Plasmon excitations with a semi-integer angular momentum

We provide an explicit model for a spin-1/2 quasi-particle, based on the superposition of plasmon excitations in a quantum plasmas with intrinsic orbital angular momentum. Such quasi-particle solutions can show remarkable similarities with single electrons moving in vacuum: they have spin-1/2, a finite rest mass, and a quantum dispersion. We also show that these quasi-particle solutions satisfy a criterium of energy minimum.


Plasmon Quasi-Particle
We consider high-frequency waves in a quantum plasma. Assuming the ions at rest, we describe the plasma behavior using the electron quantum fluid equations 13

B
They determine the evolution of the electron density n and mean velocity v, in the presence of an electrostatic potential V. This potential, and the Bohm potential (also sometimes referred as the quantum pressure) V B are determined by We assume small perturbations = −  n n n 0 around the equilibrium plasma density n 0 , and neglect the nonlinear terms. We also use for the pressure a simple equation of state satisfying ∇ = ∇  P mv n F 2 , where v F is the electron Fermi velocity. Assuming an uniform and infinite medium, we can easily get from the above equations a wave equation of the form where ω p is the electron plasma frequency, and the perturbed Bohm potential ∼ V B and quantum factor β 2 are determined by In order to solve this equation, we assume simple wave solutions of the form , describing propagation along the arbitrary Oz-direction, with frequency ω and wavenumber k. We also assume that the wave amplitude n k slowly varies along propagation (∂ ∂  n z kn / k k ), and it also slowly varies in the transverse direction ⊥ r , in such a way that the Helmholtz equation is satisfied Replacing this in eq. (3), we obtain the well known dispersion relation for electron plasma waves in a quantum plasma Following the usual concepts of plasma turbulence, we can say that an arbitrary spectrum of electron plasma waves is equivalent to an ensemble of quasi-particles, or plasmons. Defining the plasmon energy as ω = k k ε , and introducing the plasmon effective mass m f , we can obtain from the above dispersion relation the following relation between the energy and the plasmon momentum p = ħk, as where W is formally identical to the plasmon energy in a classical plasma (apart from the replacement of the thermal velocity by v F ), as defined by The plasmon effective mass m f and the plasmon effective relativistic factor γ f , are defined as m v F f F . We can also write the effective relativistic factor in a alternative way, as We can see from the above equations that the plasmon dynamics in a quantum plasma looks formally identical to that of a single massive particle (such as an electron) moving in vacuum. Here, the Fermi velocity v F plays the role of the speed of light in vacuum c, as an asymptotic velocity limit. In a classical plasma, v F would be replaced by the thermal velocity = S T m 3 / e , where T is the electron temperature. We can see that the quantum plasma corrections, associated with the term in β 2 , shown in eq. (7), introduce a small deviation from this interesting analogy with vacuum electrodynamics. Such formal analogies and, in particular, the existence of an effective plasmon mass m f , had already been considered before for a classical plasma, when v F is replaced by the thermal velocity and the β 2 corrections are absent 14 . We should however notice that coupling between the particle spin and the electromagnetic spin, usually present in quantum electrodynamics, is missing for plasmon quasi-particles.
The analogy between a plasmon quasi-particle and an electron in vacuum can be further explored by returning to the wave equation (3). It is known that, due to electron Landau damping, the plasmon frequencies are always of the order of ω p . This means that a solution of this equation can also be written in the form p where Φ ≡ Φ(r, t) contains a small deviation of the wave frequency with respect to the plasma frequency, such that we can take Replacing this in eq. (3), and rearranging terms, we can easily derive for Φ, the envelope equation Multiplying by ħ, and introducing the above definition of the plasmon effective mass, we can then rewrite this in the standard form of a Schrödinger equation, as We recognize on the r.h.s. of eq. (13) the usual kinetic energy operator, written in therms of the effective plasmon mass m f , and a quantum correction K Q , which results from the Bohm potential and represents the quantum plasma response. This clearly shows that a plasmon can be described as a quantum quasi-particle, satisfying a Schrödinger equation, written in the usual form. Such a description is valid even in the case of a classical plasma, where the additional term K Q is absent. The derivation of this Schrödinger equation is not surprising, given the fact that our initial wave equation (3) is formally identical to a Klein-Gordon equation. The small deviation of ω with respect to ω p can therefore be seen as a small kinetic energy with respect to the rest mass (energy) of the plasmon quasi-particle. This means that such quasi-particles are always in a kind of non-relativistic regime. This has nothing to do with the plasma regime itself, which was considered from the beginning as (quantum but) non-relativistic, as described by eq. (1).

Twisted Plasmons
We now examine the spatial structure of the plasmon modes, and discuss the possible solutions compatible with the assumed Helmholtz equation (5). In homogeneous and infinite medium, we can of course use plane wave solutions such that the density amplitude perturbation is the same in any point of the perpendicular planes, corresponding to n k = const. But here we focus on pulsed or beam wave solutions, finite in the transverse direction, such that ≡ ⊥ n n z r ( , ) k k is also assumed to vary slowly along the propagation direction Oz. In this case, we can derive from (5) a paraxial equation, of the form It is well known 15 that the general solution of this equation can be represented by a superposition of orthogonal Laguerre-Gauss (LG) modes, F lp (r), defined as where L p l are the associated Laguerre polynomials of argument X = r 2 /w 2 (z), where w(z) defines the (slowly varying) transverse size of the wave beam. The normalization factors are defined as This means that a plasmon beam solution, satisfying the dispersion relation (6) and propagating along the z-axis, can be generally described as where u lp are the LG mode amplitudes and ϕ(z, t) = (kz − ωt) is the phase function. Notice that such a beam solution is compatible with the above plasmon dispersion relation, which is usually associated with simple plane waves. Such a validity is guaranteed by the use of the Helmholtz equation (5), which establishes a link between the dispersion relation and the paraxial equation.
Next, we demonstrate that, by conveniently choosing the mode coefficients u lp , it is possible to built a plasmon solution with an intrinsic angular momentum s = ±1/2. But, in order to keep our discussion on more general grounds, we consider the case where the above expression will reduce to a solution of the form where μ is a real number. For convenience, we define μ inside the interval (−1, +1). Using the orthogonality condition (18), we can see that the mode coefficients in eq. (19) are u lp = 〈F lp |n k 〉. On the other hand, multiplying eq. (20) by F lp and integrating over (r, θ), we obtain 〈F lp |n k 〉 = 〈F lp |Re iμθ 〉. This allows us to choose In order to derive an explicit expression for these quantities, we rewrite this expression in an alternative and more explicit form, as We now focus on the particular case μ = s = ±1/2. Comparison with the general case will be considered later. Noting that, for s = ±1/2, we have sin(sπ) = 2s, we can use the above results to derive the identity We can see that the explicit value of the quantities u lp allowing us to built an electrostatic vortex with topological charge s will only depend on the transverse shape of the plasmon mode, R(r, z). The two plasmon states are illustrated in Fig. 1.
Let us now consider another approach, which allows us to define the transverse shape a posteriori. For that purpose, we return to eq. (19) and rewrite it in the form where the new radial profile function R lp (r, z), which is independent of the poloidal variable θ, is related to the LG mode functions by the obvious expression We now take the complex conjugate of eq. (27) and write the new identity Multiplying this unit factor to the r.h.s. of eq. (30), we obtain But, it should be noticed that this quantity is by definition independent of θ, which means that we need to assume that l = n. If we further consider the simplest possible radial profile, which is associated with the radial index p = 0, we are then reduced to In principle, the quantity μ, could take arbitrary values inside the interval (−1, 1). But, it can easily be shown that the energy of the twisted wave solutions has a minimum for μ = s. This allows us to understand the particular relevance of the spin-s = ±1/2 case: because it minimizes the energy. This is illustrated in Fig. 2. This energy minimum is absent in the integer case, when s is replaced by l and the plasmon wave packet reduces to a single LG mode.

Conclusions
In conclusion, we have shown that plasmons behave in a way that shows strong similarities with the behavior of a relativistic particle in vacuum. In particular, an effective mass and an effective relativistic γ-factor can be defined for such quasi-particles. The Fermi velocity (or the thermal velocity, in a classical plasma) replaces the velocity of light in vacuum c, as the maximum possible speed.
We have also shown that the spin-1/2 plasmons can be built by a superposition of twisted LG modes. Furthermore, the envelope equation for the plasmon field can be written in the standard form of a Schrödinger equation. This means that, in many respects, the spin-1/2 plasmon behavior shows very strong qualitative similarities with the electron behavior in vacuum. Quantum plasma contributions to the plasmon dispersion were also retained. They introduce small deviations to this analogy. Furthermore, it was shown that the case s = ±1/2 corresponds to an energy minimum among all the possible superpositions of the LG plasmon modes. This could give special relevance to the semi-integer spin case with respect to all other possible mode configurations. The interest of such analogies is that they point to the existence of a new type of plasmonic structure, never considered before in the literature. In recent years, we have learned that intense laser pulses could excite non-trivial forms of electrostatic waves, such as donut-shape wakefields 16 . In the present paper, we show that spin-1/2 plasmons could also be excited in a plasma. This could eventually lead to new experimental studies. The next step is to understand the nonlinear regime, and the resulting wave-particle processes.
Finally, we should not forget the fundamental differences between an elementary particle in vacuum (the electron) and a quasi-particle in a medium (the plasmon). In particular, in contrast with the electrons, there is no intrinsic magnetic moment associated with these plasmon states, at least in the linear approximation. The magnetic properties of plasmons, in the nonlinear regime, and the possible use of laser-plasma interactions to excite spin-1/2 plasmon structures, will be addressed in the future.