Optical-cavity mode squeezing by free electrons

2.1.1 The evolution operator

Following the methods introduced in Ref. [21], we start by considering a structure characterized by a single photonic mode of energy ℏω ₀ satisfying a bosonic statistics and interacting with an incoming electron initially prepared with a wave function ψ 0 ( r , t ) = e i ( E 0 t / ℏ − k 0 ⋅ r ) ϕ 0 ( r − v t ) . This expression assumes that ψ ₀(r, t) is made of momentum components that are closely peaked around a value ℏ k ₀ corresponding to a central energy E 0 = c m e 2 c 2 + ℏ 2 k 0 2 , such that the function ϕ ₀(r − v t) varies slowly in space and time. By further assuming the lifetime of the photonic mode to be much longer than the electron–specimen interaction time, and adopting the nonrecoil approximation, the dynamics of the combined electron–cavity system can be well described by the Schrödinger equation i ℏ ∂ t | ψ ( r , t ) 〉 = ( H ̂ 0 + H ̂ int ) | ψ ( r , t ) 〉 with free and interaction Hamiltonians given by (see Methods, Section 4.1.1)

where A ̂ ( r ) = ( − i c / ω 0 ) E ⃗ 0 ( r ) a − E ⃗ 0 * ( r ) a † is the quantized vector potential operator incorporating the normalized electric field mode profile E ⃗ 0 ( r ) , the sum over i runs over Cartesian components, and the vector g = (e ²/2m _e c ² γ)(1, 1, γ ⁻²) with γ = 1 / 1 − ( v / c ) 2 introduces approximate relativistic corrections in the ponderomotive interaction. We remark that the nonrecoil approximation that we adopt to describe the electron–cavity interaction is reasonable provided the condition ℓℏω ₀/E ₀ ≪ 1 is satisfied (e.g., for keV electrons and cavities resonating at visible or lower frequencies). More precisely, as shown in Ref. [30], the solution including recoil already converges to the nonrecoil limit for ℓℏω ₀/E ₀ ∼ 0.3. For example, it is safe to ignore recoil for interaction strengths leading to ℓ ∼ 25, using E ₀ ∼ 5 eV electrons that couple to a ℏω ₀ ∼ 60 meV cavity.

As explained in detail in Methods, Section 4.1.2, given the initial condition | ψ ( r , t → − ∞ ) 〉 = ψ 0 ( r , t ) ∑ n α n 0 e − i n ω 0 t | n 〉 (i.e., uncorrelated electron and cavity states), the problem admits an analytical solution that represents a generalization to the one found in Ref. [21], including the effect of the single-mode ponderomotive force [the A ² terms in Eq. (15b)]. More precisely, taking the electron velocity as v = v z ̂ , we find

where the phases θ n = − ∑ i ∫ − ∞ z d z ′ | η i ( z ′ ) | 2 ( 2 n + 1 ) incorporate a dependence on the mode field and electron velocity through

For the sake of simplicity, we hereinafter do not explicitly indicate the dependence on the electron position vector r. The expansion coefficients f ℓ n in Eq. (2) can be written as

in terms of the matrix elements

where S ̂ ( σ 0 ) = e σ 0 a a / 2 − σ 0 * a † a † / 2 and D ̂ ( β ̃ 0 ) = e β ̃ 0 * a † − β ̃ 0 a are squeezing and displacement operators, respectively. The central operator e − i λ ( a † a + a a † ) / 2 is an exponential of the number operator, and therefore, it does not create or destroy excitations, so it just introduces a phase −(n+1/2)λ when applied on a Fock state |n⟩. This phase is often negligible when θ ₀ ∼ 0 (e.g., for a phase-matched interaction; see Section 2.1.2). Incidentally, an overall phase χ ̃ emerges in Eq. (4) as a result of the retarded elastic image interaction between the electron and the cavity [31], but such phase is irrelevant for this study. In Eq. (5), we introduce the squeezing and displacement coefficients

as well as λ = arg{μ}. These quantities are directly obtained from the solution of the Ricatti differential equation [32] ∂ _z R/2 = k* − kR ² with R = ν/μ,

and k = i ∑ i η i 2 e 4 i θ 0 . Interestingly, by differentiating Eq. (7), we readily verify that μ and ν satisfy the unitary condition of a generalized Bogoliubov transformation, |μ|² − |ν|² = 1 [33]. Also, when the ponderomotive force is neglected, the displacement factor β ̃ 0 becomes identical with the coupling factor found in Ref. [21]: β 0 = ( e / ℏ ω 0 ) ∫ − ∞ z d z ′ E 0 , z ( z ′ ) e − i ω 0 z ′ / v . We remark again that, given any distribution of the vector field E ⃗ 0 ( r ) , the function k can be computed and used to integrate the Ricatti differential equation and obtain the profile R. Then, Eq. (7) together with Eq. (6) provide a complete characterization of the electron–cavity interaction.

2.1.2 Phase-matched interaction

Although the solution in Eq. (4) can be applied to any field profile, here we focus on a simple configuration that allows us to easily understand the problem, even though less symmetric scenarios could yield more efficient ponderomotive couplings. Specifically, we consider the electron to move along a path of length L while interacting with an electric field of the form E ⃗ 0 ( r ) = E ⃗ 0 ( R ) e i q z z , such that the phase velocity matches the electron velocity (i.e., q _z = ω ₀/v). In this scenario, which requires a specimen that possesses translational invariance along the e-beam direction, the ponderomotive coupling factor becomes independent of the longitudinal position z, so we have

[see Eq. (3) for η _i ] by disregarding the small phase θ ₀. Then, the Ricatti equation admits an analytical solution that directly leads to the following compact forms of the coupling coefficients (see Methods, Section 4.2):

where β 0 = ( e L / ℏ ω 0 ) E 0 , z ( R ) and λ = 0. Equation (8a) represents a generalization of the intrinsic PINEM coupling coefficient, now also incorporating the effect of the ponderomotive interaction. Interestingly, the presence of σ ₀ can play a significant role in the determination of the linear coupling and, in turn, in the computation of the related EELS probability given by | β ̃ 0 | 2 , especially for long interaction lengths L. A quantitative estimate of such correction can be easily obtained for the special case in which all coefficients η _i are real, such that | β ̃ 0 / β 0 | = 1 + 2 i | σ 0 | . In Figure 2(a), we show that the deviation reaches ∼ 9 % for |σ ₀| ∼ 0.2. Besides the modification of the linear coupling, σ ₀ in Eq. (8a) reflects the magnitude of the effect produced by the squeezing operator [see Eq. (5)] on the electron–cavity dynamics.

Figure 2:

Linear and quadratic coupling in phase-matched interactions. (a) Deviation of the linear coupling coefficient | β ̃ 0 | from the value obtained by neglecting the ponderomotive force |β ₀| as a function of |σ ₀| for a phase-matched interaction with real η _i [see Eqs. (3) and (8a)]. (b) Squeezing parameter |σ ₀| (left vertical axis) and squeezing factor 20|σ ₀| dB (right vertical axis) as a function of η = ω ₀ a/v for different values of v ² a under the electron–cavity configuration depicted in the inset, in which the e-beam moves with velocity v along the axis of a cylindrical hole of radius a and the cavity mode consists of a polariton made of a combination of azimuthal numbers m = ±1 and phase-matched axial wave vector q _z = ω ₀/v.

A key point in this discussion refers to the normalization of the mode field E ⃗ 0 , which is determined by the integral ∫ d 3 r ϵ ( r ) | E ⃗ 0 ( r ) | 2 = 2 π ℏ ω 0 for dielectric cavities made of materials with real permittivity ϵ(r) [34]. For instance, for an electric field of real amplitude E ₀ uniformly distributed in vacuum over an area A and oriented perpendicularly with respect to the e-beam velocity, the normalization condition leads to E 0 = 2 π ℏ ω 0 / L A , which in turn yields a squeezing factor |σ ₀| = e ² π/ω ₀ m _e vA. Unfortunately, we immediately notice that the ponderomotive coupling coefficient is not proportional to the path length L, in contrast to the linear PINEM term β ̃ 0 arising when the field has a component along the z direction – a mechanism that has been argued to enable strong electron–cavity coupling and, from here, electron-mediated entanglement between different cavities [35]. Such different scaling suggests the use of modes lacking translational invariance along the electron trajectory and having a strong confinement in the tranverse directions (e.g., nanoscale gaps between two metallic nanoparticles [36] or a nanoparticle and a planar metallic surface [37]) as an approach to reach high squeezing values.

Nonetheless, the dependence of σ ₀ on the inverse of the electron velocity, the inverse mode frequency, and the field confinement (i.e., ∝ 1/A) should allow us to reach sizeable ponderomotive couplings by going to slow electrons interacting with infrared fields. Indeed, for a mode of energy ℏω ₀ = 15 meV distributed over an area A = 100 nm², a 50 eV electron produces a squeezing parameter |σ ₀| ∼ 0.1.

As a potentially practical scenario, we consider the electron to be focused at the center of a circular hole (R = 0) of radius a drilled in a polaritonic material (see inset in Figure 2(b)) under the aforementioned phase-matching condition. In particular, for a mode consisting of a linear combination of two degenerate modes with azimuthal numbers m = ±1 and a relative phase ψ, the linear coupling vanishes (i.e., E 0 z = 0 ), while the ponderomotive coupling coefficient becomes σ ₀ = i e^iψ e ²/2m _e γv ² aηI(η) with η = ω ₀ a/v and I(η) standing for a normalization integral (see Methods, Section 4.2.1, for details). This dependence clearly favors small radii (i.e., strong field confinement), small electron velocities, and small resonance energies (see Figure 2(b)), in agreement with the scaling properties presented above.

2.2.1 Cavity prepared in the ground state

We now discuss the photonic state after interaction with the electron for the cavity mode starting in the ground state (i.e., α n 0 = δ n , 0 ) and the e-beam being well focused around a transverse position R ₀, so that we can approximate |ψ ₀(r, t)|² ≈ δ(R − R ₀)|ϕ ₀(z − vt)|². In addition, because we are interested in the state of the system after the electron has abandoned the interaction region, we let the longitudinal coordinate z go to infinity in the calculation of all coupling parameters.

By starting from Eq. (2), we first obtain the full density matrix ρ ̂ ( r , r ′ , t ) = | ψ ( r , t ) 〉 〈 ψ ( r ′ , t ) | , and then, by integrating over the electron coordinates, we find that the density matrix of the cavity mode reduces to ∫ d 3 r ρ ̂ ( r , r , t ) = ∑ n n ′ e i ( n ′ − n ) ω 0 t ρ n n ′ ph | n 〉 〈 n ′ | , with matrix elements ρ n n ′ ph = M ω 0 ( n ′ − n ) / v e i θ n − θ n ′ F − n n F − n ′ n ′ * that involve the coherence factor M ω / v = ∫ − ∞ ∞ d z | ϕ 0 ( z ) | 2 e i ω z / v . The latter, which only depends on the e-beam probability density and not on the phase of the wave function, determines the ability of the system to interfere with light synchronized with the electron pulse [38, 39]. We remark that this procedure renders the correct statistical properties of the cavity when no measurement is performed on the electron [40].

Interestingly, the coefficients F − n n turn out to be the projections of the well-known two-photon coherent state defined in a seminal paper by Yuen [29], from which we find

where H _n are Hermite polynomials and we have defined

as well as the normalization constant

Equation (9a) tells us that different postinteraction cavity states can be selected by properly tuning the factors β ̃ 0 and k. For instance, in a geometry for which E ⃗ 0 ⋅ v ∼ 0 , as in the case analyzed in the previous section, where an electron traverses a circular hole, the coupling coefficient β ̃ 0 vanishes and the state assumes the form of a squeezed vacuum containing only an even number of photons in the diagonal of the density matrix. This conclusion can be easily verified by evaluating the Hermite polynomials at zero argument or, alternatively, directly from Eq. (4). In contrast, the off-diagonal elements bear a more intricate dependence on the incoming electron wave function given by the coherence factor M ω 0 m / v , which determines the possibility of creating squeezed states with the correct coherences. For example, an electron that has undergone a PINEM interaction with an optical field of photon energy ℏω ₀ and coupling strength |β _PINEM|, followed by free propagation along a macroscopic distance d (see Figure 3(a)), has an associated probability density consisting of a train of pulses described by the coherence factor

where the phase of the PINEM interaction has been set to zero, z _T = 4πm _e v ³ γ ³/ℏω ₀ is the so-called Talbot distance [38, 41], and J _ℓ is the Bessel function of order ℓ. We show the resulting coherence factor in Figure 3(b). The corresponding state left in the cavity after interacting with the electron (top row of Figure 3(c)) comprises off-diagonal elements that become visible only when | M ω 0 m / v | takes values around its maximum along the propagation distance ( | M ω 0 m / v | ∼ 0.58 for m = 1). Despite this limitation, values of | M ω 0 m / v | ∼ 1 have been shown to be realizable by means of a sequence of several cycles of PINEM interaction and free-space propagation [42]. From the expression of σ ₀ related to such interaction, and by using the state in Eq. (9a) with full electron coherence ( M ω 0 m / v = 1 ) as well as by setting β ̃ 0 = 0 , λ = 0 and θ _n = 0, we compute the variance ⟨ Δ X ̂ 2 ⟩ = ⟨ X ̂ 2 ⟩ − ⟨ X ̂ ⟩ 2 with X ̂ = ( a + a † ) / 2 – a quantity of utmost importance, which represents the fluctuations in the position of the harmonic oscillator (HO) for the cavity mode. By choosing ψ = −π/2 to make σ ₀ real, we obtain ⟨ Δ X ̂ 2 ⟩ = e − 2 | σ 0 | / 4 . In Figure 2(b), we present the degree of squeezing in decibel (vertical right axis) through the relation − 10 ⁡ log 4 ⟨ Δ X ̂ 2 ⟩ = 20 | σ 0 | . Interestingly, the squeezing approaches values as high as ∼ 10 dB for a 5 eV electron, a ∼ 3 nm, and a resonance energy ℏω ₀ ∼ 60 meV; this squeezing value represents a threshold of particular importance to grant us access into fault-tolerant quantum algorithms by means of GKB codes [43]. High values of |σ ₀| can also be obtained for larger electron energies (e.g., ≳70 eV for a ≳ 1 nm) by targeting mid-infrared excitations (e.g., phonon polaritons in hexagonal boron nitride, which exhibit high quality factors even when structured with a lateral size of a few nanometers [44]).

Figure 3:

Postinteraction cavity-mode population. (a) Sketch of an electron wave packet undergoing a classical PINEM interaction of strength β _PINEM, and subsequently propagating in free-space for a distance d, where it develops a sequence of probability-density pulses and interacts with a cavity with coupling coefficients σ ₀ and β ̃ 0 . (b) Coherence factor | M m ω 0 / v | , determining the amount of coherence left in the two-photon coherent state [see Eq. (9a)] for the scenario depicted in panel (a) with β _PINEM = 4. We plot the result as a function of the normalized propagation distance d/z _T for several harmonics m. (c) Position variance ⟨ Δ X ̂ 2 ⟩ for a PINEM-compressed electron [solid curves; see panel (a)] and for a perfectly coherent electron (dashed curves). We consider different real values of σ ₀ with β ̃ = 0 , θ _n = 0, and λ = 0. (d) Elements ρ n n ′ ph of the density matrix associated with the cavity state after interaction with a compressed electron [see panel (a)]. The interaction region is taken to be placed at the propagation distances highlighted by the color-coordinated vertical dashed lines in panel (b) assuming pairs of coupling parameters | β ̃ 0 | = 0.1 and |σ ₀| = 1 (top row); or | β ̃ 0 | = 2 , |σ ₀| = 0.1 (bottom row). All plots in panel (d) are calculated for ϕ = 0.

We also consider the effect of having a partially coherent electron. In particular, by compressing the electron through free-space propagation following a single PINEM interaction (setting again β ̃ 0 = 0 , λ = 0, and θ _n = 0), we directly compute ⟨ Δ X ̂ 2 ⟩ by using the matrix elements in Eq. (9a) and find the results presented in Figure 3(c), which clearly show an increase in the position uncertainty with respect to the values obtained for perfectly coherent electrons (e.g., when treated as point particles). In addition, it is interesting to note that ⟨ Δ X ̂ 2 ⟩ increases with |σ ₀| as a consequence of the vanishing coherence factor M ω 0 m / v at the harmonics connected to the populated Fock states |n⟩.

In the opposite extreme (k → 0), from Eqs. (6a) and (7), we obtain |σ ₀| → 0, which leads to a Poissonian distribution in P n = ρ n n ph by applying the asymptotic form of H _n (ζ) or, again, by simply taking the same limit in Eq. (4). This behavior is particularly evident in the bottom row of Figure 3(c), where we plot the mode population obtained upon direct evaluation of Eq. (9a) for different propagation distances from the first PINEM interaction.

For parameters β ̃ 0 and σ ₀ lying in between the two extremes under consideration, the cavity state varies in a continuous fashion, spanning values of the zero-delay second-order correlation function g ( 2 ) ( 0 ) = ∑ n n ( n − 1 ) P n ∑ n n P n 2 ranging from a super-Poissonian statistics with g ⁽²⁾(0) ∼ 3 + sinh⁻²(|σ ₀|) to a Poissonian statistics with g ⁽²⁾(0) ∼ 1 when the phase shift is ϕ = π/2 [see Eq. (9d)], as well as to a sub-Poissonian statistics for ϕ = 0. This is a consequence of the decreasing variance of the distribution of photon numbers, Δ n 2 = 2 sinh 2 ( | σ 0 | ) cosh 2 ( | σ 0 | ) + | β ̃ 0 | 2 [ c o s h ⁡ ( 4 | σ 0 | ) − cos ( 2 ϕ ) ⁡ s i n h ⁡ ( 4 | σ 0 | ) ] . We remark that the exponential dependence of the variance on the squeezing parameter renders the postinteraction cavity state extremely sensitive to any change in the phase ϕ, even for small |σ ₀|, provided the interaction is assisted by a high coupling with the electric field component along the e-beam direction.

2.2.2 Cavity under laser illumination

Although the optical cavity mode may in principle be chosen such that it constrains the phase ϕ [Eq. (9d)], in practical configurations this parameter would be hard to change in a single experimental setup. A more feasible route to gain control over the phase ϕ consists in aligning the e-beam such that E 0 , z = 0 while the cavity is irradiated by a laser. Intuitively, the postinteraction cavity state should take a form that resemblances the one in Eq. (9a), but with β ̃ 0 changed to a factor α _L proportional to the incident laser amplitude and carrying its phase. More precisely, the probability to leave n photons in the cavity is

In contrast to the scenario with no external pumping, the probabilities P _n in Eq. (10) are strongly dependent on the electron coherence factor. Specifically, Eq. (10) reduces to Eq. (9a) only if the electron bears full coherence with respect to the laser [i.e., for M ω 0 ( ℓ ′ − ℓ ) / v = 1 ].

4.1.1 The effective Hamiltonian

We start by considering a fast electron interacting with a classical electromagnetic field described in the temporal gauge (i.e., with zero scalar potential) in the presence of a material structure. In our analysis, the initial electron wave function is assumed to be in a superposition of momenta concentrated around a value ℏ k ₀, which corresponds to a relativistic energy E ₀ and velocity v = ℏ k ₀ c ²/E ₀. Under these conditions, and for optical excitations in the infrared-visible range, the system dynamics can be modeled by means of the effective Schrödinger equation i ℏ ∂ t ψ ( r , t ) = H ( t ) ψ ( r , t ) with the minimal-coupling Hamiltonian [51]

where A(r, t) is the time-dependent vector potential associated with the external illumination in the presence of the cavity. The coefficient vector g = (e ²/2m _e c ² γ)(1, 1, γ ⁻²) in the ponderomotive term approximately incorporates relativistic corrections through the Lorentz factor γ = 1 / 1 − v 2 / c 2 . The electron motion is then described by the scalar wave function ψ(r, t).

In this work, we are interested in a quantum description of the cavity and, thus, adopt a quantum-optics approach [34] with the vector potential treated as an operator: A ̂ ( r ) = ∑ j ( − i c / ω j ) E ⃗ j ( r ) a j − E ⃗ j * ( r ) a j † , where j runs over cavity modes, E ⃗ j ( r ) are the corresponding normalized electric field distributions, and we introduce bosonic annihilation and creation operators a _j and a j † . By further assuming the sample to undergo negligible inelastic losses, and focusing on the interaction with a spectrally dominant and well-defined mode (labeled by j = 0, and for simplicity we dismiss the j label in the ladder operators), the electron–cavity system can be described by the modified Schrödinger equation i ℏ ∂ t | ψ ( r , t ) 〉 = ( H ̂ 0 + H ̂ int ) | ψ ( r , t ) 〉 with

where we have added the noninteracting Hamiltonian of the cavity mode ℏω ₀ a ^† a and included the photonic degrees of freedom in the combined state | ψ ( r , t ) 〉 = ∑ n = 0 ∞ ψ n ( r , t ) e − i ω 0 n t | n 〉 .

4.1.2 Exact analytical solution

The interaction Hamiltonian in Eq. (15b) presents a quadratic term in the mode ladder operators; therefore, generating a dynamics that substantially differs from the one addressed in previous works [16, 35] where only the linear coupling with the electromagnetic field was taken into account. Despite this difference, by following similar steps as those used in Ref. [21], we find an analytical solution for the combined wave function.

We start by inserting the ansatz ψ n ( r , t ) = ψ 0 ( r , t ) ∑ ℓ = − ∞ ∞ f ℓ n ( r ) e i θ n e i ω 0 ℓ ( z / v − t ) into the Schrödinger equation, where ψ 0 ( r , t ) = e − i E 0 t / ℏ + i k o ⋅ r ϕ 0 ( r − v t ) is the initial electron wave function, θ n = − ∑ i ∫ − ∞ z d z ′ | η i | 2 ( 2 n + 1 ) , η i = ℏ / 2 m e v γ μ i , and μ i = ( e / ℏ ω 0 ) E 0 , i e − i ω 0 z / v . This leads to the set of one-dimensional differential equations

with p = μ z e 2 i θ 0 and k = i ∑ i η i 2 e 4 i θ 0 .

Importantly, Eq. (16) conserves the total number of excitations, and thus, the set of coefficients defined by ℓ + n = s for each choice of an integer s evolves separately. This property allows us to map our problem onto a quantum harmonic oscillator under nonlinear coupling. Indeed, given the Hamiltonian H ̂ HO = H ̂ HO 0 + g 1 ( t ) a + g 1 * ( t ) a † + g 2 ( t ) a 2 + g 2 * ( t ) a † 2 , with H ̂ HO 0 = ℏ ω 0 a † a , and plugging the general state | ψ ( t ) 〉 = ∑ n c n ( t ) e − i n ω 0 t | n 〉 into the associated Schrödinger equation, we obtain

which connects to Eq. (16) by means of the transformation c n → f s − n n , t → z, i g 1 e − i ω 0 t / ℏ → p , and − i g 2 e − 2 i ω 0 t / ℏ → k . The coefficients c _n (t) can also be expressed in terms of the scattering operator S ̂ ( t , − ∞ ) as c n ( t ) = ∑ m = 0 ∞ ⟨ n | S ̂ ( t , − ∞ ) | m ⟩ c m ( − ∞ ) , assuming an initial state of the harmonic oscillator written here in the interaction picture [52] as |ψ(−∞)⟩_I = ∑ _m c _m (−∞)|m⟩. Consequently, we proceed by evaluating the matrix elements of the scattering operator.

To find the scattering operator, we first notice that an analytical solution to the equation i ℏ ∂ t U ̂ ( t , t 0 ) = H ̂ HO U ̂ ( t , t 0 ) was found by Yuen [29] and is given by [33]

where we have defined the operators S ̂ ( z ) = e z a a / 2 − z * a † a † / 2 and D ̂ ( z ) = e z * a † − z a , the linear coupling β ̃ 0 = ( i / ℏ ) ∫ t 0 t d t ′ ( g 1 ( t ′ ) e i ω 0 t ′ − g 1 * ( t ′ ) e i ω 0 t ′ ν * ) , and the quadratic coupling σ ₀ = arcsinh(|ν|)e^{i(arg{μ}−arg{ν})}. The coefficients μ = e^iϕ and ν = ( 2 i / ℏ ) ∫ t 0 t d t ′ g 2 * ( t ′ ) e 2 i ω 0 t ′ μ ( t ′ ) , with ϕ = ( − 2 / ℏ ) ∫ t 0 t d t ′ g 2 ( t ′ ) e − 2 i ω 0 t ′ R ( t ′ ) , satisfy the relation |μ|² − |ν|² = 1 at every time, while the function R = ν/μ obeys the Ricatti equation [32] ∂ _t R/2 = k* − k R ² solved with the initial condition R(t ₀) = 0. We have also defined the phase λ = arg{μ} and introduced the phase χ ̃ = ( i / 2 ) ∫ t 0 t d t ′ β ̃ 0 * ∂ t ′ β ̃ 0 − β ̃ 0 ∂ t ′ β ̃ 0 * , which is a generalization of the Berry phase found in a forced quantum harmonic oscillator [31, 53].

At this point, we invoke the relation S ̂ ( t , t 0 ) = e i H ̂ HO 0 t / ℏ U ̂ ( t , t 0 ) e − i H ̂ HO 0 t 0 / ℏ , as well as e i H ̂ HO 0 t 0 / ℏ D ̂ β ̃ 0 e i ω 0 t 0 e − i H ̂ HO 0 t 0 / ℏ = D ̂ β ̃ 0 and e i H ̂ HO 0 t / ℏ S ̂ σ 0 e 2 i ω 0 t e − i H ̂ HO 0 t / ℏ = S ̂ σ 0 , which can be easily verified by using ordering theorems for the displacement operator D ̂ β ̃ 0 = e − | β ̃ 0 | 2 / 2 e β ̃ 0 * a † e − β ̃ 0 a and the squeezing operator S ̂ ( σ 0 ) = exp − tanh ( | σ 0 | ) e − i arg { σ 0 } a † a † / 2 exp − log ⁡ cosh ⁡ ( | σ 0 | ) ( a † a + a a † ) / 2 exp tanh ⁡ ( | σ 0 | ) e i arg { σ 0 } ⁡ a a / 2 , respectively [54]. By employing these expressions, we find

Finally, once the aforementioned transformation is applied to Eq. (18) and the t ₀ → −∞ limit is taken, the coefficients f ℓ n in Eq. (4) of the main text are found by imposing the boundary condition ψ n ( z → − ∞ , t ) = ψ 0 ( r , t ) ∑ n = 0 α n 0 e − i n ω 0 t | n 〉 , which is equivalent to f ℓ n ( z → − ∞ ) = δ ℓ , 0 α n 0 .

4.2.1 Hole in a metallic slab

In order to estimate the squeezing coupling parameter σ ₀ in a scenario of practical interest, we consider a circular hole of radius a and length L extending along the z direction and drilled in a homogeneous metallic slab of permittivity ϵ(ω), with the electron traveling in vacuum parallel to the hole axis. For ω ₀ a/c ≪ 1, we can neglect retardation in the description of the mode, which for an azimuthal number m, has an associated electric potential inside the hole (R < a) of the form ϕ 0 ( r ) = A I m ( q z R ) e i q z z + i m φ , and ϕ 0 ( r ) = B K m ( q z R ) e i q z z + i m φ in the material region (R > a). These expressions are given in cylindrical coordinates r = (R, φ, z) and involve the modified Bessel functions K _m and I _m . Also, the constants A and B must satisfy the condition B / A = I m ( q z a ) / K m ( q z a ) = ϵ ( ω ) I m ′ ( q z a ) / K m ′ ( q z a ) to guarantee the continuity of the potential and the normal electric displacement at the surface of the hole. We then calculate the corresponding electric field E ⃗ 0 ( r ) = − ∇ ϕ 0 ( r ) and normalize the field as indicated in Section 2.1.2.

As an example, we take the electron to be focused at the hole center (R = 0, approaching with an azimuthal angle φ). Since excitations of symmetries ±m are degenerate, we can choose the mode as any linear combination of these two. In particular, we take m = ±1 waves combined with a relative phase ψ, such that the mode field becomes E ⃗ 0 m = ± 1 ( R = 0 , z ) = − E 0 / 2 3 / 2 R ̂ ( 1 + e i ( ψ − 2 φ ) ) + i φ ̂ ( 1 − e i ( ψ − 2 φ ) ) e i q z z + i φ , with E ₀ = q _z A acting as a normalization constant. We estimate E ₀ from the normalization condition ∫ d 2 R | E ⃗ 0 ( R ) | 2 = 2 π ℏ ω 0 / L , neglecting the field inside the metal, which should be small because of the condition |ϵ(ω ₀)| ≫ 1 required to have strong confinement. We obtain E 0 2 = ℏ ω 0 / L a 2 I ( η ) , where η = ω ₀ a/v and I ( η ) = ∫ 0 1 x d x [ I 0 ( x η ) + I 2 ( x η ) ] 2 / 4 + I 1 2 ( x η ) ( 1 / x 2 η 2 + 1 ) . Plugging the field into the expressions given above for k and σ ₀, we obtain the result shown at the end of Section 2.1.2.