The Dynamics of Compact Laser Pulses

We discuss the use of a class of exact finite energy solutions to the vacuum source-free Maxwell equations as models for multi- and single cycle laser pulses in classical interaction with relativistic charged point particles. These compact solutions are classified in terms of their chiral content and their influence on particular charge configurations in space. The results of such classical interactions motivate a phenomenological quantum description of a propagating laser pulse in a medium in terms of an effective quantum Hamiltonian.

mation science. Three-level bosonic quantum systems composed of two photons in the same spatial and temporal configuration have also been contemplated [11] in attempts to construct more efficient quantum gates for quantum communication. If one regards a free propagating classical single-cycle (therefore non-time-harmonic) laser pulse in vacuo as a spatially compact classical electromagnetic configuration with definite energy J and temporal width T 0 one expects that when J T 0 ∼ its dynamical evolution should be controlled by a quantum Hamiltonian rather than a classical one [12]. Such quantized collective states could then be entangled with other quantized pulses or free photon states and their interaction with classical or quantized states of electrically neutral continua (e.g. optically inhomogeneous and anisotropic dielectrics or plasmas) or charged matter (e.g. trapped ions [11]) may be worthy of investigation for technological applications.
In this Letter we first discuss a viable methodology for parameterizing a particular class of propagating solutions to the source free classical Maxwell equations in vacuo that offers an efficient means to explore the classical effects of compact laser pulses on free electrons in dynamical regimes where quantum effects are absent. The parameterization is based on a remarkable class of explicit solutions of the scalar wave equation found by Ziolkowski [13,14,15,16,17] following pioneering work by Brittingham [18]. Such solutions can be used to construct classical Maxwell solutions with bounded total electromagnetic energy and fields bounded in all three spatial directions. With simple analytic structures their diffractive properties can be readily determined together with the behaviour of charged particle-pulse interactions over a broad parameter range without recourse to expensive numerical computation.
Based on a classical analysis of this particular class of laser pulse configurations possessing chiral states that can be distinguished experimentally by such interactions, an effective phenomenological quantum Hamiltonian is proposed that can be used to evolve general non-stationary pulse states from an initially prepared collective quantum state of the electromagnetic field. Such collective photon states are constructed in terms of a pair of three intrinsic quantized angular momentum components rather than the two polarisation states associated with single photons.
If the complex scalar field α satisfies α = 0 and Π µν is any covariantly constant (degree 2) anti-symmetric tensor field on spacetime (i.e. Π µν;δ = 0) for all µ, ν, δ = 0, 1, 2, 3, then the complex tensor field F βδ = ∂ β A δ − ∂ δ A β satisfies the source free Maxwell equations in vacuo with: where A is an arbitrary constant, |g| is the determinant of the spacetime metric and γµβ δ denotes the Levi-Civita alternating symbol. In the following g refers to the Minkowski metric tensor field, in which case the components Π µν can be used to encode three independent Hertz vector fields and their duals 1 .
General solutions to α = 0 can be constructed by Fourier analysis. In cylindrical polar Minkowski coordinates {t, r, z, θ}, axially symmetric solutions propagating along a z−axis have, for z ≥ 0, the double integral representation: in terms of the zero order Bessel function and the speed of light in vacuo c. Conditions on the Fourier amplitudes f ω (k) can be given so that the Hertz procedure above gives rise to real singularity free electromagnetic fields with finite total electromagnetic energy. A particularly simple class of pulses that can be generated in this way follows from the complex axi-symmetric scalar solution: where 0 , ψ 1 , ψ 2 are arbitrary real positive definite parameters with physical dimensions of length. The relative sizes of ψ 1 and ψ 2 determine both the direction of propagation along the z−axis of the dominant maximum of the pulse profile and the number of spatial cycles in its peak magnitude. When ψ 1 ψ 2 , the dominant maximum propagates along the z−axis to the right. The parameter 0 determines the magnitude of such a maximum. The structure of such solutions has been extensively studied in [19,20] in conjunction with particular choices of Π µν together with generalizations discussed in [21,22].
In general the six anti-symmetric tensors with components δ µ [γ δ ν σ] in a Minkowski Cartesian coordinate system are covariantly constant and can be used to construct a complex eigen-basis of antisymmetric chiral tensors Π s, κ , with s ∈ {CE, CM} and κ ∈ {−1, 0, 1}, satisfying where the operator O z represents θ rotations about the z−axis generated by −i∂ θ on tensors 2 . These in turn can be used to construct a complex basis of chiral eigen-Maxwell tensor fields F s, κ . The index s indicates that the CE (CM) chiral family contain electric (magnetic) fields that are orthogonal to the z−axis when κ = 0. The chiral eigen-fields F s, 0 inherit the axial symmetry of α(t, r, z) while those with κ = ±1 do not. The directions of electric and magnetic fields in any of these Maxwell solutions depend on their location in the pulse and the concept of a pulse polarisation is not strictly applicable. The chiral content as defined here can be used in its place. Non-chiral pulse configurations can be constructed by superposition s κ F s, κ C s, κ with arbitrary complex coefficients C s, κ . The energy, linear and angular momentum of the pulse in vacuo can be calculated from the components T µν of the Maxwell stress-energy tensor . If e and b denote time-dependent real electric and magnetic 3vector fields associated with any pulse solution, its total electromagnetic energy J , for a fixed set of parameters and any z, is calculated from where S can be any plane with constant z = z 0 > 0. For spatially compact pulse fields in vacuo this coincides with the total pulse electromagnetic energy To correlate J with other laser pulse properties and the choice of parameters, we bring the pulse into classical interaction with one or more charged point particles. The world-line of a single particle, parameterized in arbitrary coordinate as x µ = ξ µ (τ ) with a parameter τ , is taken as a solution of the coupled non-linear differential equations in terms of the particle charge q and rest mass m 0 , for some initial conditions ξ(0), V (0), where the particle 4-velocity satisfies V ν V ν = −1 and its 4-acceleration is expressed in terms of the Christoffel symbols Γ δ β µ as ). In the following, radiation reaction and inter-particle forces are assumed negligible. From the solution ξ(τ ) one can determine the increase (or decrease) in the relativistic kinetic energy transferred from the electromagnetic pulse to any particle and the nature of its trajectory in the laboratory frame. This information can then be used to correlate the dynamical properties of the interaction with the laser pulse properties fixed by the parameters. To facilitate this exercise, it proves important to reduce the above equations of motion to dimensionless form and fix the physical dimensions of the fields involved. The Minkowski metric tensor field g = g µν dx µ dx ν (with g µν = diag(−1, 1, 1, 1)) in inertial coordinates Then with the dimensionless complex scalar field α(T, R, Z) = α(t, r, z) and greek indices ranging over {T, R, Z, θ} with T,R,Z,θ = 1 , we write for a choice of dimensionless covariantly constant tensor Π µβ so that The parameter Λ controls the strength of all electric and magnetic fields in F βδ for fixed values of the parameters Ψ 1 , Ψ 2 , Φ, Ξ and the overall scale 0 will be fixed in terms of the total electromagnetic energy of the pulse. For a choice of such parameters the real fields e and b enable one to calculate a numerical value Γ such that J = 0 Γ. The diffraction of the pulse peak along the z−axis can be used to define a pulse range relative to the maximum of the pulse peak at z = 0. To this end, the density E(T, Z) defines the dimensionless range Z rg by E(0, 0)/ E(T 1 , Z rg ) = 2, where the peak at Z = Z rg > 0 and T = T 1 > 0 is half the height of the peak at Z = 0, T = 0. If during the interval [0, T 1 ] the pulse propagates with negligible deformation in Z, one may estimate its width Z w at half height and the dimensionless pulse axial speed β = Z rg /T 1 . This yields the dimensionless pulse duration or temporal width T 0 = Z w /β. From these dimensionless values one deduces the pulse MKS characteristics in terms of 0 and hence J . If the picosecond is used as a unit of time, the pulse duration becomes t 0 = 0 T 0 /c = 0 Z w /(βc) = N 10 −12 sec for some value N and hence 0 = (cβN/Z w ) 10 −12 metres, J = (ΓβcN/Z w ) 10 −12 Joules, z rg = 0 ΞZ rg = (ΞβcN Z rg /Z w ) 10 −12 metres and z w = Ξcβn 10 −12 metres. A dimensionless spot-size of the pulse at Z = Z 0 > 0, T = Z 0 /β is then determined by the behaviour of P (R, Z 0 /β, Z 0 , θ). At each value of Z 0 this function of R and θ has a clearly defined principal maximum. If one associates a circle of dimensionless radius R s (Z 0 ) with such a maximum locus it can be used to define a spot-size with radius r s (z 0 ) = 0 ΦR s (Z 0 ) = (cβ N ΦR s (Z 0 )/Z w ) 10 −12 metres at z = z 0 . Figure 1 displays a clearly pronounced principle maximum in the power density profile P as a function of X = R cos(θ) and Y = R sin(θ) at Z = 0, T = 0 for a specific choice of the parameters (Λ, Ψ 1 , Ψ 2 , Φ, Ξ) . The same parameter set is used to numerically solve (6) for a collection of trajectories for charged particles, each arranged initially around the circumference of a circle in a plane orthogonal to the propagation axis of incident CM type laser pulses with different chirality. The resulting space curves in 3-dimensions, displayed in figure 2, clearly exhibit the different responses to CM pulses with distinct chirality values. The instantaneous specific relativistic kinetic energy of a particle with laboratory speed v is γ − 1 in terms of the Lorentz factor γ given by γ −1 = 1 − v 2 c 2 . In figure 3, this quantity is displayed on the left for a charged particle accelerated by a fixed chirality (CM,−1) type pulse where the pulse energy is varied by changing Λ. On the right the energy transfer dependence on pulse chirality for both CE and CM type pulses with fixed laser energy is displayed. We deduce that the momentum and angular momentum [23] in the propagation direction can transfer an impulsive force and torque respectively to charges lying in an orthogonal plane. More generally, the classical configurations of a high energy pulse labelled CE and CM can be distinguished by their interaction with different arrangements of charged matter. Furthermore, by a suitable choice of parameters, (CE, κ) type modes can be constructed that yield the same physical properties (J , z rg , z w , β) for all κ. Similarly the (CM,κ) type modes yield a κ independent set with physical properties distinct from those determined by the (CE,κ) modes. The pulse group speed magnitudes (as defined above) of all these configurations are determined numerically and are bounded above by the value c. To illustrate some of these statements, table 1 summarizes the MKS laser pulse characteristics for a specific choice of {Ψ 1 , Ψ 2 , Φ, Ξ} and various values of Λ.
These observations suggest that, for narrow electromagnetic micro-pulses with J T 0 , a parameterized effective quantum Hamiltonian H describing a particular pair of noninteracting massive point particles may provide an effective description of general nonstationary quantum states of free quantized laser pulses in vacuo. Such states are then defined as elements of a complex Hilbert space where I denotes the identity operator and, for real parameters µ n > 0 with n = 1, 2 For quantum pulses that are deemed relativistic one can include relativistic corrections by replacing the first factor on the right hand side of (9) with {P 2 n c 2 + µ 2 n c 4 } 1/2 − µ n c 2 , (P n = ( /i)∇ xn ), and work in a momentum representation.
The six classical chiral states labelled (s, κ) that evolve according to the classical vacuum Maxwell equations are now replaced by quantized (bi-qutrit [11]) elements Ψ t ∈ H satisfying the Schrödinger equation: with Ψ t 0 prescribed at any time t 0 and satisfying Ψ t 0 , Ψ t 0 = 1. The vacuum Hamiltonian H may offer an alternative method of encoding information into the pair of triplet angular momentum states of a quantum pulse Ψ t 0 rather than in the two polarization states of an elementary photon [24]. The evolution of a non-stationary quantum pulse in an optically inhomogeneous anisotropic medium could be modelled by adding to H n an interaction Hamiltonian I ⊗ V n (x n ) in terms of a Hermitian operator V n (x n ) with a 3 × 3 matrix representation on R 3 . An extension describing quantized multi-bi-qutrit states follows by constructing them as elements of a bosonic Fock space based on H in the standard manner [25]. A discussion of how bi-qutrit information could be addressed and controlled by bringing quantized electromagnetic pulses into interaction with classical and quantised matter systems that arrest their unitary evolution will be discussed elsewhere.