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Excitation of a Uniformly Moving Atom Through Vacuum Fluctuations

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Excitation of a Uniformly Moving Atom Through Vacuum Fluctuations PHYSICAL REVIEW RESEARCH 1, 033027 (2019) Excitation of a uniformly moving atom through vacuum fluctuations Anatoly A. Svidzinsky Department of Physics & Astronomy, Texas A&M University, College Station, Texas 77843, USA (Received 7 August 2019; published 16 October 2019) In systems with broken Lorentz invariance the vacuum state of the field can depend on the choice of the inertial reference frame. This leads to observable effects. For example, the motion of an atom with a constant velocity above a flat metal surface can yield atomic excitation with simultaneous emission of a surface plasmon, while uniform motion of the atom through an optical cavity can yield atomic excitation with emission of a photon into cavity modes. We calculate the probability of these processes and make a connection with the Cherenkov radiation of a neutral atom uniformly moving through a medium and Unruh radiation of the atom accelerated in free space. In all these cases the moving atom becomes excited and the field quantum is emitted through vacuum fluctuations at the expense of the kinetic energy of the atom. DOI: 10.1103/PhysRevResearch.1.033027 I. INTRODUCTION Strong enhancement of radiation can be produced by a fast nonadiabatic switch of the interaction of atoms with the field In free space, the quantum vacuum is Lorentz invariant, so at the boundaries of a cavity [13]. a uniformly moving observer would not see any effect due Dependence of the field vacuum state on a choice of to motion, but an accelerated observer would. For example, noninertial reference frame leads to the acceleration radiation. an atom accelerated through Minkowski vacuum can become In one frame the field could be in a vacuum state, but in excited. This is known as the Unruh effect [1] (or Fulling- another frame the same state of the field can contain particles Davies-Unruh effect in full [1–3]). Interpretation of the effect which can excite the atom. depends on the choice of the reference frame. In systems with broken Lorentz invariance the vacuum A noninertial observer having a proper constant accelera- state can also depend on a choice of the inertial reference tion a, i.e., a Rindler observer [4], sees that space is filled with frame. The presence of matter breaks Lorentz invariance of thermal photons with Unruh temperature T proportional to U the quantum vacuum. As a consequence, an atom uniformly the acceleration [1] moving relative to matter can become excited by a mecha- ha¯ nism similar to that of the Unruh effect. Cherenkov radiation TU = . (1) 2πkBc produced by a uniformly moving atom through a medium is From the perspective of the accelerated observer the ground- an example. It can be understood based on the energy and state atoms accelerated through the Minkowski vacuum will momentum conservation during photon emission [14,15]. V be promoted to an excited state by absorption of the Rindler For a nonrelativistic motion of the atom with velocity n particles (Unruh effect) [1]. However, an inertial observer through a medium with the refractive index the conservation interprets the absorption of a Rindler particle as the emission equations in the laboratory frame read of a Minkowski particle [5], which is known as acceleration h¯k + p = 0, (2) radiation. A similar mechanism yields excitation of an atom freely falling in gravitational field [6] or a fixed atom in V · p + Eatom + h¯ν = 0, (3) Minkowski space in the presence of an accelerated mirror [7]. It is necessary to reach accelerations on the order of whereh ¯k andh ¯ν = hck¯ /n are the momentum and energy of ∼1020 m/s2 to obtain an Unruh radiation corresponding to the emitted photon, p is the change of the atom’s momen- a temperature of a few kelvins. That is effect is very small. tum, V · p is the change of the atom’s kinetic energy, and As a consequence, direct experimental evidence for the Unruh Eatom is the change of the atom’s internal energy. effect is still lacking. One closely related observation in Equations (2) and (3)give laboratories is the electron depolarization in storage rings [8], c which can be connected to the “circular Unruh effect” [9–12]. E = hkV¯ cos θ − , (4) atom Vn where θ is the photon emission angle relative to V. Equation (4) shows that if the photon is emitted inside the Cherenkov Published by the American Physical Society under the terms of the cone (cos θ>c/Vn) then Eatom > 0; that is, the atom be- Creative Commons Attribution 4.0 International license. Further comes excited [14–16]. This is analogous to the Unruh ac- distribution of this work must maintain attribution to the author(s) celeration radiation for which an inertial observer sees that and the published article’s title, journal citation, and DOI. the accelerated atom becomes excited by emitting a photon. 2643-1564/2019/1(3)/033027(8) 033027-1 Published by the American Physical Society ANATOLY A. SVIDZINSKY PHYSICAL REVIEW RESEARCH 1, 033027 (2019) In both cases the energy is gained from the atom’s kinetic energy. If emission occurs outside the Cherenkov cone then Eatom < 0 and the atom must go from the excited to the ground state. In the reference frame moving with velocity V the photon frequency shifts and becomes ck − V · k ν = n . (5) 1 − V 2 FIG. 1. Atom is moving with constant velocity V above metal c2 surface and becomes excited by emitting a surface plasmon. Elec- tromagnetic field of the surface plasmon exponentially decays away If V > c/n there are photons with negative frequency. The from the metal surface. wave frequency becomes negative if it is measured in a coor- dinate system which is moving faster than the phase velocity II. EXCITATION OF AN ATOM UNIFORMLY MOVING of the wave. A change in the sign of the frequency of the ABOVE A METAL SURFACE wave between two inertial reference frames corresponds to a reversal of the phase velocity. Yet from the point of view of Here we consider a flat interface between a metal and the relation E = h¯ν, a positive quantum of energy apparently vacuum (see Fig. 1). In such system there exist collective exci- becomes a negative-energy one [17]. tations of the electromagnetic field and metal electrons which The behavior of the negative-energy quanta is essential propagate along the surface, known as surface plasmons [34]. to a kinematics of amplification of waves which can explain Electromagnetic field in the surface plasmon exponentially certain amplification phenomena [18]: a variety of traveling- decays away from the surface as shown in Fig. 1. For example, wave-tube-type amplifiers [19], a resistive-wall amplifier [20], in the Drude model of metal if we omit dissipation the and amplification of ultrasound waves in semiconductors [21]. frequency-dependent dielectric function reads The negative-energy quanta also play a role in Cherenkov ω2 p radiation, quantum friction, and the Hawking radiation. The ε(ν) = 1 − , ν2 quantum friction is predicted to occur between relatively moving dielectrics at zero temperature [22,23] and stems and the field outside the metal is given by [34] from the mixing of positive- and negative-frequency waves −iνt+ikz−αx φν (t, r) = e , (7) in the two materials [24,25]. Quantum friction can also occur in rotating dielectric bodies [26–28]. At zero temperature, where ν is the mode frequency which is related to the plasmon vacuum friction transforms mechanical energy into light emis- wave number k through the dispersion relation sion and heats the rotating body. Hawking radiation [29] and ω2 ω4 its laboratory analogs [30,31] originate from the change in p p ν2 = + c2k2 − + c4k4, (8) the sign of the frequency of a wave as it crosses the event 2 4 horizon. 2 ω = nee In the uniformly moving frame the field Hamiltonian con- p ε is the electron plasma frequency, ne is the density + 0me sistent with the canonical commutation relations [bˆν , bˆ ] = 1 ν of free electrons of mass me, and α is the inverse of the decay reads [16,32] length which depends on k, + ω4 ω2 ˆ = ν ˆ ˆ , p p H h¯ bν bν (6) α2 = + k4 − . (9) ν 4c4 2c2 We assume that in the laboratory frame the plasmon field is in + where bˆν is the creation operator for a photon in the medium a vacuum state. with the Doppler-shifted frequency ν. Since in the frame In the inertial frame moving with velocity V along the moving with V > c/n there are photons with negative fre- metal surface the mode frequency is Doppler shifted and quencies, in this frame there is no lower bound on the photon becomes energy. ν − ν = kV . From the perspective of the moving observer, photons (10) − V 2 emitted inside the Cherenkov cone have negative energy [33] 1 c2 which excites the atom, while photons emitted outside the In the moving frame the plasmon frequency is negative for Cherenkov cone have positive energy which deexcites the > c2−V 2 ω / atom. modes with k 2c2−V 2 p V . When phase velocity of the Next we investigate a similar problem with translational wave is smaller than medium velocity the energy of the symmetry along the y, z, and t axes which somewhat resem- quantum which propagates opposite to the medium motion bles Cherenkov radiation in the medium and Unruh acceler- is negative. Negative frequency implies that the vacuum ation radiation of an atom moving through the Minkowski state in the moving frame is different from the vacuum in vacuum. the laboratory frame, which yields observable effects.
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