## Producing high fidelity single photons with optimal brightness via waveguided parametric down-conversion

Optics Express, Vol. 17, Issue 25, pp. 22823-22837 (2009)

http://dx.doi.org/10.1364/OE.17.022823

Acrobat PDF (906 KB)

### Abstract

Parametric down-conversion (PDC) offers the possibility to control the fabrication of non-Gaussian states such as Fock states. However, in conventional PDC sources energy and momentum conservation introduce strict frequency and photon number correlations, which impact the fidelity of the prepared state. In our work we optimize the preparation of single-photon Fock states from the emission of waveguided PDC via spectral filtering. We study the effect of correlations via photon number resolving detection and quantum interference. Our measurements show how the reduction of mixedness due to filtering can be evaluated. Interfering the prepared photon with a coherent state we establish an experimentally measured fidelity of the produced target state of 78%.

© 2009 Optical Society of America

## 1. Introduction

1. A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. **87**, 050402 (
2001). [CrossRef] [PubMed]

5. V. Parigi, M. Kim, and M. Bellini, “Probing quantum commutation rules by addition and substraction of single photons to/from a light field,” Science **317**, 1890–1893 (
2007). [CrossRef] [PubMed]

6. Y.-H. Kim and W. P. Grice, “Measurement of the spectral properties of the two-photon state generated via type II spontaneous parametric down-conversion,” Opt. Lett. **30**, 908–910 (
2004). [CrossRef]

9. J. Chen, A. J. Pearlman, A. Ling, J. Fan, and A. Migdall, “A versatile waveguide source of photon pairs for chip-scale quantum information processing,” Opt. Express **17**, 6727–6740 (
2009). [CrossRef] [PubMed]

10. M. Hendrych, X. Shi, A. Valencia, and J. P. Torres, “Broadening the bandwidth of entangled photons: a step towards the generation of extremely short biphotons,” Phys. Rev. A **79**, 023817 (
2009). [CrossRef]

12. M. Avenhaus, M. V. Chekhova, L. A. Krivitsky, G. Leuchs, and Ch. Silberhorn, “Experimental verification of high spectral entanglement for pulsed waveguided spontaneous parametric down-conversion,” Phys. Rev. A **79**, 043836 (
2009). [CrossRef]

15. T. E. Keller and M. H. Rubin, “Theory of two-photon entanglement for spontaneous parametric down-conversion driven by a narrow pump pulse,” Phys. Rev. A **56**, 1534–1541 (
1997). [CrossRef]

16. W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A **64**, 063815 (
2001). [CrossRef]

17. P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, Ch. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. **100**, 133601 (
2008). [CrossRef] [PubMed]

21. M. Halder, J. l, B. Cemlyn, A. Clark, C. Xiong, W. J. Wadsworth, and J. G. Rarity, “Nonclassical 2-photon interference with separate intrinsically narrowband fibre sources,” Opt. Express **17**, 4670–4676 (
2009). [CrossRef] [PubMed]

*et al*. [22

22. W. Wasilewski, P. Kolenderski, and R. Frankowski, “Spectral density matrix of a single photon measured,” Phys. Lett. **99**, 123601 (
2007). [CrossRef]

23. D. Achilles, Ch. Silberhorn, and I. A. Walmsley, “Direct, loss-tolerant characterization of nonclassical photon statistics,” Phys. Rev. Lett. **97**, 043602 (
2005). [CrossRef]

24. M. Avenhaus, H. B. Coldenstrodt-Ronge, K. Laiho, W. Mauerer, I. A. Walmsley, and Ch. Silberhorn, “Photon number statistics of multimode parametric down-conversion,” Phys. Rev. Lett. **101**, 053601 (
2008). [CrossRef] [PubMed]

25. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. **59**, 2044–2046 (
1987). [CrossRef] [PubMed]

12. M. Avenhaus, M. V. Chekhova, L. A. Krivitsky, G. Leuchs, and Ch. Silberhorn, “Experimental verification of high spectral entanglement for pulsed waveguided spontaneous parametric down-conversion,” Phys. Rev. A **79**, 043836 (
2009). [CrossRef]

26. J. G. Rarity and P. R. Tapster, “Quantum interference: experiments and applications,” Phil. Trans. R. Soc. Lond. A **355**, 2267–2277 (
1997). [CrossRef]

31. K. Laiho, M. Avenhaus, K. N. Cassemiro, and Ch. Silberhorn, “Direct probing of the Wigner function by time-multiplexed detection of photon statistics,” New. J. Phys. **11**, 043012 (
2009). [CrossRef]

32. M. Fiorentino, S. M. Spillane, R. G. Beausoleil, T. D. Roberts, P. Battle, and M. W. Munro, “Spontaneous parametric down-conversion in periodically poled KTP waveguides and bulk crystals,” Opt. Express **15**, 7479–7488 (
2007). [CrossRef] [PubMed]

## 2. Properties of the spectral correlation function

*ζ*|

^{2}≪1 is the pair creation probability proportional to the pump power, and the labels

*s*,

*i*refer to signal and idler. The spectral correlation function

*ϕ*(

*ω*,

_{s}*ω*) arises as a consequence of energy and momentum conservation. The former is constrained by the spectrum of the pump pulse

_{i}*α*(

*ω*,

_{s}*ω*) and the latter, being governed by the dispersion of the nonlinear optical medium of the length

_{i}*L*, is characterized by the phase-matching (PM) function

*ω*

^{0}

_{µ}(

*µ*=

*p*,

*s*,

*i*), at which

*h*̄

*ω*

^{0}

*=*

_{p}*h*̄

*ω*

^{0}

_{s}+

*h*̄

*ω*

^{0}

*and*

_{i}*ν*=

_{µ}*ω*-

_{µ}*ω*

^{0}

*is the detuning from the central wavelength. In addition, we expand the phase-mismatch Δ*

_{µ}*k*(

*ω*,

_{s}*ω*) appearing in the PM function in first order Taylor series around

_{i}*ω*

^{0}

*and use the Gaussian approximation for the sinc-function. Thus, the PM function can be estimated by*

_{µ}*κ*=

_{µ}*k*′

*(*

_{µ}*ω*

^{0}

*)-*

_{µ}*k*′

*(*

_{p}*ω*

^{0}

*) is determined by the group-velocity mismatch of the non-linear medium and the parameter*

_{p}*γ*=0.193 adapts the width of the Gaussian to the width of the original sinc-function. Finally, the amplitude of the spectral correlation function is written as

*𝓜*and are equal to

^{2}≈0) the tilt

*θ*is given by tan (

*θ*)=

*κ*/

_{s}*κ*and the minor axis of the ellipse is purely determined by the PM properties,

_{i}*L*considering the expected material properties of our WG. Even though this theoretical prediction gives an indication about the PM properties, the actual spectral correlation function needs to be accurately characterized. This is especially important for waveguided PDC as the non-ideal properties of the guide typically influence the PM function significantly.

## 2.1. Simple estimation of the ellipse’s angle and axis

*ω*(

_{µ}*µ*=

*i*,

*s*) is the FWHM of the marginal. However, several spatial modes can propagate in theWGand an accurate determination of the tilt via this method becomes difficult [33

33. A. Christ, K. Laiho, A. Eckstein, T. Lauckner, P. J. Mosley, and Ch. Silberhorn, “Spatial modes in waveguided parametric down-conversion,” Phys. Rev. A **80**, 033829 (
2009). [CrossRef]

*θ*=54.7°±1.5°.

*ω*/cos(

_{s}*θ*)≈Δ

*ω*/sin(

_{i}*θ*). Our results indicate that pumping the WG with 1–1.5nm broad pulse leads to an approximately 35–40nm broad major axis, which increases to values between 55–60nm, if the pump is 2–2.5nm broad.

*ω*=

_{i}*ω*of the (

_{s}*ω*,

_{s}*ω*)-space, as illustrated in Fig. 3(a). In other words, tuning the frequency of the fundamental probe and measuring the respective SH response allows us to characterize the width of the PM function. If the fundamental probe is narrow with respect to the PM bandwidth σPM, and the tilt of the PM only slightly deviates from 45°, the envelope of the SH response is approximately equal to

_{i}## 3. Decorrelation of signal and idler

34. W. P. Grice and I. A. Walmsley, “Spectral information and distinguishability in type-II down-conversion with broadband pump,” Phys. Rev. A **56**, 1627–1634 (
1997). [CrossRef]

12. M. Avenhaus, M. V. Chekhova, L. A. Krivitsky, G. Leuchs, and Ch. Silberhorn, “Experimental verification of high spectral entanglement for pulsed waveguided spontaneous parametric down-conversion,” Phys. Rev. A **79**, 043836 (
2009). [CrossRef]

*θ*and aspect ratio

*𝓐*, defined as the ratio between the major and minor axes. They can be expressed as

*V*=(

*C*-

_{max}*C*)/(

_{min}*C*+

_{max}*C*), where

_{min}*C*indicates the amount of coincident counts. In terms of the overlap factor, the visibility can also be expressed as

*V*=(1+

*𝒪*)/(3-

*𝒪*). The classical case, in which signal and idler are completely distinguishable, corresponds to the situation of zero overlap, i.e.

*V*=1/3. We measured, without any background subtraction, the following visibilities:

*V*(

*I*)=0.34±0.02,

*V*(

*II*)=0.58±0.02, and

*V*(

*III*)=0.81±0.03, corresponding to each of the above mentioned cases. The corresponding overlap factors must be compared with the theoretical prediction given in Eqs (9) and (10). For this we approximate the aspect ratios in a straightforward manner. When no filtering is applied,

*𝓐*is estimated as the ratio of the experimentally extracted major and minor axes, i.e,

*𝓐*(

*I*) ~90–100. When spectral filtering is applied, we determine

*𝓐*as the ratio of the filter bandwidth to the PM width i.e.,

*𝓐*(

*II*) ~4.2 and

*𝓐*(

*III*) ~1.7. As expected, the spectral overlap alone does not explain our results, as can be seen in Fig. 6. By including the temporal overlap in the analysis, the employed model predicts the right tendency, even though in the experiment we do not have any compensation for dispersion or chirp.

## 4. Measurement of the heralded statistics

*L*(

*η*) and convolution matrices

*C*[35]. The signal statistics

*ρ*⃗ was then deduced by inverting the click statistics

*p*⃗

*=*

_{click}*CL*(

*η*)

*ρ*⃗ with a maximum-likekihood technique [36

36. K. Banaszek, “Maximum-likelihood estimation of photon-number distribution from homodyne statistics,” Phys. Rev. A **57**, 5013–5015 (
1998). [CrossRef]

16. W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A **64**, 063815 (
2001). [CrossRef]

*n*photons in signal and detecting a click from idler is given by

*P*

_{n∩click}=

*P*[1-(1-

^{PDC}_{n}*η*)

_{t}*], where*

^{n}*η*is the efficiency at which the filtered mode is detected and

_{t}*P*the probability of creating

^{PDC}_{n}*n*signal (idler) photons, which always occur in pairs for PDC. Since in the experimental realization losses are very high, the joint probability can be approximated by

*P*

_{n∩click}≈

*η*. The overall click probability is estimated as

_{t}nP^{PDC}_{n}*P*=∑

_{click}

_{n}*P*

_{n∩click}≈

*η*〈

_{t}*n*〉, where 〈

*n*〉 indicates the mean photon number of the marginal PDC statistics. In this loss regime, the heralded statistics become independent of the trigger efficiency

*η*and its mean photon number can be evaluated as 〈

_{t}*n*

_{heralded}〉=∑

_{n}*nP*

_{n|click}=〈

*n*

^{2}〉/〈

*n*〉, with 〈

*n*

^{2}〉 being the second moment of the marginal PDC statistics.

*P*is given by the convolution of the modes, each having the mean photon number 〈

^{PDC}_{n}*n*〉 and the second moment 〈

_{sm}*n*

^{2}

*〉. For*

_{sm}*M*excited modes we find that 〈

*n*〉=

*M*〈

*n*〉 and 〈

_{sm}*n*

^{2}〉=

*M*〈

*n*

^{2}

*〉+*

_{sm}*M*(

*M*-1) 〈

*n*〉

_{sm}^{2}, from which we can calculate the multimode mean photon number of the heralded statistics. Considering a thermal single mode marginal distribution,

*ρ*=|

^{sm}_{n}*χ*|

^{2n}(1-|

*χ*|

^{2}) with |

*χ*| being the gain of the PDC mode, the mean photon number of the heralded statistics can be approximated by

*χ*|

^{2}∝ pump power, the slope of the mean photon number depends on the number of modes. In Fig. 7(c) we show the mean photon number of the measured heralded statistics with respect to the pump power. By comparing the slopes, we estimate that the reduction in the number of modes due to the filter at signal is on the order of 30. We emphasize that the model used is an approximation, where all modes are treated similarly, even though the trigger mode is a special one.

## 5. Interference between independent sources

*a*and

*b*of a 50/50 beam splitter. The density matrix of the input state

*ρ*̂

*can be written as*

_{ab}*p*(

_{n}*n*=0,1,2) corresponds to the statistics of the photon-number mixed signal mode with ∑

*npn*=1. The reference field has amplitude

*β*and spectral distribution

*u*(

*ω*), such that ∫

*dω*|

*βu*(

*ω*)|

^{2}=|

*β*|

^{2}. Since mode “

*a*” is spectrally pure, its one- and two-photon components can be respectively described by |1〉a=∫

*dω*

*f*(

*ω*)

*a*̂

^{†}(

*ω*) |0〉

*and |2〉*

_{a}*a*=(1/√2)∫

*dω*∫

*dω*̃

*f*(

*ω*)

*f*(

*ω*̃)

*a*̂

^{†}(

*ω*)

*a*̂

^{†}(

*ω*̃) |0〉

*, in which the spectral distribution*

_{a}*f*(

*ω*) is normalized, i.e. ∫

*dw*|

*f*(

*ω*)|

^{2}=1.

*τ*. In the frequency domain, the beam splitter relations are given by

*c*and

*d*are the two output modes. These equations allow us to calculate the evolved state

*ρ*̂

*. Finally, the probability of a coincident click can be calculated via*

_{cd}*P*=

*T*{

_{r}*ρ*̂

_{cd}*⊗*

_{c}*, where*

_{d}*⊗*

_{c}*=(*

_{d}*𝟙*-|0〉

_{c}*〈0|)⊗(𝟙-|0〉*

_{c}*〈0|) describes the POVM of two simultaneous clicks in the APDs. Using the bosonic commutation relation [*

_{d d}*a*̂(

*ω*),

*a*̂

^{†}(

*ω*̃)]=

*δ*(

*ω*-

*ω*̃) and the fact that

*b*̂(

*ω*)|

*β*〉=

*βu*(

*ω*)|

*β*〉, the probability of detecting a coincident click is given by

*β*′=

*β*/√2. The functions

*T*(

*τ*) and

*T*′(

*τ*) have the form

*g*(

*ω*

_{1},

*ω*

_{2})=

*f*(

*ω*

_{1})

*f**(

*ω*

_{2}) and

*h*(

*ω*

_{1},

*ω*

_{2},

*ω*

_{3},

*ω*

_{4})=

*f*(

*ω*

_{1})

*f*(

*ω*

_{2})

*f**(

*ω*

_{3})

*f**(

*ω*

_{4}). At origin (

*τ*=0) these factors determine the spectral overlap of the one- and two-photon wave-packets with respect to the reference. Finally, the visibility of the interference dip is defined by

*β*|

^{2}compatible with the experimental realization, we estimate that the interference of the one-photon component would have a visibility of above 98%, whereas the one of the two-photon component would be less than 5%. Therefore, in the regime of power considered here, the two-photon component of signal gives only a background contribution to the coincidences. Thus, we consider the spectral degree of freedom exclusively for the one-photon component of signal and reference. Further, we approximate

*a*” plus “

*b*”). The remaining contributions are shown in Fig. 8 and they can be retrieved from Eq. (15) as follows. The first term corresponds to accidental counts due to the two-photon component of the reference and it can be re-expressed as

*p*

_{1}

*T*(

*τ*)|

*β*′|

^{2}. The fourth term takes into account accidental counts coming from signal,

*β*|

^{2}as

*g*(

*ω*

_{1},

*ω*

_{2}) by a new one, in which a trace operation over the idler mode is realized. The spectral density of the PDC state after filtering is given by

*t*(

_{µ}*ω*) is the intensity transmission profile of the signal and idler filters employed in the state generation and

*N*accounts for the normalization.

*, signal filter*

_{β}*, as well as by the ellipse’s tilt*

_{pm}*θ*. The amplitude

*T*

^{max}describes the maximal expected value of the overlap, ranging from zero to one. We emphasize that the two- and three-fold interference dips have the same temporal width.

*𝓥*

^{max}

_{2-fold}=0.46 and

*𝓥*

^{max}

_{3-fold}=0.75. In these values we disregard the spectral mismatch, i.e.

*T*=1, and consider only the degradation due to the two-photon component. Our experimental results are shown in Fig. 9, where we plot the measured visibilities with respect to the mean photon number of the reference beam |

*β*|

^{2}. The latter can be found by measuring the single-counts rate of the reference in one of the detectors while blocking the signal,

*T*from the measurement by fitting Eq. (19) to the experimental data. The obtained overlap values are

*T*

_{2-fold}=0.41±0.02 and

*T*

_{3-fold}=0.65±0.03. This clearly shows the increase in purity achieved via filtered herald, i.e the degree of spectral correlation between the twins is reduced for the selected conditioned events. Finally, the fidelity of the prepared state is given by

*T*is the spectral overlap and

*ρ*

_{1}is the one-photon component of the inverted statistics shown in Table 2. Combining the results of both measurements we find that the fidelity of the heralded state was

*𝓕*

_{3-fold}=0.78±0.03.

*β*|

^{2}

_{max}in Fig. 9. The measured visibilities were

*𝓥*

_{2-fold}=0.18±0.02 and

*𝓥*

_{3-fold}=0.48±0.03, respectively. The temporal width of the interference dips was 2.0±0.2 ps. Following Eq. (21) we theoretically study how this width changes as a function of the PM bandwidth [Fig. 10(c)]. The comparison of the measurement with our model suggests a FWHM of 0.5±0.1nm for the PM, which is in good agreement with the value extracted from SH measurements. Further exploring our knowledge about the spectral correlation function, we use Eqs. (16) and (20) in order to estimate the maximal spectral overlap

*T*

^{max}in the two- and three-fold cases. In Fig. 10(c) we illustrate the theoretical predictions and compare them with the measured overlap values. In the three-fold case we find a good agreement with the model. However, the measured value of the two-fold overlap deviates largely from the expected one. As no selection of the detection events by triggering is applied in the two-fold case, we believe that the properties of non-ideal WG, such as fluorescence and higher order spatial modes with complex spectral structure, degrade the result. In addition to this, the expected values have not been achieved due to other experimental imperfections, in particular mechanical instabilities and spatial mode mismatch.

## 6. Conclusions

## Acknowledgments

## References and links

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**OCIS Codes**

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

(270.1670) Quantum optics : Coherent optical effects

(270.5290) Quantum optics : Photon statistics

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: September 30, 2009

Revised Manuscript: October 23, 2009

Manuscript Accepted: November 21, 2009

Published: November 30, 2009

**Citation**

K. Laiho, K. N. Cassemiro, and Ch. Silberhorn, "Producing high fidelity single photons with optimal brightness via waveguided parametric down-conversion," Opt. Express **17**, 22823-22837 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-25-22823

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