Nonspreading relativistic electron wavepacket in a strong laser field

A solution of the Dirac equation in a strong laser field presenting a nonspreading wave packet in the rest frame of the electron is derived. It consists of a generalization of the self-accelerating free electron wave packet [Kaminer et al. Nature Phys. 11, 261 (2015)] to the case with the background of a strong laser field. Built upon the notion of nonspreading for an extended relativistic wavepacket, the concept of Born rigidity for accelerated motion in relativity is the key ingredient of the solution. At its core, the solution comes from the connection between the self-accelerated free electron wave packet and the eigenstate of a Dirac electron in a constant and homogeneous gravitational field via the equivalence principle. The solution is an essential step towards the realization of the laser-driven relativistic collider [Meuren et al. PRL 114, 143201 (2015)], where the large spreading of a common Gaussian wave packet during the excursion in a strong laser field strongly limits the expectable yields.

Introduction.Recent advances in ultrastrong laser technology [1][2][3] provide bright prospects for laser-driven particle acceleration techniques.Especially successful are laser-driven plasma-based accelerators [4], which raised hopes to develop further the technique to compete with conventional electronpositron colliders [5,6], reducing the scale of the accelerating device.Even more dramatic scale change promises the idea of the laser-driven coherent microscopic collider [7][8][9][10], where the electron and positron generation, acceleration, and collision are realized within a single stage in a microscopic scale, providing high luminosity due to the coherently controlled electron-positron recollision.The bottleneck of this idea is the large spreading of a single electron wave packet in the rest frame of the electron during the excursion in the laser field within one laser period, which significantly restrains the luminosity of the collision.Thus, the covet is the overriding of the wavepacket spreading for the electron motion in the continuum.Nonspreading free electron wave packets via interference of different momentum components in the wavepacket, so-called particle Airy beams, are known for the Schrödinger equation [11,12], which generalize the similar idea for optical beams [13][14][15][16][17].However, Airy beams are not normalizable, i.e., span the whole space.Because of the infinite extension of such wavepackets in space, they are not applicable for a laser-driven collider, as the luminosity of the collider should be quenched.
In the nonspreading wavepacket, the distance between two points remains constant during the motion.While the latter has a well-defined meaning in nonrelativistic mechanics, in the relativistic case, surprises arise, particularly involving Bell's paradox [18].In this Gedankenexperiment two points connected by a thread move with a constant acceleration keeping a constant distance between them in the Lab-frame, however, the thread between the points is broken because of the contracted length of the thread in the Lab-frame [19].Then, how do the two points have to move to avoid breaking the thread connecting them?This question is resolved by the Born rigidity concept [20], defining the notion of a rigid body in a relativistic setting: The wordlines of the rigid body points have to be equidistant curves in spacetime.Or in more simple terms, the space distance between two infinitesimally close points measured simultaneously in the co-moving inertial frame (rest frame) should be constant.In particular this will be the case, and the thread will not break in Bell's paradox, if the points move with different constant accelerations along hyperbolic trajectories [21].In the Lab-frame the space distance between the infinitesimally close points will decrease, fitting to the Lorentz contraction, while the distance between them in the rest frame will remain constant.Note that for the luminosity of the laser-driven collider, namely the rest frame size of the electron and positron wave packets matters at the recollision.
Although seemingly unrelated, generating nonspreading wavepackets in relativistic quantum mechanics shares a common thread with the resolution of Bell's paradox through the concept of Born rigidity and hyperbolic motion.In both cases, the motion of different points of the objects is crucial, whether it is the motion of the points of the rigid body or the dynamics of interference fringes of the electron wavepacket along hyperbolic trajectories in the case of quantum mechanics.
In this Letter, inspired by the geometrical concept of Born rigidity, we use the Covariant Relativistic Dynamical Inversion (CRDI) technique [22] to demonstrate the existence of nonspreading wavepackets in a laser field fulfilling the Born rigidity requirements.These wavepackets in the local rest frame of the electron feature interference fringes with a constant distance between them due to the fringes' dynamics along the hyperbolic trajectories [23].Employing the CRDI technique, we develop a procedure to transform the wavepacket in the laser field to the local rest frame of the electron, where it evolves into a free electron wavepacket.To impose nonspreading property on the wavepacket fringes, we invoke the equivalence principle, which tells us that the hyperbolic trajectories, i.e., trajectories corresponding to a motion with constant acceleration are similar to those in a constant gravitational field.The latter allows us the construction of the nonspreading free electron wavepacket via mimicking locally the exact solution of the Dirac equation for the electron in a constant and homogeneous gravitational field [24].We have identified the finite lifetime of the nonspreading wavepacket because of the leaking from the Rindler space and proved that it is sufficient to allow recollision in a laser-driven collider.
Born's rigidity Our main aim is to create relativistic nonspreading wavepackets in a sense that the distance between the wavepacket's fringes remains constant with time in the electron's local rest frame.The Born rigidity concept tells us that this aim will be realized if the dynamics of fringes of the wavepacket manifests hyperbolic trajectories along the so-called Rindler coordinates [21] g sinh(gt) + z sinh(gt), with the coordinates (T, X, Y, Z) and (t, x, y, z) in the Lab-and comoving frames, respectively.When the system of the points, moving with a constant proper acceleration g is rigid [i.e., dx 2 + dy 2 + dz 2 = const in the co-moving rest frame measured in time coincidence dt = 0], in the Lab-frame they will represent a family of hyperbolic trajectories with a constant interval Thus, the family of the hyperbolic trajectories given by Rindler coordinates represents rigid dynamics.Quantum dynamics of an accelerating electron.As previously mentioned, the nonspreading wavepacket is closely related to the confined Dirac solution of Greiner [24] for the electron in a constant gravitational field and, due to the equivalence principle, can be deduced from it.In the chiral representation [25], the eigenspinor of the Greiner's solution for a spin-up electron reads: where K ν (x) is a Bessel function, N is a normalization constant, m is the electron mass, γ 5 = iγ 0 γ 1 γ 2 γ 3 , and Ω the eigenenergy.(η, u) are defined as the comoving coordinates of an inertial observer momentarily at rest with respect to the electron.Hence, using the Rindler coordinates we have η ≡ gt and u ≡ z + 1/g = (Z + 1/g) 2 − T 2 .The spinor (1) can be cast in the following form, see Eq. ( 13) in Sec.B: with the momentum parameterized by the rapidity b as p = m sinh b and w = tanh −1 (T/Z).The wavefunction of Eq. ( 1) is an eigenstate and is confined in the coordinate u.Note that only for gravitational fields can an accelerated electron be described as a superposition of plane waves.This is a direct consequence of the equivalence principle.In fact, only gravity induced acceleration can be transformed away by a coordinate transformation in the immediate vicinity of the particle.
The confined solution for the eigenstate ψ R to the free Dirac equation with respect to the accelerated frame (η, u) of Eq. ( 2) can be mimicked by a superposition ψ of the Dirac solutions for a free electron with respect to the Lab-frame (T, Z) (see Sec. B): where the free wavepacket ψ should have a momentum chirp via the phase φ(b) = −αb, with α = Ω, according to Eq. ( 2).When additionally we use the momentum distribution in the free wavepacket h(p) = e −aE p , with the constant a > 0 characterizing the momentum spread of the wavepacket, we get the following dispersionless free spinorial wavepacket (hereinafter, overbar stands for the correspondingly dimensionless parameters): While the wavepacket (4) is discussed already in Ref. [23], the emphasis here is its direct relation to the nonspreading concept.
Nonspreading wave packet in a strong laser field.Our aim is to construct a solution of the Dirac equation in a laser field in the form of a wavepacket and to show, using the CRDI technique, that it represents a nonspreading spinor in the local rest frame of the electron.We construct the desired wavepacket from the Volkov solutions ψ p (T, X) for an electron in a plane wave laser field eA µ = (0, ḟ1 (ξ), ḟ2 (ξ), 0): where with 2 ]dϕ, and the laser field phase ξ = ω(T − Z).For the superposition coefficients in Eq. ( 5) we use those which yield the free wavepacket ψ, see Eqs. ( 2

)-(3). After performing the change of variables
the closed expression for the integral in ( 7) is where ζ′ = i (ā K ±1/2+iα ( ζ′ ).Let us transform the spinorial wavepacket (8) to the rest frame, which is defined as the space-time dependent frame in which the spatial components of the electron's four-current vanish at the given space-time point, and demonstrate its nonspreading property.In the free-electron case, such Lorentz transformation is the matrix e −γ 0 γ 3 w/2 on the left of the spinor (2).An equivalent transformation is now needed for the case in which the electron is interacting with a plane wave field.In order to construct the desired Lorentz transformation, we make use of the CRDI technique.In the Hestenes formulation (see, for instance, section 3 of Ref. [27]), the spinor (6) can be written as As discussed in [22] the matrix e n /∧A / n µ pµ ≡ R is, in fact, a Lorentz transformation.In the chiral representation it is given by However, the Lorentz transformation ( 10) is valid only for the wavefunction (9) but not (8).Moreover, it does not account for the transformation to the Rindler (accelerated) frame.In order to encompass both transformations, we start with the following ansatz where d * and η ′ are free functions to be found by the requirements that the resulting spinor is of the same form as Eq. ( 2) and that the electron's current vanishes.These requirements are fulfilled by the following functions with superscript * standing for complex conjugation.Applying R to (5), R ψ L , leads to the following spinor describing the electron in its rest frame: The final step of the transformation consists of the following coordinate transformation One now needs to prove that the constructed wave function ψR obeys the Dirac equation in the plane wave field given by A /.In order to do so one must do the following: where ∇ µ = ∂/∂X µ + Ω µ , X µ = (T, X, Y, Z), and describes the electron in its rest frame.The matrix Ω µ is the spinor connection and is given by 2Ω µ = Ω i jµ σ i j , 2σ i j = γ i γ j and Ω i jµ = −e ν j e i σ e σ a ∂ µ e a ν .The wavepacket in the rest frame will be nonspreading when the spinor of Eq. ( 5) is represented in exactly the same form as the free spinor of Eq. ( 13).This is achieved by the consecutive application of the coordinate transformation Z ′ = Z − Φ, T ′ = T + Φ.With the change of vierbein e ′α µ = ∂X ′α ∂X µ , e ′µ α = ∂X µ ∂X ′α , and with the new gammamatrices γ ′α = e ′α µ γµ , γ ′ α = e ′µ α γµ , the transformed spinor (12) satisfies the Dirac equation in the new frame The variable ξ(ξ ′ ) is then given by inverting the coordinate transformation.
Thus, the constructed wavepacket of the electron in a laser field in the form of Eq. ( 5) [or Eq. ( 8)] coincides, up to a boost, see Eq. ( 12), with the free self-accelerating nonspreading wavepacket [cf.Eq. ( 4)] in the local rest frame of the electron at each (Z ′ , T ′ ).Note that the exact Lorentz transformation of the electron Dirac wave function in a laser field to the electron rest frame is essentially facilitated by application of the CRDI technique [22,28].
There is an important deviation of the nonspreading wave packet Eq. ( 12) from the accelerating electron solution Eq. ( 1).While in the latter a = 0, the former has a finite size of the wavepacket a 0 which has essential implications.To discuss this, consider the spinor (12) in the lab frame (hereinafter we drop the primes in order to simplify the notation) The impact of the wave packet size a is given by the prefactor in  (defined as Region I in Ref. [24]) representing the nonspreading wavepacket.The latter shows interference fringes, each lobe corresponding to a hyperbolic trajectory.Such a feature is entirely due to the chirping parameter α.From the rest frame perspective [Fig.1(c,d)] the wavepacket is nonspreading (the width of the wavepacket at each instant of the Rindler's time η remains constant), while in the Lab-frame the width of the wavepacket is contracting with time T according to the Lorentz-transformation.
There is a significant effect stemming from the value of the wavepacket size parameter a, cf.Panels (a,c) with (b,d) in Fig. 1.During evolution, the nonspreading part of the wavepacket is gradually leaking out into the normal one.The parameter a controls the balance between the nonspreading and normal parts of the wavepacket, see section C, and determines the lifetime of the nonspreading part of the wavepacket.Such leaking is responsible for washing out the interference fringes: the smaller the value of a, the slower is the interference fringes extinction [Fig.1].There is no extinction in the case of the accelerating electron solution of Eq. ( 1) with a = 0. We can estimate the lifetime of the nonspreading wave packet using asymptotic expressions of the wavefunctions at η ≫ 1, see section C: Both asymptotic expansions will coincide, if e η ā ∼ ū, or Z − T ≳ ā.Taking into account that the equation for the rightmost hyperbolic trajectory ( Z0 ≈ α) as a function of time is Z( T ) ≈ √ α 2 + T 2 , we have an estimate for the lifetime of the nonspreading wavepacket: which indicates that large α and small ā are beneficial for the extension of the lifetime.For instance, T l ≲ 1 fs when using α = 30 and ā = 0.001.
We also analyze the balance of the nonspreading and normal parts of the wavepacket introducing the asymmetry parameter via , calculating densities of these parts via Eqs.(2), (13), see section D for a definition.The example of the dependence of A on the parameters α and ā is shown in Fig. 2(a) for T = 1 fs.The wavepacket is nonspreading if |ψ| 2 ≈ | ψR | 2 |, i.e. at A → 0, while at A → 1, the nonspreading wave packet is fully extinguished |ψ| 2 → 0. A is very sensitive to ā.Smaller ā is preferred for A → 0, however, the larger α allows larger ā at a given A [Fig. 2(a)].
We numerically evaluated the wavepacket spatial size via the accelerated frame spinor, see Sec.D, i.e., considering only the parts outside the light-cone.The standard deviation δū = ⟨ū 2 ⟩ − ⟨ū⟩ 2 is calculated with ā = 10 −6 and α = 40 [Fig.2(b)].With the rightmost hyperbolic trajectory the trans-formation between the time in the accelerated frame and in the Lab-frame is then T = α sinh η.As seen in Fig. 2(b), the wavepacket spreading δū stays constant up to η ≈ 11 which corresponds to the Lab-time T ≈ 2 fs.
Considering that the fringes of the self-accelerating part of the wavepacket must last for at least one period of the laser field, let us estimate the maximum value for the parameter ā.For a full cycle of the laser field in the electron's rest frame, one has ω(T −Z) = 2π.The latter combined with the condition | Z − T | ≳ ā, we have a ≲ λ ′ , where λ ′ is the laser wavelength in the electron rest frame.
We consider applications of the nonspreading relativistic wavepackets to a laser-driven collider [10].Here electrons and positrons are created from vacuum by high-energy gammaphotons counterpropagating an ultrastrong laser field.They are accelerated by the laser field and collide within a cycle of the field.The rest frame of the created pairs depends on the γ-photon energy (Ω 0 ) and the laser strong field parameter (a 0 ≡ eE 0 /(mω), with the laser field amplitude E 0 ).We estimated F that at Ω 0 ∼ 1 GeV and a 0 = 10 2 (the laser intensity of 10 22 W/cm 2 ), the rest frame of the pair moves with γ ≈ 30.The laser period in this frame T ′ = T L /γ ∼ 3 × 10 −2 fs (with the laser period T L in the Lab-frame) which is less than the wavepacket leaking time T l ∼ 1 fs (for α = 30 and ā = 0.001), i.e., the recollision time is short enough to maintain the nonspreading character of the wavepacket.The next point is how to create the nonspreading wavepacket (see Sec. F).For the latter the specially tailored momentum chirping of the wavepacket given by the phase φ(p) is essential.This chirping induces a spatial shift of each momentum component in the laser field δx(p) = ∂φ(p)/∂p.The created wavepacket of the electron (positron) will be chirped if the particle with the corresponding momentum value is created with the corresponding spatial delay δx(p).The created particle momentum in the Lab-frame is determined either by the laser field intensity, or by the γ-photon energy.Tailoring specifically the laser intensity in space according to the function δx(p), one can achieve chirping of the created wavepackets of the electron and positron.Another possibility is to use a chirped γ-photon beam.The approach based on the Dirac equation is applicable here because radiation reaction is negligible, as shown in Sec.F, for the typical parameters of the laser-driven collider.
Conclusion.We have shown the existence of nonspreading relativistic wavepackets in a laser field, which in the local rest frame of the electron is similar to a self-accelerating nonspreading free wavepacket.We have established that there is a finite lifetime for the self-accelerating wavepacket and found that the wavepacket chirping and extension parameters impose strict restrictions on the lifetime duration.The nonspreading feature of the relativistic wave packet represents the essential property to permit an efficient laser-driven high-energy collider.
Moreover, the nonspreading free electron wavepacket is obtained via mimicking the confined Dirac eigenstate in a constant gravitational field.This idea based on the equivalence principle can be further developed, constructing different free electron wavepackets emulating electron bound states in a gravitational field of various configurations.This would provide a new avenue in laboratory astrophysics/cosmology: to investigate particle quantum dynamics in gravitational fields via the dynamics of their physical counterpart with specially engineered free electron wavepackets [29][30][31][32][33].
Hence, from A's perspective each part of the accelerated frame R undergo a Lorentz contraction.
Going back to the motion of the particle in the accelerated frame R from the point of view of A, consider the velocity v for the particle located permanently at position z, that is, it is at rest with respect to R. By definition, the proper time of the particle can be calculated from Eq. (A4) as given that dx/dt = dy/dt = dz/dt = 0.
Appendix B: Construction of the free Dirac spinor 1. Exact solution of the Dirac equation for an electron in a frame moving with constant four-acceleration.
Here we rewrite the spinor solution for an electron in a homogeneous and constant gravitational field as an integral in order to demonstrated its relation with the self-acceleration spinor.The concept described here holds for any spatial dimension.Let us first describe the case of a single spatial dimension as discussed in the previous section.In the Chiral representation, the Dirac spinor for a spin-up electron in a reference frame undergoing constant four-acceleration is which by using the connection formula H (1)  ν (ix) = 2K ν (x) πie iπν/2 and defining κ = mc/ℏ can be rewritten as where K ν (x) is a Bessel function, N a normalization constant, Ω > 0 the electron's kinetic energy, γ 5 = γ 5 = iγ 0 γ 1 γ 2 γ 3 and g the constant four-acceleration proper length.The (η, u) are defined as the comoving coordinates of an inertial observer momentarily at rest with respect to the electron.Hence, from Eqs. (A5) after shifting the origin we have η = gt/c, u = z + c 2 /g, Z = u cosh η and cT = u sinh η.Let us massage Eq. (B2) a bit more to gain intuition on how to build it from a wavepacket.First, note that along with cosh(t) = i sinh(t − iπ/2).Combining both identities, (B2) becomes The spinor (B4) is the desired result.

The Dirac equation in the accelerated frame
Here we show how the spinor discussed in the previous section is connected with the self-accelerated spinor in the lab frame by a Lorentz transformation.From the coordinate relations Z = u cosh η and cT = u sinh η we have Eq. ( B5) is exactly the Dirac equation in the lab frame.It can be transformed to the accelerated frame as follows which can be rewritten in the more compact form with ψ R = e −γ 0 γ 3 η 2 ψ, where ψ is the solution of the free Dirac equation in the lab frame while ψ R is the solution in the Rindler (a.k.a accelerated) reference frame.

Constructing the superposition for a free particle
Equipped with the spinor (B4) and the relationship ψ R = e −γ 0 γ 3 η 2 ψ, here we will build the self-accelerating wavepacket in the lab frame.In the Chiral representation, the Dirac spinor for a spin up electron in its rest frame with respect to a global inertial frame is where h(0) is some momentum dependent envelop function.Let us apply a boost to a frame moving along the Z-axis with momentum p with E p = m 2 c 2 + p 2 .Now we build a wavepacket by integrating over p the spinor(B8) in which cd p 2E p renders the integral Lorentz invariant.Let us now choose the following envelop function where N is a normalization constant and a > 0 is a constant with units of length.Upon making the variable substitution p = mc sinh(b) and including the phase factor e iαb with α being a arbitrary real number in (B9) one ends up with the desired superposition Due to the particular form of h(b), the b integration in (B11) can be performed exactly.In order to see this, first note that, for a > 0 Before continuing, note that ix = iκ (a + icT ) 2 + Z 2 and H (1)  ν (ix) = 2K ν (x) πi 1+ν .Hence, by performing the b integration one gets where Appendix C: Asymptotic expansions: wavepacket leaking Let us begin with the spinor in the accelerated frame, which is related to (B13) in the same way as the spinor (B4) is related to the one in the Lab.frame.We have which is independent of time., B = e −w(V 1 γ 0 γ 1 +V 2 γ 0 γ 2 +V 3 γ 0 γ 3 ) (E1) where ḟ1 (ξ) 2 + ḟ2 (ξ) 2 cos θ, Incidentally, the boost B leads to the following proper velocity . This is expected since the solutions to the classical and quantum equations of motion for an electron in a laser field are, up to the phase factor to the right of the matrix spinor (i.e., Ψ on the main text), the same.

Figure 1 .
Figure 1.Spacetime profile of the electron density for the free electron modulated wavepacket of Eq. (13): (a,b) in the Lab-frame; (c,d) in the accelerated frame (i.e., in Rindler coordinates); (a,c) α = 30 and ā = 0.005; (b,d) α = 30 and ā = 2.Note that the Rindler coordinates only cover the region outside the light-cone to the right.
B3) Since Z = u cosh η, cT = u sinh η, by defining the momentum p = mc sinh b for b real and making the change of coordinates t = η − b in (B3), we finally have

Appendix E : 3 2
Decomposition of R into boosts and rotationsIn the simplified case with a = 0 and α = 0 it is straightforward to see the following relationship R = e − η ′ γ 0 γ