Physics Letters A 368 (2007) 442–449
www.elsevier.com/locate/pla

The effect of dissipation on quantum transmission resonance
Kohkichi Konno a,∗ , Munehiro Nishida b , Satoshi Tanda a , Noriyuki Hatakenaka ca Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japanb Graduate School of Advanced Sciences of Matter, Hiroshima University, Higashi-Hiroshima 739-8530, Japanc Graduate School of Integrated Arts and Sciences, Hiroshima University, Higashi-Hiroshima 739-8521, Japan

Received 12 January 2007; received in revised form 16 April 2007; accepted 17 April 2007
Available online 19 April 2007
Communicated by P.R. Holland

Abstract
Quantum transmissions of a free particle passing through a rectangular potential barrier with dissipation are studied using a path decomposition
technique. Dissipative processes strongly suppress the transmission probability at resonance just above the barrier resulting in an unexpected
reduction of the mean traversal time through the potential barrier.
© 2007 Elsevier B.V. All rights reserved.
PACS: 03.65.Xp; 03.65.Yz; 03.75.Lm

1. Introduction
Quantum mechanics successfully describes physical processes at the microscopic scale and sometimes exhibits uniquecounter-intuitive phenomena. One such example is quantummechanical tunneling that cannot be explained in terms of classical mechanics. In search of an applicability for quantum mechanics on a macroscopic scale [1], the effect of dissipation
on quantum tunneling has been studied in respect to systems
such as superconducting devices [2–4] because dissipation isinherent and inevitable in macroscopic scale. The observationof quantum-tunneling-rate reduction was a first important result to be associated with quantum mechanics on a macroscopic
scale.
Quantum tunneling, however, is not the only intrinsic feature of quantum mechanics. Another unconventional example
in a classical sense is quantum reflection. Suppose that there
is a particle incident to a potential barrier, which has a heightslightly lower than the particle’s energy, as shown in Fig. 1(a).
According to classical mechanics, the particle overcomes the* Corresponding author.

E-mail addresses: konno@topology.coe.hokudai.ac.jp (K. Konno),
tanda@eng.hokudai.ac.jp (S. Tanda).
0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.physleta.2007.04.055

barrier and is never reflected back. In quantum mechanics, thereis the possibility that the particle cannot go over the barrier. This
leads to a remarkable phenomenon called a transmission resonance. That is due to multiple quantum reflections between
potential boundaries. In other words, the resonance occurs
due to interference associated with the back-and-forth motionsof the particle. Quantum tunneling only shows its quantummechanical feature at the moment of the tunneling event, while
the interference requires for a certain definite period to retain
quantum coherence based on the quantum-mechanical superposition of states. Therefore, quantum interference appears amore convincing quantum-mechanical effect than does quantum tunneling. Indeed, the effect of dissipation on the quantummechanical superposition of macroscopically distinguishable
states, so-called Schrödinger’s cat in a fundamental problemof quantum mechanics, has been studied along these lines inrespect to superconducting nanodevices [5,6] for example. Recently, this cat state has been utilized as a building block fora quantum computer in quantum information science. Furthermore, direct observations of matter wave interference revealed
that inevitable sources of dissipation for large molecules, e.g.
collisions to external molecules [7] or internal vibrations resulting in thermal photon radiations [8], play a key role forthe quantum-to-classical transition of “free” particle. Even thegravitational waves were suggested to become a source of de-K. Konno et al. / Physics Letters A 368 (2007) 442–449

443

Fig. 1. Schematic diagrams of (a) three decomposed regions for a rectangular potential barrier, (b) typical possible paths, (c) uncertainty of energy, and (d) uncertaintyof potential height, where the diagram (c) is equivalent to (d).

coherence of matter waves [9]. Thus, macroscopic transmission resonance originated from quantum interference of macroscopic object also provides an alternative platform to that ofquantum tunneling for testing the validity of quantum mechanics on a macroscopic scale.
The transmission of a particle through a rectangular potential barrier in the absence of dissipation has been well studied
within the framework of Schrödinger’s wave mechanics at the
textbook level. A simple extension for incorporating dissipation in that framework was made by Cai et al. [10] to deal
with the problem that an electron propagates above a quantum well with dissipation due to the electron–optical-phonon
interaction. They succeeded in revealing the electron-capture
process in a quantum well that involves a loss of electron energy
via phonon emission. However, in order to investigate the problem addressed here, we need an alternative approach in which
the influence of the particle motion on the environment is also
included. In particular, our interest is directed to the transmission resonance formed just above the potential barrier where theWentzel–Kramers–Brillouin (WKB) method adopted in previous studies [11,12] breaks down.
In this Letter, we employ a path decomposition expansion
method [13–16] based on the path-integral approach, and develop it to incorporate dissipative processes. Then we discuss
the effect of dissipation on quantum transmission resonance.
Since the resonance may be attributed to the back-and-forth motions of a particle between the potential boundaries, as pointed
out by Bohm in his seminal book [17], the particle will stayin region II for a longer time at resonance. Thus the resonancecould be characterized by the time spent in the potential barrier
region, i.e., the traversal time. Therefore, we investigate dissipative quantum transmission resonance in terms of traversal timeon the basis of Bohm’s interpretation.

This Letter is organized as follows. In Section 2, we formulate the transmission probability through a rectangular potential
barrier with dissipation by using a path decomposition technique. We also perform numerical calculations of the transmission probabilities. Then we will find the non-uniform reductionof transmission probability even though we assume an energyindependent damping processes. In Section 3, we introduce thetraversal time under dissipation to explain the unexpected reduction on the basis of Bohm’s interpretation. We also provide
further evidence for our interpretation by using traversal time
distribution, which results in an unexpected shortened mean traversal time. In Section 4, we provide a summary and propose apossible experimental setup.
2. Transmission probability in the presence of dissipation
2.1. Analytical description of transmission probability
We briefly review a path decomposition expansion developed by Auerbach and Kivelson [13]. This enables us to deal
with quantum transmission in terms of the path integral approach. In the path decomposition technique, the summationof possible paths is decomposed into certain groups as shownin Fig. 2. The first group is composed of all possible paths thatpass straight over regions I, II, and III in that order. The nextgroup includes all paths that go from region I to region II, thenreturn to region I once, and go from region II to III. In the sameway, one can consider all other groups. Thus, the total summation of paths is given by an infinite series of groups as shownin Fig. 2. Each group is completely expressed by the propagators K (I) (x  , x), K (II) (x  , x) and K (III) (x  , x) defined in the
restricted regions I, II and III, respectively. Since all possible
paths are taken for summation, no approximation is used in this444

K. Konno et al. / Physics Letters A 368 (2007) 442–449

Fig. 2. Illustration of the path decomposition technique. Several groups of summations of all possible paths are shown (see also text). Each group is classified
according to which regions the paths have passed over in terms of temporal development. Two examples of the possible paths are shown by solid and dashed curvesin each figure.

technique, the point of which prevails against the WKB method.
The propagator K(xT , x0 ; T ) from x0 (< a) at t = 0 to xT (> b)
at t = T is then decomposed as [13]iK(xT , x0 ; T )
T−t1
T


= dt1
dt2 iK (I) (a, x0 ; t1 )Σxa iK (II) (b, x; t2 )
0

(II)

0



× Σxb iK (III) (xT , x; T − t1 − t2 )
T−t1

T
+

dt1
0

dt2

T −t
 1 −t2dt3

T −t
1 −t2 −t3

dt4

0

0
0
 (II)
(I)a× iK (a, x0 ; t1 )Σx iK (a, x; t2 ) 
× Σxa iK (I) (a, x; t3 ) Σxa iK (II) (b, x; t4 )× Σxb iK (III) (xT , x; T − t1 − t2 − t3 − t4 )+ ···,(1)

where Σxa denotes a derivative operator defined by





h̄ ∂  (II)Σxa iK (II) (b, x; t) ≡ abiK (b, x; t)  .
2m ∂x
x=a

(2)

Here, m is the particle mass, and
1(a < b),ab ≡−1 (a > b)(3)(see Ref. [13] for details).
In the absence of dissipation, the propagator K (II) in region II is expressed as [18]K (II) (x  , x; t)

∞ 

m
im(2nd + x  − x)2exp=
2πi h̄t n=−∞
2h̄t
− exp

im((2n + 1)d − (x  − b) − (x − a))22h̄t,

negative n. Eq. (4) indicates that the propagator in region II
is essentially expressed by the summation of a free particle’s
propagator;
mim(x  − x)2
exp.
K0 (x  , t; x, 0) =(5)
2πi h̄t2h̄t(4)where d is defined by d = b − a, and n characterizes different classical paths in each term. The first term is composed of
all paths with even numbers of reflections at the boundaries of
x = a and x = b, and the second term corresponds to the odd
numbers of reflections. Paths that start to move in a negative
direction in relation to the initial position are characterized by

Now let us consider the propagator KD in the presence ofdissipation. Dissipation in quantum mechanics has long been
discussed since it cannot be included as a form of analytical mechanics. It remains unresolved. However, several aspects suchas dissipative quantum tunneling and quantum Brownian motion have been presented. Here we employ a phenomenologicalmodel successfully introduced by Caldeira and Leggett [19], todescribe dissipation in a study of quantum Brownian motion ofa particle in harmonic potential. They modeled an environment
as a set of a huge number of harmonic oscillators that produces
a classical equation of motion with dissipation. We apply theirmodel to a free particle coupled to the environment.
(II)
According to their model, the effect of the propagator KD
is included in the expression for the time evolution of the system
of interest coupled to environment described byρ(x, y, t) = dxi dyi J (x, y, t; xi , yi , 0)ρ(xi , yi , 0),(6)where J (x, y, t; x  , y  , 0) is the propagator for the density matrix ρ(x, y, t) = x|ψ(t)ψ(t)|y of the free particle, and the
autocorrelation of stochastic force Fcl (τ )Fcl (τ )Fcl (s) = 2ηkB T  δ(τ − s)(7)

is imposed on it. Indeed, in the absence of dissipation, the above
expression for a free particle includes the free particle’s propagator K0 asρ(x, y, t) = dxi dyi K0∗ (y, t; yi , 0)K0 (x, t; xi , 0)ρ(xi , yi , 0).
(8)In particular, when the initial state is given by x|ψ(0) = δ(x −x0 ), we simply haveρ(x, x, t) = K0∗ (x, t; x0 , 0)K0 (x, t; x0 , 0) =m.
2πi h̄t(9)

In the case of a free particle coupled to a set of harmonic
oscillators, with x|ψ(0) = δ(x − x0 ), the density matrix is

K. Konno et al. / Physics Letters A 368 (2007) 442–449

given bym
ρ(x, x, t) =(10)f (t),
2π h̄twithσ teγ t,f (t) ≡
(11)
sinh σ twhere
γ is a relaxation rate, and σ is defined by σ ≡2γ + (4γ Ω/π)2 . Here Ω is the cutoff frequency for thefrequency distribution of harmonic oscillators. The function
f (t) can be obtained by integrating the degrees of freedomof the environment and, therefore, includes the effect of theassociated harmonic oscillators. Since the cutoff frequency Ω
is sufficiently large compared with γ , the function f (t) becomes a monotonically decreasing function, which decays exponentially, i.e., f (t) ∝ e−(σ −γ )t . It should also be noted thatf (0) = 1 and f (t) > 0 for t > 0. In the limit of γ → 0, since
f (t) → 1, we retrieve the result for the non-dissipative case|K0 (x, t; x0 , 0)|2 = m/2π h̄t, where K0 is the free particle’s
propagator shown in Eq. (5). Therefore, we can consider the
propagator for the dissipative case to be effectively expressedas
KD (x, t; x0 , 0) = f (t)K0 (x, t; x0 , 0),(12)which gives the same result as Eq. (10).
The propagator in region II is obtained by summing up thefree particle’s propagators along possible classical paths. Thus,
we obtain the effective propagator in region II with dissipation
as

(II)
KD (x  , x; t) = f (t)K (II) (x  , x; t).
(13)
The full propagator, taking account of the dissipative effect, isgiven byiKD (xT , x0 ; T )
T−t1
T



dt2 f (t2 )iK (I) (a, x0 ; t1 )Σxa iK (II) (b, x; t2 )
= dt1
0


b

0


× Σx iK (III) (xT , x; T − t1 − t2 )+T −t
 1 −t2

T−t1

T
dt1
0

dt2

dt3

Utilizing the expressionf (Σi ti )∞ = dτ f (τ )δ(τ − Σi ti )
0

∞
=

dτ



1
f (τ )
2π0∞dω e−iω(τ −Σi ti ) ,

(16)

−∞

we finally obtain the Green function including the dissipative
effect as
GD (xT , x0 ; E) = wD (E, V0 )G0 (xT , x0 ; E),(17)

where G0 (xT , x0 ) is the Green function when the barrier is absent, and the transmission amplitude wD (E, V0 ) is given by∞wD (E, V0 ) =

dτ
0



∞
f (τ )−∞dω −iωτw(E, V0 − h̄ω). (18)e
2πHere, w(E, V0 ) is the transmission amplitude in the absence ofdissipationw(E, V0 ) = −2ikκe−ikd,(k 2 + κ 2 ) sin κd + 2ikκ cos κd

(19)

where
√

√
2mE
2m(E − V0 )
k≡
(20)
,
κ≡
.
h̄
h̄
The paths in an energy representation are introduced by the potential deviations in terms of an energy quantum h̄ω as shownin Fig. 1(c) and (d). Therefore, the transmission probability
including the effect of dissipation is given by |wD (E, V0 )|2 .
Eqs. (17) and (18) are our main result.
2.2. Numerical estimates of transmission probabilities
In our numerical calculations, we restrict ourselves to a constant γ damping for any E even though the relaxation rate γ
may depend on the incident energy E of the particle in general. Fig. 3 shows the transmission probabilities for different γ

dt4

0


a
× f (t2 + t4 )iK (a, x0 ; t1 )Σx iK (II) (a, x; t2 ) 
× Σxa iK (I) (a, x; t3 ) Σxa iK (II) (b, x; t4 )× Σxb iK (III) (xT , x; T − t1 − t2 − t3 − t4 )



0

T −t
1 −t2 −t3

445

0
(I)

+ ···,(14)

where since f (t) behaves as an exponentially decaying function, we have used the approximationf (t)f (t ) ≈ f (t + t ),whose iterative use leads to i f (ti ) ≈ f ( i ti ).
To convert Eq. (14) into an energy representation, we perform a Fourier transform of the propagator as∞GD (xT , x0 ; E) ≡ i

dT KD (xT , x0 ; T )e0iET /h̄.

(15)

Fig. 3. Transmission probabilities as a functionof E/V0 . γ τ∗ = 5 × 10−3 ,
1 × 10−3 and 0 are plotted, where τ∗ ≡ md 2 /2V0 . In the calculations, we
adopted d/λ0 = 5 and Ωτ∗ = 100.

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K. Konno et al. / Physics Letters A 368 (2007) 442–449

values. Two dimensionless parameters characterize the transmission probability, i.e., the particle energy normalized by thebypotential height, E/V0 , and the potential width normalized√
the typical scale of length, d/λ0 , where λ0 ≡ h̄/ 2mV0 . The
dashed curve shows the transmission probabilities for a nondissipative case. The oscillatory structure is caused by transmission resonance, which arises as a result of quantum reflection
at the barrier edges. From Eq. (19), perfect transmission occurs
whenever the barrier contains an integer number of a half wavelength, i.e., κd = nπ (n = 1, 2, . . .).
In the presence of dissipation, the transmission probabilities
are greatly suppressed. The blurred oscillatory structure implies
that the suppression is not uniform with respect to energy, evenfor the energy-independent relaxation rate γ . Indeed, significant suppression occurs around the resonance conditions.
3. Traversal Time through Barrier

Under the resonance (the constructive interference) conditions κd = nπ (n = 1, 2, . . .), Eq. (23) is reduced to the expressionτ  =(25)

Using the inequalityα+β  αβ (∀α, β > 0),(26)
2
we can prove that for any condition, τ  is larger than the classical traversal time τcl defined by τcl ≡ md/h̄κ. That is, τ   τcl .
As well, the imaginary part of the mean traversal time vanishes
under these conditions.
On the other hand, destructive interference occurs at κd =(n + 1/2)π . Under these conditions, the mean traversal time isτ  =

3.1. Traversal time and transmission probability

mdA.
2h̄kκ 2mB
2mkd
−i 2 .
h̄A
h̄κ A

(27)

In particular, we have

In a classical description transmission resonance can be attributed to the back-and-forth motion of a particle between theedges of the potential barrier. This will lead to a longer stay inregion II, equivalent to a longer traversal time. Hence we caninvestigate the transmission probabilities in the presence of dissipation in terms of the traversal time through region II.
In terms of the path integral formalism [14–16], the traversal
time has a distribution, because all possible paths are considered and a certain weight is assigned to each path in the path
integral. According to Fertig [16], the probability amplitude ofa particle spending time τ in region II is defined by
iS(C)/h̄ δ(τ − τ (C))
II
path:C e,
F (τ ) ≡(21)
iS(C)/h̄epath:C
where C denotes a path, S denotes the action, and τII (C) isthe traversal time through region II along path C. The denominator in Eq. (21) is by definition equivalent to the propagator
K(xT , x0 ; T ). The delta function in the numerator extracts thepaths with traversal time τ . In the following discussion, we assume a constant energy state within the limits of x0 → −∞ andxT → ∞.
First, we investigate the relationship between the transmission resonance and the traversal time in the absence of dissipation. The mean traversal time τ  is defined by  2mkd
Re τ  =
 τcl .
h̄A

(28)

Hence, the real part of τ  is smaller than the classical traversal
time for any case of destructive resonance.
Fig. 4 shows |τ | − τcl as a function of E/V0 . There areseveral peaks around the resonance conditions. The absolutevalue |τ | approaches the classical traversal time for larger E.
Thus, the particle experiences a longer traversal time under resonance conditions as a result of multiple quantum reflections
at the boundaries. This explains why non-uniform suppressionsof transmission probabilities in the presence of dissipation occur. That is, the longer traversal time leads to larger dissipation
because the damping factor f (t) depends on the traversal timethrough region II. The most significant reduction in the transmission probability is then expected to occur at around the firstresonance resulting from the longest traversal time compared tothe classical one as shown in Fig. 4.

∞τ  ≡

τ F (τ ) dτ.

(22)

0

In the case of a rectangular potential barrier it is given by [16]τ  =

m 2k Aκd − B sin κd cos κdh̄ κ B 2 sin2 κd + 4k 2 κ 2
m Bκd cos κd − A sin κd B+isin κd,h̄ B 2 sin2 κd + 4k 2 κ 2 κ 2

(23)

where
A ≡ k2 + κ 2,

B ≡ k2 − κ 2.

(24)

Fig. 4. Deviation of the mean traversal time from the classical traversal time,|τ | − τcl , in the absence of dissipation. In the calculation, we adopted
d/λ0 = 5. The resonance points are given by E/V0 = 1 + n2 π 2 /(d/λ0 )2
1.39, 2.58, 4.55, . . . .

K. Konno et al. / Physics Letters A 368 (2007) 442–449

447This can be seen in the formula of the transmission amplitude wD (E, V0 ) with dissipation Eq. (18). It can be rewrittenas∞wD (E, V0 ) = w(E, V0 )dτf (τ )F (τ ).

(29)

0

In the absence of dissipation,
 ∞we can reproduce the nondissipative expression since 0 dτ F (τ ) = 1. The functionf (τ ) is a positive-definite√ decaying function as mentioned
above. Hence, the factor f (τ ) in Eq. (29) is considered to
weaken the contribution from paths with a longer traversal timethrough region II. Therefore, paths with a shorter traversal time
mainly contribute to the probability amplitude in the presenceof dissipation.
3.2. Traversal time distribution in the presence of dissipation
The dissipative effect on the traversal time also appears intraversal time distribution. Based on the same formulation [16],we can evaluate the traversal time distribution in the presence ofdissipation by replacing w with wD . The probability amplitudeis then given by
FD (τ ) =1
wD (E, V0 )∞−∞dω −iωτwD (E, V0 − h̄ω).
e
2πCD (τ ) =4. Summary and discussiondτ  FD (τ  )01=
wD (E, V0 )

∞
dω
−∞

sin ωτ
wD (E, V0 − h̄ω),πωfaster than the curve for γ = 0. This means that the traversaltime distribution becomes narrower and the mean value becomes smaller when the dissipative effect is taken into account.
This reduced mean value arises from the path selection causedby the dissipative effect, whereby paths taking a longer time inregion II are effectively discarded. Therefore, the mean traversal time becomes shorter.

(30)

Note that the traversal time distribution is related to the potential variation given by the right-hand side in Eq. (30), which
implies an equivalence between Fig. 1(c) and Fig. 1(d). Aftersome calculations, we obtain√f (τ )F (τ ).
FD (τ ) =  ∞ √(31)f (τ )F (τ ) dτ0√
From the above-mentioned properties of f (τ ), the center ofthe distribution function FD (τ ) is relatively shifted to the direction of smaller τ when compared with F (τ )without dissi∞
pation. Namely, the mean traversal time τD  = 0 τ FD (τ ) dτis considered to become smaller than τ . This feature may be
counterintuitive with respect to the idea that dissipation causes
particle slow down.
This feature can be also found in the cumulative probability
amplitude CD (τ ) defined asτFig. 5. The absolute value of the cumulative probability amplitude CD (τ ). We
adopted E/V0 = 1.3, d/λ0 = 5, Ωτ∗ = 100 and γ τ∗ = 5 × 10−3 or 0.

(32)

which describes the probability amplitude for the traversal timetaking a value between 0 and τ . The function CD (τ ) asymptotically approaches unity for a larger τ . Fig. 5 shows examplesof the cumulative probability amplitude CD (τ ) for γ = 0 and
γ = 0. Two curves converge to 1 by definition as τ becomeslarger. In particular, CD (τ ) with a non-zero γ converges to 1

We have studied the effect of dissipation on the quantum
transmission of a particle through a rectangular potential barrier, especially focusing on transmission resonance. We extended the path decomposition method to incorporate the dissipative effect into the calculations of the quantum transmission. The transmission probabilities are always suppressed by
the effect of dissipation, especially at the first resonance because of the longer traversal time. As well, the mean traversal
time in the presence of dissipation becomes smaller than thatin a non-dissipative case. This is the result of path selections
due to dissipation. We have not restricted ourselves to a specific scale. Thus our theory is applicable to any scale, including
a macroscopic one. The study of transmission resonance of amacroscopic object is useful to test an applicability of quantum
mechanics on a macroscopic scale.
Finally, we discuss an experimental setup for such a macroscopic object to test our theory. Such an experiment could be realized in a specific macroscopic system. A promising candidate
is a system consisting of a fluxon in a long Josephson junction.
The fluxon is a topological soliton excitation with a quantum
unit of magnetic flux produced by a circulating supercurrent,i.e., a vortex, and is regarded as a single free macroscopic particle characterized by a huge number of microscopic degrees of
freedom [20]. It also behaves like a quantum particle [21] in amesoscopic Josephson junction with small capacitance per unit
area. In fact, the quantum tunneling of a fluxon has recently
been observed in a long annular Josephson junction [22].
Under these circumstances, a fluxon transmission experiment is possible. The potential barrier for the fluxon can bemade of a microshort [23], which is a part made of a thinner insulator than the other part. The study of fluxon transmission will

448

K. Konno et al. / Physics Letters A 368 (2007) 442–449

complement that of quantum tunneling in research on macroscopic quantum phenomena. Moreover, a fluxon transmission
experiment will also provide an important basis for implementing quantum computation in superconducting nanocircuits. Indeed, a qubit using superposition states of fluxons or breathers
has recently been considered [24].
Acknowledgements
This work was supported in part by a Grant-in-Aid for Scientific Research from The 21st Century COE Program “Topological Science and Technology”, by a Grant-in-Aid for Scientific Research (18540352) from the Ministry of Education
Culture, Sports, Science and Technology of Japan and by JSPS
KAKENHI (17740267). One author (K.K.) thanks Y. Asano for
useful conversations. The numerical calculations were carriedout on computers at YITP in Kyoto University.
Appendix A. Propagator and its Green’s function
In the energy representation, the propagator is expressed by
the Fourier transformation as∞G(xT , x0 ; E) ≡ i

dT K(xT , x0 ; T )eiET /h̄ .

(A.1)

0

Eq. (1) is now reexpressed in this representation as
G(xT , x0 ; E) = G(I) (a, x0 ; E)Σxa G(II) (b, x; E) Σxb G(III) (xT , x; E) + G(I) (a, x0 ; E)Σxa G(II) (a, x; E) Σxa G(I) (a, x; E) × Σxa G(II) (b, x; E) Σxb G(III) (xT , x; E) + · · · . (A.2)

In the case of the rectangular potential, the Green functions are
given by [16]G(I) (x  , x) = −2m −ik(x−a)
sin k(x  − a),eh̄k2m sin κ(x − a) sin κ(x  − b),sin κdh̄κ
2m ik(x  −a)sin k(x − a),eG(III) (x  , x) =h̄k
G(II) (x  , x) = −where x < x  , d ≡ b − a and√√2m(E − V0 )
2mE
k≡
,
κ≡
.
h̄
h̄

(A.3)

(A.4)

2ikκe−ikd.
(A.9)(k 2 + κ 2 ) sin κd + 2ikκ cos κd
The usual expression for the transmission probability is obtained as
4k 2 κ 2
w(E, V0 )2 =(A.10).
(k 2 − κ 2 )2 sin2 κd + 4k 2 κ 2Thus the exact transmission probability can also be obtainedusing the path decomposition expansion.
w(E, V0 ) = −Appendix B. Free particle coupled to harmonic oscillators
Caldeira and Leggett [19] published a result for the quantum
Brownian motion in a harmonic potential. The result is modified in the following way for a free particle,
J (xf , yf , t; xi , yi , 0)

N (t)
i
=
exp
K(t)(ζf ξf + ζi ξi ) − L(t)ζi ξf
π h̄
h̄
m
− N (t)ζf ξi − γ (ζf ξf − ζi ξi )2
1
× exp − A(t)ξf2 + B(t)ξf ξi + C(t)ξi2 ,
h̄
where
m
K(t) = σ coth σ t,
2m σ e−γ t
L(t) =,
2 sinh σ t
m σ eγ t,
N (t) =
2 sinh σ t
Ω
t
τ
h̄ω
2mγ
dω ω cothdτdsA(t) =π
2kB T 0×e0sinh2 σ t
Ω
t
τ
h̄ω
2mγ
dω ω cothdτ ds
B(t) = −π
2kB T 

(A.5)

(A.6)

(A.7)

where G0 (xT , x0 ) is the Green function when the barrier is absent
m ik(xT −x0 ),G0 (xT , x0 ) = −
(A.8)
e
i h̄k

×e

sinh2 σ t + sinh σ (τ − t) sinh σ s ,2mγC(t) =
π

Ω
0

×

0


γ (τ +s−t) cos ω(τ − s)h̄ωdω ω coth
2kB T 

(B.1)

(B.2)
(B.3)
(B.4)

0

γ (τ +s−2t) cos ω(τ − s) sinh σ τ sinh σ s0Here m is the particle mass. Using these Green functions,Eq. (A.2) can be calculated in the form [16]G(xT , x0 ; E) = w(E, V0 )G0 (xT , x0 ),and w(E, V0 ) is the transmission amplitude given by

,

(B.5)

0

sinh σ τ sinh σ (s − t)
(B.6)

t

τ
dτ

0

ds eγ (τ +s)

0cos ω(τ − s) sinh σ (τ − t) sinh σ (s − t)(B.7),
sinh2 σ twhere J (xf , yf , t; xi , yi , 0) is the propagator for the density matrixy, t) of the free particle, i.e., ρ(x, y, t) =  ρ(x,dx dy  J (x, y, t; x  , y  , 0)ρ(x  , y  , 0). Here, kB is the Boltzmann constant, T  is the temperature, ζ ≡ x + y, ξ ≡ x − y,and the boundary condition is given by ζ (0) = ζi , ζ (t) = ζf ,ξ(0) = ξi and ξ(t) = ξf .

K. Konno et al. / Physics Letters A 368 (2007) 442–449

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