1. Introduction

A beam-splitter (BS) is one of the most widely used optical components that is understood totally by classical treatment. However, as the use of non-classical sources of light in experiments increases [1, 2, 3], it becomes essential to understand the behavior of all the components used, in a purely quantum mechanical sense, in order to interpret the results of these experiments as well as the limitations of these results. Several authors have considered the behavior of the lossless quantum mechanical beam-splitter [4, 5, 6]. Unlike the classical case where the energy in one beam is merely split into two parts, a quantum-mechanical analysis shows that the BS modifies the basic statistical properties of the beams [6, 7]. Thus a simple BS can be used to probe the quantum nature of light by simple yet subtle experiments [8, 9]. Recently the use of a BS for generating Schrödinger Cat-like states has been proposed [10]. A BS offers one of the simplest interaction of a light mode with an external environment. It serves as a model for the interaction of a radiation mode with an external environment attenuating the field and the entering of external fluctuations via the second port of the beam-splitter is in accordance with the fluctuation-dissipation theory. Thus the effects of an external environment on the light modes can be environment by observing the effects on the statistical properties of the modes caused by the BS.

But any real BS is also expected to be lossy. It is well known that, when the dielectric matter, with small imaginary part of the permitivity, is considered in free space, the input-output relations correspond to unitary transformation between the operators of input and output modes. However, these concepts fail when the effect of absorption on radiation passing through the dielectric medium is taken into account. In addition, any losses would also render it dispersive in accordance with the Kramers-Kronig relations. The losses in addition to the attenuation of the input fields, would be expected to affect the photon statistics as the loss would couple the fields to a reservoir whose oscillators act as noise sources. It is for this reason that quantization in dispersive, lossy media is considerably complicated.

Extensive work has been carried out to model the losses in linear media. A relatively simple representation of the loss mechanism is provided by a continuous distribution of fictitious optical beam-splitters in an otherwise homogeneous and lossless medium[11, 12]. There have been attempts to microscopically model the losses and dispersion in homogeneous media in terms of the polariton and reservoir operators [13, 14, 15]. The quantization of the radiation field has been approached by using Langevin forces to represent the noise, based on a development of the familiar formalism of quantum noise theory [16]. This approach has been successfully applied to the description of dielectric slabs [17]. A comprehensive treatment of field quantization in dispersive media is given in Ref[18], which gives consistent expressions for the macroscopic quantized fields in infinite dielectric media and dielectric slabs. All the above works take either a macroscopic or microscopic approach to the quantization of the field operators. The microscopic models [13, 14, 15] have the advantage that they explicitly obtain the diagonalization of the coupled system of electromagnetic filed, dielectric oscillator and the reservoir. The dielectric function is expressed in terms of the parameters of the models. However the Langevin force calculation to which most of the macroscopic schemes resort to are much simpler but do not yield information in terms of the material properties of the system. Recently quantum input-output relations were derived for dispersive, multilayer plates using the Green’s function approach to quantization of the phenomenological Maxwell’s equations to show that the absorption introduces an additional noise to the vacuum noise [19].

In this paper, a microscopic model of a dissipative BS is presented. In order to apply it to two mode interactions, we have simplified the picture by neglecting the dielectric polariton excitations and have instead directly coupled the electro magnetic field directly to the thermal excitations of the medium. In the model the losses are caused by the resonant absorption of light due to optical phonons in the medium. It is true in the mid infra-red to the microwave region of the spectrum (5μm onwards) in host of wide band dielectric media where the atomic and band absorption is negligible. Hence the dielectric function of the medium in which the major contribution stems from the electron scattering can be assumed to be almost dispersion-less. This assumption is justified to the extent that the absorption is reasonably small i.e., Im_χ ≪Re_χ. Our main concern here is to investigate the effect of losses on the output modes of the BS. The model treats the losses as a perturbation on the BS transformation and the effects on the output modes is calculated up to the second order of a perturbative expansion. The BS is considered as a reservoir of phonons at some finite temperature and it is assumed that the photon-phonon interaction does not disturb the thermal equilibrium of the phonon system. Thus the model provides a simplified picture to analyze a lossy beam-splitter. Some of the effects on the output modes is expected such as attenuation and increase in the noise. However, the analysis shows that losses also depend upon quantum phase correlations in the input fields, which means that in spite of having a lossy medium, lossless cases arise under certain conditions. To the best of our knowledge, such a loss quenching due to phase correlations is being reported for the first time.

 

Figure 1. A beam-splitter with input and output light beams

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The paper is organized as follows. In section II the lossless quantum mechanical beam-splitter is reviewed following the approach of Prasad et al. [4]. In section III, a model for the photon-phonon interaction and the consequent losses is developed. An expression for the transformation of the reduced density matrix for the radiation field is derived using the radiation phonon interaction as a perturbation. In Section IV, the formalism developed is applied to a lossy medium and then to a lossy BS. It is found that additional fluctuations of a greater order enter the modes due to the losses in the BS. We discuss our results and analyze the limitations of our approach in the last section.

2. The lossless quantum mechanical beam-splitter

The general approach consists of breaking up the annihilation operators of the two interacting modes into two parts to correspond to the splitting of the beam in a manner that conserves the commutation relations. This causes the BS to couple the light modes. In Fig. 1, we show a lossless BS with the two light waves E⃗ ₁ and E⃗ ₂ falling on it from the two sides of the BS. Now both E⃗ ₁ and E⃗ ₂ give rise to the output waves E⃗′ ₁ and E⃗′ ₂. Hence, for the positive components of the electromagnetic fields,

where α₁₁ and α₂₁ are transmission and reflection coefficients for mode 1, α₁₂ and α₂₂ are reflection and transmission coefficients for mode 2. The coefficients could be dependent on direction (k⃗) or polarization (ê _λ) of the light mode. In the second quantized notation for the electromagnetic field, the E ⁽⁺⁾ in the equations go over directly to the annihilation operators for the light modes. More conveniently in matrix form we can write,

Similarly, one can obtain two more sets of equations for the creation operators. Conservation of energy and preservation of the commutation relations between α′ ₁ and α′ ₂ demands the following relations between the matrix coefficients.

implying the transformation is unitary. Hence we look for a unitary operator 𝒖 such that

The operator 𝒖 turns out to be

where ξ ^* = η for 𝒖 to be unitary. This relates the observable quantities α_ij to the physical parameters ξ and η as

We notice that α₁₁ = α₂₂ and ∣α₁₂∣ = ∥α₂₁∥ i.e. the transmittance and reflectance are the same regardless of the wave vector k⃗ with the crystal axis of the BS as these relations were derived entirely from the boundary conditions. It was also de facto assumed that the interaction due to the BS was isotropic and that the polarizations were not rotated either.

The action of the BS can be viewed in two ways. One, the Heisenberg picture in which the field operators a and a ^† get transformed by the BS transformation while the state vectors evolve freely in time. The second is that what Prasad et al. call the interaction picture where the state vectors get transformed by the BS operator while the field operators evolve freely in time. In the following, we shall work in the interaction picture. We represent the photon state to be a product number state ∣n ₁,n ₂〉, where n ₁ and n ₂ are the number of photons in the light modes 1 and 2 respectively

i.e., the final state is a coherent superposition of two single photon states in the two modes. The operation of 𝒖 on the state ∣n ₁,n ₂〉 can be written as [4],

The calculated fluctuations for pure number states ∣n ₁,n ₂〉 and coherent states ∣α,β〉 are given by

One can see that the coherent input modes are transformed into two completely uncorrelated coherent output beams by the action of the BS. This is due to the inherent SU(2) symmetry of the transformation [20].

3. Radiation field–phonon interaction

It is recognized that the BS action occurs mainly due to lossless scattering by oscillating dipoles and bound currents due to orbital (bound) electrons in the atoms. The energy losses of the radiation field in the BS is due to the excitation of optical phonons in the BS i.e., the interaction of the photon field with the phonons in the BS. This causes energy to be irretrievably passed into the phonon energies. The BS is considered to be a reservoir of phonons at some finite temperature.

 

Figure 2. Lattice displacements due to optical phonons : (a) transverse, (b) longitudinal.

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The phonon system is described by the phonon creation and annihilation operators b_k and $b_k^{†}$ defined as [21]

These satisfy the commutation relation [b_k, $b_{k\prime }^{†}$] = δ_kk′ . The unperturbed Hamiltonian for the phonon reservoir can be written as

Here M is the mass, u⃗(R⃗) is the displacement about the mean position, P⃗(R⃗) is the momentum of the ion in the lattice; N is the total number of atoms in the crystal; ω_k and k⃗ are the frequency and the wave vector of the phonon. For simplicity only a single branch of the phonon modes is considered.

Classically the interaction energy of a charged particle with an electromagnetic field in the minimal coupling form is given by

We can write the interaction energy for m charged ions in the lattice as V_I = mV_i . This can be done only for optical phonons and is justified as following. If we observe the motion of the ions in the lattice due to optical phonons, they look as in Fig. 2. If we consider any two adjacent ions, their charges are opposite (q ₁ = -q ₂) and their momenta are approximately equal in magnitude but opposite in direction i.e. p⃗ ₁ = -p⃗ ₂ and hence their energies add up. Here an approximation is made that the positive and negative ions have approximately the same mass. A more rigorous analysis of which optical phonon modes will contribute effectively to the process is given in Ref.[22].

One has to exercise caution in replacing A⃗ by the creation and annihilation operators as the R⃗ is a label for the second quantized electromagnetic field while it is the position operator for the particle in the coordinate space. We replace A⃗(R⃗) by the requisite operators in the Fock space, p⃗ by the operator in the coordinate space and the compound operator acts on the states spanning the product space, thus

Here A⃗ acts on the ∣j〉 states and p_μ acts on the f_i (x^μ ). A⃗ can be expressed in terms of the creation and annihilation operators of the field

where R̂ is the position operator for the particle. A⃗(R̂) means summation over the k⃗ and nothing else. V_int is the volume of interaction of the beam inside the BS medium. We note that the role of e ^ik⃗·R̂ and e ^-ik⃗·R̂ are merely translations in the momentum space and preserve the conservation of momentum i.e., each time a photon of momentum ħk⃗ is annihilated, the crystal momentum is increased by the same amount. But generally the photon momenta are small compared to the phonon momenta. In addition the BS splitter is kept clamped. Hence we neglect the conservation of momenta and retain only the first term in the exponential i.e., unity. This gives the so-called dipole approximation.

The momentum p⃗ of the particle can be expressed in terms of the b_k and $b_k^{†}$ operators

We write the interaction energy of the charged particle with the two modes of the electromagnetic field as

where $\gamma =\frac{\mathit{iqħm}}{2}\sqrt{\frac{{\omega }_l}{\mathit{∊M}{V}_{\mathit{int}}{N}_{\omega }}}$. Now we make the rotating wave approximation (RWA) and neglect the fast varying sum frequency terms and seek the interaction averaged in time over several periods so that over this time interval the phonon reservoir has attained equilibrium. The full Hamiltonian of the radiation field and phonon system can be written as

where H_o = ħω($a_1^{†}$ + $a_2^{†}$ a ₂ + 1) is the free Hamiltonian and R and V_I are given by equations 11 and 16.

If S_d is the density matrix for the coupled system in the interaction picture, where the state vectors evolve in with V_I , then the reduced density matrix given by taking the trace over the reservoir states s = Tr_R [S_d ], and S_o = s_of_o (R), before interaction of the photon and the phonon systems, where f_o (R) is the equilibrium distribution of the phonon states. Following the conventional treatment [23], one can write down the transformation of the reduced density matrix for the light field in terms of the ensemble averages of the phonon reservoir. In doing so, one makes the Markoff approximation by assuming the interaction time scales much larger than the correlation time of the phonon reservoir and much smaller than the cavity mode decay time, i.e. ${\tau }_{\mathit{\text{corr}}}^{\left(R\right)}$ ≪ t_int ${\mathrm{\tau }}_{\mathit{\text{decay}}}^{\left(s\right)}$. In order to take a trace over the reservoir states, we note the following ensemble averages

where n̅_ph is the mean occupancy of the phonon levels which is simply the Bose-Einstein distribution ${\overline{n}}_{\mathit{ph}}\left({\omega }_l\right)={\left[\exp \left(\frac{\mathit{ħ}\omega }{\mathit{kT}}\right)-1\right]}^{-1}$. The transformation for the density matrix can be written as

where, L_ij = ∑_l,k γ_il (ω_l )${\gamma }_{\mathit{\text{jk}}}^{*}$ (ω_k )(ω_lω_k /ω ²)^1/2 〈cosθ_il cosθ_jk 〉 δ(ω - ω_l ) δ(ω - ω_k ) δ_lk with γ_il = γê_λi ∙ ê_l . The averaging of (cosθ_il cosθ_jk ) is over a sphere. As ω_l are closely spaced the summation over l goes over into an integral ∑_l - ${\int }_0^{\infty }$ g(ω_l )dω_l , where g(ω_l ) is the density of modes for the phonons, we obtain

This gives the transformation of the reduced density matrix due to phonon interaction up to second-order.

4. Losses due to phonons – the lossy beam-splitter

4.1 Lossy medium

The analysis of a simple transmission of a beam through the phonon medium using our formalism is instructive. Let us for the time-being assume that there is no resonant scattering of light by electrons i.e., the only interaction the light has with the BS is through the phonons. Then the term 𝒖 drops out. For a mixed state we have to calculate the trace over the photon states as 〈Ô〉=Tr [sÔ]. The ensemble averages for different transformed operators derived from the equation (19) due the lossy BS (medium) in terms of the lossless BS (medium) expectation values are given in appendix-A. However, for a pure number state ∣n ₁, n ₂〉 as the initial state ,the density matrix is reduced to a single element s_o = ∣n ₁,n ₂〉〈n ₁,n ₂∣. The expectation values for the transformed beams for number states come out to be

We note that L ₁₁ and L ₂₂ are the total absorption coefficients for the two modes. S ₁₁ and S ₂₂ represent the spontaneous emission (black-body radiation) by phonons in the two modes. Here we have no cross-terms in intensities of the modes due to phonon interaction alone. That is because such a process would correspond to annihilation of a photon in one mode, the corresponding creation of a phonon, annihilation of a phonon and creation of a photon in the other mode which would be a fourth order process and we have considered processes only up to the second order. However if one observes the transformation of the operator a₁ due to losses, as the phonon field couples all the modes of the radiation field, one finds the contribution of the other modes in it,

For the current case we have only two modes.

The number fluctuations in the transmitted beam-1 is,

Note that the contribution of absorption is in the coefficient of n ² while spontaneous emission appears only in the coefficient of n.

We know that for small losses, Beer’s law of absorption reduces to

where α is the absorption coefficient per unit length. Considering the transmission of any one of the modes through a length l, the number of photons absorbed is

i.e., proportional to the area of the beam and length traversed in the medium. The term in the square bracket in equation 25 is identified as the absorption coefficient. This provides a microscopic description of the absorption coefficient in our model. The dependence of the absorption coefficient on the frequency in our model comes merely through the dependence on the density of the phonon modes. The dispersion in the real part due the interaction has to be imposed by the Kramers-Kronig relations. This is also the way the real part is experimentally determined. It is well known that optical modes are better represented by the Einstein model while the Debye model are more adapted only to acoustical modes [22]. So we take the density of modes for the phonons as $g\left(\omega \right)=3\left(n-1\right)\frac{V_x}{v_a}\delta \left(\omega -{\omega }_E\right)$ where ω_E is the Einstein frequency, n is the number atoms per primitive unit cell and v_a is the volume of the primitive unit cell. This results in a single resonance form for the absorption. This is typical of alkali halides where there is only one infra-red active mode.

4.2 Lossy beam-splitter

When the BS scatter the initial modes to the final modes, then the operators in the interaction picture will evolve according to the unitery BS evolution given by equations 4–6. Taking the initial state to be pure number states, ∣n ₁,n ₂〉, we get

and a similar expression for 〈n ₁,n ₂∣$a_2^{†\prime }$a′₂∣n ₁,n ₂〉. Putting $L_{11}=L_{22}=\frac{L}{2}$ and realizing $L_{12}=\frac{L}{2}\sin \theta$ , we get the total loss as

where we have used ξ = ∣ξ∣eiδ_ξ . The last equation shows that the radiation field cannot gain energy from the medium. The cross-term which is dependent on the incidence angle of the beams, is zero when the intensities are equal and maximum when one of them is zero. This resembles the situation in scattering in a four-wave mixing process. When the intensity of one of the beams is zero we can adjust the parameters of the second term such that the loss is equal to zero. This happens only when $\theta =\frac{\pi }{2}$, i.e., it is a perfect 50-50 beam-splitter at normal incidence. ξ is dependent on the material properties. The cross-term which involves the incidence angles of the beams, is zero when the intensities are equal and maximum when one of them is zero. The no loss situation arises due to some kind of a quantum interference between the input beam and vacuum entering from the other port. To get a further insight, we calculate the losses if the initial states are coherent states ∣α, β〉,

We can see that the two output beams do not go into completely uncorrelated coherent states. The losses break the inherent SU(2) symmetry of the BS transformation. The quantum interference terms in the losses bring about cross-correlations in the two modes. The total loss, taking into account both the beams and neglecting the spontaneous emission, comes out to be

with the motivation to minimize losses we find that when α = β and with the optimal conditions $\theta =\frac{\pi }{2}$, the loss goes to zero unlike the earlier case when we could have perfect BS action only for a single input number state (vacuum at the other port ). This is a consequence of the quantum interference between the two modes, with the phonon medium providing the coupling between the two modes. This can be clearly seen by putting the loss in the following form,

The difference in the behavior of the losses for the pure number states and the coherent states is thought to arise from phase correlation between the two input beams. In case of initial number states, the phase is completely random. Hence the averaging of the 〈cosθ_il cos θ_jl 〉_sphere in the interference terms causes average to go to zero when there are two equivalent number states incident upon the two input ports. In the case of coherent input states two equal incident modes cause a constructive interference eliminating the losses. These interferences are caused by the non-local equal time excitations of the medium. When we calculate the fluctuations we find the absorption terms in the coefficients of the quadratic terms of the number of photons and spontaneous emission terms in the coefficients of the linear terms of the number of photons. This shows that the effect of the losses on the photon statistics is greater than that of spontaneous emission.

5. Conclusions

A lossy beam-splitter with the losses being due to the excitation of optical phonons in the BS medium has been modeled. The advantage of using a microscopic approach is that one can get explicit expressions for the absorption coefficients and the fluctuations in terms of the material properties of the BS. A transformation equation is derived for the density matrix of the output field in terms of the input field operators and the phonon reservoir operator ensemble averages. The interaction of the radiation with the phonon field is modeled as a first order Markoff process. The main drawback of the model is that a rigorous field quantization in a dispersive, lossy slab has not been carried out. The perturbative approach is justified only in the limit of small dispersion and losses. The model also by itself predicts only the absorption and does not yield the dispersive behavior typical of a lossy medium. But instead the dispersion would have to be imposed on it through the Kramers-Kronig relations. We have assumed the losses due to electron conduction negligible. Hence this model applies to a class of insulating ionic crystals with a wide electronic bandgap. The analysis holds for low and moderate intensities of light with small losses so that the thermal equilibrium of the phonon reservoir is not disturbed. Moreover the mode of interaction described holds true only in the mid IR to Microwave regions of the spectrum.

Due to the inclusion of losses the expectation values of the output operators is found to be related to the input field operators and noise sources from the phonon reservoir which causes the absorption. We have shown how fluctuations increase as compared to the lossless case. Even at zero temperature when there are no phonons present in the medium, the radiation field interacts with the vacuum states of the phonon system, undergoing losses and the fluctuations in the two modes get coupled. The spontaneous emission of light by the phonons comes out naturally in our model which also contributes to the fluctuations though its contribution is lesser than that due to absorption. It turns out there exist certain conditions under which the lossy medium still behaves as a lossless beam-splitter. These arise due to correlations between the input modes. We have argued how the total loss can be considered as a sum of losses in reflection, transmission and due to a quantum mechanical interference between the modes. The choice of the material properties of the BS could be prescribed to get the lossless case after considering the losses. Such lossless cases could have potential applications in laser optics and dielectric coated mirrors.

The mixing of the modes due to loss, irrespective of BS mixing, is interesting. In consequence, the noise in a beam of light, passing through a dielectric (even with 100% transmission), will be affected due to the mixing with the other vacuum mode due to the quantum correlation between the modes and their interaction with the phonons. The choice of suitable parameters, depending both on BS mixing and losses in the dielectric material, to reduce the noise in the modes is yet to be investigated. It would also be interesting to study the quasi-probability distributions of the output modes for both coherent and squeezed input modes. The numerical results on reduction of noise and the effect of the lossy BS in the quasi-probability distribution of the output modes will be reported elsewhere.