A Monte Carlo Method for Modeling Thermal Damping: Beyond the Brownian-Motion Master Equation

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by  Kurt Jacobs

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A Monte Carlo Method for Modeling Thermal Damping: Beyond
the Brownian-Motion Master Equation
Kurt Jacobs
9
0Department of Physics, University of Massachusetts at Boston, 100 Morrissey Blvd, Boston, MA 02125, USA
0
2

n
a PACS 05.40.Jc–Brownian motion
J PACS 03.65.Yz–Decoherence; open systems; quantumstatistical methods
6 PACS 03.65.Ta–Foundations of quantummechanics; measurement theory
PACS 85.85.+j–Micro- and nano-electromechanical systems (MEMS/NEMS) and devices

]
h Abstract.-The“standard”Brownianmotionmasterequation,usedtodescribethermaldamping,
p
is not completely positive, and does not admit a Monte Carlo method, important in numerical
-
t simulations. To eliminate both these problems one must add a term that generates additional
n positiondiffusion. HeweshowthatonecanobtainacompletelypositivesimplequantumBrownian
a
motion, efficiently solvable, without any extra diffusion. This is achieved by using a stochastic
u
Schr¨odinger equation (SSE), closely analogous to Langevin’s equation, that has no equivalent
q
Markovian master equation. Considering a specific example, we show that this SSE is sensitive
[
tononlinearities in situations in which themaster equation is not,and may therefore be abetter

3 model of damping for nonlinear systems.
v
1
1
2
4
.
7 All mechanical oscillators experience frictional damp- et al. [7], building on the work of Unruh and Zurek [8].
0ing, in which they lose energy to their environment. This They showed that an exact master equation for the oscil-
8
damping is accompanied by thermalization of the oscil- lator alonecouldbe derived,andthat anynon-markovian
0
lator, since the environment is a large system, and thus effects of the bath appear merely as time-dependent coef-
:
va thermal bath. One would naturally like to model the ficients in this equation. This master equation, the exact
i
Xeffects of this damping on the oscillator, and indirectly BME, is [7]
on any other devices to which it is coupled, without hav-
r
aing to describe the motion of the environment, with its ρ˙ = iΓ(t)[x, p,ρ ] ξ(t)[x,[x,ρ]]+ζ(t)[x,[p,ρ]]. (1)
− { } −
many degrees of freedom. Frictional damping of quantum
Here ρ is the density matrix for the oscillator, p,ρ =
systems by an environment, often referred to as “quan- { }
pρ+ρpistheanti-commutator,andxandparethedimen-
tum Brownian motion”, has many applications. In par-
sionless position and momentum of the oscillator, defined
ticular, it is essential in describing the behavior of nano-
as
mechanical resonators, and is therefore of current inter-
est in quantum nano-electro-mechanics (QNEMS) [1,2] x = (a+a†)/√2, (2)
and quantum opto-mechanics [3,4]. For classicalsystems,
p = i(a a†)/√2, (3)
Langevin’s equation [5], containing only a deterministic − −
frictional force and a Gaussian white noise source, gives
whereaistheoscillator’sannihilationoperator. Thethree
a simple and excellent model of damping and thermal-
functionsoftime,Γ,ξ andζ,arethetime-dependentcoef-
ization. The situation is much less simple for quantum
ficients thatencode allnon-Markovianeffects ofthe bath,
systems, however.
and are determined by the frequency of the resonator, ω,
The standard approach to obtaining a quantum equa- thecouplingtothebath(whichsetsthedampingrate,γ),
tion to model damping from a thermal bath is to derive a the structure of the bath (ohmic, super-ohmic, etc.), and
master equation for a linear oscillator by coupling it to a the initialbathstate (whichis determinedbythe temper-
continuum of oscillators, and then trace out these oscilla- ature, T). For reference, the physical position and mo-
tors. This was firstdone by Caldeira andLeggett [6], and mentum are X = ¯h/(ωm)x and P =√¯hωmp, where m
was brought to its completion in the tour-de-force by Hu is themassandωpthefrequencyofthe oscillator. Wenote
p-1
K. Jacobs
that the master equation that results from using a bath They also generate a non-physicalterm in the fluctuation
oftwo-levelsystems, rather than oscillators,has the same spectrum of the resonator [14] (due to the fact that the
form as Eq.(1) [9]. evolutionisnotcompletelypositive[15,16]),althoughthis
There are three issues with using the BME to model does not usually cause problems in the time domain.
damping. Thefirstisthatthe time-dependentcoefficients As firstpointed out by Diosi[17], one canobtaina new
areingeneralrathercomplex,andmustbe calculatednu- BME that does have an equivalent SSE by adding a term
merically for a given problem [7]. The second is that this to the LBME or SBME. The low-temperature version,
equationcannotbewrittenintheLindbladform[10]. Be- which we will refer to here as the “completely positive”
cause of this it has no equivalent stochastic Schr¨odinger BME (PBME) is
equation (SSE, also referred to as an “unravelling”) [11],
γ γ
andthereforecannotbesimulatedusingthewave-function ρ˙ = i [x, p,ρ ] (2n +1)[x,[x,ρ]]
T
− 4 { } − 4
Monte Carlo method [12]. This method significantly re-
γ
duces the numericaloverheadinvolvedin simulating mas- [p,[p,ρ]]. (7)
−16(2n +1)
ter equations, and is therefore of considerable practical T
importance. Thirdly, this master equation only generates Theextratermgeneratesdiffusioninposition. Thisequa-
the correctthermalsteady-state for linear oscillators,and tion is a considerable improvement, as it is now a valid
many questions of interest for mesoscopic oscillators in- quantumevolution,and canbe simulatedusing the wave-
volve non-lineairites of the oscillators themselves, or in- function Monte Carlo method. However, the PBME no-
teractions with other nonlinear devices (such interactions longer gives the correct thermal steady-state for a linear
effectively turn an oscillator into a nonlinear system). As oscillator. Thisisbecausethesteady-stateaveragephonon
far as the author is aware,no master equations havebeen number is increased by the position diffusion. Note that
derived for damping of nonlinear oscillators. Instead, the the position diffusion term in the PBME is not part of
problemis avoided: since it is usually the case in practice the exact BME derived by Hu et al. Nevertheless, a mas-
thatthenonlinearitiesaresmallcomparedtotheharmonic ter equation of the form given in Eq.(7) can be derived
motion, one assumes that the BME (and the SSE to be as an approximation to the dynamics induced by a colli-
presentedshortly)willgivereasonableresults. Ideallyone sional bath (rather than a linear bath of harmonic oscil-
would like to improve upon this situation, however, and lators) [18–22,24]), and the high temperature version of
withthisinmindweconsiderthebehaviorofvariousBME Eq.(7) has also been derived as an approximate limit of a
variants,aswellthatoftheSSE,foranexamplenonlinear system interacting with a bath of oscillators [23].
system in detail below.
We now present a quite different approach to mod-
Thestandardapproachtoaddressingthefirstissueisto
eling damping and thermal noise in mesoscopic oscilla-
chooseconstantvaluesforthetime-dependentcoefficients,
tors. This involves directly constructing a stochastic
so as to keep the resulting BME accurate to a good ap-
Schr¨odingerequationforthispurpose,ratherthanfirstde-
proximation. Different choices for these values produce
riving a master equation. In fact, while the resulting SSE
different variants. For oscillators at low temperatures,
is straightforwardto simulate [25], it does not correspond
therelevantregimeforQNEMS,areasonableapproximate to any Markovian master equation. The SSE satisfies all
version is [13] the properties required for the thermal damping of a lin-
γ γ ear system, in particular generating the correct thermal
ρ˙ = i [x, p,ρ ] (2n +1)[x,[x,ρ]]. (4)
− 4 { } − 4 T steady-state. Since this equation has no additional posi-
tion diffusion, it can be regarded as a genuine quantum
Heren istheaveragenumberofphononsintheresonator
T
version of simple damped Brownian motion. Further the
when it is at temperature T, being
motion of the mean values of position and momentum for
1 a given trajectory are precisely those given by Langevin’s
n = , (5)
T
exp[h¯ω/(k T)] 1 equation (They contain a fluctuating force that gener-
B
−
ates momentum diffusion.) The SSE may therefore be
withk Boltzmann’sconstant. Notethatinwhatfollows,
B regardedasa quantumequivalentofLangevin’sequation.
n will always be defined as the specific function of T
T Performing simulations of a particular example, we show
given in Eq.(5). We will refer to Eq.(4) as the “low tem-
thatwhiletheSSEdoesnotproducethecorrect(thermal)
peratureBME”(LBME).Forcompletenesswenotethata
steady-statefornonlinearsystems,itdoesbetterthanthe
similar variant,sometimes refereedto in the quantumop-
variousversionsofthe Brownianmotion masterequation.
tics literature as the “standard” BME (SBME), is given
The SSE we propose to model thermal damping is
by
γ k T
ρ˙ = i [x, p,ρ ] γ B [x,[x,ρ]]. (6) see Eq.(9). (8)
− 4 { } − (cid:18)2h¯ω(cid:19)
This SBME is a reasonable approximation to the BME Here H is the oscillator Hamiltonian, and the incre-
in the joint regime of high temperature and weak damp- ments dW and dW are mutually independent Wiener
1 2
ing [13]. These two variants still have no equivalent SSE. noiseincrementssatisfyingtheItocalculusrelationdW2 =
1
p-2
A Monte Carlo Method for Modeling Thermal Damping: Beyond the Brownian-Motion Master Equation
H γn i
dψ = i T x2 p2 ψ dt + 2γ n x x+n p p+ p x ψ dt
T T
| i (cid:20)− ¯h − 2 − (cid:21)| i (cid:18) h i h i 4h i (cid:19)| i
(cid:0) (cid:1)
x 1 p
+2√γn +iV p i C x ψ dW + 2√γn iV x+iC p ψ dW . (9)
T x xp 1 T p xp 2
(cid:18)2 − (cid:20) − 2(cid:21) (cid:19)| i (cid:16)2 − (cid:17)| i
dW2 =dt [11,26]. As is usual, the means, variances, and oscillator, if we make two simultaneous continuous mea-
2
covarianceofthepositionandmomentumforthestate ψ surements, one of x and the other of p, this gives an SSE
| i
are denoted by x = ψ xψ , p = ψ pψ , V = x2 whosesteady-stateisacoherentstatewithrandomlyfluc-
x
h i h | | i h i h | | i h i−
x 2, V = p2 p 2, and C = xp+px /2 x p . tuatingmeanpositionandmomentum. TheSSE,normal-
p xp
h i h i−h i h i −h ih i
To simplify the presentation, the above SSE is unnor- ized this time, for this joint measurement is [11]
malized [11]. This means that to simulate the SSE, at
each time-step dt one adds the increment dψ , and then see Eq.(15). (14)
| i
must normalize the state before adding the next incre-
where k determines the rate at which the measure-
ment. To obtain the density matrix for an oscillator un-
mentextractsinformation(oftencalledthe“measurement
dergoingBrownianmotion,onesimulatestheaboveequa-
strength”). The equations of motion for the means that
tion for many realizations of the noise processes dW . If
i
result from this SSE are
welabelthestatesresultingfromeachoftheN noisepro-
cess as ψ (t) , j = 1,...,N, then the density matrix for
theTshyestS|eSmEj a(tEitqi.m(e9)t)ihsaρs(tt)he=fo(1ll/oNwi)nPgjp|rψopj(etr)tiiheψs:j(t)|. ddhxpi == √√88kkVVxpddWW21++√√88kkCCxxppddWW12,, ((1167))
h i
1. It gives a valid quantum evolution.
andtheequationsforthecovariances,whenthewavefunc-
2. It gives the Langevin equations for the mean values
tion for the state ψ is Gaussian, are
of position and momentum: | i
V˙ = 8kV2 8kC2 +2k, (18)
d x = i [x,H] dt, (10) x − x − xp
h i − h i γ V˙ = 8kV2 8kC2 +2k, (19)
d p = i [p,H] dt p dt+√γnTdW2. (11) p − p − xp
h i − h i − 2h i C˙ = 8k(V +V )C . (20)
xp x p xp
−
3. For a harmonic oscillator, the SSE reduces every
For Brownian motion we require that the mean position
initialstatetoacoherentstate(withrandomlyfluctuating
is not drivenby any noise, andthat the meanmomentum
mean position and momentum) as t .
The first two properties confirm t→hat∞the SSE realizes hastheequationdhpi=−(γ/2)hpidt+√γnTdW2. Wecan
produce the correct equations for the means by adding a
a quantum equivalent of classical damped Brownian mo-
linear feedback term, applied by a fictitious observer who
tion. The third shows that this evolution also gives the
has access to the stream of measurement results from the
samefinalstatesasthestandardBrownianmotionmaster
continuous measurements of x and p [27]. We choose the
equation. Together these show us that the SSE provides
feedbackHamiltoniantocancelthenoisedrivingthemean
a good description of simple damped quantum Brownian
position, and to add damping to the mean momentum.
motion. The finalpropertyisnowexpectedtofollow,and
The required feedback Hamiltonian is
we show below that it does:
4. Fora harmonicoscillatorthe SSEgivesthe expected
dW 1 dW
2 1
thermal steady-state density matrix, being Hfb = 2h¯√γnT Vp + Cxp x
(cid:18) dt (cid:20) − 2(cid:21) dt (cid:19)
∞ n γ p dW dW
1 n 1 2
T + h ix 2h¯√γn V +C p.
ρ = n n, (12) T x xp
ss 2 − (cid:18) dt dt (cid:19)
(cid:18)n +1(cid:19) (cid:18)n +1(cid:19) | ih |
T nX=0 T
With the addition of this fictitious feedback, there is only
where the states n are the phonon number states of the
| i one more thing we need to do to obtain the Brownian
resonator. This canalsobe written as a GaussianWigner
motion SSE. This is to fix the value of the measurement
function. The steady-state covariances are
strength, k. Since it is the thermal bath that causes
both the measurement (decoherence) and the damping,
1
Vxss =Vpss = 2 +nT, (13) k should be related to the damping rate, γ. To get this
relationship right, we thus choose k so that the quadratic
and Css =0. termsinourSSE(thetermsthatgivethedecoherencedue
xp
Constructing the SSE: To obtain an SSE with the four to the measurement) match those of the simple forms of
propertieslistedabove,webeginbynotingthatforalinear the standard Brownian motion master equation that give
p-3
K. Jacobs
dψ = k (x x )2+(p p )2 dtψ +√2k[(x x )dW +(p p )dW ] ψ (15)
1 2
| i − −h i −h i | i −h i −h i | i
(cid:2) (cid:3)
the same damping rate [13,18]. (These quadratic terms
also match the Lindblad master equation for the damp-
ing of an opticalcavity with the same damping rate [28]).
This sets k = (γn )/2, and results in the SSE given by
T
Eq.(9).
Byconstruction,theSSEgivesavalid(completelyposi-
tive)quantumevolution(property1above),andtheequa-
tionsofmotionfortheexpectationvaluesofxandpsatisfy
property 2. It is well-established that continuous mea-
surement of any linear combination of x and p produces
Gaussian states as t [29,30]. While to the authors
→ ∞
knowledge no formal proof of this has been given to date,
the proof is fairly straightforward, and follows from the
results in [31]. Once the wave-function is Gaussian, the
equations of motion for the covariancesare those givenin
Fig. 1: Here we show the evolution of the mean phonon num-
Eqs.(18)-(20),withtheadditionoftheharmonicoscillator
ber of a harmonic oscillator, hni, under three evolutions that
dynamics, given by
model damping and thermalization (Brownian motion). The
V˙ = 2ωC , (21) temperature parameter in the equations is chosen so that the
x xp
V˙ = 2ωC , (22) correctthermalsteady-statevalueofhniis1. a)Solidline: the
p − xp low temperature Brownian motion master equation (LBME);
C˙xp = ω(Vp Vx). (23) Dashed line: thecompletely positive version of thesame mas-
−
ter equation. b) Dark solid line: The stochastic Schr¨odinger
The the linear feedback Hamiltonian has no effect on the
equation described here, averaged over approximately 20,000
covariances. The resulting steady-state is V =V =1/2,
x p trajectories; LightSolidLine: TheresultfortheLBME,which
so the SSE satisfies property 4. is largely obscured behind.
To show the the SSE satisfies property 3, we first note
that for a harmonic oscillator, in the long-time limit, the
linearoscillator,andcomparethiswiththebehaviorofthe
densitymatrixisamixtureofcoherentstates,duetoprop-
BMEvariants. AsareferencewefirstsolvetheLBMEand
erty 4. The steady-state density matrix can therefore be
PBMEmasterequationsnumerically,alongwiththe SSE,
obtainedbycalculatingtheprobabilitydensityfor x and
h i foralinearoscillator. Measuringtimeinunitsoftheoscil-
p generated by Eqs.(10) and (11). Solving these equa-
h i lationperiodoftheoscillator,wehaveω =2π. Wefurther
tions is straightforward, and the resulting steady-state
choose the damping rate γ = 4, the temperature so that
probabilitydensityfor x and p isaGaussianwithzero
h i h i n =1 (this means that T =k /(h¯ωln2)), and start the
mean and variances V = V = n . The steady-state T B
hxi hpi T
oscillator in the ground state (n = 0). In Fig. 1 we plot
forthesystemisthereforeaGaussianmixtureofcoherent
the time evolutionof the mean phonon number under the
states,and has the variancesV =V =1/2+n . This is
x p T
two master equations and the SSE. The evolution of the
precisely the thermal state given by Eq.(12).
SSE is indistinguishable from that of the LBME, both of
As mentioned above, the SSE does not correspond to
which settle down to the expected steady-state value of
any Markovian master equation. The reason for this is
the averagephonon number, n =n =1.
the feedback. Without feedback one can average the SSE T
h i
We now consider a χ(3) nonlinear oscillator, which has
over many trajectories and obtain a Markovian master
the Hamiltonian [33]
equation for the density matrix of the system. However,
withfeedbackthedynamicsofeachtrajectorydependson H =h¯ω(a†a)2. (24)
nl
the state of the system during that trajectory in a way
that cannot be reduced to a function of the density ma- This Hamiltonian has the same eigenstates as the har-
trix(beingtheaverageoveralltrajectories)atthecurrent monic oscillator, n , n=0,1,2,..., but the energy levels
| i
time. Thedynamicsofthedensitymatrixattimetcannot arenowE =h¯ωn2. Itisonlythetwolowestenergylevels
n
thereforebe written purely interms ofthe density matrix that the χ(3) oscillator has in common with the harmonic
att,makingadescriptionintermsofaMarkovianmaster oscillator (up to a shift of h¯ω/2).
equation impossible. Simulating the nonlinear oscillatorwith the two master
Damping of Nonlinear systems: We will now consider equations, with ω, γ and T (and thus n ) as before, one
T
thebehavioroftheSSEforaparticularexampleofanon- learns immediately that these give the same steady-state
p-4
A Monte Carlo Method for Modeling Thermal Damping: Beyond the Brownian-Motion Master Equation
the weighting of higher energy levels is suppressed as one
would like. However, it also does not achieve the Boltz-
mann distribution. In addition, the steady-state value of
n2 , while usually closer to the thermal value than that
h i
given by the master equations, depends on the damping
rate. In Fig. 2(a) we plot the steady-state value of n2
h i
as a function of the damping rate, with the parameter
n =1asbefore. Forlargedampingthedistributiongiven
T
by the SSE tends to that given by the master equation
(being the Boltzmann distribution for the harmonic oscil-
lator). As the damping rate is reduced, n2 decreases,
h i
and, after crossing the thermal value, settles into a limit-
ing value, becoming independent of the damping rate for
Fig.2: Hereweshowthesteady-statemeanenergy,hEi∝hn2i,
weak damping. For n = 1, the SSE gives the thermal
foraχ(3) nonlinearoscillator, aswellasthesteady-stateprob- T
value for n2 when γ 0.8. For weak damping the SSE
abilitydistribution fortheenergylevels, n,for twoevolutions: h i ≃
gives n2 =0.24whichis almostexactlyhalfthe thermal
the low temperature Brownian motion master equation, and h i
value.
theSSEdescribed here. The momentum damping rateis γ/2,
andthetemperatureischosensothatnT =1,andaharmonic In Fig. 2 (b)-(d) we show the steady-state distribution
oscillator with thesamelowest twoenergylevelshas hn2i=3. over the energy levels given by the SSE for three values
(a) circles: the value of hn2i given by the SSE; dark dashed of the damping, and compare this to the correct thermal
line: thermal value for hn2i; light dashed line: hn2i = 3. (b)- distribution for the nonlinear oscillator, and the thermal
(d): The distribution overthe energy levels for threedamping distribution for the harmonic oscillator. We see that for
rates. Bars: the distribution given by the SSE; solid line: the small damping, the SSE actually does the opposite of the
thermal (Boltzmann) distribution; dashed line: the distribu-
LBME,inthatitweightsthelowerenergylevelsmorethan
tion given by theBrownian motion master equations.
the thermaldistribution, andthus underestimates the av-
erage energy. As the damping increases, the distribution
changes and tends towards the thermal distribution for
distribution over the eigenstates, n , as they do for the
| i the harmonicoscillator,thusgivingthesameresultasthe
linear oscillator, and thus the same average value of n.
LBME in the limit of large damping. We see from Fig. 2
That is, so long as the eigenstates are those of the har-
(c) that at no point does the distribution match the ther-
monic oscillator, the steady-states of these master equa-
mal distribution: when the averageenergy is equal to the
tions know nothing aboutthe energy levels of the system.
thermal value, the SSE weights both n < 1 and n > 1
This behavior is easily understood: the master equations
more than the thermal distribution.
eliminate the off-diagonal elements of the density matrix
The case of distant systems: It is worth noting that
in the energy eigenbasis, and once this has happened the
there is a wrinkle to using the SSE for simulating damp-
Hamiltonian has no effect on the evolution. Since the en-
ing on systems that are entangle with distant systems,
ergy eigenvalues only act on the dynamics via the Hamil-
andinwhichonewishesexplicitlytoexaminequestionsof
tonian, they can have no effect on the steady-state.
non-locality. In order to provide the feedback required to
The average energy for the nonlinear oscillator is no generate the damping, the bath (the fictitious observer)
longer given by h¯ω( n + 1/2). The energy levels are must have maximal knowledge of the state of the sys-
h i
now proportional to n2, and the average energy is E = tem. This is already implied by the fact that the SSE
h i
¯hω n2 . The steady-state distribution for the nonlinear propagates a pure state, and causes no issue if the sys-
h i
oscillator given by the LBME is not the thermal (Boltz- tem is coupled only to other local systems. However, if
mann) distribution. As noted above it remains the ther- one wishes to describe a thermal resonator that is entan-
maldistributionfortheharmonicoscillator,andtherefore gled with a second, distant system, then the fact that we
gives too much weight to the higher energy levels. For propagate a pure state for the resonator and the second
T = kB/(h¯ωln2) (that is, nT = 1), the thermal value of systemmeansthatwegivethe bathfullknowledgeofthis
n2 for the nonlinear oscillator is 0.49, and that given second system. This is also fine, unless we have an ob-
h i
by the LBME is 3. To use the LBME to model damp- server performing operations on the distant system while
ing of the nonlinear oscillator at a given temperature, we the resonator is evolving. In this case, allowing the bath
could always alter the parameter nT so that the LBME access to the joint pure-state of both systems means that
gives the correct thermal value for n2 at that temper- we are assuming infinitely fast communication from the
h i
ature (and thus for the average energy). Of course this distant observer to the local bath.
does not achieve the Boltzmann distribution. If one does wish to use the SSE in such a situation,
The behavior of the SSE for the nonlinear oscillator then one can preserve locality by modifying the simula-
is quite different to that of the master equations. The tion in the following way. One first includes the measure-
SSE is sensitive to the energy levels of the oscillator, and ments of the distant observer in the SSE. Secondly, one
p-5
K. Jacobs
averages over these measurements to obtain a stochastic REFERENCES
master equation (SME) that gives the state of knowledge
of the bath. The feedback applied by the bath is now de- [1] M. P. Blencowe, Phys.Rep. 395, 159 (2004).
termined by the state-of-knowlegde of the bath, and not [2] A. Naik, O. Buu, M. D. LaHaye, A. D. Armour, A. A.
by the SSE. The SSE and SME are then propagated to- Clerk, M. P. Blencowe, and K. C. Schwab, Nature 443,
193 (2006).
gether, both containing the feedback terms that are now
[3] S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser,
determined by the SME. This procedure preserves local-
G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bauerle,
ity since the bath, and thus the feedback, only ever has
M. Aspelmeyer,andA.Zeilinger, Nature444, 67 (2006).
access to local information. However, it should be noted
[4] J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Mar-
that doing this does modify the feedback, and therefore quardt,S.M.Girvin,andJ.G.E.Harris,Nature452,72
the motion induced by the bath. That is, the motion (2008).
is no-longer exactly the simple Brownian motion that we [5] P. Langevin, C. R. Acad. Sci. 146, 530 (1908).
wish to simulate, at least while the state-of-knowledge of [6] A. O. Caldeira and A. J. Leggett, Physica A 121, 587
thebathremainsappreciablymixed. TheSSEistherefore (1983).
not alwaysappropriate for problems involving explicit in- [7] B.L.Hu,J.P.Paz,andY.Zhang,Phys.Rev.D45,2843
vestigations of nonlocality. (1992).
[8] W. G. Unruh and W. H. Zurek, Phys. Rev. D 40, 1071
Conclusion: We have shown how to obtain a quantum
(1989).
model of simple damped Brownian motion that gives a
[9] M. Schlosshauer, A. P. Hines, and G. J. Milburn, Phys.
completely positive evolution. This is achieved by con- Rev.A 77, 022111 (2008).
structing a stochastic Schr¨odinger equation rather than a [10] G. Lindblad, Comm. Math. Phys. 48, 119 (1976).
masterequation,andbyusingthephysicalnotionoffeed- [11] K. Jacobs and D. Steck, Contemporary Physics 47, 279
back from a measurement process, a concept borrowed (2006).
from the subject of quantum feedback control [27]. [12] K.Molmer, Y.Castin,andJ.Dalibard,J.Opt.Soc.Am.
B 10, 524 (1993).
While both the SSE and the Brownian motion master
[13] A.O.Caldeira,H.A.Cerdeira,andR.Ramaswamy,Phys.
equationsgivethecorrectthermaldistributionforthehar- Rev.A 40, 3438 (1989).
monic oscillator, interesting nano-mechnical devices are
[14] K. Jacobs, I. Tittonen, H. M. Wiseman, and S. Schiller,
usually nonlinear, even if relatively weakly. We therefore Phys. Rev.A 60, 538 (1999).
explore using the LBME and the SSE to model damping [15] W.J.MunroandC.W.Gardiner,Phys.Rev.A53,2633
of an oscillator in an extreme case, in which the Hamil- (1996).
tonian is a χ(3) nonlinearity. While the LBME is com- [16] S.GnutzmannandF.Haake,Z.Phys.B101,263(1996).
pletely insensitive to the energy levels ofthis system(and [17] L. Di´osi, Europhys.Lett. 22, 1 (1993).
thus, completely insensitive to the nonlinearity) the SSE [18] L. Di´osi, Europhys.Lett. 30, 63 (1995).
[19] I. Oppenheim and V. Romero-Rochin, Physica A, 147,
is not. For weak damping the distribution given by the
184 (1987).
SSE is sensitive to the nonlinearity (and independent of
[20] F.PetruccioneandB.Vacchini,Phys.Rev.E71,046134
the damping rate) although it overestimates the proba-
(2005).
bilities of the lower energy levels. In the limit of strong [21] K. Hornberger, Phys.Rev. Lett. 97, 060601 (2006).
damping it reproduces the results of the LBME. For in- [22] K. Hornberger, Europhys.Lett. 77, 50007 (2007).
termediate damping the distribution of the energy levels, [23] C. Presilla, R.Onofrio, and M.Patriarca, J. Phys.A 30,
andthustheaverageenergy,isdependentuponthedamp- 7385 (1997).
ing rate. These results suggest that the SSE is certainly [24] S.M.BarnettandJ.D.Cresser,Phys.Rev.A72,022107
no worse,and probably a better, model of thermal damp- (2005).
ing for nonlinear systems than the various forms of the [25] Code for simulating the SSE using a spectral split-
Brownian motion master equation. operator method can beobtained free of charge from the
author’s website.
The present work highlights the fact that current tech- [26] D. T. Gillespie, Am.J. Phys. 64, 225 (1996).
niques for modeling damping and thermalization in non- [27] A. C. Doherty and K. Jacobs, Phys. Rev. A 60, 2700
linear systems are poor. It may be that stochastic (1999).
Schr¨odinger equations, and the concept of feedback, will [28] C.W.GardinerandP.Zoller, Quantum Noise (Springer,
be a useful tool in developing more accurate models that New York,2004).
are also efficient to implement. [29] A.C.Doherty,S.M.Tan,A.S.Parkins,andD.F.Walls,
Phys. Rev.A 60, 2380 (1999).
Acknowledgments: I thank Lajos Diosi for stimulat-
[30] T. Bhattacharya, S. Habib, and K. Jacobs, Phys. Rev.
ing discussions, and for raising the question of simulating
Lett. 85, 4852 (2000).
Brownianmotionwithdistantsystems. Iamalsoindebted
[31] K. Jacobs and D. A.Steck,(in preparation).
toSalmanHabibformanyenlighteningdiscussionsregard- [32] Z. Bialynicka-Birula, Phys. Rev. 173 1207 (1968).
ing Brownian motion over a number of years. This work [33] F. L. Semi˜ao, K. Furuya, and G. J. Milburn, Eprint:
was performed with the supercomputing facilities in the arXiv:0808.0743.
School of Science and Mathematics at UMass Boston.
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