WEN GE WANGCASATI, GIULIOB. LImostra contributor esterni nascondi contributor esterni 

Stability of quantum motion in regular systems: A uniform semiclassical approach

Wen-ge Wang,^1,2G. Casati,^3,4,1and Baowen Li^1,5,6

1Department of Physics and Centre for Computational Science and Engineering, National University of Singapore, 117542, Republic of Singapore

2Department of Physics, Southeast University, Nanjing 210096, China

3Center for Nonlinear and Complex Systems, Università degli Studi dell’Insubria, Via Valleggio 11, 22100 Como, Italy

4CNR-INFM and Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Italy

5Laboratory of Modern Acoustics and Institute of Acoustics, Nanjing University, 210093, China

6NUS Graduate School for Integrative Sciences and Engineering, National University of Singapore, 117597, Republic of Singapore sReceived 27 November 2005; revised manuscript received 29 November 2006; published 5 January 2007d

We study the stability of quantum motion of classically regular systems in the presence of small perturba- tions. On the basis of a uniform semiclassical theory we derive the fidelity decay which displays a quite complex behavior, from Gaussian to power law decay t^−a, with 1 ø a ø 2. Semiclassical estimates are given for the time scales separating the different decaying regions, and numerical results are presented which confirm our theoretical predictions.

DOI:10.1103/PhysRevE.75.016201 PACS numberssd: 05.45.Mt, 03.65.Sq

Stable manipulation of quantum states is of importance in many research fields such as in quantum information pro- cessing and in Bose-Einstein condensation. A measure of the stability of quantum motion is the so-called fidelity or quan- tum Loschmidt echof1g, which characterizes the stability of quantum dynamics under small perturbations of the Hamil- tonian that may derive from static imperfections or from in- teraction with an external environment. From a more general point of view it is interesting to understand the behavior of fidelity in relation to the dynamical properties of the system.

While fidelity decay in classically chaotic systems has been extensively studiedf2–11g, the situation in regular sys- tems is much less clearf5,10–19g and only for the particular case of vanishing time-average perturbations has a clear un- derstanding been achievedf12g. Moreover, for a general per- turbation and for a single initial Gaussian wave packet, a Gaussian decay has been predicted in some time interval which is not exactly specified f5g. On the other hand, nu- merical investigations show a much richer behavior of the fidelity decay, ranging from power law to exponential up to a Gaussian decay, depending on initial conditions, perturbation strength, and time intervalf15g. In addition, a somehow un- expected regime in which the fidelity decay in regular sys- tems is faster than in classically chaotic systems has been found in f5g. All of the above call for a theory which can account for these diverse analytical and numerical findings.

We would like to draw the reader’s attention to the fact that in the general theory of dynamical systems, integrability is the exception rather than the rule and it is therefore ex- tremely rare. However, the theory we present in this paper applies also to integrable regimes of systems with divided phase space in which both chaotic and integrable compo- nents are present and this is indeed the typical situation.

Moreover, as shown in f20g, for a proper operability of a quantum computer it is desirable to remain below the border for transition to quantum chaos, a situation which is likely to correspond to quasi-integrable behavior. Finally, some quantum algorithm can have a phase space representation and, in particular, the Grover algorithm can be interpreted as

a simple quantum map which turns out to be a regular map f21g.

The above considerations have motivated our interest in the stability of integrable motion. In this paper, we develop a uniform semiclassical approach to the fidelity decay in regu- lar systems and we provide a unified description together with the corresponding time scales. Numerical computations confirm our analytical estimates.

Quantitatively, the fidelity for an initial state uC0l is de- fined as Mstd = umstdu², where

mstd = kC0uexpsiHt/"dexps− iH0t/"duC0l. s1d

Here H₀ and H = H₀+eVare the unperturbed and perturbed Hamiltonians, withe a small quantity and V a generic per- turbing potential.

Consider an initial Gaussian wave packet in a 2d-dimensional phase space, centered atsr˜0, p˜₀d,

c₀sr0d = spj²d^−d/4expfip˜₀· r₀/" −sr0− r˜₀d²/s2j²dg. s2d

For a sufficiently narrow, initial Gaussian packet, the semi- classical approximation to the fidelity amplitude is

m_scstd . spw_p²d^−d/2

E

F

G

where w_p= " / j and DS is the action difference between the two nearby trajectories of the two systems H and H₀starting atsp0, r˜₀d f7g. We mention that, for not very narrow initial Gaussian packets, Eq.s3d may still hold with a redefinition of w_pf9g. For more general initial states, the fidelity ampli- tude can be expressed in terms of the Wigner function of the initial state in a uniform semiclassical approachf10g.

The action difference can be calculated in the first-order classical perturbation theory:

DS.e

E

with V evaluated along one of the two trajectories. Equations s3d and s4d give quite accurate predictions even for relatively long times, much more accurate than that usually expected for a first-order perturbation treatment f7,8g. The reason for the unexpected accuracy is explained inf10g by making use of the shadowing theorem, the two trajectories for DS being in fact the so-called shadowing trajectories of the two sys- tems with slightly different initial conditions.

Equations3d shows that the behavior of fidelity is mainly determined by two factors: sid DS as a function of p0 and t andsiid the Gaussian term, which specifies an effective win- dow in the p₀ space with size w_p. For simplicity, in what follows, we consider kicked systems with d = 1 and set the domains of r and p to be f0 , 2pd.

For an integrable system or a regular region of a mixed system, at least locally there exist action-angle variables su, Id, connected to the variables sr , pd by a canonical trans- formation. The integrand in Eq.s4d can then be written as

V_t; VfIstd,ustdg, withustd =u₀+ nt, s5d Istd . I0, and n; ]HsId / ]I.

The main features of DS as a function of t and p₀can be seen by substituting Eq. s5d into Eq. s4d, replacing nt byf

= nt, and noting the periodicity of the angleu. This gives

DS.esUIt+ S_fd, where U_I; 1

2p

E

S_f;1 n

F ^E

b

VsI,u₀+fddf− bU_I

G

Here b; nt − 2pnt, where n_t is the integer part of nt / 2p.

Therefore, for a fixed p₀, b is a sawtooth-type function of time oscillating between 0 and 2p with a frequency n / 2p.

As a result S_foscillates correspondingly; hence, DS oscillates around its linearly increasing part eU_It. On the other hand, for a fixed t, U_Ias a function of p₀changes almost linearly in the neighborhood of p˜₀while S_f oscillates with a frequency n

˜8^t/ 2p, where n8^{; ]n / ]p0}. Therefore the average slope of DS with respect to p₀iseU_I8^{t, where U}I8^{; ]U}I/ ]p₀.fWe use a tilde above a quantity to indicate its value taken at the centersr˜0, p˜₀d.g

We first study the fidelity decay for times t ,t₁, wheret₁ is the time scale before which the right-hand side of Eq.s3d can be calculated by a linear approximation to DS with re- spect to p₀− p˜₀. This gives

M_sc1std . exp

F

pd²

G

]p₀, s8d where s =e/ " is the strength of perturbation. The explicit dependence of k_p on time t can be calculated by using Eq.

s6d. The leading terms give

k_p. sU_I8^{+ U}udt, where U_u;1 t

]S_f

]p₀. sVt− U_Idn8 n .

s9d

Since U_u oscillates in time around zero, the fidelity has, on average, an initial Gaussian decay with a rate depending on initial conditions, with larger values ofuU_I8u implying faster decay.

To test the above predictions, we consider the kicked ro- tator model H =¹₂p²+ k₀cos ro_n=0^` dst − nTd. The quantized system has a finite Hilbert space with dimension N. We take k₀= kN / 2p and T = 2p / N, with k = k₀Tindependent of N.

The classical limit corresponds to N → `. The one-period quantum evolution is given by the Floquet operator U

= expf−ipˆ²T/ 2gexpf−ik0cossrˆdg and is numerically computed by the method of fast Fourier transform. Here 2p / N serves as an effective Planck constant. The inset of Fig.1shows an example of DS vs p₀, in which the values ofuU_I8u are large at the borders while quite small in the central, oscillating region of p₀. In agreement with our theory, under the perturbation k → k +e, the fidelity for initial states lying in the two regions of p₀ has a quite different decaying rate, as seen in Fig. 1.

The agreement with the theory is particularly good in the case where the oscillations of DS are not too strong.

The time scalet₁can be estimated by the time at which the second-order term in the Taylor expansion of DS / " is order of 1 at the point p₀= p˜₀+ w_p—i.e., sw_p²u k˜ppstd u , 1 for t=t₁, where

FIG. 1. sColor onlined Comparison between the numerically computed fidelity decaysblack circles and open squaresd and the semiclassical prediction M_sc1std in Eq. s8d, in the kicked rotator model with k = 0.3, s = 1.5, N = 2¹⁷, and j²= " / 20. Centers of the initial Gaussian packets aresr˜₀, p˜₀d = s1.2p , 0.6pd for the circles and s1.2p , 0.2pd for the squares. The solid and dashed curves represent the semiclassical predictions, respectively, with kp evaluated nu- merically in the classical systems. Inset: DS / 2p vs p₀/ 2p, for r₀

= 1.2p at t = 20. The average slope of DS in the monotonically in- creasing part is much larger than that in the central part; therefore, the fidelity decay at p˜₀= 0.2p is faster than that at p˜₀= 0.6p. The oscillations of DS in the central region imply larger second- and higher-order terms in the expansion of DS around p˜₀= 0.6p, hence larger deviations of the fidelity from M_sc1std.

k_ppstd ;]k_p ]p₀=sn8^d2

n ]V ]u^t

2+ Ostd. s10d

A numerical check of this prediction is presented in Fig.2.

When the t²term dominates,t₁~Î1 / sw_p².

Beyondt₁, a direct analytical computation of the fidelity is difficult since higher- and higher-order terms in the Taylor expansion of DS with respect to p₀− p˜₀ need to be consid- ered. We take therefore the following approach: we divide the domain of p₀into segments separated by points p_0jswith p₀₀= p˜₀d, in such a way that Sf completes one oscillation period within each segment.sWe will use the subscript j to indicate quantities taken at the point p_0j.d It is easy to see that sp0j+1− p_0jdn8_j^t. 2p and therefore the number of segments increases linearly with time t. The quantity m_scstd in Eq. s3d can now be written as a sum of contributions m_jstd of differ- ent segments: namely, m_scstd = ojm_jstd.

Let us introduce the time scalet_s, at which S_fcompletes one oscillation within the window W_p=sp˜₀− w_p, p˜₀+ w_pd.

From Eq.s7d,t_s< p / sn˜8^wpd. For t @t_s, there are many seg- ments within W_p; hence, within each segment, the variation of the Gaussian term on the right-hand side of Eq. s3d is negligible. Therefore one may write

m_jstd .e^{−sp0j− p˜0d2/wp2}

Î_pwpnj8t e^iDSj/"Fstd, s11d Fstd =

E

H

^E

u˜std+f

VsI˜,u8^ddu8

J

Indeed for p_0jnot far from p˜₀, Fstd is independent of j, and sinceuFstdu does not decay with time, we will not consider it further.

First we notice that Eq. s11d predicts a plateau in the fidelity decay when the change of DS_j/ " within the window W_p is negligible. The plateau disappears when sw_pU˜

I8^{t, p.}

If, for example, U_I= 0 for the system H₀, then U_I~e for the system H, and the plateau will end at a time proportional to

"^1/2e⁻²for j ~ "^1/2in agreement with the result of f12g.

Due to the Gaussian term on the right-hand side of Eq.

s11d, the main contribution to mscstd comes from p0jnot far from p˜₀. For these p_0j, n8_j is almost a constant, as well as the value of S_f; hence, apart from a common phase, DS_jin Eq.

s11d can be approximated by eU_Ijt. For a sufficiently large number of segments, the sumojm_jstd can be replaced by an integral over p₀. Then, setting q = p₀− p˜₀, we have

m_scstd . Fstd

2pÎpw_p

E

Expanding U_I to the second-order terms in q, we have

M_sc2std . 2c

Î_{4 +sw}²_psU^˜_I9td^2exp

F

I9td²

G

where U_I9^{; ]U}I8^{/ ]p}0and c is a constant with c< 1 for suffi- ciently small s.

From Eq.s14d it is seen that for sw²_psU˜

I9^td2! 4, the fidelity has a Gaussian decay e^{−swpsU˜I8td2/2}, which agrees with the one given in f5g for weak perturbations. On the other hand, if sw_p²sU˜

I9^td2@ 4—i.e., for larger time t—Msc2std in Eq. s14d has a power law decay 1 / t. In the transition region between the two decays, the fidelity may have an approximate expo- nential decayssee Fig.3d which can explain the exponential- like decay found numerically inf15g.

With further increasing time t, higher-order terms in the Taylor expansion of U_I will become important and will modify the 1 / t decay. In order to evaluate the effect of higher-order terms, we divide the interval W_pinto subinter- vals labeled by l, in such a way that inside each of them the linear approximation of U_I can be used. Thus their width FIG. 2.sColor onlined The time scale t₁versus p˜₀, for the same

parameters as in Fig.1. The circles give the values of t₁calculated by the first time at whichufMstd − M_sc1stdg / Mstdu . 0.1 fsee Eq. s8dg.

The solid curve is the semiclassical estimate given by sw_p²uk˜ppst₁du = 1, with k˜ppstd numerically computed in the corre- sponding classical system.

FIG. 3. sColor onlined Solid curve: fidelity decay for s = 0.2, N= 2¹², k = 0.3, j²= " / 20, andsr˜₀, p˜₀d = s1.2p , 0.8pd. Dashed curve:

the prediction of Eq.s14d with c = 1 and U˜

I8< −0.019, U˜

I9^{< 0.042} numerically computed from the classical system. Dotted curve: the Gaussian decay e^{−swpsU˜I8td2/2}.fFor small values of s, the difference between this decay and M_sc1 in Eq.s8d is small before t₁.g The dash-dotted line shows the approximate, intermediate, exponential decay. Inset: long time decay for the same case with r˜₀= 0.6psev- ery 500 steps shownd. The dashed line gives the 1 / t^1.1decay.

must be of the orderdq. 1 /Î_uUI9u st and their number in the region W_pis l_m~Îst. For sufficiently long time t, l_m@ 1 and the width of each subinterval is much smaller than W_p. As a result, the Gaussian term in Eq. s13d can be regarded as constant within each subinterval. Then, the integral s13d re- duces to m_scstd . olm_lstd = olc_le^iflstd/ st, with some coeffi- cients c_land phases f_lstd. The detailed behavior of this sum depends on the properties of the function U_I. However, one can give estimates in some limiting cases: sid Since uolc_le^iflstdu ø oluclu ~ sstd^1/2, the slowest decay is 1 /Îst. siid The fastest decay is obtained when uolc_le^iflstdu is a constant swhich may happen due to mutual cancellation of phasesf_ld.

In this case, m_sc, 1 / st. siiid In the case of random phasesf_l, one hasuolc_le^iflstdu ~ sstd^1/4, then m_sc, 1 / sstd^3/4, which coin- cides with the result given in Ref.f13g for averaged fidelity.

(For an analysis of Gaussian and power law decay of fidelity, which is based on statistics of action difference, see Ref.

f10scdg).

Therefore, in general, the decay of Mstd has the power law dependence set/ "d^−a with 1 øaø 2. This is in agree- ment with numerical results in f15g, as well as with our ex- tensive numerical simulationsssee, e.g., Fig.4 and the inset of Fig. 3d. The cases of a. 1 or 2 have been found to be quite rare in our simulations.

It is important to remark that, contrary to chaotic systems for which the decay rate depends on the strength but not on the shape of the perturbation, integrable systems lack of such generic behavior. This is typical in the general theory of dynamical systems, and it is due to the peculiarity of inte- grability. In the present case the numerical value of a de- pends on the particular shape of U_I.

Our approach can also explain an interesting feature of fidelity decay in regular systems observed numerically in f15g, that is the fact that, in some cases, the decay rate may decrease with increasing the perturbation strength s. First we note that Eq.s14d is valid only when uDSj+1− DS_ju / " is small compared with p, since it is obtained by replacing the sum ojm_jstd by an integral over p0. Since U_I8 is given bysDSj+1

− DS_jd / sp0j+1− p_0jdet, then uDSj+1− DS_ju / " . 2psuU_Ij8^{/ n}j8u, which may exceed p for sufficiently large s. In this case, we can replace DS_j/ " byc_j; DSj/ " − 2pmj, where m is an in- teger such thatuc_j+1−c_ju ø p. Then using the fact that, for p0j

close to p˜₀,sp0j− p˜₀dn˜8^t. 2pj, we find that the term sU˜

I8ⁱⁿ Eq.s14d can be replaced byb; usU˜

I8^{− mn˜}8u. With increasing s, the value of b, which gives the decay rate, oscillates be- tween 0 andun˜8u / 2, with a period un˜8^{/ U˜}I8u. The results of Fig.

4 nicely confirm the above analysis.

Finally a word of comment on the comparison of fidelity

decay in classically regular and chaotic systems. As shown in f5g, in a sufficiently weak perturbation regime s , sc~Î", the Gaussian decay e^{−swpsU˜I8td2/2} in the regular case can be faster than the Fermi-golden-rule decay in the chaotic case.

This is not surprising as it may look at first sight, since in this regime, w_psc~ " can be quite small and the fidelity may remain close to 1 for a time comparable to the Heisenberg time,1 / ", which can be quite long. Moreover, the above Gaussian decay in the regular case is followed by a power law decay, which is slower than the exponential decay in the chaotic case.

We thank T. Prosen for useful discussions. This work was supported in part by the Academic Research Fund of the National University of Singapore and the Temasek Young Investigator AwardsB.L.d of DSTA Singapore under Project Agreement No. POD0410553. Support was also given by the EC RTN Contract No. HPRN-CT-2000-0156, the NSA and ARDA under ARO Contract No. DAAD19-02-1-0086, the project EDIQIP of the IST-FET programme of the EC, the PRIN-2002 “Fault tolerance, control and stability in quantum information processing,” and Natural Science Foundation of China Grant No. 10275011.

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I8

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