1. Introduction

The length of time during which quantum particles tunnel through a barrier has attracted considerable attention for both fundamental and technological reasons since the 1930s [1–8]. Hartman calculated the tunneling of a wavepacket through a rectangular potential barrier [3] and found that group delay becomes constant as barrier length increases. This phenomenon, known as the Hartman effect, implies that for sufficiently large barriers, the effective group velocity of a particle may be superluminal. Although a number of experiments have reported that the group delays saturate with barrier width thus confirming the Hartman effect [5–7], the definition of tunneling time and its exact physical meaning based on experimental results remain under heated debate [4, 5]. A large number of tunneling time definitions have been proposed, including group delay or phase time [3], dwell time [5], Larmor times [9, 10], and Büttiker-Landauer time [11]. Winful recently proposed that the group delay in tunneling represents a lifetime of stored energy escaping through both sides of the barrier and does not represent a transit time [5, 12]. Thus, the issues of superluminality do not even arise.

Beyond Schrödinger’s nonrelativistic quantum mechanics, the group delay for Dirac particles traveling through a potential well was also studied by using Dirac’s fully relativistic quantum theory [13]. The behavior of Dirac particles is found to be the same as that in nonrelativistic quantum mechanics. Liu et al. recently studied the effect of the electromagnetic waves (EW) on the group delay of electrons and found that the group delay of the transmitted wave packet increases linearly with barrier length [14]. This peculiar tunneling effect is attributed to current leakage in a time-dependent barrier generated via the EW [11, 14]. If the zero-point field is considered, all potential barriers are combined with electromagnetic fields. Thus in the framework of quantum field theory, there is no Hartman effect.

However, in the quantum tunneling of photons, a number of theories based on the special relativistic covariant Maxwell equations have proven that the group delay of photons becomes constant as the length of optical structures increases [4–7, 15–19]. However, whether the Hartman effect still exists with a more accurate theory [e.g. general relativity theory (GRT)] remains unclear. One of the unique predictions in GRT is the existence of GW [20–23]. Thus, within the GRT framework, the optical structures such as MDM may be irradiated by GW. The center frequency and the width of the photonic bandgap (PBG) of the MDM will vary with the GW. Similar to the electron tunnelling in a time-dependent barrier [11], variations of PBG will result in an additional leak EW. Such EW may propagate at the speed of light. Thus, the Hartman effect may be absent in photon tunnelling within the framework of GRT.

GW modulation on the group delay of photons may also provide a different method for the detection of GW. Although there is indirect evidence for GW existence, direct detection of the gravitational wave is both important to the understanding of the cosmology and the general relativity [20–25]. These second generation observatories based on the laser interferometry, Advanced LIGO [23], GEO-HF [24], and Advanced Virgo [25], may allow for the direct detection of GW with the precision $h<{10}^{-20}/\sqrt{Hz}$.

In this paper, we investigate the effect of GW on the group delay of photon passing through MDM. Our simulation shows that with a thick MDM, group delay increases linearly with increasing barrier width. The group velocity is slightly less than the speed of light in in vacuum. We also find that the group delay of tunneling photons is sensitive to GW, which may provide a different method for the detection of GW. In particularly, the MDM comprises alternating dielectric layers and vacuum layers [see Fig. 1(a)]. All layers are nonmagnetic, and the thicknesses of the dielectric layers (vacuum layers) satisfy $D_1={\lambda }_0/4\sqrt{{\epsilon }_1}$ (D₂ = λ₀/4 + ζ₂λ₀/2), where ε₁ = 2.25 is the relative dielectric constant of dielectric layers, λ₀ is the center wavelength of the input electromagnetic pulse, and ζ₂ is a positive integer. The group delay of tunneling photons is generally more sensitive to GW at large ζ₂. We set ζ₂ = 5 in this paper, unless otherwise specified. The electromagnetic pulse (the plane-polarized GW) is incident perpendicular (parallel) to the surface of MDM.

 

Fig. 1 (a) Schematic diagram of the tunneling process in an MDM structure. (b) Group delay for the reflected wave packet and (c) that for the transmitted wave packet as a function of the number of MDM periods. Incident (black lines), tunneled (red lines) and reflected (green lines) pulses with GW amplitude A_GW = 1 × 10⁻⁴ for the following number of MDM periods: (d) n_MDM = 3, (e) n_MDM = 19, and (f) n_MDM = 30. The inset shows the normalized tunneled pulse overlaid with the incident pulse.

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2. Theory

Similar as in the laser interferometer GW detection, the influence of the GW on the EW in MDM can be attributed to the displacement of the dielectric layers [20,26,27]. Thus the proper relative displacement z_R can be given by

3. Numerical results

The propagation of a wave packet through a barrier can be demonstrated by numerically solving Eq. (2) directly. To ensure high precision, the space increment Δx = λ₀/1.5 × 10³ and the time increment Δt = 2 × 10⁻⁴T₀ are used. When the space and time increments are increased or reduced 10 times, the error is less than 3%. Numerical results of the group delay τ_DR (τ_DT), i.e., the delays of the peaks of the reflected (transmitted) pulses, are shown in Figs. 1(b)–(f). This time definition can be easily verified in the experiment. Figure 1(b) shows the group delay for the reflected wave packet as a function of the number of MDM periods n_MDM. Similar to the traditional quantum tunneling, the group delay is saturated by increasing n_MDM, and the saturated group delay is identical to the dwell time. Meanwhile, the group delay for the reflected wave packet is unaffected by the extrinsic GW.

However, the influence of the GW on the group delay of a transmitted wave packet is different. As shown in Fig. 1(c), the group delay for the transmitted wave packet increases linearly with large n_MDM. This result can be explained by the variations of PBG attributed to the GW. Similar to the electron tunnelling in a time-dependent barrier [11, 14], the variations of PBG will result in an additional leak EW. Given that the amplitude of the additional leakage EW attributed GW is quite small, for the case of a small number of MDM periods, e.g., n_MDM < 15, the tunneling EW is significantly larger than the additional leakage EW. The group delay is unaffected by the extrinsic GW [see Fig. 1(c)] and the distortion is minimal [see Fig. 1(d)]. As the number of MDM periods increases, the tunneling rates decrease rapidly. For a large number of MDM periods, e.g, n_MDM = 19, as shown in Fig. 1(e) the amplitude of the additional leakage EW and that of the tunneling EW are comparable, a serious distortion of transmitted wave packet occurs [29]. However, for the case of large number of MDM periods, e.g., n_MDM > 24, the tunneling EW is significantly weaker than the additional leakage EW. No distortion occurs at this scale [see Fig. 1(f)]. Similar to the electron tunnelling in a dynamic barrier [11], the dynamic PBG caused by the GW is not a complete gap and the leak EW is no longer exponential decay with increasing barrier width. Thus, the additional leakage EW will travel at the speed of light and the group delay increases with the MDM width.

Under a weaker GW, the additional leakage EW will determine the group delay for the transmitted wave packet with a relatively larger n_MDM. Specifically, for A_GW = 1 × 10⁻⁴ (A_GW = 2 × 10⁻⁵), the group delay increases linearly when n_MDM > 24 (n_MDM > 26). Thus, if n_MDM is sufficiently large, even under a relatively weak GW (e.g., the GW background radiation [30–32]), no Hartman effect occurs. On the other hand, for a non-strictly periodic GW emitted by various sources(e.g., the chaos compact binary system [33]), the time-dependent variations attributed to the GW will also result in an additional leak EW, thus modifying the group delay. From Fig. 1(c), we can also find that the group velocity of the additional leakage EW is independent of the amplitude of GW. The effective group velocity V_l = L_1o/Δτ_DT ≈ 2.95×10⁸ m/s, where L_1o is the optical path length of each MDM period, Δτ_DT is the corresponding time increment. No superluminal appears.

However, we have to determine whether the group delay of the transmitted wave packet in a thick MDM with GW equates to tunneling time. In traditional quantum tunneling, the consensus is that the group delay does not equate to a tunneling time. However, the group delay of EW with GW is different. The group delay with GW cannot be explained by the dwell time because such delay is considerably larger than the dwell time. For n_MDM = 30, the full width at half maximum (FWHM) of the injected wave packet is approximately 66T₀, which is smaller than the total optical path length of MDM. Meanwhile, the group delay is approximately 80T₀, which is larger than the FWHM of the injected wave packet. The peaks of the injected wave packet and that of the transmitted wave packet are distinguishable. Thus, the group delay of the transmitted wave packet in a thick MDM with GW may be regarded as the tunneling time.

Notably, the group delay in the tunneling process also shows good sensitivity to the GW. For instance, for n_MDM = 25 and A_GW = 1 × 10⁻⁴, the group delay without (with) GW is approximately 5.3T₀ (65T₀). The group delay is increased by approximately 12 times. In the Michelson interferometer GW detection, although the interferometer measures the intensity rather than the time delay of the interference light, we can still make a comparison with Michelson interferometer. For a Michelson interferometer with an arm length of L_o = 75λ₀ (same as the total optical path length of MDM with n_MDM = 25), only when the GW amplitude A_GW is approximately 1.6 × 10⁻³ (i.e., the GW amplitude satisfies 4πA_GWL_o/λ₀ ≈ π/2), the intensity of the interference light with and without GW can vary by approximately 12 times.

The sensitivity of the group delay to the GW depends on the frequency of the GW and the thickness of the vacuum layers [see Fig. 2]. For high-frequency GW, e.g., ω_GW = ω₀, the variations of PBG occur too rapidly, the additional leak EW is relatively weak. The group delay increases remarkably only when n_MDM > 22. However, for a relatively low-frequency GW, the additional leak EW is enhanced. The group delay increases remarkably when n_MDM > 19 for ω_GW = 0.1ω₀. For an extremely low-frequency GW, e.g., ω_GW = 0.005ω₀, the period of the GW becomes larger than the FWHM of EW, and the effect of the GW on the group delay becomes small. For n_MDM < 25, the group delay does not change significantly. For MDM with thicker vacuum layers (i.e., large ζ₂), the GW-induced variation of layer spacing is enhanced, and a larger additional leak EW can be achieved. The group delay increases remarkably when n_MDM > 12 (n_MDM > 19) for ζ₂ = 25 (ζ₂ = 5). On the other hand, the relative group delay τ_DT/L_1o of the additional leakage EW is independent of ζ₂ [see the inset of Fig. 2(b)], which indicates that the pulses propagate with the same group velocity for different ζ₂.

 

Fig. 2 (a) Group delay of the transmitted wave packet as a function of the number of MDM periods for different GW frequencies with A_GW = 1 × 10⁻⁴. (b) Group delay of the transmitted wave packet as a function of the number of MDM periods for different vacuum layer thicknesses with A_GW = 1 × 10⁻⁴, ω_GW = 0.1ω₀, and τ₀ = 2000T₀. The inset shows the relative group delay as a function of the number of MDM periods.

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Finally, we discuss the experimental realization of our theoretical predication. Although a strong GW is used in the numerical calculation, our results show that the sensitivity of the group delay to the GW may be better than the Michelson interferometer. Thus, the detection for the variety of group delay induced by the weak GW (h ≈ 10⁻²⁰) may be feasible in very thick MDM. However, as a new detection method, there still have some problems need to overcome. The group delay of a Gaussian EW pulse is investigated to show that even in the GW background radiation (ω_GW ∼ 10¹⁰Hz) there is no Hartman effect, but the relative short EW pulse is more sensitive to a high-frequency GW (ω_GW > 10⁴Hz, e.g, the GW emitted from supernova explosions or GW background radiation [34]). To detect the low-frequency GW, the low frequency GW modulation on the phase time of continuous laser beams [11] and the influence of noises should be investigated carefully. Another interesting question is that can we detect the GW modulation on the group delay through astronomical observation, e.g., when the pulsar electromagnetic radiation tunnels through a strong GW radiation source such as the black hole binary, the variation of the group delay in this tunneling process may be detectable.

4. Conclusion

In conclusion, we have calculated the group delay of optical pulses through MDM combined with GW. We found that the group delay increases linearly with MDM length for the transmitted wave packet. The Hartman effect disappears. This peculiar tunneling effect is attributed to the additional EW leakage attributed to the GW-induced variations of PBG. Thus, the issues of superluminality in traditional quantum tunneling do not even arise when the quantum theory is combined with the general relativity theory. We also show that the group delay of the tunneling photons is sensitive to GW. For a relatively low-frequency GW or thick vacuum layers, the sensitivity can be enhanced remarkably. Our study provides insight into the nature of the quantum tunnelling as well as a novel process by which to detect the GW.