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arXiv:2412.13139 · Physical Review Letters

Physical Interpretation of Imaginary Time Delay

Isabella L. Giovannelli, Steven M. Anlage

Published Jun 3, 2025• 10 citations
arXivHTMLPDF

Abstract

The scattering matrix S linearly relates the vector of incoming waves to outgoing wave excitations, and contains an enormous amount of information about the scattering system and its connections to the scattering channels. Time delay is one way to extract information from S, and the transmission time delay τ_{T} is a complex (even for Hermitian systems with unitary scattering matrices) measure of how long a wave excitation lingers before being transmitted. The real part of τ_{T} is a well-studied quantity, but the imaginary part of τ_{T} has not been systematically examined experimentally, and theoretical predictions for its behavior have not been tested. Here we experimentally test the predictions of Asano et al. [Nat. Commun. 7, 13488 (2016)NCAOBW2041-172310.1038/ncomms13488] for the imaginary part of transmission time delay in a nonunitary scattering system. We utilize Gaussian time-domain pulses scattering from a two-port microwave graph supporting a series of well-isolated absorptive modes to show that the carrier frequency of the pulses is changed in the scattering process by an amount in agreement with the imaginary part of the independently determined complex transmission time delay, Im[τ_{T}], from frequency-domain measurements of the subunitary S matrix. Our results also generalize and extend those of Asano et al., establishing a means to predict pulse propagation properties of non-Hermitian systems over a broad range of conditions.

DOI
10.1103/nnk7-xy4v

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The scattering matrix S linearly relates the vector of incoming waves to outgoing wave excitations, and contains an enormous amount of information about the scattering system and its connections to the scattering channels. Time delay is one way to extract information from S, and the transmission time delay τ_{T} is a complex (even for Hermitian systems with unitary scattering matrices) measure of how long a wave excitation lingers before being transmitted. The real part of τ_{T} is a well-studied quantity, but the imaginary part of τ_{T} has not been systematically examined experimentally, and theoretical predictions for its behavior have not been tested. Here we experimentally test the predictions of Asano et al. [Nat. Commun. 7, 13488 (2016)NCAOBW2041-172310.1038/ncomms13488] for the imaginary part of transmission time delay in a nonunitary scattering system. We utilize Gaussian time-domain pulses scattering from a two-port microwave graph supporting a series of well-isolated absorptive modes to show that the carrier frequency of the pulses is changed in the scattering process by an amount in agreement with the imaginary part of the independently determined complex transmission time delay, Im[τ_{T}], from frequency-domain measurements of the subunitary S matrix. Our results also generalize and extend those of Asano et al., establishing a means to predict pulse propagation properties of non-Hermitian systems over a broad range of conditions.
The analogy between quantum tunneling of particles and evanescent electromagnetic waves can be used to study particle tunneling. Possibilities for the measurement of the time delay resulting from the transmission of waves through an evanescent region, say with lower dielectric constant, are discussed. In contrast to particle tunneling, an electromagnetic pulse can consist of many photons and can be probed in a noninvasive way. We emphasize the delay of the centroid of an electromagnetic pulse transmitted below cutoff through a portion of a waveguide with the same cross section as the adjacent propagating guides. The boundary conditions at the interface between the propagating and the evanescent region lead to the same transmission and reflection coefficients as for a square-barrier tunneling problem. For a pulse restricted to a narrow frequency range, the time delay depends only on the frequency derivative of the phase shift associated with transmission. The delay of a centroid is just that; there is no deeper physical sense which links the incoming centroid to the outgoing centroid. For sufficiently long evanescent regions, the delay is independent of thickness.
The quantum tunneling process is a fundamental concept that presents the unconventional nature of quantum dynamics. The main anomaly of this effect revolves around the choice of imposing the classical 4-momentum to become imaginary during the barrier traversal. This paper addresses this paradox by introducing a complex-time parametrization, which uses a complex temporal coordinate, τ=t+iκτ= t + iκτ=t+iκ. Here, the real axis (t)(t)(t) represents standard thermodynamic time, while the imaginary axis (iκ)(iκ)(iκ) is correlated to spatial evanescence. We demonstrate that evaluating this extended complex-time framework yields a hyperbolic section of the causal lightcone, proving that the traversal is strictly bounded by the speed of light and preserves causal disconnection. Applied to quantum tunneling, we resolve the anomaly of imaginary momentum through a continuous geometric signature flip of the effective complex-time momentum norm from a timelike to a spacelike state. This theoretical model finds a physically phenomenological analogy in microwave scattering experiments using open, non-Hermitian systems, which link complex transmission time delay to spatial wave decay.
To overcome the fast oscillatory behavior of correlation functions for extracting scattering phase shift in real-time quantum simulations encountered in Ref.\cite{Guo:2026qkx}, we propose and test two solutions in the present work. One is to simulate Hermitian systems in imaginary time, the other is to simulate non-Hermitian systems in real time. We demonstrate that both approaches lead to the problem of non-unitary quantum evolution which can be solved by combining two quantum algorithms: block encoding and Hadamard test. The combined quantum algorithm does not require mid-circuit measurements or adjustment of the input parameters of the Hamiltonian, and can be easily implemented on quantum computers. Numerical tests on quantum simulators show that both approaches agree with exact solutions for a sufficiently long time before the signal is lost in statistical fluctuations. The results bode well for using non-Hermitian and imaginary-time simulations to circumvent oscillations inherent in real-time simulation of other quantum systems. In particular, the non-Hermitian approach shows a decisive advantage over the imaginary-time one on the number of required ancillary qubits, and hence is more practical to scale up.
Time-independent scattering methods are widely employed to analyze transport in non-Hermitian systems. Their application, however, rests on a critical yet often overlooked assumption: that an incident wave is a pure superposition of scattering states. In practice, any physically realistic, spatially localized wave packet will generally have a nonzero overlap with the system's bound states, thereby violating this premise. While this violation is inconsequential in Hermitian systems, it can invalidate the conventional scattering picture in their non-Hermitian counterparts. The underlying cause is the emergence of time-growing bound states, which manifest as poles of the scattering matrix (SSS matrix) in the first quadrant of the complex wave-number plane. Any initial overlap with these states becomes exponentially amplified, eventually dominating the long-time dynamics. Consequently, the actual evolution of a wave packet diverges dramatically from the conventional scattering picture, rendering the transmission and reflection coefficients derived from time-independent scattering methods unphysical. Using tight-binding models with non-Hermiticity introduced via imaginary on-site potentials or asymmetric hopping, we demonstrate that parameter regimes supporting such growing states are common. We therefore conclude that an analysis of SSS-matrix poles is an indispensable step to confirm the physical relevance of time-independent scattering calculations in non-Hermitian systems.
Measures of pulse delay in microwave propagation, in open air and for short distances (not much greater than 1 m), were made by using launcher and receiver horns. When these are facing each other we observe a delay time corresponding to a speed equal to c while, if the receiver horn is shifted or tilted with respect to the launcher horn, the delay time decreases showing a superluminal behavior. In other words the modulation phase shift, interpreted as a propagation time, turns out to be surprisingly smaller than the one relative to the light speed. This effect, which disappears for longer distances, is here interpreted on the basis of the existence of a special kind of evanescent waves (leaky waves). Just the presence of evanescent waves allows one to make a comparison with the tunneling processes where superluminal transport properties have been theoretically predicted.
We consider a scattering process inside a dispersive and absorptive medium, and derive an analytical expression for the scattering-time parameter that characterizes the duration of this process. We consider a microwave wave packet that scatters from a dielectric inhomogeneity inside an otherwise homogeneous waveguide. The waveguide is filled with a dielectric medium with a finite inhomogeneity region in the direction of the wave-packet propagation. The permittivity of the entire medium is described by a complex function of the frequency. Hence both absorption and dispersion affect the propagation of the wave packet. We define the scattering-time parameter to be the difference in the ``average arrival time'' between the free and actual propagation. We show that this time difference (in the limit of a narrow-band wave packet) is given by a linear combination of the real and imaginary parts of the ``complex tunneling time.'' In the special case where there is no dissipation or attenuation outside the scatterer, we recover the well-known result that the scattering-time parameter is identical to the real part of the complex tunneling time. The contribution of the imaginary part of the complex tunneling time to the scattering-time parameter can, therefore, be attributed to absorption and attenuation outside the scatterer. We further show that the real and imaginary-time parameters satisfy dispersion relations. Causality in this process is briefly discussed.