DOI 10.1007/s11141-015-9567-4

Radiophysics and Quantum Electronics, Vol. 57, No. 11, April, 2015 (Russian Original Vol. 57, No. 11, November, 2014)

ANISOTROPY-INDUCED TRANSPARENCY IN OPTICALLY DENSE MEDIA M. D. Tokman and M. A. Erukhimova

UDC 535.44+537.876+535.14

The effect of anisotropy-induced transparency, which is analogous to electromagnetically induced transparency in the three-level medium located in a resonance field, is predicted and studied theoretically. This effect is connected with destructive interference between oscillations in different degrees of freedom of an anisotropic medium, which are connected with each other, as radiation propagates at an angle to one of the optical axes in a triaxial or uniaxial crystal. In this case, a hybrid-type polariton is formed in the “transparency window,” which combines the quasilongitudinal polarization with the “vacuum” refractive index. Such a wave is excited easily by radiation incident from the vacuum and should have enhanced impedance of coupling with active or nonlinear elements, which can be useful for the creation of small-size optical systems. Due to the interest in quantum-optical effects displayed recently, the regime of anisotropy-induced transparency is considered within the framework of the quantum theory of radiation in an optically dense medium.

1.

INTRODUCTION

Studying of variants of the effect of electromagnetically induced transparency (EIT) is a fast-growing domain of nonlinear and quantum optics. The best-known variant is the effect of electromagnetically induced transparency in quantum three-level systems controlled by resonance pumping (see review works [1–3]). In slightly different terms, the corresponding effects are called the Fano resonances (see review work [4]). In all cases, the effect is manifested as destructive interference of two resonances in the process of parametric interaction of two oscillatory systems (quantum or classical ones) which are linked by the control field (see [5, 6]). As of now, a group of closely related effects have been found, specifically, acoustically induced transparency (AIT [7]), parametrically induced transparency (PIT [8–10]), plasmon-induced transparency [4], and undulator-induced transparency (UIT [11–13]). A macroscopic manifestation of the effect of electromagnetically induced transparency is formation of a “steep” dispersion dependence, which corresponds to strong group deceleration, for probe radiation in a narrow frequency transparence band. At present, similar effects are implemented in such high-technology objects and new artificial optical media as plasmon nanostructures, photon crystals, quantum points and filaments, and electromagnetic metamaterials. Primarily, these works serve the interests of precision spectroscopy, although their use in some important fields of coherent and nonlinear optics, as well as quantum computer science, are also widely discussed (see [4, 14–33]). In all these systems, small-size electromagnetic elements (quantum or classical ones) model polarization properties of three-level atoms placed in the control resonance pumping field. We turn our attention to the effect of undulator-induced transparency (UIT) in magnetized plasma [11–13]. Within the UIT scheme, the pumping is produced by a magnetostatic undulator, and the “transparency window” is formed within the frequency range of cyclotron absorption when the cyclotron and plasma frequencies coincide. As electromagnetic radiation is injected from free space into a plasma Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 57, No. 11, pp. 917–933, November 2014. Original article submitted June 25, 2014; accepted September 17, 2014. c 2015 Springer Science+Business Media New York 0033-8443/15/5711-0821  821

with undulator-induced transparency, the incident transverse wave is transformed to the quasi-longitudinal (hybrid) wave [12, 13]. In such a wave, the Poynting vector is related to the relatively small field component, which is transverse to the wave vector, therefore, the group velocity turns to be much slower than the velocity of light. In the presence of additional active or nonlinear elements, such a mode would interact with them much stronger compared with the transverse wave, since it would have a higher coupling impedance. Note that this property of the quasi-longitudinal mode is used in lasers based on surface plasmons (see, e.g., [34– 36]): in such systems, the field energy is concentrated in a region which has a nanoscale size determined by the thickness of the skin layer for optical radiation. It is evident that the formation of a quasi-longitudinal mode with strong group slowdown in the bulk (and not on the surface) is a rather attractive scheme. Similarly to the regime of undulator-induced transparency, one can assume that, technically, one can convert the radiation, which is incident from vacuum, to the quasi-electrostatic mode in any anisotropic medium containing electrostatic polaritons or plasmons. To do this, it is necessary to ensure that the wave propagates at a certain angle to one of the optical axes of the crystal. We will call this regime the “anisotropy-induced transparency” (AnIT) and study this effect herein. Allowing for the great interest in quantum-optical effects, which has been taken in recent years, we develop here the quantum theory of the anisotropy-induced effect, which is closely related to a more general problem of developing the quantum description of radiation in an optically dense medium. The quantum description of the field in a medium is usually based on applying of the standard quantization procedure to phenomenological equations of the classical electrodynamics of continuous media [37, 38]. The phenomenological character of this theory is determined by that a preset dielectric permittivity or magnetic permeability are assigned to the medium [39, 40]. Evidently, the above-specified procedure is not always adequate to the problem of self-consistent description of the field and the medium. Therefore, we will use self-consistent Heisenberg equations for the field and the medium, which allow, in particular, for the effect of the acting field. The criteria for realization of the effects, which are related to the electromagnetically induced transparence, always depend strongly on dissipation in the medium. Due to this, we will have to dwell here also on some aspects of quantum description of the field in a dissipative medium. The text is structured as follows. Section 2 deals with the formulation of the self-consistent quantum description of the field in the medium: subsection 2.1 formulates the basic initial equations for the field and the medium, a material equation within the linear approximation is obtained in subsection 2.2, subsection 2.3 describes the procedure of second quantization of the radiation field in a transparent anisotropic medium, and subsection 2.4 presents the quantum description of the field in a dissipative medium. Section 3 considers the anisotropy-induced transparency effect proper for the hybrid modes of the anisotropic medium: subsection 3.1 studies transverse photons and longitudinal polaritons propagating along the optical crystal axes, subsection 3.2 analyzes quasi-electrostatic (hybrid) waves propagating at a low angle to the crystal axis, and subsection 3.3 discusses the analogy between the effects of anisotropy-induced transparency and electromagnetically induced transparency. 2.

QUANTUM DESCRIPTION OF THE FIELD IN AN ANISOTROPIC DIELECTRIC MEDIUM

2.1.

Initial relations

Consider a continuous dielectric medium characterized by the energy spectrum Wn and the set fmn of matrix elements for all physical values f of interest (for simplicity, we assume that the indices n and m are discrete, although this does not have any fundamental importance). In the case of a medium without spatial dispersion, the spatiotemporal dependence of the average f (r, t) can be determined by the relation  f (r, t) = n,m fnm ρmn (r), where ρmn (r, t) is the corresponding “local” density matrix [9, 41]. Treating the field-medium interaction in the quantum way, it is convenient to introduce the Heisenberg operator of the distributed value fˆ(r, t), which is determined by means of the corresponding Heisenberg density operator:  ˆmn (r, t). The operators fˆ(r, t) and, correspondingly, σ ˆmn (r, t) are usually introduced fˆ(r, t) = n,m fnm σ 822

as the values of the corresponding operators for individual atoms1 averaged over a physically small volume. (j) In this case, it is insignificant whether the density operator σ ˆmn for an individual (jth) atom is determined (j) (j)† (j) as the product of the operators for quantum-state creation and annihilation σ ˆ mn = a ˆn a ˆm (here, the index (j) “†” means the creation operator) [42], or as the projection operator σ ˆmn = |nj m| [2, 3]. We turn our attention to some of the properties of the above-introduced density operator σ ˆmn (r, t), which are important for the further consideration. This operator satisfies the following commutation relation [42]: ˆmn (r , t)] = δ(r − r ) [δqn σ ˆmp (r, t) − δmp σ ˆqn (r, t)]. (1) [ˆ σqp (r, t), σ Each element of the matrix σ ˆmn (r, t) is an operator, and after averaging over the constant (initial) quantum state of the system |Ψ, which appears in the Heisenberg approach, it should coincide with the ˆmn (r, t) |Ψ. In this case, one corresponding element of the “standard” density matrix: ρmn (r, t) = Ψ| σ can always use the corresponding unitary transformations to modify the initial vector of the state |Ψ and 0 (r) simultaneously in such a way as to ensure that the average values the initial values of the operators σ ˆmn 0 (r) |Ψ = ρ of these parameters stay unchanged. For example, Ψ| σ ˆmm mm (r) is the initial spatial popula0 tion distribution, Ψ| σ ˆm, n=m (r) |Ψ = ρm, n=m (r) is the initial spatial distribution of quantum coherences, 0 (r)ˆ 0 (r) |Ψ is the initial value of the spatial correlator, etc. Thus, when the Heisenberg approach Ψ| σ ˆmn σpq is used, physically measurable average values at the initial time moment, which are required for analysis of this or that problem, can be specified as the initial conditions. In this case, the choice of the initial vector of the system state is purely symbolic, in a way, and is determined by the ease of description. This approach is similar to the method which is frequently used in the quantum field theory when the vector of the initial state is specified as a result of the effect of the unitary operator on the “universal” initial Fock state |0 [43, 44]. We specify the quantum Hamiltonian of the “field+medium” system in the volume V as follows:    ˆ2 ˆ2 α E + B 2 ˆ +w ˆ = − P ˆ d3 r, (2) H 8π 2 V

ˆ t) and B(r, ˆ t) are operators of the electric and magnetic fields, respectively, where E(r,  Wn σ ˆnn w ˆ=

(3)

n

is the internal-energy density operator, ˆ = P



dnm σ ˆnm

(4)

n,m

is the operator of polarization of a unit medium volume, and dnm is the corresponding matrix element of the dipole moment. That formula (2) includes a term proportional to the parameter α is connected with the allowance for the energy of the near dipole–dipole interaction in a polarized medium [45–47]. The procedure of isolating the energy of quasistatic interaction of elementary dipoles within the Hamiltonian of the “field+medium” system is described in depth in [48] for an ensemble of magnetic dipoles. The corresponding procedure can also be generalized for the case of electric dipoles. The corresponding constant within the prevailing Lorentz— Lorentz model is α = 4π/3. Applicability of this model within framework of the quantum-electrodynamic description was demonstrated in [47] for the two-level medium. Within the framework of the electrodipole field—medium interaction, the canonically conjugate pair ˆ (here, B ˆ = ∇ × A) ˆ and the generalized momentum F ˆ which is of field variables is the vector potential A 1

Or other elementary objects.

823

proportional to the electric-displacement vector [37]: ˆ ˆ =− D , F 4πc

ˆ =E ˆ + 4π P. ˆ D

ˆ and D, ˆ Hamiltonian (2) has the following form: In the variables A

  ˆ2  ˆ 2 D + (∇ × A) α ˆP ˆ + 2π − ˆ 2 d3 r. ˆ = +w ˆ−D P H 8π 2

(5)

(6)

V

We will use the standard commutation relation for the canonically conjugate pair “momentum— coordinate” [43, 45, 49]: [Fˆp (r , t), Aˆq (r, t)] = −iδpq δ(r − r ) (7) (here, the subscripts p, q = x, y, z and  is the Planck constant). Since the Heisenberg operators retain the commutation properties of the constant Schr¨odinger operators [43], the field operators and the medium operators commute with each other: ˆmn (r, t)] = [Aˆp (r , t), σ ˆnm (r, t)] = 0. [Fˆp (r , t), σ

(8)

Together with constraints (3), (4), (5) and commutation relations (1), (7), and (8), quantum Hamiltonian (6) yields a self-contained description of the field in the dielectric. ˆ˙ = i [H, ˆ F]/ ˆ From the Heisenberg equation F and commutation relation (7), we find that ˆ˙ = c ∇ × ∇ × A, ˆ D

(9)

(the dot means the time derivative, and c is the speed of light), wherefrom it follows that ˆ = 0. ∇D

(10)

Further, we represent the polarization vector as a sum of the vortex and potential components ˆ  , where ∇ × P ˆ  = ∇P ˆ ⊥ = 0. Using relation (10), one can prove the following equality: ˆ ˆ P = P⊥ + P   3 ˆ ˆ ˆP ˆ ⊥ ) d3 r. (DP) d r = (D (11) V

V

The latter relation can be proven in the easiest way by using periodic boundary conditions for the vector ˆ t) and D(r, ˆ t) over a fields on the surface of the quantization volume V and expanding the operators P(r, 2 ˆ set of discrete spatial Fourier harmonics allowing for the property ∇D = 0. Allowing for relation (11), one can make sure that Hamiltonian (6) is equivalent to the formula presented in [45], which was obtained for the special case of a two-level system. ˆ˙ = i [H, ˆ A]/ ˆ In view of Eq. (11), it follows from the Heisenberg equation A that ˆ˙ = −c (D ˆ − 4π P ˆ ⊥ ). A

(12)

ˆ from Eq. (5) to Eq. (12), we obtain Substituting the formula for the operator D ˆ˙ − 4π P ˆ . ˆ t) = −c−1 A E(r, δ



2

V

824

(13)

For classical fields, one can prove the equality of variations of the functionals for fixed boundary values of the vector fields:  (DP) d3 r = δ V (DP⊥ ) d3 r.

Further, substituting Eq. (13) into Eq. (9), we find the operator wave equation ¨ˆ ˆ = 0. D + c2 ∇ × ∇ × E

(14)

The field equations should be supplemented with equations for the medium dynamics. Using ˆ σ the Heisenberg equation σ ˆ˙ mn = i [H, ˆmn ]/ and allowing for relations (1), (3), (4), and (8), we obtain (see [42, 50]) i σ ˆ˙ mn = − 

 ˆ mν σ ˆ νn ), (h ˆνn − σ ˆmν h

ˆ mν (r, t) = Wm δmν − E ˆ a (r, t) dmν , h

(15)

ν

ˆ a (r, t) [45–47, 51] of the acting field is written as where the operator E ˆa = E ˆ + αP ˆ =D ˆ − (4π − α)P. ˆ E

(16)

Equation (15) for the medium density operator has the form of the standard von Neumann equation despite the fact that the initial motion equation for the Heisenberg operator has the opposite sign in front of the corresponding term with the commutator. This fact was pointed out in [42, 50]. Operator equations (14) and (15) with constraints (3), (4), (5), and (16) yield a self-contained description of the Hamiltonian system “quantum field+dielectric quantum medium”. Sometimes, it is necessary to generalize the obtained equations for the case of dissipative systems. This can be done by supplementing additively the operators of relaxation and Langevin noises (see, e.g., [2, 3, 42, 49, 50, 52–55]) to Eq. (16) of the atomic-system dynamics: i σ ˆ˙ mn = − 

 ˆ mν σ ˆ νn ) + R ˆ mn + L ˆ mn . (h ˆνn − σ ˆmν h

(17)

ν

ˆ mn is the Langevin noise operator which describes fluctuations ˆ mn is the relaxation operator and L Here, R in the atomic system. Obviously, the interaction of the field with the medium results in entanglement of the quantum states of the corresponding subsystems. Within the framework of the Heisenberg description, to which the fixed state vector corresponds, each Heisenberg operator, generally speaking, is a function of constant (Schr¨odinger) operators of both subsystems and time. Development of the entanglement enters this approach as a time-dependent functional relation between the operators of the subsystems. It is important to note that in this case, the commutation relations for the time-dependent Heisenberg operators are still identical to those for the Schr¨odinger operators (see, e.g., [43]).

2.2.

Material equation for a linear anisotropic dielectric

In many cases, it is convenient to use the Heisenberg operators acting only in the space of field states.3 The transition to such operators means averaging with respect to the initial quantum state of the medium; for example, the dispersion relation for the photon in the medium ω = ω(k) [37, 38, 43] should also include ˆmm . In the domain of applicability of the linear quantum-field the populations ρmn , but not the operators σ ˆ approximation E, one can perform averaging with respect to the medium variables directly in the equations for operator evolution. In this case, the diagonal elements of the density operator σ ˆ mm are replaced with ˆm n=m will the constant populations (c-numbers) ρˆ0mm , and the equations for the non-diagonal elements σ 3

In [45], the transition to the operators acting only on the state of the medium is also considered.

825

depend on the field operators and population operators4 : i ˆ a (r, t) + R ˆ mn , σ ˆ˙ mn + iωmn σ ˆmn = (ρ0nn − ρ0mm ) dmn E 

(18)

where Wm − Wn = ωmn . In Eq. (18), Langevin noise has been omitted, since within the linear problem, it is more convenient to allow for fluctuations (whenever necessary) directly in the finite expressions for the field operators (see subsection 3.4 below). The solution of operator equation (18), together with relations (4), (5), and (16), determines the constituting relation, which can formally be represented in the standard form ∞

↔

ˆ t) = εˆ E(r, ˆ t) ≡ D(r,

← → ˆ t − τ ) dτ, ε (r, τ )E(r,

(19)

0 ↔

where εˆ is the tensor operator5 of dielectric permittivity, to which the spectral transform ∞ ↔ ↔ εˆ (r, τ ) exp(iωτ ) dτ = εˆ ω (r)

(20)

0

ˆ mn = −γmn σ ˆmn as the operacorresponds [38–40]. Choosing the relaxation operator in the simplest form R 6 tor, we obtain from Eqs. (4), (5), (16), and (18) that ↔ εˆ ω (r)

↔

↔

↔

↔

= 1 +4π [ 1 −α χ ˆ aω (r)]−1 χ ˆ aω (r),

(21)

↔

↔

ˆ aω (r) is determined by its effect on a certain vector C: where 1 is the unit diagonal matrix, and the tensor χ ↔ χ ˆ aω (r)C

≡

1  ρ0nn (r) − ρ0mm (r) dnm (dmn C).  m,n ωmn − iγmn − ω

(22)

Equations (14), (19), and (20) are used, e.g., in [37], in the quantization procedure applied to the phenomenological equations of classical electrodynamics of a continuous medium with given dielectric permittivity. In contrast with the above-specified approach, we used the self-consistent Heisenberg equations for the field and the medium, which are specified by Hamiltonian (2) in the case of a closed system.

2.3.

Quantum description of the field in a transparent dielectric

In a homogeneous medium, the standard second quantization procedure can be applied to Eqs. (14), (19), and (20), if one represents the field as a set of plane monochromatic waves [37, 38]   (σ) (σ) (σ)∗ (σ)† ˆ = Ek ˆbk exp(ikr) + Ek ˆbk exp(−ikr) . (23) E k,σ

Here, the set of wave vectors k is determined by the boundary conditions on the surface of the quantization (σ) (σ)† volume V ; ˆbk and ˆbk are the operators of creation and annihilation of independent boson states marked with the indices σ and k, where the asterisk superscript means complex conjugation. The creation and 4 5

In [42, 56], this approach was used in the presence of the classical control field (pumping). ↔

To avoid a misunderstanding, it should be emphasized that appearance of the operator εˆ here is connected not with the quantum consideration, but with the specific features of formation of the electrodynamic response in a dispersive medium [37–40]. 6 For possible refinements of this simplest expression, see, e.g., [50].

826

annihilation operators satisfy the equations ˆb˙ (σ) + iω (σ)ˆb(σ) = 0, k k k

ˆb˙ (σ)† − iω (σ)ˆb(σ)† = 0. k k k

(σ)

(σ)

The frequencies ωk and the components of the normalization vector Ek relations for the anisotropic medium [37, 40]: 

c2 k2 ↔ c2 ↔ ε det 1 + 2 kk = 0, ω− ω2 ω

 2 2 2 ↔ c k ↔ c (σ) εˆ ω − 2 1 + 2 kk Ek = 0. ω ω

(24) are determined by standard

(25) (26)

Here, we use the symbolic notation kk for the matrix formed by the vector components: (kkij = ki kj , i, j = x, y, z; the superscript σ enumerates different solution branches ωk = ω(k) of Eq. (25); as a rule, different indexes σ correspond to waves with different polarizations. (σ)

Normalization conditions for |Ek |2 within the framework of the phenomenological approach [37, 38] are specified by using the standard relation for the energy WHF of the classical monochromatic field in a dispersive medium [38–40]. For the boson operators corresponding to the standard commutation relation 

(σ) (σ )† [ˆbk , ˆbk ] = δσσ δkk ,

(27)

(σ)

the normalization conditions for |Ek |2 are chosen based on the requirement that replacement of the classical field with quantum operator (23) transforms the energy WHF to the Hamiltonian corresponding to an ensemble of harmonic oscillators:  (σ)  (σ)† (σ) 1  ˆ ωk ˆbk ˆbk + . (28) HHF = 2 k,σ

ˆ HF as the difference between HamiltoWe will use a stricter approach and determine the operator H nian (2) and the energy of the unperturbed electromagnetic field of the medium:      ˆ2 ˆ2 + B α E 2 3 ˆ ˆ − P +w ˆ d r− Wm ρ0mm d3 r, (29) H= 8π 2 m V

V

where the populations ρ0mm correspond to the medium unperturbed by the electromagnetic field. For the operator w ˆ of the energy density of the medium’s internal degrees of freedom, we will use the relation which follows directly from Eqs. (15) and (16) with allowance for definitions (3), (4), and (5):

w ˆ=

 m

Wm ρ0mm

α ˆ2 1 + P + 2 4π

t −∞

ˆ2 ˆ˙ E ˆ dτ − E . D 8π

(30)

ˆ and E. ˆ Applying the method Then, we will use linear relation (19) and (20) between the operators D described in [37–40], we obtain the following result:  t V −∞



↔  V  (σ)∗ ∂(ω ε ω )  (σ) ˙ˆ ˆ 3 ˆb(σ)†ˆb(σ) + ˆb(σ)ˆb(σ)† . DE dτ d r = Ek ω=ω(σ) Ek k k k k 2 ∂ω k

(31)

k,σ

827

The magnetic-field operator is specified by the formula   c  (σ) ˆ(σ) (σ)∗ ˆ(σ)† ˆ = [k × E ] b exp(ikr) + [k × E ] b exp(−ikr) . B k k k k (σ) ω k,σ k  ˆ 2 d3 r can be transformed to the following from using Eqs. (25) and (26): The formula for the operator V B     (σ)∗ ↔  (σ) ˆb(σ)†ˆb(σ) + ˆb(σ)ˆb(σ)† . ˆ 2 d3 r = V B Ek ε ω ω=ω(σ) Ek k k k k k,σ

V

k

Allowing for the latter relation and Eqs. (29)–(31), we find     2↔  ε ∂(ω V 1 )  (σ)∗ (σ) ω ˆb(σ)†ˆb(σ) + ˆb(σ)ˆb(σ)† . ˆ HF = Ek H ω=ω(σ) Ek k k k k 4π 2ω ∂ω k

(32)

k,σ

↔

1 ∂(ω 2 ε ω ) (σ)∗ Ek > 0 (the positive-energy mode) is fulfilled, Eq. (32) yields the nor2ω ∂ω malization condition which transforms Hamiltonian (29) to standard form (28): (σ)

If the condition Ek

(σ)∗ 1 Ek 2ω

↔ (σ) 2π [ωk ]2 ∂(ω 2 ε ω )  (σ) . ω=ω(σ) Ek = ∂ω V k

(33)

Thus, we have demonstrated that Hamiltonian (28) corresponds to the energy of collective excitation of the field and the medium, which includes the energy of the electromagnetic field proper, the energy of internal degrees of freedom of atoms, and the energy of near dipole interaction in the polarized medium.  ↔ ↔ (σ) In the limiting case of vacuum ( ε = 1 ), normalization (33) has the standard form |Ek | = 2πc |k|/V according to [43, 44, 49].

2.4.

Quantum description of the field in a dissipative medium

The above-obtained equations are applicable in the case of a transparent medium when the solutions of dispersion relation (25) correspond to real wave vectors. At the same time, when non-zero relaxation constants γmn are allowed for in Eq. (22), dispersion relation (25) turns to be complex-valued. For a relatively weak dissipation, the complex solution of dispersion relation (25) can be represented in the form ω (σ) (k) = (σ) (σ) (σ) (σ) (σ) (σ) (σ) ωk − iγk for real wave vectors and k(σ) (ω) = kω + iκω , for real frequencies (here, ωk , γk , kω , (σ) and κω are real values). The temporal and spatial damping rates are connected by the standard relation (σ) (σ) (σ) (σ) (σ) γk = κω vgr according to [40], where vgr = ∂ωk /∂k is the group velocity. Let us develop a quantum consideration of narrow-band radiation being the sum of modes whose wave vectors are concentrated around a certain “central” value k in a relatively small interval |Δk| |k|. This sum can be represented as a field with a slowly varying amplitude operator [2, 3, 42, 56] (when considering the field with a certain polarization, we omit the subscript σ): ˜ ∗ cˆ† (r, t) exp(iωk t − ikr), ˆ = Eˆ ˜ c(r, t) exp(−iωk t + ikr) + E E

(34)

c/∂r where the operators cˆ(r, t) and cˆ† (r, t) are “slow” functions of the coordinates and time:7 cˆ˙ ωk cˆ, ∂ˆ ˜ in Eq. (34) in accordance with formula (33), where we set V = 1, kˆ c. We determine the normalized vector E however. In the case of such normalization, the operator n ˆ (r, t) = cˆ† cˆ has the meaning of the operator of 7 Operator inequalities are understood as the corresponding conditions for average values of the quantum-mechanical operators.

828

spatial density of the photon number [37]; certainly, the use of this notion makes sense only for quasioptical configurations with the dimensions being much wider than the radiation wavelength. Specifically, for the ray tube with the aperture area S⊥ , which is oriented along the group velocity vgr , the operators cˆ(r, t) and cˆ† (r, t) satisfy the following commutation relation (see [42, 56]): 

cˆ(r, t), cˆ† (r, t) =

|k| Δω , 2πS⊥ (vgr k)

where Δω is the frequency band under consideration. In what follows, we will present the equations for the operators cˆ(r, t) and cˆ† (r, t), which were obtained ˆ ext , which in [42, 56] and allow not only for dissipation, but also for the “external” resonance polarization δP can be connected with nonlinear effects, external sources, active elements, etc.: i ˆ ˜∗ ˆ cˆ˙ + (vgr ∇) cˆ + γk cˆ = δP 0 E + F, 

i ˆ† ˜ † E) + Fˆ † . cˆ˙ + (vgr ∇) cˆ† + γk cˆ† = − δP  0

(35)

ˆ † are “slow amplitudes” of the operator δP ˆ ext represented as a wave field with a smooth ˆ 0 and δP Here, δP 0 envelope: ˆ ext = δP ˆ 0 (r, t) exp(−iωk t + ikr) + δP ˆ † (r, t) exp(iωk t − ikr), δP (36) 0 Fˆ is the fluctuation operator (Langevin source), which should be expressed, generally speaking, in terms of ˆ mn of Langevin sources for quantum transitions [42, 50] from medium dynamics equathe operators L tions (17). The noise properties are specified by the commutation and correlation properties of the operators Fˆ and Fˆ † . Correct quantum Langevin operators should satisfy the commutation relation [Fˆ (r, t), Fˆ † (r , t )] = 2γk δ(t − t ) δ(r − r ), which ensures that the commutation relations are retained for the operators cˆ† and cˆ at γk = 0 [42, 50]. The methods of determining the correlation properties of Langevin sources in various systems were discussed, e.g., in [42, 49, 52, 53, 57, 58]. In the simplest case of the linear equilibrium system, the correlator of the fluctuation field operator can be determined by means of the fluctuation-dissipation theorem [42, 49, 56]: Fˆ † (r, t)Fˆ (r , t ) = 2γk nT (ωk ) δ(t − t ) δ(r − r ), where nT (ωk ) = [exp(ωk /T ) − 1]−1 is the average number of “thermal” quanta at the temperature T . 3.

ANISOTROPY-INDUCED TRANSPARENCY FOR HYBRID POLARITONS

3.1.

Photons in the medium and longitudinal polaritons

Let the dielectric medium under consideration correspond to a three-axis crystal having the symmetry axes x , y  , and z  . In the system of coordinates, which is formed by the symmetry axes of the crystal, the ↔ ↔ tensors χ aω and ε ω are diagonal. In the “inherent” coordinate system x , y  , z  , we will use the following ↔ ↔ ↔ ↔ notations for these tensors: χ ω for χ aω and ε ω for ε ω . Using relation (21), we obtain: ⎞ ⎛ ε1 (ω) 0 0 4πχβ (ω) ↔ εω = ⎝ 0 0 ⎠, εβ (ω) = 1 + , β = 1, 2, 3, ε2 (ω) 1 − αχβ (ω) 0 0 ε3 (ω) ⎞ ⎛ χ1 (ω) 0 0 ↔ χω = ⎝ 0 (37) χ2 (ω) 0 ⎠. 0 0 χ3 (ω) Consider first the waves propagating along one of the symmetry axes of the system, e.g., the x axis, for definiteness. We introduce a set of wave numbers k corresponding to the spatial dependences exp(±ikx ) 829

which satisfy the periodic boundary conditions. Neglecting the dissipation, one can represent the field in the form of Eqs. (23) and (24), where (1)

(2)

Ek ↑↑ x0 ,

σ = 1, 2, 3,

Ek ↑↑ y0 ,

(3)

Ek ↑↑ z0 ,

and x0 , y0 and z0 are the unit vectors of the axes x , y  , and z  . The superscript σ = 1 corresponds to the longitudinal (electrostatic) polariton mode which satisfies the dispersion relation (1)

ε1 (ωk ) = 0.

(38)

Within the framework of the model of a medium without spatial dispersion, the frequency of this mode does not depend on the wave number [40]. The superscripts σ = 2, 3 correspond to the waves with transverse polarization (“photons in a medium” [37, 38]) which satisfy the dispersion relations (2)

(2)

ε2 (ωk ) = (ck/ωk )2 ,

(3)

(3)

ε3 (ωk ) = (ck/ωk )2 .

(39)

In the foregoing, we used the term “polariton” for the electrostatic mode, which in principle cannot exist in vacuum. We will call the wave modes, for which the dispersion relations allow the passage to the limit in vacuum, “photons in a medium” according to [37, 38]. To avoid a misunderstanding, we note that there is a different terminology (see, e.g., [59]), within which a polariton is any electromagnetic wave in a medium, whose dispersion law differs from that in vacuum and depends on the closeness of the wave frequency to a certain eigenfrequency of the medium.

3.2.

Hybrid polaritons

Let us consider now propagation of waves at an arbitrary angle θ relative to the symmetry axis x in the plane that is orthogonal to the symmetry axis z  . We introduce a new coordinate system x, y, z, connected with the chosen propagation direction: x = x cos θ + y  sin θ, In the system of coordinates ⎛ ε˜1 g ↔ ε ω = ⎝ g ε˜2 0 0

y = y  cos θ − x sin θ,

z = z .

x, y, z, the dielectric-permittivity tensor is transformed to the following form: ⎞ 0 0⎠, ε˜1 = ε1 cos2 θ + ε2 sin2 θ, ε˜2 = ε2 cos2 θ + ε1 sin2 θ, ε3 ε2 − ε1 sin 2θ. (40) g= 2

A change in the propagation direction does not affect the z-polarized transverse wave: according to the standard classification, it is an “ordinary” wave [40]. The fields of the polariton and photon, which are polarized in the xy plane, turn to be coupled8 and form the so-called “extraordinary” wave. Having in mind the vicinity of the frequency of the longitudinal polariton, we will call this wave “hybrid” by analogy with the hybrid quasielectrostatic modes of magnetized plasma [60]. Let us represent the field of the hybrid mode in the standard form   ˆh = Ek ˆbk exp(ikx) + ˆb†k exp(−ikx) . (41) E k

The frequencies which determine the Heisenberg equations for the field operators ˆbk and ˆb†k are specified by 8

830

If the medium is not isotropic in the considered plane, i.e., ε2 = ε1 .

the relation following from dispersion relation (25):

 g(ωk ) ε˜1 (ωk ) det = ε˜1 (˜ ε2 − n2 ) − g2 = 0, g(ωk ) ε˜2 (ωk ) − n2

(42)

where n = ck/ωk . The components of the normalized vector Ek are determined by the formula for the ˆ = 0): polarization coefficient yielded by Eq. (26) (the same result is obtained by using the condition ∇ D (x)

Γ=

Ek

(y)

Ek

=−

g . ε˜1

(43)

Exactly as in the magnetized plasma [60], quasielectrostatic waves, which are determined by the relations ε˜1 (ωk ) = ε1 (ωk ) cos2 θ + ε2 (ωk ) sin2 θ = 0, n2 1, exist in the medium under consideration. It is evident that it is difficult to excite these waves using the radiation incident from vacuum. We would like to emphasize the fact that under certain conditions, a wave with quasi-longitudinal polarization has the “vacuum” refractive index n2 = 1, i.e., it can be excited with high efficiency and sufficient ease in the medium by the radiation incident from vacuum. We will use the “resonance” model. Within the framework of this model, each element of the polarizability tensor χ1 and χ2 is determined by only one (“its own”) transition with the orientation of the matrix element of the dipole moment along the x and y  axes, respectively. Let the factors corresponding to these transitions are the eigenfrequencies Ω1 and Ω2 , modules of non-diagonal dipole moment matrix elements d1 and d2 , lower-level populations N1 and N2 , as well as the relaxation constants γ1 and γ2 . Introducing cooperative frequencies9 ωβ = 4πd2β Nβ /, we obtain from Eq. (22) χβ =

ωβ , 4π (Ωβ − iγβ − ω)

where β = 1, 2. Then, using Eq. (37), we obtain the following formulas for the components of the dielectricpermittivity tensor: ωβ , (44) εβ = 1 + (⊥) Ωβ − iγβ − ω (⊥)

where Ωβ = Ωβ − αωβ /(4π) is the resonance frequencies shifted relative to the transition frequencies due to the collective effect of the acting field, and β = 1, 2. In this case, the frequencies of the longitudinal () polaritons propagating along the symmetry axes x and y  are equal to Ωβ = Ωβ + [1 − α/(4π)] ωβ , β = 1 and 2, respectively. Consider wave propagation in the case of small angles θ 1 in the range of frequencies of the () longitudinal polariton: |Ω1 − ω| ω1 . In this case, we assume that the cooperative frequencies exceed the relaxation constants considerably: γ1 ω1 , γ2 ω2 . Then, the formulas for the refractive index and the polarization coefficient can be transformed to the following form: 

()  ω + iγ1 − Ω1 − A2 ω1 /ω2 , n ≈ 1 + ω2 (⊥) ()  ()  (Ω2 − iγ2 − Ω1 )2 ω + iγ1 − Ω1 Aω1 , Γ ≈ −  (⊥) ()   ()  Ω2 − iγ2 − Ω1 ω + iγ1 − Ω1 2

(⊥)

Ω2

()  

− iγ2 − Ω1

(45) (46)

9 For clarity, we note that a slightly different notion of the cooperative frequency is also used in literature: ωc2 = 8πΩd2 N/, where Ω is the frequency of the resonance transition.

831

where

()

A= Given that

(⊥)

{γ1 |Ω2

()

Ω2 − Ω1 sin 2θ. 2 ()

− Ω1 |, γ1 γ2 }

near the frequency ()

ωr ≈ Ω1 + we obtain n2 ≈ 1 +

ω1 2 A ω2

(47) (48)

ω1 A2 , ω2 Ω(⊥) − Ω() 2 1

ω22 ω − ωr + iγ1 , ω1 A2

(49)

ω2 . A

(50)

vgr k c c = ≈ 2 2 k n + [ω/(2n)] (∂n /∂ω) 1 + ωr ω2 /(2ω1 A2 )

(51)

Γ≈− The “effective” group velocity veff =

and the spatial absorption coefficient Im k = κ ≈

γ1 ωr ω22 γ1 = 2cω1 A2 veff

(52)

correspond to dependence (49). The damping rate −Im ω = veff Im k coincides with the relaxation constant γ1 for the corresponding transition. In the case of a sufficiently strong dissipation for the chosen transition, where the opposite inequality is fulfilled instead of (48), deviation of the wave propagation direction away from the symmetry axis of the medium turns to be an insignificant effect against the background of dissipation.

3.3.

Effect of anisotropy-induced transparency (AnIT)

Figure 1 shows the dependences of the squared refractive index on the radiation frequencies, which are determined by Eqs. (40), (42), and (44). One can see that the excitation of a hybrid mode can ensure a transparency window in the frequency range where the “standard” photons do not propagate through the medium. In this respect, we have an evident analogy with the well-known effect of electromagnetically induced transparency (EIT) in a three-level system located in a control field [1]. However, the analogy is not restricted to this fact only: Eqs. (49) and (51) for the refractive index and the group velocity of the hybrid polariton correspond exactly to similar relations obtained for the “transparency window” of a three-level system in the regime of electromagnetically induced transparency. In this case, the parameter A determined by the system anisotropy acts as the Rabi frequency for the pumping field which controls the three-level system. Exactly as in the standard variant of the electromagnetically induced transparency, at a sufficiently small value of A, the group velocity of radiation in the window of transparency induced by the medium anisotropy is much lower than the speed of light. In both cases, the group velocity is equal to the ratio of the Poynting vector and the field energy density. However, the physical reasons of group slowdown in the systems under consideration are completely different. In the case of electromagnetically induced transparency, the effective field energy in the system increases at the cost of reversible parametric energy exchange with the pumping wave [61, 62] and variation in the population difference for the low-frequency transition. In the case of anisotropy-induced transparency, the energy of the hybrid mode remains close to the energy of the longitudinal polariton, and the group velocity turns to be low due to the smallness of the Poynting vector 832

Fig. 1. Real and imaginary parts (solid and dashed lines, respectively) of the squared refractive index as () functions of the frequency normalized with respect to Ω1 for the “hybrid” wave propagating at the angle (⊥) θ = 0.5 at Ω2 = 1.1, ω1 = ω2 = 0.2, and γ1 = γ2 = 0.01 (a). For the sake of comparison, the corresponding dependences are presented for the wave propagating along the symmetry axis, i.e., at the angle θ = 0 (b). related to the specific feature of polarization of this mode when it propagates at a low angle to the symmetry axis10 (see Eq. (50) for the polarization coefficient). 4.

CONCLUSIONS

The possibility of fulfilling the condition n2 ≈ 1 for the hybrid mode allows one to excite this mode in a medium by using photons incident from vacuum with minimum reflections (see Fig. 2). It is evident 10

To make the analysis less complex, we do not discuss here the effect of general strong deviation of the group velocity direction away from the direction of the wave vector, which is significant, in principle. However, one can easily ascertain that at θ  1, regardless of the mutual orientation of the vectors vgr and k, we always have vgr  c for the quasi-longitudinal polariton.

833

Fig. 2. Excitation of the hybrid wave in the case of radiation incidence at an angle to the crystal axis. x , y  , and z  are optical axes, and x is the direction of radiation incidence on the medium. The pulse compression due to group deceleration is shown schematically. from Eqs. (35) that the mode of this type interacts with active or nonlinear elements more efficiently than the wave propagating with group velocity close to the speed of light. For example, the same gain would be achieved along a much shorter (by c/vgr times) path. This work was supported by the Russian Foundation for Basic Research (project Nos. 13–02–97039 and 13–02–00376). REFERENCES

1. S. E. Harris, Phys. Today, 50, 36 (1997). 2. M. D. Lukin, Rev. Modern Phys., 75, 457 (2003). 3. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev. Modern Phys., 77, 633 (2005). 4. B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, et al., Nature Materials, 9, 707 (2010). 5. A. G. Litvak and M. D. Tokman, Radiophys. Quantum Electron., 44, Nos. 5–6, 375 (2001). 6. A. G. Litvak and M. D. Tokman, Phys. Rev. Lett., 88, 095003 (2002). 7. Y. V. Radeonychev, M. D. Tokman, A. G. Litvak, and O. Kocharovskaya, Phys. Rev. Lett., 96, No. 10, Art. no. 093602 (2006). 8. A. Yu. Kryachko, M. D. Tokman, and E. Westerhof, Phys. Plasmas, 13, 07210 (2006). 9. M. D. Tokman, M. A. Erukhimova, and D. O. D’yachenko, Phys. Rev. A, 78, 053808 (2008). 10. M. Erukhimova and M. Tokman, Phys. Rev. A, 83, 063814 (2011). 11. G. Shvets and J. S. Wurtele, Phys. Rev. Lett., 89, 115003 (2002). 12. M. Tushentsov, G. Shvets, A. Yu. Kryachko, and M. D. Tokman, IEEE Trans. Plasma Sci., 33, No. 1, 23 (2005). 13. M. Tushentsov, G. Shvets, A. Yu. Kryachko, and M. D. Tokman, Phys. Plasmas, 12, 056701 (2005). 14. A. C. Johnson, C. M. Marcus, M. P. Hanson, et al., Phys. Rev. Lett., 93, 106803 (2004). 15. K. Kobayashi, H. Aikawa, A. Sano, et al., Phys. Rev. B, 70, 035319 (2004). 16. A. E. Miroshnichenko, S. Flach, A. V. Gorbach, et al., Opt. Photonics News, 19, 48 (2008). 17. F. Hao, Y. Sonnefrand, P. V. Dorpe, et al., Nano Lett., 8, 3983 (2008). 18. N. Verellen, Y. Sonnefrand, H. Sobhani, et al., Nano Lett., 9, 1663 (2009). 19. G. Shvets and Y. Urzhumov, Phys. Rev. Lett., 93, 243902 (2004). 834

20. Y. S. Kivshar, Opt. Express, 16, 22126 (2008). 21. A. Christ, O. J. F. Martin, Y. Ekinci, et al., Nano Lett., 8, 2171 (2008). 22. U. Naether, D. E. Rivas, M. A. Larenas, et al., Opt. Lett., 34, 2721 (2009). 23. N. Liu, T. Weiss, M. Mesch, et al., Nano Lett., 10, 1103 (2010). 24. M. V. Rybin, A. B. Khanikaev, M. Inoue, et al., Phys. Rev. Lett., 103, 023901 (2009). 25. C. Kurter, P. Tassin, L. Zhang, et al., Phys. Rev. Lett., 107, 043901 (2011). 26. N. Papasimakis and N. Zheludev, Opt. Photonics News, 20, 22 (2009). 27. S. Zhang, Phys. Rev. Lett., 101, 047401 (2008). 28. V. A. Fedotov, Phys. Rev. Lett., 99, 147401 (2007). 29. A. Ishikawa, R. F. Oulton, T. Zentgraf, and X. Zhang, Phys. Rev. B, 85, 155108 (2012). 30. C. Kurter, P. Tassin, L. Zhang, et al., Phys. Rev. Lett., 107, 043901 (2011). 31. Y. Tamayama, T. Nakanishi, and M. Kitano, Phys. Rev. B, 85, 073102 (2012). 32. Y. Sun, H. Jiang, Y. Yang, et al., Phys. Rev. B, 83, 195140 (2011). 33. Y. Lu, H. Xu, J. Y. Rhee, et al., Phys. Rev. B, 82, 195112 (2010). 34. D. J. Bergman, and M. I. Stockman, Phys. Rev. Lett., 90, 027402 (2003). 35. N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, et al., Nature Photonics, 2, 351 (2008). 36. J. Seidel, S. Grafstr¨ om, and L. Eng, Phys. Rev. Lett., 94, 177401 (2005). 37. V. M. Fain and Ya. I. Khanin, Quantum Radiophysics. Vol. 1. Photons and Nonlinear Media [in Russian], Sov. Radio, Moscow (1972). 38. V. L. Ginzburg, Physics and Astrophysics: A Selection of Key Problems, Pergamon, Oxford (1985). 39. L. D. Landau and E. M. Lifshits, Electrodynamics of Continuous Media, Elsevier Butterworth– Heinemann, Oxford (1989). 40. V. M. Agranovich and V. L. Ginzburg, Spatial Dispersion in Crystal Optics and the Theory of Excitons, J. Wiley, New York (1966). 41. A. V. Gaponov-Grekhov and M. D. Tokman, J. Exp. Theor. Phys., 85, 640 (1997). 42. M. Tokman, X. Yao, and A. Belyanin, Phys. Rev. Lett., 110, 077404 (2013). 43. H. Haken, Quantum Field Theory of Solids, Elsevier Science Ltd. (1983). 44. V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics, Butterworth— Heinemann, Oxford (2012). 45. A. V. Andreev, V. I. Emelyanov, and Yu. A. Il’inskii, Cooperative Phenomena in Optics [in Russian], Nauka, Moscow (1988). 46. Yu. A. Il’inskii and L. V. Keldysh, Electromagnetic Response of Material Media, Plenum Press, New York (1994). 47. M. E. Crenshaw, C. M. Bowden, Phys. Rev. A, 63, 013801 (2000). 48. A. I. Akhiezer, V. G. Baryakhtar, and S. V. Peletminskii, Spin Waves, North-Holland, Amsterdam (1968). 49. M. O. Scully and M. S. Zubairy, Quantum Optics, Cambrigde Univ. Press, Cambridge (1997). 50. M. Tokman and M. Erukhimova, J. Luminescence, 137, 148 (2013). 51. J. Jackson, Classical Electrodynamics, Wiley, New York (1998). 835

52. M. Lax, Fluctuations and Coherence Phenomena in Classical and Quantum Physics, Gordon and Breach, New York (1968). 53. J. Perina, Quantum Statistics of Linear and Nonlinear Optical Phenomena, Reidel (1984). 54. L. Davidovich, Rev. Modern Phys., 68, 127 (1996). 55. L. D. Landau and E. M. Lifshitz, Statistical Physics, Elsevier Butterworth—Heinemann, Oxford (2005). 56. V. Vdovin and M. Tokman, Phys. Rev. A, 87, 012323 (2013). 57. M. Erukhimova and M. Tokman, Proc. V Intern. Conf. “Frontiers of Nonlinear Phys.,” Nizhny Novgorod—Yelabuga—Nizhny Novgorod, Russia, July 28–August 2, 2013, p. 208. 58. M. Tokman, Proc. V Intern. Conf. “Frontiers of Nonlinear Phys.,” Nizhny Novgorod—Yelabuga— Nizhny Novgorod, Russia, July 28–August 2, 2013, p. 247. 59. A. P. Sukhorukov, Nonlinear Wave Interactions in Optics and Radiophysics [in Russian], Nauka, Moscow (1988). 60. A. I. Akhiezer, Ed., Plasma Electrodynamics [in Russian], Nauka, Moscow (1974). 61. R. N. Shakhmuratov and J. Odeurs, Laser Phys., 17, 652 (2007). 62. R. N. Shakhmuratov, J. Modern Opt., 57, 1355 (2010).

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