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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 1 — Jan. 13, 2003
  • pp: 7–13
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Theory of photon statistics and squeezing in quantum interference of a sub-threshold parametric oscillator

R. Andrews, E. R. Pike, and Sarben Sarkar  »View Author Affiliations


Optics Express, Vol. 11, Issue 1, pp. 7-13 (2003)
http://dx.doi.org/10.1364/OE.11.000007


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Abstract

A multimode theory describing quantum interference of a sub-threshold optical parametric oscillator (OPO) with a coherent local oscillator (LO) in a homodyne detection scheme is presented. Analytic expressions for the count rates in terms of the correlation time and relative phase difference between the LO and OPO have been derived. The spectrum of squeezing is also derived and the threshold for squeezing obtained in terms of the crystal nonlinearity and LO and OPO beam intensities.

© 2003 Optical Society of America

1. Introduction

Recently there has been renewed interest in photon-antibunched sources of light due to their possible applications in quantum cryptographic systems [1

1. G. Brassard, N. Lutkenhaus, T. Mor, and B. Sanders, “Limitations on practical quantum cryptography,” Phys. Rev. Lett. 85, 1330–1333 (2000). [CrossRef] [PubMed]

]. Such sources contain a single photon in a given time interval and can therefore be used for secure communication purposes. Photon bunching and antibunching has been observed recently in fluorescence from a variety of sources like quantum dots [2

2. A. Kiraz, S. Falth, C. Bechner, B. Gayral, W. V. Schoenfeld, P. M. Petroff, Lidong Zhang, E. Hu, and A. Imamoglu, “Photon correlation spectroscopy of a single quantum dot,” Phys. Rev. B 65, 161303–161304 (2002). [CrossRef]

,3

3. C. Santori, M. Pelton, G. Solomon, Y. Dale, and Y. Yamamoto, “Triggered single photons from a quantum dot,” Phys. Rev. Lett. 86, 1502–1505 (2001). [CrossRef] [PubMed]

], color centers in diamond [4

4. R. Brouri, A. Beveratos, J. Poizat, and P. Grangier, “Photon antibunching in the fluorescence of individual color centers in diamond,” Opt. Lett. 25, 1294–1296 (2000). [CrossRef]

] and single molecules [5

5. L. Fleury, J. Segura, G. Zumofen, B. Hecht, and U. P. Wild, “Nonclassical photon statistics in single-molecule fluorescence at room temperature,” Phys. Rev. Lett. 84, 1148–1151 (2000). [CrossRef] [PubMed]

]. In this paper we describe theoretically from first principles the generation, via interference, of two-photon antibunched light from a sub-threshold parametric oscillator. We consider the recent experiments of Ou and Lu [6

6. Y. J. Lu and Z. Y. Ou, “Observation of nonclassical photon statistics due to quantum interference,” Phys. Rev. Lett. 88, 023601-1–023601-4 (2002).

] in which the output of a sub-threshold parametric oscillator (OPO) is mixed with a continuous-wave local oscillator at a 50/50 beam splitter. Due to quantum interference in the detection of photon pairs from the local oscillator (LO) and the OPO, it has been demonstrated experimentally that the degree of bunching or antibunching can be controlled by the relative phase between the OPO and LO. We apply the recent multimode formalism of the authors [7

7. R. Andrews, E. R. Pike, and Sarben Sarkar, “Photon correlations of a sub-threshold optical parametric oscillator,” Opt. Express 10, 461–468 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-11-461. [CrossRef] [PubMed]

] which describes the quantum correlations of the output of the OPO and examine the nonclassical effects of homodyning the OPO field with a CW laser field or local oscillator (LO). We first examine the variation of the pair count rate with phase-difference between the LO and the OPO. Our theoretical results are then to used to model the experimental results of Lu and Ou [6

6. Y. J. Lu and Z. Y. Ou, “Observation of nonclassical photon statistics due to quantum interference,” Phys. Rev. Lett. 88, 023601-1–023601-4 (2002).

]. We then examine the variation of the spectrum of squeezing [8

8. A. Gatti and L. Lugiato, “Quantum images and critical fluctuations in the optical parametric oscillator,” Phys. Rev. A 52, 1675–1690 (1995). [CrossRef] [PubMed]

] with phase-difference.

2. Amplitude for pair detection in homodyne

In the experiment of Ou and Lu [6

6. Y. J. Lu and Z. Y. Ou, “Observation of nonclassical photon statistics due to quantum interference,” Phys. Rev. Lett. 88, 023601-1–023601-4 (2002).

] a coherent state acting as the LO is injected into the OPO and is mixed with the two-photon state. The resultant field then exits the cavity and interacts with a beam splitter outside the OPO (see Fig. 1 in reference [6

6. Y. J. Lu and Z. Y. Ou, “Observation of nonclassical photon statistics due to quantum interference,” Phys. Rev. Lett. 88, 023601-1–023601-4 (2002).

]). As pointed out in [6

6. Y. J. Lu and Z. Y. Ou, “Observation of nonclassical photon statistics due to quantum interference,” Phys. Rev. Lett. 88, 023601-1–023601-4 (2002).

], the effect of injecting the coherent state into the cavity is to synchronize the two fields and for spatial filtering of the coherent state. If we assume that a perfect coherent state and a two-photon state in superposition exits the cavity, then such a procedure is equivalent to the following homodyne detection scheme in Fig. 1 since the beam splitter transformation of the electric fields of the OPO and LO is the same in both cases. Therefore, for convenience in describing the experiment of Ou and Lu, consider the following setup (see Fig. 1 below) in which the field from the OPO is mixed with the field of the LO:

Fig. 1. Schematic showing the interference of the field from a LO and OPO. BS is a 50/50 beam splitter and D1 and D2 are photon counters.

If we assume for simplicity that the photon detectors D1 and D2 are single atoms located at r1 and r2 respectively, the amplitude for pair detection in the Heisenberg representation is described by

A(2)=αLO,αpump,0EH(+)r2t2EH(+)r1t1αLO,αpump,0
(1)

A(2)=12αLO,αpump,0ELO(+)r2t2ELO(+)r1t1αLO,αpump,0
+(12εpχepd3k1d3k2d3r3Uk1λ1*(r3)Uk2λ2*(r3)Uk0λ0(r3)
×αLO,αpump0Eout(+)r2t2Eout(+)r1t1αLO,αpump,k1,k2δωk1ωk2ωk0)
(2)

EI(ri,t) are the interaction-picture electric field operators and εpp are the amplitude and phase of the pump respectively; Ukiλi(r3) are the spatial mode functions of the electromagnetic field inside the cavity [7

7. R. Andrews, E. R. Pike, and Sarben Sarkar, “Photon correlations of a sub-threshold optical parametric oscillator,” Opt. Express 10, 461–468 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-11-461. [CrossRef] [PubMed]

,9

9. F. De Martini, M. Marrocco, P. Mataloni, L. Crescentini, and R. Loudon, “Spontaneous emission in the optical microscopic cavity,” Phys. Rev. A 43, 2480–2497 (1991). [CrossRef] [PubMed]

] with k 1, k 2 as the wave vectors of the signal and idler photons and k 0 the wave vector of the input pump photon. In obtaining (2) we employed beam splitter transformations for the electric field at the detectors:

Er1t1=12[Eoutr1t1+iELOr1t1]
Er2t2=12[iEoutr2t2ELOr2t2]
(3)

A(2)εl2ei2ϕlχSεPeiϕpeτn1vd
(4)

εll are the amplitude and phase of the LO, τ is the correlation time, d is the length of the OPO cavity and v is the first-order dispersion coefficient of the nonlinear crystal; S and n 1 are defined as

S=π2Vt2o2t1p(1+r2o)n1
n1=2r2o1+r2o
(5)

where t 2o,r 2o are the amplitude transmission, reflection coefficients of the output mirror of the OPO at the signal and idler frequencies and V the volume of the irradiated crystal. We obtain the count rate which is proportional to A(2)2 as

A(2)21+q2e2τn1vd+2qeτn1vdcos(ϕp2ϕl)
(6)

where q=εpχSεl2

3. Spectrum of squeezing

Let us now determine, in general, the nature and extent of squeezing when interference effects are present. In balanced homodyne detection, one measures the difference in the powers of the two beams at r1 and r2. The difference in powers of the two beams is obtained from P(t) defined as

P(t)=E()r1tE(+)r1tE()r2tE(+)r2t
(7)

Quantum fluctuations around a mean value 〈P(t)〉 is described by

δP(t)=P(t)P(t)
(8)

and the spectrum of squeezing, Q(ω), is defined as

Q(ω)=dteiωtδP(t)δP(0)
(9)

The expected value in (9) is calculated to be

δP(t)δP(0)=βεl2[11εl24γεpcos(2ϕlϕp)F(t)+2Δnk02c(k02)2sinc(Δt)]
(10)

where

F(t)=dωeiωt1+r2oexp(i2vdω)2
(11)

Δ is the bandwidth of the detected light, nk02 is the refractive index of the nonlinear crystal at the degenerate frequency. The other constant γ in (10) is given as

γ=πVχ(nk02)2(t2o2t10)(k02)4
(12)

The variable ω in (11) is defined as ω=ωk02ωk , where ωk is the frequency of the signal photon detected at D1. Taking the Fourier transform of the R.H.S of (10) we obtain the normalized spectrum of squeezing at the degenerate frequency ωk=ωk02 as

V(0)=1+μδcos(2ϕlϕp)
(13)

where

μ=11εl22πnk02c(k02)2
δ=2Vχεpnk02c(k02)2t2o2t1o1+r2o2
(14)

4. Results

Figures 2a, 2b and 2c are plots of A(2)2 against τn1vd for q = 2, Δ = 2φl - φp = 0; q = 2 and Δφ = π and q = 1, Δφ = π respectively. These results agree qualitatively with the experimental results of Lu and Ou in Fig. 3 of their paper [6

6. Y. J. Lu and Z. Y. Ou, “Observation of nonclassical photon statistics due to quantum interference,” Phys. Rev. Lett. 88, 023601-1–023601-4 (2002).

]. We confirm with our model photon antibunching with q = 1, Δφ = π at zero time delay (Fig. 2c) and at non-zero time delays with q = 2, Δφ = π (Fig. 2b).

Fig. 2. Normalized two-photon count rates (A(2)2) against τn1vd based on Eq. (7) showing antibunching effects. (a) q = 2 and Δφ = 0, (b) q = 2 and Δφ = π and (c) q = 1 and Δφ = π.

In Section 3 of our paper we examined the spectrum of squeezing. We found that for the degenerate frequency, the spectrum was given by (12). In the RHS of (13) the first term represents the shot noise, the second is the noise contribution due to the local oscillator and the third is a noise term due to quantum interference. It is clear that noise reduction or squeezing will occur when Δφ = 0 and δ > μ. Based on the results given in Fig. 2, squeezing is then only obtained for highly bunched photon pairs. Using the expressions given in (14) we find that for a 4 mm OPO cavity, degenerate wavelength 700 nm, output mirror transmittivity of 1%, crystal refractive index 1.5, irradiated crystal volume 10-9 mm3, squeezing is obtained when εpεl2>3×106χ , where χ is the nonlinear susceptibility of the crystal. This can certainly be achieved with current technology and therefore such squeezing should be measurable.

5. Conclusion

We presented, from a first principles calculation, a fully quantized multimode theory to describe interference effects of a sub-threshold optical parametric oscillator with a local oscillator. We provided theoretical calculations which describe adequately recent experimental results [6

6. Y. J. Lu and Z. Y. Ou, “Observation of nonclassical photon statistics due to quantum interference,” Phys. Rev. Lett. 88, 023601-1–023601-4 (2002).

]. In addition we investigated the spectrum of squeezing in the presence of interference and found that a well-defined squeezing threshold exists for zero relative phase difference between the pump and local oscillator and for specific pump and local oscillator intensities given in terms of their respective amplitudes by εpεl2>3×106χ .

References and links

1.

G. Brassard, N. Lutkenhaus, T. Mor, and B. Sanders, “Limitations on practical quantum cryptography,” Phys. Rev. Lett. 85, 1330–1333 (2000). [CrossRef] [PubMed]

2.

A. Kiraz, S. Falth, C. Bechner, B. Gayral, W. V. Schoenfeld, P. M. Petroff, Lidong Zhang, E. Hu, and A. Imamoglu, “Photon correlation spectroscopy of a single quantum dot,” Phys. Rev. B 65, 161303–161304 (2002). [CrossRef]

3.

C. Santori, M. Pelton, G. Solomon, Y. Dale, and Y. Yamamoto, “Triggered single photons from a quantum dot,” Phys. Rev. Lett. 86, 1502–1505 (2001). [CrossRef] [PubMed]

4.

R. Brouri, A. Beveratos, J. Poizat, and P. Grangier, “Photon antibunching in the fluorescence of individual color centers in diamond,” Opt. Lett. 25, 1294–1296 (2000). [CrossRef]

5.

L. Fleury, J. Segura, G. Zumofen, B. Hecht, and U. P. Wild, “Nonclassical photon statistics in single-molecule fluorescence at room temperature,” Phys. Rev. Lett. 84, 1148–1151 (2000). [CrossRef] [PubMed]

6.

Y. J. Lu and Z. Y. Ou, “Observation of nonclassical photon statistics due to quantum interference,” Phys. Rev. Lett. 88, 023601-1–023601-4 (2002).

7.

R. Andrews, E. R. Pike, and Sarben Sarkar, “Photon correlations of a sub-threshold optical parametric oscillator,” Opt. Express 10, 461–468 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-11-461. [CrossRef] [PubMed]

8.

A. Gatti and L. Lugiato, “Quantum images and critical fluctuations in the optical parametric oscillator,” Phys. Rev. A 52, 1675–1690 (1995). [CrossRef] [PubMed]

9.

F. De Martini, M. Marrocco, P. Mataloni, L. Crescentini, and R. Loudon, “Spontaneous emission in the optical microscopic cavity,” Phys. Rev. A 43, 2480–2497 (1991). [CrossRef] [PubMed]

OCIS Codes
(190.4410) Nonlinear optics : Nonlinear optics, parametric processes
(270.5290) Quantum optics : Photon statistics

ToC Category:
Research Papers

History
Original Manuscript: November 12, 2002
Revised Manuscript: December 11, 2002
Published: January 13, 2003

Citation
Roger Andrews, E. Pike, and Sarben Sarkar, "Theory of photon statistics and squeezing in quantum interference of a sub-threshold parametric oscillator," Opt. Express 11, 7-13 (2003)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-1-7


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References

  1. G. Brassard, N. Lutkenhaus, T. Mor, and B. Sanders, �??Limitations on practical quantum cryptography,�?? Phys. Rev. Lett. 85, 1330-1333 (2000). [CrossRef] [PubMed]
  2. A. Kiraz, S. Falth, C. Bechner , B. Gayral, W. V. Schoenfeld, P. M. Petroff, Lidong Zhang, E. Hu and A. Imamoglu, �??Photon correlation spectroscopy of a single quantum dot,�?? Phys. Rev. B 65, 161303-161304 (2002). [CrossRef]
  3. C. Santori, M. Pelton, G. Solomon, Y. Dale, and Y. Yamamoto, �??Triggered single photons from a quantum dot,�?? Phys. Rev. Lett. 86, 1502-1505 (2001). [CrossRef] [PubMed]
  4. R. Brouri, A. Beveratos, J. Poizat, P. Grangier, �??Photon antibunching in the fluorescence of individual color centers in diamond,�?? Opt. Lett. 25, 1294-1296 (2000). [CrossRef]
  5. L. Fleury, J. Segura, G. Zumofen, B. Hecht and U. P. Wild, �??Nonclassical photon statistics in single-molecule fluorescence at room temperature,�?? Phys. Rev. Lett. 84, 1148-1151 (2000). [CrossRef] [PubMed]
  6. Y. J. Lu and Z. Y. Ou, �??Observation of nonclassical photon statistics due to quantum interference,�?? Phys. Rev. Lett. 88, 023601-1-023601-4 (2002).
  7. R. Andrews, E. R. Pike and Sarben Sarkar, �??Photon correlations of a sub-threshold optical parametric oscillator,�?? Opt. Express 10, 461-468 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-11-461">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-11-461</a>. [CrossRef] [PubMed]
  8. A. Gatti and L. Lugiato, �??Quantum images and critical fluctuations in the optical parametric oscillator,�?? Phys. Rev. A 52, 1675-1690 (1995). [CrossRef] [PubMed]
  9. F. De Martini, M. Marrocco , P. Mataloni, L. Crescentini and R. Loudon, �??�??Spontaneous emission in the optical microscopic cavity,�??�?? Phys. Rev. A 43, 2480-2497 (1991). [CrossRef] [PubMed]

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