Control of Arrival Time using Structured Wave Packets
**corresponding author [email protected] + Current Address Department of Physics, University of Notre Dame, Notre Dame, IN 46556
Control of Arrival Time using Non-Gaussian Wave Packets T. A. Saxton + and A. L. Harris* Physics Department, Illinois State University, Normal, IL, USA 61790
Abstract
Scattering dynamics are examined for Gaussian and non-Gaussian wave packets with identical momentum densities. Average arrival time delays, dwell times, and phase time delays are calculated for wave packets scattering from a square barrier, and it is shown that the non-Gaussian wave packets exhibit different average arrival time delays than the Gaussian wave packets. These differences result from the non-linear terms in the momentum wave function phase of the non-Gaussian wave packets, which alters the self-interaction times of the wave packets. Control of the average arrival time delay can be achieved through adjustment of the momentum wave function phase, independent of wave packet energy and momentum density.
1. Introduction
What features of a quantum mechanical wave packet influence its interaction time with a localized potential and can a wave packet’s arrival time at a given location be controlled? The answers to these questions are of fundamental importance and have applications in fields such as microscopy [1–3], attosecond science [4–6], materials science [7–9], and electronics [10,11]. Interaction time relates closely to the concepts of traversal time and tunneling time, in which one is interested in the duration of time a particle spends in the interaction region and/or its arrival time at a given spatial location. For potential barriers, arrival of the particle beyond the interaction region may be a result of tunneling through the potential barrier or transmission over the potential barrier. Unfortunately, there is no unique definition for a tunneling or arrival time because time is a parameter, not an observable, in quantum mechanics [12]. This has led to countless suggestions and definitions for calculating and measuring tunneling and arrival times [4,12–24]. For tunneling time, even the basic question of whether the tunneling process occurs instantaneously or over a finite time interval has not yet been answered [4–6,15,25–28]. The question has been explored in the context of attosecond tunneling spectroscopy and attoclock experiments [4,6,26,29], where some experiments yielded a tunneling time of zero [4,29,30], while others resulted in a non-zero tunneling time [6,26]. Predictions of recent theoretical models are also mixed, with models such as time-dependent Schrödinger equation and numerical attoclock simulations showing zero tunneling time [5,27,30–34], while earlier models predict non-zero tunneling times [12,18,35,36]. Tunneling time has also been investigated in condensed matter applications, which show non-zero tunneling times [37,38]. In recent years, renewed interest in tunneling and arrival time has been spurred by technological developments and applications in areas such as electronics, imaging, and quantum information. Electronic devices such as Josephson junctions, tunnel diodes, and many nanoelectronic devices have electron tunneling at their core [2,8,9], where it is directly related to their functional speed. Additionally, a thorough understanding of tunneling dynamics will be crucial to further miniaturization of electronic devices [9]. In imaging applications, electron microscopes are used to obtain detailed structural information about biological, molecular, and nanostructure samples. Enhancements in microscope resolution require tunneling techniques, as do new methods that utilize electron wave function phase [2]. As another example, the tunneling of Bose-Einstein condensates through optical lattices is being explored for use in the development of quantum computing and quantum information, and initial experiments show the possibility of controlling tunneling time in this environment [8]. Also occurring in the last decade are experiments that have generated electron wave packets with non-Gaussian spatial profiles [1,39], including electron Airy beams [1] and electron ortex beams [39,40]. These new spatially structured electron wave packets have properties such as quantized orbital angular momentum, self-acceleration, self-healing, and minimal dispersion [1,39,41]. They have proposed uses in the study of fundamental atomic properties [42–44], control and rotation of nanoparticles [42,43,45,46], and electron microscopy [42,47,48]. For example, super resolution and light sheet optical microscopy have become powerful tools that rely on the properties of spatially structured light beams to achieve resolution beyond the diffraction limit [49–51]. It is possible that spatially structured electron beams could lead to similar improvements in electron microscopy [2]. Additionally, electron vortex beams with quantized orbital angular momentum have been proposed as a means to achieve rotational control of nanoparticles or small molecules [42,43,45,46], and it has been shown that the beam’s orbital angular momentum can be transferred to both the electronic and center of mass motion the target [48,52]. Some early experiments also show qualitative signatures of nanoparticle rotation induced by electron vortex beam [46]. Likewise, there is evidence that electron vortex beams can provide fundamental information about the magnetic properties of matter [42,47], dichroism in chiral molecules [43], and electronic transitions in atoms [53,54]. The experimental realization of spatially structured electron beams, combined with the fundamental importance and numerous applications of tunneling and arrival times, leads to the unique opportunity to combine these areas of study to directly examine how individual wave function properties affect interaction times. The investigation of non-Gaussian wave packet tunneling and transmission dynamics is necessary for the proposed applications that rely on the use and control of these recently realized wave packets. In particular, the comparison of scattering dynamics for carefully chosen Airy and Gaussian wave forms can provide insight into he question of how wave function phase, spatial density, and momentum density influence arrival time. To date, most studies of tunneling and arrival time have been performed using Gaussian wave packets. In this case, any change to the spatial density of the wave packet also changes its momentum density, making direct study of how wave packet properties influence arrival time difficult. An Airy wave packet has a Gaussian momentum density, but a non-Gaussian spatial density and can therefore be designed with a momentum density identical to that of a spatial Gaussian wave packet. This allows for direct comparison of the effect of spatial and momentum density on arrival time. In addition, Airy and Gaussian wave packets differ in momentum wave function phase, with the Gaussian wave function phase being linear, while the Airy wave function phase is cubic. Comparison of Airy and Gaussian wave packets whose primary difference is phase allows for direct study of the role of phase in arrival time. Here, we use the average arrival time delay [13], phase time delay [18–20], and dwell time [14] to quantify the interaction time for over-the-barrier scattering and tunneling. We show that the average arrival time delays and phase time delays of wave packets with identical momentum densities are not necessarily identical, and we determine under what conditions the arrival time of an Airy wave packet differs from that of a Gaussian wave packet with identical momentum density. We also show that the arrival time of a general non-Gaussian wave packet can be controlled through adjustment of the momentum wave function phase, independent of the incident wave packet’s energy or momentum density.
2. Theory
Consider the one-dimensional scattering of a wave packet with mean incident momentum ℏ𝑘𝑘 from a square barrier of height 𝑉𝑉 and width 𝐿𝐿 , as shown in Figure 1. During the collision, he wave packet undergoes reflection, transmission, and tunneling. Classically, if the particle’s incident energy 𝐸𝐸 is less than the barrier height, it can only be reflected. For classical transmission beyond the barrier region to occur, the particle’s energy needs to be greater than the barrier height. However, for quantum mechanical particles, transmission can occur even when the particle’s mean energy is less than the barrier height, resulting in a non-zero probability of finding the particle beyond the barrier region. Conversely, quantum mechanical wave packets with mean energy above the barrier height will experience some reflection from the barrier, resulting in transmission probability less than 1. Figure 1 Wave packet with mean momentum ℏ𝑘𝑘 and mean position 𝑥𝑥 incident on a square barrier potential with width L and height 𝑉𝑉 . Average arrival time is calculated at 𝑥𝑥′ . Reflection and transmission coefficients can be used to quantify the probability of finding the particle to the right or left of the barrier long after the scattering has occurred, and these asymptotic quantities provide information about the spatial and momentum features of the interaction. However, they do not provide information about the time of interaction. To gain insight into the wave packets’ temporal interactions with the barrier, it is necessary to examine quantities that characterize the duration of the scattering process, which requires the calculation of time-dependent wave functions. We calculate time-dependent spatial wave functions with our Path Integral Quantum Trajectory (PIQTr) model [55] in which an initial state wave function is propagated in time by iterating the equation ( 𝑥𝑥 𝑏𝑏 , 𝑡𝑡 𝑏𝑏 ) = ∫ 𝐾𝐾 ( 𝑥𝑥 𝑎𝑎 , 𝑥𝑥 𝑏𝑏 , 𝑡𝑡 𝑎𝑎 , 𝑡𝑡 𝑏𝑏 ) ψ ( 𝑥𝑥 𝑎𝑎 , 𝑡𝑡 𝑎𝑎 ) 𝑑𝑑𝑥𝑥 𝑎𝑎∞−∞ , (1) for small time steps. The initial state wave function at position 𝑥𝑥 𝑎𝑎 and time t a is ψ ( 𝑥𝑥 𝑎𝑎 , 𝑡𝑡 𝑎𝑎 ) . The propagator 𝐾𝐾 ( 𝑥𝑥 𝑎𝑎 , 𝑥𝑥 𝑏𝑏 , 𝑡𝑡 𝑎𝑎 , 𝑡𝑡 𝑏𝑏 ) is written in terms of the classical action, and the result is the time-evolved wave function ψ ( 𝑥𝑥 𝑏𝑏 , 𝑡𝑡 𝑏𝑏 ) at position 𝑥𝑥 𝑏𝑏 and time t b . To directly determine how wave packet properties, such as spatial density, wave function phase, and spatial and momentum uncertainty affect interaction time, we compare three specific wave packets: the Gaussian, Airy, and inverted Airy wave packets. The Gaussian wave packet is the most common localized wave packet used for the study of scattering and its spatial wave function is given by ψ G ( 𝑥𝑥 , 0) = 𝑒𝑒 − ( 𝑥𝑥−𝑥𝑥0 ) / ( 𝜋𝜋𝜎𝜎 ) / 𝑒𝑒 𝑖𝑖𝑘𝑘 ( 𝑥𝑥−𝑥𝑥 ) , (2) with standard deviation 𝜎𝜎 and initial mean position 𝑥𝑥 . It is the minimum uncertainty wave packet because it satisfies the lower bound of the uncertainty relation Δ𝑥𝑥Δ𝑝𝑝 ≥ ℏ /2 . (3) For the Gaussian wave packet, the equality in Eq. (3) holds, and there is a one-to-one relationship between the spatial and momentum uncertainty. A larger spatial uncertainty always results in a smaller momentum uncertainty and vice versa. The corresponding Gaussian momentum wave function is itself a Gaussian 𝜑𝜑 𝐺𝐺 ( 𝑝𝑝 , 0) = 𝜎𝜎 / 𝜋𝜋 / 𝑒𝑒 −𝑖𝑖𝑝𝑝𝑥𝑥 𝑒𝑒 −𝜎𝜎 ( 𝑝𝑝−𝑘𝑘 ) / , (4) with linear phase 𝜙𝜙 𝐺𝐺 = −𝑝𝑝𝑥𝑥 . (5) .2 Airy Wave Packets Recently, non-Gaussian localized wave packets have been experimentally generated using electrons [1,40,56,57]. One such wave packet is the truncated Airy wave function [1,58] 𝜓𝜓 𝐴𝐴 ( 𝑥𝑥 , 0) = (8 𝜋𝜋𝜋𝜋 ) / 𝑒𝑒 −𝛼𝛼 / 𝐴𝐴𝐴𝐴 ( 𝑥𝑥 − 𝑥𝑥 ) 𝑒𝑒 𝛼𝛼 ( 𝑥𝑥−𝑥𝑥 ) 𝑒𝑒 𝑖𝑖𝑘𝑘 ( 𝑥𝑥−𝑥𝑥 ) , (6) with truncation parameter 𝜋𝜋 and Airy function 𝐴𝐴𝐴𝐴 ( 𝑥𝑥 − 𝑥𝑥 ) . The Airy wave function without the truncation term is a solution to the free particle Schrödinger equation [41], but like its more commonly used plane wave counterpart, has infinite transverse extent and infinite energy. The truncation term in Eq. (6) yields a finite width wave packet with finite energy that is square normalizable. Airy wave functions have many unique properties, including self-acceleration, self-healing, and minimal spreading [1,41,58]. While these features are intriguing by themselves, we are primarily interested in the truncated Airy wave function’s Gaussian momentum density because it provides a straightforward means to directly study the role of position and momentum uncertainty, spatial and momentum density, and momentum wave function phase in scattering dynamics through comparison of Gaussian and Airy wave packets. The Airy momentum wave function is given by 𝜑𝜑 𝐴𝐴 ( 𝑝𝑝 , 0) = ( ) / √2𝜋𝜋 𝑒𝑒 −𝛼𝛼 ( 𝑝𝑝−𝑘𝑘 ) 𝑒𝑒 −𝑖𝑖𝑝𝑝𝑥𝑥 𝑒𝑒 𝑖𝑖� ( 𝑝𝑝−𝑘𝑘0 ) −𝛼𝛼 ( 𝑝𝑝−𝑘𝑘 ) � , (7) with cubic phase 𝜙𝜙 𝐴𝐴 = 𝜙𝜙 𝐺𝐺 + � ( 𝑝𝑝−𝑘𝑘 ) − 𝜋𝜋 ( 𝑝𝑝 − 𝑘𝑘 ) � . (8) The modulus squared of Eq. (7) is clearly seen to be Gaussian, allowing for the standard deviation of the Gaussian wave packet and the truncation parameter of the Airy wave packet to be chosen in such a way that their initial momentum densities | 𝜑𝜑 ( 𝑝𝑝 , 0)| and momentum ncertainties Δ𝑝𝑝 will be identical ( 𝜋𝜋 = 𝜎𝜎 ). However, their spatial densities | 𝜓𝜓 ( 𝑥𝑥 , 0)| , position uncertainties Δ𝑥𝑥 , and momentum wave function phases 𝜙𝜙 𝐴𝐴 , 𝐺𝐺 will be different. For example, an Airy wave packet with 𝜋𝜋 = 0.317 has the same momentum density and momentum uncertainty as a Gaussian wave packet with 𝜎𝜎 = 0.795 , but a different spatial density, position uncertainty, and momentum wave function phase (see Table 1 and Figure 3). Unlike the Gaussian wave packet, the Airy wave packet is not a minimum uncertainty wave packet and there is not a one-to-one relationship between position and momentum uncertainty. For most values of 𝜋𝜋 , a given position uncertainty leads to two different momentum uncertainties, such that two Airy wave functions can be designed with identical position uncertainties Δ𝑥𝑥 , but different momentum uncertainties
Δ𝑝𝑝 . For example, Airy wave packets with 𝜋𝜋 = 0.317 and 𝜋𝜋 = 1.5 have identical position uncertainties but different momentum uncertainties, spatial densities, and momentum wave function phases. Figure 2 shows the spatial and momentum uncertainties of the Gaussian and Airy wave packets as a function of the standard deviation and truncation parameter. For large values of 𝜋𝜋 and 𝜎𝜎 , the position and momentum uncertainties of the Airy and Gaussian wave packets become identical, however their momentum wave function phases and spatial densities remain different. Figure 2 Position and momentum uncertainties for the Gaussian, Airy, and inverted Airy wave packets as a function of standard deviation 𝜎𝜎 and truncation parameter 𝜋𝜋 . Unlike the Gaussian wave function, the Airy wave packet’s spatial density is not symmetric (see Figure 3) and the spatial orientation of the wave packet relative to its momentum direction is physically important. An Airy wave packet with its largest peak to the right and momentum to the right behaves differently than an Airy wave packet with its largest peak to the left and momentum to the right. The latter case is a spatial reflection of the Airy wave packet and is referred to as the inverted Airy wave packet.
Figure 3 Spatial densities for the Gaussian ( 𝜎𝜎 = 0.795 ), Airy ( 𝜋𝜋 = 0.317 ), and inverted Airy ( 𝜋𝜋 =0.317 ) wave packets with 𝑥𝑥 = 0 a.u. The Gaussian and Airy curves have been shifted above the x-axis for clarity. The inverted Airy momentum wave function is identical to that of the Airy wave function, except for a change in sign in the non-Gaussian term of the phase 𝜑𝜑 𝐼𝐼𝐴𝐴 ( 𝑝𝑝 , 0) = ( ) / √2𝜋𝜋 𝑒𝑒 −𝛼𝛼 ( 𝑝𝑝−𝑘𝑘 ) 𝑒𝑒 −𝑖𝑖𝑝𝑝𝑥𝑥 𝑒𝑒 −𝑖𝑖� ( 𝑝𝑝−𝑘𝑘0 ) −𝛼𝛼 ( 𝑝𝑝−𝑘𝑘 ) � , (9) with 𝜙𝜙 𝐼𝐼𝐴𝐴 = 𝜙𝜙 𝐺𝐺 − � ( 𝑝𝑝−𝑘𝑘 ) − 𝜋𝜋 ( 𝑝𝑝 − 𝑘𝑘 ) � . (10) Airy and inverted Airy wave functions with identical truncation parameters will always have the same spatial uncertainties, momentum uncertainties, and momentum densities, but different momentum wave function phases and spatial densities. Comparison of Airy and inverted Airy wave packet scattering can then be used to isolate the effects of momentum wave function phase and spatial density. Unfortunately, even the simple concept of when a particle arrives at a given spatial location still has no universally accepted definition within quantum mechanics [4,12–23]. While such an arrival time definition is straightforward for classical particles, quantum particle haracteristics such as tunneling, interference, uncertainty, and dispersion complicate attempts to define a quantum mechanical arrival time. Many candidates are offered in the extensive literature that discusses the feasibility and applicability of such definitions [4,12–24], and we refer the interested reader to these more detailed analyses and references therein. Here, we have chosen three generally accepted quantities to characterize the interaction time. They are the average arrival time delay [13], the average dwell time of the particle in the barrier region [14], and the group (or phase time) delay [18–20]. The average arrival time 𝑇𝑇 ( 𝑥𝑥 ′ ) is a local quantity that averages the spatial density at a location 𝑥𝑥′ over time to yield an average time that the particle arrives at 𝑥𝑥′ . It is given by [13] 𝑇𝑇 ( 𝑥𝑥′ ) = ∫ 𝑑𝑑𝑡𝑡 𝑡𝑡𝑃𝑃 𝑥𝑥 ′ ( 𝑡𝑡 ) ∞0 , (11) where 𝑃𝑃 𝑥𝑥 ′ ( 𝑡𝑡 ) is the arrival time distribution 𝑃𝑃 𝑥𝑥 ′ ( 𝑡𝑡 ) = �𝜓𝜓�𝑥𝑥 ′ , 𝑡𝑡�� ∫ 𝑑𝑑𝑡𝑡 ′ | 𝜓𝜓 ( 𝑥𝑥 ′ , 𝑡𝑡 ′ )| . (12) Since we are interested in the duration of the interaction, the average arrival time delay between interacting and non-interacting particles can be used to quantify the interaction time. The average arrival time delay is defined as Δ𝑇𝑇 𝑖𝑖 = 𝑇𝑇 𝑖𝑖𝑖𝑖𝑡𝑡𝑖𝑖 − 𝑇𝑇 𝑓𝑓𝑓𝑓𝑒𝑒𝑒𝑒𝑖𝑖 , (13) where 𝑇𝑇 𝑖𝑖𝑖𝑖𝑡𝑡𝑖𝑖 is the average arrival time of an interacting particle, 𝑇𝑇 𝑓𝑓𝑓𝑓𝑒𝑒𝑒𝑒𝑖𝑖 is the average arrival time of an identical non-interacting free particle, and 𝐴𝐴 = 𝐺𝐺 , 𝐴𝐴 , 𝐼𝐼𝐴𝐴 refers to either a Gaussian, Airy, or inverted Airy wave packet, respectively. A positive value of
Δ𝑇𝑇 𝑖𝑖 indicates that the interacting particle was impeded by the barrier, while a negative value indicates that the particle was accelerated by its interaction with the barrier. The group delay, or phase time delay, can be found through analytical analysis using the stationary phase approximation [18–20]. In this approximation, it is assumed that the phase of the momentum wave function is stationary at each barrier edge 𝑥𝑥 = 0 and 𝑥𝑥 = 𝐿𝐿 , and a time of arrival at these two points is used to quantify the delay of the particle relative to an identical free particle. It is analogous to the average arrival time delay [13]. The phase time delays for the Airy Δ𝑡𝑡 𝜁𝜁𝐴𝐴 and inverted Airy
Δ𝑡𝑡 𝜁𝜁𝐼𝐼𝐴𝐴 wave packets have an additional term compared to the Gaussian phase time delay
Δ𝑡𝑡 𝜁𝜁𝐺𝐺 such that their phase time delay either increases or decreases relative to the Gaussian wave packet (see Appendix A). They are given by
Δ𝑡𝑡 𝜁𝜁𝐴𝐴 = Δ𝑡𝑡 𝜁𝜁𝐺𝐺 + 𝑚𝑚 ( 𝑘𝑘 𝑇𝑇 −𝑘𝑘 ) −𝛼𝛼 ℏ𝑘𝑘 𝑇𝑇 (14) for the Airy wave packet and Δ𝑡𝑡 𝜁𝜁𝐼𝐼𝐴𝐴 = Δ𝑡𝑡 𝜁𝜁𝐺𝐺 − 𝑚𝑚 ( 𝑘𝑘 𝑇𝑇 −𝑘𝑘 ) −𝛼𝛼 ℏ𝑘𝑘 𝑇𝑇 (15) for the inverted Airy wave packet, where Δ𝑡𝑡 𝜁𝜁𝐺𝐺 is the Gaussian phase time delay time, ℏ𝑘𝑘 𝑇𝑇 is the average momentum of the transmitted wave packet, and m is the mass of the particle. The additional term in the phase time delay results from the non-linear terms in these wave packets’ momentum wave function phases. Physically, the phase time delay is a result of two independent contributions [15,59] – the dwell time 𝜏𝜏 𝐷𝐷 and the self-interaction time 𝜏𝜏 𝐼𝐼 Δ𝑡𝑡 𝜁𝜁 = 𝜏𝜏 𝐷𝐷 + 𝜏𝜏 𝐼𝐼 . (16) The dwell time quantifies the total time a particle spends in the barrier region. It is a non-local quantity that averages the spatial density of the wave packet over the barrier region [14] 𝜏𝜏 𝐷𝐷 = ∫ 𝑑𝑑𝑡𝑡 ∞−∞ ∫ 𝑑𝑑𝑥𝑥 | 𝜓𝜓 ( 𝑥𝑥 , 𝑡𝑡 )| 𝐿𝐿 . (17) he self-interaction time is a result of interference between the incident and reflected parts of the wave packets in the region to the left of the barrier and can be found by subtracting the dwell time from the phase time delay.
3. Results 3.1 Wide Barrier
We begin with the specific case of Gaussian, Airy, and inverted Airy wave packets scattering from a barrier of height 𝑉𝑉 = 50 a.u. and width 𝑤𝑤 = 1 a.u. The wave packets are chosen such that they have identical momentum densities, but different spatial densities, position uncertainties, and momentum wave function phases. They are labeled as wave packets 1-3 in Table 1, which lists their position and momentum uncertainties and average arrival time delays. The initial mean position of the wave packets is 𝑥𝑥 = − a.u. and the initial mean momentum is ℏ𝑘𝑘 = 7.75 a.u. The average arrival time delays are calculated at 𝑥𝑥 ′ = 1 a.u., which is the right edge of the barrier, and 𝑥𝑥 ′ = 20 a.u., which is sufficiently far from the barrier that the scattering process has been completed. The average arrival time delays at 𝑥𝑥 ′ = 1 a.u. show that the Gaussian and inverted Airy wave packets are accelerated by their interaction with the barrier relative to an identical non-interacting free particle, which results in negative average arrival time delays. In contrast, the Airy wave packet is delayed by its interaction with the barrier. However, at the asymptotic location of 𝑥𝑥 ′ = 20 a.u., all three wave packets arrive before their free particle counterparts. This is due to an increase in the mean momentum of the transmitted wave packet relative to the mean incident momentum, as discussed below. Clearly, the average arrival time delays are different for wave packets 1-3, and we note that regardless of which position is used, the relative average arrival time delays between the wave packets are unchanged. The inverted Airy wave packet has the smallest average arrival ime delay, indicating that its interaction time with the barrier is shortest and it arrives at 𝑥𝑥′ earliest. The Airy wave packet has the largest average arrival time delay and interaction time with the barrier and therefore arrives at 𝑥𝑥′ latest. The Gaussian wave packet’s average arrival time delay lies in between the Airy and inverted Airy wave packets’ average arrival time delays. Additional evidence of the variation in average arrival time delays can be observed in the time-dependent position and momentum densities, as well as the arrival time distributions shown in Figure 4. (1) Inverted Airy 𝜋𝜋 = 0.317 (2) Gaussian 𝜎𝜎 = 0.795 (3) Airy 𝜋𝜋 = 0.317 (4) Inverted Airy 𝜋𝜋 = 1.5 (5) Gaussian 𝜎𝜎 = 1.76 (6) Airy 𝜋𝜋 = 1.5 Δ𝑥𝑥
Δ𝑝𝑝
Δ𝑇𝑇 ( 𝑥𝑥 ′ = 1) -0.8 -0.3 0.3 -0.2 -0.2 -0.2 Δ𝑇𝑇 ( 𝑥𝑥 ′ = 20) - 1.4 - 0.9 - 0.3 -0.3 -0.3 -0.3 Table 1 Position and momentum uncertainties and average arrival time delays for the wave packets discussed in the text and figures. Average arrival time delays are calculated for 𝑥𝑥 ′ = 1 a.u., 𝑥𝑥 ′ = 20 𝑎𝑎 . 𝑢𝑢 . , 𝑥𝑥 = − 𝑎𝑎 . 𝑢𝑢 . , 𝑉𝑉 = 50 𝑎𝑎 . 𝑢𝑢 . , ℏ𝑘𝑘 = 7.75 a.u., and 𝑤𝑤 = 1 a.u. The arrival time distributions 𝑃𝑃 𝑥𝑥 ′ ( 𝑡𝑡 ) (row 1) for wave packets 1-3 show signatures of the different average arrival times between the three wave packets, with the peaks of the inverted Airy wave packet distributions located at earlier times than the Gaussian and Airy wave packet distributions. This separation of the distributions leads to the quantifiable differences in average arrival time delays shown in Table 1. The shapes of the arrival time distributions are a result of the different spatial density profiles of the transmitted wave packets, which are altered from the incident wave packet densities due to the scattering process. The time-dependent spatial densities for wave packets 1-3 are shown in row 2 of Figure 4, where the inverted Airy wave packet is observed to arrive at 𝑥𝑥 ′ = 1 a.u. and 𝑥𝑥 ′ = 20 a.u. before the Gaussian wave packet, while the Airy wave packet arrives after the Gaussian wave acket. Because Figure 4 and Table 1 show clear differences in average arrival time delays between wave packets with identical momentum densities, we conclude that a wave packet characteristic(s) other than momentum density determines the interaction time. The remaining differences between the wave packets are the spatial densities, position uncertainties, and momentum wave function phases, and a pair-wise comparison of wave packet scattering dynamics can be used to identify which of these characteristics causes the average arrival time delays to be different. Figure 4 Row 1 – arrival time distributions at 𝑥𝑥 ′ = 1 a.u. and 𝑥𝑥 ′ = 20 𝑎𝑎 . 𝑢𝑢 . for inverted Airy, Gaussian, and Airy wave packets interacting with a square barrier of height 𝑉𝑉 = 50 𝑎𝑎 . 𝑢𝑢 . and width 𝑤𝑤 = 1 𝑎𝑎 . 𝑢𝑢 . he mean incident momentum is ℏ𝑘𝑘 = 7.75 a.u. and initial mean position is 𝑥𝑥 = − 𝑎𝑎 . 𝑢𝑢 . Rows 2 and 3 – time-dependent position and momentum densities for the same wave packets. The color bar corresponds to the densities | 𝜓𝜓 ( 𝑥𝑥 , 𝑡𝑡 )| (row 2) and | 𝜑𝜑 ( 𝑝𝑝 , 𝑡𝑡 )| (row 3). Row 2 insets – initial wave packet spatial densities. The Airy and inverted Airy wave packets have identical position uncertainties, but different average arrival time delays, eliminating position uncertainty as the source of the differences in average arrival time delays. Therefore, either the momentum wave function phase or the spatial density profile must cause the difference in average arrival time delays. These two characteristics of the wave functions are inextricably linked, and their effects cannot be individually isolated. A change in momentum wave function phase always results in a change in spatial density and vice versa, and so we conclude that both the momentum wave function phase and spatial density profile control the interaction time. Some insight into the role of these two features can be found by examining the phase time delay for the different wave packets, which is analogous to the average arrival time delay at the right edge of the barrier [13]. Equations (14) and (15) predict that the Airy (inverted Airy) will arrive after (before) the Gaussian wave packet with a difference in phase time delays of ± 𝑚𝑚 ( 𝑘𝑘 𝑇𝑇 −𝑘𝑘 ) −𝛼𝛼 ℏ𝑘𝑘 𝑇𝑇 . Using the mean transmitted momentum value from the numerical simulations ( 𝑘𝑘 𝑇𝑇 = 10.4 a.u. for wave packets 1-3) yields a predicted difference in phase time delays of +0.7 a.u. between the Gaussian and Airy/inverted Airy wave packets. These values are similar to the numerical results found in the simulation and shown in Table 1, confirming the relationship between average arrival time delay and phase time delay and demonstrating that the momentum wave function phase is the source of the time delay differences between the wave packets. If conditions are such that the non-Gaussian term in the phase time delay is negligible, then no difference in average arrival time delays will be observed, despite the differences in momentum wave function phase. he clear influence of the momentum wave function phase on the average arrival time delay does not provide any information regarding the role of spatial density on the interaction time. To understand the effect of spatial density profile, it is necessary to examine the dwell times and self-interaction times of the different wave packets. Recall from Section 2.4 that the phase time delay can be written as the sum of the dwell time and self-interaction time. Therefore, the increased (decreased) interaction time with the barrier for the Airy (inverted Airy) wave packet relative to the Gaussian wave packet must be due to either or both of these components. Using Eq. (17), we calculated the dwell times for wave packets 1-3 and found them to be identical with 𝜏𝜏 𝐷𝐷 = 0.03 a.u, which is at least an order of magnitude smaller than the magnitude of the average arrival time delays. This makes the effect of the dwell time on the average arrival time delay negligible and implies that the average arrival time delay is largely determined by the self-interaction time, which therefore must differ by wave packet type. The self-interaction time is a result of interference between the incident and reflected parts of the wave packet, and insight into its variation with wave packet type can be found from the time-dependent momentum density plots in row 3 of Figure 4. They show that the reflected part of the wave packet appears earliest for the inverted Airy wave packet, followed by the Gaussian and Airy wave packets. This is because the tail of the inverted Airy wave packet has significant non-zero density well to the right of the average position of the wave packet and it begins interacting with the barrier earlier than the leading edge of the Gaussian or Airy wave packets. Therefore, the reflection process begins first for the inverted Airy wave packet and last for the Airy wave packet. The momentum density plots also show that the small, positive momentum components persist for less time for the inverted Airy ave packet than for the Gaussian or Airy wave packets. This implies that for the inverted Airy wave packet, these components spend less time interacting with the barrier. Additional information about the complementary roles of the spatial density and momentum wave function phase can be found through comparison of wave packets with very similar spatial densities, but different momentum wave function phases. Three such wave packets are listed as wave packets 4-6 in Table 1 and shown in Figure 5. These wave packets have identical spatial uncertainties, identical momentum densities, and nearly identical spatial densities, but different momentum wave function phases. Table 1 shows that the average arrival time delays for wave packets 4-6 are equal at 𝑥𝑥 ′ = 1 a.u. and 𝑥𝑥 ′ = 20 a.u. Calculation of their dwell times also results in identical values ( 𝜏𝜏 𝐷𝐷 = 0.02 a.u.), and therefore, their self-interaction times must be identical. This confirms that while the momentum wave function phase is the source of the different interaction times, the wave packets must also have significantly different spatial densities in order to exhibit differences in interaction times. Figure 5 Spatial densities for wave packets 4-6 with 𝑥𝑥 = 0 a.u. astly, for the wide barrier, we investigate the cause of the negative average arrival time delays. The acceleration of a wave packet due to its interaction with the barrier can be a result of two possible scattering mechanisms: tunneling or over-the-barrier scattering. It is well-known that tunneling wave packets arrive before their identical non-interacting counterparts [13,15,36,60], however in this case, the large barrier width suppresses tunneling. This is confirmed by the presence of only momentum components corresponding to energies above the barrier in the transmitted wave packet and is observable in the momentum density plots of Figure 4. Momentum values greater than 10 a.u. correspond to energies above the barrier height and are the primary components in the transmitted wave packets. Therefore, the negative average arrival time delays are due to the over-the-barrier scattering and not tunneling. The transmitted wave packets also each exhibit an increased average velocity (inverse slope, ℏ𝑘𝑘 𝑇𝑇 = 10.4 a.u.) relative to the incident velocity (green line, ℏ𝑘𝑘 = 7.75 a.u.), as seen by the change in slope of the transmitted wave packet trajectory in the spatial density plots. This is another indication that the large momentum components of the wave packet are the dominant contribution to transmission. The barrier effectively acts as a filter, allowing only the large momentum components to be transmitted. As shown above, the interaction times of the Airy, Gaussian, and inverted Airy wave packets differed due to their momentum wave function phases and spatial densities, but this was for one particular case when transmission was caused by over the barrier components of momentum and the contribution of tunneling to transmission was negligible. To determine if differences in interaction times exist for tunneling wave packets, we calculate time-dependent wave functions and interaction times for wave packets 1-3 with the same kinematics scattering rom a narrow barrier with 𝑤𝑤 = 0.2 a.u. Figure 6 shows time dependent spatial (row 2) and momentum (row 3) densities and arrival time distributions (row 1) for wave packets 1-3 scattering from the narrow barrier. Figure 6 Same as Figure 4, but for a barrier width of 𝑤𝑤 = 0.2 a.u. The momentum densities show that for the tunneling wave packets, a larger range of momentum components are transmitted through the barrier with an average transmitted momentum of ℏ𝑘𝑘 𝑇𝑇 = 8 a.u. for all three wave packets. While the tunneling wave packets’ transmitted momenta are now more similar to the incident momentum, they are still larger, and herefore the tunneled wave packets arrive at 𝑥𝑥 ′ = 20 a.u. before their free particle counterparts with negative average arrival time delays. The wider range of transmitted momentum values and the lower mean transmitted momentum are a result of more below-the-barrier momentum components tunneling through the barrier and contributing to the transmitted momentum. The similarity of transmitted and incident momentum values is also observed in the spatial densities of Figure 6 that show the slopes (inverse velocity) of the transmitted wave packet trajectories are similar to the slopes of the incident wave packets (green lines). The momentum density plots also show that as in the case of the wide barrier, the reflected part of the wave packet appears earliest for the inverted Airy wave packet, followed by the Gaussian and Airy wave packets. This is expected because this feature is caused by the spatial profile of the wave packets and should be independent of barrier width. Unlike the arrival time distributions for the wide barrier, the arrival time distributions for the narrow barrier are overlapping and result in identical average arrival time delays that are close to zero ( Δ𝑇𝑇 ( 𝑥𝑥 ′ = 0.2) = − Δ𝑇𝑇 ( 𝑥𝑥 = −
20) = − a.u.). Calculation of the dwell times for the tunneling wave packets again yields identical values of 𝜏𝜏 𝐷𝐷 = 0.02 a.u. for each of the wave packets, which results in identical self-interaction times. At first glance, this seems contradictory to the results for scattering from the wide barrier, where the wave packets’ spatial densities caused different self-interaction times and average arrival time delays. However, no contradiction exists because the spatial density profile only influences arrival times when the non-Gaussian term in the phase time delays of Eqs. (14) and (15) is significantly non-zero. In the case of tunneling wave packets, the similarity of the mean transmitted momentum and the mean incident momentum results in the non-Gaussian term in the phase time delays being negligible. Therefore, no difference in average arrival time delay is observed and consequently, here is no difference in self-interaction times. In this case, the spatial density of the wave packet is unimportant because the difference in phases does not lead to a difference in arrival times. The results presented above were for wave packets with mean incident energy below the barrier height, and it is important to know how the interaction times change with incident energy. To this end, we calculate average arrival time delays, phase time delays, and dwell times as a function of incident momentum in three situations: (a) wave packets 1-3 scattering from the wide barrier, (b) wave packets 1-3 scattering from the narrow barrier, and (c) wave packets 4-6 scattering from the wide barrier. In case (a), the wave packets have identical momentum densities but very different spatial densities and momentum wave function phases. Because the wide barrier is used, transmission is dominated by over-the-barrier momentum components and tunneling is negligible. Figure 7a and 7b show the average arrival time delays for case (a) as a function of incident energy. Generally, for wave packets with energies below the barrier height, the average arrival time delays are negative, while wave packets with energies above the barrier height have positive average arrival time delays. The one exception is the Airy wave packet average arrival time delay at the right edge of the barrier, which is positive for all incident energies. In this case, the airy wave packet always arrives after its free particle counterpart, which is likely a result of the extended tail of the incident Airy wave packet to the left of its mean position. The negative average arrival time delays for mean incident energy below threshold are expected because below-the-barrier scattering preferentially selects large momentum components from the incident wave packet resulting in the advance of an interacting wave packet relative to a free particle [13]. Wave packets with energies above the barrier will experience a educed velocity in the barrier region, leading to positive average arrival time delays. The average arrival time delays of the wave packets are most different at low energy, with the Airy (inverted Airy) wave packet having the least (most) negative average arrival time delay. Further details on the differences in average arrival time delays can be observed in Figure 7c, which shows the relative average arrival time delays (
Δ𝑇𝑇 𝐺𝐺 − Δ𝑇𝑇 𝐴𝐴 , 𝐼𝐼𝐴𝐴 ) and relative phase time delays (
Δ𝑡𝑡 𝜁𝜁𝐺𝐺 − Δ𝑡𝑡 𝜁𝜁𝐴𝐴 , 𝐼𝐼𝐴𝐴 ) of the Airy and inverted Airy wave packets compared to the Gaussian wave packet. As incident energy increases, the mean transmitted momentum approaches the mean incident momentum, and the relative time delays go to zero, such that all three wave packets have identical interaction times. The dependence of the relative time delays on transmitted momentum is expected from the phase time delays of Eqs. (14) and (15) and is consistent with the special cases examined above. Only when the mean transmitted momentum differs from the mean incident momentum does the non-Gaussian term in the phase time delay result in non-zero relative average arrival time delays. The relative average arrival time delay curves in Figure 7c are approximately symmetric about zero, indicating that any advancement of the inverted Airy wave packet relative the Gaussian wave packet is nearly identical to the lag experienced by the Airy wave packet. Again, this is consistent with the predictions of Eqs. (14) and (15).
Figure 7 (a) Average arrival time delays for wave packets 1-3 at 𝑥𝑥 ′ = 1 a.u. as a function of incident energy 𝐸𝐸 relative to barrier height 𝑉𝑉 . The barrier width is 𝑤𝑤 = 1 a.u. (b) Same as (a), but at 𝑥𝑥 ′ = 20 a.u. (c) Relative average arrival time delays (lines) and relative phase time delays (points) of the Airy and inverted Airy wave packets as a function of incident energy. The top horizontal axis shows transmitted wave packet mean momentum ℏ𝑘𝑘 𝑇𝑇 relative to mean incident momentum ℏ𝑘𝑘 . The dwell times for wave packets 1-3 are identical at each energy and shown as a function of incident energy in Figure 8. For case (a), the dwell time is a maximum for
𝐸𝐸𝑉𝑉 = 1.2 , where it is of similar magnitude to the average arrival time delay. Near threshold, the dwell time is the dominant contribution to the time delays and the self-interaction time is negligible. This is caused by reduced reflection probability for wave packets with larger incident energies, resulting in reduced interference between the incident and reflected parts of the wave packet. For wave packets with mean incident energies below the barrier height, the opposite is true. The dwell time is negligible and the self-interaction time is the dominant contribution to the time delays due to the increased probability of reflection that increases interference between the incident and reflected parts of the wave packet. Figure 8 Dwell time as a function of incident energy for wave packets 1-3 interacting with barriers of width 𝑤𝑤 = 0.2 and 𝑤𝑤 = 1 a.u. For case (b), we performed calculations of average arrival time delays, phase time delays, and dwell times for wave packets 1-3 scattering from the narrow barrier while varying the incident momentum ( 𝑎𝑎 . 𝑢𝑢 . ≤ ℏ𝑘𝑘 ≤ 𝑎𝑎 . 𝑢𝑢 . ). In these cases, tunneling is the dominant transmission mechanism and the average arrival time delays are close to zero ( | Δ𝑇𝑇 ( 𝑥𝑥 ′ = 1)| <0.02 a.u., | Δ𝑇𝑇 ( 𝑥𝑥 ′ = 20)| < 0.05 a.u.). The ratio of the mean transmitted momentum to mean incident momentum is between 1 and 1.05 for all incident energies, which results in relative average arrival time delays and relative phase time delays also near zero. The dwell times for wave packets 1-3 were again identical at each incident energy and, as shown in Figure 8, exhibit a similar energy dependence as those of the wide barrier, except that they are an order of magnitude smaller. This again results in dwell time being the dominant contribution to average arrival time delays for wave packets with incident energies near threshold, but an insignificant contribution at low incident energies. Consequently, the self-interaction time is most important at low energies where more reflection occurs, however, it is independent of wave packet type due to the non-Gaussian term in the phase time delays being negligible. For case (c), we performed calculations for wave packets 4-6 scattering from the wide barrier with incident momentum 𝑎𝑎 . 𝑢𝑢 . ≤ ℏ𝑘𝑘 ≤ 𝑎𝑎 . 𝑢𝑢 . As in case (a), transmission here is largely due to over-the-barrier momentum components. However, unlike case (a), wave packets -6 have very similar spatial profiles, differing primarily only in momentum wave function phase. Based on the results above, we expect that the average arrival time delays will be similar in magnitude to those in case (a) and that if the ratio of mean transmitted momentum to mean incident momentum is approximately unity, the average arrival time delays will be identical for wave packets 4-6. This is exactly what is observed. The ratios of 𝑘𝑘 𝑇𝑇 / 𝑘𝑘 are between 1 and 1.09, leading to identical average arrival time delays at 𝑥𝑥 ′ = 20 a.u. between -0.3 ( 𝐸𝐸 / 𝑉𝑉 = 0.5 ) and 0.25 ( 𝐸𝐸 \ 𝑉𝑉 = 1.1 ). The dwell times for wave packets 4-6 are also identical at each energy, closely resembling those in case (a). This again leads to the dwell time being the dominant contribution to time delays for wave packets with energies near threshold, while the self-interaction time is dominant for wave packets with energies below threshold. Overall, cases (a) – (c) confirm that when the mean transmitted momentum is significantly different than the mean incident momentum, the non-linear terms in the momentum wave function phase lead to arrival times that are different from those of a Gaussian wave packet. The relative delay or advance of the non-Gaussian wave packet is caused by its altered self-interaction time that results from the wave packet’s non-Gaussian spatial density profile. Differences in average arrival time delays are predominantly observed for wide barriers where transmission is dominated by over-the-barrier momentum components. For narrow barriers, where tunneling is the dominant transmission mechanism, no differences in average arrival time delays are observed. The influence of the wave packet spatial profile and momentum wave function phase on the average arrival time delay provides a means for temporal control of wave packets with identical energies and momentum densities. This control is only possible with wave packets aving non-linear momentum wave function phases, and in the next section, we demonstrate how such control may be achieved. The Airy and inverted Airy wave packets are two special cases of wave packets with cubic phases. To demonstrate the possibility of controlling average arrival time, we introduce a more general momentum wave function with a cubic phase containing an adjustable parameter 𝑏𝑏 𝜑𝜑 𝐶𝐶 ( 𝑝𝑝 , 0) = √𝜎𝜎𝜋𝜋 / 𝑒𝑒 −𝜎𝜎 ( 𝑝𝑝−𝑘𝑘 ) / 𝑒𝑒 −𝑖𝑖𝑝𝑝𝑥𝑥 𝑒𝑒 𝑖𝑖𝑏𝑏 ( 𝑝𝑝−𝑘𝑘 ) , (18) where 𝜎𝜎 is the standard deviation of the Gaussian wave packet. The inverse Fourier transform of this wave function yields an initial state spatial wave function (see Figure 9) that can then be time evolved to study scattering dynamics and interaction times, as above. The stationary phase approximation predicts that the relative phase time delay between a Gaussian wave function and the wave function in Eq. (18) will be − ( 𝑘𝑘 𝑇𝑇 −𝑘𝑘 ) ℏ𝑘𝑘 𝑇𝑇 , which varies linearly with 𝑏𝑏 . This implies that control of arrival time can be achieved by alteration of the cubic parameter 𝑏𝑏 , independent of wave packet energy and momentum density. In Figure 9 we show results from our numerical simulations and the stationary phase approximation for the relative average arrival time delay between the wave packet of Eq. (18) and a Gaussian wave packet as a function of b . Both wave packets have identical momentum densities with mean incident momentum ℏ𝑘𝑘 = 7.75 a.u. and mean incident position 𝑥𝑥 = − a.u. The barrier width is 𝑤𝑤 = 1 a.u. and barrier height is 𝑉𝑉 = 50 a.u. As predicted by the stationary phase approximation, Figure 9 shows that the relative average arrival time delay of the numerical calculations varies linearly with b and confirms that the arrival time of a wave packet can be controlled through adjustment of this parameter. We note that the spatial wave function corresponding the momentum wave function of Eq. (18) resembles the Airy (positive 𝑏𝑏 ) or nverted Airy (negative 𝑏𝑏 ) wave packet with multiple peaks and larger spatial uncertainty than a Gaussian with identical momentum density. As 𝑏𝑏 increases, the spatial uncertainty increases, resulting in a larger self-interaction time that in turn leads to a larger relative average arrival time delay. Figure 9 Top – initial wave packet spatial densities for the momentum wave function of Eq. (18). Bottom – relative average arrival time delays (red dashed line) and relative phase time delays (points) for the wave packet of Eq. (18). Mean incident energy is ℏ𝑘𝑘 = 7.75 a.u., mean initial position is 𝑥𝑥 = − a.u., barrier width is 𝑤𝑤 = 1 a.u., and barrier height is 𝑉𝑉 = 50 a.u. Greater control over wave packet arrival time can be achieved through additional terms in the momentum wave function phase. For example, quadratic or linear terms could be included with independent parameters that provide further means of control. It is also possible to use polynomials higher than third order in the momentum wave function phase, which will yield more control parameters, each of which could be adjusted to provide the desired arrival time.
4. Conclusion
We have examined time-dependent scattering of non-Gaussian wave packets from one-dimensional square barrier potentials. We used average arrival time delay, phase time delay, and well time to quantify the wave packet’s interaction time with the barrier. By comparing wave packets with identical momentum densities, we showed that the average arrival time delays of the transmitted wave packets are controlled by the momentum wave function phase. Under certain conditions, non-Gaussian wave packets exhibit different average arrival time delays than their Gaussian counterparts. When the mean transmitted momentum is significantly different than the mean incident momentum, the phase time delay of a non-Gaussian wave packet is shifted from that of a Gaussian wave packet due to the non-linear terms in its momentum wave function phase. Differences in the average arrival time delays are primarily observed for wave packets scattering from wide barriers, where transmission of over-the-barrier momentum components is the primary scattering mechanism. At low energies, the probability of reflection increases, which leads to an increased self-interaction time that becomes the dominant contribution to the time delay. The different spatial density profiles for Gaussian and non-Gaussian wave packets cause the self-interaction times to vary by wave packet type. We have also demonstrated that the arrival time of a wave packet can be controlled by adjustment of the momentum wave function phase. Such control is only possible with non-Gaussian wave packets and numerical and analytical calculations for a wave function with an arbitrary cubic phase showed that the relative average arrival time delay was linearly dependent on the cubic term’s coefficient, as predicted by the relative phase time delay. Thus, wave packets with identical incident energies and momentum densities can be designed to arrive on the far side of the barrier at differing and controllable times. As interest in spatially structured electron beams grows, so too will the need to understand their dynamical properties. The difference in interaction times between the Gaussian and non-Gaussian wave packets shown here adds to the list of unique features exhibited by these ave packets and may open the door to new applications. For example, current experimental techniques allow for measurements on the timescales of a few attoseconds, making the relative average arrival time delays predicted here large enough to be measured. Thus, an experiment comparing average arrival time delays of Airy and Gaussian wave packets could provide insight into the validity of the various arrival time definitions. Additionally, the possible control of arrival time may be of use in electronics and microscopy applications where electron transmission and tunneling are the dominant mechanisms, as well as in applications where electron diffraction and interference are used. Finally, one can imagine possible benefits of arrival time control in studies using pump-probe techniques, where the relative timing of the pump and probe pulses is crucial.
Acknowledgements
We gratefully acknowledge the support of the NSF under Grant No. PHY-1912093.
Appendix A
We are interested in calculating the phase time delays of Airy, inverted Airy, and Gaussian wave packets using the stationary phase approximation. Following the derivation of [20], a time-evolved wave packet initially centered at 𝑥𝑥 = 𝑥𝑥 may be written as 𝜓𝜓 ( 𝑥𝑥 , 𝑡𝑡 ) = ∫ 𝜙𝜙 ( 𝑝𝑝 , 0) ∞−∞ 𝑢𝑢 𝑝𝑝 ( 𝑥𝑥 ) 𝑒𝑒 −𝑖𝑖 ℏ𝑝𝑝22𝑚𝑚 𝑡𝑡 𝑑𝑑𝑝𝑝 (A1) where 𝜙𝜙 ( 𝑝𝑝 , 0) is the initial momentum wave function and 𝑢𝑢 𝑝𝑝 ( 𝑥𝑥 ) is an energy eigenstate with momentum p . For the transmitted wave packet to the right of the barrier, the energy eigenstate can be written as 𝑢𝑢 𝑝𝑝 ( 𝑥𝑥 ) = 𝐶𝐶𝑒𝑒 𝑖𝑖𝑝𝑝𝑥𝑥 , (A2) here 𝐶𝐶 = − 𝑒𝑒 −𝜌𝜌𝜌𝜌 𝑒𝑒 −𝑖𝑖𝑝𝑝𝑖𝑖 𝑒𝑒 −𝑖𝑖𝑖𝑖 and | 𝐶𝐶 | is the transmission coefficient. The quantities 𝜌𝜌 , 𝑅𝑅 , 𝜃𝜃 , and 𝛽𝛽 are expressed in terms of 𝑝𝑝 and incident wave number 𝑘𝑘 𝜌𝜌 = 𝑘𝑘 − 𝑝𝑝 (A3) cos 𝜃𝜃 = 𝑝𝑝𝑘𝑘 (A4) sin 𝜃𝜃 = 𝜌𝜌𝑘𝑘 (A5) 𝑅𝑅𝑒𝑒 −𝑖𝑖𝑖𝑖 = 𝑒𝑒 𝑒𝑒 −2𝜌𝜌𝑖𝑖 − 𝑒𝑒 −2𝑖𝑖𝜃𝜃 (A6) 𝑅𝑅 = √ 𝑒𝑒 −4𝜌𝜌𝑖𝑖 − 𝑒𝑒 −2𝜌𝜌𝑖𝑖 cos 4 𝜃𝜃 (A7) tan 𝛽𝛽 = coth 𝜌𝜌𝐿𝐿 tan 2 𝜃𝜃 . (A8) Inserting the momentum wave function for the Gaussian wave packet of Eq. (4) into Eq. (A1) yields 𝜓𝜓 𝐺𝐺 ( 𝑥𝑥 , 𝑡𝑡 ) = −1√2𝜋𝜋 ∫ 𝜎𝜎 𝜋𝜋 / 𝑒𝑒 −𝑖𝑖𝑝𝑝𝑥𝑥 𝑒𝑒 −𝜎𝜎 ( 𝑝𝑝−𝑘𝑘 ) / −2𝜌𝜌𝜌𝜌 𝑒𝑒 −𝑖𝑖𝑝𝑝𝑖𝑖 𝑒𝑒 −𝑖𝑖𝑖𝑖 𝑒𝑒 𝑖𝑖𝑝𝑝𝑥𝑥 𝑒𝑒 −𝑖𝑖 ℏ𝑝𝑝22𝑚𝑚 𝑡𝑡 𝑑𝑑𝑝𝑝 . (A9) By assuming that the phase 𝑓𝑓 of the wave packet in Eq. (A9) is stationary at the right edge of the barrier ( 𝑥𝑥 = 𝐿𝐿 ), an equation of motion can be found that yields a time of arrival at the right edge of the barrier, i.e. 𝑑𝑑𝑓𝑓𝑑𝑑𝑝𝑝 � 𝑘𝑘 𝑇𝑇 = 0 , (A10) where ℏ𝑘𝑘 𝑇𝑇 is the mean momentum of the transmitted wave packet. For the Gaussian wave packet, the phase is 𝑓𝑓 = 𝑝𝑝 ( 𝑥𝑥 − 𝑥𝑥 − 𝐿𝐿 ) − 𝜋𝜋2 − 𝛽𝛽 − ℏ𝑝𝑝 𝑡𝑡2𝑚𝑚 (A11) and the corresponding equation of motion is 𝑥𝑥 = 𝑥𝑥 + 𝐿𝐿 + 𝑑𝑑𝑖𝑖𝑑𝑑𝑝𝑝 � 𝑘𝑘 𝑇𝑇 + ℏ𝑘𝑘 𝑇𝑇 𝑡𝑡𝑚𝑚 . (A12) Setting 𝑥𝑥 = 𝐿𝐿 and solving for t gives 𝑖𝑖𝐺𝐺 = −𝑚𝑚�𝑥𝑥 + 𝑑𝑑𝑑𝑑𝑑𝑑𝑝𝑝 � 𝑘𝑘𝑇𝑇 �ℏ𝑘𝑘 𝑇𝑇 . (A13) Similar derivations for the Airy and inverted Airy wave packets yield 𝑡𝑡 𝑖𝑖𝐴𝐴 = 𝑚𝑚�−𝑥𝑥 − 𝑑𝑑𝑑𝑑𝑑𝑑𝑝𝑝 � 𝑘𝑘𝑇𝑇 + ( 𝑘𝑘 𝑇𝑇 −𝑘𝑘 ) −𝛼𝛼 �ℏ𝑘𝑘 𝑇𝑇 = 𝑡𝑡 𝑖𝑖𝐺𝐺 + 𝑚𝑚� ( 𝑘𝑘 𝑇𝑇 −𝑘𝑘 ) −𝛼𝛼 �ℏ𝑘𝑘 𝑇𝑇 (A14) and 𝑡𝑡 𝑖𝑖𝐼𝐼𝐴𝐴 = 𝑚𝑚�−𝑥𝑥 − 𝑑𝑑𝑑𝑑𝑑𝑑𝑝𝑝 � 𝑘𝑘𝑇𝑇 − ( 𝑘𝑘 𝑇𝑇 −𝑘𝑘 ) +𝛼𝛼 �ℏ𝑘𝑘 𝑇𝑇 = 𝑡𝑡 𝑖𝑖𝐺𝐺 − 𝑚𝑚� ( 𝑘𝑘 𝑇𝑇 −𝑘𝑘 ) −𝛼𝛼 �ℏ𝑘𝑘 𝑇𝑇 . (A15) These are the stationary phase times for wave packet arrival at the right edge of the barrier. The corresponding phase time delays can be found by subtracting the non-interacting wave packet arrival time at 𝑥𝑥 = 𝐿𝐿 from the scattered wave packet arrival time Δ𝑡𝑡 𝜁𝜁𝑖𝑖 = 𝑡𝑡 𝑖𝑖𝑖𝑖 − 𝑚𝑚 ( 𝑖𝑖−𝑥𝑥 ) ℏ𝑘𝑘 , (A16) where 𝐴𝐴 = 𝐺𝐺 , 𝐴𝐴 , 𝐼𝐼𝐴𝐴 for Gaussian, Airy, and inverted Airy wave packets. Using Eqs. (A13) - (A16), the phase time delays can be written as in Eqs. (14) and (15)
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