≡ e h ( m ) xω are essential. At the same time, we sup-pose that the scale of correlations is negligible in com-parison to other scales ( a , ℓ , and the diffusion length ℓ ω ).Similarly to Eq. (15), we determine random sources inthe right-hand side of Eq. (26) through the short-rangecorrelation function given by D ζ ( m ) xω ζ ( m ′ ) x ′ ω ′ E = δ mm ′ δ ( ω + ω ′ )(2 π ) (cid:18) w m + w m ∂ ∂x∂x ′ (cid:19) δ ( x − x ′ ) . (27)Here, w m and w m stand for strengths of (a)- and (b)-type of noises along the m -th wire. [20] The non-uniformsolution of Eq. (26), e h ( m ) xω = − Z ℓ/ − ℓ/ dx ′ G ( m ) xx ′ ζ ( m ) x ′ ω /D m , (28)is written through the Green’s function with the m -dependent complex wave vector κ ω = p ( iω + ν m ) /D m , G ( m ) xx ′ = [ κ ω sinh( κ ω ℓ )] − (29) × (cid:26) cosh κ ω ( ℓ/ x ) cosh κ ω ( ℓ/ − x ′ ) , x < x ′ cosh κ ω ( ℓ/ x ′ ) cosh κ ω ( ℓ/ − x ) , x > x ′ , and a jump of the derivative of G ( m ) xx ′ takes place at x = x ′ .After averaging over σ , Eq. (24) for a random field h kτω takes the form (cid:18) d dτ + V τ (cid:19) h kτω = 0 , V τ = ( ka/ (cosh τ ) + 1 / h k ± τ w ω = e h ( ± ) kω . Here, thepotential V τ is localized around zero, where | τ | < τ ≥ τ ≪ τ w , and the region ( − τ , τ ) with V τ = 0 canbe replaced by the boundary conditions dh kτω dτ (cid:12)(cid:12)(cid:12)(cid:12) τ = τ τ = − τ ≈ τ Z − τ dτ V τ h kτ =0 ω ≡ v k h kτ =0 ω (31)and h kτω | τ = τ τ = − τ ≈
0. Integration over τ in Eq. (31) givesthe coefficient v k ≈ . τ w ( ka/ . The linear-dependenton τ solution of the problem involving (30) and (31)should be substituted to Eq. (25) and after integrationsover τ and x one connects the Fourier transform of theflux, δ Φ ω , and the random fields at the interfaces (28) asfollows: δ Φ ω = − c Z dk ℓe − ikx (cid:12)(cid:12) ℓ/ − ℓ/ + ℓ/ Z − ℓ/ dxe − ikx × Z τ w − τ w dτ g ττ h kτω = − aℓc Z dkα k (cid:16)e h (+) kω + e h ( − ) kω (cid:17) , (32) α k = (cid:18) kℓ − i (cid:19) sin (cid:18) kℓ (cid:19) . v k /τ w v k . Here, ℓ -dependent factors in α k appear due to integrationalong x and e h ( m ) kω should be written through the Fouriertransform of the Green’s function (29): G ( m ) kx = Z ℓ/ − ℓ/ dx ′ π e ikx ′ G ( m ) x ′ x = e ikx − F ( m ) kx π ( k + κ ω ) , (33) F ( m ) kx = ik cosh κ ω ( x + x ) κ ω sinh κ ω ℓ e ikx (cid:12)(cid:12) x = ℓ/ x = − ℓ/ , which is a continuous function of x (it is convenient toreplace x ↔ x ′ here and below).Furthermore, the Fourier transform of the flux-flux cor-relator h δ Φ δ Φ i ω is obtained after substitution of Eqs.(28), (33) into (32) and averaging according to Eq. (27).The result takes the form h δ Φ δ Φ i ω = X m ℓ/ Z − ℓ/ dxℓ/ (cid:18) A m (cid:12)(cid:12)(cid:12) G ( m ) xω (cid:12)(cid:12)(cid:12) + B m (cid:12)(cid:12)(cid:12) G ( m ) xω (cid:12)(cid:12)(cid:12) (cid:19) , (34) FIG. 3: (a) Density of the flux-flux correlator along a TL forcase (Ba) versus 2 x/ℓ at different ωτ D (marked) and τ m = 3 . K ω / A versus ωτ D at ντ D = 0(the dotted curve) and 0.1 (the gray solid curve). The asymp-totes ∝ ω − and ∝ ω − . are shown by dashed red lines forthe low- and high-frequency regions, respectively. where the coefficients A m = w m ℓ ( a/ πD m c ) and B m = w m ℓ ( a/πD m c ) are similar to the ones used in Sect.IIIA after replacements W → w m ℓ and W → w m ℓ .The distribution of h δ Φ δ Φ i ω along the TL is determinedby the dimensionless factors (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G ( m ) xω G ( m ) xω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = Z dkα k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G ( m ) kx /ℓ∂G ( m ) kx /∂x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (35)with ∂G ( m ) kx /∂x given by the derivative of Eq. (33). Be-low, we consider the case of identical wires, when D , w and w are not dependent on m = ± , so P m A m or P m B m should be replaced by the doubled coefficients, A = 2 wℓ ( aℓ/Dc ) or B = 2 wℓ (2 a/Dc ) . After the in-tegration over the complex k -plane in Eq. (35) we trans-form the distributions (35) into (cid:12)(cid:12)(cid:12)(cid:12) |G xω | |G xω | (cid:12)(cid:12)(cid:12)(cid:12) ≃ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) κ ω ℓ ) + .
69 sinh( κ ω x ) τ w cosh( κ ω ℓ/ (cid:12)(cid:12)(cid:12) . τ w (cid:12)(cid:12)(cid:12) cosh( κ ω x )cosh( κ ω ℓ/ (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (36)and these distributions are not dependent on the aspectratio p explicitly, see Appendix B for details. Thus, thefinal result h δ Φ δ Φ i ω = K ω + K ω is obtained after straight-forward integrations along the TL in the form K ω = A (cid:20) ωτ D ) + 0 . τ w r ωτ D D − ( √ ωτ D ) (cid:21) , K ω = B . τ w r ωτ D D + ( √ ωτ D ) , (37)where D ± ( s ) = (sinh s ± sin s ) / (cosh s + cos s ) and wherewe restrict ourselves to the collisionless case ν/ω → ντ D → FIG. 4: (a) The same as in Fig. 3a for the case (Bb).(b) K ω / B versus ωτ D at ντ D = 0 (the dotted curve) and 1(the gray solid curve). High-frequency asymptote ∝ ω − . and low-frequency saturation are indicated by the dashed redlines for ντ D = 0. noise for ω ≤ ν are shown in Figs. 3b and 4b). Distribu-tion of the correlator density along the TL, |G xω | , versusthe dimensionless coordinate 2 x/ℓ is plotted in Fig. 3a forcase (Ba). If √ ωτ D ≫
1, the density of noise decreasesexponentially around | x | ∼
0, but it is ∼ (0 . /τ m ) in narrow regions near | x | ∼ ℓ/
2. If √ ωτ D ≤ x -independent and noise increases ∝ ( ωτ D ) − . Due to this, spectral dependency takes form K ω / A ≈ . / ( ωτ D ) in the low-frequency region while,in the high-frequency region, √ ωτ D ≫
1, noise decreasesslower K ω / A ≈ . /τ m √ ωτ D because only narrow re-gions | x | ∼ ℓ/ A and B as well as τ D ∝ ℓ and τ w = ln( a/r w ). One obtains that K ω ∝ ( a/ℓ ) is only de-pendent on the aspect ratio for the low-frequency regionwhile, for √ ωτ D ≫
1, the result K ω ∝ a ℓ/τ w increaseswith the sizes of the TL.Spectral and size dependencies for the case (Bb) areshown in Figs. 4a and 4b. At √ ωτ D >
1, the distributionof the correlator along the TL, which is symmetric now( G − xω = G xω ) due to the absence of an x -independentcontribution in Eq. (36), decreases exponentially around | x | ∼
0. If √ ωτ D <
1, this distribution becomes constant, |G xω | ≈ . /τ m . As a result, K ω / B ≈ . /τ w in thelow-frequency region, while, in the high-frequency region, √ ωτ D ≫
1, noise decreases as K ω / B ≈ . /τ w √ ωτ D ,cf Figs. 3b and 4b. Size dependencies of these asymp-totic relations are determined with the use of the abovecoefficients. One obtains a value proportional to thesquare of the area dependency, K ω ∝ ( aℓ ) /τ w , in thelow-frequency region and K ω ∝ a ℓ appears for the highfrequencies, the same as for case (Ba). h δ Φ δ Φ i ω ℓ ≪ ℓ ω ℓ ≫ ℓ ω (Aa) ∝ ( aℓ ) ln( ℓ ω /ℓ ) ∝ aℓ (Ab) ∝ aℓ ln( u max a/ ℓ ) ∝ aℓ ln( u max ℓ ω /ℓ )(Ba) ∝ ( a/ℓ ) ∝ a ℓ/ [ln( a/r w )] (Bb) ∝ ( aℓ ) / [ln( a/r w )] ∝ a ℓ/ [ln( a/r w )] TABLE I: Size dependencies of h δ Φ δ Φ i ω for the cases underconsideration. Here ℓ ω = 2 p D/ω is the diffusion length dur-ing times ∼ ω − . IV. CONCLUSIONS AND OUTLOOK
To summarize, we find that weak external noisesshould give a dominant contribution in long enough TLs.The flux-flux correlators considered in Sects. IIIA andIIIB show visible modifications of spectral dispersion ina wide spectral region around τ − D , and noise effect in-creases with the size of the TL, see asymptotics collectedin Table I for short and long TLs at fixed ω . The asymp-totic behavior of h δ Φ δ Φ i ω in the low- and high-frequencyregions is as follows: h δ Φ δ Φ i ω ≈ A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (cid:16) √ ωτ D (cid:17) . p/ωτ D (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + B . p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ln (cid:16) u max e u (cid:17) ln (cid:16) u max √ ωτ D (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + A (cid:12)(cid:12)(cid:12)(cid:12) . / ( ωτ D ) . τ w √ ωτ D (cid:12)(cid:12)(cid:12)(cid:12) + B (cid:12)(cid:12)(cid:12)(cid:12) . /τ w . τ w √ ωτ D (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) √ ωτ D ≪ √ ωτ D ≫ (cid:12)(cid:12)(cid:12)(cid:12) , (38)where the phenomenological parameters ( W , W , w , w , D , and so on) should be determined from experimentaldata or microscopic models. These spectral and size de-pendencies enable us to separate different mechanisms ofnoise and opens the way for an optimization and miti-gation of relaxation effects on the coherent dynamics ofsuperconducting devices used for quantum informationprocessing.The results obtained are based on several assumptions,which are discussed below: • Phenomenological consideration of 1/f type noises inSects. IIIA and IIIB are based on the diffusive Langevinequations (14) and (26) with the scalar coefficients D and ν as well as with random sources which are deter-mined through short-range correlators. External noisesare considered without the effect of currents in the TLon random magnetization. A self-consistent microscopicstudy of these mechanisms and extension beyond the hy-drodynamics approach require a special consideration. • In spite of we described the regime of classical noisein Sects. IIIA and IIIB, a similar consideration can beapplied to the analysis of quantized random fields afterreplacement of the right-hand parts of Langevin equa-tions (14) and (26) by quantized random sources. • Here, we restricted ourselves to the case of a TLshorted at x = − ℓ/
2; an open TL (with zero currentsat x = ± ℓ/
2) or the general case of a TL shunted bytwo-pole circuits at both ends should be checked sepa-rately using suitable boundary conditions for the tele-graph equations. • We have considered here a simplest geometry of thedirect two-wire TL (similar to the TL with fixed ℓ used in[7] for characterization of spin defects at the interfaces).For a non-direct TL, the equation of motion in Sect IIAmust be modified by taking into account any curvaturesor angles along the TL. A coplanar TL requires addi-tional numerical calculations, which can change the re-sults quantitatively. • Equations of motion of the LC-shunted qubit inAppedix A are analyzed for frequencies lower than thecharacteristic frequency of the TL, ω LC . For frequenciescomparable to ω LC , the problem of quantization cannotbe based on the non-local equation of motion (A1), evenwhen the classical regime of response becomes compli-cated and it should be analyzed starting with differentassumptions. [21] • The consideration performed here is irrelevant to a C-shunted qubit, [22] where the inductance contribution isnegligible and the effect of flux noise on the dynamics ofthe qubit is reduced. However, TLs, which form quantumhardware based on these qubits, can be analyzed usingthe above results. • Finally, we have considered only low-frequency noisesaroused from external magnetization described by the setof phenomenological coefficients, without an assumptionof the thermodynamics equilibrium. A few additionalmechanisms (noise due to dielectric losses, [23] fluctua-tions of charge at the Josephson junction, [24] interact-ing two-level defects, [7, 25] and fluctuations of currentin the TL caused by thermal generation of electron-holepairs [26, 27]) are possible for specific devices. The omit-ted mechanisms and the above-listed assumptions restrictthe area of applicability of this paper but do not affecton the main result: the non-trivial spectral and size de-pendencies of the noise effect.We turn now to a discussion of possibilities for ex-perimental verification of the results obtained. Directmeasurements of size-dependent flux-flux correlations arecomplicated but one can analyze modifications of thespectral dependencies of a voltage-voltage correlator (11)or population relaxation time (A6) for long TLs in region ωτ D ∼
1. Using a maximal value of diffusion coefficient D ∼ nm/s reported in [12] and ℓ ∼ µ m, we esti-mate τ D ∼
25 ms so that frequency dispersion takes placein the spectral region ≤
100 Hz. Such measurements al-low one to separate different mechanisms of noise dueto the dependencies on the length of the TL (for thecase of a narrow TL). It is also possible to mitigate re-laxation effects, e.g. one can use a smaller TL area atfixed ω LC . The paper presents analytical results for atwo-wire TL, which give explicate dependencies on thephenomenological parameters of noise and open the wayfor the qualitative verification of noise parameters in theTL via random voltage at the open end of a TL or via thepopulation relaxation rate in a qubit shunted by a TL.Quantitative estimates for the structures used in current experiments, see [12, 13] and the references therein, or fora specially designed structure require additional numeri-cal simulations for the verification of noise effects becausefield distributions around two-wire and coplanar TLs aredifferent.In closing, the importance of noise effects in ∼ ∼ Appendix A: LC-shunted qubit
A convenient way for verification of TL with noisesis to study flux qubit formed by SQUID loop shuntedby this TL, where the nonlinear current-voltage char-acteristics of Josephson junction provide a double-wellpotential [28] with frequency level splitting in the GHzregion. [12, 13, 29] Equation of motion for such a qubit isdetermined from the Kirchhoff’s requirement of currentconservation, I Jω = I + ,ω , where V + ,ω in Eq. (9) shouldbe replaced through flux at ℓ/ iω Φ ω . This edge con-dition is transformed into an equation of motion for aqubit by the use of the current through the Josephsonjunction I Jt = I c sin(2 π Φ Jt / Φ ), where I c is the criticalcurrent, Φ is the flux quantum, and V Jt + V x =0 t = 0 or d Φ Jt /dt = − d Φ x =0 t /dt or Φ Jt = − Φ x =0 t (under a suit-able initial condition). Note, that here we use a single-junction model with a variable effective critical current, I c , instead of a description of a SQUID with the totalcurrent through a loop which is controlled by externalflux. In the t -domain one arrives to a non-local equationof motion I Jt = I + ,t which is written in the form I c sin 2 π Φ t Φ = Z ∞−∞ dt ′ Y t − t ′ d Φ t ′ dt ′ − δ J t (A1)with the kernel Y t − t ′ and random current δ J t determinedby the Fourier transforms of Eqs. (9) and (10), respec-tively.Within the low-frequency limit, ω ≪ ω LC , the equa-tion of motion (A1) takes the form: C d Φ t dt + Φ t L + I c sin 2 π Φ t Φ + δ Φ t L = 0 (A2)and only third part of the TL’s capacitance is involvedhere. The classic Hamiltonian, which results in this equa-tion of motion, can be written through charge-flux vari-ables, Q t and Φ t , as follows H t = 32 Q t C + Φ t L − I c Φ π cos 2 π Φ t Φ + Φ t δ Φ t L . (A3)
FIG. 5: Contours C + (a) and C − (b) used for integrationsof the ∝ exp( ± kℓ/
2) contributions in Eq. (35).
After the standard quantization procedure of H t whichis based on the commutation relation h Φ , ˆ Q i = i ¯ h , onearrives to the flux qubit Hamiltonian [13, 29]ˆ H = 32 C d d Φ + Φ L − I c Φ π cos 2 π ΦΦ + Φ δ Φ t L , (A4)where an effective flux noise δ Φ t is dependent on theTL’s parameters according to Eq. (12). Using the noise-less wave functions, which are determined by the eigen-state problem ˆ H δ Φ=0 | r i = ε r | r i with r = 0 , , . . . , wedescribe the qubit-environment interaction through theflux matrix elements Φ rr ′ = h r | Φ | r ′ i . Within the two-level approach ( r = 0 ,
1) we use the 2 × H t = ˆ σ z ε / σ x Φ δ Φ t /L written through the Paulimatrices, ˆ σ , and the level splitting energy, ε = ε − ε .Here we consider the symmetric qubit, without any tiltflux applied through the TL, and restrict ourselves to theclassical noise regime, supposing that an effective temper-ature of noise δ Φ t is greater than ε .In addition to a direct examination of h δ Φ δ Φ i ω viathe voltage fluctuations (11), one can study relaxation ofpopulation in the flux qubit described by the Hamiltonian(A4). After averaging of the density-matrix equation overrandom noise, the balance equation for the populationsof upper ( r = 1) and lower ( r = 0) levels takes the form[30] dn rt dt = ( − r Φ L Z −∞ d ∆ te λ ∆ t h δ Φ t +∆ t δ Φ t i (A5) × (cid:0) e iω ∆ t + e − iω ∆ t (cid:1) ( n t +∆ t − n t +∆ t ) , where ω = ε / ¯ h . Within the Markov approximation n rt +∆ t ≈ n rt , the population re-distribution δn t = n t − n t is governed by the first-order equation dδn t /dt = − δn t /T with the population relaxation rate1 T = 2Φ L Z ∞−∞ d ∆ te iω ∆ t h δ Φ t +∆ t δ Φ t i . (A6)Thus T − = (2Φ /L ) h δ Φ δ Φ i ω and the relaxationtime T is connected with the correlator (11) accordingto the relation h δV δV i ω = ( ωL/ Φ ) / T . Because thedecoherence time T is shorter than T and characteriza-tion of the noise through T is more complicated, we donot consider relaxation of the non-diagonal part of thedensity matrix. Appendix B: Contour integration
Here we calculate distributions (35) performing inte-grations over a complex k -plane with the use of the con- tours C ± shown in Fig. 5. We separate the ∝ exp( ± kℓ/ α k in Eq. (32) and close integrals over k in the upper/lower half-spaces, respectively. Under thesubstitution G kx given by Eq. (33) into G xω one shouldtake into account that there are no poles at k → ± iκ ω ,i.e. G ± iκ ω x < const. The only pole appears because α k ∝ /k at | k | →
0, so Z C ± dkiℓ e ± ikℓ/ α k G kx = ± πℓ G k =0 x = ± ( κ ω ℓ ) − (B1)and G xω is determined from the Cauchy’s theorem G xω +∆ G xω = 2 / ( κ ω ℓ ) where ∆ G xω involves integrals over theupper and lower half-circles. Replacing k by k c e iθ andtaking into account that k c a → ∞ in the factor α k wecalculate ∆ G xω as follows∆ G xω = i . τ w k c π Z dθe iθ e ikℓ G kx (B2) − − π Z dθe iθ e − ikℓ G kx k → k c e iθ ≈ . τ w sinh( κ ω x )cosh( κ ω ℓ/ . Because of k c → ∞ , only the regions θ ∼ ∼ ± π are essential here. Collecting these integrals, one obtainsthe |G xω | given by Eq. (36).Distribution G xω is determined by Eq. (35) throughthe derivative ∂G kx ∂x = ik e ikx − e F kx π ( k + κ ω ) (B3) e F kx = sinh κ ω ( x + x ) κ ω sinh κ ω ℓ e ikx (cid:12)(cid:12) x = ℓ/ x = − ℓ/ , and there are no poles at ± iκ ω (similar to the above case)as well as at | k | →
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