## Sampling quantum phase space with squeezed states

Optics Express, Vol. 3, Issue 4, pp. 141-146 (1998)

http://dx.doi.org/10.1364/OE.3.000141

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### Abstract

We study the application of squeezed states in a quantum optical scheme for direct sampling of the phase space by photon counting. We prove that the detection setup with a squeezed coherent probe field is equivalent to the probing of the squeezed signal field with a coherent state. An example of the SchrÖdinger cat state measurement shows that the use of squeezed states allows one to detect clearly the interference between distinct phase space components despite losses through the unused output port of the setup.

© Optical Society of America

## 1. Introduction

1. K. E. Cahill and R. J. Glauber, “Density operators and quasiprobability distributions,” Phys. Rev. **177**, 1882–1902 (1969). [CrossRef]

2. D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum,” Phys. Rev. Lett. **70**, 1244–1247 (1993). [CrossRef] [PubMed]

3. S. Wallentowitz and W. Vogel, “Unbalanced homodyning for quantum state measurements,” Phys. Rev. A **53**, 4528–4533 (1996). [CrossRef] [PubMed]

4. K. Banaszek and K. Wódkiewicz, “Direct sampling of quantum phase space by photon counting,” Phys. Rev. Lett. **76**, 4344–4347 (1996). [CrossRef] [PubMed]

5. R. Loudon and P. L. Knight, “Squeezed light,” J. Mod. Opt. **34**, 709–759 (1987). [CrossRef]

6. U. Leonhardt and H. Paul, “High-accuracy optical homodyne detection with low-efficiency detectors: ‘Preamplification’ from antisqueezing,” Phys. Rev. Lett. **72**, 4086–4089 (1994). [CrossRef] [PubMed]

7. M. S. Kim and B. C. Sanders, “Squeezing and antisqueezing in homodyne measurements,” Phys. Rev. A **53**, 3694–3697 (1996). [CrossRef] [PubMed]

## 2. Experimental scheme

*T*, of a transmitted signal mode and a reflected probe mode. The statistics of the detector counts {

*p*

_{n}} is used to calculate an alternating series

^{n}

*p*

_{n}. In terms of the outgoing mode, this series is given by the expectation value of the parity operator:

*â*

_{out}is a linear combination of the signal and the probe field operators:

4. K. Banaszek and K. Wódkiewicz, “Direct sampling of quantum phase space by photon counting,” Phys. Rev. Lett. **76**, 4344–4347 (1996). [CrossRef] [PubMed]

*â*

_{p}|

*α*〉

_{p}=

*α*|

*α*〉

_{p}. The quantum expectation value over the probe mode can be easily evaluated in this case using the normally ordered form given in Eq. (3). Thus the measured observable is given by the following operator acting in the Hilbert space of the signal mode:

8. U. Leonhardt and H. Paul, “Realistic optical homodyne measurements and quasidistribution functions,” Phys. Rev. A **48**, 4598–4604 (1993). [CrossRef] [PubMed]

9. K. Banaszek and K. Wódkiewicz, “Operational theory of homodyne detection,” Phys. Rev. A **55**, 3117–3123 (1997). [CrossRef]

*T*→ 1, when the complete signal field is detected, we measure directly the Wigner function, corresponding to the symmetric ordering.

## 3. Sampling with squeezed states

*S*

_{p}(

*r*,

*φ*)|

*α*〉

_{p}enters through the probe port of the beam splitter. We use the following definition of the squeezing operator for an

*i*th mode:

*r*. This change of the sign swaps the field quadratures that get squeezed or antisqueezed under the squeezing transformation.

*s*= -(1 -

*T*)/

*T*-ordered quasidistribution function at a phase space point

*squeezed*signal field.

## 4. Detection of SchrÖdinger cat state

10. W. Schleich, M. Pernigo, and F. LeKien, “Nonclassical state from two pseudoclassical states,” Phys. Rev. A **44**, 2172–2187 (1991). [CrossRef] [PubMed]

*κ*is a real parameter. The Wigner function of such a state contains, in addition to two positive peaks corresponding to the coherent states, an oscillating term originating from quantum interference between the classical-like components. This nonclassical feature is extremely fragile, and disappears very quickly in the presence of dissipation [11

11. V. Bužek and P. L. Knight, “Quantum interference, superposition states of light, and nonclassical effects,” in *Progress in Optics XXXIV*, ed. by E. Wolf (north-Holland, Amsterdam, 1995), 1–158. [CrossRef]

*α*〉

_{p}and squeezed coherent states

*Ŝ*

_{P}(

*r*= 1,0)|

*α*〉

_{p}. The beam splitter transmission is

*T*= 80%. When coherent states are used, only a faint trace of the oscillatory pattern can be noticed due to losses of the signal field. In contrast, probing of the SchrÖdinger cat state with suitably chosen squeezed states yields a clear picture of quantum coherence between distinct phase space components. This effect is particularly surprising if we realize that 20% of the signal field power is lost through the unused output port of the beam splitter.

*r*. This can be most easily understood using the Wigner phase space description of the discussed scheme [4

4. K. Banaszek and K. Wódkiewicz, “Direct sampling of quantum phase space by photon counting,” Phys. Rev. Lett. **76**, 4344–4347 (1996). [CrossRef] [PubMed]

## 5. Conclusions

## Acknowledgements

## References

1. | K. E. Cahill and R. J. Glauber, “Density operators and quasiprobability distributions,” Phys. Rev. |

2. | D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum,” Phys. Rev. Lett. |

3. | S. Wallentowitz and W. Vogel, “Unbalanced homodyning for quantum state measurements,” Phys. Rev. A |

4. | K. Banaszek and K. Wódkiewicz, “Direct sampling of quantum phase space by photon counting,” Phys. Rev. Lett. |

5. | R. Loudon and P. L. Knight, “Squeezed light,” J. Mod. Opt. |

6. | U. Leonhardt and H. Paul, “High-accuracy optical homodyne detection with low-efficiency detectors: ‘Preamplification’ from antisqueezing,” Phys. Rev. Lett. |

7. | M. S. Kim and B. C. Sanders, “Squeezing and antisqueezing in homodyne measurements,” Phys. Rev. A |

8. | U. Leonhardt and H. Paul, “Realistic optical homodyne measurements and quasidistribution functions,” Phys. Rev. A |

9. | K. Banaszek and K. Wódkiewicz, “Operational theory of homodyne detection,” Phys. Rev. A |

10. | W. Schleich, M. Pernigo, and F. LeKien, “Nonclassical state from two pseudoclassical states,” Phys. Rev. A |

11. | V. Bužek and P. L. Knight, “Quantum interference, superposition states of light, and nonclassical effects,” in |

**OCIS Codes**

(270.5570) Quantum optics : Quantum detectors

(270.6570) Quantum optics : Squeezed states

**ToC Category:**

Focus Issue: Quantum noise reduction in optical systems

**History**

Original Manuscript: March 27, 1998

Published: August 17, 1998

**Citation**

Konrad Banaszek and Krzysztof Wodkiewicz, "Sampling quantum phase space with squeezed states," Opt. Express **3**, 141-146 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-3-4-141

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### References

- K. E. Cahill and R. J. Glauber, "Density operators and quasiprobability distributions," Phys. Rev. 177, 1882-1902 (1969). [CrossRef]
- D. T. Smithey, M. Beck, M. G. Raymer and A. Faridani, "Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum," Phys. Rev. Lett. 70, 1244-1247 (1993). [CrossRef] [PubMed]
- S. Wallentowitz and W. Vogel, "Unbalanced homodyning for quantum state measurements," Phys. Rev. A 53, 4528-4533 (1996). [CrossRef] [PubMed]
- K. Banaszek and K. Wodkiewicz, "Direct sampling of quantum phase space by photon counting," Phys. Rev. Lett. 76, 4344-4347 (1996). [CrossRef] [PubMed]
- R. Loudon and P. L. Knight, "Squeezed light," J. Mod. Opt. 34, 709-759 (1987). [CrossRef]
- U. Leonhardt and H. Paul, "High-accuracy optical homodyne detection with low-efficiency detectors: 'Preamplification' from antisqueezing," Phys. Rev. Lett. 72, 4086-4089 (1994). [CrossRef] [PubMed]
- M. S. Kim and B. C. Sanders, "Squeezing and antisqueezing in homodyne measurements," Phys. Rev. A 53, 3694-3697 (1996). [CrossRef] [PubMed]
- U. Leonhardt and H. Paul, "Realistic optical homodyne measurements and quasidistribution functions," Phys. Rev. A 48, 4598-4604 (1993). [CrossRef] [PubMed]
- K. Banaszek and K. Wodkiewicz, "Operational theory of homodyne detection," Phys. Rev. A 55, 3117-3123 (1997). [CrossRef]
- W. Schleich, M. Pernigo and F. LeKien, "Nonclassical state from two pseudoclassical states," Phys. Rev. A 44, 2172-2187 (1991). [CrossRef] [PubMed]
- V. Buzek and P. L. Knight, "Quantum interference, superposition states of light, and nonclassical effects," in Progress in Optics XXXIV, ed. by E. Wolf (North-Holland, Amsterdam, 1995), 1-158. [CrossRef]

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