Analogue Hawking temperature of a laser-driven plasma
aa r X i v : . [ g r- q c ] F e b Analogue Hawking temperature of alaser-driven plasma
C. Fiedler D. A. BurtonFebruary 5, 2021
Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK
Abstract
We present a method for exploring analogue Hawking radiationusing a laser pulse propagating through an underdense plasma. Thepropagating fields in the Hawking effect are local perturbations of theplasma density and laser amplitude. We derive the dependence of theresulting Hawking temperature on the dimensionless amplitude of thelaser and the behaviour of the spot area of the laser at the analogueevent horizon. We demonstrate one possible way of obtaining theanalogue Hawking temperature in terms of the plasma wavelength,and our analysis shows that for a high intensity near-IR laser theanalogue Hawking temperature is less than approximately 25 K for areasonable choice of parameters.
Hawking radiation [1] is a well established theoretical prediction that a blackhole will evaporate, emitting a thermal spectrum of radiation in the process.The temperature of this radiation, called Hawking temperature, is inverselyproportional to the mass of a black hole and thus it is very small, eluding ex-perimental searches. However, it has been discovered that black holes are notthe only phenomena that can generate Hawking radiation. Analogue grav-ity [2] investigates non-gravitational analogues of general relativity in variousphysical systems such as fluids [3–5], dielectric media [6,7], and Bose-Einsteincondensates [8]. In these systems, the field equations can be written as amassless wave equation on an effective background spacetime. Experimentsperformed in the context of analogue gravity include water waves [9–14],ultra-cold quantum gases [15–18], optics [19–23], and polaritons [24]. Since1xperimental study of black holes is not feasible, analogue models provide theopportunity to test theoretical aspects of black hole physics in a laboratoryenvironment.The interest in analogue gravity is not only to reproduce gravitationalphenomena in some analogue model, but to indirectly probe the simplifica-tions that underpin calculations of black hole evaporation. Deviations of realanalogue models from their simplest forms may shed light on the deviationsthat are likely to appear in the corresponding gravitational systems. Seeref. [2] for a detailed discussion of the applicability of analogue models inthis context.To our knowledge, the prospects of using laser-driven plasma to studyanalogue black holes has not been thoroughly investigated. In this articlewe obtain the analogue Hawking temperature of a laser-driven underdenseplasma. In the second section, we briefly describe how to obtain the non-linear field equations that underpin the work. In the third section we showthat, under certain conditions, two effective Lorentzian metrics emerge whenthe field equations are linearised about an exact solution. In the fourthsection we determine required properties of the fields in order to attain aneffective metric analogous to the Schwarzschild metric, and we calculate theassociated analogue Hawking temperature corresponding to using an intensenear-IR laser. In the final section we summarise presented work.
Quantities with tilde over them denote dimensionful variables which willbe made dimensionless for convenience in the analysis. To first approxi-mation, the properties of the plasma electrons can be described by a localenergy-momentum relation and a continuity equation arising from chargeconservation, whilst the laser pulse satisfies a local dispersion relation andconservation of wave action (i.e. conservation of classical photon number).This approach is underpinned by a separation of the laser-plasma dynamicsinto slowly evolving parts and rapidly oscillating parts that arises becausethe length scale of the internal oscillations of the laser pulse (i.e. the wave-length of the laser) is much shorter than the envelope of the laser pulse andthe plasma wavelength. The rapidly oscillating quantities enter the slowdynamics via their local averages.The relativistic energy-momentum relation E − c p − c e h A i = m e c (1)is satisfied by the averaged motion of the plasma electrons, where E is the2ocal relativistic energy of the averaged motion, p is the local relativistickinetic momentum of the averaged motion and A is the vector potentialof the laser pulse. The angle brackets in (1) denote averaging over the fastinternal oscillations of the laser pulse, and the third term on the left-hand sidearises because e A can be identified with the momentum of fast oscillatorymotion in the plane orthogonal to the direction of propagation of the laserpulse. The dispersion relation ω − c k = ω p (2)is satisfied by the local angular frequency ω and the local wave vector k ofthe laser pulse. The local plasma frequency ω p is given by ω p = s e nε m e γ (3)where γ is the local Lorentz factor of the averaged motion (i.e. E = m e c γ ),and n is the averaged number density, of the plasma electrons. Note that E , p , h A i , ω , k , ω p , n , and γ are fields.The above physical quantities are coupled together by the conservationof wave action ∂ ˜ t ( h A i ω ) + c e ∇ · ( h A i k ) = 0 (4)and the continuity equation ∂ ˜ t n + e ∇ · ( n v ) = 0, i.e. ∂ ˜ t ( ω p E ) + c e ∇ · ( ω p p ) = 0 (5)where p = m e γ v . The symbol e ∇ denotes the gradient operator with respectto ˜ x , ˜ y , ˜ z . The system of field equations (1), (2), (4), (5) was previously usedin a self-consistent model [25] of the interaction between the laser pulse andthe plasma wake it excites, in the context of electron acceleration.If the forces due to the magnetic field sourced by the averaged current arenegligible in comparison to the forces produced by the laser pulse then theflow of electrons can be chosen to be irrotational; thus, p can be expressed interms of a potential e Ψ as p = e ∇ e Ψ. Furthermore, if the electric field sourcedby the averaged charge density can be neglected then one can determine E using E = − ∂ ˜ t e Ψ. The expression c e h A i = ( ∂ ˜ t e Ψ) − c ( e ∇ e Ψ) − m e c (6)for h A i is then obtained from (1). For completeness, it is straightforward toshow that (1) alongside the expressions E = − ∂ ˜ t e Ψ, p = e ∇ e Ψ, v = p / ( m e γ )3olve the local equation of momentum balance ∂ ˜ t p + ( v · e ∇ ) p = − e m e γ e ∇ h A i (7)where the force on the electrons is due entirely to the laser pulse. The termon the right-hand side of (7) is the relativistic version of the “ponderomotiveforce” exerted by the laser pulse, which is a fundamental concept in laser-driven plasma-based electron acceleration [26]. See ref. [27] for a derivationof the relativistic ponderomotive force using the method of multiple scales.The conservation of wave action, equation (4), emerges when the vec-tor potential of the laser pulse is subject to the eikonal approximation [25].Hence, the local angular frequency and local wave vector of the laser pulsecan be expressed in terms of the phase e Φ of the laser pulse as k = e ∇ e Φ, ω = − ∂ ˜ t e Φ; hence the expression ω p = ( ∂ ˜ t e Φ) − c ( e ∇ e Φ) (8)for ω p follows from (2).Expressing (4) and (5) entirely in terms of e Φ, e Ψ, and fundamental con-stants, gives the Euler-Lagrange equations for e Φ and e Ψ obtained from theLagrangian density L = ε ω p h A i /
2. Thus, we have the action S [ ˜Φ , ˜Ψ] = ε e c Z d ˜ x (cid:8) (( ∂ ˜ t ˜Φ) − c ( e ∇ ˜Φ) )(( ∂ ˜ t ˜Ψ) − c ( e ∇ ˜Ψ) − c ) (cid:9) . (9)In cases where it is possible to express ω p h A i as a product of a functionof (˜ t, ˜ z ) and a function of (˜ x, ˜ y ), where the laser pulse propagates in the˜ z -direction, the approximation R f (˜ x, ˜ y ) d ˜ xd ˜ y ≈ max( f ) R S d ˜ xd ˜ y , for a suffi-ciently well behaved function f , allows the introduction of˜Λ = max[ ω p h A i ] ˜ x, ˜ y R S d ˜ xd ˜ y [ ω p h A i ] | ˜ x =˜ y =0 , (10)where S is the region on which ω p h A i is non-zero and max[ . . . ] ˜ x, ˜ y denotesthe maximum value of the bracketed function in (˜ x, ˜ y ). Thus the actionbecomes S [ ˜Φ , ˜Ψ] ≈ ε q c Z d ˜ td ˜ z ˜Λ (cid:8) (( ∂ ˜ t ˜Φ) − c ( ∂ ˜ z ˜Φ) )(( ∂ ˜ t ˜Ψ) − c ( ∂ ˜ z ˜Ψ) − m c ) (cid:9) | ˜ x =˜ y =0 , (11)4here the derivatives in the ˜ x , ˜ y plane are assumed to be negligible. Theline ˜ x = ˜ y = 0 has been chosen to lie along the centre of the laser pulse. Thequantity R S d ˜ xd ˜ y is the cross-sectional area of the laser pulse (spot area), andthus ˜Λ will depend on (˜ z, ˜ t ).The above considerations suggest a relativistic bi-scalar field theory givenby the action S [Φ , Ψ] = 12 ~ Z d x √− η Λ η µν ∂ µ Φ ∂ ν Φ( η στ ∂ σ Ψ ∂ τ Ψ + 1) , (12)where η µν is the background Minkowski metric with signature ( − , +), and µ, ν = 0 ,
1. The fields Φ, Ψ, Λ, η µν and the coordinates x = t , x = z aredimensionless. The action (11) is obtained from (12) using the substitutions t = c ˜ tl ∗ , z = ˜ zl ∗ , ˜Λ = l ∗ Λ˜Φ | ˜ x =˜ y =0 = s ~ e ε m e c l ∗ Φ , ˜Ψ | ˜ x =˜ y =0 = m e cl ∗ Ψ , (13)where the length scale l ∗ has been introduced to facilitate the non-dimensionalisationand has no direct physical significance. The field equations arising from thevariation of the action (12) are ∂ µ (Λ ∂ ν Φ ∂ ν Φ ∂ µ Ψ) = 0 , (14) ∂ µ (Λ( ∂ ν Ψ ∂ ν Ψ + 1) ∂ µ Φ) = 0 , (15)where indices are raised using the background metric. Eqs. (14), (15) allowus to derive two effective metrics through a linearisation process. Consider the perturbed fields Ψ = Ψ + ǫ Ψ + O ( ǫ ), Φ = Φ + ǫ Φ + O ( ǫ ),where ǫ is the perturbation parameter and Φ , Ψ solve eqs. (14), (15)exactly. Field eqs. (14), (15) in first order of ǫ give ∂ µ (2Λ ∂ ν Φ ∂ ν Φ ∂ µ Ψ + Λ ∂ ν Φ ∂ ν Φ ∂ µ Ψ ) = 0 , (16) ∂ µ (2Λ( ∂ ν Ψ ∂ ν Ψ ) ∂ µ Φ + Λ( ∂ ν Ψ ∂ ν Ψ + 1) ∂ µ Φ ) = 0 , (17)respectively. The perturbations Φ , Ψ are coupled and, in general, their fieldequations (16), (17) cannot be readily expressed in a manner that reveals oneor more effective metrics. However, a pair of effective metrics follows from5onsidering high frequency perturbations of the form Ψ = Re( a exp( iK/ ˇ ǫ )),Φ = Re( b exp( iK/ ˇ ǫ )), where ˇ ǫ is a parameter that facilitates the approxi-mation. In this case, the lowest order of ˇ ǫ yieldsΛ ∂ µ K [2 b∂ µ Ψ ∂ ν Φ ∂ ν K + a∂ µ K∂ ν Φ ∂ ν Φ ] = 0 . (18)Λ ∂ µ K [2 a∂ µ Φ ∂ ν Ψ ∂ ν K + b∂ µ K ( ∂ ν Ψ ∂ ν Ψ + 1)] = 0 . (19)These equations can be written as the matrix (cid:18) Λ ∂ µ K∂ µ K∂ ν Φ ∂ ν Φ ∂ µ K∂ µ Ψ ∂ ν K∂ ν Φ ∂ µ K∂ µ Ψ ∂ ν K∂ ν Φ Λ ∂ µ K∂ µ K ( ∂ ν Ψ ∂ ν Ψ + 1) (cid:19) (20)acting on ( a, b ) T . The determinant of the matrix in (20) must be zero so that a , b are non-zero. The determinant can be factorised to give two effectivemetrics g eff µν, + , g eff µν, + whose inverses are g µν eff , + = s Λ { η µσ η ντ ( ∂ σ Φ ∂ τ Ψ + ∂ τ Φ ∂ σ Ψ )+ q ( η στ ∂ σ Ψ ∂ τ Ψ + 1) η γδ ∂ γ Φ ∂ δ Φ η µν } , (21) g µν eff , − = s Λ { η µσ η ντ ( ∂ σ Φ ∂ τ Ψ + ∂ τ Φ ∂ σ Ψ ) − q ( η στ ∂ σ Ψ ∂ τ Ψ + 1) η γδ ∂ γ Φ ∂ δ Φ η µν } , (22)where the constant s satisfies s = 1 and has been introduced for conveniencelater in the analysis. It is straightforward to confirm that the determinant ofthe matrix (20) can be written as g µν eff , + K µ K ν g στ eff , − K σ K τ , where K µ = ∂ µ K .Note that raising and lowering indices is done with the background metric η µν . The properties of the effective metrics depend on the properties of thefields Φ , Ψ . We will present a regime in which one of these effective metricsis conformally related to the Schwarzschild metric. To obtain an effective metric that is conformally related to the Schwarzschildmetric, the ratio of the diagonalised effective metric components must beidentified with the ratio of the Schwarzschild metric components (1 − GM/ ( c r ))and (1 − GM/ ( c r )) − . The first step in achieving this is to assume thatΛ is a function of z only, and the fields are of the form Φ = γ Φ t + h Φ ( z ),Ψ = γ Ψ t + h Ψ ( z ) for some constants γ Φ , γ Ψ and some functions h Φ , h Ψ .Introducing two transformations τ ± = a ± t ± f ± ( z ) for some constants a ± ,6nd choosing f ± ( z ) such that the effective metrics become diagonal, gives therequirement − (cid:16) − z S z (cid:17) = a − ± (cid:16) h ′ Φ h ′ Ψ ± p ( h ′ − γ )( h ′ − γ + 1) (cid:17) γ Φ γ Ψ ∓ p ( h ′ − γ )( h ′ − γ + 1) − ( γ Ψ h ′ Φ + γ Φ h ′ Ψ ) h ′ Φ h ′ Ψ ± √ ( h ′ − γ )( h ′ − γ +1) (23)where the dimensionless quantity z S corresponds to the horizon in the Schwarzschildmetric, prime denotes derivative with respect to z , and ± corresponds to g µν eff , ± . Eq (23) suggests using scaled variables h Φ = γ Φ ˇ h Φ , h Ψ = γ Ψ ˇ h Ψ . Noweqs. (14), (15) yield the relationship β ( ε − (ˇ h ′ Ψ ) )ˇ h ′ Φ = (1 − (ˇ h ′ Φ ) )ˇ h ′ Ψ , (24)where β is a constant of integration and ε = ( γ − /γ . Note that(ˇ h ′ Ψ ) < ε < ε , whilethe lower bound is required so that the effective metric components do notbecome imaginary. Introducing the field h = ˇ h ′ Ψ /ε and choosing β = 1 /ε results in ˇ h ′ Φ = h from eq. (24). Furthermore h < t , z coordinates which are given by g , ± = s Λ γ Φ γ Ψ (2 ∓ | ε | (1 − h )) , (26) g , ± = s Λ γ Φ γ Ψ (2 εh ± | ε | (1 − h )) , (27) g , ± = − s Λ γ Φ γ Ψ (1 + ε ) h. (28)Note that by definition Λ >
0, thus the choice s = − g , ± component and the off-diagonal terms, as well as Λ, will be non-zero forall values within the constraints (25). However equating g , ± to zero andsolving for h leads to a horizon if ε is chosen appropriately. The value of h when (27) equals zero satisfies h = | ε | / ( | ε | ∓ ε ). When ε is positive, therewill be no horizon in g µν eff , + , while there will be a horizon at h = 1 / g µν eff , − .The converse is true for ε <
0. Since these two outcomes are equivalent, thecase of ε < g µν eff , ± ) < − < ε < / < h < g µν eff , + is of interest and will7e explored further, as the other effective metric does not contain a horizon.Let ν = − ε for convenience. All of the above considerations let us write theright-hand side of eq. (23) as a − (1 − h ) ν ν (1 − h )(2 − ν (1 − h )) − (1 − ν ) h . (29)Note that the numerator is always positive, and the denominator is propor-tional to the determinant of g µν eff , + and as such (29) is always negative for1 / < h < < ν <
1. Thus eq. (23) will always have a solution for h ( z ) in the specified range. An expression for a + is obtained by matchingthe limit of z → ∞ to h →
1, yielding a = (cid:18) ν ν (cid:19) . (30)An algebraic solution to eq. (23) can be found since it is a quadratic equationin h , however the solution is cumbersome and a simpler approach is availablefor establishing the behaviour of the laser-driven plasma. By taking thesquare root of eq. (23), differentiating with respect to z and evaluating at z = z S gives h ′ | S = − a + (1 − ν ) / (6 νz S ), where | S indicates evaluation at z = z S , and h | S = 1 / √ a + must be negativebecause h > / h ′ | S >
0, thus the negative root of eq. (30) is required,and hence h ′ | S = 13 1 − ν ν z S . (31)Introducing the dimensionless amplitude of the laser pulse a given by a = e p h A i / ( m e c ) and using eq. (6) gives an expression for ν : ν = √ a p a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S . (32)Note that Λ ∝ [ h (1 − h )] − follows from eqs. (14), (15). It follows thatΛ ′ | S = 0 and it is straightforward to show that (Λ ′′ / Λ) | S = 9 h ′ | S , which canbe used to obtain ˜ z S = 1 − ν ν s ˜Λ d ˜Λ /d ˜ z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S . (33)The effective metric g µν eff , + is conformally related to the Schwarzschild met-ric. However, the conformal factor is regular at the event horizon, and the sur-face gravity and Hawking temperature are independent of this conformal fac-tor [28]. By construction ˜ z S = GM/c and thus, using T H = ~ c / (8 πk B GM ),8he analogue Hawking temperature is given by T H = ~ c πk B ν − ν s d ˜Λ /d ˜ z ˜Λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S . (34)The Hawking temperature can be calculated if the dimensionless amplitude a and the laser cross-sectional area ˜Λ near ˜ z = ˜ z S are known. Since Λ ′ | S = 0,˜Λ can be expressed as˜Λ = ˜Λ | S (cid:18) z − ˜ z S ) ˜ l + O (cid:0) (˜ z − ˜ z S ) (cid:1)(cid:19) , (35)where ˜ l has dimensions of length. It follows that ( ˜Λ / ( d ˜Λ /d ˜ z )) | S = ˜ l , andthus the Hawking temperature is given by T H = ~ c πk B ν − ν l . (36)For practical reasons, ˜ l cannot be less than approximately the plasma wave-length λ p , and the dimensionless amplitude should satisfy a ≤
1. As anexample λ p ≈ µ m is achievable [26] for maintaining an intense near-IRlaser pulse propagating through a plasma. This results in ν . .
77, the massof the effective black hole satisfies M & . × kg, and the associatedHawking temperature satisfies T H .
25 K.
We obtained the analogue Hawking temperature of a laser-driven plasmasystem. Perturbed field equations governing a laser-driven plasma were lin-earised, and the perturbations were assumed to have high frequency in orderto derive two effective metrics. The required properties of the fields werefound such that one of the effective metrics is conformally related to theSchwarzschild metric. An expression for the Hawking temperature associatedwith the analogue black hole has been derived. This temperature dependson the values of the dimensionless amplitude and the laser spot area near theanalogue event horizon. We have presented one possible way of determiningthe spot area in terms of the plasma wavelength, with which we demonstratedthat for a high-intensity near-IR laser the analogue Hawking temperature isless than approximately 25 K.In common with standard analytical treatments of laser-driven plasmaaccelerators, our results are based on a ‘cold’, collisionless, model of the9lasma electrons. However, a comparison of our results and typical plasmatemperatures suggests that a detailed model of the laser-driven plasma isneeded to confidently identify signatures of the analogue Hawking effect.The temperature of the plasma electrons in a laser-driven plasma acceleratoris ∼ × K [26], which is ∼ × times larger than the expected analogueHawking temperature.Even so, for comparison, it is claimed [29] that an analogue Hawking tem-perature of 1 . This work was supported by the UK Engineering and Physical Sciences Re-search Council grant EP/N028694/1 (D.A.B.), and the Lancaster UniversityFaculty of Science and Technology (C.F.). All of the results can be fullyreproduced using the methods described in the article.