Vortex particles in axially symmetric fields and applications of the quantum Busch theorem

3.1.1. Emittance and the Schrödinger uncertainty relation

Let us consider a non-relativistic and not necessarily Gaussian packet of a freely propagating massive particle in 1 + 1 dimensional space-time. Its mean coordinate follows the classical trajectory,

$$\begin{equation}\langle x\rangle =\langle u\rangle t,\quad \langle u\rangle =\frac{\langle \hat{p}\rangle }{m},\end{equation} \tag{ 17 }$$

where $\hat{p}=-\mathrm{i}{\partial }_{x}$ in the coordinate representation, and the Hamiltonian is

$$\begin{equation}{\hat{H}}_{\text{fr}}=\frac{{\hat{p}}^{2}}{2m}.\end{equation} \tag{ 18 }$$

The latter does not commute with x² and the Heisenberg's equation of motion is

$$\begin{equation}\frac{{\partial }^{2}\langle {x}^{2}\rangle \left(t\right)}{\partial {t}^{2}}=-\left\langle \left[\left[{\hat{x}}^{2},{\hat{H}}_{\text{fr}}\right],{\hat{H}}_{\text{fr}}\right]\right\rangle =2\langle {u}^{2}\rangle =2\frac{\langle {\hat{p}}^{2}\rangle }{{m}^{2}}.\end{equation} \tag{ 19 }$$

The general solution to this equation reads (cf [44, 55])

$$\begin{equation}\langle {x}^{2}\rangle \left(t\right)=\langle {x}^{2}\rangle \left(0\right)+\frac{\partial \langle {x}^{2}\rangle \left(0\right)}{\partial t}t+\langle {u}^{2}\rangle {t}^{2},\end{equation} \tag{ 20 }$$

and describes spreading of a packet of an arbitrary shape with time and distance.

Let the packet be in a quantum state described with a wave function ψ or with a quantum phase-space distribution (Wigner) function. We denote a quantum mechanical averaging as ⟨...⟩ and also

$$\begin{equation}{q}_{i}=\left\{x,u\right\},\quad i=1,2.\end{equation} \tag{ 21 }$$

Let us define a matrix of central moments as follows:

$$\begin{equation}{Q}_{ij}=\langle {q}_{i}{q}_{j}\rangle -\langle {q}_{i}\rangle \langle {q}_{j}\rangle .\end{equation} \tag{ 22 }$$

The square root of its determinant,

$$\begin{equation}{{\epsilon}}_{x}=\sqrt{\mathrm{d}\mathrm{e}\mathrm{t}\enspace {Q}_{ij}}=\sqrt{\left(\langle {x}^{2}\rangle -{\langle x\rangle }^{2}\right)\left(\langle {u}^{2}\rangle -{\langle u\rangle }^{2}\right)-{\left(\langle xu\rangle -\langle x\rangle \langle u\rangle \right)}^{2}}\end{equation} \tag{ 23 }$$

we call a normalized emittance of the packet. This general definition is Lorentz invariant and it also holds for the particle in external electromagnetic fields. For a free particle, we take equation (20) into account and have

$$\begin{equation}{{\epsilon}}_{x}^{\text{free}}=\sqrt{\langle {x}^{2}\rangle \left(0\right)\left(\langle {u}^{2}\rangle -\langle {u\rangle }^{2}\right)-{\left(\langle xu\rangle \left(0\right)\right)}^{2}},\end{equation} \tag{ 24 }$$

i.e. the emittance is conserved in time.

The usefulness of the emittance (23) is that it explicitly measures non-classicality of the packet via the Schrödinger uncertainty relation ¹ [47]. Indeed, let us define the following quality factor of a packet as a measure of non-classicality:

$$\begin{equation}M=\frac{2{{\epsilon}}_{x}}{{\lambda }_{\mathrm{c}}}{\geqslant}1,\end{equation} \tag{ 25 }$$

where

$$\begin{equation}{\lambda }_{\mathrm{c}}=\frac{1}{m}\equiv \frac{\hslash }{mc}\end{equation} \tag{ 26 }$$

is a Compton wavelength of the particle with a mass m, which is 3.9 × 10⁻¹¹ cm for electron. We will also need an electron's Compton time,

$$\begin{equation}{t}_{\mathrm{c}}={\lambda }_{\mathrm{c}}/c\approx 1.3{\times}1{0}^{-21}\enspace \mathrm{s},\end{equation} \tag{ 27 }$$

which is a typical lifetime of a virtual electron-positron pair. Then we have (cf [57])

$$\begin{align}\hfill & \frac{M}{2}=\frac{{{\epsilon}}_{x}}{{\lambda }_{\mathrm{c}}}=\sqrt{\left(\langle {x}^{2}\rangle -{\langle x\rangle }^{2}\right)\left(\langle {p}^{2}\rangle -{\langle p\rangle }^{2}\right)-{\left(\langle xp\rangle -\langle x\rangle \langle p\rangle \right)}^{2}}{\geqslant}\frac{1}{2},\hfill \\ \hfill & \text{or}\;{{\epsilon}}_{x}{\geqslant}\frac{{\lambda }_{\mathrm{c}}}{2},\hfill \end{align} \tag{ 28 }$$

which is just the Schrödinger uncertainty relation [47]. The quality factor M defines the conserved total number of the mutually orthogonal states in the particle's phase space. The higher this number is, the larger is the rhs of the uncertainty relation and the 'more quantum' is the packet. Clearly, even for the simplest free packet it is the Schrödinger uncertainty relation that adequately measures the packet's non-classicality and not the Heisenberg relation without the correlation term.

3.1.2. Gaussian packet and the Gouy phase

As a simplest example, we take a Gaussian packet with a mean momentum ⟨p⟩ and its uncertainty σ_p ,

$$\begin{equation}\psi \left(p\right)=N\enspace \mathrm{exp}\left\{-\frac{{\left(p-\langle p\rangle \right)}^{2}}{2{\sigma }_{p}^{2}}\right\}.\end{equation} \tag{ 29 }$$

In the coordinate space we have

$$\begin{align}\hfill & \psi \left(x,t\right)=\int \frac{\mathrm{d}p}{2\pi }\enspace \psi \left(p\right)\mathrm{exp}\left\{-\mathrm{i}t\frac{{p}^{2}}{2m}+\mathrm{i}px\right\}\hfill \\ \hfill & \qquad =N\enspace \frac{1}{\sqrt{{\sigma }_{x}\left(t\right)}}\mathrm{exp}\left\{-\mathrm{i}t\frac{{\langle p\rangle }^{2}}{2m}+\mathrm{i}\langle p\rangle x-\frac{\mathrm{i}}{2}\mathrm{arctan}\frac{t}{{t}_{\mathrm{d}}}-\frac{{\left(x-\langle u\rangle t\right)}^{2}}{2{\sigma }_{x}^{2}\left(t\right)}\left(1-\mathrm{i}\frac{t}{{t}_{\mathrm{d}}}\right)\right\},\hfill \end{align} \tag{ 30 }$$

where

$$\begin{equation}{\sigma }_{x}\left(t\right)={\sigma }_{p}^{-1}\sqrt{1+\frac{{t}^{2}}{{t}_{\mathrm{d}}^{2}}},\qquad {t}_{\mathrm{d}}=\frac{m}{{\sigma }_{p}^{2}},\qquad \langle {x}^{2}\rangle =\frac{{\sigma }_{x}^{2}\left(t\right)}{2},\qquad \frac{\partial \langle {x}^{2}\rangle \left(0\right)}{\partial t}=0.\end{equation} \tag{ 31 }$$

The normalization constant N can be found from the condition

$$\begin{equation}\int \mathrm{d}x\enspace \vert \psi \left(x,t\right){\vert }^{2}=\int \frac{\mathrm{d}p}{2\pi }\enspace \vert \psi \left(p\right){\vert }^{2}=1.\end{equation} \tag{ 32 }$$

Equation (30) represents an exact non-stationary solution to the Schrödinger equation,

$$\begin{equation}\left(\mathrm{i}{\partial }_{t}-\frac{{\hat{p}}^{2}}{2m}\right)\psi \left(x,t\right)=0.\end{equation} \tag{ 33 }$$

The Gouy phase in (30)

$$\begin{equation}{\varphi }^{\mathrm{G}}=\mathrm{arctan}\enspace t/{t}_{\mathrm{d}}\end{equation} \tag{ 34 }$$

comes about due to the packet's spreading with time, even at rest with ⟨u⟩ = 0. Note the following identity:

$$\begin{equation}1-\mathrm{i}\frac{t}{{t}_{\mathrm{d}}}=\sqrt{1+\frac{{t}^{2}}{{t}_{\mathrm{d}}^{2}}}\mathrm{exp}\left\{-\mathrm{i}\enspace \mathrm{arctan}\frac{t}{{t}_{\mathrm{d}}}\right\}.\end{equation} \tag{ 35 }$$

Importantly, the diffraction (spreading) time ${t}_{\mathrm{d}}=m/{\sigma }_{p}^{2}$ and the Gouy phase itself are Lorentz invariant because the momentum uncertainty σ_p and the time t are defined in the packet's rest frame. An inversion of time t → −t implies ⟨p⟩ → −⟨p⟩ and so (see also [18])

$$\begin{equation}t\to -t:\psi \left(x,t\right)\to {\psi }^{{\ast}}\left(x,t\right),\end{equation} \tag{ 36 }$$

in accord with the Wigner theorem and the general charge, parity, and time reversal (CPT)-theorem [58].

3.1.3. Elegant Hermite–Gaussian packet

One can 'excite' the Gaussian states by adding a factor (p − ⟨p⟩)ⁿ with n = 0, 1, 2, ... to equation (29):

$$\begin{equation}{\psi }_{n}\left(p\right)=N\enspace {\left(p-\langle p\rangle \right)}^{n}\enspace \mathrm{exp}\left\{-\frac{{\left(p-\langle p\rangle \right)}^{2}}{2{\sigma }_{p}^{2}}\right\}.\end{equation} \tag{ 37 }$$

The corresponding wave function of the coordinate representation reads

$$\begin{align}\hfill {\psi }_{n}\left(x,t\right)& =N\frac{1}{{\left(1+{t}^{2}/{t}_{\mathrm{d}}^{2}\right)}^{\frac{n+1}{4}}}\enspace {H}_{n}\left(\left(x-\langle u\rangle t\right)\sqrt{\frac{1-\mathrm{i}t/{t}_{\mathrm{d}}}{2{\sigma }_{x}^{2}\left(t\right)}}\right)\hfill \\ \hfill & \quad {\times}\mathrm{exp}\left\{-\mathrm{i}t\frac{\langle {p\rangle }^{2}}{2m}+\mathrm{i}\langle p\rangle x-\frac{\mathrm{i}}{2}\left(n+1\right)\mathrm{arctan}\frac{t}{{t}_{\mathrm{d}}}-\frac{{\left(x-\langle u\rangle t\right)}^{2}}{2{\sigma }_{x}^{2}\left(t\right)}\left(1-\mathrm{i}\frac{t}{{t}_{\mathrm{d}}}\right)\right\},\hfill \end{align} \tag{ 38 }$$

where H_n are Hermite polynomials and a prefactor of the Gouy phase now depends on n. One can calculate the following averages:

$$\begin{align}\hfill & \langle {x}^{2}\rangle ={\sigma }_{x}^{2}\left(t\right)\enspace \left(n+1/2\right)-{\sigma }_{x}^{2}\left(0\right)\enspace \frac{n\left(n-1\right)}{n-1/2}+{\langle u\rangle }^{2}{t}^{2},\hfill \\ \hfill & \langle {p}^{2}\rangle ={\langle p\rangle }^{2}+{\sigma }_{p}^{2}\enspace \left(n+1/2\right),\qquad \langle \varepsilon \rangle =\frac{\langle {p}^{2}\rangle }{2m}=\frac{{\langle p\rangle }^{2}}{2m}+\frac{{\sigma }_{p}^{2}}{2m}\enspace \left(n+1/2\right),\hfill \\ \hfill & \langle xu\rangle =\frac{1}{2}\frac{\partial \langle {x}^{2}\rangle }{\partial t}=\frac{1}{m}\enspace \frac{t}{{t}_{\mathrm{d}}}\enspace \left(n+1/2\right)+{\langle u\rangle }^{2}t,\quad \langle xu\rangle \left(0\right)=0.\hfill \end{align} \tag{ 39 }$$

Note that the use of momentum representation is somewhat easier here, and the coordinate operator in momentum space is

$$\begin{equation}\hat{x}=\mathrm{i}\frac{\partial }{\partial p}+\frac{p}{m}t.\end{equation} \tag{ 40 }$$

Although this packet is non-stationary and spreads, its mean energy ⟨ɛ⟩ is conserved.

The emittance of this packet is

$$\begin{equation}{{\epsilon}}_{x}=\frac{{\lambda }_{\mathrm{c}}}{2}\enspace \sqrt{\frac{\left(2n+1\right)\left(4n-1\right)}{2n-1}},\end{equation} \tag{ 41 }$$

whereas the quality factor reads

$$\begin{equation}M=\frac{2{{\epsilon}}_{x}}{{\lambda }_{\mathrm{c}}}=\sqrt{\frac{\left(2n+1\right)\left(4n-1\right)}{2n-1}}=\begin{cases}1,\quad \hfill & \quad n=0,\hfill \\ 2\sqrt{n},\quad \hfill & \quad n\gg 1.\hfill \end{cases}\end{equation} \tag{ 42 }$$

Equation (38) also represents an exact non-stationary solution to the Schrödinger equation and in optics it is often called an elegant HG packet [59–61]. The set of these functions, however, is not orthogonal,

$$\begin{equation}\int \mathrm{d}x\enspace {\psi }_{n}^{{\ast}}\left(x,t\right){\psi }_{m}\left(x,t\right)/\propto {\delta }_{nm},\end{equation} \tag{ 43 }$$

but rather biorthogonal as the function adjoint to ψ_n (x, t) is not ${\psi }_{n}^{{\ast}}\left(x,t\right)$ [59, 60].

3.1.4. Standard Hermite–Gaussian packet

There exists a very similar but not identical set of functions, which is complete and orthogonal and it can be obtained by the following substitution in equation (37):

$$\begin{equation*}{\left(p-\langle p\rangle \right)}^{n}\to {\left(p-\langle p\rangle -\mathrm{i}\text{const}\enspace \frac{\partial }{\partial p}\right)}^{n}.\end{equation*}$$

The corresponding coordinate function

$$\begin{align}\hfill {\psi }_{n}\left(x,t\right)& =N\enspace \frac{1}{{\left(1+{t}^{2}/{t}_{\mathrm{d}}^{2}\right)}^{1/4}}\enspace {H}_{n}\left(\frac{x-\langle u\rangle t}{{\sigma }_{x}\left(t\right)}\right)\hfill \\ \hfill & \quad {\times}\mathrm{exp}\left\{-\mathrm{i}t\frac{\langle {p\rangle }^{2}}{2m}+\mathrm{i}\langle p\rangle x-\frac{\mathrm{i}}{2}\left(2n+1\right)\mathrm{arctan}\frac{t}{{t}_{\mathrm{d}}}-\frac{{\left(x-\langle u\rangle t\right)}^{2}}{2{\sigma }_{x}^{2}\left(t\right)}\left(1-\mathrm{i}\frac{t}{{t}_{\mathrm{d}}}\right)\right\}\hfill \end{align} \tag{ 44 }$$

satisfies the Schrödinger equation exactly and it is called a standard HG packet. Compared to equation (38), a prefactor of the Gouy phase has now 2n + 1 instead of n + 1, which is closely connected with the quality factor.

Indeed, calculating the averages we arrive at

$$\begin{equation}\langle {x}^{2}\rangle ={\sigma }_{x}^{2}\left(t\right)\enspace \left(n+1/2\right)+\langle {u\rangle }^{2}{t}^{2},\qquad \langle {p}^{2}\rangle =\langle {p\rangle }^{2}+{\sigma }_{p}^{2}\enspace \left(n+1/2\right),\end{equation} \tag{ 45 }$$

and so

$$\begin{equation}{{\epsilon}}_{x}={\lambda }_{\mathrm{c}}\enspace \left(n+\frac{1}{2}\right),\qquad M=\frac{2{{\epsilon}}_{x}}{{\lambda }_{\mathrm{c}}}=2n+1.\end{equation} \tag{ 46 }$$

Thus the quality factor of the orthogonal (standard) HG packet grows linearly with n, while for the non-orthogonal (elegant) packet it does only as $\sqrt{n}$.

For a packet with the focus at t = 0—that is, ⟨xu⟩(0) = 0—we have

$$\begin{equation}\sqrt{\langle {x}^{2}\rangle -{\langle x\rangle }^{2}}{\vert }_{t=0}\enspace \sqrt{\langle {p}^{2}\rangle -{\langle p\rangle }^{2}}=\frac{{{\epsilon}}_{x}}{{\lambda }_{\mathrm{c}}}=\frac{M}{2}\approx \begin{cases}n,\quad \hfill & \quad \text{standard}\;\text{HG},\hfill \\ \sqrt{n},\quad \hfill & \quad \text{elegant}\;\text{HG},\hfill \end{cases}\end{equation} \tag{ 47 }$$

so the quality factor M is just a total number of the mutually orthogonal quantum states in the particle's phase space. Analogously to photons, when the beam consists of many identical packets, the number M/2 defines information capacity of the beam. As the quality factor of the non-orthogonal HG packet scales as $\sqrt{n}$, such a beam clearly has a lower information capacity than the standard one.

Note that the number of states grows with time together with the corresponding entropy S while the packet spreads,

$$\begin{equation}S=\mathrm{ln}\left(\sqrt{\langle {x}^{2}\rangle -\langle {x\rangle }^{2}}\enspace \sqrt{\langle {p}^{2}\rangle -\langle {p\rangle }^{2}}\right)\propto \mathrm{ln}\enspace t.\end{equation} \tag{ 48 }$$

This entropy actually is an attribute of a closed system, which includes the packet and an observer, the latter being a macroscopic classical system. The observer performed two measurements of ⟨x²⟩ at t = 0 and then after the time t. As there were no measurements in between, the growth of the entropy is due to the loss of information by the observer. However, 'the entropy' of the packet alone does not grow, which is connected to conservation of the emittance.

3.1.5. Courant–Snyder formalism

An arbitrary packet can also be described by the Courant–Snyder or Twiss parameters $\hat{\alpha },\hat{\beta },\hat{\gamma }$ used in accelerator physics ² [29], while the emittance defines a conserved area π_x of an ellipse in the particle's phase space (see figure 1),

$$\begin{align}\hfill & \hat{\beta }\enspace {\left(\dot {X}\right)}^{2}+2\hat{\alpha }X\dot {X}+\hat{\gamma }{X}^{2}={{\epsilon}}_{x}{\geqslant}\frac{{\lambda }_{\mathrm{c}}}{2},\hfill \\ \hfill & X{:=}\sqrt{\langle {x}^{2}\rangle -{\langle x\rangle }^{2}},\hfill \\ \hfill & \text{or}\enspace \frac{\hat{\beta }}{2}\left(\langle {u}^{2}\rangle -{\langle u\rangle }^{2}\right)+\hat{\alpha }\left(\langle xu\rangle -\langle x\rangle \langle u\rangle \right)+\frac{\hat{\gamma }}{2}\left(\langle {x}^{2}\rangle -{\langle x\rangle }^{2}\right)={{\epsilon}}_{x},\hfill \end{align} \tag{ 49 }$$

where

$$\begin{align}\hfill & \hat{\beta }=\frac{\langle {x}^{2}\rangle -{\langle x\rangle }^{2}}{{{\epsilon}}_{x}}=\frac{{X}^{2}}{{{\epsilon}}_{x}},\qquad \hat{\alpha }=-\frac{1}{2}\frac{\partial \hat{\beta }}{\partial t}=-\frac{\langle xu\rangle -\langle x\rangle \langle u\rangle }{{{\epsilon}}_{x}}=-\frac{X\dot {X}}{{{\epsilon}}_{x}},\hfill \\ \hfill & \hat{\gamma }=\frac{\langle {u}^{2}\rangle -{\langle u\rangle }^{2}}{{{\epsilon}}_{x}}=\frac{X\ddot {X}+{\left(\dot {X}\right)}^{2}}{{{\epsilon}}_{x}},\qquad \hat{\beta }\hat{\gamma }-{\hat{\alpha }}^{2}=1.\hfill \end{align} \tag{ 50 }$$

The area of the ellipse is limited from below by the Schrödinger uncertainty relation, π_x ⩾ πλ_c/2, the slope with respect to a coordinate system is defined by the correlation parameter $\hat{\alpha }\left(\delta \right)$, and the Twiss parameters at the angle δ are defined only by $\hat{\beta }\left(0\right)$ as follows:

$$\begin{equation}\left(\begin{matrix}\hfill \hat{\alpha }\left(\delta \right)\hfill \\ \hfill \hat{\beta }\left(\delta \right)\hfill \\ \hfill \hat{\gamma }\left(\delta \right)\hfill \end{matrix}\right)=\left(\begin{matrix}\hfill 0\hfill & \hfill -\mathrm{sin}\enspace \delta \enspace \mathrm{cos}\enspace \delta \hfill & \hfill \mathrm{sin}\enspace \delta \enspace \mathrm{cos}\enspace \delta \hfill \\ \hfill 0\hfill & \hfill {\mathrm{cos}}^{2}\enspace \delta \hfill & \hfill {\mathrm{sin}}^{2}\enspace \delta \hfill \\ \hfill 0\hfill & \hfill {\mathrm{sin}}^{2}\enspace \delta \hfill & \hfill {\mathrm{cos}}^{2}\enspace \delta \hfill \end{matrix}\right)\left(\begin{matrix}\hfill 0\hfill \\ \hfill \hat{\beta }\left(0\right)\hfill \\ \hfill {\hat{\beta }}^{-1}\left(0\right)\hfill \end{matrix}\right),\end{equation} \tag{ 51 }$$

where we have taken into account that $\hat{\alpha }\left(0\right)=0,\enspace \hat{\gamma }\left(0\right)={\hat{\beta }}^{-1}\left(0\right)$.

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One can describe the packet's rms size in terms of the $\hat{\beta }$-function. For orthogonal packets, the $\hat{\beta }$-function does not depend on the quality factor. For instance, for the standard HG packets

$$\begin{equation}{\hat{\beta }}^{\text{HG}}\left(t\right)=\frac{{\sigma }_{x}^{2}\left(t\right)}{{\lambda }_{\mathrm{c}}}\gg {\sigma }_{x}\left(t\right),\end{equation} \tag{ 52 }$$

while for the non-orthogonal packets it can still depend on this factor (say, for the elegant HG packets $\hat{\beta }\propto \sqrt{n},\enspace n\gg 1$). That is why the use of the Courant–Snyder parameters is more convenient for orthogonal packets. Moreover, the Gouy phase in equation (44) can also be written as follows:

$$\begin{equation}{\varphi }^{\mathrm{G}}=\int \frac{\mathrm{d}t}{\hat{\beta }\left(t\right)},\end{equation} \tag{ 53 }$$

while this is not the case for the elegant beam (38). Thus the Gouy phase of an arbitrary orthogonal packet is analogous to the Courant–Snyder phase advance employed in classical beam physics [29, 40]. Its overall factor M measures the packet's 'quantumness' according to the Schrödinger uncertainty relation.

3.2.1. Hermite–Gaussian packets

In a 3 + 1 dimensional space-time, a wave function of the HG packet simply factorizes,

$$\begin{equation}{\psi }_{n,j,k}\left(\boldsymbol{r},t\right)={\psi }_{n}\left(x,t\right){\psi }_{j}\left(y,t\right){\psi }_{k}\left(z,t\right),\end{equation} \tag{ 54 }$$

where each of the factors is either the elegant or the standard HG packet. So, the dynamics along each of the coordinates is independent. Let the packet propagate along the z axis on average,

$$\begin{equation*}\langle \boldsymbol{r}\rangle =\langle \boldsymbol{u}\rangle t=\left\{0,0,\langle u\rangle t\right\},\quad \langle x\rangle =\langle y\rangle =0.\end{equation*}$$

Then a transverse wave function of the standard HG packet is

$$\begin{align}\hfill {\psi }_{n,j}\left(x,y,t\right)& ={\psi }_{n}\left(x,t\right){\psi }_{j}\left(y,t\right)=\frac{N}{{\left(1+{t}^{2}/{t}_{\mathrm{d},x}^{2}\right)}^{1/4}{\left(1+{t}^{2}/{t}_{\mathrm{d},y}^{2}\right)}^{1/4}}\enspace {H}_{n}\left(\frac{x}{{\sigma }_{x}\left(t\right)}\right)\enspace {H}_{j}\left(\frac{y}{{\sigma }_{y}\left(t\right)}\right)\hfill \\ \hfill & \quad {\times}\mathrm{exp}\left\{-\frac{\mathrm{i}}{2}\left(2n+1\right)\mathrm{arctan}\frac{t}{{t}_{\mathrm{d},x}}-\frac{\mathrm{i}}{2}\left(2j+1\right)\mathrm{arctan}\frac{t}{{t}_{\mathrm{d},y}}\right.\hfill \\ \hfill & \quad \left.-\frac{{x}^{2}}{2{\sigma }_{x}^{2}\left(t\right)}\left(1-\mathrm{i}\frac{t}{{t}_{\mathrm{d},x}}\right)-\frac{{y}^{2}}{2{\sigma }_{y}^{2}\left(t\right)}\left(1-\mathrm{i}\frac{t}{{t}_{\mathrm{d},y}}\right)\right\}.\hfill \end{align} \tag{ 55 }$$

The coefficients at the Gouy phases are simply the packet's quality factors,

$$\begin{equation}{M}_{x}=2n+1,\qquad {M}_{y}=2j+1,\end{equation} \tag{ 56 }$$

whereas both the Gouy phases themselves can be written as follows:

$$\begin{equation}\mathrm{arctan}\frac{t}{{t}_{\mathrm{d},x}}=\int \frac{\mathrm{d}\left(t/{t}_{\mathrm{d},x}\right)}{1+{t}^{2}/{t}_{\mathrm{d},x}^{2}}=\int \frac{\mathrm{d}t}{{\hat{\beta }}_{x}\left(t\right)},\end{equation} \tag{ 57 }$$

where we have used the fact that

$$\begin{equation}{\hat{\beta }}_{x}\left(t\right)=\frac{\langle {x}^{2}\rangle }{{{\epsilon}}_{x}}={\hat{\beta }}_{x}\left(0\right)\left(1+{t}^{2}/{t}_{\mathrm{d},x}^{2}\right).\end{equation} \tag{ 58 }$$

3.2.2. Laguerre–Gaussian packets

As known from the works by Allen et al [62], one can make a packet with a coupling in the transverse plane, which is a hallmark of the OAM with respect to the propagation axis,

$$\begin{align}\hfill & \langle {\hat{L}}_{z}\rangle =\int {\mathrm{d}}^{3}x\enspace {\psi }^{{\ast}}\left(\boldsymbol{r},t\right)\enspace {\hat{L}}_{z}\enspace \psi \left(\boldsymbol{r},t\right)=\ell ,\quad \ell =0,{\pm}1,{\pm}2,\dots ,\hfill \\ \hfill & \int {\mathrm{d}}^{3}x\vert \psi \left(\boldsymbol{r},t\right){\vert }^{2}=1,\quad {\hat{L}}_{z}={\left[\boldsymbol{r}{\times}\hat{\boldsymbol{p}}\right]}_{z}=-\mathrm{i}\frac{\partial }{\partial \phi }.\hfill \end{align} \tag{ 59 }$$

For this, one needs to change in the momentum space

$$\begin{equation*}{\left({p}_{x}-\mathrm{i}\enspace \text{const}\enspace \frac{\partial }{\partial {p}_{x}}\right)}^{n}{\left({p}_{y}-\mathrm{i}\enspace \text{const}\enspace \frac{\partial }{\partial {p}_{y}}\right)}^{j}\to {\left({p}_{x}^{2}+{p}_{y}^{2}\right)}^{\vert \ell \vert /2}\enspace {L}_{n}^{\vert \ell \vert }\left(\frac{{p}_{x}^{2}+{p}_{y}^{2}}{{\sigma }_{p}^{2}}\right),\end{equation*}$$

where ${L}_{n}^{\vert \ell \vert }$ are associated Laguerre polynomials. The corresponding transverse function

$$\begin{align}\hfill {\psi }_{\ell ,n}\left(\boldsymbol{\rho },t\right)& =N\frac{{\rho }^{\vert \ell \vert }}{{\left({\sigma }_{\perp }\left(t\right)\right)}^{\vert \ell \vert +1}}\enspace {L}_{n}^{\vert \ell \vert }\left(\frac{{\rho }^{2}}{{\left({\sigma }_{\perp }\left(t\right)\right)}^{2}}\right)\mathrm{exp}\left\{\mathrm{i}\ell {\phi }_{r}-\mathrm{i}\left(2n+\vert \ell \vert +1\right)\mathrm{arctan}\left(t/{t}_{\mathrm{d}}\right)\right.\hfill \\ \hfill & \quad \left.-\frac{{\rho }^{2}}{2{\left({\sigma }_{\perp }\left(t\right)\right)}^{2}}\enspace \left(1-\mathrm{i}t/{t}_{\mathrm{d}}\right)\right\},\qquad \int {\mathrm{d}}^{2}\rho \enspace \vert {\psi }_{\ell ,n}\left(\boldsymbol{\rho },t\right){\vert }^{2}=1,\hfill \\ \hfill & \quad {\sigma }_{\perp }\left(t\right)={\sigma }_{\perp }\left(0\right)\sqrt{1+\frac{{t}^{2}}{{t}_{\mathrm{d}}^{2}}},\qquad {\sigma }_{\perp }\left(0\right)=1/{\sigma }_{p},\hfill \end{align} \tag{ 60 }$$

is an exact non-stationary solution to the two-dimensional Schrödinger equation, called a standard or a generalized [16, 18] LG packet ³ . The set of these functions is complete and orthogonal.

Analogously to the HG packets, there is also an elegant LG packet,

$$\begin{align}\hfill {\psi }_{\ell ,n}\left(\boldsymbol{\rho },t\right)& =N\frac{{\rho }^{\vert \ell \vert }}{{\left({\sigma }_{\perp }\left(t\right)\right)}^{n+\vert \ell \vert +1}}\enspace {L}_{n}^{\vert \ell \vert }\left(\frac{{\rho }^{2}}{2{\left({\sigma }_{\perp }\left(t\right)\right)}^{2}}\left(1-\mathrm{i}t/{t}_{\mathrm{d}}\right)\right)\hfill \\ \hfill & \quad {\times}\mathrm{exp}\left\{\mathrm{i}\ell {\phi }_{r}-\mathrm{i}\left(n+\vert \ell \vert +1\right)\mathrm{arctan}\left(t/{t}_{\mathrm{d}}\right)-\frac{{\rho }^{2}}{2{\left({\sigma }_{\perp }\left(t\right)\right)}^{2}}\left(1-\mathrm{i}t/{t}_{\mathrm{d}}\right)\right\},\hfill \end{align} \tag{ 61 }$$

which also satisfies two-dimensional Schrödinger equation, has an OAM $\langle {\hat{L}}_{z}\rangle =\ell$ with its vanishing uncertainty, but the set is not orthogonal. One can also combine LG and HG packets for instance as follows:

$$\begin{equation}{\psi }_{n,\ell ,k}\left(\boldsymbol{r},t\right)={\psi }_{\ell ,n}\left(\boldsymbol{\rho },t\right){\psi }_{k}\left(z,t\right),\end{equation} \tag{ 62 }$$

where ψ_ℓ,n ( ρ , t) is either a standard or an elegant LG packet, while the longitudinal function ψ_k (z, t) represents one of the HG packets.

3.2.3. Transverse phase space and the Courant–Snyder formalism

Regardless of the specific model, quantum dynamics of a packet in a three-dimensional space-time is governed by the same Hamiltonian ${\hat{H}}_{\text{fr}}={\hat{\boldsymbol{p}}}^{2}/2m$ and the mean-square radius of the packet ⟨ρ²⟩ is not conserved,

$$\begin{equation}\frac{{\partial }^{2}\langle {\rho }^{2}\rangle \left(t\right)}{\partial {t}^{2}}=-\left\langle \left[\left[{\rho }^{2},{\hat{H}}_{\text{fr}}\right],{\hat{H}}_{\text{fr}}\right]\right\rangle =2\langle {\boldsymbol{u}}_{\perp }^{2}\rangle .\end{equation} \tag{ 63 }$$

The general solution to this Heisenberg equation,

$$\begin{equation}\langle {\rho }^{2}\rangle \left(t\right)=\langle {\rho }^{2}\rangle \left(0\right)+\frac{\partial \langle {\rho }^{2}\rangle \left(0\right)}{\partial t}\enspace t+\langle {\boldsymbol{u}}_{\perp }^{2}\rangle \enspace {t}^{2},\end{equation} \tag{ 64 }$$

is analogous to equation (20). Note that ⟨ρ²⟩(t) is connected to an intrinsic electric quadrupole moment, which is generally not vanishing for a non-Gaussian charged packet [17, 21]. As its third derivative vanishes, the quadrupole moment does not emit electromagnetic waves [54]. If the focus is at t = 0, the correlation term in (64) vanishes and we recover the familiar result,

$$\begin{equation}\langle {\rho }^{2}\rangle \left(t\right)=\langle {\rho }^{2}\rangle \left(0\right)+\langle {\boldsymbol{u}}_{\perp }^{2}\rangle \enspace {t}^{2}=\langle {\rho }^{2}\rangle \left(0\right)\left(1+\frac{{t}^{2}}{{t}_{\mathrm{d}}^{2}}\right)=\langle {\rho }^{2}\rangle \left(0\right)\left(1+\frac{\langle {z\rangle }^{2}}{{z}_{\mathrm{R}}^{2}}\right),\end{equation} \tag{ 65 }$$

where

$$\begin{equation}{t}_{\mathrm{d}}=\sqrt{\frac{\langle {\rho }^{2}\rangle \left(0\right)}{\langle {\boldsymbol{u}}_{\perp }^{2}\rangle }},\qquad {z}_{\mathrm{R}}=\langle u\rangle {t}_{\mathrm{d}}\end{equation} \tag{ 66 }$$

are a diffraction time and a Rayleigh length, respectively.

Similar to before, one can define a matrix of central moments,

$$\begin{equation}{Q}_{ij}=\langle {\boldsymbol{q}}_{i}\cdot {\boldsymbol{q}}_{j}\rangle -\langle {\boldsymbol{q}}_{i}\rangle \cdot \langle {\boldsymbol{q}}_{j}\rangle ,\qquad {\boldsymbol{q}}_{i}=\left\{\boldsymbol{\rho },{\boldsymbol{u}}_{\perp }\right\},\quad i=1,2.\end{equation} \tag{ 67 }$$

The square root of its determinant,

$$\begin{equation}{{\epsilon}}_{\rho }=\sqrt{\mathrm{d}\mathrm{e}\mathrm{t}\enspace {Q}_{ij}}=\sqrt{\langle {\rho }^{2}\rangle \left(t\right)\langle {\boldsymbol{u}}_{\perp }^{2}\rangle -{\left(\langle \boldsymbol{\rho }\cdot {\boldsymbol{u}}_{\perp }\rangle \left(t\right)\right)}^{2}}=\sqrt{\langle {\rho }^{2}\rangle \left(0\right)\langle {\boldsymbol{u}}_{\perp }^{2}\rangle -{\left(\langle \boldsymbol{\rho }\cdot {\boldsymbol{u}}_{\perp }\rangle \left(0\right)\right)}^{2}},\end{equation} \tag{ 68 }$$

is called a normalized radial emittance of a packet propagating along the z axis. If the focus is at t = 0, the result

$$\begin{equation}{{\epsilon}}_{\rho }=\sqrt{\langle {\rho }^{2}\rangle \left(0\right)\enspace \langle {\boldsymbol{u}}_{\perp }^{2}\rangle }{\geqslant}{\lambda }_{\mathrm{c}}\end{equation} \tag{ 69 }$$

obeys the Heisenberg uncertainty relation. One can define a transverse quality factor as follows:

$$\begin{equation}M=\frac{{{\epsilon}}_{\rho }}{{\lambda }_{\mathrm{c}}}{\geqslant}1.\end{equation} \tag{ 70 }$$

A packet with M = 1 minimizes the Schrödinger uncertainty relation and it can be called a fundamental (or Gaussian) mode.

So the diffraction time and the Rayleigh length become

$$\begin{equation}{t}_{\mathrm{d}}=\frac{\langle {\rho }^{2}\rangle \left(0\right)}{{{\epsilon}}_{\rho }}=\frac{\langle {\rho }^{2}\rangle \left(0\right)}{M{\lambda }_{\mathrm{c}}},\qquad {z}_{\mathrm{R}}=\langle u\rangle {t}_{\mathrm{d}}.\end{equation} \tag{ 71 }$$

It might seem that these quantities decrease with the growth of the quality factor M. If this were the case, the diffraction time could drop below the Compton time t_c (27) for M ≫ 1. However, for a single-particle state with a stable vacuum of quantum electrodynamics (QED) the coordinate uncertainty $\sqrt{\langle {\rho }^{2}\rangle \left(0\right)}$ must be larger than λ_c, the momentum uncertainty $\sqrt{\langle {\boldsymbol{p}}_{\perp }^{2}\rangle \left(0\right)}$ must be smaller than m for the fundamental mode, and all the time intervals must be larger than the lifetime of a virtual electron-positron pair t_c. This implies that

$$\begin{equation}{t}_{\mathrm{d}}{\geqslant}{t}_{\mathrm{c}},\quad \sqrt{\langle {\rho }^{2}\rangle \left(0\right)}{\geqslant}{\lambda }_{\mathrm{c}}\enspace {M}^{\left(1+\zeta \right)/2},\qquad \sqrt{\langle {\boldsymbol{p}}_{\perp }^{2}\rangle \left(0\right)}{\leqslant}{\lambda }_{\mathrm{c}}^{-1}\enspace {M}^{\left(1-\zeta \right)/2},\end{equation} \tag{ 72 }$$

where ζ = 0 corresponds to the generalized coherent states (say, the LG packets), while ζ > 0 corresponds to a squeezed state of a particle. In the former case, the diffraction time does not depend on the packet's quality at all, ${t}_{\mathrm{d}}=\mathcal{O}\left({M}^{0}\right)$, while for the squeezed state such a dependence generally exists, t_d ∝ M^ζ .

It is natural to introduce the following Courant–Snyder variables:

$$\begin{align}\hfill & \hat{\beta }\left(t\right)=\frac{\langle {\rho }^{2}\rangle \left(t\right)}{{{\epsilon}}_{\rho }},\qquad \hat{\alpha }\left(t\right)=-\frac{1}{2}\frac{\partial \beta \left(t\right)}{\partial t}=-\frac{\langle \boldsymbol{\rho }{\boldsymbol{u}}_{\perp }\rangle \left(t\right)}{{{\epsilon}}_{\rho }},\qquad \hat{\gamma }=\frac{\langle {\boldsymbol{u}}_{\perp }^{2}\rangle }{{{\epsilon}}_{\rho }},\hfill \\ \hfill & \hat{\beta }\hat{\gamma }-{\hat{\alpha }}^{2}=1,\qquad \hat{\beta }\left(0\right)={t}_{\mathrm{d}}={z}_{\mathrm{R}}/\langle u\rangle ,\hfill \end{align} \tag{ 73 }$$

so the conservation of the packet's emittance can be written as an ellipse equation

$$\begin{align}\hfill & \hat{\gamma }\enspace {P}^{2}+2\hat{\alpha }\enspace P\dot {P}+\hat{\beta }\enspace {\left(\dot {P}\right)}^{2}={{\epsilon}}_{\rho }{\geqslant}{\lambda }_{\mathrm{c}},\hfill \\ \hfill & P{:=}\sqrt{\langle {\rho }^{2}\rangle },\hfill \\ \hfill & \text{or}\;\frac{\hat{\gamma }}{2}\enspace \langle {\rho }^{2}\rangle +\hat{\alpha }\enspace \langle \boldsymbol{\rho }{\boldsymbol{u}}_{\perp }\rangle +\frac{\hat{\beta }}{2}\enspace \langle {\boldsymbol{u}}_{\perp }^{2}\rangle ={{\epsilon}}_{\rho }{\geqslant}{\lambda }_{\mathrm{c}}.\hfill \end{align} \tag{ 74 }$$

Clearly, for the generalized coherent states with ζ = 0 these functions do not depend on M. They are most convenient for an orthogonal set, which we imply here, while for a non-orthogonal set (say, the elegant LG packets) the Courant–Snyder parameters can depend on the beam quality factor, which is not related to squeezing. Analogously to equation (57), one can also define a Gouy phase of an arbitrary packet as follows

$$\begin{equation}{\varphi }^{\mathrm{G}}=\mathrm{arctan}\frac{t}{{t}_{\mathrm{d}}}=\int \frac{\mathrm{d}t}{\hat{\beta }\left(t\right)}.\end{equation} \tag{ 75 }$$

We exemplify these definitions with the standard LG packet (60) with ζ = 0 for which we find the following averages:

$$\begin{align}\hfill & \langle {\rho }^{2}\rangle ={\sigma }_{\perp }^{2}\left(t\right)\enspace M,\qquad \langle {\boldsymbol{u}}_{\perp }^{2}\rangle =\frac{{\sigma }_{p}^{2}}{{m}^{2}}\enspace M=\frac{{\lambda }_{\mathrm{c}}^{2}}{{\sigma }_{\perp }^{2}\left(0\right)}\enspace M,\hfill \\ \hfill & {{\epsilon}}_{\rho }={\lambda }_{\mathrm{c}}\enspace \left(n+\vert \ell \vert +1\right){\geqslant}{\lambda }_{\mathrm{c}},\qquad M=n+\vert \ell \vert +1,\hfill \\ \hfill & \hat{\beta }=\frac{{\sigma }_{\perp }^{2}\left(t\right)}{{\lambda }_{\mathrm{c}}}=\frac{\langle {\rho }^{2}\rangle {\left(t\right)}_{M=1}}{{\lambda }_{\mathrm{c}}}.\hfill \end{align} \tag{ 76 }$$

Now we return to the spreading law of an arbitrary quantum packet, equation (65), and note that when the spreading is essential—i.e. the detector is in the far field—we have

$$\begin{equation}\sqrt{\langle {\rho }^{2}\rangle \left(t\right)}=\sqrt{\langle {\boldsymbol{u}}_{\perp }^{2}\rangle }\enspace t=\frac{\langle z\rangle }{\sqrt{\langle {\rho }^{2}\rangle \left(0\right)}}\enspace M\enspace \frac{{\lambda }_{\text{dB}}}{2\pi },\quad {\lambda }_{\text{dB}}=\frac{2\pi }{\langle p\rangle }.\end{equation} \tag{ 77 }$$

As a result, there is a simple relation between the registered width of the packet at the detector $\sqrt{\langle {\rho }^{2}\rangle \left(\langle z\rangle \right)}$ and the initial (emitted) width $\sqrt{\langle {\rho }^{2}\rangle \left(0\right)}$ at the source:

$$\begin{equation}\sqrt{\langle {\rho }^{2}\rangle \left(0\right)}=\frac{\langle z\rangle {\lambda }_{\text{dB}}}{2\pi \sqrt{\langle {\rho }^{2}\rangle \left(\langle z\rangle \right)}}\enspace M.\end{equation} \tag{ 78 }$$

For the fundamental mode with M = 1, this is the well-known van Cittert–Zernike theorem for a uniform source [48–52] where the transverse coherence length ${\xi }_{\perp }=\sqrt{2}\enspace \sqrt{\langle {\rho }^{2}\rangle \left(\langle z\rangle \right)}$ and the effective source radius ${r}_{\text{eff}}=\sqrt{2}\enspace \sqrt{\langle {\rho }^{2}\rangle \left(0\right)}$. Equation (78) is also applicable for a 'more quantum' packet with M > 1, that is, not Gaussian one.

Moreover, the general spreading law (65) is applicable not only in the far field, but also nearby the source. In particular, one can keep the first correction to the van Cittert–Zernike theorem when the detector is in a Fresnel zone,

$$\begin{equation}\sqrt{\langle {\rho }^{2}\rangle \left(\langle z\rangle \right)}=\frac{\langle z\rangle {\lambda }_{\text{dB}}}{2\pi \sqrt{\langle {\rho }^{2}\rangle \left(0\right)}}\enspace M\enspace \left(1+\frac{1}{2}{\left(\frac{2\pi \langle {\rho }^{2}\rangle \left(0\right)}{\langle z\rangle {\lambda }_{\text{dB}}}\frac{1}{M}\right)}^{2}\right),\end{equation} \tag{ 79 }$$

where the second term in the rhs vanishes in the far field when (see equation (71))

$$\begin{equation}\langle z\rangle \gg {z}_{\mathrm{R}}=\frac{2\pi }{M}\frac{\langle {\rho }^{2}\rangle \left(0\right)}{{\lambda }_{\text{dB}}}.\end{equation} \tag{ 80 }$$

For electrons with $\sqrt{\langle {\rho }^{2}\rangle }\sim 1\enspace \text{nm},\enspace \beta \sim 1{0}^{-3}\text{--}1{0}^{-1}$, the Rayleigh length z_R lies between 1 nm and 100 nm, see equation (159) below.

Here a note on coherence is in order. Although equation (65) was derived quantum-mechanically for a single-particle packet, the very same spreading law takes place for a classical beam of many point particles [29, 40] where ⟨...⟩ is an average over a particle distribution in the beam ⁴ . In an accelerator, such a beam usually represents an incoherent mixture of particles and the packets do not interfere, in accord with the fact that the van Cittert–Zernike theorem implies an incoherent source. Remarkably, the quantum derivation of equation (65) explicitly shows that the packets inside the beam or the plane-wave components of a single packet can interfere constructively. Moreover, the packet of a massive particle is fully coherent with itself when $\sqrt{\langle {\rho }^{2}\rangle }\gg {\lambda }_{\mathrm{c}}$. This coherence can only be violated for such tightly focused quantum superpositions with $\sqrt{\langle {\rho }^{2}\rangle }\gtrsim {\lambda }_{\mathrm{c}}$ as, for instance, the Schrödinger cat states with a not-everywhere positive Wigner function [63].

Thus the generalized van Cittert–Zernike theorem (78) or (79) holds valid both for coherent superpositions of quantum states with a possible fermionic repulsion (e.g. for electrons emitted from cold cathodes and for beams of electron microscopes) and for such incoherent mixtures as the accelerator beams. Finally, note that semi-classically one can derive the far-field van Cittert–Zernike theorem (78) and its M scaling from a Bragg condition of constructive interference of two waves in a beam, that is, implying full coherence.

We start with the constant and homogeneous magnetic field (2) and define an operator of the canonical momentum

$$\begin{equation}{\hat{\boldsymbol{p}}}^{\text{can}}={\hat{\boldsymbol{p}}}^{\text{kin}}+e\boldsymbol{A}=-\mathrm{i}\nabla .\end{equation} \tag{ 81 }$$

There exist two angular momenta,

$$\begin{equation}{\hat{\boldsymbol{L}}}^{\text{can}}=\boldsymbol{r}{\times}{\hat{\boldsymbol{p}}}^{\text{can}}\quad \text{and}\quad {\hat{\boldsymbol{L}}}^{\text{kin}}=\boldsymbol{r}{\times}{\hat{\boldsymbol{p}}}^{\text{kin}}.\end{equation} \tag{ 82 }$$

The z-component of the former, ${\hat{L}}_{z}^{\text{can}}=-\mathrm{i}\partial /\partial \phi$, commutes with the Hamiltonian

$$\begin{equation}\hat{H}=\frac{{\left({\hat{\boldsymbol{p}}}^{\text{kin}}\right)}^{2}}{2m}=\frac{{\left({\hat{\boldsymbol{p}}}^{\text{can}}\right)}^{2}}{2m}-{\omega }_{\mathrm{L}}{\hat{L}}_{z}^{\text{can}}+\frac{m}{2}\enspace {\omega }_{\mathrm{L}}^{2}\enspace {\rho }^{2}\end{equation} \tag{ 83 }$$

and therefore $\langle {\hat{L}}_{z}^{\text{can}}\rangle$ is conserved in the field, exactly like for the free twisted particle. Let the packet propagate along the z-axis on average,

$$\begin{align}\hfill & \langle \boldsymbol{r}\rangle \left(t\right)=\left\{0,0,\langle u\rangle t\right\},\qquad \langle \boldsymbol{u}\rangle =\frac{\langle {\hat{\boldsymbol{p}}}^{\text{kin}}\rangle }{m},\hfill \\ \hfill & \langle {\hat{\boldsymbol{p}}}_{\perp }^{\text{kin}}\rangle =\langle {\hat{\boldsymbol{p}}}_{\perp }^{\text{can}}\rangle =0,\qquad \langle {\hat{p}}_{z}^{\text{kin}}\rangle =\langle {\hat{p}}_{z}^{\text{can}}\rangle =\langle u\rangle \enspace m,\hfill \end{align} \tag{ 84 }$$

and be twisted,

$$\begin{equation}\langle {\hat{L}}_{z}^{\text{can}}\rangle =\ell ,\quad \ell =0,{\pm}1,{\pm}2,\dots \end{equation} \tag{ 85 }$$

While we start with the field which is homogeneous in all space, we will later consider the particle entering a solenoid of a finite length at a certain moment of time. As the initial transverse momentum is vanishing, the packet's centroid does not rotate and the mean path stays rectilinear. The canonical AM of the incoming free packet is not vanishing and its conservation in such a field means that if a twisted particle enters a solenoid, moves along the field lines, and then leaves it, its final OAM stays the same. In other words, vortex particles can be trapped and focused by the solenoids and magnetic lenses without deteriorating their quantum states. However, any particle that is off the solenoid axis will get a transverse kick when entering a magnetic lens, so the mean kinetic momentum will no longer be parallel to the quantization axis of the canonical AM (z). That is why for a vortex beam of many particles a thick lens will inevitably lead to broadening of an OAM spectrum (an OAM bandwidth), even though the mean OAM is conserved.

Unlike $\langle {\hat{L}}_{z}^{\text{can}}\rangle$, its kinetic counterpart,

$$\begin{equation}\langle {\hat{L}}_{z}^{\text{kin}}\rangle =\ell -m{\omega }_{\mathrm{L}}\langle {\rho }^{2}\rangle =\ell -2\enspace \mathrm{s}\mathrm{g}\mathrm{n}\left(e\right)\frac{\langle {\rho }^{2}\rangle }{{\rho }_{\mathrm{H}}^{2}},\end{equation} \tag{ 86 }$$

is not generally conserved in time, as ρ² does not commute with the Hamiltonian. Here

$$\begin{equation}{\rho }_{\mathrm{H}}=\sqrt{\frac{4}{\vert e\vert H}}=2{\lambda }_{\mathrm{c}}\sqrt{\frac{{H}_{\mathrm{c}}}{H}}\end{equation} \tag{ 87 }$$

is denoted (recall that H > 0 and H_c is from equation (14)). For typical field strengths of H ∼ 0.1–10 T, we have

$$\begin{equation}{\rho }_{\mathrm{H}}\sim 10\text{--}100\enspace \text{nm}.\end{equation} \tag{ 88 }$$

There exists an exact stationary solution to the Schrödinger equation in the given field, which satisfies the above requirements—the so-called Landau states [6, 64, 65],

$$\begin{equation}{\Psi}\left(\boldsymbol{r},t\right)=\text{const}{\left(\frac{\rho }{{\rho }_{\mathrm{H}}}\right)}^{\vert \ell \vert }{L}_{{n}_{\mathrm{H}}}^{\vert \ell \vert }\left(\frac{2{\rho }^{2}}{{\rho }_{\mathrm{H}}^{2}}\right)\mathrm{exp}\left\{-\frac{{\rho }^{2}}{{\rho }_{\mathrm{H}}^{2}}+\mathrm{i}\ell \phi +\mathrm{i}{p}_{z}z-\mathrm{i}\varepsilon t\right\},\end{equation} \tag{ 89 }$$

where ${L}_{{n}_{\mathrm{H}}}^{\vert \ell \vert }$ are the generalized Laguerre polynomials with n_H = 0, 1, 2, ..., and

$$\begin{align}\hfill & \enspace \varepsilon ={\varepsilon }_{\perp }+\frac{{p}_{z}^{2}}{2m},\hfill \\ \hfill & {\varepsilon }_{\perp }=\frac{\left\langle {\left({\hat{\boldsymbol{p}}}_{\perp }^{\text{kin}}\right)}^{2}\right\rangle }{2m}=\vert {\omega }_{\mathrm{L}}\vert \left(2{n}_{\mathrm{H}}+\vert \ell \vert -\mathrm{s}\mathrm{g}\mathrm{n}\left(e\right)\ell +1\right){\geqslant}\vert {\omega }_{\mathrm{L}}\vert .\hfill \end{align} \tag{ 90 }$$

The mean-square radius ${\langle {\rho }^{2}\rangle }_{\text{LG}}$ calculated with these states is

$$\begin{equation}{\langle {\rho }^{2}\rangle }_{\text{LG}}=\frac{{\rho }_{\mathrm{H}}^{2}}{2}\enspace \left(2{n}_{\mathrm{H}}+\vert \ell \vert +1\right){\geqslant}\frac{{\rho }_{\mathrm{H}}^{2}}{2}.\end{equation} \tag{ 91 }$$

As it is independent of time, the kinetic AM

$$\begin{equation}\langle {\hat{L}}_{z}^{\text{kin}}\rangle =\ell -\mathrm{s}\mathrm{g}\mathrm{n}\left(e\right)\enspace \left(2{n}_{\mathrm{H}}+\vert \ell \vert +1\right)\end{equation} \tag{ 92 }$$

is also conserved and its minimum value is ${\langle {\hat{L}}_{z}^{\text{kin}}\rangle }_{\mathrm{min}}=-\mathrm{s}\mathrm{g}\mathrm{n}\left(e\right)$. However, this conclusion only holds in the stationary (thick-lens) regime—that is, if the time Δt during which the particle stays in the solenoid is at least one rotation period,

$$\begin{equation}{\Delta}t={T}_{\mathrm{c}}=\frac{2\pi }{{\omega }_{\mathrm{c}}}\quad \text{or}\quad {\Delta}t\gg {T}_{\mathrm{c}}.\end{equation} \tag{ 93 }$$

So the stationary Landau states are not applicable for describing quantum dynamics of the packet in a thin-lens regime when Δt ≪ T_c.

To describe this dynamics for arbitrary Δt, we need to solve the Heisenberg equation,

$$\begin{align}\hfill & \frac{{\partial }^{2}\langle {\rho }^{2}\rangle \left(t\right)}{\partial {t}^{2}}=-\left\langle \left[\left[{\rho }^{2},\hat{H}\right],\hat{H}\right]\right\rangle \hfill \\ \hfill & \qquad =-{\omega }_{\mathrm{c}}^{2}\enspace \left(\langle {\rho }^{2}\rangle \left(t\right)-{\langle {\rho }^{2}\rangle }_{\text{st}}\right),\hfill \end{align} \tag{ 94 }$$

where

$$\begin{equation}{\langle {\rho }^{2}\rangle }_{\text{st}}=\frac{1}{m{\omega }_{\mathrm{L}}}\left(\frac{{\varepsilon }_{\perp }}{{\omega }_{\mathrm{L}}}+\ell \right).\end{equation} \tag{ 95 }$$

The general solution to this equation contains two integration constants α₁ and α₂,

$$\begin{equation}\langle {\rho }^{2}\rangle \left(t\right)={\langle {\rho }^{2}\rangle }_{\text{st}}+{\alpha }_{1}\enspace \mathrm{cos}\left({\omega }_{\mathrm{c}}t+{\alpha }_{2}\right).\end{equation} \tag{ 96 }$$

Thus although the packet's centroid moves rectilinearly, the mean-square radius ⟨ρ²⟩ oscillates with the period of T_c. This periodic behavior is sometimes called quantum broadening of the classical trajectories [65, 66]. This rms radius also defines an intrinsic electric quadrupole moment of the packet in the transverse plane [17, 21]. As its third derivative does not vanish, this packet emits electromagnetic waves [54].

Averaging equation (96) over at least one period, we return to the stationary (thick lens) regime with

$$\begin{equation}\bar{\left\langle {\rho }^{2}\right\rangle \left(t\right)}={\langle {\rho }^{2}\rangle }_{\text{st}}={\langle {\rho }^{2}\rangle }_{\text{LG}},\end{equation} \tag{ 97 }$$

for which the Landau states are applicable.

The equations of motion for ρ² can also be obtained by the Ehrenfest theorem,

$$\begin{equation}\frac{\mathrm{d}\langle \boldsymbol{\rho }\rangle }{\mathrm{d}t}=\langle {\boldsymbol{u}}_{\perp }\rangle ,\qquad \frac{{\mathrm{d}}^{2}\langle \boldsymbol{\rho }\rangle }{\mathrm{d}{t}^{2}}=\frac{e}{m}\enspace \left\langle \boldsymbol{u}{\times}\boldsymbol{H}\right\rangle ,\end{equation} \tag{ 98 }$$

and so

$$\begin{equation}\frac{{\partial }^{2}\langle {\rho }^{2}\rangle \left(t\right)}{\partial {t}^{2}}=2\langle {\boldsymbol{u}}_{\perp }^{2}\rangle +2\frac{e}{m}\enspace \langle \boldsymbol{\rho }\cdot \left[\boldsymbol{u}{\times}\boldsymbol{H}\right]\rangle =2\langle {\boldsymbol{u}}_{\perp }^{2}\rangle +2\frac{{\omega }_{\mathrm{c}}}{m}\enspace \langle {\hat{L}}_{z}^{\text{kin}}\rangle \left(t\right).\end{equation} \tag{ 99 }$$

One can rewrite equation (94) as follows:

$$\begin{align}\hfill & \frac{{\partial }^{2}\langle {\rho }^{2}\rangle \left(t\right)}{\partial {t}^{2}}+2K\left(t\right)\langle {\rho }^{2}\rangle \left(t\right)=2\langle {\boldsymbol{u}}_{\perp }^{2}\rangle ,\hfill \\ \hfill & K\left(t\right)=-\frac{1}{\langle {\rho }^{2}\rangle \left(t\right)}\enspace \left\langle \boldsymbol{\rho }\frac{{\mathrm{d}}^{2}\boldsymbol{\rho }}{\mathrm{d}{t}^{2}}\right\rangle =-\frac{{\omega }_{\mathrm{c}}}{m}\frac{\langle {\hat{L}}_{z}^{\text{kin}}\rangle \left(t\right)}{\langle {\rho }^{2}\rangle \left(t\right)}.\hfill \end{align} \tag{ 100 }$$

Taking $\langle {\hat{L}}_{z}^{\text{kin}}\rangle \left(t\right)$ from equation (86), we recover the rhs of equation (94). When $\langle {\hat{L}}_{z}^{\text{kin}}\rangle \left(t\right)\to 0$ this equation coincides with equation (63) for the free packet. However, equation (96) does not continuously transform to equation (64) when H → 0 before the boundary conditions are applied.

Importantly, equation (94) and the conservation of $\langle {\hat{L}}_{z}^{\text{can}}\rangle$ also hold for a solenoid of a finite length L, which is described by the potential

$$\begin{equation}\boldsymbol{A}=\frac{H\left(z\right)}{2}\left\{-y,x,0\right\},\end{equation} \tag{ 101 }$$

where H(z) = H when 0 ⩽ z ⩽ L and zero otherwise (a hard-edge approximation). Such a field has a linear radial component

$$\begin{equation}{H}_{\rho }\left(\rho ,z\right)=-\frac{\rho }{2}\quad {H}^{\prime }\left(z\right),\enspace {H}^{\prime }\left(z\right)\equiv \frac{\partial H\left(z\right)}{\partial z},\end{equation} \tag{ 102 }$$

which vanishes at the solenoid axis and leads to focusing and rotation of a particle that is off this axis.

Moreover, although the field of a real solenoid is not radially homogeneous, its dependence on ρ can be neglected, H(ρ, z) ≈ H(0, z) ≡ H(z), when the field does not change over the packet's transverse coherence length, which is

$$\begin{equation}\sqrt{{\langle {\rho }^{2}\rangle }_{\text{LG}}}\sim {\rho }_{\mathrm{H}}\sqrt{2{n}_{\mathrm{H}}+\vert \ell \vert +1}\sim \left(10\text{--}100\enspace \text{nm}\right)\enspace \sqrt{2{n}_{\mathrm{H}}+\vert \ell \vert +1}.\end{equation} \tag{ 103 }$$

Clearly, the radius $\sqrt{{\langle {\rho }^{2}\rangle }_{\text{LG}}}$ is usually much smaller than the typical widths of the multi-particle beams, which are of the order of 10 μm–10 mm. So while the approximation of a radially homogeneous field may fail for the beam as a whole, it can still hold for each individual packet in the beam. Indeed, for the inhomogeneous field we have instead of equation (86):

$$\begin{align}\hfill & \langle {\hat{L}}_{z}^{\text{kin}}\rangle =\ell -e\langle \rho {A}_{\phi }\rangle =\ell -\frac{e}{2}\langle {\rho }^{2}H\left(\rho ,z\right)\rangle ,\hfill \\ \hfill & \langle {\rho }^{2}H\left(\rho ,z\right)\rangle =\int {\mathrm{d}}^{3}r\enspace {{\Psi}}^{{\ast}}\left(\boldsymbol{r},t\right)\enspace {\rho }^{2}H\left(\rho ,z\right)\enspace {\Psi}\left(\boldsymbol{r},t\right),\hfill \end{align} \tag{ 104 }$$

where ${\Psi}\left(\boldsymbol{r},t\right)\propto \mathrm{exp}\left\{-{\rho }^{2}/{\rho }_{\mathrm{H}}^{2}\right\}$ are the Landau states (89). Therefore, one can put H(ρ, z) ≈ H(z) when it is nearly constant over the distance of ρ_H. The solenoid fields can generally be presented in a form of series [29],

$$\begin{align}\hfill & {H}_{z}\left(\rho ,z\right)={H}_{z}\left(0,z\right)-\frac{{\rho }^{2}}{4}\enspace {H}_{z}^{\prime \prime }\left(0,z\right)+\mathcal{O}\left({\rho }^{4}\right),\hfill \\ \hfill & {H}_{\rho }\left(\rho ,z\right)=-\frac{\rho }{2}\enspace {H}^{\prime }\left(z\right)+\mathcal{O}\left({\rho }^{3}\right),\hfill \end{align} \tag{ 105 }$$

and so the non-linear terms are generally less important for single quantum wave packets than for wide beams of accelerators.

Let the standard LG packet (60) enter the solenoid at t = 0, ⟨z⟩ = 0. The central moments in the transverse plane,

$$\begin{equation}{Q}_{ij}\left(t\right)=\langle {\boldsymbol{q}}_{i}\cdot {\boldsymbol{q}}_{j}\rangle -\langle {\boldsymbol{q}}_{i}\rangle \cdot \langle {\boldsymbol{q}}_{j}\rangle ,\qquad {\boldsymbol{q}}_{i}=\left\{\boldsymbol{\rho },{\boldsymbol{u}}_{\perp }\right\},\quad i=1,2,\end{equation} \tag{ 106 }$$

are continuous at the boundary together with the transverse emittance ${{\epsilon}}_{\rho }=\sqrt{\mathrm{d}\mathrm{e}\mathrm{t}\enspace {Q}_{ij}}$ and with the packet's transverse quality factor M = _ρ /λ_c,

$$\begin{equation}{Q}_{ij}^{\text{free}}\left(0\right)={Q}_{ij}^{\text{field}}\left(0\right),\qquad {{\epsilon}}_{\rho }^{\text{free}}={{\epsilon}}_{\rho }^{\text{field}}\left(0\right),\qquad {M}^{\text{free}}={M}^{\text{field}}\left(0\right).\end{equation} \tag{ 107 }$$

This implies the following boundary conditions:

$$\begin{equation}{\langle {\rho }^{2}\rangle }^{\text{free}}\left(0\right)={\langle {\rho }^{2}\rangle }^{\text{field}}\left(0\right),\qquad {\langle \boldsymbol{\rho }{\boldsymbol{u}}_{\perp }\rangle }^{\text{free}}\left(0\right)={\langle \boldsymbol{\rho }{\boldsymbol{u}}_{\perp }\rangle }^{\text{field}}\left(0\right),\qquad {\langle {\boldsymbol{u}}_{\perp }^{2}\rangle }^{\text{free}}\left(0\right)={\langle {\boldsymbol{u}}_{\perp }^{2}\rangle }^{\text{field}}\left(0\right).\end{equation} \tag{ 108 }$$

Finding α₁, α₂ from these conditions, the rms-radius (96) takes the following form (cf [9, 10]):

$$\begin{equation}\langle {\rho }^{2}\rangle \left(t\right)={\langle {\rho }^{2}\rangle }_{\text{st}}+\left({\langle {\rho }^{2}\rangle }^{\text{free}}-{\langle {\rho }^{2}\rangle }_{\text{st}}\right)\mathrm{cos}\left({\omega }_{\mathrm{c}}t\right),\end{equation} \tag{ 109 }$$

where ${\langle {\rho }^{2}\rangle }^{\text{free}}\equiv {\langle {\rho }^{2}\rangle }^{\text{free}}\left(0\right)$. Now when we switch off the field, H → 0, this solution continuously transforms to ${\langle {\rho }^{2}\rangle }^{\text{free}}$. As ⟨ρ²⟩(t) must be positive, we have the following inequalities:

$$\begin{align}\hfill & {\langle {\rho }^{2}\rangle }_{\text{st}}={\langle {\rho }^{2}\rangle }_{\text{LG}}{ >}\frac{{\langle {\rho }^{2}\rangle }^{\text{free}}}{2},\hfill \\ \hfill & {{\epsilon}}_{\rho }^{\text{LG}}=\sqrt{\mathrm{d}\mathrm{e}\mathrm{t}\enspace \bar{{Q}_{ij}\left(t\right)}}=\sqrt{{\langle {\rho }^{2}\rangle }_{\text{LG}}\enspace \langle {\boldsymbol{u}}_{\perp }^{2}\rangle }{ >}\frac{{{\epsilon}}_{\rho }^{\text{free}}}{\sqrt{2}}{\geqslant}\frac{{\lambda }_{\mathrm{c}}}{\sqrt{2}}.\hfill \end{align} \tag{ 110 }$$

The last condition in equation (108) connects the principal quantum number n_H in the magnetic field and the radial quantum number n of the free LG packet,

$$\begin{equation}\sqrt{\frac{2{n}_{\mathrm{H}}+\vert \ell \vert -\mathrm{s}\mathrm{g}\mathrm{n}\left(e\right)\ell +1}{n+\vert \ell \vert +1}}=\frac{1}{2}\enspace \frac{{\rho }_{\mathrm{H}}}{{\sigma }_{\perp }\left(0\right)},\end{equation} \tag{ 111 }$$

and therefore when both n and n_H are not vanishing

$$\begin{equation}\frac{{n}_{\mathrm{H}}}{n}\sim {\left(\frac{{\rho }_{\mathrm{H}}}{{\sigma }_{\perp }\left(0\right)}\right)}^{2}.\end{equation} \tag{ 112 }$$

When the distance from the solenoid to the electron source is shorter than the Rayleigh length (say, for relativistic electrons), the width of the fundamental mode σ_⊥ does not exceed a few nanometers [49, 51, 52] and so σ_⊥ ≪ ρ_H, n_H ≫ n. The opposite regime is realized when the packet is non-relativistic and it can spread so that ⁵ σ_⊥ ≫ ρ_H, n_H ≪ n.

The function ⟨ρ²⟩(t) from equation (109) has two extrema: at t = 0 and at t = T_c/2 = π/ω_c. However, which of them is the maximum and which is the minimum depends on whether ${\langle {\rho }^{2}\rangle }_{\text{st}}{ >}{\langle {\rho }^{2}\rangle }^{\text{free}}$ or ${\langle {\rho }^{2}\rangle }_{\text{st}}{< }{\langle {\rho }^{2}\rangle }^{\text{free}}$. The former case corresponds to ρ_H > σ_⊥, n_H > n and the lens can be called defocusing at short times (shown in figure 2), while the latter regime is realized when ρ_H < σ_⊥, n_H < n, that is, the lens is focusing at short times and we have

$$\begin{equation}{\langle {\rho }^{2}\rangle }^{\text{free}}/2{< }{\langle {\rho }^{2}\rangle }_{\text{st}}{< }{\langle {\rho }^{2}\rangle }^{\text{free}}\end{equation} \tag{ 113 }$$

due to (110). In other words, the rms-radius and the emittance are nearly constant for the focusing lens. At large times, ⟨ρ²⟩(t) oscillates with a period T_c around ${\langle {\rho }^{2}\rangle }_{\text{LG}}$ together with the emittance ${{\epsilon}}_{\rho }\left(t\right)=\sqrt{\mathrm{d}\mathrm{e}\mathrm{t}\enspace {Q}_{ij}\left(t\right)}$ with its mean value

$$\begin{align}\hfill & {{\epsilon}}_{\rho }^{\text{LG}}=\sqrt{\mathrm{d}\mathrm{e}\mathrm{t}\enspace \bar{{Q}_{ij}\left(t\right)}}=\sqrt{{\langle {\rho }^{2}\rangle }_{\text{LG}}\enspace \langle {\boldsymbol{u}}_{\perp }^{2}\rangle }\hfill \\ \hfill & \quad ={\lambda }_{\mathrm{c}}\enspace \sqrt{2}\enspace \sqrt{\left(2{n}_{\mathrm{H}}+\vert \ell \vert +1\right)\left(2{n}_{\mathrm{H}}+\vert \ell \vert -\mathrm{s}\mathrm{g}\mathrm{n}\left(e\right)\ell +1\right)}.\hfill \end{align} \tag{ 114 }$$

The packet's effective quality factor $M={{\epsilon}}_{\rho }^{\text{LG}}/{\lambda }_{\mathrm{c}}$ can be an irrational number and it can be much larger than the quality factor of the incoming LG packet M = n + |ℓ| + 1 when n_H ≫ n, i.e. for the defocusing-at-short-times lens. In this case, the mean emittance in the field ${{\epsilon}}_{\rho }^{\text{LG}}$ is larger than ${{\epsilon}}_{\rho }^{\text{free}}$, so the packet becomes 'more quantum' ⁶ . In contrast to the free LG packet, the mean emittance and the quality factor depend on the sign of the OAM ℓ and of the particle's charge. In particular, when sgn(e)ℓ < 0 the emittance grows as ${{\epsilon}}_{\rho }^{\text{LG}}\propto \vert \ell \vert$ for large ℓ, while it is only ${{\epsilon}}_{\rho }^{\text{LG}}\propto \sqrt{\vert \ell \vert }$ for sgn(e)ℓ > 0. In other words, the electron state with ℓ > 0 (along the magnetic field) has a higher quality factor than the electron state with ℓ < 0 (opposite to the magnetic field).

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In a thin-lens regime with Δt ≪ T_c, we can represent (109) in a form, similar to that of a free packet:

$$\begin{align}\hfill & \langle {\rho }^{2}\rangle \left(t\right)\approx {\langle {\rho }^{2}\rangle }^{\text{free}}+\frac{{\omega }_{\mathrm{c}}^{2}}{2}\enspace \left({\langle {\rho }^{2}\rangle }_{\text{st}}-{\langle {\rho }^{2}\rangle }^{\text{free}}\right){t}^{2}\hfill \\ \hfill & \qquad ={\langle {\rho }^{2}\rangle }^{\text{free}}\left(1{\pm}\frac{{t}^{2}}{{t}_{\mathrm{d}}^{2}}\right)\equiv {\langle {\rho }^{2}\rangle }^{\text{free}}\left(1{\pm}\frac{\langle {z\rangle }^{2}}{{z}_{\mathrm{R}}^{2}}\right),\quad {z}_{\mathrm{R}}=\langle u\rangle {t}_{\mathrm{d}},\hfill \end{align} \tag{ 115 }$$

where the upper sign corresponds to ${\langle {\rho }^{2}\rangle }_{\text{st}}{ >}{\langle {\rho }^{2}\rangle }^{\text{free}}$ (defocusing) and the lower sign—to the opposite case (focusing). The diffraction time is

$$\begin{equation}{t}_{\mathrm{d}}=\frac{{T}_{\mathrm{c}}}{\pi \sqrt{2}}\enspace \sqrt{\frac{{\langle {\rho }^{2}\rangle }^{\text{free}}}{\vert {\langle {\rho }^{2}\rangle }_{\text{st}}-{\langle {\rho }^{2}\rangle }^{\text{free}}\vert }}.\end{equation} \tag{ 116 }$$

Accordingly, one can define a Lorentz invariant Gouy phase and a $\hat{\beta }$-function of the electron packet in a thin magnetic lens:

$$\begin{equation}{\varphi }^{\mathrm{G}}={\pm}\mathrm{arctan}\frac{t}{{t}_{\mathrm{d}}}={\pm}\int \frac{\mathrm{d}t}{\hat{\beta }\left(t\right)}\approx {\pm}\frac{t}{{t}_{\mathrm{d}}}.\end{equation} \tag{ 117 }$$

One can also derive this phase from a wave function, which is a solution to the paraxial wave equation in the magnetic field [22]. We see that the condition of paraxiality,

$$\begin{equation}\sqrt{\langle {\left({\boldsymbol{p}}_{\perp }^{\text{kin}}\right)}^{2}\rangle }\ll \langle {\hat{p}}_{z}^{\text{kin}}\rangle \end{equation} \tag{ 118 }$$

is intimately connected with the thin-lens condition, Δt ≪ T_c. Both of them imply a non-vanishing longitudinal momentum and the finite time of flight through the field. These conditions may not be met in a thick lens or in a storage ring.

Let us estimate how realistic the thin-lens (paraxial) approximation is. The cyclotron period T_c = 2π/ω_c can be presented as follows:

$$\begin{equation}{T}_{\mathrm{c}}=2\pi {t}_{\mathrm{c}}\frac{{H}_{\mathrm{c}}}{H},\end{equation} \tag{ 119 }$$

where t_c is the Compton time (27). For the field strengths of H ∼ 0.1–10 T, we have

$$\begin{equation}{T}_{\mathrm{c}}\sim 1{0}^{-12}\text{--}1{0}^{-10}\enspace \mathrm{s}.\end{equation} \tag{ 120 }$$

In the laboratory frame of reference this time is $\gamma =1/\sqrt{1-{\beta }^{2}}$ times larger. The time of flight Δt = L/⟨u⟩ through the lens of the length L ∼ 1 cm–1 m is

$$\begin{equation}{\Delta}t\sim 1{0}^{-10}\text{--}1{0}^{-8}\enspace \mathrm{s}.\end{equation} \tag{ 121 }$$

We see that the paraxial regime Δt ≪ T_c can be realized for ultrarelativistic electrons with γ ∼ 10²–10³, the thin lens of L < 10 cm, and not very high field strengths, H ∼ 0.1 T. A more realistic scenario with H > 0.1 T, γ ∼ 1–10, L > 10 cm is not compatible with the paraxial approximation and in this (thick-lens) regime the Gouy phase and the $\hat{\beta }$-function (117) do not make sense because the packet does not spread on average.

Now we return to the kinetic AM (86), which is

$$\begin{align}\hfill & \langle {\hat{L}}_{z}^{\text{kin}}\rangle \left(t\right)=\ell -m{\omega }_{\mathrm{L}}\left({\langle {\rho }^{2}\rangle }_{\text{st}}+\left({\langle {\rho }^{2}\rangle }^{\text{free}}-{\langle {\rho }^{2}\rangle }_{\text{st}}\right)\mathrm{cos}\left({\omega }_{\mathrm{c}}t\right)\right)\hfill \\ \hfill & \qquad =-\frac{{\varepsilon }_{\perp }}{{\omega }_{\mathrm{L}}}-m{\omega }_{\mathrm{L}}\left({\langle {\rho }^{2}\rangle }^{\text{free}}-{\langle {\rho }^{2}\rangle }_{\text{st}}\right)\mathrm{cos}\left({\omega }_{\mathrm{c}}t\right).\hfill \end{align} \tag{ 122 }$$

In the hard-edge approximation, it experiences a sudden change when the particle enters the solenoid at t = 0,

$$\begin{equation}\langle {\hat{L}}_{z}^{\text{kin}}\rangle \left(0\right)-{\langle {\hat{L}}_{z}\rangle }^{\text{free}}=-m{\omega }_{\mathrm{L}}{\langle {\rho }^{2}\rangle }^{\text{free}}=-2\enspace \mathrm{s}\mathrm{g}\mathrm{n}\left(e\right)\enspace \frac{{\langle {\rho }^{2}\rangle }^{\text{free}}}{{\rho }_{\mathrm{H}}^{2}}.\end{equation} \tag{ 123 }$$

If the incoming particle is a Gaussian packet with ${\langle {\hat{L}}_{z}\rangle }^{\text{free}}=\ell =0$, then equation (123) yields the kinetic AM that the packet acquires when entering the solenoid. In full analogy to a classical beam [29], all the Bohmian trajectories of the packet that are off the axis start to rotate, which results in a finite azimuthal component of the current density.

The sudden change of the kinetic AM can also be represented as a quantum average of an Aharonov–Bohm phase,

$$\begin{equation}\left\langle e\oint \frac{\mathrm{d}\boldsymbol{\rho }}{2\pi }\enspace \boldsymbol{A}\right\rangle =\langle {\hat{L}}_{z}^{\text{kin}}\rangle \left(0\right)-{\langle {\hat{L}}_{z}\rangle }^{\text{free}}.\end{equation} \tag{ 124 }$$

Importantly, as the particle's rectilinear path is not changed by the field on the solenoid axis, the classical trajectory is not closed and the flux of the field is calculated through the transverse area of the wave-packet itself. In other words, the integral is taken not over the cyclotron radius but over all the Bohmian trajectories with a subsequent quantum averaging. The corresponding mean flux through the packet's area

$$\begin{equation}\langle {\Phi}\rangle =\pi \enspace H\enspace {\langle {\rho }^{2}\rangle }^{\text{free}}=-\left\langle \oint \mathrm{d}\boldsymbol{\rho }\enspace \boldsymbol{A}\right\rangle \end{equation} \tag{ 125 }$$

is quantized when the incoming state is not the fundamental mode—for instance, it is the standard LG packet with M > 1,

$$\begin{equation}\langle {\Phi}\rangle =\pi \enspace H\enspace {\langle {\rho }^{2}\rangle }^{\text{free}}=\pi \enspace H\enspace {\sigma }_{\perp }^{2}\enspace \left(n+\vert \ell \vert +1\right).\end{equation} \tag{ 126 }$$

One can generalize these results when the incoming transverse momentum $\langle {\hat{\boldsymbol{p}}}_{\perp }^{\text{kin}}\rangle$ and the cyclotron radius are not vanishing. Then the AM (86) contains two contributions: (i) ${L}_{z}^{\text{cyclo}}$ comes from the cyclotron motion of the packet's centroid and (ii) ${L}_{z}^{\text{w}-\text{p}}$ is purely quantum and connected with the current circulating around this centroid:

$$\begin{align}\hfill & \langle {\hat{L}}_{z}^{\text{kin}}\rangle ={L}_{z}^{\text{cyclo}}+{L}_{z}^{\text{w}-\text{p}},\quad {L}_{z}^{\text{cyclo}}=-m{\omega }_{\mathrm{L}}{\langle \boldsymbol{\rho }\rangle }^{2},\hfill \\ \hfill & {L}_{z}^{\text{w}-\text{p}}=\ell -m{\omega }_{\mathrm{L}}\left(\langle {\rho }^{2}\rangle -{\langle \boldsymbol{\rho }\rangle }^{2}\right)\equiv \langle {\hat{L}}_{z}^{\text{can}}\rangle +{L}_{z}^{\text{dia}}.\hfill \end{align} \tag{ 127 }$$

A part of the latter, ${L}_{z}^{\text{dia}}\left(t\right)=-m{\omega }_{\mathrm{L}}\left(\langle {\rho }^{2}\rangle \left(t\right)-{\langle \boldsymbol{\rho }\rangle }^{2}\right)$, is sometimes called the diamagnetic AM [10]. The mean path ⟨ r ⟩(t) coincides with equation (3) and so (cf equation (11)) ${\langle \boldsymbol{\rho }\rangle }^{2}={\rho }_{\mathrm{c}}^{2},\enspace {L}_{z}^{\text{cyclo}}={L}_{z}^{\text{int}}/2$. In this case $\bar{\langle {\rho }^{2}\rangle \left(t\right)}$ is to be calculated over the states with $\langle {\hat{\boldsymbol{p}}}_{\perp }^{\text{kin}}\rangle \ne 0$.

We would like to emphasize that the stationary Landau states are not the most general solution to the problem and there is a family of other states with different quantum numbers, including the non-stationary states that also have $\langle {\hat{L}}_{z}\rangle =\ell$ [67]. Nevertheless, in order to fully describe quantum dynamics of all the vortex electron's observables in a magnetic lens it is enough to know the incoming free states, the stationary states, and to apply the boundary conditions together with the quantum equations of motion.

The conservation of the canonical AM ℓ takes place for more general axially symmetric and not necessarily homogeneous fields such as a combination of magnetic and electric lenses [29]. Indeed, the condition $\left[\hat{H},{\hat{L}}_{z}^{\text{can}}\right]=0$ holds when neither A nor A⁰ depends on ϕ. The simplest example is when along with the solenoid field (2) we also have a constant and homogeneous longitudinal electric field E = {0, 0, E}. Clearly, the rms-radius ⟨ρ²⟩(t) from equation (96) is not changed, so such a field can accelerate vortex particles. This rms-radius is changed, however, when the electric field has a radial component E_ρ (electric lens).

Let us suppose that along with the longitudinal magnetic field and the longitudinal (accelerating) electric field we have an inhomogeneous radial component of the electric field,

$$\begin{equation}\boldsymbol{E}={E}_{\rho }\left(\rho \right)\enspace {\boldsymbol{e}}_{\rho }+{E}_{z}\enspace {\boldsymbol{e}}_{z},\qquad \boldsymbol{H}=\left\{0,0,H\right\}.\end{equation} \tag{ 128 }$$

where e _ρ = ρ /ρ, e _z = {0, 0, 1}. In a linear approximation, we suppose that the radial field is absent on the axis, E_ρ (0) = 0, and so (see section 3.3.2 in reference [29])

$$\begin{equation}{E}_{\rho }\left(\rho \right)\approx \rho {E}_{\rho }^{\prime },\end{equation} \tag{ 129 }$$

where E_ρ ' ≡ ∂E_ρ (0)/∂ρ can be either positive or negative. In the same approximation, a radial component of the magnetic field (102) can also be added to (128) for a solenoid of a finite length, which does not however change the main conclusions below. An example of the linear field (128) and (129) is an azimuthally symmetric Penning trap (as well as its generalizations such as a Penning–Malmberg trap) with the following scalar potential (see, for instance, [68]):

$$\begin{equation}{A}^{0}\left(\rho ,z\right)=a\enspace \left({\rho }^{2}-2{z}^{2}\right),\quad a=\text{const},\quad {E}_{\rho }\left(\rho \right)=-2a\rho .\end{equation} \tag{ 130 }$$

The quantum equations of motion look now as follows (cf equation (94)):

$$\begin{align}\hfill & \frac{{\partial }^{2}\langle {\rho }^{2}\rangle \left(t\right)}{\partial {t}^{2}}=2\langle {\boldsymbol{u}}_{\perp }^{2}\rangle \left(t\right)+2\frac{{\omega }_{\mathrm{c}}}{m}\enspace \ell -\langle {\rho }^{2}\rangle \left(t\right)\left({\omega }_{\mathrm{c}}^{2}-\frac{2e{E}_{\rho }^{\prime }}{m}\right),\hfill \\ \hfill & \frac{{\partial }^{2}\langle {\boldsymbol{u}}_{\perp }^{2}\rangle \left(t\right)}{\partial {t}^{2}}=\frac{e{E}_{\rho }^{\prime }}{m}\enspace \frac{{\partial }^{2}\langle {\rho }^{2}\rangle \left(t\right)}{\partial {t}^{2}}.\hfill \end{align} \tag{ 131 }$$

where there is an additional term with E_ρ ' and $\langle {\boldsymbol{u}}_{\perp }^{2}\rangle$ is no longer constant. The second equation together with the condition $\langle {\boldsymbol{u}}_{\perp }^{2}\rangle \left(t\right)=\langle {\boldsymbol{u}}_{\perp }^{2}\rangle$ at E_ρ = 0 yield

$$\begin{equation}\langle {\boldsymbol{u}}_{\perp }^{2}\rangle \left(t\right)=\langle {\boldsymbol{u}}_{\perp }^{2}\rangle +\frac{e{E}_{\rho }^{\prime }}{m}\enspace \langle {\rho }^{2}\rangle \left(t\right).\end{equation} \tag{ 132 }$$

Then the first equation looks as follows:

$$\begin{align}\hfill & \frac{{\partial }^{2}\langle {\rho }^{2}\rangle \left(t\right)}{\partial {t}^{2}}=2\langle {\boldsymbol{u}}_{\perp }^{2}\rangle +2\frac{{\omega }_{\mathrm{c}}}{m}\enspace \ell -\langle {\rho }^{2}\rangle \left(t\right)\left({\omega }_{\mathrm{c}}^{2}-\frac{4e{E}_{\rho }^{\prime }}{m}\right)\hfill \\ \hfill & \qquad ={\omega }_{\mathrm{c}}^{2}{\langle {\rho }^{2}\rangle }_{\text{st}}-\langle {\rho }^{2}\rangle \left(t\right)\left({\omega }_{\mathrm{c}}^{2}-\frac{4e{E}_{\rho }^{\prime }}{m}\right),\hfill \end{align} \tag{ 133 }$$

where ${\langle {\rho }^{2}\rangle }_{\text{st}}$ is from equation (95).

The solution to this equation crucially depends on the sign of ${\omega }_{\mathrm{c}}^{2}-4e{E}_{\rho }^{\prime }/m$. For a simplest electrostatic lens with H = 0 (say, an Einzel lens), the solution is oscillatory (the lens is focusing) only when

$$\begin{equation*}e{E}_{\rho }^{\prime }{< }0,\end{equation*}$$

i.e. either e < 0, E_ρ ' > 0 or vice versa. It is

$$\begin{align}\hfill & \langle {\rho }^{2}\rangle \left(t\right)=\frac{{\varepsilon }_{\perp }}{\vert e{E}_{\rho }^{\prime }\vert }+{\alpha }_{1}\enspace \mathrm{cos}\left(t\sqrt{\frac{4}{m}\enspace \vert e{E}_{\rho }^{\prime }\vert }+{\alpha }_{2}\right),\hfill \\ \hfill & \text{or}\enspace \langle {\rho }^{2}\rangle \left(t\right)=\frac{{\varepsilon }_{\perp }}{\vert e{E}_{\rho }^{\prime }\vert }+\left({\langle {\rho }^{2}\rangle }^{\text{free}}-\frac{{\varepsilon }_{\perp }}{\vert e{E}_{\rho }^{\prime }\vert }\right)\mathrm{cos}\left(t\sqrt{\frac{4}{m}\enspace \vert e{E}_{\rho }^{\prime }\vert }\right)\hfill \end{align} \tag{ 134 }$$

when a free particle enters the lens at t = 0, see figure 2. In the stationary regime (a thick lens), we have

$$\begin{equation}\bar{\langle {\rho }^{2}\rangle \left(t\right)}={\varepsilon }_{\perp }/\vert e{E}_{\rho }^{\prime }\vert .\end{equation} \tag{ 135 }$$

In the opposite case with eE_ρ ' > 0, the rms-radius grows with time—the lens is globally defocusing.

When the magnetic field is not vanishing, the lens is focusing and the mean emittance is conserved only when (in accord with equation (25.35) in reference [65])

$$\begin{equation}{\omega }_{\mathrm{c}}^{2}-4e{E}_{\rho }^{\prime }/m{ >}0,\end{equation} \tag{ 136 }$$

which can be realized even for eE_ρ ' > 0. The corresponding solution is

$$\begin{align}\hfill & \langle {\rho }^{2}\rangle \left(t\right)=\frac{{\omega }_{\mathrm{c}}^{2}}{{\omega }_{\mathrm{c}}^{2}-4e{E}_{\rho }^{\prime }/m}{\langle {\rho }^{2}\rangle }_{\text{st}}+{\alpha }_{1}\enspace \mathrm{cos}\left(t\sqrt{{\omega }_{\mathrm{c}}^{2}-4e{E}_{\rho }^{\prime }/m}+{\alpha }_{2}\right),\hfill \\ \hfill & \text{or}\enspace \langle {\rho }^{2}\rangle \left(t\right)=\frac{{\omega }_{\mathrm{c}}^{2}}{{\omega }_{\mathrm{c}}^{2}-4e{E}_{\rho }^{\prime }/m}\enspace {\langle {\rho }^{2}\rangle }_{\text{st}}\hfill \\ \hfill & +\left({\langle {\rho }^{2}\rangle }^{\text{free}}-\frac{{\omega }_{\mathrm{c}}^{2}}{{\omega }_{\mathrm{c}}^{2}-4e{E}_{\rho }^{\prime }/m}\enspace {\langle {\rho }^{2}\rangle }_{\text{st}}\right)\mathrm{cos}\left(t\sqrt{{\omega }_{\mathrm{c}}^{2}-4e{E}_{\rho }^{\prime }/m}\right),\hfill \end{align} \tag{ 137 }$$

when a particle enters the field at t = 0. For a thin lens, the time interval during which the particle stays in the field is shorter than the oscillation period

$$\begin{equation}{T}_{\mathrm{H}\mathrm{E}}=\frac{2\pi }{\sqrt{{\omega }_{\mathrm{c}}^{2}-4e{E}_{\rho }^{\prime }/m}},\end{equation} \tag{ 138 }$$

and we also have an effective diffraction time t_d, a $\hat{\beta }$-function, and a Gouy phase ${\varphi }^{\mathrm{G}}={\pm}\mathrm{arctan}t/{t}_{\mathrm{d}}={\pm}\int \mathrm{d}t/\hat{\beta }\left(t\right)\approx {\pm}t/{t}_{\mathrm{d}}$,

$$\begin{equation}\langle {\rho }^{2}\rangle \left(t\right)={\langle {\rho }^{2}\rangle }^{\text{free}}\left(1{\pm}\frac{{t}^{2}}{{t}_{\mathrm{d}}^{2}}\right),\quad {t}_{\mathrm{d}}=\frac{{T}_{\mathrm{H}\mathrm{E}}}{\pi \sqrt{2}}\frac{\sqrt{{\langle {\rho }^{2}\rangle }^{\text{free}}}}{\sqrt{\left\vert {\langle {\rho }^{2}\rangle }^{\text{free}}-\frac{{\omega }_{\mathrm{c}}^{2}}{{\omega }_{\mathrm{c}}^{2}-4e{E}_{\rho }^{\prime }/m}\enspace {\langle {\rho }^{2}\rangle }_{\text{st}}\right\vert }}.\end{equation} \tag{ 139 }$$

Importantly, at ${\omega }_{\mathrm{c}}^{2}-4e{E}_{\rho }^{\prime }/m\to +0$ the oscillations get 'frozen', T_HE → ∞, and the Gouy phase becomes positive, which means that the packet spreads and the lens is defocusing not only for ${\omega }_{\mathrm{c}}^{2}-4e{E}_{\rho }^{\prime }/m{< }0$.

To estimate the emittance and the quality factor, we also need the solutions of the equations of motion

$$\begin{equation}\frac{{\partial }^{2}\langle \boldsymbol{\rho }\rangle }{\partial {t}^{2}}=\frac{e}{m}\enspace \langle {\boldsymbol{E}}_{\rho }\rangle +\frac{e}{m}\enspace \langle \boldsymbol{u}{\times}\boldsymbol{H}\rangle ,\quad \langle {\boldsymbol{E}}_{\rho }\rangle ={E}_{\rho }^{\prime }\enspace \langle \boldsymbol{\rho }\rangle .\end{equation} \tag{ 140 }$$

For the focusing lens, they are

$$\begin{align}\hfill & \langle x\rangle \left(t\right)={\delta }_{1}\enspace \mathrm{cos}\left(\frac{t}{2}\left(-{\omega }_{\mathrm{c}}+\sqrt{{\omega }_{\mathrm{c}}^{2}-4e{E}_{\rho }^{\prime }/m}\right)+{\varphi }_{1}\right)\hfill \\ \hfill & \qquad +{\delta }_{2}\enspace \mathrm{cos}\left(\frac{t}{2}\left(-{\omega }_{\mathrm{c}}-\sqrt{{\omega }_{\mathrm{c}}^{2}-4e{E}_{\rho }^{\prime }/m}\right)+{\varphi }_{2}\right),\hfill \\ \hfill & \langle y\rangle \left(t\right)={\delta }_{1}\enspace \mathrm{sin}\left(\frac{t}{2}\left(-{\omega }_{\mathrm{c}}+\sqrt{{\omega }_{\mathrm{c}}^{2}-4e{E}_{\rho }^{\prime }/m}\right)+{\varphi }_{1}\right)\hfill \\ \hfill & \qquad +{\delta }_{2}\enspace \mathrm{sin}\left(\frac{t}{2}\left(-{\omega }_{\mathrm{c}}-\sqrt{{\omega }_{\mathrm{c}}^{2}-4e{E}_{\rho }^{\prime }/m}\right)+{\varphi }_{2}\right),\hfill \end{align} \tag{ 141 }$$

where δ₁, δ₂, φ₁, φ₂ are defined by the initial conditions. The matrix of central moments

$$\begin{equation}{Q}_{ij}\left(t\right)=\left(\begin{matrix}\hfill \langle {\rho }^{2}\rangle -\langle {\boldsymbol{\rho }\rangle }^{2}\hfill & \hfill \langle \boldsymbol{\rho }\cdot {\boldsymbol{u}}_{\perp }\rangle -\langle \boldsymbol{\rho }\rangle \cdot \langle {\boldsymbol{u}}_{\perp }\rangle \hfill \\ \hfill \langle \boldsymbol{\rho }\cdot {\boldsymbol{u}}_{\perp }\rangle -\langle \boldsymbol{\rho }\rangle \cdot \langle {\boldsymbol{u}}_{\perp }\rangle \hfill & \hfill \langle {\boldsymbol{u}}_{\perp }^{2}\rangle -{\langle {\boldsymbol{u}}_{\perp }\rangle }^{2}\hfill \end{matrix}\right)\end{equation} \tag{ 142 }$$

becomes diagonal after averaging over the oscillation period,

$$\begin{equation}\bar{{Q}_{ij}\left(t\right)}=\left(\begin{matrix}\hfill \bar{\langle {\rho }^{2}\rangle }-\bar{\langle {\boldsymbol{\rho }\rangle }^{2}}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \bar{\langle {\boldsymbol{u}}_{\perp }^{2}\rangle }-\bar{{\langle {\boldsymbol{u}}_{\perp }\rangle }^{2}}\hfill \end{matrix}\right),\end{equation} \tag{ 143 }$$

where

$$\begin{align}\hfill & \bar{\langle {\rho }^{2}\rangle \left(t\right)}=\frac{{\omega }_{\mathrm{c}}^{2}}{{\omega }_{\mathrm{c}}^{2}-4e{E}_{\rho }^{\prime }/m}\enspace {\langle {\rho }^{2}\rangle }_{\text{st}},\qquad \bar{{\langle \boldsymbol{\rho }\rangle }^{2}}={\delta }_{1}^{2}+{\delta }_{2}^{2},\hfill \\ \hfill & \bar{\langle {\boldsymbol{u}}_{\perp }^{2}\rangle \left(t\right)}=\langle {\boldsymbol{u}}_{\perp }^{2}\rangle +\frac{e{E}_{\rho }^{\prime }}{m}\enspace \frac{{\omega }_{\mathrm{c}}^{2}}{{\omega }_{\mathrm{c}}^{2}-4e{E}_{\rho }^{\prime }/m}\enspace {\langle {\rho }^{2}\rangle }_{\text{st}},\hfill \\ \hfill & \bar{{\langle {\boldsymbol{u}}_{\perp }\rangle }^{2}}=\frac{1}{4}\left({\delta }_{1}^{2}\enspace {\left({\omega }_{\mathrm{c}}-\sqrt{{\omega }_{\mathrm{c}}^{2}-4e{E}_{\rho }^{\prime }/m}\right)}^{2}+{\delta }_{2}^{2}\enspace {\left({\omega }_{\mathrm{c}}+\sqrt{{\omega }_{\mathrm{c}}^{2}-4e{E}_{\rho }^{\prime }/m}\right)}^{2}\right).\hfill \end{align} \tag{ 144 }$$

The effective emittance ${{\epsilon}}_{\rho }=\sqrt{\mathrm{d}\mathrm{e}\mathrm{t}\enspace \bar{{Q}_{ij}\left(t\right)}}$ and the quality factor M = _ρ /λ_c are conserved (see figure 3), they grow with ℓ together with ${\langle {\rho }^{2}\rangle }_{\text{st}}$, while the latter radius is to be calculated by using the quantum states in the field (128) given in reference [65]. Remarkably, the use of the more intricate non-stationary states [67] is not needed to fully describe quantum dynamics in the focusing electric and magnetic lenses.

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Thus the twisted wave packets can be safely transmitted through a combination of axially symmetric and linear magnetic and electric lenses, while being simultaneously accelerated without changing their mean OAM, see figure 2. The rms radius and the rms transverse velocity can be changed by the lenses and their final values depend on the moment of time, i.e. on the phase, at which the particle leaves the lens.

Let us suppose that at t = 0, ⟨z⟩ = 0 a non-rotating wave-packet with a vanishing kinetic AM

$$\begin{equation}\langle {\hat{L}}_{z}^{\text{kin}}\rangle =0\end{equation} \tag{ 145 }$$

is generated inside a solenoid—say, emitted from a thermocathode or a photocathode. As can be seen from equation (86), the canonical AM ℓ is necessarily not vanishing and it has the same sign as eH (H can be negative in this section),

$$\begin{equation}\ell =\frac{eH}{2}\enspace \langle {\rho }^{2}\rangle =2\enspace \mathrm{s}\mathrm{g}\mathrm{n}\left(e\right)\enspace \frac{\langle {\rho }^{2}\rangle }{{\rho }_{\mathrm{H}}^{2}}=\frac{1}{2}\enspace \mathrm{s}\mathrm{g}\mathrm{n}\left(e\right)\enspace \frac{\langle {\rho }^{2}\rangle }{{\lambda }_{\mathrm{c}}^{2}}\enspace \frac{H}{{H}_{\mathrm{c}}},\end{equation} \tag{ 146 }$$

where $\sqrt{\langle {\rho }^{2}\rangle }\equiv \sqrt{\langle {\rho }^{2}\rangle \left(0\right)}$ is the packet's rms-radius in the region of generation—say, on a surface of the cathode, see figure 4. The flux of the magnetic field ⟨Φ⟩ = Hπ⟨ρ²⟩ through an area π⟨ρ²⟩ of the wave packet at a point of generation is quantized, analogously to equation (125).

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An instantaneous emission of an electron implies that the duration of the process δt ∼ 1/δɛ is much shorter than the cyclotron period T_c. So the energy uncertainty must be large (see equation (119)),

$$\begin{equation}\delta \varepsilon \gg m\enspace \frac{H}{{H}_{\mathrm{c}}}\sim \left(1{0}^{-4}\text{--}1{0}^{-2}\right)\text{eV}\end{equation} \tag{ 147 }$$

for H ∼ 0.1–10 T. The typical values of δɛ ∼ 0.1–0.5 eV for field emission and photoemission [52] satisfy this condition.

In accelerator physics, equation (146) is called the Busch theorem [29, 37, 38] and it relates the field strength H on a magnetized cathode, the beam rms-radius at the point of generation, and the resultant non-integer AM. Here, we deal with a single-particle wave packet and ⟨...⟩ is a quantum averaging over the Bohmian trajectories, and equation (146) can be called the quantum Busch theorem. The presence of an electric field does not influence this theorem and, therefore, if a vortex packet with the canonical AM ℓ is born inside a solenoid it can further be accelerated by the field E_z to an anode and its emittance stays nearly the same. When leaving the field—say, through a round aperture in the anode—the particle stays twisted due to the conservation of the canonical AM.

When emitted from a thermocathode or a photocathode, the electrons' transverse coherence length at room temperature is

$$\begin{equation*}\sqrt{\langle {\rho }^{2}\rangle }\sim 1\enspace \text{nm}\end{equation*}$$

for tungsten [49, 51, 52], so the minimal field strength to get an electron with ℓ = 1 is

$$\begin{equation}H\sim 1{0}^{2}\enspace \text{T},\end{equation} \tag{ 148 }$$

while it is H ∼ 1 T for $\sqrt{\langle {\rho }^{2}\rangle }\sim 10$ nm. The transverse coherence of the order of 10–100 nm can be achieved by using cryogenically cooled field emitters, photocathodes or cold-atom electron sources [49, 50, 52, 69]. The cold-atom electron sources reveal the Maxwellian momentum distribution of the emitted electrons [69] with

$$\begin{equation}\sqrt{\langle {\rho }^{2}\rangle }=\frac{1}{\sqrt{m{k}_{\mathrm{B}}T}},\end{equation} \tag{ 149 }$$

where k_B is the Boltzmann constant. This yields some 1.7 nm at room temperature, in accord with the results for field emitters and photocathodes [49, 51, 52]. For lower temperatures, however, the temperature dependence of the transverse coherence length for field emission reveals an approximately inverse $\sqrt{\langle {\rho }^{2}\rangle }\propto 1/T$ dependence rather than the Maxwellian $\sqrt{\langle {\rho }^{2}\rangle }\propto 1/\sqrt{T}$ one. Indeed, decrease of the temperature from the room value to 78 K resulted in a corresponding increase of the electron coherence in reference [49],

$$\begin{equation}{\sqrt{\langle {\rho }^{2}\rangle }}_{T=78\enspace \text{K}}/{\sqrt{\langle {\rho }^{2}\rangle }}_{T\approx 295\enspace \text{K}}=295/78\approx 3.8.\end{equation} \tag{ 150 }$$

This hints that the Fermi–Dirac distribution should be used for such partially coherent sources as field emitters and photocathodes.

A minimal inelastic mean free path of electrons in a 2D electron gas in metals varies from 10 nm to some 100 nm in the temperature range of 3.5–178 K, and it reveals an inverse 1/m*k_B T dependence due to the Fermi–Dirac statistics with an effective mass m* [70]. One can expect, therefore, that the similar 1/T dependence (150) can take place for the coherence length of electrons emitted from the surface of metals, which would yield

$$\begin{equation*}\sqrt{\langle {\rho }^{2}\rangle }\sim 30\enspace \text{nm}\end{equation*}$$

at T = 10 K. In this case, the field of H ∼ 0.1 T would suffice for creation of an electron with ℓ ∼ 1. In an optimistic scenario, an electron with $\sqrt{\langle {\rho }^{2}\rangle }\sim 100$ nm and the kinetic energy of several eV (u ∼ 10⁻³ c) is coherently emitted from a cooled source. Its emission time is roughly 100 nm/u ∼ 10⁻¹³ s, which is still smaller than T_c from (120), and the process can be treated as instantaneous. The inverse of this time defines the minimum energy width δɛ ≲ 10⁻² eV and so δɛ/ɛ < 10⁻³ at the emission region.

The requirement to have a cathode at low temperature can be circumvented by using a photocathode at room temperature illuminated by a LG beam of optical twisted photons as we discuss in reference [41]. If a detector does not differentiate the azimuthal angles at which an electron is emitted, the electrons can be coherent over all azimuthal angles and the electron coherence length can match that of the incident photon beam, which can easily be much larger than 1 μm. Thus the generation of vortex electrons with ℓ ≫ 1 and the subsequent acceleration of them to relativistic energies seem feasible in this scheme and it is somewhat similar to the generation of classical angular-momentum-dominated beams.

The quantum Busch theorem implies that the magnetized cathode technique can be adapted to generate other types of charged particles with OAM, such as protons and anti-protons, light and heavy ions, positrons, muons, etc. For a particle with a charge q = Ze, the Busch theorem is obtained from equation (146) by e → Ze and it can be presented as follows:

$$\begin{equation}\vert \ell \vert \approx 1.5{\times}1{0}^{-3}\enspace \vert Z\vert \enspace \langle {\rho }^{2}\rangle \left({\text{nm}}^{2}\right)\enspace \vert H\vert \left(\text{T}\right),\end{equation} \tag{ 151 }$$

where $\sqrt{\langle {\rho }^{2}\rangle }$ and H are measured in nanometers and in tesla, respectively. So the OAM grows with |Z|. If the transverse coherence length of an electron emitted from a cathode at a certain temperature scales as

$$\begin{equation}\sqrt{\langle {\rho }^{2}\rangle }\left(m\right)\propto 1/mT,\end{equation} \tag{ 152 }$$

then for a source of fermions of a different mass at the same temperature we have

$$\begin{equation}{T}_{1}={T}_{2}:\frac{\sqrt{\langle {\rho }^{2}\rangle }\left({m}_{1}\right)}{\sqrt{\langle {\rho }^{2}\rangle }\left({m}_{2}\right)}=\frac{{m}_{2}}{{m}_{1}}.\end{equation} \tag{ 153 }$$

In particular, if m₂ = m_p is a proton mass then the proton wave packet is some 1840 times narrower than that of electron. At room temperature, this yields

$$\begin{equation}\sqrt{\langle {\rho }^{2}\rangle }\left({m}_{\mathrm{p}}\right)\lesssim 1\enspace \text{pm}.\end{equation} \tag{ 154 }$$

The scaling (153) and the estimate (154) coincide with those obtained for an ultrarelativistic bunch of non-interacting fermions. Indeed, equation (61) in reference [19] yields

$$\begin{equation}\frac{\delta {\varepsilon }_{1}}{{\varepsilon }_{1}}=\frac{\delta {\varepsilon }_{2}}{{\varepsilon }_{2}}:\frac{\sqrt{\langle {\rho }^{2}\rangle }\left({m}_{1}\right)}{\sqrt{\langle {\rho }^{2}\rangle }\left({m}_{2}\right)}{\geqslant}\frac{{\lambda }_{\mathrm{c}}\left({m}_{1}\right)}{{\lambda }_{\mathrm{c}}\left({m}_{2}\right)}=\frac{{m}_{2}}{{m}_{1}},\end{equation} \tag{ 155 }$$

and so

$$\begin{equation}\sqrt{\langle {\rho }^{2}\rangle }\left({m}_{\mathrm{p}}\right){\geqslant}\frac{{\lambda }_{\mathrm{c}}\left({m}_{\mathrm{p}}\right)}{\delta \varepsilon /\varepsilon }\sim 0.2\enspace \text{pm}\quad \text{for}\enspace \delta \varepsilon /\varepsilon \sim 1{0}^{-3},\end{equation} \tag{ 156 }$$

i.e. the packet is paraxial, $\sqrt{\langle {\rho }^{2}\rangle }\left({m}_{\mathrm{p}}\right)\gg {\lambda }_{\mathrm{c}}\left({m}_{\mathrm{p}}\right)=\hslash /{m}_{\mathrm{p}}c$. The lower temperatures correspond to lower values of δɛ/ɛ and hence to the larger coherence lengths.

One can employ a stripping foil immersed in the solenoid field at some point ⟨z⟩, applied when changing the charge state of ion beams [34–37]. It can be used to produce both the vortex ions—from H⁻ to uranium—and the vortex protons. Neglecting multiple scattering in the foil, the second beam moments are continuous together with the emittance, which is often the case for low-intensity ion beams [35, 37], so the rms width $\sqrt{\langle {\rho }^{2}\rangle }$ of the outgoing ion packet is defined by that of the incoming ion. The canonical AM experiences a jump from 0 to ℓ at the foil due to the change of charge from q_in = Z_in e to q_out = Z_out e, while the kinetic AM stays the same, see figure 5. So the quantum Busch theorem in this case is

$$\begin{equation}\ell =\frac{{q}_{\text{out}}-{q}_{\text{in}}}{2}H\langle {\rho }^{2}\rangle =\left({Z}_{\text{out}}-{Z}_{\text{in}}\right)\frac{eH}{2}\langle {\rho }^{2}\rangle .\end{equation} \tag{ 157 }$$

where ⟨ρ²⟩ is taken at the point ⟨z⟩ where the foil is installed. Compared to the Busch theorem for electrons (146), we have an enhancement due to the factor Z_out − Z_in but the transverse coherences seem to be smaller than those for electrons.

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The non-relativistic packets rapidly spread while propagating, which is why $\sqrt{\langle {\rho }^{2}\rangle }\left(\langle z\rangle \right)$ at the foil can be much larger than $\sqrt{\langle {\rho }^{2}\rangle }\left(0\right)$ at an ion source if the distance ⟨z⟩ between the source and the foil is much larger than the Rayleigh length (see equation (71))

$$\begin{equation}{z}_{\mathrm{R}}=\beta \frac{\langle {\rho }^{2}\rangle \left(0\right)}{M{\lambda }_{\mathrm{c}}},\quad {\lambda }_{\mathrm{c}}=\hslash /{m}_{\mathrm{i}}c,\end{equation} \tag{ 158 }$$

where m_i is the ion mass, M is the packet's quality factor. For electrons with $\sqrt{\langle {\rho }^{2}\rangle }\sim 1$ nm and the velocity β = ⟨u⟩/c, the Rayleigh length is

$$\begin{equation}{z}_{\mathrm{R}}^{\left(e\right)}\approx 2.5\beta \enspace \left(\mu \text{m}\right),\end{equation} \tag{ 159 }$$

whereas for protons with $\sqrt{\langle {\rho }^{2}\rangle }\sim 1$ pm we have

$$\begin{equation}{z}_{\mathrm{R}}^{\left(p\right)}\approx 5\beta \enspace \left(\text{nm}\right),\end{equation} \tag{ 160 }$$

and yet smaller values for heavy ions as

$$\begin{equation}{z}_{\mathrm{R}}\propto \langle {\rho }^{2}\rangle /{\lambda }_{\mathrm{c}}\propto 1/{m}_{\mathrm{i}}\end{equation} \tag{ 161 }$$

if the scaling (152) holds valid. Note that the Maxwellian scaling (149) predicts that z_R does not depend on the mass at all, which is definitely wrong. Clearly, for any macroscopic distance—say, of several mm and larger—the incident packet with β ∼ 10⁻³–10⁻¹ can be considered to be spatially coherent over the entire width of the foil (see figure 5), provided that no accelerating or focusing fields are applied between the source and the foil with a possible exception of a collimator. The use of a collimator could guarantee that the beam intensity is low and no separation of the final ions of different charges is required. Indeed, as seen from the previous sections the quantum packet does not spread on average in axially symmetric electric and magnetic fields, which is why any accelerating section between the source and the foil would leave the packet's rms radius nearly the same. To avoid this, one could install a short drift section with no fields whatsoever after the collimator and the accelerating section to let the accelerated ion packet freely propagate and spread before hitting the foil.

Thus twisting heavy charged particles with a magnetized stripping foil can be even more effective than it is for electrons with a magnetized cathode. To produce individual packets of vortex particles, the beam intensity and the current must be low enough to exclude the space charge effects. One can also envisage a foil made in a form of a spiral or a ring, which would facilitate the production of vortex ions. Alternatively, gas and liquid type strippers can also be employed—see, for instance, reference [36].

Let us briefly discuss how scattering in the foil could alter these predictions. The spreading packet with the OAM ℓ can be characterized with a following opening angle:

$$\begin{equation}\mathrm{tan}\enspace \alpha \approx \alpha =\frac{\sqrt{\langle {\rho }^{2}\rangle }\left({z}_{\mathrm{R}}\right)-\sqrt{\langle {\rho }^{2}\rangle }}{{z}_{\mathrm{R}}}\approx 0.41\frac{\sqrt{\langle {\rho }^{2}\rangle }}{{z}_{\mathrm{R}}}\sim \frac{\vert \ell \vert }{\beta }\enspace 1{0}^{-4},\end{equation} \tag{ 162 }$$

for ${\lambda }_{\mathrm{c}}/\sqrt{\langle {\rho }^{2}\rangle }\sim 1{0}^{-4}$. If β < 10⁻¹ this angle is larger than the angular straggling in the foil of 0.19 mrad reported in reference [36]. The angular straggling of δα ∼ 1 mrad results in the variation of the OAM δ|ℓ|∼ 1 for β ∼ 0.1. So scattering in a foil, gas, or liquid stripper leads to broadening of the OAM spectrum, which can be safely neglected for |ℓ| ≫ 1, provided that the beam quality does not degrade much and the space-charge effects can be neglected.

Other possibilities to prepare and store vortex ions, positrons, anti-protons, and other exotic particles include the use of a Penning trap and its modifications such as a Penning–Malmberg trap because the canonical AM is also an integral of motion for them (see section 4.3). One can employ a cathode with a large spatial coherence of the emitted electrons exposed to the magnetic field stronger than 1 T like in a Penning negative ion source or in a Penning ion gauge source used, for instance, at GSI in Darmstadt. A magnetized photocathode with an incident beam of twisted photons can also be used.

If a vortex particle is generated inside the trap, its OAM would hold provided that collisions in the plasma can be neglected. So the number of twisted particles stored in a trap cannot be large due to quantum decoherence. Another problem is how to extract such particles from the trap. For negative ions, the use of a caesium-coated ring-shaped cathode can be envisaged. The ions can be extracted through a round aperture made in the center of the ring and so the vortex ions, which have larger spatial coherence, can be separated from the untwisted ions by varying the radius of the slit and letting the untwisted particles leave the trap.

Finally, note that the Busch theorem (151) does not specify the physical process that led to the creation of a particle with the canonical AM ℓ. Therefore it holds valid for other processes in magnetic fields, for instance, for ionization of cold atoms in magneto-optical traps. One can photoionize cold Rydberg atoms with the radii up to 1 μm in a magnetic field and the resulting electron and the Rydberg ion will acquire a canonical AM. However a dedicated study of the spatial coherence of photoelectrons emitted from such atoms is needed as the results of reference [71] hint that the electron coherence length can be but a few nanometers, which would imply low values of the OAM.