Entropy production in open quantum systems: exactly solvable qubit models
We present analytical results for the time-dependent information entropy in exactly solvable two-state (qubit) models. The first model describes dephasing (decoherence) in a qubit coupled to a bath of harmonic oscillators. The entropy production for this model in the regimes of "complete"...
Збережено в:
| Опубліковано в: : | Condensed Matter Physics |
|---|---|
| Дата: | 2012 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут фізики конденсованих систем НАН України
2012
|
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/120301 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Entropy production in open quantum systems: exactly solvable qubit models / V.G.Morozov, G. Röpke // Condensed Matter Physics. — 2012. — Т. 15, № 4. — С. 43004:1-9. — Бібліогр.: 15 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860122715021639680 |
|---|---|
| author | Morozov, V.G. Röpke, G. |
| author_facet | Morozov, V.G. Röpke, G. |
| citation_txt | Entropy production in open quantum systems: exactly solvable qubit models / V.G.Morozov, G. Röpke // Condensed Matter Physics. — 2012. — Т. 15, № 4. — С. 43004:1-9. — Бібліогр.: 15 назв. — англ. |
| collection | DSpace DC |
| container_title | Condensed Matter Physics |
| description | We present analytical results for the time-dependent information entropy in exactly solvable two-state (qubit) models. The first model describes dephasing (decoherence) in a qubit coupled to a bath of harmonic oscillators. The entropy production for this model in the regimes of "complete" and "incomplete" decoherence is discussed. As another example, we consider the damped Jaynes-Cummings model describing a spontaneous decay of a two-level system into the field vacuum. It is shown that, for all strengths of coupling, the open system passes through the mixed state with the maximum information entropy.
Ми представляємо аналiтичнi результати для часовозалежної iнформацiйної ентропiї в двостанових точно розв’язних (кубiт) моделях. Перша модель описує дефазування (декогеренцiю) в кубiтi, який є зв’язаний з резервуаром гармонiчних осциляторiв. Обговорюється продукування ентропiї для цiєї моделi у режимах “повної” та “неповної” декогеренцiї. Як iнший приклад ми розглядаємо задемпфовану модель Джейнса-Каммiнгса, яка описує спонтанне затухання дворiвневої системи в польовому вакуумi. Показано, що для всiх сил зв’язку, вiдкрита система переходить через змiшаний стан з максимумом iнформацiйної ентропiї.
|
| first_indexed | 2025-12-07T17:39:58Z |
| format | Article |
| fulltext |
Condensed Matter Physics, 2012, Vol. 15, No 4, 43004: 1–9
DOI: 10.5488/CMP.15.43004
http://www.icmp.lviv.ua/journal
Entropy production in open quantum systems:
exactly solvable qubit models
V.G. Morozov1, G. Röpke2
1 Moscow State Technical University of Radioengineering, Electronics, and Automation,
Vernadsky Prospect 78, 119454 Moscow, Russia
2 University of Rostock, FB Physik, Universitätsplatz 3, D–18051 Rostock, Germany
Received July 3, 2012, in final form August 23, 2012
We present analytical results for the time-dependent information entropy in exactly solvable two-state (qubit)
models. The first model describes dephasing (decoherence) in a qubit coupled to a bath of harmonic oscillators.
The entropy production for this model in the regimes of “complete” and “incomplete” decoherence is discussed.
As another example, we consider the damped Jaynes-Cummings model describing a spontaneous decay of a
two-level system into the field vacuum. It is shown that, for all strengths of coupling, the open system passes
through the mixed state with the maximum information entropy.
Key words: information entropy, open quantum systems, qubit models, decoherence, quantum entanglement
PACS: 03.65.Ud, 03.65.Yz
1. Introduction
The notion of information entropy has attracted a renewed interest over the last few decades in con-
nection with fundamental problems in the theory of open quantum systems and quantum entanglement
(see, e.g., [1–4] and references therein). Although the literature concerning diverse aspects of this topic
is now quite voluminous, not much is known about the entropy behavior in concrete open systems ex-
hibiting especially intriguing features of quantum dynamics: memory effects, dephasing (decoherence),
entanglement, etc.
The simplest models describing many fundamental dynamic properties of open quantum systems
are two-state systems. Such systems themselves deserve thorough studies as the elementary carriers of
quantum information (qubits) [5, 6]. It is also important to note that some of two-state models admit exact
solutions. The latter fact allows one to gain a valuable insight into general properties of open quantum
systems. From this point of view, it is of interest to analyze the time behavior of information entropy in
exactly solvable models.
In this paper we present exact analytic results for the time-dependent information entropy in two
physically reasonable qubit models which are frequently used in discussing different problems in the
theory of open quantum systems.
2. Entropy of a qubit
We consider a two-state quantum system (qubit) coupled to a reservoir. In what follows, the qubit
and the reservoir will be referred to as subsystems A and B, respectively.
Suppose that at time t the state of the combined system (qubit plus reservoir) is described by some
density matrix ̺AB (t). Then, the reduced density matrix of the qubit is defined as
̺A(t)= TrB̺AB (t), (1)
© V.G. Morozov, G. Röpke, 2012 43004-1
http://dx.doi.org/10.5488/CMP.15.43004
http://www.icmp.lviv.ua/journal
V.G. Morozov, G. Röpke
where TrB denotes the trace over the reservoir degrees of freedom. The von Neumann (information)
entropy of the qubit is given by
S A(t) =−TrA
[
̺A(t) ln̺A(t)
]
. (2)
It is convenient to use the “spin” representation for a qubit by writing its orthonormal basis |0〉 and
|1〉 as
|0〉 =
(
0
1
)
, |1〉 =
(
1
0
)
. (3)
Then, all operators referring to a qubit can be expressed in terms of the Pauli matrices ~σ =
{
σ1,σ2,σ3
}
.
In particular, the density matrix (1) can be written in the form [7, 8]
̺A(t)= 1
2
[1+~σ ·~v (t)], (4)
where
~v(t)= TrA
[
~σ̺A(t)
]
(5)
is the so-called Bloch vector. In [9] we have shown that there exists another representation for the qubit
density matrix, which is better suited to calculate ln̺A(t) in (2):
̺A(t)= 1
2
√
1− v2(t) exp[~σ ·~u(t)], u = 1
2
ln
(
1+ v
1− v
)
. (6)
Here, v(t) is the modulus of the Bloch vector and ~u⇈~v . Strictly speaking, the representation (6) is valid
only for a mixed state of a qubit with v < 1. Note, however, that the limit v → 1 can be taken directly in
the entropy (2) after calculating the trace. Using expressions (6), one easily derives from (2)
S A(t) = ln 2−
1
2
(1+ v) ln (1+ v)−
1
2
(1− v) ln (1− v) . (7)
For a pure state (v → 1), we have S A = 0, as it should be. The entropy has its maximum S A = ln2 in the
mixed state with v = 0, when the density matrix (4) is diagonal and 〈0|̺A |0〉 = 〈1|̺A |1〉 = 1/2.
The square modulus of the Bloch vector (5) can in general be written as
v2(t) = 4v+(t)v−(t)+ v2
3(t), (8)
where
v±(t)= TrA[σ±̺A(t)], (9)
andσ± =
(
σ1 ± iσ2
)
/2. The two terms in (8) have different physical interpretations. The quantities v+(t)=
〈0|̺A(t)|1〉 and v−(t) = 〈1|̺A(t)|0〉 are often referred to as the coherences. They describe the environ-
mentally induced dephasing [1]. On the other hand, the time behavior of the component of the Bloch
vector v3 = 〈1|̺A (t)|1〉− 〈0|̺A(t)|0〉 is determined by energy exchange between an open system and its
environment, which is responsible for complete statistical equilibrium in the combined system. Thus, for-
mulas (7) and (8) are convenient for studying the role of different relaxation mechanisms in the entropy
production.
3. Entropy production in a dephasing model
We start with a simple spin-boson model describing a qubit coupled to a reservoir of harmonic oscil-
lators [1, 10–12]. The Hamiltonian of the model is (in our units ħ= 1)
H = HA +HB +HI =
ω0
2
σ3 +
∑
k
ωk b†
k
bk +σ3
∑
k
(
gk b†
k
+ g∗
k bk
)
, (10)
where ω0 is the energy difference between the excited state |1〉 and the ground state |0〉 of the qubit.
Bosonic operators b†
k
and b
k
correspond to the kth reservoir mode with frequency ω
k
.
43004-2
Entropy production in open quantum systems: exactly solvable qubit models
Note that σ3 commutes with the Hamiltonian (10). As a consequence, the populations 〈0|̺A(t)|0〉
and 〈1|̺A (t)|1〉 do not depend on time. In other words, there is no relaxation to a complete equilibrium
between the qubit and the environment; that is, themodel is nonergodic. However, we shall see below that
thismodel exhibits a dephasing relaxation and entropy productionwithout energy exchange between the
qubit and the environment.
Let us first specify the initial density matrix ̺
AB
(0) of the combined system. Usually (see, e.g., [1,
10–12]) it is assumed that the subsystems A and B are uncorrelated, and the reservoir is in thermal
equilibrium at some temperature T . In this paper we will concentrate on the entropy production in time-
dependent entangled quantum states of the combined system. We assume that at time t = 0, the combined
system is prepared in a pure quantum state which is a direct product
|ψAB (0)〉 =
(
a0|0〉+a1|1〉
)
⊗|0B 〉, (11)
where |a0|
2 +|a1|
2 = 1, and |0B 〉 denotes the ground state of the reservoir. The initial state |0B 〉 is chosen
only for simplicity’s sake. The subsequent discussion may easily be extended to the case of an arbitrary
initial state |ψB (0)〉 of the reservoir. The density matrix corresponding to (11) is
̺AB (0) = |ψAB (0)〉〈ψAB (0)|. (12)
Since the evolution of the combined system is unitary, the initial state (11) evolves after time t into the
pure state
|ψAB (t)〉 = exp(−iH t) |ψAB (0)〉, (13)
so that the density matrix of the combined system is given by
̺AB (t) = |ψAB (t)〉〈ψAB (t)|. (14)
In principle, the qubit density matrix ̺A(t), and then the entropy S A (t) can be calculated by using (14).
It is more convenient, however, to calculate the modulus of the Bloch vector, v(t), and then apply for-
mula (7). Since v3 is constant and is determined by the amplitudes a0 and a1 in (11), we need only to
consider the coherences v±(t). Note that expression (9) can be rewritten as
v±(t)= TrAB
[
σ±(t)̺AB (0)
]
, (15)
where σ±(t) are the Heisenberg picture operators and the trace is taken over all degrees of freedom of
the combined system. In the model (10), equations of motion for σ±(t) can be solved exactly. The result
reads [9]
σ±(t)= exp
[
±iω0t ∓R(t)
]
σ± , (16)
where the operator R(t) acts only on the reservoir states and is given by
R(t) =
∑
k
[
αk (t)b†
k
−α∗
k (t)bk
]
, αk (t)= 2gk
1−eiω
k
t
ω
k
. (17)
Substituting the expression (16) into (15) and recalling formula (12), we find
v±(t) = v±(0)exp
[
±iω0t −γvac(t)
]
(18)
with the so-called vacuum decoherence function [1]
γvac(t)=− ln〈0B |exp[R(t)]|0B 〉 =−
∑
k
ln〈0B |exp
[
αk (t)b†
k
−α∗
k (t)bk
]
|0B 〉. (19)
After simple algebra which we omit, we obtain
γvac(t)=
∞∫
0
dω J (ω)
1−cosωt
ω2
, (20)
43004-3
V.G. Morozov, G. Röpke
where the continuum limit of the reservoir modes is performed, and the spectral density J (ω) is intro-
duced by the rule
∑
k
4|gk |
2 f (ωk ) =
∞∫
0
dω J (ω) f (ω). (21)
Now using the solution (18) and taking into account that v(0) = 1, we find from equation (8)
v2(t) = v2
3 +
(
1− v2
3
)
exp
[
−2γvac(t)
]
. (22)
Formulas (7) and (22) determine the time evolution of the qubit entropy. To go beyond these formal
relations, one needs some information on the spectral density J (ω). In many cases of physical interest
(see, e.g., [12, 13]), J (ω) may be considered to be a reasonably smooth function which has a power-law
behavior J (ω) ∝ ωs (s > 0) at frequencies much less than some “cutoff” frequency Ω, characteristic of
the reservoir modes. In the limit ω→ ∞, J (ω) is assumed to fall off at least as some negative power of
ω. For the spectral density, we shall take the expression which is most commonly used in the theory of
spin-boson systems [10–13]:
J (ω)= λsΩ
1−s ωs e−ω/Ω , (23)
where λs is a dimensionless coupling constant. The case s = 1 is usually called the “Ohmic” case, the case
s > 1 is “super-Ohmic”, and the case 0< s < 1 is “sub-Ohmic”.
Substituting the spectral density (23) into (20) and doing standard integrals, one gets
γvac(t)=λsΓ(s −1)
{
1−
cos[(s −1) arctan(Ωt)]
(
1+Ω2t 2
)(s−1)/2
}
, (s , 1),
γvac(t)=
λ1
2
ln
(
1+Ω
2t 2
)
, (s = 1),
(24)
where Γ(s) is the Euler gamma function. Some important properties of γvac(t) can easily be seen directly
from the above expressions. First, γvac(t) is a monotonously increasing function of time for s É 1. Second,
in the super-Ohmic case γvac(t) has a long-time limit:
γvac(∞) ≡ lim
t→∞
γvac(t)= λsΓ(s −1), (s > 1). (25)
Finally, the γvac(t) monotonously saturates to γvac(∞) for 1 < s É 2 and is a nonmonotonous function of
time for s > 2.
In discussing the properties of the qubit entropy in the model (10), it is necessary to distinguish two
cases: the regime of “complete decoherence”, and the regime of “incomplete decoherence”. In the former
case (s É 1) we have γvac(t) →∞ as t →∞, and hence v±(t)→ 0. Then, it follows directly from (7) and (22)
that the limiting value of the qubit entropy is given by
S A(∞)= ln2−
1
2
(
1+|v3|
)
ln
(
1+|v3|
)
−
1
2
(
1−|v3|
)
ln
(
1−|v3|
)
, (s É 1). (26)
The maximum qubit entropy Smax(∞)= ln2 corresponds to the initial state with equal populations (v3 =
0). In the case of “incomplete decoherence”, we have
S A(∞)= ln2−
1
2
(
1+ v∞
)
ln
(
1+ v∞
)
−
1
2
(
1− v∞
)
ln
(
1− v∞
)
, (s > 1), (27)
where
v∞ =
{
v2
3 +
(
1− v2
3
)
exp
[
−2γvac(∞)
]}1/2
. (28)
Figure 1 illustrates the time behavior of the qubit entropy for different values of the parameter s and for
different coupling strengths. It should be emphasized at once that, for the case under consideration here,
the entropy production has no “thermodynamic” meaning. Indeed, at any time t , the combined system is
in the pure quantum state (13). Note, however, that this state is entangled [1], i.e., it cannot be written as
a direct product of states of the subsystems. Thus, the information entropy S A(t) may be regarded as a
measure of entanglement.
43004-4
Entropy production in open quantum systems: exactly solvable qubit models
s=0.5
Wt
SAHtL�ln 2
1 - Λ=0.5
2 - Λ=1
3 - Λ=2
4 - Λ=5
1
2
3
4
0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
1.0
s=1
Wt
SAHtL�ln 2
1 - Λ=0.5
2 - Λ=1
3 - Λ=2
4 - Λ=5
1
2
34
0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
1.0
s=3
Wt
SAHtL�ln 2
1 - Λ=0.3
2 - Λ=0.5
3 - Λ=1
4 - Λ=2
1
2
34
0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
1.0
Figure 1. Time evolution of the qubit entropy in the dephasing model (10). In all cases λ≡ λs . The qubit
is initially in the pure state with equal populations (v3 = 0).
Within the framework of the model (10) with the initial condition (11), the structure of the entan-
gled state (13) can easily be established. Using the fact that the Hamiltonian (10) contains only the qubit
operator σ3, we obtain
|ψAB (t)〉 = a0e−iω0t/2|ψ(−)
B
(t)〉⊗ |0〉+a1eiω0t/2|ψ(+)
B
(t)〉⊗ |1〉 (29)
with
|ψ(±)
B
(t)〉 = exp
(
− iH (±)
B
t
)
|0B 〉, H (±)
B
=
∑
k
ωk b†
k
bk ±
∑
k
(
gk b†
k
+ g∗
k bk
)
. (30)
The reservoir states |ψ(±)
B
(t)〉 are normalized but 〈ψ(+)
B
(t)|ψ(−)
B
(t)〉, 0. It is possible, however, to write the
states appearing in (29) as linear combinations (the argument t is omitted for brevity)
|ψ(±)
B
〉 =β(±)
0 |Φ0〉+β(±)
1 |Φ1〉, |0〉 =α0|φ0〉+α1|φ1〉, |1〉 =α′
0|φ0〉+α′
1|φ1〉, (31)
where the amplitudes may be determined from the following conditions: a) the pairs of states (|Φ0〉,|Φ1〉)
and (|φ0〉,|φ1〉) are orthonormal; b) the terms with |Φ0〉⊗|φ1〉 and |Φ1〉⊗|φ0〉 cancel when expressions (31)
are inserted into (29). Then, the state of the combined system takes the form
|ψAB (t)〉 = A0(t)|Φ0(t)〉⊗ |φ0(t)〉+ A1(t)|Φ1(t)〉⊗ |φ1(t)〉 (32)
with amplitudes satisfying |A0(t)|2+|A1(t)|2 = 1. Explicit expressions for A0(t) and A1(t) are rather cum-
bersome and are not given here.
Formula (32) is an example of the so-called Schmidt decomposition of quantum states of a combined
system [1]. Amplitudes A0(t) and A1(t) play a role of the corresponding Schmidt coefficients. From (32) it
follows that the reduced density matrices of the subsystems may be written as
̺A(t)= |A0|
2|φ0〉〈φ0|+ |A1|
2|φ1〉〈φ1|, ̺B (t) = |A0|
2|Φ0〉〈Φ0|+ |A1|
2|Φ1〉〈Φ1|. (33)
Using these formulas, it is easy to show that
S A(t)= SB (t) =−|A0(t)|2 ln |A0(t)|2 −|A1(t)|2 ln |A1(t)|2, (34)
which is the well-known consequence of the Schmidt decomposition theorem for entangled quantum
states [1].
4. Spontaneous decay of a qubit
Nowwe consider the entropy production in an exactly solvable model which describes a spontaneous
decay of a two-level system into a field vacuum [1, 14, 15]. The total Hamiltonian of the model is
H = HA +HB +HI =
ω0
2
σ3 +
∑
k
ωk b†
k
bk +
∑
k
(
gk b†
k
σ−+ g∗
k bkσ+
)
, (35)
43004-5
V.G. Morozov, G. Röpke
where the index k labels the photon modes with frequencies ω
k
.
The initial state of the combined system is again assumed to be given by (11), i.e., v(0) = 1 and, conse-
quently, S A(0) = 0. To obtain the entropy of the qubit at times t > 0 we need to calculate v(t). To do this,
we will closely follow the approach taken in the works [14, 15].
It is convenient to work in the interaction picture with the unperturbed Hamiltonian H0 = HA +HB .
Introducing the interaction picture state vector |ψ̃
AB
(t)〉 = exp
(
iH0t
)
|ψ
AB
(t)〉, and applying the standard
procedure, one obtains
|ψ̃AB (t)〉 = exp+
−i
t∫
0
dt ′ HI (t ′)
|ψAB (0)〉, (36)
where exp+ [· · · ] is the chronologically ordered exponent, and
HI (t)=
∑
k
(
gk e−i(ω0−ωk )t b†
k
σ−+ g∗
k ei(ω0−ωk )t bkσ+
)
(37)
is the interaction picture Hamiltonian. As noted in the work [14] (see also [1, 15]), there exists a simple
representation for the state (36), which can be derived using the following properties of the Hamilto-
nian (37):
HI (t) |0〉⊗ |0B 〉 = 0, HI (t) |1〉⊗ |0B 〉 =
∑
k
gk e−i(ω0−ωk )t |0〉⊗b†
k
|0B 〉,
HI (t) |0〉⊗b†
k
|0B 〉 = g∗
k ei(ω0−ωk )t |1〉⊗ |0B 〉.
(38)
Recalling the expression (11), one can easily show that at any time t , the state (36) is a superposition of
|0〉⊗ |0
B
〉, |1〉⊗ |0
B
〉, and |0〉⊗b†
k
|0
B
〉:
|ψ̃AB (t)〉 =
[
a0|0〉+c1(t)|1〉
]
⊗|0B 〉+
∑
k
ck (t)|0〉⊗b†
k
|0B 〉. (39)
The amplitudes c1(t) and c
k
(t) satisfy the initial conditions c1(0) = a1, c
k
(0) = 0, and the normalization
condition
|a0|
2 +|c1(t)|2 +
∑
k
|ck (t)|2 = 1. (40)
The qubit density matrix in the interaction picture can now be calculated using (39):
˜̺A(t) = TrB
{
|ψ̃AB (t)〉〈ψ̃AB (t)|
}
= 1
2
+ 1
2
(
2|c1(t)|2 −1
)
σ3 +a∗
0 c1(t)σ++a0c∗1 (t)σ− , (41)
where the amplitudes c
k
(t) have been eliminated with the help of (40). Expressions for v±(t) and v3(t)
follow immediately from (5) and the relation ̺
A
(t) = exp(−iHA t)˜̺
A
(t)exp(iHA t):
v+(t) = eiω0t a0c∗1 (t), v−(t) = e−iω0t a∗
0 c1(t), v3(t)= 2|c1(t)|2 −1. (42)
Substituting these expressions into (8) and using |a0|
2 = 1−|c1(0)|2, we finally obtain
v2(t) = 1−4|c1(t)|2
[
|c1(0)|2 −|c1(t)|2
]
. (43)
Thus, the modulus of the Bloch vector and, consequently, the qubit entropy S
A
(t) are completely deter-
mined by |c1(t)|2 which is the time-dependent population of the excited state |1〉. It is important to note
that the probability amplitude c1(t) obeys a closed equation [1, 15]
ċ1(t)=−
t∫
0
dt ′ f (t − t ′)c1(t ′) (44)
with the kernel
f (t) =
∑
k
|gk |
2 ei(ω0−ωk
)t ≡
∫
dω J (ω)ei(ω0−ω)t , (45)
43004-6
Entropy production in open quantum systems: exactly solvable qubit models
where J (ω) is the spectral density of the field. In some important special cases of J (ω), equation (44) can
be solved to give exact analytic solutions for c1(t). We shall restrict ourselves to the so-called damped
Jaynes-Cummings model (see, e.g., [1, 15]) which describes the resonant coupling of a two-level atom to a
single cavity mode of the field. In this model, the effective spectral density has the form
J (ω)=
1
2π
γ0λ
2
(
ω0 −ω
)2 +λ2
, (46)
where λ is a spectral width of the coupling, and the parameter γ0 defines the characteristic time scale
τA = 1/γ0 on which the state of the qubit changes. For the details of solving the equation (44) we refer
to [1, 15]. Here, we quote the resulting expression for the population |c1(t)|2:
|c1(t)|2 = |c1(0)|2e−λt
[
cosh(Λt/2)+
λ
Λ
sinh (Λt/2)
]2
, (47)
where Λ=
(
λ2 −2γ0λ
)1/2
. Substituting the above expression into (43) and introducing the dimensionless
coupling parameter
K = 2γ0/λ, (48)
we obtain
v2(t) = 1−4|c1(0)|4 e−λt FK (t)
[
1−e−λt FK (t)
]
. (49)
The form of the function FK (t) depends on the value of K :
FK (t)=
[
cosh
(p
1−K λt/2
)
+ 1
p
1−K
sinh
(p
1−K λt/2
)]2
, K < 1,
FK (t)=
[
cos
(p
K −1λt/2
)
+ 1
p
K −1
sin
(p
K −1λt/2
)]2
, K > 1,
FK (t)= (1+λt/2)2 , K = 1.
(50)
Formulas (7) and (49) completely determine the qubit entropy in the Jaynes-Cummings model. Figure 2
a)
SAHtL � ln 2
Γ0 t
K = 0.2
K = 0.5
K = 0.9
0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
0.8
1.0
1.2
b)
SAHtL � ln 2
Γ0 t
K = 2
K = 5
K = 10
0 5 10 15 20 25 30
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Figure 2. Time evolution of the qubit entropy in the damped Jaynes-Cummings model: a) Weak and mod-
erate coupling (K < 1); b) Strong coupling (K > 1). In all cases the qubit is initially in the excited state |1〉,
i.e., |c1(0)|2 = 1.
illustrates the time behavior of S A(t). We take the situation where the initial pure state |1〉 of the qubit
evolves into the final pure state |0〉. Note that for all strengths of coupling, at some time tm, the qubit
passes through the mixed state with the maximum entropy S A(tm) = ln2, i.e., with v(tm) = 0. It is inter-
esting that this “maximally entangled” intermediate state of the combined system appears only in the
case of the maximum initial population of the excited qubit state |1〉 (|c1(0)|2 = 1), as may be seen directly
from (43). The time behavior of the qubit entropy for smaller values of |c1(0)|2 is shown in figure 3.
43004-7
V.G. Morozov, G. Röpke
K=0.5
Γ0t
SAHtL�ln 2
1 - Èc1H0L
2=1
2 - Èc1H0L
2=0.7
3 - Èc1H0L
2=0.5
1
2
3
0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
0.8
1.0
1.2
K=5
Γ0t
SAHtL�ln 2
1 - Èc1H0L
2=1
2 - Èc1H0L
2=0.7
3 - Èc1H0L
2=0.5
1
2
3
0 5 10 15
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Figure 3. Time evolution of the qubit entropy in the damped Jaynes-Cummings model for different initial
populations of the excited qubit state |1〉 in the cases of weak and strong coupling.
5. Conclusion
In this paper we have considered the time evolution of information entropy in two exactly solvable
models of two-state open quantum systems (qubits). Calculations were based on the simple but quit gen-
eral representation (7) for the qubit entropy.
Our discussion was restricted to the special case of the qubit and the reservoir being initially uncor-
related and the reservoir being in the ground state, i.e., at zero temperature. This simplifying assumption
allows one to study in detail the entropy production in entangled time-dependent quantum states of a
combined system. It should be noted, however, that in many situations of physical interest, a factorized
(uncorrelated) initial state at T = 0 cannot always be realized, so that the dynamics of thermal and cor-
related initial states, including the entropy behavior, is of great significance. For instance, it was shown
in [9] that the dephasing model (10) admits exact solutions for a large class of physically reasonable cor-
related initial states at finite temperatures. It was found that, for a sufficiently strong coupling, initial
qubit-environment correlations have a profound effect on the dephasing process and on the time be-
havior of entropy. It would be of interest to study the entropy behavior in further examples of exactly
solvable models of open quantum systems in the presence of initial system-environment correlations.
Acknowledgement
This work was supported by DFG (Deutsche Forschungsgemeinschaft), SFB 652 (Sonderforschungs-
bereich— Collective Research Center 652).
References
1. Breuer H.-P., Petruccione F., The Theory of Open Quantum Systems, Oxford University Press, Oxford, 2002.
2. Zurek W.H., Rev. Mod. Phys., 2003, 75, 715; doi:10.1103/RevModPhys.75.715.
3. Amico L., Fazio R., Osterloh A., Vedral V., Rev. Mod. Phys., 2008, 80, 517; doi:10.1103/RevModPhys.80.517.
4. Horodecki R., Horodecki P., Horodecki M., Horodecki K., Rev. Mod. Phys., 2009, 81, 865;
doi:10.1103/RevModPhys.81.865.
5. The Physics of Quantum Information. Eds. D. Bouwmeester, A. Ekert, A. Zeilinger, Springer-Verlag, Berlin, 2000.
6. Valiev K.A., Phys.-Usp., 2005, 48, 1; doi:10.1070/PU2005v048n01ABEH002024.
7. Allen L., Eberly J.H., Optical Resonance and Two-Level Atoms, Dover, New York, 1975.
8. Cohen-Tannoudji C., Diu B., Laloë F., Quantum Mechanics, Wiley, New York, 1977.
9. Morozov V.G., Mathey S., Röpke G., Phys. Rev. A, 2012, 85, 022101; doi:10.1103/PhysRevA.85.022101.
10. Łuczka J., Physica A, 1990, 167, 919; doi:10.1016/0378-4371(90)90299-8.
11. Unruh W.G., Phys. Rev. A, 1995, 51, 992; doi:10.1103/PhysRevA.51.992.
12. Palma G. M., Suominen K.-A., Ekert A.K., Proc. R. Soc. London, Ser. A, 1996, 452, 567; doi:10.1098/rspa.1996.0029.
13. Leggett A.J., Chakravarty S., Dorsey A.T., Fisher M.P.A., Garg A., Zwerger W., Rev. Mod. Phys., 1987, 59, 1;
doi:10.1103/RevModPhys.59.1.
43004-8
http://dx.doi.org/10.1103/RevModPhys.75.715
http://dx.doi.org/10.1103/RevModPhys.80.517
http://dx.doi.org/10.1103/RevModPhys.81.865
http://dx.doi.org/10.1070/PU2005v048n01ABEH002024
http://dx.doi.org/10.1103/PhysRevA.85.022101
http://dx.doi.org/10.1016/0378-4371(90)90299-8
http://dx.doi.org/10.1103/PhysRevA.51.992
http://dx.doi.org/10.1098/rspa.1996.0029
http://dx.doi.org/10.1103/RevModPhys.59.1
Entropy production in open quantum systems: exactly solvable qubit models
14. Garraway B.M., Phys. Rev. A, 1997, 55, 2290; doi:10.1103/PhysRevA.55.2290.
15. Breuer H.P., Kappler B., Petruccione F., Phys. Rev. A, 1999, 59, 1633; doi:10.1103/PhysRevA.59.1633.
Продукування ентропiї у вiдкритих квантових системах:
точно розв’язнi кубiт моделi
В.Г. Морозов1 , Г. Репке2
1 Московський державний технiчний унiверситет радiотехнiки, електронiки та автоматики,
проспект Вернадського, 78, 119454 Москва, Росiя
2 Унiверситет м. Росток, Унiверситетська площа, 3, D–18051 м. Росток, Нiмеччина
Ми представляємо аналiтичнi результати для часовозалежної iнформацiйної ентропiї в двостанових то-
чно розв’язних (кубiт) моделях. Перша модель описує дефазування (декогеренцiю) в кубiтi, який є зв’я-
заний з резервуаром гармонiчних осциляторiв. Обговорюється продукування ентропiї для цiєї моделi у
режимах “повної” та “неповної” декогеренцiї. Як iнший приклад ми розглядаємо задемпфовану модель
Джейнса-Каммiнгса, яка описує спонтанне затухання дворiвневої системи в польовому вакуумi. Показа-
но, що для всiх сил зв’язку, вiдкрита система переходить через змiшаний стан з максимумом iнформацiй-
ної ентропiї.
Ключовi слова: iнформацiйна ентропiя, вiдкритi квантовi системи, кубiт моделi, декогеренцiя, квантове
заплутування
43004-9
http://dx.doi.org/10.1103/PhysRevA.55.2290
http://dx.doi.org/10.1103/PhysRevA.59.1633
Introduction
Entropy of a qubit
Entropy production in a dephasing model
Spontaneous decay of a qubit
Conclusion
|
| id | nasplib_isofts_kiev_ua-123456789-120301 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| language | English |
| last_indexed | 2025-12-07T17:39:58Z |
| publishDate | 2012 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Morozov, V.G. Röpke, G. 2017-06-11T14:51:27Z 2017-06-11T14:51:27Z 2012 Entropy production in open quantum systems: exactly solvable qubit models / V.G.Morozov, G. Röpke // Condensed Matter Physics. — 2012. — Т. 15, № 4. — С. 43004:1-9. — Бібліогр.: 15 назв. — англ. PACS: 03.65.Ud, 03.65.Yz DOI:10.5488/CMP.15.43004 arXiv:1212.6135 https://nasplib.isofts.kiev.ua/handle/123456789/120301 We present analytical results for the time-dependent information entropy in exactly solvable two-state (qubit) models. The first model describes dephasing (decoherence) in a qubit coupled to a bath of harmonic oscillators. The entropy production for this model in the regimes of "complete" and "incomplete" decoherence is discussed. As another example, we consider the damped Jaynes-Cummings model describing a spontaneous decay of a two-level system into the field vacuum. It is shown that, for all strengths of coupling, the open system passes through the mixed state with the maximum information entropy. Ми представляємо аналiтичнi результати для часовозалежної iнформацiйної ентропiї в двостанових точно розв’язних (кубiт) моделях. Перша модель описує дефазування (декогеренцiю) в кубiтi, який є зв’язаний з резервуаром гармонiчних осциляторiв. Обговорюється продукування ентропiї для цiєї моделi у режимах “повної” та “неповної” декогеренцiї. Як iнший приклад ми розглядаємо задемпфовану модель Джейнса-Каммiнгса, яка описує спонтанне затухання дворiвневої системи в польовому вакуумi. Показано, що для всiх сил зв’язку, вiдкрита система переходить через змiшаний стан з максимумом iнформацiйної ентропiї. This work was supported by DFG (Deutsche Forschungsgemeinschaft), SFB 652 (Sonderforschungs-bereich — Collective Research Center 652). en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Entropy production in open quantum systems: exactly solvable qubit models Продукування ентропiї у вiдкритих квантових системах: точно розв’язнi кубiт моделi Article published earlier |
| spellingShingle | Entropy production in open quantum systems: exactly solvable qubit models Morozov, V.G. Röpke, G. |
| title | Entropy production in open quantum systems: exactly solvable qubit models |
| title_alt | Продукування ентропiї у вiдкритих квантових системах: точно розв’язнi кубiт моделi |
| title_full | Entropy production in open quantum systems: exactly solvable qubit models |
| title_fullStr | Entropy production in open quantum systems: exactly solvable qubit models |
| title_full_unstemmed | Entropy production in open quantum systems: exactly solvable qubit models |
| title_short | Entropy production in open quantum systems: exactly solvable qubit models |
| title_sort | entropy production in open quantum systems: exactly solvable qubit models |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/120301 |
| work_keys_str_mv | AT morozovvg entropyproductioninopenquantumsystemsexactlysolvablequbitmodels AT ropkeg entropyproductioninopenquantumsystemsexactlysolvablequbitmodels AT morozovvg produkuvannâentropiíuvidkritihkvantovihsistemahtočnorozvâznikubitmodeli AT ropkeg produkuvannâentropiíuvidkritihkvantovihsistemahtočnorozvâznikubitmodeli |