## Experimental generation of multi-photon Fock states |

Optics Express, Vol. 21, Issue 5, pp. 5309-5317 (2013)

http://dx.doi.org/10.1364/OE.21.005309

Acrobat PDF (1995 KB)

### Abstract

We experimentally demonstrate the generation of multi-photon Fock states with up to three photons in well-defined spatial-temporal modes synchronized with a classical clock. The states are characterized using quantum optical homodyne tomography to ensure mode selectivity. The three-photon Fock states are probabilistically generated by pulsed spontaneous parametric down conversion at a rate of one per second, enabling complete characterization in 12 hours.

© 2013 OSA

## 1. Introduction

1. M. J. Holland and K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. **71**, 1355–1358 (1993). [CrossRef] [PubMed]

8. T. J. Bartley, G. Donati, J. B. Spring, X.-M. Jin, M. Barbieri, A. Datta, B. J. Smith, and I. A. Walmsley, “Multi-photon state engineering by heralded interference between single photons and coherent states,” Phys. Rev. A **86**, 043820 (2012). [CrossRef]

9. P. van Loock, “Optical hybrid approaches to quantum information,” Laser Photon. Rev. **5**, 167–200 (2011). [CrossRef]

10. 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]

12. S. R. Huisman, N. Jain, S. A. Babichev, F. Vewinger, A. N. Zhang, S. H. Youn, and A. I. Lvovsky, “Instant single-photon fock state tomography,” Opt. Lett. **34**, 2739–2741 (2009). [CrossRef] [PubMed]

13. A. Ourjoumtsev, R. Tualle-Brouri, and P. Grangier, “Quantum homodyne tomography of a two-photon Fock state,” Phys. Rev. Lett. **96**, 213601 (2006). [CrossRef] [PubMed]

14. A. Zavatta, V. Parigi, and M. Bellini, “Toward quantum frequency combs: boosting the generation of highly nonclassical light states by cavity-enhanced parametric down-conversion at high repetition rates,” Phys. Rev. A **78**, 033809 (2008). [CrossRef]

15. K. J. Resch, J. S. Lundeen, and A. M. Steinberg, “Quantum state preparation and conditional coherence,” Phys. Rev. Lett. **88**, 113601 (2002). [CrossRef] [PubMed]

16. A. I. Lvovsky and J. Mlynek, “Quantum-optical catalysis: generating nonclassical states of light by means of linear optics,” Phys. Rev. Lett. **88**, 250401 (2002). [CrossRef] [PubMed]

7. E. Bimbard, N. Jain, A. MacRae, and A. I. Lvovsky, “Quantum-optical state engineering up to the two-photon level,” Nat. Photonics **4**, 243–247 (2010). [CrossRef]

17. A. M. Lance, H. Jeong, N. B. Grosse, T. Symul, T. C. Ralph, and P. K. Lam, “Quantum-state engineering with continuous-variable postselection,” Phys. Rev. A **73**, 041801 (2006). [CrossRef]

18. A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical Schrödinger cats from photon number states,” Nature **448**, 784–786 (2007). [CrossRef] [PubMed]

*n*-photon Fock states and continuous-variable post-selection exemplify the importance of reaching into higher-order Fock layers for quantum state engineering.

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

4. M. S. Kim, W. Son, V. Bužek, and P. L. Knight, “Entanglement by a beam splitter: Nonclassicality as a prerequisite for entanglement,” Phys. Rev. A **65**, 032323 (2002). [CrossRef]

5. J. K. Asbóth, J. Calsamiglia, and H. Ritsch, “Computable measure of nonclassicality for light,” Phys. Rev. Lett. **94**, 173602 (2005). [CrossRef] [PubMed]

7. E. Bimbard, N. Jain, A. MacRae, and A. I. Lvovsky, “Quantum-optical state engineering up to the two-photon level,” Nat. Photonics **4**, 243–247 (2010). [CrossRef]

8. T. J. Bartley, G. Donati, J. B. Spring, X.-M. Jin, M. Barbieri, A. Datta, B. J. Smith, and I. A. Walmsley, “Multi-photon state engineering by heralded interference between single photons and coherent states,” Phys. Rev. A **86**, 043820 (2012). [CrossRef]

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

## 2. Experimental setup

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

**100**, 133601 (2008). [CrossRef] [PubMed]

*λ*is the squeezing parameter for the particular optical modes collected, and |

*n*,

*m*〉 is a two-mode state with

*n*(

*m*) photons in the signal (trigger) mode. The intrinsic photon-number correlations between the two modes, in our case distinct polarization modes, enable the preparation of Fock states in one mode, which we shall call the signal, upon projection onto photon number in the conjugate mode, which we shall call the trigger mode.

*= |*

_{n}*n*〉〈

*n*|}. Since such a detector is not readily available we instead use a pseudo-number-resolving detector by employing a spatially-multiplexed detector (SMD) comprising an array of three APDs (Perkin-Elmer SPCM-AQ4C) and two 50:50 fiber beam splitters (FBS). An incoming wave packet in the trigger mode is probabilistically split between the three APDs. The signal mode is then prepared in a one-, two-, or three-photon Fock state conditioned on obtaining exactly one, two, or three clicks respectively from the SMD.

## 3. Fock-state tomography

21. D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, and I. A. Walmsley, “Fiber-assisted detection with photon number resolution,” Opt. Lett. **28**, 2387–2389 (2003). [CrossRef] [PubMed]

22. R. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics **3**, 696–705 (2009). [CrossRef]

*η*

_{mm}=

*V*

^{2}, where

*V*is the classical interference visibility [23

23. T. Aichele, A. Lvovsky, and S. Schiller, “Optical mode characterization of single photons prepared by means of conditional measurements on a biphoton state,” EPJ. D **18**, 237–245 (2002). [CrossRef]

*V*. In order to achieve the initial overlap between LO and signal we are able to seed the down-converter with a pick-off from the original 830 nm oscillator beam, shown as the dashed red line in Fig. 1. When this seed beam is properly overlapped with the blue pump inside the KDP, difference frequency generation (DFG) results allowing access to a bright alignment beam in a mode approximately the same as the conditionally prepared Fock states [23

23. T. Aichele, A. Lvovsky, and S. Schiller, “Optical mode characterization of single photons prepared by means of conditional measurements on a biphoton state,” EPJ. D **18**, 237–245 (2002). [CrossRef]

24. C. Kim, R.-D. Li, and P. Kumar, “Deamplification response of a traveling-wave phase-sensitive optical parametric amplifier,” Opt. Lett. **19**, 132–134 (1994). [CrossRef] [PubMed]

*η*

_{apd}is the APD efficiency,

*R*

_{C}is the raw coincidence rate and

*R*

_{trigger}is the raw singles rate in the trigger mode. By ensuring near-perfect coupling of the LO into this overlap fiber, the measured heralding efficiency then serves as a direct measure of the degree of spatial overlap between the LO and signal-mode photon. We commonly achieve a heralding efficiency in excess of

*η*

_{he}= 0.65 after correcting for

*η*

_{apd}= 0.45.

^{−1}were routinely observed, thus allowing real-time computer-controlled optimization of parameters such as the temporal overlap of the signal and LO. In general the reconstructed state, neglecting higher-order terms, will be an admixture of vacuum and single-photon components where

*η*is the overall homodyne efficiency for the measurement. This comprises four main contributions: 1) the detector efficiency

*η*

_{bhd}, 2) the mode-matching between LO and the conditionally-prepared single photon

*η*

_{mm}, 3) the non-unit purity of the heralded state

26. F. Grosshans and P. Grangier, “Effective quantum efficiency in the pulsed homodyne detection of a n-photon state,” EPJ. D **14**, 119–125 (2001). [CrossRef]

*η*

_{dc}.

*η*

_{bhd}was determined to be 0.85 by performing tomography of well-calibrated coherent state probes.

*η*

_{mm}is estimated to be 0.66 based on the measured heralding efficiency and spectral overlap between LO and the heralded state and the JSI measurement gives an estimate of

^{−1}on a trigger rate of 180, 000 s

^{−1}. This gives an overall estimate of

*η*

_{est}=

*η*

_{bhd}

*η*

_{mm}

*η*

_{p}

*η*

_{dc}= 0.54.

27. A. I. Lvovsky, “Iterative maximum-likelihood reconstruction in quantum homodyne tomography,” J. Opt. B: Quantum Semiclass. Opt. **6**, S556–S559 (2004). [CrossRef]

*η*= 0.545. This value is clearly in excellent agreement with the estimated value of

*η*

_{est}= 0.54. The marginal distribution

*P*(

*X*), photon number statistics

*P*(

*n*) and Wigner function

*W*(

*X,P*) of the detected state, conditioned on the one-click event are shown in Fig. 3(a–c). The single-photon Wigner function, which is corrected for the detector efficiency

*η*

_{bhd}= 0.85, is strongly negative at the origin of phase space,

*W*(0, 0) = −0.095.

^{−1}(1 s

^{−1}) for the two- (three-) photon state, thus allowing acquisition of a sufficient data in about 5 minutes (12 hours). When performing tomography of the three-photon state we utilized the maximum available 415 nm pump power of 800 mW, whereas only a fraction was used for producing the one- and two-photon states in order to minimize higher-order terms. The squeezing parameter

*λ*in Eq. (1) is estimated to be 0.071 during the one- and two-photon state experiments and 0.087 for the three-photon experiment, when utilizing the full pump power.

10. 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]

*P*(

*X*) is consistent with the predicted distribution, shown as the red curve in Fig. 3(d), which we determine by calculating the marginal distribution of a pure two-photon state subject to preparation with overall efficiency

*η*= 0.545. Furthermore, the two-photon state Wigner function exhibits negativity after correction for the detector efficiency

*η*

_{bhd}= 0.85.

*η*= 0.545, which was determined from the single-photon reconstruction. The assumption made is that the overall detection efficiency due to mode-matching and the detector efficiency is the same for the single-photon and three-photon state. It is evident that the measured quadrature statistics of the pulse containing the heralded three-photon state correspond well with the predicted marginal, with small errors introduced most likely due to the limited digitizer resolution of 8-bits. The error bars are calculated based on the expected uncertainty,

*N*events. Thus the bin size is chosen so as to show enough detail in the marginal distribution whilst ensuring statistical errors are not overly significant.

*η*

_{bhd}= 0.85. Fig. 4(b) shows a plot of the reconstructed photon number statistics

*P*(

*n*) for both the predicted state (green) and the reconstructed state (red). We find an excellent correspondence between the predicted and measured states and suspect the discrepancies are partly due to higher order terms in the measured state. The Wigner function of the state reconstructed from the homodyne data, after correcting for the detector efficiency, is presented in Fig. 5(a) and exhibits negativity near the origin of phase space, as seen in the cross section, Fig. 5(b). We calculated the fidelity between the predicted and reconstructed three-photon state defined as where

*ρ*̂

*and*

_{m}*ρ*̂

*are the density matrices of the measured and predicted states respectively. The fidelity is found to be 99.7% which represents a remarkable correspondence between the experimentally reconstructed thee-photon state and the predicted state, thus giving further credence to characterization of such states by homodyne tomography.*

_{p}## 4. Conclusion

## Acknowledgments

## References and links

1. | M. J. Holland and K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. |

2. | K. Banaszek and P. L. Knight, “Quantum interference in three-photon down-conversion,” Phys. Rev. A |

3. | J.-W. Pan, D. Bouwmeester, M. Daniell, H. Weinfurter, and A. Zeilinger, “Experimental test of quantum nonlocality in three-photon Greenberger-Horne-Zeilinger entanglement,” Nature |

4. | M. S. Kim, W. Son, V. Bužek, and P. L. Knight, “Entanglement by a beam splitter: Nonclassicality as a prerequisite for entanglement,” Phys. Rev. A |

5. | J. K. Asbóth, J. Calsamiglia, and H. Ritsch, “Computable measure of nonclassicality for light,” Phys. Rev. Lett. |

6. | K. Bencheikh, F. Gravier, J. Douady, A. Levenson, and B. Boulanger, “Triple photons: a challenge in nonlinear and quantum optics,” Comptes Rendus Physique |

7. | E. Bimbard, N. Jain, A. MacRae, and A. I. Lvovsky, “Quantum-optical state engineering up to the two-photon level,” Nat. Photonics |

8. | T. J. Bartley, G. Donati, J. B. Spring, X.-M. Jin, M. Barbieri, A. Datta, B. J. Smith, and I. A. Walmsley, “Multi-photon state engineering by heralded interference between single photons and coherent states,” Phys. Rev. A |

9. | P. van Loock, “Optical hybrid approaches to quantum information,” Laser Photon. Rev. |

10. | 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. |

11. | A. Zavatta, S. Viciani, and M. Bellini, “Tomographic reconstruction of the single-photon Fock state by high-frequency homodyne detection,” Phys. Rev. A |

12. | S. R. Huisman, N. Jain, S. A. Babichev, F. Vewinger, A. N. Zhang, S. H. Youn, and A. I. Lvovsky, “Instant single-photon fock state tomography,” Opt. Lett. |

13. | A. Ourjoumtsev, R. Tualle-Brouri, and P. Grangier, “Quantum homodyne tomography of a two-photon Fock state,” Phys. Rev. Lett. |

14. | A. Zavatta, V. Parigi, and M. Bellini, “Toward quantum frequency combs: boosting the generation of highly nonclassical light states by cavity-enhanced parametric down-conversion at high repetition rates,” Phys. Rev. A |

15. | K. J. Resch, J. S. Lundeen, and A. M. Steinberg, “Quantum state preparation and conditional coherence,” Phys. Rev. Lett. |

16. | A. I. Lvovsky and J. Mlynek, “Quantum-optical catalysis: generating nonclassical states of light by means of linear optics,” Phys. Rev. Lett. |

17. | A. M. Lance, H. Jeong, N. B. Grosse, T. Symul, T. C. Ralph, and P. K. Lam, “Quantum-state engineering with continuous-variable postselection,” Phys. Rev. A |

18. | A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical Schrödinger cats from photon number states,” Nature |

19. | P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. |

20. | M. Cooper, C. Söller, and B. J. Smith, “High-stability time-domain balanced homodyne detector for ultrafast optical pulse applications,” arXiv :1112.0875 [quant-ph] (2011). |

21. | D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, and I. A. Walmsley, “Fiber-assisted detection with photon number resolution,” Opt. Lett. |

22. | R. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics |

23. | T. Aichele, A. Lvovsky, and S. Schiller, “Optical mode characterization of single photons prepared by means of conditional measurements on a biphoton state,” EPJ. D |

24. | C. Kim, R.-D. Li, and P. Kumar, “Deamplification response of a traveling-wave phase-sensitive optical parametric amplifier,” Opt. Lett. |

25. | Y.-H. Kim and W. P. Grice, “Measurement of the spectral properties of the two-photon state generated via type-II spontaneous parametric downconversion,” Opt. Lett. |

26. | F. Grosshans and P. Grangier, “Effective quantum efficiency in the pulsed homodyne detection of a n-photon state,” EPJ. D |

27. | A. I. Lvovsky, “Iterative maximum-likelihood reconstruction in quantum homodyne tomography,” J. Opt. B: Quantum Semiclass. Opt. |

**OCIS Codes**

(270.0270) Quantum optics : Quantum optics

(270.5290) Quantum optics : Photon statistics

(270.5570) Quantum optics : Quantum detectors

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: December 28, 2012

Revised Manuscript: February 7, 2013

Manuscript Accepted: February 7, 2013

Published: February 25, 2013

**Citation**

Merlin Cooper, Laura J. Wright, Christoph Söller, and Brian J. Smith, "Experimental generation of multi-photon Fock states," Opt. Express **21**, 5309-5317 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-5-5309

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### References

- M. J. Holland and K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett.71, 1355–1358 (1993). [CrossRef] [PubMed]
- K. Banaszek and P. L. Knight, “Quantum interference in three-photon down-conversion,” Phys. Rev. A55, 2368–2375 (1997). [CrossRef]
- J.-W. Pan, D. Bouwmeester, M. Daniell, H. Weinfurter, and A. Zeilinger, “Experimental test of quantum nonlocality in three-photon Greenberger-Horne-Zeilinger entanglement,” Nature403, 515–519 (2000). [CrossRef] [PubMed]
- M. S. Kim, W. Son, V. Bužek, and P. L. Knight, “Entanglement by a beam splitter: Nonclassicality as a prerequisite for entanglement,” Phys. Rev. A65, 032323 (2002). [CrossRef]
- J. K. Asbóth, J. Calsamiglia, and H. Ritsch, “Computable measure of nonclassicality for light,” Phys. Rev. Lett.94, 173602 (2005). [CrossRef] [PubMed]
- K. Bencheikh, F. Gravier, J. Douady, A. Levenson, and B. Boulanger, “Triple photons: a challenge in nonlinear and quantum optics,” Comptes Rendus Physique8, 206–220 (2007). [CrossRef]
- E. Bimbard, N. Jain, A. MacRae, and A. I. Lvovsky, “Quantum-optical state engineering up to the two-photon level,” Nat. Photonics4, 243–247 (2010). [CrossRef]
- T. J. Bartley, G. Donati, J. B. Spring, X.-M. Jin, M. Barbieri, A. Datta, B. J. Smith, and I. A. Walmsley, “Multi-photon state engineering by heralded interference between single photons and coherent states,” Phys. Rev. A86, 043820 (2012). [CrossRef]
- P. van Loock, “Optical hybrid approaches to quantum information,” Laser Photon. Rev.5, 167–200 (2011). [CrossRef]
- 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]
- A. Zavatta, S. Viciani, and M. Bellini, “Tomographic reconstruction of the single-photon Fock state by high-frequency homodyne detection,” Phys. Rev. A70, 053821 (2004). [CrossRef]
- S. R. Huisman, N. Jain, S. A. Babichev, F. Vewinger, A. N. Zhang, S. H. Youn, and A. I. Lvovsky, “Instant single-photon fock state tomography,” Opt. Lett.34, 2739–2741 (2009). [CrossRef] [PubMed]
- A. Ourjoumtsev, R. Tualle-Brouri, and P. Grangier, “Quantum homodyne tomography of a two-photon Fock state,” Phys. Rev. Lett.96, 213601 (2006). [CrossRef] [PubMed]
- A. Zavatta, V. Parigi, and M. Bellini, “Toward quantum frequency combs: boosting the generation of highly nonclassical light states by cavity-enhanced parametric down-conversion at high repetition rates,” Phys. Rev. A78, 033809 (2008). [CrossRef]
- K. J. Resch, J. S. Lundeen, and A. M. Steinberg, “Quantum state preparation and conditional coherence,” Phys. Rev. Lett.88, 113601 (2002). [CrossRef] [PubMed]
- A. I. Lvovsky and J. Mlynek, “Quantum-optical catalysis: generating nonclassical states of light by means of linear optics,” Phys. Rev. Lett.88, 250401 (2002). [CrossRef] [PubMed]
- A. M. Lance, H. Jeong, N. B. Grosse, T. Symul, T. C. Ralph, and P. K. Lam, “Quantum-state engineering with continuous-variable postselection,” Phys. Rev. A73, 041801 (2006). [CrossRef]
- A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical Schrödinger cats from photon number states,” Nature448, 784–786 (2007). [CrossRef] [PubMed]
- P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett.100, 133601 (2008). [CrossRef] [PubMed]
- M. Cooper, C. Söller, and B. J. Smith, “High-stability time-domain balanced homodyne detector for ultrafast optical pulse applications,” arXiv :1112.0875 [quant-ph] (2011).
- D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, and I. A. Walmsley, “Fiber-assisted detection with photon number resolution,” Opt. Lett.28, 2387–2389 (2003). [CrossRef] [PubMed]
- R. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics3, 696–705 (2009). [CrossRef]
- T. Aichele, A. Lvovsky, and S. Schiller, “Optical mode characterization of single photons prepared by means of conditional measurements on a biphoton state,” EPJ. D18, 237–245 (2002). [CrossRef]
- C. Kim, R.-D. Li, and P. Kumar, “Deamplification response of a traveling-wave phase-sensitive optical parametric amplifier,” Opt. Lett.19, 132–134 (1994). [CrossRef] [PubMed]
- Y.-H. Kim and W. P. Grice, “Measurement of the spectral properties of the two-photon state generated via type-II spontaneous parametric downconversion,” Opt. Lett.30, 908–910 (2005). [CrossRef] [PubMed]
- F. Grosshans and P. Grangier, “Effective quantum efficiency in the pulsed homodyne detection of a n-photon state,” EPJ. D14, 119–125 (2001). [CrossRef]
- A. I. Lvovsky, “Iterative maximum-likelihood reconstruction in quantum homodyne tomography,” J. Opt. B: Quantum Semiclass. Opt. 6, S556–S559 (2004). [CrossRef]

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