The search for a consistent definition of the time it takes for a particle to traverse a classically forbidden region is an enduring problem in quantum mechanics (see, e.g., [1, 2] and references therein). The difficulties involved in the problem are both experimental and theoretical. The time scales involved, typically of the order of attoseconds, only recently became accessible experimentally [3–5], and the interpretation of the experimental results is not yet free of controversy [5, 6]. From the theoretical point of view, the difficulties are related to the impossibility of defining a self-adjoint (observable) time operator in quantum mechanics. Thus, there arises the need for operational definitions of quantum tunneling time.

Several competing definitions of tunneling time exist, among which the most well-known and useful are the phase time [7], the dwell time [8], the Larmor time [9, 10], and the Salecker-Wigner-Peres (SWP) clock time [11–14]. However, all of these tunneling times can lead to counter-intuitive results in specific situations. In particular, all of them can result in superluminal speeds for particles tunneling through an opaque barrier, the so-called Hartman effect (HE) [15], and thus their status as traversal times is controversial.

The HE can be generalized for multiple potential barriers, in which case it has been established that in the opaque limit (low probability of transmission) the phase time becomes independent of the barrier spacing [16]; this so-called generalized Hartman effect (GHE) has been verified both in non-relativistic [16–19]) and relativistic [20] quantum mechanics and it is usually assumed to hold for other time scales, such as the dwell time [21]. At first sight the GHE seems to be even more troublesome than the original version of the HE, since one would expect that for the free region in between the barriers the traversal time would be well described by the phase time.

The vast literature discussing tunneling times and the HE, as well as its generalized version, focuses mostly in one-dimensional square potential barriers, with few other potentials such as the sech-square potential [22] and Dirac delta barriers (see, e.g., [21, 23, 24] and references therein) having been considered. From a practical point of view, the understanding of one-dimensional tunneling may help to understand phenomena in nanoscale systems such as, for example, conductance in nanowires (see [25] and references therein). However, to reach a better understanding of tunneling times and the associated questions, it is important and useful to study a larger variety of potentials, such as the generalized one-dimensional point interactions.

Point interactions in one-dimensional quantum mechanics have been the source of significant theoretical and mathematical interest for providing the opportunity to investigate regularization and renormalization schemes in the framework of quantum mechanics (see, e.g., [26–29]). There have also been controversies in the literature due to conflicting regularization-dependent results obtained in the treatment of the so-called δ′-interaction [30–33]. These controversies, however, were clarified by using mathematically rigorous methods such as self-adjoint extensions (SAE) [34–37] and Schwartz's distribution theory [38, 39] (there are also alternative approaches using non-standard distribution theory, e.g., [40, 41]), both of which unequivocally demonstrate the existence of a four-parameter family of point interactions in one dimensional quantum mechanics—these include, but are not restricted to, the δ interaction as well as other interactions commonly associated with the δ′ in the literature. It should be noticed that the distributional approach [39] makes it unambiguously clear, from parity considerations, that a “genuine” δ′ interaction does not exist (for a particular regularization of the δ′ interaction displaying even behavior under parity transformations see [42]).

From the point of view of applications, point-like interactions have long provided models of significant experimental relevance for lower dimensional systems [43, 44], such as the Girardeau-Tonks gas [45, 46] and the Lieb-Liniger model [47, 48] and, more recently, as a zero-range model of ultra-thin heterostructures [49, 50]. The theoretical study of tunneling across point barriers is another important application of these interactions, since it has the important feature that one does not need to be concerned with above-the-barrier components when considering tunneling of wave packets, while for barriers of finite height such components dominate the transmitted packet in the case of opaque barriers and lead to difficulties in the interpretation of tunneling times—tunneling across point barriers is, therefore, free of such difficulties.

The general four-parameter family of interactions in one dimension has been used to investigate tunneling times for a non-relativistic particle incident on a single point scatterer [51]. However, in order to generate a richer tunneling structure and to investigate some of the more counterintuitive phenomena associated with tunneling times it is necessary to consider (at least) two point scatterers. For example, it is not difficult to convince oneself that for a single point barrier the SWP clock time (and the dwell time) must be trivially zero, while for two point barriers it is non-trivial. Futhermore, in addition to its relevance to the study of resonant tunneling in heterostructures in the zero-range limit [49, 50], the presence of two point barriers allows us to investigate the GHE—in contradistinction, the HE cannot be addressed by considering only one point scatterer, since the distance “traveled” in that case vanishes (the support of a group of point interactions is a set of zero measure). Therefore, in this work we investigate the scattering of a non-relativistic particle by two point scatterers given by the most general potential [34–36, 39]. Given the relevance of parity-symmetric considerations in the controversial discussion of regularized implementations of point interactions, and the related discussion of the δ′-interaction, we start by introducing the singular interaction following the distributional approach introduced in Lunardi and Manzoni [39], and analyzing the behavior of such potential under parity transformations, thus obtaining the necessary conditions to have even and odd potentials under such transformations. We also analyze the limits in which the separation between the barriers shrinks to zero and extend the work of Šeba [35] to include the limit for a renormalized¹ odd-arrangement of two δ′-interactions². Then, we proceed to calculate the phase and SWP clock times for barriers with well defined parity symmetry, analyze the behavior of these times in the opaque regime and discuss whether GHE occurs or not for these potentials (a preliminary report on tunneling times, not accounting for the symmetry of the interactions, appeared in Lee et al. [52]).

Let us consider the scattering of a non-relativistic particle of mass m and energy E=ℏ2k22m by two general point interactions localized at x = x₁ and x = x₂, with x₁<x₂. According to Calçada et al. [39] the distributional Schrödinger equation can be written as (throughout this paper we will adopt the atomic Rydberg units such that ℏ = 1 and 2m = 1, as usual when dealing with point interactions)

where the prime indicates a space derivative, and the interaction distributions are given by

or

with a_j, b_j, c_j, d_j ∈ ℝ, a_jd_j − b_jc_j = 1 and θ_j ∈ [0, π), j = 1, 2. In these expressions we used the notation ψ(xj±)=limε→0+ψ(xj±ε), and similarly for ψ′(xj±). Such interaction terms correspond to non-separated solutions, and will be the only type of interactions considered in this work. The above distributional Schrödinger equation implies the same boundary conditions at the point singularities as those obtained from the SAE method [39, 53] (also see [54] for a rigorous study of the boundary conditions for point interactions):

The general solution for the Schrödinger equation in the region outside the double potential is

where

with k and q being the wave numbers in the region outside and inside the double barrier, respectively, and³

with the transfer matrix given by:

and

From Equations (5) and (6) we can write

From the above expressions we observe that when x₂→x₁ the double point interaction converges to a single point interaction, with the transfer matrix given by Γ = Λ₂Λ₁, i.e., in this limit the composition law for the parameters has the U(1) × SL(2, R) group structure (see [55] for the group structure of a restricted class of double point interactions). The explicit form of the interaction term s[ψ] in Equation (1) is suitable to study its symmetry properties [39]. To this end, and without losing generality, let us assume x₂ = −x₁ ≡ L > 0. Then, under a parity transformation this term changes as

where

The interaction is even if s~[ψ]=s[ψ] and it is odd if s~[ψ]=-s[ψ] (for details see [39]). These conditions, taking Equation (3) into account, imply that the double general interactions with well defined parity symmetry are characterized by the following constraints:

– Even Interactions

which is equivalent to the two b.c. matrices (redefining the parameters as θ₁ ≡ θ, d₁ ≡ d, b₁ ≡ b, c₁ ≡ c and a₁ ≡ a)

From the above, the even arrangements of two general point interactions constitute a four parameter family of interactions.

– Odd Interactions

and the conditions a_id_i − b_ic_i = 1 (i = 1, 2) further imply a₂ = a₁ (thus d₂ = d₁), with b₁c₁ ≤ 0.

Redefining b₁ ≡ b and c₁ ≡ c, odd interactions can be conveniently characterized by the b.c. matrices

with bc ≤ 0 and s = ±1. Thus, the odd arrangements of two general point interactions constitute a three parameter family of interactions.

From Equation (14) one can see that, given an arbitrary point interaction, it is always possible to find a suitable second point interaction resulting on an even arrangement⁴; on the other hand, odd double point interactions are more restrictive and can only be built by combining very specific kinds of point interactions, as revealed by the constraints on the parameters of the two b.c. matrices indicated in Equation (16). Particularly interesting among the even arrangements of double-interactions are those built from two delta (δ) interactions (θ = 0, a = d = 1, b = 0, c arbitrary), two “delta prime” (δ′) interactions (θ = 0, a = d = 1, c = 0, b arbitrary) and two δ⁽¹⁾ interactions (θ = 0, b = c = 0, a = d⁻¹ arbitrary). Among the odd arrangements we are interested in those built from two δ or two δ′ (it is not possible to built an odd arrangement from two δ⁽¹⁾ interactions). As already mentioned in Section 1, neither δ′ nor δ⁽¹⁾ stand for a “genuine” derivative of a δ interaction—they simply refer to particular choices of parameters in the general point interaction [34, 35, 36] (a recent proposal for physically interpreting the boundary conditions given by single point interactions is given in Kulinskii and Panchenko [58]).

It is well known that single point interactions are either even under parity or have no defined symmetry, that is, there exists no single point interaction which is odd under parity [39] (for instance, both a single δ and a single δ′ are even interactions, whereas a single δ⁽¹⁾ has no defined parity). Thus, there naturally arises the question: what is the resulting symmetry of an interaction obtained by shrinking the inter-barrier spacing to zero for an initial double point barrier with well defined odd symmetry? It is simple to show that in this limit, characterized by taking x₂ → x₁ in Equation (8), the resulting interaction either does not have a defined symmetry or vanishes (when bc = 0), a result in accordance with the conclusion that there is no possible regularization capable of producing an odd single point interaction [39]. The equivalent limit for an even double point barrier results, unsurprisingly, in an even point interaction.

Šeba [35] investigated the limit L → 0 of the renormalized dipole interaction (1/2L^ν)[δ(x + L) − δ(x − L)], in the context of SAE, and found that for ν<12 this renormalized odd interaction converges to the free case, for ν=12 it converges to the (even) delta interaction and for ν>12 such a limit results in separated solutions (i.e., free Hamiltonians on the two half-lines defined from the origin). Šeba's results can be immediately reproduced in the formalism developed above by assuming the parameters for an odd arrangement of delta interactions in Equation (16), b = 0 and c ≡ λL^−ν, and taking the limit L → 0 of Γδν(Λ1(o),Λ2(o),k,- L,L), where we introduced the indices δ and ν to indicate the odd arrangement of deltas and the dependence on the parameter ν, respectively.

Let us now extend the above analysis to the case of a renormalized odd arrangement of “delta prime” interactions. Thus, choosing b ≡ λL^−ν and c = 0 in Equation (16) we obtain

where we followed the same notation of the previous paragraph and the subscript δ′ indicates the odd arrangement of “delta prime” interactions. From Equation (17) we have the limits

where 1 indicates the 2 × 2 identity matrix. For ν ≥ 1, similarly to Equation (20), the limit will result in matrices whose elements have improper limits [albeit different from Equation (20)].

From Equations (18) to (20), it follows that the renormalized odd arrangement of δ′ interactions converges to the free case for ν<12. For ν=12 the limit (Equation 19) seems, at first, to result in a single (even) δ′ interaction. However, the k²-dependence in the limiting transfer matrix would imply an interaction distribution with singular order +3 in Equation (1) and would violate the tenets of the distributional approach, which allows a maximum singular order of +1 [39]—otherwise one would have singular wave functions from which no physically meaningful solutions of the Schrödinger equation can be obtained⁵. Therefore, interactions such as Equation (19) are not allowed by the distributional approach to general singular interactions (nor are they allowed by the SAE method), and for ν=12 the zero range limit of a renormalized odd arrangement of δ′ interactions does not converge to a well defined (in the sense of distributions) singular interaction. In the same way, the k²-dependence in Equation (20), and indeed for all ν>12, means that for these values of ν the odd arrangement of δ′ interactions does not converge to a generalized point interaction.

In this section we will consider the tunneling times for a non-relativistic particle scattered by two point barriers. We are mainly interested in the time associated with the transmission across the double barrier as given by both a SWP clock, which runs only when the particle is in this region, and by the phase time.

In order to calculate the SWP clock time, it is sufficient for the purposes of this work to notice that the effect of the particle-clock coupling is equivalent to introduce a weak potential V(x) equal to a constant V₀ (V₀ ≪ E) in the region (x₁, x₂) and zero otherwise (for an in depth account of the SWP clock see [12, 13]). Introducing this coupling into the Schrödinger Equation (1) it becomes

where θ(x) is the Heaviside theta function.

Assuming a particle incident upon the barriers from the left, the reflection and transmission amplitudes [R(k, q) and T(k, q), respectively] can be immediately obtained from the formalism developed in Section 2. Then, by considering a unit incident flux, we have A₁ = 1, A₂ = R(k, q), C₁ = T(k, q) and C₂ = 0, and using Equation (6) the incident and reflection amplitudes can be written in terms of the matrix elements of Equation (7) as

where in the last equation we have used the fact that det M=e2i(θ1 + θ2), as can be promptly verified from the conditions on the parameters in Equation (4) and from Equations (7) to (9).

For the purpose of calculating tunneling times the transmission and reflection amplitudes given by Equation (22), with k≡E and q≡k2-V0, are more conveniently expressed as

where φ_T(k, q) is the phase difference, at any time t, between the transmitted wave at x₂ and the incident wave function at x₁, whereas φ_R(k, q) is the phase difference, at any time t, between the reflected and the incident waves at x₁. Then, the SWP clock times for transmission and reflection (sometimes called the conditional dwell times [21]) are given by [12, 13]

and satisfy the identity τD(k)=|T(k,k)|2tcT(k)+|R(k,k)|2tcR(k), where τ_D(k) is the dwell time [13].

Similarly, the transmission and reflection phase times are given by [10, 15]

Since the point interaction parameters θ₁ and θ₂ cause a simple shift in the phase difference, and due to its independence of the energy E and the perturbation potential V₀, it follows that both the clock and the phase times are independent of these parameters—for even and odd arrangements, Equations (13) and (15), there is, in fact, no change in phase due to θ₁ and θ₂ (an in-depth analysis of the consequences of the phase factor, arising at the boundary conditions, for one-dimensional quantum tunneling is presented in Furuhashi et al. [59]). In what follows we consider only arrangements of point interactions with well-defined symmetry under parity transformations, according to Equations (13)–(16), and obtain explicit results for the clock and phase times. The tunneling times are further analyzed for special cases of one-parameter arrangements.

An analysis of the symmetry under parity transformation and of the tunneling times was provided for a one-dimensional non-relativistic particle scattered by a double barrier composed of two general point potentials.

Our general analysis of the symmetry shows that it is possible to construct potentials with well-defined parity, both odd and even, for such double barriers of point interactions. This is in stark contrast with the case of a single point interaction, for which no odd interaction exists [37, 39, 64]—as illustrated in Section 2 by the fact that the limit L → 0 of a renormalized δ-dipole interaction, an odd arrangement of two δ [35], is either trivial or results in a point interaction which is even or has no defined parity. Furthermore, we extended Šeba's [35] procedure to the case of a renormalized odd arrangement of δ′ interactions and found that in the limit L → 0 this arrangement is either trivial (for ν < 1/2) or it does not converge to a general singular interaction (for ν ≥ 1/2). In other words, for ν ≥ 1/2 such limit is not of the form (Equation 2) [equivalently, (Equation 4)] and has no self-adjoint extension as well.

We then proceeded to calculate the quantum tunneling times for the double point barriers with well defined parity. We calculated the phase and SWP clock times for three special cases of even arrangements [pairs of δ, δ′ or δ⁽¹⁾ satisfying the constraints (Equations 13 and 14)] as well as for two odd arrangements of double point barriers [pairs of δ or δ′ satisfying (Equation 15)–(Equation 16)]. Even though the reflection tunneling times were also provided for completeness, our main concern was the behavior of the transmission times in the opaque regime and the existence of the so-called generalized Hartman effect [16].

We found that, for all interactions considered, both the phase and SWP clock times approach zero in the extreme opaque limit. However, in all cases, for any finite values of the interaction strength (for the even arrangement of δ⁽¹⁾s, also γ≠0) the tunneling times do depend on the interbarrier spacing, and such dependence is essentially linear for off-resonance values of the parameters. It is only strictly for the interaction strength at the extreme opaque value (|γ| = ∞ or γ = 0 for the even double-δ⁽¹⁾; |γ| = ∞ for all other interactions) that the times do not depend on L, since in this case their off-resonance value is zero. However, for these extreme values of the parameters the very concept of tunneling time looses its meaning, since there is no transmission (the barriers become impenetrable and thus the solutions in both sides of each barrier are separable). Therefore, for the potentials analyzed the GHE is just a mathematical artifact of such an extreme limit and devoid of any physical meaning. As a final remark we stress that even if the GHE is an artifact, its more striking consequence, which is the apparent superluminality of tunneling, is not precluded—in fact, superluminality does not require GHE (or HE), and the slow growth of the tunneling times with respect to the interbarrier distance in the opaque regime could result, for sufficiently separated barriers, in an apparent superluminal transmission. However, any apparent violation of causality is avoided by interpreting the superluminal phase and clock times respectively in terms of wave packet reshaping and weak measurements [21, 23] (for a recent treatment of tunneling times from the point of view of weak measurement theory see [65]). It should also be noticed that apparent superluminality is more properly addressed in a relativistic theory. The above results on the GHE reinforce similar conclusions reached in Winful [1], Lunardi and Manzoni [20], Aharonov et al. [21] and Lunardi et al. [24]—an analysis of the GHE is, unavoidably, case-by-case and, thus, this work adds to the body of results clarifying such effect.

JL and LM contributed equally and in all stages of this work. ML and EN contributed in the calculation of tunneling times.