Shu Zheng, Hao Xiaolei, Li Weidong, Chen Jing. General way to define tunneling time. Chinese Physics B, 2019, 28(5): 050301
Permissions
General way to define tunneling time
Shu Zheng1, Hao Xiaolei2, †, Li Weidong2, Chen Jing1, 3, 4, ‡
Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
Institute of Theoretical Physics and Department of Physics, State Key Laboratory of Quantum Optics and Quantum Optics Devices, Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China
High Energy Density Physics Simulation (HEDPS), Center for Applied Physics and Technology, Peking University, Beijing 100084, China
Center for Advanced Material Diagnostic Technology, ShenzhenTechnology University, Shenzhen 518118, China
Project supported by the National Key Research and Development Program of China (Grant No. 2016YFA0401100) and the National Natural Science Foundation of China (Grant Nos. 11425414, 11504215, and 11874246).
Abstract
Abstract
With the development of attosecond science, tunneling time can now be measured experimentally with the attoclock technique. However, there are many different theoretical definitions of tunneling time and no consensus has been achieved. Here, we bridge the relationship between different definitions of tunneling time based on a quantum travel time in one-dimensional rectangular barrier tunneling problem. We find that the real quantum travel time tRe is equal to the Bohmian time tBohmian, which is related to the resonance lifetime of a bound state. The total quantum travel time τt can perfectly retrieve the transversal time tx and the Büttiker–Landauer time τBL in two opposite limits, regardless of the particle energy.
The tunneling time problem is almost as old as quantum mechanics itself[1] and has been a subject of intense theoretical debate for many years.[2–15] Recently, progress in attosecond science has allowed for a measurement of tunneling time in the so-called attoclock experiments.[16–26] However, there are considerable controversies in the interpretation of the attoclock experiments. Some investigations have supported a nonzero tunneling delay time,[17,19,26] while others supported a zero tunneling delay time.[27–30]
On the theoretical side, there is still no consensus on the definition of tunneling time. The Larmor clock was proposed by Baz’[3] and Rybachenko[4] to measure tunneling time. This idea is to use spin polarized electrons and a potential barrier with a constant magnetic field inside, considering the rotation of the spin in the plane that is perpendicular to the magnetic field can define a time. Büttiker[6] recognized that the Larmor time is equal to the dwell time calculated by the ratio of integrated probability density over the barrier region to the incident flux. The identity relation between Larmor time and dwell time is also fulfilled for a potential barrier of arbitrary shape[11] and in the relativistic case.[10] Büttiker also pointed out that the main effect of the magnetic field in lamor precession is to align the spin with the field. Thus, the barrier preferentially transmits a particle with spin parallel to the magnetic field, which can be described by the Büttiker–Landauer time.[6] The Büttiker–Landauer time τBL[5] describes the time spent by the particle to travel from the entrance point x1 to the exit point x2 under the barrier V(x). Within the Wentzel–Kramers–Brillouin (WKB) approximation, the general definition of Büttiker–Landauer time is ,[5] where is the momentum of particle with the kinetic energy E under the barrier. Afterwards, Sokolovski and Baskin constructed a single complex time by Feynman path-integral technique, which can elegantly combine the Larmor time and the Büttiker–Landauer time.[31] Very recently, we proposed a definition of quantum travel time which provides a reasonable interpretation of the tunneling delay time measured by attoclock experiment and bridges the gap between it and the Büttiker–Landauer time.[32] This quantum travel time can also provide a reasonable description in the case of very thin barrier where the Büttiker–Landauer time is not well defined.
Although there are many conflicting definitions of tunneling time in conventional quantum mechanics, Bohmian mechanics[33,34] privileges the time that the Bohmian trajectory spends between the entrance and exit points of the potential barrier.[35] Recently, it has been found that the Bohmian time does not correspond to the tunneling time, but agrees with the resonance lifetime of a bound state.[35]
In this paper, we further explore the relationship between the newly introduced quantum travel time with other definitions of tunneling time (Larmor time, Büttiker–Landauer time, and Bohmian time) in one-dimension rectangular tunneling process. In Section 2, we give a brief calculation about the one-dimension rectangular barrier tunneling and show the different definitions of tunneling time (quantum travel time τt, tRe, tIm; the Larmor time tLM and associated times (tx, ty, and tz)). In Section 3, we investigate the relationship between different tunneling times and find that the real quantum travel time tRe is equal to the Bohmian time tBohmian, and the total quantum travel time τt can bridge the connection between the time tx and the Büttiker–Landauer time τBL.
2. Theoretical definition of tunneling time
2.1. Rectangular barrier tunneling
Consider a particle with kinetic energy moving along the x-axis and interacting with a rectangular barrier of height V0 and width d centered at x = 0, as shown in Fig. 1. The wave function of the particle within different regions (I, II, and III) can be written as
where and A, B, C, and D are coefficients determined by the continuity of wavefunction and its first derivative at the edge of the rectangular barrier. These coefficients can be obtained as[5]
Fig. 1. Schematic diagram: a particle with kinetic energy E moves along the x-axis and interacts with a rectangular barrier with height V0 and width d centered at x = 0.
2.2. Quantum travel time
In quantum mechanics, the momentum operator is well defined, thus the quantum travel time is introduced by analogy with the classical travel time[32]
where m is the mass of the particle and means making the modula of “”. The average momentum of the particle during its stay within the barrier region can be obtained by calculating the expected value of the momentum operator
where is the wave function of the particle within the region .
By combining Eqs. (5)–(9) and substituting Eq. (2) into Eq. (11), we can obtain the average momentum of the particle within the barrier region[32]
where
Here, is the ratio of the transmission probability to the reflection probability. Substituting Eqs. (12)–(14) into Eq. (10), we can obtain
The average momentum divided by the mass of particle gives the average velocity of the particle within the barrier region. We define the average velocity as follows:
According to the definition of quantum travel time in Eq. (10), we can also define another two different times
We can easily obtain the relationship between the three times, tRe, tIm, and τt, from Eq. (16)
2.3. Larmor time and associated times
Considering an incident particle with x-direction polarized spin tunneling through a barrier region within a constant z-direction magnetic field B, the particle will experience a Larmor precession.[6] For the transmitted wavefunction of the particle, we can calculate the expectation values of spin in three different directions: , , and . The Larmor time tLM is the average time spent by the particle inside the barrier, which is given by the degree of precession in the y-direction Sy[10,14]
where is the Larmor frequency (the atomic units (a.u.) are used). By combining the definitions of r and , we can write the Larmor time tLM as follows:
Since the particle tunneling through the barrier can also acquire a spin component parallel to the magnetic field,[6] the z-direction precession of spin can also define a time tz[6,14]
This time tz can be rewritten as
For the expectation value of spin in the x-direction , a time tx is also defined,[6] which satisfies the relationship as follows:
3. Relationship between different tunneling times
3.1. The quantum travel times tRe, tIm, and τt
First, we discuss the behavior of the quantum travel time in two opposite limiting cases. In the limit of the very thin barrier , the coefficients on the right side of Eq. (12) can be approximated to and . In the opposite limit of opaque barrier , we have and . Accordingly, we can obtain
where is the Büttiker–Landauer time. Thus, for the very thin barrier , the time τt is mainly determined by the time tRe, i.e., , which means that a classical particle passes through a distance d with momentum . While in the case of opaque barrier , approaches the B üttiker–Landauer time which is also obtained in the limit of opaque barrier.[5] These behaviors of the three quantum travel times can be clearly seen in Fig. 2.
Fig. 2. A comparison between time tRe, tIm, and τt as a function of with E=0.1 a.u. and V0=1.0 a.u. The atomic units are used with .
3.2. The Bohmian time tBohmian, the Lamor time tLM, and time tRe
The Bohmian time is defined as the time required for a Bohmian trajectory to pass the region between the two classical turning points x1 and x2[34,35]
where is the stationary probability flux. After substituting Eq. (2) into Eq. (29), we obtain
The Bohmian time was found to be related to the resonance lifetime of a bound state,Ref. [35] which can be calculated from the resonance widths taken from Ref. [36]. Comparing Eq. (21) with Eq. (30), we can obtain . Thus, the time tRe is also related to the decay rates of quasistationary states and reflects the resonance lifetime.
3.3. The Büttiker–Landauer time τBL, the times tx, and τt
The Büttiker–Landauer time τBL is based on the onset of the “cross-over” regime between pure tunneling and tunneling while absorbing one or more photons from the oscillating field.[5] Büttiker and Landauer argued that this time τBL can determine the actual barrier transversal time. In Section 3.1, we show that the time τt is mainly determined by the time tIm for an opaque barrier, which is equal to the Büttiker–Landauer time τBL. In Fig. 3, we show the comparison between and tIm for different kinetic energies E of the particle. It can be seen that the time tIm perfectly coincides with the Büttiker–Landauer time τBL in the condition of . However, there is obvious difference between them in the region . The time tIm, in the limit of thin barrier, i.e., , approaches a finite value 1/V0 regardless of the kinetic energy of the particle. This can be understood from the uncertainty relation of the time and energy considering that for a particle to pass through the barrier with a height V0, the time it needs is about 1/V0. It is noted that the Büttiker–Landauer time τBL is obtained in the opaque barrier approximation,[5] while in the limit of very thin barrier, there is no well-defined Büttiker–Landauer time.
Fig. 3. A comparison between time tIm and the Büttiker–Landauer time τBL as a function of κ d with E=0.5 a.u. and V0=1.0 a.u. The atomic units are used with .
Büttiker proposed that the time tx (Eq. (24)) can also be used as the barrier traversal time. In Fig. 4, we show a comparison between the times tx (red line), τBL (blue line), and τt (black line) under different kinetic energies of the particle. Although both the time tx and the Büttiker–Landauer time τBL can be treated as barrier traversal time, there is a difference between them, especially when the dimensionless parameter κd is not very large. The difference increases as the kinetic energy of the particle decreases. It is interesting that the quantum travel time τt can perfectly retrieve tx and τBL in two opposite limits, regardless of the particle energy. As seen in Fig. 4, τt is equal to the time tx when κd is small. And τt has an asymptotic behavior similar to the time tx, i.e., approaching the finite value 1/V0 in the limit that the kinetic energy of the particle and the width of the potential barrier tends to zero at the same time. In the opposite limit, τt perfectly coincides with the Büttiker–Landauer time τBL when . While in the range , the value of time τt just falls between the two times: tx and τBL. The quantum travel time τt is composed of two parts: time tRe and time tIm. The time tIm characterizes the quantum property of the tunneling process. The time tRe reflects the time needed by the entire wavefunction of the particle to tunnel through the barrier, which includes the influence of the transmission probability. In the condition that the width of barrier approaches zero, the transmission probability tends to one. Thus, the time tRe tends to the classical time, which means the time that a classical particle needs to pass through a distance d. The time tx is also composed of two parts: ty and tz. The time ty is related to the time tRe (). The time tz is derived under infinitesimal field, which can be approximated to τBL under opaque barrier approximation. While in our definition of quantum travel time, the time tIm is equivalent to the Büttiker–Landauer time τBL under opaque barrier approximation. Thus, we bridge the connection between the time tx and the Büttiker–Landauer time τBL through the quantum travel time τt.
Fig. 4. The time tx, the time τt, and the Büttiker–Landauer time τBL as a function of κ d with (a) E=0.5 a.u., (b) E=0.2 a.u., (c) E=0.1 a.u., and (d) E=0.01 a.u. The height of the potential barrier V0=1.0 a.u. is the same in (a), (b), (c), and (d). The atomic units are used with .
4. Conclusion
In conclusion, based on a new definition of quantum travel time, we bridge the connections between different tunneling times (τBL, tBohmian, and tx) in the one-dimensional rectangular barrier tunneling. The time tRe is equal to the Bohmian time tBohmian, which is related to the resonance lifetime of a bound state. The time τt is a generalized traversal time which cannot only retrieve the Büttiker–Landauer time for but also equal the time tx when κd is small. In the case where the kinetic energy of the particle and the width of the barrier tend to zero at the same time, then time tx and time τt have the same limiting value 1/V0.
