Quantum particles can be found in regions in which classical particles are not allowed to exist. An example of this is evanescent (exponentially decaying) quantum states at potential barriers and steps. When a stream of particles encounters a potential step that is reflective from a classical viewpoint, there is a non-vanishing probability of finding particles within the potential step. This intriguing behaviour arises from the ability of quantum mechanical states and their associated wave functions ψ(x) to generate domains of negative kinetic energy1, where \(\psi (\bfx)^-1\widehat\bfp^2\psi (\bfx)/2m < 0\). Here, \(\widehat\bfp\) and m denote the momentum operator and mass, respectively, and ψ(x) is considered an energy eigenstate. This ability enables quantum particles to migrate into regions considered energetically forbidden in classical terms while maintaining energy conservation. Although this is well known, the kinematic interpretation of these states raises questions that have not yet been conclusively answered. This includes questions about whether a speed can be assigned to these states of motion, and if so, how it depends on energy.
The kinematic interpretation of classically forbidden states of motion is intrinsically linked to the longstanding problem of tunnelling times. This problem refers to the question of how long a particle needs to tunnel through a (finite) potential barrier2,3,4,5,6. Despite the controversy surrounding this topic, important insights, methodologies10,11,12,13,14,15,16,17,18 and experiments19,20,21,22,23,24,25,26,27 have emerged from this debate. Investigating energy–speed relationships in evanescent phenomena can contribute to the discussion by offering a complementary perspective on this problem. Speed measurements in quantum mechanics can also serve as tests for the Bohmian interpretation of quantum mechanics, in which particle velocities play a very prominent part7,28,29,30. Apart from the Schrödinger equation, a guiding equation is assumed that assigns a velocity field vS(x, t) to a wave function \(\psi (\bfx,t)=\sqrtn(\bfx,t)\exp (\rmiS(\bfx,t))\) using
$$\bfv_\rmS(\bfx,t)=\frac\hbar m\nabla S(\bfx,t).$$
(1)
Here, the real-valued functions n(x, t) and S(x, t) describe the particle density and phase of the wave function, respectively. The velocity field vS(x, t) gives rise to trajectories along which the particles are expected to move. If the initial particle positions of these trajectories are chosen according to the Born rule (quantum equilibrium hypothesis), the resulting probability density aligns with the predictions of standard quantum mechanics. Whether this implies that both theories generally make identical predictions remains a matter of debate. Discrepancies are sought, for example, in temporal quantities such as tunnelling or arrival times31,32,33,34. Although measurement results in standard quantum mechanics are considered inherently random, the existence of Bohmian mechanics—a fully deterministic dynamical system—demonstrates that quantum phenomena do not necessarily rely on randomness at their core. These sharply divergent views on the fundamental processes of nature motivate the search for experimental tests to distinguish between the two theories.
In our experiments, we investigate states of motion that arise when a particle is reflected from a potential step. Standard quantum mechanics does not incorporate particle velocities into its formalism in the same explicit way as Bohmian mechanics. Moreover, conventional measures of motion—such as group or phase velocity—do not yield physically meaningful results in this scenario. To measure particle speeds associated with these states, we compare translational motion within a waveguide to population hopping between waveguides. Initially, consider a system of two degenerate localized quantum states that are coupled to each other with a coupling constant J0 > 0. If the probability amplitude is initially concentrated entirely in one of the states (with a value set to 1), the population in the initially unoccupied state follows sin2(J0t) as a function of time t and, consequently, for small times, increases as (J0t)2. Provided the coupling constant is known, the respective populations of the two states can be understood as a measure of time. We then translate this behaviour to a system of two coupled waveguides. Consider two parallel waveguides, each capable of restricting the motion of particles to one dimension. When these two waveguides are brought in proximity, tunnelling processes allow particle exchange between the waveguides, which can be described by a coupling constant J0 as introduced above. If at a certain point x = 0 the entire population is concentrated in one waveguide, then, sufficiently close to x = 0, the population of the other waveguide is expected to grow as (J0x/v)2 (replacing t with x/v). Thus, provided that the coupling constant J0 is known, we can infer the speed v by measuring the spatial population build-up in the initially unoccupied state. For a more formal justification of this measurement scheme, based on modelling the system using coupled Schrödinger equations, we refer to the Methods and ref. 35. We emphasize that this measurement scheme does not determine the direction of motion of particles; thus, it is not a velocity measurement. For more details, see the Methods.
To implement this scheme experimentally, we use a quantum-confined photon gas in a planar optical microcavity36,37,38,39. In this system, the photons essentially behave like massive particles in two dimensions. More specifically, the mirrors of the microcavity restrict the allowed values for the photon wavevector k = (kx, ky, kz) to those that satisfy a resonance condition along the optical axis, that is kz = qπ/D0. Here, q = 1, 2, 3, … is the longitudinal mode number and D0 is the spatial separation of the mirrors. If the photons travel predominantly in the direction of the optical axis (paraxial limit), the projection of the photon motion onto the resonator plane creates a two-dimensional non-relativistic gas of particles with wavevectors (kx, ky). In this projection, the photons acquire an effective mass m, whose value follows the energy–mass equivalence \(E_=m\widetildec^2\), where E0 is the energy of the transverse ground mode and \(\widetildec\) is the speed of light in the resonator medium36. The latter follows from the photonic energy–momentum relationship \(E_\rmph=\hbar \widetilde\rmc| \bfk| \), which, assuming a paraxial limit \(|k_x,y|\ll |k_z|\), can be approximated by \(E_\rmph\approx m\widetildec^2+\hbar ^2(k_x^2+k_y^2)/2m\), with the photon mass \(m=\hbar \widetildecq\rm\pi /D_=E_/\widetilde\rmc^2\). In our experiments, we have m ≈ 6.95 × 10−36 kg. In this system, the transverse motion of the photons can be precisely controlled by nanostructured mirror surfaces. It can be shown39 that a local change in the distance between the mirrors Δd(x, y) introduces a potential energy V(x, y) for the particles that follows \(V(x,y)\,=\) \(-m\widetildec^2\Delta d(x,y)/D_\). In our experiment, this effect is used to capture photons in one-dimensional waveguide potentials and to couple waveguide pairs in a controlled manner. Owing to a small amount of mirror transmission with a rate of γ ≈ 3.7 GHz, corresponding to a particle lifetime of τ = 1/γ ≈ 270 ps, particle densities in the resonator plane can be fully reconstructed by imaging the transmitted light onto a camera. This is true regardless of whether the transverse state of the emitted light represents evanescent or propagating solutions, allowing us to study evanescent phenomena at semi-infinitely extended step potentials.
The core of our experimental setup, as described in Fig. 1a, consists of a high-finesse microcavity with a dye medium. One of the mirrors is nanostructured to create waveguiding potentials for photons in the transverse plane of the resonator. The main waveguide begins with a linear ramp, allowing us to vary the initial potential energy of the photons by changing the point along the ramp at which we optically pump the system with a non-resonant laser pulse. The pulse duration of 26 ns (full width at half maximum (FWHM)) is considerably longer than all relevant time scales in the system, creating a quasi-stationary scenario for all our experiments. This pumping triggers a lasing process that favours the creation of photons with low kinetic energy at the position of the pump, maximizing the overlap of their wave function with the optical gain. The potential energy of the photons created at a certain position in the ramp is then converted into kinetic energy as the photons move down and reach a region of constant potential. The photons then impinge on the step potential, as shown in Fig. 1b, in which a second waveguide opens up. We refer to this second waveguide as the auxiliary waveguide in the subsequent discussion. Three examples showing the photon densities in the microcavity for different pump spot positions are shown in Fig. 1c. The densities are recorded by imaging light transmitted through one of the mirrors onto a camera. The auxiliary waveguide allows us to determine the particle speed by comparing the translational motion along the waveguide axis with the hopping between waveguides. A prerequisite for this is determining the coupling constant J0 between the waveguides. This can be performed through a wavenumber analysis of observed mode patterns in the classically allowed regime, yielding J0 = 2π(6.34 ± 0.01) GHz or \(\hbar J_=(26.22\pm 0.04)\,\rm\mu \rmeV\). For further details, see the Methods. In the same way, we can also determine all other quantities of interest, such as the height of the step potential V0 = (0.538 ± 0.003) meV and the kinetic energy E of the incoming particles.
a, Schematic of the experimental setup. The microcavity consists of two planar mirrors, one of which is nanostructured, with an optically active dye in between. The optical medium can be non-resonantly pumped to induce a lasing process, generating microcavity photons at the point of pumping. A small amount of the light circulating in the microcavity is transmitted through the mirrors and can be imaged onto a camera. b, Height map and cross-sections (see colour coding) of the nanostructured mirror. The height profile of the nanostructured mirror effectively induces a potential energy landscape that confines and guides the photons in the transverse plane of the resonator. Specifically, the light is guided in a waveguiding potential from x = −900 μm to the end of the structure at x = 500 μm. In the first section, from x = −900 μm to x = −600 μm, a linear ramp potential is superimposed on the waveguide potential. By changing the position of the non-resonant optical pumping along the ramp with a spatial light modulator (SLM), we can set the initial potential energy of photons. At x = 0, a step potential of height V0 = (0.538 ± 0.003) meV is superimposed on the waveguide that runs from x = 0 μm to x = 500 μm. Apart from the main waveguide described above (red cross-section), an auxiliary waveguide (green cross-section) is introduced at the start of the step potential at x = 0. This effectively creates a double-well potential in the direction orthogonal to the waveguide axis (grey cross-section), with a coupling between the respective ground states of the wells of J0 = 2π(6.34 ± 0.01) GHz. c, Camera images showing the photon populations in the coupled waveguides for three distinct energy regimes associated with three different dynamics: population oscillations in the classically allowed regime \((\varDelta > \hbar J_)\), long-range non-oscillatory propagation \((|\varDelta |\le \hbar J_)\) and evanescent decay \((\varDelta < -\hbar J_)\). Scale bar, 50 μm (b, height); 100 μm (b, length).
Figure 2a,b shows the populations |ψm,a|2 in the main (ψm) and the auxiliary (ψa) waveguides as a function of position x for specific values of \(\varDelta =E-V_+\hbar J_\), which corresponds to the kinetic energy of the particles in the step potential. For \(\varDelta > \hbar J_\) (Fig. 2a), we observe that the population oscillates between the waveguides as the particles propagate in the step potential. The observed population decay over several hundred micrometres is because of mirror transmission and can be used to experimentally determine the photon loss rate γ introduced before. For \(\varDelta < -\hbar J_\) (Fig. 2b), evanescent types of wave functions are observed for which the populations in the waveguides relax to a common value. Figure 2c shows the decay constant of the population in the main waveguide, denoted by κ, as a function of Δ (Methods). For sufficiently large |Δ|, the measurement is in excellent agreement with the expected decay constant \(\kappa =\sqrt/\hbar \).
a, Measurement example of the particle densities along the waveguide axis in the main waveguide |ψm|2 (blue) and auxiliary waveguide |ψa|2 (green) in the classically allowed parameter regime \((\varDelta > \hbar J_)\). The fast periodic modulations on top of the population curves stem from a residual reflection at the end of the coupled waveguide system. b, Particle densities in the classically forbidden regime. c, Decay constant κ of the population in the main waveguide for different energies Δ (see the Methods for further details). The observed decay constant closely follows \(\kappa =\sqrt/\hbar \). Error bars indicate the standard error of the mean. d, Examples of measured relative populations in the auxiliary waveguide ρa = |ψa|2/(|ψm|2 + |ψa|2) for classically allowed states. A parabolic fit ρa = (J0x/v)2 is used to determine the speed of the motion. e, Relative populations for evanescent states in which the step is fully reflective. In these states, particles enter the step potential and eventually exit it with their direction of motion reversed. As in the case of the propagating states, a parabolic fit ρa = (J0x/v)2 is used to determine the speed of the motion. All data shown in this figure, except those in the inset of c, are derived from single data-taking events, each comprising 20 optical excitations of the system.
Figure 2d,e shows the relative population in the auxiliary waveguide ρa = |ψa|2/(|ψm|2 + |ψa|2) as a function of distance to the potential step for different values of Δ. Calculating the relative populations eliminates common factors affecting both waveguides, such as the exponential decay in the case of negative Δ. In all cases, the data are compatible with a quadratic increase in relative population. By modelling our data with ρa = (J0x/v)2, we obtain the particle speed v shown in Fig. 3a. It is to be noted that, within the experimental uncertainties, v is a mirror-symmetrical function of Δ. For negative Δ, this means, rather counterintuitively, that the lower the energy of the particles, the faster the particles move within the step potential. We emphasize that the measured speed represents the local speed of particles directly behind the potential step. In the coupled waveguide system, a position-dependent state of motion can arise from an interplay with the coupling energy, even if the potential energy remains constant. However, these effects are not expected when the coupling energy is negligible, specifically when \(|\varDelta |\gg \hbar J_\). In this limit, our data align well with \(v=\sqrt/m\) for both negative and positive values of Δ (Fig. 3a). The residual differences are probably caused by a systematic overestimation of the speed in the fitting procedure of up to 6.7% (see the Methods for details). Note that for negative Δ, the speed \(v=\sqrt/m\) is proportional to the decay constant \(\kappa =\sqrt\varDelta /\hbar \), following \(v=\hbar \kappa /m\). Introducing the decay length λ = 1/κ, this relation can also be stated as \(\lambda =\hbar /mv\), which represents a de Broglie relation for evanescent particle states, connecting the decay length to the speed.
a, Particle speed v at the potential step as a function of energy Δ. Blue markers indicate the speed values obtained from the parabolic fitting of the population increase in the auxiliary waveguide. The speed obtained in this way is found to be a mirror-symmetric function of the energy Δ. For sufficiently large |Δ|, the data align well with \(v=\sqrt \varDelta \) (green line). For negative Δ, this means that the lower the speed of the incoming particles, the faster the particles move within the step potential. The lowest speed values are found in the vicinity of Δ = 0 and are close to the theoretical prediction35 \(v_\min =\sqrt\hbar J_/m\approx 777\,\rmkm\,\rms^-1\) for J0 = 2π × 6.34 GHz. Error bars indicate the standard error of the mean. b, Dwell time as a function of energy. The dwell time τdwell (squares) describes how long particles are stored in the step potential and can be determined using τdwell = N/jin, where N is the number of stored particles and jin is the incoming particle current. Based on the energy–speed relation in a, a semiclassical estimate of τdwell can be introduced using τλ = λ/v, where λ is the decay length (circles). This estimate is found to be in good agreement with τdwell for a wide range of energies. The two solid lines give the expected behaviour for the dwell time and its semiclassical estimate. The given formulas are derived for a single waveguide step potential. Error bars for τdwell indicate the standard error of the mean. Error bars for τλ are derived from error propagation.
As an application of the measured energy–speed relation, we consider the dwell time of particles in the step potential13. The definition of the dwell time is based on the idea that, in a steady state, the storage time of particles in a reservoir should be given by τdwell = N/jin, where N is the number of stored particles and jin is the incoming particle current. For a fully reflecting step potential—producing a standing wave pattern in front of the step and an exponentially decaying wave function within it—the incoming and outgoing particle currents can be assumed to have equal magnitudes but opposite directions. In analysing the experiment, the incoming current is taken as jin = (|ψmax|2/4)vin, where ψmax is the maximum amplitude of the wave function in front of the step. The factor of 4 arises from both the standing wave interference effect and the division into incoming and outgoing components. Alternatively, we could use the average density in front of the step along with a division factor of 2. The velocity vin is taken as \(v_\rmin=\hbar k_/m\), where k0 is determined from the periodicity of the observed standing wave interference pattern. Note that this determination of jin does not hold in the framework of Bohmian mechanics, as will be discussed below. The dwell time based on this procedure is observed to be within 1–2 ps for a wide range of energies (Fig. 3b). Using the measured energy–speed relation, we can give a semiclassical estimate of τdwell by considering τλ = λ/v, where the decay length λ corresponds to a typical travel distance into and out of the step potential. The latter is derived from \(2\langle \widehatx\rangle _x > 0=2\varrho _^-1\int _^\infty \varrho (x)x\rmdx=\lambda \), where \(\varrho (x)=(\rme^-x/\lambda )^2\) and \(\varrho _=\int _^\infty \varrho (x)\rmdx\). The quantity τλ is found to be close to the dwell time for a wide energy range (Fig. 3b). We consider this to be further evidence for the physical significance of the found energy–speed relation.
In further measurements, shown in Fig. 4, we interfere quantum states in the step potential with their spatially mirrored images to detect phase gradients (see the Methods for further details). These phase gradients can then be converted into velocities following equation (1). For Δ > 0 (Fig. 4b, left), we obtain velocities that are compatible with the results derived from the population transfer measurements shown in Fig. 3a. For Δ < 0, the phase gradients are found to be close to zero (Fig. 4b, right). If we translate the residual phase gradients into velocities using equation (1), we obtain on average vS = (59 ± 42) km s−1, as shown in Fig. 4c, which is consistent with zero. The vanishing of vS indicates a vanishing net particle current, meaning that the motion becomes non-directional as the step potential becomes fully reflective. The Bohmian interpretation of equation (1) goes further by postulating that vS is a complete characterization of the state of motion of the particles. Therefore, a vanishing vS not only indicates non-directionality but also implies that the particles are at rest. Note that this implication is in contradiction with the conclusions drawn from our speed measurements (Fig. 3a).
a, To measure the phase gradient of quantum states in the step potential, we incorporate a Mach–Zehnder interferometer into our experimental setup. A Dove prism is placed in one arm of the interferometer to create a mirrored image of the input state. The wavenumber associated with the observed interference pattern at the camera, ktot = kMZI + kS, has two contributions: a component related to the alignment of the interferometer, kMZI, which can be determined in an independent measurement, and a component associated with the intrinsic phase gradient of the quantum state kS. b, The top row shows sketches of wave functions and their mirror images, ψ and ψM, interfering at the output of the interferometer in the classically allowed (left) and forbidden (right) regimes, respectively. The middle row displays images of experimentally obtained interference patterns. The bottom row shows the corresponding line-integrated densities. From the periodicity of the obtained interference patterns, we extract ktot. This is then converted to the velocity \(v_\rmS=\hbar k_\rmS/m=\hbar (k_\rmtot-k_\rmMZI)/m\). For the two shown cases, we obtain vS ≈ 2,400 km s−1 for Δ ≈ 0.1 meV and vS ≈ (31 ± 42) km s−1 for Δ ≈ −0.08 meV. c Blue markers indicate the velocity vS in the classically forbidden regime, derived using the procedure described above. The green dashed line shows the average value, ⟨vS⟩ = (59 ± 42) km s−1, of all measurements with Δ ≤ −0.05 meV. Error bars indicate the standard error of the mean.
The assumption that particles in the step potential are at rest also implies a fundamentally different behaviour of other quantities. The dwell time, for example, can be identically defined by τdwell = N/jin in both standard quantum mechanics and Bohmian mechanics. As the wave functions are identical, the particle number N—defined as the integral of the modulus of the wave function squared within the step potential—is the same in both cases. However, the incoming particle current jin cannot, in general, be assumed to be identical. Scattering at a fully reflective potential step—when the energy of the particle is lower than the step potential—gives rise to eigenfunctions that can be chosen to be purely real-valued. The corresponding absence of phase gradients indicates, in the Bohmian framework, that the particles are at rest everywhere in the system. In other words, there are no particle currents—specifically, no incoming current into the step potential. Consequently, the ratio N/jin diverges, implying an infinite dwell time. Note that jin = 0 is not merely a consequence but also a requirement within Bohmian mechanics to remain consistent with the guiding equation inside the potential step, which also predicts the absence of motion—and thus an infinite dwell time. Thus, the dwell time for scattering at a semi-infinite step potential is an example of a physical quantity that, although identically defined and measurable in both Bohmian mechanics and standard quantum mechanics, yields different values in the two theories.
Furthermore, an absence of particle motion in the step potential—and consequently an infinite dwell time—implies that losses are essential to the scattering process observed in our experiment, as every particle entering the potential step would eventually be lost because of the dwell time exceeding the finite lifetime of the particles of 270 ps. By contrast, the wave function observed in the vicinity of the potential step closely matches the wave function expected for a lossless system (see, for example, Fig. 2c), thereby implying that losses are not significant in the scattering process. Thus, the wave picture based on the Schrödinger equation and the particle picture based on the guiding equation offer inconsistent accounts of the significance of losses in the scattering process, within the Bohmian interpretation of our experiment. Even independently of the speed measurements presented in this work, this discrepancy casts doubt on whether the guiding equation accurately captures the temporal dynamics of the scattering process.
In this work, we have measured an energy–speed relationship for tunnelling particles based on the analysis of two orthogonal motions in a system of two coupled waveguides: translational motion along the waveguides and population hopping between the waveguides. Quantitatively, we see agreement with an energy–speed relation \(v=\sqrt \varDelta \). For Δ < 0, this implies that particles move faster the less kinetic energy they have, that is, the more negative their local kinetic energy becomes. If the speed v is translated into the time τλ = λ/v, for E < V0 and the decay length λ, this accurately reproduces the experimentally observed dwell time of the particles over a wide energy range. If v is translated into the time τb = b/v, for E < V0 and the barrier width b, the result is the tunnelling time proposed by Büttiker and Landauer12,13,40. These straightforward applications of the measured energy–speed relation to evanescent phenomena suggest that this relation plays a fundamental part in quantum tunnelling. More broadly, our work indicates that, in addition to phase gradients, amplitude decays in quantum mechanical wave functions serve as indicators of motion, following a de Broglie relation \(\lambda =\hbar /mv\), where λ is the decay length and v denotes the nondirectional particle speed measured in this work. This contrasts with Bohmian mechanics, which considers phase gradients as the sole indicators of motion.
