Double occupancy errors in quantum computing operations: Corrections to adiabaticity

Double occupancy errors in quantum computing operations: Corrections to adiabaticity

Ryan Requist,^1,*John Schliemann,^2,†Alexander G. Abanov,^1,‡and Daniel Loss^2,§

1Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3840, USA

2Department of Physics and Astronomy, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland 共Received 14 September 2004; published 21 March 2005兲

We study the corrections to adiabatic dynamics of two coupled quantum dot spin qubits, each dot singly occupied with an electron, in the context of a quantum computing operation. Tunneling causes double occu- pancy at the conclusion of an operation and constitutes a processing error. We model the gate operation with an effective two-level system, where nonadiabatic transitions correspond to double occupancy. The model is integrable and possesses three independent parameters. We confirm the accuracy of Dykhne’s formula, a nonperturbative estimate of transitions, and discuss physically intuitive conditions for its validity. Our semi- classical results are in excellent agreement with numerical simulations of the exact time evolution. A similar approach applies to two-level systems in different contexts.

DOI: 10.1103/PhysRevB.71.115315 PACS number共s兲: 03.67.Lx, 03.65.Sq, 73.23.Hk

I. INTRODUCTION

Quantum information processing is an active and fascinat- ing direction of research with participation from various fields of physics and neighboring scientific disciplines.¹This extraordinary interest has generated a fairly vast amount of theoretical and experimental studies. Possible experimental realizations of quantum information processing are presently being investigated. Among the different approaches, those in a solid-state setting are attractive, because they offer the po- tential of scalability—the integration of a large number of quantum gates into a quantum computer once the individual gates and qubits are established. With that in mind, several proposals for using electron and/or nuclear spins in solid- state systems have been put forward in recent years.^2–7Spe- cifically, in Ref. 2 it was proposed to use the spin of electrons residing in semiconductor quantum dots as qubits.^8–15In this paper we revisit the quantum dynamics of gate operations between qubits of this type. Such two-qubit operations are performed by varying the amplitude of electron tunneling between the dots via external electric potentials. In a generic scenario, the tunneling amplitude between the dots is zero 共or, more precisely, exponentially small兲before and after the gate operation, while it is finite and appreciable during such a process. Thus, the typical time dependence of the tunneling amplitude is a pulse roughly characterized by its duration, amplitude, and ramp time共see Fig. 1兲. During such a pulse, the tunneling amplitude is finite and essentially constant, and both electrons can explore the total system of two quantum dots. Therefore, their indistinguishable fermionic character is of relevance.^10,16,17 In particular, in such gate operations entanglement-like quantum correlations arise which require a description different from the usual entanglement between distinguishable parties共Alice, Bob,…兲in bipartite共or mul- tipartite兲systems. In such a case the proper statistics of the indistinguishable particles has to be taken into account.^10,16,17 Another important aspect of having a finite共as opposed to infinitely high兲 tunneling barrier between the dots is that it necessarily leads to共partially兲doubly occupied states in the two-electron wave function, i.e., contributions to the wave

function where both electrons are on the same dot 共having different spins兲occur with finite amplitude. Doubly occupied states which arise as the result of a measurement after the gate operation destroy the information in those qubits and lead to errors in the information processing. Therefore, it is desirable to reduce the probability of such errors, i.e., the occurrence of doubly occupied states, in the resulting two- electron state after the gate operation, while it is necessarily finite during the operation.^9,10If the error probability can be sufficiently reduced, error events can be tolerable and handled with quantum error correction schemes. An effective way of guaranteeing error suppression is to maintain nearly adiabatic time evolution. Doubly occupied states then corre- spond to corrections to adiabatic evolution, which are often called “nonadiabatic transitions.” Numerical simulations¹⁰ have shown that the adiabatic region, in terms of the pulse parameters such as ramp time and amplitude, is rather large.

On a heuristic level, this numerical result is plausible on the basis of the classic papers on adiabatic quantum motion in two-level systems by Landau,¹⁸Zener,¹⁹ Stueckelberg,²⁰and Rosen and Zener.²¹For an overview see Ref. 22.

In this work we study the quantum dynamics of the two- qubit gate operations described above and use Dykhne’s semiclassical result to estimate the probability of nonadia- batic transition.²³ The applicability of Dykhne’s formula is analyzed from the standpoint of the theory of semiclassical approximations. These semiclassical estimates are found to be in excellent agreement with numerical simulations of the exact time evolution. Moreover, in a certain limit our model is integrable, allowing us to explicitly calculate and interpret the corrections to Dykhne’s formula.

This paper is organized as follows. Section II reviews the derivation¹⁰ of an effective two-level model. In Sec. III, we present our main result—the asymptotic estimate of double occupancy, which in Sec. IV is compared with an integrable model and a numerical integration of the Schrödinger equa- tion. In the Appendix, we construct the scattering matrix for the integrable model, which has three independent param- eters.

II. MAPPING TO AN EFFECTIVE TWO-LEVEL SYSTEM For the purpose of studying double occupancy it is prac- tical to examine the dynamics of the quantum gate operation in a subspace spanned by singly and doubly occupied states.

Following Ref. 10 with only minor changes of notation, we now detail how to reduce the description of a system of two coupled quantum dot spin qubits to an effective two-level Hamiltonian. The system is described by a Hamiltonian of the formH= T + C, where C denotes the Coulomb repulsion between the electrons and T =兺i=1,2h_i is the single-particle part with

h_i= 1

2m

冉

^{pជi+ecAជ共rជi兲}

冊

^{2+ V共rជi兲. 共1兲}

The single-particle Hamiltonian h_idescribes electron dynam- ics confined to the xy plane in a perpendicular magnetic field Bជ. The effective mass m is a material dependent parameter.

The coupling of the dots共which includes tunneling兲is mod- eled by a quartic potential

V共rជ兲= V共x,y兲=m␻0 2

2

冉

^{4a12共x2− a2兲2+ y2}

冊

^{, 共2兲}

which separates into two harmonic wells of frequency ␻0

共one for each dot兲 in the limit aⰇa₀, where a is half the distance between the dots and a₀=

冑

ប/ m␻0 is the effective Bohr radius of a dot.

Following Burkard et al.⁸we employ the Hund-Mulliken method of molecular orbits to describe the low-lying spec- trum of our system. This approach concentrates on the lowest orbital states in each dot and is an extension of the Heitler- London method.⁸关In the following, we assume for simplicity that ប␻ⰇU_H i.e., 共single particle orbital level spacing兲 Ⰷ 共quantum dot charging energy兲, so that orbital excitations can be safely neglected. Such a situation is reached for suffi- ciently small quantum dots.¹²兴The Hund-Mulliken approach accounts for the fact that both electrons can, in the presence of a finite tunneling amplitude, explore the entire system of the two dots, and therefore adequately includes the possibil- ity of doubly occupied states. In the usual symmetric gauge

Aជ_{= B共−y , x , 0兲}/ 2 the Fock-Darwin ground state of a single dot with harmonic confinement centered around rជ

=共±a , 0 , 0兲reads

␸±a共x,y兲=

冑

^m␲^␻ប exp

冉

^−m2ប^␻关共x⫿a兲²+ y²兴

冊

⫻exp

冉

^⫿2iylaB

2

冊

^{, 共3兲}

where l_B=

冑

បc / eB is the magnetic length, and the frequency

␻ is given by ␻²⁼␻0

2+共␻L/ 2兲² where ␻L= eB / mc is the usual Larmor frequency. From these nonorthogonal single- particle states we construct the orthonormalized states 兩A典 and兩B典 with wave functions

具rជ兩A典= 1

冑

1 − 2Sg + g²共␸+a− g␸−a兲, 共4兲 具rជ兩B典= 1

冑

1 − 2Sg + g²共␸−a− g␸+a兲, 共5兲 with S being the overlap between the states 共3兲 and g =共1

−

冑

1 − S²兲/ S. For appropriate values of system parameters such as the interdot distance and the external magnetic field, the overlap S becomes exponentially small.⁸In this limit an electron in one of the states兩A典,兩B典is predominantly local- ized around rជ⁼共±a , 0 , 0兲. In the following we consider this case and use these states as basis states to define qubits, i.e., qubits are realized by the spin state of an electron in either orbital兩A典or orbital兩B典.

An appropriate basis set for the six-dimensional two- particle Hilbert space is given 共using standard notation兲 by the three spin singlets,

兩S₁典= 1

冑

^{2共cA+↑cB+↓− cA+↓cB+↑兲兩0典, 共6兲}

兩S₂典= 1

冑

2共c_A⁺_↑c_A⁺_↓+ c_B⁺_↑c_B⁺_↓兲兩0典, 共7兲 兩S₃典= 1

冑

2共c_A⁺_↑c_A⁺_↓− c_B⁺_↑c_B⁺_↓兲兩0典, 共8兲 and the triplet multiplet,

兩T⁻¹典= c_A⁺_↓c_B⁺_↓兩0典, 共9兲 兩T⁰典= 1

冑

2共c_A⁺_↑c_B⁺_↓+ c_A⁺_↓c_B⁺_↑兲兩0典, 共10兲 兩T¹典= c_A⁺_↑c_B⁺_↑兩0典. 共11兲 As the Hamiltonian conserves spin, the three triplet states are degenerate eigenstates共typically we can ignore possible Zee- man splittings⁸兲and have the eigenvalue

␧trip= 2␧1+ V₋, 共12兲 where we have defined

FIG. 1. A realistic profile of the tunneling pulse共23兲, labeled with the characteristic duration共T⬇13␶兲and ramp time scales. The pulse shown has dimensionless strength␦⁼¹₂^.

␧1=具A兩h₁兩A典=具B兩h₁兩B典 共13兲 and the expectation value of Coulomb energy,

V₋=具T^␣兩C兩T^␣典, V₊=具S₁兩C兩S₁典. 共14兲 An important further observation is that, as a consequence of inversion symmetry along the axis connecting the dots, the Hamiltonian does not have any nonzero matrix elements between the singlet state兩S₃典and other states. Hence,兩S₃典is, independently of the system parameters, an eigenstate. The eigenvalues of the triplet and兩S₃典, however, do depend on system parameters. The Hamiltonian acting on the remaining space spanned by兩S₁典and兩S₂典 can be written as

H= 2␧1+1

2U_H+ V₊−U_H

2

冉

^t1H − 1^tH

冊

^{, 共15兲}

where

t_H= − 4

U_H

冉

^{具A兩h1兩B典+12具S2兩C兩S1典}

冊

^共16兲

and

U_H=具S₂兩C兩S₂典− V₊. 共17兲 The nontrivial part of Eq.共15兲 is a simple Hubbard Hamil- tonian on two sites and can be identified as the Hamiltonian of a pseudospin-¹₂ object in a pseudomagnetic field having a component U_Hin the zˆ direction and U_Ht_Hin the xˆ direction of pseudospin space.共Note that this pseudospin is not related to the spin degree of freedom which constitutes the qubit.兲 The basis states themselves are eigenstates only in the case of vanishing tunneling amplitude t_Hwhere兩S₁典is the ground state and 兩S₂典 is a higher lying state due of the Coulomb 共Hubbard兲energy. In all other cases, the ground state has an admixture of doubly occupied states contained in 兩S₂典. The energy gap between the triplet and the singlet ground state is

␧trip−␧gs= V₋− V₊−U_H 2 +U_H

2

冑

1 + t_H². 共18兲 A key challenge for state-of-the-art quantum information processing is the construction of systems composed of two coupled quantum dots which can be coupled to perform swap operations USW, i.e., unitary two-qubit operations which in- terchange the spin states共qubits兲of the electrons on the two dots. By combining the “square root” USW

1/2 of such a swap with other isolated-qubit manipulations one can construct a quantumXORgate. A quantumXORgate, along with isolated- qubit operations, has been shown to be sufficient for the implementation of any quantum algorithm.²⁴Hence a practi- cal and reliable realization of a swap gate would be an im- portant step toward the fabrication of a solid-state quantum computer. A swap operation in the present system is a unitary transformation which turns a state having the qubits in dif- ferent states, say,

c_A⁺_↑c_B⁺_↓兩0典= 1

冑

2共兩T⁰典+兩S₁典兲, 共19兲 into a state where the contents of the qubits are interchanged,

c_A⁺_↓c_B⁺_↑兩0典= 1

冑

2共兩T⁰典−兩S₁典兲. 共20兲 These two states are eigenstates in the case V₊= V₋ and t_H

= 0 for which the singlet-triplet splitting vanishes.

As discussed in Refs. 2, 8, and 10, swapping may be achieved by the action of a gate that lowers the potential barrier between the quantum dots. If the duration and ampli- tude of a tunneling pulse are adjusted appropriately, the rela- tive dynamical phase between the singlet and the triplet states accumulates a shift of␲^,

1 ប

冕

−⬁

⬁

dt关␧trip共t兲−␧gs共t兲兴=␲ 共21兲 and the swapping operation between states共19兲and共20兲 is performed. However, during the operation the state 兩S₁典 is coupled to 兩S₂典, and they evolve according to Eq. 共15兲. Double occupancy errors are thus generically introduced.

The reduction of the dynamics to the time evolution of a two-level system relies on the fact that the system has inver- sion symmetry along the xˆ axis in real space connecting the dots. This symmetry can be broken if odd powers of the particle coordinates x_iare added to the Hamiltonian共1兲, for example, the potential of a homogeneous electric field. The breaking of inversion symmetry introduces additional matrix elements between 兩S₃典 and the other two singlets leading to an effective three-level Hamiltonian. However, as it was shown in Ref. 10, this more inclusive Hamiltonian has quali- tatively the same properties concerning nonadiabatic dynam- ics as the two-level system on which we shall concentrate in the following.

So far we have not considered a possible Zeeman cou- pling to the electron spin. This would not change the situa- tion essentially since all states involved in the swapping pro- cess 共兩T⁰典, 兩S₁典, 兩S₂典, and possibly 兩S₃典兲 have the total spin quantum number S^z= 0.

III. ANALYSIS OF NONADIABATIC TRANSITIONS In this section we use Dykhne’s formula for nonadiabatic transitions to derive an asymptotic expression for the prob- ability of final double occupancy, given physically motivated properties of the two-qubit operation.

As described in the preceding section, the modulation of the tunneling barrier during the swapping process induces a coupling between the singly occupied qubit state兩S₁典and the doubly occupied state兩S₂典. Their dynamics are governed by the effective Hamiltonian

Heff= −U_H

2

冉

^t1H − 1^tH

冊

^共22兲

in the兩S_1,2典 basis. The terms omitted from Eq. 共15兲 do not contribute to transitions, because the identity operator in the 兩S_1,2典basis commutes with the remainder of the Hamiltonian.

The large energy offset U_Hbetween singly and doubly occu- pied states, primarily due to the Coulomb repulsion, is per- turbed only by an exponentially small additive quantity共pro- portional to the overlap, S兲 during the swapping operation

and is hereafter assumed to be a constant. Our specification of the pulse共Fig. 1兲

t_H共t兲= ␦ 1 + cosh共t/␶兲

cosh共T/2␶兲

共23兲

with dimensionless strength ␦ is considered to realistically reflect the tunneling amplitude that would arise from a modulation of the gate potential.¹⁰ The exponential depen- dence of the ramping near t = ± T / 2 has its origin in the ex- ponential sensitivity of the coupling to the gate voltage and in turn the exponential decay of the single-particle wave functions 共3兲 in the interdot region.²⁵ The pulse mimics a step of duration T and magnitude␦^UH/ 2, whose ramping on and off has a characteristic time ␶. The perturbation of the instantaneous eigenvalues by the pulse is shown in Fig. 2.

The Schrödinger equation is iបd

dt兩␺共t兲典=Heff共t兲兩␺共t兲典. 共24兲 Our task is to find the component of double occupancy in the final state,具S₂兩␺共⬁兲典, given that the prepared state is purely singly occupied,兩具S₁兩␺共−⬁兲典兩= 1.

Our model involves three dimensionless scales, assigned for our purposes as follows: ␦^, ␭⬅U_H␶^{/ 2}ប, and ␩^{= T /}␶^. Presently, the case of interest is

␭ Ⰷ1, ␩Ⰷ1. 共25兲 The first of these conditions reflects the adiabaticity of the problem. The second requires that the ramping on and ramp- ing off of the pulse be temporally well-separated and distinct events.

Let us pause and for this paragraph review the familiar notions of transitions under the action of a time-dependent perturbation. The pulse acts as a transient perturbation and otherwise the Hamiltonian共22兲is diagonal. By force of the adiabatic theorem, the probability of transition among eigen- states vanishes in the limit␶→⬁, where the ramping on and off of the pulse is adiabatic. In the zeroth order of adiabatic

perturbation theory, there are no transitions, and the leading behavior of the general solution is simply the dynamical phase of each component eigenstate

兩␺共t兲典 ⬇exp

冋

^បi

^冕

^−t⬁dt

^⬘

^␧共t

^⬘

^兲

册

^{兩␰1共t兲典具␰1共−⬁兲兩␺共−⬁兲典}

+ exp

冋

^−បi

^冕

^−t⬁dt

^⬘

^␧共t

^⬘

^兲

册

^{兩␰2共t兲典具␰2共−⬁兲兩␺共−⬁兲典,}

共26兲 where 兩␰1,2共t兲典 are the instantaneous eigenstates 关given ex- plicitly in Eq. 共54兲兴 of Hamiltonian 共22兲 corresponding to eigenvalues

⫿␧共t兲= ⫿U_H

2

冑

1 + t_H², 共27兲 respectively. In general, the approximate solution could also include a factor representing Berry phase. However, for a real symmetric Hamiltonian such as Eq.共22兲, Berry phase is irrelevant, because the Hamiltonian has an inherent planarity.

In pseudospin one-half notation,Heff= Hជ_共t兲·␴ជ, the time evo- lution of the pseudomagnetic field Hជ_共t兲 is in a plane. If the azimuthal axis共north pole兲is chosen to lie within that plane, the solid angle subtended by the pseudomagnetic field van- ishes identically. Although Berry phase is out of consider- ation, there are interesting circumstances where Berry phase is relevant to transitions. It can correct the transition amplitude²⁶and produce topological selection rules for spin tunneling.^27,28Our problem is one of a class initiated by the work of Landau, Zener, and Stueckelberg.^18–20However, we emphasize that for our model关with the pulse specified as Eq.

共23兲兴the linearization of Hamiltonian matrix elements near the times where adiabaticity is most severely violated is not applicable and leads to an incorrect result. As we will see the shape of the pulse is important.

A. Application of Dykhne’s formula

Returning to our model, we observe that if the time inter- val t苸共−⬁,⬁兲 is divided into two domains t⬍0 and t⬎0, and in the limit␩⬅T /␶Ⰷ1, the pulse共23兲is approximated by

t_H共t兲 ⬇

冦

^{1 + e1 + e−共t/共␦␦t/␶兲␶兲−−共T/2共T/2␶兲␶兲,, tt⬍⬎00.}

冧

^共28兲

In each domain the pulse behaves as a step, and the dynam- ics are integrable 共see Sec. IV兲. We will focus first on the interval t⬍0, where the probability of transition to a doubly occupied state P_⬍ may be estimated with Dykhne’s formula²³

FIG. 2. A profile of the instantaneous eigenvalues ±␧共t兲 corre- sponding to␭= 2 and the pulse shown in Fig. 1.

P_⬍=兩具S₂兩␺共0兲典兩²⬃exp

冋

^−4បIm

^冕

^{Ret1共t1兲dz␧共z兲}

册

^{, 共29兲}

where the approximation共28兲 is used implicitly for the in- stantaneous eigenenergies⫿␧共t兲 defined above by Eq.共27兲. The turning point t = t₁, given explicitly below, is a complex root of the function␧共t兲; in other words, it is an intersection of the energy surfaces of the two instantaneous 共“frozen”兲 eigenstates. Our model is the patching together of two do- mains of time, and transitions that occur during t⬍0 and t

⬎0 interfere. The expression for the probability of transition during the time evolution from t = −⬁to t =⬁is

P =兩具S₂兩␺共⬁兲典兩²

⬃

冏

^exp

冋

^បi

^冕

^{Cadz␧共z兲}

册

^{+ exp}

冋

^បi

^冕

^{Cbdz␧共z兲}

册 ^冏

²

⬃

冏

^exp

冋

^{បi Re}

^冕

^{Cadz␧共z兲}

册

⫻exp

冋

^{−ប2 Im}

^冕

^{Ret1共t1兲dz␧共z兲}

册

^{+ exp}

冋

^{បi Re}

^冕

^{Cbdz␧共z兲}

册

⫻exp

冋

^{−ប2 Im}

^冕

^{Ret2共t2兲dz␧共z兲}

册 ^冏

^{2 共30兲}

=4 sin²

冉

^{ប1 Re}

^冕

^t1 t₂

dz␧共z兲

冊

^{P⬍, 共31兲}

where the contoursCa,bare shown in Fig. 3, and according to the sign of the integration variable, sgn共Re z兲, one or the other of the approximations共28兲is used. The turning points t = t_1,2appearing in the limits of integration of Eq. 共30兲are chosen as the two roots of␧共t兲that are closest to and above the real time axis共see Fig. 3兲,

t_1,2= ⫿

冉

^T2+␶ln共

^冑

^{1 +␦2兲}

冊

^{+ i␶关␲− arctan共␦兲兴. 共32兲}

They are nonreal because the Hamiltonian共22兲is nondegen- erate for real times. Equation 共31兲 follows from Eq. 共30兲, because the symmetry of the pulse implies Im共t₁兲= Im共t₂兲

and P_⬍= P_⬎. The oscillatory first factor of Eq. 共31兲 is the interference of the dynamical phase of each term of Eq.共30兲. The magnitude of P is dominated by the second factor P_⬍ whose exponent is given by the following integral:

− 4␭Im

冕

ln共冑1+␦²兲

ln共冑1+␦²兲+i␲−i arctan共␦兲

dz

冋

^{1 +}

^冉

^{1 + e␦ z}

^冊

²

册

^1/2

= − 2␲␭共1 +

冑

1 +␦²⁻␦兲. 共33兲 Substituting this result in Eq.共29兲we have

P_⬍⬃e^{−2␲␭共1+}冑1+␦²⁻␦兲. 共34兲 From Eq.共31兲, we have our main result, an asymptotic esti- mate for the probability of final double occupancy,

P⬃4 sin²

冉

^1បRe

^冕

^t1 t₂

dz␧共z兲

冊

^{e−2␲␭共1+冑1+␦2−␦兲, 共35兲}

which is shown as a function of␦in Fig. 4. The probability P is characteristically nonperturbative in the adiabatic limit

␶→⬁with U_Hfixed, or equivalently ␭→⬁. Hence, the di- mensionless quantity associated with the exponential sup- pression is ␭ and has been called the “adiabaticity param- eter.” For␩⬅T /␶Ⰷ1, the approximation 共28兲allows us to estimate the argument of the prefactor of Eq.共35兲 to expo- nential accuracy,

1 បRe

冕

t₁

t₂

dz␧共z兲=

冑

1 +␦²␭␩^{− 2}␭兵ln共

冑

1 +␦^{2+ 1}兲

−

冑

1 +␦^2ln关2共1 +␦²兲兴+␦^ln共

冑

1 +␦²⁺␦兲 +共

冑

1 +␦^{2− 1}兲ln共␦兲其+ O共e^−␩/2兲. 共36兲 The oscillation with respect to the duration of the pulse T is reminiscent of a similar factor in the Rosen-Zener model.

The phenomenon of pulsed perturbations that return the full amplitude/occupation to the initial state has been studied in the context of atom-laser interactions.^29–32 In Figs. 4–6, we compare our semiclassical estimate 共35兲 with results from numerical simulations of the exact quantum mechanical time FIG. 3. The analytic structure of the function␧共t兲shown only in

a segment of the upper half plane. The contourCais associated with transitions that occur due to the ramping on of the pulse, while contourCb is associated with the ramping off. By Cauchy’s theo- rem, an integral on the contourCais equal to the integral on the contourC˜

a. Bold lines represent branch cuts, dots represent branch points, and poles are denoted with a⫻.

FIG. 4. The probability for nonadiabatic transitions for␭= 2 and

␩= 50 as a function of ␦. We compare our semiclassical estimate according to expression 共35兲 with results from numerical simula- tions of the exact quantum mechanical time evolution as done in Ref. 35. The results are in excellent agreement.

evolution, following Ref. 10. Both results are in excellent agreement and differ only at very small ␦, i.e., for weak pulses. Of course, the nonadiabatic transition probability vanishes in this limit, whereas the semiclassical approxima- tion breaks down共see Sec. III C兲. This regime is beyond the exponential accuracy of Dykhne’s formula. The integrability of our model allows us to make precise statements about the form and magnitude of the corrections to Dykhne’s formula 共see Sec. IV兲. For example, in the limit␭Ⰷ1 and␦␭Ⰶ1 we have from the expansion共63兲that P_⬍⬃共2␲␦␭兲²e^−4␲␭, while in the same limit the result of Dykhne’s formula共35兲gives only the exponential factor e^−4␲␭ without information about the prefactor. This explains a trend among Figs. 4–6, namely, the increasing range, in terms of␦, of validity of Dykhne’s formula with increasing ␭. The value for the adiabaticity parameter␭= 2, represented in Fig. 4, corresponds to a ramp time␶^{= 4}ប/ U_H, which was identified in Ref. 10 as a practical lower bound to ensure sufficient adiabatic behavior in a gate operation between two quantum dot spin qubits. It is inter- esting that Dykhne’s formula remains accurate for smaller values of␭in particular␭= 1 as seen in Fig. 6. The reason is that the results共35兲and共61兲have an incidental factor of 2␲ in the exponent, giving in practical terms the requirement for exponential suppression 2␲␭Ⰷ1.

The expressions共34兲and共35兲, along with Figs. 4–6, com- prise our main results. For the remainder of this section, we will address the justification and limitations of these results.

B. Origin of Dykhne’s formula

Dykhne derived a concise expression for nonadiabatic transitions from a local analysis of the Schrödinger equation in the vicinity of the turning point.²³ Dykhne’s formula can be viewed as a semiclassical approximation, and an elegant interpretation and proof was given by Hwang and Pechukas³³ 共see also Ref. 34兲. We will briefly discuss the key elements and scope of the proof. Their method was to study the solu- tion of the Schrödinger equation in the complex plane of the independent variable, time. According to the adiabatic theo- rem, the projection of the solution onto any eigenstate other than the initial eigenstate approaches zero in the adiabatic limit. One might suppose that weak statement is all the adia- batic theorem can tell us about transitions; however, it does

not exhaust its capacities. The reason lies in the following: a basis of eigenstates兩␰1,2共t兲典, when extended into the complex time plane, is multivalued. In particular, as a basis state is analytically continued across a branch cut of the function

␧共t兲, its long-time asymptotics are discontinuously changed.

In accord with our above two-level problem, we uniquely specify the basis by its asymptotics,

兩␰1,2共t兲典→兩S_1,2典 as t→ ±⬁. 共37兲 The multivalued nature is not manifest on the real time axis, because owing to the nondegeneracy of the spectrum⫿␧共t兲, the branch points are nonreal. We can choose a single-valued basis兩^˜␰1,2共t兲典, which makes reference to兩␰1,2共t兲典but has fixed asymptotics, by defining rules for continuing the basis states across branch cuts. Equivalently, this new basis is said to be defined over a Riemann surface with sheets 共copies of the complex time plane兲 corresponding to each of the two branches of the function关␧共t兲²兴^1/2.共The Riemann surfaceR1

for the eigenstate basis 兩^˜␰1,2共t兲典 has four sheets, while the Riemann surface R2 for the function 关␧共t兲²兴^1/2 has two sheets. Of the four sheets of R1, two correspond to one branch of关␧共t兲²兴^1/2and the other two correspond to the other branch of 关␧共t兲²兴^1/2. Therefore, the phases ␣1,2 and ␤1,2 of Eq.共38兲, though constant on each sheet, can assume different values on different sheets of R1.兲 Crossing a branch cut means passing to the other sheet of the Riemann surface. We assign the following relations among the eigenstates:

兩^˜␰1,2共t兲典= e^i␣1,2兩␰1,2共t兲典, t苸sheet 1 of关␧共t兲²兴^1/2, 兩^˜␰1,2共t兲典= e^i␤1,2兩␰2,1共t兲典, t苸sheet 2 of关␧共t兲²兴^1/2, 共38兲 where␣1,2and␤1,2are phase definitions that are chosen to maintain continuity of the basis兩^˜␰1,2典across the branch cut.

Given兩具^˜␰1共−⬁兲兩␺共−⬁兲典兩= 1, the conclusion of the adiabatic theorem may be restated on a Riemann surface as

兩具^˜␰1共t兲兩␺共t兲典兩→1 ∀t as ␶→⬁, 共39兲 where ␶ is the characteristic time scale for variation of Heff共t兲. The only exception to Eq. 共39兲 is for times within FIG. 6. The probability for nonadiabatic transitions for␭= 1 and

␩= 50 as a function of␦^. FIG. 5. The probability for nonadiabatic transitions for␭= 4 and

␩= 50 as a function of␦^.

O共␶␭^−2/3兲of a turning point, for there the semiclassical cri- terion 共45兲 is invalid. As remarked above, the zeroth-order approximation共26兲of the solution as␶→⬁is the dynamical phase. The zeroth-order approximation may be extended into the complex plane by evaluating the dynamical phase on a contourC. Continuing with the above example, a state that is purely singly occupied at t = −⬁is for complex time given by 兩␺共t兲典 ⬇exp

再

^−បi

^冕

^{Cdz关−␧共z兲兴}

冎

^{兩˜␰1共t兲典, 共40兲}

whereC is a contour from z = −⬁ to z = t. The amplitude of transition is readily obtained as the projection of the solution onto the doubly occupied state兩S₂典, as t→⬁ on the second Riemann sheet共see Fig. 7兲of关␧共t兲²兴^1/2, i.e.,

具S₂兩具␺共⬁兲典= e^−i␣1具^˜␰1共⬁兲兩␺共⬁兲典

⬇e^−i␣1exp

再

^−បi

^冕

^{Cdz关−␧共z兲兴}

冎

^{, 共41兲}

where the contourCcrosses the branch cut emanating from the branch point that is closest to the real axis. Dykhne’s formula is simply the square modulus of this amplitude.

In the adiabatic regime, in contrast to the perturbative regime, the leading contribution to transitions comes from the zeroth-order term of perturbation theory instead of the first-order term. By retaining only the zeroth-order term, it appears that we have neglected completely the coupling t_H between states. However, the coupling enters implicitly in the multivalued function关␧共t兲²兴^1/2and influences the location of the turning points—the complex roots of␧共t兲. Transition amplitudes are obtained by carefully considering the differ- ent branches of this function. In the following section, we consider the validity of keeping only the zeroth-order term.

C. Validity and accuracy of Dykhne’s formula The theory of semiclassical approximations, especially WKB analysis, provides a foundation from which to evaluate the validity of Dykhne’s formula. The calculation of nona- diabatic transitions is closely related to the semiclassical approximation³³because the semiclassical limitប→0 can be mathematically equivalent to the adiabatic limit ␶→⬁. An essential element of the proof by Hwang and Pechukas is the

existence of a complex time contour that 共1兲 connects the two sheets of the Riemann surface and 共2兲 on which the zeroth-order approximation of adiabatic perturbation theory is the correct leading behavior of the solution in the adiabatic limit. These are sufficient conditions for Dykhne’s formula to give the correct asymptotic form of the transition probability in the adiabatic limit␭→⬁. Having established the existence of such a contour, one can calculate a more precise value for the prefactor of Dykhne’s formula by applying time- dependent perturbation theory along the contour. We expect Dykhne’s formula to break down when the contour ceases to exist. At the limit of its range of validity, the higher-order terms become comparable to the zeroth- order term. Intro- ducing the unitary transformation U that diagonalizes the Hamiltonian, i.e., U^†HU =␧␴3, we can write the Schrödinger equation in the basis of instantaneous eigenstates,

iបd

dt兩␰共t兲典=关␧共t兲␴3+បaˆ共t兲兴兩␰共t兲典 共42兲 with the off-diagonal perturbation aˆ共t兲= −U^†i⳵tU. A domi- nancy balance among the terms gives the condition for the accuracy of the zeroth-order approximation,

兩␧共t兲兩

ប Ⰷ兩aˆ共t兲兩 共43兲 or in scaled time x = t /␶^,

兩␧共x兲兩␶

ប Ⰷ兩aˆ共x兲兩. 共44兲 For our model of the dynamics,␭⬀␶is the largest scale and 兩aˆ兩⬃1. The condition Eq. 共44兲 must be maintained at all points on the contour. Applying Eq. 共44兲 on the real axis, where兩␧共x兲兩⬃U_H, gives the adiabaticity condition␭Ⰷ1. Ad- ditionally, in order to connect two Riemann sheets, the con- tour must pass between two turning points 共see Fig. 3兲, where兩␧共x兲兩⬃␦^UH, giving the condition␦␭Ⰷ1.

Beginning instead from an intuitive approach, we can evaluate the adiabaticity of the dynamics along a given con- tour. To test whether a given contour is adequate, we can exploit the analogy between quasiadiabatic dynamics and semiclassical scattering. Recall the semiclassical criterion

⌬共⌳兲

⌳ ⬃d⌳

dt Ⰶ1. 共45兲

The analog of the de Broglie wavelength␭共x兲= 2␲ប/ p共x兲in scattering problems is the period ⌳共t兲⬅2␲ប/␧共t兲. In other words, the condition共45兲says that the change of period over the course of one period is small. We now require that the semiclassical criterion be obeyed everywhere along an ad- missible contour, i.e., one that connects the two Riemann sheets. To find an admissible contour, we must appeal to the analytic structure of the eigenenergy ␧共t兲; see Fig. 3. For clarity we will focus on the time interval t⬍0 and operate under the approximation 共28兲. The singularities of ␧共t兲 are branch points at t = −共T / 2兲−␶^ln共−1 ± i␦兲 and poles at t =

−共T / 2兲−␶^ln共−1兲. If we agree to define a branch cut connect- FIG. 7. An example of a Riemann surface with two sheets, a

branch point at t = t₀and a contourCcorresponding to a transition.

ing the nearest and next nearest branch points to the real time axis,

t₁= −T

2−␶^ln

冑

1 +␦^{2+ i}␶关␲^{− arctan}共␦兲兴, t₃= −T

2−␶^ln

冑

1 +␦^{2+ i}␶关␲^{+ arctan}共␦兲兴,

respectively, then an admissible contour is one that crosses this branch cut exactly once. For the semiclassical criterion to be obeyed, the admissible contour cannot pass too close to a branch point. In essence, if ␦ is too small, the contour is pinched between the branch points t₁ and t₃. Evaluating the maximum of d⌳/ dt on a contour C that crosses the branch cut between t = t₁ and t = t₃ we arrive at the condition ␦ Ⰷ␭⁻¹. Together with the adiabatic limit␭Ⰷ1, we have the following conditions on the interdependence of the physical parameters:

␭⬃U_H␶

ប Ⰷ1, 共46兲

␦␭⬃␦^UH␶

ប Ⰷ1. 共47兲

Each of these dimensionless quantities is a product of a char- acteristic energy and time scale. If these conditions are not satisfied, there does not exist a contour on which the motion is adiabatic. The integrability共Sec. IV兲of our model allows us to investigate the intermediate regime␭Ⰷ1 and ␦␭Ⰶ1, where Dykhne’s formula cannot be justified with the analysis of Hwang and Pechukas.

IV. IDENTIFICATION WITH AN INTEGRABLE MODEL The result obtained by Dykhne’s formula in Sec. III A is now shown to be equivalent to the exact result for an inte- grable model in the appropriate limit.

Under the approximations 共28兲 for the time intervals t

⬍0 and t⬎0, the Hamiltonian Heff= −U_H

2

冉

^t1H − 1^tH

冊

^共48兲

is approximated by

H_eff⬇

再

^{HH⬍⬎,, tt⬎⬍0,0}

冎

^共49兲

with

H_⬍= −U_H

2

冢

^{1 + e−共t/1␦␶兲−共T/2␶兲 1 + e−− 1共␦t/␶兲−共T/2␶兲}

冣

^{, 共50兲}

H_⬎= −U_H

2

冢

^{1 + e共t/1␦␶兲−共T/2␶兲 1 + e共− 1t/␦␶兲−共T/2␶兲}

冣

^{. 共51兲}

The HamiltoniansH_⬍andH_⬎can be obtained as a special case of

Hexact=

冢

^{a + c tanhb x2 a + c tanh− b 2x}

冣

^共52兲

by identifying ±c = a = −␦␭/ 2, b = −␭ and rescaling time x

= t /␶^{± T / 2}␶, respectively. The Schrödinger equation

i⳵x兩␺共x兲典=Hexact兩␺共x兲典 共53兲 is exactly solvable³⁵共see the Appendix兲.

In analogy with one-dimensional scattering, the transition amplitude from a singly occupied state兩S₁典 to a doubly oc- cupied state兩S₂典may be viewed as an off-diagonal element of the scattering matrixSthat connects the coefficients of the asymptotic final states to the asymptotic initial states. The asymptotic states are the limit as t→±⬁of the instantaneous eigenstates 兩␰1,2共t兲典 of Heff corresponding to eigenvalues

⫿␧共t兲, respectively,

兩␰1共t兲典= 1

冑

2␧

冉

⁻

^{冑 冑}

^␧␧⁻+^␭␭

冊

^,

兩␰2共t兲典= 1

冑

2␧

冉 ^{冑 冑}

^{␧␧+−␭␭}

冊

^{. 共54兲}

The leading behavior of the long-time asymptotics of a gen- eral solution has the form

兩␺共t兲典=

冦

^ab11expexp

^{再 再}

^{−− បបii}

^{冕 冕}

^ttdtdt

^{⬘ ⬘}

^{关关−−␧共␧共tt}

^{⬘ ⬘}

^兲兴兲兴

^{冎 冎}

^{兩兩␰␰11共共tt兲典兲典+ a+ b22expexp}

^{冋 冋}

^{−−បបii}

^{冕 冕}

^ttdtdt

^{⬘ ⬘}

^{␧共␧共tt}

^{⬘ ⬘}

^兲兲

^{册 册}

^{兩兩␰␰22共共tt兲典兲典 as tas t→→−⬁⬁,}

冧

^共55兲

and the scattering matrix relates the coefficients,

冉

^bb12

冊

^=S

冉

^aa12

冊

^{. 共56兲}

Referring to the statement of our problem in Sec. III, the amplitude of final double occupancy is the elementS21of the scattering matrix, which we parametrize as

S=

冉

^−SS1112 ^S12

* S11

*

冊

^{. 共57兲}

The scattering matrices S_⬍ and S_⬎ associated with the HamiltoniansH_⬍ andH_⬎ may be obtained by substitution from the exact scattering matrixW共derived in the Appendix兲 associated with the HamiltonianHexact. By the symmetry of the pulse, we haveS_⬎=S_⬍^†. Patching together the two do- mains of time evolution共49兲, we find the scattering matrix

S=S_⬍^† exp

冉

^បi␴3Re

^冕

^t1 t₂

dt␧共t兲

冊

^{S⬍, 共58兲}

where the integral of the exponent has been estimated in Eq.

共36兲and the elements ofS_⬍are obtained fromW, 共S_⬍兲11=

冑

_␮²+^␮␭

⌫共i2␭兲⌫共i2␮兲

⌫共i␮^{+ i}␭+ i␦␭兲⌫共i␮^{+ i}␭− i␦␭兲, 共59兲

共S_⬍兲12=

冑

_␮²−^␮␭

⌫共i2␭兲⌫共− i2␮兲

⌫共− i␮^{+ i}␭+ i␦␭兲⌫共− i␮^{+ i}␭− i␦␭兲, 共60兲 where␮⁼␭

冑

1 +␦². Dykhne’s formula共34兲for P_⬍is recov- ered as exactly the leading term of 兩共S_⬍兲21兩² in the limit

␭,␦␭Ⰷ1, P_⬍=兩共S_⬍兲21兩²

=sinh关␲␭共

冑

1 +␦^{2− 1 +}␦兲兴sinh关−␲␭共

冑

1 +␦^{2− 1 −}␦兲兴 sinh共2␲␭兲sinh共2␲␭

冑

1 +␦²兲

共61兲

⬃e^{−2␲␭共1+}冑1+␦²⁻␦兲共1 − e^{−2␲␭共␦+1−}冑1+␦²兲+ ¯兲. 共62兲 The nonperturbative corrections are typically very small. For

␭= 2 and␦= 1 / 2, the relative contribution of the second term of Eq.共62兲 is less than 1%. This accounts for the excellent agreement in Figs. 4–6, between the probability of double occupancy as given by the semiclassical result共35兲based on Dykhne’s formula and the result of a numerical integration of the Schrödinger equation. We can interpret the subleading term in the parentheses of Eq.共62兲as the contribution from the contourC of Fig. 8, which crosses the branch cut three times. The sign of the correction is negative and arises from the factor e^−i␣1 associated with matching the basis 共38兲 across the branch cut. Similarly, it may be possible to obtain further subdominant corrections to Dykhne’s formula by summing over all inequivalent complex paths that give dis- tinct positive values for Im兰_Cdt␧共t兲.³⁶ For many physical

problems this type of nonperturbative correction is domi- nated by perturbative corrections along the contour—those mentioned in Sec. III C. The striking absence of perturbative corrections in the limit␭,␦␭Ⰷ1, is a unique artifact of the integrability of our model.

With a knowledge of the exact result, we can also inves- tigate the intermediate regime ␭Ⰷ1 and ␦␭Ⰶ1, where the analysis of Hwang and Pechukas共Sec. III B兲cannot be used to prove Dykhne’s formula. In this limit, the transition prob- ability共61兲becomes

P_⬍⬃ 共2␲␦␭兲²e^−4␲␭. 共63兲 Although Dykhne’s formula does not apply in this limit be- cause␦^{−1 and not}␭is the largest scale, it nevertheless gives the correct controlling factor e^−4␲␭of Eq.共63兲, except for␦ that are exponentially small with respect to␭. This exponen- tial factor is resilient and remains the controlling factor for a range of parameters beyond the naive expectation based on the arguments of Sec. III C.

V. CONCLUSIONS

The dynamics of two coupled quantum dot spin qubits can be mapped to an effective two-level system, where nona- diabatic transitions correspond to double occupancy. We have estimated the probability of final double occupancy with Dykhne’s formula. In the adiabatic regime, the perva- sive feature of transitions is their exponential suppression by a dimensionless adiabaticity parameter ␭. Our main result 共35兲was expressed in terms of the dimensionless quantities

␭, ␦^{, and} ␩. An integral constraint 共21兲 on the swapping operation gives one relation among the three dimensionless parameters. The problem is uniquely defined by specifying any two, and in a solid-state setting, conservative estimates are␭⬇2 and␦⬇1 / 2. The probability of double occupancy P⬇10⁻¹⁰is sufficiently rare that the operation of a quantum gate will not be obstructed by this type of error. It is note- worthy that the probability of double occupancy 共35兲 has nodes for

k␲^{= Re}

冕

t₁ t₂

dt␧共t兲 ⬇

冑

1 +␦^2␭␩^{, k苸Z. 共64兲} However, this property is not immediately relevant to the suppression of transitions, because the oscillatory factor sin²共

冑

1 +␦²␭␩兲of Eq. 共35兲vanishes algebraically and for it FIG. 8. The contour C gives a subdominant correction to Dykhne’s formula.