We investigate the sensitivity of the time evolution of semiclassical wave
packets in two-dimensional chaotic billiards with respect to local
perturbations of their boundaries. For this purpose, we address, analytically
and numerically, the time decay of the Loschmidt echo (LE). We find the LE to
decay exponentially in time, with the rate equal to the classical escape rate
from an open billiard obtained from the original one by removing the
perturbation-affected region of its boundary. Finally, we propose a principal
scheme for the experimental observation of the LE decay.Comment: Final version; 4 pages, 3 figure

Arseni Goussev

H. De Raedt

Klaus Richter

M. Brack

English

arXiv

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Loschmidt-echo decay from local boundary perturbations

Goussev, Arseni

Richter, Klaus

Portsmouth University Research Portal (Pure)

We investigate the sensitivity of the time evolution of semiclassical wave packets in two-dimensional chaotic billiards with respect to local perturbations of their boundaries. For this purpose, we address, analytically and numerically, the time decay of the Loschmidt echo (LE). We find the LE to decay exponentially in time, with the rate equal to the classical escape rate from an open billiard obtained from the original one by removing the perturbation-affected region of its boundary. Finally, we propose a principal scheme for the experimental observation of the LE decay

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Loschmidt echo decay from local boundary perturbations

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Loschmidt-echo decay from local boundary perturbationsArseni Goussev and Klaus RichterInstitut für Theoretische Physik, Universität Regensburg, 93040 Regensburg, GermanyReceived 23 October 2006; published 9 January 2007We investigate the sensitivity of the time evolution of semiclassical wave packets in two-dimensionalchaotic billiards with respect to local perturbations of their boundaries. For this purpose, we address, analyti-cally and numerically, the time decay of the Loschmidt echo LE. We find the LE to decay exponentially intime, with the rate equal to the classical escape rate from an open billiard obtained from the original one byremoving the perturbation-affected region of its boundary. Finally, we propose a principal scheme for theexperimental observation of the LE decay.DOI: 10.1103/PhysRevE.75.015201 PACS numbers: 05.45.Mt, 03.65.SqThe study of the sensitivity of the quantum dynamics toperturbations of the system’s Hamiltonian is one of the im-portant objectives of the field of quantum chaos. An essentialconcept here is the Loschmidt echo LE, also known as fi-delity, that was first introduced by Peres 1 and has beenwidely discussed in the literature since then 2. The LE,Mt, is defined as an overlap of the quantum state e−iHt/0obtained from an initial state 0 in the course of its evolu-tion through a time t under a Hamiltonian H, with the statee−iH˜ t/0 that results from the same initial state by evolvingthe latter through the same time, but under a perturbedHamiltonian H˜ different from H:Mt = 0eiH˜ t/e−iHt/02. 1It can be also interpreted as the overlap of the initial state0 and the state obtained by first propagating 0 throughthe time t under the Hamiltonian H, and then through thetime −t under H˜ . The LE equals unity at t=0, and typicallydecays further in time.Jalabert and Pastawski have analytically discovered 3that in a quantum system, with a chaotic classical counter-part, Hamiltonian perturbations sufficiently week not to af-fect the geometry of classical trajectories, but strong enoughto significantly modify their actions result in the exponentialdecay of the average LE Mt, where the averaging is per-formed over an ensemble of initial states or system Hamil-tonians: Mte−t. The decay rate  equals the averageLyapunov exponent of the classical system. This decay re-gime, known as the Lyapunov regime, provides a strong,appealing connection between classical and quantum chaos,and is supported by extensive numerical simulations 4. Fordiscussion of other decay regimes consult Ref. 2.In this paper we report a regime for the time decay of theunaveraged, individual LE for a semiclassical wave packetevolving in a two-dimensional billiard that is chaotic in theclassical limit. We consider the general class of strong per-turbations of the Hamiltonian that locally modify the bil-liard’s boundary: the perturbation only affects a boundarysegment of length w small compared to the perimeter P, seeFigs. 1 and 2. Both w and the perturbation length scale in thedirection perpendicular to the boundary are considered to bemuch larger than the de Broglie wavelength , so that theperturbation significantly modifies trajectories of the under-lying classical system, see Fig. 1. Our analytical calculations,confirmed by results of numerical simulations, show that theLE in such a system follows the exponential decayMte−2t, with  being the rate at which classical particleswould escape from an open billiard obtained from the origi-nal, unperturbed billiard by removing the perturbation-affected boundary segment. The LE decay is independent ofthe shape of a particular boundary perturbation, and onlydepends on the length of the perturbation region. Further-more, our numerical analysis shows that for certain choicesof system parameters the exponential decay persists for timest even longer than the Heisenberg time tH.We proceed by considering a Gaussian wave packet,0r =1	exp ip0 · r − r0 −r − r0222  , 2centered at a point r0 inside the domain A of a two-dimensional chaotic billiard e.g., the solid-line boundary inFig. 1, and characterized by an average momentum p0 thatdefines the de Broglie wavelength of the moving particle,= / p0. The dispersion  is assumed to be sufficientlysmall for the normalization integral Adr0r2 to be closeFIG. 1. An unperturbed, chaotic billiard solid line, togetherwith the perturbation dashed line. The boundary of the unper-turbed billiard consists of two segments, B0 and B1. The perturba-tion replaces the latter segment by B˜ 1, rendering the perturbed bil-liard to be bounded by B0 and B˜ 1. The initial Gaussian wave packetis centered at r0. Three possible types of trajectories, s0, s1, and s˜1,leading from r0 to another point r, are shown.PHYSICAL REVIEW E 75, 015201R 2007RAPID COMMUNICATIONS1539-3755/2007/751/0152014 ©2007 The American Physical Society015201-1to unity. We let the wave packet evolve inside the billiardthrough a time t according to the time-dependentSchrödinger equation with hard-wall Dirichlet boundaryconditions. This evolution yields the wave function tr= re−iHt/0, where H stands for the Hamiltonian of thebilliard. Then, we consider a perturbed billiard obtained fromthe original one by modifying the shape of a small segmentof its boundary. Figure 1 illustrates the perturbation: the un-perturbed billiard is bounded by segments B0 and B1,whereas the boundary of the perturbed billiard is composedof B0 and B˜ 1. The perturbation, B1→B˜ 1, is assumed to besuch that the domain A˜ of the perturbed billiard entirelycontains the domain A of the unperturbed one. The timeevolution of the initial wave packet, Eq. 2, inside the per-turbed billiard results to ˜ tr= re−iH˜ t/0, with H˜ beingthe Hamiltonian of the perturbed billiard. Then, the LE, de-fined in Eq. 1, readsMt = Adr˜ t*rtr2, 3where the asterisk denotes complex conjugation.We now present a semiclassical calculation of the overlapintegral Eq. 3. As the starting point we take the expression3,4 for the time evolution of the small such that  is muchsmaller than the characteristic length scale of the billiardGaussian wave packet, defined by Eq. 2:tr  421/2 sr,r0,tKsr,r0,te−2ps − p02/22. 4This expression is obtained by applying the semiclassicalVan Vleck propagator 5, with the action linearized in thevicinity of the wave packet center r0, to the wave packet0r. Here, the sum goes over all possible trajectoriessr ,r0 , t of a classical particle inside the unperturbed billiardleading from the point r0 to the point r in time t e.g., tra-jectories s0 and s1 in Fig. 1, andKsr,r0,t =	Ds2iexp iSsr,r0,t − is2  , 5where Ssr ,r0 , t denotes the classical action along the paths. In a hard-wall billiard Ssr ,r0 , t= m /2tLs2r ,r0, whereLsr ,r0 is the length of the trajectory s, and m is the mass ofthe moving particle. In Eq. 5, Ds= det−2Ss /rr0 is theVan Vleck determinant, and s is an index equal to twice thenumber of collisions with the hard-wall billiard boundarythat a particle, traveling along s, experiences during time t6. In Eq. 4, ps=−Ssr ,r0 , t /r0 stands for the initialmomentum of a particle on the trajectory s. The expressionfor the time-dependent wave function ˜ tr is obtained fromEq. 4 by replacing the trajectories sr ,r0 , t by pathss˜r ,r0 , t that lead from r0 to r in time t within the bound-aries of the perturbed billiard e.g., trajectories s0 and s˜1 inFig. 1.The wave functions of the unperturbed and perturbed bil-liards at a point rA can be written astr = t0r + t1r ,˜ tr = t0r + ˜ t1r , 6where t0r is given by Eq. 4 with the sum in the right-hand side RHS involving only trajectories s0, which scatteronly off the part of the boundary, B0, that stays unaffected bythe perturbation, see Fig. 1. On the other hand, the wavefunction t1 ˜ t1 involves only such trajectories s1 s˜1 thatundergo at least one collision with the perturbation-affectedregion, B1 B˜ 1, see Fig. 1. The LE integral in Eq. 3 hasnow four contributions:Adr˜ t*t = Adrt02 + Adrt0*t1+ Adr˜ t1*t0 + Adr˜ t1*t1. 7We argue that the dominant contribution to the LE overlapcomes from the first integral in the RHS of the last equation.Indeed, all the integrands in Eq. 7 contain the factorexpiSs−Ss /− is−s /2, where the trajectory s is ei-ther of the type s0 or s1, and s is either of the type s0 or s˜1,see Fig. 1. An integral vanishes if there is no correlationbetween s and s, since the corresponding integrand is a rap-idly oscillating function of r. This is indeed the case for thelast two integrals: they involve such trajectory pairs s ,sthat s is of the type s0 or s1, and s is of the type s˜1, so thatthe absence of correlations within such pairs is guaranteed bythe fact that the scale of the boundary deformation is muchlarger than . Then, we restrict ourselves to the diagonalapproximation, in which only the trajectory pairs with s=ssurvive the integration over r. The second integral in theRHS of Eq. 7 only contains the trajectory pairs of the types0 ,s1, and therefore vanishes in the diagonal approxima-tion. Thus the only nonvanishing contribution readsFIG. 2. Forward-time wave packet evolution in the unperturbedDD billiard, followed by the reversed-time evolution in the per-turbed billiard. The initial Gaussian wave packet is characterized bythe size =12 and de Broglie wavelength =15/; the arrowshows the momentum direction of the initial wave packet. The DDbilliard is characterized by L=400 and R=200	10. The perturbationis defined by w=60 and r=30. The propagation time corresponds toapproximately ten collisions of the classical particle.ARSENI GOUSSEV AND KLAUS RICHTER PHYSICAL REVIEW E 75, 015201R 2007RAPID COMMUNICATIONS015201-2Adrt02 22Adrs0Ds0 exp− 22 ps0 − p02 PtAdp22exp− 22p − p02 , 8where Ds0 = detps0 /r, with ps0 being the initial momen-tum on the trajectory s0r ,r0 , t, serves as the Jacobian of thetransformation from the space of final positions rA to thespace of initial momenta pPtA. Here, PtA is the set ofall momenta p such that a trajectory, starting from the phase-space point r0 ,p, arrives at a coordinate point rA afterthe time t, while undergoing collisions only with the bound-ary B0 and thus avoiding B1, see Fig. 1. Thus Adrt02 ismerely the probability that a classical particle, with the initialmomentum sampled from the Gaussian distribution, experi-ences no collisions with B1 during the time t. Therefore if theboundary segment B1 is removed, this integral correspondsto the survival probability of the classical particle in the re-sulting open billiard. In chaotic billiards the survival prob-ability decays exponentially 7 as e−t, with the escape rate given by = vwA, 9where v= p0 /m is the particle’s velocity, and A stands forthe area of the billiard. Equation 9 assumes that the char-acteristic escape time 1/ is much longer that the averagefree flight time tf. In chaotic billiards the latter is given by 8tf =A /vP, where P is the perimeter of the billiard. Condi-tion tf	1/ is equivalent to w	P.In accordance with Eqs. 3 and 7 the LE decays asMt  exp− 2t . 10Equation 10 constitutes the central result of the paper. To-gether with Eq. 9 it shows that for a given billiard the LEmerely depends on the length w of the boundary segmentaffected by the perturbation and on the de Broglie wave-length = /mv. It is independent of the shape and area ofthe boundary perturbation, as well as of the position, size,and momentum direction of the initial wave packet. We ex-clude initial conditions for which the wave packet interactswith the perturbation before having considerably exploredthe allowed phase space.The decay rate , and thus the LE, are also related toclassical properties of the chaotic set of periodic trajectoriesunaffected by the boundary perturbation, i.e., to properties ofthe chaotic repellor of the open billiard 9: = r − hKS, 11where r is the average Lyapunov exponent of the repellor,and hKS is its Kolmogorov-Sinai entropy. Thus Eqs. 10 and11 provide an interesting link between classical and quan-tum chaos.In order to verify the analytical predictions we simulatedthe dynamics of a Gaussian wave packet inside a desymme-trized diamond DD billiard, defined as the fundamental do-main of the area confined by four intersecting disks centeredat the vertices of a square. According to the theorem of Ref.10 the DD billiard is chaotic in the classical limit. It can becharacterized by the disk radius R and the length L of thelongest straight segment of the boundary, see Fig. 2. Weconsider the Hamiltonian perturbation that replaces a straightsegment of length w of the boundary of the unperturbed bil-liard by an arc of radius r, see Fig. 2. In general, w2r.To simulate the time evolution of the wave packet insidethe billiard we utilize the Trotter-Suzuki algorithm 11. Fig-ure 2 illustrates the time evolution of a Gaussian waveFIG. 3. Color online The Loschmidt-echoLE decay in the DD billiard for four differentvalues of the curvature radius r of the arc pertur-bation. The width of the perturbation region isfixed, w=60. The other system parameters are L=400, R=200	10, =3, and =5/. The solidstraight line gives the trend of the exp−2t de-cay, with  given by Eq. 9. The inset presentsthe decay of the average LE, with averaging per-formed over individual LE curves correspondingto different values of r.LOSCHMIDT-ECHO DECAY FROM LOCAL BOUNDARY… PHYSICAL REVIEW E 75, 015201R 2007RAPID COMMUNICATIONS015201-3packet in the DD billiard followed by the time-reversed evo-lution inside the perturbed billiard. The parameters charac-terizing the system are L=400, R=200	10, w=60, and r=30. The Gaussian wave packet is parametrized by =12,=15/; the arrow shows the momentum direction of theinitial wave packet. The evolution time t in Fig. 2 corre-sponds to some ten free flight times of the correspondingclassical particle, i.e., t=10tf.Figure 3 shows the time dependence of the LE computedfor the DD billiard system characterized by L=400, R=200	10, =3, and =5/. The initial momentum directionis the same as in Fig. 2. The different LE decay curves cor-respond to different shapes of the local boundary perturba-tion: the width of the perturbation region stays fixed, w=60,and the curvature radius of the perturbation arc takes thevalues r=30, 35, 40, and 45. In all four cases the LE displaysthe exponential decay for times t up to 40tf–45tf followed byLE fluctuations around a saturation value, Ms. The thicksolid straight line shows the trend of the e−2t exponentialdecay, with  given by Eq. 9. One can see strong agree-ment between the numerical and analytical LE decay rates.We have also verified numerically that the LE decay rate isindependent of the momentum direction of the initial wavepacket.The inset in Fig. 3 presents the time decay of the averageLE Mt, with the averaging performed over 16 individualdecay curves Mt corresponding to different values of thearc radius r, ranging from r=30 to 45. The saturation mecha-nism for the LE decay was first proposed by Peres 1 andlater discussed in Ref. 4. The LE saturates at a value Msinversely proportional to the number N of energy levels sig-nificantly represented in the initial state. If the areas of theunperturbed and perturbed billiards are relatively close, thenNA /2 and Ms2 /A. We have verified the latter rela-tion by computing the LE saturation value for billiards ofdifferent area. Thus one might expect the exponential decayof the LE to persist for times t ts, with the saturation timets= 1/2vln N. The latter can be longer than the Heisenbergtime tH=A /2v for a system with sufficiently large effectiveHilbert space, since ts / tH /wln N. Indeed, for the systemcorresponding to Fig. 3 one has tH29tf, whereas the expo-nential decay persists for times t40tf.Finally, we sketch a principal experimental scheme formeasuring the LE decay regime proposed in this paper. Con-sider a two-dimensional, AlGaAs-GaAs heterojunction-basedballistic cavity with the shape of a chaotic billiard, e.g., Fig.1. Let the initial electron state be given by 0= 0 ,where 0 is the spatial part defined by Eq. 2, and=2−1/2↑ + ↓  represents a spin-1 /2 state. Here, ↑ and ↓  are the eigenstates of the spin-projection operator inthe z-direction, perpendicular to the billiard plane. Then, is the eigenstate of the spin-projection operator sx= 2x, withx= ↑ ↓+ ↓ ↑, in some x-direction, fixed in the billiardplane. Suppose now that a half-metallic ferromagnet, magne-tized in the z-direction, is attached to the boundary of theballistic cavity. One may consider the region bounded by B0and B1 in Fig. 1 to represent the ballistic cavity, and theregion bounded by B1 and B˜ 1 to represent the ferromagnet.Then the ferromagnet-cavity interface will reflect the↑ -component of the state, but will transmit the↓ -component. As a result, the two components will evolveunder two different spatial Hamiltonians, H and H˜ , corre-sponding to the geometry of the ballistic cavity and the ge-ometry of the cavity-ferromagnet compound, respectively.Then 0 will evolve tot =1	2 e−iHt/0  ↑ + e−iH˜ t/0  ↓ . 12The expectation value of the projection of the spin in thex-direction is related to the LE overlap bys¯xt  tsxt =2Re0eiH˜ t/e−iHt/0 , 13where Re denotes the real part. As we have shown above,this overlap is real and decays exponentially in time. There-fore the average spin projection in the x-direction will alsorelax exponentially with time, i.e., s¯xt2 exp−t, withthe relaxation rate  determined by Eq. 9. This result pro-vides a link between the spin relaxation in chaotic, mesos-copic structures 12 and the LE decay due to local boundaryperturbations.The authors would like to thank Inanc Adagideli, ArndBäcker, Fernando Cucchietti, Philippe Jacquod, Thomas Se-ligman, and Oleg Zaitsev for helpful conversations. A.G. ac-knowledges the Alexander von Humboldt FoundationGermany and K.R. acknowledges the DeutscheForschungsgemeinschaft DFG for support of the project.1 A. Peres, Phys. Rev. A 30, 1610 1984.2 See, e.g., T. Gorin, T. Prosen, T. H. Seligman, and M. Znidaric,Phys. Rep. 435, 33 2006; C. Petitjean and Ph. Jacquod, Phys.Rev. E 71, 036223 2005, and references therein.3 R. A. Jalabert and H. M. Pastawski, Phys. Rev. Lett. 86, 24902001.4 F. M. Cucchietti, H. M. Pastawski, and R. A. Jalabert, Phys.Rev. B 70, 035311 2004.5 M. Brack and R. K. Bhaduri, Semiclassical Physics, Frontier inPhysics, Vol. 96 Westview Press, Boulder, 1997.6 P. Gaspard and S. A. Rice, J. Chem. Phys. 90, 2242 1989.7 See, e.g., O. Legrand and D. Sornette, Phys. Rev. Lett. 66,2172 1991, and references therein.8 S. F. Nielsen, P. Dahlqvist, and P. Cvitanović, J. Phys. A 32,6757 1999.9 H. Kantz and P. Grassberger, Physica D 17, 75 1985; J.-P.Eckmann and D. Ruelle, Rev. Mod. Phys. 57, 617 1985.10 A. Krámli, N. Simányi, and D. Szász, Commun. Math. Phys.125, 439 1989.11 H. De Raedt, Annu. Rev. Comput. Phys. 4, 107 1996.12 C. W. J. Beenakker, Phys. Rev. B 73, 201304R 2006.ARSENI GOUSSEV AND KLAUS RICHTER PHYSICAL REVIEW E 75, 015201R 2007RAPID COMMUNICATIONS015201-4

Loschmidt echo decay from local boundary perturbations

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