Optomechanical multistability in the quantum regime

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by  C. Schulz

epl draft
Optomechanical multistability in the quantum regime
C. Schulz, A. Alvermann(a), L. Bakemeier and H. Fehske
Institute of Physics, Ernst-Moritz-Arndt-University, 17487 Greifswald, Germany
6 PACS 42.50.Ct–Quantum description of interaction of light and matter; related experiments
1 PACS 37.10.Vz–Mechanical effects of light on atoms, molecules, and ions
0 PACS 07.10.Cm–Micromechanical devices and systems
2
r Abstract – Classical optomechanical systems feature self-sustained oscillations, where multiple
p periodicorbitsatdifferentamplitudescoexist. Westudyhowthismultistabilityisrealizedinthe
A quantum regime, where new dynamical patterns appear because quantum trajectories can move
between different classical orbits. We explain the resulting quantum dynamics from the phase
7
space point of view, and provide a quantitative description in terms of autocorrelation functions.
1
Inthiswaywecanidentifycleardynamicalsignaturesofthecrossoverfromclassicaltoquantum

] mechanics in experimentally accessible quantities. Finally, we discuss a possible interpretation
h of our results in the sense that quantum mechanics protects optomechanical systems against the
p chaotic dynamics realized in the classical limit.
-
t
n
a
u
q
[
Introduction. – The interaction of light with me- The prototypical optomechanical system is a vibrating

2 chanical objects [1,2] enjoys continued interest due to the cantileversubjecttotheradiationpressureofacavitypho-
v successful construction and manipulation of optomechan- ton field, for which the Hamilton operator reads [3,4,17]
0
ical devices over a wide range of system sizes and pa-
62 rameter combinations (see the recent reviews [3,4] and (cid:126)1H =(cid:2)Ωcav−Ωlas+grad(b†+b)(cid:3)a†a (1)
4 references cited therein). With these devices both clas- +Ωb†b+α (a†+a),
las
0 sical non-linear dynamics such as self-sustained oscilla-
1. tions[5–8]andchaos[9–11]aswellasquantummechanical where b(†) and a(†) are bosonic operators for the vibra-
0 mechanisms such as cooling into the groundstate [12,13] tional mode of the cantilever (frequency Ω) and for the
6 and quantum non-demolition measurements [14–16] can
cavity photon field (Ω ), respectively. This Hamilton
1 be studied in a unified experimental setup. cav
operator applies to any generic optomechanical system,
:
v This raises the question whether it might be possible but we adopt the cavity-cantilever terminology through-
Xi todetectthecrossoverfromclassicaltoquantummechan- out this paper. For our theoretical analysis we use the
ics directly in the dynamical behaviour of optomechanical quantum optical master equation [18]
r
a systems. In a previous paper [11] we observed that the
classical dynamical patterns, which are characterized by i
∂ ρ=− [H,ρ]+ΓD[b,ρ]+κD[a,ρ] (2)
the multistability of self-sustained oscillations, change in t (cid:126)
acharacteristicwayifonemovesintothequantumregime.
for the cantilever-cavity density matrix ρ(t), with the dis-
Previously stable orbits become unstable, the system os-
sipative terms
cillates at a new amplitude, and especially the classical
chaotic dynamics is almost immediately replaced by sim- 1
ple periodic oscillations. In this paper we explain this D[L,ρ]=LρL†− (L†Lρ+ρL†L) (3)
2
behaviour from the point of view of classical and quan-
tum phase space dynamics. Most importantly, we will that account for cantilever damping (∝ Γ) and radiative
show that the dynamical patterns do not change at ran- losses (∝ κ). Note that the above Hamilton operator is
dom but that clearly identifiable and new signatures can given in a frame that rotates with the frequency Ωlas of
be observed. the external pump laser such that only the cavity-laser
detuning Ω −Ω appears, and that we assume zero
cav las
(a)E-mail: [email protected] temperature in the master equation.
p-1
C. Schulz et al.
-4 -2 0 2 4 -4 -2 0 2 4 cl. limit qm
4(a) (b) 4 1 1 4
2 2 3
p 0 0 x0 0 σσxp2 3
-2 -2 -1 -1 1
-4 -4
100 110
42(c) (d) 24 16 17 18 19τ/2π271 272 273 274 00 100 τ2/020π 300 400
p 0 0
-2 -2 Fig.2: (coloronline)Leftpanel: Cantileverpositionx(τ)from
-4 -4 the classical equations of motion (5) and from the quantum
-4 -2 x0 2 4 -4 -2 x0 2 4 mechanical master equation (2) at σ = 0.1, for P = 1.5, ∆ =
−0.4 (case (a) in Fig. 1). Right panel: Cantilever position-
Fig. 1: (color online) Left panel: Chart of self-sustained oscil- momentumuncertaintyproductσxσpforthesameparameters.
lationsintheclassicallimitforP =1.5. Self-sustainedoscilla-
tionsoccurforamplitudesAwherethepowerbalancebetween
gains from the radiation pressure (P =P(cid:104)|α|2Imβ(cid:105) ) and The classical equations of motion predict the onset of
rad avg
lossesduetofriction(Pfric =Γ¯(cid:104)|β|2(cid:105)avg)changesfrompositive self-sustained cantilever oscillations x(τ) = x0 +Acosτ
tonegativevalueswithincreasingA[5,6]. Rightpanels: Classi- as the pump power P is increased. Figure 1 shows the
calorbitsinthe(x,p)cantileverphasespace,for(a)∆=−0.4,
possible amplitudes A of these oscillations, which are ob-
(b)∆=−1.1,(c)∆=−0.85,and(d)∆=−0.7,asmarkedby
tainedwiththeansatzfromRef.[5],forthevalueP =1.5.
vertical lines in the left panel. In case (a), the two innermost
We keep this value fixed throughout the paper, as the be-
orbitshaveamplitudesA ≈1.2andA ≈2.7. Incases(b),(c)
1 2 haviour discussed here does not depend on it. Note in
the innermost orbit shows a few period doubling bifurcations
Fig. 1 that several stable oscillatory solutions at different
that occur on the route to chaos [11], in case (d) it is chaotic.
amplitudes A can coexist for one parameter choice. This
classical multistability of self-sustained oscillations is the
Now introduce the five dimensionless parameters [5,6] origin of the quantum multistability analyzed next.
Ω −Ω 8α2 g2 g Quantum multistability. – We now move into the
∆= las cav , P = las rad , σ = rad , (4)
Ω Ω4 κ quantum regime by letting σ become finite. In all our
examples the quantum system is initially prepared in the
andκ¯ =κ/(2Ω),Γ¯ =Γ/(2Ω),andmeasuretimeasτ =Ωt.
pureproductstateofacoherentcantileverandcavitystate
Theparameter∆givesthedetuningofthepumplaserand
at α=β =0, i.e., in the state that is closest to a classical
cavity, while P gives the strength of the laser pumping.
state at these coordinates. The cantilever-cavity density
Forlaternumericalresultswesetthedampingparameters
matrix is then evolved according to Eq. (2).
κ¯ =0.5, Γ¯ =5×10−4 to typical experimental values [4].
Figure 2 shows the cantilever position x and the
The quantum-classical scaling parameter σ is the ra-
position-momentum uncertainty product σ σ , with the
x p
tio of the quantum mechanical quantity g , which is of
rad uncertainty σ = ((cid:104)Oˆ2(cid:105)−(cid:104)Oˆ(cid:105)2)1/2 of an observable O.
order (cid:126)1/2 because the quantum mechanical position op- O
Thequantumdynamicsatfiniteσ closelyfollowstheclas-
erator xˆ∝(cid:126)1/2(b†+b) of the cantilever enters the expres-
sicaloscillationsforaninitialperiodoftime,beforeitdevi-
sion for the radiation pressure, to the classical quantity κ
ates significantly at later times. Deviations occur because
that measures the cavity quality. The parameter σ thus
the quantum state spreads out in phase space, as wit-
controls the crossover from classical (σ = 0) to quantum
nessedbythegrowthoftheuncertaintyproduct, whereby
(σ > 0) mechanics [6]. In the following we will increase
the cantilever position is smeared out.
σ to move into the quantum regime, but keep σ (cid:28) 1 in
The full phase space dynamics in Fig. 3, which we dis-
ordertoremaininthevicinityoftheclassicallimitσ =0.
play with the Wigner function W(x,p) of the cantilever
Classical multistability. – Our analysis begins in mode (see, e.g., Ref. [19] for the definitions), reveals a
the limit σ = 0, where the optomechanical system is de- moredefinitedynamicalpattern. Forearlytimes(τ (cid:39)16)
scribed by the classical equations of motion [6] the Wigner function retraces the classical orbit with am-
plitude A ≈ 1.2 from case (a) in Fig. 1. At later times
1
∂τα=(i∆−κ¯)α−i(β+β∗)α− 21i, (5a) (τ (cid:39)64)theWignerfunctionshowsacontributionfroma
∂ β =(−i−Γ¯)β− 1iP|α|2 (5b) second circular orbit with larger amplitude, before almost
τ 2
all weight is concentrated on the new orbit (τ (cid:39) 270). In
for the cavity and cantilever phase space variables comparison to case (a) in Fig. 1 this orbit is identified as
α = (Ω/(2αlas))(cid:104)a(cid:105), β = (grad/Ω)(cid:104)b(cid:105). We also use the second classical orbit with amplitude A2 ≈ 2.7. Dur-
the cantilever position and momentum operator xˆ = ing time evolution the quantum state spreads out along,
√ √
(1/ 2)(g /Ω)(b†+b), pˆ= (i/ 2)(g /Ω)(b†−b), with but not perpendicular to, these two classical orbits.
rad rad √
corresponding phase space variables x=(cid:104)xˆ(cid:105)=1/ 2(β+ The classical multistability of the optomechanical sys-
√
β∗) and p=(cid:104)pˆ(cid:105)=(i/ 2)(β∗−β). temthushasadirectcounterpartinthequantumdynam-
p-2
Optomechanical multistability in the quantum regime
4 σ=0.1 1 (i) (ii)
0.5 2
2
δτ()τ0 p-0 .05 xk 0
R -2
-2 - 11 0 0
(iii) (iv)
-4 0.5 2σκ)1.5
16 17(τ+δ1τ8)/2π19 p 0 /(p 1
σ
-0.5 x
4 σ=0.1 -1-1 -0.50 0 0.5 - 110-0.5 0 0.5 1 σ 0.5 45 50 55 60
2 x x τ/2π
)
τ
δ( 0 Fig. 4: (color online) Left panels: Wigner function W(x,p)
τ
R
for a single quantum trajectory starting from a “Schro¨dinger
-2
cat” state at (i) τ = 0, and at later times (ii) τ = 0.001, (iii)
-4 τ =0.008, and (iv) τ =0.4. Right panels: Cantilever position
64 65 66 67
(τ+δτ)/2π xk and uncertainty product σxσp (see Eq. (8)) for a single
quantum trajectory at later times, in the situation of Fig. 5.
4 σ=0.1
All results are for case (a) from Fig. 1 and σ=0.1.
2
)
τ
δ(τ0 all times. In this way, the multistability of the quantum
R
-2 dynamicsisnotonlyobservableduringashortinitialtime
period but during extended periods of time.
-4
271 272 273 274
(τ+δτ)/2π Multistability of quantum trajectories. – The
mechanism behind the quantum multistability can be un-
Fig. 3: (Color online) Wigner function W(x,p) in cantilever derstood through the phase space dynamics of individual
phase space (left panels) and cantilever position autocorrela- quantum trajectories, as they arise in the quantum state
tionfunctionR (δτ)(rightpanels)forcase(a)fromFig.1,at
τ diffusion (QSD) approach [20] to the solution of Lindblad
σ =0.1 slightly away from the classical limit. The autocorre-
master equations such as Eq. (2).
lationfunctionsforthetwoinnerclassicalorbitsatamplitudes
In QSD the density matrix is represented by an ensem-
A are included as dashed curves.
1/2
ble of quantum trajectories |ψ (τ)(cid:105), from which it is ob-
k
tained as an average
icsatsmallσ,wherethesystemmovesbetweenthediffer-
(cid:110) (cid:111)
ent classical orbits. This kind of quantum multistability ρ(τ)=mean |ψ (τ)(cid:105)(cid:104)ψ (τ)| . (7)
k k k
leadstodistinctdynamicalfeaturesbecausetheoscillatory
nature of the different orbits is preserved. Accordingly,expectationvaluesarecomputedasensemble
(cid:110) (cid:111)
Thequantummultistabilityisclearlydetectedwiththe averages O(τ) = tr[ρ(τ)Oˆ] = mean (cid:104)ψ (τ)|Oˆ|ψ (τ)(cid:105) .
k k k
cantilever position autocorrelation function
Each quantum trajectory |ψ (τ)(cid:105) follows a stochastic
k
τ+π equation of motion that combines the Hamiltonian and
(cid:90)
R (δτ)= (cid:104)xˆ(τ(cid:48))xˆ(τ(cid:48)+δτ)(cid:105)dτ(cid:48) , (6) dissipative dynamics with a noise term [20]. Numerically,
τ
the density matrix is obtained through Monte Carlo sam-
τ−π
plingofthetrajectoriesfordifferentnoiserealizations. We
instead of the position expectation value that averages use the QSD implementation from Ref. [21], and typically
over the phase space distribution. We choose this func- average over (cid:39) 3000 trajectories to obtain the results in
tion because the dynamics is best described in cantilever Figs. 2–4, 7, 8. Although a single quantum trajectory is
phase space. Autocorrelation functions for the cavity not observable by itself, the phase space dynamics of in-
modecould be usedas wellandshould be moreaccessible dividual trajectories as shown in Figs. 5, 6 allows us to
to experimental measurements, but their interpretation is deduce the properties of the entire density matrix.
lessstraightforwardbecauseoftheadditionalsidebandsat Close to the classical limit quantum trajectories evolve
multiples of the fundamental oscillation frequency. rapidly into localized phase space states as a consequence
The autocorrelation function in Fig. 3 is the weighted ofdissipation[22–24]. ThisisillustratedinFig.4forasin-
sumoftheoscillatorymotiononthetwoorbitsseeninthe gle trajectory that starts from a “Schr¨odinger cat” state,
Wigner function. The frequency of the two orbits is iden- givenasthesuperpositionoftwocoherentstates,withthe
tical (essentially, the cantilever frequency Ω), such that characteristicinterferencepatternintheWignerfunction.
onlyoneoscillationisvisibleinR (δτ). Theamplitudeof In less than one oscillation period (τ = 0.4 < 1) the tra-
τ
R (δτ) increases as weight is transferred from the inner jectoryevolvesintoanearlycoherentstatewithapositive
τ
to the outer orbit. Noteworthy, the oscillations persist at Wignerfunction,whichshowstherapiddecoherence. The
p-3
C. Schulz et al.
τ/2π=0..40 τ/2π=40..80 τ/2π=80..270 second orbit. Consequently, all oscillations are averaged
out in expectation values such as the cantilever position
2
x(τ) in Fig. 2. Such values are, therefore, not the right
p 0 quantities to detect the quantum multistability.
Instead, successful detection requires autocorrelation
-2
functionssuchasR (δτ)fromEq.(6). Similartotheden-
τ
-2 0 2 -2 0 2 -2 0 2 sity matrix the function R (δτ) can be expressed (drop-
x x x τ
ping the τ(cid:48)-integration here) as an ensemble average
Fig. 5: (color online) Stroboscopic (x,p)-phase space plot of a (cid:88)
R (δτ)= x (τ)x (τ +δτ)
single quantum trajectory (red dots), for case (a) from Fig. 1 τ k k
andσ=0.1,atearly(leftpanel),intermediate(centralpanel), k
(cid:88)(cid:10)(cid:0) (cid:1)(cid:0) (cid:1)(cid:11)
andlater(rightpanel)timesτ asindicated. Theinitialcondi- + xˆ(τ)−x (τ) xˆ(τ +δτ)−x (τ +δτ) , (9)
k k k
tions are x(0)=p(0)=0, the quantum system is prepared in
k
a coherent state at these coordinates. The two classical orbits
wheretheexpectationvalue (cid:104)·(cid:105) =(cid:104)ψ |·|ψ (cid:105)iscomputed
at amplitudes A are depicted with dashed curves. k k k
1/2
for each individual quantum trajectory. The correlation
function in the second line is bounded by
quantumtrajectoryremainsinsuchastateduringthesub-
sequenttimeevolution,andtheuncertaintyproductstays (cid:12)(cid:12)(cid:10)(cid:0)xˆ(τ)−x (τ)(cid:1)(cid:0)xˆ(τ +δτ)−x (τ +δτ)(cid:1)(cid:11) (cid:12)(cid:12)2
(cid:12) k k k(cid:12)
close to its minimal value
(cid:10)(cid:0) (cid:1)2(cid:11) (cid:10)(cid:0) (cid:1)2(cid:11)
≤ xˆ(τ)−x (τ) xˆ(τ +δτ)−x (τ +δτ) .
σqσp ≥ 21(grad/Ω)2 = 12(σκ¯)2 (8) k k k k(10)
given by the Heisenberg uncertainty relation for the xˆ, pˆ Whenever the position uncertainty (cid:104)(xˆ −x )2(cid:105) of each
k k
operators (here, the quantum-classical scaling parameter trajectory becomes small, as it is the case for σ (cid:28) 1,
σ comes into play). Notice that phase space localization the autocorrelation function R (δτ) is thus given by the
τ
occursonlyinthevicinityoftheclassicallimit,forσ (cid:28)1. ensemble average of the autocorrelation functions of the
It also explains the transition into the classical limit: For individualtrajectories,i.e.,bythefirstlineinEq.(9). Ac-
σ →0 the quantum trajectories evolve infinitely fast into cordingly,theoscillationsseeninx (τ)foreachindividual
k
minimaluncertaintystates,andatthesametimethelower trajectory(cf. Fig.4)arepreservedintheautocorrelation
boundinEq.(8)goestozero. Then,everytrajectoryoccu- function in spite of the ensemble average. Furthermore,
pies one point in phase space, i.e., it has become classical. R (δτ) is the weighted sum of the autocorrelation func-
τ
Under this condition the classical equations of motion (5) tions for the different classical orbits, which are directly
can be derived directly from the master equation (2). related to the orbit amplitudes A as seen in Fig. 3.
1/2
Becauseaquantumtrajectoryisverylocalizedinphase Noticethatthebehaviourdescribedhere—themotionof
space it is well represented by a single phase space point, quantum trajectories between different classical orbits—
similar to a classical trajectory. In Fig. 5 this represen- emerges only because the trajectory states |ψ (cid:105) deviate
k
tation is used for a stroboscopic phase space plot of a from coherent states. The noise terms in the QSD equa-
single quantum trajectory that contributes to the Wigner tionshavetheformΓ¯(b−(cid:104)b(cid:105) )|ψ (cid:105)dξ,hereforthemechan-
k k
functions in Fig. 3. This plot clearly shows the multi- icaldamping,witharandomvariabledξ ∝dτ1/2 fromthe
stability of the quantum trajectory, which initially fol- underlying Wiener process [20]. If |ψ (cid:105) is exactly a coher-
k
lows the inner orbit before it moves towards the outer ent state, such that (b − (cid:104)b(cid:105) )|ψ (cid:105) = 0, the noise term
k k
orbit. During the time evolution the quantum trajectory will vanish identically. This observation explains why the
followstheoscillatorymotionofthetwoorbitsatthecan- “quantum noise” disappears in the classical limit σ = 0,
tilever frequency, and because the trajectory state is well andthequantumtrajectoriesfollowthedeterministicclas-
localizedinphasespacetheseoscillationsarenotaveraged sical equations of motion (5). At finite but small σ (cid:28) 1
out but appear directly in the position expectation value trajectories are almost but not exactly in coherent states.
x (τ)=(cid:104)ψ (τ)|xˆ|ψ (τ)(cid:105) that is depicted in Fig. 4. The noise terms become effective but remain small, such
k k k
Since every individual trajectory shows this type of that the quantum trajectories still follow the classical dy-
quantum multistability it is also seen in the entire den- namics but are subject to a small stochastic correction.
sity matrix, given as the ensemble average of all trajec- Thissmallcorrectioncanchangethelong-timestabilityof
tories. Because of the noise term in the stochastic QSD classical orbits and their basin of attraction but does not
equation of motion the quantum trajectories are not ex- destroy the classical dynamical patterns. Consequently,
actlyatthesamephasespacepointbutatdifferentpoints thequantumtrajectoriesdonotmovearbitrarilyinphase
on the respective orbits. This results in the broad distri- spacebutfollowaclassicalorbitforsometimebeforethey
bution of the relative angle in phase space seen in the leave the orbit with a finite probability. Afterwards, the
Wigner functions in Fig. 3 especially at later times, when trajectoriescansettleonadifferentattractiveorbitifsuch
the quantum trajectories are spread out fully along the an orbit exists at larger amplitudes.
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Optomechanical multistability in the quantum regime
Fig. 7: (color online) Wigner function W(x,p) in cantilever
phase space for case (d) from Fig. 1, for τ and σ as in Fig. 6.
stillasecondsimpleperiodicorbitsatlargeramplitudeex-
ists such that oscillations can be observed after the quan-
tum trajectories have left the inner orbit.
This is illustrated for case (d) in Figs. 7, 8. First, we
observe again that the relevant time scale changes signifi-
cantlywithσ. Ifσisincreasedfrom0.05to0.1inFig.7al-
mostallweightoftheWignerfunctionistransferredfrom
the inner to the outer orbit. Second, the Wigner func-
tions themselves look quite similar to those for case (a)
in Fig. 3. In agreement with this, well-defined oscillations
areobservedinthecantileverpositionandautocorrelation
function in Fig. 8, and the respective amplitudes can be
related to those of the classical orbits in Fig. 1.
Fig. 6: (color online) Phase space plot of many quantum tra-
The present data might suggest a more ambitious in-
jectories in the QSD ensemble (red points) for cases (a)–(d)
from Fig. 1, different times τ, and values of σ as indicated. terpretation. Apparently, all curves at finite σ in Fig. 8
In all cases, the two innermost classical orbits from Fig. 1 are show simple periodic oscillations even if (at σ = 0.05)
includedassolidcurves. Incase(b)thesecondorbitismissing. most weight in the Wigner function is still on the inner—
classically chaotic—orbit. To a certain extent, quantum
mechanics protects the optomechanical system against
Quantum multistability and classical orbits. –
classical chaotic dynamics. Initially, the quantum state
The quantum multistability observed for case (a) from
cannot follow the intricate chaotic orbit curve because it
Fig. 1 depends on the presence of at least two classical
occupies a finite part of phase space. Because of phase
orbits between which the quantum trajectories can move.
space averaging the chaotic motion is replaced by simple
The remaining cases (b)–(d) are variations of this situa-
oscillations at the fundamental system (i.e., cantilever)
tion, where either the second orbit is missing (case (b))
frequency. Later, the quantum trajectories move to the
or the nature of the first orbit has changed (cases (c),
second—classically simple periodic—orbit. At all times,
(d)). The four cases are compared in Fig. 6 with phase
the chaotic classical dynamics is replaced by clearly de-
space plots of many quantum trajectories that represent
fined simple oscillations in the quantum regime. Notice
the QSD ensemble for the density matrix.
that we here discuss possible signatures of classical chaos
In all cases the time scale relevant for quantum mul-
in the associated dissipative quantum dynamics and not
tistability shortens with increasing σ because the quan-
in quantities such as the level statistics that are defined
tumtrajectoriesleavethefirstclassicalorbitmorerapidly
for conservative Hamiltonian systems only [25,26].
whenthenoisetermsbecomelarger. Fortoolargeσ (e.g.,
σ = 0.3 in case (a)) the clear dynamical pattern of quan- Conclusions. – In this paper we establish the quan-
tum multistability—the movement between different clas- tum mechanical counterpart of the classical multistability
sical orbits—disappears altogether. of optomechanical systems. While classical multistabil-
In case (b) the quantum trajectories cannot settle on a ity corresponds to the coexistence of self-sustained oscil-
nearby classical orbit once they left the first orbit. Quan- lations at multiple amplitudes, quantum multistability is
tum multistability, which is characterized by the preva- a dynamical effect in which the amplitude of oscillations
lence of oscillatory motion over random diffusion, cannot changesovertime. Thechangecanbedetectedwithphase
be observed in such a situation. space techniques such as the Wigner function, and ana-
Incases(c),(d)theinnerorbitisnolongersimplerperi- lyzed quantitatively with autocorrelation functions.
odic but a period-two orbit after the first period doubling Quantum multistability is observed close to the classi-
bifurcation on the route to chaos (case (c)) or a chaotic cal limit. There, the quantum trajectories in the QSD
orbit (case (d)). Quantum multistability is not affected picture of dissipative dynamics are well localized in phase
bythedifferentnatureoftheinnerclassicalorbit,because space. Quantum multistability results from corrections
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C. Schulz et al.
σ=0 1 ∗∗∗
1 )
τ
x0 δ( This work was supported by Deutsche Forschungsge-
0 τ
R
-1 meinschaft via Sonderforschungsbereich 652 (project B5).
σ=0.05 2
1 )
τ
x0 0δ( REFERENCES
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