The lognormal-like statistics of a stochastic squeeze process

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by  Dekel Shapira

The lognormal-like statistics of a stochastic squeeze process
Dekel Shapira, Doron Cohen
Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
Weanalyzethefullstatisticsofastochasticsqueezeprocess. Themodel’stwoparametersarethe
bare stretching rate w, and the angular diffusion coefficient D. We carry out an exact analysis to
determinethedriftandthediffusioncoefficientoflog(r),whereristheradialcoordinate. Theresults
gobeyondtheheuristiclognormaldescriptionthatisimpliedbythecentrallimittheorem. Contrary
tothecommon”QuantumZeno”approximation,theradialdiffusionisnotsimplyD =(1/8)w2/D,
r
but has a non-monotonic dependence on w/D. Furthermore, the calculation of the radial moments
is dominated by the far non-Gaussian tails of the log(r) distribution.
7
1
0
2 I. INTRODUCTION it agrees with numerical simulations only for extremely
short times for which (w2/D)t(cid:28)1. The semiclassical
n
a Inthispaperweanalyzethefullstatisticsofastochas- explanation is as follows: In each τ-step of the evolution
J tic squeeze process that is described by the Langevin the phase-space distribution is stretched by a random
5 equation (Stratonovich interpretation): factor λn =exp[αn], where the αn are uncorrelated ran-
dom variables. Hence by the central limit theorem the
] x˙ = wx + ω(t)y product λ=λ ...λ λ has lognormal distribution, where
t 2 1
h y˙ = −wy − ω(t)x (1) log(λ) has some average µ∝t and variance σ2 ∝t that
c
determine an A(t) that differs from the naive expression
e
m where the average rotation frequency is (cid:104)ω(t)(cid:105)=0, with of Eq.(3). The essence of the QZE is that µ and σ2
fluctuations that looks like white noise: are inversely proportional to the intensity of the erratic
-
t driving. Consequently one has to distinguish between 3
a (cid:104)ω(t(cid:48))ω(t(cid:48)(cid:48))(cid:105) = 2Dδ(t(cid:48)−t(cid:48)(cid:48)) (2)
time scales: the “classical” time for phase ergodization
t
s Accordingly the model has two parameters: the angular τ ∼D−1 which is related to the angular diffusion; the
.
at diffusion coefficient D of the polar phase, and the bare “classical” time for loss of isotropy tr ∼(w2/D)−1 that
m stretching rate w of the radial coordinate r =(cid:112)x2+y2. characterizes the radial spreading; and the “quantum”
- The squeeze operation is of interest in numerous fields coherence time tc ∼r0−2tr, after which P(t)(cid:28)1.
d of science and engineering, but our main motivation In [9] the time dependence of µ and σ has been de-
n originates from the quantum mechanical arena, where termined numerically. Here we would like to work out a
o it is known as parametric amplification. In particular properanalyticaltheory. Itturnsoutthataquantitative
c
[ it describes the dynamics of a Bosonic Josephson junc- analysisofthestochasticsqueezingprocessrequirestogo
tion, given that all the particles are initially condensed beyond the above heuristic description. The complica-
1 in the upper orbital (see [1] and references therein). tion arises because what we have is not multiplication of
v Such preparation is unstable. By introducing noise, random number, but multiplication of random matrices.
1
quantum decoherence of the initial preparation can be Furthermoreweshallseethatthecalculationofmoments
8
3 slowed down, which is the so-called “quantum Zeno ef- requirestogobeyondcentrallimittheorem,becausethey
1 fects” (QZE) [2–6]. The stochastic squeeze process has are dominated by the far tails of the distribution.
0 been recognized as a prototype example for this QZE Outline.– WeintroducethemodelinSectionII.The
1. [7, 8]. The common perspective is to say that the noise phase randomization and the radial spreading are dis-
0 is like a measurement: it randomizes the phase over a cussedinSectionsIIIandIV.Theexactcalculationofthe
7 time scale τ ∼1/D, hence introducing “collapse” of the log(r) diffusion is presented in Sections V and VI, while
1 wave-function. The succession of such interventions lead the r moments are analyzed in Sections VII and VIII.
v: to relatively slow exponential decay of the decoherence, Some extra details regarding the QZE perspective and
i namely P(t)=exp{−[A(t)−A(0)]}, where other technicalities are provided in the Appendices.
X
r (cid:18)w2(cid:19)
a A(t)−A(0) = r2 t (3)
0 D II. THE MODEL
The stronger the noise (D), the slower is the decay
Foraparticularrealizationofω(t)theevolutionthatis
ofP(t). Usingasemiclassicalperspective[9]thefunction
A(t)isidentifiedwiththe(cid:10)r2(cid:11)ofanevolvingphase-space generatedbyEq.(1)isrepresentedbyasymplecticmatrix
dofisatrmibiuntiimona.lwInavtehpeaacbkeotveinfosrcmaluedlaurn0itiss,tmheeainniitnigaltrhaadtiur02s (cid:18)xy((tt))(cid:19) = U(cid:18)xy00(cid:19) (4)
corresponds to a Planck-cell in phase space.
Ithasbeenrealized[9]thatEq.(3),inspiteofitspopu- The matrix is characterized by its trace a=trace(U). If
larity, poorly describes the decoherence process. In fact, |a|<2 it means elliptic matrix (rotation). If |a| > 2 it
2
means hyperbolic matrix. In the latter case, the radial
coordinate r is stretched in one major direction by some 1.00
factor exp(α), while in the other major direction it is
squeezed by factor exp(−α). Hence a=±2cosh(α). For 0.75
more details see Appendix A. w
/
The numerical procedure of generating a stochastic r 0.50
w
process that is described by Eq.(1) is explained in Ap-
pendixB.Rarelytheresultisarotation. Sofromnowon 0.25
we refer to it as “squeeze”. The squeeze operation is of
greatimportanceinQuantumOptics. Theimplicationof 0.00
0 10 20 30 40 50
a squeeze on the “coherence” of a minimal wavepacket,
w/D
and the relation of a stochastic squeeze process to the
QZE, are summarized in Appendix C.
Belowwearenotusingamatrixlanguage,butaddress
FIG. 1. Scaled stretching rate w /w versus w/D. The nu-
r
directly the statistical properties of an evolving distribu-
mericalresults(blacksymbols)arebasedonsimulationswith
tion. In (ϕ,r) polar coordinates Eq.(1) takes the form 2000 realizations. The lines are for the naive result Eq.(10)
(green dotted); the exact result Eq.(20) (red solid); and its
ϕ˙ = −wsin(2ϕ) + ω(t) (5)
practical approximation Eq.(21) (blue dashed-dotted). For
r˙ = [wcos(2ϕ)] r (6) large values of w/D we get w /w=1, as for a pure stretch.
r
We see the equation for the phase decouples, while for
the radius 0.20
d
ln(r(t)) = wcos(2ϕ) (7) 0.15
dt
w
The RHS has some finite correlation time τ ∼1/D, and / 0.10
therefore ln(r) is like a sum of t/τ uncorrelated random Dr
variables. It follows from the central limit theorem that
0.05
forlongtimethemainbodyoftheln(r)distributioncan
be approximated by a normal distribution, with some
average µ∝t, and some variance σ2 ∝t. Consequently 0.000 5 10 15 20
we can define a radial stretching rate w and a radial
r w/D
diffusioncoefficientD viathefollowingasymptotictime
r
dependence:
FIG. 2. Scaled diffusion coefficient D /w versus w/D. The
r
numerical results (black symbols) are based on simulations
µ = w t (8)
r with 2000 realizations. The lines are for the naive result
σ2 = 2Drt (9) Eq.(11) (green dotted); the exact result Eq.(30) (red solid);
andtheapproximationEq.(25)withτ =1/(2D)(bluedashed-
Ourobjectiveistofindexplicitexpressionforw andD ,
r r dotted), and with Eq.(31) (dashed orange line).
andalsotocharacterizethefullstatisticsofr(t)interms
ofthesetwoparameters,thataredeterminedbythebare
modelparameters(w,D). Weshallseethatthestatistics
III. PHASE ERGODIZATION
of r(t) is described by a bounded lognormal distribution.
Some rough estimates are in order. For large D
The Fokker-Planck equation that is associated with
one naively assumes that due to ergodization of the
Eq.(5) is
phase µ=(cid:104)cos(2ϕ)(cid:105)w is zero, while σ2 ∼(wτ)2(t/τ).
Hence one deduces that wr →0 while Dr ∝w2/D. A ∂ρ ∂ (cid:20)(cid:18) ∂ (cid:19) (cid:21)
more careful approach that takes into account the non- = D +wsin(2ϕ) ρ (12)
∂t ∂ϕ ∂ϕ
isotropic distribution of the phase gives [9] the asymp-
totic results
It has the canonical steady state solution
w2
w ∼ (10) (cid:104) w (cid:105)
r 4D ρ∞(ϕ) ∝ exp 2D cos(2ϕ) (13)
w2
D ∼ (11)
r 8D If we neglect the cosine potential in Eq.(12) then the
time for ergodization is τ ∼1/D. But if w/D is large
The dimensionless parameter that controls the accuracy erg
we have to incorporate an activation factor, accordingly
of this result is w/D. These approximations are satis-
factory for w/D (cid:28)1, and fails otherwise, see Fig.1 and 1 (cid:104)w(cid:105)
Fig.2. For large w/D we get wr →w, while Dr →0. τerg = D exp D (14)
3
1.00 1.0
(a) (b)
n0.5
X
0.75
0.0
) 0.4
φ 0.50
(
ρ n
Δ0.2
0.25
0.0
4×10-4
0.00-π -π/2 0 π/2 π -π/2 0 π/2 Δn 2×10-4
0×10-4
φ φ 1 2 3 4 5
n
FIG. 3. (a) Phase distribution for (w/D) = 10/3 after
time (wt) = 6, with initial conditions ϕ = 0 (filled, yellow) FIG. 4. (a) The values of X versus n for some values of
n
and ϕ = π/2 (green bars) with 2000 realizations. For larger w/D. Frombottomtotopw/D=1,2,3,4,5. (b)Thevalues
times,bothreachthesteadystateofEq.(13)(redline). (b) of∆ versusnforthesamevaluesofw/D,fromtoptobottom
n
The distributions of ϕ modulo π. at n=1. (c) ∆ versus n for large w/D. Here w/D = 400.
n
The asymptotic approximation Eq.(19) is indicated by blue
line.
Fig.3(a)showsthedistributionofthephasefortwodiffer-
entinitialconditions,asobtainedbyafinitetimenumeri-
calsimulation. Itiscomparedwiththesteadystatesolu-
IV. RADIAL SPREADING
tion. Thedynamicsofrdependsonlyon2ϕ,andisdom-
inated by the distribution at the vicinity of cos(2ϕ)∼1.
If follows from Eq.(7) that the radial stretching rate is
We therefore display in Fig.3(b) the distribution of ϕ
modulo π. We deduce that the transient time of the w = w(cid:104)cos(2ϕ)(cid:105) = X w (20)
r ∞ 1
ln(r) spreading is much shorter than τ .
erg
For the later calculation of w we have to know the AroughinterpolationforX thatisbasedontheasymp-
r 1
moments of the angular distribution, namely, toticexpressionsfortheBesselfunctionsinEq.(15)leads
I (cid:0) w (cid:1) to the following approximation
X ≡ (cid:104)cos(2nϕ)(cid:105) = n 2D (15)
n ∞ I (cid:0) w (cid:1) (cid:104) (cid:16) w (cid:17)(cid:105)
0 2D wr ≈ w 1−exp −4D (21)
Here I (z) are the modified Bessel functions. For small
n
z wehaveI (z)≈[1/n!](z/2)n,whileforlargez wehave The exact result as well as the approximation are illus-
n
I (z)≈(2πz)−1/2ez. ThedependenceoftheX onnfor trated in Fig.1 and compared with the results of numer-
n n
representative values of w/D is illustrated in the upper ical simulations.
panel of Fig.4. For the second moment it follows from Eq.(7) that the
For the later calculation of D we have to know also radial diffusion coefficient is
r
the temporal correlations. We define (cid:90) ∞
D = w2 C (t)dt = c w2 (22)
C (t)=(cid:104)cos(2nϕ )cos(2ϕ)(cid:105) −X X (16) r 1 1
n t ∞ n 1 0
whereaconstantissubtractedsuchthatC (∞)=0. We
n If we assume that the ergodic angular distribution is
use the notations
isotropic, the calculation of C (t) becomes very simple,
(cid:90) ∞ 1
namely,
c ≡ C (t)dt (17)
n n
0
1 1
and C (t) = (cid:104)cos2(ϕ −ϕ )(cid:105) = e−4D|t| (23)
1 2 t 0 2
1
∆ ≡ C (0) = (X +X )−X X (18)
n n 2 n+1 n−1 n 1 This expression implies a correlation time τ =1/(2D),
such that c =(1/2)∆ τ is half the “area” of the corre-
In order to find an asymptotic expression we use 1 1
lationfunctionwhose“height”is∆ =1/2. Thusweget
I (z)≈ √ez (cid:20)1− 4n2−1 + (4n2−1)(4n2−9)(cid:21) for the radial diffusion coefficient D1r =w2/(8D).
n 2πz (8z) 2(8z)2 But in fact the ergodic angular distribution is not
and get isotropic, meaning that X1 is not zero, and ∆1 <1/2.
If w is not too large we may assume that the correlation
(cid:16)w(cid:17)−2 (cid:16)w(cid:17)
∆ ≈ 2 n2 for (cid:29)1 (19) time τ is not affected. Then it follows that a reasonable
n D D
approximation for the correlation function is
Thedependenceofthe∆ onnforrepresentativevalues
n
of w/D is illustrated in the lower panel of Fig.4. C (t) ≈ ∆ e−2|t|/τ (24)
1 1
4
leading to therecursionbackwardsasexplainedinthenextsection.
The bottom line is the following expression
1 w2
D ≈ ∆ τw2 = ∆ (25)
r 2 1 14D (cid:88)∞ (−1)n
D = c w2 = − ∆ X w (30)
This approximation is compared to the exact result that r 1 n n n
n=1
wederivelaterinFig.2. Unliketheroughapproximation
D =w2/(8D), it captures the observed non-monotonic where X and ∆ are given by Eq.(15) and Eq.(18) re-
r n n
dependence of D versus w, but quantitatively it is an spectively.
r
over-estimate. The leading term approximation D ≈∆ X w is con-
r 1 1
sistentwiththeheuristicexpressionD ≈(1/2)∆ τw2 of
r 1
Eq.(25) upon the identification
V. THE EXACT CALCULATION OF THE
2 (cid:104) (cid:16) w (cid:17)(cid:105)
DIFFUSION COEFFICIENT
τ = 1−exp − (31)
w 4D
Propagatinganinitialdistributionρ0(ϕ)withtheFPE This expression reflects the crossover from diffusion-
Eq.(12) we define the moments: limited (τ ∝1/D) to drift-limited (τ ∝1/w) spreading.
Fig.2 compares the approximation that is based on
x = (cid:104)cos(2nϕ )(cid:105) = (cid:104)cos(2nϕ)(cid:105)
n t 0 t Eq.(25) with Eq.(31) to the exact result Eq.(30).
(cid:90)
= cos(2nϕ) ρ (ϕ)dϕ (26) In the limit (w/D) → 0 the asymptotic result for the
t
radial diffusion coefficient is D =w2/(8D). We now
r
turn to figure out what is the asymptotic result in the
The moments equation of motion is [10]:
other extreme limit (w/D)→∞. The large w/D ap-
d proximation that is based on the first term of Eq.(30),
x = −Λ x + W (x −x ) (27)
dt n n n n n−1 n+1 with the limiting value X1 =1, provides the asymptotic
estimate D ≈2D2/w. This expression is based on the
where Λ =4Dn2 and W = wn. Due to Λ =W =0 r
n n 0 0 asymptotic result Eq.(19) for ∆ with n=1. In fact we
thezerothmomentx =1doesnotchangeintime. Thus n
0 can do better and add all the higher order terms. Using
the rank of Eq.(26) is less than its dimension reflecting
Abel summation we get
the existence of a zero mode x = X that corresponds
n n
tothesteadystateoftheFPE.Weshallusethesubscript D2 (cid:88)∞ 1D2
”∞”toindicatethesteadystatedistribution. Anyother Dr = 2 w (−1)n−1n = 2 w (32)
solutionxn(t)goestoXn inthelongtimelimit,whileall n=1
the other modes are decaying. To find X the equation
n Thus the higher order terms merely add a factor 1/4 to
should be solved with the boundary condition X =0,
∞ theasymptoticresult. IfweusedEq.(25),wewouldhave
and normalized such that X =1. Clearly this is not
0 obtained the wrong prediction D ≈ D/2 that ignores
requiredinpractice: becausewealreadyknowthesteady r
the τ dependence of Eq.(31).
state solution Eq.(12), hence Eq.(15).
We define x (t;ϕ ) as the time-dependent solution for
n 0
an initial preparation ρ (ϕ)=δ(ϕ−ϕ ). Then we can
0 0
VI. DERIVATION OF THE RECURSIVE
express the correlation function of Eq.(16) as follows:
SOLUTION
C (t) = (cid:104)x (t;ϕ)cos(2ϕ)(cid:105) −X X (28)
n n ∞ n 1
We describe the procedure to solve the equation
By linearity the C (t) obey the same equation of motion
n
asthatofthexn(t),butwiththespecialinitialconditions −Wn+cn+1+Λncn−Wn−cn−1 = ∆n (33)
C (0)=∆ . Note that C (t)=0 at any time. In the
n n 0
infinite time limit C (∞)=0 for any n. A similar problem was solved in [11], here we present a
n
Our interest is in the area c as defined in Eq.(17). simpler derivation. First we solve the associated homo-
n
Writing Eq.(27) for C (t), and integrating it over time geneous equation. The solution c =X satisfies
n n n
we get
−W+X +Λ X −W−X = 0 (34)
n n+1 n n n n−1
Λ c − W (c −c ) = ∆ (29)
n n n n−1 n+1 n
andonecandefinetheratiosR =X /X . Notethat
n n n−1
Thisequationshouldbesolvedwiththeboundarycondi- these ratios satisfies a simple first-order recursive rela-
tions c0 =0 and c∞ =0. The solution is unique because tion. However we bypass this stage because we can ex-
then=0sitehasbeeneffectivelyremoved,andthetrun- tract the solution from the steady state distribution.
cated matrix is no longer with zero mode. One possible We write the solution of the non-homogeneous equa-
numericalprocedureistostartiteratingwithc1 asinitial tion as
condition, and to adjust it such that the solution will go
to zero at infinity. An optional procedure is to integrate c := X c˜ (35)
n n n
5
and we get the equation 2.0
−Wn+Xn+1c˜n+1+ΛnXnc˜n−Wn−Xn−1c˜n−1 = ∆n )t 1.5
w
Clearly it can be re-written as /(
> 1.0
2
−W+X (c˜ −c˜ )+W−X (c˜ −c˜ ) = ∆ <r
n n+1 n+1 n n n−1 n n−1 n
n 0.5
L (a)
We define the discrete derivative
0.0
a˜n := c˜n−c˜n−1 (36) 5
And obtain a reduction to a first-order equation: ) 4
t
w
−Wn+Xn+1a˜n+1+Wn−Xn−1a˜n = ∆n (37) >/( 3
4
r 2
This can be re-written in a simpler way by appropriate <
n
definition of scaled variables. Namely, we define the no- L 1 (b)
tations
0
W+ ∆ 0 5 10 15 20
R˜ = n R ∆˜ = n (38)
n W− n n W+ w/D
n n
and the rescaled variable FIG. 5. Scaled moments versus w/D. The red solid lines
are the exact results for the 2nd and 4th moments, given by
a := X a˜ (39)
n n n Eq.(45) and Eq.(60), and the large w/D asymptotic values
are at 2 and 4, respectively. These are compared with the
and then solve the a recursion in the backwards direc-
n numerical results (black symbols), and contrasted with the
tion:
Lognormal prediction (orange dashed lines). The later pro-
(cid:104) (cid:105) vides an overestimate for intermediate values of w/D.
a =0; a =R˜ ∆˜ +a (40)
∞ n n n n+1
If all the Rn were unity it would imply that a1 − a∞ expected results, which are w2/D and 3w2/D respec-
(cid:80)
equals ∆ . Soitisimportanttoverifythatthe”area”
n tively. For large w/D the dynamics is dominated by the
converges. Nextwecansolveintheforwarddirectionthe
stretching,meaningthatw ≈w,whileD →0,soagain
r r
c recursion for the non-homogeneous equation, namely,
n we have a trivial agreement. But for intermediate values
of w/D the lognormal moments constitute an overesti-
c =0; c =R c +a (41)
0 n n n−1 n
mate when compared with the exact analytical results
In fact we are only interested in that we derive in the next section. In fact also the exact
analytical result looks like an overestimate when com-
c1 = a1 = R˜1∆˜1+R˜1R˜2∆˜2+... (42) pared with the results of numerical simulations. But the
latter is clearly a sampling issue that is explained in Ap-
NotethatinourcalculationtheR˜ =−R ,andtherefore
n n pendix D.
R˜ ···R˜ =(−1)nX .
1 n n The deviation of the lognormal moments from the ex-
act results indicates that the statistics of large devia-
tions is not captured by the central limit theorem. This
VII. THE MOMENTS OF THE RADIAL
point is illuminated in Fig.6. The Gaussian approxima-
SPREADING
tion constitutes a good approximation for the body of
the distribution but not for the tails that dominate the
The moments of the lognormal distribution are given
moment-calculation. Clearly, the actual distribution has
by the following expression
a natural cutoff that is implied by the strict inequality
w <w. The stretching rate cannot be faster than w.
1 r
ln(cid:104)rn(cid:105) = µn + σ2n2 (43) Butinfact,asobservedinFig.6b,thedeviationfromthe
2
lognormal distribution happens even before the cutoff is
Hence,ifweassumethatthelognormalapproximationis reached.
valid, we deduce that Below we carry out an exact calculation for the 2nd
and 4th moments. In the former case we show that
d
ln(cid:104)rn(cid:105) = nw +n2D (44)
dt r r d ln(cid:104)r2(cid:105) ∼ 2(cid:16)(w2+D2)1/2−D(cid:17) (45)
dt
In Fig.5 we plot the lognormal-based expected growth-
rate of the 2nd and the 4th moments as a function of Thisagreewiththelognormal-basedpredictionw2/Dfor
w/D. Forsmallw/D thereisagoodagreementwiththe (w/D)(cid:28)1, and goes to 2w for (w/D)(cid:29)1, as could be
6
form
0.02
(a)
ρ 0.01 x˙j = vj +Gjω(t) (48)
0.00 (cid:104)ω(t)ω(t(cid:48))(cid:105)=2Dδ(t−t(cid:48)) (49)
0
The Stratonovich interpretation is used in order to as-
-2
) sociate with it an FPE, from which the Ito equation of
P
(n -4 motion for the moments can be derived, namely,
L -6 (b) (cid:28) (cid:18) (cid:19)(cid:29)
d ∂X ∂G
-8 (cid:104)X(cid:105) = v + jDG
dt ∂x j ∂x i
1250 1300 j i
Ln(r) (cid:28) ∂2X (cid:29)
+ G DG (50)
∂x ∂x j i
i j
FIG.6. (a)Distributionofln(r)forw/D=10/3aftertime
For the first moments we get
wt=2000withinitialconditionsr=1andϕ=0. Numerical
results(greenhistogram),thatarebasedon2000realizations,
(cid:104)x˙(cid:105)=w(cid:104)x(cid:105)−D(cid:104)x(cid:105) (51)
are fitted to a Gaussian distribution (blue line). (b) In-
verse cumulative probability of the same distribution. The (cid:104)y˙(cid:105)=−w(cid:104)y(cid:105)−D(cid:104)y(cid:105) (52)
black dotted line indicate the numerically determined value
ln(cid:10)r2(cid:11)1/2 ≈1323. This value is predominated by the tail with the solution
of the distribution. The Gaussian fit fails to reproduce this (cid:104)x(cid:105)=x exp[−(D−w)t] (53)
value, and provides a gross over-estimate ln(cid:10)r2(cid:11)1/2 ≈1701. 0
(cid:104)y(cid:105)=y exp[−(D+w)t] (54)
0
For the second moments
anticipated. d (cid:18)(cid:10)x2(cid:11)(cid:19) (cid:104) (cid:105)(cid:18)(cid:10)x2(cid:11)(cid:19)
Beforewegothederivationofthisresultwewouldlike dt (cid:10)y2(cid:11) = −2D+2Dσ1+2wσ3 (cid:10)y2(cid:11) (55)
to illuminates its main features by considering a simple-
d
minded reasoning. Let us ask ourselves what would be (cid:104)xy(cid:105) =−4D(cid:104)xy(cid:105) (56)
the result if the spreading was isotropic (w = 0). In dt
r
such case the moments of spreading can be calculated where σ are Pauli matrices. The solution is:
anstsuempifbiwesresτ.a=rNe1ad/m(e2aeDlliyn),g,aaswnsiudtmhtirtnehgaettimhnagutltttiphalseicadautdiroiasntcirooenfteroafinneddaoecmxh, (cid:10)(cid:10)xy22(cid:11)(cid:11)=(cid:20)e−2D0tM e−04Dt(cid:21) xy0220  (57)
(cid:104)xy(cid:105) x y
Eq.(6)impliesthatthespreadingisobtainedbymultipli- 0 0
cation of uncorrelated stretching factors exp[wτcos(ϕ)].
where M is the following matrix:
Each stretching exponent has zero mean and dispersion
σ12 =(1/2)[wτ]2, which implies Dr = σ12/(2τ). Conse- cosh[2(w2+D2)1/2t]+sinh[2(w2+D2)1/2t]D√σ1+wσ3
quently we get for the moments w2+D2
(cid:104)rn(cid:105) = (cid:104)(cid:68)enwτcos(2ϕ)(cid:69)(cid:105)t/τ rn (46) For an initial isotropic distribution we get (cid:104)r2(cid:105)t =Mr02,
0 where
leading to M =e−2Dtcosh[2(w2+D2)1/2t] (58)
D
d 1 (cid:104) (cid:16)√ (cid:17)(cid:105) +√ e−2Dtsinh[2(w2+D2)1/2t]
ln(cid:104)rn(cid:105) = ln I 2nσ (47) w2+D2
dt τ 0 1
Theshorttimetdependenceisquadratic,reflecting“bal-
This gives a crossover from n2D for σ (cid:28)1 to nw for
r 1 listic” spreading, while for long times
σ (cid:29)1, reflecting isotropic lognormal spreading in the
1
former case, and pure stretching in the latter case. So (cid:10)r2(cid:11) ≈ r02 (cid:18)1+ √ D (cid:19)×
again we see that the asymptotic limits are easily under- t 2 w2+D2
stood, butforthederivationofthecorrectinterpolation, (cid:104) (cid:16) (cid:17) (cid:105)
exp 2 (w2+D2)1/2−D t (59)
say Eq.(45), further effort is required.
FromherewegetEq.(45). Forthe4thmomentstheequa-
tions are separated into two blocks of even-even powers
VIII. THE EXACT CALCULATION OF THE
and odd-odd powers in x and y. For the even block:
MOMENTS
 (cid:10)x4(cid:11)   (cid:10)x4(cid:11) 
We turn to perform an exact calculation of the mo- d (cid:10)x2y2(cid:11) = 2M˜ (cid:10)x2y2(cid:11) (60)
ments. The Langevin equation Eq.(1) is of the general dt (cid:10)y4(cid:11) (cid:10)y4(cid:11)
7
where
t 1000 (a)
n
  u
2(w−D) 6D 0 o 500
M˜ =  D −6D D  (61) C 0
0 6D −2(w+D)
30 (b)
)r 25
(
The eigenvalues of this matrix are the solution of λ3 + n 20
10Dλ2 +(16D2 −4w2)λ−24Dw2 = 0. There are two L
15
negativeroots,andonepositiveroot. Forsmallw/D the
15 20 25 30
latter is λ≈(3/2)(w2/D), and we get that the growth-
Ln(|a|)
rateis3w2/D asexpectedfromthelog-normalstatistics.
FIG.7. Weconsider2000realizationsofastochasticsqueeze
process. For each realization the trace a=trace(U) is calcu-
lated. (a)Thecumulativecountoftheavalues. Greenpoints
IX. DISCUSSION
are for positive values, while blue rectangles are for negative
values. Here (w/D)=10/3 and wt=40. For simulations
Inthisworkwehavestudiedthestatisticsofastochas- with longer times the distribution of positive and negative
tic squeeze process, defined by Eq.(1). Consequently values become identical (not shown). (b) Scatter plot of |a|
we are able to provide a quantitatively valid theory for versus the radial coordinate r. For simulations with longer
times we get full correlation.
the description of the noise-affected decoherence pro-
cess in bimodal Bose-Einstein condensates, aka QZE.
As the ratio w/D is increased, the radial diffusion co-
Appendix A: The squeeze operation
efficient of ln(r) changes in a non-monotonic way from
D =w2/(8D) to D =D2/(2w), and the non-isotropy
r r
isenhanced,namelytheaveragestretchingrateincreases Thesqueezeoperationisdescribedbyarealsymplectic
from w =w2/(4D) to the bare value w =w. The ana- matrixthathasunitdeterminantandtrace|a|>2. Any
r r
lyticalresultsEq.(20)andEq.(30)areillustratedinFig.1 such matrix can be expressed as follows:
and Fig.2, (cid:18) (cid:19)
a b
Additionallywehavesolvedforthemomentsofr. One U = = ±eαH [ad−cb=1] (A1)
c d
observes that the central limit theorem is not enough for
this calculation, because the moments are predominated where H is a real traceless matrix that satisfies H2 =1.
by the non-Gaussian tails of the ln(r) distribution. In Hence it can be expressed as a linear combination of the
particular we have derived for the second moment the three Pauli matrices:
expression(cid:104)r2(cid:105) =Mr2 withM thatisgivenbyEq.(58),
t 0
or optionally one can use the practical approximation H = n1σ1+in2σ2+n3σ3 (A2)
Eq.(45).
with n2−n2+n2 =1. Consequently
The introduction of the present paper is in fact some- 1 2 3
what misleading. The standard literature of QZE fo- U = ±[cosh(α)1+sinh(α)H] (A3)
cuson“fringefeasibility”measurementthatcorresponds,
Wedefinethecanonicalformofthesqueezeoperationas
from a theoretical perspective, to the expectation value
S of some operator, let us call it xˆ. This leads to (cid:18) (cid:19)
x exp(α) 0
the definition of P(t) in terms of the Bloch vector, Λ = (A4)
0 exp(−α)
see Appendix C. In a semi-classical perspective (Wigner
function picture) the phase-space coordinate x satisfies Then we can obtain any general squeeze operation via
Eq.(1), where ω(t) arises from frequent interventions, or similarity transformation that involves re-scaling of the
measurements, or noise that comes from the surround- axes and rotation, and on top an optional reflection.
ing. Hence each realization of x can be regarded as the
resultofonerunoftheexperiment,andP(t)isidentified
with the second moment of the spreading. But it is im- Appendix B: Numerical simulations
plied by our discussion of the sampling problem that it is
impracticaltodeterminethissecondmoment, andhence
For the purpose of simulations we divide the time into
it is impractical to extract a result for P(t) from any re-
small dt intervals (dt (cid:28) τ), and regard t as a discrete
alistic experiment. The reliable experimental procedure
index. The evolution of a vector r = (x ,y ) during
t t t
would be to keep the full probability distribution of the
each interval is given by r =U r , where
t+dt t t
measuredxvariable,andtoextracttheµandtheσ that
characterize its lognormal statistics. For the latter we (cid:18)cosα −sinα (cid:19) (cid:18)ewdt 0 (cid:19)
U = t t (B1)
predict non-trivial dependence on w/D. t sinα cosα 0 e−wdt
t t
8
The uncorrelated random variables αt have ze√ro mean, 500
andaretakenfrom aboxdistribution ofwidth 24Ddt,
suchthattheirvarianceis2Ddt. Theevolutionovertime 400
t is obtained by multiplication of t/dt matrices. The
radial coordinate r is calculated under the assumption 2> 300
r
that the the preparation is (x =1,y =0. Accordingly, <
0 0 n 200
what we calculate for each realization is L
(cid:113)
100
r = U2 +U2 (B2)
xx yx
In Fig.7a we display the distribution of the trace a 0
0 50 100 150 200 250
for many realizations of such stochastic squeeze process.
σ2
Rarely the result is a rotation, and therefore in the main
text we refer to it as “squeeze”. From the trace we get
FIG.8. ln(cid:10)r2(cid:11)versusσforLognormaldistribution. Without
the squeeze exponent α, and from Eq.(B2) we get the
lossofgeneralityµ=0. Thetrueresultisrepresentedbyred
radial coordinate r. The correlation between these two
line. Numericalestimatebasedon102and105realizationsare
squeeze measures is illustrated in Fig.7b. For the long
indicated by green crosses and blue rectangles, respectively.
timesimulationsthatweperforminordertoextractvar-
For the latter set of realization we get an en estimate using
ious moments, we observe full correlation (not shown). anoptionalprocedure(blackdots). Namely,wecalculatethe
In order to extract the various moments, we perform the sample average and the sample variance of the lnr values in
simulation for a maximum time of wt = 7500, with the ordertodetermineµandσ,andthenuseEq.(43)toestimate
initial condition r =(1,0). the moments.
0
We note that the results of Section VIII for the evolu-
tion of the moments can be recovered by averaging over
productoftheevolutionmatrices. Forthefirstmoments Appendix D: Sample moments of a lognormal
distribution
we get the linear relation (cid:104)r (cid:105)=(cid:104)U(cid:105)r , where
t 0
(cid:104)U(cid:105) = (cid:104)... Ut3 Ut2 Ut1(cid:105) = [(cid:104)Ut(cid:105)]t/dt Consider a lognormal distribution of r values. This
(cid:18)e−(D+w)t 0 (cid:19) mean that the lnr values have a Gaussian distribution.
= (B3)
0 e−(D−w)t For a finite sample of N values, one can calculate the
sample average and the sample variance of the lnr val-
Similar procedure can be applied for the calculation of
ues in order to get a reliable estimate for µ and σ, and
the higher moments. then calculate the moments (cid:104)rn(cid:105) via Eq.(43). But a di-
rectcalculationofthesemomentsprovidesagrossunder-
estimateasillustratedinFig.8. Thisisbecausethedirect
Appendix C: Relation to QZE
averageispredominatedbyrarevaluesthatbelongtothe
tail of the distribution.
In[9]thedecayofthecoherenceofabosonicJosephson The lesson is that direct calculation of moments for
junction has been characterized by P(t)=|S(t)|2, where log-widedistributioncannotbetrusted. Itcanprovidea
S(t) is the length of the Bloch vector. For a noiseless lower bound to the true results, not an actual estimate.
canonical squeeze operation with α=wt one can show
that
(cid:26) (cid:27)
1
|S(t)|=exp − [A(t)−A(0)] (C1)
2
where A(t)=(cid:104)r2(cid:105) . Setting x=x eα and y =x e−α we
t 0 0
get (cid:104)r2(cid:105)=r2cosh(2α) leading to
0
A(t) = cosh(2wt) r2 (C2)
0
Hence[A(t)−A(0)]=[2sinh2(wt)]r2, whichisquadratic
0
for short times.
In a stochastic squeeze process one assumes that for
strong D the spreading A(t) is a multiplication of t/τ
factors cosh(2α)≈1+2α2 with α=wτ. The time step
for phase randomization is τ =1/(2D), leading to
A(t) = exp[(w2/D)t] r2 (C3)
0
The standard QZE expression Eq.(3) is recovered in the
shorttimeregimeτ (cid:28)t(cid:28)t ,duringwhichthedeviation
r
fromisotropycanbetreatedasafirst-orderperturbation.
9
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