# Effect of transport coefficients on the time-dependence of density matrix

**ABSTRACT** For Lindblad's master equation of open quantum systems with a general quadratic form of the Hamiltonian, the propagator of the density matrix is analytically calculated by using path integral techniques. The time-dependent density matrix is applied to nuclear barrier penetration in heavy ion collisions with inverted oscillator and double-well potentials. The quantum mechanical decoherence of pairs of phase space histories in the propagator is studied and shown that the decoherence depends crucially on the transport coefficients. Comment: 13 pages, 4 figures, submitted to J.Phys.A

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**ABSTRACT:**Using a simple model potential, we study the effects of weak Markovian dissipation on the quantum arrival time. The interaction with the environment is incorporated into the dynamics through a Markovian master equation of Lindblad type, which allows us to compare time-of-arrival distributions and approximate crossing probabilities for different dissipation strengths and temperatures. We also establish a connection to an earlier study where quantum tunneling with dissipation was investigated, which leads us to some conclusions concerning the formulation of the continuity equation in the Lindblad theory.Physical Review A 01/2009; 80:052112. · 3.04 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The generalized Langevin approach is suggested to describe the capture inside of the Coulomb barrier of two heavy nuclei at bombarding energies near the barrier. The equations of motion for the relative distance (collective coordinate) between two interacting nuclei are consistent with the generalized quantum fluctuation-dissipation relations. The analytical expressions are derived for the time-dependent non-Markovian microscopic transport coefficients for the stable and unstable collective modes. The calculated results show that the quantum effects in the diffusion process increase with increasing friction or/and decreasing temperature. The capture probability inside of the Coulomb barrier is enhanced by the quantum noise at low energies near the barrier. An increase of the passing probability with dissipation is found at sub-barrier energies.Physical Review C 02/2008; 77(2). · 3.72 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**With the quantum diffusion approach, the probability of passing through the parabolic barrier and the quasistationary thermal decay rate from a metastable state are examined in the limit of linear coupling both in momentum and in coordinate between a collective subsystem and the environment. An increase of passing probability with friction coefficient is demonstrated to occur at subbarrier energies.Physical Review A 09/2011; 84(3). · 3.04 Impact Factor

Page 1

arXiv:cond-mat/9911015v1 2 Nov 1999

Effect of transport coefficients on the time-dependence

of density matrix

Yu.V.Palchikov1, G.G.Adamian1,2, N.V.Antonenko1,2and W.Scheid2

1Joint Institute for Nuclear Research, 141980 Dubna, Russia

2Institut f¨ ur Theoretische Physik der Justus–Liebig–Universit¨ at, D–35392 Giessen, Germany

(February 1, 2008)

Abstract

For Lindblad’s master equation of open quantum systems with a general

quadratic form of the Hamiltonian, the propagator of the density matrix is

analytically calculated by using path integral techniques. The time-dependent

density matrix is applied to nuclear barrier penetration in heavy ion collisions

with inverted oscillator and double-well potentials. The quantum mechanical

decoherence of pairs of phase space histories in the propagator is studied and

shown that the decoherence depends crucially on the transport coefficients.

PACS: 03.65.-w, 05.30.-d, 24.60.-k

Typeset using REVTEX

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I. INTRODUCTION

In many problems of nuclear physics and quantum optics, where one deals with open

quantum systems, the memory time of the environment is very short and a Markovian

approximation is suitable. Disregarding the averaging over the intrinsic degrees of freedom,

one can consider the open system starting from the general Markovian master equation for

the reduced density matrix of the collective degrees of freedom as given by Lindblad [1]

dˆ ρ(t)

dt

= −i

¯ h[ˆH0, ˆ ρ] +1

2¯ h

?

j

?

[ˆVjˆ ρ,ˆV+

j] + [ˆVj, ˆ ρˆV+

j]

?

.(1)

Here,ˆH0is the Hamiltonian of the collective subsystem andˆVjare operators acting in the

Hilbert space of the subsystem. The terms in the sum of Eq. (1) are responsible for the

friction and diffusion and supply the irreversibility of the dynamics of the open quantum

system. Omitting these terms we get a standard form for the evolution equation of the

density matrix for closed systems. This equation and similar equations were used e.g. in

Refs. [2–12].

Path integral methods are a conventional tool to describe open quantum systems [10–18].

Here, we use results of Strunz [10] elaborated with path integral techniques and derive

analytical expressions for the time-dependent density matrix for a general Hamiltonian of

quadratic form with an inverse oscillator potential which can be applied to the description

of fission and fusion through potential barriers in nuclear physics. The decoherence of pairs

of phase space trajectories will be studied in the semiclassical limit for different choices of

the effects of the environment on the system. As was shown in [11], the initial Gaussian

distribution remains to be Gaussian in an oscillator potential. We extend this statement

for any quadratic form of the Hamiltonian of the subsystem. By a direct numerical solution

of Eq.(1) we consider the evolution of the density matrix in time in a double-well potential

under various sets of transport coefficients. Such potentials are more useful and realistic to

investigate nuclear fission problems than inverse oscillator potentials.

II. PATH INTEGRAL PROPAGATOR AND DECOHERENCE

With the propagator G(q,q′,t;q0,q

matrix < q|ˆ ρ(t)|q′> (in coordinate representation) at any time from the initial one < q|ˆ ρ(t =

0)|q′>:

′

0,0) of the density matrix one can find the density

< q|ˆ ρ(t)|q′>=

?

dq0

?

dq

′

0G(q,q′,t;q0,q

′

0,0) < q0|ˆ ρ(t = 0)|q

′

0> .(2)

In the one-dimensional case an expression for the phase space path integral of the propagator

corresponding to (1) was derived in [10] as

G(q,q′,t;q0,q

′

0,0) =

(q,t)

?

(q0,0)

D[α]

(q′,t)

?

(q′

0,0)

D[α′]exp(i

¯ hS[α;α′]),

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S[α;α′] = S[q,p;q′,p′] =

t

?

0

dτ{˙ q(τ)p(τ) − Heff(q(τ),p(τ))}

−

t ?

0

dτ{˙ q′(τ)p′(τ) − H∗

eff(q′(τ),p′(τ))}

− i

?

j

t

?

0

dτ{Vj(q(τ),p(τ))V∗

j(q′(τ),p′(τ))},(3)

with phase space paths [α] = [q,p], where q and p are the position and momentum, respec-

tively. The effective Hamiltonian is given by

Heff= H0−i

2

?

j

|Vj|2.

Here, the quantities H0, |Vj|2, Vj and V∗

ˆV+

Choosing an inverse oscillator potential, we write the Hamiltonian of the collective sub-

system in a more general quadratic form

2mˆ p2−mω2

2

The environment operators are assumed as linear

ˆVj= Ajˆ p + Bjˆ q,

ˆV+

jare the Wigner transforms of the operatorsˆH0,

jˆVj,ˆVjandˆV+

j in (1), respectively.

ˆH0=

1

ˆ q2+µ

2(ˆ pˆ q + ˆ qˆ p).(4)

j = A∗

jˆ p + B∗

jˆ q,j = 1,2.(5)

As shown in [10] for the similar case of an harmonic oscillator, the integrals in (3) over the

momentum yield Gaussian integrals and can be evaluated. Then the propagator is reduced

to path integrals in coordinate space [10]:

G(q,q′,t;q0,q

′

0,0) =

q(t)

?

q0

D[q]

q′(t)

?

q′

0

D[q′]exp(i

¯ hS[q;q′]),

S[q;q′] = Scl[q] − Scl[q′] − i¯ hλt + Φ[q,q′] +i

2D[q,q′]2.(6)

In Eq. (6) the classical action of the isolated system Scl, the phase function Φ[q,q′] and the

square of the decohering amplitude D[q,q′] can be expressed as

Scl=

t ?

0

dτ

?1

2m˙ q2+m

2ω2q2

?

, (7)

Φ[q,q′] = mλ

t

?

0

dτ(˙ qq′− q ˙ q′) + mλp− λq

2

t ?

0

dτ(q′˙ q′− q ˙ q) − mλpλq

2

t ?

0

dτ(q2− q′2),(8)

D[q,q′]2=2

¯ h

(Dpp+ m2λ2

− 2m(Dpq+ mλpDqq)

pDqq+ 2mλpDpq)

t ?

0

dτ(q − q′)2

t ?

0

dτ(q − q′)(˙ q − ˙ q′) + m2Dqq

t ?

0

dτ(˙ q − ˙ q′)2

.(9)

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The quantum mechanical diffusion coefficients are Dqq =

¯ h

2

?

j|Aj|2for coordinate, Dpp=

jBj for the mixed case.

¯ h

2

damping rate λ = −Im?

DppDqq− D2

density matrix at any time. The values λp= λ + µ and λq= λ − µ (λp+ λq= 2λ) are fric-

tion coefficients for coordinate and momentum, respectively. Both position and momentum

undergo a direct damping and diffusion process in contrast to the classical case. If D[q,q′]

increases with time for q ?= q′, then the propagator suppresses the non-diagonal components

of the density matrix. Thus, the interference between different positions q and q′becomes

weaker.

Since Heff depends at most quadraticly on p and q, the path integrals are Gaussian.

In that case a semiclassical solution of the path integrals with the method of stationary

phases leads to an exact analytical evaluation of the propagator. First, equations of motion

along the path trajectories q(τ) and q′(τ) (complex trajectories) are calculated with the

condition of stationary phase δS[q,p;q′,p′] = 0 with S of Eq. (3). The following equations

for Q1= q + q′, Q2= q − q′, P1= p + p′and P2= p − p′result, which are solved with the

boundary conditions q = (q(0) = q0,q(t),q′(0) = q0′,q′(t)).

?

j|Bj|2for momentum and Dqp = −¯ h

2Re?

jA∗

The frictional

jA∗

jBj and the diffusion coefficients must satisfy the constraint:

pq≥ λ2¯ h2/4 with Dqq > 0, Dpp > 0 which secures the non-negativity of the

˙Q1

˙P1

˙Q2

˙P2

=

−λq

mω2

0

0

m−1

−λp

0

0

4iDpq

¯ h

4iDpp

¯ h

λp

mω2

−4iDqq

¯ h

−4iDpq

¯ h

m−1

λq

Q1

P1

Q2

P2

(10)

Next, the solutions q(τ) and q′(τ) of Eq.(10) depending on the parameters q0, q(t), q0′and

q′(t) are inserted into the action function S[q;q′] of Eq.(6) and integrated over τ. The square

of the decohering amplitude is found as:

D[q,q′]2= sinh−2[ψt](At(q0− q′

0)2− A−t(q(t) − q′(t))2+ Bt(q0− q′

0)(q(t) − q′(t))),(11)

where ψ =

?

ω2+ (λq− λp)2/4 and

At=Dpp− m(mω2Dqq+ (λq− λp)Dpq)

2¯ hλ

+ exp[2λt]ψ2Dpp− m(−2Dpqλp+ mDqq(ω2− 2λλp))

2¯ hλ(λpλq− ω2)

ψ

2¯ h(ω2− λpλq)sinh[2ψt]{Dpp+ m(2Dpqλp+ mDqq(ω2− λp(λq− λp))}

1

2¯ h(ω2− λpλq)cosh[2ψt]{−λDpp+ m[(−2Dpq+ m(λq− λp)Dqq)ψ2

− λ(−Dpq(λq− λp) + mDqq(ω2+ 0.5(λq− λp)2))]},

Bt=ψsinh[(ψ − λ)t]

¯ hλ(ω2− λpλq){ψ[−Dpp+ m(−2Dpqλp+ mDqq(ω2− 2λλp))]

− λ[Dpp+ m(2Dpqλp+ mDqq(ω2− λp(λq− λp)))]}

+

−

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−ψsinh[(ψ + λ)t]

¯ hλ(ω2− λpλq){ψ[−Dpp+ m(−2Dpqλp+ mDqq(ω2− 2λλp))]

+ λ[Dpp+ m(2Dpqλp+ mDqq(ω2− λp(λq− λp)))]}. (12)

Similar analytical expressions are obtained for the action Scland phase Φ[q,q′]. Then, the

propagator (6) is finally evaluated as

G(q,q′,t;q0,q

′

0,0) =

mψ

2π¯ hsinh(ψt)exp(λt)exp(iSR/¯ h)exp(−D[q,q′]2/(2¯ h)),

SR= Scl[q] − Scl[q′] + Φ[q,q′]

mω

2sinh(ψt){cosh(ψt − φ)[q2

=

0− q

′

0

2] + cosh(ψt + φ)[q2− q

′2]

− 2cosh(φ)cosh(λt)[q0q − q

′

0q′] − 2cosh(φ)sinh(λt)[q0q′− q

′

0q]},(13)

where sinhφ = (λq− λp)/(2ω). This propagator is correct for any quadratic Hamiltonian

and is a generalization of the results of Refs. [10–12] where propagators were obtained for

harmonic and inverted oscillators only. For the initial density matrix (¯ q(0) and ¯ p(0) are

mean values)

< q|ˆ ρ(0)|q′> = (2πσqq(0))−1/2

× exp[−

1

4σqq(0){(q − ¯ q(0))2+ (q′− ¯ q(0))2} −i

¯ h¯ p(0)(q′− q)],(14)

the density matrix at time t is calculated with (2) and (13) as follows

< q|ˆ ρ(t)|q′> =

1

?

2πσqq(t)

exp

?

−

1

2σqq(t)

?q + q′

2

− ¯ q(t)

?2

−

1

2¯ h2

?

σpp(t) −σ2

pq(t)

σqq(t)

?

(q − q′)2

+iσpq(t)

¯ hσqq(t)

?q + q′

2

− ¯ q(t)

?

(q − q′) +i

¯ h¯ p(t)(q − q′)

?

(15)

or in explicit form

< q|ˆ ρ(t)|q′> =

mψ

2sinh(ψt)exp(λt)

1

?

2πσqq(0)

1

?

4f3f6− f2

5

× exp

?

−−f2f4f5+ f1f2

5+ f2

4f3f6− f2

2f6+ f3(f2

4− 4f1f6)

5

?

, (16)

where

f1= sinh−2[ψt](−¯ h¯ q2(0)

2σqq(0)sinh2[ψt] +1

2A−t(q − q′)2

+1

2imω cosh[ψt + φ]sinh[ψt](q2− q′2))

f2= sinh−2[ψt](2iσqq(0)¯ p(0) + ¯ h¯ q(0)

2σqq(0)

sinh2[ψt] −1

2Bt(q − q′)

− imω cosh[φ]sinh[ψt](q cosh[λt] + q′sinh[λt]))

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f3= sinh−2[ψt]1

2(imω cosh[ψt − φ]sinh[ψt] −

¯ h

2σqq(0)sinh2[ψt] − At)

sinh2[ψt] +1

2Bt(q − q′)

f4= sinh−2[ψt](−2iσqq(0)¯ p(0) + ¯ h¯ q(0)

2σqq(0)

+ imω cosh[φ]sinh[ψt](q sinh[λt] + q′cosh[λt]))

f5= sinh−2[ψt]At

f6= −sinh−2[ψt]1

2(imω cosh[ψt − φ]sinh[ψt] +

¯ h

2σqq(0)sinh2[ψt] + At).(17)

Here, ¯ q(t) and ¯ p(t) are the mean values of ˆ q and ˆ p, respectively, and σqq(t), σpp(t) and σpq(t)

the corresponding variances [8,12]. Explicit expressions for these mean values and variances

are given in Ref. [12]. The diagonal part of the density matrix (15) yields a Gaussian

distribution at time t

ρ(q,t) =< q|ˆ ρ(t)|q >= (2πσqq(t))−1/2exp[−

1

2σqq(t)(q − ¯ q(t))2],(18)

where

¯ q(t) = e−λt

?

¯ q(0)

?

cosh(ψt) +λp− λq

ψ

sinh(ψt)

?

+

1

mψ¯ p(0)sinh(ψt)

?

,

σqq(t) =

1

2m2λ(ω2− λpλq)

?

1

2mω2[(λq− λp)C3+ 2C2ψ]sinh(2ψt)

?

m2(ω2− 2λpλ)Dqq− Dpp− 2mλpDpq

?

+ e−2λt

2C1

m(λq− λp)−

1

2mω2[(λq− λp)C2+ 2C3ψ]cosh(2ψt)

+

?

(19)

with the following notations

C1=mω2(λq− λp)

4ψ2

?

σqq(0) −

1

m2ω2σpp(0) +λq− λp

mω2σpq(0)

−1

λDqq+

1

4ψ2

1

m2ω2λDpp−(λq− λp)

?λq− λp

m

1

ω2− λpλq

1

2mψ

mω2λ

Dpq

?

,

C2=

(σpp(0) − m2ω2σqq(0)) + 4ω2σpq(0)

+

?2ω2− λpλq

m

m2ω2σqq(0) + σpp(0) +

[Dpp+ m2ω2Dqq] +λ2

q

mDpp+ λ2

pmω2Dqq+ 4λω2Dpq

??

,

C3= −

?

1

ω2− λpλq(λqDpp+ 2mω2Dpq+ m2ω2λpDqq)

?

.

For the values λp= λq= 0, Dpp= Dqq= Dpq= 0, σpp(0) = ¯ h2/(4σqq(0)) and σqp(0) = 0, we

obtain the same result with these expressions as in Refs. [19,20]. For λq= 0, Dpp= Dqq=

Dpq= 0 and σpp(0) = ¯ h2/(4σqq(0)) and σqp(0) = 0, our results coincide in the underdamped

limit with the results of Ref. [21], where tunneling was studied with the inverted Caldirola-

Kanai Hamiltonian. For λq= 0 and Dqq= 0, our result can be transformed to the one of

Ref. [22].

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III. CALCULATED RESULTS

An influence of the friction and diffusion coefficients on the tunneling was considered in

Refs. [11,12,23]. Here, we study the time-dependence of the decohering amplitude and the

non-diagonal part of the density matrix for different sets of the transport coefficients. In

order to demonstrate the effect of the diffusion and friction coefficients of the coordinate,

Dqqand λq, on the change of the distance between phase space trajectories, we take a simple

expression for the diffusion coefficients:

Dpp= (1 + κ)λm¯ hωeffcoth(¯ hωeff/(2kT))/2,

Dqq= (1 − κ)¯ hλcoth(¯ hωeff/(2kT))/(2mωeff),Dpq= 0. (20)

Here, κ is an adjustable parameter. The parameter ωeffcould be found from a microscopic

consideration of the open system. With κ = 1 and Dqq= Dpq= 0 we obtain the ”classic”

set of diffusion coefficients which does not preserve the non-negativity of the density matrix

at all times [5,7,8].

As an example we consider the relative motion of the two nuclei

the Coulomb barrier which is approximated by the inverted oscillator. Fig. 1 shows the

time-dependence of the density matrix ρ(q,q′) for κ = 0 and 1 in (20), λp = (1 + κ)λ

and λq = (1 − κ)λ. Since with κ = 0 and 1 the density matrix is practically diagonal

after a short time interval of about 5 × 10−22s, semiclassical methods work quite well in

heavy ion collisions. The density matrix becomes faster diagonal in the case κ = 1 than

for κ = 0. The time behaviour of the non-diagonal components of the density matrix

is evidently correlated with the time-dependence of the decoherence D which is shown in

Fig. 2. After a decrease of D during a short time interval the decoherence increases indicating

a depression of the interference between different states (trajectories). The decoherence

increases slowest for κ = 0 and more rapidly for higher temperatures. Further, Fig. 2 shows

that the decoherence amplitude decreases with increasing λq. This has the consequence that

the penetrability through the barrier increases due to a larger interference between different

states (trajectories) [11,12].

In order to show the role of Dqqin distorting the coherence between states, we compare

the time-dependence of D in Fig. 3 for κ = 0, 0.5 and 1 in (20) with λp= 2λ and λq= 0.

For times t > 5 × 10−22s, which are of interest for physical observables, the decoherence

increases fastly for Dqq= 0 (κ = 1). With Dqq?= 0 (κ < 1) the interference between different

states survives a longer time.

Eq. (1) can also be solved by rewriting it in a system of equations for the matrix elements

of ˆ ρ in some basis [23]. These equations can be numerically treated for arbitrary potentials.

With complete orthogonal set of basis functions |n > we obtain from Eq.(1) the system of

equations for the matrix elements of density matrix ˆ ρ:

76Ge and

170Er at

dρmn

dt

=

?

l

{−i

¯ h(< m|ˆH0|l > ρln− < l|ˆH0|n > ρml)

+ ρmlBln+ ρlnCml+

?

l′

ρll′Amll′n}, (21)

where the coefficients are defined as follows

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Bln=

?

?

l′

(D−

1∆+

ll′∆+

l′n+ D+

1

∗∆−

ll′∆−

l′n− D−

2∆−

ll′∆+

l′n− D+

2

∗∆+

ll′∆−

l′n),

Cml=

l′

(D+

1∆+

l′l∆+

ml′ + D+

1

∗∆−

l′l∆−

ml′ − D+

2∆−

l′l∆+

ml′ − D−

2

∗∆+

l′l∆−

ml′),

Amll′n= −(D−

+ (D+

1+ D+

2+ D+

1)∆+

∗)∆−

ml∆+

ml∆+

l′n− (D−

l′n.

1

∗+ D+

1

∗)∆−

ml∆−

l′n+ (D−

2+ D−

2

∗)∆+

ml∆−

l′n

2

(22)

Here, ∆−

and annihilation a operators, and D±

For the basis related to the eigenfunctions of harmonic oscillator with the frequency ω,

D1= (mωDqq− Dpp/mω + 2iDpq)/¯ h and D2= (mωDqq+ Dpp/mω)/¯ h. For calculations,

either the eigenfunctions of harmonic oscillator or the eigenfunctions of potential U(ˆ q) are

convenient as complete orthogonal set of basis functions |n > With the initial state of the

open system determined by the wave function Ψ(q) the initial density matrix is calculated

as ρmn(t = 0) =< m|Ψ >< Ψ|n >. With this initial condition we can solve Eq.(21) and find

the time dependence of average value F = Tr(ˆ ρ(t)ˆF) of any operatorˆF and of diagonal and

nondiagonal elements of the density matrix.

Let us consider a system with mass m=53m0(m0is the mass of a nucleon) in a symmetric

double-well potential

mn=< m|a|n > and ∆+

mn=< m|a+|n > are the matrix elements of the creation a+

1= 0.5(D1±0.5(λp−λq)), D±

2= 0.5(D1±0.5(λp+λq)).

U(q) = −8∆U

L2q2+16∆U

L4

q4

(23)

with ∆U=1.5 MeV, L=3 fm and start with an initial Gaussian state for the density matrix

with a variance σqq(0) = 0.14 fm2in the left well at ¯ q(0) = −1.5 fm. The calculated time-

dependence of ρ(q,q′) is presented in Fig. 4 for κ = 1 and κ = 0 in (20) with λp= 2λ and

λq = 0. The transition of the system to the right well mainly occurs along the direction

q = q′. At the same time the non-diagonal part of the density matrix is larger with Dqq?= 0

than with Dqq= 0. For Dqq?= 0, the distribution in the right well is wider and the transition

rate between the two wells is larger [23,24].

IV. SUMMARY

Using the path integral method for master equations of general Lindblad form for Marko-

vian open quantum systems, we obtained an analytical expression for the propagator of the

density matrix of a general quadratic Hamiltonian coupled linearly (in coordinate and mo-

mentum) with the environment. The time-dependent diagonal and nondiagonal elements

of the density matrix in coordinate representation were calculated as a function of different

sets of transport coefficients for the inverted oscillator and double-well potentials. At times

of interest for heavy ion collisions at the Coulomb barrier, the density matrix is practically

diagonal which justifies the use of semiclassical methods. The time behaviour of the decoher-

ence crucially depends on the choice of the friction and diffusion coefficients. With diffusion

coefficients preserving the non-negativity of the density matrix at any time, the decoherence

increases slower than in the classical case with Dqq= 0. Therefore, the penetrability of a

barrier is larger in the case of Dqq?= 0 due to a stronger coherence between different states.

G.G.A. is grateful to the Alexander von Humboldt-Stiftung (Bonn) for support. This

work was supported in part by DFG and RFBR.

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Page 9

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Phys. A 3 635; Isar A., Sandulescu A. and Scheid W. 1993 J. Math. Phys. 34 3887

[9] Antonenko N.V., Ivanova S.P., Jolos R.V. and Scheid W. 1994 J. Phys. G:

Nucl.Part.Phys. 20 (1994) 1447

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9

Page 10

FIGURES

-2

0

q

2

-2

0

2

q'

0.0

0.2

0.4

κ = 0

κ = 1

t = 0.0

|ρ|

-2

0

2

-2

0

2

0.0

0.2

0.4

t = 0.5

|ρ|

q'

q

-2

0

2

-2

0

2

q'

0.0

0.2

0.4

t = 2.5

|ρ|

q

-4

-2

0

2

4

-4-2024

0.0

0.2

0.4

t = 1.0

|ρ|

q'

q

-4

-2

0

2

4

-4-2024

0.0

0.2

0.4

t = 5.0

|ρ|

q'

q

FIG. 1. Calculated time-dependence of the module of the density matrix |ρ(q,q′)| in the inverted

oscillator potential, reproducing the Coulomb barrier in a76Ge+170Er collision, for κ = 1 (left side)

and 0 (right side) in (20), λp= (1+κ)λ and λq= (1−κ)λ. The parameters are ¯ q(0) = 0, ¯ p(0) = 0,

¯ hω = ¯ hωeff= 2.0 MeV, σqq(0) = 0.7 fm2, m = 53m0(m0is the mass of nucleon), ¯ hλ = 2 MeV

and T = 0 MeV. The initial density matrix is presented on the top. The time is given in units of

6.582 × 10−22s.

10

Page 11

0.0

0.2

0.4

0.6

0.8

0.0

0.2

0.4

0.6

0.8

0

10

20

30

40

T = 0 MeV

D-1

κ

t

0.0

0.2

0.4

0.6

0.8

0.0

0.2

0.4

0.6

0.8

0

10

20

T = 5 MeV

D-1

κ

t

FIG. 2. Dependence of the inverse decoherence D−1(units ¯ h−1/2) on time and κ, used in the

definitions of Dppand Dqqin (20), λp= (1 + κ)λ and λq= (1 − κ)λ at T = 0 and 5 MeV. The

initial values are q(0) − q′(0) = 0.01 fm and p(0) = p′(0) = 0. The other parameters are the same

as in Fig.1. The time is given in units of 6.582 × 10−22s.

11

Page 12

0.0

0.5

1.0

1.5

2.0

D

0.00.20.40.60.81.0

0.0

0.5

1.0

1.5

T = 5 MeV

T = 0 MeV

t

FIG. 3. Time-dependence of the decoherence D (units ¯ h1/2) for λp= 2λ, λq= 0, T = 0 and

5 MeV, and κ = 0 (solid lines), 0.5 (dashed lines) and 1 (dotted lines). The initial values are

q(0) − q′(0) = 0.01 fm and p(0) = p′(0) = 0. The other parameters are the same as in Fig.1. The

time is given in units of 6.582 × 10−22s.

12

Page 13

-2

0

2

-2

0

2

q'

0.0

0.5

1.0

κ = 0

κ = 1

t = 0 s

|ρ|

q

-2

0

2

-2

0

2

q'

0.0

0.2

0.4

0.6

t = 1.0×10

-20s

|ρ|

q

-2

0

2

-2

0

2

q'

0.0

0.2

0.4

0.6

t = 2.0×10

-20s

|ρ|

q

-2

0

2

-2

0

2

q'

0.0

0.2

0.4

0.6

|ρ|

q

-2

0

2

-2

0

2

q'

0.0

0.2

0.4

0.6

|ρ|

q

FIG. 4. Calculated time-dependence of the module of the density matrix |ρ(q,q′)| in a dou-

ble-well potential (see text) for κ = 1 (left side) and 0 (right side) in (20) at T = 0 MeV and

λp= 2λ, λq= 0. The initial Gaussian distribution (top part) with σqq(0) = 0.14 fm2, ¯ p(0) = 0

starts with ¯ q = −1.5 fm in the left well. The further parameters are ¯ hωeff= 2.0 MeV, ¯ hλ = 2

MeV and m = 53m0. The time is indicated.

13

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