Quasiparticles in an Interacting System of Charge and Monochromatic Field

Quasiparticles in an interacting system of charge and monochromatic ﬁeld

P. Mati∗ ELI-ALPS, ELI-Hu NKft, Dugonics tér13, Szeged 6720, Hungary (Dated: May 16, 2017) The spectrum of an exactly solvable non-relativistic system of a charged particle interacting with a quantized electromagnetic mode is studied with various polarizations. Quasiparticle dispersion relations can be derived from the diagonalized Hamiltonian which, in the case of a linearly polarized field, indicates a bulk plasmon excitation, whereas in the elliptically and circularly polarized cases the dispersions exhibit a global minimum and show a singular behavior as the wavenumber tends to zero. These new type of dispersion relations lead to modified plasma frequencies and reflectivities, as well as to negative group velocities. It is shown that the zero-point energy of the system implies a repulsive force between two parallel plates, which vanishes when the charge is set to zero.

I. INTRODUCTION II. THE MODEL

Achieving exact solutions of a problem related to an The Hamiltonian of the simple interacting system of interacting physical system is rather rare (e.g., [1–7]). the charged particle with the quantized mode has the These systems are usually simplifications of more com- following form plicated situations and are analyzed in order to capture the entire physics and mathematics of the problem under 1 e 2 1 H = p − A + ω + a†a , (1) consideration. The physical relevance of the simplified 2m c ~ 2 models are limited; however, sometimes it is highly valu- able to trade the range of answerable questions for a non- where p, m, and e are the momentum, the mass, and the approximated insight on specific properties of the given charge of the particle; ω is the angular frequency of the problem. In this paper, a non-relativistic system consist- electromagnetic (EM) mode; c is the speed of light; and ing of a charged particle and a quantized mode of radi- A is the vector potential. Likewise in [1], the polarization ation field is studied. Considering this interaction with of the electromagnetic field is treated as a parameter of only one mode could seem to be an oversimplification; the Hamiltonian and thus two essentially different cases however, there are physical situations where indeed an can be distinguished: approximation of the radiation field with one monochro- ∗ † † matic component is well justified, e.g., in lasers. In fact, Ac = α(εa + ε a ), Al = α a + a , (2) the same approximations are used in the two fundamen- where A and A correspond to the CP and LP cases, tal models of quantum optics: the Jaynes-Cummings and c l respectively. The parameter α = cp2π /V ω, with quan- the Dicke models [6, 7]. In the work of Bergou and Varró ~ tization volume V and Planck constant . In the case of [1], the corresponding Hamiltonian is exactly diagonal- ~ CP, the polarization vectors are complex valued with the ized which provides the non-perturbative description of criteria εε = ε∗ε∗ = 0 and εε∗ = 1, whereas is a real- its spectrum: It describes a free charged particle and a valued unit vector for the LP field. quasi excitation. The relativistic treatment of the same In addition to these two, a third case can be introduced problem can be found in [2]. Varróalso studied the en- corresponding to the EP field tangled photon-electron states of this system in [3, 4]. In the present work it is shown that the dispersion rela- ∗ † Ae = α(E a + E a ), (3) tion of the quasi excitations qualitatively depends on the polarization of the field. For linear polarization (LP) it is where E must be parametrized so that in certain limits described by a bulk plasmon dispersion whereas for cir- it gives the polarization vectors of the CP and LP cases, arXiv:1611.02255v3 [quant-ph] 15 May 2017 cular polarization (CP) and elliptic polarization (EP) the i.e., E(ξ → 1) = ε and E(ξ → 0) = with some pa- dispersion relation has a global minimum but is singular rameter ξ ∈ [0, 1]. In particular, the parametrization, for vanishing wavenumber. The analysis of the dispersion E = p1/(1 + ξ2)( Re u + iξ Im u) with u = (1, i) gives relations shows that the plasma frequency can be differ- the right limits and its norm EE∗ = 1, independently ent for various polarizations and the group velocities for of ξ. In the following, indices like those in (2) and (3) non-LP cases can acquire negative values. Associated to are suppressed for the sake of clarity, and the EP case is the minimum of the dispersion relations, the zero-point understood unless stated otherwise. Even though in the energy of the system can be found, which implies a re- definition of the vector potential in (2) and (3) a dipole pulsive force between two parallel plates. approximation is used, all the results about the dispersion relations can be generalized to plane waves for specific circumstances, as shown in Appendix B. However, it is only important when the properties that are related ∗ [email protected]; [email protected] to the wave propagation are considered. 2

A displacement and a Bogoliubov transformation of is considered because the spectrum is the focus of the the creation and annihilation operators are used for the present study. It reads diagonalization of the Hamiltonian in (1) in order to elim- HΨ = E Ψ , (9) inate the linear and the quadratic terms in a and a† [1, 2]. p,n p,n p,n In the following, only the final result, i.e. the diagonal- with the energy levels ized Hamiltonian, is presented. The details of the com- p2 1 E = + Ω n + − |σ|2 , (10) putation can be found in Appendix A. The EP case is the p,n 2m ~ b 2 most general and is the focus of all the results presented but both the CP and the LP results can be obtained by and the eigenstates having the form of taking the appropriate limits in ξ. The Hamiltonian after Ψp,n = |pi ⊗ Dσ |nib , (11) the transformation reads which is the direct product of the momentum eigen- p2 1 state of the free charge and the shifted/Bogoliubov- H = + Ω b†b + − σ†σ , (4) 2m ~ 2 transformed Fock state of the photons [1]. Any state vector of the transformed Fock space can be described where Ω plays the key role in the differences between the by a superposition of state vectors from the original spectra for the differently polarized modes: Fock space and thus the two spaces can be considered to be equivalent. The solution of the time dependent v ! u 2 ω2 Schrödingerequation can be found in [1]. u 2 2 ξ p Ω = tω + ωp 1 + , (5) (ξ2 + 1)2 ω2

2 2 III. QUASIPARTICLE EXCITATIONS where ωp = 4πe /mV was introduced as ”plasma frequency” [1, 8]. The plasma oscillation usually is considered a collective phenomenon, where a number of N Quasiparticles are collective excitations of a given in- charged particles define the given plasma frequency with teracting system [12]. Inside plasmas or metals, oscilla- 2 2 tion of the electron density can produce plasma oscilla- ωp = 4πe ne/m, where ne = N/V is the electron density [9]. However, in the present case N = 1. The operators, tions. The quanta of such oscillations are called plasmons b and b†, are obtained through a Bogoliubov transforma- [9] and it was shown that for a system of free electron gas, tion (C = exp{ 1 θ(a†a† + aa)}), which is necessary in interacting with a coherent LP electromagnetic field can θ 2 produce such quasi mode [13]. However, in the current order to eliminate the quadratic terms in a and a†: case, only a single charged particle is present; therefore, −1 † the charge density is set as ne = 1/V . On the other hand, b = Cθ bCθ = cosh θa + sinh θa , † −1 † † (1) can be modified so that it describes an interaction of b = Cθ b Cθ = cosh θa + sinh θa. (6) N point charges with the electromagnetic field:

The argument of the hyperbolic functions is defined N 2 1 X e 1 † through the relation H = pi − A + ω + a a . (12) 2m c ~ 2 i=1 e2α2 1 1 − ξ2 tanh 2θ = . (7) Such a Hamiltonian describes a plasma where the mc2 e2α2 1 + ξ2 ~ω + mc2 electron-electron interactions were neglected due to the Debye screening, and, further assuming low velocities for The transformation in (6) for the special case of the CP the electrons the effects of the electron-ion and electron- (ξ → 1) coincides with the identity transformation. The atom collisions are also absent [13]. By using the same transformation Cθ is the squeeze operator that gener- method as above, the Hamiltonian can be diagonalized ates a squeezed state from the vacuum (in the original with the following modifications in (4): p(2) → P p(2) Fock space) [10] which will be the vacuum state in the i i and e2 → Ne2. Apart from these differences, the shape transformed Fock space. There is an additional shift to of the Hamiltonian remains the same as in the single- the Bogoliubov-transformed number operator (b†b =n ˆ ) b electron case and the eigenstates are the direct product of in (4), which consists of the displacement transforma- N free-electron momentum eigenstates and the displaced tion parameter D−1b(†)D = b(†) + σ(†) that performs σ σ photon number state with the appropriate modifications a shift on the raising and lowering operators: 2 P 2 in the shift parameter, i.e., σ(p, e ) → σ i pi, Ne . cosh 2θ αe p −θ θ In this way, ωp can be truly considered as the plasma σ = e Re u − e iξ Im u .(8) 1 e2α2 mc p 2 frequency with charge density of ne = N/V . The re- ~ω + 2 (1 + ξ ) mc maining part of the Hamiltonian describes a quantum † † oscillator with the quasi mode Ω. The operator Dσ = exp(σb − σ b) generates a coherent state from the vacuum |0ib [1, 11]. Acting with both of the operators on the vacuum of the original Fock space will define a coherent squeezed state. 1 The author thanks to S. Varróthe discussions about the many In the following, the stationary Schrödingerequation electron scenario and plasma frequency at this point. 3

persion relations are shown in Fig. 1. The only func- 2.5 tion from the dispersion relations that has the mini- ∗ mum at k = 0 is for LP and Ωl(0) = ωp. In general, 2.0 ∗ ∗ p 2 Ω(k ) = Ω = ωp (1 + ξ)/ 1 + ξ at the minimum. ωp is the quantity that determines the frequency below which p 1.5 light waves are fully reﬂected in the case of plasmas or Ω/ω metals [9]. Indeed, considering Ωl and expressing k, the 1.0 EP,ξ∈(0,1) condition of the solutions for k ≥ 0 is Ωl ≥ ωp. Otherwise CP,ξ=1 the wavenumber takes imaginary values which cannot be 0.5 LP,ξ=0 associated to any traveling wave. A similar analysis can Free photon also be performed for the other polarizations. In general, 0.0 0.0 0.5 1.0 1.5 expressing the wavenumber from the dispersion relation

k/kp reads v s FIG. 1: Dispersion relations of the quasi mode Ω for u 4ξ2ω4 ± 1 u 2 2 2 22 p various polarizations (13). The frequency and the k = √ tΩ − ωp ± Ω − ωp − . (15) 2c (ξ2 + 1)2 wavenumber are rescaled by ωp and kp, respectively. The black dots represent the minimum of the functions. Except for the LP, where this relation reduces to k = The dispersion of the free photon is shown for reference. q 2 2 1/c Ω − ωp, two distinct branches define the wavenumber, denoted by k+ and k−. Strictly speaking, (15) with A. Dispersion relations an overall negative sign also gives the right dispersion in (13), however, it would define negative wavenumbers, By taking the limit p → 0 the system is governed only which is physically meaningless. In order to discuss prop- by the quantum oscillator with frequency Ω. In fact, this erties that are related to wave propagation the dipole limit can be thought of as a uniformly distributed charge approximation of the vector potential might not be sat- in a cube of volume V , due to the uncertainty principle isfactory. However, as it was mentioned earlier, the same dispersion relation can be derived for plane wave vec- in quantum mechanics. In the case of N charges, all pi can be taken to zero, which smears the net charge of Ne tor potential in the p → 0 limit (see Appendix B), in in the volume V . Hence, in this limit the Hamiltonian a similar manner as it was done in [2]. Thus, by con- represents the behavior of EM waves in a plasma where sidering a plane wave, the positivity of the real part the negative net charge is uniformly distributed. The of the wavenumber indicates a propagating wave in the corresponding dispersion relation can be obtained by us- medium. However, as soon as the imaginary part devel- ing ω = ck, where k = |k| is the wavenumber, i.e., the ops a nonzero value, a damping of the oscillatory wave free photon case. The dispersion relation thus reads occurs, corresponding to a finite penetration depth. The real and the imaginary parts of the wavenumber for var- v ! ious polarizations are shown in Fig. 2a and Fig. 2b, re- u ξ2 ω2 u 2 2 2 p spectively. It is clear from the figures that the above Ω = tc k + ωp 1 + . (13) (ξ2 + 1)2 c2k2 statement about the total reflection is only true for the ∗ LP case (dashed red line): For Ω < Ω = ωp the real part This expression takes the form for CP and LP, respec- of the wavenumber vanishes and the imaginary part be- tively, comes finite. For all the other cases the real part remains finite even for values Ω < Ω∗, and it only disappears at a ! ω2 q ˜ p 2 1 p 2 2 2 distinguished value Ω = ωp (1 − ξ)/ ξ + 1, where from Ωc = ck 1 + 2 2 and Ωl = c k + ωp. (14) ± p 2 2 c k (15), k (Ω)˜ = kp −ξ/(ξ + 1). Thus, for frequencies Ω < Ω˜ the wavenumber becomes imaginary just like for It is clear that Ωl describes a bulk plasmon (or bulk the LP, and hence a complete reflection of EM waves is plasmon polariton) [9, 13], with the plasma frequency present. Therefore, Ω˜ can be considered as a modified limk→0 Ωl = ωp. More precisely, Ωl is the dispersion of a plasma frequency for polarizations different from LP. In transverse electromagnetic wave in a plasma, in contrast other words, the propagation of EM waves in a plasma to the longitudinal wave which has a constant dispersion highly depends on its polarization. The following state- ωp. However, for Ω in general, i.e., for 0 < ξ ≤ 1, the dis- ment can be formulated: An EM wave in a plasma is persion relation becomes more structured, since besides • a traveling wave for frequencies Ω ≥ Ω∗, the constant ωp, it contains a singular term ∝ 1/(ck), too. It is not hard to see that Ω restores the free pho- • a decaying traveling wave for Ω˜ < Ω < Ω∗, ton dispersion at the V → ∞ limit, as ωp vanishes. All the functions defined in (13) exhibit a global minimum • and a decaying standing wave for Ω ≤ Ω˜ (evanes- ∗ p 2 at k = kp ξ/(1 + ξ ), where kp = ωp/c. The dis- cent wave). 4

EP,k + 1.0 EP,k + 2.0 EP,k - EP,k - ▲ ▲ ▲ ▲ + ▲ ▲ CP,k + CP,k ▲ ▲ - 0.5 CP,k - 1.5 CP,k ▲ ▲

p LP

LP p

1.0 Im k/ Re k/ 0.0 ● ● ● ● ● ●●●●●●▲

●●● ●● ● 0.5 ● ● -0.5

0.0 ▲ ▲▲▲▲▲ ●▲▲▲▲▲ 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5

Ω/ωp Ω/ωp (a) (b)

FIG. 2: The (a) real and (b) imaginary parts of the wavenumber k for various polarizations. In both panels the k+ and the k− branches are shown from (15). The black dots indicate the value where the wavenumber develops an imaginary part (at Ω = Ω∗). The black triangles show where the real part of the wavenumber vanishes and hence k becomes purely imaginary (at Ω = Ω).˜

Fig. 2b shows that for Im k besides the two branches that For frequencies Ω ≤ ωp the EM waves are reflected com- develop for Ω < Ω∗, the k− branch has a jump discontinu- pletely, i.e., R = 1. This is in accordance with what − p 2 − ity at Ω = Ω˜ from Im k = −kp ξ/(1 + ξ ) to Im k = (15) predicts for the LP case. On the other hand, for all p 2 + kp ξ/(1 + ξ ), whereas the k branch has a cusp. The the other polarizations (0 < ξ ≤ 1) the reflectivity func- two branches then bifurcate again at this point and they tion behaves differently: As it could be extracted from + p 2 (15), transmission of the EM wave still occurs below ω terminate at Ω = 0, where k (0) = kpξ/ 1 + ξ and p − p 2 and only at Ω˜ does it reflect completely. Another inter- k (0) = kp/ 1 + ξ . The CP case behaves somewhat esting behavior can be identified at Ω∗: For frequencies differently. The frequency where√ the imaginary part de- ∗ Ω > Ω∗ the reflectivity function bifurcates similarly to velops a finite value is at Ωc = 2ωp, and that where the real part vanishes is at Ω˜ = 0. This means that for CP Fig. 2a. In fact, this clearly occurs as a consequence of c the degeneracy of the frequency Ω in the wavenumber. there is never a total reflection of the EM waves; how- − ever, a damping still occurs as the imaginary part of the It is apparent that the waves corresponding to the k √ branch have a higher reflectivity, and as Ω → ∞ they re- wavenumber is nonzero for frequencies below 2ωp. For + the CP case there is only one bifurcation of the imagi- flect completely, whereas the reflectivity of the k waves falls off rapidly and tends to zero as Ω → ∞. nary part√ of the wavenumber: The two branches depart from Ω = 2ωp and continue√ all the way to Ω = 0, where ± limΩ→0 Im k = ±kp/ 2. 1.0 ▲▲▲▲▲▲ ●▲▲▲▲▲ In order to give a more detailed insight in the reflectivity property of such a system, the dielectric function (or 0.8 relative permittivity) must be given " s !# 0.6 1 ω2 Ω2 2 4ξ2 ± p + ζ = 1 ± 1 − − ∓ 1 . (16) R EP,R 2 Ω2 ω2 2 2 p (ξ + 1) 0.4 EP,R - The ”±” sign corresponds to the two branches in (15). By ● CP,R + 0.2 ● CP,R - ● taking ξ → 0 (LP) the well-known result for the plasma ● + 2 2 ●●●● LP dielectric function is obtained: ζ = 1 − ωp/Ω for Ω ≥ − 2 2 + − 0.0 ωp and ζ = 1−ωp/Ω for Ω < ωp, hence ζl = ζ ∪ζ = 2 2 √ 0.0 0.5 1.0 1.5 2.0 2.5 3.0 1 − ωp/Ω [9]. The refractive index is η = ζ, which can be used to compute the normal incidence reflectivity: Ω/ωp 2 η − 1 FIG. 3: Reflectivity of EM waves in plasma. The R = . (17) well-known curve is obtained for the LP case. For all η + 1 the other polarizations the complete reflection is at Ω˜ This function is shown in Fig. 3 for various polarizations. and the curves bifurcate at Ω∗, likewise in Fig. 2a. The LP case (dashed red line) shows the reflectivity associated to the well-known plasma relative permittivity: 5

B. Phase and group velocities speed of the information when the dispersion is anoma- lous [14, 15]. Experimental observations of ”fast” and It is instructive to study the phase and group velocities backward-propagating light pulses are reported in [16]. of the corresponding quasi mode. The former is obtained For the LP no negative group velocity is observed, since by dividing the dispersion by the wavenumber: the numerator in (19) cannot be negative for ξ = 0. The phase and group velocities as a function of the wavenum- s 2 2 4 ber for various polarizations are shown in Fig. 4. Ω kp ξ kp vph = = c 1 + + . (18) k k2 (ξ2 + 1)2 k4 IV. ENERGY SPECTRA AND REPULSIVE It is apparent that the phase velocity is always greater FORCE than the speed of light since c has a factor which is greater than one independently of k and ξ. In particular, at the minimum, vc (k∗) = 2c (for CP) and vl (k∗) = ∞ In the following, only the energy spectrum of the CP ph ph and LP cases are analyzed in full details. Evaluating (10) (for LP). The latter is characteristic of the longitudinal for these two separate cases plasma oscillation. The group velocity is deﬁned as the wavenumber derivative of the dispersion relation: 2 2 2 2 2 ! 2p ω + pzωp ωp 1 Ec = + ω 1 + + na , (20) 2 4 2 2 ~ 2 ξ kp 2m 2ω + ωp 2ω 2 ∂Ω 1 − 2 2 k4 (ξ +1) ! vg = = cr . (19) p2 ω2 cos2(φ) q 1 ∂k k2 2 k4 p 2 2 p ξ p El = 1 − + ω + ω + nb , 1 + 2 + 2 2 4 2 2 ~ p k (ξ +1) k 2m ω + ωp 2

where φ in El is the angle between the momentum p 3 and the polarization vector . Like the dispersion relation Ωc, the energy spectrum Ec also diverges as ω → 0

2 – a characteristic for all non-LP cases. For every ﬁxed p and na value a minimum of the energy can be found with respect to ω which√ for p → 0 with na = 0 gives 1 ∗ ∗ Ec = ~Ωc /2 = ~ωp/ 2. This can be considered as the lowest value of the zero-point energy of the system. In 0 ∗ p 2 general E = ~ωp(1 + ξ)/2 1 + ξ , which of course re- e e ∗ v g v ph duces to El = ~ωp/2 in the case of the LP light. Looking velocities(in units of c) c c -1 v g v ph at El in (20), an interesting case can be observed, when

l l p is parallel with the polarization vector . Considering v g v ph -2 this situation with ω → 0 the first term of El yields zero. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 This means that independently of the momentum p, the k/kp ∗ energy of the system is El = ~ωp/2. The energy spec- FIG. 4: The phase and group velocities above and below trum as a function of the momentum p = |p| and ω for light speed (solid black line), respectively. The EP cases the CP and LP cases are shown in Fig. 5. In both panels are in the shaded area, and likewise for the gray line for the excitation number na(b) = 0; however, the character which ξ = 0.2, bounded by the CP and LP velocities. of the plot would have the same features for finite na(b) values, too. For the non-LP cases, besides the singular nature, the degeneracy is also transferred to the energy At the minimum of the dispersion relation the group ve- spectrum from the dispersion relation, meaning that two ∗ ± ± locity clearly vanishes, i.e. vg(k ) = 0, describing a lo- distinct frequencies ω (= ck ) are associated to a par- calized oscillation for all polarizations. A more inter- ticular energy level above the corresponding minimum. esting observation can be made for non-LP cases be- The presence of the zero-point energy is the most sig- ∗ p 2 low their k = kp ξ/(1 + ξ ) minimum: As the dis- nificant difference between the quantized system pre- persion relation is a decreasing function of the wavenum- sented above and its classical equivalent – where the ber in this region, the group velocity exhibits negative operators become c-numbers and hence the Hamilto- values. Moreover, its value even can exceed the speed nian becomes classical. The zero-point energy min- p of light in absolute value. Note that, in this situa- imum can be rewritten as E∗ = κ~e π/mV , with tion both the group and the phase velocities are larger κ ≡ (1 + ξ)/p1 + ξ2. From this form it is clear that than the light speed, but their orientation is opposite. the finite E∗ is a consequence of the finite quantiza- In particular for the CP, this threshold is k = kp/2, tion volume, i.e., it vanishes in the V → ∞ limit, and 2/3p 2/3 4/3 2 and for EP it is k = kp ξ −ξ + ξ + 1/ ξ + 1 . the zero-point energy becomes the usual expression of Even though this seems to violate causality, in fact, the the free quantum harmonic oscillator, i.e., E = ~ω/2. group velocity cannot be identified with the propagation There is another well-known finite-volume phenomenon, 6

(a) (b)

FIG. 5: The energy spectrum of (a) the CP and (b) the LP case in a (p, ω) plot, with the oscillator excitation number set to na(b) = 0. The momentum is chosen so that in (20) (a) pz = 0 in Ec and (b) p is parallel to and hence φ = 0 in El. The most apparent difference between the two plot is in the ω → 0 limit: (a) exhibiting a singular behavior and (b) having a finite value El = ~ωp/2. The axes have been rescaled appropriately for clarity. the Casimir effect, that deals with the zero-point energy Alternatively, it is possible to express the force by using [17–19]. In contradistinction to the current monochro- the plasma frequency matic case, however, in order to obtain the Casimir effect, all the modes of the EM field must be summed, re- P ∗ ∗ κ ~ ωp(d, A) κ ~ ωp sulting in a divergent series ECas = k ~ωk/2, which can F = F A = ≈ . (22) be evaluated by using appropriate regularization schemes 4 d 4 d (see, e.g., [19]). In this case the zero-point energy implies an attractive force between two parallel perfectly conduc- However, when using this representation, it must be tive plates with a surface area A. The result was derived borne in mind that the plasma frequency itself depends first by Casimir [17]; he obtained the force per unit area 2 4 on the distance and the surface area, which is indi- FCas ≡ FCas/A = − cπ /240 d , where d is the distance ~ cated in the argument ωp(d, A). The simplified scaling between the plates. The same lines of thought can be of F ∗ ∝ d−1 is valid only when the ∆d change in the applied to the present model, where the finite-volume distance makes a negligible correction to the plasma fre- dependence is carried by the plasma frequency ωp(V ), quency, i.e., ωp(d + ∆d) ≈ ωp. It is apparent that the and the volume of the box (in which a total number of e implied force depends on the charge, too. In fact, no charge is smeared uniformly) is defined with the param- such repulsive force would be present if the charge van- eters of the two-plate layout (V = A d), likewise in the ished, since E∗ → 0 as e → 0. On the other hand, the Casimir effect experiment. The force between the two Casimir force seemingly does not depend on the charge plates is defined as the derivative of the negative energy at all. However, Jaffe showed the contrary in [20]: In his with respect to their distance, i.e., F = −∂E/∂d. Unlike argument the Casimir-force originates from the interac- in the case of the Casimir effect, the current force is fre- tion between the EM modes and the conducting plates; quency dependent due to the lack of mode summation. thus, it can be shown that it also vanishes as e → 0. In However, the minimum of the zero-point energy can be that sense, this property agrees with the findings about considered as a distinguished point and hence the force the present repulsive force. By replacing the vacuum for this special case is presented here. The general case with appropriate dielectric materials, Lifshitz et al. in with frequency dependence, as well its derivation, can be [21] showed that a repulsive type of Casimir-force can found in Appendix C. The force implied by the minimum be achieved. This effect was experimentally verified by of the zero-point energy reads Munday et al. [22]. r √ κ πR e2 κ πR e2 The above results equally hold the for Ne charges and n F ∗ = B , F ∗ = B , (21) 2 A d3/2 2 A3/2 d3/2 (quasi)photons (i.e., non-vacuum states) that could en- hance the magnitude of the force: 2 2 where RB =√~ /me was introduced as the Bohr radius and κ ∈ [1, 2]. F ∗ represents a repulsive force, scal- −3/2 r 2 ing with the distance as ∝ d . Moreover, the force ∗ πRB Ne 1 ∗ F = κ + n . (23) per unit area F , unlike FCas, is not independent of the A d3/2 2 surface area: The repulsion of the plates increases as the surfaces decrease. Tuning the characteristic length scale√ of the system to the order of the Bohr radius, i.e., It could be of interest to measure such a force in an ap- ∗ 2 2 d ∼ A ∼ RB, the force behaves as F ∝ e /RB. propriate experimental setup. 7

V. CONCLUSION the right limits and its norm EE∗ = 1, independently of ξ. The Hamiltonian will have the following form after In summary, a non-relativistic quantum mechanical substituting (A2): model has been considered consisting of a charged parti- p2 1 eα H = + ω + a†a − p(Ea + E∗a†) cle interacting with one electromagnetic radiation mode. 2m ~ 2 mc The diagonalization of the Hamiltonian leads to plasmon- 2 2 like quasiparticle excitations from which modified plasma e α ∗ † 2 + 2 (Ea + E a ) frequencies (Ω˜ ≤ ωp) can be found for the non-LP cases. 2mc The reflectivity function also shows a strong dependence p2 eα = − p(Ea + E∗a†) (A3) on the polarization of the EM wave: total reflection can 2m mc only be found for frequencies Ω < Ω.˜ For instance, in the e2α2 1 + ω + + a†a case of CP, there is no complete reflection of the wave ~ mc2 2 at any finite frequency since Ω˜ = 0. The phase and c e2α2 1 − ξ2 group velocities are also determined for different polar- + a2 + (a†)2 . izations. It is found that the phase velocity of the quasi 2mc2 1 + ξ2 modes always exceeds the light speed, independently of The p-independent part of the Hamiltonian can be the polarization. The group velocities vanish at the posi- rewritten in terms of the operators b and b† obtained tion of the minimum of the dispersions which for the LP by Bogoliubov transformation: coincides with zero wavenumber. For non-LP cases, the b = cosh θa + sinh θa†, b† = cosh θa† + sinh θa. (A4) group velocities can take negative values and even exceed the light speed in absolute value. The zero-point energy The Hamiltonian reads of the system is derived which implies a repulsive force p2 eα H = − p(Ea + E∗a†) between two parallel plates. The force scales as ∝ 1/d 2m mc with the distance between the plates when the change in e2α2 + ω + the plasma frequency, due to the volume increasement, is ~ mc2 negligible. If the change in the plasma frequency is con- sinh 2θ 1 siderable with respect to the volume growth, the force, × − b2 + (b†)2 + cosh 2θb†b + sinh2 θ + at the minimum of zero-point energy, scales as ∝ d−3/2. 2 2 e2α2 1 − ξ2 + 2mc2 1 + ξ2 ACKNOWLEDGMENTS × cosh 2θ b2 + (b†)2 − sinh 2θ(2b†b + 1) . (A5) The author thanks S. Varró,A. Jakovác,H. Gies, M. In order to eliminate the quadratic terms in b(†) the fol- Horváth,and G. Sárosifor useful discussions and com- lowing definition for θ is required: ments. The ELI-ALPS project (GINOP-2.3.6-15-2015- e2α2 1 − ξ2 1 00001) is supported by the European Union and co- tanh 2θ = 2 2 . (A6) mc2 1 + ξ2 ω + e α financed by the European Regional Development Fund. ~ mc2 Concerning the CP and LP the following expressions for θ can be found:

Appendix A: Diagonalization of the Hamiltonian for tanh 2θ|ξ=1 = 0, various polarizations e2α2 1 (A7) tanh 2θ| = . ξ=0 mc2 ω + e2α2 In the following the details of the diagonalization of ~ mc2 the Hamiltonian For the CP case [the ﬁrst equation in (A7)], it also means (†) 2 that the Bogoliubov transformation is trivial, i.e., b = 1 e 1 † (†) H = p − A + ~ω + a a (A1) a . Thus, what remained from the Hamiltonian is the 2m c 2 following expression: are presented for an elliptically polarized ﬁeld along the p2 eα H = − p(Ea + E∗a†) lines applied in [1] for the circularly polarized and linearly 2m mc polarized cases. Here e2α2 + ~ω + ∗ † mc2 Ae = α(E a + E a ), (A2) 1 × cosh 2θb†b + sinh2 θ + (A8) where E must be parametrized so that in certain lim- 2 its it gives the polarization vectors of the CP and LP e2α2 1 − ξ2 cases, i.e., E(ξ → 1) = ε and E(ξ → 0) = with some + parameter ξ ∈ [0, 1]. In particular, the parametrization 2mc2 1 + ξ2 E = p1/(1 + ξ2)( Re u + iξ Im u) with u = (1, i) gives × − sinh 2θ(2b†b + 1) . 8

After some algebra and using the hyperbolic function At this point, the parameter σ must be defined so that identities, the linear terms cancel out. Therefore, the following re- 2 lationship must hold: p eα ∗ † H = − p(Ea + E a ) 2 2 2 1−ξ 2m mc e α 2 αe p (A9) 1+ξ σ = e−θ Re u − eθiξ Im u . 2 2 2 2 p e α 1 − ξ 1 † 1 mc sinh 2θ mc (1 + ξ2) + 2 2 b b + . mc 1 + ξ sinh 2θ 2 (A18) As the quadratic terms have been eliminated, the focus This sets σ to can be shifted to the linear terms which also contain the c sinh 2θ p momentum dependence. σ = e−θ Re u − eθiξ Im u , eα 1−ξ2 p 2 The second term in the Hamiltonian that consists of lin- 1+ξ2 (1 + ξ ) ear terms in the creation and annihilation operators can (A19) be rewritten in terms of b and b†, giving or equivalently αe p † p ( Re u + iξ Im u)a + ( Re u − iξ Im u)a cosh 2θ αe p mc (1 + ξ2) σ = e−θ Re u − eθiξ Im u . e2α2 p 2 αe p ω + 2 mc (1 + ξ ) = e−θ(b + b†) Re u + eθ(b − b†)iξ Im u . ~ mc mc p(1 + ξ2) (A20) (A10) The parameter σ for the CP and LP cases can be ob- After this transformation the Hamiltonian reads tained by taking the limits ξ → 1 and ξ → 0, respectively: p2 αe p H = − αe ∗ 1 2m mc p(1 + ξ2) lim σ = pε = σc, ξ→1 mc ~Ωc −θ † θ † × e (b + b ) Re u + e (b − b )iξ Im u (A11) αe −θ 1 lim σ = pee = σl. (A21) e2α2 1 − ξ2 1 1 ξ→0 mc ~Ωl + b†b + . mc2 1 + ξ2 sinh 2θ 2 Here, the effective frequencies are defined for the two separate cases: For the elimination of the linear terms the displacement ! ω2 q operator is used: 1 p 2 2 Ωc = ω 1 + 2 , Ωl = ω + ωp, (A22) † † 2 ω Dσ = exp(σb − σ b), (A12) and the polarization vectors are = Re u, ε = Re u + with σ being arbitrary at this point, but requiring 2 2 i Im u, and ωp = 4πe /mV . The Hamiltonian now ap- [σ, σ(†)] = [σ, b(†)] = [σ, p] = 0. (A13) proaches its final form:

2 † 2 2 2 1−ξ Acting on b and b adds a shift to the operator, p e α 2 1 H = + 1+ξ b†b + + σ†σ (A23) −1 (†) (†) (†) 2m mc2 sinh 2θ 2 Dσ b Dσ = b + σ . (A14) αe p −θ θ The transformation must also be unitary: − e Re u + e iξ Im u σ mc p(1 + ξ2) −1 † αe p Dσ = Dσ, (A15) − e−θ Re u − eθiξ Im u σ†. mc p(1 + ξ2) hence −1 † † † † † The third and fourth terms can be rewritten by using the Dσ b bDσ = b b + σ b + b σ + σ σ. (A16) deﬁnition of σ in (A20), Applying the transformation to the Hamiltonian yields 2 2 2 2 1−ξ p e α 1+ξ2 † 1 † 2 H = + b b + + σ σ 2 2 2 1−ξ 2m mc2 sinh 2θ 2 p e α 1+ξ2 † 1 † H = + b b + + σ σ 2 2m mc2 sinh 2θ 2 2 2 1−ξ e α 1+ξ2 † 2 2 −2 σ σ 2 2 1−ξ 2 2 1−ξ mc2 sinh 2θ e α 1+ξ2 † e α 1+ξ2 † + σ b + b σ 2 2 2 2 2 2 1−ξ mc sinh 2θ mc sinh 2θ p e α 2 1 αe p = + 1+ξ b†b + − σ†σ , (A24) − e−θ Re u + eθiξ Im u (b + σ) 2m mc2 sinh 2θ 2 mc p(1 + ξ2) αe p or equivalently, by using the relation in (A6) − e−θ Re u − eθiξ Im u (b† + σ†). p 2 2 mc (1 + ξ2) 2 e α p ~ω + 2 1 H = + mc b†b + − σ†σ . (A25) (A17) 2m cosh 2θ 2 9

After some algebra the following expression for the eﬀec- where X and Y are operators. Applying the lemma ﬁrst tive angular frequency can be found: to the kinetic term gives v ! 1 u 2 ω2 2 † 2 † 2 u 2 2 ξ p Up U = p + ik a a + r, p Ω(ξ) = tω + ωp 1 + , (A26) 2 (ξ2 + 1)2 ω2 2 2 (ik) † 1 2 2 2 + a a + r, r, p + ··· .(B6) where ωp = 4πe /mV . Hence the Hamiltonian reads 2 2

2 2 2 p † 1 † The commutators give r, p = 2i p and r r, p = H = + ~Ω(ξ) b b + − σ σ . (A27) ~ 2m 2 (i~)2. The higher-order terms are identically zeros since the second commutator results in a c-number. Thus, (Here H is used for the final form of the transformed " # Hamiltonian in order to distinguish it from the original p2 p2 kp 1 2k2 12 H.) Taking the limits ξ → 1 and ξ → 0, the Hamiltonians U U † = − ~ a†a + + ~ a†a + 2m 2m m 2 2m 2 for the CP and LP cases are obtained, with Ωc and Ωl, respectively. These coincide with the results in [1]. 1 12 = p − k a†a + . (B7) 2m ~ 2 Appendix B: Extension to plane waves The term proportional to UpEaU † can be rewritten as UpU †EUaU †; hence, the transformation for p and a can In the following, it is shown that by using an appropri- be considered separately: ate unitary transformation, as applied in [2], the Hamil- tonian with a plane wave vector potential becomes equiv- † † 1 † 1 UpU = p + ik a a + [r, p] = p − ~k a a + . alent to the one with a dipole approximation up to a term 2 2 of order O(~2) when the limit p → 0 is taken. The vector (B8) potential is defined as Higher-order terms vanish identically since [r, p] gives al- i(kr−ωt) ∗ † −i(kr−ωt) Ae = α E ae + E a e . (B1) ready a c-number. In fact the product with the polarization is In this case the Hamiltonian in (1) becomes UpU †E = pE, (B9) p2 eα H = − p Eaei(kr−ωt) + E∗a†e−i(kr−ωt) pw 2m mc since the wavenumber vector is orthogonal to the polar- † † e2α2 1 ization, i.e., kE = 0. Hence UpEaU = pEUaU . + ω + + a†a (B2) The remaining terms contain a(†) in linear and quadratic ~ mc2 2 order. Their transformation is discussed in the following. e2α2 1 − ξ2 h i + a2e2 i(kr−ωt) + (a†)2e−2 i(kr−ωt) . The transformation of a reads 2mc2 1 + ξ2 UaU † = a + i(kr − ωt)[a†a, a] In order to eliminate the exponential position depen- [i(kr − ωt)]2 dence, the following unitary transformation is used: + [a†a, [a†a, a]] + ··· . (B10) 2 i(kr−ωt)(a†a+ 1 ) U = e 2 . (B3) Since [a†a, a] = −a, by using induction it is not hard to Acting with the transformation U on H results in see that the nth term is (−1)na. Thus, the expression in (B10) collapses to p2 eα UH U † = U U † − Up EaU †ei(kr−ωt) pw 2m mc ( 2 ) † [−i(kr − ωt)] e2α2 1 UaU = a 1 + [−i(kr − ωt)] + + ··· +E∗a†U † e−i(kr−ωt) + ω + + a†a 2 ~ mc2 2 = ae−i(kr−ωt). (B11) e2α2 1 − ξ2 h + Ua2U †e2 i(kr−ωt) 2mc2 1 + ξ2 The two exponential factors in (B11) and (B4) cancel out i each other, leaving only the operator a behind. Using the +U(a†)2U †e−2 i(kr−ωt) . (B4) same procedure for a† gives † The term containing the number operator a a transforms Ua†U † = a† + i(kr − ωt)[a†a, a†] (B12) trivially. All the other terms are considered separately in [i(kr − ωt)]2 the following. In order to compute the action of U the + [a†a, [a†a, a†]] + ··· = a†ei(kr−ωt). Baker-Hausdorff-Campbell identity is used, i.e., 2 1 The commutator in this case was [a†a, a†] = a†, and for eX Y e−X = Y + [X,Y ] + [X, [X,Y ]] + ··· , (B5) † 2 the nth term a . Again, the exponential factors in (B12) 10

and (B4) cancel out each other. of this modiﬁed dispersion is beyond the scope of the Regarding the quadratic terms in a(†), in these cases the present paper. On the other hand, there are circum- exponential factors of e±2i(kr−ωt) must be eliminated: stances where (13) remains valid. The term ∝ kp van-

2 † 2 † 2 ishes if k is perpendicular to p or when the p → 0 limit Ua U = a + i(kr − ωt)[a a, a ] is taken. The latter scenario describes a system where [i(kr − ωt)]2 the EM mode interacts with a charge that is uniformly + [a†a, [a†a, a2]] + ··· , 2 distributed in the volume V . This limit is the subject of U(a†)2U † = (a†)2 + i(kr − ωt) a†a, (a†)2 (B13) the analysis performed in Sec. III. The remaining Hamil- 2 tonian reads [i(kr − ωt)] † † † 2 + a a, a a, (a ) + ··· . 2 2 2 2 2 † e α ~ k 1 1 lim UHpwU = ~ω + 2 + nˆ + nˆ + And the commutators are p→0 mc 2m 2 2 2 2 2 e α 1 − ξ 2 † 2 [a†a, a2] = [a†, a]a + a[a†, a] a = −2a2, + a + (a ) . (B19) 2mc2 1 + ξ2 a†a, (a†)2 = a† a†[a, a†] + [a, a†]a† = 2(a†)2. (B14) The second new term is of O(~2); hence this can be con- n 2 It is not hard to see that the nth term gives (±2) a . sidered negligible compared to the O(~) terms for not Substituting these findings back to the sum gives very large wavenumber and photon number, for which it could dominate. Altogether the Hamiltonian is [−2i(kr − ωt)]2 Ua2U † = a2 1 + [−2i(kr − ωt)] + + ··· 2 2 2 † e α 2 1 lim UHpwU = ~ω + + O(~ ) nˆ + = a2e−2i(kr−ωt), p→0 mc2 2 [2i(kr − ωt)]2 e2α2 1 − ξ2 U(a†)2U † = (a†)2 1 + [2i(kr − ωt)] + + ··· + a2 + (a†)2 , (B20) 2 2mc2 1 + ξ2 = (a†)2e2i(kr−ωt). (B15) which reproduces limp→0 H in (A3) up to a negligible 2 The exponential factors in (B15) and (B4), like for the term of O(~ ). This shows that all the conclusions drawn linear terms, cancel out each other. for the dispersion relation obtained from the Hamilto- Hence, collecting all the terms together, the transformed nian with dipole approximation remains valid for the Hamiltonian reads plane wave vector potential, too, under the condition that p → 0 and for not too large wavenumber and photon 2 1 1 eα number. It is interesting to note that the zero-momentum UH U † = p − k a†a + − p Ea + E∗a† pw 2m ~ 2 mc limit and the action of U are not interchangeable: 2 2 e α 1 † † + ω + + a a (B16) U lim HpwU = lim H, ~ mc2 2 p→0 p→0 2 2 2 † 2 e α 1 − ξ lim UHpwU = lim H + O(~ ) ≈ lim H. (B21) + a2 + (a†)2 . p→0 p→0 p→0 2mc2 1 + ξ2 In the first case the equality is exact, whereas in the It is clear that the transformation canceled all the expo- second case it is only approximate. nentials; however, the kinetic term of the charge has been modified, too. It is possible to rearrange the Hamiltonian in the following way (ˆn = a†a): Appendix C: Derivation of the repulsive force p2 eα between two parallel plates UH U † = − p Ea + E∗a† (B17) pw 2m mc e2α2 kp 2k2 1 1 The vacuum energy for the general EP case reads as + ω + − ~ + ~ nˆ + nˆ + ~ mc2 m 2m 2 2 v ! 2 2 2 1 u ξ2 ω2 e α 1 − ξ 2 † 2 u 2 2 p + a + (a ) . E = ~tω + ωp 1 + , (C1) 2mc2 1 + ξ2 2 (ξ2 + 1)2 ω2 Comparing (B17) to (A3), two new terms appear in the coefficient of (ˆn + 1/2) in (B17), namely, which simplifies to the second term in Ec(na = 0) and El(nb = 0) in (20) when taking the limits ξ → 1 and ξ → 2 2 kp k 1 0, respectively. The plasma frequency ωp encapsulates −~ and ~ nˆ + . (B18) m 2m 2 the finite-volume dependence √ These two terms clearly modify the dispersion relation 2 πe ωp = √ , (C2) that was found in (13). However, the detailed analysis m A d 11 where A is the surface area of the two parallel plates and Here, it must be remembered that the plasma frequency d is their distance from each other. The force between depends on the geometric parameters d and A, i.e., ωp = the two plates is computed as ωp(d, A). By substituting (C2) into (C5) the explicit distance and surface area dependence can be obtained. ∂E F = − . (C3) However, if the ∆d change in the distance does not mod- ∂d ify the plasma frequency considerably, i.e., ωp(d + ∆d) ≈ ω , then (C5) gives the frequency-dependent repulsive The only term that depends on the distance is the plasma p force between the plates that scales as ∝ 1/d with the frequency. Its derivative reads distance. For fixed d the limit limω→∞ F = 0 for all ξ, √ on the other hand, the limit lim F = ∞ for 0 < ξ ≤ 1. ∂ ωp 2 πe 1 1 ωp ω→0 = −√ = − . (C4) In the case of LP (ξ = 0), the limit lim F = ω /4d, ∂ d d3/2 2 d ω→0 ~ p m A which is the maximum of the force F for LP. ∗ Thus, the force by using (C3) is At the minimum of the zero-point energy, ω = p 2 ωp ξ/(1 + ξ ), the repulsive force in (C5) takes the form 2 ω2 ω ∂ ωp 1 + 2ξ p r ~ p ∂ d (ξ2+1)2 ω2 κ ω (d, A) κ πR e2 F = − F ∗ = ~ p = B , (C6) r 3/2 2 ω2 4 d 2 A d 2 ω2 + ω2 1 + ξ p p (ξ2+1)2 ω2 p 2 2 2 where κ = (1 + ξ)/ 1 + ξ and RB = ~ /me is the 2 ω2 ω 1 + 2ξ p Bohr radius. For ξ = 0 this coincides with the force in ~ p (ξ2+1)2 ω2 1 = r . (C5) the ω → 0 limit. 2 2 d ω2 ξ ωp 4 2 + 1 + 2 2 2 ωp (ξ +1) ω

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