Signature of non-Markovianity in time-resolved energy transfer

0.21 MB·

by  Junjie Liu

Signature ofnon-Markovianityintime-resolved energy transfer
Junjie Liu,1,∗ Hui Xu,1 and Chang-qin Wu1,2
1StateKeyLaboratoryofSurfacePhysicsandDepartmentofPhysics,FudanUniversity,Shanghai 200433,China
2CollaborativeInnovationCenterofAdvancedMicrostructures, FudanUniversity,Shanghai 200433, China
(Dated:January19,2017)
Weexploresignaturesofthenon-Markovianity inthetime-resolvedenergytransferprocessesforquantum
opensystems. Focusingontypical systemssuchastheexact solvabledamped Jaynes-Cummings model and
thegeneralspin-bosonmodel,weestablishquantitativelinksbetweenthetime-resolvedenergycurrentandthe
symmetriclogarithmicderivativequantumFisherinformation(SLD-QFI)flow,oneofmeasuresquantifyingthe
non-Markovianity,withintheframeworkofnon-Markovianmasterequationsintime-localforms.Wefindinthe
dampedJaynes-CummingsmodelthattheSLD-QFIbackflowfromthereservoirtothesystemalwayscorrelates
7
withanenergybackflow,thuswecandirectlywitnessthenon-Markovianityfromthedynamicsoftheenergy
1
0 current. Inthespin-boson model, therelationisbuiltontherotating-waveapproximation, calibratedagainst
2 exactnumericalresults,andprovenreliableintheweakcouplingregime. Fromtherelation, wedemonstrate
thatwhetherthenon-Markovianityguaranteestheoccurrenceofanenergybackflowdependsonthebathspectral
n
function.FortheOhmicandsub-Ohmiccases,weshowthatnoenergybackflowoccursandtheenergycurrent
a
alwaysflowoutofthesystemeveninthenon-Markovianregime. Whileinthesuper-Ohmiccase,weobserve
J
thatthenon-Markoviandynamicscaninduceanenergybackflow.
8
1
PACSnumbers:03.65.Yz,03.67.-a,66.70.-f

]
h
I. INTRODUCTION backflowobtainedinthefullcountingstatisticformalismand
p
- the non-Markovianitymeasuredby the trace distance [12] is
t builtinthespin-bosonmodel[25]. However,anindependent
n The unavoidable interaction of the quantum system with
a qualitative study [26] demonstrated that the backflow of the
its environment leads to dissipation and decoherence pro-
u trace distance did not necessarily correlate with the energy
cesses which strongly modify the dynamics of the system.
q backflowinthespin-bosonmodelbasedontheenergycurrent
[ In the well-established frameworkof quantumopen systems
obtained from the definition of work via the power operator
[1, 2], the dynamics of the system is totally determined by
2 [27,28]. SuchabehaviorisalsofoundinthequantumBrow-
the reduced density matrix and the environment-inducedef-
v nian motion by utilizing the Gaussian interferometric power
fectsmanifestthemselvesinthenonunitarytimeevolutionof
0 asanon-Markovianitymeasure[29]. Nevertheless,aquanti-
7 thereduceddensitymatrix. Undertheconditionofshorten-
tative study addressing the role of non-Markovianity in the
5 vironmental correlation times, one can formulate the evolu-
energy transfer process is still absent, explicit and analyti-
4 tion of the reduced density matrix by means of a dynamical
calconnectionsbetweenthenon-Markovianityanddynamics
0 semigroup with a correspondingtime-independentgenerator
. of energytransfer are called for in order to make conclusive
1 in Lindblad form [1, 3], a so-called Markovian dynamics is
statements.
0 thus defined. If the dynamics of an open system substan-
7 tially deviates from that of a dynamical semigroup, one en- In this paper, among the different criteria and measures
1 counteranon-Markovianprocess.Duetoanimportantroleof that quantify quantum non-Markovianity [11–17], we focus
v: non-Markovianeffectsinmanyrealisticexperimentalscenar- ontheonewhichutilizesthesymmetriclogarithmicderivative
i ios[4–10], a seriesofmeasuresareproposedtoquantifythe quantumFisherinformation(SLD-QFI)flowasthefingerprint
X
degree of quantum non-Markovianity[11–17]. In despite of [14]. As a promising measure, the SLD-QFI flow coincides
r distinctformsofthosemeasures,non-Markovianopenquan- withthetimederivativeofthetracedistanceinexactsolvable
a
tumsystemscanbebasicallycharacterizedviatheircapability models[14, 30]. However,the SLD-QFI schemeonlyneeds
to gain back informationpreviously lost due to decoherence anoptimalinputstateinsteadofoptimalstatepairsinobtain-
[18]. ingthetrace distance[31]. Furthermore,theSLD-QFI isdi-
rectlyrelatedtotheimaginarypartofthedynamicalsuscepti-
Besides the reduced dynamics of open systems, the ex-
bility[32],thelatterisshowntodeterminetheenergytransfer
change of energy between the open system and its environ-
properties in the spin-boson model [23], such a connection
ment also draw a great deal of attention during recent years
impliesquantitativelinksbetweenthenon-Markovianityand
(see[19–24]andreferencestherein).Notingtheenergytrans-
theenergycurrentfromtheSLD-QFIpointofview.
ferpropertiesarecloselyrelatedtothedynamicalcharacteris-
tics of open systems, a natural question about the signature Weconcentrateourattentiontotheexactsolvabledamped
of quantum non-Markovianity in energy transfer processes Jaynes-Cummingsmodelandthespin-bosonmodel,bothde-
arises. Recently, a qualitativeconnectionbetweenan energy scribedwithinnon-Markovianmasterequationsintime-local
forms. We obtain explicit relations between SLD-QFI flow
andtime-resolvedenergycurrentforthesetwomodels. Not-
ingthemodelsweconsideredareparadigmsinthetheoretical
∗Electronicaddress:[email protected] studiesofdynamicsofquantumopensystems[1,2],thusour
2
resultspossessgoodadaptability. rates and Lindblad operators, respectively, {P,Q} is the an-
Fromtheso-obtainedrelationships,wefindinthedamped ticommutatorforarbitraryoperatorsP andQ. Master equa-
Jaynes-Cummings model that the SLD-QFI backflow from tions in such time-local forms can be derived by employing
the reservoirto thesystem alwayscorrelateswiththe energy thetime-convolutionlessprojectionoperatortechnique[1]. If
backflow, thus we can witness quantum non-Markovianity theHamiltonianH˜ (t),therelaxationratesγ (t)andLindblad
s i
directly from the dynamics of the energy current. In the operatorsA (t)aretimeindependent,andallγ arepositive,
i i
spin-boson model, the relation is built on the rotating-wave- the above master equation reduces to the well-known Lind-
approximation(RWA),benchmarksagainstthequasiadiabatic bladequationdescribingtheconventionalMarkovianprocess
propagator path integral (QuAPI) shows that our theory is [3]. However, in the time-dependentcase γ (t) can become
i
reliable in the weak coupling regime. We demonstrate that temporarily negative without violating complete positivity,
whether the non-Markovianity guarantees the occurrence of thusleadingtothenon-Markoviandynamics.Inthisstudy,we
an energy backflow depends on the bath spectral function. focusontwospecificinteractionmodelswhosemasterequa-
For the Ohmic as well as sub-Ohmiccases, no energyback- tions have the time-local forms and for which signatures of
flowoccursandtheenergycurrentalwaysflowoutofthesys- non-Markovianity in the energy transfer process can be re-
temeveninthenon-Markovianregime. Forthesuper-Ohmic vealedexplicitly.
case, weobservethatthenon-Markoviandynamicscanhave
anenergybackflow. Comparedwithpreviousstudies,ourde-
ductions are completely analytical and hence the results are A. TheDampedJaynes-Cummingsmodel
more convincing. Moreover, our scheme is helpful for an
experimental determination of SLD-QFI as well as the non- ThefirstoneisthedampedJaynes-Cummingsmodel[1,14,
Markovianity since the energy current is an experimentally 18, 36]. The system Hamiltonian H and the environmental
s
measurablequantity. HamiltonianH aregivenby
B
The article is organized as follows. In Sec. II, we intro-
duceinteractionmodels,thecorrespondingtime-localmaster H = ω σ σ , H = ω b†b , (4)
s 0 + − B k k k
equationsandtheirsolutionsarealsorecalled. InSec. III,we Xk
firstbrieflyreviewtheSLD-QFIanditsproperties,thenderive
whereω istheenergyspacingbetweengroundstate|giand
explicitrelationsbetweenSLD-QFIflowandtransientenergy 0
excitedstate|eiofthesystem,σ = |gihe|andσ = |eihg|
current for models we considered. Finally, in Sec. IV, we − +
discuss the results in detail and reveal the signature of non- aretheloweringandraisingoperatorsofthesystem,bkandb†k
Markovianity in energy transfer processes. Conclusions are are annihilationandcreationoperatorsforthe bosonicreser-
drawninSec. V. voir mode labeled by k with frequency ωk. The interaction
termistakentobe
H = (σ g b +σ g∗b†) (5)
II. MODELANDMASTEREQUATIONS I + k k − k k
Xk
Generalquantumopensystemsaregovernedbythefollow- withcouplingconstantsg .
k
ingHamiltonian Due to the conservationof the numberof excitations, this
modelcanbeexactlysolvable. Forthesakeofcompleteness,
H = Hs+HI +HB, (1) we briefly recall the exactresults. In the interaction picture,
theexactmasterequationforρ(t)hasthetime-localformof
whereH andH are theHamiltoniansofsystem andenvi-
s B Eq. (2)[1]
ronment,respectively,andH denotesasystem-environment
I
interaction Hamiltonian. Given a factorized initial system- d i
environmentstate,theevolutionofthereduceddensitymatrix ρs(t) = − S(t)[σ+σ−,ρs(t)]+γ(t)[σ−ρs(t)σ+
dt 2
ρ canbedescribedbythetime-localmasterequations[1,33]
s 1
− {σ σ ,ρ (t)} , (6)
d 2 + − s (cid:21)
ρ (t) = K(t)ρ (t), (2)
s s
dt
where the time-dependentLamb shift H (t) = S(t)σ σ
LS + −
in order to preserve the Hermiticity and trace of the density
with S(t) = −2Im G˙(t)/G(t) and the relaxation rate
matrix, the time-dependentgeneratorK mustbe of the form
h i
intheSchro¨dingerpicture[12,18,33–35] γ(t) = −2Re G˙(t)/G(t) , the function G(t) is totally de-
h i
terminedasthesolutionoftheintegralequation
K(t)ρ = −i[H˜ (t),ρ ]+ γ (t) A (t)ρ A†(t)
s s s i i s i
Xi h d t
G(t) = − dτf(t−τ)G(τ) (7)
1
− 2{A†i(t)Ai(t),ρs}(cid:21), (3) dt Z0
withf(t−τ)= dωJ(ω)exp[i(ω −ω)(t−τ)]theinverse
0
whereH˜s(t)=Hs+HLS(t)withHLS(t)thetime-dependent Fourier transformRof the bath spectral density J(ω) and the
LambshiftHamiltonian,γ (t)andA (t)denotetherelaxation initialconditionG(0)=1.
i i
3
Themodelwasfrequentlyusedtodescribetheatom-cavity We assume thatthe system-environmentcouplingis weak
system[1],soweusuallychoosetheLorentzianspectralfunc- and employ the second-order time-convolutionless master
tion equation[1]withintheRWA todescribetheevolutionofthe
reduceddensitymatrix. Withintheapproximation,theLamb
1 γ λ2
J(ω) = 0 (8) shiftisnegligible,wehavethefollowingSchro¨dingerpicture
2π(ω0−ω)2+λ2 masterequation[42–44]
ewnivthiroλnmtehnetcwouidptlhinga.nTdheγn0thtehefunsctrtieonngtGh(to)ftaktheesthsyesftoermm- ddtρs(t) = −iω20[σz,ρs(t)]+ γm(t) σmρs(t)σm†
mX=± (cid:2)
dt λ dt 1
G(t) = e−λt/2 cosh + sinh , (9) − {σ† σ ,ρ (t)} , (14)
(cid:20) (cid:18) 2 (cid:19) d (cid:18) 2 (cid:19)(cid:21) 2 m m s (cid:21)
where d = λ2−2γ λ. Since G(t) is real, we will obtain wheretherelaxationratesaredeterminedby
0
avanishingLpambshiftS(t)= 0,andtherelaxationrateγ(t)
1 sin(ω±ω )t
reads 0
γ (t) = J(ω) (1+n )
± B
2Z (cid:20) ω±ω
0
2γ λsinh(dt/2)
γ(t) = 0 . (10) sin(ω∓ω )t
dcosh(dt/2)+λsinh(dt/2) +n 0 dω (15)
B
ω∓ω (cid:21)
0
Forweakcouplings,correspondingtoγ < λ/2,Eq. (10)is
0 with n the Bose-Einstein distribution characterized by the
always positivesuch that the dynamicsis Markovian. In the B
temperatureT.
limit of γ ≪ λ/2 the rate γ(t) becomes time independent,
0 In order to solve the abovemaster equation, we can write
thenthemasterequationEq. (6)isoftheLindbladformand
itintermsoftheBlochvectorB~(t) = Tr [~σρ (t)]with~σ =
leadstoadynamicalsemigroup. However,inthestrongcou- s s
(σ ,σ ,σ )[1]
pling regime, namely, for γ > λ/2, γ(t) will take negative x y z
0
values,implyingthatanon-Markoviandynamicsemerges.
d
Considering the general initial condition for the reduced B~(t) = M(t)B~(t)+~b(t), (16)
dt
densitymatrix
thematrixM(t)isgivenby
1 cosη+1 sinη
ρ (0) = (11)
s 2(cid:18) sinη 1−cosη (cid:19) −1γ (t) −ω 0
2 s 0
with η a real parameter. We can obtain an explicit form for M(t) =  ω0 −21γs(t) 0 , (17)
0 0 −γ (t)
ρs(t)fromthemasterequationEq. (6)asfollows  s 
1 (cosη+1)G2(t) sinηG(t) and the vector~b(t) = (0,0,γd(t))T, where we have intro-
ρs(t) = 2(cid:18) sinηG(t) 2−(cosη+1)G2(t)(cid:19). duced
(12) 1 t
γ (t) = γ (t)+γ (t)= dscos(ω s)D (s),(18)
s + − 0 1
2Z
0
B. Thespin-bosonmodel 1 t
γ (t) = γ (t)−γ (t)=− dssin(ω s)D(s)(19)
d + − 0
2Z
0
The second model is the spin-boson model [2, 37]. As a
minimal prototype in studying nontrivial effects induced by with D1(s) = 2 0∞dωJ(ω)coth2ωT cosωs and D(s) =
∞
the environment, it can describe many physical realizations, 2 dωJ(ω)sinωRs the noise and dissipation kernel of the
0
such as electron-transfer reactions [38], biomolecules [39], mRodel,respectively.
superconductingcircuits[2],tunnelinglightparticlesinmet- WiththeinitialconditionEq. (11),wefindthesolutionof
als [40], anomalous low temperature thermal properties in Eq. (16)as
glasses [41], to mention just a few. The environmentis still
takentobeabosonicbathandthecorrespondingHamiltonian B~(t) = (e−Λsinηcosω t,e−Λsinηsinω t,e−Γ(cosη+δ)),
0 0
isagaingivenbyEq. (4),butthesystemHamiltonianandthe (20)
interactiontermarereplacedby where Γ(t) = tγ (τ)dτ, Λ(t) = Γ(t)/2 and δ(t) =
0 s
Hs = ω20σz, HI = σx (gkbk+gk∗b†k), (13) Rb0eteexΓp(τr)eγsdse(τd)adsτ.RTherefore, the reduced density matrix can
Xk
1
whereσz,xdenotetheusualPaulimatrices,othernotationsre- ρs(t)= 2(I+B~(t)·~σ) (21)
mainthesamemeaningswiththedampedJaynes-Cummings
model. withIthe2×2identitymatrix.
4
III. NON-MARKOVIANITYANDTIME-RESOLVED (26)whichcorrespondstoanoptimalexperimentalsetup(see
ENERGYCURRENT the details in the AppendixA), then adoptan optimalinitial
spinstatetofurthermaximizetheSLD-QFI.Bydoingso,the
In this section, we will build explicit relations between maximumSLD-QFIofTLSsreads
thenon-Markovianityandthetransientenergycurrentforthe
2
abovetwomodels. F (t)= B~ (t) (27)
M M
(cid:12) (cid:12)
(cid:12) (cid:12)
withB~ (t)theBlochvectorE(cid:12)q. (20)(cid:12)obtainedfromtheop-
A. QuantumFisherinformationandnon-Markovianity M
timal initial spin state. Since the initial condition Eq. (11)
corresponds to a pure state, then the initial maximum SLD-
Atfirst,forthesakeofcompleteness,webrieflyreviewthe
QFI F (0) is equal to 1 according to the property of the
M
properties of the SLD-QFI and especially its application in Bloch sphere. If the dynamics is Markovian, the evolution
quantifyingthenon-Markovianity.Byapplyingaphasetrans- oftheopensystemwillbegovernedbyacompletelypositive
formationtothereduceddensitymatrixsuchthatρs(θ;t)con- andtrace-preservingmap(oradynamicalsemigroup)andthe
tainsarealparameterθ. ThentheSLD-QFIisdefinedas[45] maximumSLD-QFIF (t)ofthesystemwillmonotonically
M
decreasesfrom1[56],meaningthatinformationcontinuously
F(θ;t) = Tr[ρ (θ;t)L2(θ;t)], (22)
s flowoutofthesystemtowardstheenvironment.Analogously,
where L(θ;t) is the symmetric logarithmic derivative deter- a temporary increase of the maximum SLD-QFI FM(t) can
minedby be ascribed to a backflow of information from the environ-
menttothesystemagain.Non-Markovianquantumdynamics
∂ 1 areaccordinglydefinedasthosewhichshowanonmonotonic
ρ (θ;t)= [L(θ;t)ρ (θ;t)+ρ (θ;t)L(θ;t)]. (23)
∂θ s 2 s s behaviorofthemaximumSLD-QFIF (t),similartothesit-
M
uationof the trace distance [12]. Thenwe can introducethe
The SLD-QFI plays a vital role in quantum metrology [46]
SLD-QFIflow
and quantum estimation theory [45]. An equivalence to the
fidelity susceptibility even enables the SLD-QFI to analyze d
quantumphasetransitionsinthegroundstate[47–49]andin IQ(t) = FM(t) (28)
dt
dissipative systems [50, 51]. Furthermore, multiparticle en-
tanglementcanbedetectedviatheSLD-QFI[52]. to quantify the non-Markovianity [14]. Once the SLD-QFI
According to the quantum Crame´r-Rao theorem for the flow becomes positive at some time interval, it signifies the
mean-square error h(δθ)2i [53] of estimation results for the emergenceofthenon-Markoviandynamics.
parameterθ[45]
h(δθ)2i> 1 (24) B. Informationflowandenergycurrent
NF(θ;t)
In this subsection, we will establish explicit relations be-
withN thetimesofindependentmeasurements,weknowthat
tweentheSLD-QFIflow[Eq. (28)]andthetime-resolveden-
theSLD-QFIimposesanupperboundontheprecisionofpa-
ergy current. Since we focus on the dynamics of the open
rameterestimation.AlargerSLD-QFIimpliesthattheparam-
system and note that the maximum SLD-QFI is totally de-
eterθcanbeestimatedwithhigherprecision.
terminedbythereduceddensitymatrix,correspondingly,we
Hereweconsidertheparameterθ introducedontothesys-
considerthefollowingdefinitionforthetime-resolvedenergy
temthroughthefollowinginterferometer[54]
current
ρ (θ;t) = eiθJnρ (t)e−iθJn, (25)
s s d
I = hH i=Tr[ρ˙ (t)H ], (29)
E s s s
where J = ~n · J~ with ~n an arbitrary direction and J~ the dt
n
angularmomentum. For two-levelsystems (TLSs), we have where H is the system Hamiltonian. If I > 0, meaning
s E
J~ = ~σ/2. Under such a transformation, the resulting SLD- that we have a backflow of energy from the environment to
QFIisindependentoftheparameterθandtakesasimpleform thesystem.
inTLSs[55] ForthedampedJaynes-Cummingsmodel,wehavealready
knownthat[14]
2
F(θ;t)=F(t)= ~n×B~(t) (26)
(cid:12) (cid:12) IQ(t) = 2G(t)G˙(t) (30)
(cid:12) (cid:12)
with B~(t)theBloch vectorfromt(cid:12)hereduce(cid:12)ddensitymatrix. with the optimal initial condition corresponds to sinη = 1
Therefore,theresultsweobtainbelowisindependentofextra in Eq. (11). Under the same initial condition, the reduced
parametersand just manifests intrinsic propertiesof the sys- densitymatrixEq. (12)reads
temsweconsidered.
Since only the maximum SLD-QFI is of physical signifi- 1 G2(t) G(t)
ρ (t) = . (31)
cance,weshouldfirstlychooseanoptimaldirection~noinEq. s 2(cid:18) G(t) 2−G2(t)(cid:19)
5
Thenthetime-resolvedenergycurrentEq.(29)takestheform A. TheDampedJaynes-Cummingsmodel
I (t) = ω G(t)G˙(t). (32)
E 0 WefirstlookatthedampedJaynes-Cummingsmodel. The
physicalmeaningoftherelationEq. (33)isquiteapparent.In
Compared the above result with Eq. (30), we immediately
theweakcouplingregimewherethedynamicsisMarkovian,
obtain
I andI arealwaysnegativeascanbeseenfromFig. 1(a),
Q E
I = ω0I (33) implyingtheSLD-QFIandenergyarecontinuouslylostdur-
E 2 Q ingthetimeevolutionoftheopensystem. Atsufficientlarge
timeswhena thermalequilibriumis establishedbetweenthe
forthedampedJaynes-Cummingsmodel.
systemandthebosonicbath,theyapproachzero.
Forthespin-bosonmodel,we findtheoptimalinitialstate
In the strong coupling regime in which the dynamics is
thatmaximizestheSLD-QFI[Eq.(27)]tobethespin-upstate
non-Markovian, the relaxation rate takes on negative values
of σ whichcorrespondsto cosη = 1in Eq. (11), therefore
z in some intervalsof time as shown in the inset of Fig. 1(b).
theBlochvector[Eq.(20)]becomesB~M(t)=(0,0,e−Γ(1+ Withinthoseintervals,bothIQandIEbecomepositive.Thus
δ))and in this exactsolvable model, the SLD-QFI backflow or non-
Markovianityalwaysguaranteestheoccurrenceoftheenergy
FM(t) = e−2Γ(t)(1+δ(t))2. (34) backflow and we can witness the non-Markovianitydirectly
fromthedynamicsoftheenergyexchange.
HencetheSLD-QFIflowtakestheform
IQ(t) = 2e−2Γ(t)(1+δ(t)) δ˙(t)−Γ˙(t)(1+δ(t)) . 0 0.5
(cid:16) (cid:17)(35) (a) IQ (b) IQ
IE IE
Fromthereduceddensitymatrix,wehave
IE = ω20e−Γ(t) δ˙(t)−Γ˙(t)(1+δ(t)) , (36) urrent-0.05 0
(cid:16) (cid:17) c
y
g
Thenwecanobtainforthespinbosonmodelthat er
en 0.25 -0.5 40
d
I = ω0 I , (37) w an 0.2 20
E 4hσzi Q FI flo -0.1 tγ()00.1.15 tγ() 0
Q -1 -20
wherewehaveusedthefactthatB isjustthespinpopulation 0.05
3
0 -40
hσzi. 0 10λt20 30 0 2 λt 4 6
Eqs. (33) and(37) are the main results of this study. It is
worthwhileto remarkthatalthoughthedefinitionsEqs. (28) -0.15 -1.5
0 10 20 30 0 2 4 6
and (29) are quite general, a universal link between IQ and λt λt
I isabsentsincetheoptimalinitialsystemstateinobtaining
E
themaximumSLD-QFIismodel-dependentaswesawinthe FIG. 1: (Color online)SLD-QFI flow (solid red line) and transient
present two models. However, we expect model-dependent energy current (dashed-dotted green line) as functions of rescaled
connectionstoexistinopensystemsbeyondparadigmscon- timefor the damped Jaynes-Cummings model. (a) Weak coupling
sidered here, such as the open multi-level system which de- regimewithγ0 = 0.2λ;(b)Strongcouplingregimewithγ0 = 5λ.
siresafurtherinvestigation. Insets show the relaxation rate γ as functions of rescaled time for
Nevertheless, the identification given by Eqs. (33) and bothregimes.Wehavechoseω0 =1astheunit.
(37) still haveseveralconceptualimplications. First, we can
studythenon-Markovianityfromtheenergytransferperspec-
tive. Experimentaladvancesinthefieldofenergytransferat
nanoscale[57–60]evenenableustoexperimentallydetermine B. Thespin-bosonmodel
SLD-QFIaswellasnon-Markovianity. Second,throughrig-
orous relations we show that the dynamics of open systems
Comparedwiththeaboveexactsolvablesystem,theinter-
inevitablymanifestsintheenergytransferprocess. playbetweenI andI in thespin-bosonmodeliscompli-
Q E
catedduetotheevolutionofthespinpopulationasillustrated
byEq.(37).
IV. DISCUSSIONS Inthefollowingstudy,wechoosethespectraldensityas[2]
In this section, we will study the signature of the non- J(ω) = παωsω1−se−ω/ωc, (38)
c
Markovianity measured by the SLD-QFI flow in the energy
transfer process for the two models by using Eqs. (33) and whereα is the dimensionlesssystem-bathcouplingstrength,
(37). ω denotes the cut-off frequency. The case s > 1(s < 1)
c
6
corresponds to super-Ohmic (sub-Ohmic) dissipation, and by taking the possible experimental relevance into account,
s=1representstheimportantcaseoffrequency-independent wechooses=3sinceitcanapplytoadefecttunnelinginthe
(Ohmic)dissipation. Since we have utilized the RWA in the b.c.c. materialsNbandFewithcouplingtoacousticphonons
master equation Eq. (14), the results we obtained should be [62]. Sinceourtheoryiscorrectintheweakcouplingregime,
validintheweakcouplingregime[1]. weonlypresenttheoreticalresultsinFig. 3forsimplicity.
We first focus on the Ohmic case. Noting SLD-QFI [Eq.
(27)]andI [Eq. (29)]aretotallydeterminedbythereduced
E
densitymatrixρ (t),wecancarryoutnumericalsimulations 1 0.5
s (a)s=0.8 (b)s=3
tocalculatethemforcomparisons. Amongvariousnumerical IQ IQ
methods,weutilizetheQuAPIwhichservesasabenchmark IE IE
hσzi
forthespin-bosonmodel[61]toobtainexacttemporalbehav-
0.5
iors for the reduced density matrix. The initial spin state in
0
numericalsimulationsistaken to bethe optimalinitialstate,
namely, the spin-upstate of σ . Once the exactevolutionof
z
the reduced density matrix ρ (t) is obtained, we can get the 0
s
correspondingnumericalresultsforI (t)andI (t)accord-
Q E 1
ingtodefinitionsEqs. (28)and(29),respectively.Theresults
adreeedsucmapmtuarreizseedssinenFtiiagl.fe2a.turAess ocafnqubaenstieteiens, wouerctohnesoirdyeriend- -0.5 -0.5 σhiz0.50
in the weak coupling regime, although minor deviations ex- -0.5
0 1000 2000
istbetweenthetheoryandexactnumericalresults. Fromthe t
-1 -1
0 10 20 30 40 0 0.2 0.4 0.6 0.8 1
t t
1 1
(a) Theory IQ (b) QuAPI IQ
IE IE FIG.3:(Coloronline)TheoreticalpredictionsfortheSLD-QFIflow
hσzi hσzi
IQ(solidredline,Eq.(35)),thetransientenergycurrentIE(dashed-
0.5 0.5 dottedgreenline,Eq. (36))andthespinpopulationhσzi(solidblue
line,Eq. (20))inthespinbosonmodelwithnon-Ohmicspectrums:
(a)asub-Ohmicspectrumwiths=0.8,(b)asuper-Ohmicspectrum
withs=3. Wehavechoseω0 =1astheunit,ωc =10,α=0.02
0 0 andT =0.5forbothcases.
As can be seen from Fig. 3(a), the evolutionbehaviorsof
-0.5 -0.5 IQandIE inthesub-Ohmiccasearequitesimilartothosein
theOhmiccase,thuswecanconcludethatfors ≤ 1thereis
noenergybackfloweveninthenon-Markovianregime.While
forthesuper-OhmiccaseasshowninFig. 3(b),bothI and
Q
-1 -1 I depictoscillatingbehaviorsasafunctionoftime.Forpos-
0 10 20 30 40 0 10 20 30 40 E
t t itivespinpopulations,thoseoscillationsensurethattheSLD-
QFIbackflowalwaysaccompaniesanenergybackflow. Not-
ingthetimefromwhichthespinpopulationbecomesnegative
FIG.2:(Coloronline)EvolutionsoftheSLD-QFIflowIQ(solidred isquitelargecomparedwiththetimescaleω−1ofthesystem
line),thetransientenergycurrentIE (dashed-dottedgreenline)and 0
asshownintheinset,thuswecanobservetheenergybackflow
the spin population hσzi (solid blue line) in the spin boson model
inthenon-Markovianregimeforalongtime.
withanOhmicspectrum.(a)TheoryresultsbasedonEqs.(35),(36)
and(20); (b)ExactnumericalresultsusingQuAPI.Wehavechose
ω0 =1astheunit,ωc =10,α=0.02andT =0.01.
V. SUMMARY
figure,weobservethattheSLD-QFIflowI isnegativeonly
Q
at short time scales, after a certain time, a backflow will oc- Wehavestudiedsignaturesofthenon-Markovianityinthe
cur and the non-Markovian dynamics emerges. In contrast, transientenergycurrentfortypicalquantumopensystems.By
thetransientenergycurrentI isalwaysnegativeevenatthe utilizingtheSLD-QFI flow, oneof measuresquantifyingthe
E
non-Markovianregimeduetothesignchangeofthespinpop- non-Markovianity,weestablishexplicitrelationsbetweenthe
ulation.Thusinthiscasethenon-Markovianitydoesnotguar- non-Markovianityandtransientenergycurrentforthedamped
antees the occurrenceof the energy backflow, in accordance Jaynes-Cummingsmodelandthespin-bosonmodel,bothde-
withthefindinginRef.[26]. scribedwithinnon-Markonianmasterequationsintime-local
Wethenturntothenon-Ohmiccase. Forsub-Ohmicspec- forms.
trums, we take s = 0.8 as an illustration, other values with IntheexactsolvabledampedJaynes-Cummingsmodel,the
s < 1depictsimilarbehaviors. Forsuper-Ohmicspectrums, relationbetweenSLD-QFIflowandthetransientenergycur-
7
rentclearlyrevealsthattheSLD-QFIbackflowalwayscorre- eigenvector of a matrix C with the maximal eigenvalue, the
lates with an energy backflow, thus we can witness the non- entriesofthematrixC reads[44,54]
Markovianitydirectlyfromthedynamicsofenergyexchange.
By contrast, in the spin-bosonmodel, we focuson the weak
(p −p )2
couplingregimewherethetheoryisvalidasconfirmedbythe [C] = i j [hi|J |jihj|J |ii
kl k l
p +p
quasiadiabatic propagatorpath integral and demonstrate that Xi6=j i j
whether the non-Markovianity guarantees the occurrence of
+hi|J |jihj|J |ii], (A1)
l k
anenergybackflowdependsonthebathspectralfunction. In
theOhmicaswellassub-Ohmiccase,thereissimplynoen-
ergybackflowoccursandtheenergycurrentalwaysflowout whereJkisthekthcomponentoftheangularmomentumvec-
ofthesystemeveninthenon-Markovianregime.Whileinthe torJ~.
super-Ohmiccase,weobservethatanenergybackflowoccurs Noting for the TLS, ρ (t) = (I + B~ · ~σ)/2 with B~ =
s
whenevera non-Markoviandynamicsemergesduringa long (B ,B ,B ),thustheeigenvectorsofρ taketheform
1 2 3 s
timeinterval.
In perspective, it would be interesting for us to go be-
ψ ψ
yond the rotating-wave-approximation and investigate the |1i = cos e−iξ/2|ei+sin eiξ/2|gi, (A2)
strong couplingregime of the spin-bosonmodel where non- 2 2
Markovian effects are significant. The generality of the |2i = −sinψe−iξ/2|ei+cosψeiξ/2|gi (A3)
presentresultalsoneedsathoroughinvestigation. 2 2
with tanξ = B /B , tanψ = B2+B2/B and |gi,|ei
Acknowledgments 2 1 1 2 3
the eigenvectors of σz. The corpresponding eigenvalues are
givenby
Support from the National Nature Science Foundation of
ChinawithGrantNo. 11574050isgratefullyacknowledged.
1 1
p = + B2+4(B2+B2), (A4)
1 2 2q 3 1 2
AppendixA:Optimaldirections 1 1
p = − B2+4(B2+B2). (A5)
2 2 2q 3 1 2
Withthediagonalformofthereduceddensitymatrixρ =
s
p |iihi|, the optimal direction ~n is determined by the Therefore,thematrixC hasthefollowingexplicitform
i i o
P
2(cos2ξcos2ψ+sin2ξ) −sin2ξsin2ψ −cosξsin2ψ
(p −p )2
C = 1 2  −sin2ξsin2ψ 2(cos2ξ+sin2ξcos2ψ) −sin2ψsinξ . (A6)
4
−cosξsin2ψ −sin2ψsinξ 2sin2ψ
 
A straightforward calculation shows that the eigenvector It can be easily checked that the so-obtained optimal direc-
correspondstothemaximumeigenvalueofthematrixC has tionsareorthogonaltotheBlochvectorB~.
atwo-folddegeneracy. UsingtheSchmidtorthogonalization,
wecanobtaintwooptimaldirectionswiththeform
~n1 = (−sinξ,cosξ,0), (A7)
o
~n2 = (cosψcosξ,cosψsinξ,−sinψ). (A8)
o
[1] H.-P.BreuerandF.Petruccione,TheTheoryofOpenQuantum [5] S.Cialdi,D.Brivio,E.Tesio,andM.G.A.Paris,Phys.Rev.A
Systems(OxfordUniversityPress,Oxford,2007). 83,042308(2011).
[2] U.Weiss,QuantumDissipativeSystems(WorldScientific,Sin- [6] B.-H.Liu,L.Li,Y.-F.Huang,C.-F.Li,G.-C.Guo,E.-M.Laine,
gapore,2012). H.-P.Breuer,andJ.Piilo,Nat.Phys7,931(2011).
[3] G.Lindblad,Commun.Math.Phys.48,119(1976). [7] A.Smirne,D.Brivio,S.Cialdi,B.Vacchini,andM.G.A.Paris,
[4] C.-F.Li,J.-S.Tang,Y.-L.Li,andG.-C.Guo,Phys.Rev.A83, Phys.Rev.A84,032112(2011).
064102(2011). [8] J. T.Barreiro, M.Muller, P.Schindler, D.Nigg, T.Monz, M.
8
Chwalla,M.Hennrich,C.F.Roos,P.Zoller,andR.Blatt,Na- [35] H.-P.Breuer,Phys.Rev.A70,012106(2004).
ture470,486(2011). [36] H.-S.Zeng,N.Tang,Y.-P.Zheng,andG.-Y.Wang,Phys.Rev.
[9] M.Gessner, M.Ramm, T.Pruttivarasin, A.Buchleitner, H.-P. A84,032118(2011).
Breuer,andH.Haffner,Nat.Phys10,105(2014). [37] A.J.Leggett,S.Chakravarty,A.T.Dorsey,M.P.A.Fisher,A.
[10] J.-S.Tanget.al.,Optica2,1014(2015). Garg,andW.Zwerger,Rev.Mod.Phys.59,1(1987).
[11] M.M.Wolf,J.Eisert,T.S.Cubitt,andJ.I.Cirac,Phys.Rev. [38] M. Merkli, G. P. Berman, and R. Sayre, J. Math. Chem. 51,
Lett.101,150402(2008). 890(2013).
[12] H.-P. Breuer, E.-M. Laine, and J. Piilo, Phys. Rev. Lett. 103, [39] A.Garg,J.N.Onuchic,andV.Ambegaokar,J.Chem.Phys.83,
210401(2009). 4491(1985).
[13] A.Rivas,S.F.Huelga,andM.B.Plenio,Phys.Rev.Lett.105, [40] Y.Fukai,TheMetal-HydrogenSystem(Springer-Verlag,Berlin,
050403(2010). 2005).
[14] X.-M. Lu, X. Wang, and C. P. Sun, Phys. Rev. A 82, [41] P.W.Anderson,B.I.Halperin,andC.M.Varma,Philos.Mag.
042103(2010). 25,1(1972).
[15] D. Chrus´cin´ski, A. Kossakowski, and A. Rivas, Phys. Rev. A [42] P.Haikka,Phys.Scr.2010,014047(2010).
83,052128(2011). [43] G.ClosandH.-P.Breuer,Phys.Rev.A86,012115(2012).
[16] S.Luo,S.Fu,andH.Song,Phys.Rev.A86,044101(2012). [44] X.Hao,N.-H.Tong,andS.Zhu,J.Phys.A:Math.Theor.46,
[17] D. Chrus´cin´ski and S.Maniscalco, Phys. Rev. Lett. 112, 355302(2013).
120404(2014). [45] C. W. Helstrom, Quantum Detection and Estima-
[18] H.-P.Breuer,E.-M.Laine,J.Piilo,andB.Vacchini,Rev.Mod. tion(Academic,NewYork,1976).
Phys.88,021002(2016). [46] V.Giovannetti,S.Lloyd,andL.Maccone,Phys.Rev.Lett.96,
[19] D.SegalandA.Nitzan,Phys.Rev.Lett.94,034301(2005). 010401(2006).
[20] J. Ren, P. Ha¨nggi, and B. Li, Phys. Rev. Lett. 104, [47] W. L. You, Y. W. Li, and S. J. Gu, Phys. Rev. E 76,
170601(2010). 022101(2007).
[21] N.BoudjadaandD.Segal,J.Phys.Chem.A118,11323(2014). [48] L.C.VenutiandP.Zanardi,Phys.Rev.Lett.99,095701(2007).
[22] C.Wang,J.Ren,andJ.Cao,Sci.Rep.5,11787(2015). [49] P. Zanardi, P. Giorda, and M. Cozzini, Phys. Rev. Lett. 99,
[23] J.Liu,H.Xu,B.Li,andC.Wu,arXiv:1609.05598(2016). 100603(2007).
[24] M.Carrega, P.Solinas, M. Sassetti, andU. Weiss, Phys.Rev. [50] L. Banchi, P. Giorda, and P. Zanardi, Phys. Rev. E 89,
Lett.116,240403(2016). 022102(2014).
[25] G.Guarnieri,C.Uchiyama,andB.Vacchini,Phys.Rev.A93, [51] U.MarzolinoandT.Prosen,Phys.Rev.A90,062130(2014).
012118(2016). [52] P. Hyllus, W. Laskowski, R. Krischek, C. Schwemmer, W.
[26] R.Schmidt,S.Maniscalco,andT.Ala-Nissila,Phys.Rev.A94, Wieczorek,H.Weinfurter,L.Pezze´,andA.Smerzi,Phys.Rev.
010101(2016). A85,022321(2012).
[27] P. Solinas, D. V. Averin, and J. P. Pekola, Phys. Rev. B 87, [53] S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72,
060508(2013). 3439(1994).
[28] R. Schmidt, M. F. Carusela, J. P. Pekola, S. Suomela, and J. [54] P. Hyllus, O. Gu¨hne, and A. Smerzi, Phys. Rev. A 82,
Ankerhold,Phys.Rev.B91,224303(2015). 012337(2010).
[29] G. Guarnieri, J. Nokkala, R. Schmidt, S. Maniscalco, and B. [55] J.Suzuki,Phys.Rev.A94,042306(2016).
Vacchini,Phys.Rev.A94,062101(2016). [56] A.Fujiwara,Phys.Rev.A63,042304(2001).
[30] P. Haikka, J. Goold, S. McEndoo, F. Plastina, and S. Manis- [57] Y.K.KohandD.G.Cahill,Phys.Rev.B76,075207(2007).
calco,Phys.Rev.A85,060101(2012). [58] S.Lefe`vreandS.Volz,Phys.Sci.Instrum.76,033701(2005).
[31] Z. He, J. Zou, L. Li, and B. Shao, Phys. Rev. A 83, [59] E. Cha´vez-A´ngel and et al., Appl. Phys. Lett. Mater. 2,
012108(2011). 012113(2014).
[32] P.Hauke,M.Heyl,L.Tagliacozzo,andP.Zoller,Nat.Phys.12, [60] J.Reparazandetal.,Phys.Sci.Instrum.85,034901(2014).
778(2016). [61] N.MakriandD.E.Makarov,J.Chem.Phys.102,4600(1995).
[33] D. Chrus´cin´ski and A. Kossakowski, Phys. Rev. Lett. 104, [62] R. Go¨rlich, M. Sassetti, and U. Weiss, Europhys. Lett. 10,
070406(2010). 507(1989).
[34] V.Gorini, A. Kossakowski, and E. C. G. Sudarshan, J. Math.
Phys.(N.Y.)17,821(1976).