arXiv:1702.04239v1 [quant-ph] 14 Feb 2017
Production of Entanglement Entropy by Decoherence
in Biological Systems
M. Merkli∗
G.P. Berman†
R.T. Sayre‡
A.I. Nesterov¶
X. Wang§
February 15, 2017
LA-UR-17-20823
Abstract
We present a theoretical analysis of the entanglement entropy of an open dimer,
such as a photosynthetic biological system of two excited states of chlorophyll
molecules interacting with a protein-solvent environment, modeled by a field of
oscillators. We show that the entanglement entropy of the chlorophyll dimer is generated by its decoherence, caused by the environment interaction. Depending on the
type of interaction, not all environment oscillators carry significant entanglement
entropy. We identify the frequency regions in the environment which contribute to
the production of entanglement entropy. We do not make any smallness assumption
on the dimer-environment interaction and we analyze the dynamics of all parts of
the interacting system. Our results are mathematically rigorous.
1
Introduction
It is known that quantum coherent effects can survive for relatively long times in biological
systems, even at ambient conditions and for strong interactions (large reconstruction energy)
between the system and the protein-solvent environment it is embedded in [1, 2, 3, 4, 5, 6].
In particular, in photosynthetic organisms the energy (exciton) transfer (ET) between lightsensitive molecules (chlorophylls and carotenoids) is accompanied by quantum coherent effects
∗
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL,
Canada A1C 5S7; [email protected]
†
Bioscience Division, B-11, Los Alamos National Laboratory and the New Mexico Consortium, Los
Alamos, NM, 87544, USA; [email protected]
‡
Bioscience Division, B-11, Los Alamos National Laboratory and the New Mexico Consortium, 100
Entrada Dr., Los Alamos, NM 87544, USA; [email protected]
§
Quantum Strategics, San Diego, California, USA; [email protected]
¶
Departamento de Fisica, CUCEI, Universidad de Guadalajara, Av. Revolución 1500, Guadalajara,
CP 44420, Jalisco, México; [email protected]
1
which reveal themselves in a relatively slow decoherence, i.e., a slow decay of the non-diagonal
elements of the reduced system density matrix in the diagonal representation of the system
Hamiltonian. Typical decoherence time scales are τD ∼ 300 fs and there are many explanations
for this quantum “memory” effect in bio-systems [4]. In some situations, especially for energy
conserving interactions where the dimer energy is not changed during the dynamics, we do
not have true decoherence at all, but only partial decoherence takes place, meaning that the
non-diagonal density matrix elements do not decay to zero for large times [7, 8]. The standard
theory to analyze decoherence is the theory of open quantum systems, which are systems (dimer)
which interact with environments (protein-solvent). In the theory of open quantum systems, it
is most commonly assumed that the environment is in a mixed quantum state, given by a density
matrix (and not a wave function, which would represent a pure state) [9]. Indeed, traditionally,
environments are usually considered to be in, or close to, thermal equilibrium (the Gibbs state)
which is a mixed state given by a density matrix that is diagonal in the environment energy basis
and has thus no coherence [8, 9, 10]. This description of the system-environment complex is thus
not purely of quantum nature. Moreover, there are many arguments indicating that proteinsolvent environments may not actually be in equilibrium, but rather in states including noisy
and non-equilibrium components [8, 11]. In particular, it is argued in [12] that the temperature
of a plant leaf containing the light harvesting complexes can differ from the temperature of its
environment. Actually, it is not even clear that the concept of “temperature” can be introduced
in this case. It is therefore important to understand the quantum coherent properties of biosystems embedded in protein-solvent environments when both the bio-system and its environment
are initially prepared in a pure quantum state (in particular, not in equilibrium).
Due to decoherence caused by the interaction with its environment, an initially pure system
state becomes mixed, hence ‘more classical’ [13]. This loss of purity (or, ‘quantumness’) is mediated by quantum correlations, called entanglement, between the system and its environment.
A good measure for it turns out to be the so-called entanglement entropy (EE). First introduced
in [14], the examination of EE is receiving enormous attention from researches in different fields
of science, ranging from solid state physics and the theory of quantum computation and information to quantum field- and black hole theory [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25] (see
also references therein).
Goals, general setup. The main goal of our current paper is to introduce and examine
the notion of EE for biological systems and to elucidate precisely its relation to decoherence.
To be concrete, we consider a dimer based on two chlorophyll molecules, Chlb (donor) and Chla
(acceptor), embedded in a protein-solvent environment, see Figure 1. The dimer is described by
a two-level system representing the two excited energy levels of the chlorophyll molecules. The
ground states of the chlorophylls are not included in the description, as it is assumed that the
time of the excitation transfer between the two excited levels is much shorter than either of the
time of fluorescence and the time of decay of the exciton to the environment [4]. Initially, the
chlorophyll dimer is excited by the absorption of a photon from the sun, so the initial dimer state
is a pure state (possibly a superposition of Chla and Chlb excitations). As motivated above and
contrary to previous considerations [8, 26], we assume here that the protein-solvent environment
is initially in a pure state as well, and not in equilibrium (or any other mixed state). For
concreteness, we consider that the environment is represented by a collection of non-interacting
harmonic oscillators (modeling atomic vibrational modes) which are initially in coherent states.
2
Figure 1: Schematic of the model.
The total system is then described by a wave function, i.e., a pure state. The interaction between
the dimer and the environment is considered to be (almost) energy-conserving. Corrections due
to energy-exchange terms are estimated. We analyze the EE of the reservoir as well and we
exhibit oscillator frequency domains corresponding to parts of the reservoir which contribute to
the EE production, and others which do not. In this work, the interaction strength between
the dimer and the environment is not assumed to be weak as compared to the dimer energy
difference (in fact, it can take arbitrary values). This makes our work relevant for biological
systems [8] even though our results are valid for many systems in solid state physics and quantum
optics. We give mathematical proofs of all our results. Our numerical simulations support the
analytical results.
Entanglement entropy (EE). The notion of EE is central in the present work and we
give its motivation and definition here. A quantum state (density matrix) ρ is called a pure state
if it has rank one, or equivalently, if ρ = |ψihψ| for some vector ψ. A state which is not pure is
called a mixed state. The von Neumann entropy of a state ρ is defined by S(ρ) = −Tr(ρ ln ρ),
which is a non-negative number. The equivalence
ρ is pure ⇐⇒ S(ρ) = 0
makes the von Neumann entropy a good measure for the purity of a state.
Given a bipartite quantum system A + B, a pure state determined by a vector ψ is called
entangled if it is not possible to find vectors ψA and ψB of the individual subsystems, such that
ψ = ψA ⊗ ψB . Equivalently, ψ is called disentangled (or not entangled), or a product state, if
ψ = ψA ⊗ ψB for some subsystem vectors ψA , ψB .
The following is a basic relation between purity and entanglement. Let ψ be a pure state
of a bipartite quantum system A + B. The reduced states (density matrices) of subsystems A
and B are defined by ρA = TrB |ψihψ| and ρB = TrA |ψihψ| (partial traces). Then the following
equivalence holds,
3
ρA is pure ⇐⇒ ψ is not entangled.
This is also equivalent to ρB being a pure state. It is important that the state of the system
A + B be pure for the above equivalence to hold; indeed if ρ = σA ⊗ σB is the product of two
mixed states, then the reduced state of A, ρA = σA , is not pure. Combining the above two
equivalences leads to the following result.
Let ψ be a pure state of a bipartite quantum system A + B and let ρA = TrB |ψihψ|. Then
S(ρA ) = 0 ⇐⇒ ψ is not entangled.
This motivates the definition of entanglement entropy [14, 28] to be S(ρA ), the von Neumann entropy of the reduction of the pure state of a bipartite system to one of its parts. As
it turns out, when ρA and ρB are the reductions to the subsystems A and B, obtained from a
pure state of A + B, then S(ρA ) = S(ρB ). So it does not matter which subsystem state we take
the von Neumann entropy of, in the definition of entanglement entropy.
Structure of the paper. In Section 1.1 we outline the main results. In Section 2 the
dynamics of the dimer and of the reservoir is described for a finite number of atoms (oscillators)
in the reservoir. The dynamics in the continuous frequency limit is then analyzed in Section
3. In Section 4 we discuss the relation between processes of energy exchange, dissipation and
equilibration. Our numerical simulations are presented in Section 5. In Sections 6 and 7 we give
the mathematical proofs for the X- and D-interaction models, respectively. The Conclusion at
the end of the paper briefly summarizes our results.
1.1
Outline of main results
We consider a dimer (a two-level system) in contact with N oscillators. The initial state of the
entire system is
|Ψ0 i = |ψS i ⊗ |α1 i ⊗ |α2 i · · · ⊗ |αN i,
(1.1)
where |ψS i = a| ↑i + b| ↓i (with |a|2 = p, |b|2 = 1 − p) is an arbitrary pure state of the dimer and
|αj i = e−|αj |
2 /2
X αnj
√ |ni
n!
n≥0
is a single-oscillator coherent state, αj ∈ C. While we discuss here an environment of oscillators
initially in coherent states, many of our formulas are readily expressed for general, pure or mixed
initial states for the oscillators (see for instance Section 6.1). We analyze two different models.
In the first one, called the X-interaction model, the total Hamiltonian is given by
HX =
1
2 Ωσz
+
N
X
ωj a†j aj
j=1
+ λσz ⊗
N
X
gj (a†j + aj ),
(1.2)
j=1
where Ω, ωj > 0, λ ∈ R and σz is the Pauli z-matrix. The operators a†j and aj are the creation
and annihilation operators associated with the oscillator labeled by j, satisfying the canonical
commutation relations aj a†k − a†k aj = δkj (Kronecker delta). The collection of complex numbers
{gj }N
j=1 is called the form factor. The size of |gj | measures how strongly the oscillator labeled
4
by j is coupled to the dimer. The second model we discuss is described by a Hamiltonian with
a density-, or D-interaction, given by
HD = 12 Ωσz + 12 V σx +
N
X
j=1
ωj a†j aj + λσz ⊗
N
X
j=1
gj a†j aj + µσx ⊗
N
X
fj a†j aj .
(1.3)
j=1
Here, σx is the Pauli x-matrix and we assume in this model that Ω, ωj , λ, µ, gj , fj > 0.
Note that the dimer energy, 12 Ωσz , is a conserved quantity for the dynamics generated by
HX as well as for that generated by HD with V = µ = 0. Models
the dimer energy
P in which
†
is conserved are called energy conserving. The oscillator energy, N
ω
a
a
,
j=1 j j j is also conserved
for HD , but not for HX .
We denote by ρS (t) be the reduced dimer density matrix, obtained by tracing out all
oscillator degrees of freedom. It is given by
ρS (t) = Troscillators e−ıtH |Ψ0 ihΨ0 | eıtH ,
(1.4)
where Ψ0 is the initial pure dimer-reservoir state (1.1). The matrix elements are denoted by
[ρS (t)]ij (i = 1 being associated with the eigenvector | ↑i of σz with eigenvalue +1). The reduced
oscillator density matrix is defined as follows. Let J ⊆ R be a window of frequencies of interest.
For a given distribution of the oscillator frequencies {ω1 , . . . , ωN }, we have, say, K discrete
frequencies lying inside J. Denote by ρJ (t) the reduced oscillator density matrix of all
oscillators with frequencies inside J. Similarly to (1.4), ρJ (t) is obtained from the full density
matrix by tracing out the degrees of freedom of the dimer and of all oscillators with (the N − K
discrete) frequencies lying outside J,
ρJ (t) = Trsystem + oscillators
ℓ with ωℓ 6∈J
e−ıtH |Ψ0 ihΨ0 | eıtH .
(1.5)
The sets of non-vanishing eigenvalues of ρS (t) and of ρJ (t) are called the entanglement spectra
of the respective reduce density matrices [27].
1.1.1
Dimer decoherence
For finitely many oscillators (frequencies), the dynamics of the dimer and of the oscillators is
quasi-periodic in time. As the oscillator frequency spectrum becomes denser, recurrence times
are longer and in the (idealized) limit of a continuum of oscillator frequencies (see Section 1.2),
one observes irreversible dynamics. Since the system energy is conserved for the Hamiltonians
(1.2) and (1.3) with V = µ = 0 (the Hamiltonian 12 Ωσz commutes with the total Hamiltonian),
the populations of the system – the diagonal matrix elements in the energy basis – are constant
in time. In Theorems 2.1 and 2.2 we analyze the dimer energy conserving dynamics of ρS (t) and
ρJ (t), for any frequency window J. We show that
• The oscillator density matrix ρJ (t) has rank two (t > 0). Its entanglement spectrum
consists of two eigenvalues which we find explicitly for any t > 0 and any N .
• The dimer off-diagonal dimer density matrix element evolves according to the law
[ρS (t)]12 = |s[0,∞) (t)| [ρS (0)]12 ,
(1.6)
5
where, for any J ⊆ [0, ∞),
(
R
1−cos(ωt)
2
dω
J hX (ω)
ω2
e−4λ
, X-interaction model
R
sJ (t) =
2ıtλg(ω)
− J hD (ω) [1−e
] dω
e
, D-interaction model
(1.7)
Here,
hX (ω) =
dn
dω
|g(ω)|2 ,
and
hD (ω) =
dn
dω
|α(ω)|2
(1.8)
dn
(ω) is the frequency density of the reservoir initial state at the frequency ω. In the
and dω
continuous mode limit, the decoherence factor 0 ≤ |s[0,∞)(t)| ≤ 1 satisfies (see Propositions
3.1 and 3.2)
(
2 h (0) [1+o ]
t
X
e−2πtλ
,
X-interaction model
R
∞
|s[0,∞) (t)| =
(1.9)
− 0 hD (ω)dω
2
e
1 + O (tpd /t) , D-interaction model
and ot is a quantity converging to zero as t → ∞. Relations (1.6) and (1.9) show that
decoherence is full for the X-interaction (provided hX (0) 6= 0), meaning that [ρS (t)]12 → 0
as t → ∞. For the D-interaction, decoherence is only partial and happens on a partial
decoherence time scale tpd , explicitly given in (2.11).
• While (1.6) and (1.9) hold for dimer energy conserving models, we examine in Theorems
2.3 and 7.1 the evolution generated by (1.3) when additionally, V /Ω takes small nonvanishing values. We isolate the main term in the dynamics as well as the first order
correction ∝ V /Ω, and we control the error on time scales up to tpd .
1.1.2
Dynamics of the entanglement entropy
We show in Theorems 2.1 and 2.2 that the (nonzero) spectrum of the reduced density matrices
ρS (t) and ρJ (t), in the energy-conserving situations, are given by
(1.10)
spec ρJ (t) = 21 ± 21 rJ (t)
spec ρS (t) = spec ρJ=[0,∞) (t) ,
(1.11)
where
rJ (t) =
p
1 − 4p(1 − p)(1 − |sJ (t)|2 ),
(1.12)
with p = |h↑ |ψS i|2 . Note that (1.11) follows immediately from (1.10). Indeed, by partitioning the whole system into parts A=dimer and B=oscillators, the entanglement entropy of A
automatically equals that of B (see the introduction). The entanglement entropy is thus
SJ (t) ≡ S ρJ (t) = − 21 + 21 rJ (t) ln 12 + 21 rJ (t) − 12 − 12 rJ (t) ln 21 − 12 rJ (t) .
(1.13)
This gives us the following results.
1. The entanglement entropy SJ is a monotonically decreasing function of the
decoherence factor |sJ |. The parameter |sJ | (see (1.7)) takes values in the closed
interval [0, 1]. The entanglement entropy SJ is monotonically decreasing in the difference
of the eigenvalues, |rJ |, hence by (1.12), SJ is a monotonically decreasing function of |sJ |
6
(meaning that d|sdJ | SJ (|sJ |) < 0). The value of SJ is maximal (SJ = ln 2) when both
eigenvalues are equal, for rJ = 0. It is minimal (SJ = 0) if rJ = 1. In particular, if p = 0
or p = 1, then rJ (t) = 1 for all J and all t and no entanglement is ever created. We have
sJ (0) = 1 = rJ (0) and SJ (0) = 0, reflecting the fact that the initial state is pure and
entirely disentangled.
2. Dimer decoherence and dimer EE production. According to (1.6) and (1.9), we say
the dimer decoherence is large provided |s[0,∞) (t)| (off-diagonal) is small. This shows that
an increase in the dimer entanglement entropy is necessarily accompanied by an increase in
the dimer decoherence. We say the dimer undergoes full decoherence if limt→∞ |s[0,∞) (t)| =
0 and partial decoherence if limt→∞ |s[0,∞)(t)| > 0. If the dimer undergoes full decoherence,
p
then its final EE, S[0,∞) (∞), is given by (1.13) with rJ = r(p) = 1 − 4p(1 − p). As a
function of p ∈ [0, 1], S[0,∞)(∞) is symmetric around the point p = 1/2 and increases
monotonically from the value 0 for p = 0 to the value ln 2 for p = 1/2. Full decoherence
happens in the energy conserving X-interaction system (Proposition 3.1), while for the
energy-conserving D-interaction, we have partial decoherence (Proposition 3.2).
3. Localization of EE in the oscillators. Consider first the X-interaction model. We
show in Proposition 3.1 (B) below that the value of |sJ (t)| for J = [0, ω1 ] with arbitrary
ω1 > 0, and hence the entanglement entropy S(ρJ (t)), are determined for large times
entirely by the coupling function at the frequency zero, hX (0) (see (1.8)). In Proposition
3.1 (C) we show that as the interval J moves away from the low frequency oscillator
region, the entanglement entropy of ρJ decreases. More precisely, let J = [ω0 , ω1 ] with
0 < ω0 < ω1 . When ω0 in increases, the quantity |sJ (t)| increases as well and hence the
entanglement entropy of ρJ (t) decreases. This shows that for the X-interaction, the
entanglement entropy in the oscillators is concentrated in the low-frequency
region. For the D-interaction, a different picture emerges from our Proposition 3.2 below.
By point 1. above, SJ decreases monotonically in sJ , which by (1.7) itself is a decreasing
function in the size of the considered frequency window. Assuming for the sake of the
argument that hD (ω) ≈ hD,0 is roughly constant over a region ω ∈ [0, ωc ], we see from
(1.9) that for large times, we have sJ ≈ e−|J|hD,0 , where |J| is the length of the interval J ⊂
[0, ωc ]. It follows that the oscillator entanglement entropy S(ρ0J ) is translation invariant
for large times. Namely, for the D-interaction, the entanglement entropy in the
oscillators with frequencies in J only depends on the size of J, not on the
location of J. This suggests that in the D-interaction model, all oscillators contribute
to the dynamical process.
1.1.3
A conjecture
A mathematically controllable theory for the dynamics in the presence of an additional term
V σx in the Hamiltonian (1.2) is not available yet, to our knowledge. We conjecture that for any
small nonzero V and any initially pure system-reservoir state, the dimer is driven to a Gibbs
state for large times, regardless of the dimer-reservoir interaction strength λ (see also Section
4.2). In view of the relation of decoherence and entanglement entropy production as discussed
above, we further anticipate that the oscillators associated to a frequency region around ω = Ω
will also show significant entanglement entropy.
7
Conjecture. Consider H = HX + V σx , where HX is given in (1.2), and take any pure
dimer-reservoir initial state. Consider the model in the continuous mode limit.
1. For any value of the system-reservoir coupling constant λ and for V 6= 0 small, in the limit
of large times, the reduced dimer density matrix reaches an asymptotic state which is diagonal in
the energy basis (w.r.t. σz ), modulo O(V ) corrections. In this asymptotic state, the populations
are given by p± ∝ e∓βΩ/2 , where β is a parameter only depending on the initial reservoir state,
not on the initial dimer state.
2. The entanglement entropy of ρJ (t), for large times, will be significant not only for frequency windows J containing ω = 0 but also for an interval J localized around the dimer Bohr
frequency ω = Ω.1
1.2
Continuous mode limit
The dynamics generated by either of HX or HD is quasi-periodic, and stays so even when
N → ∞, if the oscillator frequencies ωj form a discrete set. Irreversible dynamics (of the dimer
and of the oscillators) emerges only in the presence of a continuum of oscillator frequencies. We
will refer to the procedure of taking the oscillator frequencies closer and closer together as the
continuous mode limit.2 The continuous mode limit is taken as follows. Let fj = f (ωj ) be a
function of the frequency ωj . Then
N
X
j=1
′
′
f (ωj ) =
N
X
j=1
n(ωj )f (ωj ) =
N
X
n(ωj )
j=1
∆ωj
f (ωj )∆ωj −→
Z
0
∞
dn(ω)
f (ω)dω,
dω
(1.14)
where N ′ is the number of distinct, discrete frequencies and n(ωj ) is the number of oscillators
having the same frequency ωj , ∆ωj = ωj+1 − ωj and dn(ω)
dω is the density of oscillators at the
given continuous frequency ω. In what follows, when we refer to functions of the continuous
frequency, α(ω), f (ω), g(ω), we assume that the continuous mode limit has been performed.
The procedure (1.14) tells us how to obtain the continuous mode limit of suitable sums of
a frequency dependent function. However, not all quantities of interest are of this form! For
example, the reduced density matrix ρJ (t) of all oscillators with frequencies lying inside J is an
operator acting on the Hilbert space ⊗K
j=1 Hosc , where Hosc is the pure state space of a single
oscillator and K counts the number of elements in J. We derive in Lemma 6.2 an explicit form
of ρJ (t). It cannot be expressed merely in terms of quantities of the form (1.14). What then
is the continuous mode limit of the state ρJ (t)? This is not an easy question. It is not even
clear on what Hilbert space this continuous mode state should act. Indeed, as we make the
oscillator frequency spectrum denser and denser, the window J will contain more and more of
the discrete frequencies. In the wanted limit, the cardinality K of the number of modes inside
J will increase to c = |R|, the cardinality of R. Then we do not even have a candidate for a
1
More generally, Ω should be replaced by the Bohr frequency of a renormalized dimer whose energies
are shifted by the interaction with the oscillators. If both dimer levels are coupled to the environment
in an equal way, as is the case of (1.2), then both energy levels are shifted by the same amount and the
original Bohr frequency coincides with the one of the renormalized dimer, see [8].
2
This does not mean that we create ‘modes’ in the sense of delocalized standing waves in the reservoir.
Rather, it means that we increase the number of atoms at varying frequencies making up the environment,
thus getting a very dense frequency spectrum.
8
Hilbert space, since we do not know how to make sense of ⊗K
j=1 Hosc in this limit! (How should
we take a tensor product of uncountably many Hilbert spaces?) Nevertheless, there is a method
of constructing the state and its Hilbert space in the continuous mode limit. It is based on
the Gelfand-Naimark-Segal construction for linear functionals on C ∗ -algebras (i.e., states). This
construction has been carried out in [21] for a coherent state reservoir. However, in the present
context, as we will show below, the entanglement spectrum of ρJ (t) can be expressed purely in
terms of quantities to which we can apply the procedure (1.14). Therefore, the mathematically
more involved formalism of [21] is not needed for the considerations of the present work.
Dynamics of subsystems for finite N
2
2.1
Energy conserving interactions
The structure of the reduced density matrices is easy to understand in general for the energy
conserving situation as follows. Suppose a d-dimensional system is coupled to a reservoir, described by an interacting Hamiltonian H = HS + HR + HSR , where HSR commutes
P with HS .
Let |ψi = |ψS i ⊗ |ψR i be a pure system-reservoir initial state. Expanding |ψS i = dk=1 ck |φk i,
where {|φk i}dk=1 is an orthonormal basis of eigenvectors of HS , we get
|ψ(t)i = e−ıtH |ψi =
d
X
k=1
ck |φk i ⊗ e−ıtHk |ψR i,
(2.1)
with Hk a Hamiltonian acting on the reservoir alone. Indeed, because HS and HSR commute,
the eigenvectors |φk i of HS also diagonalize the system part of the operator HSR , meaning that
′
′
HSR (|φk i ⊗ |ψR i) = |φk i ⊗ (HSR,k
|ψR i) for any reservoir state |ψR i, and where HSR,k
is an
3
′
operator acting purely on the reservoir Hilbert space. Then Hk = Ek + HR + HSR,k , where Ek
is the energy of |φk i (i.e., HS |φk i = Ek |φk i).
It follows from (2.1) that the reduced reservoir density matrix has the form
ρR (t) = TrS |ψ(t)ihψ(t)| =
d
X
k=1
|ck |2 e−ıtHk |ψR ihψR |eıtHk .
(2.2)
Even though ρR (t) is a density matrix acting on an (in general) infinite-dimensional reservoir
Hilbert space, (2.2) shows that its rank is at most d, the dimension of the small system. Since
the initial state is pure, this fact is also immediately implied by the Schmidt decomposition
theorem, which shows that the ranks of ρS (t) and ρR (t) are equal, where ρS (t) is the reduced
system density matrix. Assume now
PN that the reservoir consists of N uncoupled subsystems
(e.g., oscillators), so that HR =
j=1 HR,j , and that the coupling to the system is of the
PN
form j=1 Gj ⊗ Φj , where Gj and Φj are coupling operators acting on the system and on the
jth reservoir
subsystem (oscillator), respectively. Then the operator Hk in (2.1) has the form
P
Hk = N
H
j=1 k,j , where Hk,j acts on the jth reservoir subsystem only (see also footnote 3). For
3
If HSR = G ⊗ Φ with G and Φ acting on the system and on the reservoir, respectively, then
= gk Φ, where G|φk i = gk |φk i.
′
HSR,k
9
unentangled initial states |ψR i = |ψR,1 i ⊗ · · · ⊗ |ψR,N i expression (2.2) becomes
ρR (t) =
d
X
k=1
|ck |2
NN
ıtHk,j .
−ıtHk,j |ψ
R,j ihψR,j |e
j=1 e
(2.3)
Therefore, reducing ρR (t) to a collection J of reservoir subsystems (oscillators) results in the
density matrix
ρJ (t) = Tr{1,...,N }\J ρR (t) =
d
X
k=1
|ck |2
N
j∈J
e−ıtHk,j |ψR,j ihψR,j |eıtHk,j .
(2.4)
This shows that the reduced density matrix of any part of the reservoir has rank at most d as well.
For general systems with energy exchange, starting with a pure initial state, it is still true that
whole reservoir density matrix ρR (t) has rank ≤ d (again, just by the Schmidt decomposition
theorem). However, the rank of ρJ (t) might exceed d. Indeed, ρJ (t) is obtained from ρR (t) by
tracing out parts of the reservoir degrees of freedom, an operation which may increase the rank
(for instance, the partial trace of a pure state is generally a mixed state, hence taking the partial
trace generally increases the rank). This can also be understood from a different point of view:
switching on an energy exchange term (even a small one) shifts the spectrum of ρJ (t) and may
cause some of its zero eigenvalues to become nonzero, hence increasing the rank of ρJ (t).
We examine in this section the reduced dynamics of the dimer and (parts of) the reservoir
under the dynamics generated by (1.2) and (1.3) for V = µ = 0. Given J ⊆ [0, ∞), we define
the quantity (the discrete frequency analogue of (1.7))
i
h

P
2
2 1−cos(ωj t) , X-interaction model

exp
−
4λ
|g
|

j
2
{j : ωj ∈J}
ωj

(2.5)
sJ (t) =
h P
i


 exp −
|αj |2 (1 − e2ıtλgj ) ,
D-interaction model
{j : ωj ∈J}
We will prove in Theorems 2.1, 2.2 below that the reduced reservoir density matrix of all oscillators with frequencies inside J is a rank-two operator (see also the discussion at the beginning
of this section), described in a suitable basis (consisting of vectors which depend on time t) by
the matrix
p
p + (1 −p
p)|sJ (t)|2
(1 − p)sJ 1 − |sJ (t)|2
ρJ (t) =
,
(2.6)
(1 − p)s̄J (t) 1 − |sJ (t)|2
(1 − p)(1 − |sJ (t)|2 )
with p and sJ (t) given after (1.1) and in (2.5), respectively. The reservoir entanglement spectrum
is entirely determined by (2.6).
Theorem 2.1 (Energy conserving X-interaction) Consider the system with the X-interaction,
(1.2) for a fixed, finite N .
(A) Let ρS (t) be the reduced dimer density matrix. Its diagonal is time-independent and the
off-diagonal is given by
P
1−eıωj t
[ρS (0)]12 ,
(2.7)
[ρS (t)]12 = e−ıΩt s[0,∞) (t) exp 4ıλ Im N
j=1 ᾱj gj ωj
where s[0,∞)(t) is given in (2.5) (X-interaction) with J = [0, ∞).
10
(B) Let ρJ (t) be the reduced density matrix of all oscillators having frequencies inside J. The
rank of ρJ (t) is two (t > 0) and in a suitable orthonormal basis, it has the matrix representation (2.6).
The basis in which ρJ (t) takes the form (2.6) depends on time t, the number N of oscillators
and the frequency window J. In Section 4 we show that setting αj = 0 in (2.7) gives the dimer
dynamics for a reservoir of oscillators, each in thermal equilibrium, see (4.5).
For the energy conserving D-interaction (V = µ = 0) we denote the reduced dimer and
oscillator density matrices by ρ0S (t) and ρ0J (t). The superscript 0 indicates that these are the
unperturbed states, as later on, we will switch on the energy exchange interactions.
Theorem 2.2 (Energy conserving D-interaction) Let N be fixed and finite. Denote by
ρ0S (t) and ρ0J (t) the reduced system and oscillator density matrices evolving according to the
D-interaction, (1.3), for V = µ = 0.
(A) The diagonal of ρ0S (t) is time-independent and the off-diagonal is
[ρ0S (t)]12 = e−ıΩt s̄[0,∞)(t) [ρS (0)]12 ,
(2.8)
where s̄[0,∞)(t) is the complex conjugate of s[0,∞)(t), given in (2.5) (D-interaction) with
J = [0, ∞).
(B) The rank of ρ0J (t) is two (t > 0) and in a suitable orthonormal basis, it has the matrix
representation (2.6).
D-interaction: partial decoherence
2.2
As mentioned in Section 1.2, irreversible dynamics emerges in the continuous mode limit only.
An example is the evolution of coherences, encoded in the time dependence of sJ (t), (2.5). For
the D-interaction model, we obtain in the continuous mode limit (1.14),
|sJ (t)| = e−
R
J
hD (ω)dω
eRe
R
J
hD (ω) e2ıtλg(ω) dω
where
−→ e−
R
J
hD (ω)dω
,
t → ∞,
dn(ω)
dω
|α(ω)|2 .
(2.9)
R
The convergence is due to fast oscillations, causing J hD (ω)e2ıtλg(ω) dω → 0 as t → ∞, a
process we can control by a ‘Riemann-Lebesgue Lemma’ argument as follows. For definiteness,
consider h to be a function of ω ∈ [0, ωc ], where ωc is some cutoff (at and beyond which h and
h′ vanish) and suppose that the coupling function g is invertible on ω ∈ [0, ωc ]. Using that
1
∂ω e−2ıtλω and integrating by parts, we expand
e−2ıtλω = −2ıtλ
hD (ω) =
Z
∞
h(ω)e−2ıtλg(ω) dω =
0
=
Z
g(ωc )
g(0)
h(g−1 (ω)) −2ıtλω
e
dω
g′ (g−1 (ω))
h(g −1 (ω)) ′ 1
ı h(0)
−3
+
+
O
(tλ)
.
2tλ g′ (0) 4t2 λ2 g′ (g−1 (ω)) ω=g(0)
11
(2.10)
It follows from (2.10) and (2.8) that |[ρ0S (t)]12 | decays partially (but not monotonically in time)
over the partial decoherence time
p
|ξ|
,
(2.11)
tpd =
λ
with ξ given by
h (g −1 (ω)) ′ D
.
(2.12)
ξ=
g′ (g−1 (ω))
ω=g(0)
We point out that the decoherence is only partial, since
R∞
0
[ρS (t)]12 = [ρS (0)]12 e− 0 hD (ω)dω 1 + O(t2 /t2 )
pd
(2.13)
does not vanish as t → ∞. In contrast, we will show that for the X-interaction model, we have
full decoherence (see Proposition 3.1).
2.3
D-interaction: almost energy conserving regime
Even for V and µ small (not vanishing), the energy exchange interaction terms in HD , (1.3),
have a big influence over long time scales. For instance, they are responsible for the process of
relaxation (thermalization, if the reservoir is in a thermal state). We consider here time scales
of the order tpd , (2.11), over which partial decoherence happens and we show that the effects of
the non-energy conserving terms are merely a perturbation of this dynamics for small V, µ. We
call this the almost energy conserving regime.
In the result below, we use the notation o(x) for a real number s.t. lim√
x→0 o(x)/x = 0. Also,
we denote by k · k1 the trace norm, namely, kAk1 = Tr|A|, where |A| = A∗ A is the absolute
value of an operator A.
Theorem 2.3 (Energy exchange D-interaction) Let N be finite and consider the D-interaction
with µ = 0 and V ≥ 0. Suppose that V /Ω << 1, V /λ << 1 and that Ω/λ << min1≤j≤N gj . Then,
for times 0 ≤ t ≤ tpd , we have the following.
(A) The dimer density matrix satisfies
ρS (t) − ρ0S (t) −
V
Ω
e−
PN
j=1
|αj |2 1
ρS (t)1
= o(V /Ω),
where ρ0S (t) is given in (2.8) and
Re 1 − e−ıΩt [ρS (0)]12
(p − 21 )(1 −e−ıΩt )
1
ρS (t) =
.
(p − 21 )(1 − eıΩt )
−Re 1 − e−ıΩt [ρS (0)]12
(2.14)
(2.15)
The remainder is uniform in t for 0 ≤ t ≤ tpd .
(B) The oscillator density matrix satisfies
ρJ (t) − ρ00
J (t) −
V
Ω
−
e
P
{j : ωj ∈J }
12
|αj |2 1
ρJ (t)1
= o(V /Ω),
(2.16)
where ρ00
J (t) is the 3 × 3 matrix obtained by adding a (third) row and a (third) column
consisting of zeroes to the matrix in (2.6), and, expressed in the same basis, the hermitian
ρ1J (t) has matrix elements
[ρ1J (t)]11 = −[ρ1J (t)]22 = 2 Re v̄J (1 − sJ )
[ρ1J (t)]33 = 0
[ρ1J (t)]12 = (vJ − v̄J sJ ) √1−sJ
1−|sJ
[ρ1J (t)]13
[ρ1J (t)]23
−1/2
vJ (t) ≡ vJ
NJ
− vJ
p
1 − |sJ |2
= (δJ )
NJ (vJ − sJ v J )
p
= − 1 − |sJ |2 v J (δJ )−1/2 NJ .
The parameters are given by
δJ
|2
−
= e
P
{j : ωj ∈J }
(2.17)
|αj |2
−
−ıΩt
1
) [ρS (0)]12 e
2 (1 − e
q
sJ
1 − 2δJ 1−Re
=
.
1−|sJ |2
=
P
(2.18)
{j : ωj 6∈J }
|αj
|2
(2.19)
(2.20)
We refer to Theorem 7.1 below for results when in addition, µ > 0.
3
3.1
Dynamics in the continuous mode limit
Dynamics of the dimer and of sJ
In the continuous mode limit (1.14), the disctrete mode expressions for sJ (t), (2.5), become
(1.7).
Proposition 3.1 (X-interaction) The dynamics of ρS and sJ for the X-interaction model,
in the continuous mode limit, satisfies the following. In what follows, ot denotes a real number
satisfying limt→∞ ot = 0.
(A) Suppose that hX (ω), (1.8), is continuous as ω → 0+ . Then
[ρS (t)]12 = [ρS (0)]12 e−2πtλ2 hX (0)[1+ot ] .
(3.1)
(B) Let J = [0, ω1 ], where 0 < ω1 ≤ ∞. Then for all t ≥ 0,
2h
|sJ (t)| = e−2πtλ
X (0) [1+ot ]
.
(3.2)
(C) Let J = [ω0 , ω1 ], where 0 < ω0 < ω1 < ∞ and set mJ = inf ω∈J hX (ω), MJ = supω∈J hX (ω).
We have for all t ≥ 0,
−4λ2 MJ
e
ω1 −ω0 +2/t
2
ω0
−4λ2 mJ
≤ |sJ (t)| ≤ e
ω1 −ω0 −2/t
2
ω1
.
(3.3)
According to (A), the dimer undergoes full decoherence for couplings hX (0) 6= 0, exponentially
quickly in time.
13
Proposition 3.2 (D-interaction, V = µ = 0) The dynamics of ρS and sJ for the energy conserving D-interaction model, in the continuous mode limit, satisfies the following.
(A) We have
R∞
0
[ρS (t)]12 = [ρS (0)]12 e− 0 hD (ω)dω 1 + O t2 /t2 ,
pd
(3.4)
where tpd ∝ 1/λ is the partial decoherence time (c.f. (2.11)).
(B) For any J ⊆ [0, ∞), the quantity sJ (t), given in (1.7), satisfies
R
sJ (t) = e− J hD (ω)dω 1 + O t2pd /t2 .
(3.5)
The expansions are correct provided tpd > 0, which translates into a condition of effective
coupling, see (7.7), (7.8). The dimer undergoes partial decoherence.
We close the discussion by explaining how the energy-exchange term ∝ V of the D-interaction
model, (1.3), influences the entanglement entropy. For some results including additionally the
term ∝ µ, see Section 7.
Proposition 3.3 (Entanglement entropies, D-interaction, V > 0) Consider the dynamics associated with the D-interaction Hamiltonian, for V > 0 (and µ = 0), in the continuous
mode limit. Define
R
− R hD (ω)dω
rJ1 (t) = 2(1 − 2p) e
rJ (t)
Re [ρS (0)]12 (1 − e−ıtΩ )(1 − s̄J (t)).
(3.6)
Suppose that V /Ω << 1, V /λ << 1 and that Ω/λ << inf ω≥0 g(ω). Then, for times 0 ≤ t ≤ tpd ,
we have:
(A) The eigenvalues of ρS (t) are
spec(ρS (t)) =
1
±
1
2
r[0,∞) (t) −
V 1
Ω r[0,∞) (t)
+ o(V /Ω).
(3.7)
(B) The eigenvalues of ρJ (t) are, in the continuous mode limit,
spec(ρJ (t)) = 12 ± 12 rJ (t) − VΩ rJ1 (t) ∪ {0} + o(V /Ω).
(3.8)
2
The small-o notation means that o(x) is a real number s.t. limx→0 o(x)/x = 0. Also, rJ (t) is
given in (1.12).
Discussion. 1. Theorem 2.3 and Proposition 3.3 show that the energy-exchange perturbation ∝ V in (1.3) leaves the rank of the oscillator density matrices ρJ (t) unchanged, up to
order V /Ω (see (3.8)).
2. For p = 1/2, the term on the right side of (3.7), which is linear in V /Ω, vanishes.
The entanglement entropy of the dimer decreases if the gap between the two eigenvalues of
ρS (t) increases, which happens exactly if the term ∝ V /Ω in the parentheses in (3.7) has a
positive sign. This shows that the energy-exchange interaction produces a correction to the
(strictly positive) entanglement entropy caused by the energy conserving interaction, whose sign
fluctuates in time and depends on the initial dimer state.
14
4
Energy exchange, dissipation, equilibration
4.1
Systems close to thermal equilibrium
In the theory of open quantum systems one studies the dynamics of ‘small systems’ coupled to
‘environments’ (also called reservoirs or quantum noises). The environment itself is a quantum
system but it shows irreversible dynamical effects, both in its own evolution and in that of a
small system coupled to it. A reservoir is obtained by a limiting procedure from a ‘normal’
(finite) quantum system, for instance by taking a continuous mode limit. The limits turn purely
discrete energy spectrum of finite systems into continuous spectrum, which causes irreversibility
in the dynamics. It is plausible to expect that a system with continuous energy (frequency)
spectrum shows dissipative dynamics, suggesting that reservoirs have a unique stationary state.
The same thing happens when a transition of discrete to continuous frequencies is caused by a
thermodynamic limit, where the volume of a reservoir increases, see [21, 30, 31].
A non-interacting system environment complex, described by a Hamiltonian H0 = HS +
HR , has a manifold of stationary states, spanned by |ψE ihψE | ⊗ ρR , where ψE are the system
eigenfunctions and ρR is the reservoir invariant state. What happens as the system and reservoir
start to interact, due to a term HSR in the Hamiltonian? In the situation where the reservoir
is in (or close to) thermal equilibrium (the thermodynamic limit of finite-volume Gibbs states),
one can show that, for very generic interactions HSR , the interacting system has a unique
stationary state, which is the coupled (interacting) equilibrium state. The fact that such a
coupled equilibrium exists is guaranteed by the stability theory of KMS states.4 This theory
says, quite generally, that if an unperturbed Hamiltonian H0 generates a dynamics which has an
equilibrium (KMS) state ρ0 then a perturbation H0 + HSR generates a dynamics with associated
equilibrium state ρSR , which varies continuously in HSR . The fact that ρSR is the only stationary
state can be proven for many open systems, provided an efficient coupling condition is satisfied.
The latter requires that energy exchange takes place between the system and the environment,
and it will not be satisfied in general for models where the system energy is conserved.5 We sum
up the situation close to equilibrium as follows:
1. The coupled system-reservoir complex has an equilibrium state.
2. Due to efficient coupling, requiring in particular energy-exchange, the coupled equilibrium
is the only stationary state.
3. The coupled dynamics is dissipative and drives any initial state towards the coupled
equilibrium.
4
For continuous modes (frequencies) systems, the Hamiltonian HR has continuous spectrum and one
does not know how to interpret the candidate e−βHR for an equilibrium density matrix, since its trace is
infinite. A better characterization of equilibrium is the KMS characterization (Kubo-Martin-Schwinger)
[31], which reduces to the usual Gibbs state formulation for equilibrium in finite systems.
5
Suppose H = HS + HR + HSR is energy conserving, meaning that [HS , HSR ] = 0. Then all initial
states |ψE ihψE | ⊗ ρR , where HS ψE = EψE and ρR is an arbitrary reservoir density matrix produce
stationary system states, TrR {e−ıtH (|ψE ihψE | ⊗ ρR )eıtH } = |ψE ihψE |. This observation hints at the need
of energy exchange to guarantee uniqueness of a stationary state for the whole, coupled system-reservoir
complex.
15
For small HSR the reduction of the coupled equilibrium to the system alone is the usual Gibbs
state ∝ e−βHS (modulo O(HSR ) terms) at the temperature inherited from the reservoir. For
strong system-reservoir coupling, the reduced state is a Gibbs state (again at inverse temperature
β) with respect to a Hamiltonian having renormalized energies [8]. The renormalizations are
∝ (HSR )2 and can be large. In either case, the small system becomes equilibrated due to the
contact with a thermal reservoir, in the course of time.
We have developed the dynamical resonance theory [10], allowing a rigorous proof of 1.3., for initial states of the form ρS ⊗ ρR (or perturbations thereof), where ρR is the reservoir
equilibrium, hence a mixed state. However, for the analysis of entanglement entropy, one has
in mind pure system-reservoir initial states. A modification at the very core of the resonance
theory is necessary to make it applicable to the study of entanglement entropy. While we are
working to expand the theory in this direction, the present paper investigates the almost energy
conserving situation only.
4.2
Emergence of Gibbs states from reducing pure states
In the “standard setup of open systems” just described, the starting point is a state where the
reservoir is in (or close to) equilibrium. A different question is how a system equilibrium (Gibbs)
state emerges due to the interaction with a reservoir when initially the system-reservoir state is
pure.
Suppose H = HS + HR + HSR where the reservoir has discrete, but close lying energy levels,
spaced by maximally ∆B, while the system energy spacing is ∆ǫ. Denote by ρ(B) the reservoir
d
ln ρ(B). In the regime
density of states and define the effective temperature by β ≡ β(B) = dB
∆B << kHSR k << ∆ǫ (and under additional assumptions), it is shown in [23, 29] that for an
initial pure state Φ(0) of the system plus reservoir, having a (coupled) energy distribution peaked
at a value E, one has
hAit ≡ hΦ(t), (A ⊗ 1lR )Φ(t)i ≈ hAican
(4.1)
β(E) ,
= TrS (e−βHS A)/TrS e−βHS is
for all system observables A, where Φ(t) = e−ıtH Φ(0) and hAican
β
the system Gibbs canonical equilibrium state. Relation (4.1) is valid for sufficiently large, typical
times t (it does not hold for all times, since the reservoir is not taken to be in the continuous
mode limit and hence the overall dynamics is quasi-periodic). It would be interesting to give
a proof of (4.1) for our X-interaction model with a coherent state initial reservoir, especially
for strong system-reservoir couplings relevant in quantum biological processes. In view of the
techniques developed in [8], we might expect such a result to hold, with HS replaced with a
renormalized Hamiltonian.
A proof of the emergence of the system Gibbs state, obtained by reducing the full density
matrix to the system part, can be based on the following two arguments.
1. (Decoherence in the energy basis.) After the system and reservoir have interacted for
some time, the system will be in a state which is diagonal in its (uncoupled) energy basis,
and which does not depend on the initial system state. It follows from this (assumption,
or principle) that the system state is entirely described by the populations (occupation
probabilities) pi of the energy levels, and that those populations are functions of the energy
only, pi = f (Ei ). It is implicitly assumed here that the system and reservoir exchange
16
energy since if they were not, then the system populations would be time-independent
and they would not settle to values independent of the initial system state.
2. When coupling two identical, independent systems to the same reservoir (no direct interaction), the population probabilities of the joint two-system complex should split into
products of single system probabilities, due to their independence. Here, it is assumed
that the systems-reservoir interaction is weak, so that independence of the two systems
is approximately preserved even while they are in contact with the same reservoir. This
leads to the constraint f (x + y) = f (x)f (y), whose solution has the Gibbs distribution
form f (x) ∝ e−βx , where β is a reservoir-dependent constant.
In the arguments of [23, 29], decoherence is derived by observing that for the models considered
there, the stationary Schrödinger equation of the interacting system can be interpreted as that of
a single quantum particle in a potential. A weak system-energy interaction makes this potential
vary slowly in space and as a consequence, eigenstates of the interacting system associated with
energies differing by more than the interaction energy have roughly disjoint support (quasiclassical analysis), which leads to decoherence (c.f. equations (6) in [23]). It is not clear to us if this
reasoning is applicable to situations we are interested in here, namely a dimer strongly coupled
to an environment.
While the above considerations are based on the dynamics, there is also a ‘static’ link between
reduced states of a large system and Gibbs states. In particular, for a collection of solvable manyparticle free lattice particle systems and spin chain models in their ground state, the reduced
density matrices of parts of the system and their entanglement spectra have been analyzed
in [22, 26, 28, 32]. The key finding is that the reduced density matrix ρα (where α labels a
lattice subregion) is of thermal form, ρα ∝ e−Hα , where Hα is an effective, local, free particle
Hamiltonian, but Hα is not the original Hamiltonian restricted to the region α. In these models,
a thermal structure thus emerges generically by reduction of the ground state to a subregion of
space.
4.3
Energy change in the reservoir
Consider H = HS +HR +HSR . It is easy to see that for any energy conserving H, i.e. [HS , H] = 0,
we have TrSR {HS e−ıtH (ρS ⊗ ρR )e−ıtH } = TrS {HS ρS }. However, the energy of the reservoir,
TrR (HR ρ′R (t)), changes in time. Here, ρ′R (t) = TrS e−ıtH (ρS ⊗ ρR )eıtH is the reduced reservoir
density matrix.
For the energy conserving X-interaction model, (1.2), with initial state (1.1), the energy
difference ∆ER (t) := TrR (HR ρ′R (t)) − TrR (HR ρ′R (0)) in the continuous mode limit is given by
∆ER (t) = ∆ER,therm (t) + ∆ER,coh (t),
(4.2)
where
Z
∞
1 − cos ωt
dn
|g(ω)|2
dω
dω
ω
0
Z ∞
1 − e−ıωt
dn
ḡ(ω)α(ω)
dω.
∆ER,coh (t) = 2λ(1 − 2p)Re
dω
ω
0
2
∆ER,therm (t) = 2λ
17
(4.3)
(4.4)
The first contribution, ∆ER,therm (t), is positive and does not depend on the initial condition
of the reservoir nor the dimer. It equals the difference in the energy of the reservoir at time t
minus the energy at time t = 0, in the situation when each oscillator in the reservoir starts out
in thermal equilibrium at inverse temperature β. This change in energy is actually independent
of the temperature. The second contribution, ∆ER,coh (t), depends on the initial condition of the
dimer (p) and of the reservoir (α) and can be negative. It vanishes for an initially completely
mixed dimer state, p = 1/2.
Using Lemma 6.1 below, we calculate the evolution of the dimer density matrix coupled to
a thermal reservoir (each oscillator in thermal equilibrium at equal temperature 1/β) by the
X-interaction. The diagonal matrix elements are constant and the off-diagonal evolves as
[ρS (t)]12 = e−ıΩt s[0,∞) (t) [ρS (0)]12
(4.5)
where s[0,∞)(t) is given in (2.5), and is actually independent of the temperature of the oscillators.
Comparing with the dynamics of the dimer coupled to the coherent state reservoir, given in (2.7),
we see that having the initial reservoir in a coherent state versus a thermal one is simply to mulP
1−eıωj t
tiply the off-diagonal dimer density matrix element by the phase exp{4ıλ Im N
j=1 ᾱj gj ωj }.
This also shows that the dimer dynamics obtained from an energy conserving X-interaction with
a reservoir in its ground state (or, the coherent state with αj = 0) is the same as that obtained
from a thermal reservoir at any temperature.
In the situation where the reservoir is initially in thermal equilibrium at inverse temperature
β, the Landauer Principle says that ∆SS (t) ≤ β∆ER (t), [33, 34, 35], where ∆SS (t) = SS (0) −
SS (t) is the entropy loss of the dimer. Starting from a pure dimer state, the contact with the
thermal reservoir will create generically positive entropy in the dimer, so that ∆SS (t) < 0 and,
since ∆ER (t) = ∆ER,therm (t) ≥ 0, see (4.3), the Landauer inequality is trivially satisfied.
5
Numerical simulations
In our numerical simulations we use Eq. (1.13) for entanglement entropy of the environment,
written as
SE (t) = − 21 (1 + r(t)) ln 21 (1 + r(t)) − 12 (1 − r(t)) ln 12 (1 − r(t))
where r(t) =
p
1 − 4p(1 − p)(1 − |s(t)|2 ) and
Z
|s(t)| = exp − 4λ2
∞
0
h(ω)
1 − cos(ωt)
dω
.
ω2
(5.1)
(5.2)
We adopt units in which ~ = 1.
5.1
X-interaction
Let us consider the function hX (ω) ≡ h(ω), given in (1.8), of the form
h(ω) = Aq ω 2q+2 e−ω/ωc ,
18
(5.3)
(a)
(b)
Figure 2: (Color online) X-interaction. The entanglement entropy SE as a function of the
donor population p and the dimensionless time τ = ωc t. (a) q = 1, ε = 0.1; (b) q = −1,
ε = 0.1. Note: the maximal possible value of SE is ln 2 ≈ 0.69.
with an exponential high frequency cutoff ωc > 0, where Aq > 0 and where p ∈ R determines
the low frequency behavior. It is convenient to introduce a new variable, z = ω/ωc , and a
dimensionless time, τ = ωc t. Then, Eq. (5.2) can be written as,
|s(t)| = exp − εQ2 (τ ) ,
(5.4)
where ε = 4λ2 Aq ωc2q+1 and
Q2 (τ ) =
Z
∞
0
z 2q e−z (1 − cos(τ z))z..
(5.5)
Below, we consider two cases: q > −1/2 and q = −1. Performing the integration in (5.5) gives
the following.
• Case: q > −1/2
where Γ(u) =
• Case: q = −1
R∞
0
1
,
Q2 (τ ) = Γ(2q + 1)ℜ 1 −
(1 + iτ )2q+1
(5.6)
e−z z u−1 z. is the gamma function, defined for u ∈ C with ℜu > 0.
Q2 (τ ) = τ arctan τ − 21 ln(1 + τ 2 ).
(5.7)
For both these cases, the EE, SE (τ ), is presented in Fig. 2. For q = 1, we have: h(0) = 0, and we
see from Fig. 2a that only partial decoherence occurs in the X-interaction model. The function
19
|s(t)| does not decay to zero for large times. Correspondingly, the EE SE (t) does not approach
its maximum for a given value of p. In particular, for p = 1/2, limτ →∞ SE (τ ) ≈ 0.3. At the
same time, for q = −1 and p = 1/2, the EE reaches its maximum: limτ →∞ SE (τ ) = ln 2 ≈ 0.7,
as in this case full decoherence takes place (limτ →∞ s(τ ) = 0).
5.1.1
Contribution of low and high frequencies to the EE
To estimate the contribution of low and high frequencies, we consider lower, ω0 , and upper, ωc ,
cutoff frequencies in the function Q2 (τ ) by introducing,
Z z0
z 2q e−z (1 − cos(τ z))z.,
(5.8)
Q2 (τ, z0 ) =
0
and
Q̃2 (τ, z0 ) =
Z
∞
z0
z 2q e−z (1 − cos(τ z))z.,
(5.9)
where z0 = ω0 /ωc . We have Q2 (τ ) = Q2 (τ, z0 ) + Q̃2 (τ, z0 ), but the EE, SE (τ ), is of course not
a linear function of Q2 .
In the following, we will evaluate SE (τ ), given in (5.1), in which the parameter |s(τ )| is
chosen to be either |s(τ )| = e−εQ2 (τ,z0 ) or |s̃(τ )| = e−εQ̃2 (τ,z0 ) . We call the former the low
frequency contribution to the EE (it is associated with (5.8)) and the latter the high frequency
contribution to the EE (it is associated with (5.9)). In the following figures, the high and low
frequency contributions are represented by red and cyan colored surfaces, respectively. It is not
the case that the total EE is the sum of the low and high frequency contributions (as EE is not
a linear function of the Q2 ). Rather, if the coupling function h(ω), on top of its shape (5.3),
has an additional high frequency suppression for frequencies ω > ω0 , for instance because the
dn
of the reservoir or the coupling function g is small at high frequencies (see
frequency density dω
(1.8)), then the high frequency part Q̃2 (τ, z0 ) vanishes and the EE is given by the above-defined
low frequency contribution. An analogous discussion is valid if low frequencies are suppressed,
in which case the EE is determined using the quantity Q̃2 (τ, z0 ). A computation yields the
following results:
• Case: q > −1/2, h(ω) = Aq ω 2q+2 e−ω/ωc . We have,
1
,
Q2 (τ, z0 ) = γ(2q + 1, z0 )ℜ 1 −
(1 + iτ )2q+1
1
Q̃2 (τ, z0 ) = Γ(2q + 1, z0 )ℜ 1 −
,
(1 + iτ )2q+1
where
γ(x, α) =
Z
α
0
e−z z x−1 z.
(5.10)
(5.11)
(5.12)
is the incomplete gamma function, and Γ(x, α) denotes its complement, so that γ(x, α) +
Γ(x, α) = Γ(x). Our results of the numerical simulations for q = 1 are presented in
Fig. 3. The red surfaces show the contributions to the dynamics of the EE coming
20
(a)
(b)
(c)
(d)
Figure 3: (Color online) X-interaction. The entanglement entropy, SE , as a function of p
and τ : q = 1, ε = 0.1. Red surfaces correspond to high frequency contributions ω > ω0 .
Cyan surfaces give the low frequencies contributions, ω < ω0 . (a) z0 = 0.1, (b) z0 = 1, (c)
z0 = 2. (d) z0 = 5.
21
(a)
(b)
(c)
(d)
Figure 4: (Color online) X-interaction. The entanglement entropy, SE , as a function of p
and τ : q = −1, ε = 0.1. The red surfaces show the high frequency contributions, ω > ω0 .
The cyan surfaces show the low frequency contributions, ω < ω0 . (a) z0 = 0.001, (b)
z0 = 0.01, (c) z0 = 0.1, (d) z0 = 1.
22
from the high frequency region, ω0 ≤ ω < ∞, or z0 = ω0 /ωc ≤ z = ω/ωc < ∞. The
cyan surfaces correspond to the low frequency contribution. Because q = 1, the small
frequencies are suppressed (h(0) = 0), and the EE does not reach its maximum for any
value of p. Fig. 3a shows that for small values of z0 , the cyan surface (small frequency
contribution) is significantly suppressed. As z0 increases (Figs. 3b,c), the contribution
from the small frequencies becomes more significant, and the cyan surface grows. Finally,
when z0 reaches relatively large values, the small frequencies saturate at the total value
of the EE. Correspondingly, the high frequency contribution is small (red surface in Fig.
3d).
• Case: q = −1, h(ω) = Aq ω 2q+2 e−ω/ωc . We have
1
1
1
ln(1 + τ 2 ) − E2 (z0 ) + ℜE2 ((1 + iτ )z0 ),
2
z0
z0
1
1
Q̃2 (τ, z0 ) = E2 (z0 ) − ℜE2 ((1 + iτ )z0 ).
z0
z0
Q2 (τ, z0 ) =τ arctan τ −
(5.13)
(5.14)
Here, En (z) is the exponential integral
En (z) =
Z
1
∞
e−zt
t.
tn .
(5.15)
The results of the numerical simulations are presented in Fig. 4. Since q = −1, we have:
h(0) = const > 0. So, the small frequencies influence the EE. As one can see, even for
small values of z0 = 0.001 (small region of integration in the vicinity of ω = 0), the cyan
surface reaches its asymptotic value of ln 2 for p = 1/2. For z0 = 0.1 the small frequency
contribution to the EE is large (Fig. 4d), and that of the high frequencies (red surface) is
significantly suppressed.
• Case: h(ω) = const inside of the frequency interval (0, ωc ). We consider the function h(ω)
given by
h(ω) = h0 (Θ(0) − Θ(ωc )),
(5.16)
where h0 > 0 and where Θ(x) is the Heaviside step function. Then (5.2) takes the form
(5.4) with ε = 2λ2 πh0 /ωc and Q2 (τ ) = Q2 (τ, z0 ) + Q̃2 (τ, z0 ), with
Z
2 z0 (1 − cos(τ z))
z.,
Q2 (τ, z0 ) =
π 0
z2
Z
2 1 (1 − cos(τ z))
z..
Q̃2 (τ, z0 ) =
π z0
z2
We rewrite the integrals as
2τ
4 sin2 (z0 τ /2)
Si(z0 τ ) −
,
π
πz0
4
2τ
sin2 (z0 τ /2) − sin2 (τ /2) .
si(τ ) − si(z0 τ ) +
Q̃2 (τ, z0 ) =
π
πz0
Q2 (τ, z0 ) =
23
(5.17)
(5.18)
Here Si(z) and si(z) are given by
Z z
sin x
x,
Si(z) =
x .
0
si(z) = −
Z
0
z
sin x
x.
x .
(5.19)
Using the identity, si(z) = Si(z) − π/2, we obtain
Q2 (τ, z0 ) =τ +
4 sin2 (z0 τ /2)
2τ
si(z0 τ ) −
.
π
πz0
(5.20)
The results of our numerical simulations are presented in Fig. 5. They show that the small
frequencies (cyan surfaces) give the main contribution to the dynamics of the EE.
5.1.2
Contribution of low and high frequencies for h(ω) = h0 inside of the
frequency interval (ω0 , ω1)
Let us consider the function h(ω) given by,
h(ω) = h0 (Θ(ω0 ) − Θ(ω1 )),
(5.21)
where Θ(x) is the Heaviside step function. We define ωh = 2λ2 πh0 , and introduce the following
dimensionless variables: ν0 = ω0 /ωh , ν1 = ω1 /ωh and τ = ωh t. In the new variables we have
|s(τ )| = e−(1/ν0 )Q2 (τ ) , where
Z
2 ν1 (1 − cos(τ z))
z.
Q2 (τ ) =
π ν0
z2
4 sin2 (ν0 τ /2) sin2 (ν1 τ /2) 2τ
=
−
Si(ν1 τ ) − Si(ν0 τ ) +
.
π
π
ν0
ν1
We use the asymptotic formula
Si(z) ∼
to obtain that for ν0 τ, ν1 τ ≫ 1,
Q2 (τ ) ∼
π cos z sin z
−
− 2 ,
2
z
z
z ≫ 1,
1 sin(ν0 τ ) sin(ν1 τ ) 2 ν1 − ν0
+
−
.
π ν0 ν1
τ
ν02
ν12
(5.22)
(5.23)
For fixed δ = ν1 −ν0 , we study numerically the behavior of the functions |s(τ, ν0 )| = e−(1/ν0 )Q2 (τ,ν0 )
and Q2 (τ, ν0 ), where
Q2 (τ, ν0 ) ∼
2
δ
1 sin(ν0 τ ) sin((ν0 + δ)τ ) −
+
.
π ν0 (ν0 + δ) τ
(ν0 + δ)2
ν02
(5.24)
Fig. 6 shows the result for δ = 0.1. The entanglement entropy as a function of τ and p is
depicted in Fig. 7, for a fixed value of the frequency domain, δ = ν1 − ν0 , and for increasing
value of ν0 . These results show hat the small frequencies, near ω = 0 give the main contribution
to the EE in the X-interaction model.
24
(a)
(b)
(c)
(d)
Figure 5: (Color online) X-interaction. The entanglement entropy, SE , as a function of
p and τ : ε = 0.1, h(ω) = h0 (Θ(0) − Θ(ωc )). Red surfaces correspond to the EE for the
frequencies inside the interval, ω0 ≤ ω ≤ ωc . Cyan surfaces correspond to the EE for the
frequencies inside the interval, 0 ≤ ω ≤ ω0 . (a) z0 = 0.001, (b) z0 = 0.1, (c) z0 = 0.5, (d)
z0 = 1.
25
(a)
(b)
Figure 6: (Color online) X-interaction. For the fixed value δ = 0.1: (a) The function
Q2 (τ, ν0 ) vs. τ and ν0 . (b) The function |s(τ, ν0 )| = e−(1/ν0 )Q2 (τ,ν0 ) vs. τ and ν0 .
5.2
D-interaction
For the D-interaction model, the function, s(t), given by Eq. (1.7), is written here as,
Z ∞
hD (ω) [1 − e2ıtλg(ω) ] dω ,
s(t) = exp −
(5.25)
0
For illustrative purposes, in our numerical simulations we choose hD (ω) = h0 (Θ(0) − Θ(ωc )) and
g(ω) = g0 ω k , where k > 0. Performing the integration in (5.25), we obtain,
h
n
io
|s(τ )| = exp − εµk 1 − k(µτ1)1/k ℜ e−iπ/2k γ(1/k, iµτ )
,
(5.26)
where µ = 2λg0 h0 ωck , ε = h0 2λg0 , τ = h0 t is the dimensionless time and γ(x, α) denotes the
incomplete gamma function (5.12). Writing γ(x, α) = Γ(x) − Γ(x, α) and using the asymptotic
formula for Γ(x, α),
Γ(a, z) ∼ z a−1 e−z ,
(5.27)
we obtain
h
|s(t)| ∼ exp − εµ
k
Γ(1/k) cos π/2k
cos(µτ ) i
,
+
1+
k(µτ )2
k(µτ )1/k
(µτ → ∞).
(5.28)
The entanglement entropy vs τ and p is depicted in Fig. 8. As one can see, in this case, only
partial decoherence occurs, and the function, s(t), does not decay to zero. Correspondingly, the
EE does not reach its potential maximum ln 2 ≈ 0.7 for any values of parameters.
26
(a)
(c)
(b)
(d)
Figure 7: (Color online) X-interaction. The entanglement entropy SE as a function of p
and τ : δ = 0.1; (a) ν0 = 0.1, (b) ν0 = 0.5, (c) ν0 = 1, (d) ν0 = 2.
27
(a)
(b)
(c)
(d)
Figure 8: (Color online)D-interaction. The entanglement entropy, SE , as a function of p
and τ (ε = 0.1). (a) k = 1/2, µ = 1/2; (b) k = 1, µ = 2; (c) k = 1, µ = 2; (d) k = 2,
µ = 2.
28
5.3
1/f noise
To estimate the impact of 1/f noise on the entanglement entropy, we assume that for both,
X-interaction and D-interaction, the function h(ω) in the interval J = (0, ωc ) can be written as,
(
h0 /ω0 , 0 < ω < ω0 ,
h(ω) =
(5.29)
h0 /ω, ω0 < ω < ωc .
Further we set z0 = ω0 /ωc and τ = ωc t.
• X-interaction
Rω
dω), we obtain
Performing integration in the formula |s(t)| = exp(−4λ2 0 c h(ω) 1−cos(ωt)
ω2
−εQ(τ
)
2
|s(t)| = e
, where ε = 2πλh0 /ωc and
τ2
τ2
τ
τ
2 τ
2 τ
.
Si(z0 τ ) + Ci(τ ) − Ci(z0 τ ) +
sin(z0 τ ) − sin τ − sin
Q(τ ) =
π z0
2
2
2z0
2
2
(5.30)
Here Ci(z) denotes the “cosine integral”,
Ci(z) = γ + ln z +
Z
0
z
cos x − 1
x..
x
The asymptotic behavior of Q(τ ) for τ ≫ 1,
τ τ
2 cos(z0 τ )
Q(τ ) ∼
+ z0 sin2
,
1−
z0
πτ
z0
2
yields
ετ
1 + O(1/τ ) .
|s(t)| = exp −
z0
(5.31)
(5.32)
(5.33)
We conclude that full decoherence takes place and the EE approaches it maximum ln 2 ∼
0.7 for large times.
• D-interaction
For illustrative purposes, we choose, g(ω) = g0 ω. Performing integration in Eq. (5.25),
we obtain, |s(t)| = exp(−h0 QD (τ )), where
QD (τ ) = 1 − ln z0 −
sin(z0 ετ )
+ ℜ E1 (iετ ) − E1 (iz0 ετ ) .
z0 ετ
(5.34)
Here ε = 2λg0 , and E1 (z) denotes the exponential integral in (5.15). Using the asymptotic
formula for E1 (z),
e−z
,
(5.35)
E1 (z) ∼
z
we obtain
h
sin(ετ ) i
|s(τ )| ∼ exp − h0 1 − ln z0 +
.
(5.36)
ετ
It follows that only partial decoherence occurs, the function s(τ ) does not decay to zero.
Therefore the EE does not achieve its maximum for large times.
29
6
6.1
X-interaction, proofs
Finite mode reservoirs
The Hamiltonian HX , (1.2), leaves the dimer populations stationary. Namely, setting
σz = P1 − P2 = |ϕ1 ihϕ1 | − |ϕ2 ihϕ2 |,
(6.1)
where P1 is the projection onto the ‘donor level’ (state ϕ1 with eigenvalue +1 of σz ) and P2 is
the projection onto the ‘acceptor level’ ϕ2 , the commutators [H, P1 ] = [H, P2 ] = 0 vanish. We
take an initial state of the combined system which is pure and completely disentangled, of the
form
Ψ(0) = ψS ⊗ ψ1 ⊗ ψ2 ⊗ · · · ⊗ ψN .
(6.2)
Here,
ψS = aϕ1 + bϕ2
(6.3)
is an arbitrary pure initial state of the dimer, with |a|2 + |b|2 = 1 and ψj is a pure initial state
of the jth oscillator. Below, we specialize to the coherent state reservoir initial state (1.1). We
set
p = |a|2 ,
1 − p = |b|2 .
(6.4)
The diagonal entries of the density matrix |ψS ihψS | are p and 1 − p.
6.1.1
Reduced states
The state of the total system at time t is
+
−
ıΩt/2
Ψ(t) = e−ıtH Ψ(0) = a e−ıΩt/2 P1 ⊗N
P2 ⊗N
j=1 Uj (t)ψj + b e
j=1 Uj (t)ψj ,
where
Uj± (t)
−ıt ωj (a†j aj +1/2)±λgj (a†j +aj )
=e
.
(6.5)
(6.6)
By a polaron transformation, we can diagonalize Uj± (t) as follows,
†
2 |g
Dj (λgj /ωj ) Uj+ (t) Dj (λgj /ωj )∗ = Sj (t) ≡ e−ıt(ωj aj aj −λ
j|
2 /ω
j)
Dj (λgj /ωj )∗ Uj− (t) Dj (λgj /ωj ) = Sj (t)
(6.7)
where Dj (z), z ∈ C, are the (coherent state) displacement operators
†
Dj (z) = ezaj −z̄aj .
(6.8)
(Note that the second line in (6.7) is obtained from the first one by switching λ → −λ.)
Remark. The transformation (6.7) is only valid for ωj 6= 0, which is of course the case in
the discrete mode situation. When we will perform the continuous mode limit, then ω becomes
a variable with values in [0, ∞) (c.f. Section 6.2.1) and in particular, the mode ω = 0 will play
a role (see also Lemmas 3.3, 3.1).
30
Lemma 6.1 (Reduced dimer state) The reduced dimer density matrix has constant diagonals and (with [ρS (0)]12 = ab̄),
[ρS (t)]12 = e−ıΩt [ρS (0)]12
N
Y
j=1
ψj , Dj 2λgj (1 − eıωj t )/ωj ψj .
(6.9)
Proof. The reduced density matrix of the dimer is obtained immediately from (6.5),
ρS (t) = Trj=1,...,N |Ψ(t)ihΨ(t)|
= |a|2 P1 + |b|2 P2 + ab̄ e−ıΩt |ϕ1 ihϕ2 |
N D
E
Y
ψj , (Uj− (t))∗ Uj+ (t)ψj + h.c.
(6.10)
j=1
Using (6.7) and simply writing Dj for Dj (λgj /ωj ) and Sj for Sj (t),
E D
ψj , (Uj− (t))∗ Uj+ (t)ψj = ψj , Dj Sj∗ (Dj∗ )2 Sj Dj ψj .
(6.11)
The displacement operators satisfy
Dj (z)Dj (ζ) = e2ıImz ζ̄ Dj (z + ζ)
†
(6.12)
†
so that (Dj∗ )2 = Dj (−2λgj /ωj ). Next, eıωj taj aj Dj (z)e−ıωj taj aj = Dj (eıωj t z) and hence Sj∗ (Dj∗ )2 Sj =
Dj (−2eıωj t λgj /ωj ). Then using again (6.12) twice we find
Dj Sj∗ (Dj∗ )2 Sj Dj = Dj 2λgj (1 − eıωj t )/ωj .
(6.13)
Using expression (6.13) in (6.11) gives (6.9). This shows Lemma 6.1
Next we obtain the reduced density matrix of the collection of operators with given indices.
Lemma 6.2 (Reduced oscillators state) Let J ⊆ R+ and denote by ρJ (t) the reduced density matrix of the oscillators having frequencies inside J, obtained from the total density matrix
by tracing out all other oscillator degrees of freedom as well as those of the dimer. Then
−
−
+
ρJ (t) = p |Ψ+
J (t)ihΨJ (t)| + (1 − p) |ΨJ (t)ihΨJ (t)|,
(6.14)
where p = [ρS (0)]11 and
±
Ψ±
J (t) = ⊗j∈Je Uj (t)ψj ,
where
Je = j ∈ {1, . . . , N } : ωj ∈ J .
Depending on time t, the matrix ρJ (t) has rank either one or two. It has rank one exactly when
e
| ψj , Dj (2λgj (1 − eıtωj t )/ωj )ψj | = 1
∀j ∈ J.
(6.15)
Apart from t = 0, condition (6.15) is difficult to satisfy, so generically, the rank of ρJ (t) is two.
Proof. The reduced density matrix in question is obtained from (6.5),
ρJ (t) = TrS, j∈{1,...,N }\Je |Ψ(t)ihΨ(t)|
= |a|2 ⊗j∈Je |Uj+ (t)ψj ihUj+ (t)ψj | + |b|2 ⊗j∈Je |Uj− (t)ψj ihUj− (t)ψj |.
This shows the form of ρJ (t) as given in Lemma
6.2. Since
the vectors Ψ±
J (t) are normalized,
−
+
their span is one-dimensional exactly if | ΨJ (t), ΨJ (t) | = 1. This equation is equivalent to
e
(c.f. (6.11), (6.13)) | ψj , Dj (2λgj (1 − eıtωj t )/ωj )ψj | = 1, ∀j ∈ J.
31
6.1.2
Entanglement entropy
Lemma 6.3 (Entanglement entropy) The entanglement entropy of ρS (t) and ρJ (t), given
in Lemmas 6.1 and 6.2, respectively, are as follows.
(6.16)
S(ρJ (t)) = − 12 + 21 rJ (t) ln 12 + 21 rJ (t) − 12 − 21 rJ (t) ln 12 − 12 rJ (t)
S(ρS (t)) = S(ρJ={ω1 ,...,ωN } (t))
(6.17)
where p is the (time-independent) population probability of the first level of the dimer and
Y p
ψj , D 2(1 − eıωj t )λgj /ωj ψj .
rJ (t) = 1 − 4p(1 − p)(1 − |sJ (t)|2 ) and |sJ (t)| =
j∈Je
Remarks. 1. The nonzero eigenvalues of ρJ (t) are 21 ± 21 rJ (t). The entropy of ρJ (t)
is maximal when the nonzero eigenvalues are all equal, i.e. for rJ (t) = 0 or p(1 − p)(1 −
|sJ (t)|2 ) = 14 . However, p(1 − p) ≤ 41 with equality exactly for p = 12 and (1 − |sJ (t)|2 ) ≤ 1 with
equality exactly for sJ (t) = 0. It follows that the entanglement entropy of ρJ (t) and of ρS (t)
increases
if the entropy of the
initial dimer state is increased, and it also increases if the overlaps
| ψj , D 2(1 − eıtωj t )λgj /ω ψj | decrease.
2. If the dimer is populated entirely on one level, i.e., p = 0 or p = 1, then rJ (t) = 1 and
S(ρJ (t)) = 0 regardless of J and t. That is, no entanglement entropy is created during the
dynamics in any subsystem. This can be understood in the following way: for ΨS = ϕ1 the
evolution of Ψ(0), (6.2), under the dynamics generated by HX , (1.2), is
Ψ(t) = e−ıtH Ψ(0) = e−ıΩt/2 ϕ1 ⊗ ψ1 (t) ⊗ ψ2 (t) · · · ⊗ ψN (t).
Therefore, the state is of product form for all times, so the reduction to any subsystem is again
a pure state and has zero entropy.
3. The initially pure dimer state has maximal entropy for p = 1/2. Then rJ (t) = |sJ (t)|.
Proof of Lemma 6.3. The entanglement entropy S(ρJ (t)) = −TrρJ (t) log ρJ (t) of the state
ρJ (t), (6.14), is expressed purely through the two non-zero eigenvalues of ρJ (t). (The operator
ρJ (t) acts on an infinite-dimensional Hilbert space of |Je| independent harmonic oscillators,
however, all but two of its eigenvalues are zero.)
−
−
+
To find the two eigenvalues, we write ρJ = p|Ψ+
J ihΨJ | + (1 − p)|ΨJ ihΨJ | (not exhibiting the
t dependence for simplicity of notation) and decompose
+
−
−
with sJ = Ψ+
Ψ−
J , ΨJ and ηJ ⊥ ΨJ .
J = sΨJ + ηJ ,
Then we express ρJ in the ordered orthonormal basis {Ψ+
J , η̂J }, where η̂J = ηJ /kηJ k. Namely,
it has the form (2.6). The nonzero eigenvalues of ρJ are then immediately found,
q
o
n
(6.18)
spec(ρJ ) = 21 ± 14 − p(1 − p)(1 − |sJ |2 ) ∪ {0}.
While p is the time-independent population of the first level of the dimer, the overlap sJ depends
on time,
Y
ψj , D 2(1 − eıωj t )λgj /ωj ψj .
(6.19)
|sJ | =
j∈Je
Note that |sJ (t)| ≤ 1 and |sj (t)| = 1 ⇔ ρJ (t) has rank one (c.f. Lemma 6.2). This completes
the proof of Lemma 6.3.
32
6.2
Coherent state reservoir
Consider the initial reservoir state (1.1). Since D(ξ)|αi = eıImξ ᾱ |α+ξi we obtain hαj , D(ξj )αj i =
1
2
eıImξj ᾱj hαj , αj + ξj i = e−2ıImαj ξ̄j e− 2 |ξj | , where ξj = 2(1 − eıωj t )λgj /ωj . It follows that (see
Lemma 6.3 and (6.19))
1
|sJ (t)| = e− 2
P
j∈Je
|ξj |2
X |gj |2
ıωj t 2
|1
−
e
|
.
= exp − 2λ2
ωj2
(6.20)
j∈Je
Remarks. 1. We note that |sJ (t)| is independent of the α1 , . . . , αN determining the initial
reservoir state! Moreover, since the entanglement entropy only depends on this quantity, it too
will not depend on the αj .
2. The interval of frequencies J samples a window of all frequencies present in the reservoir.
Assuming a roughly homogeneous coupling, say |gj | ∼ 1, the exponent in (2.7) becomes large if
J contains low frequencies ωj → 0, making |sJ (t)| small. As discussed after Lemma 6.3, small
values of |sJ (t)| give large values for the entanglement entropy. We conclude that entanglement
entropy in the reservoir is concentrated in the low frequency oscillators. A more precise statement
of this fact is given in Lemma 3.3.
6.2.1
Proofs of Propositions 3.3 and 3.1
The results of Proposition 3.3 follow immediately from the expressions for the discrete oscillators
by using (1.14). We now turn to the proof of Proposition 3.1.
In (2.7), we write 1 − cos(ωt) = 2 sin2 (ωt/2) and use the fact that
1
t→∞ t
lim
Z
∞
hX (ω)
0
sin2 (ωt/2)
dω = πhX (0)/4
ω2
to arrive at (3.1). This proves (A). The proof of (B) is obtained in the same way. To prove (C)
we use the estimate
−4λ2 MJ
e
Rω
1 1−cos(ωt) dω
ω0
ω2
−4λ2 mJ
≤ |sJ (t)| ≤ e
Rω
1 1−cos(ωt) dω
ω0
ω2
.
(6.21)
Next
Z
ω1
ω0
1 − cos(ωt)
dω ≥
ω2
=
Z ω1
1
1 − cos(ωt) dω
2
ω 1 ω0
1 1 sin(ω1 t)−sin(ω0 t)
ω
−
ω
−
ω
−
ω
−
2/t
.
≥
1
0
1
0
t
ω12
ω12
Combining this with (6.21) yields the upper bound in (3.3). To get the lower bound in (3.3), we
Rω
dω ≤ ω12 (ω1 − ω0 + 2/t) in (6.21). This completes the proof of Proposition 3.1.
use ω01 1−cos(ωt)
ω2
0
33
7
7.1
D-interaction, proofs
Dimer dynamics for V, µ > 0, small
Theorem 7.1 below describes the change to this process under the influence of (small) energyexchange interaction, i.e., for V and µ in (1.3) small. The smallness of the energy-exchange
terms is measured by the parameter
 µ sup f (ω)
ω

if µΩ supω f (ω) ≥ λV inf ω g(ω)
 λ inf ω g(ω)
(7.1)
η≡

 V +2µ supω f (ω) if µΩ sup f (ω) < λV inf g(ω)
ω
ω
Ω+2λ inf ω g(ω)
We consider here the coupling function to satisfy
m ≡ inf g(ω) > 0.
(7.2)
ω≥0
If µ = V = 0 then η = 0.
R∞
Theorem 7.1 (Dynamics of the dimer) Let Cα = 0 hD (ω)f (ω)dω, where hD is given in
(1.8) and f is the coupling function in (1.3). The reduced dimer density matrix ρS (t) has the
expansion
R∞
ρS (t) − ρ0S (t) − V e− 0 h(ω)dω ρ1S (t) ≤ C V 2 /Ω2 + η + tηV + tηµCα + tV 3 /Ω2 .
(7.3)
Ω
Here, ρ0S (t) is given in (2.8) (with constant diagonal) and
ρ1S (t) =
| ↑ih↑ | − | ↓ih↓ | Re [ρS (0)]12 1 − e−ıΩt
+ | ↑ih↓ |(p − 21 ) 1 − e−ıtΩt + | ↓ih↑ | (p − 12 ) 1 − eıtΩt .
In (7.3), C is a constant not depending on any of the parameters V, Ω, η, t, µ, nor on the functions
h, α.
The remainder in (7.3) is not uniform in time. However, up to times t ∼ tpd , when the
partial decoherence is completed (c.f. (2.11)), the remainder is at most O(B1 ), where
p
p
B1 = V 2 /Ω2 + η + |ξ|ηV /λ + |ξ|ηµCα /λ.
(7.4)
p
(use t = |ξ|/λ in the right side of (7.3)). The expansion (7.3) is meaningful provided the
remainder is smaller than the main term. Therefore, in order to resolve the effect of the term
∝ VΩ on the dimer dynamics (the left side of (7.3)) on the time-scale tpd , we need B1 << VΩ .
Using the definition of η, (7.1) (parameter regime of the second line), we see that
p
i
1 + 2 Vµ supω f (ω) h
|ξ|
B1
V
=
+
(V
+
µC
)
<< 1
(7.5)
1
+
α
λ
V /Ω
Ω
λ
1 + 2Ω
inf ω g(ω)
provided that
V << Ω,
p
|ξ| V 2 << λΩ
and
This shows the following result.
34
µΩ sup f (ω) << λV inf g(ω).
ω
ω
(7.6)
Corollary 7.2 (Validity of the perturbation expansion) In the parameter regime (7.6),
the right side of (7.3) is << V /Ω. This means that the perturbation expansion (7.3) resolves the
partial decoherence process and the O(V /Ω) corrections to the dynamics, and in particular, the
associated change in the dimer entanglement entropy.
Remarks. 1. If g ′ (0) 6= 0 then
ξ=
h′ (0)g′ (0)(g−1 )′ (g(0)) − h(0)g ′′ (0)(g−1 )′ (g(0))
[g′ (0)]2
2. If ε > 0, δ ≥ 0 and g(ω) = ε + ω δ as ω ∼ 0, then we obtain
′
h (0) if δ = 1
ξ=
2h(0) if δ = 1/2 and h′ (0) = 0.
(7.7)
(7.8)
Note that strictly speaking, we need ε > 0 to have inf ω g(ω) > 0, however, the value
of ξ does not depend on ε. This independence can be understood as follows: shifting the
coupling function g(ω) in (1.3) by a constant value ε amounts to renormalizing the frequency
P
†
Ω as Ω + 2λε N
j=1 aj aj . However, the energy-conserving partial decoherence process does not
depend on that frequency.
3. If one is only interested in the closeness of ρS (t) to ρ0S (t) then the term O(V /Ω) can be
included in the remainder, and the remainder stays small for a wider parameter regime than
(7.6).
4. We derive a more precise estimate than (7.3). Namely, we can omit the remainder term
tV 3 /Ω2 on the right side of (7.3) by replacing ρ1S (t) in the left side of (7.3) by
√
2
2 (7.9)
| ↑ih↑ | − | ↓ih↓ | Re [ρS (0)]12 1 − e−ıt Ω +V
ρe1S (t) =
√
√
2
2
2
2 + | ↑ih↓ | e−it Ω +V − e−iΩt [ρS (0)]12 + (p − 12 ) 1 − e−ıt Ω +V
√ 2 2
√
2
2 + | ↓ih↑ | eit Ω +V − eiΩt [ρS (0)]21 + (p − 12 ) 1 − eıt Ω +V
√
Due to the bound e−it
7.2
Ω2 +V 2
= e−iΩt (1 + O(tV 2 /Ω)), we have ρe1S (t) = ρ1S (t) + O(tV 2 /Ω).
Dynamics of the oscillators for V > 0 small, µ = 0
For simplicity of the presentation, we consider in this section the Hamiltonian (1.3) with µ = 0
only. Let J ⊆ [0, ∞) be a window of continuous oscillator frequencies of interest.
Theorem 7.3 (Dynamics of the oscillators) Consider the system for finite N , with Hamiltonian (1.3) and µ = 0. Denote by ρJ (t) the reduced density matrix of the oscillators having
(discrete) frequencies inside J ⊆ R. We have
ρJ (t) − ρ00 (t) − V ρ1 (t) ≤ C V 2 + tV 2 + V
(7.10)
J
J
Ω
Ω
Ω
Ω+λm
1
where C is a constant independent of N, J, t, V, Ω, λ. Here, m is given in (7.2) and ρ00
J and
1
ρJ are as in Theorem 2.3, (B). On the left side of (7.10), k · k1 denotes the trace norm of
operators acting on the Hilbert space of K harmonic oscillators, where K is the number of
discrete frequencies ωj lying inside J.
35
7.3
Proof of Theorem 7.1
The reduced dimer density matrix is
ρS (t) = TrR e−ıtH |ψS ⊗ α1 · · · ⊗ αN ihψS ⊗ α1 · · · ⊗ αN |eıtH
X
=
F (n1 , . . . , nN ) e−ıtH̄ |ψS ihψS |eıtH̄ ,
(7.11)
n1 ,...,nN ≥0
where
N
Y
F (n1 , . . . , nN ) =
e−|αj |
2
j=1
and
H̄ =
1
2
ω̄
V̄
V̄
,
−ω̄
ω̄ = Ω + 2λ
N
X
|αj |2nj
(nj )!
gj nj ,
(7.12)
V̄ = V + 2µ
N
X
fj nj .
(7.13)
j=1
j=1
The spectral representation of e−ıtH̄ is
e−ıtH̄ = e−ıtr |χ+ ihχ+ | + eıtr |χ− ihχ− |,
where
r=
√
1
2
and, with N = 2(ω̄ 2 + V̄ 2 + ω̄ V̄ 2 + ω̄ 2 ),
√
2
2
−1/2 ω̄ + V̄ + ω̄
,
χ+ = N
V̄
p
(7.14)
V̄ 2 + ω̄ 2
χ− = N
(7.15)
−1/2
√−V̄
.
ω̄ + V̄ 2 + ω̄ 2
(7.16)
In the sum in (7.11) we split off the term with n1 = . . . = nN = 0,
−ıtH 0
ρS (t) = F (0) e
where F (0) =
ıtH 0
|ψS ihψS |e
QN
+
X∗
n1 ,...,nN ≥0
−|αj |2 ,
j=1 e
H0 =
1
2
F (n1 , . . . , nN ) e−ıtH̄ |ψS ihψS |eıtH̄ ,
Ω V
V −Ω
(7.17)
(7.18)
and where the ∗ indicates that we omit the term in which all n1 , . . . , nN vanish in the summation.
The point of this splitting is that in the remaining sum at least one of the n1 , . . . , nN is ≥ 1, so
that we can estimate the parameter
V̄
η̄ ≡
(7.19)
ω̄
by (see also (7.1))
η̄ ≤ η ≡
=
V + 2µν maxj fj
ν=1,2,... Ω + 2λν minj gj
( µ maxj fj
if µΩ maxj fj ≥ λV minj gj
λ minj gj
sup
V +2µ maxj fj
Ω+2λ minj gj
if µΩ maxj fj < λV minj gj
36
.
(7.20)
Then we use the decomposition (7.14) and the expansions
ı
ı
e±ıtr = e± 2 tω̄ 1 + O(tω̄ η̄ 2 ) = e± 2 tω̄ 1 + O(tη V̄ )
r = 21 ω̄ 1 + O(η̄ 2 ) ,
and
1
χ+ = | ↑i + O(η) ≡
+ O(η),
0
to estimate
ı
ı
e−ıtH̄ = e− 2 tω̄ | ↑ih↑ | + e 2 tω̄ | ↓ih↓ | + O η + tη V̄ .
It follows that
X∗
n1 ,...,nN ≥0
=
0
χ− = | ↓i + O(η) ≡
+ O(η)
1
n1 ,...,nN ≥0
+Rem.,
(7.22)
(7.23)
F (n1 , . . . , nN ) e−ıtH̄ |ψS ihψS |eıtH̄
X∗
(7.21)
(7.24)
ı
ı
ı
ı
F (n1 , . . . , nN ) e− 2 tω̄ | ↑ih↑ | + e 2 tω̄ | ↓ih↓ | |ψS ihψS | e 2 tω̄ | ↑ih↑ | + e− 2 tω̄ | ↓ih↓ |
where the remainder has the estimate (c.f. (7.12))
|Rem.| ≤ C η + tη
X∗
n1 ,...,nN ≥0
N
X
fj nj
F (n1 , . . . , nN ) V + 2µ
j=1
N
X
2
fj |αj | ,
≤ C η + tηV + 2tηµ
(7.25)
j=1
and where C is a constant independent of t, η, V, µ, αj , N . Finally we ‘remove the *’ in the sum
on the right side of (7.24) by adding (and subtracting) the summand for n1 = . . . = nN = 0,
and we obtain
X∗
n1 ,...,nN ≥0
F (n1 , . . . , nN ) e−ıtH̄ |ψS ihψS |eıtH̄
00
= −F (0) e−ıtH |ψS ihψS |eıtH
PN
00
+ | ↑ih↑ | p + | ↓ih↓ |(1 − p)
|2 (1−e−2ıtλgj )
+ | ↑ih↓ | ab̄ e−ıtΩ e− j=1 |αj
N
X
fj |αj |2 .
+O η + tηV + tηµ
+ | ↓ih↑ | āb eıtΩ e−
(7.26)
PN
j=1
|αj
|2 (1−e2ıtλgj )
j=1
Here we have set
H
00
=
1
2
Ω 0
0 −Ω
37
(7.27)
and ψS = a| ↑i + b| ↓i, p = |a|2 . Combining (7.17) and (7.26) yields
ρS (t) =
N
Y
2
0
0
00
00
e−|αj |
e−ıtH |ψS ihψS |eıtH − e−ıtH |ψS ihψS |eıtH
j=1
+| ↑ih↑ | p + | ↓ih↓ |(1 − p)
PN
|2 (1−e−2ıtλgj )
+ | ↑ih↓ | ab̄ e−ıtΩ e− j=1 |αj
N
X
fj |αj |2 .
+O η + tηV + tηµ
+ | ↓ih↑ | āb eıtΩ e−
(7.28)
PN
j=1
|αj
|2 (1−e2ıtλgj )
j=1
Note that the main term has trace one and, for t = 0, it reduces to the correct value ρS (0).
We may now diagonalize H 0 given in (7.18) exactly analogously to (7.14)-(7.16). However, we
resolve the O(V /Ω)-term in the expansion, namely,
χ+ = | ↑i +
V
2Ω | ↓i
+ O(V 2 /Ω2 ),
χ− = | ↓i −
V
2Ω | ↑i
0
+ O(V 2 /Ω2 ).
0
(7.29)
0
Using these√expressions in the spectral representation e−ıtH = e−ıtr |χ+ ihχ+ | + eıtr |χ− ihχ− |
with r0 = 12 Ω2 + V 2 , an easy calculation leads to
0
0
00
00
e−ıtH |ψS ihψS |eıtH − e−ıtH |ψS ihψS |eıtH
0
V
| ↑ih↑ | − | ↓ih↓ | Re [ρS (0)]12 1 − e−2ıtr
=
Ω
0
0
V
+ | ↑ih↓ | e−2ıtr − e−ıΩt [ρS (0)]12 + (p − 12 ) 1 − e−2ıtr
Ω
V
0
0
+ | ↓ih↑ | e2ıtr − eıΩt [ρS (0)]21 + (p − 12 ) 1 − e2ıtr
Ω
+O(V 2 /Ω2 ).
(7.30)
Combining (7.28) with (7.30) and taking the continuous mode limit shows (7.3). This finishes
the proof of Theorem 7.1.
7.4
Proofs of Theorem 7.3 and Proposition 3.3
We examine the reduced oscillator density matrix ρr for oscillators with indices j = 1, . . . , r,
where 1 ≤ r ≤ N .
Let H be the Hamiltonian (1.3) with µ = 0 and let |M1 , . . . , Mr i and |M̄1 , . . . M̄r i be number
states of the first r oscillators (Mj , M̄j ∈ N). The matrix elements of ρr (t) are
X
M1 , . . . , Mr , ρr (t)M̄1 , . . . , M̄r =
F ψS , eıtHM̄ e−ıtHM ψS ,
(7.31)
nr+1 ,...,nN ∈N
where we recall that the initial state is (1.1) and where we have set
F =
r
Y
j=1
hMj , αj i αj , M̄j
N
Y
j=r+1
38
Pr
| hnj , αj i |2 e−ıt j=1 ωj (Mj −M̄j )
(7.32)
and
HM
1
=
2
Ω + λTM
V
V
−Ω − λTM
,
TM = 2
r
X
gj Mj + 2
j=1
N
X
gj nj .
(7.33)
j=r+1
0, the scalar
Of course, HM̄ is (7.33) with Mj replace with M̄j . ItPis readily seen that for V =
P
−ıλt rj=1 gj (Mj −M̄j )
ıλt rj=1 gj (Mj −M̄j )
product factor in the sum of (7.31) is simply pe
+ (1 − p)e
(where p = | hψS , ↑i |2 ). A little algebra on (7.31) then gives the following evolution of ρr , for
V = 0:
=0)
(7.34)
ρ(V
(t) = p Ut+ ρr (0)(Ut+ )∗ + (1 − p) Ut− ρr (0)(Ut− )∗
r
with ρr (0) = |α1 , . . . , αr ihα1 , . . . , αr | and
Ut±
−ıtH ±
=e
,
H
±
r
X
(ωj ± λgj )a†j aj .
=
(7.35)
j=1
(V =0)
In particular, ρr
(t) acts non-trivially on span{Ut+ |α1 , . . . , αr i, Ut− |α1 , . . . , αr i} and has rank
two (t 6= 0). By the Gram-Schmidt procedure we construct an orthonormal basis of this span
(V =0)
and express ρr
(t) as a 2 × 2 matrix (see (7.52) below). To deal with the situation V 6= 0,
we diagonalize the matrices HM and HM̄ in (7.31),
M
ıtrM M
|χ− ihχM
e−ıtHM = e−ıtrM |χM
− |.
+ ihχ+ | + e
We may use the formulas (7.14)-(7.16) with suitable substitutions. In particular,
!
!
V
1
−
2(Ω+λTM )
+ O ( VΩ )2
χM
+ O ( VΩ )2 ,
χM
V
− =
+ =
1
2(Ω+λTM )
with a remainder uniform in the values of M1 , . . . , MN , and
1
V2
.
e±ıtrM = e± 2 ıt(Ω+λTM ) 1 + O t Ω+λT
M
(7.36)
(7.37)
(7.38)
Our strategy is now similar to what we did in the derivation of the dimer density matrix. We
split off the term in (7.31) in which nr+1 = · · · = nN = 0, which we can calculate explicitly
(to its main contribution). The remaining terms in the sum have at least one nj ≥ 1 for some
j ≥ r + 1, and this will make those terms small, provided that minj≥r+1 gj is relatively large.
Using formula (7.36) for each exponential and expanding the eigenvectors as in (7.37), it is
not hard (albeit a bit tedious) to obtain the following bound,
ψS , eıtHM̄ e−ıtHM ψS |nr+1 =···=nN =0
1
V
V
= e− 2 ıtλ(WM −WM̄ ) p + 12
[ρS (0)]21
[ρS (0)]12 + 21
Ω + λWM̄
Ω + λWM
1
V
V
[ρS (0)]12
[ρS (0)]21 − 12
+e 2 ıtλ(WM −WM̄ ) 1 − p − 21
Ω + λWM̄
Ω + λWM
1
1
1
V
−
[ρS (0)]21
−eıΩt e 2 ıtλ(WM +WM̄ )
2 Ω + λWM
Ω + λWM̄
1
1
1
V
+e−ıΩt e− 2 ıtλ(WM +WM̄ )
−
[ρS (0)]12
2 Ω + λWM
Ω + λWM̄
2
1
1
.
(7.39)
+
+O VΩ + tV 2 Ω+λW
Ω+λW
M
M̄
39
Here, we have set WM = 2
Pr
j=1 gj Mj .
Noticing that
1
1
e 2 ıtλWM |M1 , . . . , Mr i = e 2 ıtλŴ |M1 , . . . , Mr i,
where
Ŵ = 2
Pr
†
j=1 gj aj aj ,
(7.40)
we see that the term with nr+1 = · · · = nN = 0 on the right side of (7.31) is the hM1 , . . . , Mr | ·
|M̄1 , . . . , M̄r i-matrix element of the selfadjoint operator
n PN
2
V [ρS (0)]21
[ρS (0)]12
V
+
ρr (0) (Ut+ )∗
e− j=r+1 |αj | Ut+ pρr (0) + ρr (0)
2
2 Ω + λŴ
Ω + λŴ
V
V
[ρS (0)]12
[ρ
(0)]
S
21
+Ut− (1 − p)ρr (0) − ρr (0)
−
ρr (0) (Ut− )∗
2
2 Ω + λŴ
Ω + λŴ
1
V [ρS (0)]21 −
1
Ut
−eıΩt
ρr (0) − ρr (0)
(Ut+ )∗
2
Ω + λŴ
Ω + λŴ
1
1
V [ρS (0)]12 + ρr (0) − ρr (0)
(Ut− )∗
Ut
+e−ıΩt
2
Ω + λŴ
Ω+
λ
Ŵ
2
o
2
+O VΩ + tVΩ
(7.41)
So far, the remainder in (7.41) is estimated in the maximum norm for matrices. More precisely,
let ρr,0 (t) be the operator obtained by taking only the term nr+1 , . . . , nN = 0 in (7.31), and let
µ(t) be the operator (7.41), without the error term O. What we have shown so far is that
sup
hM1 , . . . , Mr | ρr,0 (t) − µ(t) |M̄1 , . . . M̄r i ≤ C(1 + t)V 2 /Ω,
(7.42)
M1 ,...,Mr ,M̄1 ,...M̄N ∈N
where C is a constant independent of time t and of r. We show now that we actually have the
stronger estimate kρr,0 − µ(t)k1 ≤ C(1 + t)V 2 /Ω for the same constant C, where
√ kAk1 = Tr|A|
is the trace norm, with the absolute value of an operator A defined by |A| = A∗ A. To do so,
ıtHM̄ and e−ıtHM in the term
that the error
we recall
contains terms brought about by replacing e
ıtH
−ıtH
Mψ
ψS , e M̄ e
S in (7.31) by using (7.37) and (7.38). It is then not difficult to see that this
remainder is a sum of at most 16 terms, each of the form A|αihα|B, for some operators A and
B satisfying kAkkBk ≤ C(1 + t)V 2 /Ω, for a C independent of t, r. To estimate this remainder
in trace norm, we note that
p
p
A|αihα|B = B ∗ |αihα| A∗ A|αihα| B = kA|αik |B ∗ αihB ∗ α| = kA|αik kB ∗ |αik |U ihU |,
where |U i is the normalization of B ∗ |αi. Hence |(A|αihα|B)| is rank-one and thus its trace,
being the sum of its eigenvalues, equals kA|αik kB ∗ |αik ≤ kAk kB ∗ k ≤ C(1 + t)V 2 /Ω. We
conclude that
kρr,0 (t) − µ(t)k1 ≤ C(1 + t)V 2 /Ω.
(7.43)
In other words, the remainder term in (7.41) is understood in the trace-norm topology.
Let
P0 = |0, . . . , 0ih0, . . . , 0|
(7.44)
be the orthogonal projection onto the state |n1 = 0, . . . , nr = 0i. We have Ut± P0 = P0 , Ŵ P0 = 0
and P0⊥ Ŵ ≥ mP0⊥ , where m = 2 minj≥1 gj . (Here, P0⊥ = 1l − P0 .) Therefore,
1
Ω + λŴ
=
1
1
1
P0 +
P0⊥ = P0 + O
Ω
Ω
Ω + λŴ
40
1
Ω+λm
.
Using this bound in (7.41), we arrive at
PN
2
(7.41) = e− j=r+1 |αj | ×
n
V [ρS (0)]21
V [ρS (0)]12 +
Ut ρr (0)P0 +
P0 ρr (0) (Ut+ )∗
Ut+ pρr (0)(Ut+ )∗ +
2Ω
2Ω
V [ρS (0)]12
V [ρS (0)]21 −
Ut ρr (0)P0 −
P0 ρr (0) (Ut− )∗
+Ut− (1 − p)ρr (0)(Ut− )∗ −
2Ω
2Ω
V [ρS (0)]21 P0 ρr (0)(Ut+ )∗ − Ut− ρr (0)P0
−eıΩt
2Ω
V
[ρ
S (0)]12
+e−ıΩt
P0 ρr (0)(Ut− )∗ − Ut+ ρr (0)P0
2Ω
o
+O
V2
Ω
+
tV 2
Ω
+
V
Ω+λm
.
(7.45)
V
will be small compared to V /Ω for sizable m. Again, this error is estimated in
The error Ω+λm
the trace-norm topology (as in (7.43)).
we treat the
terms in the sum (7.31) in which at least one nj is ≥ 1. We expand
Now
ψS , eıtHM̄ e−ıtHM ψS using (7.36)-(7.38), as we did to arrive at (7.39). We allow for errors of
the order V /(Ω
+λTM ) ≤ V /(Ω + λm), where m = 2 min1≤j≤N gj . For instance, we use (c.f.
1
(7.37)) χM
+ O(V /(Ω + λm)). We then obtain, whenever at least one nj ≥ 1 for some
+ =
0
j = r + 1, . . . , N ,
1
1
ψS , eıtHM̄ e−ıtHM ψS = e− 2 ıλt(WM −WM̄ ) p + e 2 ıλt(WM −WM̄ ) (1 − p) + O
V +tV 2
Ω+λm
The right side of (7.46) is independent of nj with j ≥ r + 1. Moreover,
X
∗
nr+1 ,...,nN ∈N
N
Y
hnj , αj i 2 =
j=r+1
X
.
N
N
Y
Y
hnj , αj i 2 −
h0, αj i 2
nr+1 ,...,nN ∈N j=r+1
P
2
− N
j=r+1 |αj |
= 1−e
(7.46)
j=r+1
,
(7.47)
where the star ∗ indicates that we sum over all terms except the one with all nj equal to zero.
Combining (7.46), (7.47) and the definition of F , (7.32), we see that the contribution to the sum
(7.31) coming from all terms where at least one nj ≥ 1, is the hM1 , . . . , MN | · |M̄1 , . . . , M̄N imatrix element of the operator
PN
2
+tV 2
.
(7.48)
1 − e− j=r+1 |αj | Ut+ pρr (0)(Ut+ )∗ + Ut− (1 − p)ρr (0)(Ut− )∗ + O VΩ+λm
As above, the error is estimated in the trace-norm topology. The whole sum in (7.31) is thus the
hM1 , . . . , MN | · |M̄1 , . . . , M̄N i-matrix element of the sum of the two operators (7.45) plus (7.48).
We conclude that
ρr (t) = Ut+ ρr (0) 21 p(Ut+ )∗ + vP0 + Ut− ρr (0) 12 (1 − p)(Ut− )∗ − v̄P0 + h.c.
2
2
V
,
(7.49)
+O VΩ + tVΩ + Ω+λm
41
(error estimated in trace-norm topology) where we have defined
v=
2
V 1 − e−ıΩt − PN
j=r+1 |αj | [ρ (0)]
e
S
12 .
2
Ω
(7.50)
This shows that, modulo the remainder in (7.49),
ρr (t) = |e+ ihf+ | + |e− ihf− | + h.c.,
(7.51)
where |f+ i = ( 21 pUt+ + v̄P0 )|α1 , . . . , αr i, |f− i = ( 12 (1 − p)Ut− − vP0 )|α1 , . . . , αr i and
|e± i = Ut± |α1 , . . . , αr i.
(7.52)
Since P0 |α1 , . . . , αr i ∝ |0i ≡ |0, . . . , 0i this shows that (the main part of) ρr (t) acts nontrivially
on the three-dimensional space spanned by the vectors |e± i and |0i ≡ |0, . . . , 0i. By the GramSchmidt procedure we construct an orthonormal basis of span{|e+ i, |e− i, |0i} and then express
ρr (t) as a 3 × 3 matrix. The orthonormal basis is given by {|e+ i, |η̂i, |χ̂i}, where
|η̂i = (1 − |s|2 )−1/2 |e− i − s |e+ i
−1/2 1−s̄
√
|η̂i
.
(7.53)
|0i
−
δ|e
i
−
δ
|χ̂i = 1 − 2δ 1−Res
+
2
2
1−|s|
1−|s|
Here we have set
s = he+ , e− i = e−
Pr
j=1
|αj |2 (1−e2itλgj )
,
δ = e−
Pr
j=1
|αj |2
.
(7.54)
It is now a straightforward calculation to express ρt (t), (7.51), as a matrix which in the continuous
mode limit is precisely (2.17). Note that for simplicity of notation, we have derived the reduced
density matrix of the oscillators with indices j = 1, . . . , r in this proof; the general case resulting
in the reduced density matrix of oscillators with frequencies in a window J ⊆ R is directly
obtained from the expressions derived here, just as we did it in Lemma 6.2. This completes the
proof of Theorem 7.3.
Proof of Proposition 3.3. We write the 3 × 3 matrix associated to the operator ρ0J (t) +
(c.f. Theorem 7.3) as M0 + VΩ M1 and solve the equation det(M0 + VΩ M1 − z) = 0 for z.
For V = 0 we obtain the three values z0 = 0, 12 ± 21 rJ , where rJ is given in (1.12). Then, for each
2
z0 , we make the Ansatz z = z0 + VΩ t + VΩ2 t′ + · · · and solve the equation det(M0 + VΩ M1 − z) = 0
for t. We obtain one value for each z0 .
V 1
Ω ρJ (t)
Conclusion
We give a theoretical analysis of the production of entanglement entropy (EE) in the simplest
photosynthetic bio-complex, consisting of a chlorophyll based dimer interacting with a proteinsolvent reservoir. The dimer is modeled by a two-level quantum system, with a donor and an
acceptor level representing the (first) excited state of each chlorophyll molecule. In contrast to
previous considerations, we consider the initial state of the reservoir to be a pure quantum state
(not a mixed one, such as the Gibbs equilibrium state). The notion of EE, first introduced in [14]
42
in the field of quantum computation, has recently been used in many fields of science, including
solid state physics, chemistry, quantum field theory, thermodynamics, statistical physics, and
quantum gravity. EE plays an important role for better understanding the fundamental laws of
quantum physics, including Carnot cycles, the Landauer principle, and quantum chaos. In the
present work, we do not assume any restrictions on the strength of the dimer-reservoir interaction. This is important because the interaction (which is proportional to the reconstruction
energies) can be large in photosynthetic bio-systems. We consider two models, characterized by
an X-, or dipole-dipole- and a D-, or dipole-density dimer-reservoir interaction. The main results
for the production of the EE entropy are derived analytically, for energy conserving interactions.
We demonstrate that for the X-model, only the small frequencies of the reservoir significantly
contribute to the increase of the EE. For the D-model, all frequencies give a mostly similar
contribution to the EE production. We also analyze the corrections related to the influence
of non-energy conserving effects on the EE dynamics. We discuss a conjecture related to the
appearance of the Gibbs state for the dimer in our system. We provide numerical calculations
which support our analytical results.
Acknowledgment
This work was carried out under the auspices of the National Nuclear Security Administration
of the U.S. Department of Energy at Los Alamos National Laboratory under Contract No.
DE-AC52-06NA25396. M.M. has been supported by NSERC through a Discovery Grant and a
Discovery Accelerator Supplement. M.M. is grateful for the hospitality and financial support
of LANL, where part of this work was carried out. A.I.N. acknowledges support from the
CONACyT, Grant No. 15349 and partial support during his visit from the Biology Division,
B-11, at LANL. G.P.B. and R.T.S. acknowledge support from the LDRD program at LANL.
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