Table Of ContentInfluence of measurements on the statistics of work performed on a quantum system
Michele Campisi, Peter Talkner, and Peter Ha¨nggi
Institute of Physics, University of Augsburg, Universita¨tsstrasse 1, D-86135 Augsburg, Germany
(Dated: January 13, 2011)
Therecentlydemonstratedrobustnessoffluctuationtheoremsagainstmeasurements[M.Campisi
et al., Phys. Rev. Lett. 105 140601 (2010)] does not imply that the probability distributions of
nonequilibrium quantities, such as heat and work, remain unaffected. We determine the impact of
1 measurements that are performed during a running force protocol on the characteristic function of
1 work. The results are illustrated by means of theLandau-Zener(-Stu¨ckelberg-Majorana) model. In
0 the limit of continuous measurements the quantum Zeno effect suppresses any unitary dynamics.
2 It is demonstrated that thecharacteristic function of work is the same as for an adiabatic protocol
when the continuously measured quantity coincides with the Hamiltonian governing the unitary
n
dynamicsof thesystem in the absence of measurements.
a
J
PACSnumbers: 05.70.Ln,05.40.-a05.30.-d03.65.Ta03.65.Xp
2
Keywords: QuantumZenoEffect,Fluctuation Theorems,Landau-Zener
1
]
h I. INTRODUCTION dynamicsisinterruptedbyprojectivequantummeasure-
c ments.
e
The Jarzynski equality and the fluctuation theorems Based on the fact that the process of measurement of
m
aregeneralandsurprisinglyrobustexactrelationsofnon- a quantum system is rather invasive due to the collapse
t- equilibrium thermodynamics. The validity of these rela- of the wave function, it has been argued [11, 12] that,
a tions was rigorouslyestablishedirrespective of the speed although the value of the ratio of backward and forward
t
s of the externally applied forcingprotocolfor drivenclas- pdf’s inEq. (1)remainsunchanged,additionalmeasure-
t. sical [1–3] and quantum systems [4, 5], staying either in mentsaffectthevaluesoftheindividualworkpdf’s. One
a complete isolation or in weak [6] or even strong contact must expect that, in a driven-measured quantum sys-
m
with an environment [7, 8]. See also the reviews [9, 10] tem, the work done is not only determined by the in-
- and references therein. In this work we will focus on the teractionwith the manipulated externalfield λ, but also
d
n Tasaki-Crookswork fluctuation theorem, reading: by the measurements themselves, which are physically
o realized by a measurement apparatus. In Sec. III we
c p (w) calculate the statistics of work in a prototypical model
[ F =eβ(w−∆F), (1) of driven quantum system, namely the Landau-Zener(-
pB(−w) Stu¨ckelberg-Majorana)[13–16]model,andillustratehow
1
v where p (w) is the probability density function (pdf) of itisinfluencedbyprojectivequantummeasurements. We
4 F will also draw the attention on interesting features re-
workperformedbyaforcingprotocoldenotedasλ . This
0 t lated to the quantum Zeno effect appearing when the
forcing acts on the system between the times t = 0 and
4 measurements become very frequent. As we will see, if
2 t = τ. Accordingly, pB(w) is the pdf of work performed theobservablethatismeasuredattimet isthe Hamilto-
1. on the system when the backward(B) protocol λτ−t de- nianH(t), thenin the Zenolimit the workcharacteristic
scribesthe forcingofthe system. The forwardandback-
0 function approaches the same expression as for an adia-
ward processes start in the Gibbs equilibrium states at
1
batic protocol with no intermediate measurements.
1 the same inverse temperature β and at the initial pa-
Sec. IV closes the paper with some concluding re-
: rameter values λ and λ , respectively. The free energy
v 0 τ marks.
difference between these two states is denoted by ∆F.
i
X In the derivations [4, 5] of the quantum work fluc-
r tuation theorem, Eq. (1), the energy of the system is
a
measured at times t = 0 and t = τ, and the work w is
II. FLUCTUATION THEOREMS FOR
determinedbythedifferenceofthe obtainedeigenvalues.
DRIVEN-MEASURED QUANTUM SYSTEMS
Recently,weshowedthattheTasaki-Crooksworkfluctu-
ationtheorem,Eq. (1),aswellasotherquantumfluctua-
tiontheorems,remainunaffectedforotherscenariosthan We consider a quantum system that, in the time span
this two-measurement scheme: the ratio of forward and [0,τ], is thermally isolated, and interacts with the exter-
backwardpdf’s inEq. (1)staysunalteredevenif further nal world only through a mechanical coupling to a time
projectivequantummeasurementsofanysequenceofar- dependent external force field and a measurement ap-
bitrary observables are performed while the protocol is paratus. The information regarding this interaction is
in action [11]. In Sec. II we provide an alternative proof encoded in a protocol which we denote as
of this result, based on the calculation of the character-
istic function of workof a drivenquantum systemwhose σ ={H(λ ),(t ,A )}. (2)
t i i
2
Itspecifiesa)thesystemHamiltonianH(λ )ateachtime Thus,immediatelybeforethemeasurementofA ,occur-
t 1
t in terms of the external forces λ , and b) the times ring at t , the density matrix is:
t 1
t ∈(0,τ), i=1...N,atwhichmeasurementsofthe ob-
i
servables A occur. For the sake of simplicity, in the fol- ρ (t−)=U ρ (0+)U† , (10)
i n 1 t1,0 n t1,0
lowing we will adopt the notation H(t) for H(λ ). As in
t
the two-measurementscheme [4, 5] of the Tasaki-Crooks and the subsequent measurement projects it into
theorem, we assume that, besides the N intermediate
measurements of A1,...AN, the energies determined by ρn(t+1)= Π1rUt1,0ρn(0+)Ut†1,0Π1r. (11)
H(0)andH(τ),aremeasuredattimest0 =0andtf =τ. Xr
For simplicity we then set f =N +1 and
Likewise, just before the measurement of A = H(τ) at
f
A =H(0) A =H(τ). (3) time τ, the density matrix becomes
0 f
Thehremriteemanaionpinegraotobrsseravraebrleeqsu,irAe1d,t.o..hAavNe,drisecprreetseesnpteedctrbay, ρn(τ−)=Uτ,t1ρn(t+1)Uτ†,t1, (12)
but otherwise can be chosen arbitrarily. andtheprobabilitythattheoutcomeofthemeasurement
For times t ≤ 0 the system is assumed to stay in the of A =H(τ) at time τ is αf is:
f m
Gibbs thermal state at inverse temperature β:
p (m,τ|n,0)=TrΠf U ρ (t+)U† . (13)
ρβ =e−βH(0)/Z , (4) σ m τ,t1 n 1 τ,t1
0 0
Finally, the pdf of work w, P (w), is obtained as the
σ
where
sum of the joint probability p (m,τ|n,0)p restricted to
σ n
Z =Tre−βH(0) (5)
0 w =αf −α0 , (14)
n n
is the canonical partition function.
and hence becomes
We denote the orthogonal eigenprojectors of the ob-
servable A by Πi. These satisfy the eigenvalue equa-
i k P (w)= δ[w−αf −α0]p (m,τ|n,0)p . (15)
tions: σ n n σ n
Xn,m
A Πi =αiΠi , (6)
i k k k
where αi are the eigenvalues of A . Hence, with the A. The characteristic function of work
k i
above choice of the first and last measured observables,
α0 and αf represent the instantaneous eigen-energies of Next we focus on the characteristic function of work
k k
the system at t=0 and t=τ, respectively. Gσ(u), given by the Fourier transform of the work pdf
In the following we compute the conditional probabil- Pσ(w):
ity p (m,τ|n,0) to find the eigenvalue αf obtained in
σ m
the lastmeasurementattime τ, providedthatthe eigen- G (u)= dwP (w)eiuw. (16)
value α0 was the result of the first measurement, under σ Z σ
n
thoseconditionsthatarespecifiedbytheprotocolσ. We
begin our discussion considering only one intermediate Substituting (15)into (16) weobtainforthe characteris-
measurement (N =1) of the observable A . tic function:
1
According to the von Neumann postulates, immedi-
ately after the eigenvalue α0 is measured at t = 0, the Gσ(u)= eiu[αfm−α0n]pσ(m,τ|n,0)e−βα0n/Z0
n
system density matrix becomes: Xm,n
ρn(0+)=Π0nρβ0Π0n/pn, (7) =mX,n,reiu[αfm−α0n]TrΠfmUτ,t1Π1rUt1,0
×Π0ρβΠ0U† Π1U†
where n 0 n t1,0 r τ,t1
pn =TrΠ0nρβ0/Z0 =e−βα0n/Z0 (8) =Xr TreiuH(τ)Uτ,t1Π1rUt1,0e−iuH(0)ρβ0
is the probability to find the system initially in the state ×U† Π1U†
t1,0 r τ,t1
with energy α0. Since we assumed that the system is
n =Tr eiuH(τ) e−iuH(0)ρβ, (17)
thermally isolated, it subsequently evolves until time t 0
1 h iσ
accordingtothe unitarytime evolutionU thatisgov-
t1,0
erned by the Schr¨odinger equation: where [X]σ denotes the time evolution of an operator X
from t = 0 to t = τ in presence of the protocol σ, that
i~∂ U =H(t)U , U = . (9) impliesaunitaryevolutiongovernedbyH(t)interrupted
t t,0 t,0 0,0
1
3
by a measurementof an observableA at time t=t . It where the tilde (σ˜) indicates the temporal inversion of
1 1
takes the form the protocol σ, that is:
[X] = U† Π1U† XU Π1U , N =1. (18) σ˜ ={H(τ −t),(τ −ti,Ai)}. (23)
σ t1,0 r τ,t1 τ,t1 r t1,0
Xr Hence, σ˜ specifies the succession of force values and
measurements in the reversed order, specifically with
In the case of a protocol σ containing N interrupting
the measurement of the observables A at times τ −t .
i i
measurementsthe formalexpressionofthecharacteristic
In particular it implies that at time 0, the observable
functionisthesameasforoneinterruptingmeasurement,
A =H(τ) andattime τ, the observableA =H(0)are
f 0
Eq. (17) with the time evolution [X] given by
σ measured. Accordingly, the initial state of the backward
process is given by the Gibbs state ρβ, Eq. (21), i.e. the
[X] = U† Π1 U† Π2 ... f
σ t1,0 r1 t2,t1 r2 system is at equilibrium with inverse temperature β and
r1X...rN force value λ .
τ
...U† ΠN U† XU ΠN U ... Eq. (22) holds under the assumptions that both H(t)
tN,tN−1 rN τ,tN τ,tN rN tN,tN−1
and A commute with the quantum mechanical anti-
...Π2 U Π1 U . (19) i
r2 t2,t1 r1 t1,0 unitary time reversal operator Θ [18]. That is, for all
t∈[0,τ], and i=0...f, we assume
B. The Jarzynski equality H(t)Θ=ΘH(t) (24)
A Θ=ΘA . (25)
i i
Putting u=iβ, one recovers the Jarzynskiequality:
We prove Eq. (22) for the simplest case of a single
intermediatemeasurement,N =1. Thegeneralizationto
G (iβ)=he−βwi =Tr e−βH(τ) eβH(0)ρβ
σ σ 0 many measurements is straightforward. From Eq. (17)
h iσ
we have
=Z−1Tr e−βH(τ)
0
h iσ G (u)=Tr U† Π1U† eiuH(τ)
=Z0−1Tre−βH(τ) =Zf/Z0 =e−β∆F , (20) σ Xr t1,0 r τ,t1
×U Π1U ei(iβ−u)H(0)/Z . (26)
because Tr[X] = TrX for any trace class operator τ,t1 r t1,0 0
σ
X [17]. This follows from the cyclic invariance of the
IntroducingthenotationUt′,t forthetimeevolutiongov-
trace, the unitarity relation U† U = , and the com-
t,s t,s 1 erned by H(t) ≡ H(τ −te), the backward characteristic
Zplfete=neTssreo−fβtHh(eτ)prdoejnecottieosntohpeepraatrotirtsioPnrfuΠnirct=ion1.ofHtehree functGion(oufe)w=oTrkrcanUb†e writΠte1nU†as: eiuHe(τ)
Gibbs state at the initial temperature and final parame- σ˜ τ−t1,0 r τ,τ−t1
ter values: Xr e e
e
×U Π1U e−iuH(0)e−βH(τ)/Z
ρβ =e−βH(τ)/Z , (21) τ,τ−t1 r τ−t1,0 f
f f =Tr Θe Θ†U† eΘΘ†Π1ΘΘ†U† Θ
τ−t1,0 r τ,τ−t1
and ∆F = −β−1lnZ0/Zf is the difference of the free Xr he e e
energies of the thermal equilibrium states ρβf and ρβ0. ×Θ†eiuH(τ)ΘΘ†Uτ,τ−t1ΘΘ†Π1rΘ
The symbol h·iσ denotes an average with respect to ×Θ†U ΘΘ†ee−i(u+iβ)He(0)Θ Θ†/Z , (27)
the work pdf Pσ(w). Eq. (20) says that the Jarzynski τ−t1,0 i f
equality holds irrespective of the details of the interac- where we useed the antiunitarity ΘΘ† = . From Eq.
tion protocol σ. Independent of number and nature of (25) it follows that all eigenprojection op1erators com-
the measuredobservablesas wellas strength,speed, and mute with the time-reversal operator, i.e. Π1Θ = ΘΠ1.
r r
functional form the driving force the average exponenti- The time reversal invariance, expressed by Eq. (24) im-
ated work is solely determined by the free energy differ- plies Θ†esH(t)Θ = es∗H(t) for any C-number s. Further,
ence [8]. microreversibility of driven systems [10, 19] implies
Θ†U Θ=U
τ−t,0 t,τ
C. The work fluctuation theorem Θ†Ueτ,τ−tΘ=U0,t. (28)
Using these relationseand recalling that, for any trace
Besides the Jarzynski equality also the Tasaki-Crooks
class operator X, TrΘXΘ† =TrX†, one ends up with:
theorem is robust under repeated quantum measure-
ments. In the presence of many intermediate measure- G (u)=Tr ei(iβ−u)H(τ)U Π1U
ments it reads: σ˜ τ,t1 r t1,0
Xr
Pσ(w)=Pσ˜(−w)eβ(w−∆F), (22) ×eiuH(0)Ut†1,0Π1rUτ†,t1/Zf. (29)
4
By comparison with Eq. (26), we finally find evidently changes when p(0,τ/2|0,−τ/2) is replaced by
p1(0,τ/2|0,−τ/2). Thesamecanbesaidforthestandard
Z G (iβ−u)=Z G (u), (30) deviation of work that reads
f σ˜ 0 σ
hence, by means of an inverse Fourier transform the h∆w2i=(2Eτ/2)2[1−p(0,τ/2|0,−τ/2)]−hwi2. (34)
1
searched fluctuation theorem (22).
We thus have proved that the fluctuation theorem of Notably, the introduction of an intermediate measure-
Tasaki-Crooks remains unchanged if additionally to the ment may lower the averagework.
measurementsofenergyattimet=0andt=τ,interme- Figure 1b) shows the N dependence of the survival
diatemeasurementsoftimereversalinvariantobservables probability pN(0,τ/2|0,−τ/2) for N equally spaced in-
A areperformed attimes t , providedthe orderof mea- termediatemeasurementsofH(t), andtwosetsofmodel
i i
surements in the backward protocol is properly changed parameters. Oscillatory behavior is observed for small
in accordance with the corresponding times τ −t . values ofN, while asN increasesthe asymptotic value 1
i
isapproached,seetheinset. Whenthe measurementfre-
quencyishighenoughtheunitarydynamicsbetweensub-
sequent measurements becomes increasingly suppressed
III. EXAMPLE
untilthedynamicsiscompletelyfrozenandconsequently
the survival probability reaches the asymptotic value 1
As mentioned in the introduction, one expects mea-
surementstostronglyinfluence the pdfofworkalthough
theJarzynskiequalityandtheTasaki-Crooksrelationare
a)
insensitive to intermediate measurements. To illustrate 0.9
this point in more detail, we consider the example of the
Landau-Zener(-Stu¨ckelberg-Majorana)[13–16]modelde- τ/2)τ/2) 0.8
scribed by the Hamiltonian: /2|0,/2|0, 0.7
ττ
−−
H(t)= v2tσz +∆σx. (31) 11p(0,p(0, 0.6
0.5 τ=h¯/∆,v=5∆2/¯h
It governs the dynamics of a two-level quantum system
whose energy separation, vt, varies linearly in time, and
0 2 4 6 8 10
whose states are coupled via the interaction energy ∆.
tt11 [[∆∆//¯h¯h]]
Here, σ and σ denote Pauli matrices.
x z
The Landau-Zener model is one of the few time-
dependent quantum mechanical problems that have an
analytic solution. The elements of the 2 × 2 unitary b) 11
time evolution matrix U can be expressed in terms
t,s 1 00..7755
of parabolic cylinder functions [20]. The instantaneous
eigenvalues of H(t) are: /2)/2) 00..55
2|0,τ2|0,τ 0.75 110022 110033 110044 110055
Et =(n−1/2) v2t2+4∆2, n=0,1. (32) τ/τ/
n −− 0.5
p 0,0,
((
Since these energies are symmetric with respect to an NNpp
inversionaboutt=0wechoosetheinitialandfinaltimes 0.25 τ=20h¯/∆,v=5∆2/¯h
t0 andtf,as−τ/2andτ/2,respectively,insteadof0and τ=10h¯/∆,v=20∆2/¯h
τ, as in the previous discussion. 0
Figure 1a) depicts the survival probability 0 20 40 60 80 100
p1(0,τ/2|0,−τ/2) as a function of the instant t of NN
1
a single intermediate measurement of H(t ) for positive
1
times t and fixed length τ of the protocol. As an even FIG. 1: (Color online) Survival probability of the Landau-
1
function of t this also specifies p1(0,τ/2|0,−τ/2) for Zener system, Eq. (31), interrupted by measurements. a)
1
Survival probability as a function of the time t1 of a sin-
negative t . The straighthorizontalline shows the value
1 gle energy measurement of H(t1). The parameter values are
of the survival probability p(0,τ/2|0,−τ/2) without τ = 20~/∆, v = 5∆2/~. The horizontal line is the survival
intermediate measurement. It is obvious that the
probability p(0,−τ/2|0,τ/2) with no intermediate measure-
intermediate measurement in general alters the survival ments. b) Survival probability as a function of the number
probability thus affecting the work pdf. For example, N ofequallyspacedintermediatemeasurementsofH(t). The
the average work, parameter values are τ = 20~/∆, v = 5∆2/~ (black circles),
and τ = 10~/∆, v = 20∆2/~ (red triangles). Inset: the sur-
hwi=2Eτ/2tanh[βEτ/2][1−p(0,τ/2|0,−τ/2)], (33) vivalprobabilities approach thevalue 1 in the Zeno-limit.
1 1
5
for N → ∞. This phenomenon is known as the quan- which consequently can be neglected, i.e., states on dif-
tum Zeno effect [21–24]. We investigate it further in the ferentadiabaticenergybranchesatneighboringtimesare
following section. almost orthogonal
hψ (t )|ψ (t )i=δ +O(ε2). (39)
rk k rk+1 k+1 rk,rk+1
A. Quantum Zeno Effect
Herewiththe lefthandsideofEq. (38)simplifiesto read
in the limit of infinitely many dense measurements
To formally elucidate the quantum Zeno effect ob-
servedinthisparticularexample,andundermoregeneral
eiuH(τ) = |ψ (t )ihψ (t )|eiuH(tf)|ψ (t )ihψ (t )|.
conditions as well, we analyze the form of the character- h iν Xr r 0 r f r f r 0
istic function given by Eq. (17) in the limit of infinitely (40)
many measurements of energy. We approach this limit For the generating function this yields
by considering a finite number of N intermediate mea-
surements of energy that take place at equally spaced G∞(u)= eiu[αr(tf)−αr(t0)]e−βαr(t0)/Z . (41)
ν 0
instants Xr
t =t +kε, k =1...N, (35) WethereforefindthatintheZeno-limitthecharacteristic
k 0
functionofworkandaccordinglythepdfofworkcoincide
where ε = (t −t )/(N +1) denotes the time elapsing withthe respectiveexpressionsobtainedforanadiabatic
f 0
between two subsequent measurements. We denote the protocolintheabsenceofanyintermediatemeasurement.
corresponding protocol with symbol ν, that is: Thesamelineofreasoningalsoappliesforthecasethat
an observablethat does not change in time is repeatedly
ν ={H(t),(tk,H(tk))}. (36) measured, i.e., for a protocol
Since we are interested in the limiting case of infinitely µ={H(t),(t ,A)}. (42)
k
many measurements, we may choose N sufficiently large
suchthattheHamiltonianbetweentwosubsequentmea- For suchanobservableAwith eigenprojectionoperators
surementscansafelybeapproximatedbyitsvalueatthe Πr one then obtains in the Zeno limit for the character-
later measurement, i.e. H(t) ≈ H(t ) for t ∈ (t ,t ) istic function
k k−1 k
with t = t +kε being the instant of the kth measure-
k 0 G∞(u)=Tr Π eiuH(tf)Π e−iuH(t0)e−βH(t0).
ment. Thetimeevolutionwithinsuchashortperiodthen µ r r
becomes Xr (cid:16) (cid:17)
(43)
U ≃e−iH(tk)ε/~. (37)
tk,tk−1
1. Landau-Zener
According to Eq. (19) the time evolution U† acts
tk,tk−1
ontheprojectionoperatorΠ (t )resultinginthephase-
rk k In the case of the Landau-Zener problem studied in
factoreiαrk(tk)ε/~ whilethecomplexconjugatephasefac- the previous section, Eq. (31), the eigenvalues on each
tor is obtained from the product Π (t )U which
rk k tk,tk−1 adiabatic branch coincide at the beginning and the end
appearsrightoftheoperatorX [30]. Hence,thesefactors
of the protocol and therefore the coefficients of u in the
cancel each other and one obtains for the ν-propagated
exponential terms on the right hand side of Eq. (41)
exponential operator X =eiuH(τ) the expression
vanish leading to
eiuH(τ) = Π (t )Π (t )... G∞(u)=1. (44)
r1 1 r2 2 ν
h iν r1,rX2,...rN
(38) This leads to the expected result that under permanent
×Π (t )eiuH(τ)Π (t )...
rN N rN N observation of the Hamiltonian no work is done, i.e.,
×Π (t )Π (t ).
r2 2 r1 1
P∞(w)=δ(w). (45)
ν
Assuming first that H(t) is non-degenerate at any time
t ∈ (t ,t ) all projection operators are of the one- TheZenoeffect setsinwhenthe measurementfrequency
0 f
dimensional form Π (t ) = |ψ (t )ihψ (t )| with in- is so large that any unitary evolution within the time
rk k rk k rk k
stantaneous eigenfunctions |ψ (t )i of the Hamiltonian leaps between two subsequent measurements can be ne-
H(t ). Thelabelingoftheinsrtkantkaneouseigenstatescan glected, hence for ε v2τ2/4+4∆/~ ≪ 1, or equiva-
k
be arrangedin sucha waythat the eigenvalues α (t) are lently if N ≫ τ v2τp2/4+4∆/~. With the parameters
k
continuous functions of time. In other words, each adia- used in Fig. 1b)p, the Zeno effect sets in for N ≫103, as
batic energy branch is labeled by an index r. The scalar shown in the inset of Fig. 1b).
products of eigenfunctions at neighboringtimes then de- Inthecaseofdegeneracy,Eq. (41)continuestoholdif
viate from a Kronecker delta by terms of the order ε2 the instantaneous energy branches are labeled such that
6
the corresponding instantaneous eigenvalues smoothly IV. DISCUSSION
vary along each branch. For the Landau-Zener problem
a degeneracy happens at t = 0 if the coupling strength We provided an alternative proof of the robustness of
∆ vanishes. One then finds the characteristic function the Tasaki-Crooks fluctuation theorem against repeated
measurements [11]. The present proof however is more
cosh[(β+2iu)vτ/2]
G∞(u)= [∆=0], (46) general in regard to the fact that it allows for possi-
ν cosh(βvτ/2)
bly degenerate eigenvalues of the measured observables.
Further it was found that the characteristic function of
yielding for the pdf of work
workassumestheformofaquantumtwo-timecorrelation
P∞(w)=p δ(w−vτ)+p δ(w+vτ) [∆=0], (47) function of the exponentiated initial Hamiltonian, and
ν − +
the exponentiated final Hamiltonian in the Heisenberg
where p = e∓βvτ/2/[2cosh(βτ/2)] denote the thermal picture generated by the unitary evolution interrupted
±
populations of the ground- and the excited states at the by quantum collapses.
initial time t=−τ/2, respectively. Ourproofkeepsholdingalsoifquantummeasurements
In the case of countinuous measurement of the Pauli are done with respect to positive operator valued mea-
operators σ , i = x,y,z, the driving-measuremet proto- sures (POVM) [25] which are less invasive than those
i
cols are described by von Neumann projection valued measures.
For the case of measurements with respect to a POVM,
µ ={H(t),(t ,σ )}, i=x,y,z. (48)
i k i M M† = (53)
r r 1
The characteristic functions of work then become Xr
the interruptedevolution[X] is obtainedfromEq. (19)
1 ∆2 σ
G∞µx(u)= 2(cid:18)1− q2 (cid:19) p−e2iqu+p+e−2iqu with the projection operators Πr on the left hand side
(cid:0) (cid:1) ofX being replacedby the weak measurementoperators
1 ∆2 M , and those right of X by the adjoint operators M†.
+ 1+ r r
2(cid:18) q2 (cid:19) DuetoEq. (53),Tr[X]σ =TrX continuestoholdimply-
ing the validity of the Tasaki-Crooks relation, Eq. (22),
1
G∞µy(u)= 2 p−e2iqu+p+e−2iqu+1 (49) for weak measurement of time reversal invariant observ-
(cid:0) (cid:1) ables with M Θ=ΘM .
1 ∆2 k k
G∞(u)= 1+ p e2iqu+p e−2iqu In provingEq. (22) we assumed that the Hamiltonian
µz 2(cid:18) q2 (cid:19) − +
(cid:0) (cid:1) and the measured observables commute with the time
1 ∆2 reversal operator. This assumption may be relaxed and
+ 1− ,
2(cid:18) q2 (cid:19) the Tasaki-Crooksfluctuation theoremcontinues to hold
if the backwardprotocol is defined as:
where p denotes the populations of the exited and the
± σ˜ ={ΘH(τ −t)Θ†,(τ −t ,ΘA Θ†)}. (54)
ground states of the initial Hamiltonian, respectively, i i
given by Notwithstanding the robustness of the Tasaki-Crooks
theorem, we found, in accordance with the intuitive ex-
p =e∓βq/ eβq+e−βq , (50)
± pectation, that the interruption of the dynamics of a
(cid:0) (cid:1) drivenquantum system by means of projective measure-
with
ments alters the statistics of work performed on the sys-
tem. Weillustratedtheinfluenceofmeasurementsonthe
q = (vτ/2)2+∆2. (51)
work distribution by the example of the Landau-Zener(-
p
Stu¨ckelberg-Majorana)model. We noticed that depend-
TheworkpdffollowsthenbymeansofaninverseFourier
ing on the positions in time or frequency of one or more
transform to read
interruptingmeasurements,theaverageworkhwimaybe
1 ∆2 lowered, Fig. 1.
p∞ = 1− (p δ(w−2q)+p δ(w+2q))
µx 2(cid:18) q2 (cid:19) − + Schmiedl et al. [26] studied the problem of design-
ing optimal protocols that minimize the average work
1 ∆2
+ 1+ δ(w) spent during the forcing, without considering interme-
2(cid:18) q2 (cid:19) diate measurements. In order to further minimize the
1 average work, one might expand the optimization pa-
p∞ = (p δ(w−2q)+p δ(w+2q)+δ(w)) (52)
µy 2 − + rameter space and include the possibility of performing
1 ∆2 intermediate measurements. This opens for the possibil-
p∞ = 1+ (p δ(w−2q)+p δ(w+2q))
µz 2(cid:18) q2 (cid:19) − + ity of a much greater control over the energy flow into
the system.
1 ∆2
+ 1− δ(w). In the limit of very frequent measurements the uni-
2(cid:18) q2 (cid:19) tary dynamics becomes completely suppressed due to
7
the quantum Zeno effect. If at each measurement time spectively. These are crucial prerequisite for the imple-
the instantaneous Hamiltonian is measured, then in the mentation of working quantum computers. The com-
Zeno limit the work characteristic function approaches bined study of dissipative Landau-Zener dynamics with
the same form that it would assume if the protocol was frequent observations, could unveil yet new practical
adiabatic. methods for quantum state control and manipulation.
The result of an intermediate measurement may be
used to alter the subsequent force protocol. In this way
a feedback control can be implemented for classical [27]
as well as for quantum systems [12]. In both cases the
Acknowledgments
Jarzynski equality only holds in a modified form.
Landau-Zener dynamics and frequent quantum mea-
surements of a two-level system coupled to a thermal This work was supported by the cluster of excellence
bath were recently shown to provide efficient means for Nanosystems Initiative Munich (NIM) and the Volkswa-
quantum-state preparation [28] and purification [29], re- gen Foundation (project I/80424).
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