Abstract

1. Introduction
Quantum mechanics successfully describes physical proc- esses at t the microscopic scale and sometimes exhibits unique counter-intuitive phenomena. One such example is quantum- mechanical tunneling that cannot be explained in terms of clas- sical mechanics. In search of an applicability for quantum me- chanics on a macroscopic scale [1], the effect of dissipation on quantum tunneling has been studied in respect to systems such as s superconducting devices [2–4] because dissipation is inherent and inevitable in macroscopic scale. The observation of quantum-tunneling-rate reduction was a first important re- sult to be associated with quantum mechanics on a macroscopic scale. Quantum tunneling, however, is not the only intrinsic fea- ture 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 height slightly lower than the particle's energy, as shown in Fig. 1(a). According to classical mechanics, the particle overcomes the barrier and is never reflected back. In quantum mechanics, there is the possibility that the particle cannot go over the barrier. This leads to a remarkable phenomenon called a transmission res- onance . 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 motions of the particle. Quantum tunneling only shows its quantum- mechanical feature at t he moment of the tunneling event, while the interference requires for a certain definite period to retain quantum coherence based on the quantum-mechanical super- position of states. Therefore, quantum interference appears a more convincing quantum-mechanical effect than does quan- tum tunneling. Indeed, the effect of dissipation on the quantum- mechanical superposition of macroscopically distinguishable states, so-called Schrödinger's cat in a fundamental problem of quantum mechanics, has been studied along these lines in respect to superconducting nanodevices [5,6] for example. Re- cently, this cat state has been utilized as a building block for a quantum computer in quantum information science. Further- more, 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 re- sulting in thermal photon radiations [8], play a key role for the quantum-to-classical transition of "free" particle. Even the gravitational waves were suggested to become a source of de- coherence of matter waves [9]. Thus, macroscopic transmis- sion resonance originated from quantum interference of macro- scopic object also provides an alternative platform to that of quantum tunneling for testing the validity of quantum mechan- ics on a macroscopic scale. The transmission of a particle through a rectangular poten- tial 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 dissipa- tion in that framework was made by Cai et al. [10] to deal with the problem that an electron propagates above a quan- tum 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 prob- lem 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 transmis- sion resonance formed just above the potential barrier where the Wentzel–Kramers–Brillouin (WKB) method adopted in previ- ous studies [11,12] breaks down. In this Letter, we employ a path decomposition expansion method [13–16] based on the path-integral approach, and de- velop it t o 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 mo- tions of a particle between the potential boundaries, as pointed out by Bohm in his seminal book [17], the particle will stay in region II for a longer time at resonance. Thus the resonance could be characterized by the time spent in the potential barrier region, i.e., the traversal time. Therefore, we investigate dissipa- tive quantum transmission resonance in terms of traversal time on the basis of Bohm’s interpretation. This Letter is organized as follows. In Section 2, we formu- late the transmission probability through a rectangular potential barrier with dissipation by using a path decomposition tech- nique. We also perform numerical calculations of the transmis- sion probabilities. Then we will find the non-uniform reduction of transmission probability even though we assume an energy- independent damping processes. In Section 3, we introduce the traversal time under dissipation to explain the unexpected re- duction 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 tra- versal time. In Section 4, we provide a summary and propose a possible experimental setup.
2. Transmission probability in the presence of dissipation
2.1. Analytical description of transmission probability
We briefly review a path decomposition expansion devel- oped by Auerbach and Kivelson [13]. This enables us to deal with quantum transmission in terms of the path integral ap- proach. In the path decomposition technique, the summation of possible paths is decomposed into certain groups as shown in Fig. 2. The first group is composed of all possible paths that pass straight over regions I, II, and III in that order. The next group includes all paths that go from region I to region II, then return to region I once, and go from region II to III. In the same way, one can consider all other groups. Thus, the total summa- tion of paths is given by an infinite series of groups as shown in Fig. 2. Each group is completely expressed by the propa- gators K (1)(x',x), K (11)(x',x) and K (111)(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 technique, the point of which prevails against the WKB method. The propagator K ( x T , x 0 ; T ) from x 0 << a ) at t = 0 to x T (> b ) at t = T is then decomposed as [13]
where Σ α  denotes a derivative operator defined by Here, m is the particle mass, and
(see Ref. [13] for details). In the absence of dissipation, the propagator K (ll) in re- gion II is expressed as [18]
where d is defined by d = b − a , and n characterizes differ- ent 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 negative n . Eq. (4) indicates that the propagator in region II is essentially expressed by the summation of a free particle's propagator;
(5) Now let us consider the propagator K (II) in the presence of dissipation. Dissipation in quantum mechanics has long been discussed since it cannot be included as a form of analytical me- chanics. It remains unresolved. However, several aspects such as dissipative quantum tunneling and quantum Brownian mo- tion have been presented. Here we employ a phenomenological model successfully introduced by Caldeira and Leggett [19], to describe dissipation in a study of quantum Brownian motion of a 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 their model to a free particle coupled to the environment. According to their model, the effect of the propagator Ki is included in the expression for the time evolution of the system of interest coupled to environment described by
where J (x, y,t; x', y', 0) is the propagator for the density ma- trix ρ(x, y, t) = ⟨ x | ψ (t) 〈 ψ (t) 〉 of the free particle, and the autocorrelation of stochastic force F c1 (τ)
is imposed on it. Indeed, in the absence of dissipation, the above expression for a free particle includes the free particle's propa- gator Ko as
In particular, when the initial state is given by ⟨ x | ψ (0)} = δ(x xo), we simply have
In the case of a free particle coupled to a set of harmonic oscillators, with 〈 x 〉 (0) ＝ δ ( x − x 0 ), the density matrix is
given by
Utilizing the expression
we finally obtain the Green function including the dissipative effect as
where G 0 ( x T , x 0 ) is the Green function when the barrier is ab- sent, and the transmission amplitude wD(E,Vo) is given by
Here, w(E , V 0 ) is the transmission amplitude in the absence of dissipation
where
The paths in an energy representation are introduced by the po- tential deviations in terms of an energy quantum  ĥω as shown in Fig. 1(c) and (d). Therefore, the transmission probability including the effect of dissipation is given by | w D ( E , V 0 ) | 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 con- stant γ  damping for any E even though the relaxation rate γ may depend on the incident energy E of the particle in gen- eral. Fig. 3 shows the transmission probabilities for different γ Fig. 3. Transmission probabilities as a function of E/Vo. γτ* = 5 × 10 −3, 1 × 10~3 and 0 are plotted, where τ。 ≡ √md²/2V 0. In the calculations, we adopted d / λ 0 = 5 and Ω τ * = 100.
where γ is a relaxation rate, and σ is defined by σ ≡ √γ² + (4γΩ/π)². Here Ω  is  the  cutoff frequency for the frequency distribution of harmonic oscillators. The function f ( t ) can be obtained by integrating the degrees of freedom of the environment and, therefore, includes the effect of the associated harmonic oscillators. Since the cutoff frequency Ω is sufficiently large compared with γ , the function f(t) be- comes a monotonically decreasing function, which decays ex- ponentially, i.e., f (t) å e-(σ-γ) . It should also be noted that f (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 |K 0 (x, t; x 0, 0)|² = m /2π ht, where K 0 is the free particle's propagator shown in Eq. (5). Therefore, we can consider the propagator for the dissipative case to be effectively expressed as
which gives the same result as Eq. (10). The propagator in region II is obtained by summing up the free particle's propagators along possible classical paths. Thus, we obtain the effective propagator in region II with dissipation as
The full propagator, taking account of the dissipative effect, is given by
where since f ( t ) behaves as an exponentially decaying func- tion, we have used the approximation f(t)f(t') ≈ f(t + t'), whose iterative use leads to Π i f (t i ) ≈ f ( ∑ i t i ). To convert Eq. (14) into an energy representation, we per- form a Fourier transform of the propagator as
(11) values. Two dimensionless parameters characterize the trans- mission probability, i.e., the particle energy normalized by the potential height, E / V 0 , and the potential width normalized by the typical scale of length, d/λ 0 , where λ 0 ≡ ћ/ √2mV 0 . The dashed curve shows the transmission probabilities for a non- dissipative case. The oscillatory structure is caused by transmis- sion 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 wave- length, i.e., κd = ππ (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, even for the energy-independent relaxation rate γ . Indeed, signifi- cant suppression occurs around the resonance conditions.
3.  Traversal Time through Barrier
3.1. Traversal time and transmission probability In a classical description transmission resonance can be at- tributed to the back-and-forth motion of a particle between the edges of the potential barrier. This will lead to a longer stay in region II, equivalent to a longer traversal time. Hence we can investigate the transmission probabilities in the presence of dis- sipation 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 consid- ered and a certain weight is assigned to each path in the path integral. According to Fertig [16], the probability amplitude of a particle spending time τ in region II is defined by
where C denotes a path, S denotes the action, and τ II (C) is the traversal time through region II along path C . The denom- inator in Eq. (21) is by definition equivalent to the propagator K ( x 7 , x 0 ; T ). The delta function in the numerator extracts the paths with traversal time τ . In the following discussion, we as- sume a constant energy state within the limits of x 0 → 一∞ and XT → ∞. First, we investigate the relationship between the transmis- sion resonance and the traversal time in the absence of dissipa- tion. The mean traversal time ⟨τ⟩ is defined by
In the case of a rectangular potential barrier it is given by [16]
where (24) Under the resonance (the constructive interference) condi- tions κd = ππ (n = 1, 2, . . ), Eq. (23) is reduced to the expres- sion
we can prove that for any condition, {τ} is larger than the classi- cal traversal time τ c1 defined by τ c1 ≡ md / hκ . That is, 〈τ〉 ≥ τ c1. 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
Hence, the real part of ⟨τ⟩ is smaller than the classical traversal time for any case of destructive resonance. Fig. 4 shows | { τ } | — τ c | as a function of E / V 0 . There are several peaks around the resonance conditions. The absolute value |<τ}| approaches the classical traversal time for larger E . Thus, the particle experiences a longer traversal time under res- onance conditions as a result of multiple quantum reflections at the boundaries. This explains why non-uniform suppressions of transmission probabilities in the presence of dissipation oc- cur. That is, the longer traversal time leads to larger dissipation because the damping factor f ( t ) depends on the traversal time through region II. The most significant reduction in the trans- mission probability is then expected to occur at around the first resonance resulting from the longest traversal time compared to the classical one as shown in Fig. 4.
This can be seen in the formula of the transmission ampli- tude wD(E, Vo) with dissipation Eq. (18). It can be rewritten as
In the absence of dissipation, we can reproduce the non- dissipative expression since f ∞ d τ F ( τ ) = 1. The function f(τ) is a positive-definite decaying function as mentioned above. Hence, the factor √ (τ) in Eq. (29) is considered to weaken the contribution from paths with a longer traversal time through region II. Therefore, paths with a shorter traversal time mainly contribute to the probability amplitude in the presence of dissipation.
3.2. Traversal time distribution in the presence of dissipation
The dissipative effect on the traversal time also appears in traversal time distribution. Based on the same formulation [16], we can evaluate the traversal time distribution in the presence of dissipation by replacing w with wD . The probability amplitude is then given by
Note that the traversal time distribution is related to the poten- tial variation given by the right-hand side in Eq. (30), which implies an equivalence between Fig. 1(c) and Fig. 1(d). After some calculations, we obtain
From the above-mentioned properties of  // ( τ ), the center of the distribution function FD(τ) is relatively shifted to the di- rection of smaller τ when compared with F(τ) without dissi- pation. Namely, the mean traversal time ⟨τD⟩ = ∫∞ τ FD(τ) dτ is 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 as
which describes the probability amplitude for the traversal time taking a value between 0 and τ . The function CD(τ) asymptot- ically approaches unity for a larger τ . Fig. 5 shows examples of the cumulative probability amplitude CD(τ) for γ ≠ 0 and γ = 0. Two curves converge to 1 by definition as τ becomes larger. In particular, CD(t) with a non-zero γ converges to 1 faster than the curve for γ = 0. This means that the traversal- time distribution becomes narrower and the mean value be- comes smaller when the dissipative effect is taken into account. This reduced mean value arises from the path selection caused by the dissipative effect, whereby paths taking a longer time in region II are effectively discarded. Therefore, the mean traver- sal time becomes shorter.
4. Summary and discussion
We have studied the effect of dissipation on the quantum transmission of a particle through a rectangular potential bar- rier, especially focusing on transmission resonance. We ex- tended the path decomposition method to incorporate the dis- sipative effect into the calculations of the quantum transmis- sion. The transmission probabilities are always suppressed by the effect of dissipation, especially at the first resonance be- cause of the longer traversal time. As well, the mean traversal time in the presence of dissipation becomes smaller than that in a non-dissipative case. This is the result of path selections due to dissipation. We have not restricted ourselves to a spe- cific scale. Thus our theory is applicable to any scale, including a macroscopic one. The study of transmission resonance of a macroscopic object is useful to test an applicability of quantum mechanics on a macroscopic scale. Finally, we discuss an experimental setup for such a macro- scopic object to test our theory. Such an experiment could be re- alized 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 par- ticle characterized by a huge number of microscopic degrees of freedom [20]. It also behaves like a quantum particle [21] in a mesoscopic 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 experi- ment is possible. The potential barrier for the fluxon can be made of a microshort [23] , which is a part made of a thinner in- sulator than the other part. The study of fluxon transmission will complement that of quantum tunneling in research on macro- scopic quantum phenomena. Moreover, a fluxon transmission experiment will also provide an important basis for implement- ing quantum computation in superconducting nanocircuits. In- deed, 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 Sci- entific Research from The 21st Century COE Program "Topo- logical Science and Technology", by a Grant-in-Aid for Sci- entific 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 carried out 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
Eq. (1) is now reexpressed in this representation as
given by [16]
where x < x ', d ≡ b
Here m is the particle mass. Using these Green functions, Eq. (A.2) can be calculated in the form [16]
where G 0 ( x τ , x 0 ) is the Green function when the barrier is ab- sent
and w ( E , V 0 ) is the transmission amplitude given by
using the path decomposition expansion.
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 modi- fied in the following way for a free particle,
trix ρ(x, y, t) of the free particle, i.e., ρ(x, y, t) = f dx' dy' J(x, y, t; x', y', 0)ρ(x', y', 0). Here, kB is the Boltz- mann 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.