Efficiency of quantum controlled non-Markovian thermalization

0.29 MB·English

by  Victor Mukherjee

Efficiency of quantum controlled non-Markovian thermalization
V. Mukherjee,1,2 V. Giovannetti,1 R. Fazio,1,3 S. F. Huelga,4 T. Calarco,2 and S. Montangero2
1NEST, Scuola Normale Superiore & Istituto Nanoscienze-CNR, I-56126 Pisa, Italy
2 Institute for Complex Quantum Systems & IQST, Universita¨t Ulm, D-89069 Ulm, Germany
3Centre for Quantum Technologies, National University of Singapore, 117543, Singapore
4 Institut fu¨r Theoretische Physik & IQST, Universita¨t Ulm, D-89069 Ulm, Germany
We study optimal control strategies to optimize the relaxation rate towards the fixed point of a
quantum system in the presence of a non-Markovian dissipative bath. Contrary to naive expecta-
tionsthat suggest that memory effectsmight beexploited toimproveoptimal control effectiveness,
non-Markovian effects influence the optimal strategy in a non trivial way: we present a necessary
5 conditiontobesatisfiedsothattheeffectivenessofoptimalcontrolisenhancedbynon-Markovianity
1 subject to suitable unitary controls. For illustration, we specialize our findings for the case of the
0 dynamics of single qubit amplitude damping channels. The optimal control strategy presented
2 here can be used to implement optimal cooling processes in quantum technologies and may have
implications in quantumthermodynamicswhen assessing theefficiency of thermalmicro-machines.
n
u
J
I. INTRODUCTION namics of open quantum systems is linked to the possi-
4
bilityofidentifyingwellseparatedtime scalesinthe evo-
2
lutionofsystemandenvironment. Recently,anumberof
Controllingquantumsystemsbyusingtime-dependent proposals have been put forward to quantitatively char-
]
h fields[1]isofprimaryimportanceindifferentbranchesof acterize this effect in terms of explicit non-Markovianity
p science, rangingfrom chemicalreactions [2, 3], NMR [4], measures [37–40]. In this light, one can define an evo-
- molecularphysics[5]tothe emergentquantumtechnolo-
t lution to be Markovian if described by a quantum dy-
n gies [6–8]. Investigations on optimal control of open
namical semigroup (time-homogeneous Lindbladian evo-
a quantum systems mostly focus on memoryless environ-
lution)[41],whichwouldbethetraditionalMarkovianity
u
ments [9–11] and specifically on those situations where
q consideredinmostpreviousworkonopensystemcontrol.
the reduced dynamics can be described by a Markovian
[ However, other definitions encompass this as a special
master equation of the Lindblad form [12]. In this con-
case while allowing for more general, non-homogeneous
2 text, optimal control applications to open quantum sys-
generators,albeit still ensuring the divisibility of the as-
v
tems have been explored in different settings [5, 13–18]
sociated dynamical map and the unidirectionality of the
2
and recently the ultimate limits to optimal control dic-
8 system-environment information flow, and therefore the
tated by quantum mechanics in closed and open sys-
3 absence of memory effects in the dynamics of the sys-
7 tems [19–22] and the complexity of dealing with many- tem [42–45]. Relevant for our analysis is the definition
0 body systems [23–25] have been determined. Time- of Markovianevolution in terms of the divisibility of the
. optimal quantum control has been extensively discussed
1 associated dynamical map [46]. When the dynamics is
foronequbitsystemsinadissipativeenvironment[9,26]
0 parametrizedusing a time-local master equation, the re-
5 and the optimal relaxation times determined in [27]. quirement of trace and hermiticity preservation, yields a
1 These studies might have both fundamental and prac-
generator of the form
: tical applications, for example in assessing the ultimate
v
i efficiency of quantum thermal machines [28], or to im- ρ˙s(t) = i[Hs(t),ρ(t)]+ ¯(t)(ρ(t))
X plement fast cooling schemes which have already proven − L
= i[H (t),ρ(t)]+ γ (t)(A (t)ρ(t)A†(t) (1)
r to be advantageous [29, 30]. − s k k k
a
However, introducing a Markovian approximation re- Xk
1
quires some constraints on system and environment, A†(t)A (t)ρ(t) ),
whichmaynotbevalidingeneral[31,32]. Consequently, − 2{ k k }
incorporating non-Markovian (NM) effects of the envi- where ¯(t) is a time dependent Lindblad superopera-
ronment, in a sense that will be defined more precisely L
tor, γ (t) are generalized (i.e. not necessarily positive)
below,mightbeanecessityinamanyexperimentalsitu- k
decay rates, the A (t)’s form an orthonormal basis for
ations. Recently,thepossibleinfluenceofmemoryeffects k
the operators for the system, see e.g. Ref. [47] (here-
onthe orthogonalitycatastrophe[33],onquantumspeed after ~ has been set equal to one for convenience) and
of evolution [34] and on quantum control [35, 36] have
H (t) is the effective Hamiltonian acting on the system.
been analyzed. Here, we present a study of the optimal s
Equation (1) generalizes the familiar Lindbladian struc-
controlstrategies to manipulate quantum systems in the
turetoincludeNMeffectswhilemaintainingatime-local
presence of NM dissipative baths and compare the per-
structure. However, apart from same special cases, it
formance of optimal control with the case of operating
not known which are the conditions which H (t), A (t),
subject to a Markovian(M) environment. s k
andγ (t) haveto satisfy inorderto guaranteeComplete
k
Intuitively, the absence of memory effects in the dy- Positivity (CP) [47–51], i.e. the fundamental prerequi-
2
site which under fairly general assumptions is needed instantaneous control pulse, there is absolutely no clear
to describe a proper quantum evolution [31, 32]. In evidencethattheopendynamicsfort t shouldbestill
1
≥
whatfollows we will focus ona simplified scenariowhere describedbythesamegeneralizedLindbladianγ(t) ,the
L
the γ (t)’s either are null or coincide with an assigned system environment being highly sensitive to whatever
k
function γ(t), and where the A (t)’s are explicitly time- the system itself has experienced in its previous history.
k
independent. Accordingly, in the absence of any control We note that in a more realistic scenario, any control
Hamiltonian applied during the course of the evolution, pulse will have a non-zero width δt in time. Clearly, a
we assume a dynamicalevolutiondescribedby the equa- sufficiently large δt can invalidate the assumption of ap-
tion plying control pulses only at the very beginning and the
veryend,therebymodifyingthedynamicssignificantlyas
ρ˙(t)=γ(t) (ρ(t)), (2)
L described above. However, one can expect the assump-
tion of instantaneous pulses to be valid as long as δt is
where isa(time-independent)Lindbladgeneratorchar-
L negligiblecomparedtothetimescaleassociatedwiththe
acterizedby havinga unique fixedpointρ (i.e. (ρ)=
fp
L dynamics in absence of any control.
0 iff ρ = ρ ). For this model, in the absence of any
fp
Keepinginmindtheabovelimitations,inthisworkwe
Hamiltonian term (i.e. H (t)=0) CP over a time inter-
s
attempt for the first time a systematic study of optimal
val [0,T] is guaranteed when [48]
controlprotocols which would allow one to speed up the
t drivingofageneric(butknown)initialstateρ(0)toward
γ(t′)dt′ 0, t [0,T], (3)
≥ ∀ ∈ the fixed point ρfp of the bare dissipative evolution for
Z0 themodelofEq.(2)whichexplicitlyincludesNMeffects.
whiledivisibility(i.e,Markovianity)istantamounttothe We arrive at the quantum speed limit times when appli-
positivity of the single decay rate at all times [47]: if cation of only two control pulses and , at initial
in fin
there existsa time intervalwhere γ(t)becomesnegative, and final times respectively, is enUough toUfollow the op-
the ensuing dynamics is no longer divisible and the evo- timal trajectory. On the other hand, we present lower
lution is NM. In this context we will assume a control bounds to the same when optimal control strategy de-
Hamiltonian Hs(t) to represent time-localized infinitely mands unitary pulses at intermediate times as well. We
strongpulses, which induce instantaneousunitary trans- show that the efficiency of optimal control protocols is
formationsat specific controltimes. This correspondsto notdeterminedbytheM/NMdividealonebutitdepends
writing H (t) = δ(t t )Θ(j), where Θ(j) are time drastically on the behaviour of the NM channel: if the
s j − j s s
independent operators which act impulsively on the sys- system displays NM behavior before reaching the fixed
P
tem at t = t (δ(t) being the Dirac delta-function), at point for the first time, NM effects might be exploited
j
whichinstants onecanneglectthe contributionfromthe to obtain an increasedoptimal controlefficiency as com-
non-unitary part, and represent the master equation by pared to the M scenario. On the contrary, NM effects
ρ˙(t) i[H (t),ρ(t)]. Therefore the resulting dynamics are detrimental to the optimal controleffectiveness if in-
s
≈−
is described by a sequence of free evolutions induced by formationback-flowoccursonly after the systemreaches
thenoiseovertheintervalst [t ,t ]interweavedwith the fixed point (see Fig. 1). These results are valid ir-
j j+1
∈
unitary rotations U =exp[ iΘ(j)], i.e. respective of the detailed description of the system, i.e.
j s
− its dimension, Hamiltonian, control field, or the explicit
ρ(t) = fin j j j−1 form of the dissipative bath.
U ◦D ◦U ◦D ◦···
(ρ(0)), (4)
1 0 in
···◦U ◦D ◦U
where =exp tj+1 ¯(t) , ( )=U( )U† and “ ” II. THE MODEL
Dj tj L U ··· ··· ◦
is the compositihoRn of supeir-operators. When only two
controlpulsesareapplied(thefirst attheverybegin- ThedivisibilitymeasureforthemodelEq.(2)isequiv-
in
U
ningandthesecond attheveryendofthetemporal alentto the characterizationofmemoryeffects by means
fin
U
evolution),the non-unitaryevolutionis describedbyEq. of the time evolution of the trace distance [52]. This
(2)andCPofthetrajectory(4)isautomaticallyguaran- provides an intuitive characterization of the presence of
teed by Eq. (3), the scenario corresponding to the real- memory effects in terms of a temporary increase in the
istic case where one acts on the system with very strong distinguishability of quantum states as a result of an in-
control pulses at the state preparation stage and imme- formation back-flow from the system and into the envi-
diately before the measuring stage. When more ’s are ronment that is absent when the evolution is divisible
j
U
present, the situation however becomes more complex. [53]. As a result, a divisible evolution for which the sin-
Thereisnoclearphysicalprescriptionwhichonecanfol- gle decay rate γ(t) 0 at all times will exhibit a mono-
≥
low to impose the associated dynamics on the system at tonic decrease of the trace distance of any input state
leastwhenthedissipativeevolutionisassumedtobeNM. towards a (assumed to be unique) fixed point ρ of the
fp
Considerforinstancethecase (ρ(0)) Lindblad generator [54]. On the contrary, as illus-
fin 1 1 0 in
U ◦D ◦U ◦D ◦U L
where is a non trivial unitary. Even admitting that trated in Fig. 1, the behaviour of the trace distance can
1
U
the latter is enforced by applying at time t a strong benon-monotonicwhenthedynamicsisNM.Inthiscase,
1
3
1
We assume full knowledge of the initial state and we
A allowforanerrortoleranceof0<ǫ 1,consideringthat
d(t) R ≪
M > 1 the target is reached whenever the condition d(t) ǫ
| | ≤
M RA is satisfied. To obtain a lower bound on the minimum
NM
time T needed to fulfill such constraint we restrict
QSL
our analysis to the ideal limit of infinite control which
allowsustocarryoutanyunitarytransformationinstan-
taneously along the lines of (and with all the limitations
associated with) the formalism detailed in Eq. (4). In
NMA NMB the limit of infinite control an important role is played
0 by the Casimir invariants Γ (j = 2,3,...,N for a N
TF t level system). The Casimir jinvariants of a state ρ are
relatedtothetraceinvariantsTr(ρj)(j =2,3,...,N)and
they cannot be altered by unitary transformation alone
FIG.1: (Coloronline)SchematicdiagramofNMdynamicsin [10, 59]. For example, a two level system has a single
ClassA(redline)andClassB(blackline). Theinstantaneous Casimirinvariant–itspurityP =Tr ρ2 ,whichremains
tracedistanced(t)=||ρ(t)−ρ ||startsincreasingonlyafter
fp unchanged under any unitary transformation. Conse-
the system reaches the fixed point when |γt| → ∞ in case quently,anyoptimalstrategywithth(cid:0)eco(cid:1)ntrolsrestricted
of Class A, while it shows oscillatory behavior even before
tounitarytransformationsonly,wouldbetoreachastate
it reaches the fixed point in case of Class B. In comparison
ρ characterired by all Casimir invariants same as those
dynamicsforaMchannelisshownbythebluelinewhered(t)
of ρ in the minimum possible time. Following this we
decreases monotonically and assymptotically to d(t→∞)= fp
0. ThespeedupobtainedbyMdynamicsisalwaysbiggerthan canapplyaunitarypulsetoreachthefixedpointinstan-
thatobtainedfortheNMone,i.e., RA/RA ≥1incasethe taneously. Clearly, any constrained control will at most
M NM
NMevolutionisofClassA,whiletheMarkovianlimitcanbe be as efficient as the results we present hereafter, based
surpassed by NMof class B. onthe analysiswehavepresentedpreviouslyforthe case
of M dynamics [5, 27].
Inwhatfollows,wewillanalyseClassAandBchannels
there existtime interval(s)where γ(t) becomesnegative. independently.
Denoting by d(t) = ρ(t) ρ the trace distance be-
|| − fp|| Class A: As shown in Fig. 1, in the NM regime d(t)
tween ρ(t) and the fixed point ρ , it straightforwardly
fp goes to zero at t = T when ρ = ρ and (ρ ) = 0.
follows that d˙(t) 0 t in the M limit. Looking at this F fp L fp
≤ ∀ At the same time we expect γ(t) at t TF in
quantity one can classify NM dynamics into two distinct | | → ∞ ≈
orderto havefinite ρ˙(t)=γ(t) (ρ(t)) evenfor (ρ(t))
classes (see Fig. 1): the first one (Class A) is defined by L L ≈
(ρ ) = 0, as is required for a non-monotonic d of the
fp
thosedynamics wherethe systemreachesthe fixedpoint L
formshowninFig.1. Noticethatγ(t)andhencethetime
at time T before γ(t) changes sign i.e., γ(t) 0 and
d˙(t) 0 foFr 0 t < TF. In this case, the NM d≥ynamics tρ.=CToFnsaetqwuehnictlhyγa(nty)→opt∞imaarlecionntgreonlepraroltinocdoelpwenhdicehntino-f
≤ ≤
reaches the fixed point ρfp and then start to oscillate. volvesunitary transformationofρ(t) generatedby Hs(t)
On the other hand, Class B dynamics is characterized atearliertimest<T followedbynon-unitaryrelaxation
by γ(t)that changessign(andcorrespondinglyd˙(t)>0) to ρ is expected toF be ineffective in this case and we
fp
at some time t < TF, that is the solutions of the equa- haveTQSL =TF. Thatis,thegain(orefficiency)ofopti-
tion γ(ts) = 0 are such that ts < TF for at least one maltrajectory in the NM class is RNAM =TF/TQSL =1.
s. In contrast, in the M dynamics d(t) always decreases One can easily see T /T = 1 implies absence of any
F QSL
monotonically and asymptotically to d(t )=0. speed up, whereas any advantage one gains by optimal
→∞
NMchannelsofclassA/Barisefromdifferentphysical controlcanbequantifiedbyT /T >1. Ontheother
F QSL
implementations. As an illustration, the damped Jaynes hand in the M limit γ(t)=γ is finite and constant,and
0
Cummings model exemplifies a Class A dynamics. Here the system relaxes asymptotically to the fixed point in
aqubitiscoupledtoasinglecavitymodewhichinturnis the absence of any control. In this case we introduce an
coupledtoareservoirconsistingofharmonicoscillatorsin error tolerance ǫ 1, such that we say the target state
≪
the vacuum state (see Eq. (6))[31, 55–57]. On the other is reached if d(T ) ǫ. Clearly, T increases with de-
F F
hand, dynamics similar to Class B can arise for example creasingǫdiv|erging|to≤T inthelimitǫ 0,ascan
F
→∞ →
in a two level system in contact with an environment be expected for finite γ . Therefore the above argument
0
made of another two level system, as realized recently in of γ(t) at t T does not apply in this case and
F
| |→∞ ≈
an experimental demonstration of NM dynamics [58]. ingeneralonecanexpectthetimeofevolutiontodepend
As we will see hereafter, the difference between Class onthe initialstate. Consequentlythe quantumspeedup
A and B appears to drastically affect the performance ratio RA can exceed RA 1, as is explicitly derived
M NM ≈
of any possible optimal control strategy to improve the below in the case of a two level system in presence of an
speedofrelaxationofthesystemtowardsthefixedpoint. amplitude damping channel. Similar arguments apply
also in the case when an additional unitary transforma-
4
tion is needed at the end of the evolution to reach ρ , 1/β gives the temperature of the bath. The system evo-
fp
where RA for ǫ 0 [27]. lution is given by Eq. 2, and we will focus on two dif-
M →∞ →
Our above result RA RA can be expected ferent functional dependence of the parameter γ(t) cor-
M ≥ NM
to be valid in a more generic scenario with ρ˙(t) = responding to the Class A and B dynamics. We will
γ (t) (ρ(t)) as well, where not all γ ’s (= 0) are analyze the system evolution following the Bloch vector
k k Lk k 6
same, ’sarethetimeindependentLindbladgenerators ~r representingthestateρ=(I+~r.~σ)/2insidethe Bloch
k
P L
andthe unique fixedpointρ is definedby (ρ p)=0 sphere, where the unitary part of the dynamics gener-
fp k f
L
for all k. In this case at least one of the γ ’s can be ex- ated by H induce rotations, thus preserving the purity
k s
pectedtodivergeattime t=T inordertoensureClass P =(1+ ~r 2)/2. Incontrast,ingeneraltheactionofthe
F
| |
ANMdynamicsasshowninFig(1),thusmakinganyop- noise is expected to modify the purity as well.
timal controlineffective as detailed above. We note that Class A: An example of this class of dynamics is ob-
one can have dynamics with time dependent Lindblad tained under the assumption
generators and uncontrollable drift Hamiltonians acting
on the system during the course of the evolution, in ad- γ(t)= 2λγ0sinht2g ; g = λ2 2γ λ. (6)
dition to the instantaneous control pulses, as well. The gcoshtg +λsinhtg − 0
2 2
drift Hamiltonians can be expected to modify the Lind- p
bladgeneratorsthus making the problemmore complex; In the above expression λ and γ0 are two positive con-
however the analysis in this case is beyond the scope of stantswhoseratiodeterminesthebathbehavior: λ>2γ0
our present work. corresponds to a M bath, whereas g becomes imaginary
Class B: Here we focus on systems of Class B where in the limit γ0 > λ/2 resulting in NM dynamics. In the
as already mentioned γ(t) changes sign for ts <TF with NM limit of γ0 ≫λ,1 the bath time scale is determined
s = 1,...,Ns. Clearly, in this case γ(t) does not nec- by the product λγ0 and is independent of the specific
| |
essarily diverge for any t. Consequently the arguments form of the super-operator . It can be easily seen that
L
presented above for class A fails to hold any longer and γ(t) increases monotonically from 0 to γ(t) at
→∞
thetimeofrelaxationtothefixedpointcaningeneralbe
π
expected to depend on ρ(t) (and hence on H ). Further- lim T , (7)
more,itmightbe possibletoexploitthe NMseffects such γ0/λ→∞ F ≈ √2λγ0
thateventhoughd˙(t)>0fort t t onecan,by ap-
1 2 where T is independent of the initial state and the sys-
≤ ≤ F
plicationofoptimalcontrol,makesurethatΓ˙j(t)>0and temreachesthefixedpointwhenγ(t)diverges. Withthis
maximum∀t,j (wherewehaveassumedΓj(t=0)<Γjf choiceofγ(t)inEq. (6)thetimescaleisgivenby√2λγ0
fi∀xjedanpdoinΓtjfρfdpe)n.oTtehsistphreesjetnhtsCtahseimpiorssinibvialirtiyanotfsefxoprlothite- sincatlheeinNtMheliMmitlimγ0it≫λ λ, wγ0h.ileThγe(tr)ef≈oreγ0thseettsimtheetatikmene
ing NM effects to achieve better control as opposed to to reach the fixed point≫can be expected to decrease as
the M dynamics, as is presented below for the case of a 1/√λγ in the NM limit while it scales as 1/γ in the
0 0
two level system in the presence of an amplitude damp- M limit. ∼
ing channel. Howeverwe stress that this is not a general As mentioned above, this form of γ(t) arises in the
result and explicit examples can be constructed where damped Jaynes-Cummings model at absolute zero tem-
this is actually not true. perature, where one considers only a single excitation in
thequbit-cavitysystemandEq. (5)reducesto (ρ(t))=
L
(ρ(t)). However,here we consider a phenomenological
2
A. Generalized amplitude damping channel Lform of the Lindblad generator (5) with arbitrary β to
showthegeneralityofourresults. Inthiscontextthepa-
Letusnowanalyzeindetailthegenericformalismout- rameter λ in γ(t) denotes the spectral width of the cou-
lined above for the specific case of a two level system in plingtothereservoir,whileγ characterizesthestrength
0
contactwithNMamplitudedampingchannelsofthetwo of the coupling.
classes introduced before. In the absence of any control the qubit relaxes to
We consider the non-unitary dissipative dynamics de- a fixed point ρ characterized by the Bloch vector
fp
srcerdiubceeddbdyetnhseittyimmealtorciaxlρm(at)stoefreaqquuabtiiotnaancdtinwgeocnonas2id×e2r 0,0,11−+eeββ andtheoptimalcontrolweanalyzehereaims
the time independent Linbladian given by (cid:16)to accelera(cid:17)te the relaxation towards this state with un-
L constrained unitary control. Following the strategy pro-
(ρ(t)) = (ρ(t))+eβ (ρ(t)), posed above, we look for the extremal speed v = ∂P/∂t
1 2
L L L
of purity change for every r. For this model, the speed
1
1(ρ(t)) = σ+ρ(t)σ− σ−σ+,ρ(t) , of change of purity is given by v(r,θ,t) = γ(t)(eβ
L − 2{ } − −
(cid:18) (cid:19) 1)r cosθ+ r 1+cos2θ : notethatapositive(neg-
1 2rfp
(ρ(t)) = σ ρ(t)σ σ σ ,ρ(t) , (5)
L2 − +− 2{ + − } ativhe) v denotes(cid:0)increasing(cid:1)(idecreasing) purity. The two
(cid:18) (cid:19) strategiesdiffer slightly in caseof coolingor heating (i.e.
with σ being the raising/lowering qubit operators and the final purity is lower or higher than the initial one);
±
5
FIG. 3: (Color online) (a) Plot showing the variation of
the ratio of the gains RB /RB of the optimal trajectory
NM M
as a function of Ω in case of Class B (cooling) with γ(t) =
FIG. 2: (Color online) (a) Parametric plot showing variation exp(−t)cos(Ωt),β = 2,ǫ = 0.01 and initial state given by
of time TQcoSoLl of reaching the fixed point with λ and γ0 for rxi = 0.3,ryi = 0,rzi = 0.4. Clearly gain RMB (= 3.4) in
β = 2, ri = 0.5 and ǫ = 0.01. The Markovian (M) and non- the M limit Ω = 0, shown by the green dot, is less than
Markovian(NM)regionsareseparatedbythebluelineonthe that (RNBM) in the NM limit Ω > 0. (b) Schematic diagram
λ−γ0plane. (b)Plotshowingvariationofquantumspeedup showing optimal path in the x−z plane of the Bloch sphere
ratioRAwithλandγ0forβ =2,rxi=0.3,ryi=0,rzi=0.4 (red curve) in case of cooling a qubit in Class B (Eq. (8)),
and ǫ = 0.01. RA saturates to RMA ≈ 2 (RNAM ≈ 1) in the when we start from an arbitrary stateρi. Thefixed point ρf
extreme M(NM) limit. is denoted by brown star.
evolution described by a master equation Eq. (2) with
but both cases correspond to applying unitary rotations
given by Eq. (5); however for our present purpose we
at the beginning and at the end (for heating) of the L
formulate a γ(t) given by
dynamical evolution, thus yielding a trajectory of the
form Eq. (4) which is fully compatible with the CP re- γ(t)=e−ζtcos(Ωt), (8)
quirement and which doesn’t pose any problem in terms
with ζ,Ω being two positive constants satisfying the CP
of physical implementation (see discussion in Sec. II).
conditionEq.(3). Withthischoice,intheabsenceofthe
Specifically we need to apply unitary control so that the
controlHamiltonian, NM effects manifest themselves for
systemevolvesalongθ =π tillthefinalpurityisreached
(2n+1)π/2 < Ωt < (2n+3)π/2 for integer n 0 as
in the case of cooling, while θ =0 is the optimal path in ≥
γ(t) changes sign at Ωt = (2n+1)π/2, simultaneously
the caseofheating [27], in agreementwitha recentwork
altering the sign of d˙(t) to d˙(t) > 0. With a proper
on quantum speed limit in open quantum systems [34]
choice of parametersone canmake γ(t) (and hence d(t))
(seeAppendixIVfordetails). Ouranalysisclearlyshows
exhibit oscillatory dynamics for 1 > d > 0. As for the
that Tcool decreases with increasing λ as 1/√λ for
QSL ∼ previous example, also in this case the extremals of v
small λ, finally saturating to λ independent constant
are independent of γ(t) and determined by (ρ(t)) only.
values in the M limit (λ 1). However, it would be L
≫ Therefore they occur at exactly the same points as for
misleading to conclude about the role of Markovianity
class A (6), i.e., at θ = 0,π and arccos(r/r ). In this
on T from this alone, since both Tcool and Theat de- f
QSL QSL QSL caseaninstantaneouspulsewouldcorrespondtoitstime
creasewithincreasingγ aswell. Inparticular,theyscale
0 width δt min 1/ζ,1/Ω,1/ eβ +1 .
as 1/γ0 in the M limit of small γ0, while the scaling ≪ { }
∼ The unconstrained optimal strategies are now modi-
changes to 1/√γ0 for λ/γ0 0. Indeed, the behavior (cid:0) (cid:1)
→ fied as follows. In the case of cooling, the optimal strat-
depends more on the specific path in the (γ ,λ) plane
0 egy is to follow the path θ = π for 0 t < Tcool,
ratherthanwhetherthesystemisMorNM(seeFig.2a). ≤ QSL
during which time the purity increases monotonically,
However,ifoneanalyzesthe speedup obtainedby means
where we have assumed the system reaches the target
of optimal control strategies, the scenario changes: in
at t = Tcool < π/(2Ω) for simplicity. Consequently the
this case (as shown in Fig. 2b) the ratio RA =T /Tcool QSL
F QSL optimal strategy demands a single pulse at time t = 0
clearly distinguishes between M and NM dynamics with
only to make θ = π, which corresponds to an evolution
typical limiting values given by lim RA 2
ǫ→0,λ/γ0→∞ M → operator of the form D . Clearly, this evo-
and limǫ→0,λ/γ0→0RNAM →1 (see Appendix for details). lution is CPT as alreaUdfyind◦iscus◦seUdinin section I with D
Finally, one can show that a control pulse can be con-
depending on γ and (see Eqs. (5) and (8)) and is
sidered to be instantaneous as long as its width in time t L
thus possible to implement physically. In Fig. 3(a) we
δt 1/ γ (eβ +1) in the M limit of λ γ , while in
≪ 0 ≫ 0 report the speedup obtained by such optimal strategy
the NM (cid:2)limit λ≪γ(cid:3)0 one has δt≪1/ λγ0(eβ +1). for different values of Ω in Eq. (8), where we have taken
Class B: Finally, we investigate a particular case be- ǫ = 0.01 large enough so that ΩTcool < π/2. In this
p QSL
longing to the class B dynamics and compare it to the case we arrive at the M limit by setting Ω = 0; as can
previouscase. Asinthepreviouscaseweconsideratime be clearlyseenthe speedupinthe NMlimitis suchthat
6
RB > RB, Ω > 0 showing that there exist scenarios contrastto the cooling problem, now v >0 θ making it
NM M ∀ ∀
where NM effects can be exploited to improve the con- impossible to take advantage of the NM effects to accel-
trol effectiveness. On the contrary, if the system does erate the evolution. However, even in this case one can
notreachthetargetforΩTcool <π/2,theoptimalstrat- alwaysminimizetheunwantedeffectofinformationback-
QSL
egy changes: at Ωt=π/2, γ(t) and hence v change sign, flow for γ(t) < 0 by increasing θ to θ = arccos( rfp/r)
−
leadingtodecrease(increase)ofpurityforθ =π (θ =0). where v has a minimum.
Interestingly,wecantakeadvantageofthiseffectbymak-
ingθ =0att=π/(2Ω),wherevexhibitsamaximumfor
γ(t) < 0. As mentioned before, application of a unitary
III. CONCLUSION
pulse during the course of an evolution may change the
formofγ(t)and (seeEq. (4)). However,forsimplicity
L
let us assume v changes sign at t = (2n+1)π/2 and as- We have studied the effectiveness of unconstrained
sumes extremum values at θ = 0,π and arccos( r/rfp) optimal control of a generic quantum system in the
−
eveninpresenceofunitarycontrol. Onecaneasilyextend presence of a non-Markovian dissipative bath. Contrary
ouranalysistoamoregenericcasewherethesimplifying to common expectations, the speedup does not crucially
assumptions do not hold by following the path of max- depend on the Markovian versus non-Markovian divide,
imum (minimum) v for cooling (heating). Let us first but rather on the specific details of the non-Markovian
consider the case where the system reaches the target evolution. We showed that the speed up drastically
rz = rf −ǫ at t = TQcoSoLl < 3π/(2Ω). In such a scenario, depends on whether the system dynamics is monotonic
as depicted in Fig. 3 (b), we let the system evolve freely or not before reaching the fixed point for the first
for π/(2Ω)<t Tcool, following which we take the sys- time, as determined by the trace distance to the fixed
≤ QSL
temtoθ =πandr = r +ǫ,thusobtainingthedesired point (class A and class B dynamics respectively).
z f
−
goal. Clearly,anoptimalpathexistsincaseoftheclassB Indeed, in the former case, the speed up obtained
non-Markovian channel, which if possible to be followed via optimal control is always higher in presence of
by applicationofsuitable unitary controls,helps incool- a Markovian bath as compared to a non-Markovian
ing andin particularmightmakeit possible to reachthe one, while the reverse can be true in the latter case.
fixed point in finite time (if ǫ = 0). Generalization to Finally, we have presented some specific examples of
the case ΩTcool > 3π/2 where multiple π rotations are these findings for the case of a two level system subject
QSL
needed is straightforward. However, we emphasize that to an amplitude damping channel. Note that, in the
the strategy presented above for ΩTcool > π/2 follows more realistic scenario where one can apply control
QSL
an evolution operator of the form Eq. (4) with unitary pulses of finite strength only, the presented results serve
pulses applied at intermediate times. Consequently our astheoreticalboundstotheoptimalcontroleffectiveness.
analysis gives a lower bound to Tcool only, achieved by
QSL
followingthe optimal pathshownin Fig. 3(b), for which Acknowledgements –TheauthorsacknowledgeAndrea
at present we do not have any implementation strategy. Mari, Andrea Smirne, Alberto Carlini and Eric Lutz for
Finally, we address the problem of heating the system helpful discussions. We ackowledge support from the
intheshortestpossibletime,whichamountstominimiz- Deutsche Forschungsgemeinshaft(DFG) within the SFB
ing v t. Therefore, the optimal path dictates to set TR21 and the EU through EU-TherMiQ (Grant Agree-
∀
θ =0 at t=0 and then let it evolve freely till Ωt=π/2, ment 618074), QUIBEC, SIQS, the STREP project PA-
where γ(t) and hence v change sign. Unfortunately in PETS and QUCHIP.
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Quantum Systems (Oxford University Press, Oxford,
2002). A. Class A optimal strategy
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tems: AnIntroduction(Springer,Heidelberg,2011),and
Analysis of v(r,θ,t) = γ(t)(eβ
arXiv:1104.5242.
− −
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[34] S. Deffner and E Lutz, Phys. Rev. Lett. 111, 010402 egyhin case of c(cid:0)ooling is t(cid:1)oi apply a unitary pulse at
t = 0 so as to rotate ~r to θ = π. Following this we
(2013).
switch off the control and allow the qubit to relax by
[35] D. M. Reich, N. Katz and C. P. Koch, arXiv:1409.7497
(2014). the application of the dissipative bath alone for a time
[36] C. Addis, F. Ciccarello, M. Cascio, G. M. Palma and S. t= TQcoSoLl, till it reaches r(TQcoSoLl)= rfp−ǫ. In contrast,
Maniscalco, arXiv:1502.02528 (2015). while considering the problem of minimizing the time
[37] T. J. G. Apollaro, C. DiFranco, F. Plastina and M. Pa- taken to heat the qubit, the optimal strategy is to first
ternostro, Phy.Rev.A 83, 032103 (2011). rotate the Bloch vector to θ = 0. As before, we then
[38] S. Lorenzo, F. Plastina, M. Paternostro, Phys. Rev. A turn off the unitary controland let it relax till it reaches
(R) 88, 020102 (2013).
(0,0,r ), following which we apply a second unitary
fp
[39] A. Rivas, S. F. Huelga and M. B. Plenio, Rep. Prog.
pulse to take the system to the fixed point~r [27].
Phys.77, 094001 (2014). fp
Weusetheoptimalstrategyformalismpresentedabove
[40] C. Addis, B. Bylicka, D.Chruscinski and S. Maniscalco,
Phys.Rev.A 90, 052103 (2014). to arrive at the minimum time TQcoSoLl needed to cool the
[41] M.M.Wolf,J.Eisert,T.S.Cubitt,andJ.I.Cirac,Phys. systeminthedifferentlimits. Thetimeforcoolinginthe
Rev.Lett. 101, 150402 (2008). M limit λ/γ0 is given by
→∞
[42] D. Chruscinski, A. Kossakowski, and A. Rivas, Phys.
Rev.A 83, 052128 (2011). lim Tcool 1 lnrfp−ri, (9)
[43] N.K.Bernardes, A.R.R.Carvalho, C.H.Monken,and λ/γ0→∞ QSL ≈ γ0(1+eβ) ǫ
8
whereas the same in the NM limit is where we have assumed r ǫ, which is typically the
xi
≫
case for ǫ 0. In the case of cooling, as can be seen
λ/lγim0→0TQcoSoLl ≈rλ2γ0 "π2 −(cid:18)rfpǫ−ri(cid:19)[2(exp(1β)+1)]#.(10) lforonmgaEsqr.xi,(→9(r)fapn−drEi)q≫. (ǫ1a3n),dRǫMA→≈0.2OinntthheeoMthelrimhiatnads,
the NM limit of λ/γ 0 yields
0
Ontheotherhand,followingtheoptimalstrategytoheat →
the qubit one gets
2/(exp(β)+1)
2 π ǫ
T , (14)
1 r +r F
λ/γli0m→∞TQheSaLt ≈ γ0(1+eβ)ln(cid:20)2rffpp+iǫ(cid:21) (11) ≈rλγ0 "2 −(cid:18)rx0(cid:19) #
in the M limit, while in the NM limit it is thus reducing the gain to RA 1 in the limit of ǫ 0
NM ≈ →
(see Fig. (2b)). In case of heating we have
1
lim Theat 2 cos−1 2rfp+ǫ 2(exp(β)+1) (.12)
λ/γ0→0 QSL ≈rλγ0 "(cid:18)ri+rfp(cid:19) # limRA 2 |lnǫ| (15)
As canbe easily seen, one needs to considera non-zeroǫ ǫ→0 M ≈ ln rfp+ri
2rfp
in order to keep the time of cooling finite in the M limit h i
whilewecansetitexactlyto0intheothercasesandyet
in the M limit, while the same in the NM limit is
reach the fixed point in finite time.
Incontrasttotheresultsderivedabove,theadvantage
π/2
one gains by application of optimal control presents a limRA , (16)
completely different picture. As before, one can under- ǫ→0 NM ≈ cos−1 2rfp 2(exp(1β)+1)
stand this from the the quantum speed-up ratio RA = ri+rfp
T /T . In the M limit λ/γ , γ(t) γ one gets (cid:20)(cid:16) (cid:17) (cid:21)
F QSL 0 0
→∞ ≈
which again implies the gain is much higher in the M
2 r
xi
T ln| |, (13) limit for ǫ 0.
F ≈ γ (1+eβ) ǫ →
0