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1、arXiv:quant-ph/9906001v2 16 Jul 1999 QED in the Presence of Arbitrary KramersKronig Dielectric Media Stefan Scheel and DirkGunnar Welsch Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universit at Jena, Max-Wien-Platz 1, D-07743 Jena, Germany The phenomenological Maxwell fi eld is quantized
2、 for arbitrarily space- and frequency-dependent complex permittivity. The formalism takes account of the KramersKronig relation and the dissipation-fl uctuation theorem and yields the fundamental equal-time commutation relations of QED. Applica- tions to the quantum-state transformation at absorbing
3、 and amplifying four- port devices and to the spontaneous decay of an excited atom in the presence of absorbing dielectric bodies are discussed. I. INTRODUCTION Quantization of the electromagnetic fi eld in dispersive and absorbing dielectrics requires a concept which is consistent with both the pri
4、nciple of causality and the dissipation fl uctuation theorem and which necessarily yields the fundamental equal-time commutation relations of QED. In order to achieve this goal, several approaches are possible. The micro- scopic approach starts from the exact Hamiltonian of the coupled radiationmatt
5、er system and integrates out, in some approximation, the matter degrees of freedom to obtain an ef- fective theory for the electromagnetic fi eld. Since the procedure can hardly be performed for arbitrary media, simplifi ed model systems are considered. A typical example is the use of harmonic-oscil
6、lator models for the matter polarization and the reservoir variables together with the assumption of bilinear couplings 1. In the macroscopic approach, the phenomeno- logical Maxwell theory, in which the eff ect of the medium is described in terms of constitutive equations, is quantized. Since this
7、concept does not use any microscopic description of the medium, it has the benefi t of being universally valid, at least as long as the medium can be regarded as a continuum. Here we study the problem of quantization of the phenomenological Maxwell theory for nonmagnetic but otherwise arbitrary line
8、ar media at rest, starting from the classical Green function integral representation of the electromagnetic fi eld. The method was fi rst established for one-dimensional systems 2 and simple three-dimensional systems 3 and later generalized to arbitrary inhomogeneous dielectrics described in terms o
9、f a spatially varying permittivity which is a complex function of frequency 4. In Sec. II we briefl y review the quantization scheme and give an extension to anisotropic dielectrics (including amplifying media), which complete the class of nonmagnetic (local) media. In Sec. III we apply the method t
10、o the problem of quantum-state transformation at absorbing and amplifying four-port devices, and in Sec. IV we give an application to the problem of spontaneous decay of an excited atom in the presence of absorbing bodies. II. QUANTIZATION SCHEME Let us fi rst consider the electromagnetic fi eld in
11、isotropic dielectrics without external sources. The (operator-valued) phenomenological Maxwell equations in the temporal Fourier space read B(r,) = 0, E(r,) = iB(r,),(1) 0(r,)E(r,) = (r,), B(r,) = i(/c2)(r,)E(r,) + 0j(r,).(2) From the principle of causality it follows that the complex-valued permitt
12、ivity (r,) = R(r,)+iI (r,) satisfi es the KramersKronig relations. Hence, it is a holomorphic func- tion in the upper complex frequency plane without poles and zeros and approaches unity in the high-frequency limit. Consistency with the dissipationfl uctuation theorem requires the introduction of an
13、 operator noise charge density (r,) and an operator noise current den- sityj(r,) satisfying the equation of continuity. Quantization is performed by introducing bosonic vector fi elds f(r,), j(r,) = q h0I(r,)/f(r,),(3) which play the role of the fundamental variables of the theory. All relevant oper
14、ators of the system such as the electric and magnetic fi elds and the matter polarization can be constructed in terms of them. For example, the operator of the electric fi eld is given by the integral representation Ek(r) = i0 q h0/ Z 0 d Z d3r2 q I(r,)Gkk(r,r,)fk(r,) + H.c.,(4) with Gkk(r,r,) being
15、 the classical dyadic Green function. This representation together with the fundamental relation Z d3s(/c)2I(s,)Gik(r,s,)G jk(r ,s,) = ImGij(r,r,), (5) which follows directly from the partial diff erential equation for the dyadic Green function, leads to the equal-time commutation relation 3 h 0Ek(r
16、), Bl(r) i = ( h/)lmkr m Z d (/c2)Gkk(r,r,).(6) Using general properties of the Green function, it can be shown 4 that Eq. (4) reduces, for arbitrary (r,), to the well-known QED commutation relation h 0Ek(r), Bl(r) i = i hklmr m(r r ) (7) The extension to anisotropic and amplifying media is straight
17、forward, since we may assume the medium to be reciprocal, so that the permittivity tensor ij(r,) is necessarily symmetric. In particular, ij(r,) can be diagonalized by an orthogonal matrix Okl(r,). With regard to amplifying media, we note that amplifi cation requires the role of the noise creation a
18、nd annihilation operators to be exchanged. The calculation then shows that the fundamental relation (3) can be generalized to ji(r,) = q h0/ h ij(r,) fj(r,) + + ij(r,) f j(r,) i ,(8) with ij(r,) = Oik(r,) q | kl I(r,)| O1 lj (r,) kl I(r,),(9) ij I(r,) = ij(i) I (r,) = O1 ik (r,)kl I(r,)Olj(r,).(10)
19、Equation (8) completes the quantization scheme for the electromagnetic fi eld in arbitrary linear, nonmagnetic (local) media. III. QUANTUM-STATE TRANSFORMATIONS BY ABSORBING AND AMPLIFYING FOUR-PORT DEVICES Let us fi rst apply the theory to the problem of quantum-state transformation at absorbing an
20、d amplifying four-port devices such as beam-splitter-like devices. Specifying the formulas to the one-dimensional case for simplicity and rewriting the integral representation (7) in terms of amplitude operators aj() andbj() for the incoming and outgoing waves (j=1,2), the action of an absorbing dev
21、ice can be given by the (vector) operator transformation b() = T() a() + A() g(),(11) where gj() are the operators of device excitations and T() and A() are the characteristic transformation and absorption matrices of the device given in terms of its complex refractive- index profi le 5. Note that a
22、j() and gj() are independent bosonic operators. Further, it can be shown that the relation T()T+()+ A()A+ ()= I is satisfi ed, which ensures bosonic commutation relations forbj(). In order to construct the unitary transformation, we introduce some auxiliary (bosonic) device variables hj(), combine t
23、he two-vectors a() and g() to the four-vector (), and accordingly b() and h() to (). The four-vectors () and () are related to each other as () = () (),() SU(4).(12) Introducing the positive Hermitian matrices C()= q T()T+() and S()= q A()A+(), the four-matrix () can be written in the form 6 () = T(
24、)A() S()C1()T() C()S1()A() ! (13) (=1). The inputoutput relation (12) can then be expressed in terms of a unitary operator transformation ()= U ()U. Equivalently, U can be applied to the density operator of the input quantum state in (), (), and tracing over the device variables yields (Field) out =
25、 Tr(Device) n in h +() (),T() () io .(14) To give an example, let us consider the case when one input channel is prepared in an n-photon Fock state and the device and the second input channel are left in vacuum, i.e., in= |n,0,0,0ihn,0,0,0|. Applying Eq. (14), after some algebra we derive for the de
26、nsity operator of the i-th output channel (Field) out,i = n X k=0 n k ! |Ti1|2k ? 1 |Ti1|2 ?nk |kihk|.(15) Next, let us assume that the two input channels are prepared in single-photon Fock states, i.e., in=|1,1,0,0ih1,1,0,0|. We derive for the density operator of the i-th output channel (Field) out
27、,i = h 1 |Ti1|2(1 |Ti2|2) |Ti2|2(1 |Ti1|2) i |0ih0| + ? |Ti1|2+ |Ti2|2 4|Ti1|2|Ti2|2 ? |1ih1| + 2|Ti1|2|Ti2|2|2ih2|.(16) The extension to amplifying devices is straightforward. One has to replace the annihila- tion operators gj() in Eq. (11) by the corresponding creation operators g j(). This leads
28、again to an input-output relation of the form (12) but with =1 in Eq. (13), the matrix () being now an element of the noncompact group SU(2,2). IV. SPONTANEOUS DECAY NEAR DIELECTRIC BODIES Spontaneous decay of an excited atom is a process that is directly related to the quantum vacuum noise, which i
29、n the presence of absorbing bodies is drastically changed and so is the rate of spontaneous decay, because of the additional noise introduced by absorption. To study a radiating (two-level) atom in the presence of dielectric media, we start from the following Hamiltonian in dipole and rotating wave
30、approximations: H = Z d3r Z 0 d hf(r,) f(r,) + 2 X =1 hA h i21A21A(+)(rA) d21+ H.c. i . (17) Here, the atomic operators A= |ih| are introduced, and A(+)(rA) is the (positive- frequency part of the) vector potential (in Weyl gauge) at the position of the atom. Note that the fi rst term in Eq. (17) is
31、 the (diagonal) Hamiltonian of the system that consists of the electromagnetic fi eld and the medium (including the dissipative system) and is expressed in terms of the fundamental variables f(r,). Solving the resulting equations of motion in Markov approximation, the well-known Bloch equations for
32、the atom are recognized, where the decay rate is given by 7 = 22 Akk/( h0c 2)ImGkk(rA,rA,A) (18) k(d21)k, A21. Note that from Eq. (4) together with Eq. (5) it follows that ImGkk(r,r,)( ) = 0c2/( h2)h0|Ek(r,), E k(r ,)|0i (19) in full agreement with the dissipation-fl uctuation theorem. Equation (18)
33、 is valid for any absorbing dielectric body. For example, when the atom is suffi ciently near to an absorbing planar interface, then purely nonradiative decay is observed, with 8 = 0 1 + 2 z 2 ! I(A) |(A) + 1|2 3c3 (2Az)3 ,(20) where z is the distance between the atom and the interface, and 0is the
34、spontaneous emission rate in free space (for a guest atom embedded in an absorbing dielectric, see 7). Acknowledgement This work was supported by the Deutsche Forschungsgemeinschaft. 1 B. Huttner and S.M. Barnett, Phys. Rev. A 46, 4306 (1992). 2 T. Gruner and D.-G. Welsch, Phys. Rev. A 53, 1818 (199
35、6). 3 Ho Trung Dung, L. Kn oll, and D.-G. Welsch, Phys. Rev. A 57, 3931 (1998). 4 S. Scheel, L. Kn oll, and D.-G. Welsch, Phys. Rev. A 58, 700 (1998). 5 T. Gruner and D.-G. Welsch, Phys. Rev. A 54, 1661 (1996). 6 L. Kn oll, S. Scheel, E. Schmidt, D.-G. Welsch, and A.V. Chizhov, Phys. Rev. A 59, 4716 (1999). 7 S. Scheel, L. Kn oll, D.-G. Welsch, and S.M. Barnett, Phys. Rev. A 60 (1999), in press; S. Scheel, L. Kn oll, and D.-G. Welsch, submitted to Phys. Rev. A. (quant-ph/9904015). 8 S. Scheel, L. Kn oll, and D.-G. Welsch, Acta Phys. Slov. 49 (special issue), 585 (1999).