Table Of ContentAnomalous slow fidelity decay for symmetry breaking perturbations
T. Gorin1, H. Kohler2, T. Prosen3, T. H. Seligman4, H.-J. St¨ockmann5 and M. Zˇnidariˇc3,6
1 Max-Planck Institute fu¨r Physik der Komplexer Systeme, D-01187 Dresden, Germany
2 Institut fu¨r theoretische Physik, Universit¨at Heidelberg, D-69120 Heidelberg, Germany
3 Department of Physics, Faculty of Mathematics and Physics,
University of Ljubljana, SI-1000 Ljubljana, Slovenia
4 Centro de Ciencias F´ısicas, UNAM, Cuernavaca, Mexico
5 Fachbereich Physik der Philipps-Universit¨at Marburg, D-35032 Marburg, Germany
6 Abteilung fu¨r Quantenphysik, Universit¨at Ulm, D-89069 Ulm, Germany
(Dated: February 8, 2008)
Symmetries as well as other special conditions can cause anomalous slowing down of fidelity
6
decay. These situations will be characterized, and a family of random matrix models to emulate
0
them generically presented. An analytic solution based on exponentiated linear response will be
0
given. Foronerepresentativecasetheexactsolutionisobtainedfromasupersymmetriccalculation.
2
The results agree well with dynamical calculations for a kicked top.
n
a PACSnumbers: 05.45.Mt,03.65.Sq,03.65.Yz
J
0
1 Sensitivity to perturbations as measured by fidelity ear respons regime, as well as an exact analytical result
decay has received a great deal of attention both in obtained by a supersymmetry method [18] for the antiu-
]
D the context of quantum and classical dynamical sys- nitary symmetry-breaking case. Our results display ex-
tems [1, 2, 3, 4] and as a benchmark for the stability cellent agreement with numerical experiments for quan-
C
of possible quantum information devices [5, 6]. Fidelity tum kicked tops.
.
n may be described as the cross-correlation function be- To study fidelity decay, we consider the perturbed
li tween unperturbed and perturbed evolution of a quan- Hamiltonian H = H0 +λV. If U(t) and U0(t) are the
n
tumorclassicalwavesystem. Linearresponsetheoryhas unitary propagators under H and H , respectively, we
[ 0
been particularly successful in describing fidelity decay, define the fidelity amplitude as
1 by relating it to the correlation decay of the perturbing
v operator in the interaction picture [2, 3, 6]. For chaotic f(t)=hΨ0(t)|Ψ(t)i=hΨ(0)|U0(−t)U(t)|Ψ(0)i, (1)
0
2 systems a recently introduced random matrix model [7] whereΨ(t)=U(t)Ψ(0)andΨ0(t)=U0(t)Ψ(0). Fidelity
0 is in excellent agreement with experiments [8, 9]. is defined as F(t) = |f(t)|2. In Refs. [2, 3], it has been
1 It has been shown, that so-called residual perturba- shown that within second order time-dependent pertur-
0 tions that have vanishing diagonal matrix elements in bation theory (Born series) the fidelity amplitude, aver-
6 the eigenbasis of the unperturbed Hamiltonian, lead to agedoverrandominitialstates,canbeexpressedinterms
0
/ very slow fidelity decay known also as quantum freeze of the two-point time correlation integral C(t):
n of fidelity [11, 12]. Fidelity freeze is a pure wave phe-
li nomenon without classical analogue [12]. In the field of hf(t)iE =1−4π2λ2 C(t)+O(λ4), (2)
n
quantum computation, it is known as dynamical decou-
v: pling [13, 14]. One can identify four different physical where h...iE denotes the average over random initial
states,whichareconcentratedonasmallenergyinterval
i situationsinwhichquantumfreezeoranomalousslowfi-
X that contains many levels N . For the fidelity ampli-
E
delitydecayoccurs. Thecasewhentheperturbationcan
r tude, this averagingamounts to taking a restrictedtrace
a be written as a time derivative (or commutator with the
oftheechooperator,intheeigenbasisoftheunperturbed
unperturbed Hamiltonian) is treated in Refs. [11, 12].
Hamiltonian H . Therefore, we may write:
0
Inthisletterwetreatthethreeotherandindeedphys-
ically most important cases: The first one corresponds t t′
C(t)= dt′ dt′′hV(t′)V(t′′)i (3)
toaperturbationwhichbreaksanantiunitarysymmetry E
(e.g. time-reversal)in an optimal way, meaning that the Z0 Z0
perturbationanticommuteswiththeantiunitarysymme- hV(t′)V(t′′)iE = 1 ′ |Vαβ|2e2πi(Eα−Eβ)(t′−t′′),
N
try, e.g. switching on the magnetic field. The other two E α β
X X
cases correspond to a mean field approach in which the
where the E are the eigenvalues of H , and V(t) =
α 0
diagonal part of the perturbation is moved to the un- U (t)†V U (t) is the perturbation in the interaction pic-
0 0
perturbed Hamiltonian. We will consider unperturbed
ture. ThematrixelementsV aretakenintheeigenbasis
αβ
Hamiltonians, with and without an antiunitary symme-
of H . The primed sum runs over N eigenstates of H .
0 E 0
try. To obtain a generic understanding, all cases are
Often, the exponentiated version of Eq. (2),
considered in the framework of random matrix theory
(RMT). We presentgeneraltheoreticalresultsinthe lin- hf(t)i =e−εC(t)+O(λ4), ε=4π2λ2, (4)
E
2
is able to describe the fidelity decay from the perturba- This yields a logarithmically slow decay of fidelity
tive to the golden rule regime well [7, 8, 9, 10]. This is ε
true,inparticular,if the systemis chaotic,suchthatthe hf(t)i≈1− [2+ln(2t)]+O(λ4t2). (10)
12
perturbation can be described [7] by one of the random
Both results for the plateau value of the fidelity ampli-
Gaussian ensembles (RMT approach) [15].
tude, Eq. (8) and Eq. (10), follow from Eq. (6). They
We now consider the situation of a residual perturba-
arethusvalidforanyofthe three ensemblesusedforthe
tion, i.e., one with vanishing diagonal, V ≡ 0 within
αα
perturbation, (i), (ii), and (iii). Indeed, a similar result
the RMT models. Thus, we assume that the non-zero
couldbe obtainedfor the Gaussiansymplectic ensemble.
matrix elements of this perturbation are independent
In [7] the averages hF(t)i and |hf(t)i|2 were shown to
normalized Gaussian random variables with variance
differ only in a term proportional to t2, which is absent
h|V |2i=1−δ , (5) in the quantum freeze case. Thus, the eqs. (2) and (6)
αβ αβ
yieldthelinearresponseapproximationforfidelitydecay.
whereh...idenotesanensembleaverage. Forthepertur- Numerics indicate that hF(t)i ≈ |hf(t)i|2, in general, in
bation matrix V, we consider three different ensembles,
accordance with an argument given in [4].
which fulfill Eq. (5): (i) an ensemble of imaginary anti-
Forlongtimesandsmallperturbations,thefidelityam-
symmetric matrices, (ii) an ensemble of real symmetric
plitude can be expressed in terms of level shifts. Within
matrices with deleted diagonal, and (iii) an ensemble of
the second order stationary perturbation theory, its av-
Hermitean matrices with deleted diagonal.
erage is then given by the Fourier transform of the level
TheaverageofthecorrelationintegralC(t)overanyof
curvature distribution [13] which was obtained analyti-
those ensembles gives:
cally in [19]. The final result is surprisingly simple:
t t t′
hC(t)i= − dt′ dt′′b(t′′) (6) τ K1(τ) :GOE εt
2 hf(t)i= τ = . (11)
Z0 Z0 ((1+τ)e−τ :GUE 2
where b(t) is the two-point spectral form factor of H .
0
where K is the modified Bessel function of first or-
Here, as well as throughout the rest of the letter, we use 1
der. The GOE branch is valid for an unperturbed GOE
units, where the Heisenberg time is equal to one. If N
E
Hamiltonian,andaperturbationmatrixoftype(i)or(ii).
is sufficiently large, b(t) tends to a well defined smooth
TheGUEbranchisvalidforanunperturbedGUEHamil-
function (self-averaging); else we average over a random
tonian,andaperturbationmatrixoftype(iii). Morede-
matrixensembleforH aswell. ForGaussianorthogonal
0
tails onthe asymptoticbehavioroffidelity decayinsitu-
(GOE) and unitary (GUE) ensembles, the form factors
ationsofquantumfreezewillbepublishedelsewhere[13].
aregiveninRef.[15]. Notethatthetermproportionalto
Oneshouldstressthatdiagonalelementsofthepertur-
t2 ismissingascomparedtothelinearresponseresultfor
bationvanishalsointhepresenceofadiscreteorcontin-
a generic perturbation [7]. For what follows, we assume
uous unitary symmetry R, of H , which anti-commutes
that the average h...i includes such a procedure. 0
withV,RV =−VR. However,itturnsoutthatitseffect
Let us first concentrate on the special case of unper-
on fidelity enhancement is less drastic than the predic-
turbed Hamiltonians H taken from the GUE. Then, we
0
tions of Eqs. (7) and (9), because of the lack of corre-
have b(t)=max{1−t,0}, and
lations between different subspectra of H . As a result,
0
t − t2 + t3 :t≤1, theasymptoticgrowthofthecorrelationintegralislinear
hC(t)i=CGUE(t)=(12 2 6 :t>1 (7) C(t)∝t, for times before and after the Heisenberg time.
6 For the case of H taken from the GOE and a purely
0
As a result, for times longer than the Heisenberg time, imaginary antisymmetric perturbation the average fi-
Eq.(2)predictsthefidelityto“freeze”onaminimalvalue delity can be obtained exactly by supersymmetry tech-
niques in the limit of large dimension N. The calcula-
ε tion is technically much more involved than for the case
fplateau =1− . (8) of a GOE perturbation [16] and will be presented else-
6
where [18]. The result again is a VWZ-like integral (see
Since the next correction term grows quadratically in
Ref. [17], Eq. (8.10)) and is given by
time,hf(t)i=f +O(λ4t2),wefindthattheplateau
plateau
ends at a time of order t∗ ∼1/λ. t u
vdv
Second,wechooseH fromtheGOE.Alsointhiscase, hf(t)i=2 du
0
[u2−v2][(u+1)2−v2]
the integrals in equation (6) can be performed analyti- Z Z
Max(0,t−1) 0
cally (see Ref. [7]). Here we just give the the leading p
(t−u)(1−t+u)
asymptotics for t≫1: × [1+ε(t2−v2)]
(v2−t2)2
CGOE(t)= ln(2t)+2 +O(t−1 lnt) (9) ×[t(2u+1−t)+v2]e−ε2[t(2u+1−t)−v2]. (12)
12
3
1 merical random matrix simulations, where we computed
(a) averages over 104 samples of 100×100 matrices. Only
0.1 the10centralstateshavebeentakenintoaccount. Inthe
GOEcase,wehavealsoperformednumericalsimulations
)i 0.01 for symmetric perturbations with deleted diagonal (case
t
( (ii)) and we have not found significant deviations from
f
h
− 10−3 the antisymmetric case (i). For strong perturbations the
1 plateaudisappearsandwefindapartialrevivaloffidelity
near the Heisenberg time, similar as in Ref. [16].
10−4
We have concentrated on the unitary antisymmetric
perturbation, because of its invariance properties, which
1 allow to obtain results in closed form. Yet the linear
(b) response resultin Eq.(2) can be carriedto higher order,
0.1 andat leastup to sixth orderthey coincide with the one
for symmetric perturbations with deleted diagonal [13].
i
t) 0.01 The RMT model can also be compared to dynamical
(
hf systems with chaotic classical limit. For this purpose we
− 10−3 have considered a quantized kicked top [20].
1
In the firstexample, we choose a one step propagator
10−4
10−5 Uλ =P21e−iγSyP21e−iλSx, γ =π/2.4 (13)
0.1 1 10 100 1000 10000
t with P = e−iαSz2/2Se−iSz and Sx,y,z being standard spin
operators. U is time-reversal invariant, and the pertur-
0
FIG.1: Thecomplementoftheaveragefidelityamplitude,for bation S is antisymmetric in the eigenbasis of U . The
a weak perturbation, ε=5×10−5. Part (a) shows the GOE y 0
“symmetrization”ofU isessentialforV toanticommute
case with a purely antisymmetric random Gaussian pertur- 0
with the time-reversalsymmetry.
bation. Part (b) shows the GUE case, with an independent
GUEperturbationwith deleteddiagonal. Theexponentiated We choose the spin S = 200, one initial random state
linearresponseapproximationsareplottedwithshortdashed andaveragethefidelityover400realizationsoftheprop-
lines(forcomparison,theexactresultsforfullGOEandGUE agator U where for each realization we draw a param-
λ
perturbationsareshownwithchaincurves). Theresultsfrom eter α from a Gaussian distribution of width 1 centered
time-independent perturbation theory are plotted with long
around 30. The results of fidelity decay for different
dashed lines. In the GOE case, the exact analytical result,
strengthsofperturbationareshowninFig.2(a). Wefind
Eq.(12)isplottedwithasolidline. Therandommatrixsim-
good agreement with the square of the theoretical result
ulations are plotted with points.
(12) for the fidelity amplitude, which in turn agrees well
withRMTsimulationsforthefidelity(notshown). With-
outaveragingoveranensemble ofdynamicalsystems we
The only difference to Ref. [16], where a GOE perturba-
get considerable fluctuations around RMT curves. Note
tionwasconsidered,istheadditionalfactor[1+ε(t2−v2)]
that we do not use any fit parameters. The dimension-
in the integrand, and a minus sign with the v2 term in
less perturbation strength ε in Eq. (12) is obtained as
theexponent,whereintheGOEcasethereisaplussign. ε = 2Nσ (Sλ)2 = 4λ2S3σ , where σ = 0.153 is an
cl cl cl
Figure 1 shows the fidelity decay both for H taken integral of the classical correlation function calculated
0
from the GOE (a) and the GUE (b), for a small pertur- using the corresponding classical map, see e.g. Ref. [3]
bation ε = 5×10−5. It compares different theoretical for more details. Heisenberg time is t =N =2S.
H
approaches. To this end, one minus the average fidelity We also consider an unperturbed propagator without
amplitude is plotted on a double-log scale. The figure time-reversalsymmetry, that corresponds to GUE case,
shows the exact result calculated from Eq. (12) (in the
GOE case only), together with the results from the ex- Uλ =Pe−iγSye−iµSx2/2Se−iξSxe−iλSx′ . (14)
ponentiated linear response approximation (4) and the
asymptotic expression (11). For comparison the fidelity withγ =π/2.4,µ=10,ξ =1. Herewehavesetdiagonal
decay for the case of a GOE perturbation [16] is shown matrix elements of the perturbation in the eigenbasis of
as well. We see that the linear response approximation U to zero by hand, S′ =S −diagS . We take S =200
0 x x x
isabletodescribethe fidelitydecayforquite alongtime andaveragethefidelityover100samples,similarlyasfor
very well. Immediately beyond the time when the linear GOE case. As above we determine ε from the classical
response formulae fail, the asymptotic results (11) de- correlation integral σ = 0.168. In Fig. 2b the results
cl
scribe fidelity decay quite well. We also compare to nu- of the numerical simulation are plotted, together with
4
(a) 100 the perturbation in the unperturbed Hamiltonian, and
ε=4.9 thus the quantum freeze sheds new light on the surpris-
10-1 ε=0.049 ingsuccessofsuchtheories. Ontheotherhandthisresult
shows that for a quantum information process to be ef-
10-2
fective beyond the Heisenberg time, one has to suppress
10-3 ε=4.9•10-4 the diagonal part of any static perturbation.
F
1-
10-4 ε=4.9•10-6 We thank H. Schomerus for drawing our attention
to Ref. [19]. THS acknowledges support by the grants
10-5
No.10803(DGAPA)andNo.41000-F(CONACyT).MZ
10-6 thanks the Alexander von Humboldt Foundation. TP
acknowledgessupportby the grantP1-0044ofthe Slove-
10-7 nian Research Agency.
10-3 10-2 10-1 100 101 102 103 104 105 106
τ=t/t
H
(b) 100
ε=4.9
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