Tunneling time in coupled-channel systems

This paper presents a coupled-channel formalism to describe the tunneling time of a quantum particle through composite compounds with multiple energy levels or complex structures, which can be modeled as quasi-one-dimensional multi-channel systems.

The Gist

The Big Question: How Long Does a "Ghost" Take to Walk Through a Wall?

In the quantum world, particles like electrons can do something impossible in our daily lives: they can "tunnel" through a barrier they shouldn't be able to cross, like a ghost walking through a brick wall. For decades, physicists have argued about one specific question: How long does this tunneling take?

Is it instant? Does it take a nanosecond? The paper you are asking about tackles this mystery, but with a twist. Instead of looking at a simple, flat wall, the authors look at a "complex compound"—a barrier that is more like a busy train station with multiple tracks and waiting rooms than a simple brick wall.

The Main Idea: It's Not Just One Path

Imagine you are trying to get through a crowded airport security checkpoint.

• The Old Way (Single Channel): In the past, scientists mostly studied a scenario where you walk through a single, straight line. You enter, you pass through, you exit. The time it takes is calculated based on how fast you move and how long the line is.
• The New Way (Coupled-Channel): This paper says, "Wait, real life is messier." Sometimes, while you are in line, you might get distracted, step into a side room to check your phone, or even switch lines entirely. The barrier isn't just a wall; it has internal "rooms" (energy levels) that the particle can visit.

The authors created a new mathematical "map" (a formalism) to track how long a particle spends in this complex system, accounting for the fact that it might bounce between different "rooms" or "tracks" while trying to get through.

The "Two-Stopwatch" Analogy

One of the most famous ideas in this field is that tunneling time isn't just one number; it's actually two numbers mixed together, like a complex number (a real part and an imaginary part).

Think of it like this:

1. Stopwatch A (The Real Time): This measures the actual "delay" or how long the particle seems to hang around in the barrier.
2. Stopwatch B (The "Wobble" Time): This measures the uncertainty or the "fuzziness" of the measurement.

In the past, scientists had a formula for Stopwatch A that worked well for simple, elastic collisions (where the particle hits the wall and bounces off without changing its internal state). However, this paper argues that when the particle interacts with a complex object (like an atom that can get excited), the old formula breaks.

The authors show that you have to look at the phase (the timing of the wave) of the particle as it moves through these different channels. They prove that the "time" is related to how the particle's wave changes as it navigates these internal rooms.

The Two Examples They Used

To prove their theory works, the authors looked at two specific scenarios:

1. The "Bouncy Ball" and the "Springy Box"
Imagine an electron (the ball) hitting a composite object (the box).

• Scenario A: The ball hits the box, and the box stays the same. The ball bounces off. (Elastic scattering).
• Scenario B: The ball hits the box, and the box gets excited (like a spring compressing) before the ball leaves. (Inelastic scattering).
  The authors' new math handles both scenarios at once, showing how the "time" changes when the box starts vibrating or changing states.

2. The "Narrow Hallway" (Waveguide)
Imagine a long, narrow hallway (a waveguide) where people (electrons) can only walk forward. However, the hallway has a width, and people can wiggle side-to-side.

• If the hallway is wide enough, people can walk in different "lanes" (modes).
• Sometimes, people get stuck in a "dead end" lane (evanescent modes) where they don't move forward but still affect the people in the main lanes.
  The paper shows that even these "dead end" lanes, where no current flows, secretly influence how long it takes for the main group to get through. It's like a crowd in a hallway: even if someone is just standing in a side alcove, their presence changes how the main crowd flows.

The "Sum of Parts" Discovery

A key finding in the paper is about how to calculate the total time.

• Old Intuition: You might think that if a particle takes a detour into a side room, the total time is just the sum of the time in the main room plus the time in the side room.
• The Paper's Result: They confirmed that for the "real" part of the time, you can add up the times spent in each specific channel (each lane or room) to get the total time.
• The Catch: However, when you look at the "imaginary" part (the uncertainty/wobble), things get weird. If you try to calculate the time spent specifically between two different channels (like the time spent switching from Lane 1 to Lane 2), the math can give you a negative number.

The authors explain that a negative time here doesn't mean the particle traveled back in time. It just means that this specific "cross-channel" time isn't a physical clock you can hold in your hand. It's a mathematical artifact that describes the complex relationship between the different paths, not a duration you can measure with a stopwatch.

Summary

In simple terms, this paper provides a new, more flexible rulebook for calculating how long quantum particles take to tunnel through complex, multi-layered barriers.

• It moves beyond simple "straight-line" tunneling to include complex interactions where particles change states or get distracted.
• It confirms that total tunneling time is the sum of the times spent in all the different "lanes" the particle could take.
• It clarifies that while some parts of this time calculation make physical sense, other parts (specifically the "cross-talk" between lanes) are mathematical tools rather than measurable clock times.

The authors hope this new framework will help experimentalists better interpret data from modern "attoclock" experiments, which try to measure these incredibly fast tunneling events with extreme precision.

Technical Summary

1. Problem Statement

The paper addresses the long-standing debate regarding the definition and measurement of quantum tunneling time, particularly in systems where inelastic scattering and internal excitations play a significant role.

• Context: Traditional tunneling time theories (e.g., Büttiker-Landauer time) often rely on single-channel, elastic scattering approximations. In these cases, the tunneling time is typically derived from the energy derivative of the scattering phase shift ($\tau = d\delta/dE$).
• The Gap: Many physical systems, such as electrons scattering off molecules with internal energy levels, atoms in hyperfine states, or particles in quasi-one-dimensional (Q1D) waveguides, involve coupled channels where transitions between ground and excited states (inelastic scattering) occur.
• Challenge: Existing single-channel formalisms fail to properly account for inelastic effects (characterized by an inelasticity parameter $\eta < 1$) and the complex interplay between propagating and evanescent modes in multi-channel systems. The authors aim to generalize the tunneling time formalism to these coupled-channel scenarios.

2. Methodology

The authors develop a rigorous coupled-channel formalism based on scattering theory and Green's functions.

• Theoretical Framework:
  • They utilize the Lippmann-Schwinger equations for multi-channel systems to define the wave functions and scattering matrices ($S$-matrix).
  • They employ the Muskhelishvili-Omnès (MO) representation, which relates the determinant of the transmission amplitude matrix to the $S$-matrix via a dispersion relation.
  • They generalize the Friedel formula, connecting the integrated Green's function to the energy derivative of the determinant of the $S$-matrix.
• Definition of Tunneling Time:
  • The traversal time is defined as a complex quantity $\tau_E = \tau_2 + i\tau_1$, where $\tau_1$ corresponds to the Büttiker-Landauer time (related to the density of states) and $\tau_2$ relates to the Landauer resistance/uncertainty.
  • The total time is derived from the trace of the integrated Green's function over the barrier region: $\tau_E = -\int_{-L}^{L} dx \, \text{Tr}[\langle x | \hat{G}(E) | x \rangle]$.
• Specific Models Analyzed:
  1. Exactly Solvable Two-Channel Model: A system with contact interactions (delta potentials) representing an electron scattering off a composite compound with a ground state ($X$) and an excited state ($X^*$). This allows for analytical derivation of transmission/reflection amplitudes and phase shifts.
  2. Quasi-One-Dimensional (Q1D) Waveguide: A 2D/3D system confined in one direction (e.g., a waveguide with impurities). This reduces the problem to a multi-channel scattering problem where transverse confinement creates discrete sub-bands (channels). The authors analyze how evanescent modes (non-propagating states) influence the tunneling time.

3. Key Contributions

• Generalization of Tunneling Time: The paper provides a formal definition of tunneling time for systems with inelastic scattering. It demonstrates that in the presence of inelasticity ($\eta < 1$), the tunneling time is not simply the derivative of the scattering phase shift ($\delta$), but rather the derivative of the phase of the transmission amplitude ($\phi$).
  • Relationship: $\tau_1 = d\phi/dE$, where $\phi$ depends on both $\delta$ and $\eta$.
• Decomposition of Time Components: The authors introduce a decomposition of the total traversal time into:
  • Diagonal components ($\tau^{(ii)}_E$): Time spent within a specific channel $i$ (elastic-like contribution within the channel).
  • Off-diagonal components ($\tau^{(ij)}_E$): Time associated with transitions between different channels $i$ and $j$.
• Role of Evanescent Modes: In Q1D systems, the authors highlight that evanescent modes (which do not carry current) significantly influence the scattering matrix and the tunneling time, particularly through correction terms involving reflection amplitudes. These effects are absent in simple 1D models but crucial in mesoscopic systems.
• Additivity of Time: They show that the total traversal time in a coupled-channel system is the sum of the traversal times of individual channels: $\tau_E = \sum_i \tau^{(ii)}_E$.

4. Key Results

• Analytical Expressions: For the two-channel contact interaction model, the authors derived explicit formulas for the transmission matrix determinant and the complex traversal time.
  • The determinant of the transmission matrix is expressed as: $\det[t(E)] = \frac{\cos(\delta_1 + \delta_2) + \eta \cos(\delta_1 - \delta_2)}{2} e^{i(\delta_1+\delta_2)}$.
  • The traversal time includes terms dependent on the system size ($L$) and reflection amplitudes, highlighting finite-size effects.
• Inelasticity Impact:
  • As inelasticity increases ($\eta \to 0$), the off-diagonal transmission amplitudes dominate, and the simple phase-shift derivative formula fails.
  • The imaginary part of the off-diagonal time component ($\text{Im}[\tau^{(12)}_E]$) can be negative, indicating that these components cannot be interpreted as "time" in the classical sense, unlike the diagonal components which remain positive.
• Hartman Effect in Inelastic Scattering: The study suggests that the Hartman effect (saturation of tunneling time with barrier width) persists even in inelastic scattering scenarios where the potential becomes non-Hermitian and the $S$-matrix is non-unitary.
• Q1D Waveguide Application: In the waveguide model, the tunneling time expression includes a correction term involving reflection amplitudes ($r_{nn}$). This term vanishes at Fano resonances due to evanescent modes, a behavior distinct from pure 1D systems.

5. Significance and Implications

• Experimental Relevance: The formalism provides a theoretical basis for interpreting recent attoclock experiments and scattering experiments involving hyperfine splitting (e.g., in $^{87}\text{Rb}$ atoms), where inelastic effects are non-negligible.
• Complex Potentials: The work supports the use of complex potentials to model absorption and inelasticity, linking tunneling time to the propagation of quantum waves in absorbing media.
• Mesoscopic Physics: By clarifying the role of evanescent modes and off-diagonal time components, the paper offers new insights into quantum transport in disordered systems, conductance, and the Hall effect in Q1D structures.
• Theoretical Unification: It unifies the description of tunneling time across elastic and inelastic regimes, extending the Büttiker-Landauer and Friedel formalisms to a broader class of quantum systems.

In conclusion, this paper establishes a robust mathematical framework for calculating tunneling times in complex, multi-level quantum systems, resolving ambiguities regarding inelastic scattering and providing tools to analyze time delays in modern quantum transport experiments.