## A dissipative quantum mechanical beam-splitter

Optics Express, Vol. 2, Issue 2, pp. 29-39 (1998)

http://dx.doi.org/10.1364/OE.2.000029

Acrobat PDF (261 KB)

### Abstract

A dissipative beam-splitter (BS) has been analyzed by modeling the losses in the BS due to the excitation of optical phonons. The losses are obtained in terms of the BS medium properties. The model simplifies the picture by treating the loss mechanism as a perturbation on the photon modes in a linear, non-lossy medium in the limit of small losses, instead of using the full field quantization in lossy, dispersive media. The model uses second order perturbation in the Markoff approximation and yields the Beer’s law for absorption in the first approximation, thus providing a microscopic description of the absorption coefficient. It is shown that the fluctuations in the modes get increased because of the losses. We show the existence of quantum interferences due to phase correlations between the input beams and it is shown that these correlations can result in loss quenching. Hence in spite of having such a dissipative medium, it is possible to design a lossless 50–50 BS at normal incidence which may have potential applications in laser optics and dielectric-coated mirrors.

© Optical Society of America

## 1. Introduction

11. J. R. Jeffers, N. Imoto, and R. Loudon, “Quantumum optics of traveling wave attenuators and amplifiers”, Phys. Rev. A **47**, 3346 (1993). [CrossRef] [PubMed]

12. U. Leonhardt, “Quantumum statistics of a lossless beam splitter : SU(2) symmetry in phase space”, Phys. Rev. A **48**, 3265 (1993). [CrossRef] [PubMed]

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

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

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

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

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

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

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

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

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

*μ*m 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.

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

## 2. The lossless quantum mechanical beam-splitter

*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,

_{11}and α

_{21}are transmission and reflection coefficients for mode 1, α

_{12}and α

_{22}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,

*α′*

_{1}and

*α′*

_{2}demands the following relations between the matrix coefficients.

*𝒖*such that

*𝒖*turns out to be

*ξ*

^{*}=

*η*for

*𝒖*to be unitary. This relates the observable quantities

*α*

_{ij}to the physical parameters

*ξ*and

*η*as

_{11}= α

_{22}and ∣α

_{12}∣ = ∥α

_{21}∥

*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.

*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

*𝒖*on the state ∣

*n*

_{1},

*n*

_{2}〉 can be written as [4

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

*n*

_{1},

*n*

_{2}〉 and coherent states ∣α,

*β*〉 are given by

## 3. Radiation field–phonon interaction

*b*

_{k}and

*b*

_{k},

*δ*

_{kk′}. The unperturbed Hamiltonian for the phonon reservoir can be written as

*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.

*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].

*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

*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

*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.

*p⃗*of the particle can be expressed in terms of the

*b*

_{k}and

*H*

_{o}=

*ħω*(

*a*

_{2}+ 1) is the free Hamiltonian and

*R*and

*V*

_{I}are given by equations 11 and 16.

*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*

_{o}

*f*

_{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.

*t*

_{int}

*n̅*

_{ph}is the mean occupancy of the phonon levels which is simply the Bose-Einstein distribution

*L*

_{ij}= ∑

_{l,k}

*γ*

_{il}(

*ω*

_{l})

*ω*

_{k})(

*ω*

_{l}

*ω*

_{k}/

*ω*

^{2})

^{1/2}〈cos

*θ*

_{il}cos

*θ*

_{jk}〉

*δ*(

*ω*-

*ω*

_{l})

*δ*(

*ω*-

*ω*

_{k})

*δ*

_{lk}with

*γ*

_{il}=

*γê*∙

_{λi}*ê*

_{l}. The averaging of (cos

*θ*

_{il}cos

*θ*

_{jk}) is over a sphere. As

*ω*

_{l}are closely spaced the summation over

*l*goes over into an integral ∑

_{l}-

*g*(

*ω*

_{l})

*dω*

_{l}, where

*g*(

*ω*

_{l}) is the density of modes for the phonons, we obtain

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

### 4.1 Lossy medium

*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

*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

*a*

_{1}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,

*n*

^{2}while spontaneous emission appears only in the coefficient of

*n*.

*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

*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

*n*

_{1},

*n*

_{2}〉, we get

*n*

_{1},

*n*

_{2}∣

*a*′

_{2}∣

*n*

_{1},

*n*

_{2}〉. Putting

*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

*α*,

*β*〉,

*α*=

*β*and with the optimal conditions

*θ*

_{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

## Acknowledgments

## References

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

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

3. | Also see the special issue of J. Opt. Soc. Am. B4 (1987). |

4. | S. Prasad, M. O. Scully, and W. Martienssen, “A quantum description of the beam-splitter”, Opt. Commun. , |

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

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 |

7. | B. Huttner and Y. Ben-Aryeh, “Influence of a beam splitter on photon statistics”, Phys. Rev. A |

8. | J. Brendel, S. Schutrumpf, R. Lange, W. Martienssen, and M. O. Scully, “A beam splitting experiment with correlated photons”, Europhys. Lett. , |

9. | C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of sub-picosecond time intervals between two photons by interference”, Phys. Rev. Lett. , |

10. | M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D.-G. Welsh, “Generating Schrodinger Cat-like states by means of conditional measurements on a beam-splitter”, Phys. Rev. A |

11. | J. R. Jeffers, N. Imoto, and R. Loudon, “Quantumum optics of traveling wave attenuators and amplifiers”, Phys. Rev. A |

12. | U. Leonhardt, “Quantumum statistics of a lossless beam splitter : SU(2) symmetry in phase space”, Phys. Rev. A |

13. | S.-T. Ho and P. Kumar, “Quantumum optics in a dielectric: macroscopic electromagnetic field and medium operators for a linear dispersive lossy medium - a microsopic derivation of the operators and their commutation relations”, J. Opt. Soc. Am. B |

14. | B. Huttner and S. M. Barnett, “Dispersion and loss in a Hopfield dielectric”, Europhys. Lett . |

15. | B. Huttner and S. M. Barnett, “Quantumization of the electromagnetic field in dielectrics”, Phys. Rev. A |

16. | C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems : quantum stochastic differential equation and the master equation”, Phys. Rev. A |

17. | U. Leonhardt, “Influence of a dispersive and dissipative medium on spectral squeezing”, J. Mod. Opt. |

18. | R. Matloob and R. Loudon, “Electromagnetic field quantization in absorbing dielectrics”, Phys. Rev. A |

19. | T. Gruner and D.-G. Welsch, “Quantumum optical input-output relations for dispersive and lossy multilayer dielectrics”, Phys. Rev. A |

20. | Y. Aharanov, D. Falkoff, E. Lerner, and H. Pendleton, “A quantum characterization of classical radiation”, Ann. Phys. |

21. | N. W. Ashcroft and N. D. Mermin, |

22. | P. Bruesch, |

23. | W. H. Louisell, |

**OCIS Codes**

(230.1360) Optical devices : Beam splitters

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: September 16, 1997

Revised Manuscript: September 3, 1997

Published: January 19, 1998

**Citation**

S. Anantha Ramakrishna, Abir Bandyopadhyay, and Jagdish Rai, "A dissipative quantum mechanical beam-splitter," Opt. Express **2**, 29-39 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-2-2-29

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### References

- For a review see the tutorial by M. C. Teich and B. A. E. Saleh, "Squeezed states of light", Quantum. Opt. 1, 151(1989). [CrossRef]
- Also see the special issue of J. Mod. Opt. 34 (1987).
- Also see the special issue of J. Opt. Soc. Am. B 4(1987).
- S. Prasad, M. O. Scully and W. Martienssen, "A quantum description of the beam-splitter", Opt. Commun., 62, 139 (1987). [CrossRef]
- B. Yurke, S. L. McCall and J. R. Klauder, "SU(2) and SU(1,1) interferometers", Phys. Rev. A 33, 4033 (1986). [CrossRef] [PubMed]
- 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). [CrossRef] [PubMed]
- B. Huttner and Y. Ben-Aryeh, "In uence of a beam splitter on photon statistics", Phys. Rev. A 38, 204 (1988). [CrossRef] [PubMed]
- J. Brendel, S. Schutrumpf, R. Lange, W. Martienssen and M. O. Scully, "A beam splitting experiment with correlated photons", Europhys. Lett., 5, 223 (1988). [CrossRef]
- C. K. Hong, Z. Y. Ou and L. Mandel, "Measurement of sub-picosecond time intervals between two photons by interference", Phys. Rev. Lett., 59, 2044 (1987). [CrossRef] [PubMed]
- M. Dakna, T. Anhut, T. Opatrny, L. Knoll and D.-G. Welsh, "Generating Schrodinger Cat-like states by means of conditional measurements on a beam-splitter", Phys. Rev. A 55, 3184 (1997). [CrossRef]
- J. R. Jeers, N. Imoto and R. Loudon, "Quantumum optics of traveling wave attenuators and ampliers", Phys. Rev. A 47, 3346 (1993). [CrossRef] [PubMed]
- U. Leonhardt, " Quantumum statistics of a lossless beam splitter : SU(2) symmetry in phase space", Phys. Rev. A 48, 3265 (1993). [CrossRef] [PubMed]
- S.-T. Ho and P. Kumar, "Quantumum optics in a dielectric: macroscopic electromagnetic field and medium operators for a linear dispersive lossy medium - a microsopic derivation of the operators and their commutation relations", J. Opt. Soc. Am. B 10, 1620 (1993). [CrossRef]
- B. Huttner and S. M. Barnett, "Dispersion and loss in a Hopeld dielectric", Europhys. Lett. 18, 487 (1992). [CrossRef]
- B. Huttner and S. M. Barnett, "Quantumization of the electromagnetic field in dielectrics", Phys. Rev. A 46, 4306 (1992). [CrossRef] [PubMed]
- C. W. Gardiner and M. J. Collett, "Input and output in damped quantum systems : quantum stochastic dierential equation and the master equation", Phys. Rev. A 31, 3761 (1985). [CrossRef] [PubMed]
- U. Leonhardt, "In uence of a dispersive and dissipative medium on spectral squeezing", J. Mod. Opt. 42, 1165 (1995).
- R. Matloob and R. Loudon, "Electromagnetic field quantization in absorbing dielectrics", Phys. Rev. A 52, 4823 (1995). [CrossRef] [PubMed]
- T. Gruner and D.-G. Welsch, "Quantumum optical input-output relations for dispersive and lossy multilayer dielectrics", Phys. Rev. A 54, 1661 (1996). [CrossRef] [PubMed]
- Y. Aharanov, D. Falko, E. Lerner and H. Pendleton, "A quantum characterization of classical radiation", Ann. Phys. 39, 498 (1966). [CrossRef]
- N. W. Ashcroft and N. D. Mermin, Solid State Physics, International ed., (Saunders College, Philadelphia, 1976), Appendix-L .
- P. Bruesch, Phonons : Theory and Experiments, Vol-I and II, (Springer-Verlag, Heidelberg, 1983).
- W. H. Louisell, Quantumum Statistical Properties of Radiation, (John Wiley and Sons, NY, 1973).

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