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Exploring QED with ultra intense lasers

J.T.Mendon¸ca

CFP and CFIF, Instituto Superior T´ecnico, 1049-001 Lisboa, Portugal

Abstract

We consider the possible use of ultra-intense laser fields to probe the nonlinear properties of the electromagnetic quantum vacuum. Several new processes asso- ciated with the nonlinear optical properties of vacuum, as predicted by are described here. First, we consider the presence of a static and a rotating magnetic field. A second type of processes is acceleration in vacuum, and corresponds to an adiabatic shift of a probe photon beam, moving in vacuum in the presence of an ultra-intense burst of radiation. Finally, quantum is considered inside an empty optical cavity, and can be linked with a dynamical Casimir process. The possible observation of these nonlinear vacuum processes with the existing Peta-Watt laser systems is discussed.

1 Introduction

Vacuum is one of the most fascinating physical concepts. Several aspects of nonlinear vacuum have been studied in recent years, in the frame of quantum electrodynamics or QED. It is well known from QED that photon-photon interactions in vacuum are possible, due to the existence of virtual - pairs [1, 2]. This leads to nonlinear corrections of the photon dispersion relation in vacuum, and vacuum becomes a nonlinear . However, such QED corrections are extremely small, and only recently, with the advent of very intense laser systems, in the Peta-watt regime, the prospect for experimental observation of such nonlinearities started to be seriously considered.

1 2

A large variety of different effects has been discussed, including photon splitting [3, 4], harmonic generation [6], self-focusing [5], nonlinear [7] or sideband generation by rotating magnetic fields. Also, attention has been paid to collective photon phenomena [8], such as the electromagnetic wave collapse [9, 10] as well as the formation of photon bullets [11] and wedges [12]. Recently, we have explored another process associated with the collective photon interactions, related to possible frequency shift of test immersed in a modulated radiation background [13]. This new process could be called photon acceleration in vacuum, because of its obvious analogies with the well known photon acceleration processes that can occur in a or in an optical medium [14].

Another kind of nonlinear optical QED effects is associated with magnetized vacuum. Or recent work [15] considers a rotating magnetic field. In this case birrefringence is inhibited, but optical sidebands are excited, providing a possible signature to the vacuum nonlinearity.

Finally, we consider vacuum detuning of an optical cavity, produced by an intense laser pulse. We assume that this intense laser beam crosses the empty cavity in the perpendicular direction.The optical length of the cavity is then slightly changed, due to the nonlinear QED perturbation of vacuum, and the cavity modes are frequency shifted (or frequency modulated) and detuned. This sudden change of the of a cavity can be seen as a time refraction process [16], or alternatively as a dynamical Casimir effect [17]. What is new and relevant here is that such processes are not taking place in a variable optical medium, or in a variable cavity, but in pure vacuum. The basic principles behinds the proposed optical configuration rely on the phase shift amplification associated with resonant optical cavities.

2 QED in vacuum

The nonlinear QED effects associated with the creation of virtual electron-positron pairs in vacuum can be described by the Heisenberg–Euler Lagrangian density, which can be written as a nonlinear quantum correction δL to the usual classical electromagnetic La- 3

2 2 grangian density L0, in the form [18]: L = L0 + δL = 0F + ζ(4F + 7G ), where the 1 2 2 2 ~ ~ quantities F and G are determined from F ≡ L0/0 = 2 (E − c B ) and G = c(E · B). Here E~ and B~ are the electric and magnetic fields, respectively. The nonlinear parameter 2 2 3 4 5 2 appearing in δL is ζ = 2α 0h¯ /45mec , where α = e /20hc ≈ 1/137 is the fine structure constant. If we consider a given photon state in this modified vacuum, with the frequency ω and the wavevector ~k, we can derive the following nonlinear dispersion relation [3, 7]  1  ω = kc 1 −  λ f(~k,~k0)|E(~k0)|2 , (1) 2 0 ± where E~ (~k0) is the electric field of the background radiation, f(~k,~k0) is a geometric factor, and λ+ = 14ζ or λ− = 8ζ, according to the state. The resulting vacuum refractive index n = kc/ω is

2¯hλ Z d~k0 n(ω,~k, t) = 1 + ± f(~k,~k0)k0N(~r,~k0, t) , (2) c (2π)3 where we have introduced the photon occupation , or number of photons per field ~ 0 0 ~ 0 2 mode, N(k ) = (0c/4¯hk ) |E(k )| .

3 Birefringence

~ ~ Let us start with a static magnetic field Be ≡ B0 in vacuum. In this case, the dispersion relation for a photon state with frequency ω and wave vector ~k also depends on the direc- tion of its unit polarization vectore ˆ = E/~ |E|. For polarization parallel and perpendicular to the static magnetic field, the photon refractive index will be

4 2 4 2 2 nk = 1 + 7µ0c ζB0 , n⊥ = 1 + 4µ0c ζB0 sin α (3) ~ where α is de angle between the static field B0 and the direction of propagation, defined by ~k. This is the well known vacuum birefringence [3, 4]. We notice here that, for propagation along the static field we are reduced to nk = n⊥ = 1, which means that Faraday rotation is forbidden.

The situation suffers an important qualitative change if we consider a rotation mag- ~ netic field. Let us then assume the case of a static but rotating magnetic field Be(t) = 4

ˆ ˆ B0 b0(t) with a constant amplitude B0, where the unit vector b0(t) rotates with a very ~ small angular frequency ω0  ω in the plane perpendicular to the photon wavevector k. In other words, it rotates in the plane of wave polarization.

In this case the dispersion relation for non-interacting photon modes is given by

2 ! 2 2 ω ω 2 k − 2 = 11 ζ|B0| (4) c 0 which corresponds to a modified refractive index in the magnetized vacuum of 11 n = 1 + µ c4ζ|B2| (5) 2 0 0 Comparing with equations (??) we see that rotation of the magnetic field inhibits bir- refringence. But, on the other hand, this result is in complete agreement with those for a static field, because in a rotating field the polarization states change with time and are mixed together. Therefore, if we replace B0 by Be(t) in equations (3), and average over a rotation period 2π/ω0, we will get equation (5).

4 Sidebands

We now consider another effect that originates from the second order nonlinear terms ~ with respect to the rotating magnetic field Be(t). These terms will couple photon modes with ωn = ω + 2nω0, and will generate a cascade of sidebands. The different mode amplitudes are determined by

d   E = iw E ei2φ + E e−i2φ (6) dτ n n n−1 n+1 where the phase mismatch is determined by φ = −ω0z/c, and the coupling constants are

2 3 c 2 wn = ζωn|B0| (7) 4 0

For small interacting distances, z  λ(2ω0/ω), we can neglect the phase mismatch and obtain simple analytical solutions for the electric field amplitudes En. Solutions compat- ible with initial conditions such that E(ω, τ = 0) ≡ E0 and no sidebands initially exist 5

0.6

0.5

0.4

0.3

0.2

0.1

0 -2 -1 0 1 2

Figure 1: (a) Laser sidebands generated in vacuum: relative intensity versus sideband frequency, for the extreme case wτ = 1.

En(τ = 0) = 0, are then given by

inπ/2 En(τ) = E0 Jn(wτ)e (8)

where Jn are the Bessel functions of the first kind. Using the asymptotic expressions for the Bessel functions will small arguments we get for the first sideband amplitude

 2 E1 3 α B0 ∆z = (9) E0 8 45 Bcrit λ

2 2 where ∆z is the optical path, α is the fine structure constant, and Bcrit = m c /eh¯ the critical field. This could be applied to the PVLAS configuration [19], where an harmonic sideband E1 is observed. However, the observed amplitude lies a few orders above the expected QED signal and is due to some spurious signal or to a new physical effect, eventually associated to a pseudo-scalar particle like an axion. With an improved accuracy, the observation of the second sideband E2 would eventually help to clarify the observations. 6

5 Photon acceleration in vacuum

In the geometric approximation, the dynamics of photons can be described by the ray equations, which can be stated in the following canonical form

d~r ∂ω d~k ∂ω = , = − , (10) dt ∂~k dt ∂~r where the Hamiltonian ω is determined by Eq. (1). Let us consider the important particular case where the background radiation is dominated by a single photon beam ~ 0 0 propagating in a direction Oz. This means that we can make k = k ~ez. If the probe photon described by these equations of motion propagates at a given angle θ with respect to the z-axis, we have (~n · ~n0) = cos θ and (~n · ~e0) = sin θ cos ψ, where ψ is the second angle necessary to define the direction of ~n with respect to the background electric field. In this case, the geometric factor is f(~k,~k0) ≡ f(θ) = [2(1 − cos θ) − sin2 θ]. The photon dispersion relation becomes  1  ω(~r,~k, t) = kc 1 − λ∗f(θ)I(~r, t) , (11) 2

∗ where λ = 4¯hλ±/c, the angle θ can vary in and time according to the photon dynamics described by Eq. (10), and the background field intensity I(~r; t) is determined by the integral Z d~k0 I(~r, t) = k0N(~r,~k0, t) . (12) (2π)3 Let us assume that the beam is modulated in intensity along the propagation direction, and that we can write I(z − ut, ~r⊥). For a very large beam waist, we can neglect the dependence over the transverse direction and assume that u = c. From the equations of motion (10), we can establish a relation between he initial value of the frequency

ω0 ≡ ω(t = 0), and a subsequent value ω(t), such that

1 − δn(t) (1 − cos θ0 − δn0) ω(t) = ω0 . (13) (1 − δn0) 1 − cos θ(t) − δn(t) This expression will be very useful to calculate the frequency shift of a test photon due to its interaction with the radiation background, Such a frequency shift can be associated 7 with a process of photon acceleration. Notice that the group velocity of the photon as determined by the nonlinear dispersion relation, and given by [3] ~v = ~nc[1 − δn(t)], is independent of the photon frequency, but changes with time due to the change in the angle between the photon and the background beam. In order to determine how this process evolves with time, we would have to go back to the parallel photon dynamical equations.

The very weak nonlinear QED force acting on the test photon and due to the gradient of the beam intensity (represented by the derivative of background intensity profile) acts on the probe photon over very large distances, because it travels with a parallel velocity nearly equal to the light c, eventually leading to a non-negligible frequency up- shift, which could eventually be observed in future experiments using Peta-Watt laser intensities.

6 Time refraction in vacuum

Let us now consider time refraction due to the temporal change in the refractive index of the optical medium, and apply it to wave propagating in an unbounded perturbed vacuum. In contrast with the previous case, no gradients of intensity are considered here and the medium is uniform. Conversion to a bounded medium, is straightforward, and the case of an empty cavity will be briefly discussed below.

Let us also assume that the refractive index of the optical medium starts changing at time t = 0, due to some external agent. We can describe the variation of the refractive index by a generic function of time, n(t), which can be approximated by a sequence of discrete steps of duration τ, where the continuous limit of τ → 0 can be taken. This discretization process is analytically described by n(t) = nj, for (j − 1)τ < t ≤ jτ, with j integer, where we can determine the successive values of the refractive index as nj = nj−1 + (dn(t)/dt)τ, for τ → 0. For a given mode of the electromagnetic propagating along the arbitrary Ox-direction in the changing medium, the electric field 8 is determined by h i E~ (x, t) = E~ (t)e−iφ(t) + E~ 0(t)eiφ(t) eikx + c.c., (14) with the phase function Z t φ(t) = ω(t0)dt0. (15) 0 Here, E~ and E~ 0 are the field amplitudes for waves propagating in the positive and negative Ox-directions, respectively. The time dependent value for the mode frequency ω will have to obey the linear instantaneous dispersion relation

ω(t) = kc/n(t). (16)

This expression can be seen as the temporal Snell’s law [14], because it relates the wave frequencies at two different t1 6= t2, as ω(t1)n(t1) = ω(t2)n(t2), whereas the usual Snell’s law for (space) refraction relates the wavevectores in two different media. Accord- ing to our previous work [16], the temporal evolution of the mode electric field in a time varying medium is determined by dE 1 dn   Z t  = − 3E + E0 exp +2i ω(t0)dt0 (17) dt 2n dt and dE0 1 dn   Z t  = − 3E0 + E exp −2i ω(t0)dt0 (18) dt 2n dt In order to understand the physical meaning of these equations, let us consider the special case where initially we have a wave propagating along the positive direction that dominates over the wave propagating in the opposite direction, |E|  |E0|, and also that the temporal changes in the medium are very slow, which means that there is a weak coupling between these two waves. In this approximation, we get, for a slowly varying medium " 3 Z t 1 dn # E(t) ' 1 − dt0 E(0) ≡ T (t)E(0) (19) 2 0 n(t0) dt0 where the temporal transmission coefficient is T (t) ∼ 1. Considering now equation (18), and assuming that E(t) ∼ E(0) = const., we can easily obtain the reflected field resulting from the non-stationarity of the medium, of the form E0(t) = R(t)E(0), with E(0) Z t 1 dn Z t0 ! R(t) ' − exp −2i ω(t”)dt” dt0 (20) 2 0 n(t0) dt0 9

Int 1

0.8

0.6

0.4

0.2

x -4 -2 2 4

Figure 2: Time refraction in vacuum: an initial gaussian pulse (at t = 0) suffers a temporal split that can be observed (at a time t > 0), as shown by the curve in bold. For illustration, the amplitude of the reflected signal is multiplied here by a very large factor.

These expressions for the temporal transmission and reflection coefficients, T (t) and R(t), are formally analogous to the well known reflection coefficient for stationary by non- homogenous media T (x) and R(x) [20], with the space coordinate along the gradient of the refractive index replaced by the time coordinate.

We can now consider the case where the waves are propagating in vacuum, and the changes in refractive index are simply due to the nonlinear QED effects considerer in the previous section. Assuming that the probe photons propagate perpendicularly to the strong laser pulse that is responsible for the nonlinear vacuum perturbation, two different effects can take place: frequency shift, and photon reflection. These effects are extremely small, but if observed they would give us direct information of the nonlinear vacuum. The frequency shift is determined by the temporal Snell’s law (16), or δω(t) λ∗ = δn(t) = − f(θ)I(~r, t) (21) ω(0) 2 This is a very tiny frequency shift, which can however be amplified with interferometric techniques. If the frequency shifted signal propagates along one of the arms of an interfer- 10 ometer, with a total length L, and assuming that the reference signal propagating in the other arm was not frequency shifted, we will obtain a total amount of fringe displacement given by L 2πL δφ(t) = δω(t) = δn(t) (22) c λ(0) Notice that we have assumed that the frequency shift only occurs locally, at the intersec- tion of the probe and the intense laser beam, and not along the whole propagation length. We can see that amplification factor resulting from the interferometric technique is of or- der L/λ(0) and can be very large. For L of the order of 1 meter, for near photons this factor can be as large as 106. This would give some hope on the possible observation of the nonlinear quantum vacuum. Finally, another amplification technique can be used, in order to improve the signal, by using a symmetric experimental scheme, where the two arms of interferometer are successively disturbed by the ultra-intense laser pulse. Time correlation techniques will eventually allow to improve the signal by an additional factor of 103. Limitations of this scheme are mainly due to the residual gas inside the empty cavity.

7 Conclusions

We have described here several new processes associated with the nonlinear optical prop- erties of the electromagnetic vacuum, as predicted by quantum electrodynamics. We have assumed that light propagates in pure vacuum, with no material support of the interac- tion between different photon modes, except for the QED nonlinearities of vacuum. These processes include, optical birefringence in a static magnetic field, and it possible disap- pearance in a rotating magnetic field, optical sideband cascades due to the rotating field, photon acceleration of probe photons interacting with an intense radiation background radiation, and time refraction in the perturbed vacuum of an optical cavity. Such a vari- ety of nonlinear effects shows that QED vacuum can be seen as a virtual plasma, where the nonlinear effects are due to the existence of virtual electron-positron pairs, excited by intense fields. This is a difficult but promising new area, which can help us to understand 11 the physical background of particles and fields. Possible experimental configurations us- ing Peta-Watt laser systems can be seen as promising candidates for the observation of nonlinear effects in the QED vacuum.

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