Exact Fidelity and Full Fidelity Statistics in Regular and Chaotic Surroundings
Heiner Kohler, Hans-Juergen Sommers, Sven Aberg, Thomas Guhr
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec Exact Fidelity and Full Fidelity Statistics in Regular and Chaotic Surroundings
Heiner Kohler (1) , Hans–J¨urgen Sommers (1) , Sven ˚Aberg (2) and Thomas Guhr (1) (1)
Fakult¨at f¨ur Physik, Universit¨at Duisburg-Essen,47057 Duisburg, Germany. (2)
Mathematical Physics, LTH,Lund University, P.O. Box 118, S-221 00 Lund, Sweden
For a prepared state exact expressions for the time dependent mean fidelity as well as for the meaninverse paricipation ratio are obtained analytically. The distribution function of fidelity in the longtime limit and of inverse participation ratio are studied numerically and analytically. Surprisingfeatures like fidelity revival and enhanced non–ergodicity are observed. The role of the couplingcoefficients and of complexity of background is studied as well.
Keywords: Fidelity, Loschmidt echo, survival probability, random matrix theory, Doorway states
In quantum information it is crucial to know how wella prepared state can be isolated, and how the unavoid-able mixing with the surrounding behaves. The fidelity(often called quantum Loschmidt echo or survival prob-ability) describes how the purity of a prepared state de-creases due to the interaction with surrounding states[1]. It serves as a benchmark test of prepared states,e.g. a set of qubits in quantum information [2]. Of in-terest is to describe the explicit time-dependence of thefidelity, and how the decay depends on the coupling tothe environment, as well as the role played by the com-plexity of background states. Time-scales, energy scalesand explicit shapes of these functions are thus frequentlystudied [3, 4]. Survival probability of a state weakly cou-pled to a background has been subject of considerableresearch in mesoscopics and in semiclassics [5, 6, 7, 8].F(ermis) G(olden) R(ule) predicts an exponential de-cay of the prepared state with decay rate Γ = 2 πλ D ,where D is the mean level spacing of the backgroundand λ is the average coupling strength. Deviations fromthis behavior become important when Γ ≃ D [5, 6]. Cor-rections to the FGR, which are similar to weak local-isation corrections in Quantum Transport can lead tonon–ergodicity, i. e. the prepared state will never decaycompletely[5, 6].The observed saturation of fidelity in the longtime limit allows us to connect it to the I(nverse)P(articipation) R(atio), which is a time independentquantity. Usually average behavior is considered. Butthe fidelity in an explicit situation can deviate much fromthe average behavior[9]. When constructing quantum in-formation and other devices, requiring high fidelity, oneis interested in a high probability to have states with fi-delity superior to some minimal value, above which errorcorrection is possible [10]. In this Letter we thereforestudy fluctuations of the fidelity and the full fidelity dis-tribution. We find that the latter in the long time limittends to a stable distribution. This distribution is differ-ent from the (time–independent) IPR distribution, bothcoincide in the weak coupling limit. In this limit we pro-vide an analytical solution. This allows us to study howthe distribution depends on coupling strength, coupling type as well as on dynamics of the background.In addition we provide exact solutions for the meanfidelity for all times and all coupling strength as well asfor the mean IPR. We report on nonperturbative featuressuch as a fidelity recovery and enhanced non–ergodicity.We model the coupling between the prepared statewith surrounding complexity by the Hamiltonian H λ = H s + H b + λV = E s | s ih s | + N X ν =1 E ν | b ν ih b ν | + λ N X ν =1 ( V ν | s ih b ν | + h.c. ) . (1)where H s represents a special, pure state, that is coupledto a background of complex states described by H b , andwhere the coupling, λV , is controlled by the sortless pa-rameter λ . Without loss of generality we may put theunperturbed energy of the special state to zero, E s = 0.The Schr¨odinger equations for the uncoupled Hamilto-nians are H s | s i = 0 and H b | b ν i = E ν | b ν i . (2)The eigenvalue problem for the coupled Hamiltonian is H λ | n i = E n | n i . (3)The N + 1 eigenfunctions are expressed in a basis of thespecial state and the background states | n i = c ns | s i + N X ν =1 c nν | b ν i , (4)where c ns = h n | s i and c nν = h n | b ν i . We model thecomplex surrounding of background states by randommatrix theory, and describe generic chaotic states withan ensemble of Gaussian random matrices that can ei-ther show time-reversal invariance (GOE, β = 1) or not(GUE, β = 2). Regularity of the background states ismodeled by assuming Poisson statistics for H b . In allcases the spectrum is unfolded so the mean level spacingequals one, D = 1, at least in a surrounding of the spe-cial state. This implies that the energy scale, includingthe coupling strength, is always expressed in units of themean level spacing, while the time scale is expressed inunits of the Heisenberg time, τ H = ~ /D . In the following ~ = 1.Matrix elements of the operator V between the spe-cial state and the complex surrounding, V ν = h s | V | b ν i ,are taken as Gaussian distributed random numbers withzero mean and variance one. As we will see, it is impor-tant to distinguish between real coupling ( V ν ∈ R ) andcomplex coupling ( V ν ∈ C ). The size N of the Hilbertspace describing the complex surrounding is in principleinfinite in RMT. In the numerical simulations it is takensufficiently large to achieve convergence.We first focus on fidelity decay. The fidelity ampli-tude f λ ( t ) = h Ψ ( t ) | Ψ λ ( t ) i is the overlap, between thepure state, time developed under the influence of thepure Hamiltonian, | Ψ ( t ) i = exp ( − i ( H s + H b ) t ) | Ψ (0) i ,and under the influence of the perturbed Hamiltonian, | Ψ λ ( t ) i = exp ( − iH λ t ) | Ψ (0) i . It is a measure of howthe pure initial state, | Ψ(0) i , gets disturbed or mixeddue to the (unavoidable) coupling to the complex sur-rounding a time t later. We are interested in the decayof the special state | s i , i. e. | Ψ(0) i = | s i . We expand itin a basis of eigenstates to H λ . The fidelity amplitudebecomes f λ ( t ) = X n | c ns | · exp ( iE n t ) . (5)It is seen that f λ ( t ) is the Fourier transform of the localdensity of states (LDOS), ρ ( E ) = X n | c ns | δ ( E − E n ) . (6)The smooth part of the LDOS follows a Breit-Wigner dis-tribution, with the width given by Γ = 2 πλ , as obtainedfrom FGR under very general assumptions. Thereforethe mean fidelity amplitude of the special initial state | s i will unavoidably decay exponentially f λ ( t ) = exp( − Γ t/ , (7)where the bar denotes average over background and overcoupling matrix elements.The fidelity (survival probability), of the special stateis defined as, F λ ( t ) = | f λ ( t ) | . In a Drude–type approxi-mation for the mean fidelity F λ = f λ = exp( − Γ t ) (8)FGR is recovered. This result is also obtained by secondorder perturbation theory. We write fidelity as the sum F λ ( t ) = IPR + F fluc ( t ) , (9)of a constant term and a term which is fluctuating on atimescale comparable to Heisenberg time. The constantterm IPR = X n | c ns | = D Z dEρ ( E ) , (10) is the inverse participation ratio of the special state inthe basis of the eigenvectors of the full Hamiltonian. Thefluctuating term F fluc ( t ) = 2 X n,m | c ns | | c ms | cos(( E n − E m ) t ) (11)vanishes, if we average fidelity over a time window,large compared with Heisenberg time. Using a Breit–Wigner distribution for ρ ( E ) and the Drude approxima-tion Eq. (8), IPR λ = D/ ( π Γ) is found [6], which canobviously not hold for small Γ.For complex coupling we were able to calculate F λ ( t )exactly for a regular and for a GOE/GUE background.For a regular background the result [11] is given by F λ ( t ) = 1+ λ √ π Z dx √ x e − xπ λ − x ) (cid:26) π √ − x (cid:18) e − t λ x cosh (cid:18) πλ t √ − x (cid:19) − (cid:19) − tx e − t λ x sinh (cid:18) πλ t √ − x (cid:19)(cid:27) . (12)For a GUE/GOE background the corresponding morecomplicated expressions can be found in [11]. For smalltimes this function decays exponentially according to theFGR law. Surprisingly, after some characteristic timefidelity reaches a minimum and increases afterwards to a λ dependent saturation value F λ ( ∞ ) = IPR λ . We findfor a regular backgroundIPR λ = 1 − √ π λ (cid:18) πλ (cid:19) , (13)with D( ω ) = exp( ω )erfc( ω ). IPR λ is a monotonouslydecreasing function. For small coupling the saturationvalue behaves as IPR λ ≃ − π / λ . For large values of λ it decays algebraically as ≃ / ( π λ ) which is four timesthe value, obtained by the Drude approximation, Eq. (8).This is a striking enhancement of non–ergodicity. Thecorresponding expresssion for a GUE background is F λ ( ∞ ) = 1 − π λ − √ π λ (cid:0) − π λ (cid:1) D ( πλ ) . (14)This function decays algebraically as ≃ / ( π λ ) for large λ which is twice the value, predicted by the Drude ap-proximation.In Fig.1 fidelity F λ ( t ) is plotted for λ = 0 . f i d e li t y , < F > λ = 0.1 Poisson C Poisson RGUE CGOE R GOE CGUE R
FIG. 1: (Color online): Evolution in time of fidelity for cou-pling strength λ = 0 .
1. The full lines describe top downfidelity decay for real coupling to a Poissonian (green), to aGOE (red) and to a GUE (black) background. The dashedlines describe top down fidelity decay for complex coupling toa Poissonian (green), to a GOE (red) and to a GUE (blue)background. for a regular background. This is in accordance with theoriginal perturbative arguments by Peres [1].Quite remarkably the decay of fidelity is much less sen-sitive to the complexity of the background than to thestructure of the coupling. There is practically no dif-ference between a time reversal invariant chaotic back-ground and a background with broken time–reversal in-variance. Nevertheless the difference between a real cou-pling of the special state to the background and a cou-pling which breaks time reversal symmetry is sizeable.For one reason, because the return probability from thebackground into the special state is suppressed by a cou-pling, which breaks time reversal invariance.We now turn to full fidelity statistics. We introducethe full distribution function P F ( c, t ) = δ ( c − F λ ( t )) . (15)The distribution of the fidelity P F ( c, t ) is calculated nu-merically and plotted for different times in Fig. 2. Asaturation of the distribution function is found for timeslarger than a ( λ –dependent) saturation time. In Fig. 2this saturation time is about 20 times Heisenberg time.The saturated distribution is shown in Fig. 3 for GOEstatistics and for coupling strength λ = 0 .
05. It can becompared to the IPR–distribution P IPR ( c ) = δ c − X n | c ns | ! . (16)We see that the two distributions are similar but dif-ferent. The reason lies in the fluctuating term F fluc (see Eq. (9)). Although the ensemble average of F fluc vanishes, its variance does not. Ultimately F fluc ( t ) con-tributes substantially to the full fidelity distribution. As D i s t r i bu t i on λ = 0.1 t=1t=5 t=20t=2.5t=10 FIG. 2: (Color online): Fidelity distributions at times t = 1(pink), t = 2 . t = 5 (red), t = 10 (green) and t = 20(black) for coupling strength λ = 0 .
1. For times t >
20 thedistribution is stable. a result the stable fidelity distribution and IPR distribu-tion are different.We are interested mainly in cases of high fidelity, thatis, when the coupling strength, λ , is small. In this limit P F ( c, ∞ ) should be better and better approximated by P IPR ( c ). In this limit an analytic result for P IPR ( c ) canbe obtained. By solving the Schr¨odinger equations (2)and (3) an expression for the component of the specialstate in the eigenstates | n i of H λ is obtained as | c ns | = λ N X µ =1 | V ν | ( E n − E µ ) ! − . (17)This exact expression for the components contains eigen-value solutions, E n , as well as input matrix elements V ν and energies E ν . For small λ we may approximate theexact eigenvalues E n by their unperturbed value E ν ( n ) .If we denote by | i the eigenstate of H λ which has evolvedfrom the special state | s i , we see that in this approxima-tion all | c ns | but | c s | vanish. This means that for smallcouplings the IPR will be dominated by only one term | c s | . We define the distribution P ( c ) = δ ( c − | c s | ) . (18)Then for small λ , P F ( c, ∞ ) ≃ P IPR and likewise P IPR ≃ P . We could perform the ensemble average of P exactly. For detail of the calculation see [12]. For aregular surrounding (Poisson statistics) we find: P ( c ) = 14 √ c λa β (1 − √ c ) / e − ( λa β ) π √ c −√ c . (19)where a = p /π for real coupling and a = p π/ d i s t r i bu t i on GOE R fidelityIPR c λ = 0.05 FIG. 3: (Color online): Comparison of the distributions P obtained from Eq. (20) (dashed red line) with P IPR (full greenline) and P F ( c, ∞ ) (full black line) obtained from Monte–Carlo simulations for a coupling constant λ = 0 . time-reversal symmetry (GOE): P ( c ) = 14 √ c s π λ c − √ c ) e − X ( K ( X ) + K ( X ))(20)where K n are modified Bessel functions of second kind.For a chaotic background without time reversal symme-try (GUE) we get: P ( c ) = 14 √ c s πλ (1 − √ c ) e − X (1 + 2 X ) , (21)with X β = β π λ √ c −√ c ) . In Fig. 3 we compare the ana-lytical expressions for the distribution P to numericalsimulations of P IPR and of P F ( c, ∞ ) for λ = 0 .
05 in thecase of a GOE background. We observe good agreementof the analytically calculated P ( c ) with the distribution P IPR ( c ) obtained by simulations. Both curves are in-distinguishable at least in the range c ≥ .
6. There is asmall but notable difference to the distribution P F ( c, ∞ ),which vanishes if we go to smaller values of λ .In conclusion, we studied fidelity decay for a specialstate coupled to a regular or chaotic environment. Wefound a saturation in the long time limit and a revival.In [13] a fidelity freeze was predicted. For a purelyoff–diagonal perturbation, after an initial decay fidelityfreezes on a plateau for some time and decays afterwardsto zero. In the model considered here, the perturbation V is purely off–diagonal as well. The saturation, we foundhere might thus be considered an extreme case of fidelityfreeze. The fidelity revival found here is genuinely dif-ferent to the one reported earlier [14]. There, a satisfac-tory explanation was given by the spectral rigidity of theGUE/GOE[14, 15]. The fact that the revival occurs fora regular background as well encumbers such an expla-nation in the present case. The saturation of fidelity is a direct consequence of thefact that the fidelity distribution relaxes in the long timelimit into a stable distribution. In the small couplinglimit it becomes the distribution of the IPR. In this limitwe found an analytic expression. Both distributions havea rich structure and are highly sensitive to even smallchanges in the coupling strength. Their relation to max-imum strength distribution, introduced recently [16, 17]will be discussed elsewhere [12]. Their calculation forarbitrary coupling strength is a challenge for the future.Our results yield an important benchmark for the de-cay of a prepared quantum state. It might be probednumerically and experimentally in chaotic quantum sys-tems [18, 19, 20] or on quantum information devices. Oneinstance for a possible numerical experiment is the decayrate of a regular state in a mushroom billiard due todynamical tunnelling into the chaotic part of the phasespace.We thank B. Gutkin, R. Oberhage, P. Mello and T. H.Seligman for useful discussions. We acknowledge supportfrom Deutsche Forschungsgemeinschaft by the grantsKO3538/1-2 (HK), Sonderforschungsbereich Transregio12 (TG, HK, HJS). [1] A. Peres, Phys. Rev. A , 1610 (1984).[2] M. A. Nielsen and I. L. Chuang, Quantum Computationand Quantum Information (Cambridge University Press,Cambridge, 2000).[3] T. Gorin, T. Prosen, T. H. Seligman, and M. Znidaric,Phys. Rep. , 33 (2006).[4] P. Jaquod and C. Petitjean, Adv. Phys. , 67 (2009).[5] V. N. Prigodin, B. L. Altshuler, K. B. Efetov, and S. Iida,Phys. Rev. Lett. , 546 (1994).[6] J. L. Gruveret al., Phys. Rev. E , 6370 (1997).[7] J. Vanicek and D. Cohen, J. Phys. A , 9591 (2003).[8] M. Guti´errez, D. Waltner, J. Kuipers, and K. Richter,Phys. Rev. E , 046212 (2009).[9] T. Gorin, T. Prosen, and T. H. Seligman, New Jour.Phys. , 20 (2004).[10] A. Preskill, Proc. Roy. Soc. Lond. , 385 (1998).[11] H. Kohler, H. J. Sommers, and S. ˚Aberg, to be submitted.[12] H. Kohler, T. Guhr, and S. ˚Aberg, to be submitted.[13] T. Prosen and M. Znidaric, Phys. Rev. Lett. , 044101(2005).[14] H. J. St¨ockmann and R. Sch¨afer, New J. Phys. , 199(2004); H. J. St¨ockmann and R. Sch¨afer, Phys. Rev. Lett. , 244101 (2005).[15] H. Kohler et al., Phys. Rev. Lett. , 190404 (2008).[16] E. Bogomolny et al., Phys. Rev. Lett. , 254102 (2006).[17] S. ˚Aberg, T. Guhr, M. Miski-Oglu, and A. Richter, Phys.Rev. Lett. , 204101 (2008).[18] J. Feist et al., Phys. Rev. Lett. , 116804 (2006).[19] A. B¨acker, R. Ketzmerick, S. L¨ock, and L. Schilling,Phys. Rev. Lett. , 104101 (2008).[20] A. B¨acker et al. , Phys. Rev. Lett.100