NSQUID Arrays as Conveyers of Quantum Information

8/11/2019 NSQUID Arrays as Conveyers of Quantum Information

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NSQUID arrays as conveyers of quantum information

Qiang Deng and Dmitri V. AverinDepartment of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, NY 11794-3800

(Dated: September 5, 2014)

We have considered the quantum dynamics of an array of nSQUIDs – two-junction SQUIDs withnegative mutual inductance between their two arms. Effective dual-rail structure of the array createsadditional internal degree of freedom for the uxons in the array, which can be used to encode and

transport quantum information. Physically, this degree of freedom is represented by electromagneticexcitations localized on the uxon. We have calculated the spatial prole and frequency spectrumof these excitations. Their dynamics can be reduced to two quantum states, so that each uxonsmoving through the array carries with it a qubit of information. Coherence properties of such apropagating qubits in the nSQUID array are characterized by the dynamic suppression of the low-frequency decoherence due to the motion-induced spreading of the noise spectral density to a largerfrequency interval.

PACS numbers:

I. INTRODUCTION

Coherence properties and precision of control over the

dynamics of superconducting qubits (see, e.g., recentexperiments [ 1–5]) have reached the level when it be-comes possible and interesting to discuss potential ar-chitecture of the superconducting quantum computingcircuits, either within the gate-model paradigm [6–9] orthe adiabatic ground-state approach [10, 11]. Besides theformidable problem of maintaining the level of qubit co-herence with increasing circuit complexity, the centralissue that needs to be addressed by any architectureof scalable quantum computing devices is the require-ment of rapid transfer of quantum information among alarge number of qubits with sufficient delity. So far, thesuggested solutions to the problem of transfer of quan-

tum information were based on controllable direct cou-pling of qubits or coupling through a common resonator.These solutions, while working nicely for the circuits of few qubits, can not be scaled easily to larger circuits.The purpose of this work is to suggest another approachto the problem of information transfer along a quantumcircuit of the superconducting qubits utilizing quantumdynamics of magnetic ux, in which the quantum infor-mation is transported along the circuit by propagatingclassical pulses. This approach uses the arrays of two- junction SQUIDs, where each of them has a negative mu-tual inductance between its two arms. Dynamics of such“nSQUIDs” [12] can be represented in terms of the twodegrees of freedom, the “differential” mode and the “com-mon” modes, with very different properties. The formercan be used to encode quantum information, while thelatter – to transport it. Then, the overall architecture of aquantum computing circuit built of nSQUIDs is very sim-ilar to superconducting classical reversible circuits alsobased on nSQUIDs [13, 14], in which the computation isorganized around the information-carrying pulses propa-gating along the circuit.

Existence the two degrees of freedom with differentproperties makes nSQUID arrays qualitatively and ad-

vantageously different from previously considered arraysof superconducting qubits – see, e.g., [15–17], or spins[18–20], as tools for quantum information transfer. Ingeneral, a physical variable encoding quantum informa-tion and the one used to carry it should satisfy com-pletely different sets of requirements, and in the caseof nSQUIDs, the common and the differential dynamicmodes can be optimized separately to satisfy these differ-ent requirements. Most importantly, the common mode,i.e. the degree of freedom transporting the quantum in-formation does not necessarily need to be itself quan-tum, since a classical dynamics is sufficient to transporta qubit along the array. This makes potential opera-tion of nSQUID arrays as conveyers of quantum informa-tion considerably more straightforward, since it avoids achallenging problem of maintaining quantum coherenceof uxons, supported by the common mode, along a largearray of junctions.

II. BASIC MODEL OF NSQUID ARRAY

We begin our detailed discussion with a description of the elementary cell of the arrays considered in this work– nSQUID [12], a two-junction SQUID with a negativeinductance between its two arms (Fig. 1). Dynamics of this structure can be separated naturally into the dynam-ics of two degrees of freedom: common mode representingthe total current owing through the two junctions of theSQUID, and the differential mode which represents thedifference of the two junction currents, i.e., the currentcirculating along the SQUID loop. In the congurationof a one-dimensional array, these two degrees of freedomgive rise to the two different excitation modes of the ar-ray. The common mode corresponds to excitations prop-agating along the array and, in the appropriate regime,takes the form of individual uxons.

The main difference between the nSQUID and theusual two-junction SQUID can be seen if one thinks verycrudely about two arms of a SQUID as two parallel wires.

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FIG. 1: Equivalent circuit of an nSQUID: two-junctionSQUID with junction capacitances C and Josephson cou-pling energies E J and the negative mutual inductance − M between its inductive arms with inductances L. The nega-tive mutual inductance makes the effective inductance of thecommon mode of the SQUID dynamics much smaller thanthe inductance of the differential mode. Also shown are thephase bias χ e of the common mode and φe of the differentialmode.

For the plain wires, the mutual inductance M betweenthe wires is positive, M > 0, ensuring that the effectiveinductance of the differential mode is always smaller thanthat of the common mode. In this case, the differentialmode can have a non-trivial dynamics only together withthe common mode. By contrast, the negative mutual in-ductance −M between the SQUID arms makes the effec-tive inductance of the differential mode larger than theinductance of the common mode. As a result, one canrealize a situation when the differential mode exhibitsa non-trivial, e.g. bi-stable, dynamic at low frequencies

without exciting the common mode which supports onlythe excitations with larger frequencies. The differentialmode can then be used to encode information, while thedynamics of the common mode is optimized separatelyfor transfer of this information along the array. If thedynamics of both modes is classical, nSQUID structuresprovide a convenient basis for implementation of classi-cal reversible computing [13, 14]. If, however, parame-ters of the differential mode are such that its behavioris quantum, it can be used to encode quantum informa-tion, which can then be transported along the array bythe evolution of the common mode.

Hamiltonian H of the individual nSQUID (Fig. 1)is given by the standard expression which includes thecharging energies and the Josephson coupling energies of the two SQUID junctions. Adding the magnetic energyof the two inductive arms of the SQUID coupled by thenegative mutual inductance, one obtains the followingexpression for the nSQUID Hamiltonian

H = K 2

2C t+

Q2

4C −2E J cos χ cos φ

+Φ0

2π

2 (χ −χ e )2

L −M +

(φ −φe )2

L + M . (1)

Here Φ0 = π /e is the magnetic ux quantum, K and χare the variables of the common mode: K = Q1 + Q2 isthe total charge on the two capacitances of the SQUID junctions, and χ = ( φ1 + φ2 )/ 2 is the average Josephsonphase difference across the junctions. The common modehas effective inductance ( L −M )/ 2 and capacitance C t ,which in the case of the circuit in Fig. 1, is equal to thetotal capacitance 2 C of the two junctions, but in general

can include additional contributions from the externalbiasing circuit (as, e.g., in Fig. 2 below). In quantumdynamic, K and χ are canonically conjugated variablesthat satisfy the commutation relation [ χ, K ] = 2ei, stan-dard for the charge and phase of a Josephson junction.The corresponding variables of the differential mode arethe charge difference Q = Q1 −Q2 and the phase differ-ence φ = ( φ1 −φ2 )/ 2 which have the same commutationrelation [ φ, Q] = 2ei. The effective inductance and capac-itance of this mode are 2( L + M ) and C/ 2 respectively.(Their apparent values in Eq. ( 1) are different because of the chosen normalization of Q and φ.)

As mentioned above, qualitative effect of the nega-tive mutual inductance is to make the dynamic prop-erties of the common and the differential mode in theHamiltonian ( 1) of one nSQUID very different. If oneneglects for a moment the Josephson coupling, the reso-nance frequencies of the two modes are [2 / (L −M )C t ]1/ 2

and 1 / [(L + M )C ]1/ 2 ≡ ω p, and in the large-negative-inductance limit, M → L, the excitation frequency of the common mode becomes much larger than that of thedifferential mode, making it possible to clearly separatethe frequency ranges of the dynamics of the two modes.As a result, when the nSQUID cells are connected in anarray as in Fig. 2, the two modes can be used to per-form different functions. In particular, if the coupling

inductances LC are designed to have negative mutual in-ductance −M C between them, the main feature of thenSQUID dynamics is preserved: the common mode re-mains rigid, i.e., it is not affected by the evolution of thedifferential mode, and is essentially xed at some valuewhich is either applied externally or generated dynami-cally. This phase plays then the role of the qubit controlsignal which is distributed along the array through the“clock” line (upper horizontal line in Fig. 2). The differ-ential mode can be used to encode a classical or quantumbit of information in the current circulating along thecoupled SQUID loops. Dynamics of the common modeensures then that the information encoded by the differ-

ential mode is transported along the array.Quantitatively, we consider the arrays with no bias

phases applied externally either for the common or differ-ential mode; rather the bias phases χ e,j for the commonmodes χ j of the array cells, where the index j numbersthe cells, are generated dynamically by the array struc-ture. The phase χ e,j is the phase of the superconduct-ing order parameter at the bias node point of the j thnSQUID, i.e., the points connected by inductance L0 inFig. 2. The Hamiltonian H 0 of such an array, a segment



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of which is shown in Fig. 2, can be written as

H 0 =j

H ( j ) +

2e

2 (χ e,j −χ e,j − 1 )2

2L0

+(φj −φj − 1 )2

LC + M C +

(χ e,j −χ j + χ j − 1 −χ e,j − 1)2

LC −M C , (2)

where the sum runs over all nSQUID cells of the array.Each term H ( j ) in this expression,

H ( j ) =K 2j4C

+Q2

j

4C +

K 2e,j

2C 0+

2e

2 (χ e,j −χ j )2

L −M

+φ2

j

L + M −2E J cos χ j cos φj , (3)

is the Hamiltonian of the j th cell, while the rest of the terms in Eq. ( 2) describe the coupling between thenearest-neighbor cells due to inductances L0 and LC . Inaddition to the Hamiltonian (1), each term H ( j ) includesnow the charging energy of the charge K e,j on the capaci-tance C 0 of the j th cell. The charge K e,j is the conjugatevariable to the phase χ e,j , with the standard commuta-tion relation [ χ e,j , K e,j ] = 2ei. As shown explicitly inthe Appendix, in the limit of strong negative couplingof the array inductances, M → L, M C → LC , when thecharacteristic features of the nSQUID dynamics manifestthemselves most prominently, the common phase of eachcell of the array becomes pinned down to the correspond-ing bias phase χ e,j , χ j ≃χ e,j , and the array HamiltonianH 0 reduces to

H 0 =j

K 2j2C t

+Q2

j

4C −2E J cos χ j cos φj

+ 2e

2 (χ j −χ j − 1)2

2L0+ φ

2j

2L + (φj −φj − 1)2

2LC , (4)

where C t = 2C + C 0 . Note that although we included foruniformity the negative coupling M C in the requirementsof the strong-negative-coupling limit, in principle, as fol-lows from the discussion in the Appendix – cf. Eq. ( A.7),the condition M →L alone is sufficient for the reductionof the array Hamiltonian to form ( 4) with different effec-tive coupling inductance.

In what follows, we consider the situation when boththe clock phases χ j , and the information phases φj canbe described in the continuous approximation, χ j , φj →χ (x), φ(x), where x is a continuous dimensionless positionalong the array (which can be understood as the realspatial coordinate in units of the size of the elementarycell of the array). Such an approximation is strictly validif the characteristic length of variation of each phase,“Josephson penetration length”, is large, λ0 , λC ≫ 1,where

λ0 ≡( / 2e)/ [2E J L0]1/ 2 , λC ≡( / 2e)/ [2E J LC ]1/ 2 .

In the continuous approximation, the Hamiltonian ( 4) of

FIG. 2: Dual-rail Josephson array made of nSQUID cellsshown in Fig. 1. The array cells are coupled by inductancesLC , with negative mutual inductance − M C between them.No bias phases are applied externally either for the common ordifferential mode; the bias for the common mode is generatedself-consistently by the array dynamics. In this dynamics, thecommon mode plays the role of the qubit control signal propa-gating along the control line with specic capacitance C 0 andinductance L 0 , whereas the differential mode encodes a qubit

of quantum information that is being transported along thearray.

the array can be written as

H 0 = dx

2e

2C t χ 2 +

(χ ′ )2

2L0+ C φ2 +

φ2

2L

+(φ′ )2

2LC −2E J cos χ (x, t )cos φ(x, t ) , (5)

where the prime denotes the derivative with respect tox, and all the parameters, C t , L0 ,C ,L C , E J , are denednow per unit length, with the exception of inductanceL, for which the inverse inductance is proportional tolength, and one denes 1 /L per unit length.

In the situation of interest for quantum informationtransfer, we can also adopt an assumption that the dy-namics of φ is restricted to the regime of small phases,

|φ| ≪ 1, since the qubit designs aim typically at thisregime to minimize the decoherence effects. In this case,one can expand the Hamiltonian ( 5) in φ and express itas a sum of the parts governing the evolution of the clockand information phases:

H 0 = H (χ ) + H (φ) ,

where

H (χ ) = dx

2e

2 C t χ 2

2 +

(χ ′ )2

2L0 −2E J cos χ , (6)

and, in the quadratic approximation,

H (φ) = dx

2e

2C φ2 +

(φ′ )2

2LC +

φ2

2L

+ E J cos χ (x, t )φ2 . (7)



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The χ -part ( 6) is equivalent to the sine-Gordon Hamil-tonian of a regular long Josephson junction. For the pur-pose of our discussion, the main feature of this Hamil-tonian is that it supports propagation of the individualuxons (see, e.g. [21]):

χ (x, t ) = 4 tan − 1[exp((x −vt)/λ 0)] . (8)

This expression describes the uxon propagating alongthe array with a small velocity v ≪ (L0C t )− 1/ 2 . In thesituation desired in the context of quantum computa-tion, the energy losses in the array are negligible. Theuxon motion in this regime is ballistic, i.e., the velocityv is determined by the process of uxon injection intothe array. In the ballistic regime, uxon propagation canbe used for measurements of superconducting qubits [ 22–25]. If small energy losses in the dynamics of the commonmode are non-negligible, velocity v of the uxon motionis established by the balance between these losses andthe driving force created by the applied bias current [ 26–28]. In the present context, such a weakly-dissipativeregime of the uxon motion does not prevent uxonsfrom serving as carriers of quantum information, as longas dissipation is conned to the common mode dynamics.In both regimes, velocity v is directly related to the dcbias voltage across the nSQUIDs of the array, which canbe used to control and monitor the uxon motion. Inlong nSQUID arrays we consider in this work, propagat-ing uxon described by Eq. (8) serves as the clock pulsetransporting along the array the excitations of the differ-ential phase φ(x, t ) that encodes quantum information.

III. LOCALIZED EXCITATIONS ASINFORMATION CARRIERS

The Hamiltonian H (φ) (7) describing the dynamics of the phase φ(x, t ) is roughly similar to the Hamiltonian of an individual SQUID that are used in typical supercon-ducting qubits. The main new feature of the phase φ(x, t )in comparison to the phase in usual qubits is that φ(x, t )is now a eld with dependence on coordinate x, and ac-cordingly, the eigenstates of H (φ) have a spatial structurebeing distributed along the array. This spatial structureis controlled by the variation of the clock phase χ (x, t )in the uxon ( 8) which modulates the energy density inH (φ) (7). The lowest-energy excitations of the informa-tion phase φ(x, t ) are localized in the region where χ ≃πand the effective Josephson coupling energy 2 E J cos χ isnegative and largest in absolute value. These excitationscan be used to encode information, e.g., by serving asthe basis state of a qubit localized on the uxon. As onecan see from Eq. ( 7), depending on the relative magni-tude of the Josephson coupling strength and inductanceL, dynamics of the phase φ(x, t ) in the χ ≃ π region isgoverned by either a bi-stable potential as required forencoding the qubit of information in two different uxstates in the ux qubits [ 29, 30], or a monostable po-tential in which the information can be encoded in two

different energy states, similarly to the phase qubits [31].In either case, in the nSQUID array, the qubit is attachedto the uxon and moves with it along the array.

Naturally, to account for the bi-stable dynamics of φone needs to keep higher-order terms in φ2 in the Hamil-tonian H (φ) (7). In the following, we consider quanti-tatively the situation similar to the phase qubits, whenthe relative magnitude of the Josephson coupling is such

thatβ ≡2E J L/ ( / 2e)2 < 1 ,

and the effective potential for φ(x, t ) in the Hamilto-nian ( 7) is monostable. In this case, and for β not tooclose to 1, one can use quadratic approximation, as inEq. (7), to determine the space structure and frequenciesof information-encoding excitations of the phase φ(x, t ).

For the purpose of quantum information transfer, oneis interested in the regime of the sufficiently slow evolu-tion of χ (x, t ), when the characteristic frequencies of thedynamics of φ, which are on the order of ω p = (2 LC )− 1/ 2 ,are much larger than the frequency associated with theuxon propagation, ˙χ ≃ v ≪ ω p. In this, adiabatic,regime, the qubit transfer process along the array thatis driven by the time evolution of χ (x, t ) in the mov-ing uxon ( 8), with exponential accuracy does not af-fect the states of phase φ. Quantitatively, for adiabaticevolution of χ , one can neglect the time dependence inEq. (8) when calculating the structure of the excitationspectrum. Then, substituting the uxon shape ( 8) intoEq. (7), one gets the following equation of motion for thephase φ(x, t ) from the resulting Hamiltonian:

(L/L C )φ′′ = ω− 2 p

φ + 1 + β − 2β

cosh2 x/λ 0φ . (9)

As a rst step, we solve this equation classically byseparating the time and space dependence of φ(x, t ), andexpressing it as a sum of different excitation modes withsome coefficients cj

φ(x, t ) =j

cj φj (x)e− iω j t . (10)

After this substitution, Eq. ( 9) takes the form of anexactly solvable (see, e.g., [ 32]) stationary Schr¨ odingerequation that determines the spatial prole φj (x) andfrequencies ωj of the excitation modes:

−(L/L C )φ′′ − 2β cosh2 x/λ 0

φ = − 1 + β −(ωj /ω p)2 φ ,

(11)We are interested in the discrete part of the spectrum

of this system which consists of the modes localized onthe uxon, in the χ ≃ π, i.e. x ≃ 0, region. Thequalitative features of Eq. ( 11) as Schr odinger equation:potential that has the minimum value −2β at x = 0and approaches 0 at x → ∞, imply that there is a con-tinuous spectrum of frequencies of delocalized modes at



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ω ≥ ω p(1 + β )1/ 2 , while the frequencies of the modeslocalized on the uxon lie within the range

ω p(1 −β )1/ 2 ≤ωj ≤ω p(1 + β )1/ 2 . (12)

Substituting into Eq. ( 11) the prole of the lowest-frequency mode

φ0(x) = A0 cosh(x/λ 0 ) − ν 0 , (13)

one sees that this equation is satised for

ν 0 = 12

(1 + 2 /l )1/ 2 −1 ,

where l ≡L0 / 4LC , and gives the frequency

ω0 = ω p 1+ β (1−δ 0) 1/ 2 , δ 0 = (2+ l)1/ 2−l1/ 2 2 . (14)

For the subsequent quantization, it is convenient to havethe mode prole normalized by the condition

dx[φ0(x)]2 = 1 . (15)

This condition determines the normalization constant forthe zeroth mode ( 13) as

A0 = 1π1/ 4

Γ(ν 0 + 1 / 2)λ0Γ(ν 0 )

1/ 2.

Depending on the inductances ratio l, the frequency ( 14)spans the whole interval ( 12). For l →0 (i.e., L0 ≪LC ),ω0 →ω p(1 −β )1/ 2 , qualitatively because ν 0 is large andthe mode is strongly localized at x ≃ 0, where effectiveJosephson coupling reaches minimum, −E J . For largel, the mode is weakly localized, ν 0 → 0, and the fre-quency ω

0 approaches the edge of the continuous spec-

trum ω p(1 + β )1/ 2 . The middle of the interval, ω0 = ω pis achieved for L0 = LC , when ν 0 = 1.

This discussion implies that the zeroth localized modewith frequency ( 14) exists for arbitrary values of the cir-cuit parameters. For sufficiently small inductance L0(making the characteristic width λ0 of potential welllarge), Eq. ( 11) has other localized modes with higherfrequencies. For any given value of the inductance rationl, the j th localized mode exists if

j < 12

(1 + 2 /l )1/ 2 −1 ,

and has the frequency

ωj = ω p 1+ β (1−δ j ) 1/ 2 , δ j = l (1+2 /l )1/ 2−(2 j +1) 2 .

As one can see from Eqs. ( 9) and ( 10), dynamicsof each localized mode is equivalent to that of a har-monic oscillator, and can be quantized in the standardway by expressing the amplitudes of the frequency com-ponents of φ(x, t ) in terms of the usual bosonic cre-ation/annihalation operators aj , a†

j . Substituting expan-sion (10) into the Hamiltonian ( 7) and evaluating the

integral by making use of the general properties of themode functions φj (x): Eq. ( 11), orthogonality, and nor-malization ( 15), one transforms the Hamiltonian ( 7) intothe standard form:

H (φ) =j

ωj (a†j a j + 1 / 2) ,

and obtains the quantum version of the classical modeexpansion ( 10) of the phase eld φ(x, t ):

φ(x, t ) =j

e2

Cω j

1/ 2φj (x) (a j + a†j ), (16)

and the charge density Q(x, t ) associated with the dy-namics of the differential mode on the junction capaci-tance C of the nSQUIDs:

Q(x, t ) = C

eφ(x, t ) = i

j

( Cω j )1/ 2 φj (x) (a†j −a j ).

(17)

In principle, the sums in all these expressions shouldrun also over the continuous part of the excitation spec-trum, but for the information-encoding purposes dis-cussed in this work, only the discrete modes that arelocalized on the moving uxons are of interest. The lo-calized modes can be used in a variety of ways to encodequantum information. The most direct approach is to usethe similarity of the considered systems with the conven-tional phase qubits in individual SQUIDs, and use as thetwo basis states of the qubit of information the groundstate |0 of the dynamics of the differential phase φ andthe rst excited state of the lowest-frequency mode:

{|0 , a†

0

|0

}. (18)

As in the phase qubits, nonlinearity of the Josephsoncoupling energy in the array Hamiltonian ( 5) makes itpossible then to conne dynamics of φ to the two states(18) producing controllable qubit of information, whichnow, in the nSQUID conguration, is carried along thearray by propagating uxon. In the situation withoutthe external bias for φ considered above, the lowest-ordernonlinear perturbation V of the quadratic Hamiltonian(7) is created by the fourth-order terms in the expansionof the Josephson energy:

V = V 0(a0 + a†0)4 , (19)

V 0 = E J

12 e2

Cω0

2 dxφ40 (x)

2cosh2 x/λ 0 −1 .

Here, we took into account that for the purpose of us-ing this expression in the rst-order perturbation theory,one can keep in it the expansion ( 16) of the phase φtruncated to include only the same zeroth mode that de-nes the basis states ( 18). Also, for the adiabatic uxonmotion, v ≪ ω p, the time dependence of χ (x, t ) was



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neglected making the nonlinearity V (19) a static per-turbation. The perturbation V creates the rst-ordercorrections to the harmonic oscillator energies of the ex-citations of the zeroth mode obtained in the quadraticapproximation ( 11). These corrections make the energygap between the two qubit basis states ( 18) different byδE from the gap separating the upper qubit state fromthe next energy level. Using the mode function ( 13) to

calculate the spatial integral in Eq. ( 19), we nd δE fromthe standard rst-order perturbation theory in V :

δE = E J √ πλ 0

e2Γ(ν 0 + 1 / 2) Cω0Γ(ν 0 )

2 (2ν 0 −1/ 2)Γ(2 ν 0)Γ(2ν 0 + 3 / 2)

. (20)

To give a numerical example, we take L0 = LC , whenν 0 = 1 and ω0 = ω p = (2 LC )− 1/ 2 . Then the relativemagnitude of δE (20) can be expressed as δE/ ω p =(β/ 10λ0)(e2/ )(L/ 2C )1/ 2 , and can be estimated to beon the order of few tenths of a percent for typical val-ues of parameter. The magnitude of nonlinearity can beincreased by introducing the external bias into the dif-ferential mode, which decreases the order of perturbationfrom the forth-order term ( 19) to the third-order term.Finite nonlinearity is needed to operate the qubit ( 18)similarly to the phase qubits, by controlling it with RFpulses. The pulse frequency is tuned to the energy dif-ference between the basis states ( 18) of the φ0 mode, i.e.,approximately to the frequency ω0 (14), while nonlinear-ity ensures that these pulses do not drive the system tothe higher excitation states, limiting its dynamics to thetwo basis states ( 18). In this regime, the magnitude of δE determines the the required qubit operation time.

IV. DECOHERENCE PROPERTIES OFMOVING QUBITS

Although the structure of the nSQUID arrays is opti-mized for quantum information transfer, to be potentiallyuseful, the moving qubits supported by the arrays shouldat least preserve the coherence properties of the current“static” qubit designs, making it important to under-stand these properties of the nSQUID qubits. NSQUIDarrays share two main physical mechanisms of decoher-ence with other superconducting qubits: low-frequency,typically 1 /f , noise produced by the two-level uctua-tors in the materials surrounding the qubits (see, e.g.,[33–35]), and electromagnetic uctuations in the controllines of the device. Although the low-frequency noise canbe expected to play an even stronger role in the nSQUIDarrays because of their more complex, multilayer, struc-ture, decoherence effects of this noise are suppressed bythe mechanisms inherent in the nSQUID dynamics. Partof this suppression is due to the same mechanism as inmost current superconducting qubits: making the energydifference between the qubit basis states only weakly (ei-ther linearly with small coefficient, or quadratically) de-pendent on the external control parameters, e.g. mag-netic ux or electric charge, with their low-frequency

uctuations. In the nSQUID qubits, this is the mech-anisms suppressing the decoherence by the noise in thecommon phase χ (x, t ), i.e. uctuations in the position orshape of the uxon ( 8) carrying the qubit. As discussedin the previous section, the basis states of this qubit areautomatically localized in the region where χ ≃ π. Forsuch χ , ∂ cos χ/∂χ ≃ 0, and the Hamiltonian ( 7) thatgoverns dynamics of the differential mode depends only

quadratically on the uctuations of χ . In the regimeof small uctuations of χ relevant for quantum compu-tation, quadratic coupling suppresses their decoherenceeffects.

The low-frequency noise also affects the dynamics of the information-encoding differential mode φ(x, t ) di-rectly. In the situation of the quasiclassical dynamicsof the Josephson phases of the nSQUIDs that is of inter-est for our discussion, the dominant low-frequency noiseis the noise Φ(x, t ) of the magnetic ux in the nSQUIDs.One can view this noise as the uctuating part of theux, which provides the phase bias φe (t) for the differen-tial phase φ of each individual nSQUID, and couples tothis phase as in the Hamiltonian ( 1). In the continuouslimit, and for strong negative coupling of inductances,this noise results then in the following perturbation termthat should be added to the basic Hamiltonian ( 5):

U = −

2eL dxφ(x, t )Φ(x, t ) . (21)

Since the magnetic ux noise is produced presum-ably by the microscopic two-level systems localized atthe superconductor-dielectric interfaces of the nSQUIDstructure [36–38], the noise uxes are uncorrelated amongdifferent nSQUIDs [ 39]. In the continuous approxima-tion, this means that the noise Φ( x, t ) is δ -correlated:

Φ(x, t )Φ(x ′ , t ′ ) = δ (x −x ′ ) dω2π

S 0(ω)eiω ( t − t ′ ) , (22)

where the spectral density S 0(ω) of noise in one nSQUIDshould have the 1 /f prole:

S 0(ω) = A/ |ω| , (23)

with some low- and high-frequency cutoffs, ωl and ωh .In the situation of the monostable potential for the dif-

ferential mode φ(x, t ) (β < 1), and absence of the exter-nal bias for it, φe (x, t ) ≡0, that we focus on in this work,perturbative corrections due to U (21) to the energies of the qubit basis states vanish in the rst order, as one canargue, e.g., from the φ → −φ symmetry of potential inthe Hamiltonian ( 5). Therefore, similarly to the uctu-ations of the common mode χ (x, t ), the perturbation U (21) associated with the uctuations of the differentialmode, produces only the second-order, quadratic, non-vanishing corrections to the qubit energies, reducing thedecoherence effects of these uctuations.

Besides this suppression of the low-frequency decoher-ence by reduced coupling to the noise source, which is thesame as in the static qubits, qubits transported along



7

the nSQUID arrays should exhibit another suppressionmechanism resulting from the dynamic spreading of thenoise spectrum due to the qubit motion. Qualitatively, if the qubit is transported with velocity v along the array,where the low-frequency noise is not correlated in dif-ferent nSQUIDs, the qubit sees the effective noise withthe correlation time on the order of 1 /v , the time of thequbit motion between the nearest-neighbor cells of the

array. This mean that the spectral density of the noiseas seen by the qubit is changed from the low-frequencynoise spectrum of one nSQUID into the noise distributeduniformly over the frequency range limited by v. Suchspreading of the low-frequency noise over a large fre-quency range results in the change of the coherence timeof the qubit from some inhomogeneous dephasing timetd into homogeneous dephasing time t∼t2

d v and can besignicantly increased by the qubit motion. Such a sup-pression of the low-frequency qubit decoherence is qual-itatively similar to the motional narrowing of the NMRlines (see, e.g., [40]), with the main difference that itshould happen not due to random thermal motion butcontrolled uniform propagation of the qubit.

To make this description more quantitative, we con-sider specically the qubit discussed in the previous Sec-tion, with the basis ( 18) spanned by the two lowest energystates of the lowest-frequency localized excitation of thedifferential mode. The qubit decoherence depends on thespectral density S (ω) of the ux noise Φn (t) as seen bythe qubit. For the qubit ( 18) transported with velocityv, this noise is determined by the spatial prole φ0(x)(13) of the lowest-frequency excitation of the differentialmode:

Φn (t) = dxφ0 (x −vt)Φ(x, t ) . (24)

This expression gives the following result for the noisespectral density S (ω) seen by the qubit:

S (ω) = dτ Φn (t + τ )Φn (t) e− iωτ

= dω′ S 0(ω′ )f (ω −ω′ ) , (25)

where the function f (ω) is dened as

f (ω) = 12π dt dxφ0 (x)φ0 (x −vt)eiωt . (26)

Taking into account the normalization condition ( 15),one can see directly from Eq. ( 26) that f (ω) satises thetwo basic conditions:

dωf (ω) = 1 , f (0) = 12πv dxφ0 (x)

2. (27)

Qualitatively, this function describes how the noisespectral density is spread over the large frequency rangeby qubit motion. For stationary qubit, v = 0, normal-ization condition ( 15) shows that f (ω −ω′ ) = δ (ω −ω′ ),

FIG. 3: Effective frequency spectrum of the 1 /f noise actingon the qubit transported along the nSQUID array with veloc-ity v. The noise is redistributed over frequencies by the qubitmotion. The dashed line shows the original 1 /f noise (23).

For discussion, see main text.

and the qubit states distributed over several nSQUIDs inthe array see the same noise ( 25) as in one nSQUID:

S (ω) = S 0(ω) .

For rapid qubit motion, when the frequency range up toa large frequency proportional to v encloses all intensityof the low-frequency noise, ωh ≪v, Eq. (25) reduces to:

S (ω) = W f (ω), W ≡ dωS 0(ω) . (28)

In general, the qubit motion does not change the totalnoise intensity, as one can see from Eqs. (25) and ( 27):

dωS (ω) = W ,

but it changes it distribution over frequencies. In partic-ular, for the rapid motion, the total intensity of the orig-inal noise S 0(ω) is distributed over the whole frequencyrange given by the qubit velocity, and the resulting noisespectrum S (ω) is completely determined by the motion,as in Eq. ( 28). As one can see from the second equationin (27), an important consequence of this is that S (ω)

becomes at, S (ω) ≃ S (0) in the large frequency rangeω ≪ v, and its magnitude is suppressed by velocity as1/v :

S (0) = W 2πv dxφ0 (x)

2.

Such suppressed spectral density of effectively white noisealso implies similarly suppressed qubit decoherence rate.

To see this more explicitly, we consider the case whenthe two inductances in the nSQUID array cell (Fig. 2) are



8

equal, LC = L0 . The spatial prole ( 13) of the lowest-frequency excitation of the differential mode then is:

φ0(x) = [ 2λ0 cosh(x/λ 0 )]− 1 .

From this expression, one obtains explicitly f (ω) (26):

f (ω) = πλ 0

4vcosh

πλ 0 ω

2v

− 2 . (29)

Making use of this f (ω) in Eq. ( 25) one can nd the noiseS (ω) for the qubit moving along the nSQUID array withan arbitrary velocity v, which results from the dynamicspreading of the 1 /f noise (23). Figure 3 shows S (ω) ob-tained in this way for several velocities v assuming a rea-sonable frequency range of the noise, ωh /ω l = 10 7 . Theplot illustrates the transition from the rapid to slowerqubit motion. The lowest curve (largest v) coincides withEq. ( 28) with f (ω) given by Eq. ( 29). In agreement withthe discussion above, the main feature of all curves inFig. 3 is that the noise spectrum becomes at for fre-quencies below the characteristic frequency ν

≡ 2v/πλ 0

associated with the qubit motion, inverse of the timeit takes a qubit to move by a distance on the order of its (dimensionless) “length”. For slower qubit motion,ωl ≪ ν ≪ ωh , the magnitude of S in this low-frequencyrange can be estimated as

S (0)≃ Aν

ln ν ωl

. (30)

Also, for ν ≪ ωh , the spreading function f (ω) becomesnarrow, and S (ω) coincides with S 0(ω) at the higher-frequency end of the spectrum. (The sharp feature of S (ω) at ω = ωh in this regime is explained by the crudemodel of hard high-frequency cutoff used in the calcula-tion.)

Dynamic transfer of the noise spectrum from low tohigh frequencies increases the qubit dephasing time. Toestimate the magnitude of this increase, we assume thatthere is a nonvanishing differential phase bias for thenSQUIDs in the array, so that the qubit is coupled di-rectly (linearly) to the noise, but this bias is weak and wecan use quantitatively all the previous results on the noisespectrum spreading. In this case, the dephasing time tdof the stationary qubit, in the regime when it limited bythe low-frequency noise producing inhomogeneous Gaus-sian decoherence, is t d = (4 π/W )1/ 2 (including the rele-vant coupling constants into the denition of noise). Thedephasing time t(m )d of the same, but rapidly moving,qubit, which sees as a result the white noise spectrumproducing the exponential homogeneous decoherence, ist (m )

d = 2 /S (0). Using Eqs. ( 28) and ( 29) to nd S (0), weobtain the relation between the two dephasing times:

t (m )d = νt2

d /π . (31)

From a conservative estimate of decoherence time of phase qubits, when it is dominated by the low-frequency

noise, td ∼ 200 ns, and a reasonable expected propaga-tion frequency ν ∼1 GHz, we see that dynamic spreadingof the low-frequency noise spectrum should increase thedephasing time of a qubit propagating in the nSQUID ar-ray roughly by a factor of 50, to about t (m )

d ∼10 µs, thevalue comparable to the present-day decoherence timesof superconducting qubits.

V. SUMMARY AND OUTLOOK

In summary, we have derived the continuous modelof an nSQUID array in the limit of large negative in-ductances of the nSQUIDs, and calculated the frequencyspectrum and the spatial prole of the excitations of thedifferential mode in the array. Reduced to the two-statelimit, dynamics of these excitations can be use to en-code quantum information, which is transported alongthe array by propagating uxons supported by the com-mon mode. Qualitatively, qubit of the differential modethat is attached to the uxon turns it into a particle with

articial spin 1/2. We have calculated the decoherenceproperties of these qubits which are characterized by thedynamic spreading of the noise spectrum due to qubitmotion. The spreading reduces the decoherence rate as-sociated with the low-frequency noise.

In this work, we have considered only the case of longnSQUID arrays, the common mode of which can fully ac-commodate individual uxons, and made the assumptionthat the uxon dynamics is classical. NSQUID arrays canbe used to transport quantum information beyond thisregime. The theory presented above is not directly appli-cable to the case of short arrays, arguably more accessibleexperimentally. For short arrays, the distribution of thecommon phase over the length of the array is differentfrom that in a uxon ( 8), and continuous approximationmay not be valid any longer. Still, even the shortestpossible “arrays” of two nSQUIDs should exhibit the in-formation transfer dynamics qualitatively similar to theone considered above. Another interesting question ispresented by the possibility of quantum motion of uxonsalong the array, which is not completely out of the reachexperimentally [ 41]. In this regime, when not only thedifferential, but also the common mode of the array be-haves according to the quantum mechanics, the informa-tion propagates along the array quantum mechanically.If, in addition, the dynamics of the differential mode isbistable as in the ux qubits, the nSQUID arrays shouldprovide then a convenient way of implementing the uni-versal adiabatic quantum algorithms. All these possibleuses of the nSQUID arrays as conveyers of quantum in-formation merit further investigation.

VI. ACKNOWLEDGEMENTS

The authors acknowledge useful discussions with S.Han, K.K. Likharev, and V.K. Semenov. This work was



9

supported in part by the NSF grant PHY-1314758 andthe European Union Seventh Framework Programme IN-FERNOS (FP7/2007-2013) under the grant agreementno. 308850.

Appendix: Reduction of the array Hamiltonian

In this Appendix, we show explicitly how the Hamilto-nian ( 2) of the nSQUID array, segment of which is shownin Fig. 2, is reduced to the form given by Eq. ( 4) in thelarge-negative-inductance limit M → L, M C → LC . Inorder to do this, we start by transforming two of theinitial degrees of freedom of each cell of the array, thecommon phase χ j of the j th nSQUID, and the supercon-ducting phase χ e,j at bias node point of this nSQUID(Fig. 2), into two new variables ηj , µj dened as

ηj =

2e(χ j −χ e,j ) , µj =

2eC t(2Cχ j + C 0χ e,j ) . (A.1)

The corresponding canonical “momenta” conjugated tothese new coordinates are the following combinations of charges K j and K e,j :

q j = ( C 0 K j −2CK e,j )/C t , pj = K j + K e,j . (A.2)

One can see directly that they indeed satisfy the neces-sary commutation relations:

[ηj , q j ] = i , [µj , pj ] = i , [ηj , pj ] = 0 , [µj , q j ] = 0 .

This transformation is performed to separate explicitlythe relative dynamics of the two phases, χ j and χ e,j ,which will be quenched in the limit M →L, M C →LC .To see this, we consider the relevant, diverging in thislimit, part H ′ of the total array Hamiltonian H 0 [givenby Eqs. ( 2) and ( 3)]:

H ′ =j

K 2j4C

+K 2e,j

2C 0+

2e

2 (χ e,j −χ j )2

L −M

+(χ e,j −χ j + χ j − 1 −χ e,j − 1 )2

LC −M C . (A.3)

To make the subsequent calculation explicit, we need tospecify the array structure. We assume the array of N cells with periodic boundary conditions at the ends, e.g.,η0 ≡ηN . Then, inverting the relations ( A.2):

K j = q j + 2 Cp j /C t , K e,j = ( C 0 /C t ) pj −q j ,

we express Eq. ( A.3) in terms of new variables:

H ′ =N

j =1

p2j

2C t+

q 2j2C r

+η2

j

L −M +

(ηj −ηj − 1 )2

LC −M C , (A.4)

where C r = 2CC 0 / (2C + C 0). The last three termsin this expression form the Hamiltonian H d which gov-erns the relative dynamics of the phases χ j and χ e,j .

As usual, this Hamiltonian can be diagonalized by in-troducing plane-wave excitations with wavevectors k =2πn/N, n = −(N −1)/ 2, −(N −1)/ 2+1 ,... (N −1)/ 2 (weassume for simplicity that N is odd), with correspondingcreation/annihalation operators bk , b†

k :

ηj =k

2C r ωk N

1/ 2(bk + b†

− k )eikj , (A.5)

q j = −ik

C r ωk

2N

1/ 2(bk −b†

− k )eikj . (A.6)

In terms of these plane waves, the Hamiltonian has theusual “phonon” form:

H d =k

ωk (b†k bk + 1 / 2) ,

with the following frequency spectrum:

ωk = 2C

r

1L

−M

+ 4sin2(k/ 2)

LC −

M C

1/ 2, (A.7)

which satises the condition ωk = ω− k .The main feature of this spectrum is that all frequen-

cies become very large, ωk → ∞ in the large-negative-inductance limit. This means that these phonon modescan not be excited by the dynamics of the nSQUID arraywhich occurs at much smaller frequencies, and the modesremain in the ground state throughout the time evolutionof the array. The Hamiltonian H d of the relative dynam-ics of the phases χ j and χ e,j effectively reduces in thisregime to a constant, H d = k ωk / 2, irrelevant for thesystem dynamics. Then, the part H ′ (A.3) of the arrayHamiltonian considered in this Appendix reduces to

H ′ = 12C t

N

j =1

p2j +

12

k

ωk . (A.8)

As one can see from Eq. ( A.5), suppression of the dy-namics of the ωk modes to their ground states, withadded condition ωk → ∞, also implies that ηj → 0.Indeed, simple estimate of the uctuations of ηj usingEq. (A.5) shows that η2

j < [(L −M )/ 8C r ]1/ 2 →0 forM → L, effectively imposing the constraint χ j = χ e,jon the system dynamics. This constraint means that the“center-of-mass” coordinate µj (A.1) of these two phasesalso coincides with them: µj = ( / 2e)χ j , and the mo-mentum pj in the Hamiltonian ( A.8) conjugate to thiscoordinate acts simultaneously as the momentum associ-ated with the common phase χ j . (The difference betweenthe two momenta can be seen only at the large frequen-cies ωk not present in the system dynamics.) Takingthis into account, and inserting the part H ′ in the form(A.8), but without the irrelevant constant, back into thetotal array Hamiltonian H 0 , we obtain H 0 in the large-negative-inductance limit M → L, M C → LC as givenby Eq. ( 4).



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NSQUID Arrays as Conveyers of Quantum Information

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