Wigner time delays and Goos-Hänchen shifts of 2D quantum vortices scattered by potential barriers

Throughout this work we deal with the 2D Schrödinger equation using the units $\hbar=m=1$:

where $\psi$ is the wavefunction, ${\bf r}=(x,y)$, ${\bm{\nabla}}=\partial/\partial{\bf r}$, and $V(x)$ is the potential. We consider three examples of potential barriers:

1. (A)

   Step potential $V(x)=V_{0}\,\Theta(x)$, where $\Theta(x)$ is the Heaviside step-function;

2. (B)

   Delta-function barrier $V(x)=W_{0}\,\delta(x)$, where $\delta(x)$ is the Dirac delta-function;

3. (C)

   Rectangular barrier $V(x)=V_{0}\left[\Theta(x)-\Theta(x-a)\right]$.

We will consider reflection and transmission of plane waves and wavepackets incident from the $x<0$ half-plane at the angle $\theta$ with respect to the $x$-axis, see Fig. 1.

Consider first a single incident plane wave, $\psi\propto\exp\left(-i\,\omega\,t+i\,{\bf k}\cdot{\bf r}\right)$, $x\leq 0$, with energy (frequency) $E=\omega$, momentum (wave vector) ${\bf p}={\bf k}\equiv(k_{x},k_{y})=k(\cos\theta,\sin\theta)$, and dispersion $\omega=k^{2}/2$ following from Eq. (1). In the step-potential case (A), the reflected and transmitted waves, $\psi^{r}\propto R\exp\left(-i\,\omega\,t+i\,{\bf k}^{r}\cdot{\bf r}\right)$, $x\leq 0$, and $\psi^{t}\propto T\exp\left(-i\,\omega\,t+i\,{\bf k}^{t}\cdot{\bf r}\right)$, $x\geq 0$, have the same energies and wavevectors ${\bf k}^{r}=k(-\cos\theta,\sin\theta)$ and ${\bf k}^{t}=k^{\prime}(\cos\theta^{\prime},\sin\theta^{\prime})$, where $k^{\prime}=k\sqrt{1-V_{0}/E}$ and $\sin\theta^{\prime}=\sin\theta/\sqrt{1-V_{0}/E}$ [19]. Applying the appropriate boundary conditions, i.e., the continuity of the total wavefunction $\psi$ and its normal derivative $\partial\psi/\partial x$ at $x=0$, we obtain the scattering (reflection and transmission) amplitudes [see Fig. 2(a)]:

where $k^{\prime}_{x}=k^{\prime}\cos\theta^{\prime}$. Note that the reflected and transmitted waves exist and acquire real amplitudes (2) in the regime of partial reflection/transmission, $\theta<\theta_{c}=\arccos\sqrt{V_{0}/E}$, while only the reflected propagating wave is generated in the regime of total reflection $\theta>\theta_{c}$, where one should use $k^{\prime}_{x}=i\sqrt{2(V_{0}-E\cos^{2}\theta)}$, $T=0$, and $R$ becomes complex.

In the delta-potential case (B), the incident wave is partially reflected/transmitted for all angles $\theta\in(0,\pi/2)$, whereas the transmitted wave has the same wavevector as the incident one, ${\bf k}^{t}={\bf k}$. The boundary conditions are: the continuity of the total wavefunction at $x=0$ and relation $\left.\dfrac{\partial\psi}{\partial x}\right|_{x=+0}-\left.\dfrac{\partial\psi}{\partial x}\right|_{x=-0}=2W_{0}$ which follows from the integration of the Schrödinger equation (1) across the delta-barrier, $x\in(-0,+0)$. This results in the complex reflection and transmission amplitudes [see Fig. 2(b)]

Finally, in the case (C) of a finite rectangular barrier, consisting of two potential-step interfaces, the complex reflection and transmission amplitudes read [see Fig. 2(c)]

where $k^{\prime}_{x}=k^{\prime}\cos\theta^{\prime}$ is the normal wavevector component inside the barrier. Note that the scattering coefficients (2)–(4) satisfy the conservation law $|R|^{2}+|T|^{2}=1$, and scattering by a rectangular barrier (C) exhibits resonances with total transmission $|T|=1$, $|R|=0$, Fig. 2(d).

We now consider reflection and transmission of a Gaussian wavepacket at the potential barriers of types (A)–(C). The incident wavepacket can be characterized by its Fourier spectrum (momentum representation)

Here and hereafter we use the coordinate frame $(X,Y)$ with the $X$-axis oriented along the propagation of the incident wavepacket, i.e., rotated by the angle $\theta$ with respect to the $(x,y)$ frame (see Fig. 1), $k_{0}$ is the central wavevector in the wavepacket, $\Delta$ is the $Y$-width of the packet, whereas $\gamma\Delta$ is its $X$-length. We assume the wavepacket (5) to be paraxial and large as compared with the wavelength (i.e., semiclassical): $\{k_{0}\Delta,\gamma k_{0}\Delta\}\gg 1$. In the case (C) of a rectangular barrier we also assume that the wavepacket is much larger than the barrier width: $\{\Delta,\gamma\Delta\}\gg a$; this is to avoid numerous maxima in the reflected/transmitted signal due to the multiple reflections of a short pulse inside the barrier [9].

The real-space (coordinate) representation of the Gaussian wavepacket (5) is given by the Fourier integral

where $\omega\left({\bf{k}}\right)=k^{2}/2$, $\xi=X-v_{g}\,t$ is the wavepacket-accompanying coordinate with the group velocity $v_{g}=k_{0}$, and we neglected diffraction effects in the paraxial approximation. An example of the momentum-space and real-space wavefunctions (5) and (6) is shown in Fig. 3(a).

Spatial dispersion of the scattering coefficients (2)–(4), i.e., their dependence on the angle of incidence, $\theta$, causes the GH shifts of the scattered wavepackets [21, 22, 10]. The linear (position) and angular (direction) GH shifts can be characterized by the mean transverse position $\langle Y\rangle$ and transverse momentum $\langle k_{Y}\rangle$ (with respect to the expected geometrical-optics or semiclassical trajectory) of the scattered wavepackets. The generalized Artmann formula [22, 10] yields

Here and hereafter the coordinates and wavevectors of the reflected and transmitted wavepackets are written in the coordinate frames $(X^{r,t},Y^{r,t})$ accompanying the reflected and refracted wavepackets [10], the subscript “0” indicates that we deal with a Gaussian (no-vortex) wavepacket, $\theta^{t}=\theta^{\prime}$ in the case of step potential (A), and $\theta^{t}=\theta$ in the cases (B) and (C). The coordinate frames and schematics of some of the GH shifts are shown in Fig. 1. The linear shifts are non-diffractive effects appearing right after the scattering event, while the angular shifts accumulate additional spatial shifts proportional to the propagation distances $X^{r,t}$ [10, 43, 44, 45]. The GH shifts are well studied in optics and also in quantum systems [46, 47, 48, 49], although, to the best of our knowledge, the angular GH effect has never been discussed for quantum systems.

In analogy to the GH effect, the temporal dispersion of the scattering coefficients (2)–(4), i.e., their dependence on the energy of the incident wave, $E$, results in the Wigner time delays of the scattered wavepackets [20, 7, 8, 9, 18].

Here $\langle\tau\rangle$ denotes the time delay of the scattered wavepacket as compared to its expected arrival time, whereas $\langle\epsilon\rangle$ is the shift of its mean energy/frequency with respect to the central energy of the incident wavepacket, $E_{0}=\omega_{0}=k_{0}^{2}/2$. These are temporal counterparts of the linear and angular GH shifts, so that the energy shift accumulates an additional time delay proportional to the propagation time of the wavepacket. The Wigner time delays (9) and (10) can be equally regarded as spatial and angular shifts in the longitudinal wavepacket-accompanying coordinate $\langle\xi^{r,t}\rangle=-v^{r,t}_{g}\langle\tau^{r,t}\rangle$ and the corresponding wavevector component $\langle\delta k^{r,t}_{X}\rangle=(v^{r,t}_{g})^{-1}\langle\epsilon^{r,t}\rangle$, where $\xi^{r,t}=X^{r,t}-v^{r,t}_{g}t$, $\delta k^{r,t}_{X}=k^{r,t}_{X}-k^{r,t}_{0}$, $v^{t}_{g}=k^{t}_{0}=k^{\prime}_{0}$ in the step-potential case (A) and $v^{r,t}_{g}=k^{r,t}_{0}=k_{0}$ in the other cases.

The coordinate/momentum GH shifts, Eqs. (7) and (8), and the Wigner time/frequency shifts, Eqs. (9) and (10), can be associated with the real/imaginary parts of complex values $i\partial\ln\{R,T\}/\partial k_{y}$ and $-i\partial\ln\{R,T\}/\partial\omega$. This duality and connections between the GH and Wigner effects were discussed in various contexts in Refs. [50, 51, 18] and is related to manifestations of complex quantum weak values [11, 12, 52, 18, 53, 6].

The GH shifts and Wigner time delays, Eqs. (7)–(10) versus the angle of incidence $\theta$ in specific cases of barriers (A)–(C) are shown in Figures 4–6 by dotted curves. Notice that the linear quantities $\langle Y\rangle$ and $\langle\tau\rangle$, Eqs. (7) and (9), vanish in the case of partial reflection and transmission at the step potential (A), $\theta<\theta_{c}$, because the scattering coefficients (2) are purely real in this case. The angular quantities $\langle k_{Y}\rangle$ and $\langle\epsilon\rangle$, Eqs. (8) and (10), are generally nonzero in all cases (A)–(C). Importantly, the analytical expressions for the shifts and time delays can diverge at zeros of scattering coefficients: e.g., at the critical angle of incidence, $\theta_{c}$, in the case (A) and near zeros of the reflection coefficient (transmission resonances) in the case (C). Approximate formulas (7)–(10) become inapplicable near such singularities, whereas the shifts/delays can exceed their typical values by several orders of magnitude [43, 24, 45, 54, 18, 55, 36]. This amplification of otherwise small effects is a manifestation of quantum weak measurements and superweak values [56, 18].

To check the above theoretical derivations, we performed numerical calculations of the reflection/transmission of a Laguerre-Gaussian vortex wavepacket by potential barriers (A)–(C) by applying exact boundary conditions and wavevector-dependent scattering amplitudes (2)–(4) to each plane wave in the incident wavepacket spectrum (11). This provides transformations of the wavefunctions in the momentum representation: $\tilde{\psi}^{r}({\bf k})=R({\bf k})\,\tilde{\psi}({\bf k})$ and $\tilde{\psi}^{t}({\bf k})=T({\bf k})\,\tilde{\psi}({\bf k})$. In the paraxial approximation, the scattering amplitudes can be expanded in the Taylor series near the central wavevector:

These equations should be supplied with the corresponding transformations of each wavevector [10, 11, 12, 36]. Using the dispersion relations together with the conservation of the energy and $y$-components of the momentum (wavevector) at interfaces, we derive the following relations between the deflections of the wavevectors from their central values:

Here $\delta k_{X}=k_{X}-k_{0}$, Eqs. (4.1) are derived in the second-order approximation in ${\bf k}-{\bf k}_{0}$, and we introduced auxiliary quantities

The spatial and angular shifts are calculated as expectation values of the corresponding operators in the momentum representation:

where the inner product involves integration over the corresponding wavevector components $\left(\delta k^{r,t}_{X},k^{r,t}_{Y}\right)$. Substituting the wavefunctions and wavevector components of the scattered wavepackets, Eqs. (17)–(4.1) into Eqs. (4.1), we obtain the formalism for calculations of all of the required shifts. In doing so, all quantities are expressed via the parameters of the incident wavepacket. In particular, the derivatives with respect to the wavevector components in the first two Eqs. (4.1) are expressed, using relations (4.1) and (4.1), as

Importantly, for reflected wavepackets and transmitted wavepackets with $\theta^{t}=\theta$, i.e., in all cases apart from the refraction at the step potential, case (A), the coefficients (4.1) vanish: $A=B=C=D=0$. In such case, Eqs. (4.1) and (4.1) are simplified dramatically and become equivalent to standard equation for optical beams [10, 11, 12, 36]. The results of numerical calculations of Eqs. (17)–(4.1) with $A=B=C=D=0$ are depicted by symbols in Figs. 4–6. One can see that they perfectly agree with the analytical expressions (7)–(10) and (14)–(16) everywhere apart from the vicinity of resonant singularities. Analytical description of the resonant behaviour of wavepacket shifts, which regularizes divergencies of the Artmann and Wigner formulas, can be constructed within the nonlinear quantum-weak-weasurement approach [52, 18].

The nontrivial case of the transmitted wavepacket at a step potential, case (A), when the coefficients (4.1) do not vanish, requires additional considerations. We have shown that calculations without the $A$, $B$, $C$, $D$ terms in Eqs. (4.1) and (4.1) yield the shifts described by equations (7)–(10) and (14)–(16). Hence, substituting the $A$, $B$, $C$, $D$ terms of Eqs. (4.1) and (4.1) into Eqs. (4.1), we obtain corrections to the results of previous sections. For angular shifts, this yields

For linear shifts, we obtain

Remarkably the $A$, $B$, $D$ corrections in Eqs. (22)–(25) affect even the shifts of Gaussian wavepackets, $\ell=0$. Thus, that the standard Artmann and Wigner formulae (7)–(10) become inaccurate in the case of the refraction at a step potential.

Figure 7 shows the linear and angular shifts of transmitted wavepackets at a step potential, both analytical expressions (22)–(25) and numerical calculations using Eqs. (17)–(4.1). By comparing these plots with Fig. 4 one can see that the $A$, $B$, $C$, $D$ corrections can considerably modify the values of the shifts and even change their signs.