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arXiv:0802.1596v1 [quant-ph] 12 Feb 2008
COHERENCE CONSERVATION OF A QUBIT
COUPLED TO A DISSIPATING THERMAL ENVIRONMENT
ROBABEH RAHIMI
Interdisciplinary Graduate School of Science and Engineering, Kinki University,
Higashi-Osaka, Osaka 577-8502, Japan
rahimi@alice.math.kindai.ac.jp
AKIRA SAITOH
Department of Systems Innovations, Graduate School of Engineering Science, Osaka University
Toyonaka, Osaka, 560-8531, Japan
saitoh@qc.ee.es.osaka-u.ac.jp
MIKIO NAKAHARA
Department of Physics, Kinki University
Higashi-Osaka, Osaka 577-8502, Japan
nakahara@math.kindai.ac.jp
It is shown that quantum coherence is conserved in a principal system in the case
that the system is coupled to a fast dissipating environment [arXiv:0709.0562]. The
phenomenon is called the quantum wipe effect. Here, this effect is reviewed and the
analytical proof for a model system consisting of a one-qubit system coupled to a fast
dissipating environment is extended to an environment at a thermal equilibrium.
Keywords: decoherence conservation; dissipating environment.
1. Introduction
A conventional approach in facing the problem of decoherence in a system, most
usually, is by concentrating on the principal system itself. It is somehow implicitly
assumed that there is no control over the environmental system 1−7. Then, by
operating with the system itself, for instance, by applying on it, fast and strong
multi-pulses for a dynamical control of the system 2,3, the decoherece is aimed to
be suppressed. On the other hand, the situation could be entirely different if one
could have access to the environment for controlling the decoherence of the system.
From the first point of view, the idea of suppressing decoherence of a principal
system by controlling the environment instead of working directly with the princi-
pal system might seem not to be reasonable. This is true specially when one tries
to decrease the noisy behavior of the environment for stabilizing the principal sys-
tem. However, a control of the decoherence of a system is attained by making the
environment even noisier, then this approach should turn out to be feasible. In a
1
2Robabeh Rahimi, Akira SaiToh, Mikio Nakahara
numerical simulation of bang-bang control of entanglement in a spin-bus model,
decoherence in the principal system is shown to be suppressed if the environment is
made to be rapidly dissipating to a very large bath environment 8. In fact. simillar
phenomena in different models are known in the community 9,10.
We have investigated the concept of controlling the decoherence of a system
while the corresponding environment dissipates fast into another larger environment
that can be a bath system 11. This phenomenon is called the “quantum wipe effect”.
This effect is proved analytically for the case of a one-qubit principal system when
it is coupled to a maximally mixed state environment that is dissipating to a large
bath environment. Numerical evaluations for a single-qubit principal system coupled
to a dissipating bosonic environment is conducted in addition to the example of an
entangled two spin state as a principal system coupled to the environment 11. Here,
we extend the analytical proof of the previous work to the case where the principal
system is a one-qubit system coupled with a fast dissipating environment that is
initially in a Boltzmann distribution rather than a simple maximally mixed state.
In the following section the model for this study is introduced. In section 3
the analytical proof is given. In section 4, findings of the mathematical proof is
discussed in more details, giving an sketched overview on the phenomenon itself
and explaining the conditions under which the phenomenon can be effective.
2. Model
We assume a model system involving a principal system (system 1) coupled to an
environment (system 2). The system is represented by ρ[1,2]. We further assume
a large thermal environment system surrounding the systems 1 and 2, such that
the state of the system 2 is replaced by that of the thermal environment with
probability p(namely, with some dissipation rate) per unit time interval τ. The
thermal environment is represented by the density matrix σ.
The Hamiltonian affecting the time evolution is reduced to the one consisting
only of the time-independent Hamiltonian Hthat governs systems 1 and 2 including
their interaction. This model is illustrated in Fig. 1. For a small time interval ∆t,
the evolution of the systems 1 and 2 obeys the equation
ρ[1,2](˜
t+∆t) = e−iH∆tx∆tρ[1,2](˜
t) + (1 −x∆t)Tr2ρ[1,2](˜
t)⊗σeiH∆t,(1)
where x= (1 −p)1/τ and ˜
tdenotes a certain time step. The dissipation rate pcan
be modified by changing the experimental setup under a static control.
For this model it is shown 11 that coherence conservation is achieved by pvery
close to 1, meaning that the environmental system 2 is rapidly dissipating to the
thermal environment. For the case that the principal system is a one-qubit system
and the system 2 is a maximally mixed state, the analytical proof is previously
given, in addition to numerical evaluations of the effect for other cases of which the
principal system is a two-qubit system or an entanglement of two spins 11. Here,
Coherence Conservation of a Qubit Coupled to a Thermal Dissipating Environment 3
Principal System
(System 1) Environmental
System (System 2)
p
Environmental
System at the
Thermal State
HHamiltonian
Fig. 1. Model for the system consisting of the principal system (system 1) and the environmental
system (system 2) whose time evolution is governed by the Hamiltonian H. System 2 is replaced
with a thermal environmental system with the dissipation probability pfor the time interval τ.
we extend the analytical proof to a more operationally reasonable case where the
system 2 is in the Boltzmann thermal state.
3. Qubit-qubit coupling with a thermal environmental qubit
Let us consider the following setting. The principal system is originally represented
by a density matrix
ρ[1](0) = a b
b∗1−a(2)
with 0 ≤a≤1 and 0 ≤ |b| ≤ pa(1 −a). The environmental system at thermal
equilibrium is represented by the thermal density matrix (under the high temper-
ature approximation)
ρ[2](0) = σ=(1 + ǫ)/2 0
0 (1 −ǫ)/2
with ǫ= tanh[E∆/(2kBT)], the polarization for the Zeeman energy E∆and tem-
perature T(kBis the Boltzmann constant). The initial state of the total system
is set to ρ[1,2](0) = ρ[1] (0) ⊗ρ[2] (0).The Hamiltonian His set to cIz⊗Iz=
diag(c/4,−c/4,−c/4, c/4) [here, Iz= diag(1/2,−1/2)].
Under these conditions, ρ[1,2] at time t=m∆t(m∈ {0,1,2, . . .}) is given as
ρ[1,2](m∆t) =
a(1 + ǫ)/2 0 fm0
0a(1 −ǫ)/2 0 gm
f∗
m0 (1 −a)(1 + ǫ)/2 0
0g∗
m0 (1 −a)(1 −ǫ)/2
,
with functions fmand gmdepending on m, satisfying the system of recurrence
formulae as follows
fm+1 =e−ic∆t/2x∆tfm+1+ǫ
2(1 −x∆t)(fm+gm)
gm+1 =eic∆t/2x∆tgm+1−ǫ
2(1 −x∆t)(fm+gm)
4Robabeh Rahimi, Akira SaiToh, Mikio Nakahara
with f0=b(1 + ǫ)/2 and g0=b(1 −ǫ)/2. This leads to the following recurrence
formula:
κm+2 −κm+1 1 + ǫ
2+1−ǫ
2x∆te−ic∆t/2+1−ǫ
2+1 + ǫ
2x∆teic∆t/2
+κmx∆t= 0,
(3)
where κm=fmor gmwith f0and g0as introduced above, and f1=b(1 +
ǫ)e−ic∆t/2/2, and g1=b(1 −ǫ)eic∆t/2/2.
One can derive functions f(t) = lim∆t→0,m∆t=tfmand g(t) = lim∆t→0,m∆t=tgm
in the following way. By linearization, Eq. (3) is put in the form:
κm+2 −2κm+1 +κm−∆tln x(κm+1 −κm) + (∆t)2
2(c2/2−icǫ ln x)κm+1
−(∆t)2
2(ln x)2(κm+1 −κm) + O[(∆t)3] = 0.
Dividing this equation by (∆t)2and taking the limit ∆t→0 lead to
∂2
∂t2κ(t)−ln x∂
∂t κ(t) + c2
4−icǫ ln x
2κ(t) = 0,
where κ(t) = lim∆t→0,m∆t=tκm. The solution of this differential equation is
κ(t) = uκe−r+t+vκe−r−t
with constants uκand vκ(κ=for g), and the complex decoherence factor
r±=−1
2hln x±p(ln x)2−c2+i2cǫ ln xi.
We need to impose the conditions that f(0) = b(1 + ǫ)/2, g(0) = b(1 −ǫ)/2,
and κ′(0) = lim∆t→0(κ1−κ0)/∆t. The latter condition can be written as −r+uf−
r−vf=−ibc(1+ǫ)/4 and −r+ug−r−vg=ibc(1−ǫ)/4, for κ=fand g, respectively.
Thus we obtain
uf=−b(1 + ǫ)
2(r+−r−)r−−ic
2, vf=b(1 + ǫ)
2(r+−r−)r+−ic
2,
ug=−b(1 −ǫ)
2(r+−r−)r−+ic
2, vg=b(1 −ǫ)
2(r+−r−)r++ic
2.
Consequently, we have
f(t) = b(ic −2r−)(1 + ǫ)
4(r+−r−)e−r+t+b(−ic + 2r+)(1 + ǫ)
4(r+−r−)e−r−t,
g(t) = b(−ic −2r−)(1 −ǫ)
4(r+−r−)e−r+t+b(ic + 2r+)(1 −ǫ)
4(r+−r−)e−r−t.
One can now write the reduced density matrix of the principal system at tas
ρ[1](t) = a η(t)
η(t)∗1−a(4)
Coherence Conservation of a Qubit Coupled to a Thermal Dissipating Environment 5
0
0.25
0.5
0.75
1
1.25
1.5
0 1 2 3 4
-(ln x) / c
Re r /c (ε=0.0)
−
Re r /c (ε=0.0)
+
Re r /c (ε=0.25)
−
Re r /c (ε=0.8)
−
Re r /c (ε=0.25)
+
Re r /c (ε=0.8)
+
(a)
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
0 1 2 3 4
-(ln x) / c
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
0 1 2 3 4
-(ln x) / c
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
0 1 2 3 4
-(ln x) / c
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
0 1 2 3 4
-(ln x) / c
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
0 1 2 3 4
-(ln x) / c
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
0 1 2 3 4
-(ln x) / c
Im r /c (ε=0.25)
+
Im r /c (ε=0.25)
−
Im r /c (ε=0.8)
+
Im r /c (ε=0.8)
−
Im r /c (ε=0.0)
−
Im r /c (ε=0.0)
+
(b)
Fig. 2. (a) Plots of Re r±/c as functions of −(ln x)/c for different values of polarization ǫ. For
ǫ= 0, the decoherence factors Re r±increases until they reach c/2 as the dissipation rate increases;
the factor Re r+starts decreasing at the point of −ln x=c(i.e., p= 1 −e−cτ ) while the factor
Re r−starts increasing rapidly at this point. For other values of ǫ, the behavior is similar to that
of ǫ= 0. (b) Plots of Im r±/c as functions of −(ln x)/c for different values of polarization ǫ.
with
η(t) = b−r−+icǫ/2
r+−r−
e−r+t+r+−icǫ/2
r+−r−
e−r−t.(5)
It is possible to realize that if pincreases to reach one, then |η(t)|converges to
b, giving the coherence of ρ[1] (t), Eq. (4), equal to that of ρ[1](0), Eq. (2), ignoring
an unimportant phase factor. This is clear if one investigates the behavior of r±in
details in relation to pand ǫ. Figure 2, (a) and (b), shows the real and imaginary
parts of r±as functions of −(ln x)/c for several different values of ǫ. It is clear that
the total behavior of Re r±does not very much depend on different values of ǫ.
However, by increasing −(ln x)/c, namely by increasing p, the decoherence factors
Re r±increase until they reach a certain value (e.g. c/2 for the case of ǫ= 0) then
the factor Re r+starts decreasing while the factor Re r−starts increasing very
rapidly. Fig. 2 (b) shows that as −(ln x)/c increases, namely as papproaches to
unity, Im r±do not have any large change.
These plots help us to depict the overall behavior of η(t), Eq. (5), for pclose to
one. The imaginary terms, for such p, contribute to the phase factor of η(t) mainly
which is not a significant factor of coherence. Among real factors, Re r+contributes
to η(t) through the first term of Eq. (5), since Re r+converges to zero for pclose to
one. However, if Re r−becomes very large then the second term of Eq. (5) does not
have a big contribution. Then |η(t)|in the limit of pclose to one converges to band
gives ρ[1](t), Eq. (4), equal to ρ[1](0), Eq. (2), ignoring the phase of η(t). One can
conclude that for large dissipation rate p, decoherence does not have effect on the
principal system. The convergence of |η(t)|to bfor large dissipation rate is clearly
depicted in Fig. 3 in which the time evolution of |η(t)|is shown for several different
values of pwhen ǫ= 0.0 (Fig. 3 (a)) and ǫ= 0.25 (Fig. 3 (b)), and ǫ= 0.8 (Fig. 3
6Robabeh Rahimi, Akira SaiToh, Mikio Nakahara
0
0.2b
0.4b
0.6b
0.8b
b
0 5 10 15 20
|η(t)|
t [ms]
p=0.0
p=0.25
p=0.50
p=0.75
p=0.95
p=1.0
(a)
0
0.2b
0.4b
0.6b
0.8b
b
0 5 10 15 20
|η(t)|
t [ms]
p=0.0
p=0.25
p=0.50
p=0.75
p=0.95
p=1.0
(b)
0
0.2b
0.4b
0.6b
0.8b
b
0 5 10 15 20
|η(t)|
t [ms]
p=0.0
p=0.25
p=0.50
p=0.75
p=0.95
p=1.0
(c)
Fig. 3. (a) Time evolution of |η(t)|for several
different values of p(0.0, 0.25, 0.5, 0.75, 0.95,
and 1.0) when ǫ= 0.0, τ= 1.0×10−3s and
c= 1.0×103Hz. (b) Time evolution of |η(t)|for
several different values of p(0.0, 0.25, 0.5, 0.75,
0.95, and 1.0) when ǫ= 0.25, τ= 1.0×10−3s
and c= 1.0×103Hz. (c) Time evolution of |η(t)|
for several different values of p(0.0, 0.25, 0.5,
0.75, 0.95, and 1.0) when ǫ= 0.8, τ= 1.0×
10−3s and c= 1.0×103Hz.
(c)).
4. Conclusion
We consider decoherence in a model that involves a one-qubit principal system cou-
pled to an environment in the Boltzmann distribution, in which the environment
itself rapidly dissipates to a large bath environment. We have shown that deco-
herence of the principal system is suppressed for very large dissipation rates from
the Boltzmannian environmental to the large bath environment. This phenomenon
is called the quantum wipe effect11 and can be understood as follows. If the dis-
sipation rate is very large then the environmental system does not have enough
time to affect the principal system for absorbing coherence information from the
principal system. Thus the environmental system is wiped out and the decoherence
of the principal system is suppressed without touching the principal system. It is
hoped that this effect will be investigated extensively to ease the static control of
decoherence.
Acknowledgments
RR is supported by the Grant-in-Aid for JSPS fellows (Grant No. 1907329). AS is
supported by the Grant-in-Aid for JSPS fellows (Grant No. 1808962). MN would
Coherence Conservation of a Qubit Coupled to a Thermal Dissipating Environment 7
like to thank for partial support of the Grant-in-Aid for Scientific Research from
JSPS (Grant No. 19540422). This work is partially supported by “Open Research
Center” Project for Private Universities: matching fund subsidy from MEXT.
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