Zeno and anti-Zeno dynamics in spin-bath models

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by  Dvira Segal

Zeno and anti-Zeno dynamics in spin-bath models
Dvira Segal, David R. Reichman
Department of Chemistry, Columbia University, 3000 Broadway, New York, NY 10027
WeinvestigatethequantumZenoandanti-Zenoeffectsinspinbathmodels: thespin-bosonmodel
7 andaspin-fermionmodel. WeshowthattheZeno-anti-Zenotransitioniscriticallycontrolledbythe
0 system-bath coupling parameter, the same parameter that determines spin decoherence rate. We
0 also discuss thecrossover in a biased system, at high temperatures, and for a nonequilibrium spin-
2 fermion system, manifesting the counteracting roles of electrical bias, temperature, and magnetic

n field on thespin decoherencerate.
a
J PACSnumbers: 03.65.Xp,03.65.Yz,05.30.-d,31.70.Dk

6
2
ThequantumZenoeffect(QZE)describesthebehavior mous interestdue to significantexperimentalprogressin

] of a quantum systemwhen frequent shorttime measure- mesoscopic physics [10]. For both models the prototype
l
l ments inhibits decay [1, 2]. In some cases, however, the Hamiltonian includes three contributions
a
h anti-Zeno effect (AZE), namely an enhancement of the
H =H +H +H . (1)
- decay due to frequent measurements, is observed [3, 4]. S B SB
s
The QZE can be easily obtained for an oscillating (re-
e The spin system includes a two level system (TLS) with
m versible)quantumsystem. Whenthe systemisunstable,
a bare tunneling amplitude ∆ and a level splitting B,
the situation is more involved, and one can obtain both
.
at the QZE and AZE, depending on the interaction Hamil- B ∆
H = σ + σ . (2)
m tonian [4], as well as the measurement interval [5]. A S 2 z 2 x
crossoverfrom QZE to AZE behavior has been observed
- In what follows we refer to the bias B as a magnetic
d inanunstabletrappedcoldatomicsystemviaatuningof
n field in order to distinguish it from potential bias in a
the measurement frequency [6]. Recently, Maniscalco et
o nonequilibrium system. In the bosonic case the thermal
al. [7]havetheoreticallyinvestigatedthe Zeno-anti-Zeno
c bath includes a set of independent harmonic oscillators,
[ crossover in a model of a damped quantum harmonic
and the system-bath interaction is bilinear
oscillator. These authors demonstrated the crucial role
1
v playedbytheshorttimebehavioroftheenvironmentally H(b) = ǫ b†b ,
B j j j
5 induced decoherence.
j
5 X
6 Inthisletter wefocusonspin-bathmodels,paradigms H(b) = λj(b†+b )σ . (3)
1 of quantum dissipative systems, and analyze the condi- SB 2 j j z
j
0 tions for the occurrence of the Zeno and anti-Zeno ef- X
7 fects. We show that the crossover between the two pro- b†,b arebosoniccreationandannihilationoperators,re-
0 j j
cesses is critically controlled by the dimensionless cou- spectively. For a fermionic system we employ the model
/
t pling strength, as well as the temperature and the en-
a
m ergy bias between the spin states. We also analyze the HB(f) = ǫka†k,nak,n,
- Zenodynamicsforout-of-equilibrium,electricallybiased, Xk
nd swithuicahtiodnetse.rOmuinremthaeinexretseunlttoifsqtuhaantttuhme scaomheerepnacreamfoerttehres HS(fB) = Vk,n2;k′,n′a†k,nak′,n′σz. (4)
o transientpopulationvariable,alsocontrolthe Zeno-anti- k,kX′,n,n′
c
: Zeno transition. Therefore, the nature of spin decoher- Here a†,a arefermionic creationandannihilationoper-
v ence can be predicted from the Zeno dynamics. k k
i ators, and the spin interacts with n reservoirs. We also
X
The models of interest here are the spin-boson model define two auxiliary Hamiltonians H± as
r
and a nonequilibrium steady state spin-fermion model,
a B
which is a simplified variant of the Kondo model. The H =± +H (σ =±)+H . (5)
± SB z B
2
spin-boson model, describing a two-levelsystem coupled
to a bath of harmonic oscillators, is one of the most NotethattheHamiltonian(1)doesnotincludeexplicitly
important models for elucidating the effect of environ- the measuring device. While in Refs. [11] (boson bath)
mentally controlled dissipation in quantum mechanics and[12](fermionbath)thereservoirsserveascontinuous
[8]. The Kondo model, possibly the simplest model of a detectors, here they are part of the dynamical system
magnetic impurity coupled to an environment, describes under measurement.
the coupling of a magnetic atom to the conductionband Before studying the Zeno dynamics of the dissipative
electrons [9]. This system has recently regained enor- systems,webrieflyreviewtheresultsforanisolatedTLS.
2
For zero magnetic field, the probability to remain in the 15
0.6
initialpreparedstateisgivenbyW(t)=cos2(∆t/2).The
short-time dynamics (∆t≪1 ) can be approximated by τ)0.3 ξ=0.1
W(t) ∼ 1−(∆/2)2t2. If measurements are performed γ(
at regular intervals τ, the survival probability at time
10 0 ξ=0.2
t=nτ becomesW(t)∼1−n(∆/2)2τ2 ∼exp(−∆2τt/4). 0 ω20 τ 40
c
Atshorttimes aneffectiverelaxationrateisidentifiedas τ)
γ(
γ (τ)=(∆/2)2τ. (6)
0
5
As τ goes to zero, decay is inhibited, and the dynamics
is frozen. This is the quantum Zeno effect.
We consider next the influence of the environment on
this behavior. Within perturbation theory the dynamics
of a TLS coupled to a general heat bath is given as a 0
0 5 10 15 20
power series of ∆ terms, W(t) = ∞ (∆)2nΦ (t) [13]. ω τ
n=0 2 n c
We have numerically verified that for ω τ . 20, (ω is
c c
P
the reservoircutoff frequency) the series can be approxi- FIG.1: Theeffectivedecayrateforaspincoupledtoabosonic
mated by the first two terms bathatzerotemperatureforωc=1,Es=0,ξ=0.1,0.2,0.3...1,
top to bottom. The appearance of a maximum point in γ(τ)
W(τ) ∼ 1−2ℜ ∆ 2 τ dt t1dt K(t ), aroundωcτ =2forξ>1/2marksthetransitionfromQZEto
2 1 2 2 AZE.Theinsetdepictstherateforξ=0.4(full),0.5(dashed),
K(t) = he−iH+t(cid:18)eiH−(cid:19)ti,Z0 Z0 (7) 0.6 (dotted),ωc=10.
even for strong system-bath coupling [14], provided that
where ξ is a system-bath dimensionless coupling param-
∆τ < 1 [15]. Here ℜ denotes the Real part, the Hamil-
eter[8]. Atzerotemperatureandforzeromagneticfield,
tonians H are defined in Eq. (5), and the trace is
±
disregardingthe energyshift, the effective decay rate(8)
doneoverthebathdegreesoffreedom,irrespectiveofthe
is given by (ξ 6= 1,1)
statistics. Similarly to the isolated case, we can identify 2
an effective decay rate at short times 1−(1+ω2τ2)1−ξcos[2(1−ξ)atan(ω τ)]
γ(τ) = c c .
∆2 τ t1 τω2(1−ξ)(1−2ξ)
γ(τ)= ℜ dt K(t′)dt′. (8) c
2τ 1 (11)
Z0 Z0
In what follows we disregard the multiplicative factor The limiting behavior of this expression is
(∆/2)2. Thebasicquestiontobe addressedinthisletter
is how does this relaxation rate depend on the system τ ω τ <1
c
γ(τ)∝ (12)
bath coupling. It is clear that when the system is de- (τ1−2ξ ωcτ >1.
coupled from the environment, Eq. (8) reproduces the
result of the isolated system, Eq. (6). For a dissipative Therefore, while at very short times (ωcτ < 1) the sys-
system the central object of our calculation is therefore tem always shows the QZE, irrespective of the system-
the correlationfunction K(t) defined in Eq. (7). For the bathcoupling,atintermediatetimes1/ωc <τ <1/∆the
bosonic system (3), a standard calculation yields [16] qualitative behavior of the decay rate crucially depends
on the dimensionless coupling coefficient. The decay is
1 ∞ J(ω)
K (t) = e−iEstexp − dω inhibited (QZE) for ξ < 1, and accelerated (AZE) for
b (cid:26) π Z0 ω2 ξ > 1. At ξ = 1 the de2cay rate does no depend on
2 2
× (nω +1) 1−e−iωt +nω 1−eiωt .(9) the measurement interval, γ(τ) −l−og−ω(−cωτc−τ−)<−→1 π∆2. Corre-
(cid:27) spondingly, at zero temperature an unbiased4ωscpin oscil-
(cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3)
Here nω = [eβω − 1]−1 is the Bose-Einstein distribu- lates coherently at ξ =0, shows damped harmonic oscil-
tion function, β = 1/T is the inverse temperature, and lation for 0 < ξ < 1, and decays incoherently at strong
Es = B + Jπ(ωω)dω is the polaron shift. The spectral dissipation, 12 <ξ <21. At the crossovervalue ξ = 12 the
density J(ω) includes the informationabout the system- system can be mapped onto the Toulouse problem, de-
R
bathinteractionJ(ω)=π jλ2jδ(ω−ωj). Foranohmic scribinganimpuritycoupledtoaFermibathwithacon-
model, J(ω)=2πξωe−ω/ωc, Eq. (9) becomes [17] stantdensityofstates. ThisleadstoapurelyMarkovian
P
exponential decay which is unaffected by measurements
2ξ
2ξ πt [18]. We therefore find that the dimensionless coupling
Kb(t)=(cid:18)1+1iωct(cid:19) sin(cid:16)hβ (cid:17)πt  ×e−iEst (10) dcoentesrtmanitneξswthheichocccounrtrreonlcsethoef tehxeteQntZoEfaspnidntdheecoAhZerEe.nce,
β
 (cid:16) (cid:17)
3
8 F (ω); τ=1
τ
2
7γτ() 1.5 Fτ(ω); τ=5
1
6 K(ω); ξ=0.2
5 0 K(ω); ξ=0.8
2 4ω τ6 8 10 1
τ)4 c
γ(
3
0.5
2
1
0
0 0 1 2 3 4
0 2 4 6 8 10 ω
ω τ
c
FIG. 3: The QZE and the AZE, as observed through the
FIG.2: TheZeno-anti-Zenotransitionofaspin-bosonmodel
under a magnetic field, T = 0K, ωc=1, ξ = 0.1, B=0, 0.2, s0p(esceteraEl qfu.nc(t1i3o)nsfoKr(dωe)finaintdionFsτ)(.ω)InfcorreaTsin=g0ξ,,ωKc=(ω1), Eshsift=s
B0.4=,00,.6ξ,=to0p.1to(fbuoltl)t,oξm=. I0n.s4et(:dTasehmedp)e,raξt=ure0.e7ff(edcto,ttβeωdc).=0.5, towards higher frequencies overlapping poorly with Fτ=5(ω),
and strongly with Fτ=1(ω),which leads to the AZE.
Besides the occupation probability W(t), the sym-
place, while for a bath with narrow coupling spectrum
metrized equilibrium correlation function C(t) is also of
and for |E −ω |≪1/τ the QZE occurs [4].
interest. It is unclear whether the coherent-incoherent s m
Fig. 3 displays K(ω) at two values: ξ=0.2, 0.8. We
transition of this quantity occurs at ξ = 1 [19], or be-
2 also show the measurement function F (ω) for ω τ=1,
low, at ξ = 1 [20]. Yet, since the short-time dynamics τ c
3 5. We find that at weak coupling, the overlap between
of the two-spincorrelationfunction C(t) is controlledby
these twofunctions decreasesuponshorteningτ, leading
the same function K(t) [Eq. (7)] [21], its Zeno-anti-Zeno
totheQZE.Incontrast,atstrongcoupling,sincethecen-
behavior exactly corresponds with W(t), and thus rigor-
tralfrequencies ofF (ω) andK(ω)aredetuned, increas-
ously undergoes a Zeno-anti-Zeno transition at ξ = 1. τ
2 ingthe widthofF (ω)enhancesthe overlapbetweenthe
Fig. 1depictsthedecayrateγ(τ)forξ =0.1−1,man- τ
functions, therefore the decay rate, leading to the AZE.
ifesting the crossoverfromthe QZEto the AZEatξ = 1
2 Thisisclearlyseenmathematically: Atzerotemperature
(inset). The transition between the Zeno and anti-Zeno
2ξ−1
regimes can be manipulated by applying a finite mag- K(ω)= ωcΓ1(2ξ) ωωc e−ω/ωc, with ωm ∼(2ξ−1)ωc.
netic field and by employing a finite temperature bath. This argument a(cid:16)lso(cid:17)explains the AZE observed at weak
Fig. 2 shows that at weak coupling, a biased system coupling for finite magnetic fields. Since F (ω) is cen-
τ
B/ωc ∼1 tends towards a pronounced anti-Zeno behav- tered around B, and ωm ∼ 0 for ξ ≤ 0.5, the AZE is
ior,incontrasttothezeromagneticfieldcase. Theinflu- expected to prevail for Bτ >1, as seen in Fig. 2.
ence of finite temperatures is radically different (inset): We turn nowto the fermionic bath. Fora single reser-
It drives the system into the Zeno regime, for all cou- voir at zero temperature, and for times Dτ > 1, the
pling strengths. This can be deducted fromEq. (9): For fermionic correlation function can be calculated in the
βωc <1,K(t)∼e−2πξt/β,andtheresultingdecayrateis context of the Fermi-edge singularity problem [22],
proportional to τ at short times, saturating at ω τ >1.
c
Our results can be elucidated by recasting the decay e−iBt
K (t)∼ . (14)
rate as a convolution of two memory functions, f (1+iDt)δ2/π2
∆ 2 ∞
γ(τ)=2 dωK(ω)F (ω), (13) Here δ = atan(πρV), ρ is the density of states, D is
τ
(cid:18)2(cid:19) Z0 the bandwidth, the equivalent of the cutoff frequency ωc
in the bosonic case, and we used a constant coupling
where the measurement function is given by F (ω) =
τ sinc2 (ω−Es)τ , and K(ω)=ℜ ∞eiωtK(t)dt,τcan be cmoomdienlgVfkr,okm′ =thVe.diWageonhaalvceoduipslrienggarVded,tahseitencaenrgaylwshaiyfts
2π 2 0 k,k
interprehted as thie reservoir couplinRg spectrum [4]. Note be accommodated into the external magnetic field.
thatK(t)isredefinedherewithouttheenergyshift. The Comparing Eq. (14) to the bosonic expression (10),
short-time behavior is therefore determined by the over- leads to the conclusion that an unbiased spin coupled to
lapofthesetwomemoryfunctions. For|E −ω |≫1/τ, a fermionic bath can only manifest the QZE: Since the
s m
with ω as the central frequency of K(ω), AZE takes phase shift cannot exceed π/2, the exponent in (14) is
m
4
always ≤ 1/4, while according to Eq. (12), the AZE 4
takesplace only for largervalues whichareprohibitedin 0.4
3.5
tahttisaicnasael.aArgesrpienxpcoounpenletd,wtohimchorceanthlaenadatsointghleeAleZaEdm[23a]y. 3 ρε() V00..23
The nonequilibrium spin-fermion system, where the
0.1
spin couples to two fermionic baths (Vk,n;k′n′ = V, 2.5 −5 0 5
n=1,2)with chemicalpotentials shifted by δµ, is much τ) 2 ε
more interesting and involved. In this case we numer- γ(
ically calculate the correlation function K (t), since its 1.5
f
analyticformisnotknownatshorttimesDt≤1andfor
1
arbitrary voltages [24]. This is done by expressing the
zero temperature many body average as a determinant 0.5
of the single particle correlationfunctions [16]
0
0 2 4 6 8 10
Kf(t) = he−iH+teiH−ti=det[φk,k′(t)]k,k′<kf , τ
φk,k′(t) = hk|e−ih+teih−t|k′i. (15)
FIG. 4: The effective decay rate for a spin coupled to two
Here H = h are the single particle Hamiltonians, fermionic baths demonstrating the counteracting roles of the
± ±
|ki are the single particle eigenstates of H(f) and the magnetic field and electrical bias. B=0, δµ=0 (full); B=0.5,
P B δµ=0(dashed);B=0.5,δµ=0.2(dotted). Theinsetshowsthe
determinant is evaluated over the occupied states.
Lorentzian shaped density of states.
For a short-time evolution, it is satisfactory to model
the fermionic reservoirsusing 200states per bath, where
biasisappliedbydepopulatingoneofthereservoirswith
a double-wellpotential, interactingwith a currentcarry-
respect to the other. The decay rate γ(τ), (Eq. (8)), is
ing quantum point contact (QPC) [28]. Measurements
presented in the main plot for three situations: In the
ofthe spinstatecanbe doneeithercontinuouslywithan
absence of both magnetic field and electric bias (full),
additionalQPC,servingthis time asadetector,orusing
including a finite magnetic field (dashed), and under an
laser radiation,directly detecting the population in each
additional potential bias (dotted). We find that the po-
of the wells.
tential bias in the fermionic system plays the role of the
This work was supported by NSF (NIRT)-0210426.
temperatureinthebosonicenvironment[24],drivingthe
anti-Zeno behavior into a Zeno dynamics.
Summarizing,we find that the same coupling parame-
terthatmonitorsspindecoherenceandrelaxation,deter-
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