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RESEARCH 1, 033027 (2019)

Excitation of a uniformly moving through vacuum fluctuations

Anatoly A. Svidzinsky Department of & 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 surface can yield atomic excitation with simultaneous emission of a , while uniform motion of the atom through an optical cavity can yield atomic excitation with emission of a 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 is emitted through vacuum fluctuations at the expense of the kinetic 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 with the field In free space, 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 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 breaks Lorentz invariance of thermal 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 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 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 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 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 of metal if we omit dissipation the and amplification of ultrasound waves in [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 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 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. For

033027-2 EXCITATION OF A UNIFORMLY MOVING ATOM THROUGH … PHYSICAL REVIEW RESEARCH 1, 033027 (2019) example, a ground-state atom moving above the metal surface the atom with probability with a constant velocity can become excited. A classical P = g2e−2αxt 2, (15) analogy of this effect has been discussed in Ref. [35], while the rate of spontaneous emission by a moving two-level atom where t is the time of the atom’s motion above the metal interacting with the near field of a plasmonic slab has been surface. calculated in Ref. [36]. Optical instabilities in moving media In order to increase excitation probability, instead of atoms, are linked to a spontaneous parity-time-symmetry breaking one can shoot polar above the metal surface. The of the system [37]. rotational excitations of molecules are in the Interpretation of the effect depends on the choice of the regime and have an anharmonic energy spectrum. The an- inertial reference frame. In the laboratory frame the atom harmonicity allows us to pick out a two-level subspace in the becomes excited by emitting a surface plasmon with positive rotational spectrum and treat the molecular ensemble as a two- energy at the expense of kinetic energy of the atom. In level system. Trapping molecules close to the metal surface the moving frame the atom becomes excited by emitting a yields a strong electric dipole with g ∼ 104 Hz [38]. 10 3 surface plasmon with negative energy which insures energy For ωa = 10 Hz and velocity V = 10 m/s the field de- conservation. cay length outside the metal for the resonant mode is V/ωa = Next we calculate the excitation probability. We will per- 100 nm. So, in order to achieve strong coupling, one should form calculations in the laboratory frame. In this frame the shoot the molecules closer than 100 nm to the surface. For atom’s trajectory is these parameters the will be excited with almost τ V τ unit probability after moving 10 cm above the metal. Thus, t =  , z =  , (11) the effect is not very small. − V 2 − V 2 1 c2 1 c2 One should mention that the molecule experiences the Casimir-Polder interaction with the metal surface [39]. This τ where is the proper time of the atom. The interaction interaction has the same origin as the of energy Hamiltonian between the atom moving at a distance x above levels due to the zero-point fluctuations of the electromagnetic ν the metal and the surface plasmon with frequency reads field. The presence of the surface alters the spectrum of the − ω τ i a vacuum fluctuations. As a consequence, the molecule’s energy Hˆint(τ ) = hg¯ {aˆν φν [t(τ ), r(τ )] + H.c.}(ˆσe + H.c.), shift depends on the distance to the surface, which results in (12) an interaction potential. The Casimir-Polder force acting on wherea ˆν is the plasmon annihilation operator,σ ˆ is the atomic the molecule is perpendicular to the flat surface. It decays as 1/x4 for short distances x and as 1/x5 for large separation due lowering operator, ωa is the atomic frequency in the atom’s frame, and g is the atom-plasmon coupling constant. Since the to retardation. atom feels the local value of the field, the field mode function For the molecule moving parallel to the metal surface in Eq. (12) is taken at the atom’s location t(τ ), x, z(τ ). the vacuum fluctuations can yield molecule excitation with Excitation of the atom with simultaneous emission of a simultaneous emission of the surface plasmon. This occurs + + at the expense of the molecule’s kinetic energy which yields plasmon is due to the counterrotating terma ˆν σˆ in the interaction Hamiltonian. The probability P that the atom with the appearance of a friction force opposite to the direction of the motion. Contrary to the Casimir-Polder interaction, the transition frequency ωa becomes excited and a plasmon with frequency ν is emitted is given by the integral from the matrix friction force decays exponentially with the distance to the element between the initial state of the system in which the surface x. atom is in the ground state |b and there are no plasmons and the final state in which the atom is in the excited state |a and III. EXCITATION OF OPTICAL VORTICES IN A CAVITY one surface plasmon with frequency ν is present: BYAMOVINGATOM    t 2 1   Next we investigate a system with no translational sym- = τ ν , | ˆ τ | ,  P 2  d 1 a Hint( ) 0 b  metry. Namely, we consider a cylindrically symmetric cavity h¯ 0   or cavity with a rectangular symmetry for which normal  t 2 − α ν τ − τ ω τ modes of the field are Laguerre-Gaussian (optical vortices) or = 2 2 x τ i t( ) ikz( ) i a  . g e  d e e  (13) Hermite-Gaussian, respectively. The Laguerre-Gaussian and 0 Hermite-Gaussian modes form a complete set. The presence Inserting here Eq. (11) we obtain of the cavity breaks the Lorentz invariance of the field equa-    t 2 tions because the boundary conditions at the cavity mirrors 2 −2αx i(ν+ω )τ  P = g e  dτe a  , (14) depend on the reference frame. 0 We assume that the cavity is fixed in Minkowski space-time where ν is the plasmon frequency in the reference frame of and a two-level (a and b) atom is moving through the cavity the moving atom. with a constant velocity V along a straight line so that the  The mode for which ν =−ωa is in resonance. For this atomic trajectory is given by the equations mode the integral in Eq. (14) linearly grows with time. If x = Vt, y = d, z = 0, ωa  ωp and V  c then for the resonant mode k = ωa/V . The motion of the atom yields emission of the surface plas- where d is the impact parameter and z is the cavity axis (see mon into this resonant mode with simultaneous excitation of Fig. 2). The atom is moving from infinity and passes through

033027-3 ANATOLY A. SVIDZINSKY PHYSICAL REVIEW RESEARCH 1, 033027 (2019)

radius of curvature,      z 2 z 2 w(z) = σ 1 + , R(z) = z 1 + R , zR z

while σ is the beam radius at the waist. The optical vortex carries orbital angular momentum mh¯ per photon. We assume that the normalization factor in φν is subsumed under the coupling constant g. FIG. 2. A ground-state atom is moving through a cavity with For the modes (18) at the atomic position the mode func- a constant velocity at impact parameter d. Virtual transitions yield tion is excitation of the atom and emission of a photon in cavity modes. + |m| [d i sgn(m)Vt] −(V 2t 2+d2 )/σ 2 −iνt φν (t, r(t )) = e e , (19) σ |m|+1 the center of the cavity in the direction perpendicular to the cavity axis. This effectively causes a change of the coupling and Eq. (17) yields   between the atom and the cavity modes with time. 2 2 V g 2 2 We assume that initially the atom is in the ground state |b P = 1 − e−2d /σ m c2 σ 2|m|+2 and there are no photons in the cavity modes. In the interaction   picture the interaction Hamiltonian between the atom and a  ∞ 2 ×  − |m| −V 2t 2/σ 2+i(ω+ν)t  . photon with frequency ν is given by Eq. (12) in whicha ˆν is the  dt[d i sgn(m)Vt] e  (20) −∞ photon annihilation operator, τ = t 1 − V 2/c2 is the proper time for the atom, and g is the coupling constant between For m = 0 the integration over time reduces to the follow- the atom and the cavity mode φν (t, r). Since the atom feels ing integral: the local value of the field, φν is taken at the atom’s location  ∞ √ 2 2 π 2 2 t(τ ), r(τ ). dte−α t +ibt = e−b /4α . (21) The change of the coupling between field and the atom with −∞ |α| time can result in atomic excitation with simultaneous photon To find the time integral for arbitrary m, namely, emission into the cavity modes due to the counterrotating  +σ + ∞ termsa ˆν ˆ in the interaction Hamiltonian. The probability −α2t 2+ibt |m| ω dte (d − iαt ) , P that the atom with transition frequency a becomes excited −∞ and a photon in the mode φν (t, r) is generated is given by the integral we rewrite Eq. (21)as  √   ∞ π  ∞ 2 −α2t 2+ibt+γ (d−iαt ) = −b2/4α2 −γ 2/4+γ (d+b/2α). 2 ∗ iωaτ  dte e e P = g  dτφν [t(τ ), r(t(τ ))]e  . (16) −∞ |α| −∞ Differentiating over γ from both sides |m| times we obtain Changing the integration variable to t we obtain  ∞     −α2t 2+ibt − α |m| 2  ∞ 2 dte (d i t ) V 2 ∗ iωt  −∞ P = 1 − g  dtφν [t, r(t )]e  , (17) √  c2 −∞ |m|  π 2 2 d 2 = e−b /4α e−γ /4+γ (d+b/2α) . |α| γ |m|  2 2 d γ =0 where ω = 1 − V /c ωa. Next we discuss excitation of the optical vortex cavity modes by the uniformly moving atom. Taking into account that Hermite polynomials Hm(x)aregiven We assume that the cavity can support the Laguerre- by the exponential generating function, namely, Gaussian modes. For a cavity with a cylindrical symmetry, the ν ∞ m Laguerre-Gaussian modes of frequency can be written as a 2 γ e−γ /4+xγ = H (x) , function of the polar radius ρ = x2 + y2 and the azimuthal m 2mm! angle ϕ. We consider a subset of the Laguerre-Gaussian cavity m=0 modes in the form of optical vortices we obtain  |m|  imϕ |m| d 2 H| | x e ρ 2 2 −γ /4+xγ  m ( ) φ , = −ρ /w (z) e  = . ν (t r) | | e γ |m| |m| w(z) w m (z) d γ =0 2   νρ2 z × − | |+ −iνt , As a result, the integral over time is given by cos ( m 1) arctan e  2cR(z) zR ∞ −α2t 2+ibt |m| (18) dte (d − iαt ) −∞ √   π where m = 0, ±1, ±2, ... is the azimuthal parameter, zR is the −b2/4α2 b = e H| | d + . Rayleigh length, w(z) and R(z) are the beam radius and the 2|m||α| m 2α

033027-4 EXCITATION OF A UNIFORMLY MOVING ATOM THROUGH … PHYSICAL REVIEW RESEARCH 1, 033027 (2019)

angular momentum of the m =±1 vortex pair is   V 2 πhg¯ 2(ω + ν)d hP¯ + − hP¯ − = 2 1 − 1 1 c2 V 3   2d2 (ω + ν)2σ 2 × exp − − . σ 2 2V 2 This expression is positive for d > 0 and negative otherwise. The excitation probability (22) is governed by an exponen- tial factor   2d2 (ω + ν)2σ 2 exp − − , σ 2 2V 2 which is usually small. To avoid exponential suppression of FIG. 3. Excitation probability of an optical vortex with the right vorticity m =+1 in a cavity by a uniformly moving atom as a excitation one should satisfy conditions function of the impact parameter d for σ (ω + ν)/2V = 0.5. For d  σ, V  (ω + ν)σ. d > 0(d < 0) the atom moves in the same (opposite) direction as the energy flow in the vortex. The probability is normalized by its The latter condition can be approximately written as maximum value. V σ  2π , c λ This yields the following answer for the probability that the where λ = 2πc/(ω + ν) is the wavelength corresponding to atom becomes excited and a photon in the mode m is emitted: the net excitation energy of the atom and the fieldh ¯(ω + ν).     ∼ σ ∼ λ 2 π 2 σ ω + ν The excitation probability is largest if V c and .For V g 2 d ( ) P = 1 − H| | + sgn(m) such parameters m c2 22|m|V 2 m σ 2V   ℘2 ν2 2d2 (ω + ν)2σ 2 P ∼ ab ∼ 0.001, × exp − − . (22) max hc¯ 3 σ 2 2V 2 where ℘ab is the atomic dipole moment matrix element. Equation (22) shows that modes with opposite direction Ground-state atoms emit photons into the cavity modes + + of the orbital angular momentum are excited with different due to the counterrotating termsa ˆν σˆ in the Hamiltonian and probability. For example, for m =±1 the Hermite polynomial absorb photons already present in the cavity by means of the + reduces to H1(u) = 2u and usual resonant absorption described by the termsa ˆν σˆ .The     probability of photon absorption Pabs is given by Eq. (16)in 2 π 2 σ ω + ν 2 ∗ V g d ( ) which φν is replaced with φν . This is equivalent to change P± = 1 − ± 1 c2 V 2 σ 2V ν →−ν, m →−m in Eqs. (22) which yields the following   2 2 2 relation between probability of absorption and emission: 2d (ω + ν) σ   × − − . σ ω−ν   exp (23) 2 d ( ) 2 σ 2 2 H| | − sgn(m) 2ωνσ 2V abs = m  σ 2V  exc. P σ ω+ν exp P (24) H 2 d + sgn(m) ( ) V 2 If dV > 0 the optical vortex with the right vorticity is |m| σ 2V = σ 2 ω + excited with greater probability. Moreover, if dV ( If the atomic beam is injected into the cavity then after ν / =− ) 2 the left vorticity mode m 1 is not becoming excited a sufficiently long time the cavity field will reach a steady at all. state. Statistics of photons in the cavity modes can be obtained Figure 3 shows the probability of excitation of an optical using the quantum master equation technique, as developed vortex with the right vorticity P+1 as a function of the impact in the quantum theory of the laser [40]. If atoms are ejected σ ω + ν / = . > parameter d for ( ) 2V 0 5. For d 0 the atom randomly the equation of motion for the density matrix of the moves in the same direction as the energy flow in the vortex cavity field mode is [13] and excitation probability is larger. This is intuitive from the ρ perspective of the angular momentum conservation. Namely, d n,n =−e[(n + 1)ρn,n − nρn−1,n−1] angular momentum of the translational atomic motion is dt transferred into the orbital angular momentum of the field. − a[nρn,n − (n + 1)ρn+1,n+1], (25) Nevertheless, even for d < 0 when the atom moves in the   direction opposite to the energy flow in the vortex the excita- where e and a are emission and absorption rates of the  / exc/ abs tion probability of the vortex mode is nonzero. This, however, photon in the cavity. The ratio e a is equal to P P . does not contradict the conservation of the angular momentum The steady-state solution of Eq. (25) exists when absorption abs > exc because vortex modes are excited in pairs with positive and is greater than emission (P P ) and is given by thermal negative m and the average orbital angular momentum of distribution n the generated vortex pair is in the direction of the angular ρ = n¯ν , n,n + (26) momentum of the atom’s motion. For example, the average (¯nν + 1)n 1

033027-5 ANATOLY A. SVIDZINSKY PHYSICAL REVIEW RESEARCH 1, 033027 (2019)

For the Hermite-Gaussian mode (30) the field mode func- tion at the atomic position is √  √  −iνt e 2Vt 2d −(V 2t 2+d2 )/σ 2 φν (t, r(t )) = Hm Hn e σ w0 w0 (31) and Eq. (17) yields for the probability of atom excitation with photon emission in the Hermite-Gaussian mode   √  2 2 V g −2d2/σ 2 2 2d P , = 1 − e H m n c2 σ 2 n σ  √    ∞ 2  2Vt −V 2t 2/σ 2+i(ω+ν)t  ×  dtHm e  . (32)  −∞ σ  FIG. 4. Probability of emission and absorption into the vortex mode with m =+1 as a function of the impact parameter d for Time integration can be calculated using the formula σ (ω + ν)/V = 2.6andσ (ω − ν)/V = 0.2. The plots are normal-  √ abs ∞ m 2 2 m/2 ized to the maximum value of the probability of absorption P . 2 2 (−1) π(α − c ) 2 2 max −α t +ibt = −b /4α dtHm(ct )e + e −∞ αm 1   where the average photon number is bc × Hm √ , = 1 . 2α c2 − α2 n¯ν h¯ν−μ (27) e kBT − 1 which gives In this formula   √    2 π 2 ω + ν σ 2 V g 2 2d 2 ( ) hV¯ P , = 1 − H H √ T = (28) m n c2 V 2 n σ m 2k ωσ 2 2V B   is the effective temperature which is independent of the impact 2d2 (ω + ν)2σ 2 μ × exp − − . (33) parameter d, and is the effective chemical potential. In σ 2 2V 2 particular, for the modes with m =±1 The probability of excitation is governed by a similar expo- hV¯ 2 2Vd ± σ 2(ν + ω) μ = ln . (29) nential factor as in the case of Laguerre-Gaussian modes. 2ωσ 2 2Vd ± σ 2(ν − ω) One can vary the effective chemical potential at fixed T by V. SUMMARY changing the impact parameter d.Hereμ = h¯ν is a point of . At this point Pexc = Pabs.IfPexc > Pabs there Virtual transitions governed by the counterrotating terms is no steady-state solution of Eq. (25) and field in the cavity in the interaction Hamiltonian yield observable effects; e.g., grows with time indefinitely if we disregard cavity loses. they shift frequency of atomic or nuclear transitions (the In Fig. 4 we plot probability of emission and absorption Lamb shift). Here we discuss another manifestation of the into the vortex mode with m =+1 as a function of the impact virtual transitions, namely, excitation of a uniformly moving parameter d for σ (ω + ν)/V = 2.6 and σ (ω − ν)/V = 0.2. atom with simultaneous emission of the field quantum. We For d = (ω − ν)σ 2/2V the probability of absorption into the consider an atom moving with a constant velocity above cavity mode vanishes. As a result, in the vicinity of this point a metal surface or through an optical cavity, calculate the Pexc > Pabs and the field in the cavity grows with time. excitation probability, and show that for realistic parameters the effect can be large enough to be observed experimentally. Instead of a single atom one can use atomic ensembles which IV. EXCITATION OF HERMITE-GAUSSIAN could enhance excitation by orders of magnitude due to the MODESINACAVITY collective (superradiant) contribution. Analogous calculations can be done for Hermite-Gaussian Interpretation of the effect depends on a choice of the refer- modes that are relevant to the rectangular geometry of the ence frame. In the laboratory frame the moving atom becomes cavity and have field mode functions excited and the field quantum is emitted at the expense of     √ √ the kinetic energy of the atom. Excitation of the uniformly 1 2x 2y −ρ2/w2 (z) φν (t, r) = Hm Hn e moving atom through the optical cavity with simultaneous w(z) w(z) w(z) emission of a photon into the cavity can be understood as   a sum combination resonance. Such resonance occurs when νρ2 z × cos − (m + n + 1) arctan e−iνt , system parameters are modulated with the frequency equal to 2cR(z) zR the sum of two normal-mode frequencies. Taking into account (30) that the mode function of the photon in the cavity has the form −iνt where m, n = 0, 1, 2,.... φν (t, r) = φν (r)e ,

033027-6 EXCITATION OF A UNIFORMLY MOVING ATOM THROUGH … PHYSICAL REVIEW RESEARCH 1, 033027 (2019) the counterrotating term in the interaction Hamiltonian can be observer the atom cannot become excited at the expense of its written as kinetic energy and the energy conservation must be satisfied ∗ ν+ω + + by other means. For example, in the reference frame of an hg¯ φ [r(t )]ei( )t aˆ σˆ , ν ν atom moving above the metal surface the energy of surface where r(t ) is the trajectory of the atom. One can interpret plasmons is negative for large enough wave vectors. From ∗ the factor gφν [r(t )] as an effective coupling constant which the perspective of the moving observer the atom becomes depends on time due to atomic motion relative to the cavity. excited by emitting a resonant surface plasmon with nega- ∗ −i(ν+ω)t The Fourier component of gφν [r(t )] proportional to e tive energy which insures energy conservation. This example is in the sum combination resonance with the atom-field demonstrates that in systems with broken Lorentz invariance system. This resonance yields simultaneous excitation of the the vacuum state of the field can depend on the choice of the atom and the field mode from the vacuum. inertial reference frame. The effect is somewhat similar to the Unruh acceleration radiation. A neutral atom accelerated in Minkowski vacuum ACKNOWLEDGMENTS in free space can become excited by emitting a photon (from the perspective of an inertial observer). In this case the effec- This work was supported by the Air Force Office of Sci- tive coupling between light and the atom changes with time entific Research (Award No. FA9550-18-1-0141), the Office because of acceleration. of Naval Research (Awards No. N00014-16-1-3054 and No. In the frame moving with the atom the kinetic energy of the N00014-16-1-2578), the Robert A. Welch Foundation (Award atom is equal to zero. Thus, from the perspective of a moving No. A-1261), and Fujian 100-talents program.

[1] W. G. Unruh, Notes on black hole evaporation, Phys. Rev. D 14, Capasso,E.Fry,M.S.Zubairy,andM.O.Scully,Quantum 870 (1976). electrodynamics of accelerated atoms in free space and in [2] S. A. Fulling, Nonuniqueness of canonical field in cavities, Phys.Rev.A74, 023807 (2006). Riemannian space-time, Phys. Rev. D 7, 2850 (1973). [14] V. L. Ginzburg and V. P. Frolov, Excitation and emission of [3] P. C. W. Davies, Scalar production in Schwarzschild and a “detector” in accelerated motion in a vacuum or in uniform Rindler metrics, J. Phys. A 8, 609 (1975). motion at a velocity above the velocity of light in a medium, [4] W. Rindler, Kruskal space and the uniformly accelerated frame, Pis’ma Zh. Eksp. Teor. Fiz. 43, 265 (1986) [JETP Lett. 43, 339 Am.J.Phys.34, 1174 (1966). (1986)]. [5] W. G. Unruh and R. M. Wald, What happens when an acceler- [15] V. P. Frolov and V. L. Ginzburg, Excitation and radiation of an ating observer detects a Rindler particle, Phys.Rev.D29, 1047 accelerated detector and anomalous Doppler effect, Phys. Lett. (1984). A 116, 423 (1986). [6] M. O. Scully, S. Fulling, D. M. Lee, D. N. Page, W. P. Schleich, [16] I. Brevik and H. Kolbenstvedt, Quantum point detector mov- and A. A. Svidzinsky, Quantum approach to radiation ing through a dielectric medium, Nuovo Cimento B 102, 139 from atoms falling into a black hole, Proc. Natl. Acad. Sci. USA (1988). 115, 8131 (2018). [17] J. M. Jauch and K. M. Watson, Phenomenological quantum [7] A. A. Svidzinsky, J. S. Ben-Benjamin, S. A. Fulling, and D. N. electrodynamics, Phys. Rev. 74, 950 (1948). Page, Excitation of an Atom by a Uniformly Accelerated Mir- [18] T. Musha, Unified consideration of wave amplification in mov- ror through Virtual Transitions, Phys. Rev. Lett. 121, 071301 ing media, J. Appl. Phys. 35, 3273 (1964). (2018). [19] A. H. W. Beck, Space-Charge Waves and Slow Electromagnetic [8] A. A. Sokolov and I. M. Ternov, On polarization and spin Waves (Pergamon Press, Inc., New York, 1958). effects in the theory of synchrotron radiation, Sov. Phys. Dokl. [20] C. K. Birdsall, G. R. Brewer, and A. V. Haeff, The resistive-wall 8, 1203 (1964). amplifier, Proc. IRE 41, 865 (1953). [9] J. S. Bell and J. M. Leinaas, Electrons as accelerated thermome- [21] A. R. Hutson, J. H. McFee, and D. L. White, Ultrasonic ters, Nucl. Phys. B 212, 131 (1983). Amplification in CdS, Phys. Rev. Lett. 7, 237 (1961). [10] J. S. Bell and J. M. Leinaas, The Unruh effect and quantum [22] J. B. Pendry, Shearing the vacuum: Quantum friction, J. Phys.: fluctuations of electrons in storage rings, Nucl. Phys. B 284, Condens. Matter 9, 10301 (1997). 488 (1987). [23] A. I. Volokitin and B. N. J. Persson, Near-field radiative heat [11] W. G. Unruh, Acceleration radiation for orbiting electrons, transfer and noncontact friction, Rev. Mod. Phys. 79, 1291 Phys. Rep. 307, 163 (1998). (2007). [12] E. T. Akhmedov and D. Singleton, On the relation between [24] S. A. R. Horsley, Canonical quantization of the electromagnetic Unruh and Sokolov-Ternov effects, Int. J. Mod. Phys. A 22, field interacting with a moving dielectric medium, Phys. Rev. A 4797 (2007). 86, 023830 (2012). [13] M. O. Scully, V. V. Kocharovsky, A. Belyanin, E. Fry, and [25] Y. Guo and Z. Jacob, Singular evanescent wave resonances in F. Capasso, Enhancing Acceleration Radiation from Ground- moving media, Opt. Express 22, 26193 (2014). State Atoms via Cavity Quantum Electrodynamics, Phys. Rev. [26] A. Manjavacas and F. J. Garcia de Abajo, Vacuum Friction in Lett. 91, 243004 (2003);A.Belyanin,V.V.Kocharovsky,F. Rotating Particles, Phys.Rev.Lett.105, 113601 (2010).

033027-7 ANATOLY A. SVIDZINSKY PHYSICAL REVIEW RESEARCH 1, 033027 (2019)

[27] M. F. Maghrebi, R. L. Jaffe, and M. Kardar, Spontaneous [35] M. G. Silveirinha, Optical Instabilities and Spontaneous Light Emission by Rotating Objects: A Scattering Approach, Phys. Emission by Polarizable Moving Matter, Phys. Rev. X 4, Rev. Lett. 108, 230403 (2012). 031013 (2014). [28] R. Zhao, A. Manjavacas, F. J. Garcıa de Abajo, and J. B. Pendry, [36] S. Lannebère and M. G. Silveirinha, Negative spontaneous Rotational Quantum Friction, Phys.Rev.Lett.109, 123604 emission by a moving two-level atom, J. Opt. 19, 014004 (2012). (2017). [29] S. W. Hawking, Black hole explosions? Nature (London) 248, [37] M. G. Silveirinha, Spontaneous parity-time-symmetry breaking 30 (1974). in moving media, Phys.Rev.A90, 013842 (2014). [30] C. Barcelo, S. Liberati, and M. Visser, Analogue gravity, Living [38] M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, and P. Zoller, Rev. Relativ. 14, 3 (2011). Hybrid quantum devices and quantum engineering, Phys. Scr. [31] T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. Konig, and T137, 014001 (2009). U. Leonhardt, Fiber-optical analog of the event horizon, Science [39] H. B. G. Casimir and D. Polder, The influence of retardation 319, 1367 (2008). on the London–van der Waals forces, Phys. Rev. 73, 360 [32] T. Dodo, Quantum theory of negative energy in microwave (1948). tubes, Jpn. J. Appl. Phys. 3, 16 (1964). [40] For the density matrix formulation of the quantum theory of [33] C. Luo, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, the laser, see M. Scully and W. Lamb Jr., Quantum Theory Cherenkov radiation in photonic crystals, Science 299, 368 of an Optical Maser, Phys. Rev. Lett. 16, 853 (1966). For (2003). pedagogical treatment and references, see E. R. Pike and S. [34] J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, Sarkar, The Quantum Theory of Radiation (Oxford University Theory of surface plasmons and surface-plasmon , Press, New York, 1996); M. Scully and S. Zubairy, Quantum Rep. Prog. Phys. 70, 1 (2007). Optics (Cambridge University Press, Cambridge, 1997).

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