The quantum radiation through the multilayer structures containing the left-handed materials is investigated based on the Green-function approach to the quantization of the phenomenological Maxwell theory. Emphasis is placed on the effect of randomness on the generation and transmission of entangled-states. It is shown that some unusual properties appear for the present systems in comparison with those of the conventional dielectric structures. The quantum relative entropy is always enhanced with the increase of random degree due to the existence of nonlocalized mode in the present systems, while the maximal entanglement can be observed only at some certain randomness for the conventional dielectric structures. In contrast to exponential decrease in the conventional systems, the entanglement degrades slowly with the increase of disorder and thickness of the sample near the nonlocalized mode after transmission through the present systems. This will benefit the quantum communication for long distances.
©2008 Optical Society of America
In the last decade a significant effort has been devoted to the study on the quantum information and communication [1, 2]. One of the problems of quantum communication is to generate nearly perfect entangled-states between distant sites. Such states can be used, for example, to implement secure quantum cryptography or faithfully transfer quantum states via quantum teleportation [3–7]. Many realistic schemes for quantum communication are at present based on the use of photonic channels. Dielectric four-port devices as basic elements for constructing optical quantum channels have been widely discussed and the multilayer dielectric structures may serve as a model for such devices [8–14]. The generation and transformation of entanglement through the multilayer structure have been investigated in recent years . It is shown that the degree of entanglement decreases exponentially with the increase of the channel length, because of optical absorption and other channel noise. This limits largely the efficiency of quantum communication for long distances. Thus, how to realize robust quantum communication for long distances becomes another important problem in the field of quantum information.
Recently, left-handed materials (LHMs) have attracted a great deal of attention from both theoretical and experimental sides [15–20]. These materials, which are characterized by simultaneous negative permittivity and permeability, possess a number of unusual electromagnetic effects [15–20]. Multilayer structures that include the LHMs have also been analyzed [21–28]. It is found that one-dimensional stack of layers with alternating positive and negative refraction materials displays some unusual transmission properties of classical wave [21–28]. It is natural to ask whether these unusual transmission properties can play positive roles in the generation of entangled-states and realization of quantum communication for long distances? That is to say, what kind of phenomenon will occur when the entangled-states transport through the multilayer structures containing the LHMs?
Motivated by these problems, in this paper we will investigate quantum radiation through the multilayer structures containing the LHMs. Our studies are based on the quantization of the phenomenological Maxwell theory. We first extend the Green function approach, which has been developed in Ref.  for the multilayer dielectric plates, to the multilayer structures containing the LHMs. The generation and transformation of the entanglement in these structures will be studied in detail. Because nonuniformities inevitably occur in the fabrication of the microstructures, especially when the structures are of micrometer and submicrometer sizes, the effect of randomness on the efficiency of entanglement generation and transformation is emphasized. Some unusual properties are found in these systems in comparison with the conventional dielectric multilayer structures.
2. Theory of quantum state transformation
We consider a multilayer structure composed of different materials with various frequency-dependent permittivity (ε(ω)) and permeability (µ(ω)), as shown in Fig. 1. The LHMs are such a kind of materials, which ε(ω) and µ(ω) are both negative in a certain frequency region and have the following forms :
where ωp is plasma frequency, γ and Γ are the respective electric and magnetic loss terms. ω 0 is the magnetic resonance frequency. Thus, the multilayer structure containing the
LHMs is a special case of the present structure. In order to study quantum state and entanglement transformation through the structures, we first need to obtain input-output relations for quantum radiation at these systems.
2.1 Basic equations for operator input-output relations
The input-output relations of quantized radiation through a pure dielectric multilayer structure with absorption have been given in Ref. . Here we generalize them to the present systems. Let us first focus on linearly polarized radiation propagating in homogeneous media with frequency-dependent permittivity and permeability along the x direction. Introducing the vector potential Â(x,ω), the Maxwell equation in the frequency domain can be written as
with κ(x,ω)=µ-1(x,ω). According to the Green function scheme of the electromagnetic field quantization in the causal permeable dielectric background media , the “current” operator ĴN(x,ω) is expressed in the following form:
where G(x,x′,ω) is the classical Green function obeying the following equation
In the homogeneous materials where the permeability and permittivity are independent on the space, the solution of the Eq. (6) is given by
with being the complex refractive index of the medium.
where the amplitude operators
Here + and - represent the waves propagating to the right and the left, respectively. The operators â±(x,ω) obey the following quantum Langevin equations in the space domain,
Now we turn to the problem of propagation of quantized radiation through multilayer structure as shown in Fig. 1. The vector potential Âj(x) for the jth layer (xj-1≤x≤xj) can be obtained from Eq. (8) by replacing ζ(ω), µ(ω), n(ω) and β(ω) with the corresponding parameters ζj(ω), µj(ω), nj(ω) and βj(ω) in the jth layer
so that âj±(x,ω) can be represented in the form of Eq. (9) as
The relations between âj+1±(xj,ω) and âj± (xj-1,ω) are determined by Eq. (14) and the boundary continuity conditions of the vector potential Â(x) at the interface x=xj. Successive application of Eq. (14) and the continuity conditions, the input-output relations for the amplitude operators through the multilayer structure shown in Fig. 1 can be obtained as 
The Eq. (16) can be written in a compact form as
where T(ω) and A(ω) are transformation and absorption matrices, which can be obtained exactly, and corresponds to amplitude operators of the output fields, to amplitude operators of the input fields from the two sides and represents the noise operators owing to absorption. The definitions of the noise operators ĝ(N-2)±(ω) are different from those in the Ref.  because of the introduction of the noise magnetic losses. For example, for a single-slab structure with N=3, ĝ(1)±(ω) is expressed as
2.2 Quantum-state transformation and entanglement measure
After the operator input-output relations are obtained, the transformations of quantum-state and entanglement can be obtained directly according to Ref. [12, 13]. The process is summarized in the following. In order to calculate the density operator of the outgoing field for the absorbing devices, we combine the two-dimensional vector operators â(ω) and b̂(ω) to define four-dimensional input vectors and , where ĥ(ω) is auxiliary two-vector operator. Then the input-output relations (Eq. (17)) can be extended to the four-dimensional transformation [12, 13]
where the unitary operator Û(ω), which corresponds to the 4×4 unitary matrix Λ(ω), can be given by
Let the density operator of the input quantum state be a function of (ω) and †(ω), in=in[ (ω),†(ω)]. The density operator of the quantum state for the outgoing fields can be given by 
where Tr(D) means trace with respect to the structure variables. Equation (24) can be used when knowledge of the transformed quantum state as a whole is required, for example, to calculate the total correlation and entanglement transformation. The total correlation Ic is defined by 
where Si is the von Neumann entropy of the ith single-channel output state (F)out,
and S 12 is the entropy of the density matrix of the two-channel output state,
with σ̂ and S being, respectively, the bipartite quantum state under study and the set of all separable quantum states. For the system where the absorption is sufficiently weak, the output state is almost pure, the von Neumann entropy of the composite two-channel output state S 12=0 and E((F)out)≈Si. This means that the total correlation is two times of the quantum entanglement (Ic=S1+S2=2E((F)out))
3. Generation of photon entangled-state by multilayer system
Now let us first discuss the generation of photon entangled-states at the above multilayer systems. We consider the radiation prepared in a single-photon Fock state
travels through the multilayer structure (Structure I) which consists of the alternate layer of air (ε=1.0,µ=1.0) and the LHMs with ε(ω) and µ(ω) given by Eqs. (1) and (2), respectively. Here, we take ω 0=0.4ω p for the magnetic resonance frequency . For the damping parameters, the values γ=0.003ωp and Γ=0.003ω 0 are assumed and F=0.8. The material is negative-n between 0.9 and 2.4 (ω/ω 0). In order to make a comparison, we also consider the multilayer structure consisting of the air and the pure dielectric materials (Structure II). That is to say, the negative-n layers in the Structure I are replaced by the dielectric layers with ε=3.4.  The transmission coefficients for two kinds of structure with N=22 are plotted as solid lines in Fig. 2(a) and Fig. 3(a), respectively. The band-gap features are shown clearly for both of them.
where the complex transmission coefficients T 11 and T 21 are the matrix elements of T(ω).
Solid lines in Fig. 2(b) and Fig. 3(b) represent the calculated results of Ic for two kinds of structure, respectively. They exhibit similar features. The output states are not correlated at gap regions because the input photons are reflected completely and there is no mixing of their states in the structure. At the same time, the correlation is weak in the band region because the input photons are transmitted and the mixing of states is weak in the structure. The maximal entanglement only appears in the vicinity of the band edge. However, the situation becomes different with the introduction of disorder.
In general, there are two typical kinds of randomness in the multilayer structures, the thickness variations of the layers (thickness randomness) and varied dielectric constants of the layers (dielectric randomness). Here, we consider the case of thickness randomness. Such a disordered system can be produced by varying the width of layer randomly. That is to say, the width li of the ith layer (negative-n or dielectric) is taken to be random variable in the interval [l0(1-δ),l0(1+δ)], where δ is a random number between (-1.0, 1.0) and the reduced width ω0l0/c=0.5 (c is the speed of the light in the vacuum) is taken. Therefore, the value of δ represents the degree of the random. The transmission coefficients and the total correlation for two kinds of random structure with different δ are plotted in Figs. 2 and 3, respectively. All results are obtained by averaging 1000 different configurations for each frequency. Dashed lines correspond to the case with δ=0.2 and dotted line to that with δ=1.0 (complete random). Because we aim at the transformation of quantum state, in the following we focus our discussion on the band region. It can be seen from the figures that the transmittances decrease with the increase of random degree for two kinds of structure in the band regions. In contrast, the total correlations increase. It is interesting that the change feature of total correlation as a function of disorder degree is different for two kinds of structure. For the Structure I, the total correlations always increase with the increase of δ, while it can appear maximal value for the Structure II at some random degree. For example, the total correlations are bigger at δ=0.2 than those at δ=1.0. These features can be seen more clearly from Fig. 4.
Figures 4(a) and 4(b) display the entanglement measure E(ρ(F)out) as a function of the random strength at different frequencies for Structure I and Structure II, respectively. The results are obtained by the average of 200 random configurations. It is shown clearly that the maximal values at some random degrees are always observed at different frequencies on the band region for the Structure II. In contrast, E(ρ(F)out) always increases monotonously with the increase of δ for the Structure I. That is to say, the randomness always enhances the quantum relative entropy for the structure I at different frequencies on the band region. These results are only for the case of thickness randomness. In fact, similar phenomena can be found for the case of dielectric randomness. The reason to appear such a phenomenon is due to the unusual transmission feature of photon in the random multilayer structure containing the LHMs . The previous research  has shown that the nonlocalized modes exist for the wave transmission through the multilayer structures containing the LHMs. Such a nonlocalized mode is originated from the zero reflection of wave on the interface of the dielectric and negative-n layers, and it is insensitive to the disorder. The high-transmission peak at ω/ω 0=2.58 in Fig. 2(a) corresponds to such a case. The total correlation and quantum relative entropy at such a frequency do not change with the change of the disorder. However, they are always enhanced for other frequencies in the band region due to the existence of such a point. Such a nonlocalized state can not only play an important role in the generation of the entanglement, it can also have important influence on the transmission of entangled-state.
4. Transformation of entangled-state by random multilayer structures
In this part, we discuss the question of entanglement degradation during the propagation through the above two kinds of random multilayer structure. Two modes, each of which propagates through the structures, are considered. Assuming the incoming modes are prepared in a maximally entangled Bell-type states
where the complex transmission coefficients T(1) 21 and T(2) 21 are the elements of matrices T (1)(ω) and T(2)(ω), respectively. Here T(1)(ω) and T(2)(ω) represent the transformation matrices of the random multilayer structures for two modes. Then the quantum relative entropy of the output state can be calculated in a straightforward way.
The calculated results for the quantum relative entropy of the output state by two kinds of structure with N=22 as a function of the random degree are plotted in Fig. 5. Here the quantum relative entropy at ω/ω 0=2.58 (corresponding to the nonlocalized mode) for the Structure I is only shown. It is seen clearly that E(ρ(F)out) is insensitive to the randomness and it keeps nearly constant for the entangled Bell-type state transmitting through the structure. In fact, the entanglement transmissions at other frequencies near the nonlocalized mode posses the similar property. However, such a property can not be found in the Structure II. The quantum relative entropies for all frequencies in the Structure II always decrease exponentially with the increase of the random degree as shown in Fig. 5 for the case at ω/ω 0=3.0.
The entanglement transmission not only exhibits different feature as a function of δ for two kinds of structure, it also appears some difference for the case as a function of the sample thickness. Figure 6 shows the comparison of entanglement degradation for two kinds of structure as a function of the sample thickness (total layer number N) at complete random (δ=1.0). As can be seen from the figure, the entanglement of the output state almost disappears for the structure II when the number of the layer reaches 40 due to the localization of the photon. In contrast, the quantum relative entropy near the nonlocalized mode in the Structure I decrease slowly with the increasing of the layers even in the presence of the absorption. The above results are for a kind of incoming Bell-type state. For other Bell-type states such as (|00〉±|nn〉)/√2, similar phenomena can be observed. This means that a good result can be obtained by using the Structure I instead of the Structure II for robust quantum communication of long distances.
Based on the Green-function approach to the quantization of the phenomenological Maxwell theory, we have investigated the quantum radiation through the random multilayer structures containing the LHMs. The total correlation of the quantum state and the quantum relative entropy of the entanglement have been discussed in detail. We have found that some unusual properties in the generation and transmission of entangled-state can appear for the Structure I in comparison with those of the Structure II. Although the randomness can enhance the entanglement of the output state for both structures, the change feature for them is different. The maximal entanglements at some random degrees can be always observed at different frequencies on the band region for the Structure II, while the quantum relative entropy always increases monotonously with the increase of the random degree due to the existence of the nonlocalized mode in the Structure I. Such a nonlocalized mode not only plays an important role in the generation of the entanglement, it can also influence largely the transmission of the entangled-state. The quantum relative entropy decreases exponentially with the increase of the sample thickness after the entangled-state transmits through the Structure II. In contrast, the entanglement degrades slowly with the increase of disorder and thickness of the sample near the nonlocalized mode after transmission through the Structure I, which benefits the quantum communication for long distances.
We wish to thank Dr. S. Scheel for help in the numerical calculations. This work was supported by the National Natural Science Foundation of China (Grant No. 10674017) and the National Key Basic Research Special Foundation of China under Grant 2007CB613205.
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