## Generating superposition of up-to three photons for continuous variable quantum information processing |

Optics Express, Vol. 21, Issue 5, pp. 5529-5535 (2013)

http://dx.doi.org/10.1364/OE.21.005529

Acrobat PDF (1045 KB)

### Abstract

We develop an experimental scheme based on a continuous-wave (cw) laser for generating arbitrary superpositions of photon number states. In this experiment, we successfully generate superposition states of zero to three photons, namely advanced versions of superpositions of two and three coherent states. They are fully compatible with developed quantum teleportation and measurement-based quantum operations with cw lasers. Due to achieved high detection efficiency, we observe, without any loss correction, multiple areas of negativity of Wigner function, which confirm strongly nonclassical nature of the generated states.

© 2013 OSA

## 1. Introduction

1. W. K. Wooters and W. H. Zurek, “A single quantum cannot be cloned,” Nature **299**, 802–803 (1982) [CrossRef] .

2. T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature **464**, 45–53 (2010) [CrossRef] [PubMed] .

3. D. Gottesman and I. L. Chuang, “Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations,” Nature **402**, 390–393 (1999) [CrossRef] .

5. J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nature Photon . **3**, 687–695 (2009) [CrossRef] .

*resource*state, simple operations, measurement, and feedforward. In this way, the task of performing a universal quantum operation is translated to the task of generating a specific quantum state. This is usually much less of an issue, especially since the state can be, in principle, prepared by probabilistic means and then stored until it is needed.

5. J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nature Photon . **3**, 687–695 (2009) [CrossRef] .

6. C. Weedbrook, S. Pirandola, R. Garcia-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys . **84**, 621–669 (2012) [CrossRef] .

7. 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] .

9. J. S. Neergaard-Nielsen, B. M. Nielsen, H. Takahashi, A. I. Vistnes, and E. S. Polzik, “High purity bright single photon source,” Opt. Express **15**(13), 7940–7949 (2007) [CrossRef] [PubMed] .

10. M. Dakna, J. Clausen, L. Knöll, and D.-G. Welsch, “Generation of arbitrary quantum states of traveling fields,” Phys. Rev. A **59**, 1658–1661 (1999) [CrossRef] .

11. J. Fiurás̆ek, R. García-Patrón, and N. J. Cerf, “Conditional generation of arbitrary single-mode quantum states of light by repeated photon subtractions,” Phys. Rev. A **72**, 033822 (2005) [CrossRef] .

12. P. Marek, R. Filip, and A. Furusawa, “Deterministic implementation of weak quantum cubic nonlinearity,” Phys. Rev. A **84**, 053802 (2011) [CrossRef] .

13. 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] .

14. 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] .

15. R. Filip, P. Marek, and U.L. Andersen, “Measurement-induced continuous-variable quantum interactions,” Phys. Rev. A **71**, 042308 (2005) [CrossRef] .

15. R. Filip, P. Marek, and U.L. Andersen, “Measurement-induced continuous-variable quantum interactions,” Phys. Rev. A **71**, 042308 (2005) [CrossRef] .

19. T.C. Ralph, A. Gilchrist, G.J. Milburn, W.J. Munro, and S. Glancy, “Quantum computation with optical coherent states,” Phys. Rev. A **68**, 042319 (2003) [CrossRef] .

20. S. Deleglise, I. Dotsenko, C. Sayrin, J. Bernu, M. Brune, J.-M. Raimond, and S. Haroche, “Reconstruction of non-classical cavity field states with snapshots of their decoherence,” Nature **455**, 510–514 (2008) [CrossRef] [PubMed] .

*n*〉 as where the characters

*s*and

*i*denote the signal and idler modes, respectively. The quantity

*q*(0 ≤

*q*< 1) depends on the pump power and the nonlinear coefficient of the nonlinear crystal. Linear optics is now used to split the idler mode into three, and to displace each of these modes

*i*

_{1},

*i*

_{2}, and

*i*

_{3}by coherent amplitudes

*β*

_{1},

*β*

_{2}, and

*β*

_{3}. The idler modes are then measured by single-photon detectors and when the three-fold coincidence occurs, the signal mode is projected into the desired superposition state. In the limit of small pump power and small displacements, we can represent the projection process by where

*a*is an annihilation operator acting on the idler mode, which represents the single-photon detection. The factor

## 2. Experimental setup

_{4}crystal as an optical nonlinear crystal. The pump beam is generated by second harmonic generation of the fundamental beam, and frequency-shifted with an acousto-optic modulator by around 600 MHz (equal to free spectral range of NOPO, Δ

*ω*). As a result, photon pairs of frequency

*ω*(signal) and

*ω*+ Δ

*ω*(idler) are obtained. The output photons are spatially separated by a split cavity whose free spectral range is 2Δ

*ω*. The photons of frequency

*ω*+ Δ

*ω*passing through the split cavity are sent to two frequency filtering cavities [21

21. K. Wakui, H. Takahashi, A. Furusawa, and M. Sasaki, “Photon subtracted squeezed states generated with periodically poled KTiOPO_{4},” Opt. Express **15**, 3568–3574 (2007) [CrossRef] [PubMed] .

22. 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] .

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

## 3. Results and discussions

*ρ*

_{33}= 0.33 plays a significant role and the whole state is fairly well contained in the three photon subspace, with higher photon numbers populated only in 10 percent of the cases. Note that

*ρ*

_{33}is equal to the fidelity of the state, which is defined as the overlap

*F*= 〈

*ψ*|

*ρ*

_{exp}|

*ψ*〉 of the experimentally generated state

*ρ*

_{exp}with the ideal state |

*ψ*〉. The two photon and one photon contributions are caused by the experimental imperfections, such as optical losses and dark counts of the photon detectors, while the presence of higher photon numbers is caused by the strong pump power, which needed to be large enough to allow for a sufficient count rate (20 counts per minute). Despite the imperfections, the Wigner function of the three photon state, also shown in Fig. 2(a), displays all the features one would expect from the three photon Fock state: it is spherically symmetrical and along any cut in the phase space it exhibits three distinctive regions of negativity.

*β*

_{3}= 0. For suitably selected parameters, this state is a good approximation of the coherent state superposition |CSS〉 ∝ |

*α*〉 − |−

*α*〉, which can play a very important role in quantum information processing [19

19. T.C. Ralph, A. Gilchrist, G.J. Milburn, W.J. Munro, and S. Glancy, “Quantum computation with optical coherent states,” Phys. Rev. A **68**, 042319 (2003) [CrossRef] .

21. K. Wakui, H. Takahashi, A. Furusawa, and M. Sasaki, “Photon subtracted squeezed states generated with periodically poled KTiOPO_{4},” Opt. Express **15**, 3568–3574 (2007) [CrossRef] [PubMed] .

24. A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating Optical Schrödinger Kittens for Quantum Information Processing,” Science **312**, 83–86 (2006) [CrossRef] [PubMed] .

25. J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a Superposition of Odd Photon Number States for Quantum Information Networks,” Phys. Rev. Lett . **97**, 083604 (2006) [CrossRef] [PubMed] .

*α*= 1.3. The fidelity with the ideal coherent state superposition was found to be

*F*= 0.6. However, the state is actually a squeezed coherent state superposition - subsequent antisqueezing could increase the amplitude to

*α*= 1.6 while simultaneously increasing the state fidelity to

*F*= 0.61. In this sense, what was actually generated was the non-Gaussian keystone for a larger coherent state superposition [26

26. D. Menzies and R. Filip, “Gaussian-optimized preparation of non-Gaussian pure states,” Phys. Rev. A **79**, 012313 (2009) [CrossRef] .

27. 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] .

*α*= 1.6. The presence of elements corresponding to Fock states 0 and 2 is again caused by losses at various stages of the experiment. By obtaining the multiple areas of negativity resulting from higher interference effects of the coherent states, we have reached a quality of state preparation previously obtained only for field in a cavity [20

20. S. Deleglise, I. Dotsenko, C. Sayrin, J. Bernu, M. Brune, J.-M. Raimond, and S. Haroche, “Reconstruction of non-classical cavity field states with snapshots of their decoherence,” Nature **455**, 510–514 (2008) [CrossRef] [PubMed] .

28. M. Hofheinz, H. Wang, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. OfConnell, D. Sank, J. Wenner, J. M. Martinis, and A. N. Cleland, “Synthesizing arbitrary quantum states in a superconducting resonator,” Nature **459**, 546–549 (2009) [CrossRef] [PubMed] .

*s*is the displacement amplitude. Such the state is a good approximation of a different kind of coherent state superposition -

*α*, but similarly to the coherent state qubit basis, there is also a completely orthogonal basis formed of superpositions of Fock states invariant to 2

*π*/3 phase space rotation:

*π*/3 symmetry, which is exactly as predicted by the theory. The fidelity with the ideal state is

*F*= 0.61. Recently, an alternative procedure of similar state preparation for a field stored in a cavity has been suggested [29

29. J. M. Raimond, P. Facchi, B. Peaudecerf, S. Pascazio, C. Sayrin, I. Dotsenko, S. Gleyzes, M. Brune, and S. Haroche, “Quantum Zeno dynamics of a field in a cavity,” Phys. Rev. A **86**, 032120 (2012) [CrossRef] .

14. 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] .

## 4. Conclusion

15. R. Filip, P. Marek, and U.L. Andersen, “Measurement-induced continuous-variable quantum interactions,” Phys. Rev. A **71**, 042308 (2005) [CrossRef] .

20. S. Deleglise, I. Dotsenko, C. Sayrin, J. Bernu, M. Brune, J.-M. Raimond, and S. Haroche, “Reconstruction of non-classical cavity field states with snapshots of their decoherence,” Nature **455**, 510–514 (2008) [CrossRef] [PubMed] .

## Acknowledgments

## References and links

1. | W. K. Wooters and W. H. Zurek, “A single quantum cannot be cloned,” Nature |

2. | T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature |

3. | D. Gottesman and I. L. Chuang, “Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations,” Nature |

4. | S. D. Bartlett and W. J. Munro, “Quantum teleportation of optical quantum gates,” Phys. Rev. Lett . |

5. | J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nature Photon . |

6. | C. Weedbrook, S. Pirandola, R. Garcia-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys . |

7. | 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 . |

8. | A. Ourjoumtsev, R. Tualle-Brouri, and P. Grangier, “Quantum Homodyne Tomography of a Two-Photon Fock State,” Phys. Rev. Lett . |

9. | J. S. Neergaard-Nielsen, B. M. Nielsen, H. Takahashi, A. I. Vistnes, and E. S. Polzik, “High purity bright single photon source,” Opt. Express |

10. | M. Dakna, J. Clausen, L. Knöll, and D.-G. Welsch, “Generation of arbitrary quantum states of traveling fields,” Phys. Rev. A |

11. | J. Fiurás̆ek, R. García-Patrón, and N. J. Cerf, “Conditional generation of arbitrary single-mode quantum states of light by repeated photon subtractions,” Phys. Rev. A |

12. | P. Marek, R. Filip, and A. Furusawa, “Deterministic implementation of weak quantum cubic nonlinearity,” Phys. Rev. A |

13. | A. I. Lvovsky and J. Mlynek, “Quantum-Optical Catalysis: Generating Nonclassical States of Light by Means of Linear Optics,” Phys. Rev. Lett . |

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

15. | R. Filip, P. Marek, and U.L. Andersen, “Measurement-induced continuous-variable quantum interactions,” Phys. Rev. A |

16. | J. Yoshikawa, T. Hayashi, T. Akiyama, N. Takei, A. Huck, U. L. Andersen, and A. Furusawa, “Demonstration of deterministic and high fidelity squeezing of quantum information,” Phys. Rev. A |

17. | J. Yoshikawa, Y. Miwa, A. Huck, U. L. Andersen, P. van Loock, and A. Furusawa, “Demonstration of a Quantum Nondemolition Sum Gate,” Phys. Rev. Lett . |

18. | Y. Miwa, J. Yoshikawa, N. Iwata, M. Endo, P. Marek, R. Filip, P. van Loock, and A. Furusawa, “Unconditional conversion between quantum particles and waves,” arXiv: 1209.2804[quant-ph]. |

19. | T.C. Ralph, A. Gilchrist, G.J. Milburn, W.J. Munro, and S. Glancy, “Quantum computation with optical coherent states,” Phys. Rev. A |

20. | S. Deleglise, I. Dotsenko, C. Sayrin, J. Bernu, M. Brune, J.-M. Raimond, and S. Haroche, “Reconstruction of non-classical cavity field states with snapshots of their decoherence,” Nature |

21. | K. Wakui, H. Takahashi, A. Furusawa, and M. Sasaki, “Photon subtracted squeezed states generated with periodically poled KTiOPO |

22. | 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 . |

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

24. | A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating Optical Schrödinger Kittens for Quantum Information Processing,” Science |

25. | J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a Superposition of Odd Photon Number States for Quantum Information Networks,” Phys. Rev. Lett . |

26. | D. Menzies and R. Filip, “Gaussian-optimized preparation of non-Gaussian pure states,” Phys. Rev. A |

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

28. | M. Hofheinz, H. Wang, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. OfConnell, D. Sank, J. Wenner, J. M. Martinis, and A. N. Cleland, “Synthesizing arbitrary quantum states in a superconducting resonator,” Nature |

29. | J. M. Raimond, P. Facchi, B. Peaudecerf, S. Pascazio, C. Sayrin, I. Dotsenko, S. Gleyzes, M. Brune, and S. Haroche, “Quantum Zeno dynamics of a field in a cavity,” Phys. Rev. A |

**OCIS Codes**

(270.0270) Quantum optics : Quantum optics

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: January 23, 2013

Revised Manuscript: February 21, 2013

Manuscript Accepted: February 21, 2013

Published: February 27, 2013

**Citation**

Mitsuyoshi Yukawa, Kazunori Miyata, Takahiro Mizuta, Hidehiro Yonezawa, Petr Marek, Radim Filip, and Akira Furusawa, "Generating superposition of up-to three photons for continuous variable quantum information processing," Opt. Express **21**, 5529-5535 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-5-5529

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### References

- W. K. Wooters and W. H. Zurek, “A single quantum cannot be cloned,” Nature299, 802–803 (1982). [CrossRef]
- T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature464, 45–53 (2010). [CrossRef] [PubMed]
- D. Gottesman and I. L. Chuang, “Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations,” Nature402, 390–393 (1999). [CrossRef]
- S. D. Bartlett and W. J. Munro, “Quantum teleportation of optical quantum gates,” Phys. Rev. Lett. 90, 117901 (2003). [CrossRef] [PubMed]
- J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nature Photon. 3, 687–695 (2009). [CrossRef]
- C. Weedbrook, S. Pirandola, R. Garcia-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012). [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. Ourjoumtsev, R. Tualle-Brouri, and P. Grangier, “Quantum Homodyne Tomography of a Two-Photon Fock State,” Phys. Rev. Lett. 96, 213601 (2006). [CrossRef] [PubMed]
- J. S. Neergaard-Nielsen, B. M. Nielsen, H. Takahashi, A. I. Vistnes, and E. S. Polzik, “High purity bright single photon source,” Opt. Express15(13), 7940–7949 (2007). [CrossRef] [PubMed]
- M. Dakna, J. Clausen, L. Knöll, and D.-G. Welsch, “Generation of arbitrary quantum states of traveling fields,” Phys. Rev. A59, 1658–1661 (1999). [CrossRef]
- J. Fiurás̆ek, R. García-Patrón, and N. J. Cerf, “Conditional generation of arbitrary single-mode quantum states of light by repeated photon subtractions,” Phys. Rev. A72, 033822 (2005). [CrossRef]
- P. Marek, R. Filip, and A. Furusawa, “Deterministic implementation of weak quantum cubic nonlinearity,” Phys. Rev. A84, 053802 (2011). [CrossRef]
- 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]
- 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]
- R. Filip, P. Marek, and U.L. Andersen, “Measurement-induced continuous-variable quantum interactions,” Phys. Rev. A71, 042308 (2005). [CrossRef]
- J. Yoshikawa, T. Hayashi, T. Akiyama, N. Takei, A. Huck, U. L. Andersen, and A. Furusawa, “Demonstration of deterministic and high fidelity squeezing of quantum information,” Phys. Rev. A76, 060301(R)(2007). [CrossRef]
- J. Yoshikawa, Y. Miwa, A. Huck, U. L. Andersen, P. van Loock, and A. Furusawa, “Demonstration of a Quantum Nondemolition Sum Gate,” Phys. Rev. Lett. 101, 250501 (2008). [CrossRef] [PubMed]
- Y. Miwa, J. Yoshikawa, N. Iwata, M. Endo, P. Marek, R. Filip, P. van Loock, and A. Furusawa, “Unconditional conversion between quantum particles and waves,” arXiv: 1209.2804[quant-ph].
- T.C. Ralph, A. Gilchrist, G.J. Milburn, W.J. Munro, and S. Glancy, “Quantum computation with optical coherent states,” Phys. Rev. A68, 042319 (2003). [CrossRef]
- S. Deleglise, I. Dotsenko, C. Sayrin, J. Bernu, M. Brune, J.-M. Raimond, and S. Haroche, “Reconstruction of non-classical cavity field states with snapshots of their decoherence,” Nature455, 510–514 (2008). [CrossRef] [PubMed]
- K. Wakui, H. Takahashi, A. Furusawa, and M. Sasaki, “Photon subtracted squeezed states generated with periodically poled KTiOPO4,” Opt. Express15, 3568–3574 (2007). [CrossRef] [PubMed]
- 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]
- A. I. Lvovsky, “Iterative maximum-likelihood reconstruction in quantum homodyne tomography,” J. Opt. B6, S556–S559 (2004). [CrossRef]
- A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating Optical Schrödinger Kittens for Quantum Information Processing,” Science312, 83–86 (2006). [CrossRef] [PubMed]
- J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a Superposition of Odd Photon Number States for Quantum Information Networks,” Phys. Rev. Lett. 97, 083604 (2006). [CrossRef] [PubMed]
- D. Menzies and R. Filip, “Gaussian-optimized preparation of non-Gaussian pure states,” Phys. Rev. A79, 012313 (2009). [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]
- M. Hofheinz, H. Wang, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. OfConnell, D. Sank, J. Wenner, J. M. Martinis, and A. N. Cleland, “Synthesizing arbitrary quantum states in a superconducting resonator,” Nature459, 546–549 (2009). [CrossRef] [PubMed]
- J. M. Raimond, P. Facchi, B. Peaudecerf, S. Pascazio, C. Sayrin, I. Dotsenko, S. Gleyzes, M. Brune, and S. Haroche, “Quantum Zeno dynamics of a field in a cavity,” Phys. Rev. A86, 032120 (2012). [CrossRef]

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