Table Of ContentQuantum decoherence reduction by increasing the thermal bath temperature
A. Montina, F.T. Arecchi
Dipartimento di Fisica, Universit`a di Firenze, Via Sansone 1, 50019 Sesto Fiorentino (FI), Italy
(Dated: February 4, 2008)
The well-known increase of the decoherence rate with the temperature, for a quantum system
coupled to a linear thermal bath, holds no longer for a different bath dynamics. This is shown by
meansofasimpleclassicalnon-linearbath,aswellasaquantumspin-bosonmodel. Theanomalous
effectisduetothetemperaturedependenceofthebathspectralprofile. Thedecoherencereduction
viathe temperature increase can be relevant for the design of quantum computers.
8 The rapid development of nanotechnology has opened fiers, this effect is due to the non-linearity, which makes
0 thepossibilityofassemblingmaterialsanddevicesatthe theshapeofthebathspectrumdependentonthetemper-
0 atomic scale, such as nanomaterials and nano electro- ature. Forourpurpose,weconsideraclassicalnon-linear
2
mechanical systems (NEMS) [1]. Because of quantum bathmodel, consistingof a one-dimensionalparticle in a
n mechanicalandstatistical effects, the properties ofthese double-well potential, strongly coupled to a linear bath.
a
systems can be considerably different from their macro- Furthermore, we also consider a full quantum mechan-
J
scopiccounterpart. Electronsinnanofabricatedquantum ical bath model, consisting of a boson bath coupled to
4
dotsaswellasnanomechanicaldevicesarepossiblecandi- a two-state system, which is the low temperature limit
2
dates for quantum bits at nanometer scale, as suggested of a double-well system under suitable conditions [5]. A
] by recent experiments [2]. Quantum computers could probe quantum system is coupled to the non-linear bath
h
take advantage of quantum superpositions in order to and its decoherence rate is evaluated.
p
- perform highly efficient parallel calculations. The main The equation of motion of a one-dimensional particle
nt difficulty in the implementation is represented by deco- coupledtoawhitenoisethermalbathismx¨= V′(x)
− −
a herence, due to the coupling with the environment [3]. mγ¯x˙+√2Dη(t),wherem,V(x),γ¯andDaretheparticle
u This effect increases with the number of quantum bits mass,theexternalpotential,thedissipationanddiffusion
q
and is devastating even with a small thermal bath cou- coefficients, respectively. The function η(t) is a Gaus-
[
pling. Thus, in order to reduce the decoherence of quan- sian noise with η(t)η(t′) = δ(t t′). The fluctuation-
h i −
1 tum superpositions, a sufficiently small thermal distur- dissipationtheoremestablishesthat D =γ¯mk T,k and
b b
v
bance and environmentcoupling are required. The ther- T being the Boltzmann constant and the thermal bath
4
mal disturbance is expected to be reduced by decreas- temperature, respectively. We consider the double-well
4
8 ing the bath temperature. This is true when the quan- potential V(x) = ax2/2 + bx4/4. By means of the
3 tum system is coupled with a high number of degrees of rescaling x (a/b)−1/2x, t (m/a)1/2t and T a2 T,
1. freedom and the bath can be described as a linear noise the equation→of motion beco→mes → bkb
0 source. On the contrary, at the nanometer scale, the
8 quantum system can be directly coupled to few modes, x¨=x x3 γ x˙ + 2γ Tη(t), (1)
1 1
0 whose power spectrum does not necessarily grow with − −
v: temperature. Thus, it is possible to have a reversed ef- where γ1 γ¯(m/a)1/2. The indeppendent parametersare
≡
i fect, i.e., the reduction of decoherence by increasing the the rescaled dissipation coefficient and temperature.
X
thermal bath temperature. At finite temperature, the particle has a finite prob-
r
ability to jump from a well to the other one. The hop-
a A similar counterintuitive behavior occurs for exam-
pingrateis[6]R (√2πγ )−1exp[ (4γ T)−1](Kramers
ple in thermal rectifiers [4], where the heat flux can be ≡ 1 − 1
rate). In the limit of very low temperature, the hop-
suppressed by increasing the temperature difference of
pingratebecomes verysmallandthe particleremainsin
the junctions. The rectifier consists of two or three mu-
one well for a large time. As a consequence, the spec-
tually coupled layers, whose phononic bands depend on
trum of x(t) has a high zero-frequency peak. As the
thetemperaturevianon-lineareffects. Astheendsofthe
temperature is increased, the hopping rate between the
chain are coupled with two heat baths at different tem-
twowellsgrows,thecentralpeakofthespectrumreduces
peratures T and T , the overlaps among the phononic
1 2 and two smooth peaks around the frequencies πR are
bands in each layer become functions of T1 and T2 and generated. This behavior is shown in Fig. 1, w±here we
are not identical when the thermal baths are exchanged. reportthe spectrumI(ω,T)=2Re ∞ x(0)x(t) eiωtdt,
As a consequence,the thermalflux canchange when the 0 h iT
forγ =0.4andsomevaluesofthetemperature. I(ω,T)
rectifier is reversed. 1 R
has been evaluated numerically over a time interval of
In this paper, we show that the decoherence rate of 200, in the non dimensional units, and 50000 ensemble
a quantum system coupled to a non-linear thermal bath realizations. We have verified that the time interval is
can decrease with the temperature. As in thermal recti- sufficiently large to ensure a negligible time cut-off of
2
the integral. In Fig. 2, we plot I(0,T) (solid line) and Schr¨odinger equation i∂|ψi = (Ω/2)σˆ1 +ǫσˆ3x(t) ψ ,
∂t A A | i
where σˆk are the Pauli matrices in the orthonormal ba-
A (cid:2) (cid:3)
5 sis 1 , 1 . Ω is the energy difference between the
{|− i | i}
T=0.5 bare energy levels [1 1 ]/√2 and ǫ is the coupling
4 T=1 | i±|− i
T=4 coefficient between the quantum and classical systems.
3 The variable x satisfies the classical Eq. (1). We have
)
ω seen that the spectrum I(ω,T) of the anharmonic os-
(
I2 cillator is a decreasing function of temperature only for
sufficiently small values of ω . If the quantum system
1 | |
hasadynamicssufficientlyslowerthanthe classicalhop-
0 ping rate (Ω R), then it is sensitive only to the low
-4 -2 ω0 2 4 ≪
frequency components of the bath spectrum. We sat-
isfy the condition Ω R by assuming that the two lev-
FIG. 1: Power spectrum of x(t) for some values of T. The ≪
els are nearly degenerate. For the sake of simplicity we
vertical lines are placed at πRfor each spectrum.
set Ω = 0. If ψ(0) = ( 1 + 1 )/√2 is the quan-
| i |− i | i
tum state at time t = 0, then at a subsequent time t
I(0,T)/T K(0,T)(dashedline)asfunctionsofT. The
≡ the density matrix associatedwith a single realizationof
decreasing behavior of K(0,T) suggests a simple model
of thermal rectifier, as will be discussed in an extended x(t)isρˆ[x](t)= 1 1 e−2iǫR0tx(t)dt anditis
paper. HerewewillusethedecreasingbehaviorofI(0,T) 2 e2iǫR0tx(t)dt 1 !
to show the anomalous behavior of decoherence induced a functional of the thermal bath variable. This depen-
by the non-linearity. We consider a two-state quantum dence is removed by averaging over every realization of
system (A) coupled to the double-well oscillator (B), as x(t). The reduced density matrix is < ρˆ[x] >x ρˆA and
≡
sketchedin Fig. 3. The overallsystem B-(linear bath) is the modulus of its off-diagonal elements
the non-linear bath seen by A. The linear bath and the e2iǫR0tx(t)dt (t). (2)
anharmonicity of B guarantee that the overall system is |h i|≡C
a reservoir and a non-linear system, respectively. is a measure of the coherence between the superposed
states 1 and 1 . is a good indicator since 1
|− i | i C |− i
and 1 aretwoperfectpointerstatesofapuredephasing
10 I(0,T) | i
process. It ranges between the values 0 (no coherence)
K(0,T)
and 1 (complete coherence).
We will show that the decoherence rate of A is pro-
portionalto I(0,T), providedthat ǫ is sufficiently small.
1
Let t be the correlation time of x(t) and δt t be
c c
≫
a finite time interval. Let us consider the finite differ-
ence (t+δt) (t)= (t) e2iǫRtt+δtx(t)dt 1 . Fora
0 1 2 3 4 5 6 C −C C |h i|−
T largetemperaturerange,wehcanassume x2 1,ithus,if
h i≈
2ǫδt 1,thenwecanapproximatethepreviousequation
FIG. 2: I(0,T)and K(0,T) as functions of thetemperature, as (≪t+δt) (t) 2ǫ2 (t) t+δtdt′ t+δtdt¯x(t′)x(t¯) .
in logarithmic scale. C −C ≃− C t t h i
Since δt t , we can replace the second integral with
c
I(0,T) a≫nd have ˙ = 2ǫ2IR(0,T) R D(T) . The
C − C ≡ − C
decoherence rate D(T), being proportional to I(0,T), is
non−linear bath a decreasing function of T (see Fig. 2). This equation
is correct when the integration step δt is much smaller
system A B T than 1/(2ǫ) and much larger than t , thus the condition
c
1/t 2ǫ has to be satisfied. For γ = 0.4, the correla-
c 1
≫
tiontimeisabout10atT 0.25anddecreasesforhigher
∼
FIG. 3: Schematic representation of the non-linear thermal temperatures. At T 1-2, tc is for example about 5.
∼
bath,coupledtothequantumsystemA. Bistheanharmonic Thus,ǫhas tobe smallerthanabout0.1-0.05. This clas-
oscillator. sical model demonstrates that the quantum decoherence
rate is a decreasing function of the temperature when
This classical bath model is suitable for a sufficiently the quantumsystem is resonantwith bath spectralcom-
high thermal bath temperature T. A full quantum ap- ponents depleted by the thermal energy supplying. The
proach will be introduced later on. We consider an in- non-linearityisafundamentalingredient,sinceforlinear
teraction Hamiltonian linear in x and assume that the baths the spectrum grows linearly with temperature at
quantum state ψ ψ 1 + ψ 1 satisfies the any frequency.
−1 1
| i ≡ | − i | i
3
Ourclassicalmodelworkswhenhighorderenergylev- ∆ 1/τ . It is clear that for ω = ∆, the spectrum ini-
c
≫ 6
els of the double-well system are activated, i.e., for a tially increases with temperature, reaches a maximum,
sufficiently high temperature T. Now, we consider the then decreases and tends to zero for T . In the
→ ∞
opposite limit and assume that only the two lowest en- particular case ω = ∆, I(ω,T) is a monotonic decreas-
ergy states are populated. This is possible if the energy ing function for any value of temperature. With this in
difference ∆ between the ground and first excited states mind, wechoosethe frequencyΩ ofsystemAinorderto
is much smallerthan the higher transitionenergies. ∆ is maximize the effect of temperature on the decoherence
the quantum tunneling frequency between the two wells. reduction. The system A has to be resonant with the
Let the Paulimatrix σˆ3 be the position operator,whose part of the spectrum I(ω,T) which is strongly depleted
B
eigenstates 1 correspondto the particlein the left or by the temperature increase, thus we assume Ω = ∆.
|± i
right well. This quantum system (B) is coupled to a bo- This case simplifies the analysis and is sufficient for our
son bath (spin-boson model) and to the two-state probe purposes.
system(A) described the Paulimatricesσˆi . The overall At this point, we take ǫ =0 and write out the master
A 6
system B-(bosonbath) is the non-linear thermalbath of equation for A and B. If ǫ 1/τ , then it is obtained
c
≪
A (Fig. 3). fromEq.(3)byaddingthebareHamiltonianofAandthe
Moreprecisely,westudyamodelwiththeHamiltonian interaction Hamiltonian ǫσˆ3σˆ3 to the bare Hamiltonian
A B
Hˆ = Ωσˆ1 +∆σˆ1 +ǫσˆ3σˆ3 + [ησˆ3(aˆ +aˆ†)+ω aˆ†aˆ ], of B. In the rotating-wave representation (interaction
2 A 2 B A B k B k k k k k
where aˆ and aˆ† are the annihilation and creation oper- picture),theoverallbareHamiltonian(∆σˆ1 +∆σˆ1)/2is
k k P A B
ators of the bath oscillators. In the present model x is removed and the following replacements are performed:
replacedby the Paulimatrixσˆ3. Inorderto understand
how the decoherence of A depeBnds on its bare frequency σˆA3,B →σˆA−,Be−i∆t+σˆA+,Bei∆t. (6)
Ω, we have to evaluate the two-time correlationfunction
σˆ3(0)σˆ3(t) in the Heisenberg representation for ǫ=0, The condition ǫ 1/τc justifies a rotating-waveapprox-
ahsBdoneBwithi the first model. For η sufficiently small imation also for≪the interaction between A and B. Thus
(see below for a more precise condition), we can use the we obtain the master equation
rotating-wave approximation and find the master equa-
∂ρˆ
tion for B (section 10.5 of Ref. [7]) = iǫ[σˆ+σˆ−+σˆ−σˆ+,ρˆ]+ ρˆ. (7)
∂t − A B A B L
∂ρˆ ∆
B = i [σ1,ρˆ ]+ ρˆ , (3) Inordertoeliminatetheparameter∆inn¯,wedefinethe
∂t − 2 B B L B rescaled temperature T˜ T/∆.
≡
where
1
~
ρ γ n¯ σˆ−σˆ+ρˆ +ρˆ σˆ−σˆ+ 2σˆ+ρˆ σˆ− T=1
−LγBb(n¯≡+−1)b σˆ(cid:2)B+σBˆB−ρBˆBB+ρˆBσBˆB+σBˆB−B−−2σˆB−ρBˆBσBˆB+B(cid:3) (4) 0.8 TT~~==220
e ~
adnepdeσˆnB±ds≡on(σˆηB2a±(cid:2)nidσˆB3th)e/2d.eγnbsiitsyaocfotnhsetabnotscooneffimcoidee(cid:3)nstawthtihche herenc00..46 T=80
o
frequency ω =∆ and n¯(T) [exp(∆/T) 1]−1 is their C
k
≡ −
expectation value aˆ†aˆ . This equation is well-known 0.2
h k ki
since it describes the decay of a two-level atom coupled
to optical modes. 0 5 t (103 units) 15 20
From the master equation (3) and using the quantum
regressiontheorem,wehavethatthecorrelationfunction FIG. 4: Numerically evaluated coherence C as a function of
at equilibrium is [7] time for some valuesof temperature T˜.
hσˆB3(0)σˆB3(t)i=e−|t|/τc(T) e−i∆tn↑+ei∆tn↓ , (5) Inthismodel,the vectors 1 and 1 arenotperfect
| i |− i
pointerstatesbecauseofthe presenceofthe bareHamil-
where n↑ ≡ hσˆB+σˆB−i = 2coes−h∆(∆/(cid:2)2/T2T) and n↓ ≡ hσˆ(cid:3)B−σˆB+i = tonianofthe systemA,thuswecannotusetheprevious
1 n↑ are the probability weights for the energy lev- definition of coherence. We need a generalized measure
−
els of B and τc(T) [(2n¯+1)γb]−1 is the temperature- of the deviation from the unitary evolution. A good in-
≡
dependent correlation time. The corresponding spectral dicator is provided by the quantity C = 2Tr[ρˆ2] 1,
function is I(ω,T) = 2τcn↑ + 2τcn↓ , which ρˆ being the reduced density operator of the systAem−A.
1+τc2(ω−∆)2 1+τc2(ω+∆)2 A p
has two peaks centered in ∆, whose areas and widths Forpurestatesitisequalto1andtendstozeroforcom-
±
are constant and linear functions of temperature, re- pletely incoherenthigh temperature thermal states. It is
spectively. The rotating-wave approximation is suitable easy to prove that this definition is equivalent to Eq. (2)
when the two peaks have a negligible overlap, i.e., when when 1 and 1 are equally populated pointer states.
| i |− i
4
At thermal equilibrium, we have = tanh[1/(2T˜)]. Ob- thepointerstates 1 and 1 . Thus,ifτ isthe cor-
B B c
C |− i | i
viously,itisadecreasingfunctionoftemperatureandfor relationtime ofB, the probability ofthe transitionafter
T˜ 0, takesthemaximalandminimalvalues1and0, a time ∆t τ is P [ǫτ ]2∆t0 = ǫ2τ ∆t
→ ∞ 0 ≫ c (−1,1)→(1,−1) ≃ c τc c 0
respectively. The transient behavior of is numerically andweobtainheuristicallytherelaxationrateinEq.(8).
C
evaluatedbysettingtheprobesystemAinthepurestate Alternatively, the anomalous effect can be explained as
[ 1 + 1 ]/√2attheinitialtimet=0,withthesystem due to a rapid change of the phases of 1 1 and
A B
B|−theirm|aliizedatthe temperatureT˜. InFig.4wereport 1 1 whichreducesthecoherenceo|f−theitw|oistates
A B
| i |− i
(t) as a function of time for γ = 1, ǫ = 5 10−2 and and consequently their transitionprobability. It is inter-
b
CsomevaluesofT˜. Notethat,forT˜ =80,thede·coherence esting to note that the decoherence rate suppression for
rate becomes very small with respect to the value at low T inthe lastmodelcanbe obtainedbyacondition
→∞
temperature. established in Ref. [8] [equation (25) in that paper]. By
The master equation (7) has been obtained by means means of this, one finds that the Hilbert space of A is
of the conditions ǫ 1/τ ∆. Under the constraint decoherence-free, provided that n¯ 1.
c
≪ ≪ ≫
ǫ 1/τ ,itispossibletofindananalyticalrelation,since
c
≪ In conclusion, we have consider two examples of non-
somecomponentsofρˆ(t)canbeadiabaticallyeliminated.
linear baths weakly coupled to a quantum system A and
The following approximatesolution for the reduced den-
shownthat the decoherence rate ofA is a monotonic de-
sity operator of the probe system holds,
creasingfunctionoftemperature. Thisanomalousbehav-
ρˆ =ρˆ +e−ǫ2τct(c σˆ2 +c σˆ3)+e−2ǫ2τctc σˆ1, (8) ior is due to the temperature dependence of the spectral
A th 1 A 3 A 2 A
profileinnon-linearbaths. Itisimportanttorealizethat
where ρˆ is the thermalized density operator, c are at a microscopic scale non-linear effects can be relevant,
th k
constant parameters which depend on the initial condi- as it occurs with thermal rectifiers. For more realistic
tion and ǫ2τ = ǫ2γ−1tanh(1/2T˜) Γ is a decreas- models, the decoherence rate could not be a monotonic
c c ≡ d
ing function of temperature. Notice that this solution decreasing function of temperature. However our anal-
couldbeobtaineddirectlybythecorrelationfunction(5). ysis indicates that the decoherence rate is not necessar-
By means of the rotating-wave approximation and the ily minimal at T = 0, as for linear baths. Decoherence
standard hypotheses used to derive Eq. (3) and the re- reduction by the thermal energy increase can have rele-
sult of the previous model (second order expansion in vant consequences for the design of quantum computers
the coupling strength and Markov approximation), one at nanometer scale. An experimental test of this effect
finds that ρˆ is solution of a master equation identi- is feasible with the present technology and can be ob-
A
cal to Eq. (3), but with γ replaced by ǫ2τ tanh 1 = served in nanometer devices [2, 9]. For example, A and
b c 2T˜
2 B could be a superconducting qubit and a microwave
ǫ2γ−1 tanh 1 . Equation (8) is its exact solution.
c 2T˜ resonator [9]. This work was supported by Ente Cassa
This d(cid:16)erivation(cid:17)will be discussed in an expanded paper. di Risparmio di Firenze under the project ”Dinamiche
Asintheclassicalmodel,wefindthatthedecoherence cerebrali caotiche”.
andrelaxationratesaredecreasingfunctionsoftempera-
ture. Inthiscasetheeffectismuchstronger. ForT˜ 1,
≫
thedecoherencerateisinverselyproportionaltothetem-
perature and for a sufficiently high thermal noise it can
be practicallyreducedtozero. Theotherinterestingfea- [1] M. Blencowe, Phys.Rep. 395, 159 (2004).
[2] C.P.Garc´ıaetal.,Phys.Rev.Lett.95,266806(2005);M.
ture is the reduction of Γ by increasing the coupling
d D. LaHaye,O. Buu,B. Camarota, K.C. Schwab, Science
parameter γb. This effects are due to the depletion of 304, 74 (2204); A. Haik et al., Nature443, 193 (2006).
bath spectrum I(ω,T) at the frequencies resonant with [3] W. H.Zurek, Rev.Mod. Phys. 75, 715 (2003).
the quantum system. [4] M.Terraneo,M.Peyrard,G.Casati,Phys.Rev.Lett.88,
There is an interesting link between the anomalous 094302 (2002).
behaviors in the last model and the Zeno effect. As- [5] A. J. Leggett, S. Chakravarty, A. T. Dorsey, Matthew P.
sume that the two systems A and B are in the state Fisher, Anupam Garg, W. Zwerger, Rev. Mod. Phys. 59,
1 (1987).
1 1 at time t. Because of their mutual cou-
| − iA| iB [6] L. Gammaitoni, P. H¨anggi, P. Jung, and F. Marchesoni,
pling, at a subsequent time t+∆t, they evolve to the Rev.Mod. Phys.70, 223 (1998).
state 1 1 +ǫ∆t1 1 and the probabilityof
A B A B [7] D. F. Walls, G. J. Milburn, Quantum Optics (Splinder,
|− i | i | i |− i
the transition 1 A 1 B 1 A 1 B is (ǫ∆t)2. In New Yorks,1995).
|− i | i → | i |− i
the meantime, the coherence of this superposition is de- [8] S. Pascazio, J. Mod. Opt.51, 925 (2004).
stroyedbythethermalbath,whichactsasanobserverof [9] A. Wallraff et al.,Nature 431, 162 (2004).
