Simulating quantum Brownian motion with single trapped ions

0.23 MB·English

by  S. Maniscalco

Simulating quantum Brownian motion with single trapped ions
S. Maniscalco,1 J.Piilo,2,3 F. Intravaia,4 F. Petruccione,5,6 and A. Messina1
1INFM, MIUR and Dipartimento di Scienze Fisiche ed Astronomiche
dell’Universit`a di Palermo, via Archirafi 36, 90123 Palermo, Italy.∗
2Department of Physics, University of Turku, FIN-20014 Turun yliopisto, Finland
3Helsinki Institute of Physics, PL 64, FIN-00014 Helsingin yliopisto, Finland
4Laboratoire Kastler Brossel,† Case 74, 4 place Jussieu, F-75252 Paris Cedex 05, France.
5Physikalisches Institut, Albert-Ludwigs-Universit¨at,
Hermann-Herder Straße 3, D-79104 Freiburg im Breisgau, Germany
6School of Pure and Applied Physics, University of KwaZulu-Natal, Durban 4041, South Africa
(Dated: February 1, 2008)
Westudytheopensystemdynamicsofaharmonicoscillatorcoupledwithanartificiallyengineered
4 reservoir. WesingleoutthereservoirandsystemvariablesgoverningthepassagebetweenLindblad
0 type and non-Lindblad type dynamics of the reduced system’s oscillator. We demonstrate the
0 existence of conditions underwhich virtual exchanges of energy between system and reservoir take
2
place. Wepropose touse asingle trapped ion coupled toengineered reservoirs in order to simulate

n quantumBrownian motion.
a
J PACSnumbers: 03.65.Yz,03.65.Ta,32.80.Qk,32.80.Lg,05.10.Ln

3
1
I. INTRODUCTION environment [8].

2 A paradigmatic model of the theory of open systems
v is a harmonic oscillator linearly coupled with a reservoir
The dynamics ofclosedsystemsmay be calculatedex-
1
modelled as an infinite set of non interacting oscillators.
3 actly by solving directly the Schr¨odinger equation. In
Indeed this model is central in many physical contexts,
2 realistic physical conditions, however, the system one is
e.g.,quantumfieldtheory[9],quantumoptics[1,10,11],
7 interested in is coupled to its surrounding. In this case
0 thedynamicsofthetotalclosedsystemcanbeextremely and solid state physics [12].
3 In order to describe quantitatively and qualitatively
complicated. For this reason, since the very early days
0 how the reservoiraffects the system dynamics one needs
ofquantummechanics,ahugedealofattentionhasbeen
/
h devoted to the study of the dynamics of open quantum to make some assumptions on its nature and properties.
p Some of these properties, as for example the tempera-
systems [1].
- ture of the reservoir, can be experimentally measured.
t Nowadays the interest in such a broad field has no-
n Other parameters, as the reservoir spectral density or
a tablyincreasedmainlyfortworeasons. Onthe onehand the system-reservoir coupling, are assumed on the basis
u experimental advances in the coherent control of single
of physical reasonableness and deduced by the compari-
q or few atom systems have paved the way to the realiza-
son with experimental data. In this sense the model is a
:
v tion of the first basic elements of quantum computers, phenomenological one.
i c-not [2] and phase quantum gates [3]. Moreover, the
X The importance of the damped harmonic oscillator is
first quantum cryptographic [4] and quantum teleporta-
alsodue to the fact thatit is one ofthe few exactlysolv-
r tion [5] schemes have been experimentally implemented.
a able non-trivial systems. In fact the Heisenberg equa-
These technological applications rely on the persistence
tionsofmotionforthetotalsystem(oscillatorplusreser-
ofquantumcoherence. Thus,understandingdecoherence
voir)caneasilybesolved. ThesolutionoftheHeisenberg
and dissipation arising from the unavoidable interaction
equations of motion is indeed the most straightforward
between the system and its surrounding is necessary in
methodforthe descriptionofthe dynamicsofthe expec-
order to implement real-size quantum computers [6] and
tation values of observables of interest, e.g. the mean
quantum technologies. On the other hand, one of the
energy of the system. An example showing the easiness
mostdebatedaspectsofquantummechanics,namelythe
and conceptual transparency of the method is given in
quantum measurement problem, can be interpreted in
[1].
terms of environment induced decoherence [7]. Accord-
Another wayto describe the time evolutionofthe sys-
ing to this interpretation the emergence of the classical
tem is to look at the reduced density matrix which is
worldfromthe quantumworldcanbe seenasa decoher-
obtained by tracing the density matrix of the total sys-
ence process due to the interaction between system and
tem over the environmental degrees of freedom. This
procedure is motivated by the fact that, in general, one
is interested only in the dynamics of the system and not
in the state of the reservoir. An exact master equation
†E´cole Normale Sup´erieure, Centre National de la Recherche Sci-
for the reduced density matrix can be formulated and
entifique, Universit´ePierreetMarieCurie.
∗Electronicaddress: sabrina@fisica.unipa.it exactly solved [13, 14, 15, 16, 17, 18, 19].
2
During the last decade huge advances in laser cooling troduce the Master Equation for QBM and its solution
and trapping experimental techniques have made it pos- obtained by means of a superoperatorial approach. In
sibletoconfineharmonicallyasingleionandcoolitdown Sec.IIIwestudythebehavioroftheheatingfunctionfor
to very low temperatures where purely quantum mani- different values of the reservoir parameters. In Sec. IV
festations begin to play an important role [22]. A single wereviewthebasicingredientsoftheexperimentalproce-
laser cooled ion is theoretically equivalent to a particle duresforengineeringartificialamplitudereservoir,asthe
moving in a harmonic potential, whose center of mass onewestudyinthepaper,andformeasuringtheheating
motion is quantized as a harmonic oscillator. Such a function. In Sec. V we describe our experimental pro-
system is a unique experimental system since it approx- posal for revealing non-Markovian dynamics of a quan-
imates very well a closed system [22]. Indeed unwanted tum Brownian particle, simulated with a single trapped
dissipation,whichinthiscasemanifestsitselfasaheating ion. Finally in Sec. VI conclusions are presented.
process depopulating the vibrationalgroundstate of the
ion, is negligible for times much longer than the usual
times in which experiments take place [23, 24]. More- II. EXACT DYNAMICS OF A QUANTUM
over, arbitrary states of the ion motion can be prepared BROWNIAN PARTICLE
and coherently manipulated using proper laser pulses
[25,26]. EvenextremelyfragilestatesastheSchr¨odinger A. Generalized Master Equation
catstateshavebeenrealizedanddetected[27]. Quitere-
cently,byusingmultipleionsinalinearPaultrap,exper- Thedynamicsofaharmonicoscillatorlinearlycoupled
imentsonquantumnon-localityhavebeenperformed[28] with a quantized reservoir,modelled as an infinite chain
and many-particle entangled states have been realized of quantum harmonic oscillators, may be described ex-
[29]. Furthermorecoldtrappedions arethe favoritecan- actly by means of a generalized Master Equation of the
didates for a physical implementation of quantum com- form [1, 16, 35, 36]
puters [2, 6].
dρ (t) 1
The aim of this paper is to study the interaction of a S = HSρ (t) ∆(t)(XS)2 Π(t)XSPS
quantum harmonic oscillator with engineered reservoirs. dt i~ 0 S −h −
In the context of trapped ions it is possible not only to ir(t)(X2)S +iγ(t)XSPΣ ρ (t). (1)
engineer experimentally an “artificial” reservoirbut also −2 i S
to synthesize both its spectral density and the coupling
We indicate with XS(Σ) and PS(Σ) the commutator (an-
withthesystemoscillator[30,31]. Thismakesitpossible
ticommutator) position and momentum superoperators
tothinkofnewtypesofexperimentsaimedattestingthe
respectivelyandwithHS thecommutatorsuperoperator
predictionsoffundamentalmodelsastheoneofquantum 0
relative to the system Hamiltonian. The time depen-
Brownianmotion(QBM)(oritshighT limit: thefamous
dent coefficients appearing in the Master Equation can
Caldeira-Leggettmodel [15]).
be written, to the second order in the coupling strength,
Themeanvibrationalquantumnumberoftheion,also
as follows
calledheatingfunction[23],isthecentralquantitywein-
vestigate. We study the heating dynamics of the single t
∆(t) = κ(τ)cos(ω τ)dτ, (2)
oscillatorincorrespondenceto differentreservoircharac- 0
Z
0
teristicparameters. We analyzetheinfluence ofthevari-
t
ations of the engineered reservoir parameters, e.g., the γ(t) = µ(τ)sin(ω τ)dτ, (3)
0
cut-off frequency of the reservoirspectraldensity, onthe Z0
heating process. The idea of looking at the variation of t
Π(t) = κ(τ)sin(ω τ)dτ, (4)
the opensystemdynamicsinducedbythe changesofthe 0
Z
0
relevant reservoir parameters is in fact rather unusual.
t
Up to recently, indeed, only dissipation and/or decoher- r(t) = 2 µ(τ)cos(ω τ)dτ, (5)
0
enceduetointeractionwiththe“natural” reservoirwere Z0
studied [1, 10, 12, 13, 32, 33].
where
We compare our analytic results for the observable
quantities of interest, as the heating function, with κ(τ)=α2 E(τ),E(0) , (6)
h{ }i
the non-Markovian wave function (NMWF) simulations
and
[1, 34], finding a very good agreement. The main result
of the paper is the experimental proposal for observing µ(τ)=iα2 [E(τ),E(0)] , (7)
new features of quantum Brownian motion with single h i
trapped ions. We demonstrate that with currently avail- are the noise and dissipation kernels, respectively. In
abletechnologyanewregimeoftheopensystemdynam- the previous equations we indicate with α the system–
ics, characterized by virtual phonon exchanges between reservoircoupling constant,with ω the frequency ofthe
0
the system and the reservoir,may be explored. system oscillator and with E the generalized reservoir
The paper is organized as follows. In Sec. II we in- position operator.
3
The Master Equation (1) is local in time, even if The exact analytic expression for the time evolution of
non-Markovian. This feature is typical of all the gen- the heating function can be obtained from the approxi-
eralized Master Equations derived by using the time- mated RWA solution. In the RWA the QCF is [19]
convolutionless projection operator technique [1, 34] or
equivalent approaches such as the superoperatorial one χ (ξ)=e−∆Γ(t)|ξ|2χ e−Γ(t)/2e−iω0tξ , (11)
t 0
presented in [19, 20]. h i
The time dependent coefficients appearing in Eq. (1) withχ QCFoftheinitialstateofthesystem. Thequan-
0
containallthe informationaboutthe shorttime system- tities ∆ (t) and Γ(t) appearing in Eq. (11) are defined
Γ
reservoir correlation. The coefficient r(t) gives rise to a interms ofthe diffusionanddissipationcoefficients∆(t)
time dependent renormalization of the frequency of the and γ(t) respectively as follows
oscillator. The term proportional to γ(t) is a classical
damping term while the coefficients ∆(t) and Π(t) are t
Γ(t) = 2 γ(t )dt , (12)
diffusive terms. 1 1
Z
0
In what follows we study the time evolution of the
t
heatingfunction n(t) withnquantumnumberoperator. ∆ (t) = e−Γ(t) eΓ(t1)∆(t )dt . (13)
The dynamics ohf n(it) depends only on the diffusion Γ Z0 1 1
h i
coefficient ∆(t) and on the classical damping coefficient
Eq. (11) shows that the QCF is the product of an ex-
γ(t) [19]. Furthermore, the quantum number operator
ponential factor, depending on both the diffusion ∆(t)
n belongs to a class of observables not influenced by a
and the dissipation γ(t) coefficients, and a transformed
secular approximation which takes the Master Equation
initial QCF. The exponential term accounts for energy
(1) into the form [19, 37]
dissipation and is independent of the initial state of the
system. Information on the initial state is given by the
dρ (t) ∆(t)+γ(t)
S = 2aρ (t)a† a†aρ (t) ρ (t)a†a secondtermoftheproduct,thetransformedinitialQCF.
S S S
dt 2 − −
(cid:2) (cid:3) Having in mind Eq. (11) and using Eq. (10), one gets
∆(t) γ(t)
− 2a†ρ (t)a aa†ρ (t) ρ (t)aa† . the following expression for the heating function
S S S
2 − −
(cid:2) (cid:3)
(8) 1
n(t) =e−Γ(t) n(0) + e−Γ(t) 1 +∆ (t). (14)
Γ
h i h i 2(cid:16) − (cid:17)
For this reason,in order to calculate the exact time evo-
The asymptotic long time behavior of the heating func-
lution of the heating function, one can use the solution
tion is readily obtained by using the Markovian station-
of the approximated Master Equation (8). In the pre-
aryvaluesfor∆(t)andγ(t). Forathermalreservoir,one
vious equation we have introduced the bosonic annihi-
gets
lation and creation operators a = (X +iP)/√2 and
a† = (X −iP)/√2, with X and P dimensionless posi- n(t) =e−Γt n(0) +n(ω0) 1 e−Γt . (15)
tionandmomentumoperator. NotethattheaboveMas- h i h i −
(cid:0) (cid:1)
ter Equationis ofLindbladtype asfar asthe coefficients Inthenextsectionwewilldiscussindetailthe dynamics
∆(t) γ(t) are positive [21]. of the heating process and we will show the changes in
±
the short time dynamics due to the variations of typical
reservoir parameters such as its temperature and cut-off
B. Analytic solution frequency.
The superoperatorial Master Equation (1) can be ex-
actly solved by using specific algebraic properties of the III. NON-MARKOVIAN DYNAMICS OF
LINDBLAD AND NON–LINDBLAD TYPE
superoperators [19]. The solution for the density matrix
ofthesystemisderivedintermsoftheQuantumCharac-
teristic Function (QCF) χ (ξ) at time t, defined through In a previous paper we have presented a theory of
t
the equation [9] heating for a single trapped ion interacting with a nat-
ural reservoir able to describe both its short time non-
ρ (t)= 1 χ (ξ)e(ξa†−ξ∗a)d2ξ. (9) Markovian behavior and the asymptotic thermalization
S 2π Z t process[38]. Inthispaperwefocusinsteadonthecaseof
interactionwithengineeredreservoirs. Inthetrappedion
Itisworthnotingthatoneoftheadvantagesofthesuper- context, it is experimentally possible to engineer “artifi-
operatorialapproachistherelativeeasinessincalculating cial” reservoirsand couple them to the systemin a con-
theanalyticexpressionforthemeanvaluesofobservables trolledway. Sincethecouplingwiththenaturalreservoir
of interest by using the relation isnegligibleforlongtime intervals[22,23],thisallowsto
testfundamentalmodelsofopensystemdynamicsasthe
m n
a†man = d d e|ξ|2/2χ(ξ) . (10) oneforQBMweareinterestedin. Inasensethisextends
h i (cid:18)dξ(cid:19) (cid:18)−dξ∗(cid:19) (cid:12) the idea of using trapped ions for simulating the closed
(cid:12)ξ=0
(cid:12)
(cid:12)
4
dynamics of quantum optical systems [24, 39, 40] to the A further approximation to the heating function of
possibility of simulating the dynamics of an ubiquitous Eq. (19) can be obtained for times t τ :
T
≪
open system as the damped harmonic oscillator. In par-
t 2α2KT r2
ticular, by using the analytic solution, one can look for n(t) ∆(t )dt = ω t(r2+(12)0)
ranges of the relevant parameters of both the reservoir h i≃Z0 1 1 ωc (r2+1)2n c
and the system in correspondence of which deviations
(r2 1) 1 e−ωctcos(ω t) re−ωctsin(ω t)(.21)
from Markovian dissipation become experimentally ob- − − (cid:2) − 0 (cid:3)− 0 o
servable.
Thisapproximationshowsaclearconnectionbetweenthe
In the experiments on artificially engineered ampli-
sign of the diffusion coefficient ∆(t) and the time evolu-
tude reservoirs [31] the high temperature condition
tion of the heating function before thermalization. The
~ω /KT 1 is always satisfied. For this reason here
0 diffusion coefficient is indeed the time derivative of the
≪
we concentrate on this regime of the parameters. For
heating function. We remind that, since ∆(t) γ(t) for
an Ohmic reservoir spectral density with Lorentz-Drude ≫
the case considered here, whenever ∆(t) > 0 the Mas-
cut-off
ter Equation (8) is of Lindblad type, whilst the case
∆(t) < 0 corresponds to a non-Lindblad type Master
2ω ω2
J(ω)= c , (16) Equation. FromEq.(20)oneseesimmediatelythatwhile
π ω2+ω2
c for ∆(t) > 0 the heating function grows monotonically,
when ∆(t) assumes negative values it can decrease and
the dissipation and damping coefficients γ(t) and ∆(t),
present oscillations.
appearingintheMasterEquation(8),tosecondorderin
Tobetterunderstandsuchabehaviorwestudyinmore
the coupling constant, take the form
details the dynamics for three exemplary values of the
ratio r between the reservoir cut-off frequency and the
α2ω r2
γ(t)= 0 1 e−ωctcos(ω t) re−ωctsin(ω t) , (17) system oscillator frequency: r 1, r =1 and r 1. As
1+r2 h − 0 − 0 i we have alreadynoticedthe fir≫st case correspond≪sto the
assumption commonly done when dealing with natural
and
reservoirwhilethelastcasecorrespondstoanengineered
r2 “out of resonance” reservoir.
∆(t) = 2α2kT 1 e−ωct[cos(ω t)
1+r2 − 0 For r 1 the diffusion coefficient, given by Eq. (18),
(cid:8) is positiv≫e for all t and r since
(1/r) sin(ω t)] , (18)
0
− }
∆(t)∝1 e−ωctcos(ω t) 0. (22)
0
with r = ω /ω . Equation (18) has been derived in the − ≥
c 0
highT limit,whileγ(t)doesnotdependontemperature. ThereforetheMasterEquationisalwaysofLindbladtype
Comparing Eq. (18) with Eq. (17), one notices immedi- and the heating function grows monotonically from its
ately that in the high temperature regime, ∆(t) γ(t). initial null value. Equation (20) shows that, for times
Having this in mind it is easy to prove that th≫e heat- t τR and for r 1, n(t) (α2ωckT)t2, that is the
≪ ≫ h i ≃
ing function, given by Eq. (14), may be written in the initialnon-Markovianbehavioroftheheatingfunctionis
following approximated form quadratic in time.
For r = 1, a similar behavior is observed since also in
n(t) ∆ (t), (19) this case ∆(t) is positive at all times.
Γ
h i≃
Finally, in the case r 1, Eq. (18) shows that, if r is
≪
where we have assumed that the initial state of the sufficiently small, ∆(t) oscillates acquiring also negative
ion is its vibrational ground state, as it is actually the values. It is worth noting, however, that the long time
case at the end of the resolved sideband cooling process asymptotic value of ∆(t) is always positive. Whenever
[25, 26, 41]. For times much bigger than the reservoir the diffusion coefficient is negative, the heating function
correlation time τ = 1/ω the asymptotic behavior of decreases, so the overallheating process is characterized
R c
Eq. (19) is given by Eq. (15). This equation gives ev- by oscillations of the heating function. The decrease in
idence for a second characteristic time of the dynam- the population of the ground state of the system oscilla-
ics, namely the thermalization time τ = 1/Γ, with tor, after an initial increase due to the interaction with
T
Γ = α2ω r2/(r2+1). The thermalization time depends the high T reservoir, is due to the emission and subse-
0
bothonthecouplingstrengthandontheratior =ω /ω quentreabsorptionofthe samequantumofenergy. Such
c 0
between the reservoir cut-off frequency and the system an event is possible since the reservoir correlation time
oscillator frequency. Usually, when one studies QBM, τ = 1/ω is now much longer than the period of os-
R c
the condition r 1, which corresponds to a natural cillation τ = 1/ω . We underline that, although the
s 0
≫
flat reservoir, is assumed. In this case the thermaliza- Master Equation in this case is not of Lindblad type, it
tiontimeissimplyinverselyproportionaltothecoupling conserves the positivity of the reduced density matrix.
strength. For an “out of resonance” engineeredreservoir This of coursedoes notcontradictthe Lindblad theorem
with r 1, τ is notably increased and therefore the since the semigroup property is clearly violated for the
T
≪
thermalization process is slowed down. reduced system dynamics [1].
5
IV. EXPERIMENTAL TECHNIQUES previous equation. Therefore, for high T, the reservoir
discussed in the paper can be realized experimentally
This section gives a brief review of the experimental by filtering the random field, used in the experiments
procedures for engineering artificial reservoirs and for for simulating an infinite-bandwidth reservoir, with a
measuringtheheatingfunctionofthetrappedion. Start- Lorentzian shaped low pass filter at frequency ωc. The
ing from the careful analysis of the recent experiments changeofthe ratio r thus wouldbe accomplishedsimply
presented in this section, we will describe, in the Sec. V, by changing the low pass filter.
our experimentalproposalfor simulating QBM with sin-
gle trapped ions.
B. Measurements of the heating function
A. Engineering Reservoirs Inthis subsectionwefocus ontwoexperimentalmeth-
odsformeasuringtheheatingfunctionofasingletrapped
Let us begin discussing the technique used to engineer ion. The first method is based on the asymmetry in the
an artificial reservoir coupled to a single trapped ion. sideband motional spectrum of the ion and it has been
Reference[31]presentsrecentexperimentalresultsshow- used in [23] for measuring the process of thermalization
ing how to couple a properly engineered reservoirwith a of an ion, initially cooled down to its ground vibrational
quantumoscillator,namely the quantizedcenter ofmass state, due to the interaction with the natural reservoir.
(c.m.) motion of the ion. These experiments aim at Thesamemethodisusedin[26]formeasuringthecooling
measuring the decoherence of a quantum superposition dynamics of an ion subjected to sideband cooling lasers.
of coherentstates andFock states due to the presenceof The second technique allows to measure the populations
the reservoir. Several types of engineered reservoirs are of the vibrational density matrix of the ion, from which
demonstrated, e.g., thermal amplitude reservoirs, phase the heating function can be obtained. This last method
reservoirs,and zero temperature reservoirs. has been used in [31] in the case of interaction with an
A high T amplitude reservoir is obtained by applying artificial amplitude reservoir.
a random electric field E~ whose spectrum is centered on For both techniques the first step is the preparation
the axial frequency ω /2π = 11.3MHz of oscillation of of the initial vibrational and electronic ground state,
z
theion[31]. Thetrappedionmotioncouplestothisfield n = 0, n = 0 , obtained by laser cooling
| −i ≡ | i⊗|−i
due to the net charge q of the ion: H = q~x E~, with and optical pumping to the state . The mean vibra-
~x=(X,Y,Z)displacementofthec.mi.notfth−eion·fromits tional number is then measured a|−ftier fixed delay time
equilibrium position. Remembering that E~ ~ǫ (b + intervals. During the delay artificial noise, which sim-
∝ i i i ulates the amplitude reservoir, may be applied. In this
b†), withb andb† annihilationandcreationopPeratorsof
i i i way, the time evolution of n(t) is obtained.
the fluctuating field modes, and that X a+a† one h i
∝ Let us begin with the first technique. At each fixed
realizes that this coupling is equivalent to(cid:0)the bil(cid:1)inear
delay time, the ion is in its electronic ground state
one assumed to derive Eq. (1). |−i
andinacertainvibrationalstate. Alaserpulse,tunedto
Therandomelectricfieldisappliedtotheendcapelec-
a vibrational sideband, is then used to transfer the pop-
trodes through a network of properly arranged low pass
ulation to the upper electronic level + . After this, by
filters limiting the “natural” environmentalnoise but al- | i
means of an electron shelving technique, the electronic
lowing deliberately large applied fields to be effective.
state of the ion is detected in order to check wether a
This type of drive simulates an infinite-bandwidth am-
transitionto + hasoccurredornot. Repeatingthispro-
plitude reservoir [31]. It is worth stressing that, for the | i
cedure one gets the electronic excited state occupation
times of duration of the experiment, namely ∆t =20µs,
probabilityP . Theamplitudeoftheblueandredvibra-
the heating due to the natural reservoir is definitively +
tional sidebands is defined as the probability of making
negligible [31].
a transition + due to a laser pulse tuned to the
The reservoir considered in our paper is a thermal |−i→| i
blue or red sideband, respectively, and therefore is given
reservoir with spectral distribution given by
by P . This quantity depends on the mean occupation
+
I(ω) = J(ω)[n(ω)+1/2] number n . For n = 0 , only the blue sideband can be
h i | i
exited while the red one is absent. In general the asym-
ω ω2
= c coth(ω/KT), (23) metry in the amplitude of the kth red (Ik ) and blue
π ω2+ω2 red
c (Ik ) vibrational sidebands, allows to extract n [23]:
blue h i
where Eq. (16) has been used. For high T, Eq. (23)
becomes n k
Ik = h i Ik . (25)
2KT ω2 red (cid:18) n +1(cid:19) blue
I(ω)= c . (24) h i
π ω2+ω2
c A limitation of this method is given by off-resonant
The infinite-bandwidth amplitude reservoir realized in excitation via the carrier transition. If the driving field
the experiments corresponds to the case ω in the is tuned to the first lower vibrational sideband, in the
c
→ ∞
6
resolved sideband condition ω Ω, with Ω Rabi fre- variance of the random noise used to simulate an ampli-
0
quencyofthelaserpulse,process≫esinvolvingoff-resonant tude reservoiris 2 V2 =n¯γt. Practically,increasingthe
h i
transitions go as (Ω/ω )2. In order to have a sensible fluctuations of the random electric field applied at the
0
measurementoftheheatingfunction,thepopulationdue trapelectrodesisequivalenttoanincreaseintheheating
to off-resonanttransitions have to be much smaller than function n = n¯γt. Since P (t), as given by Eq. (20),
n
h i
the scale over which n(t) varies. depends only on n¯γt, one can obtain the time evolution
h i
We now focus on the second experimental method for of the populations simply by changing V2 . This is the
h i
measuring the heating function. This method actually methodusedin[31]formeasuringthepopulations,so,in
allows to measure the diagonal elements P of the vi- fact, in the experiment P ( V2 ) is recorded. In princi-
n n
brationaldensity matrix and it has been used to observe ple,byusingthe relationn¯γh= iV2 /t¯,witht¯=3µs,one
h i
their decay due to the interaction with an artificial am- could directly obtain the characteristic parameter of the
plitudereservoir,astheonedescribedinprevioussubsec- reservoir from the value of the noise voltage applied to
tion. For this type of reservoir and in the experimental the trap electrodes. However, unknown geometrical fac-
conditionsof[31],the timeevolutionofP (t) ρ (t)is torsinconvertingthe voltagetovariationsinthe secular
n nn
≡
well approximated by the law frequency prevent a direct comparison. If we indicate
with V2 = c V2 the fluctuations of the voltage
appl
1 n n¯γt j 1 2n−2j appliehd toi the electhrodies, with c R, fitting the exper-
ρnn(t)≃ 1+n¯γt (cid:18)1+n¯γt(cid:19) (cid:18)1+n¯γt(cid:19) imental data according to the th∈eoretical law given by
Xj=0 Eq.(26), allowsto extrapolatethe factor c andtherefore
∞ n¯γt l n+l j n n¯γ. It is not difficult to show that, in the experiment
(cid:18)1+n¯γt(cid:19)(cid:18) n −j (cid:19)(cid:18) j (cid:19)ρn+l−j,n+l−j(0), on the decay (heating) of a Fock state due to interac-
Xl=0 − tionwith engineeredamplitude reservoir(see Fig. [15]of
(26) [31]), c 10 and hence, for t¯= 3µs and V2 =0.25V2,
≃ h i
n¯γ 0.84 107Hz. Note that, in the experiment, V2 is
where the phenomenologicalparameters n¯ and γ are the vari≃ed from· 0 to 0.3V2. h i
meanreservoirquantumnumberandtheheatingrate,re-
Atthispointwearereadytodescribeourexperimental
spectively. Equation (26) is valid under the assumptions
proposal for observing the non-Markovian dynamics of
of high T reservoir and for times much smaller than the
the heating function and, in general, for simulating the
thermalization time, γt 1. Such conditions turn out
≪ dynamics of a quantum Brownian particle.
to be verified in the experiments described in [31]. In
this experimental situation, the Markovian behaviour of
the heating function, before thermalization takes place,
is simply given by n(t) = n¯γt. Note that in the limit V. EXPERIMENTAL PROPOSAL FOR
h i SIMULATING QBM
r 1,correspondingtoaninfinitebandwidthMarkovian
re≫servoir, Eq. (20) becomes n(t) 2α2KTt/π. Com-
h i ≃
paring this expression of the heating function with the It is well known that non-Markovian features usually
oγn=e o2fαt2hωe0/eπxp.eIrtimisewntosrtohneuncadnerildineinntgif,yhno¯w=evKerT,/thωa0taancd- oercaclu,rsiinncethωecd≫ynωa0maicnsdfotyrptiicmalelsytω≪0 ≃τR10=7H1z/fωocr.tIrnapgpeend-
cordingtoEq.(26),onlytheproductn¯γ maybededuced ions, this means that deviations from the Markoviandy-
from the experimental data. namics appear for times t 0.1µs. This is the reason
≪
Inordertodescribeourexperimentalproposalforsim- why the initial quadratic behaviour of the heating func-
ulating QBM, it is important to look in more detail at tionisnotobservedinthe experiments,whereinthe typ-
the procedure used in [31] to obtain the time evolution ical time scales go from 1 to 100µs.
of the populations P . This allows to deduce from the Awayto force non-Markovianfeatures to appearis to
n
experimental data the characteristic parameters of the ‘detune’ the trap frequency from the reservoir spectral
amplitude reservoir used in the experiments. In [31], density. This corresponds, for example, to the case in
after preparing a superposition of Fock states, the am- which r = ω /ω = 0.1. In this case the reservoir cor-
c 0
plitude noise is applied for a fixed time of t¯= 3µs af- relation time is bigger than the period of oscillation of
ter which the populations P are measured. In order the ion and this leads to the oscillatory behaviour of the
n
to measure P the ion is irradiated with a pair of Ra- heating function predicted by Eq. (20) for r 1. Under
n
≪
man beams tuned to the first blue sideband for various thisconditionτ =1µs,andthereforethenon-Markovian
c
probe times t , and P (t ) is measured from the fluo- features show up in the time evolution and can be mea-
p − p
rescence signal. From the P (t ) data the populations sured. Detuning the trap frequency from the reservoir,
− p
P (t¯) are finally extracted with a single value decom- however,decreasestheeffectivecouplingbetweenthesys-
n
position [31]. In order to observe the time evolution of tem and the environment and, for this reason, in order
P one can proceed in two equivalent ways. Either one to obtain values of the heating function big enough to
n
changestheintervaloftimet¯duringwhichtheamplitude bemeasuredweneedtoincreaseeitherthecouplingcon-
noise is applied, or one fixes t¯and varies the variance of stantα2,whichcorrespondtoanincreaseintheintensity
the applied noise V2 . In fact, as shown in [31], the of the voltage applied to the electrodes, or the strength
h i
7
0.35 10
9
0.3
8
0.25 7
6
0.2
n> n> 5
< <
0.15
4
0.1 3
2
0.05
1
0 0
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2
t (µs) t (µs)
FIG. 1: Time evolution of the heating function for FIG. 2: Time evolution of the heating function for
2α2KT/π = 0.84·109Hz, ωc = 1MHz and r = 0.1. Solid 2α2KT/π = 0.84·109Hz, ωc = 1MHz, and r = 10. Solid
line is the analytical and circles thesimulation result. line is theanalytical and circles thesimulation result.
ofthefluctuations V2 ,whichcorrespondtoanincrease
h i to increase of two order of magnitude the coefficient
in the effective temperature of the reservoir.
2α2KT/π = n¯γ. This can be done either increasing
Let us look in more detail to Eq. (20). For r 1 this
≪ the intensityorincreasingthe fluctuationsofthe applied
equation becomes
noise, or combining an increase in the intensity with an
2α2KT increase in the fluctuations. Moreover one needs to use
n(t) r2 ωct+ 1 e−ωctcos(ω0t) a low pass filter for the applied noise, as described in
h i ≃ πω −
c (cid:8) (cid:2) (cid:3) subsection IVA, having cut off frequency ω =0.1ω .
re−ωctsin(ω t) . (27) c 0
− 0 We now examine briefly the conditions for which the
(cid:9)
In the comparisonwith the experiment done in previous quadratic behaviour of the heating function, could be
section we have seen that the front factor 2α2KT/π = observed. We remind that this is the case in which
n¯γ 0.84 107Hz. Increasing of two order of magni- r 1 and the time evolution of the density matrix is
tude≃this fro·nt factor, and for r = 0.1, Eq. (27) predicts of≫Lindblad-type. In view of the considerations done
the behaviour for the heating function shown in Fig. 1. at the beginning of this section, in order to reveal non-
We believe that for this range of n and of times the Markoviandynamicsinatimescaleof1 100µsweneed
oscillations of the heating functionhcoiuld be experimen- to have τR 1µs, that is, for r = 10,−ω0 0.1MHz.
≥ ≤
tally measurable. For example, with the first technique This means that one actually needs a ‘loose’ trap. For
described in subsection IVB, using Ω = 106Hz and for example, for ω0 = 100kHz, with the same applied noise
ω = 107Hz, the ground state population transferred to usedin[31]), i.e.n¯γ =0.84 107Hz,the time evolutionof
0 ·
the excited level due to off resonant excitation is of the the heating function is the one shown in Fig 2.
order of 10−2, which is one order of magnitude smaller If one wants to perform a fast measurement of n ,
than the variation of the heating function we want to e.g.using the firstmethod andassumingΩ=106Hz,hthie
measure (see Fig. 1). Also the other method described small value of the trap frequency makes it difficult not
inprevioussubsectionseemstobeenoughaccuratetore- onlytoimplementone ofthe twomethodsformeasuring
veal the oscillatory behaviour of the heating function in the heating function, but alsoto reachthe initialground
the conditions here examined. However, it is worth not- state since the sidebands are not clearly resolved, and
ingthat,sincetheheatingfunctiondoesnotdependnow therefore resolved sideband cooling technique cannot be
onthe dimensionless variablen¯γt, butratheronω t, the applied. Itisworthstressing,however,thatcontrarilyto
c
timeevolutioncanbeobtainedonlybyvaryingthedura- thecaseofanaturalreservoir,whichisalways‘inaction’,
tionofthetimeofapplicationoftheamplitudereservoir. in the case of an engineered reservoir one can switch off
This is not equivalent to a change in the applied voltage theappliednoiseafteracertaindelaytimeand,assuming
fluctuations,asitwasintheMarkoviancasediscussedin that the effect of the naturalreservoiris negligible, mea-
subsection IVB. suretheheatingfunctionwithoutthesevererequirement
Summarizing,inordertoobservethevirtualexchanges ofbigvaluesofΩ. Ifoneassumesthatafteranamplitude
of phonons between the system and the reservoir, lead- noise pulse the state of the ion does not change, then it
ing to the oscillations of the heating function, one needs is not necessary to perform a fast measurement of n .
h i
8
Therefore one can work with smaller values of Ω, such appearance of oscillations in the heating function (non-
that Ω/ω 1. Lindblad type dynamics).
0
≪
Concluding, while measuring the quadratic behaviour Extending the ideas of using trapped ions for simulat-
oftheheatingfunction(r 1)couldbeamorechalleng- ing quantum optical systems, we have proposed to sim-
≫
ing task from the experimental point of view, revealing ulate QBM with single trapped ions coupled to artifi-
theoscillatorynon-Markovianbehaviour(r 1)appears cial reservoirs. We have carefully analyzed the possibil-
≪
to us in the grasp of the experimentalists, in the condi- ity of revealing, by using present technologies, the non-
tions we have analyzed in this section. Markovian dynamics of a single trapped ion interacting
with an engineered reservoirand we have underlined the
conditions under which non-Markovian features become
VI. CONCLUSIONS observable.
In this paper we have studied the dynamics of a single
harmonic oscillator coupled to a quantum reservoir at
VII. ACKNOWLEDGEMENTS
generic temperature T. In our analysis we have used
boththeanalyticsolutionforthereduceddensitymatrix
and the NMWF method. The authors gratefully acknowledge Heinz-Peter
We have paid special attention to the non-Markovian Breuerforhelpfulcommentsandstimulatingdiscussions.
heating dynamics typical of short times. In this regime J.P. acknowledges financial support from the Academy
the system time evolution is influenced by correlations of Finland (project no.50314) and the Finnish IT center
between the system and the reservoir. For certain val- for Science (CSC) for computer resources. J.P. and F.P.
ues of the system and reservoir parameters, virtual ex- thanktheUniversityofPalermoforthehospitality. S.M.
changes of energy between the system and its environ- acknowledges Finanziamento Progetto Giovani Ricerca-
mentbecomedominant. Thesevirtualprocessesstrongly torianno1999Comitato02forfinancialsupportandthe
affecttheshorttimedynamicsandareresponsibleforthe Helsinki Institute of Physics for the hospitality.
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