5. Fidelity 69
5.3. Fidelity of cold atoms in an optical lattice: Theoretical background
Theoretical background
In this section we present the theoretical predictions for the temporal decay of the fidelity F(t)of cold atoms loaded in an optical lattice or coupled micro-traps subject to perturba-tions of the coupling: k0→k0±δk[31]. In the context of OLs this perturbation is readily achieved by adjusting the intensity of the laser beams that create the lattice. In our analysis, we consider that
Hˆ1,2=Hˆ0∓δkBˆ. (5.13)
As in Chapters 3 and 4, the unperturbed Hamiltonian ˆH0is given by Eq. (2.39) withk=k0 and ˆBis the coupling operator (see Eq. 3.2).4 This perturbation is similar to a momentum boost which has recently been investigated [165] in the context of the fidelity. It was found that – as long as the boost is not too large – the fidelity freezes at some finite value. In the numerical analysis we will usek0≈15 and ˜U≈280. If not stated otherwise the number of particles isN =230. It is a fixed assumption throughout this work that the perturbation is classicallysmallδkδkcl, i.e., the corresponding classical Hamiltonians5
H
0,H
1, andH
2are generators of classical dynamics of the same nature. Thus we are always in theclassical LRT regime.
The main tool of the analytical considerations will be quantum linear response theory. To this end we rewrite the fidelity from Eq. (5.1) as
F(t) = up to second order we obtain
|ψ1(t)i=|ψ(0)i+iδk
4Again, we stress the fact that for an appropriate choice of energy and lattice parameters the BHH trimer can be chaotic and thus contains the main ingredient to generalize the result to larger lattices (see Section 2.6).
5These are obtained from Eq. 2.23 withvi=0 and f=3.
5.3. Fidelity of cold atoms in an optical lattice: Theoretical background 77 and using the notationh·i0=hψ(0)| · |ψ(0)iwe obtain for the fidelity:
F(t) = |hψS2(t)|ψS1(t)i|2 Expanding the wave functions ψi to higher order and summing up the terms leads to the exponentiation of the linear response result
F(t) =1− where J(t) =hJ(t)iˆ 0. Eq. (5.18) can be interpreted in terms of a fluctuation-dissipation relationship [172]. On the left hand side we have the fidelity which describes dissipation of quantum information and on the right hand side we have an integrated time-correlation function, i.e., a fluctuatingquantity. Another immediate observation by direct inspection of (5.18) is that the stronger the correlations decay the slower will the fidelity decay and vice versa [173, 101]. This is somehow counterintuitive as it states that the more chaotic a system is, the slower is the decay of the fidelity. For classically regular systems on the other hand, the correlator will generally not decay to zero and thusF(t)will decay much faster.
From Eq. (5.18) it is clear, that the only ingredient in the perturbation theory is the cor-relatorC(τ). As we have seen in Chapter 3, the information ofC(τ) is fully encoded in the bandprofile ˜C(ω)of the perturbation matrixB(see Eq. 3.14). Thus we expect that the improved random matrix theory model (see Section 3.2) which incorporates the structures in the energy landscape of the perturbation operator ˆB[59] but doesn’t contain higher or-der correlations is able to reproduce the LRT results. In Fig. 5.3 we present the results of these simulations. An excellent agreement of the IRMT data with the fidelity calculations of the BHH (2.39) in the perturbative regimeδk<δkprt is evident. This is in accord with
0 0.05
t
0.1 10-310-2 10-1 100
F(t)
Num.
Theor.
0 t2 0.01
-0.2 -0.1 0
ln[F(t)]
δk=0.5 δk=2.5
δk=0.05
Figure 5.3.:The fidelityF(t)at ˜En(0)=0.26 for three perturbation strengths:δk=0.05<δkqm(upper curves and inset),δk=0.5<δkprt(middle curves), andδk=2.5>δkprt(lower curves). Black solid lines correspond to the exact numerical result obtained from eigenstates of the Hamiltonian ˆH0 while dash-dotted lines correspond to the theoretical expectation. We have found that within the perturbative regime, the IRMT calculations coincide with the linear response theory expression from Eq. (5.18).
our findings for the IRMT in the LDoS (see Figs. 3.14a, b) and the wavepacket dynamics studies (see Figs. 4.4, 4.7a) presented in the previous two chapters. Therefore we come to the conclusion that within the perturbative regime LRT and IRMT are equivalent not only for “static” quantities (like the LDoS) but also for complex time-evolution schemes (like the fidelity). At the same time the IRMT modeling cannot describe the quantum results in the non-perturbative regime. Here, the echoes are related to subtler correlations of dynamical nature (self-trapping). These go beyond the autocorrelation functionC(τ)that determines the bandprofile and therfore cannot be captured by LRT or the IRMT modeling.
Coming back to Eq. (5.18), we mark that it is not restricted to short times. The condition for its applicability is that the perturbation δk is sufficiently small so that 1−F(t)1.
Specifically, for quantum mechanically small perturbations δk≤δkqm ∼ ∆/σ Eq. (5.18) implies a short time Gaussian decayF(t) =exp[−8C(0)(δk·t/h)¯ 2], which evolves into a long time Gaussian decay (see inset of Fig. 5.3) [101, 99, 124, 20, 66, 172]
F(t)≈ e−(σδkh¯ )2t2. (5.19) The decay ofF(t)for small perturbationsδk≤δkqm is the same irrespective of the nature (integrable or chaotic) of the underlying classical dynamics. In order to derive the above equation we used the fact that the transitions occur only between neighboring levels.6
In the case where the perturbation mixes levels in a distance larger than the mean level
6For the evaluation ofC(τ≈0) =R−ωω0
0dωC(ω)˜ in the standard perturbative regime we assume thatω0∼∆h¯.
5.3. Fidelity of cold atoms in an optical lattice: Theoretical background 79 spacing∆but still smaller than the bandwidth∆b(Wigner perturbative regime)δkqm≤δk≤ δkprt∼U˜/Nwe have to distinguish short- and long-time behavior as the correlation function decays sufficiently fast fort >τcl while it is assumed to be constant for short times. If the bandprofile of the perturbation operator B were flat, we would expect [101, 124, 20, 66, 172, 99] an exponential decayF(t)∼e−2Γt. The rateΓ∼(σδk)2/∆is given by the width of the Local Density of States (LDoS) [115] studied in Chapter 3. As a result we get
F(t) =
We note that the parabolic decay is present for any perturbation strength at sufficiently short times.
For even stronger perturbations δk>δkprt, we enter the semiclassical regime [124, 20, 66]. As we have seen in Chapter 3 the non-perturbative regime coincides with the semiclas-sical limit sinceδkprt ∼U˜/N→0 in the limitN 1 (while ˜U =const. [115]). Then for any fixed perturbationδk, eventuallyδkδkprt. In this regime (and provided that the clas-sical dynamics is chaotic) it was found recently [126, 66, 214, 124, 20] that the exponential decay persists but that the decay rate approaches some semiclassical valueγSC
F(t)∼e−γSCt . (5.21)
In some cases 7, γSC was found to be equal to the classical Lyapunov exponent Λ [126].
This result raised a lot of research activity within the last years, since on the one hand the classical dynamics (Λ) manifests itself in quantum evolution, and on the other hand suggests a perturbation independent dephasing rate. However, we would like to point out that the study of thegeneral conditions for having a fingerprint of the classical Lyapunov exponent in quantum fidelity experiments is not yet settled and surprises are possible as we will see in the next section.
In the deep non-perturbative regime the quantum-classical correspondence may become strong enough so that one can rely on purely classical calculations. The classical fidelity is then defined as
Fcl(t) = Z
Ω
dxρ−δk(x,t)ρ+δk(x,t) (5.22) whereρ±δkis the classical density function evolved under
H
(k0±δk)(see footnote 5) and the integration is performed over the whole phase space8 Ω. This means that the classical fidelityFcl(t)is defined as the fractional overlap of the two phase-space densities that result from the propagation of an initial phase-space volume underH
(k0±δk)(see Fig. 5.4).7In fact, the perturbation-independent decay ofF(t)appears only for some special sets of initial states, like narrow wave packets [66, 124]. This has to be contrasted with the perturbative regimeδk<δkprt, where theF(t)-decay is qualitatively the same for any initial preparation.
8In the actual calculation the integration is done over an energy shell of widthdE. We have checked that the numerical procedure is stable with respect todE.
H
1H
2x p
Figure 5.4.: Scheme of the classical fidelity cal-culation: an initial phase-space volume (circle) is evolved under the two HamiltoniansH1,H2. The classical fidelityFcl(t)is defined as the fractional overlap (grey shaded area) of the two phase-space volumes that result from the propagation.