Hierarchical Theory of Quantum Adiabatic Evolution

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Hierarchical Theory of Quantum Adiabatic Evolution

Qi Zhang,1, 2 Jiangbin Gong,3, 4 and Biao Wu5,6,

1College of Science, Zhejiang University of Technology, Hangzhou 310014, China2International Joint Research Laboratory for Quantum Functional Materials of Henan

Province and School of Physics and Engineering Zhengzhou University, Henan 450001,China3Department of Physics and Centre for Computational Science and Engineering,

National University of Singapore, 117542, Singapore4

NUS Graduate School for Integrative Sciences and Engineering, Singapore 117597, Singapore5International Center for Quantum Materials, Peking University, Beijing 100871, China6Collaborative Innovation Center of Quantum Matter, Beijing, China

(Dated: February 24, 2014)

Quantum adiabatic evolution is a dynamical evolution of a quantum system under slow externaldriving. According to the quantum adiabatic theorem, no transitions occur between non-degenerateinstantaneous eigen-energy levels in such a dynamical evolution. However, this is true only whenthe driving rate is infinitesimally small. For a small nonzero driving rate, there are generallysmall transition probabilities between the energy levels. We develop a theory to address the smalldeviations from the quantum adiabatic theorem order by order. A hierarchy of Hamiltonians areconstructed iteratively with the zeroth-order Hamiltonian being the original system Hamiltonian.The kth-order deviations are governed by a kth-order Hamiltonian, which depends on the timederivatives of the adiabatic parameters up to thekth-order. Two simple examples, the Landau-Zenermodel and a spin-1/2 particle in a rotating magnetic field, are used to illustrate our hierarchical

theory.

PACS numbers: 03.65.Vf, 05.40.-a, 05.45.-a, 37.10.Gh, 45.20.Jj

I. INTRODUCTION

Quantum evolution under external adiabatic drivinghas been of fundamental interests to physicists. Bornand Fock proved the quantum adiabatic theorem shortlyafter the discovery of the Schrodinger equation[1]. Thistheorem states that no transition occurs between instan-taneous eigen-energy levels in a system under adiabatic

driving. However, this is only true when the externaldriving is infinitesimally slow. With a slow but finiteexternal driving, there is generally small tunneling be-tween energy levels. There has been a great deal of effortto address this small deviation from the quantum adi-abatic theorem[27]. However, a completely successfultheory is yet to be developed. This failure has led toa controversy on the validity of the quantum adiabatictheorem [811]. With the success of the quantum adi-abatic algorithm in quantum computing, this issue hasbecome also important in a practical sense [12,13]. Ourtheory should lead to a better assessment and control ofthe errors in the quantum adiabatic computing.

In this work we present a theory to address the devia-tion from the quantum adiabatic theorem. We constructiteratively a hierarchy of Hamiltonians with the zeroth-order Hamiltonian being the original Hamiltonian. Thedeviations of thekth order are the adiabatic invariancesof thekth-order Hamiltonian while the adiabaticity of thekth-order Hamiltonian is determined by the time deriva-

Electronic address: [email protected]

tives of the external parameters (denoted R) up to thekth-order. Within this theoretical framework, the devia-tions from the quantum adiabatic theorem can be com-puted to arbitrary order iteratively. The theory breakdown at thekth-order when thekth-order time derivativeof the external parameters becomes relatively large. Weuse two simple examples, the Landau-Zener model andthe spin-1/2 under a rotating magnetic field, to illustrateour hierarchical theory.

Our hierarchical theory establishes an intuitive picturefor quantum adiabatic evolution. At the zeroth-orderthe adiabatic evolution is a smooth curve of instanta-neous eigenstates in the projective Hilbert space wherethe overall phase is removed. We call the smooth curveadiabatic trajectory(see Fig.1). At the first order, thisadiabatic trajectory is shifted by a small amount thatis proportional to the first time derivative of externaparameters (R= dR/dt). At the second order, the adiabatic trajectory is shifted again by a small amount that isproportional to R2 or other possible second-order smalparameters, such as R. This intuitive picture is illustrated in Fig.1. There may or may not be small oscilla-

tions around the shifted adiabatic trajectory dependingon the detail how R changes with time.Technically we take advantage of two facts to de

velop our theory. First, we use the superposition principle, which allows us to focus on the adiabatic evolu-tion of each individual energy eigenstate. Second, we usethe classical Hamiltonian formulation of the Schrodingerequation[14,15]. In this formalism, an energy eigenstateis mapped into an elliptic fixed point in the correspondingprojective Hilbert space. Note that this classical formulation is purely mathematical and is not the traditiona

arXiv:1402.6

431v1

[quant-ph]2

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FIG. 1: (Color online) The adiabatic trajectories of differentorders in the projective Hilbert space. The black line is forthe zeroth-order, the red line for the first-order, and the darkgreen line for the second-order. The difference between thezeroth-order and the first-order trajectories is proportional to

Rwhile the difference between the first-order and the second-order trajectories is proportional to R2. The possible smalloscillations around the adiabatic trajectories of the first andsecond orders are omitted for clarity.

semiclassical limit 0.

II. CLASSICAL HAMILTONIANFORMULATION OF THE SCHRODINGER

EQUATION

We consider a quantum system described by theHamiltonian H0(R), where R = R(t) represents time-dependent parameters in an adiabatic protocol. Asnormally assumed for quantum adiabatic evolutions [1],

H0(R) has a discrete non-degenerate spectrum duringthe entire control protocol. Further, the rate of changeinR is small as compared with the transition frequenciesof the system. Deviations from the quantum adiabatictheorem are expected so long as the protocol is not exe-cuted in the mathematical limit R 0. The aim of thiswork is to develop a general and systematic frameworkto quantitatively describe such deviations.

Though our consideration can be extended to caseswith a Hilbert space of infinite dimensions, for conve-nience we assume H0(R) lives in a finite n-dimensional

Hilbert space. H0(R) can thus be expressed as a R-dependentnnHermitian matrix. We find it mathemat-ically more convenient to use the classical Hamiltonianformulation for the Schrodinger equation [14, 15]. Weexpress the quantum state with an n-component wave-function| = (c1, c2, . . . , cn)T and Definen 1 pairs ofcanonical variables

pi = arg(ci+1) arg(c1), qi= |ci+1|2, (1)

withi = 1, 2, . . . , n1. By construction, the Schrodingerequation then yields the following Hamiltons equationsof motion,

dpidt

= H0(R)qi

, dqi

dt =

H0(R)

pi, (2

where the classical Hamiltonian H0(R) is obtained from

the quantum Hamiltonian H0(R) as

H0(R) = |H0(R)| . (3

As the overall phase is removed, the phase space in thisclassical formalism is just the projective Hilbert spaceThis alternative formalism of the Schrodinger equationwill allow us to exploit powerful and familiar tools inclassical mechanics in our analysis.

the overall phase of the wavefunction, or equivalentlyarg(c1), is removed in Eq. (2)[14,15]

It is particularly interesting to look at eigenstatesIn the original Schrodinger equation picture, an energy

eigenstate of H0(R) at a fixed R simply develops a trivial overall phase. Since the overall phase is discarded in

our formalism, such an eigenstate evolution is mappedto a fixed point in the classical phase space of H0(R)The issue of the adiabatic following with the instanta-neous energy eigenstates ofH0[R(t)] now becomes theissue of the adiabatic following with the instantaneousfixed points ofH0[R(t)].

In principle, the time evolution emanating from an ar-bitrary initial state as a superposition of different energyeigenstates can be considered. However, the linearity othe original Schrodinger equation indicates that it suf-fices to study initial states that are energy eigenstates oH0[R(0)] at t = 0. As such, in our classical formalismwe only need to consider those initial conditions that arefixed points in the phase space.

One final technical comment is in order. The mappingfrom the wavefunction componentscito phase space vari-ables (pi, qi) [see Eq. (1)] becomes ambiguous when anyone of the wavefunction componentcibecomes zero. For-tunately, this ambiguity can be easily overcome by adopt-ing a different representation to re-express the wavefunc-tion. For example, c1 in Eq. (1) is used to remove theoverall wavefunction phase. If c1 = 0, one can alwaysselect another nonzeroci to carry out a similar mapping

III. FIRST ORDER DEVIATIONS

As the generalization to arbitrary dimensions isstraightforward, we consider a quantum system with atwo-dimensional Hilbert space for the rest of the paperWithn= 2 the Hamiltons equations of motion in Eq. (2)only involve one pair of canonical variables q1 and p1The phase space is hence also two-dimensional. For clar-ity we drop the subscript 1 hereafter. A R-dependenfixed point in the phase space is denoted as [p(R),q(R)]


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There are two fixed points corresponding to two energyeigenstates ofH0(R).

According to the quantum adiabatic theorem, under asufficiently slow protocol R = R(t), the dynamics ema-nating from an energy eigenstate will follow the instanta-neous energy eigenstates. With the removal of the over-all phase, this dynamics is completely described by thesmooth curve of instantaneous energy eigenstates in the

projective Hilbert space. We shall call it adiabatic tra-jectory (see Fig. 1). In the classical formalism, this adi-abatic trajectory is the smooth curve of instantaneousfixed points in the phase space. However, in a realisticprotocol where R(t) changes slowly with a nonzero rate,there should be a deviation from this picture of perfectadiabatic following.

There were studies on the small deviations from whatthe adiabatic theorem predicts. It was done in specialclassical systems and the small deviations were found topollute the Hannays angle [1618]. Recently, the first-order deviation was studied in nonlinear quantum adia-batic evolutions [19], where the result was used success-

fully to predict a new kind of geometric phase beyond thetraditional Berry phase. As their focus was on the globaleffects of the deviations, detailed dynamics of the devia-tion was not considered. Our work conducts a systematicstudy of the quantum adiabatic evolution and reveals itshierarchical structure. Our results can be easily general-ized to classical systems and nonlinear quantum systems.

With possible deviations from the instantaneous fixedpoints [q(R),p(R)], the actual adiabatic trajectory in thephase space can be written as

p(t) = p[R(t)] +p, q (t) = q[R(t)] +q, (4)

with (p,q) being time-dependent deviations from the

ideal adiabatic trajectory [p(R),q(R)]. This section ismainly to develop a theory to understand the behaviorof (p,q) to the first order ofR.

As a preparation we first consider the case when R isfixed. Using Hamiltons equations of motion and Tay-

lor expanding H0(R)

p and H0(R)

q to the first order of

(p,q), we have dpdt

dqdt

= 0

p

q

, (5)

where

0 =

2H0qp

2H0qq

2H0pp

2H0pq

p=p,q=q

(6)

is anR-dependent matrix obtained from the second-orderderivatives ofH0(R). The terms with first-order deriva-tives ofH0(R) do not appear on the right-hand side ofEq. (5) simply because [q(R),p(R)] is a fixed point. Allhigher-order terms are neglected here.

We now consider the dynamics of (q,p) in the controprotocol where R = R(t) changes slowly with time. Inthis case, we have

dp

dt =

p(R)

RR+

d p

dtdq

dt =

q(R)

RR+

dq

dt . (7

Equation (5) consequently becomes dpdt

dqdt

= 0(R)

p

q

10 (R)

pR

qR

R

.

(8Two remarks are necessary for this equation of (p,q)

First, because it is already assumed that throughout theprotocol R = R(t) the studied energy eigenstates neverbecome degenerate, the corresponding fixed points in thephase space do not vanish or collide. It is therefore legitimate to always associate the deviations with one fixedpoint so long as (p,q) is small. Second, it can be

shown that the determinant|0| does not vanish withnon-degenerate energy eigenstates. 10 in Eq. (8) henceexists for all R.

Remarkably, Eq. (8) possesses a canonical structureThe variables (p,q) are a canonical pair and Eq. (8)can be derived from the following Hamiltonian

H1(R, R) = 1

2

2H0q2

p,q

(q B1)2

+

2H0qp

p,q

(q B1)(p A1)

+ 1

2

2H0

p2

p,q

(p

A1)

2, (9

whereA1 = A1(R, R) and B1 = B1(R, R) are defined as A1

B1

= 10 (R)

pR

qR

R . (10)

This expression was previously obtained by Fu and Liu[19]. It is clear that the first-order Hamiltonian (9) describes harmonic oscillations around the central point(A1, B1).

The first-order HamiltonianH1 generating the dynam-ics of (p,q) depends upon two parameters R(t) and

R(t). We assume that R(t) also changes slowly with timeIn this case, the dynamics of (p,q) becomes the adia-batic evolution of H1 and can be understood with thehelp of the classical adiabatic theorem. We define theaction for (p,q) as

I1 = 1

2

p d(q) . (11)

This action is the adiabatic invariant possessed byH1 [20]. (A1, B1) is the fixed point ofH1 with I1 = 0


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The dynamics of (p,q) can be viewed as a spiral motionalong the adiabatic trajectory specified by fixed point(A1, B1). The amplitude of the spiral oscillations is con-trolled by the action I1. With this analysis, it becomesclear that when both R(t) and R(t) change slowly withtime (A1, B1) describes an adiabatic trajectory shiftedfrom the ideal trajectory of fixed point [p(R),q(R)] asshown in Fig.1.

We now consider two typical cases. In the first case, Ris increased slowly from zero. In this case, asA1 andB1are zero initially, the action I1 is zero and the adiabaticevolution to the first order will follow exactly the adia-batic trajectory specified by (A1, B1). This is illustratedin Fig.2(1). In the second case, the external driving rateR is finite and small at the beginning. This means that(A1, B1) is not zero initially and the action I1 has a finiteand small value. In this second case, the adiabatic evo-lution will become a spiral motion around the trajectoryof (A1, B1) as shown in Fig. 2(2). This analysis of thesecond case in fact implies that infrequent sudden butsmall jump ofR will not break down the adiabaticity of

the evolution. Note that the smallness of the jump inR(t) is implicitly guaranteed by the slow change ofR(t).We mention it explicitly in our discussion just for clarity.

FIG. 2: (Color online) Adiabatic evolutions at first and secondorders. The black line is the zeroth-order adiabatic trajectory,

the red line the first-order adiabatic trajectory, and the darkgreen line the second-order adiabatic trajectory. (1) The evo-lution follows the first-order adiabatic trajectory when theadiabatic manipulation is gradually launched (continuous in-

creasing ofR from zero); (2) it becomes a spiral oscillatory

motion when the process is started with a finite R. (3) The

state follows the second-order adiabatic trajectory when Rischanged slowly from zero; (4) it becomes a spiral-like motion

when R is started with a finite value.

Our first-order adiabatic theory shows that a small

quantum transition to other energy eigenstates alwaysoccurs with probability proportional to R. The probability is zero only in special cases where the coefficientsin Eq. (10) vanish.

Our first-order theory offers a deep insight into thegeneric subtlety of how the adiabatic following breaksdown. Let us consider a situation where R(t) is small bu

changes with a great rate, i.e., R(t) is large. In this case

the dynamics governed by H1 is not adiabatic; I1 is notan adiabatic invariant and can not stay small for a longtime. When the evolution is long enough, the dynamical evolution of the first-order deviations (p,q) will nolonger be bounded: the small deviations (p,q) can accumulate and eventually be amplified to the zeroth-orderlevel. This breakdown due to the largeness ofR(t) clearlydepends on the detail of the protocol R(t) and the Hamil-tonian; general conclusions will be difficult to reach.

We note that our theory can be naturally extended toa Hilbert space of larger dimension n > 2, where thematrix 0 becomes 2(n 1) 2(n 1) dimension andthe first-order Hamiltonian has (n 1) pairs of canonicavariables.

IV. SECOND ORDER DEVIATIONS

In the previous section we have found that the first-order correction (p,q) evolves according to a first-orderHamiltonianH1. It is natural to wonder whether we canfind a similar Hamiltonian for the second-order devia-tions. We find that if the system follows the first-orderadiabatic trajectory (see Fig. 2(1)), we can indeed findsuch a Hamiltonian. We write

p= p+A1+2p , q= q+B1+

2q . (12)

After substituting it into H0 and with straightforwardcalculation, we obtain the second-order Hamiltonian

H2(R, R, R) =1

2

2H0qq

p+A1,q+B1

(2q B2)2

+

2H0qp

p+A1,q+B1

(2q B2)(2p A2)

+1

2

2H0pp

p+A1,q+B1

(2p A2)2 , (13)

where

A2B2

= 10 A1

R

B1R

R+ A1

R

B1R

R

12

10

A1

B1

. (14)

Here is defined as

=

p

p,q

A1+

q

p,q

B1 (15)


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with

2H0qp

2H0qq

2H0pp

2H0pq

. (16)

The detailed derivation of this second-order Hamiltonian(13) can be found in Appendix A along with some sub-

tlety involved in the derivation.The second-order HamiltonianH2 has a similar struc-ture as H1 and describes a generalized harmonic oscil-lator. The significant difference is that H2 depends onthree parameters (R, R, R) whileH1depends on only two

parameters (R, R). In the following, we conduct a similar

analysis forH2 as forH1. We focus on the case where R,along with R, R, changes slowly with time. In this case,the dynamics of the second-order deviation (2p, 2q) asgoverned by H2 is adiabatic. We define the action for(2p, 2q) as

I2 = 1

2

2p d(2q) , (17)

which is the adiabatic invariant possessed by H2 [20].(A2, B2) is the fixed point ofH2withI2 = 0. The dynam-ics of (2p, 2q) can be viewed as a spiral motion alongthe adiabatic trajectory specified by fixed point (A2, B2).The amplitude of the spiral oscillations is controlled bythe action I2. It is clear from this analysis that (A2, B2)describes an adiabatic trajectory shifted from the first-order one that is specified by [p + A1,q+ B1] (see Fig.1).

We again consider two typical cases. (i) When both R

and R are started continuous from zero, I2 is zero andthe dynamics of (2p, 2q) follows exactly (A2, B2). Thismeans that the state follows exactly the adiabatic trajec-

tory deviating from original instantaneous eigenstate by(A1+ A2, B1+ B2) (see Fig.2(3)). (ii) When the system

starts with a finite R, I2 is nonzero and the system un-dergoes a spiral motion around (A2, B2) (see Fig. 2(4)).The amplitude of the spiral motion is controlled by I2.

We can continue this procedure and construct a kth-order Hamiltonian for the kth-order deviation. The re-sult and the detailed derivation can be found in AppendixB. A general feature is that the kth-order Hamiltonianwill depend on k+ 1 parameters, R, R, R, , dkR/dtk,and the adiabaticity of its dynamics is controlled by theseparameters. We note that a kth-order Hamiltonian canbe constructed only when the dynamics of the deviations

of order (k 1) follows the (k 1)th-order adiabatic tra-jectory (the scenarios illustrated in Fig.2(1,3)).In summary, we have developed a hierarchical theory

for quantum adiabatic evolution. In this theory, a hier-archy of Hamiltonians can be constructed: thekth-orderdeviation from quantum adiabatic theorem is governedby a kth-order Hamiltonian. This theory not only of-fers explicit formula to compute the deviations of vari-ous orders but also presents an intuitive insight into theintricacy of adiabatic evolution. To illustrate the lat-ter, we use the second-order Hamiltonian H2(R, R, R) as

an example. We assume that R, R is small while R islarge. In this case the dynamics of the second-order deviation (2p, 2q) governed by H2 is not adiabatic. Asa result, the second-order deviation (2p, 2q) can growreach the first-order level, and continue to grow even big-ger. The evolution of the first-order deviations is adiabatic due to the smallness of R and R. However, thiconclusion is only true when the deviation is small. I

the second-order deviation grows so large that the devi-ation is no longer small, the adiabticity at the first-orderlevel is then broken down. Eventually, the growth start-ing from the second-order level can even break down thezeroth-order adiabaticity. This example shows that theadiabatic evolution can be maintained for an infinite longtime only when all orders of derivative ofR with respective to time are small. However, this kind of growth odeviation due to high-order non-adiabaticity takes a verylong time, which is usually beyond the physically relevanttime scale. So, we do not need to worry about it mostof time when applying in real physical problems. As theexact time scale for this growth depends on the detail of

the control protocol R(t), it can only be computed caseby case.

V. TWO EXAMPLES

We now use two simple systems to illustrate our hier-archical theory. One is a spin-1/2 particle in an externarotating magnetic field; the other is the Landau-Zenemodel. They are chosen because they are either exactlysolvable or their numerical solutions can be found withgreat accuracy. In this way, there will be no ambiguityin checking the validity of our hierarchical theory. In thissection, we always assume = 1.

A. spin-1/2 under a rotating field

In the hierarchical theory, the first-order deviation andits dynamics is of the most importance. In this subsection, we employ the simple model of a spin-1/2 particle ina rotating magnetic field to illustrate the first-order adiabatic theory. The Hamiltonian for a spin-1/2 particle inan external rotating field is

H0=1

2

0 L exp(i)

L exp(i) 0

, (18)

where(t) changes slowly with time for a rotating fieldWe use| = (c1, c2)T, where c1 andc2 are complex, todenote the quantum state of this spin-1/2 particle. Weturn to the classical formulation by introducing a pair oconjugate variables, p= arg(c2) arg(c1) and q= |c2|2The corresponding classical Hamiltonian is

H0 = |H0| =L

q q2 cos( p). (19)The classical Hamiltonian in Eq. (19) has two ellipticfixed points, namely, (q= 1/2,p= ) and (q= 1/2,p=


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FIG. 3: The first-order solution of a spin-1/2 particle in aslowly rotating magnetic field. The dots are q computedfrom the analytical solution to the first order Eq. (22). Thesolid line isB1 of the first-order fixed point. The inset showsthe first-order actionI1computed from the analytical solutionEq. (22). = 105.

+ ), corresponding respectively to the two eigenstatesof Eq. (18). We focus on the adiabatic following of thefixed point q= 1/2,p = as (rotating field) changesslowly. The conventional adiabatic theorem states thatthe actual state will accurately follow the instantaneousstate (q= 1/2,p= ).

On top of the conventional adiabatic theorem, thereare first-order corrections. To that end we now derivethe effective first-order Hamiltonian H1. According toEqs. (9), (10) and (19), one finds for fixed point (q =1/2, p= ),

H1 = L

2

2(q

2L )2

+

1

2 (p)2

. (20)

Interestingly, for this example, H1 happens to be inde-pendent of the adiabatic parameter . The first-orderfixed point is located at A1 = 0, B1 = /2L. In the fol-lowing we consider three different control protocols (t)with (0) = 0 and the initial state emanating exactlyfrom the fixed point q(0) = 1/2, p(0) =.

(i) Let us first consider the simplest protocol in which = t with being constant. At t = 0, the initialstate is q = 1/2, p = 0 while the first-order fixed pointis at (A1 = 0, B1 = /2L). So, the state starts off thefirst-order fixed point and the first-order action is

I1 = 2

4L2. (21)

According to our theory, the first-order deviation willundergo a spiral motion, similar to what is depicted inFig.2(2), with its amplitude controlled by I1.

The validity of our theory can be checked by directlyintegrating the Schrodinger equation governed by (18).This solution can be found exactly. With the omission of

FIG. 4: Numerical results for the first-order correction (p,qobtained from the Hamiltons equation of motion governed by(19). The control protocol is = 1

2at2 witha = 7.961012

The dots and circles are numerical results while the solid linesare for the first-order fixed point (A1 = 0, B1 =at/2L). Theinset shows the numerically computed I1.

higher orders, the solution can be written as

| = 12

1 2L[1 cos(Lt)]

1 + 2L (1 cos(Lt))

ei[t+L sin(Lt)]

.

(22This solution is plotted in Fig.3by mapping | to (p, q)and thus to (p,q). In this figure, we clearly see oscillations around the fixed point (A1 = 0, B1 = /2L)consistent with our first-order theory. As shown in theinset of this figure, our direct computation also confirmsthat the first-order action I1 is a constant. We point outthat this is equivalent to a system under the followingcontrol protocol

= 0 for t < 0 ; = t for t > 0 . (23)

That is, there can be a small sudden jump in at t =0. Analytically, the first-order deviation can be readilycomputed from the solution (22)

p = p p= L

sin(Lt),

q= q q= 2L

(1 cos(Lt)) , (24)

which is indeed consistent with the first-order Hamilto-nian dynamics predicted by H1 in Eq. (20).

(ii) In the second protocol, the speed increases gradually from zero. To be specific, we choose = 1

2at2 with

a= 7.96 1012. For this protocol, the first-order fixedpoint is (A1 = 0, B1 = 0) at t = 0. Therefore, according to our first-order theory, the action I1 = 0 and thedynamics of the first-order deviation (p,q) follows ex-actly the first-order fixed point (A1 = 0, B1 = at/2L)We have numerical solved the Hamiltons equations omotion governed by Eq. (19) for this second protocolThe numerical results for (p,q) and I1 are shown inFig. 4 and an excellent agreement with our first-ordertheory is found.


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FIG. 5: Numerical solution of q and I0 = 1

2

p dqwith

Hamiltons equation of motion governed by (19). For thecontrol protocol, we have ||= 105. The sign of oscillateswith the frequency = 1.

(iii) In the third protocol, we change the sign of frequently while keeping

|

|small. This is to ensure that

the second-order time derivative can be quite large.We use this protocol to illustrate an insight offered byour hierarchical theory: high-order time derivative ofRcan also lead to the breakdown of adiabaticity. For thisspin-1/2 system, the smallness of|| does not guaranteethe accuracy of the quantum adiabatic theorem. When is large, then the first-order dynamics governed by H1 isno longer adiabatic, and the accumulation of (p,q) willeventually lead to the breakdown of adiabaticity at thezeroth orders. We have solved numerically the equationsof motion governed by Eq. (19). The results are plotted inFig.5, where we see thatqcan indeed grow and destroythe adiabaticity. The solid line seen in the middle of the

pattern shown in Fig.5 demonstrates that the actionI0is no longer a constant.

B. Hierarchy of adiabatic corrections in theLandau-Zener model

In this subsection, we consider a different model, theLandau-Zener (LZ) model, and use it to demonstratehigher-order deviations. The LZ Hamiltonian can bewritten as

HLZ0 =1

2 z xx

z , (25)

where the coupling term x > 0 is a constant whereas zchanges slowly and linearly fromZ0 toZ0,

z= V t, t: Z0/V Z0/V. (26)Similarly, we define c1 =|c1|eic1 , c2 =|c2|eic2 , p =c2 c1 , and q= |c2|2, and obtain the classical Hamil-tonian (drop a constant):

H0= |HLZ0 | =x

q q2 cos(p) zq. (27)

This classical Hamiltonian has two fixed points a(p = 0,q = 12 z2x2+z2 ) and (p = ,q = 12 +

z2

x2+z2). Without loss of generality, we focus on the

fixed point ( p = , q = 1

2 + z

2

x2+z2), which corre

sponds to the eigenstate with the lower energy. According to the quantum adiabatic theorem, i.e., zeroth-order theory, when the initial state is the ground state

cos(arctan(x/Z0)

2 ), sin(arctan(x/Z0)

2 )T

at z =Z0 thesystem will follow the instantaneous eigenstate and ul-

timately reach sin( arctan(x/Z0)2 ), cos(arctan(x/Z0)2 )

Tat

z =Z0. In what follows, we will compute explicitly thefirst-order deviation and the second-order deviation, anddiscuss some general properties of the higher-order devi-ations.

According to Eqs. (9,27), the first-order HamiltonianH1 reads

H1 =

x2 +z2

1 +

z2

x2

(q)2 +

1

4x2

(p Vx2+z2 )2x2 +z2

.

(28The fixed point for the first-order deviation (p,q) (orthe first-order deviation from the zeroth-order adiabatictrajectory) is

A1B1

=

Vx2+z2

0

. (29)

The results for the second-order deviation (2p, 2q)can be computed similarly. The second-order Hamiltonian is

H2 =

x2

+z2

1 +

z2

x2

2

q 5x2zV2

4(x2 +z2)7/22

zVx2 +z2

2q 5x

2zV2

4(x2 +z2)7/2

2p

+ 1

4x2

(2p)2x

2 +z2. (30)

The fixed point (or, the deviation from the first-orderadiabatic trajectory) is

A2

B2

=

0

5x2zV2

4(x2+z2)7/2

;

We consider the limitZ0 . At this limit, we haveA1 = B1 = A2 = B2 = 0 at|z| =. This means thathe deviations of the first-order and the second-order arezero both at the beginning and at the end of the evolution. The higher-order deviations can also be computedwith the formula in Appendix B. There is no need towrite them down here. We only want to mention, for althese higher-order deviations, we also have

Ak 0; Bk 0, as |z| . (31)


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This indicates that the LZ tunneling rate at Z0 tends to zero to all orders of the small driving rate Vbased on our hierarchy theory. This is perfectly consis-tent with the standard rigorous result for the LZ tunnel-

ing rate exp(x2V ) [23], where any term in the Taylorexpansion of exp(x2V ) with respect to V is zero. Thisresult is in sharp contrast to the previous case in the lastsubsection, where the leading term of the deviation from

an ideal adiabatic behavior is proportional to 2.

FIG. 6: (color online) Schematics of different scenarios forthe first-order deviation from the zeroth-order adiabatic tra-jectory. (a) For the case of a spin-1/2 particle under a rotatingfield considered in Sec. V.A, the first-order deviation from thezeroth-order adiabatic trajectory is a constant. (b) In the caseof a LZ process considered in Sec. V.B, the average deviationin either the initial stage or the final stage approaches zero,but it can be appreciable at the intermediate stage. (c) Thefirst-order deviations (p,q) can be manipulated by design-

ing the time dependence ofR(t) and R(t).

We now summarize this section with Fig.6, where the

first-order deviations discussed above are schematicallyplotted. It is clear from our theory that the first-order de-viations can be manipulated by designing R(t) and R(t)This can be very useful to control the nonadiabatic er-ror in quantum adiabatic computation[24]. We plan topursue this issue in the near future.

VI. CONCLUSION

In this work, we have studied the quantum adiabaticevolution and developed a hierarchical theory for thesmall deviations from the conventional quantum adiabatic theorem. The claim from the theorem that nopopulation transitions between non-degenerate instantaneous energy eigenstates is true only when the externadriving is infinitely slow. For a small and nonzero driving rate of the adiabatic parameter, there are small de-viations, i.e., small transitions between different energyeigenstates. We have found that the deviations ofkth

order are governed by a kth-order Hamiltonian. Our ap-proach can be directly applied to classical adiabatic pro-cesses and nonlinear quantum adiabatic evolution on themean-field level [25,26]. It is of considerable interest toapply our findings to assist in the control of adiabaticprocesses in both classical and quantum systems[2729]

Acknowledgments

This work is supported by the NBRP of China(2013CB921903,2012CB921300) and the NSF of China

(11105123,11274024,11334001).

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Rev. A 76, 044102 (2007).[10] D. Comparat, Phys. Rev. A 80, 012106 (2009).[11] Y. Zhao, Phys. Rev. A 77, 032109 (2008); J. Ma, Y

Zhang, E. Wang, and B. Wu, Phys. Rev. Lett. 97, 128902(2006); S. Duki, H. Mathur, and O. Narayan, Phys. RevLett. 97, 128901 (2006); Z. Wu and H. Yang, Phys. RevA72, 012114 (2005); J. Du, L. Hu, Y. Wang, J. Wu, MZhao, and D. Suter, Phys. Rev. Lett.101, 060403 (2008)D. M. Tong, Phys. Rev. Lett. 104, 120401 (2010); A

Ambainis and O. Regev,quant-ph/0411152.[12] E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, and D. Preda, Science 292, 472 (2000).

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[17] A. D. A. M. Spallicci, A. Morbidelli, and G. Metris, Non-linearity18, 45 (2005).

[18] M. V. Berry and J. M. Robbins, Proc. Roy. Soc. Lond.A442, 641 (1993).

[19] J. Liu and L.B. Fu, Phys. Rev. A 81, 052112 (2010); L.B.Fu and J. Liu, Ann. Phys. 325, 2425 (2010).

[20] P.A.M. Dirac, Proc. R. Soc.107, 725 (1925).[21] Bohm Mostafazadehet al., The geometric phase in quan-

tum systems, Springer, 225-243 (2003).

[22] Q. Zhang, J. B. Gong, and C. H. Oh, Ann. Phys.327,1202 (2012); Q. Zhang, J. Phys. A 45, 295302 (2012).

[23] L.D. Landau, Phys. Z. Sowjetunion 2, 46 (1932); C.Zener, Proc. R. Soc. A 137, 696 (1932).

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arXiv:1401.1184.

Appendix A: Detailed derivations for thesecond-order theory

The premise of dealing with the second-order deviationis that the state is around the first-order fixed point. Thisallows us to express (p,q) as the following,

p =

2

p+A1; q=

2

q+B1, (A1)where 2p and 2q describe the actual dynamics of(p,q) on top of their time-averaged values (A1, B1).

Note that in deriving H1 we have only keptthe first-order term when expanding the force fieldH0(R)q , H0(R)p

. This is adequate for the first-order

theory. When considering the second-order deviation, weshould also keep the second-order terms in the expansion.Specifically, substituting Eq. (A1) into Eqs. (5,8), keep-ing the second-order expansion terms

1

2 p

p+ q

q2

H0 H0 =

H0q

or H0

p ,

(A2)and neglecting terms containing [2p]2 or [2q]2(which arefourth-order), one finds (employing Eq. (10)) d

2pdt

d2qdt

= 1

2

A1

B1

+(0+)

2p

2q

dA1dt

dB1dt

,

(A3)where is defined in Eq. (15) as the state under con-sideration shifts from (p,q) to (p+A1,q+B1).

Rearranging some terms on the right-hand side oEq. (A3), we arrive at d

2pdt

d2qdt

= (0+)

2p2q

(0+)1

dAdt

dBdt

+1

2(0+)

1

A1

B1.

The fixed-point solution for 2p and 2q can be foundfrom Eq. (A4); it is

A2B2

= 10 (R)

A1R

B1R

R+

A1V

B1V

R

12

10

A1

B1

, (A5)

where the time derivatives of the adiabatic parameter

R, R and R, are assumed to be in the same order omagnitude. All higher-order terms, such as those termof the order ofRj with j 3, are neglected. Under thitreatment, it is now seen that, in terms of their time-averaged values, a more accurate prediction of (p,q)is given by (A1 + A2, B1 + B2). Note that A2 and B2are evidently proportional to R2. Equations (A4,A5)are just the second-order dynamics and the second-orderfixed point given in the main text (see Eqs. (13) and(14)). One can now readily write down the second-orderHamiltonian

H2(R, R) = 1

2 2H0

q2 p1,q1 (2q B2)2

+

2H0qp

p1,q1

(2q B2)(2p A2)

+ 1

2

2H0p2

p1,q1

(2p A2)2, (A6)

One only need to note that (p, q) take value of (p+A1,q+B1) instead of (p,q) as we are at the second-order approx-imation.

Appendix B: High-order deviations in quantumadiabatic evolution

The dynamics of thekth-order deviation (kp, kq) canbe derived iteratively by substitutingp = A1+A2+. . .+kpand q= B1 + B2 + . . . + kqinto Eqs. (5,8) with theexpansion up to the kth-order, provided the fixed pointsof all the previous (k 1) orders have been obtainedSpecifically, (3p, 3q) can be described by a third-order

Hamiltonian H3(R, R, R, d3R

dt3 ). The kth-order deviation

(kp, kq) forms a pair of canonical variables of a kth

order Hamiltonian Hk(R, R, R , . . . ,dkR

dtk ), demonstrating
http://arxiv.org/abs/1401.1184http://arxiv.org/abs/1401.1184


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10

that thek th-order deviation will undergo adiabatic evo-lution only if the time derivatives of parameter R up tothe kth-order are all manipulated very slowly in com-parison with the intrinsic frequency k of the kth-orderHamiltonian, which is proportional to|k| |0|.

The kth order deviation consists of k terms, withthe first one associated with the ideal matrix 0 andthe adiabatic evolution of the (k 1)th-order deviation(

k1

p, k1

q), the second one associated with andthe evolution of (k2p, k2q), and the kth one associ-ated withk1 and the zeroth order adiabatic evolutionof (p,q). The sum of thek terms is the result for the dy-namical fixed point ofHk.

To illustrate that a general kth order theory is possible,we consider here only a rather simple case where R isa constant. However, even in this case our expressionsappear to be complicated and hence readers may skip thetechnical details (we present them just for completeness).In particular, thek th-order fixed point is

AkBk

=

1

0

Ak1R

Bk1R

R

1

0

k1j=1

j

Akj

Bkj

.

(B1)The deviations j in Eq. (B1) is defined as

j = Tj

ji=1

1

(i+ 1)!

j

r=1

(Ar)

p+

jr=1

(Br)

q

i

(B2)The functionTj (. . .) i n (B2) is to take the jth-order

terms in (. . .), i.e., taking the sum of all the terms of thekind AutB

vs with tu+ sv = j. For example, A2 and B2

are second-order terms in terms ofR, andA22 and A2B2become the fourth-order terms, soT2(A2 + B2 + A22+A2B2) = A2+ B2,T3(A2+ B2+ A22+ A2B2) = 0 andT4(A2 + B2 + A22 + A2B2) = A22 + A2B2, etc. Specificallywhenj = 1, = 12.

In the case of nonconstant adiabatic speed V, we

should include the derivatives of the kind (d0R/dt0 R)

k1j=0

Ak1(djR/dtj)

Bk1(djR/dtj)

dj+1R

dtj+1

for the kth-order deviation.

Generally, the hierarchy adiabatic theory can also benaturally extended to n-mode quantum system by ex-panding the matrix from dimension 2 2 to dimension2(n 1) 2(n 1).

Finally, it is necessary to make three remarks on high-order deviations. First, the deviations of all orders areobtained with respect to what the usual quantum adia-batic theorem predicts. This is the reason that Eq. (B2)looks complicated. Second, we assume that all orders otime derivatives ofR, R, R , . . . , dkR/dtk, are of the sameorder of magnitude. Thek th-order term is proportionato Rk. Third, in deriving thekth-order deviation in (B1)we have already assumed the adiabaticity holds for up tothekth-order Hamiltonian.

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Hierarchical Theory of Quantum Adiabatic Evolution