Table Of ContentDissipative tunneling through a parabolic potential
in the Lindblad theory of open quantum systems
A. Isar (a), A. Sandulescu and W. Scheid
†‡ † ‡
Department of Theoretical Physics, Institute of Physics and Nuclear Engineering
7 †
0 Bucharest-Magurele, Romania
0
Institut fu¨r Theoretische Physik der Justus-Liebig-Universit¨at
2
‡
n Giessen, Germany
a
J Abstract
8
By using the Lindblad theory for open quantum systems, an analytical ex-
1
pression of the tunneling probability through an inverted parabola is obtained.
1 Thispenetrationprobabilitydependsontheenvironmentcoefficients. Itisshown
v
thatthetunnelingprobabilityincreaseswiththedissipationandthetemperature
1
3 of the thermal bath.
1
1
PACS numbers: 03.65.Bz, 05.30.-d, 05.40.Jc
0
7 (a) e-mail address: [email protected]
0
/
h
p 1 Introduction
-
t
n
a
Quantum tunneling with dissipation has been intensively investigated in the last two
u
q decades [1–12]. Very interesting is the discussion whether the dissipation suppresses or
:
v
enhances the quantum tunneling. Caldeira and Leggett [1, 2] concluded that dissipa-
i
X
tion tends to suppress quantum tunneling. Using a different method, Schmid obtained
r
a
similar results in Ref. [13]. Widom and Clark [14] considered a parabolic potential
barrier and found that dissipation enhances tunneling. Bruinsma and Bak [15] also
considered tunneling through a barrier and found that at zero temperature the tun-
neling rate can be either increased or decreased by dissipation. Leggett [16] considered
tunneling in the presence of an arbitrary dissipation mechanism and found that, nor-
mally, dissipation impedes tunneling, but he also found an anomalous case in which
dissipation assists the tunneling process. Razavy [17] considered tunneling in a sym-
metric double-well potential and concluded that dissipation can inhibit or suppress
tunneling. Fujikawa et al. [18] also considered tunneling in a double-well potential
and found an enhancement of tunneling. Harris [19] calculated the tunneling out of
a metastable state in the presence of an environment at zero temperature and found
that quantum tunneling is enhanced by dissipation. In [20], Yu considered the tun-
neling problem in an Ohmic dissipative system with inverted harmonic potential and
he showed that while the dissipation tends to suppress the tunneling, the Brownian
motion tends to enhance it. In a series of papers [21, 22], Ankerhold, Grabert and
Ingold have studied real time dynamics of a quantum system with a potential bar-
rier coupled to a heat bath environment, employing the path integral approach. The
conclusion drawn from these papers is that different authors have studied different
problems using different methods. They obtained results which in many cases present
an evident contradiction.
In the present paper we study the tunneling in the presence of a dissipative en-
vironment in the framework of the Lindblad theory for open quantum systems, based
on completely positive dynamical semigroups [23–25]. We extend the work done in
some previous papers [26, 27]. In [27] a similar problem was treated by using the path
integral method and numerical calculations. Our study can be applied to problems of
nuclear fragmentation, fission and fusion, considered as a tunneling process through
the nuclear barrier defined in the space of collective coordinates, like charge and mass
asymmetry or the distance between the fission fragments.
For the inverted harmonic potential, the tunneling problem in the framework of
the Lindblad theory can be solved exactly. In Sec. 2 we write the basic equations
of the Lindblad theory for open quantum systems and give results for the coordinate
and momentum expectation values and variances for the damped inverted harmonic
oscillator. Then in Sec. 3 we consider the penetration of a Gaussian wave packet
through the potential barrier and define the penetration probability. In Sec. 4 we
analyze its dependence on various dimensionless parameters which enter the theory
and show that the probability increases with the dissipation and the temperature of
the thermal bath. A summary is given in Sec. 5.
2 Quantum Markovian master equation in
Lindblad theory
The simplest dynamics for an open system which describes an irreversible process is
a semigroup of transformations introducing a preferred direction in time [23–25]. In
Lindblad’s axiomatic formalism of introducing dissipation in quantum mechanics, the
usual von Neumann-Liouville equation ruling the time evolution of closed quantum
systems is replaced by the following quantum master equation for the density operator
ρ(t) in the Schr¨odinger picture [24], which is the most general Markovian evolution
2
equation preserving the positivity, hermiticity and unit trace of ρ:
dρ(t) i 1
dt = −h¯[H,ρ(t)]+ 2h¯ (2Vjρ(t)Vj† −Vj†Vjρ(t)−ρ(t)Vj†Vj). (1)
j
X
Here H is the Hamiltonian operator of the system; Vj, Vj† are bounded operators on
the Hilbert space of the Hamiltonian and model the effect of the environment. We
make the basic assumption that the general form (1) of the master equation with a
bounded generator is also valid for an unbounded generator.
As usual, we define the two possible environment operators V and V , which are
1 2
assumed linear in momentum p and coordinate q, as follows [28, 29]:
V = a p+b q, j = 1,2, (2)
j j j
with a ,b complex numbers. The Hamiltonian H is chosen of the general form
j j
µ 1
H = H + (qp+pq), H = p2 +U(q), (3)
0 0
2 2m
where U(q) is the potential. With these choices and with the notations
h¯ h¯ h¯
D = a 2,D = b 2,D = D = Re a b ,λ = Im a b , (4)
qq 2 | j| pp 2 | j| pq qp −2 ∗j j − ∗j j
j=1,2 j=1,2 j=1,2 j=1,2
X X X X
where a and b denote the complex conjugate of a and b , respectively, the master
∗j ∗j j j
equation (1) takes the following form [28, 29]:
dρ i i i
= [H ,ρ] (λ+µ)[q,ρp+pρ]+ (λ µ)[p,ρq +qρ]
0
dt −h¯ − 2h¯ 2h¯ −
D D D
pp qq pq
[q,[q,ρ]] [p,[p,ρ]]+ ([q,[p,ρ]]+[p,[q,ρ]]). (5)
− h¯2 − h¯2 h¯2
Here the quantum diffusion coefficients D ,D , D and the dissipation constant λ
pp qq pq
satisfy the following fundamental constraints [28, 29]: D > 0,D > 0 and
pp qq
λ2h¯2
D D D2 . (6)
pp qq − pq ≥ 4
In the particular case when the asymptotic state is a Gibbs state
H0 H0
ρG( ) = e−kT/Tre−kT , (7)
∞
the coefficients for a harmonic oscillator potential have the following form [28, 29]:
λ+µ h¯ω λ µ h¯ h¯ω
D = h¯mωcoth , D = − coth , D = 0, (8)
pp qq pq
2 2kT 2 mω 2kT
3
where T is the temperature of the thermal bath. The fundamental constraint (6) is
satisfied only if λ > µ.
In the following we denote by σ the dispersion (variance) of the operator A, i.e.
AA
σ =< A2 > < A >2, where < A > σ = Tr(ρA) is the expectation value of the
AA A
− ≡
operator A and Trρ = 1. By σ = 1/2 < AB+BA > < A >< B > we denote the
AB
−
correlation of the operators A and B.
From the master equation (5) we obtain the following equations of motion for the
expectation values and variances of coordinate and momentum:
dσ (t) 1
q
= (λ µ)σ (t)+ σ (t), (9)
q p
dt − − m
dσ (t) dU(q)
p
= < > (λ+µ)σ (t) (10)
p
dt − dq −
and, respectively,
dσ (t) 2
qq
= 2(λ µ)σ (t)+ σ (t)+2D , (11)
qq pq qq
dt − − m
dσ dU(q) dU(q) dU(q)
pp
= 2(λ+µ)σ (t) < p+p > +2 < > σ (t)+2D , (12)
pp p pp
dt − − dq dq dq
dσ (t) dU(q) dU(q) 1
pq
= < q > + < > σ (t)+ σ (t) 2λσ (t)+2D . (13)
q pp pq pq
dt − dq dq m −
For the harmonic oscillator with the potential U(q) = mω2q2/2, the solutions of
these equations of motion are obtained in Refs. [28, 29]. In this paper we consider the
tunneling throughapotentialbarriergivenbyaninverted harmonicpotential(inverted
parabola) with
mω2
U(q) = q2. (14)
− 2
The Hamiltonian H in equation (3) with the potential (14) can be regarded as the
0
Hamiltonian of a harmonic oscillator with an imaginary frequency iω and the equa-
tions of motion (9)–(13) for this potential are formally obtained by performing the
replacement ω iω in the corresponding equations for the harmonic oscillator. These
→
equations of motions can be solved by using the same method as in references [28, 29]
for the harmonic oscillator. Contrary to the situation of the harmonic oscillator, where
we have two cases, overdamped and underdamped, for the inverted parabola such a
4
distinction does not exist. The solutions for the expectation values and variances of
coordinate and momentum coincide formally with the solutions corresponding to the
overdamped case of the harmonic oscillator. For the expectation value of the coordi-
nate and momentum we obtain with ν √ω2 +µ2 :
≡
µ 1
σ (t) = e λt((coshνt+ sinhνt)σ (0)+ sinhνtσ (0)), (15)
q − q p
ν mν
mω2 µ
σ (t) = e λt( sinhνtσ (0)+(coshνt sinhνt)σ (0)). (16)
p − q p
ν − ν
In the following we also need the solution for the variance σ , which is given by:
qq
e 2λt ∆
σ (t) = − ∆ [(µ2 +ν2)cosh2νt+2µνsinh2νt+ω2]+ pp(cosh2νt 1)
qq 2ν2 { qq m2 −
2
+ ∆ (µcosh2νt+νsinh2νt µ) +σ ( ), (17)
pq qq
m − } ∞
where ∆ = σ (0) σ ( ), ∆ = σ (0) σ ( ), ∆ = σ (0) σ ( ) and
qq qq qq pp pp pp pq pq pq
− ∞ − ∞ − ∞
1
σ ( ) = (m2(2λ(λ+µ) ω2)D +D +2m(λ+µ)D ), (18)
qq ∞ 2m2λ(λ2 ω2 µ2) − qq pp pq
− −
1
σ ( ) = ((mω)2ω2D +(2λ(λ µ) ω2)D +2mω2(λ µ)D ),(19)
pp ∞ 2λ(λ2 ω2 µ2) qq − − pp − pq
− −
1
σ ( ) = ((λ+µ)(mω)2D +(λ µ)D +2m(λ2 µ2)D ).(20)
pq ∞ 2mλ(λ2 ω2 µ2) qq − pp − pq
− −
Note that in the case λ > ν, σ (t ) = σ (t ) = 0 and σ (t ) = σ ( ).
q p qq qq
→ ∞ → ∞ → ∞ ∞
In the case λ < ν, σ (t ),σ (t ) and σ (t ) = .
q p qq
→ ∞ → ∞ → ±∞ → ∞ ∞
3 Tunneling through an inverted parabola
In order to calculate thetunneling probability throughthe inverted harmonic potential
(14),weassumethatinitiallythewavefunctionofthesystemisaGaussianwavepacket
centered at the left of the peak of the potential at q = 0, σ (0) < 0, with a momentum
q
σ (0) > 0 towards the potential barrier peak:
p
1 1 i
ψ(q) = exp[ (q σ (0))2 + σ (0)q]. (21)
(2πσ (0))1/4 −4σ (0) − q h¯ p
qq qq
Then the corresponding initial probability density is given by:
1 1
ρ(q,t = 0) = exp[ (q σ (0))2]. (22)
(2πσ (0))1/2 −2σ (0) − q
qq qq
5
Likein[30]–[32],wecantransformthemasterequation(5)forthedensityoperator
of a particle moving in the potential (14) of an inverted parabola into the following
Fokker-Planck equation satisfied by the Wigner distribution function W(q,p,t) :
∂W p ∂W ∂W ∂ ∂
= mω2q +(λ µ) (qW)+(λ+µ) (pW)
∂t −m ∂q − ∂p − ∂q ∂p
∂2W ∂2W ∂2W
+D +D +2D . (23)
qq ∂q2 pp ∂p2 pq∂p∂q
For an initial Gaussian Wigner function, the solution of Eq. (23) is
1
W(q,p,t) =
2π σ(t)
1 q
exp [σ (t)(q σ (t))2 +σ (t)(p σ (t))2 2σ (t)(q σ (t))(p σ (t))] ,(24)
pp q qq p pq q p
× {−2σ(t) − − − − − }
which represents the most general mixed squeezed states of Gaussian form. Here
σ (t), σ (t) and σ (t), σ (t), σ (t) are the expectation values and, respectively, the
q p qq pp pq
variances corresponding to the inverted parabola as given partly in Eqs. (15)–(17) and
σ(t) = σ (t)σ (t) σ (t)2. (25)
qq pp pq
−
Since the dynamics is quadratic, then according to known general results, the initial
Wigner function remains Gaussian. The density matrix can be obtained by the inverse
Fourier transform of the Wigner function:
i q+q
′
< q ρ q >= dpexp( p(q q ))W( ,p,t). (26)
′ ′
| | h¯ − 2
Z
Using Eq. (24), we get for the density matrix the following time evolution:
1 1 q+q
< q ρ q′ >= ( )21 exp[ ( ′ σq(t))2
| | 2πσ (t) −2σ (t) 2 −
qq qq
1 σ2 (t) iσ (t) q +q i
(σ (t) pq )(q q )2 + pq ( ′ σ (t))(q q )+ σ (t)(q q )].(27)
−2h¯2 pp − σ (t) − ′ h¯σ (t) 2 − q − ′ h¯ p − ′
qq qq
The initial Gaussian density matrix also remains Gaussian, centered around the clas-
sical path, i. e. σ (t) and σ (t) give the average time-dependent location of the system
q p
along its trajectory in phase space. The wave function starts as a Glauber wave packet
at t = 0 on the left-hand side of the barrier and evolves as a mixed squeezed state at a
later time. By putting q = q in Eq. (27), we obtain the following probability density
′
of finding the particle in the position q at the moment t:
1 1
ρ(q,t) = exp[ (q σ (t))2]. (28)
(2πσ (t))1/2 −2σ (t) − q
qq qq
6
This is a Gaussian distribution centered at σ (t), which describes the classical trajec-
q
tory of a particle initially at σ (0), with initial momentum σ (0) and variance σ (t).
q p qq
Using Eq. (28), the probability for the particle to pass to the right of position q
at time t is given by
1 (q σ (t))2
P(q,t) = ∞ρ(q ,t)dq = ∞ exp( ′ − q )dq . (29)
′ ′ ′
Zq Zq 2πσqq(t) − 2σqq(t)
q
We define the tunneling probability P(t) as the probability for the particle to be at
the right of the peak at q = 0 (beyond the barrier top): P(t) = P(q = 0,t). We obtain
1 1 σ (t)
P(t) = ∞ e u2du = (1 erf( q )), (30)
−
√π −σq(t) 2 − − 2σ (t)
Z√2σqq(t) qq
q
where erf(x) is the error function with erf(x) = erf( x) and erf( )=1.
− − ∞
From Eq. (30) we see that the probability P(t) depends only upon the classical
motion of the average value of coordinate (wave packet center) and the spreading of
the wave packet in the direction of the barrier. The final tunneling probability (barrier
penetrability) is given by taking the limit t in P(t). In the present calculations
→ ∞
we ignore the fact that a part of the wave packet has already tunneled through the
barrier at t = 0. In general, this probability has a negligible value, but, in principle,
in order to find the net penetration probability, it should be subtracted from the
tunneling probability at time t.
4 Evaluation of the penetration probability and
analysis in dimensionless variables
We will show that since σ (t) and σ (t) are both proportional to the same exponen-
q qq
tial factor as time approaches infinqity, their ratio in Eq. (30) approaches a finite limit,
which determines the final tunneling probability. Indeed, as t , we see from Eqs.
→ ∞
(15) and (17) that σ (t) and σ (t) behave like
q qq
e (λ ν)t
− −
σ (t) δ (31)
q
→ 2mν
and
e (λ ν)t
σ (t) ( − − )2∆+σ ( ), (32)
qq qq
→ 2mν ∞
where we have denoted
δ m(µ+ν)σ (0)+σ (0) (33)
q p
≡
7
and
∆ m2(µ+ν)2∆ +∆ +2m(µ+ν)∆ . (34)
qq pp pq
≡
Then we obtain the following finite limit as t :
→ ∞
σ (t) δ
q
(35)
√σ → √∆
qq
if λ < ν (and the limit 0 if λ > ν) and, therefore, the expression (30) leads to the final
penetration probability (P = P(t ))
→ ∞
1 δ
P = (1 erf( )) (36)
2 − −√2∆
if λ < ν and P = 1/2 if λ > ν. In the case λ > ν, σ (t) 0 if t , that is,
q
→ → ∞
the system is located around the barrier, σ (t) tends to a finite value σ ( ) for any
qq qq
∞
initial kinetic energy and in this case P(t) = 1/2. Let us consider the other case λ < ν.
For δ = 0 the trajectory tends to the top of the potential barrier and if δ is different
from 0, then σ (t) tends to or . The trajectory which starts on the left-hand
q
∞ −∞
side of the barrier (σ (0) < 0) with a positive initial momentum σ (0) > 0 will stay
q p
on the same side for δ < 0 (and then σ ( ) ) and will cross the barrier for
q
∞ → −∞
δ > 0, i. e. if the initial kinetic energy allows to overcome the barrier (and then
σ ( ) ). For a general µ and any λ < ν, the particle crosses the barrier when
q
∞ → ∞
σ (0) > m(µ+ν)σ (0) and will stay on the same side when σ (0) < m(µ+ν)σ (0).
p q p q
− −
The barrier penetrability is larger than 1/2 if the particle classically can overcome the
top of the barrier, it is smaller than 1/2 if the particle cannot cross the barrier and it
tends to 1/2 if the position uncertainty σ is very large.
qq
For µ = 0, the values of ∆ and δ become (ν = ω):
δ = mωσ (0)+σ (0) (37)
0 q p
and
m2ω2D +D +2mωD
∆ = m2ω2σ (0)+σ (0)+2mωσ (0) qq pp pq. (38)
0 qq pp pq
− λ ω
−
The particle crosses the barrier if σ (0) > mωσ (0) and λ < ω. In this case, if
p q
−
λ increases, then the ratio δ /(2∆ )1/2 and the penetration probability P decreases.
0 0
This means that if dissipation increases, then the probability P decreases up to a value
of 1/2. If σ (0) < mωσ (0), the particle can not cross the barrier. In this case the
p q
−
8
penetration probability P increases with the dissipation λ up to 1/2. At λ ω the
≥
wave packet sticks in the barrier region.
We now introduce the dimensionless variables: z the scaled initial position, v the
scaled initial momentum, ǫ the scaled dissipation coefficient and r the scaled inverse
wave packet size, defined as follows:
h¯
σq(0) σp(0) λ s2mω
z = , v = , ǫ = , r = . (39)
σqq(0) mωσq(0) ω σqq(0)
q q
With these notations and considering a thermal bath modeled by the coefficients of
the form (8), the penetration probability takes the following form for the case µ = 0:
1 δ 1 z(1+v)
0
P = (1 erf( )) = (1 erf( )). (40)
µ=0 2 − −√2∆0 2 − − 2ǫ h¯ω
√2 1+r4 r2coth
s − ǫ 1 2kT
−
We took into account that
σ (0)σ (0) = h¯2/4, (σ (0) = 0). (41)
qq pp pq
If µ = 0, then we have 0 < ǫ < 1.
For µ = 0, in the case of a thermal bath, the expression (34) takes the form
6
(σ (0) = 0)
pq
∆ = m2(µ+ν)2σ (0)+σ (0)
qq pp
h¯m ω2[λ2 +λ(µ+ν)+µν] h¯ω
µ(µ+ν)+ coth . (42)
− ω { λ2 ω2 µ2 } 2kT
− −
With the notations (39) and introducing also the notation
µ
γ = , (43)
ω
the penetration probability takes the following form:
1 δ
P = (1 erf( )), (44)
µ=0
6 2 − −√2∆
with
δ z(γ +√1+γ2 +v)
= .(45)
√∆ (ǫ2 γ2)√1+γ2 +ǫ h¯ω
[(γ +√1+γ2)2 +r4 2(γ +√1+γ2) − r2coth ]1/2
− ǫ2 γ2 1 2kT
− −
9
If µ = 0, the inequalities λ > µ (see Eq. (8) andλ < ν lead tothe following restrictions
6
on the dimensionless variables:
γ < ǫ < 1+γ2. (46)
q
The initial energy of the particle associated with the Gaussian wave packet (21) is
1 mω2 1 mω2
E =< H > = σ (0) σ (0)+ σ2(0) σ2(0)+µσ (0)σ (0) (47)
|t=0 2m pp − 2 qq 2m p − 2 q p q
and in terms of the dimensionless variables (39) it looks
h¯ω h¯µz2v
E = [r4 1+z2(v2 1)]+ . (48)
4r2 − − 2 r2
If E < 0 it is a sub-barrier initial energy and if E > 0 it is an energy above the
barrier. In terms of the same dimensionless variables, the condition that a classical
particle does not have enough initial kinetic energy to pass the potential barrier can
be written:
v > 1, if µ = 0 (49)
−
and
v > (γ + 1+γ2), if µ = 0. (50)
− 6
q
With these two conditions and by taking 0 < r 1 (which assures that the initial
≤
fluctuation energy is negative), the total initial energy E (47) is always negative. This
corresponds to the case of the sub-barrier energy, relevant to the quantum tunneling
problem. The examples provided in the following figures reflect just this situation.
For µ = 0, Figs. 1 and 2 show the dependence of the tunneling probability on the
scaled dissipation ǫ and the temperature T of the thermal bath, via coth(h¯ω/2kT), for
fixed values of the scaled initial position z, momentum v and wave packet size r.
In the next four figures, we consider µ = 0. Figs. 3 and 4 show the dependence
6
of the penetration probability on the scaled dissipation and on the parameter γ for a
fixed scaled initial position z, momentum v and wave packet size r at the temperature
T = 0.
In Figs. 5 and 6 we give the dependence of the penetration probability on the
scaled dissipation and temperature, at fixed values of z,v,r and γ. The presented
dependence of the penetration probability on these variables can be summarized in
the following conclusions:
1) When the scaled initial momentum v is increasing, then P is increasing up to
| |
1/2 if the particle does not have enough kinetic energy to pass the potential barrier.
10
