Nonstationary Laguerre-Gaussian states vs Landau ones: choose your fighter

Although the widely used stationary Landau states describe electrons with a definite orbital angular momentum (OAM) in a magnetic field, it is the lesser known nonstationary Laguerre-Gaussian (NSLG) states that appropriately characterize vortex electrons after their transfer from free space to the field. The reason is boundary conditions lead to oscillations of the r.m.s. radius (the transverse coherence length) of the electron packet that has entered a solenoid. We comprehensively investigate properties of the NSLG states and establish their connections with the Landau states. For instance, we show that the transverse coherence length of an electron in the field usually oscillates around a value greatly exceeding the Landau state coherence length. We also discuss sensitivity of the NSLG states to a small misalignment between the propagation axis of a free electron and the field direction, which is inevitable in a real experiment. It is shown that for any state-of-the-art parameters, the corrections to the observables are negligible, and the electron OAM stays robust to a small tilt of the propagation axis. Finally, we draw analogies between a quantum wave packet and a classical beam of many particles in phase space, calculating the mean emittance of the NSLG states, which acts as a measure of their quantum nature.

Regardless of the generation method, control over the twisted beams transfer through magnetic lenses is crucial for their further use as a diagnostic tool or in other applications.There have already been attempts to investigate the propagation of electrons carrying OAM in magnetic fields [3,4,[28][29][30][31].Nonetheless, for practical applications, the transfer of a vortex electron across a boundary between free space and a solenoid (in a setup similar to that of Fig. 1) should be taken into account.The boundary conditions are defined by the state of the electron entering the magnetic field from free space or generated in the field, for example, with a magnetized cathode [29].These conditions crucially affect further propagation of the electron inside the magnetic lens.
Commonly, an electron in a magnetic field is presumed to be in a stationary Landau state [29,32,33].However, it seems highly unlikely that an electron evolves to the Landau state right after crossing the boundary in an infinitesimal period of time.Therefore, the common approach with the Landau states employed, e.g., in [29,33], seems to have limited applicability.Moreover, we can set the problem of an electron in a constant and homogeneous magnetic field using one of the two distinct gauges for the vector potential A, both leading to the same field H = {0, 0, H} [34], but to different sets of solutions: namely, Hermite-Gaussian and Laguerre-Gaussian beams.Clearly, these are two distinct physical states with different quantum numbers, and it is the boundary (or initial) conditions that determine the choice of the gauge and of the electron quantum state.Here we argue that, generally, it is nonstationary Laguerre-Gaussian (NSLG) states rather than the Landau ones that correctly describe the transition process with ap-propriate boundary conditions.Introducing a boundary makes the root-mean-square (r.m.s) radius of the electron oscillate around a value significantly larger than that predicted by the stationary Landau states.
The aim of this paper is to elaborate on the nonstationary dynamics of electrons in a magnetic field and to investigate the NSLG states in detail.In Sec.II, we introduce these states and provide their comprehensive description both in free space and in a magnetic field.We focus on the electron transverse dynamics, as the longitudinal one is not affected by the magnetic field.The transverse dynamics is supposed to be nonrelativistic and the restrictions imposed are discussed in Sec.III.In Sec.IV, we show that in the limit of H → 0 the NSLG states inside the solenoid turn into free-space Laguerre-Gaussian wave packets.Further, we consider a mismatch between a free NSLG electron propagation axis and the magnetic field direction.In Sec.V, the NSLG and the Landau states are compared, particularly, their sizes.Then we decompose the former into the superposition of the latter.Finally, in Sec.VI, analogies are drawn between a classical particle beam and a quantum wave packet.We introduce a quantum r.m.s.emittance and apply it to the NSLG states.
Electron spin has no qualitative impact on our results and is neglected.Throughout the paper, natural system of units ℏ = c = 1 is used.The electron charge is e = −e 0 , where e 0 > 0 is the elementary charge.Alongside with the electron mass, we use the Compton wavelength λ C = m −1 .

A. Longitudinal and transverse dynamics
In nonrelativistic quantum mechanics, electron dynamics is described by the Schrödinger equation Both in vacuum and inside a magnetic lens, we can single out the motion along the field and factorize the solution of Eq. (1) as Ψ(r, The longitudinal wave function is assumed to be a wave-packet solution to the one-dimensional Schrödinger equation with a nonzero average z-projection of the velocity operator −iλ C ⟨∂ z ⟩ = v.Generally, it can be presented as a superposition of plane waves with different momenta: Its explicit form does not affect the transverse dynamics.
From here on, we only discuss the transverse dynamics of twisted electrons and omit the "⊥" sign to simplify the notation.

B. General NSLG states
In the present work, we are interested in the transverse dynamics of an electron after it crosses the boundary between vacuum and a magnetic field area.In both regions, the electron can be described by the following wave function: which we call a nonstationary Laguerre-Gaussian state.
Here, L |l| n are generalized Laguerre polynomials, n = 0, 1, 2, ... is the radial quantum number, and l = 0, ±1, ±2, ... is the OAM, which is conserved in axially symmetric fields even with weak inhomogeneities [29].The difference between NSLG states in free space (NSLG f ) and in the magnetic field (NSLG H ) is determined by optical functions: dispersion σ(t), radius of curvature R(t), and Gouy phase Φ G (t).The normalization constant in Eq. ( 4) is defined by the standard condition of a single particle in the volume: The NSLG states were briefly introduced in our recent work [35] as means to account for the boundary crossing that provide consistent description of the electron state in regions with and without magnetic field.Here we dwell deeper into the dynamics of these states and discuss their properties from different angles.
The state with the transverse part (4) corresponds to an electron moving rectilinearly along the z-axis, which means that where v = −i∇ ⊥ /m − eA/m.The r.m.s.radius of the NSLG state is proportional to the dispersion: We can directly check that states (4) form an orthonormal set: The set is also complete (see the proof in the Appendix A).

C. NSLG states in free space
In this section, we derive the optical functions of the NSLG f states, which will later determine the initial conditions for the states in the field.
In free space, the transverse Hamiltonian is where the index "f" stands for "free".To derive the optical functions and then the NSLG f state, the wave function (4) can be substituted into the Schrödinger equation (1) with the Hamiltonian (9).This leads to the system of equations 1 , where the primes stand for time derivatives.Instead of R(t), we prefer using the dispersion divergence rate σ ′ (t) = σ(t)/R(t) alongside with σ(t) and Φ(t) to characterize the NSLG states.
To find the unique solution of the system (10), the initial conditions should be specified.In real experiment, twisted electrons are generated at the beam waist: where t g is the time when the twisted electron is generated and σ w is the dispersion at the waist.We set Φ f (t g ) = 0, because a constant phase factor does not change the state.The optical functions σ f (t) and Φ f (t) satisfying the system (10) with the initial conditions (11) are Here, τ d = σ 2 w /λ C is the diffraction time.The NSLG states (4) with σ(t) and Φ G (t) given by Eqs.(12) and R(t) = σ f (t)/σ ′ f (t) are the nonstationary counterparts [29,36,37] of the well-known paraxial free Laguerre-Gaussian wave packets [3,14,15,18,22].
According to Eqs. ( 12) and ( 7), the r.m.s.radius of the NSLG f state is where ρ w = σ w 2n + |l| + 1.This expression illustrates quadratic divergence of the r.m.s.radius near the beam waist and linear growth far from it.Since the NSLG states do not generally possess definite energy, we consider its expectation value.For the NSLG f state given by Eqs. ( 4) and (12), taking into account The first term in Eq. ( 14) stems from the size effect and decreases with the volume occupied by the wave packet.
The second term has a kinetic nature and is responsible for the radial divergence of the state.The free Hamiltonian (9) does not depend on time, which means that the average energy is constant.Indeed, by substituting the dispersion (12) and its derivative into Eq.( 14), we obtain We illustrate the dynamics of the NSLG f wave packet obtained in the experiment of Guzzinati et al. [15] (see Figs. 3, 4 there) in Fig. 2. The electron has the following parameters: electron energy E ∥ = 300 KeV (and the corresponding velocity v ≈ 0.78c), n = 0, l = 3 (in [15] l is designated as m), beam waist dispersion σ w = 3.25 nm (corresponding r.m.s.radius of the waist ρ w = 2n + |l| + 1 = 6.5 nm), and diffraction time τ d = 9 × 10 −5 ns.
Note that we plot the beam radius, while in the work [15] (see Fig. 4(a) there), the beam diameter is depicted.Guzzinati et al. observed several rings as they blocked half of the initial NSLG f beam and obtained a superposition of the NSLG f states.However, in this case, the original NSLG f state makes the dominant contribution, which allows us to reproduce their results.

D. Landau states
Let us now turn to a twisted electron state inside a solenoid.We describe the solenoid as a semi-infinite stationary and homogeneous magnetic field H = Hθ(z − z 0 )e z , e z = (0, 0, 1).The step function θ(z) reflects the hard-edge boundary located at z 0 .We assume the longitudinal part of the wave function to be narrow enough, so that the field can be considered to be suddenly switched on at the time t 0 .
Before moving to the NSLG H states, we would like to briefly remind the reader of the Landau ones.They are stationary solutions to the Schrödinger equation (1) with the transverse Hamiltonian Recall the aforementioned gauge issue: in the original work of Landau, the vector potential is chosen as [38] The Landau states that are the solutions of the Schrödinger equation (1) with the Hamiltonian (16) defined by the vector potential in the Landau gauge (17) are given by Hermite-Gaussian functions where σL = 1/|eH|, ω = |eH|/m is the cyclotron frequency, and s = 0, 1, 2, ... is the principal quantum number.
Alternatively, one can choose the symmetric gauge for the vector potential: where e φ = e y cos φ − e x sin φ is the azimuthal unit vector.Such a choice preserves the axial symmetry of the problem, and the corresponding solutions of the Schrödinger equation have definite values of the OAM (see, e.g., [39]): where σ L = 2/|eH| is the r.m.s.radius of the Landau state with n = l = 0.The normalization constant N nl in Eq. ( 20) is given by Eq. (5).In what follows, by Landau states, we mean the wave function (20) and not (18), which can be viewed as yet another initial condition.
The energy E L of the Landau states is where µ B = |e|/(2m) is the Bohr magneton.The last term in Eq. ( 21) is the energy of the magnetic moment −lµ B in the field H.Note that the electron energy in a Landau state is infinitely degenerate for l ≤ 0 due to the exact compensation of kinetic and magnetic "orbital motions".However, for l > 0, the two terms add up and double the contribution to the energy.The r.m.s.radius of the Landau states ( 20) is constant and equal to Note that in a given magnetic field, there is only a countable set of possible r.m.s.radii of an electron described by the Landau states.In reality, an electron enters the field from free space or is generated in the field with an arbitrary size that must evolve continuously.If this size does not fall within the countable set of possible r.m.s.radii, the free electron cannot find a suitable Landau state to transform into.Moreover, even if the r.m.s.radius of the electron equals that of the Landau state, the divergence rate must also vanish.Thus, taking into account the initial conditions, we are generally led to a non-stationary electron state in the field, which is properly described by the NSLG H state.

E. NSLG states in the field
Similarly to the NSLG f , one can derive the NSLG H states in the magnetic field.Substituting the state (4) into the Schrödinger equation (1) with the Hamiltonian ( 16) we obtain 1 This system is very similar to the set of equations for the optical functions of a free electron state ( 10), yet it results in a drastically different dynamics.
Although one can take arbitrary initial conditions to specify the unique solution of the system (23), in a real experiment, they are determined by the incoming electron state.This prompts us to use the values of the dispersion, its time derivative, and the Gouy phase of the NSLG f electron at the time t 0 when it enters the solenoid as the initial conditions for the NSLG H state: -3 -2 -1 0 1 2 3 0.00 0.02 0.04 0.06 0.08 0.10 Figure 3: The NSLG H packet r.m.s.radii (in red) for different ρ 0 ; ρ L ≈ 52.7 nm is in blue.Black dashed lines correspond to ρ st given by Eq. ( 28).In each subfigure H = 1.9 T (T c ≈ 0.02 ns.), n = 0, l = 3, Following the seminal approach of Silenko et al. [28], we derive the dispersion of the NSLG H electron from Eqs. (23) with the initial conditions (24): where the sign function is This dispersion describes the oscillations of the r.m.s.radius of the electron inside the solenoid with a period T c = 2π/ω.The value θ is the initial phase of the oscillations.
We should also note that states similar to those discussed in this section are presented in the books [40,41] as coherent states of an electron in the magnetic field with the vector potential (19).Another approach to obtaining the NSLG H wave functions using quantum Arnold transformation was recently realized in [42].
The parameter σ 2 st is the period-averaged dispersion square We further use the corresponding time-averaged radius square as a characteristic size of the oscillating wave packet.The inequalities in Eqs. ( 27), ( 28) are derived and discussed in Sec.V B.
The oscillations of the r.m.s.radius of the NSLG H states are shown in Fig. (3).We consider the magnetic field H = 1.9 T, typical for transmission electron microscopes, and quantum numbers n = 0, l = 3 (the corresponding ρ L ≈ 52.7 nm) [17].For simplicity, we set ρ ′ 0 = 0.A nonzero initial value of the divergence rate ρ ′ 0 alters the initial phase of the oscillations θ and the amplitude in accordance with Eqs. ( 25), but the picture remains qualitatively the same.We discuss how nonzero divergence rate affects the r.m.s.radius oscillations in the Appendix B. Now let us discuss the possible oscillation regimes.In Fig. 3a, the free electron size at the boundary ρ 0 = 54 nm is close to ρ L .The r.m.s.radius of the corresponding NSLG H state oscillates around approximately the same value with a negligibly small amplitude.As we will discuss later (see Sec. V C), such an electron can be considered to be in a Landau state to a good extent.In Fig. 3b, ρ 0 = 25 nm is significantly smaller than ρ L .In this case, the magnetic field "tries" to stretch the wave packet to the size of the corresponding Landau state.By the time it happens, the r.m.s.radius of the NSLG H state acquires a nonzero divergence rate and continues broadening past ρ L .In Fig. 3c, ρ 0 = 111.1 nm is larger than ρ L , and their ratio is exactly the inverse of that in 3b.Here, in contrast, the field "tries" to shrink the packet at first; as a result, the r.m.s.radius decreases past the Landau state value and oscillates.Note that for two states with initial sizes ρ 0,1 and ρ 0,2 , if ρ 0,1 /ρ L = ρ L /ρ 0,2 , the oscillations only differ by a π phase shift and are otherwise identical.Finally, in Fig. 3d, we consider an electron of the size ρ 0 = 1 µm much larger than ρ L .Then, the oscillations of the r.m.s.radius of the NSLG H electron "experience" sharp bounces from their lowest value.Similar behavior (shifted by half a period) is observed when the initial NSLG H packet size is much less than the Landau radius.
Thus, from Fig. 3, we can identify three oscillation regimes: 1. Landau-like regime: the r.m.s.radius of the NSLG H state is almost constant, 2. Sine-like regime: the stationary r.m.s.radius (28) is always larger than the Landau radius, but they have the same order of magnitude, 3. Bouncing regime: the r.m.s.radius of the NSLG H state is sharply "bouncing off" the minimal value, and its time-averaged value is much larger than that of the Landau state.
The oscillating behavior of the NSLG H states' r.m.s.radius reminds that of optical Gaussian beams in ducts or graded-index optical waveguides [43] , where λ is the beam wavelength in a medium and n 2 = d 2 n(ρ)/dρ 2 | ρ=0 is the second derivative of the refractive index with respect to the radial coordinate near the symmetry axis.Now let us consider an optical Gaussian beam with a waist dispersion distinct form σ O .In this case, the r.m.s.radius of such a beam will oscillate similar to the r.m.s.radius of the NSLG H state, whose oscillations are shown in Fig. 3.The Gouy phase of the NSLG H state is In Eq. ( 29), the arc tangent should be treated as a multivalued function for the Gouy phase to be continuous.The Gouy phase for H = 1.9 T (ρ L ≈ 64 nm), ρ 0 ≈ 122 nm, ρ ′ 0 = 0, and Φ 0 = 0 is shown in Fig. 4. The red, blue, and green lines correspond to three different pairs of quantum numbers (n, l) = {(0, 0), (0, 1), (1, 1)}, respectively.
A free Gaussian beam gains a phase factor of π while travelling from distant past to distant future [3,15,18,22,28,43,44].Most of the phase gain is accumulated around the waist of the packet.A free Laguerre-Gaussian beam acquires a phase factor of (2n + |l| + 1)π the same way, propagating near its waist.Inside the field, the dynamics are periodic, and the electron state acquires this phase factor each cyclotron period.Moreover, interaction of the OAM with the field provides an additional Zeeman-type phase lπ [3,15].Thus, the phase accumulated by the NSLG H state per T c is (2n + |l| + l + 1)π.
The average energy of the NSLG H electron is Generally, when σ 2 st /σ 2 L > 1, the kinetic rotation prevails over the magnetic one.Moreover, for OAM directed opposite to the field, the two terms do not compensate each other, which removes the degeneracy of energy levels compared to the Landau states.Note that the average energy of the NSLG H state (30) is always larger than that of the Landau one (21), and they are equal only for σ st = σ L , when the two states coincide (see Sec. V).
Although the NSLG H states have not yet been observed directly, an indirect evidence for their existence could have been obtained in the experiment of Schattschnider et al. [17].In this experiment, the authors observed a possible part of the oscillations inherent to the NSLG H states (see Fig. 2b in [17]).In Fig. 5, we reproduce the evolution of the electron r.m.s.radius with the parameters from this work: electron energy The field strength H = 1.9 T (corresponding ρ L ≈ 37 nm), ρ 0 ≈ 71 nm, ρ ′ 0 = 0, T c ≈ 0.02 ns, and Φ 0 = 0.The quantum numbers are: n = 0, l = 0 (red line); n = 0, l = 1 (blue); and n = 1, l = 1 (green).
, and H = 1.9 T. The black vertical line in Fig. 5 cuts off the z-region observed in the experiment.We extend this region a little to show the reader the subsequent bounce of the r.m.s.radius.Thus, we put forward the idea that the authors might have dealt with the NSLG H state.

III. TRANSVERSELY RELATIVISTIC WAVE PACKETS
We assume E ≪ mc 2 while investigating twisted electrons in this work, but Eqs. ( 14), (21), and (30) make it clear that this condition is no longer valid for large n and  [17].We plot σ L and not ρ L to reproduce the Fig. 2b from [17].A black vertical line marks the maximal value of z in the work [17].Longitudinal energy , and H = 1.9 T (corresponding σ L ≈ 26 nm).
|l|.Although in modern experiments, n ∼ 1, beams with OAM values of several hundred [45] and even thousand ℏ [46,47] have already been generated.The restriction E ≪ mc 2 sets the validity limits of our calculations and gives estimates of the quantum numbers that require relativistic treatment of the transverse dynamics.Furthermore, it allows considering beams that are transversely relativistic and longitudinally nonrelativistic, in contrast to those produced in accelerators nowadays.
Let us now estimate the quantum numbers n and l such that ⟨E⟩ ∼ mc 2 .We start with an NSLG f electron with energy ⟨E⟩ given by Eq. (15).Using we obtain a restriction on the quantum numbers of the free electron: Typically, twisted electrons are generated with ρ w ∼ 1 µm.For such particles, the value in the r.h.s. of Eq. ( 31) is of the order of 10 6 .However, being refocused to a 1 nm waist size, electrons with quantum numbers of the order of 10 3 become transversely relativistic.Such focusing is easily achievable with appropriate magnetic lenses [17].Thus, transversely relativistic free twisted electrons can be obtained in experiment already.
Applying the condition ⟨E⟩ ≪ mc 2 for a Landau state, we get Note that in free space, we fix the r.m.s.radius of the generated electron ρ w , but in a magnetic field, it is the dispersion σ L that is defined by the field strength.For example, if the field strength is of the order of 1 T, σ L ∼ 36 nm, and the r.h.s. of the inequality (32) is of the order of 10 5 .For negative values of l, the l.h.s. of Eq. ( 32) does not depend on OAM at all.Therefore, when the magnetic and the kinetic rotations of the Landau state compensate each other, such a state remains nonrelativistic for any attainable values of n and |l|.However, for l > 0, the relativistic regime cannot be achieved either, as it would require OAM of the order of 10 10 .
For the NSLG H states, the relativistic regime is more feasible than for the Landau counterparts, because NSLG H kinetic energy is enhanced by the factor σ 2 st /σ 2 L .Indeed, for an NSLG H wave packet, we obtain Usually, the factor σ 2 st /σ 2 L ≫ 1; for example, in the work [17], σ 2 st /σ 2 L ≈ 31.This allows us to simplify the above condition: The additional factor σ L /σ st in the r.h.s. of this inequality eases the requirements on the quantum numbers to obtain transversely relativistic states.For instance, in the experiment of Schattschnider and colleagues [17], the r.h.s. of Eq. ( 34) is of the order of 10 4 .This value can be reduced even more, for example, by increasing ρ 0 .To increase ρ 0 , one can simply move the solenoid further from the source of twisted electrons.For large wave packets with σ 0 ≫ σ L and with a sufficiently low divergence rate σ ′ 0 ≪ λ C /σ 0 , the condition (34) turns into From here it follows that for wave packets with σ 0 /σ L ≥ σ L /λ C , even a Gaussian mode with n = l = 0 is relativistic.For a field strength of the order of 1 T, this happens when σ 0 ∼ 1 mm, which can also be decreased if the divergence rate σ ′ 0 in ( 34) is taken into account.

IV. CONNECTION BETWEEN NSLG f AND NSLG H STATES
Before considering NSLG H states in detail, we should note that their explicit wave function was obtained from the continuity of the optical functions at the boundary (24).In reality, not only these functions, but also the wave function itself is continuous.This is not surprising, because electron states in free space and inside the solenoid are defined by the ansatz of the same general form (4).
We also need to make a special note about the energies of the NSLG f and NSLG H states. Generally, the quantities given by Eqs. ( 15) and (30) are not equal to each other, i.e. the energy is discontinuous at the boundary.This is a result of the energy dispersion, as the continuity of the average kinetic momentum ⟨ p⟩ does not provide that of ⟨E⟩ ∼ p2 ̸ = ⟨ p⟩ 2 .

A. Vanishing magnetic field
One of the advantages of the NSLG H states compared to the Landau ones is, they smoothly transform into free twisted electron wave packets in the vanishing magnetic field limit.To confirm this, we can find the limit of σ(t), Φ G (t) as H → 0 (σ L → ∞), see Appendix C for rigorous derivation.In Fig. 6, we show how NSLG H dispersion transforms into that of the NSLG f as the magnetic field goes to zero.
In contrast, the Landau states dispersion diverges in the vanishing magnetic field limit, and the wave functions become delocalized.

B. Off-axis injection
In a real-life setup, the propagation axis of a twisted electron wave packet cannot be perfectly aligned with the magnetic field direction.Such a misalignment can be caused by a shift of the electron source or slight inhomogeneities of the magnetic field inside the solenoid.In this section, we account for this inaccuracy by considering a twisted electron that enters the lens at a small angle α with respect to the z-axis, as shown in Fig. 7.
Imagine that by the time t 0 a free electron reaches the lens boundary at z 0 , the propagation axis of the electron is shifted by the angle α with respect to the z-axis aligned where the tilted coordinates are obtained by a rotation around the axis indicated by φ = π/2.The rotational symmetry of the problem enables an arbitrary choice of the rotation axis in the transverse plane without any influence on the results.The transverse and longitudinal parts of the wave function in Eq. ( 36) are given by Eqs. ( 4) and (3), respectively.
Let us now decompose the rotated wave function in terms of the electron states propagating along the z-axis: Here, the decomposition coefficients are (39) We are interested in the off-axis corrections to the electron state in the vicinity of the lens boundary.Therefore, we evaluate Ψnl (r, t) at z = z 0 and t = t 0 in Eq. (38).
In the first non-vanishing order in α and for z = z 0 , Eqs. (37) are simplified to (40) and the coefficients (39) take the form The integral over the transverse plane can be evaluated using Eq. ( 7.422) in [48] (there is, however, a misprint m ↔ n in the book).The absolute value of the coefficients is (42) If the longitudinal wave functions have a sufficiently narrow distribution in coordinate and momentum spaces simultaneously, we can evaluate the decomposition coefficients in a different manner.First, we can approximate the integrals over the longitudinal momentum by evaluating the integrand at the mean value p z = ⟨p z ⟩.Then, Eq. ( 38) becomes The expression (43), as compared to Eq. ( 38), does not contain the longitudinal wave function, whose entire contribution is accounted for by the average momentum ⟨p z ⟩.
Proceeding in the same manner, we get (44) From Eq. ( 44), we see that the actual dimensionless parameter defining the magnitude of the coefficients is α ⟨p z ⟩ σ(t 0 ).In real life, the value of σ(t 0 ) is of the order of several µm or less.Provided that currently n ∼ 1 , l ≲ 10 4 , even for 10 GeV-electrons with ⟨p z ⟩ ∼ 10 −3 µm −1 , we obtain |c nn ′ ll ′ | ≲ 10 −2 α.This means that the off-axis corrections are negligible for any feasible experimental scenario.

V. CONNECTION BETWEEN NSLG STATES IN SOLENOID AND LANDAU STATES
A. Landau states as a special case of NSLG states Although the Landau states ( 20) are represented by stationary wave functions, they also have the form (4).Moreover, both NSLG H and Landau states are solutions of the Schrödinger equation (1) with the same Hamiltonian (16), which leads to the same system of optical equations (23).Here, the question arises: how these two sets of states are linked?
To answer this question, one may look for a solution of the system (23) corresponding to the stationary Landau states.Such a solution exists for the unique choice of the initial conditions: This means that the Landau states are but a special case of the NSLG H ones forming when a free twisted electron with a specific size and zero divergence rate crosses the boundary.Otherwise, an electron inside the solenoid is described by general NSLG H states rather than the Landau ones.
To characterize the deviation of the NSLG H states from the Landau ones, we introduce two dimensionless parameters From Eq. ( 45), it follows that for the Landau states ξ 1 = 1, ξ 2 = 0.The more these parameters differ from 1 and 0, respectively, the more distinguishable the NSLG H and the Landau states are.This effect manifests itself most clearly in growing amplitude of the r.m.s.radius oscillations and its period-averaged value.

B. Comparison of sizes of NSLG H and Landau states
To characterize the size of an NSLG H electron, we use the stationary radius ρ st given by Eq. ( 28).Naively, it seems that this value should be equal to or at least close to ρ L [29,32,33].However, this is generally not true.In terms of the parameters (46), ρ st is expressed as From this expression, it is clear that for ξ 1 ≫ 1, ξ 1 ≪ 1, or ξ 2 ≫ 1, the relation ρ 2 st ≫ ρ 2 L holds.In contrast, for the initial conditions (45), when the electron in the field is indeed in the Landau state, the minimum value ρ st = ρ L is reached.This illustrates that boundary conditions significantly affect the electron states inside the lens.The conditions imposed on the parameters ξ 1,2 for the NSLG H state to be close to a Landau one are very specific.Unless an experimenter is intended to obtain a Landau state, an NSLG H state is almost certainly generated.For example, in the experiment of Schattschnider et al. [17], the parameters of the setup n = 0, |l| = 1, σ 0 = 4.77 × 10 −2 µm, and σ ′ 0 = −3.1 × 10 −4 lead to ξ 1 = 0.76 and ξ 2 = 29.21≫ 1.For these parameters, we find ρ st = 20.7ρL ≫ ρ L , which again supports our idea that NSLG H states were observed in the work [17].

C. Decomposition of NSLG H states in terms of Landau ones
Comparing the characteristic sizes of an NSLG H and a Landau state, we qualitatively estimate the difference between the two states.For a more substantive investigation, we should decompose an NSLG H state wave func-tion in terms of the stationary Landau ones (20): Since the evolution of both sides in Eq. ( 48) is governed by the same Hamiltonian, the decomposition coefficients do not depend on time.We present the explicit expression for a nn ′ l in the Appendix D. Note that the Kronecker delta reflects the OAM conservation.
As we have discussed in the previous section, ρ st = ρ L only when NSLG H and Landau states coincide.Indeed, from this equality, it follows that However, in experiment, it is impossible to precisely satisfy the initial conditions (45) to obtain a single Landau mode inside the solenoid.
Let us analyze what happens to the NSLG H state inside the lens when its characteristic size and ρ L with the same quantum numbers n, l are close, yet not equal: This is true when the size of the incoming packet at the boundary slightly differs from ρ L and the divergence rate is low.From Eq. ( 25), we know that in this situation, rather than being constant and equal to ρ L , the r.m.s.radius inside the lens begins oscillating around a slightly larger value, ρ st , with a small amplitude.The decomposition coefficients clearly indicate that for a small detuning, a few neighbouring Landau modes contribute to the NSLG H state: Interference of these different states results in the r.m.s.radius oscillations and a change in the periodaveraged size.
An intricate picture arises when the state inside the solenoid significantly differs from any of the Landau states, i.e. ξ 1 ≫ 1, or ξ 1 ≪ 1, or ξ 2 ≫ 1.In this case, the NSLG H state is a superposition of numerous Landau ones.The coefficients form wide, oscillating distributions as functions of the radial quantum number of the Landau states n ′ .
In Figs.8a -8d (top panel, in red), we study the distribution of |a nn ′ l | 2 for different values of l while keeping n = 0.In Fig. 8a, l = 0 and ρ L ≈ 26 nm, so the NSLG H state is wider than the Landau one with corresponding quantum numbers.As a consequence, higher-order Landau modes appear in the decomposition.Then, with increasing OAM (Figs. 8b, 8c), ρ L gets closer to ρ 0 , making the decomposition similar to δ n,n ′ .With the further increase of OAM shown in Fig. 8d, ρ L becomes larger than ρ 0 , and, once again, higher-order Landau modes appear.In this case, all the Landau states have a larger size than the NSLG H state at the boundary.However, their destructive interference results in size suppression (see.(D4)).
In Figs.8e -8h (middle panel, in blue), we set l = 0 and investigate how n affects the probability coefficients.In general, the distribution of |a nn ′ l | 2 is similar to that in Figs.8a -8d in the following sense.With increasing n, ρ L grows, and for n = 7, when ρ L ≈ ρ 0 , a δ-like peak emerges in Fig. 8g in accordance with Eq. (51).With a further increase in n, this peak vanishes, leading to numerous Landau states in Fig. 8h.
Figs. 8i -8l (bottom panel, in green) demonstrate another peculiarity of the probability coefficients distribution.Namely, for sufficiently wide distributions, the number of peaks equals n + 1.We suppose this might be connected to the number of rings of the NSLG H state; however, the true nature of this phenomenon is still unclear to us.

VI. EMITTANCE A. Emittance and the Schrödinger uncertainty relation
Classical accelerator physics mainly focuses on particle beams, described by distribution functions in phase space.At any moment of time (or any distance z along the direction of beam propagation), every particle in a beam is a point in this space.In systems with axial symmetry, dynamics in two transverse directions are independent and indistinguishable when the beam has no classical vorticity [49].This allows monitoring only one transverse coordinate x(s) and the corresponding velocity projection x ′ (s), which form two-dimensional trace space (x(s), x ′ (s)).Here, s is a variable parametrizing the particle motion, e.g., time or longitudinal coordinate.
Emittance is one of the essential measured parameters describing a beam.Depending on the problem, it can be defined in different ways; but the most common definitions are the trace space area and the r.m.s.emittance [50,51].The latter is with averaging performed over the beam distribution function, and ⟨x⟩ = ⟨x ′ ⟩ = 0 is assumed.Due to the Liouville's theorem, the phase space volume (or the trace space area) is conserved, but such a definition of emittance does not distinguish between different particle distributions in beams with the same area.Vice versa, the r.m.s.emittance is not generally constant in time, however, it is sensitive to beam distribution [50].One of the reasons why r.m.s.emittance depends on time is beam mismatch, which leads to r.m.s.radius oscillations [50,52].
We will now draw analogies between quantum mechanics and classical accelerator physics.While in the latter, particles are points in the phase space, in quantum theory, a single particle packet is smeared in the coordinate and momentum spaces.In quantum mechanics, a quantity similar to that given by Eq. ( 52) arises from the Schr 'odinger uncertainty relation [29,53] (∆â (54) Note that for ⟨â⟩ = ⟨ b⟩ = 0, the l.h.s. of Eq. ( 54) has the same form as the r.h.s. of Eq. ( 52).Thus, when â and b are the transverse coordinate and velocity operators, respectively, it is natural to call the l.h.s. of Eq. ( 54) the quantum r.m.s.emittance, see [29] for more detail.This way, we see that the r.m.s.emittance definition can be naturally extended to quantum mechanics.
In classical physics, the smaller the r.m.s.emittance is, the less disordered is the beam.In quantum mechanics, the r.m.s.emittance acquires a new meaning: it reflects non-classicality of the state.When the emittance is vanishing, the position-momentum uncertainty is minimal, similar to a classical particle, whose momentum and coordinate can both be measured with minimal error.In contrast, the larger the quantum emittance is, the more noticeable the quantum nature of the particle becomes.

B. Quantum emittance of Laguerre-Gaussian wave packets
We now derive the quantum r.m.s.emittance of the NSLG f and NSLG H states: Here, i enumerates the two transverse axes.The second equality stems from the axial symmetry, and v = −iλ C (∇ − ieA) is the kinetic velocity operator.In Eq. ( 55), the averaging is performed over the NSLG f or the NSLG H states to obtain the corresponding emittance.Using the r.m.s.emittance can be expressed through the r.m.s.radius, its derivative, and the average energy as Let us first focus on the NSLG f state.By substituting explicit expressions for the wave packet parameters from Eqs. ( 13) and ( 15) into Eq.(57), we get The r.m.s.emittance of a free particle is constant in time and minimal for the Gaussian electron state when n = l = 0. Notice that this state minimizes the Schrödinger uncertainty, not the Heisenberg one.We should note this state is a special case of the coherent states of a free particle discussed in [54].For n, |l| ∼ 1, the quantum emittance of an NSLG f state is of the order of λ C , i.e. the particle stays relatively "classical".For large quantum numbers, the emittance grows linearly, and the quantum nature of the particle becomes more pronounced.Similarly, using NSLG H optical functions and energy discussed in II E, we obtain the r.m.s.emittance of an NSLG electron inside the solenoid: (59) The r.m.s.emittance of an NSLG H state is defined by the dispersion σ(t).The time dependence stems from the mismatch at the boundary (σ 0 ̸ = σ L and/or σ ′ 0 ̸ = 0), which causes the r.m.s.radius and, hence, the r.m.s.emittance oscillations.
From Eq. ( 59), the r.m.s.emittance of the Landau state can be easily obtained by setting σ(t) = σ L : One can notice that the r.m.s.emittance is discontinuous at the boundary.This can be seen from Eq. ( 57): the dispersion and its derivative are continuous, while the average energy is not, as we discussed in the beginning of Sec.IV.The time dependence of the NSLG H emittance is shown in Fig. 9. Unlike the r.m.s.radius, it is sensitive to the OAM sign.For l < 0 (Fig. 9a), r.m.s.emittance has additional local maxima, in contrast to the case when the OAM and the magnetic field are aligned (Fig. 9b).
Following the idea that smaller quantum r.m.s.emittance corresponds to a "more classical" particle behavior, we will analyze the regime when ϵ H (t) < ϵ f .For n, |l| ∼ 1, it means that ϵ H ≲ 1. Fig. 9 shows that for some parameters of the wave packet, there are time intervals when this condition is satisfied.From Eq. (59), this is possible only for l < 0.Moreover, the following relation has to be fulfilled: Note that for n or l ≫ 1, NSLG H emittance greatly exceeds λ C when these inequalities are violated.Therefore, the emittance of an NSLG electron can be locally decreased if the electron is placed in the field.However, if we consider a finite-length solenoid, the emittance changes abruptly at both boundaries, and when the particle leaves the solenoid, the emittance is exactly the same as it was at the entrance.Thus, our findings open ways for altering the r.m.s.emittance of an electron with magnetic lenses.

VII. RESULTS AND DISCUSSION
We have analyzed the properties of nonstationary Laguerre-Gaussian (NSLG) states, which, unlike the Landau states, fully capture vortex electron dynamics both at the vacuum-solenoid boundary and inside the magnetic field.Wave functions of an electron in free space and in the magnetic field belong to the same class of functions, which enables a smooth transition between single-mode states with the same quantum numbers.
The vector potential of the magnetic field was chosen in the symmetric gauge, which has led us to the Laguerre-Gaussian states.However, an alternative choice of the vector potential gauge would result in a different family of states, such as Hermite-Gaussian states.Which gauge to use is determined by the initial state of an electron in free space and, therefore, by the boundary conditions.
The decomposition of the NSLG states in a solenoid into the conventional basis of the Landau states was performed.A wave packet slightly mismatched with a Landau state at the boundary propagates through the magnetic lens as a superposition of a few Landau states with the same OAM and neighbouring radial quantum numbers.In other cases the electron further propagates in the field as a complex superposition of Landau states with the OAM of the initial state but significantly different radial quantum numbers.
We have considered a twisted electron entering the solenoid at a small angle α to the field direction.For any sensible values of the electron energy and momentum, the condition α ≪ 1 rad is sufficient to neglect any corrections to a single NSLG state in a solenoid.Thus, the OAM of the quantum packet is robust against little deviations from the axial symmetry and small inhomogeneties of the field, which supports our previous findings [29].
Our calculations show that transversely relativistic and longitudinally nonrelativistic beams of twisted particles can be achieved in existing experimental setups.For instance, electrons with quantum numbers of the order of 10 3 , generated as NSLG states with a waist size of 1 µm and focused afterwards to 1 nm, become transversely relativistic.Such particles can be a curious object of study in accelerator physics, as their dynamics significantly differs from that of regular accelerator beams.
Finally, we have introduced the quantum analogue of beam emittance for a quantum wave packet and applied it to the NSLG state.This quantity explicitly measures the non-classicality of the state via the Schrödinger uncertainty relation, which is more general than the wellknown Heisenberg inequality.The quantum emittance of an NSLG state grows linearly with n and l for large quantum numbers.In free space, for fundamental Gaussian mode (n = l = 0), the emittance vanishes, or, equivalently, the Schrödinger uncertainty relation turns into equality.This reflects the semiclassical character of the Gaussian state and the "quantumness" of the wave packets with large quantum numbers n and l.For an electron inside the field, the emittance generally oscillates in time, and for negative OAM, it can be locally lower than the emittance of a free NSLG state that enters the lens.

VIII. CONCLUSION
Let us give a final wrap up.The Landau states play a paramount role in problems with magnetic fields.They serve as a convenient basis when studying motion of the electrons in condensed matter or radiation in the field.However, once particles are allowed to transfer between vacuum and the magnetic field region, be it free space or a crystal, the NSLG states appear as a more advantageous means for describing particle states.The nonstationary nature of the processes under study is imprinted into the time dependence of the NSLG wave functions, and continuity with the free-space states comes naturally.We hope that the next time the reader analyzes an issue of the electron injection into magnetic field, they take a moment to consider which fighter to choose.
(D1) The coefficients are independent of time and can be evaluated at t = t 0 for simplicity, when σ(t 0 ) = σ 0 , R(t 0 ) = σ 0 /σ ′ 0 and Φ G (t 0 ) = Φ 0 .The integral can be evaluated using Eq.(7.422) in [48] (there is, however, a misprint m ↔ n) and presented in the following form: In the limit which provides the following asymptotic for the decomposition coefficients:

Figure 1 :
Figure 1: Transfer of a free Laguerre-Gaussian electron through a magnetic lens.z g and z 0 are the positions of the electron source and the boundary, respectively.