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Exploring QED Vacuum with Ultra Intense Lasers

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Exploring QED Vacuum with Ultra Intense Lasers Exploring QED vacuum 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 quantum electrodynamics are described here. First, we consider the presence of a static and a rotating magnetic field. A second type of processes is photon acceleration in vacuum, and corresponds to an adiabatic frequency shift of a probe photon beam, moving in vacuum in the presence of an ultra-intense burst of radiation. Finally, quantum time refraction 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 electron-positron pairs [1, 2]. This leads to nonlinear corrections of the photon dispersion relation in vacuum, and vacuum becomes a nonlinear optical medium. 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 wave [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 light wedges [12]. Recently, we have explored another process associated with the collective photon interactions, related to possible frequency shift of test photons 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 plasma 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 refractive index 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 polarization 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 number, 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 frequencies 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 optics 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].
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