Survival Probability of a Doorway State in regular and chaotic environments
aa r X i v : . [ qu a n t - ph ] D ec Survival Probability of a Doorway State in regularand chaotic environments
Heiner Kohler (1) , Hans–J¨urgen Sommers (1) and Sven ˚Aberg (2) (1)
Fakult¨at f¨ur Physik, Universit¨at Duisburg–Essen, Duisburg, Germany (2)
Matematisk Fysik, Lunds Universitet, Lund, SwedenE-mail: [email protected]
Abstract.
We calculate survival probability of a special state which couplesrandomly to a regular or chaotic environment. The environment is modelled by asuitably chosen random matrix ensemble. The exact results exhibit non–perturbativefeatures as revival of probability and non–ergodicity. The role of backgroundcomplexity and of coupling complexity is discussed as well. urvival Probability of a Doorway State in regular and chaotic environments
1. Introduction
The stability of a special prepared quantum state weakly coupled to a continuum isa subject of considerable interest in quantum information theory [1], nuclear physics,mesoscopic, quantum chaos (see [2, 3] and references therein). Giant resonances arecollective excitations of the nucleons, which are approximate but not exact eigenstatesof the complicated many–body Hamitonian. They are an example of a special statecoupled weakly to a continuum picked from nuclear physics. Constructed superscarstates in chaotic quantum billiards, considered recently [4] fall into the same class. Thespecial state might also be implemented mesoscopically, for instance by an electronicstate on a small conducting island (quantum dot), which is weakly coupled to one (orin a non–equilibrium situation to two) continua of the leads [5]. Given a particle in aquantum state | ψ (0) i at time t = 0 the likelyhood to find the particle after some time t in the same state is measured by the survival probability P ( t ) = |h ψ ( t ) | ψ (0) i| . (1)The simplest approximation of P ( t ) is found perturbatively by F(ermi’s) G(olden) R(ule) P ( t ) = e − Γ t , where Γ is the inverse decay time, related to the mean coupling strengthof the special state to the background. Corrections to this simple exponential decaybecome important, when Γ has the same order of magnitude as the mean level spacingof the background Hamiltonian.In Ref. [6] corrections to the FGR–law, were calculated in a mesoscopic system,where the special state is sitting on a quantum dot which is weakly coupled to a reservoir.Due to these corrections the decay of the electronic state in the dot is never complete.Instead the system preserves a memory of the original state. The corrections to FGR,which are in nature very similar to the weak localisation correction to classical transport,give rise to non–ergodic behavior. In the same spirit, in Ref. [7] weak localisationcorrections to the FGR behavior were calculated using new semiclassical techniques,which became available recently [8, 9].Corrections to the FGR–law can be addressed in a more generic setting within arandom matrix model. In this model a special state | ψ (0) i = | s i couples to a largereservoir of states via randomly chosen coupling parameters V µ , µ = 1 , . . . N . Thecoupling parameters are chosen either real or complex. The dynamics of the reservoiris either chaotic, modelled by a random matrix chosen from a Gaussian RMT ensembleor regular, with Poissonian eigenvalue statistics. This RMT model has been addressedin [10, 11] and corrections to FGR regime were found.Expanding the special state in eigenstates of the Hamiltonian survival probabilitycan be written as P ( t ) = X n,m |h s | m i| |h s | n i| e i ( E n − E m ) t . (2) urvival Probability of a Doorway State in regular and chaotic environments h s | m i the ensemble average(denoted by a bar) can be performed for both sums separately. As a result P ( t ) = | p ( t ) | , (3)where p ( t ) is the Fourier transform of the local density of states LDOS ρ ( E ) = X n |h s | n i| δ ( E − E n + E s ) (4)around the energy E s of the special state. As was pointed out already by Weisskopf andWigner [12] the smooth part of ρ ( E ) is under very general assumptions of Lorentzianshape ρ ( E ) = 1 π Γ / E − E s ) + (Γ / , (5)from which FGR is recovered. We call the approximation implied in Eq. (3) D(rude)B(oltzmann)–approximation , due to its formal similarity to the approximations made inthe derivation of the Drude Boltzmann law of conductivity [5].Assuming a constant mean level spacing D in an energy region around the specialstate, from Eq. (2) one obtains P ( ∞ ) = IPR, whereIPR = X m |h s | m i| = D Z dEρ ( E ) (6)is the inverse participation ratio of the special state in the basis of the eigenstates ofthe full Hamiltonian. In the DB–approximation ρ ≃ ρ and the mean IPR can beestimated by IPR ≃ D/ ( π Γ), which shows that survival probability will not decay tozero, if Γ and D are of the same order of magnitude.Following Ref. [10] one can obtain corrections to the DB approximation due toenergy correlations as follows: Writing the expansion coefficients |h s | n i| = ρ ( E n )+ δρ ( E n ) as the sum of a smooth function of E n and of a fluctuating part, the averagedsurvival probability can be written as the sum of two contributions as well P ( t ) = Z dEdE ′ ρ ( E ) ρ ( E ′ ) e i ( E − E ′ ) t R (( E − E ′ ) /D ) + δP ( t ) , (7)where R = P n,m δ ( E − E n ) δ ( E ′ − E m ) is the averaged energy–energy correlator of thebackground Hamiltonian. The approximation made in Ref. [10] consists in neglectingthe fluctuating part δP ( t ). Using standard results of random matrix theory [13] R isthe sum of a δ –like, a connected and of an unconnected contribution. Likewise, survivalprobability is written as a sum of three contributions P ( t ) ≃ e − Γ t + Dπ Γ − ∞ Z −∞ dt ′ e − Γ | t − πD t ′ | b ( t ′ ) (8)where b is the two–level form factor. The last term in Eq. (8) accounts for the energycorrelations of the bath. The result (8) gives an intuitive insight how energy correlationsof the background Hamiltonian give rise to corrections of the FGR–law and as we will urvival Probability of a Doorway State in regular and chaotic environments ≫ D and for a chaotic background it predicts qualitativelythe correct behavior.However, it is easy to see that for small Γ or in the case where correlations areabsent Eq. (8) is not correct even qualitatively (for instance the saturation value P ( ∞ )= D/ ( π Γ) exceeds one for Γ < D/π ).Thus the question, whether Eq. (8) describes sufficiently the weak localisationcorrections to the DB–approximation has to be answered negatively. In the presentwork we therefore calculate P ( t ) for the random matrix model mentioned above exactlyfor a chaotic as well as for a regular background. We will see that the exact resultdiffers qualitatively from the predictions of Eq. (8). For instance, we find that a revivalof survival probability, predicted by Eq. (8) only for fairly strong couplings and for achaotic background, occurs for weak coupling and for a regular background as well.More general, the energy statistics of the background turn out to have little influenceon the survival probability. Instead, we find that the nature of the coupling coefficientis crucial. As a rule of thumb, for constant mean coupling strength survival probabilityis always lower for complex coupling coefficients than for real ones.We provide exact analytic expressions for the average IPR, which interpolate thepower law decay for strong couplings to the small coupling regime.On the technical level, we use powerful results for averages over characteristicpolynomials, which have become available recently [14]. This allows us to circumventa long and complicated supersymmetric calculation. This elegant shortcut is possiblefor complex couplings of the doorway state to the background, where we derive exactanalytic results. For real coupling coefficients we resort to numerics.The paper is organized as follows: In Sec. 2 and in Sec. 3 we define the randommatrix model and fix the notation. In Sec. 4 we outline how the Lorentz shape of theLDOS comes about in the present random matrix model. The calculation of survivalprobability for regular background as well as for a GUE and for a GOE background ispresented in Sec. 5.2. Finally the results are discussed and summarized in Sec. 6.
2. Definition of the Doorway model
The model to be discussed here stems from nuclear physics [15] and is also often usedin other fields [16]. For the convenience of the reader and to define our notation, wecompile its salient features.The total Hamiltonian H consists of three parts, the Hamiltonian H s for thedoorway states, the Hamiltonian H b describing the N background states, where N willeventually be taken to infinity, and the interaction V coupling the two classes of states.Often, there is only one relevant doorway state or the spacing between the doorwaystates is much larger than their spreading widths. In the present work we focus on thissituation, leaving the interesting case of many doorway states to future work. Hence,we have H = H s + H b + V urvival Probability of a Doorway State in regular and chaotic environments H = H + V = E s | s ih s | + N X ν =1 E ν | b ν ih b ν | + N X ν =1 (cid:16) V ν | s ih b ν | + h . c . (cid:17) . (9)For the matrix elements of the interaction, we make the assumptions h b ν | V | b µ i = 0 and h b ν | V | s i = V ν for any µ , ν .Resembling the situation in most systems, we put the doorway state | s i in the centerof the background spectrum. It interacts with the N surrounding states. Without lossof generality, we may set E s = 0. The eigenequations for the uncoupled Hamiltonian H are then H s | s i = 0 and H b | b ν i = E ν | b ν i . (10)We assume that the interaction matrix elements are Gaussian distributed randomvariables. We distinguish the two cases of complex ( β = 2) or real ( β = 1) matrixelements V ν . Introducing the N –component vector V , the corresponding distribution is P i ( V ) = (cid:18) β πv (cid:19) βN/ exp (cid:18) − β v V † V (cid:19) . (11)As discussed in the introduction, we are interested in the situation where the meancoupling strength is of the same order of magnitude as the mean level spacing. Wedefine the dimensionless parameter λ = p h V † V i√ N D = vD , (12)where D is the mean level spacing of the background states in the center of theband [15, 16]. The distribution P i ( V ) is chosen such that λ is independent of β .We distinguish two cases for the background dynamics. Regular dynamics of thebackground Hamiltonian H b is modelled by uncorrelated eigenvalues P b ( H b ) = N Y ν =1 p b ( E ν ) . (13)Chaotic dynamics is modelled by Gaussian random matrix ensembles given by thedistribution function P b ( H b ) = (cid:18) β b π (cid:19) N (cid:18) β b π (cid:19) βbN ( N − exp (cid:18) − β b H b (cid:19) , (14)where H b is either real symmetric ( β b = 1) modelling time reversal invariant backgrounddynamics, or Hermitean ( β b = 2), modelling background dynamics with broken timereversal invariance. The probability distribution (14) yields a mean level spacing D = p π / N in the band center, which is independent of β b . We denote the averageover both the interaction matrix elements and the background Hamiltonian by a bar( . . . ) = Z d [ H b ] P b ( H b ) Z d [ V ] P i ( V )( . . . ) . (15) urvival Probability of a Doorway State in regular and chaotic environments
3. Fidelity and Survival probability of the Doorway state
We define the echo operator M λ = e iHt e − iH t . (16)Fidelity amplitude f λ ( t ) is defined as the expectation value of the echo operator withrespect to a given initial state. Here we are interested in the Doorway state | s i as initialstate, i.e. an eigenstate of the unperturbed system. Since E s | s i = 0 the average fidelityamplitude can be written as f λ ( t ) = h s | M λ ( t ) | s i = h s | e − iHt | s i . (17)As mentioned in the introduction, fidelity amplitude is then the Fourier transform ofthe local density of states ρ ( E ). Likewise, fidelity (often called Loschmidt echo ) F λ ( t ),defined as the modulus square of the fidelity amplitude, becomes identical with thesurvival probability P ( t ) F λ ( t ) = h s | e − iHt | s i h s | e iHt | s i ≡ P ( t ) . (18)In the following we mainly stick with the notion of survival probability, keeping in mindthat in the present situation fidelity and survival probability are synonyms.When we expand fidelity in eigenstates of the full Hamiltonian according to Eq. (2),we see that in the limit of infinite large times the fidelity approaches the inverseparticipation ratio IPR of the special state in the basis of the eigenstates of the system.We therefore also define the mean inverse participation ratioIPR λ ≡ X m |h s | m i| = F λ ( ∞ ) , (19)where the sum goes over exact eigenstates of the full Hamiltonian H . The task isto calculate f λ ( t ) and F λ ( t ) exactly in the large N limit for various choices of thebackground Hamiltonian and in particular the corrections to the DB–approximation.
4. Calculation of the mean local density of states
We first prove that the average local density of states ρ ( E ) in the present case indeedhas the Lorentz shape. We write ρ ( E ) = X m |h s | m i| δ ( E − E m )= 1 π Im (cid:28) s (cid:12)(cid:12)(cid:12)(cid:12) H − E − iǫ (cid:12)(cid:12)(cid:12)(cid:12) s (cid:29) = 1 π Im det( H − E )det( H − E − iǫ ) , (20)where we used Kramer’s rule in the step from the second to the third equation. For achaotic background the average can be taken most conveniently by a mapping onto a urvival Probability of a Doorway State in regular and chaotic environments ρ ( E ) = 1 π Im Z d [ σ ] exp (cid:18) − β b σ + E ) (cid:19) Sdet − βbN ( σ )det − β b / (cid:18) V † Vσ BB + E (cid:19) , (21)where the bar now denotes an average over the coupling coefficients only. In Eq. (21) σ is a 2 × β b = 2) respectively 4 × β b = 1) supermatrix of the form a λ ∗ λ ia ! , GUE a a λ ∗ − λ a a λ ∗ − λ λ λ ia λ ∗ λ ∗ ia , GOE . (22)The matrix entries in Latin letters denote real commuting integration variables. Thematrix entries in Greek letters denote complex anticommuting integration variables. Theinfinitesimal volume elements d [ σ ] are products of the differentials of all independentintegration variables. The integration domain of the real commuting variables is thereal axis. The so–called Boson–Boson block σ BB is the upper left block of commutingvariables. The matrix integral Eq. (21) can be solved in one step with a saddlepointapproximation. For energies close to the center of the band the saddle points are σ ≃ ± i p N/ ρ ( E ) = 1 π Im (cid:18) E − i Dπ V † V (cid:19) − . (23)This result shows that LDOS has the form of a δ –spike unless the perturbation isclassically small, i. e. of the order of the mean level spacing. The Gaussian average overthe coupling coefficients finally yields the Lorentz distribution ρ ( E ) = 1 π Γ / E + (Γ / (24)with the crucial relationΓ = 2 πλ D (25)between spreading width, mean perturbation strength and mean level spacing. For aregular environment the LDOS was calculated for instance in [18] yielding the sameresult.
5. Calculation of mean Fidelity/Survival Probability
We now turn to the main task: the calculation of survival probability of the doorwaystate. In order to calculate the mean fidelity/survival probability, we write F λ ( t ) as F λ ( t ) = 1 π Z dE Z dE exp ( i ( E − E ) t ) ρ ( E + iǫ ) ρ ( E − iǫ ) urvival Probability of a Doorway State in regular and chaotic environments
8= 1 π Z dE Z dE exp ( i ( E − E ) t )det( H − E )det( H − E − iǫ ) det( H − E )det( H − E + iǫ ) , (26)where we used again Kramer’s rule. Evaluation of the determinant in the denominatoryields F λ ( t ) = Z dE Z dE exp ( i ( E − E ) t ) δ E + X µ | V µ | E µ − E ! δ E + X µ | V µ | E µ − E ! . (27)We observe that F is normalized by F λ (0) = 1 (see [10, 19]). This allows us to extractat this point of the calculation the constant term F λ (0) from the integral and to averageover F λ ( t ) − F ( t ) directly. After a Fourier transformation of the deltadistributions we find for the average F λ ( t ) = 1 + Z dE Z dE Z dk π Z dk π [exp ( it ( E − E )) − (cid:18) − i k − k E − E ) − i k + k E + E ) (cid:19) exp − i X µ | V µ | k E µ − E − i X µ | V µ | k E µ − E ! . (28)Since on the unfolded scale the mean level spacing of the background is constant, wecan assume that the unfolded average does not depend on E + E . This allows us toperform the integral over the mean energy ( E + E ) / k + k ). Wefind F λ ( τ ) = 1 + Z ds Z dk π [exp ( iτ s )) −
1] exp ( − iks ) R ( k, s ) , (29)where R ( k, s ) = exp − i X µ | V µ | ksE µ − ( Ds/ ! . (30)We introduced the dimensionless time τ = Dt , measured in units of Heisenberg time τ H = D − . It is useful to take the average over the Gaussian distributed couplingcoefficients at this stage of the calculation R ( k, s ) = Z N Y µ =1 (cid:26) d [ V µ ] β (2 πv /β ) β/ exp (cid:20) − β v (cid:18) iksD λ β ( E µ − ( Ds/ ) (cid:19) | V µ | (cid:21)) = (cid:18) det( H b − ( Ds/ )det( H b − ( Ds/ + iksD λ ) (cid:19) β/ , (31) urvival Probability of a Doorway State in regular and chaotic environments H b only, whichhas still to be performed. In the expression (31) it becomes evident why the case ofreal coupling coefficients V µ ∈ R , β = 1 is analytically more difficult than the case ofcomplex coupling β = 2. For β = 2 the expression (31) is an average over a rational ratioof products of characteristic polynomials. Much information has been gathered aboutthese averages in the last decades [20, 21, 22, 23, 24, 25, 14, 26], whereas little is knownabout averages over irrational functions of characteristic polynomials as encountered inthe case β = 1.In the subsequent analysis we therefore restrict ourselves to complex coupling andset β = 2 from now on. In the case of real coupling we recur to numerics. We distinguishthe cases of a regular background R Poisson ( k, s ), and GOE or GUE distributed chaoticbackground R GOE ( k, s ), R GUE ( k, s ). We are able to calculate all three averages exactlyin the large N limit. We first consider a regular, Poisson distributed, background. For Poisson distributedeigenvalues the average in Eq. (31) becomes a product R Poisson ( k, s ) = r ( k, s ) N (32) r ( k, s ) = D Z dxp b ( Dx ) (cid:18) x − ( s/ x − ( s/ + 2 iksλ /β (cid:19) β/ . (33)The universal final result should be independent of the distribution of the eigenvaluesof the background Hamiltonian. The simplest choice for this distribution is p b ( E ) = 1 √ N ( , | x | ≤ √ N / , | x | > √ N / , (34)where D = 1 / √ N and √ N = N D is the length of the background spectrum.For β = 2 the integral (33) can be evaluated in the large N limit by the residuetheorem r ( k, s ) = 1 − πλ | k || s | N √ s − iksλ + O (cid:18) N (cid:19) , (35) R Poisson ( k, s ) = exp (cid:18) − πλ | k || s |√ s − iksλ (cid:19) . (36)Using this result together with Eq. (32) and Eq. (29) we find F λ ( τ ) − ∞ Z dsπ [cos( τ s ) − ∞ Z dk exp (cid:18) − iks − πλ ks √ s − iksλ (cid:19) , (37)which is almost our final result. However the remaining double integral is numericallydifficult due to the oscillatory terms. We proceed by rotating the contour of k integration urvival Probability of a Doorway State in regular and chaotic environments x = ik/s onearrives at F λ ( τ ) = 1 + 1 π ∞ Z −∞ dss [cos( τ s ) − / λ Z dx exp (cid:0) − xs (cid:1) sin (cid:18) πλ xs √ − xλ (cid:19) . (38)The s integration can now be performed without difficulties. The final result is F λ ( τ ) = 1 + λ √ π Z dx √ x e − xπ λ − x ) (cid:26) π √ − x (cid:18) e − τ λ x cosh (cid:18) πλ τ − x (cid:19) − (cid:19) − τx e − τ λ x sinh (cid:18) πλ τ − x (cid:19)(cid:27) . (39)The remaining integral does not seem to have a simple analytic solution. As expectedIPR λ is not zero but saturates at a finite value, given byIPR λ = 1 − √ π λ (cid:18) πλ (cid:19) erfc (cid:18) πλ (cid:19) . (40)This function behaves for small/large values of λ as followsIPR λ ≃ − π / λ for λ ≪ π λ for λ ≫ . (41)In Figure 1 survival probability is plotted on a logarithmic and on a linear scale asa function of time in units of Heisenberg time for three different values of the meancoupling strength λ = 0 .
1, 0 . /D ≈ . , . .
3. It is seen that the survival probability reaches a minimum and increasesafterwards to its saturation value given in Eq. (40). The time evolution of survivalprobability splits into three regimes: for t ≪ τ H / Γ fidelity follows the FGR law, for t ≫ τ H / Γ survival probability has approached its saturation value and is approximatelyconstant, in the region t ≃ τ H / Γ the time evolution is a complicated smooth function,which interpolates between the two limiting regimes.In Figure 1 also the curves obtained from Eq. (8) are plotted. It is seen that for aPoisson distributed spectrum of the background, Eq. (8) is a rather poor approximationof the exact curve. In particular it predicts no revival of survival probability and thesaturation value is underestimated by a factor four for large coupling strength λ . Using the formulae of Theorem 1.3.2 of Ref. [14] ‡ we find for R ( k, s ) for a chaoticbackground Hamiltonian with broken time reversal invariance R GUE ( k, s ) = exp (cid:16) − iπ sgn ( ks ) √ s − iksλ (cid:17) ‡ For the convenience of the reader we provide Theorem 1.3.2 of Ref. [14] in the form needed here inAppendix A. urvival Probability of a Doorway State in regular and chaotic environments Τ - - - F Λ Τ F Λ Figure 1.
Plot of Eq. (39) for the values λ = 0 . λ = 0 . λ = 1 (green thick line) on a logarithmic scale (right) and on a linear scale(left). The curves obtained by Fermi’s golden rule are depicted by thinner dotted linesin all three cases. For λ = 1 and for λ = 0 . (cid:18) cos( πs ) + i sgn ( ks ) sin( πs ) s − ikλ √ s − iksλ (cid:19) . (42)We use that R ( − k, s ) = R ∗ ( k, s ) and rotate the contour of the k –integral as in thePoisson case to the negative ( s >
0) or positive ( s <
0) imaginary axis. We find for theaveraged survival probability F λ ( τ ) = 1 + 1 π ∞ Z −∞ dss [cos( τ s ) − / λ Z dx exp (cid:0) − xs (cid:1)(cid:26) sin (cid:16) πs √ − xλ (cid:17) cos( πs ) − cos (cid:16) πs √ − xλ (cid:17) sin( πs ) 1 − λ x √ − xλ (cid:27) . (43)The s integration can be performed in a tedious but straightforward way. The finalresult is again an integral expression F λ ( τ ) = 1 + λ √ π Z dx √ x √ − x exp (cid:18) − π λ W x (cid:19) (cid:16) x − W − (cid:17)(cid:26) πW + (cid:20) − exp (cid:18) − λ τ x (cid:19) cosh (cid:18) πλ τ W + x (cid:19)(cid:21) + τ exp (cid:18) − λ τ x (cid:19) sinh (cid:18) πλ τ W + x (cid:19)(cid:27) − λ √ π Z dx √ x √ − x exp (cid:18) − π λ W − x (cid:19) (cid:16) W + − x (cid:17)(cid:26) πW − (cid:20) − exp (cid:18) − λ τ x (cid:19) cosh (cid:18) πλ τ W − x (cid:19)(cid:21) + τ exp (cid:18) − λ τ x (cid:19) sinh (cid:18) πλ τ W − x (cid:19)(cid:27) , (44) urvival Probability of a Doorway State in regular and chaotic environments Τ - - - - F Λ Τ F Λ Figure 2.
Plot of Eq. (44) for the values λ = 0 . λ = 0 . λ = 1 (green thick line). The curves obtained by Fermi’s golden rule aredepicted by thinner dashed lines in all three cases. For λ = 1 and for λ = 0 . where W ± = 1 ± √ − x . (45)In Figure 2 survival probability is plotted on a logarithmic and on a linear scale asa function of time in units of Heisenberg time for three different values of the meancoupling strength λ = 0 .
1, 0 . λ correspond to a spreading widthΓ /D ≈ . , . .
3. It is seen that qualitatively the curves are quite similar to theones obtained for a regular environment. For a GUE background energy correlations arepresent and a revival of survival probability is predicted by the estimation (8). Howeverwe notice that a revival occurs also for small values of coupling strength like λ = 0 . τ → ∞ IPR λ = 1 − λ √ π Z dx √ x √ − x exp (cid:18) − π λ (2 − x ) x (cid:19)(cid:20) cosh (cid:18) π λ √ − xx (cid:19) + √ − x sinh (cid:18) π λ √ − xx (cid:19)(cid:21) . (46)The remaining integral can be simplified and expressed in terms of complementary errorfunctions akin to Eq.(40)IPR λ = 1 − π λ √ π ∞ Z e − u du ( u + π λ ) = 1 − π λ − π λ √ π (cid:0) − π λ (cid:1) exp (cid:0) π λ (cid:1) erfc( πλ ) . (47)In the limits of small and large λ we obtainIPR λ ≃ − π / λ for λ ≪ π λ for λ ≫ . (48) urvival Probability of a Doorway State in regular and chaotic environments λ is half the value obtained for a regular environment buttwice the value predicted by the Boltzmann–Drude approximation. Using Theorem 1.3.1 of Ref.[14], for details see Appendix A, an expression for R GOE ( k, s )can be derived. We find that R GOE ( k, s ) = R GUE ( k, s ) + R add ( k, s ) R add ( k, s ) = − i ks ) k sλ √ s − iksλ (cid:18) dds sin( πs ) s (cid:19) ×× ∞ Z exp (cid:16) − iπ sgn ( ks ) √ s − iksλ t (cid:17) dtt . (49)In the same fashion survival probability splits into two parts F λ, GOE = F λ, GUE + F λ, add F λ, add = Z ds Z dk π [exp ( iτ s )) −
1] exp ( − iks ) R add ( k, s ) . (50)The additional contribution to the survival probability F λ, add plays the role of aCooperon contribution. In a tedious but straightforward calculation we find for F λ, add an expression as a double integral F λ, add = Z dx ∞ Z dt √ πxλ t √ − x ( H ( τ ) + H ( − τ ) − H (0)) H ( τ ) = e − λ x ( π + W ( τ ) ) (cid:20)(cid:18) λ πx + 1 π (cid:19) W ( τ ) sinh (cid:18) π λ W ( τ ) x (cid:19) − λ x (cid:0) π + W ( τ ) (cid:1) cosh (cid:18) π λ W ( τ ) x (cid:19)(cid:21) W ( τ ) = τ + πt √ − x , (51)which can be evaluated numerically without problems. Likewise the average of the IPRobtains an additional contributionIPR λ, GOE = IPR λ, GUE + IPR λ, add IPR λ, add = − Z dx ∞ Z dt √ πxλH (0)4 t √ − x . (52)On the left hand side of Fig. 3 survival probability as given by Eq. (50) is plotted fordifferent coupling strength λ = 0 .
1, 0 . . F λ, add separately. It is plotted on the right hand side of urvival Probability of a Doorway State in regular and chaotic environments Τ F Λ Τ F add Figure 3.
Left: Plot of Eq. (50) and Eq. (51) for the values λ = 0 . λ = 0 . λ = 0 . λ with dotted lines. Right: Plot of theadditional “Cooperon” contribution to survival probability according to Eq. (51) forthe three values λ = 0 . λ = 0 . λ = 0 . Fig. 3 for the same values as before. We see that the contribution is small compared to F λ, GUE . Surprisingly, we see that it is oscillating a few times with a frequency ∝ /λ before reaching its saturation value IPR λ, add which is a non–monotonous function of λ .This contribution to the total mean IPR is depicted in the inset of Fig. 4 as a functionof the coupling strength λ . We were able to calculate time evolution of survival probability for a complex couplingof the prepared state to a Poisson, GUE or GOE environment. As explained before asimilar calculation is by now not possible for real coupling coefficents. In the latter casewe resort to numerics.In Fig. 4 time evolution of survival probability is plotted for regular and GUEbackground dynamics for real and for complex coupling coefficients (the differencebetween a GOE background and a GUE background is almost invisible on the scaleused for the plots). We see that increased background complexity reduces overallsurvival probability. This is in agreement with standard perturbative arguments [2, 3].The difference between real and complex coupling coefficients is of the same order ofmagnitude as the difference between regular and chaotic background dynamics. This issurprising inasmuch time reversal symmetry breaking in the background has practicallyno influence on survival probability.In Fig.5 the average inverse participation ratio is plotted for a complex coupling toa Poissonian background (as given by Eq. (40)) and to a GUE background (as given byEq. (47) as a function of coupling strength λ . We see that for λ & λ, add is plotted. Although negligible urvival Probability of a Doorway State in regular and chaotic environments Τ F Λ Τ F Λ Figure 4.
Left: Comparison of survival probability for real coupling to a Poissonianbackground (full red line), to a GUE background (full black line), for complex couplingto a Poissonian background (dashed red line) and to a GUE background (dashed blackline). The coupling strength is λ = 0 .
5. The inset shows the same quantities forcoupling strength λ = 0 . Λ - - - - - - IPR Λ IPR Λ ,add Figure 5.
Left: Plot of the average inverse participation ratio IPR λ as a function ofthe dimensionless mean coupling strength λ for complex interaction with a regularbackground (green curve), Eq. (40), and for complex interaction with a GUEbackground (red curve), Eq. (47). The blue dashed lines show the asymptotic behaviorfor large λ . The inset shows the additional contribution for a GOE backgroundIPR λ, add as a function of λ . for practical purposes, it is interesting to see that this contribution is a non–monotonousfunction of λ . It vanishes for λ = 0 and for λ = ∞ . It obtains its maximum for λ ≈ . urvival Probability of a Doorway State in regular and chaotic environments
6. Discussion and Summary
We calculated exactly survival probability and fidelity amplitude for a special state,which is weakly coupled to a random matrix environment. Whereas fidelity amplitudedecays exponentially according to Fermi’s golden rule, survival probability shows a richbehavior. We found a revival of survival probability after a characteristic time whichincreases with decreasing coupling strength and a saturation of survival probability at avalue given by the mean IPR of the special state in the basis of the interacting system.Our exact results largely improve existing estimates [10] of these quantities.We were able to derive analytically the full λ –dependence of the IPR in the smallcoupling regime, where the approximation IPR ∝ λ − becomes bad. It turned outthat even in the strong coupling limit IPR is largely underestimated by the Drude–Boltzmann approximation (by a factor four for a regular environment and by a factortwo for a chaotic environment).Revival of survival probability was found for all types of background complexity.The fact that it occurs also for a regular background encumbers an explanation of therevival by spectral correlations of the background energy–levels as put forward in [10].The revival is quite different in nature to the fidelity revivals reported in [27, 28, 29].There a global perturbation and fidelity of a random state was considered and a revivalof fidelity at Heisenberg time was found. An explanation of this phenomenon wasgiven by the rigidity of the spectrum of a background with chaotic dynamics. Suchan explanation obviously fails in the present case, since revival occurs even in theabsence of energy correlations of the background. The behavior rather resembles theoverdamped oscillations in a two–level system, which is coupled to a non–Markovianheat–bath (for instance one or more spin baths [30, 31]) . A possible explanation is thatfor small couplings the system effectively reduces to a two–level system involving onlythe Doorway state and its nearest neighbor (in energy). This two–level system itself isthen strongly coupled to the remaining background states.Survival probability is susceptible against changes in the background dynamics fromregularity to chaoticity, this is in agreement with the original arguments of Peres [32].It is not sensitive against time reversal symmetry breaking in a chaotic backgrounddynamics.Whereas fidelity amplitude has been calculated exactly in various random matrixmodels, this is the first exact calculation of Fidelity in a random matrix setting. Thiswas possible, due to advances in the calculation of ensemble averages of characteristicpolynomials in the last years [20, 21, 22, 23, 24, 25, 14, 26]. The exact results are limitedto the case of a complex (time reversal invariance breaking) coupling to the background.A similar calculation for real (time reversal invariant) coupling would require knowledgeof the averages of non–rational functions of characteristic polynomials. urvival Probability of a Doorway State in regular and chaotic environments Acknowledgments
We thank T. Guhr, B. Gutkin, P. Mello and R. Oberhage for useful discussions.Two of us (HK and HJS) acknowledge support from Deutsche Forschungsgemeinschaft(DFG) within Sonderforschungsbereich Transregio 12 “Symmetries and Universality inMesoscopic Systems”. HK is grateful for financial support from DFG, with grant No.Ko 3538/1-1 and 3538/1-2. SA acknowledges the hospitality of the University Duisburg–Essen.
Appendix A. Theorems 1.3.1 and 1.3.2 of Ref. [14]
For a chaotic background the ensemble average can be done using a result by Borodinand Strahov [14]. We restate Theorem 1.3.2/1.3.1 of Ref. [14] concerning the GUE/GOEensemble average of a ratio of an arbitrary number of characteristic polynomials in aform adapted to our purposes.We first state Theorem 1.3.2 concerning the GUE: Define the multivariate function C ( α, β ) of 2 n + 2 m variables α − k , α + k , β − l , β + l , 1 ≤ k ≤ n , 1 ≤ l ≤ m with α − k , β − l ∈ C and α + k , β + k ∈ C \ R as the ratio of an arbitrary number of characteristic polynomials C ( α, β ) = Q nk =1 det( H − α − k ) Q ml =1 det( H − β − l ) Q nk =1 det( H − α + k ) Q ml =1 det( H − β + l ) , (A.1)and the average over N × N random matrices chosen from GUE( . . . ) = Z d [ H ] P b ( . . . ) , (A.2)where the distribution P b is given by Eq. (14), with β = 2. Moreover, define γ = ( n + m ) + ( n − m ) and the Vandermonde Determinant ∆ n ( α ) = Q i,j ( α i − α j ).Then the following identity holdslim N →∞ C ( α/ √ N , β/ √ N ) = ( − γ/ Q nk,l ( α − k − α + l ) Q mk,l ( β − k − β + l )∆ n ( α − )∆ n ( α + )∆ m ( β − )∆ m ( β + )det[ S (2) ( α − , β + | β − , α + )] , (A.3)where S (2) ( α − , β + | β − , α + ) is a n + m matrix with rows parametrized by elements α − k , β + l and columns parametrized by elements β − k , α + l and with matrix elements § S (2) ( α − p , β − q ) = 1 π sin( α − p − β − q ) α − p − β − q , (A.4) S (2) ( α − p , α + q ) = − exp i ( α + q − α − p ) α + q − α − p , Im α + q > , exp i ( α − p − α + q ) α − p − α + q , Im α + q < , (A.5) S (2) ( β + p , β − q ) = exp i ( β + p − β − q ) β + p − β − q , Im β + p > , − exp i ( β − q − β + p ) β − q − β + p , Im β + p < , (A.6) § In the second line of Eq (A.7) there is a minus sign changed as compared to the original theorem ofRef. [14], which is apparently wrong. urvival Probability of a Doorway State in regular and chaotic environments S (2) ( β + p , α + q ) = 2 πi exp i ( β + p − α + q ) β + p − α + q , Im β + p > , Im α + q < , exp i ( α + q − β + p ) α + q − β + p , Im β + p < , Im α + q > , , in all other cases . (A.7)Since the mean level spacing is given by D = π/ √ N , we find that R ( k, s ) is exactly ofthe form (A.3) with n = m = 1 and with α − = πs , β − = − πs ,α +1 = π √ s − iksλ , β +1 = − π √ s − iksλ . (A.8)This yields Eq. (42).We now turn to Theorem 1.3.1 concerning the GOE: Define the multivariatefunction C ( α, β ) of n + m variables α k , β l , 1 ≤ k ≤ n , 1 ≤ l ≤ m with α k ∈ C and β l ∈ C \ R as the ratio of an arbitrary number of characteristic polynomials C ( α, β ) = Q nk =1 det( H − α k ) Q ml =1 det( H − β l ) , (A.9)and the average over 2 N × N random matrices chosen from GOE( . . . ) = Z d [ H ] P b ( . . . ) , (A.10)where the distribution P b is given by Eq. (14), with β = 1. Then the following identityholds k . lim N →∞ C ( α/ √ N , β/ √ N ) = Q nk =1 Q ml =1 ( α k − β l )∆ n ( α )∆ m ( β )Pf[ S (1) ( α, β | α, β )] , (A.11)where S (1) ( α − , β + | β − , α + ) is a skew–symmetric n + m matrix with rows and columnsparametrized by elements α , β and with matrix elements S (1) ( α p , α q ) = − π ∂∂α i sin( α p − α q ) α p − α q , (A.12) S (1) ( α p , β q ) = ( − exp i ( β q − α p ) β q − α p , Im β q > , exp i ( α p − β q ) α p − β q , Im β q < , (A.13) S (1) ( β p , β q ) = 2 πi R + ∞ i ( β p − β q ) t ) t dt, Im β p > , Im β q < , − R + ∞ i ( β q − β p ) t ) t dt, Im β p < , Im β q > , , in all other cases . (A.14)Setting α = πs , α = − πs ,β = π √ s − iksλ , β = − π √ s − iksλ . (A.15)in Eq. (A.11) yields Eq. (49). k a scaling factor √ urvival Probability of a Doorway State in regular and chaotic environments References [1] Nielsen M A and Chuang I L 2000
Quantum Computation and Quantum Information (Cambridge:Cambridge University Press)[2] Gorin T, Prosen T, Seligman T H and Znidaric M 2006
Phys. Rep.
Adv. Phys. et al Phys. Rev. Lett. Mesoscopic Physics of electrons and photons
Phys. Rev. Lett. Phys. Rev. E Physica Scripta
T90
Phys. Rev. Lett. Phys. Rev. E J. Phys. A Z. Phys. Random Matrices
Communication on Pure and Applied Mathematics Nuclear Structure Vol. 1
Phys. Rep
Quantum Chaos an Introduction
Mesoscopic Quantum Physics (Elsevier)[19] Kohler H, Guhr T and ˚Aberg S 2009 To be published[20] Brezin E and Hikami S 2000
Comm. Math. Phys.
Phys. Rev. E Comm. Math. Phys.
Comm. Math. Phys.
J. Math. Phys. J. Phys. A Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach
Courant lecture notes 3 (New York: New York University)[27] St¨ockmann H J and Sch¨afer R 2004
New J. Phys. Phys. Rev. Lett. Phys. Rev. E Rep. Prog. Phys. New Jour. Phys. Phys. Rev. A30