Contributions of Vacuum and Plasmon Modes to the Force on a Small Sphere near a Plate L.H. Ford 6 0 Institute of Cosmology 0 2 Department of Physics and Astronomy Tufts University, Medford, MA 02155 n a E-mail: [email protected] J 5 Abstract. The force on a small sphere with a plasma model dielectric function and 1 in the presence of a perfectly reflecting plane is considered. The contribution of both v the vacuummodesofthe quantizedelectromagneticfieldandofplasmonmodesinthe 9 3 sphere are discussed. In the case that the plasmon modes are in their ground state, 0 quasi-oscillatorytermsfromthevacuumandplasmonpartscanceloneanother,leading 1 a monotonic attractive force. If the plasmon modes are not in the ground state, the 0 6 net force is quasi-oscillatory. In both cases, the sphere behaves in the same way as 0 does an atom in either its ground state or an excited state. / h p - t n 1. Introduction a u q As is well-known, Casimir and Polder [1] calculated the interaction energy between an : v atom in its ground state and a perfectly reflecting plate and obtained the result i X 1 ∞ r V = dξα(iξ)(2z2ξ2 +2zξ +1)e−2zξ, (1) a −4πz3 Z0 where z is the distance between the atom and the plate and α(iξ) is the atomic polarizability evaluated at imaginary frequency. In the limit of large z, this potential takes the simple form 3α(0) V V = , (2) ∼ a − 8πz4 which can be interpreted as due to the interaction of the atom with the quantized electromagnetic field. In particular, the asymptotic Casimir-Polder potential can be expressed as 1 α(0) ∞ 2 2 2 V = α(0) E = dω (2ω z 1)sin2ωz+2ωzcos2ωz .(3) a −2 h i 4πz3 Z0 h − i Here E2 is the shift in the mean square of the quantized electric field due to the plate, h i and the integral on ω is understood to be evaluated by use of a convergence factor. For example, insert a factor of e−βω in the integrand, and then take the limit β 0 after → evaluation of the integral. The integrand is a function which is highly oscillatory, and Contributions of Vacuum and Plasmon Modes 2 the area under each oscillation is much larger than the finalresult for the integral. Thus, if one could slightly modify the frequency spectrum, which is the integrand in Eq. (3), both the magnitude and sign of the result could be dramatically altered. The Casimir effect between parallel plates also exhibits a quasi-oscillatory spectrum [2, 3], and the possible effects of its modification were discussed in Ref. [4]. In a previous paper, Ref. [5], the force on a small dielectric sphere in the presence of a perfectly reflecting plate was discussed. It was shown that the interaction of the sphere with the quantized electromagnetic field produces a quasi-oscillatory force whose magnitude can be much larger than the force associated with the asymptotic Casimir- Polder potential, Eq. (2). This result suggests a modification of the frequency spectrum of vacuum fluctuations. More recently, the model in Ref. [5] was extended to include finite reflectivity of the wall and finite temperature effects [6]. However, Barton [7] has suggested that the results in Refs. [5] and [6] are incomplete in that they do not include a contribution from the quantum charge fluctuations in the sphere, and that this effect will cancel the quasi-oscillatory terms when the sphere is in its ground state. This contribution, also known as radiation reaction or the effect of plasmonic modes, arises from the retarded self-interaction of quantum charge fluctuations in the presence of the mirror. As will be discussed below, Barton’s suggestion is correct; the quasi-oscillatory terms cancel for the ground state, but not for other quantum states of the plasma in the sphere, which is essentially a three dimensional quantum harmonic oscillator. In this way, the sphere behaves just as does an atom. An atom in its ground state has the attractive interaction found by Casimir and Polder, Eq. (1), but an atom in an excited state has a quasi-oscillatory interaction potential with the plate [8]. In Sect. 2, the vacuum mode contribution found in Ref. [5] will be reviewed. The contribution of the charge fluctuations will be computed in Sect. 3, and the results will be discussed in Sect. 4. 2. Contribution of the Quantized Electromagnetic Field In this section, we will summarize the results of Ref. [5] concerning the force due to the vacuum modes, or the quantized electromagnetic field. If one has a polarizable particle with dynamic polarizability α(ω) located a distance z from a perfectly reflecting plane, the force on the particle may be computed from the Maxwell stress tensor. The vacuum modes in the presence of the plate consist of a linear combination of an incident and a reflected wave. Each of these interacts with the particle and creates a radiated dipole field. The force on the particle arises from cross terms between (1) the reflected wave and the dipole field induced by the incident wave, and (2) the incident wave and the dipole field induced by the reflected wave. After integrating over all modes, one finds the net force to be 1 ∞ FV = − 4πz4 Z0 dωα1(ω) Contributions of Vacuum and Plasmon Modes 3 2 2 3 3 3sin2ωz 6zω cos2ωz 6z ω sin2ωz +4z ω cos2ωz . (4) × h − − i Here α1(ω) is the real part of α(ω). The appearance of the real part of the dynamic polarizability is a crucial feature of this result for F . Equation (4) is derived using V a dipole approximation which requires the sphere to be small compared to both the distance to the plane, z, and to the characteristic wavelengths associated with α1(ω). Now consider a sphere of radius a composed of a uniform material with dielectric function ε(ω). The complex polarizability is given by ε(ω) 1 3 α(ω) = a − . (5) ε(ω)+2 We will take the dielectric function to be that of the Drude model, ω2 ε(ω) = 1 p , (6) − ω(ω +iγ) where ω is the plasma frequency and γ is the damping parameter. From Eqs. (5) and p (6), we find that the real part of the polarizability is given by ω2 3ω2 α1 = a3ωp2 (3ω2 pω−2)2 +9ω2γ2 . (7) − p Note that although α(ω) has poles only in the lower half-ω plane, its real part α1(ω) has poles in both the upper and lower half planes. In the context of this model, the sphere is required to be small in the sense that a z and ≪ aω 1. (8) p ≪ We can evaluate the integral in Eq. (4) by rotating the contour of integration to the imaginary ω-axis. However, in this process we will also acquire a contribution from 1 the residue of the pole of α1(ω) at ω = Ω+ 2 iγ, where 1 Ω = 12ω2 9γ2. (9) 6q p − Here we are primarily interested in the limit of small damping, so we will take the limit γ 0 after the integrals are evaluated. In this limit, → ω p Ω = . (10) √3 The force can be written as F = J +P , (11) V V V where J is the net contribution from integrals along the imaginary axis, and P is that V V from the pole. The explicit forms of these two contributions for γ = 0 are a3ω2 ∞ (3ξ2 +ω2)(4z3ξ3 +6z2ξ2 +6zξ +3) J = p dξ p e−2zξ, (12) V −4πz4 Z0 (3ξ2+ω2)2 p and a3ω2 P = p 2Ωz(2Ω2z2 3)sin(2Ωz)+3(2Ω2z2 1)cos(2Ωz) .(13) V −24Ωz4 h − − i Here the positive direction is taken to be away from the plate, so a positive value for the force indicates repulsion. Contributions of Vacuum and Plasmon Modes 4 3. Contribution of the Quantum Charge Fluctuations In this section, we will calculate the average force on the sphere due to quantum fluctuations of the charge density in the sphere. Here we consider only the case where γ = 0, so the plasma oscillations are undamped, and have a resonate frequency at the pole of ǫ(ω), that is at ω = Ω where Ω is given by Eq. (10). Classically, the plasma formsathree-dimensionalharmonicoscillatorwithafrequencyΩ. Whentheelectrongas is described quantum mechanically, the plasma becomes a three-dimensional quantum harmonic oscillator. First, let us consider a classical oscillating dipole in the presence of a perfectly reflecting plate located at z = 0. Let the instantaneous dipole moment be given by p(t) = p0 cosΩt. (14) The electromagnetic field radiated by this dipole is reflected by the plate and returns to exert a reaction force on the dipole. Let E(t) and B(t) be the reflected electric and magnetic fields at the location of the dipole. The resulting force on the dipole is given by F = pj∂ E+p˙ B. (15) P j × (This form can be derived directly from the Lorentz force law, and differs slightly from the expression for the force used in Ref. [5], which was derived from the Maxwell stress tensor. However, one can show that both expressions lead to the same time-averaged force.) We are interested in the component of time-averaged force which is normal to the plate. A lengthy but straightforward calculation leads to the result p2 +p2 F = 0x 0y (8Ω2z2 3) cos(2Ωz)+2Ωz(4Ω2z2 3)sin(2Ωz) P 32z4 h − − i p2 + 0z (4Ω2z2 3) cos(2Ωz) 6Ωz sin(2Ωz) . (16) 16z4h − − i Now we need to briefly summarize the quantum mechanical description of the plasma in the sphere. Suppose that there are N free electrons in the sphere, and that each electron is displaced by u from its equilibrium position during the plasma oscillations. The Lagrangian for the plasma may be written as 1 2 2 2 = Nm(u˙ Ω u ), (17) L 2 − where m is the electron mass. Thus the system is equivalent to a single particle with mass Nm is a three-dimensional harmonic potential with resonant frequency ω. The dipole moment of the system can be expressed as p = Neu, (18) where e isthe electron charge. When thesystem isquantized, theCartesian components of u can be expressed as 1 u = (a +a†), (19) j √2NmΩ j j Contributions of Vacuum and Plasmon Modes 5 where a and a† are the annihilation and creation operators, respectively, for quanta in j j mode j. We can make the link between the previous description of a classical oscillating dipole and a dipole undergoing quantum fluctuations by identifying p2 , the quantum h ji expectation value of p2, with the time-average of the corresponding squared classical j field. Thus, we can set 2 2 p = 2 p . (20) 0j h ji If the oscillator is in its ground state, we have 1 a3ω2 hp2xi0 = hp2yi0 = hp2zi0 = 2NMΩ = 6Ωp , (21) where we used the fact that 4πNe2 3Ne2 2 ω = = , (22) p Vm a3m where V is the volume of the sphere. Thus in the ground state, we find that F = P . P V − The force due to the averaged charge fluctuations exactly cancels the quasi-oscillatory term coming from the vacuum modes. The net force is then F = J +P +F = J , (23) V V P V which is always attractive. The situation is quite different of the quantum oscillator is not in its ground state. In this case, we can write the net force as 1 2 2 2 2 2 2 F = JV + 16z4 (cid:16)hpxi−hpxi0 +hpyi−hpyi0(cid:17)h(8Ω z −3) cos(2Ωz) 2 2 + 2Ωz(4Ω z 3)sin(2Ωz) − i 1 2 2 2 2 + 8z4 (cid:16)hpzi−hpzi0(cid:17)h(4Ω z −3) cos(2Ωz)−6Ωz sin(2Ωz)i. (24) Consider, for example, the case where the state of the oscillator contains one quantum in a particular mode. For that mode, we will have hp2ji = 2hp2ji0, and F will have a quasi-oscillatory part which is just minus the vacuum contribution. Note that Eq. (24) is equivalent to Barton’s [8] Eq. (4.1). A case of special interest is when the oscillator is in a coherent state of each mode. In this case, we have a ψ = β ψ , (25) j j | i | i where ψ is the quantum state of the oscillator, and the β are three complex numbers. j | i In this case, the net force becomes 2√3a3ω p 2 2 2 2 F = J + (Reβ ) +(Reβ ) (8Ω z 3) cos(2Ωz) V 48z4 (cid:26)h x y h − 2 2 + 2Ωz(4Ω z 3)sin(2Ωz) − i 2 2 2 + 2(Reβ ) (4Ω z 3) cos(2Ωz) 6Ωz sin(2Ωz) . (26) z h − − i(cid:27) Contributions of Vacuum and Plasmon Modes 6 Thus if the quantum state of the plasma is a coherent state with β of order one, then j | | there is a quasi-oscillatory force on the sphere which is of the same order of magnitude as the vacuum contribution, F . V 4. Discussion and Conclusions We have seen that plasmonic contributions, due to quantum charge fluctuations, give a significant contribution to the force on the sphere. When these modes are in their groundstate, theyexactly cancelthequasi-oscillatory termsP arisingfromthevacuum V modes. In this case, only the vacuum mode term J which can be written as an integral V over imaginary frequency remains. Thus if one had ignored both the plasmon mode contribution and the residues of the poles of α1(ω), one would have inadvertently obtained the correct answer. When the plasmon modes are not in the ground state, the net force is quasi-oscillatory. In the context of atomic systems, there is an extensive literature on the relative effectsofvacuummodesandquantumchargefluctuations,oftencalledradiationreaction or self-reaction. See, for example, Ref. [9] and [10]. However, many of the treatments for atomic systems do not make it clear whether the separation between vacuum and self- reactioneffects isgaugeinvariant. Inthepresent model, itis obvious that thetwo effects are separately gauge invariant, as all of the calculations involve field strengths rather than potentials. When calculations are performed by doing second order perturbation theory on a Hamiltonian, as in Refs. [1] and [8], there is no need to consider the effects separately, and the cancellation is automatic. In the case of the Casimir effect between planar geometries, the role of plasmonic contributions at short distances has long been recognized [11, 12]. More recently, Intravaia and Lambrecht [13] have argued that they are significant at all separations. Itseemstobedesirabletounderstandbetterthephysicalreasonforthecancellation between vacuum and plasmonic effects. It is clear that one way to upset the cancellation is to excite the plasmon modes. However, it is less clear if this is the only way in more general geometries. Acknowledgments I would like to thank Gabriel Barton for valuable criticism and discussions. This work was supported in part by the National Science Foundation under Grant PHY-0244898. References [1] Casimir HBG and Polder D 1948 Phys. Rev. 73 360 [2] Ford LH 1988 Phys. Rev. D 38 528 [3] Hacyan S, J´auregui R, Soto F and Villarreal C 1990 J. Phys. 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