Modified radiation laws of a rectangular Kerr nonlinear blackbody

Optics Communications 284 (2011) 4190–4196

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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m

Modified radiation laws of a rectangular Kerr nonlinear blackbody Qi-Jun Zeng a, b,⁎, Ze Cheng a a b

School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China College of Physics and Electronic Engineering, Xinyang Normal University Xinyang 464000, China

a r t i c l e

i n f o

Article history: Received 2 December 2010 Received in revised form 18 April 2011 Accepted 20 April 2011 Available online 13 May 2011 Keywords: Kerr nonlinear blackbody Nonpolariton Modified radiation law

a b s t r a c t A Kerr nonlinear blackbody (KNB) is a new kind of blackbody in which bare photons with opposite wave vectors and helicities are bound into pairs and unpaired photons are transformed into nonpolaritons. In the present paper, we focus our investigation on the modified radiation laws, such as Planck and Stefan– Boltzmann radiation laws, of a rectangular KNB. Besides, the case of a KNB with no symmetry axes is also discussed. Finally, we make a numerical calculation of modified radiation laws of a cubic KNB under appropriate conditions, and we consider this work may lay the foundation for the experimental verification of the model of a KNB. © 2011 Elsevier B.V. All rights reserved.

1. Introduction It is well known that the investigation on blackbody radiation played an important role in the origins of quantum theory. Up to now, the exploration of blackbody radiation has spread over the most fields of physics including particle physics, condensed matter physics, optics and so on. Besides, the research on blackbody radiation also has lots of valuable applications such as the technique of thermal remote sensing of objects which allows us to diagnose the properties of an object without touching it. In sum, the theory of a blackbody is still a hotspot in the domain of physics [1–3]. According to the definition of a blackbody, a close approximation to the blackbody is a cavity with perfectly conducting walls, which is heated to a steady temperature T and connected to the outside by a small aperture. We shall call this system a normal blackbody. In the early time, Cheng [4,5] has studied a new kind of blackbody, named a KNB, whose interior is filled by a Kerr nonlinear crystal. Such a KNB can be regarded as a rectangular crystal having perfectly conducting walls and kept at a constant temperature T. As shown in Fig. 1, there is a very small hole in a wall, through which one can receive the thermal radiation. In the previous works [4,5], Cheng pointed out that in a KNB, bare photons with opposite wave vectors and helicities are bound into pairs and unpaired bare photons are transformed into a new kind of quasiparticle, the nonpolariton. The photon-pair system is a condensate and the nonpolariton system is a kind of boson gas. Furthermore, Cheng et al. also investigated the other properties of a KNB, including the radiation properties [6], and abnormal properties of spontaneous emission in a KNB [7], the modified dynamic Stark shift of an atom in a ⁎ Corresponding author at: School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China. Tel.: + 86 27 8754 2637. E-mail addresses: [email protected], [email protected] (Q.-J. Zeng). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.04.052

KNB [8], and the photon condensation state [9]. This new kind of condensation state can be named Bardeen–Cooper–Schrieffer (BCS) condensation state [9], and the BCS condensation of a photon system is entirely different from the Bose–Einstein condensation. In the latter case, the number of particles in the system is conserved, and the particles undergoing the condensation are all in the state with the lowest energy. But for the BCS condensation, it is not necessary to satisfy these conditions. In the BCS condensation state, we only emphasize that due to the interaction between photons and phonons, the photon system are translated into a condensate made of photon pairs and a kind of quasiparticle, the nonpolariton. In other words, the photon-pair system can be regard as a condensate of bare photons and virtual phonons, and what we accentuate is the state of pairing but not the other thing, so this kind of condensation may be named as BCS condensation. The properties of such a KNB are quite different from those of a normal blackbody and may have enormous values of theory and application. For example, in such a KNB we may predict a entirely new phenomena of condensation, i.e., the BCS condensation [9]. Moreover, this kind of photon condensation state is crucial to the theory of superlight [4], which may have an expansive application prospect. Unfortunately, up to now no experiment has been done to verify the theory of a KNB. Due to the fact that any concrete KNB used in a experiment must have finite size, thus it is necessary to investigate the modified radiation law of such a KNB, and this is also the intention of the present paper. Besides, here we must note the following point. In Ref. [10], Cheng et al. have discussed the finite-size corrections to the radiation laws of a KNB, and obtained the general expressions of the modified Planck and Stefan–Boltzmann blackbody radiation laws of a KNB. In addition, the case of a spherical KNB is studied in detail in that paper. Hence, in the present paper, we would focus our investigation on the modified radiation laws, such as Planck and Stefan–Boltzmann radiation laws, of a rectangular KNB, and the

Q.-J. Zeng, Z. Cheng / Optics Communications 284 (2011) 4190–4196

X

Y

attractive effective interaction which leads to bound photon pairs among themselves. In the standing-wave configuration a photon pair is stable only if the two photons have opposite wave vectors and helicities. Besides, unpaired bare photons in the photon system are transformed into a new kind of quasiparticle, the nonpolariton. A nonpolariton can be regard as the condensate of virtual nonpolar phonons, with a bare photon acting as the nucleus of condensation. Moreover, it is demonstrated that the pair Hamiltonian of the photon system can be given by [5] ˜ k ðT Þc†kσ ðt Þckσ ðt Þ; Hem = Ep + ∑ ℏ ω

Z

kσ

O Fig. 1. A KNB: a rectangular Kerr nonlinear crystal enclosed by perfectly conducting walls and kept at a constant temperature; there is a small hole in a wall.

case of a KNB with no symmetry axes will also be discussed. In brief, the present paper is not only a consummation but also a new development of Ref. [10]. Therefore, we prefer to directly quote the conclusions which have been reached in Ref. [10], and at the same time, for some problems which have not been explained expressly, we would try to elucidate them more clearly and deeply. As is known, Planck radiation law of a normal blackbody can be given as follows: 3

uðωÞ = ρðωÞεðωÞ = V =

ħ ω ; π2 c3 eħω = kB T −1

ð1Þ

where u(ω) is the density of energy per unit of frequency, V is the volume of the blackbody cavity, ρ(ω)dω represents the number of stationary electromagnetic modes with frequencies between ω and ω + dω, and ε(ω) = ħω/[exp(ħω/kBT) − 1] is the average energy per mode, ħ and kB are Planck constant and Boltzmann constant, respectively. Stefan–Boltzmann law can also be given by RðT Þ =

c ∞ 4 ∫ uðωÞdω = σT ; 4 0

4191

ð2Þ

where R(T) is the total energy emitted per unit of time and area, c is the velocity of light in vacuum, and σ = π 2kB4/60ℏ 3c 2 is Stefan– Boltzmann constant. Strictly speaking, the two laws above are applicable only to an ideal blackbody whose volume V is infinite. To a blackbody with finite size, the two laws must be modified and we will discuss this question in detail in the later sections. The remainder of this paper is organized as follows: In Section 2, we first briefly review the theory of a KNB mainly referring to Ref. [5], then we revisit the universal finite-size corrections to the radiation laws of a KNB referring to Ref. [10]. In addition, fluctuations of the total radiation R(T) in a KNB with no symmetry axes will also be studied in this section. In Section 3, we firstly investigate the modified radiation laws of a rectangular KNB. Secondly, after discussing the suitable conditions for experimental detection, we make a numerical calculation of the modified radiation laws of a cubic KNB. The comprehensive discussion is given in Section 4. 2. Kerr nonlinear blackbody and finite-size corrections to the radiation laws 2.1. Kerr nonlinear blackbody Because the theory of a KNB is unfamiliar to most of readers, first we would like to review some important conclusions of a KNB, which will be used in our further investigation. The crystal being filled in a KNB is determined as a specific crystal with a diamond structure, such as C. In Ref.[5], Cheng reaches the following conclusions. In a photon system, the interaction between photons and phonons can lead to an

ð3Þ

where the first term Ep is the energy of the system of photon pairs, and the second term represents the energy of nonpolaritons. In the second term, c†kσ and ckσ are, respectively, the creation and annihilation operators of nonpolaritons with wave vector k and helicity σ = ± 1, and they obey Bose equal-time commutation relations: h

i ckσ ðt Þ; c† ðt Þ = δk;k′ δσ;σ ′ ; k′ σ ′ − h i ckσ ðt Þ; ck′ σ ′ ðt Þ = 0: −

ð4Þ

˜ k ðT Þ is the frequency of a nonpolariton and it satisfies In Eq. (3), ω ω ˜ k ðT Þ = vðT Þjk j;

ð5Þ

where v(T) represents the velocity of nonpolaritons. In addition, we also know [5] that the velocity v(T) is a monotonically increasing function of temperature T and it is equal to c/n at the transition temperature Tc. Below Tc, the photons with opposite wave vectors and helicities are bound into pairs and unpaired photons are transformed into nonpolaritons. At Tc, both photon pairs and nonpolaritons are transformed into individual bare photons. When T N Tc, a KNB behaves just in the same manner as a normal blackbody. For convenience, we can introduce a dimensionless constant γ to signify the coupling strength between a bare photon and virtual nonpolar phonons. In Ref. [5], γ is defined by (namely Eq. (56) in that paper) γ=

  ħ P ð0ÞωR 2 ; 2c3 Ωn 4πε0

where Ω is the volume of a primitive cell, P(0) is the Raman coefficient at zero wave vector of phonons (referring to Eq. (21) of Ref.[5]), and so P(0) is closely related to the Raman polarizability of a Kerr crystal (referring to Eq. (15) of Ref.[5]). This is to say, γ is determined by the Raman polarizability and we can consider that γ is characteristic of a crystal. Besides, it can also be concluded that γ = nv(0)/c (referring to Eq. (55) of Ref. [5])and it is meaningful only if 0 b γ b 1. Further investigation reveals that the velocity v(T) is an increasing function of γ[5]. Because different Kerr mediums may have different values of γ, in what follows we regard γ as an adjustable parameter for the sake of studying the influence of γ on a KNB. Because a nonpolariton is made of a bare photon and virtual nonpolar phonons, and especially the photon is the nucleus of a nonpolariton, we can conclude that nonpolaritons and photons should obey the same quantum and statistical laws, and the behaviors of nonpolaritons should be similar to those of photons. Hence, in the photon system of a KNB we can define the number operators Nkσ = c†kσ ðt Þckσ ðt Þ for nonpolaritons, and the number operators Nkσ have the eigenvalues nkσ = 0; 1; 2; …. The eigenstates of number operators are given by: " # 1  † nkσ j fnkσ gi = ∏ pffiffiffiffiffiffiffiffiffiffi ckσ jGi; nkσ ! kσ

ð6Þ

where |G 〉 = U|0 〉 is the normalized state vector of photon pairs and |0 〉 is the vacuum state of the electromagnetic field. In other words, the

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vacuum state of nonpolaritons |G 〉 is a condensation state consisting of photon pairs and nonpolaritons are elementary excitations from such a condensate. Considering that the gas of nonpolaritons can constitute a thermal radiation in a KNB, we may conceive a grand canonical ensemble of nonpolaritons to characterize the thermal radiation state. Some identical systems of the ensemble may be in an eigenstate of the Hamiltonian Hem given by Eq. (3). The distribution of the ensemble over the eigenstates is described by the density operator ρ=

exp ð−Hem = kB T Þ ; Tr exp ð−Hem = kB T Þ

ð7Þ

and the ensemble average hNkσ i of the number operators of nonpolaritons in a mode kσ is hNi =

1 ; expðℏω ˜ = kB T Þ−1

ð8Þ

where we have reduced Nkσ and ω ˜ k ðT Þ to N and ω, ˜ respectively. Now in the similar way to a normal blackbody, we can get the Planck radiation law of a KNB as follows: 3

uðω ˜ Þ = ρðω ˜ Þεðω ˜Þ= V =

ħ ω ˜ ; π v ðT Þ eħ ω˜ = kB T −1 2 3

ð9Þ

where V is the volume of the blackbody cavity. Besides, in Eq. (9) the parameters ρðω ˜ Þ and εðω ˜ Þ, respectively, are determined by ρðω ˜Þ = V

ω ˜2 ; π2 v3 ðT Þ

εðω ˜ Þ = ħω ˜ hN i =

ħω ˜ ; eħ ω˜ = kB T −1

ð10Þ

ð11Þ

where ρðω ˜ Þdω ˜ represents the number of wave modes of nonpolaritons with frequencies between ω ˜ and ω ˜ + dω, ˜ and εðω ˜ Þ is the average energy per mode. Next, referring to Eq. (2) we can get the Stefan–Boltzmann law of a KNB immediately: RðT Þ =

vðT Þ ∞ 4 ∫ uðω ˜ Þdω ˜ = σ ′ ðT ÞT ; 4 0

ð12Þ

where R(T) is the total energy of the nonpolariton system emitted per unit of time and area, and σ′ = π 2kB4/60ℏ 3v 2(T). Eqs. (9) and (12) are the most important radiation laws of a KNB, and we can call them the ideal Planck and Stefan–Boltzmann radiation laws. In the following section, we will revisit the modified forms of the two laws of a KNB with finite sizes mainly referring to Ref. [10].

quantum system, or in other words, it is closely associated with the density of states ρ(E) of a given finite quantum system, where E represents the energy eigenvalue. Comprehensive theoretical investigations [11–13] on ρ(E) demonstrate that ρ(E) can always be resolved into one part ρsmooth(E) that varies smoothly with energy, and another part ρ˜ ðEÞ that oscillates with energy. This is to say that the equation ρðEÞ = ρsmooth ðEÞ + ρ˜ ðEÞ can be gotten. The term ρsmooth(E), namely Weyl expansion or ETF (extended Thomas–Fermi) expansion [13], mainly includes volume term ρV(E), surface term ρs(E), and curvature term ρC(E). The smooth term ρsmooth(E) is first studied by Weyl in 1911 [14] and now it can be derived more accurately from ETF method [13,15,16]. The term ρ˜ ðEÞ is an oscillatory term and its analytic expression can be obtained by use of the periodic orbit theory which includes Gutzwiller's trace formula [13,17] and extensions of the Gutzwiller theory [13,18,19]. Besides, these results above have also been derived by Balian and Bloch [11,12] from a multiple reflection expansion of the energy-dependent Green's function. The theory of Balian and Bloch is very similar in spirit, although quite different in the practical way of calculation, to the semiclassical approach developed by Gutzwiller. In the following discussion we would mainly adopt the theory of Balian and Bloch. The Green's function formalism developed by Balian and Bloch initially was used for analyzing scalar waves in Refs.[11,12]. Subsequently, after extending their theory and considering that the electromagnetic wave is a transverse wave, Balian and Bloch studied the smooth term ρsmooth(E) of electromagnetic waves in a cavity [20,21]. Now considering the similarity of radiation properties between the gas of free nonpolaritons and the gas of free photons, and referring to Refs. [13,20–22], we can get the following expression for a three-dimensional (3D) KNB with smooth surfaces and L ≫ λ[10]: ρðω ˜ Þ = ∑ δðω−ω ˜ ˜ Þ + ρC ðω ˜ Þ + ρ˜ðω ˜ Þ: i Þ = ρV ðω

In the expressions above, L is the characteristic length of a KNB, λ = 2π = jk j = 2πv = ω ˜ represents the wavelength, ω ˜ i stands for the ith natural mode of the cavity and ρV ðω ˜ Þ satisfies Eq. (10). The second term ρC ðω ˜ Þ and the third term ρ˜ðω ˜ Þ are curvature term and oscillatory term, respectively, and their expressions can be found in Ref. [10]. Here we note that (a) there was an error in the expression of ρC in Ref. [22] (refer to Refs. [13,20]), (b) the surface term ρs ðω ˜ Þ vanishes in Eq. (13) due to the fact that the electromagnetic wave of nonpolaritons is a transverse wave. Next, after substituting the expressions of ρV ðω ˜ Þ, ρC ðω ˜ Þ and ρ˜ðω ˜ Þ into Eq. (13), and using Eq. (11) and the equation uðω ˜ Þ = ρðω ˜ Þεðω ˜ Þ = V, we can get the following expression of modified Planck radiation law easily [10]: ( uðω ˜Þ =

1 + a1

2.2. The modified radiation laws As mentioned before, Eqs. (1), (2), (9) and (12) are applicable only to a blackbody with infinite volume V. The reason for this conclusion is as follows: from the box normalization convention for plane-wave states, we know that the distribution of wave vector k is discrete if the volume of a box V is finite. Only under the condition that V → ∞, wave vector k can change continuously, and consequently we can alter the summation to an integration under such a condition. Therefore the previous quantities uðω ˜ Þ, ρðω ˜ Þ and εðω ˜ Þ should be rewritten as uV ðω ˜ Þ, ρV ðω ˜ Þ and εV ðω ˜ Þ in the V → ∞ limit, respectively. Accordingly, the modified form of Eqs. (9) and (12) are expected since ρV ðω ˜ Þ is only the leading term of a full expansion of ρðω ˜ Þ in powers of a typical length L of the cavity, and ρV ðω ˜ Þ can be regarded as a volume term which is proportional to L 3. In order to get the modified expressions of uðω ˜ Þ and R(T), above all, we must get a more accurate expression of ρðω ˜ Þ than Eq. (10). This problem has close relation to the distribution of eigenstates of a finite

ð13Þ

i

2 2−m = 2 h i ðT Þ v ðT Þ v + ∑ DC A˜p ðω; ˜ LÞcos ω ˜ ⋅lp = vðT Þ + μ p 2 2 2−m = 2 p L ω ˜ ðLω ˜Þ

)

3

×

ħ ω ˜ ðT Þ · ; π v ðT Þ eħ ω˜ ðT Þ = kB T −1 2 3

ð14Þ

where a1 is a dimensionless coefficient that depends on the mean curvature C of the cavity, A˜p is a dimensionless amplitude corresponding to a single periodic orbit of length lp. For symmetric cavities, such as spheres (m = 3) and rectangles (m = 2), the two coefficients above can be gotten by use of the periodic orbit theory (refer to Ref. [13]). But for chaotic cavities (m = 0), it is difficult to obtain them [23]. Besides, from a physical point of view, there is a cutoff in the sum of Eq. (14). If we define lesc as the length of a ray's orbit before the ray escapes through the aperture, it is obvious that only periodic orbits whose length are shorter than the length lesc can make remarkable contributions to uðω ˜ Þ. Thereby, the sum in Eq. (14) should only run over the periodic orbits p which satisfy lp ≤ lesc, and the sum won't diverge even without the convergence factor DC. The explicit

Q.-J. Zeng, Z. Cheng / Optics Communications 284 (2011) 4190–4196

expression of the cutoff function strongly depends on the symmetries of the cavity. For symmetric cavities (m = 3, 2), the cutoff function has power-law tails but for chaotic cavities, it is quite complicated [24]. Subsequently, after inserting Eq. (14) into the expression ∞ RðT Þ = vðT Þ∫0 uðω ˜ Þdω ˜ = 4, we can immediately get the modified Stefan–Boltzman law as follows [10]: RðT Þ = RV ðT Þ + RC ðT Þ + R˜ ðT Þ;

ð15Þ

where RV(T) = σ′(T)T4, RC(T) = − C(TkB)2/36L3ℏ and this term comes from the curvature term ρC ðω ˜ Þ of the density of states. In the limit lmin ≫ ℏv(T)/kBT, where lmin represents the length of the shortest periodic orbit, the fluctuating term R˜ ðT Þ can be approximately written as [10] R˜ ðT Þ≈b0

ð16Þ

where b0 and b1 are dimensionless coefficients associated with the number m. For m= 0, 2, the two coefficients can be given as (refer to Ref. [22])   ðm + 2Þπ −2πL 1 b0 = − 4 vðT Þ

+ m=2

  Ap;t ðLÞf μ p ∑  2 + m = 2 ; p;t τp t

   Ap;t ðLÞg μ p 2πL m = 2 ∑  1 + m = 2 ; b1 = π vðT Þ p;t τp t

ð17Þ



ð18Þ

where the expressions of f(μp) and g(μp) can be found in Ref.[10]. In Eqs. (17) and (18), the amplitude Ap, t describes the contribution from a single periodic orbit p which has a period τp = lp/v(T), and t is the repeated times of τp. Here we must emphasize that (a) the largest contribution to the sum comes from the shortest periodic orbits, and so the sum needn't run over too many periodic orbits, (b) for the shortest periodic orbits, the Maslov indices μp and amplitude Ap,t can be computed analytically [13]. 2.3. Fluctuations in a KNB with no symmetry axes In order to determine the explicit analytic expressions of R˜ ðT Þ and uðω ˜ Þ, the knowledge of periodic orbits of different lengths is required. However, this way is feasible only to highly symmetric cavities with several symmetry axes. For chaotic cavities with no symmetries, this way is quite difficult. Whereas, in the latter case, we can make an estimation of the deviations by semiclassical techniques, which are based on the fact that the classical dynamics is ergodic [25]. Here we D 2E would only study R˜ ∝∑p;p′ ;t;t ′ Ap;t Ap′ ;t ′ briefly, where the average is over cavities which are characterized by the same typical length L and have no axes of symmetry. According to Eqs. (16), (17) and D(18) E 2 and setting m = 0, we can get the leading term of the double sum R˜ by taking only the diagonal p = p′ term: D

2 2 E Ap Ap 2 R˜ ≈C1 ∑  2 + C2 ∑  4 ; p p τp t τp t where

C1 =

ð19Þ

E

2 ∞ R˜ ≈C1 ∫τmin

8 0; τbτmin ; ð21Þ > > < δ; τ≈τmin ; ð22Þ K ðτÞ = > 2τ; τmin bτbτH ; ð23Þ > : τH ; τ N τH ;

K ðτÞ K ðτÞ ∞ dτ + C2 ∫τmin 4 dτ; τ3 τ

ð21Þ

˜ Þ is the Heisenberg time. where δ represents a delta function, τH = ρV ðω Subsequently, after Dsubstituting Eq. (21) into Eq. (20), we can get a more E 2 brief expression of R˜ as follow: E 2 R˜ ≈2C1 lnτH = τmin + C2 = τmin :

ð22Þ

Here we note that (a) in Eq. (22) we have neglect high-order term of 1/τH because our purpose is only to estimate the magnitude of the fluctuations; (b) Eq. (22) is valid provided that lesc ≫ lmin, otherwise one would have to consider the contribution of each periodic orbit [22]. 3. Modified Planck and Stefan–Boltzmann radiation laws of a rectangular Kerr nonlinear blackbody Because a rectangular cavity has high degree of symmetry, the modified Planck and Stefan–Boltzmann radiation laws of a rectangular KNB can be obtained exactly, and we will deal with this question in detail in the present section. 3.1. Modified Planck and Stefan–Boltzmann radiation laws In principle, we can complete this work by use of Eqs. (14), (15) and (16), which require the knowledge of parameters Ap, t and μp. However, the two physical quantities can't be gotten easily. Hence we prefer a more direct method developed by Balian and Bloch for scalar waves [11,12], namely, the time-independent or energy-dependent Green function method. From this theory, the smooth term ρsmooth(E) of the density of states results from the multiple reflections of classical orbits of ˜ E stems from the closed microcosmic particles, and the oscillatory term ρ classical trajectories. Besides, further investigation shows that the dominant parts of ρ˜ ðEÞ are determined by the shortest periodic orbits. According to the theory of Balian and Bloch, and considering that the electromagnetic wave of nonpolaritons is a transverse wave, we can reach the following conclusions. Firstly, for the case of rectangular cavities, the closed periodic orbits concerned are polygons, which have an even number of vertices, and these orbits are responsible for the oscillatory term ρ˜ ðEÞ. Secondly, with the definition that L1, L2, L3 denotes the length of each side of a rectangular cavity, respectively, we can get the expressions of ρC ðω ˜ Þ and ρ˜ ðω ˜ Þ as follows: ˜Þ = − ρC ðω

1 ðL + L2 + L3 Þ; 2πvðT Þ 1

ρ˜ ðω ˜ Þ = ρ˜ 1 ðω ˜ Þ + ρ˜ 2 ðω ˜ Þ;

ð23Þ

ð24Þ

˜ Þ and ρ˜ 2 ðω ˜ Þ,respectively, are given as where the leading terms of ρ˜ 1 ðω

π2 v2 ðT ÞkB T 2 π4 ħ2 v2 ðT Þ ; C2 = : 6 4L6 4L



Moreover, with the definition that K ðτÞ = ∑p A2p;t δ τ−tτp , we can translate Eq. (19) into the following form: D

where τmin corresponds to the period of the shortest periodic orbit and it satisfies τmin = lmin/v(T). Referring to Ref. [25,26], we know that the value of function K(τ) approximately satisfies the following expression:

D

ℏv2 ðT Þ vðT ÞkB T + b1 ; L4 L3

4193

ρ˜ 1 ðω ˜Þ =

½

ω ˜ sinð2ωm ˜ 1 L1 = vðT ÞÞ L2 L3 ∑ m1 2π2 v2 ðT Þ m1 ≠0 + L1 L3 ∑

ð20Þ

m2 ≠0



sinð2ωm ˜ 2 L 2 = vð T Þ Þ sinð2ωm ˜ 3 L3 = vðT ÞÞ + L1 L2 ∑ ; m2 m3 m3 ≠0

ð25Þ

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Vω ˜ ρ˜2 ðω ˜Þ = 2 2 π v ðT Þ +

∑

½

  sin ωL ˜ α1 = vðT Þ

∑ L α1 m2 ;m3 ≠0   sin ωL ˜ α 3 = vð T Þ

m1 ;m2 ≠0

Lα3



+

∑

m1 ;m3 ≠0

  sin ωL ˜ α2 = vðT Þ Lα2

er

ð26Þ

3.2. Numerical calculation

:

In Eq. (26), Lαj 2(j = 1, 2, 3) satisfy the following expressions: 2

2 2

Lαj = 4 ∑ mi Li ; ði; j = 1; 2; 3Þ; i≠j

where the indices mi(i = 1, 2, 3) have the values of mi = ± 2, ± 4, ±6…, i.e., mi(i = 1, 2, 3) are even numbers. In addition, ρV ðω ˜ Þ still accords with the equation ρV ðω ˜ Þ = Vω ˜ 2 ðT Þ = π2 v3 ðT Þ. With those conclusions above we can immediately derive the modified Planck law by inserting ρV ðω ˜ Þ, Eqs. (23) and (24) into the equation uðω ˜ Þ = ρðω ˜ Þεðω ˜ Þ = V: ˜ Þ + uC ðω ˜ Þ + u˜ðω ˜ Þ; uðω ˜ Þ = uV ðω

ð27Þ

where uV ðω ˜ Þ; uC ðω ˜ Þ and u˜ðω ˜ Þ, respectively, can be given as 3

ħ ω ˜ ðT Þ ; π v ðT Þ eħ ω˜ ðT Þ = kB T −1

uV ðω ˜ Þ = ρðω ˜ Þεðω ˜Þ= V =

uC ðω ˜Þ = −

u˜ ðω ˜Þ =

ð28Þ

2 3

1 ħω ˜ ðT Þ ; ðL + L2 + L3 Þ· ħ ω˜ ðT Þ = k T B −1 2πVvðT Þ 1 e

1 ħω ˜ ðT Þ : ½ρ˜ ðω ˜ Þ + ρ˜2 ðω ˜ Þ· ħ ω˜ ðT Þ = k T B −1 V 1 e

ð29Þ

ð30Þ

Eq. (27) and correlative expressions (28), (29) and (30) constitute the modified Planck radiation law of a rectangular KNB. However, the integrated expression of modified Planck radiation law is rather ∞ cumbersome. After substituting Eq. (27) into RðT Þ = vðT Þ∫ u 0

ðω ˜ Þd ω ˜ = 4, the modified Stefan–Boltzmann law can be gotten easily, but this expression will be more cumbersome. Here we would study a special case of a rectangular cavity, namely, a cubic cavity. For this case, the modified Planck and Stefan–Boltzmann radiation laws have simple expressions, and the two simplified equations still retain the characteristic features of modified radiation laws to most extend. Accordingly, by setting L1 =L2 =L3 =L, the modified Planck radiation law in Eq. (27) can be reduced to 2ωmL ˜ sin 2 ω ˜ ðT Þ 3 3ω ˜ vðT Þ − + ∑ uðω ˜Þ = 2 3 m π v ðT Þ 2πL2 vðT Þ 2π2 v2 ðT ÞL m≠0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m21 + m22 ωL ˜ sin 3ω ˜ ℏω ˜ ðT Þ vðT Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ∑ + 2π2 v2 ðT ÞL m1 ;m2 ≠0 eℏ ω˜ ðT Þ = kB T −1 m21 + m22

½

ð31Þ



Eq. (31) represents the modified Planck radiation law of a cubic ∞ ˜ Þdω ˜ =4 KNB. Subsequently, inserting Eq. (31) into RðT Þ = vðT Þ∫ uðω 0 yields the modified Stefan–Boltzmann law as follows: " # k2B π 2 3ħ v3 ðT Þ ðk πT Þ3 2 T + ∑ − B 3 cothðxT Þcsch ðxT Þ 2 2 3 3 16ℏL ℏ m≠0 4vðT ÞLπ m 8L m " # 3 3 3ħ v ðT Þ ðkB πT Þ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cothð yT Þcsch ð yT Þ ; + ∑ 2

3 = 2 − 3 3 2 ℏ m1 ;m2 ≠0 2vðT ÞLπ2 m2 + m2 8L m1 + m2

RðT Þ = σ ′ ðT ÞT − 4

1

respectively, correspond to the volume term RV(T) and curvature term RC(T), and the last two terms constitute the oscillatory term R˜ ðT Þ.

2

ð32Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where x = 2kBπLm/v(T)ħ3, y = 2kB πL m21 + m22 = vðT Þħ3 , and m, m1, m2 only take even numbers. In Eq. (32), the first term and second term,

In the above section, we have studied in detail the modified Planck and Stefan–Boltzmann radiation laws of a rectangular KNB. Especially, we gave the exact analytic expressions of the two laws of a cubic KNB. Whereas, if we want to understand the two modified laws more clearly and provide a reference for the experimental verification of the two modified laws, it will be necessary to make a numerical calculation of uðω ˜ Þ and R(T). Here we decide to only study the particular case of a cubic cavity. Above all, we must determine the suitable values of some parameters, such as the temperature T, the side length L of a cubic cavity and so on. In the following, we firstly ask about the appropriate side length L. In general, the larger the L, the smaller the deviations are, and at the same time, for a cavity with too large L, it will become difficult to keep the temperature stable. On the contrary, for a cavity with too small L, it is also difficult to manufacture the cavity perfectly, and in the meantime, the energy flux escaping through the aperture of such a cavity is too small to be measured accurately. Therefore, the appropriate side length L can be estimated to be L≈20−120(mm) [22], and we decide to choose L=25mm as a tentative length. Secondly, the suitable temperature T should be below the transition temperature Tc and is easy to be hold. For a KNB with γ=0.9, Tc is equal to 469.4K[5]. We pick up T≤5K and can obtain that v(T)≈0.9c and λm =0.92mm, where λm is the wavelength at which the energy density uðω ˜ Þ is a maximum [5]. Moreover, this condition also satisfies the limit L≫λ well and the semiclassical formalism is accurate enough. Thirdly, in the actual numerical calculation of the summation in Eq. (31), we needn't consider too many values of indices m, m1 and m2. The reason is that the dominant contribution to the sum comes from the shortest periodic orbits, and this method is in agreement with the method in Ref. [12]. Fourthly, for the actual experimental detection, we should substitute ω=ω ˜ ðT Þc = nvðT Þ for ω ˜ ðT Þ in the formulas of uðω ˜ Þ and R(T) before the numerical calculation. Finally, we need choose a particular Kerr medium for a KNB. For convenience, we take the diamond crystal whose zero wave vector frequency of the Raman-active mode is ωR =2.51×1014s− 1 and assume its refractive index n=1. After finishing these preparation, we made a numerical calculation and got the following results. First of all, the variation with ω of u(ω) can be shown in Fig. 2, where we let T = 5K and γ = 0.9. From this figure we can see that in the case of a KNB, the deviation of a modified Planck radiation law from an ideal Planck radiation law is apparent, especially at the frequency ωm = 2.05 × 10 12Hz where the energy density u(ω) is a maximum. Furthermore, the variation with ω of u(ω) is also shown in Fig. 3 under the condition that T = 5K and γ = 0.35. By comparing the two figures, we can see that the variation of parameter γ will apparently influence the deviation from an ideal Planck radiation law. Strictly speaking, the smaller the parameter γ, the larger the absolute deviations are, but the smaller the relative deviations are. Besides, we define  Rsca ðT Þ = RðT Þ = RV ðT Þ = 1 + RC + R˜ ðT Þ = RV ðT Þ, where Rsca(T) represents the relative total energy emitted per unit of time and area. Then the variation with T and γ of Rsca(T) can be shown in .4. Rsca(T) of a normal blackbody is also shown in Fig. 4 for comparison. From Fig. 4 we can find two features: (a) For fixed γ, Rsca(T) increases continuously as the temperature T increases from 0.5K to 2.5K. (b) For fixed T, Rsca(T) is a monotonically decreasing function of γ. 4. Discussions In the present paper we first briefly reviewed the theory of a KNB developed by Cheng [5], and then we revisited the finite-size corrections to the radiation laws of a KNB. Based on these conclusions, we studied the fluctuations of the total radiation R(T) in a KNB with no symmetry axes. Subsequently and especially, we focused our investigation on the case of a rectangular KNB, and derived the explicit expressions of

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Fig. 2. For fixed γ = 0.9, side length L = 25mm and temperature T = 5K, variation of u(ω) with ω, where u(ω) is the energy density per unit of frequency. The dashed and solid lines correspond to an ideal and an actual cubic KNB, respectively.

modified radiation laws. As a result, we find that the deviations of modified radiation laws from ideal radiation laws mostly result from two terms. One is named as curvature term ρC ðω ˜ Þ, and ρC ðω ˜ Þ is determined by the mean curvature C of a cavity and the velocity v(T) of nonpolaritons. Another is an oscillatory term ρ ˜ ðω ˜ Þ, which is associated with the symmetry of the cavity and the closed classical orbits. The study also shows that ρ ˜ ðω ˜ Þ is the dominant term between the two parts ρC ðω ˜Þ and ρ ˜ ðω ˜ Þ. From a physical viewpoint, the existence of ρC ðω ˜ Þ and ρ ˜ ðω ˜ Þ is due to the fact that the distribution of wave vector k in a finite-size cavity is quite different from that in an infinite-size cavity. For the former case, the wave vector k is not a continuous variable, and electromagnetic waves will experience complicated multiple reflection, interference and so on. It is evident that those factors above will yield additional corrections to the density of states ρðω ˜ Þ, and these corrections should be determined by the size and shape of a cavity. Thus the modification of radiation laws is expected. Besides, we also make a numerical calculation of modified radiation laws for the special case of a cubic cavity. From the investigations above, we can find three primary features as follows: ( a ) For fixed temperature T, by comparing Figs. 2 and 3, we can find that the dominant term of modified Planck radiation law is the oscillatory part ũ(ω), and this term is a decreasing function of v (T) and γ (because γ signifies the coupling strength between a bare

Fig. 3. For fixed γ = 0.35, side length L = 25mm and temperature T = 5K, variation of u(ω) with ω, where u(ω) is the energy density per unit of frequency. The dashed and solid lines correspond to an ideal and an actual cubic KNB, respectively. The inset in Fig. 3 is a partial magnified drawing of u(ω) to contrast with Fig. 2.

4195

Fig. 4. For three values of γ of an actual cubic KNB and a normal cubic blackbody, variation of Rsca(T) with temperature T, where Rsca(T) is the relative total energy emitted per unit of time and area. The solid line corresponds to the normal blackbody.

photon and virtual nonpolar phonons, and v(T) is an increasing function of γ[5]). From a physical point of view, with decreasing of γ, the coupling strength between a photon and phonons will become stronger and so more nonpolaritons will arise [5]. It is undoubted that this effect will lead to a result that both the terms uV(ω) and ũ(ω) will increase simultaneously as γ is decreased. It must be noted that although uC(ω) will decrease in the meantime, but its influence can be neglected in this case. Besides, it also can be found easily that the relative fluctuation △u(ω) = |u(ω) − uV(ω)|/uV(ω) will decrease with decreasing of γ. (b) For fixed γ, Rsca(T) is an increasing function of temperature T. This phenomenon can be explained as follows: when the temperature T is increased, the velocity of a nonpolariton and the average particle numbers of nonpolaritons increase correspondingly [5]. Thus RV(T), R˜ ðT Þ and the absolute value of RC(T) will increase simultaneously. Due to the fact that RC(T) is negative, the sum of RC(T) and R˜ ðT Þ will approach zero with increasing of T, and consequently Rsca(T) will increase and finally approach one. (c) For fixed T, Rsca(T) is a decreasing function of γ. From a physical viewpoint, we can interpret this result as follows: with increasing of parameter γ, the velocity v(T) of a nonpolariton increases [5], and thus the energy of the nonpolariton system and RV(T) become smaller, but R˜ ðT Þ becomes larger. According to expressions of Rsca(T), RV(T) and R˜ ðT Þ, and considering that RC(T) is a negative constant, we know that RV(T) will vary faster than the absolute value of R˜ ðT Þ + RC ðT Þ. Then considering R˜ ðT Þ + RC ðT Þ≤0, it is evident that Rsca(T) will become smaller with increasing of parameter γ. In addition, we can find that Rsca(T) of a KNB is larger than Rsca(T) of a normal blackbody, and the difference of Rsca(T) between a KNB and a normal blackbody will become larger with decreasing of parameter γ. Whereas, the differences of Rsca(T) among all these blackbodies approach zero with increasing of the temperature T. The reason is that more and more virtual nonpolar phonons will evaporate when the temperature T approach the transition temperature Tc[5], so the behavior of a nonpolariton is more and more like the behavior of a bare photon. Finally at T ≥ TC, the KNB behaves just as a normal blackbody. In addition, here we would like to compare the modified radiation laws of a rectangular KNB with those of a spherical KNB, which have been discussed in Ref. [10]. If we contrast Figs. 2, 3 and 4 in the present paper with Figs. 1, 2 and 3 in Ref. [10], it will be found that almost all the primary features of a cubic KNB are identical with those of a spherical KNB except for the following two points. First, as γ is decreased, the relative fluctuation △u(ω) of a cubic KNB will decrease apparently, whereas the relative fluctuation △u(ω) of a spherical KNB almost remains unchanged or decreases slightly. This result may be due to the fact that we have chosen different values of γ, respectively, in the two case. However, according to this comparison, at least we can conclude

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that △u(ω) of a KNB is an increasing function of γ. Second, by contrasting Fig. 2 of this paper with Fig. 1 of Ref. [10], it is clear that △u′(ω) = |u(ω) − uV(ω)| of a cubic KNB is smaller than that of a spherical KNB. Because all the parameters of the two KNBs are almost identical, it can thus be concluded that in the former case the effect of finite-size corrections to the radiation laws is not as obvious as that in the later case. Based on this conclusion, perhaps a spherical KNB is more suitable for the experimental verification of the model of a KNB. However, if we take into account the fact that a perfect cubic KNB can be manufactured more easily than a perfect spherical KNB, a cubic one should a more prior choice for the experimental verification. In belief, as mentioned before in Section 1, this paper is a perfection of Ref. [10] and the theory of a KNB, and it is expected that this work would lay the foundation for the experimental verification of the model of a KNB. In conclusion, by using semiclassical techniques, we obtain the explicit analytic expressions of radiation laws of a rectangular KNB, and these modified radiation laws can be represented as functions of the temperature and size of a cavity. Besides, the case of a KNB with no symmetry axes is also discussed. Finally, we make a calculation of modified radiation laws of a cubic KNB under appropriate conditions. We consider that this work should be helpful for the experimental verification of the model of a KNB which may have a widely prospect of application. Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant nos. 10174024 and 10474025.

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