A dissipative quantum mechanical beam-splitter

A dissipative quantum mechanical

beam-splitter

S. Anantha Ramakrishna

Laser Programme, Center for Advanced Technology, Indore - 452 013, India

[email protected]

Abir Bandyopadhyay

Physical Research Laboratory, Ahmedabad - 380 009, India

[email protected]

Jagdish Rai

Department of Physics, Indian Institute of Technology, Kanpur - 208 016, India.

[email protected]

Abstract: A dissipative beam-splitter (BS) has been analyzed bymodeling the losses in the BS due to the excitation of optical phonons.The losses are obtained in terms of the BS medium properties. Themodel simplifies the picture by treating the loss mechanism as a per-turbation on the photon modes in a linear, non-lossy medium in thelimit of small losses, instead of using the full field quantization in lossy,dispersive media. The model uses second order perturbation in theMarkoff approximation and yields the Beer’s law for absorption in thefirst approximation, thus providing a microscopic description of theabsorption coefficient. It is shown that the fluctuations in the modesget increased because of the losses. We show the existence of quantuminterferences due to phase correlations between the input beams andit is shown that these correlations can result in loss quenching. Hencein spite of having such a dissipative medium, it is possible to design alossless 50-50 BS at normal incidence which may have potential appli-cations in laser optics and dielectric-coated mirrors.c©1998 Optical Society of AmericaOCIS codes: (270.0270) Quantumum optics; (230.1360) Beam splitters

References

1. For a review see the tutorial by M. C. Teich and B. A. E. Saleh, “Squeezed states of light”,Quantum. Opt. 1, 151(1989).

2. Also see the special issue of J. Mod. Opt. 34 (1987).

3. Also see the special issue of J. Opt. Soc. Am. B 4 (1987).

4. S. Prasad, M. O. Scully and W. Martienssen, “A quantum description of the beam-splitter”,Opt. Commun., 62, 139 (1987).

5. B. Yurke, S. L. McCall and J. R. Klauder, “SU(2) and SU(1,1) interferometers”, Phys. Rev. A33, 4033 (1986).

6. R. A. Campos, B. E. A. Saleh and M. C. Teich, “Quantumum mechanical lossless beam splitter: SU(2) symmetry and photon statistics”, Phys. Rev. A 40, 1371 (1989).

7. B. Huttner and Y. Ben-Aryeh, “Influence of a beam splitter on photon statistics”, Phys. Rev.A 38, 204 (1988).

#2370 - $10.00 US Received September 3, 1997; Revised December 30, 1997

(C) 1998 OSA 19 January 1998 / Vol. 2, No. 2 / OPTICS EXPRESS 29

8. J. Brendel, S. Schutrumpf, R. Lange, W. Martienssen and M. O. Scully, “A beam splittingexperiment with correlated photons”, Europhys. Lett., 5, 223 (1988).

9. C. K. Hong, Z. Y. Ou and L. Mandel, “Measurement of sub-picosecond time intervals betweentwo photons by interference”, Phys. Rev. Lett., 59, 2044 (1987).

10. M. Dakna, T. Anhut, T. Opatrny, L. Knoll and D.-G. Welsh, “Generating Schrodinger Cat-likestates by means of conditional measurements on a beam-splitter”, Phys. Rev.A 55, 3184 (1997).

11. J. R. Jeffers, N. Imoto and R. Loudon, “Quantumum optics of traveling wave attenuators andamplifiers”, Phys. Rev. A 47, 3346 (1993).

12. U. Leonhardt, “ Quantumum statistics of a lossless beam splitter : SU(2) symmetry in phasespace”, Phys. Rev. A 48, 3265 (1993).

13. S.-T. Ho and P. Kumar, “Quantumum optics in a dielectric: macroscopic electromagnetic fieldand medium operators for a linear dispersive lossy medium - a microsopic derivation of theoperators and their commutation relations”, J. Opt. Soc. Am. B 10, 1620 (1993).

14. B. Huttner and S. M. Barnett, “Dispersion and loss in a Hopfield dielectric”, Europhys. Lett.18, 487 (1992).

15. B. Huttner and S. M. Barnett, “Quantumization of the electromagnetic field in dielectrics”,Phys. Rev. A 46, 4306 (1992).

16. C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems : quantumstochastic differential equation and the master equation”, Phys. Rev. A 31, 3761 (1985).

17. U. Leonhardt, “Influence of a dispersive and dissipative medium on spectral squeezing”, J. Mod.Opt. 42, 1165 (1995).

18. R. Matloob and R. Loudon, “Electromagnetic field quantization in absorbing dielectrics”, Phys.Rev. A 52, 4823 (1995).

19. T. Gruner and D.-G. Welsch, “Quantumum optical input-output relations for dispersive andlossy multilayer dielectrics”, Phys. Rev. A 54, 1661 (1996).

20. Y. Aharanov, D. Falkoff, E. Lerner and H. Pendleton, “A quantum characterization of classicalradiation”, Ann. Phys. 39, 498 (1966).

21. N. W. Ashcroft and N. D. Mermin, Solid State Physics, International ed., (Saunders College,Philadelphia, 1976), Appendix-L .

22. P. Bruesch, Phonons : Theory and Experiments, Vol-I and II, (Springer-Verlag, Heidelberg,1983).

23. W. H. Louisell, Quantumum Statistical Properties of Radiation, (John Wiley and Sons, NY,

1973).

1. Introduction

A beam-splitter (BS) is one of the most widely used optical components that is un-derstood totally by classical treatment. However, as the use of non-classical sources oflight in experiments increases [1, 2, 3], it becomes essential to understand the behaviorof all the components used, in a purely quantum mechanical sense, in order to inter-pret the results of these experiments as well as the limitations of these results. Severalauthors 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 twoparts, a quantum-mechanical analysis shows that the BS modifies the basic statisticalproperties of the beams [6, 7]. Thus a simple BS can be used to probe the quantumnature of light by simple yet subtle experiments [8, 9]. Recently the use of a BS forgenerating Schrodinger Cat-like states has been proposed [10]. A BS offers one of thesimplest interaction of a light mode with an external environment. It serves as a modelfor the interaction of a radiation mode with an external environment attenuating thefield and the entering of external fluctuations via the second port of the beam-splitteris in accordance with the fluctuation-dissipation theory. Thus the effects of an externalenvironment on the light modes can be environment by observing the effects on thestatistical properties of the modes caused by the BS.



But any real BS is also expected to be lossy. It is well known that, when thedielectric matter, with small imaginary part of the permitivity, is considered in freespace, the input-output relations correspond to unitary transformation between theoperators of input and output modes. However, these concepts fail when the effect ofabsorption 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 beexpected to affect the photon statistics as the loss would couple the fields to a reservoirwhose 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. Arelatively simple representation of the loss mechanism is provided by a continuous dis-tribution of fictitious optical beam-splitters in an otherwise homogeneous and losslessmedium[11, 12]. There have been attempts to microscopically model the losses anddispersion in homogeneous media in terms of the polariton and reservoir operators[13, 14, 15]. The quantization of the radiation field has been approached by usingLangevin forces to represent the noise, based on a development of the familiar for-malism of quantum noise theory [16]. This approach has been successfully applied tothe description of dielectric slabs [17]. A comprehensive treatment of field quantizationin dispersive media is given in Ref[18], which gives consistent expressions for the macro-scopic quantized fields in infinite dielectric media and dielectric slabs. All the aboveworks take either a macroscopic or microscopic approach to the quantization of the fieldoperators. The microscopic models [13, 14, 15] have the advantage that they explicitlyobtain the diagonalization of the coupled system of electromagnetic filed, dielectric os-cillator and the reservoir. The dielectric function is expressed in terms of the parametersof the models. However the Langevin force calculation to which most of the macroscopicschemes resort to are much simpler but do not yield information in terms of the ma-terial properties of the system. Recently quantum input-output relations were derivedfor dispersive, multilayer plates using the Green’s function approach to quantization ofthe phenomenological Maxwell’s equations to show that the absorption introduces anadditional noise to the vacuum noise [19].

In this paper, a microscopic model of a dissipative BS is presented. In orderto apply it to two mode interactions, we have simplified the picture by neglecting thedielectric polariton excitations and have instead directly coupled the electro magneticfield directly to the thermal excitations of the medium. In the model the losses arecaused by the resonant absorption of light due to optical phonons in the medium. It istrue in the mid infra-red to the microwave region of the spectrum (5µm onwards) inhost 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 fromthe electron scattering can be assumed to be almost dispersion-less. This assumption isjustified to the extent that the absorption is reasonably small i.e., Imχ�Reχ. Our mainconcern here is to investigate the effect of losses on the output modes of the BS. Themodel treats the losses as a perturbation on the BS transformation and the effects on theoutput modes is calculated up to the second order of a perturbative expansion. The BSis considered as a reservoir of phonons at some finite temperature and it is assumed thatthe photon-phonon interaction does not disturb the thermal equilibrium of the phononsystem. 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 increasein the noise. However, the analysis shows that losses also depend upon quantum phasecorrelations 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



BS

��>��>��

θ θ

θ θ

~E2 ~E1′

ZZZZ~ZZZZZ~ZZZZ

~E1 ~E2′

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

quenching due to phase correlations is being reported for the first time.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, amodel for the photon-phonon interaction and the consequent losses is developed. Anexpression for the transformation of the reduced density matrix for the radiation fieldis derived using the radiation phonon interaction as a perturbation. In Section IV, theformalism developed is applied to a lossy medium and then to a lossy BS. It is found thatadditional 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 twointeracting modes into two parts to correspond to the splitting of the beam in a mannerthat 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 ~E1 and ~E2 falling on it fromthe two sides of the BS. Now both ~E1 and ~E2 give rise to the output waves ~E

′1 and ~E

′2.

Hence, for the positive components of the electromagnetic fields,

E(+)1

′= α11 E

(+)1 + α12 E

(+)2

E(+)2

′= α21 E

(+)1 + α22 E

(+)2 (1)

where α11 and α21 are transmission and reflection coefficients for mode 1, α12 andα22 are reflection and transmission coefficients for mode 2. The coefficients could bedependent on direction (~k) or polarization (eλ) of the light mode. In the second quantizednotation for the electromagnetic field, the E(+) in the equations go over directly to theannihilation operators for the light modes. More conveniently in matrix form we canwrite, (

a1′

a2′

)=

(α11 α12α21 α22

)(a1a2

)(2)

Similarly, one can obtain two more sets of equations for the creation operators. Con-servation of energy and preservation of the commutation relations between a1

′ and a2′

demands the following relations between the matrix coefficients.

|α11|2 + |α21|

2 = 1

|α12|2 + |α22|

2 = 1

α11 α∗12 + α21 α

∗22 = 0 (3)



implying the transformation is unitary. Hence we look for a unitary operator U suchthat (

a1′

a2′

)= U†

(a1a2

)U =

(α11 α12α21 α22

)(a1a2

)(4)

The operator U turns out to be

U = exp[−i(ξa1

†a2+ ηa2†a1)]

(5)

where ξ∗ = η for U to be unitary. This relates the observable quantities αij to thephysical parameters ξ and η as

(α11 α12α21 α22

)=

cos√ηξ −i

√ξηsin√ηξ

−i√ηξsin√ηξ cos

√ηξ

(6)

We notice that α11 = α22 and |α12| = |α21| i.e. the transmittance and reflectance are the

same regardless of the wave vector ~k with the crystal axis of the BS as these relationswere derived entirely from the boundary conditions. It was also de facto assumed thatthe interaction due to the BS was isotropic and that the polarizations were not rotatedeither.

The action of the BS can be viewed in two ways. One, the Heisenberg picturein which the field operators a and a† get transformed by the BS transformation whilethe state vectors evolve freely in time. The second is that what Prasad et al. call theinteraction picture where the state vectors get transformed by the BS operator while thefield operators evolve freely in time. In the following, we shall work in the interactionpicture. We represent the photon state to be a product number state |n1, n2〉, where n1and n2 are the number of photons in the light modes 1 and 2 respectively

U|1, 0〉 = α11 |1, 0〉+ α21 |0, 1〉 (7)

i.e., the final state is a coherent superposition of two single photon states in the twomodes. The operation of U on the state |n1, n2〉 can be written as [4],

U|n1, n2〉 =1

√n1!n2!

(α11a

†1 + α21a

†2

)n1 (α12a

†1 + α22a

†2

)n2|0, 0〉 (8)

The calculated fluctuations for pure number states |n1, n2〉 and coherent states |α, β〉are given by

〈n1, n2|(∆n1)2|n1, n2〉 = |α11|

2|α12|2(n1 + n2 + 2n1n2)

〈α, β|(∆n1)2|α, β〉 = (|α11|

2 + |α12|2)|α11α+ α12β|

2 (9)

One can see that the coherent input modes are transformed into two completely un-correlated coherent output beams by the action of the BS. This is due to the inherentSU(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 oscillatingdipoles and bound currents due to orbital (bound) electrons in the atoms. The energylosses of the radiation field in the BS is due to the excitation of optical phonons in theBS i.e., the interaction of the photon field with the phonons in the BS. This causes



6 6 6? ? ?r r rb b b

+ + +- - -

~p1 ~p2

(a)

- - -� � �r r rb b b

+ + +- - -

~p1 ~p2

(b)

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

(b) longitudinal.

energy to be irretrievably passed into the phonon energies. The BS is considered to bea reservoir of phonons at some finite temperature.

The phonon system is described by the phonon creation and annihilation oper-ators bk and b

†k defined as [21]

bk =1√N

∑k

ei~k·~R

[√Mωk

2h~u(~R) + i

√1

2hMωk~P (~R)

]ek

b†k =1√N

∑k

e−i~k·~R

[√Mωk

2h~u(~R)− i

√1

2hMωk~P (~R)

]ek (10)

These satisfy the commutation relation[bk, b

†k′

]= δkk′ . The unperturbed Hamiltonian

for the phonon reservoir can be written as

R =∑l

hωl

(b†l bl +

1

2

)(11)

Here M is the mass, ~u(~R) is the displacement about the mean position, ~P (~R) is themomentum of the ion in the lattice; N is the total number of atoms in the crystal; ωkand ~k are the frequency and the wave vector of the phonon. For simplicity only a singlebranch of the phonon modes is considered.

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

Vi =q

M~A(~R) · ~p (12)

We can write the interaction energy for m charged ions in the lattice as VI = mVi.This can be done only for optical phonons and is justified as following. If we observethe 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 (q1 = −q2) and theirmomenta are approximately equal in magnitude but opposite in direction i.e. ~p1 = −~p2and hence their energies add up. Here an approximation is made that the positive andnegative ions have approximately the same mass. A more rigorous analysis of whichoptical phonon modes will contribute effectively to the process is given in Ref.[22].

One has to exercise caution in replacing ~A by the creation and annihilationoperators as the ~R is a label for the second quantized electromagnetic field while it isthe position operator for the particle in the coordinate space. We replace ~A(~R) by therequisite operators in the Fock space, ~p by the operator in the coordinate space and thecompound operator acts on the states spanning the product space, thus

|ψ〉 =∑j

|j〉fj(~r) (13)



Here ~A acts on the |j〉 states and pµ acts on the fi(xµ). ~A can be expressed in terms ofthe creation and annihilation operators of the field

~A(R) =∑k,λ

√h

2ωεVint

[ei~k·Rakλ + e

−i~k·Ra†kλ

]eλ (14)

where R is the position operator for the particle. ~A(R) means summation over the ~k andnothing else. Vint is the volume of interaction of the beam inside the BS medium. We

note that the role of ei~k·R and e−i

~k·R are merely translations in the momentum spaceand preserve the conservation of momentum i.e., each time a photon of momentum h~kis annihilated, the crystal momentum is increased by the same amount. But generallythe photon momenta are small compared to the phonon momenta. In addition the BSsplitter is kept clamped. Hence we neglect the conservation of momenta and retain onlythe first term in the exponential i.e., unity. This gives the so-called dipole approximation.

The momentum ~p of the particle can be expressed in terms of the bk and b†k

operators

~p(~R) =∑l

i

√hωlM

2N

(ble−iωlt − b†l e

iωlt)el (15)

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

VI =∑i=1,2

∑l

γ[aible

−i(ω+ωl)t − aib†l e−i(ω−ωl)t − a†i ble

i(ω−ωl)t +a†ib†l ei(ω+ωl)t

]eλi · el (16)

where γ = iqhm2

√ωl

εMVintNω. Now we make the rotating wave approximation (RWA)

and neglect the fast varying sum frequency terms and seek the interaction averagedin time over several periods so that over this time interval the phonon reservoir hasattained equilibrium. The full Hamiltonian of the radiation field and phonon systemcan be written as

HT = Ho +R+ VI (17)

where Ho = hω(a†1a1 + a†2a2 + 1) is the free Hamiltonian and R and VI are given by

equations 11 and 16.If Sd is the density matrix for the coupled system in the interaction picture,

where the state vectors evolve in with VI , then the reduced density matrix given by tak-ing the trace over the reservoir states s = TrR[Sd], and So = sofo(R), before interactionof the photon and the phonon systems, where fo(R) is the equilibrium distribution ofthe phonon states. Following the conventional treatment [23], one can write down thetransformation of the reduced density matrix for the light field in terms of the ensembleaverages of the phonon reservoir. In doing so, one makes the Markoff approximation byassuming 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. τ(R)corr � tint � τ

(s)decay.

In order to take a trace over the reservoir states, we note the following ensemble averages

〈bl〉R = 〈b†l 〉R = 0

〈blbk〉R = 〈b†l b†k〉R = 0

〈b†l bk〉R = nph(ωl)δlk

〈blb†k〉R = [nph(ωl) + 1]δlk (18)



where nph is the mean occupancy of the phonon levels which is simply the Bose-Einsteindistribution nph(ωl) = [exp(

hωkT )− 1]

−1. The transformation for the density matrix canbe written as

s′ = s − (1/2)∑i,j

(a†iajs− a†isaj − ajsa

†i + saja

†i )Sij

+(aia†js− aisa

†j − a

†jsai + sa

†jai)S

∗ij

+(a†iajs− ajsa†i )Lij − (aisa

†j − sa

†jai)L

∗ij (19)

where, Lij =∑l,k γil(ωl)γ

∗jk(ωk) (ωlωk/ω

2)1/2 〈cos θil cos θjk〉 δ(ω − ωl) δ(ω − ωk) δlkwith γil = γeλi · el. The averaging of 〈cos θil cos θjk〉 is over a sphere. As ωl are closelyspaced the summation over l goes over into an integral

∑l −→

∫∞0 g(ωl)dωl, where

g(ωl) is the density of modes for the phonons, we obtain

Lij = |γ(ω)|2〈cos θil cos θjl〉 g(ω)

Sij = nph(ω)Lij (20)

This gives the transformation of the reduced density matrix due to phonon interactionup 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 usingour formalism is instructive. Let us for the time-being assume that there is no resonantscattering of light by electrons i.e., the only interaction the light has with the BS isthrough the phonons. Then the term U drops out. For a mixed state we have to calculatethe trace over the photon states as 〈O〉=Tr [sO ]. The ensemble averages for differenttransformed operators derived from the equation (19) due the lossy BS (medium) interms of the lossless BS (medium) expectation values are given in appendix-A. However,for a pure number state |n1, n2〉 as the initial state ,the density matrix is reduced to asingle element so = |n1, n2〉〈n1, n2|. The expectation values for the transformed beamsfor number states come out to be

〈n1, n2|a′1†a′1|n1, n2〉 = n1(1− L11) + S11

〈n1, n2|a′2†a′2|n1, n2〉 = n2(1− L22) + S22 (21)

We note that L11 and L22 are the total absorption coefficients for the two modes. S11and S22 represent the spontaneous emission (black-body radiation) by phonons in thetwo modes. Here we have no cross-terms in intensities of the modes due to phononinteraction alone. That is because such a process would correspond to annihilation of aphoton in one mode, the corresponding creation of a phonon, annihilation of a phononand creation of a photon in the other mode which would be a fourth order processand we have considered processes only up to the second order. However if one observesthe transformation of the operator a1 due to losses, as the phonon field couples all themodes of the radiation field, one finds the contribution of the other modes in it,

〈ai〉L =∑j

(δij −

Lij

2

)〈aj〉NL (22)

For the current case we have only two modes.



The number fluctuations in the transmitted beam-1 is,

〈n1, n2|∆n12|n1, n2〉 = L11(4− L11)n

2 + [2S11 + L11(2S11 − 1)]n+ S11(1− S11) (23)

Note that the contribution of absorption is in the coefficient of n2 while spontaneousemission appears only in the coefficient of n.

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

∆I = I0[1− exp(−αl)] ≈ I0[1− (1− αl)] = I0αl (24)

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

δn1 = L11n1 =

[π2

12

q2ρg(ω)

εMVx

]An1l (25)

i.e., proportional to the area of the beam and length traversed in the medium. Theterm 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. Thedependence of the absorption coefficient on the frequency in our model comes merelythrough the dependence on the density of the phonon modes. The dispersion in the realpart due the interaction has to be imposed by the Kramers-Kronig relations. This isalso the way the real part is experimentally determined. It is well known that opticalmodes are better represented by the Einstein model while the Debye model are moreadapted only to acoustical modes [22]. So we take the density of modes for the phononsas g(ω) = 3(n − 1)Vxva δ(ω − ωE) where ωE is the Einstein frequency, n is the numberatoms per primitive unit cell and va is the volume of the primitive unit cell. This resultsin a single resonance form for the absorption. This is typical of alkali halides where thereis 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 theinteraction picture will evolve according to the unitery BS evolution given by equations4-6. Taking the initial state to be pure number states, |n1, n2〉, we get

〈n1, n2|a′1†a′1|n1, n2〉 = (|α11

2n1 + |α12|

2n2) (1− L11)

− Re[L12(α11|∗α21n1 + α∗12α22n2)] + S11 (26)

and a similar expression for 〈n1, n2|a†2

′a′2|n1, n2〉. Putting L11 = L22 =

L2 and realizing

L12 =L2 sin θ , we get the total loss as

L =L

2[(n1 + n2)− sin |ξ | sin θ sin 2δξ (n1 − n2)] (27)

where we have used ξ = |ξ|eiδξ . The last equation shows that the radiation field cannotgain energy from the medium. The cross-term which is dependent on the incidence angleof the beams, is zero when the intensities are equal and maximum when one of them iszero. This resembles the situation in scattering in a four-wave mixing process. When theintensity of one of the beams is zero we can adjust the parameters of the second termsuch that the loss is equal to zero. This happens only when θ = π

2 , i.e., it is a perfect50-50 beam-splitter at normal incidence. ξ is dependent on the material properties. Thecross-term which involves the incidence angles of the beams, is zero when the intensitiesare 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 fromthe other port. To get a further insight, we calculate the losses if the initial states arecoherent states |α, β〉,

〈α, β|a†1′a′1|α, β〉 = |α11α+ α12β|

2(1 − L11)

−Re[L12(α∗11α

∗ + α∗12β∗)(α21α+ α22β)] (28)

We can see that the two output beams do not go into completely uncorrelated coherentstates. The losses break the inherent SU(2) symmetry of the BS transformation. Thequantum 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 spontaneousemission, comes out to be

L =L

2[|α|2 + |β|2 + 2|αβ| sin θ[cos(δβ − δα) (cos

2 |ξ| − sin2 |ξ| cos 2δξ)

+ sin(δβ − δα) sin2 |ξ| sin 2δξ] (29)

with the motivation to minimize losses we find that when α = β and with the optimalconditions θ = π

2 , the loss goes to zero unlike the earlier case when we could haveperfect BS action only for a single input number state (vacuum at the other port ). Thisis a consequence of the quantum interference between the two modes, with the phononmedium providing the coupling between the two modes. This can be clearly seen byputting the loss in the following form,

L =L

2

[|α11α+ α12β|

2 + |α21α+ α22β|2 + 2 sin θ Re{(α∗11α

∗ + α∗12β∗)(α21α+ α22β)}

](30)

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

5. Conclusions

A lossy beam-splitter with the losses being due to the excitation of optical phonons inthe BS medium has been modeled. The advantage of using a microscopic approach isthat one can get explicit expressions for the absorption coefficients and the fluctuationsin terms of the material properties of the BS. A transformation equation is derivedfor the density matrix of the output field in terms of the input field operators andthe phonon reservoir operator ensemble averages. The interaction of the radiation withthe phonon field is modeled as a first order Markoff process. The main drawback ofthe model is that a rigorous field quantization in a dispersive, lossy slab has not beencarried out. The perturbative approach is justified only in the limit of small dispersionand losses. The model also by itself predicts only the absorption and does not yield thedispersive 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 lossesdue to electron conduction negligible. Hence this model applies to a class of insulatingionic crystals with a wide electronic bandgap. The analysis holds for low and moderateintensities of light with small losses so that the thermal equilibrium of the phononreservoir is not disturbed. Moreover the mode of interaction described holds true onlyin the mid IR to Microwave regions of the spectrum.

Due to the inclusion of losses the expectation values of the output operatorsis found to be related to the input field operators and noise sources from the phononreservoir which causes the absorption. We have shown how fluctuations increase ascompared to the lossless case. Even at zero temperature when there are no phononspresent in the medium, the radiation field interacts with the vacuum states of thephonon 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 modelwhich also contributes to the fluctuations though its contribution is lesser than that dueto absorption. It turns out there exist certain conditions under which the lossy mediumstill behaves as a lossless beam-splitter. These arise due to correlations between theinput modes. We have argued how the total loss can be considered as a sum of lossesin reflection, transmission and due to a quantum mechanical interference between themodes. The choice of the material properties of the BS could be prescribed to getthe lossless case after considering the losses. Such lossless cases could have potentialapplications in laser optics and dielectric coated mirrors.

The mixing of the modes due to loss, irrespective of BS mixing, is interesting. Inconsequence, 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 thequantum correlation between the modes and their interaction with the phonons. Thechoice of suitable parameters, depending both on BS mixing and losses in the dielectricmaterial, to reduce the noise in the modes is yet to be investigated. It would also beinteresting to study the quasi-probability distributions of the output modes for bothcoherent and squeezed input modes. The numerical results on reduction of noise andthe effect of the lossy BS in the quasi-probability distribution of the output modes willbe reported elsewhere.

Acknowledgments

SAR would like to thank Shiraz Minwallah and Rajan Gurjar for extremely usefuldiscussions. Both SAR and AB would like to acknowledge their studentship at IndianInstitute of Technology, Kanpur, where part of the work was carried out.



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A dissipative quantum mechanical beam-splitter