Introduction

Cherenkov radiation (ChR) by charged particles in media1,2,3,4,5,6,7,8,9,10 is the simplest example of a wide range of phenomena embracing transition radiation, diffraction, Smith–Purcell radiation, and other mechanisms of photon emission11,12,13,14. Their common microscopic origin is atomic bremsstrahlung15 due to dynamic polarization of atomic shells by the field of the charge, and the characteristic time scales are femto- and attoseconds, typical for the AC Stark effect16. Along with numerous applications of classical ChR for particle detection in acceleration experiments and cosmic rays studies17, neutrino telescopes18,19, gamma-ray astronomy20,21, and other fields10, it has also recently attracted attention of the biomedical community as a new tool for molecular imaging and therapy in cancer treatment22,23,24.

A quantum theory of ChR in momentum space was presented in 1940 by Ginzburg4 and Sokolov5 who predicted a correction to the classical Tamm-Frank result due to quantum recoil, which is negligibly small for the overwhelming majority of applications of ChR. Interest in inherently quantum features of ChR was revived in 201625,26,27 after vortex electrons with quantized orbital angular momentum projection were generated at electron microscopes28,29. Some flaws in the initial analysis of Ref. 25 were corrected in Ref. 26. More recently, it has been argued that the spatiotemporal features of ChR can be connected to the spatial coherence of the emitting charged particle30, whereas generalized measurements of either the final electron or the photon can lead to the generation of a wave packet of the other particle with the needed properties31.

The spatiotemporal characteristics of radiation in the pre-wave zone are neccesary for applications of such close relatives of ChR as transition, diffraction, and Smith–Purcell radiation, for instance, in bunch-length measurements at accelerators12,32. The classical theory of the Cherenkov wake fields in the near-field zone has also been developed for beam position monitors and radiation sources at accelerators and free-electron lasers33,34,35 and even alternative concepts of the Cherenkov wakefield acceleration have been proposed36. However, no quantum theory of ChR and its generalizations exists so far beyond the far-field approximation. The latter is tightly linked to the conventional momentum-space approach, whereas a spatiotemporal analysis of the photon field at a finite distance from the emitting particle necessitates working in real space and time, and that is always tricky in relativistic quantum theory. Although the flash duration of ChR was first estimated by Frank as early as in 1956 from classical considerations8,9, the microscopic atomic dynamics in the formation of the ChR field remains hidden even in the fully quantum treatment of ChR in momentum space.

Here we point out that one can access atomic time scales in ChR and in its generalizations by using a quantum theory in phase space where—similar to quantum optics37—we employ a Wigner function to characterize the emitted photon field. We demonstrate how to probe the field in the pre-wave—or formation—zone12,32,38 in which the partial waves interfere, the Cherenkov cone is not formed yet, and the emitted energy propagates in real space and time as a spreading wave packet. We find that our theory establishes a link between the evolving coherence length of the electron packet, the Cherenkov flash duration, and a quantum temporal delay that the photon experiences in medium and that can only be quantitatively studied beyond the far-field approximation. This delay falls within the attosecond range, a time scale characteristic of atomic excitation processes explored in attosecond spectroscopy and metrology39, and it can be either positive or negative, that is, the photon wave packet can reach the detector on average later or sooner than one can expect from the classical far-field considerations. This temporal detuning also shares some similarities with tunneling-related time effects, such as those observed in electron ionization40 and emission41. The quantum shift in the photon arrival time, coupled with the finite flash duration and the spreading dynamics, unveils intricate coherence properties that enable refined temporal control in quantum emission processes, enriching the landscape of quantum optics, ultrafast physics, and of applications of ChR and related phenomena.

Results

Photon emission in phase space

The system of units ℏ = c = 1 is used, the electron velocity is β = up/c ≡ up, m = 0.511 MeV is the electron mass, 1/m ≡ ℏ/mc = 3.86 × 10−11 cm is the reduced Compton wavelength, and the corresponding timescale is tC = 1/m ≡ ℏ/mc2 ≈ 1.3 × 10−21 s. When an electron emits a photon, the two-particle state within the first order of the perturbation theory in quantum electrodynamics is \(\left\vert {e}^{{\prime} },\gamma \right\rangle =(\hat{1}+{\hat{S}}^{(1)})\left\vert {{{\rm{in}}}}\right\rangle\), where \(\vert {{{\rm{in}}}}\rangle =\vert {e}_{{{{\rm{in}}}}}\rangle \otimes \vert {0}_{\gamma }\rangle\) and \({\hat{S}}^{(1)}=-ie\int{d}^{4}x\,{\hat{j}}^{\mu }(x){\hat{A}}_{\mu }(x)\)42. Figure 1 illustrates how the photon field gradually becomes spatially separated from the electron packet, spreads, and eventually turns into a plane wave propagating at the angles θ, ϕ. If we detect the electron in a plane-wave state \(\left\langle {{{{\bf{p}}}}}^{{\prime} },{\lambda }^{{\prime} }\right\vert\), \({\lambda }^{{\prime} }=\pm 1/2\), the state of the photon becomes

$$\left\vert \gamma \right\rangle =\left\langle {{{{\bf{p}}}}}^{{\prime} },{\lambda }^{{\prime} }\right\vert \left.{e}_{{{{\rm{in}}}}}\right\rangle \left\vert {0}_{\gamma }\right\rangle +{\sum}_{{\lambda }_{\gamma }}\int\frac{{d}^{3}k}{{(2\pi )}^{3}}\,\left\vert {{{\bf{k}}}},{\lambda }_{\gamma }\right\rangle \,{S}_{{{{\rm{fi}}}}}.$$
(1)

The second term here is a coherent superposition of plane waves with the momenta k and the helicity λγ = ±1 and a matrix element is \({S}_{{{{\rm{fi}}}}}=\left\langle {{{\bf{k}}}},{\lambda }_{\gamma };{{{{\bf{p}}}}}^{{\prime} },{\lambda }^{{\prime} }\right\vert {\hat{S}}^{(1)}\left\vert \,{{\mathrm{in}}}\,\right\rangle\). We treat the incoming electron as a Gaussian packet with a wave function \({f}_{\,{{\mathrm{e}}}}^{({{\mathrm{in}}})}({{{\bf{p}}}},\lambda )=\left\langle {{{\bf{p}}}},\lambda \right\vert \left.{e}_{{{{\rm{in}}}}}\right\rangle\), a mean momentum \(\left\langle {{{\bf{p}}}}\right\rangle\), an uncertainty σ ≪ m, which means that the rms size—also called coherence length—of the packet is much larger than the Compton wavelength, σx = 1/σ ≫ 1/m. For the moment, we take a simplified model with the packet, spherically symmetric in the laboratory frame, σ⊥ = σ∥ ≡ σx = 1/σ.

Fig. 1: Spatiotemporal dynamics of the Cherenkov radiation generated by a spreading electron wavepacket.
figure 1

σx stands for the initial size of an electron packet (blue), e, and n is the refractive index of the medium. The radiation process begins with the fundamental quantum electrodynamics interaction \(e\to {e}^{{\prime} }+\gamma\), during which the photon field (pink), γ, gradually separates from the electron packet within the formation zone and spreads. The electron is detected with a momentum \({{{{\bf{p}}}}}^{{\prime} }=| {{{{\bf{p}}}}}^{{\prime} }| \{\sin {\theta }^{{\prime} }\cos {\phi }^{{\prime} },\sin {\theta }^{{\prime} }\sin {\phi }^{{\prime} },\cos {\theta }^{{\prime} }\}\), and the Cherenkov cone is formed in the far-field only if the electron remains undetected. In phase-space picture, the photon with the momentum \({{{\bf{k}}}}=n\omega \{\sin \theta \cos \phi ,\sin \theta \sin \phi ,\cos \theta \}\) can be detected at a finite distance, meaning it does not necessarily reach the far field.

Full size image

Let us define the energy density of the photon field in real space and time, an electric part of which is (Supplementary Notes 1 and 2)

$$\frac{1}{8\pi }\left\langle \gamma \right\vert {\hat{{{{\boldsymbol{E}}}}}}^{2}({{{\bf{r}}}},t)\left\vert \gamma \right\rangle \to \frac{1}{4\pi }{\left\vert \left\langle 0\right\vert \hat{{{{\boldsymbol{E}}}}}({{{\bf{r}}}},t)\left\vert \gamma \right\rangle \right\vert }^{2}=\int\frac{{d}^{3}k}{{(2\pi )}^{3}}\,{{{\mathcal{W}}}}({{{\bf{r}}}},{{{\bf{k}}}},t),$$
(2)

where we have subtracted a contribution of the vacuum energy and \(\hat{{{{\boldsymbol{E}}}}}({{{\bf{r}}}},t)\) is a secondary-quantized electric field operator43. A contribution of the magnetic field can be written in a similar way. Here

$$ {{{\mathcal{W}}}}({{{\bf{r}}}},{{{\bf{k}}}},t)\\ =\frac{1}{4\pi }{\sum}_{{\lambda }_{\gamma },{\tilde{\lambda }}_{\gamma }}\int\frac{{d}^{3}\tilde{k}}{{(2\pi )}^{3}}\,{{{{\boldsymbol{E}}}}}_{{\tilde{\lambda }}_{\gamma }}^{* }({{{\bf{k}}}}-\tilde{{{{\bf{k}}}}}/2)\cdot {{{{\boldsymbol{E}}}}}_{{\lambda }_{\gamma }}({{{\bf{k}}}}+\tilde{{{{\bf{k}}}}}/2)\,{e}^{-it\left(\omega ({{{\bf{k}}}}+\tilde{{{{\bf{k}}}}}/2)-\omega ({{{\bf{k}}}}-\tilde{{{{\bf{k}}}}}/2)\right)+i{{{\bf{r}}}}\cdot \tilde{{{{\bf{k}}}}}}$$
(3)

is a Wigner function of the photon, and \({{{{\boldsymbol{E}}}}}_{{\lambda }_{\gamma }}({{{\bf{k}}}})=\frac{i\omega \sqrt{4\pi }}{\sqrt{2\omega {n}^{2}}}\,{{{{\boldsymbol{e}}}}}_{{{{\bf{k}}}}{\lambda }_{\gamma }}{\sum}_{\lambda }\int\frac{{d}^{3}p}{{(2\pi )}^{3}}\,{f}_{\,{{\mathrm{e}}}}^{({{\mathrm{in}}})}({{{\bf{p}}}},\lambda ){S}_{\,{{\mathrm{fi}}}}^{({{\mathrm{pw}}})},\,{{{{\boldsymbol{e}}}}}_{{{{\bf{k}}}}{\lambda }_{\gamma }}\cdot {{{\bf{k}}}}=0,\) is a positive-frequency component of its electric field.

According to the standard interpretation43, the energy density Eq. (2) defines probability of detecting a photon in a region of space centered at the point r at the moment of time t. Clearly, the second marginal distribution \(\int{d}^{3}x\,{{{\mathcal{W}}}}({{{\bf{r}}}},{{{\bf{k}}}},t)\propto {\left\vert {S}_{{{{\rm{fi}}}}}^{({{\mathrm{pw}}})}\right\vert }^{2}\) yields probability to detect a photon with the frequency ω and a wave vector k,  ∣k∣ = n(ω) ω, the standard result of the quantum theory in momentum space (see26). Therefore, it is this Wigner function Eq. (3) in phase space that contains all the information on spatial distribution of the photon energy density at a given distance r—also in the near-field zone—and on its dynamics. Consequently, emission takes place in the pre-wave zone even if the condition of ChR is not met, but the waves do not constructively interfere to form a cone in the far field.

We calculate the Wigner function in the paraxial approximation, σ ≪ m, in a medium with weak dispersion, \(\frac{\omega }{n(\omega )}\frac{dn(\omega )}{d\omega }\ll 1\), and represent the tree-level amplitude26,42 as follows:

$${S}_{\,{{\mathrm{fi}}}}^{({{\mathrm{pw}}})}=| {S}_{{{{\rm{fi}}}}}^{({{\mathrm{pw}}})}| \,{e}^{i{\zeta }_{{{{\rm{fi}}}}}}.$$
(4)

Here \(| {S}_{\,{{\mathrm{fi}}}}^{({{\mathrm{pw}}})}{| }^{2}\) defines the emission rate in momentum space where the phase28,29,44ζfi does not contribute to the probability, although it is non-vanishing even in the lowest order of the perturbation theory (see Supplementary Note 5). The result of the calculations is (see Supplementary Note 3)

$${{{\mathcal{W}}}}({{{\bf{r}}}},{{{\bf{k}}}},t)\propto \int_{0}^{\infty }d{t}^{{\prime} }\,\frac{{e}^{-{R}^{2}/{R}_{{{{\rm{eff}}}}}^{2}({t}^{{\prime} })}}{G({t}^{{\prime} })}\,\cos \left(F({t}^{{\prime} })\right),$$
(5)

where \(G({t}^{{\prime} }) \, > \, 0\) is the function particular form of which is of no concern for the following discussion, the momentum conservation \({{{\bf{p}}}}={{{{\bf{p}}}}}^{{\prime} }+{{{\bf{k}}}}\) is implied, and

$${{{\bf{R}}}}={{{\bf{r}}}}-{{{{\boldsymbol{u}}}}}_{p}t+({\partial }_{{{{\bf{p}}}}}+{\partial }_{{{{\bf{k}}}}}){\zeta }_{{{{\rm{fi}}}}}({{{\bf{p}}}},{\lambda }_{e},{{{\bf{k}}}},{\lambda }_{\gamma }).$$
(6)

Here \({{{{\boldsymbol{u}}}}}_{p}={{{\bf{p}}}}/\varepsilon ({{{\bf{p}}}}),\,\varepsilon ({{{\bf{p}}}})=\sqrt{{{{{\bf{p}}}}}^{2}+{m}^{2}},\,{{{{\boldsymbol{u}}}}}_{k}={{{\bf{k}}}}/(n| {{{\bf{k}}}}| ),| {{{{\boldsymbol{u}}}}}_{k}| =1/n\), ∂p = ∂/∂p, \(F({t}^{{\prime} })\propto \arctan {t}^{{\prime} }/{t}_{{{{\rm{d}}}}}\) contains a Gouy phase of the photon connected to its spreading with time \({t}^{{\prime} }\), and td is a diffraction time (see below).

The spatio-temporal dependence of the Wigner function Eq. (5), which is not everywhere positive even in the paraxial approximation, is governed by the ratio

$$\frac{R^2}{R_{{{{{\mathrm{eff}}}}}}^2(t^{\prime})} =\frac{1}{\sigma_x^2(t^{\prime})}\left(\underbrace{\frac{[{{{{{\bf{R}}}}}}\times ({{{{{\boldsymbol{u}}}}}}_p - {{{{{\boldsymbol{u}}}}}}_k)]^2}{({{{{{\boldsymbol{u}}}}}}_p - {{{{{\boldsymbol{u}}}}}}_k)^2}}_{{{{\rm{finite}}}}\, {{{\rm{at}}}}\, {t^{\prime}\, =\, 0}} + \, {{{{\rm{terms}}}}\, {{{\rm{vanishing}}}}\, {{{\rm{at}}}}}\, t^{\prime}= 0\right),$$
(7)

where \({\sigma }_{x}^{2}({t}^{{\prime} })={\sigma }^{-2}\left(1+{({t}^{{\prime} }/{t}_{{{{\rm{d}}}}})}^{2}\right)\) is an rms size of the electron packet. When the condition of ChR is met, up > uk, the vector uk − up is directed backwards with respect to the electron velocity up, and dependence of the Wigner function on R at small \({t}^{{\prime} }\) vanishes along uk − up, defining the Mach cone with an angle

$${\theta }_{{{{\rm{Mach}}}}}=\pi -\arcsin \left(\frac{\sin \theta }{n| {{{{\boldsymbol{u}}}}}_{k}-{{{{\boldsymbol{u}}}}}_{p}| }\right).$$
(8)

If the electron is detected in a plane-wave state, scattered at the angles \({\theta }^{{\prime} },{\phi }^{{\prime} }\), the radius \({R}_{{{{\rm{eff}}}}}({t}^{{\prime} })\) depends on the difference ϕR − ϕ between the azimuthal angle of R and that of k, so it is anisotropic. The azimuthal symmetry of the Mach cone is restored when the electron is not detected and we integrate Eq. (5) over \({{{{\bf{p}}}}}^{{\prime} }\).

At large \({t}^{{\prime} }\), we find \({R}_{\,{{\mathrm{eff}}}\,}^{2}({t}^{{\prime} })\propto {\sigma }_{x}^{2}(0)\,{t}^{{\prime} 2}/{t}_{\,{{\mathrm{d}}}\,}^{2}\) and when \({R}_{{{{\rm{eff}}}}}({t}^{{\prime} })\, \gg \, R\) dependence of the Wigner function on r and tvanishes, similar to the classical wave zone. In the other limiting case, \(R\, \gg \, {R}_{{{{\rm{eff}}}}}({t}^{{\prime} })\), the integrand in Eq. (5) is exponentially suppressed. So, an effective region where space-time correlation exists is when \({t}^{{\prime} }\, \lesssim \, {t}_{{{{\rm{d}}}}}\), and \(R \sim {R}_{{{{\rm{eff}}}}}({t}^{{\prime} })\) is where it is most pronounced, which is why one can call \({R}_{{{{\rm{eff}}}}}({t}^{{\prime} })\)the correlation radius. At \({t}^{{\prime} }\, \gg \, {t}_{{{{\rm{d}}}}}\), both the electron packet and the photon field spread and so there is no longer space-time correlation within the region \(R \, < \, {R}_{{{{\rm{eff}}}}}({t}^{{\prime} })\), which is a hallmark of the wave zone. We show in Fig. 2 that the correlation radius is orders of magnitude smaller than the distance \({u}_{p}{t}^{{\prime} }\) traveled by the electron during the time interval \({t}^{{\prime} }\) for all the angles θR of R, except for the Mach angle, θR ≈ θMach (shown in Fig. 3d).

Fig. 2: The effective correlation radius of the Cherenkov radiation.
figure 2

The ratio Reff to the distance \({u}_{p}{t}^{{\prime} }\) is given for the electron traveling times \({t}^{{\prime} }=1{0}^{7}\,{t}_{{{{\rm{C}}}}}\) (solid black line), \({t}^{{\prime} }=1{0}^{10}\,{t}_{{{{\rm{C}}}}}\) (dotted red line) and \({t}^{{\prime} }=1{0}^{14}\,{t}_{{{{\rm{C}}}}}\) (dot-dashed green line). At panel a \(\beta =0.7\,(\gamma =1.4),n=1.5,\,\theta ={\theta }_{{{{\rm{Ch.cl.}}}}}= \arccos (1/{u}_{p}n)\approx 17.{8}^{\circ },\,\omega =1{0}^{-6}m,\,\sigma =1{0}^{-5}m,\,{\phi }_{R}-\phi =0\) deg, θMach ≈ 107.8°. At panel b β = 0.9999 (γ = 70.7), n = 1.33,  θ = θCh.cl. ≈41. 2°, ω = 10−5m,  σ = 10−4m,  ϕR − ϕ = 0°, θMach ≈131. 2∘. Nearby the Mach angle θMach, space-time dependence of the Wigner function quickly vanishes within the correlation radius \(R < {R}_{{{{\rm{eff}}}}}({t}^{{\prime} })\), which is a hallmark of the wave zone.

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Fig. 3: The spreading time td of the photon field, angular spans Δθ∞ defining negative areas for td and the Mach cone of the emission.
figure 3

Panel a We take ω = 0.51 eV, σ = 10−5 and the packet width σx = 1/σ ≈38 nm. The spreading time td (in picoseconds) from Eq. (9) is displayed for the electron velocity β = 0.65 and refractive index n = 1.46 (solid black), β = 0.7 and n = 1.46 (dashed blue), β = 0.9999 and n = 1.05 (dotted green) and β = 0.9999 and n = 1.3 (dot-dashed red line). The sharp maxima are nearby the angle \({\theta }_{{{{\rm{Ch.cl.}}}}}=\arccos 1/{u}_{p}n\) as illustrated by the arrows. The Cherenkov condition is not met for the black line, which is why the photon field quickly spreads during hundreds of attoseconds. Panel b The inverse spreading time is plotted for parameters n = 1.5, β = 0.9, σ = 10−4 m and ω = 51.1 eV (solid black), ω = 5.1 eV (dashed blue) and ω = 0.51 eV (dotted green line). In the angular span of a size Δθ∞ between two points given by Eq. (10) the spreading time turns negative due to the quantum recoil (ω/ε ≠ 0, see Eq. (11)). Panel c A close-up picture of the spreading time td for parameters n = 1.5, β = 0.999,  ω = 5.1 eV, σ = 10−4 m, θCh.cl ≈48.1°. Panel d Formation of the Mach cone of the electron emission. The electron moves with velocity up exceeding the photon velocity uk. The photon field produces the Mach cone with an angle θMach according to Eq. (8).

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Spreading time and formation length

Let us discuss now the spreading time, which is found as

$${t}_{{{{\rm{d}}}}}=\frac{2}{{\sigma }^{2}}\frac{{({{{{\boldsymbol{u}}}}}_{p}-{{{{\boldsymbol{u}}}}}_{k})}^{2}}{\left(1/\omega {n}^{2}-{\varepsilon }^{-1}\right){({{{{\boldsymbol{u}}}}}_{p}-{{{{\boldsymbol{u}}}}}_{k})}^{2}+\left({\varepsilon }^{-1}-{\omega }^{-1}\right){\left[{{{{\boldsymbol{u}}}}}_{p}\times {{{{\boldsymbol{u}}}}}_{k}\right]}^{2}}.$$
(9)

where \({({{{{\boldsymbol{u}}}}}_{p}-{{{{\boldsymbol{u}}}}}_{k})}^{2}={n}^{-2}+{u}_{p}^{2}-2{u}_{p}\cos \theta /n,\,{\left[{{{{\boldsymbol{u}}}}}_{p}\times {{{{\boldsymbol{u}}}}}_{k}\right]}^{2}={u}_{p}^{2}{\sin }^{2}\theta /{n}^{2}\) with the z axis directed along the electron partial momentum p. One can also define the formation length of radiation as Lf = up∣td∣, which turns to infinity at the Cherenkov angle in the classical Tamm problem11,12,45,46.

The diffraction time and the formation length have an extremum either at the angle \(\cos {\theta }_{{{{\rm{Ch.cl.}}}}}=1/{u}_{p}n \, < \, 1\), when the Cherenkov condition up > 1/n is met, or at \(\cos \theta ={u}_{p}n \, < \, 1\) otherwise. The Wigner function and the energy density stay finite in the latter case, but the photon field rapidly spreads (see the black line in Fig. 3a). Along with the extremum, the time td has two points in which its denominator vanishes (see Fig. 3b),

$$\cos {\theta }_{\infty }\approx \frac{1}{{u}_{p}n}\left(1\mp \sqrt{\frac{\omega }{\varepsilon }}\sqrt{({n}^{2}-1)({u}_{p}^{2}{n}^{2}-1)}\right),$$
(10)

where we have kept the first correction due to quantum recoil, which is usually very small26, ω/ε ≪ 1. Clearly, the diffraction time can only turn to infinity under the condition of ChR, upn > 1. The angular width between the two points is

$$\Delta {\theta }_{\infty }\approx 2\sqrt{\frac{\omega }{\varepsilon }}\sqrt{{n}^{2}-1},$$
(11)

and it vanishes for classical emission with no recoil, ω/ε → 0. For materials like Al, Si, Be, and Ti, Cherenkov radiation can be observed in the soft X-ray range at the frequencies46,47,48ω ≈72.5, 100, 110, 453.8 eV, respectively, which for ε ~(5–20)m yields Δθ∞ < 1°–2° (cf. Fig. 3c, d).

Between the above points, the time td becomes negative—see Fig. 3b—and the Gouy phase \(\arctan {t}^{{\prime} }/{t}_{{{{\rm{d}}}}}\) changes it sign, as if the electron packet shrinks during the emission. In the classical regime with no recoil, both the points merge and so td and Lf turn to infinity at the Cherenkov angle. Indeed, in a vicinity of this angle the spreading time with the recoil kept is

$${\left.{t}_{{{{\rm{d}}}}}\right\vert }_{\cos \theta = 1/n{u}_{p}}=\frac{2\varepsilon }{{\sigma }^{2}}\frac{{n}^{2}}{1-{n}^{2}} \, < \, 0,$$
(12)

where ε = γm. We compare this with the spreading time \({t}_{\,{{\mathrm{d}}}}^{({{\mathrm{e, rest}}})}=m/{\sigma }^{2}\) of an electron packet, which is at rest on average in vacuum49. In the laboratory frame this time is γ times larger, which coincides with ∣td∣ up to the factor 2n2/(n2 − 1) > 2. So, spreading of the photon seems to reverse back in a vicinity of the Cherenkov direction and it is intimately connected with spreading of the electron packet itself.

The coherence lengths of non-relativistic electrons amount to σx(0) ~1–10 nm nearby the standard sources like cathodes of the electron guns in accelerators or electron microscopes49,50,51,52,53. These estimates can likewise be obtained by using the emission duration of photo-electrons from a tungsten tip40 for which the measured sub-femtosecond duration yields nanometer-sized packets. Therefore for γ ≳ 1–2, n ≳ 1 the electron spreading time is \({t}_{\,{{\mathrm{d}}}\,}^{(e)}\, \gtrsim \, 1{0}^{-2}-10\,\,{{\mathrm{ps}}}\,\), and the time td for the photon is of the same order of magnitude nearby the Cherenkov angle, see Fig. 3. The time of flight of an electron through a target of a few centimeters in length is roughly 0.1 ns, and therefore spreading of the electron can be relevant even not far from θCh.cl., especially for non-relativistic particles and for large Cherenkov generators employed, for instance, in neutrino telescopes.

One can use the peculiar behavior of the diffraction time nearby the Cherenkov angle as a means for detecting the quantum recoil in ChR in UV or soft X-ray range. For that, one needs to measure the size of the photon wave packet at different distances close to the electron path—that is, in the pre-wave zone—and at different polar angles θ with an angular resolution of at least 0.1°, which is definitely challenging, but not inconceivable.

Shift of the photon arrival time and flash duration

Dependence of the Wigner function Eq. (5) on the detection time t comes exclusively from the following envelope:

$$\begin{array}{l}\exp \left\{-\frac{{R}^{2}}{{R}_{\,{{\mathrm{eff}}}\,}^{2}({t}^{{\prime} })}\right\}\propto \exp \left\{-\frac{{(t-{t}_{0})}^{2}}{2{\sigma }_{t}^{2}({t}^{{\prime} })}\right\},\\ {\sigma }_{t}^{2}({t}^{{\prime} })=\frac{{\sigma }_{x}^{2}({t}^{{\prime} })}{2}\frac{{({{{{\boldsymbol{u}}}}}_{p}-{{{{\boldsymbol{u}}}}}_{k})}^{2}}{{\left[{{{{\boldsymbol{u}}}}}_{p}\times {{{{\boldsymbol{u}}}}}_{k}\right]}^{2}}.\end{array}$$
(13)

Here, natural duration of the Cherenkov flash is defined by \({\sigma }_{t}({t}^{{\prime} })\) and \({t}_{0}={{{{\boldsymbol{l}}}}}_{0}\cdot ({{{\bf{r}}}}+({\partial }_{{{{\bf{p}}}}}+{\partial }_{{{{\bf{k}}}}}){\zeta }_{{{{\rm{fi}}}}})\) is a time instant at which the probability to catch the photon around the point r is maximized, \({{{{\boldsymbol{l}}}}}_{0}=[({{{{\boldsymbol{u}}}}}_{p}-{{{{\boldsymbol{u}}}}}_{k})\times [{{{{\boldsymbol{u}}}}}_{k}\times {{{{\boldsymbol{u}}}}}_{p}]]/{[{{{{\boldsymbol{u}}}}}_{p}\times {{{{\boldsymbol{u}}}}}_{k}]}^{2}\). We will call the time t0the mean arrival time.

One can neglect the term with the phase ζfi in R Eq. (6) in the wave zone where ∣r − upt∣ ≫ ∣(∂p + ∂k)ζfi∣, and then the Wigner function Eq. (5) and the emitted energy seem to be concentrated in a vicinity of the classical electron trajectory54, r ~upt. The detector registers a photon in the far field emitted at t = 0, r0 = 0 by a classical point-like electron at the time instant

$${t}_{\,{{\mathrm{cl.}}}}^{({{\mathrm{far-f.}}})}=r/{u}_{k}=r\,n,$$
(14)

which will be called the classical arrival time. Let us compare this prediction with the above t0, derived quantum mechanically. Orienting the z axis along the electron momentum p, we find \({{{\bf{k}}}}=n\omega {{{\boldsymbol{l}}}},\,{{{\boldsymbol{l}}}}=\{\sin \theta \cos \phi ,\sin \theta \sin \phi ,\cos \theta \},\,r\equiv {{{\bf{r}}}}\cdot {{{\boldsymbol{l}}}},\,{t}_{\,{{\mathrm{cl.}}}}^{({{\mathrm{far-f.}}})}={{{\bf{r}}}}\cdot {{{\boldsymbol{l}}}}\,n,\) and \({{{{\boldsymbol{l}}}}}_{0}={\sin }^{-2}\theta \, (({u}_{p}^{-1}- n\cos \theta ) {{{\bf{p}}}}/ | {{{\bf{p}}}}| + (n- {u}_{p}^{-1}\cos \theta ) {{{\boldsymbol{l}}}})\). In a vicinity of the Cherenkov angle, we have l0 → l n and so for ζfi = 0 we get \({t}_{0}\to {t}_{\,{{\mathrm{cl.}}}}^{({{\mathrm{far-f.}}})}={{{\bf{r}}}}\cdot {{{\boldsymbol{l}}}}\,n\), in accordance with Eq. (14).

However, the phase ζficannot be ignored in the formation zone and in the near-field (in the latter case, r ≈upt, so that R ≈(∂p + ∂k)ζfi) and this makes the photon arrival time t0 different from the classical value \({t}_{\,{{\mathrm{cl.}}}}^{({{\mathrm{far-f.}}})}\). In this regime, we call tcl. = r ⋅ l0 for arbitrary emission angles and so the quantum shift is

$$\Delta t={t}_{0}-{t}_{{{{\rm{cl.}}}}}={{{{\boldsymbol{l}}}}}_{0}\cdot ({\partial }_{{{{\bf{p}}}}}+{\partial }_{{{{\bf{k}}}}}){\zeta }_{{{{\rm{fi}}}}}.$$
(15)

This shift is not necessarily positive and the physical origin of this delay or advance is the electric dipole moment density ∝ e(∂p + ∂k)ζfi induced in medium by the field of the electron. We deal with an analog of the AC Stark effect16 with the atoms being off-resonantly polarized by a broadband spectrum Δω of pseudo-photons. Similarly to the observed time delays—positive and negative—when a laser propagates in a medium55,56,57, here we encounter delays induced by the virtual photons, reemitted as real ones. Classically, one can look at this as if the photon was emitted not from a point-like electron, but from a point shifted laterally to the distance Δρ ~βγλ/2π from the electron trajectory closer or further from the detector, which is a mean free path of the virtual photon12,32. Numerically Δρ/c ~βγλ/2πc = βγ/ω ~1–100 fs for photons from IR to UV ranges and γ = ε/m ≲ 10.

When measuring the quantum shift in the photon arrival time from its classical value, the flash duration \({\sigma }_{t}({t}^{{\prime} })\) is crucial because the deviations can hardly be discerned with \({\sigma }_{t}({t}^{{\prime} })\, \gg \, | \Delta t|\) occurring for \({t}^{{\prime} }\, \gg \, {t}_{{{{\rm{d}}}}}\) far from the Cherenkov angle. This duration was estimated classically by Frank8,9 to be  ~1/Δω < 1 ps where Δω is a frequency interval for which the emission takes place. Quantum estimates from the uncertainty principle yield roughly the same result both in the macroscopic theory of ChR30 and for the microscopic off-resonant AC Stark effect in single atoms16, 1/Δω ~0.1–100 fs for Δω ~10−2–10 eV. Our quantum model predicts the following flash duration nearby the Cherenkov angle:

$${\sigma }_{t}({t}^{{\prime} })\to n\,{\sigma }_{x}(0)/\sqrt{2},$$
(16)

because in this case td → ∞. Clearly, only the electron packet’s length at the target entrance (\({t}^{{\prime} }=0\)) contributes to the flash duration nearby θCh.cl. For realistic electrons with σx(0) ~1–100 nm, we find

$${\sigma }_{t}(0) \sim 10\,\,{{\mathrm{as}}}\,-1\,\,{{\mathrm{fs}}}\,.$$
(17)

In Figs 4 and 5, we show that the typical shifts indeed belong to the attosecond range and that the flash duration is generally larger than the shift. The electron transverse momenta are chosen to be p⊥ = σ ~(10−7–10−4) m because they correspond to the spatial widths 1/p⊥ = σx(0) ≳ 1 nm–1 μm, respectively, and we also neglect the spreading. Note that within the pre-wave zone the emission does not take place only at the Cherenkov angle. The sign of the shift swaps between the two kinematic scenarios (Supplementary Notes 4 and 5), which is why the absolute value ∣Δt∣ is shown. Fixing the detector at certain angles θ, ϕ and the distance r, one would see that the photons equally probably arrive either later (time delay) than tcl. or sooner than that (an advance or negative delay), if the electron is not detected. Note that integration of the Wigner function over \({{{{\bf{p}}}}}^{{\prime} }\) puts the electron momentum to the definite value \({{{{\bf{p}}}}}^{{\prime} }={{{\bf{p}}}}-{{{\bf{k}}}}\) with subsequent integration over p with the Gaussian distribution. If one wishes to catch only the shifts with one particular sign, one should detect the photon and the electron in coincidence, which is technically more challenging.

Fig. 4: The quantum shift of the photon arrival time as a function of the emission angle.
figure 4

We take the electron energy typical for a transmission electron microscope: εc ≈200 keV, β ≈0.7, \({p}_{\perp }^{{\prime} }=0.99\times {p}_{\perp },\,{p}_{z}^{{\prime} }={p}_{z}-{k}_{z}\approx 0.98\,m,\,n=1.5\). The electron momentum uncertainty is equal to the transverse momentum: σ = p⊥. In panel a p⊥ = 10−5 m,  1/p⊥ ≳ 10 nm; panel b p⊥ = 10−6 m,  1/p⊥ ≳ 100 nm. The classical Cherenkov flash durations for the emitted photon energies ω = 10−4 m (green solid line) and 10−7 m (brown dashed line) are compared to the quantum shifts (Eq. (15)) for ω = 10−4 m (black solid), 10−5 m (red dashed), 10−6 m (blue dotted) and 10−7 m (orange dot-dashed line). The behavior at small angles is shown in the inset figures. The quantum shifts are restricted to the regions allowed by the momentum conservation law and vanish outside of them. They stay roughly the same for other values of \({p}_{\perp }^{{\prime} },\,{p}_{z}^{{\prime} }\) and for ultrarelativistic electrons, γ ≫ 1, though the Cherenkov angle grows.

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From Figs. 4 and 5 we conclude that the difference between the classical flash duration σt (green line in both figures) and the temporal shift is minimized for angles θ ≲ 10° and media with small refractive indices n ~1.01–1.5, whereas the ratio σt/∣Δt∣ ≲ 2 for all the angles θ < 10°. The argument is correct only when spreading is not taken into account because \({\sigma }_{t}({t}^{{\prime} })\) grows with time \({t}^{{\prime} }\) and the shift does not. For larger angles, θ ≳ 10°, or for n ≳ 2 the difference between the flash duration (the green line) and the shift increases. When trying to detect the quantum shift for the transmission electron microscope (TEM) energies, β ~0.7, a target made of fused silica with n ~1.44–1.47 (see, for example,58,59) in the optical range can do the job, whereas for ultrarelativistic electrons, γ ≫ 1, aerogels with n ~1.01–1.30 can be employed (see, for example,60,61), which are already used as Cherenkov generators62,63,64. The use of Brewster-Cherenkov detectors65 can also come in handy here. Note, however, that for the TEMs energies there also be transition radiation generated at a boundary of a real target, which propagates at the angles of θ ~ γ−1 and can, therefore, interfere with the Cherenkov signal. Although microscopically this type of radiation has a similar origin as ChR and, therefore, similar temporal shifts are likely to take place, one can start with studying pure Cherenkov emission. One of the alternatives would be to employ the so-called diffraction Cherenkov radiation12,46 where an electron moves close to a dielectric target but does not intersect it. In that case, diffraction radiation is generated at the angles θ ≪ γ−1 and Cherenkov emission can still be well distinguished at θ ≲ 10°.

Fig. 5: The quantum shift of the photon arrival time and the classical Cherenkov angle as a function of the refractive index.
figure 5

Here, the emission angle is chosen to be θ = 10° and the electron momentum uncertainty is σ = p⊥. We distinguish a regime of a transmission electron microscope—panel a β ≈0.7 (γ ≈1), and an accelerator regime—panel b β = 0.9999 (γ ≈70). In both panels a and b we fix \({p}_{\perp }^{{\prime} }=0.99\times {p}_{\perp },\,{p}_{z}^{{\prime} }={p}_{z}-{k}_{z}\), the classical Cherenkov flash duration is shown for ω = 10−5 m (green solid line), whereas the quantum shifts are given for ω = 10−5 m (black solid line) and 10−6 m (red dashed line). Panels c and d demonstrate dependence of the classical Cherenkov angle \({\theta }_{{{{\rm{Ch.cl.}}}}}=\arccos (1/\beta n)\) on the refractive index for two types of dielectric media—fused silica and aerogels. The blue shaded areas in c and d show the boundaries of the refractive indices due to the frequency dispersion of n(ω) for fused silica or different types of particular chemical solutions for aerogels. The minimal refractive index for which the Cherenkov condition is met in scenario a is n ≈1.444, and θCh.cl. = 10° corresponds to n ≈1.466 within the range of n from panel (c). Analogously, in scenario b θCh.cl. = 10° corresponds to n ≈1.016 within the range of n from panel (d).

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If we now go beyond the simplified model of the electron packet with σ⊥ = σ∣∣ and recall that there is Lorentz contraction, σ∣∣ = γ−1σ⊥, the flash durations can become γ times shorter than Eq. (17). Although photon spreading can be safely neglected nearby the Cherenkov angle, spreading of the electron before entering the target can significantly increase these numbers. According to the quantum dynamics of the electron coherence length \({\sigma }_{x}({t}^{{\prime} })\)—also called the generalized van Cittert-Zernike theorem49—when an electron is released from a photo-gun or a field emitter with rms sizes σ⊥ ~ σ∣∣50 of a few nanometers and an energy up to a few tens of eV, it spreads to some tens of nanometers at the distance of 1 μm and it reaches micrometer sizes already at 1 mm from the source. Clearly, electron acceleration—say, in an RF cavity—mitigates the spreading rate, but a realistic estimate of the packet length σ∣∣ at the target boundary remains unknown and it is not usually controlled. Therefore, measuring the flash duration in a vicinity of the ChR angle, one can retrieve the length of the electron packet at the target entrance, a complementary approach to interferometry that is reminiscent of bunch length measurements via classical coherence14.

In practice, achieving attosecond flash durations requires nanometer-sized electron packets generating radiation with no spreading before the target, just after the emission from a cathode. A vacuum gap of even a few millimeters after the source and before the target would yield micrometer-sized electron packets entering the target, which results in picosecond flash durations or in subpicosecond ones with acceleration to a few MeV in the gap due to the above factor γ−1, measurable by streak cameras. State-of-the-art Cherenkov counters at accelerators have picosecond time resolution64,66, the subfemtosecond resolution can be achieved at X-ray free-electron lasers67,68 or with the frequency-resolved optical gating69, whereas it is nanoseconds for Cherenkov telescopes in gamma-ray astronomy21 where spreading effects are notable. Attosecond photon pulses—including twisted photons with orbital angular momentum—are usually obtained through high-harmonic generation in the extreme-ultraviolet and even soft X-ray ranges70,71, enabling vortex electrons generation via photoionization71. The ChR can be a source of twisted photons26,31, also in the soft X-ray range, and shaping the spatial and temporal profiles of the electron wave function offers refined tuning of phase-space profile of the photons. In particular, it seems feasible to generate photon wave packets with the given spatial profile, vorticity, and the energies up to the soft X-ray range with the attosecond and even sub-attosecond durations.

Conclusions

The developed quantum framework provides visualization of the emitted photon energy in phase space and, in particular, in real space and time, including the near-field zone. This can be desirable for estimating the Cherenkov wakefields in accelerator chambers as well as for biomedical applications of ChR in radiotherapy23,24 where only the classical Tamm–Frank theory in the far field has been used so far, apparently overestimating the radiation intensity in the formation zone. We have predicted several spatiotemporal features of the quantum ChR, absent within the conventional momentum-space analysis limited to the far-field approximation. In particular, the Cherenkov flash duration close to the Cherenkov angle turns out to be determined by the electron coherence length upon entering the medium, opening avenues for controlling this duration by selecting packets of the desired length and accelerating them in the low-current regime with no space-charge effects, typical for TEMs. By using the charged particles with the Lorentz factors γ = ε/m from γ ~10 to γ ~103–104, one can generate attosecond photon flashes and even the sub-attosecond ones if the target is installed close to the particle source. Such short photon pulses can come in handy for various studies in atomic physics, including those of the temporal effects in electron excitation and photoionization.

Alternatively, measuring photon pulse durations with subpicosecond resolution can provide a technique for determining the lengths of the emitting charged-particle packets. This can be used to control the lengths of the wave packets in electron microscopes and accelerator facilities (linacs or storage rings with electrons, protons, or ions), as well as of ultrarelativistic charged particles coming from the cosmos. Naturally, ChR serves as an exemplary case within a broader class of media-induced emission phenomena where attosecond time scales are accessible via phase-space analysis.