Structure of the interaction and energy transfer between an open quantum system and its environment

0.15 MB·

by  Tarek Khalil

Structure of the interaction and energy transfer
7 between an open quantum system and its
1
0 environment
2

n
a Tarek Khalila ∗
J
and

1 Jean Richertb †
2
a
Department of Physics, Faculty of Sciences(V),

] Lebanese University, Nabatieh, Lebanon
h
b
p Institut de Physique, Universit´e de Strasbourg,
-
t 3, rue de l’Universit´e, 67084 Strasbourg Cedex, France
n
a
January 24, 2017
u
q
[

1
Abstract
v
7 Due to the coupling of a quantum system to its environment energy
2 can be transfered between the two subsystems in both directions. In the
0 presentstudyweconsiderthisprocessinageneralframeworkforinterac-
6
tions with different properties and show how these properties govern the
0
exchange.
.
1
PACS numbers: 02.50.Ey, 02.50.Ga,03.65.Aa, 05.60.Gg, 42.50.Lz
0
7
Keywords: open quantum system, Markovian and non-Markovian systems,
1
energy exchange in open quantum systems.
:
v
i
X
r 1 Introduction
a
Quantum systems are generally never completely isolated but interact with an
environment with which they may exchange energy and other physical observ-
able quantities. Their properties are naturally affected by their coupling to the
external world. The understanding and control of the influence of an environ-
ment on a given physical system is of crucial importance in different fields of
physicsandintechnologicalapplicationsandledtoalargeamountofinvestiga-
tions [1,2,3,4,5]. Amongthequantitiesofinterestenergyexchangebetweena
∗E-mailaddress: [email protected]
†E-mailaddress: [email protected]
1
systemanditsenvironmentisofprimeimportance. Thetotalsystemcomposed
oftheconsideredsystemanditsenvironmentisclosed. Itconservesthephysical
observables,inparticularthetotalenergyitcontains. However,theexistenceof
an interactionbetween its two parts leads to possible exchanges between them,
energyandotherobservablescanbetransferedbetweentheminbothdirections.
In the present work we study the conditions under which this tranfer occurs.
In order to investigate the process a number of different approaches have
been developed [6, 7, 8, 9, 10]. They make use of the cumulant method as well
as other developments [11, 12, 13]. In the following we shall use the cumulant
method in order to work out the energy exchange and the speed at which this
exchange takes place [14].
In previous studies [17, 18, 20] we defined criteria which allow to classify
open quantum systems with respect to their behaviour in the presence of an
environment. In order to do so we used a general formulation which relies on
the examination of the properties of the density operator of the system and
its environment. The dynamical behaviour of this operator is governed by the
structure of the Hamiltonians of the system, its environment and the coupling
Hamiltonian which acts between them. The method introduces a general form
of the total density operator and avoids the determination of explicit solutions
of the equation of motion which governs the time evolution of the system. We
follow this formalism in order to work out different cases of physical interest
concerning the energy exchange process.
The analysis is presented in the following way. In section 2 we define the
energy exchange and its rate starting from the characteristic function which
generates these quantities in terms of its first and second moment. Section 3
introducesageneralformalexpressionofthedensityoperatorattheinitialtime
and the structure of the total Hamiltonian of the system, the environment and
theirinteraction. Theexpressionsoftheenergyexchangeandtheexchangerate
are explicitly written out. In section 4 we analyze and work out two different
cases. They exemplify the role of the interaction between the system and the
environmentwhich may or may not induce the time divisibility property of the
open system. Conclusions are drawn in section 5.
2 Energy transfer between S and E: the cumu-
lant approach
ConsiderasystemS coupledtoanenvironmentE. Thetimedependentdensity
operatorofthetotalsystemisρˆ (t)andS andE arecoupledbyaninteraction
SE
Hˆ . The interactionmaygenerateanenergyexchangebetweenthe twoparts.
SE
This quantity can be workedout by means of the cumulant method [14] which
was developed in a series of works, first for closed driven systems [10], later
extended to open systems, see f.i. [8, 9] and further approaches quoted in [14].
Define the modified density operator
2
ρ∼ (t,0)=Tr [Uˆ (t,0)ρˆ (0)Uˆ+ (t,0)] (1)
SE E η/2 SE −η/2
where Tr is the trace over the states in E space and
E
Uˆ (t,0)=exp(+iηHˆ )Uˆ(t,0)exp(−iηHˆ ) (2)
η SE SE
HereUˆ(t,0)isthetimeevolutionoperatoroftheinteractingtotalsystemS+E.
The characteristic function obtained from the generating function reads
χ(η)(t)=Tr (ρ∼ (t,0)) (3)
S SE
From χ(η)(t) one derives the energy exchange between the environmentand
the system
dχ(η)(t)
∆E(t)= (4)
d(iη)
|η=0
The speed at which the energy flows between the system and its environment
is given by
∂χ˙(η)(t)
V (t)= (5)
E ∂(iη)
|η=0
whereχ˙(η)(t) is the time derivativeof χη(t). Ifthe energyflows fromthe sytem
to the environment V (t) is positive and negative if the flow is reversed.
E
3 Energy flow and speed of energy exchange
3.1 The density operator
At time t=0 we choose the system to stay in a mixed state
|ψ(0)i=Xcik|iki (6)
ik
where the normalized states {|i i} are eigenstates of the Hamiltonian Hˆ . The
k S
environment is described in terms of its density matrix chosen as {|αid hα|}
α,α
where{d }arethestatisticalweightsofthedensitymatrixinadiagonalbasis
α,α
of states {α}.
Given these bases of states in S space and in E space the density operator
of the total system at time t=0 is written as [29]
3
ρˆ (0)=ρˆ (0)⊗ρˆ (0)
SE S E
ρˆS(0)=X|ikicikc∗ilhil|
k,l
ρˆE(0)=X|αidα,αhα| (7)
α
The density operator ρˆ describes a system in the absence of an interaction
SE
Hˆ at t = 0, hence in the absence of an initial entanglement between S and
SE
E.
The total Hamiltonian Hˆ reads
Hˆ =Hˆ +Hˆ +Hˆ (8)
S E SE
and
Hˆ |i i=ǫ |i i
S k ik k
Hˆ |γi=E |γi (9)
E γ
where {|i i} and {|γi} are the eigenvector bases of Hˆ and Hˆ . At time t the
k S E
evolution of S+E is given by
ρˆ (t)=Uˆ(t)ρˆ (0)Uˆ+(t) (10)
S+E S+E
where Uˆ(t,0)=e−iHˆt is the evolution operator of S+E.
3.2 Explicit expressions of the energy exchange and the
speed of the flow
Using the definitions givenin Eqs. (4) and(5) the energytransfer and speed of
the energy flow read
1
∆E(t)= Tr Tr {[Hˆ ,Uˆ(t,0)]ρˆ (0)Uˆ+(t,0)}+h.c. (11)
S E E S+E
2
and
1 d
V (t)= Tr Tr {[Hˆ , Uˆ(t,0)]ρˆ (0)Uˆ+(t,0)}+h.c. (12)
E 2 S E E dt S+E
Developing these expressions in the bases of states given above they read
4
1
∆E(t)= 2X X ci1c∗i2hjγ|[HˆE,e−iHˆt]|i1γ1i
j,γ i1,i2,γ1
d hi γ |e+iHˆt|jγi+h.c. (13)
γ1,γ1 2 1
and
(−i)
VE(t)= 2 X X ci1c∗i2dα1,α1
j,γ α1,i1,i2
X{hjγ|[HˆE,HˆSE]|j1,γ1ihj1,γ1|e−iHˆt|i1α1ihi2α1||e+iHˆt|jγi
j1,γ1
+hjγ|Hˆ|j γ ihj γ |[Hˆ ,e−iHˆt]|i α ihi α |e+iHˆt|jγi
1 1 1 1 E 1 1 2 1
−hjγ|Hˆe−iHˆt|i α ihi α ||[Hˆ ,e+iHˆt]|jγi}+h.c. (14)
1 1 2 1 E
4 Properties of the interaction Hamiltonian
Weusenowthe generalexpressionsofthe energytransferanditsexchangerate
in order to test the role of the interaction Hamiltonian in this process. Since
S and E are distinct different physical systems their Hamiltonians verify the
commutationrelation[Hˆ ,Hˆ ]=0. Formerwork [17]hasshownthatonemay
S E
consider two cases which are of special interest:
(a) [Hˆ ,Hˆ ]=0
E SE
(b) [Hˆ ,Hˆ ]=0
S SE
4.1 Case (a)
It has been shown elsewhere [17, 18] that if H and H commute the evo-
E SE
lution of the system S is characterized by the divisibility property which is a
specific property of Markovian systems. Since the Hamiltonian of the system
S commutes in practice with Hˆ it follows that Hˆ commutes with the whole
E E
Hamiltonian Hˆ, hence also Uˆ(t,0) and its derivatives. Going back to the ex-
pressions of ∆E(t) and V (t) it comes out that ∆E(t)=V (t)=0.
E E
There is no energy exchange between S and E in this case. The physical
explanationis the following: the divisibility property imposes that the environ-
ment stays in a fixed state at a fixed energy which blocks any possible transfer
of energy between the two parts of the total system S+E. This is due to the
fact that Hˆ is diagonal in E space, hence the considered state |γi stays the
SE
same over any time interval.
5
4.2 Evolution of the energy: an impurity immersed in a
bosonic condensate
We introduce the Hamiltonian of a fermionic impurity interacting with a Bose-
Einstein condensate [19]. It is given by Hˆ =Hˆ +Hˆ +Hˆ where
S E SE
HˆS =Xǫ~kc~+kc~k (15)
~k
1
HˆE =Xe~ka~+ka~k+ 2V X VB(~q)(~q)a~+k1a~+k1+q~a~+k2−q~a~k2a~k1 (16)
~k ~k1~k2q~
1
HˆSE = V X c~+k3+q~c~k4a~+k4−q~a~k3 (17)
~k3~k4q~
where [c,c+] and [a,a+] are fermion and boson annihilation and creation oper-
ators.
Weconsiderthecasewherethemomentumtransfer~q =0. Thenoneexpects
that there is no energy exchange between S and E. This is indeed so since a
simple calculation shows that [Hˆ ,Hˆ ]=0 in this case.
E SE
4.3 Case (b)
Start from the general expression of ∆E(t) given by Eq.(13).
Since[Hˆ ,Hˆ ]=0allthematrixelementsarediagonalinS spaceandthe
S SE
expression takes the form
∆E(t)=X|cj|2X(Eγ −Eγ1)hjγ|e−itHˆ|jγ1idγ1γ1hjγ1|e+itHˆ|jγi (18)
j γ,γ1
Inordertofollowtheevolutionof∆E(t)intimewedeterminetheexpression
of V (t). A somewhat lengthy but straightforward calculation leads to the
E
following expression:
V (t)=V(1)(t)+c.c.+V(2)(t)+c.c. (19)
E E E
where
VE(1)(t)=(−i)X|cj|2X(Eγ −Eγ1)hjγ|HˆSE|jγ1i
j γ,γ1
Xhjγ1|e−itHˆ|jγ2idγ2,γ2hjγ2|e+itHˆ|jγi (20)
γ2
6
andV(1)∗(t)itscomplexconjugate. Itcomesoutthatthec.c. V(1)∗(t)=V(1)(t)
E E E
which means that V(1)(t) is real. The second term reads
E
VE(2)(t)=(−i)X|cj|2X(Eγ +ǫj)X(Eγ −Eγ2)
j γγ1 γ2
hjγ |e−itHˆ|jγ id hjγ |e+itHˆ|jγi (21)
1 2 γ2,γ2 2
It is easy to see that V(2)(t)+c.c.=0, hence V (t)=2V(1)(t).
E E E
For t=0
∆E(0)=X|cj|2X(Eγ −Eγ1)hjγ|jγ1idγ1γ1hjγ1|jγi=
j γ,γ1
X|cj|2X(Eγ −Eγ1)δγγ1 (22)
j γ,γ1
Hence ∆E(0)=0 which could have been anticipated from the symmetry prop-
erty of the expression of ∆E(t).
But contrary to case (a) the energy tranfer is now different from zero. It
varieswith time hence ∆E(t) increasesor de decreasesasa function ofthe sign
of V (t). This is due to the fact that now many channels can open in E space
E
because Hˆ is no longer diagonal in this space.
SE
4.4 Evolution of the energy: two examples
Inordertoillustratethiscasewedeveloptwomodelsonwhichweexemplifythe
time dependence of the energy transfer between a system and its environment
when [Hˆ ,Hˆ ]=0.
S SE
• First example: we consider a model consisting of a 2-level state system
E, [|γi = |1i,|2i]. The Hamiltonian Hˆ decomposes into two parts Hˆ =
0
Hˆ +Hˆ and Hˆ . We consider the case where
S SE E
[Hˆ ,[Hˆ ,Hˆ ]]=[Hˆ ,[Hˆ ,Hˆ ]]=0 (23)
0 0 E E 0 E
and [Hˆ ,Hˆ ]=c1 where c is a number. Then
0 E
eHˆ0+HˆE =eHˆ0eHˆEec/2 (24)
and
ei(Hˆ0+HˆE)t =eiHˆ0teiHˆEtect2/2 (25)
7
The quantities which enter the expressions which follow are defined in
Appendix A.
The expressions of ∆E(t) and V (t) read
E
∆E(t)=ect2∆12X|cj|2(d22−d11)[a(j12)2(t)+b(j12)2(t)] (26)
j
where ∆ =E −E , d ,d the weights of the states in E space and
12 1 2 11 22
VE(t)=2ect2∆12X|cj|2[I1(j2)Re(h2|Ωˆjt)|1i+R1(j2)Im(h2|Ωˆj(t)|1i] (27)
j
with
Reh2|Ωˆ (t)|1i=a11(t)[a21(t)cos(∆ t)+b21(t)sin(∆ t)]d
j j j 12 j 12 11
+a22(t)a21(t)d
j 22
Imh2|Ωˆ (t)|1i=a 11(t)[−b21(t)cos(∆ t)+a21(t)sin(∆ t)]d
j j j 12 j 12 11
+b21(t)a22(t)d (28)
j j 22
Both ∆E(t) and V (t are oscillating functions of time. ∆E(t) keeps a
E
fixedsigndependingonthesignof∆ ,V (t)maychangesignwithtime.
12 E
The energy and the speed of the energy transfer decays to zero for real c
real and negative.
• Second example: we consider the Hamiltonian Hˆ = Hˆ +Hˆ +Hˆ
S E SE
which governs the coupling of a phonon field to the electron in the BCS
theory of superconductivity.
The total Hamiltonian of the electron-phonon system reads
HˆS =Xǫ~k1c~+k1c~k1 (29)
~k1
HˆE =X~ωq~aq+~aq~ (30)
q~
HˆSE =Vph−e(~q)X(a+−q~+aq~)c~+k2+q~c~k2 (31)
~k2q~
8
where V (~q) is the phonon-electron interaction, [a,a+] and [c,c+] are
ph−e
phononandelectronannihilationandcreationoperators. Weconsiderthe
case where the phonons evolve in the zero mode ~q =0.
Then Hˆ =~ω a+a and Hˆ =V (0) (a++a )c+c .
E 0 0 0 SE ph−e P~k2 0 0 ~k2 ~k2
The density operator of the total system S +E at t = 0 is given by the
expression
ρˆ(n0,n′0) = 1 |~kn ih~k′n′| (32)
(~k,~k′) 2π(n !n′!)1/2 0 0
0 0
Working out the commutation relation between Hˆ and Hˆ leads to
S SE
[Hˆ ,Hˆ ]=0, hence the evolution operator can be written as
S SE
e−iHˆt =e−iHˆSte−i(HˆE+HˆSE)t (33)
and in explicit form
e−iHˆt =e−itPνk=1ǫ~kc~+kc~ke−i[ω0a+0a0+νVph−e(0)(a+0+a0)]t (34)
whereν isthenumberofelectronstates. Thequantityofinterestconcerns
thetimedependenceof∆E(t)andV (t)givenbyEqs.(18)and(19),hence
E
the time evolution of the matrix elements of e−iHˆt
E~k(n~k0,n′0)(t)=h~kn0|e−itPνk=1ǫ~kc~+kc~ke−i[ω0a+0a0+νVph−e(0)(a+0+a0)]t|~kn′0i (35)
These matrix elements and their hermitic conjugates which enter the
expressions of ∆E(t) and V (t) can be worked out explicitly using the
E
Zassenhaus development [30], see Appendix B and [31].
They read
E(n0,n′0)(t)=N−1e−iǫkte−iω0n0tF(n0,n′0)(t) (36)
~k~k
with
F(n0,n′0)(t)= X X (−i)n0+n3(−1)n′0+n2−n4
n2≤n0,n2≤n3n4≤n3,n4≤n′0
n0!n′0!(n2!)2(n3!)2[α(t)n0+n3−2n2][ζ(t)n′0+n3−2n4] eΨ(t) (37)
(n )2(n )2(n −n )!(n −n )!(n −n )!(n′ −n )!
2 4 0 2 3 4 3 2 0 4
and the normalizationfactor N =2π(n !n′!)1/2. The functions α(t), ζ(t)
0 0
and Ψ(t) read
νV (0)sinω t
α(t)= ph−e 0 (38)
ω
0
9
ω [1−cosνV (0)t]
ζ(t)= 0 ph−e (39)
νV (0)
ph−e
Ψ(t)=−1[ν2Vp2h−e(0)sin2(ω0t) + ω02(1−cosνVph−e(0)t)2] (40)
2 ω2 ν2V2
0 ph−e
As one can see from these expressionsE(n0,n′0)(t) are oscillating functions
~k~k
of time which leads to the conclusion that the energy transfer and the
transfer velocity oscillate continuously and stay finite over any interval of
time.
5 Conclusions, remarks
In the present work we used a cumulant approach [14] in order to study the
energy transfer between a system and its environment and the rate at which
this transfer evolves in time.
If the Hamiltonian of the environment commutes with the interaction be-
tweenthesystemandtheenvironmentthereresultsanabsenceofenergytransfer
between the two subsystems. This can be explained in the following way. As
already seen in former work the commutation property is a sufficient condition
for divisibility in the time behaviour of an open system, one of the proper-
ties which characterize Markov processes [15, 16]. In this case it has been
shown [17, 18, 20] that the environment keeps in the energy state in which it
was at the origin of time and stays there over any interval of time. Hence the
energy of the environment is blocked in a fixed state so that it cannot feed the
system and does not receive energy from it. Time delays are correlated with
the possibility of the environment to jump between different states in closed
systems as well as in open ones (see f.i. [21, 22, 23] and references quoted in
there). This is the case in non-Markoviansystems [24, 25, 27, 28].
The experimental realization of the absence of energy transfer may be ob-
tained under different conditions:
• the strengthofthe interactioncanbe chosensuchthatitkeepsveryweak
and hence does not allow any possible jump to another level in the case
of a discrete environment spectrum.
• thetemperatureoftheenvironmentiskeptclosetozerosothattheground
state is the only accessible state.
• the commutationrelationbetween the environmentand the interactionis
rigorously verified which is the case discussed in the present work.
10