## Enhancing entangled-state phase estimation by combining classical and quantum protocols |

Optics Express, Vol. 21, Issue 3, pp. 2816-2822 (2013)

http://dx.doi.org/10.1364/OE.21.002816

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### Abstract

Here we describe a laboratory procedure by which we have increased the resolution of a measurement of the position of an optical component by a factor of 16. The factor of 16 arises from a four-fold quantum enhancement through the use of an *N* = 4 *N*00*N* state and a four-fold classical enhancement from a quadruple pass through a prism pair. The possibility of achieving supersensitivity using this method is discussed.

© 2013 OSA

## 1. Introduction

*n*is the the mean number of photons involved in performing the measurement. It is possible to beat this classical limit by using nonclassical light, leading to a phenomenon known as supersensitivity [1

1. V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics **5**(4), 222–229 (2011). [CrossRef]

*ϕ*= 1/

_{HL}*n*.

*N*00

*N*states of light [2

2. A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. **85**(13), 2733–2736 (2000). [CrossRef] [PubMed]

5. T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science **316**(5825), 726–729 (2007). [CrossRef] [PubMed]

*N*00

*N*states have became a key resource for quantum interferometric lithography, quantum imaging and quantum metrology. Initially it was thought that the “superresolved” nature of interference patterns obtained beyond the Rayleigh diffraction limit was due to the quantum nature of

*N*00

*N*states. However, Korobkin

*et. al.*soon demonstrated that superresolution can be achieved with classical light [6

6. D. V. Korobkin and E. Yablonovitch, “Two-fold spatial resolution enhancement by two-photon exposure of photographic film,” Opt. Eng. **41**, 1729–1732 (2002). [CrossRef]

*et. al.*showed that the phase measurement sensitivity in

*N*00

*N*state interference is strongly dependent on experimental imperfection [7

7. N. Thomas-Peter, B. Smith, A. Datta, L. Zhang, U. Dorner, and I. Walmsley, “Real-world quantum sensors: evaluating resources for precision measurement,” Phys. Rev. Lett. **107**(11), 0113603 (2011). [CrossRef]

*N*00

*N*states and real-world detector efficiencies makes the majority of the

*N*-photon detection events lost. Further, non-unit fringe visibility decreases the measurement accuracy, and the visibility can be used to identify whether a superresolved interference pattern exhibits phase supersensitivity. In practice, this imperfection of system establishes experimental boundaries which make it essentially difficult to beat the SQL with

*N*00

*N*states.

8. G. M. Gehring, H. Shin, R. Boyd, C. M. Kim, and B. S. Ham, “Tunable optical time delay of quantum signals using a prism pair,” Opt. Express **18**(18), 19156–19162 (2010). [CrossRef] [PubMed]

*N*= 4

*N*00

*N*state. The

*N*00

*N*state undergoes four passes through a prism pair, which gives us an additional four-fold enhancement of resolution. In addition, the experimental data analysis shows that our phase measurement accuracy is also increased. The precision of our phase measurement strongly depends on the system loss, such as the absorption and reflection of optics as well as imperfect detection efficiency. We discuss the phase measurement precision achieved with our method and compare it with that obtained without. This method could find applications in phase variation measurements for the detection of gravitational waves [1

1. V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics **5**(4), 222–229 (2011). [CrossRef]

9. B. C. Barish and R. Weiss, “LIGO and the detection of gravitational waves,” Phys. Today **52**(10), 44–50 (1999). [CrossRef]

10. F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica **9**(7), 686–698 (1942). [CrossRef]

11. F. Marquardt and S. Girvin, “Optomechanics,” Physics **2**, 40 (2009). [CrossRef]

## 2. Theory

*N*00

*N*states. Assume a variable phase retarder is located in mode

*A*of the interferometer. Each photon existing in this mode will experience a phase shift

*ϕ*. As the phase of the retarder is varied, a classical interference pattern is obtained in the output ports of the interferometer. The intensity of this pattern is modulated by one period for a phase shift of 2

*π*, and is described by the sinusoidal function cos

^{2}(

*ϕ*/2).

*N*00

*N*state (|

*N*〉

*|0〉*

_{A}*+ |0〉*

_{B}*|*

_{A}*N*〉

*) is input into the interferometer, the net phase acquired by the*

_{B}*N*photons results in the output state (

*e*

^{−}

*|*

^{iNϕ}*N*〉

*|0〉*

_{A}*+ |0〉*

_{B}*|*

_{A}*N*〉

*).*

_{B}*N*-photon detection at the output of the interferometer gives an interference pattern described by the sinusoidal function cos

^{2}(

*Nϕ*/2) [2

2. A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. **85**(13), 2733–2736 (2000). [CrossRef] [PubMed]

*N*. This resolution enhancement can be mimicked by using classical protocol of multiple reflections. If we introduce multiple passes in the arm of the interferometer containing the variable phase retarder, a further enhancement of the resolution can be achieved [12

12. B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman, and G. J. Pryde, “Entanglement-free Heisenberg-limited phase estimation,” Nature **450**(7168), 393–396 (2007). [CrossRef] [PubMed]

*M*times through the same variable phase retarder, they will acquire a net phase of

*Mϕ*. Furthermore, these quantum and classical protocols can be combined. If a

*N*00

*N*state is input into such an interferometer, the output state will have the form (

*e*

^{−iNMϕ}|

*N*〉

*|0〉*

_{A}*+ |0〉*

_{B}*|*

_{A}*N*〉

*), and the interference pattern will be described by the function cos*

_{B}^{2}(

*NMϕ /*2) showing that the resolution is increased by a factor of

*NM*, compared with that of single-pass classical-light interference.

5. T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science **316**(5825), 726–729 (2007). [CrossRef] [PubMed]

15. K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O’Brien, and A. G. White, “Time-reversal and super-resolving phase measurements,” Phys. Rev. Lett. **98**(22), 0223601 (2007). [CrossRef]

*N*00

*N*state generation efficiency, which is the ratio of the number of photons generated in a particular

*N*00

*N*state to the

*N*-photon interference [5

5. T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science **316**(5825), 726–729 (2007). [CrossRef] [PubMed]

*N*= 2

*N*00

*N*state, this efficiency is equal to one. However, for

*N*> 2

*N*00

*N*states, the generation efficiency is reduced to lower than one [5

**316**(5825), 726–729 (2007). [CrossRef] [PubMed]

16. R. Campos, B. Saleh, and M. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A **40**(3), 1371–1384 (1989). [CrossRef] [PubMed]

*N*= 4

*N*00

*N*state in the next section. Taking these factors into account, our uncertainty in phase is now defined as where

*V*is the visibility of the interference pattern and

*η*is the intrinsic

_{g}*N*00

*N*state generation efficiency.

*η*is the net system efficiency and is given by

_{net}*η*is the optical transmittance of the interferometer,

_{sys}*η*is the single-pass transmittance of the prism pair, and

_{pp}*η*is the detector efficiency.

_{det}## 3. Experimental setup

*A*and

*B*at a beam splitter (BS) and each beam was incident on a β-Barium Borate (BBO) crystal. Energy-time entangled photon pairs were randomly generated in each BBO crystal by the process of spontaneous parametric down conversion (SPDC) under type-I collinear phase matching conditions [17

17. H. Shin, K. W. C. Chan, H. J. Chang, and R. W. Boyd, “Quantum spatial superresolution by optical centroid measurements,” Phys. Rev. Lett. **107**, 083603 (2011). [CrossRef] [PubMed]

*A*and

*B*after the interference filters (IF) are probabilistically given by where

*ϕ*is the phase difference between two paths,

*A*and

*B*. Note that the two-photon

*N*00

*N*state has unit intrinsic efficiency, but that of the four-photon

*N*00

*N*state is 1/4 due to the unwanted |22〉

*term in equation 3. The four-photon state at stage III in Fig. 1(b) has |13〉*

_{AB}*state with the amplitude of*

_{CD}*i*(1 −

*e*

^{i4ϕ})/4 indicating that quantum interference cancels the contribution from |22〉

*state to |13〉*

_{AB}*state via post-selection [5*

_{CD}**316**(5825), 726–729 (2007). [CrossRef] [PubMed]

18. O. Steuernagel, “de Broglie wavelength reduction for a multiphoton wave packet,” Phys. Rev. A **65**, 033820 (2002). [CrossRef]

*N*00

*N*state in this paper can be enhanced by a factor of 1.5 by using the four-photon creation method described in Ref. [5

**316**(5825), 726–729 (2007). [CrossRef] [PubMed]

*A*. Another prism pair is placed in path

*B*for the purpose of path-length matching. A triangular waveform voltage is applied to the actuator with a repetition rate of 0.004 Hz. The prism on the piezo actuator moves by 400 nm for a voltage change of 4.8 V. Both beams are combined at the second beam splitter (BS), and the output from the beam splitter is measured by single-photon counting detectors (APDs) with a coincidence circuit. The coincidence circuit has a time windows of 7 ns for two-photon coincidences and 5.4 ns for four-photon coincidences. The integration times used for detecting single-, two- and four-photon coincidence counts were 0.2, 0.2, and 100 seconds, respectively.

## 4. Experimental results

*M*= 1). Figures 2(b)–2(c) show measured two-photon coincidence count rates as a function of piezo voltage. These exhibit interference patterns obtained with an

*N*= 2

*N*00

*N*state for different numbers of multiple passes

*M*= 2 and 4 through the prism pairs. Figure 2(d) shows the measured four-photon interference pattern for

*M*= 4 multiple passes through the prism pair, and Fig. 2(e) is a zoomed-in plot of Fig. 2(d). The resolution enhancement is apparent in Figs. 2(a)–2(d) and is clearly seen to be proportional to the product of the entangled photon number

*N*and the number of multiple passes

*M*through the prism pairs. Specifically, the four-photon interference pattern with a quadruple pass through the prism pair (Fig. 2(d)) exhibits a resolution enhancement by a factor of 16 compared with that of the single-photon interference pattern with single pass (

*n̄*=

*M*= 1). These superresolution results clearly show that the resolution of interference patterns can be enhanced by combining

*N*00

*N*state interferometry with a simple double prism setup.

*V*of 85%, 70%, 65%, and 40%, respectively. Taking in account the efficiency of our single photon detectors (

*η*= 0.62) and the ratio between measured single and two-fold coincidence counts (6.3%), the net detection efficiency of our system

_{det}*η*is estimated to be 0.25 for single photon detection assuming identical losses in both pathes. The transmission loss due to multiple passes through the prism pair is negligible due to the use of anti-reflection coatings on the prisms used. The minimum phase uncertainty (in radians) for a coherent state with no multiple pass through the prism pairs is then calculated to be

_{net}*N*= 2

*N*00

*N*state with

*M*passes through the prism pairs is given by Eq. (2) as

*η*= (0.25)

_{net}^{2}= 0.0625, as we are measuring two-photon events. The minimum phase uncertainty in Figs. 2(b)–2(c) is then calculated to be Δ

*ϕ*

_{(2}

_{,}_{2)}= 1.4 and Δ

*ϕ*

_{(2,4)}= 0.8.

*N*= 4

*N*00

*N*state, note that additional beam splitters BS3:7 & BS5:5 and detectors APD3 & APD4 are added in front of APD2 reducing the system efficiency of APD2, APD3, and APD4 by 1/3 when compared to that of APD1. Then the net system efficiency is very low (

*η*= 0.25

_{net}^{4}/3

^{3}= 0.000145. This low net system efficiency as well as low visibility and non-unit intrinsic efficiency for the four-photon interference has a strong effect on the minimum phase uncertainty for the case shown in Fig. 2(d), which is calculated to be Δ

*ϕ*

_{(4,4)}= 26.0.

*η*and system

_{det}*η*efficiencies significantly lower the accuracy in phase measurement as we go to a larger number of multiple-photon events

_{sys}*N*. The use of multiple passes in the interferometer arms can be used to ameliorate the effect of low detection efficiency on the phase sensitivity. This is seen clearly as we increase the number of multiple reflections in the

*N*= 2

*N*00

*N*state interferometer from 2 to 4. The minimum phase sensitivity is lowered from 1.4 to 0.8 in this case. In principle, we could apply a similar tactic to the

*N*= 4

*N*00

*N*state interferometer to achieve high phase sensitivity in addition to superresolution.

## 5. Conclusions

*N*= 4

*N*00

*N*state and a classical enhancement of 4 from a quadruple pass through a prism pair. Further, we are able to show that the addition of multiple passes in the arms of the interferometer leads to a more accurate phase measurement than without. However, we do not achieve phase supersensitivity due to the fact that the sensitivity in phase is strongly dependent on our net system efficiency [7

7. N. Thomas-Peter, B. Smith, A. Datta, L. Zhang, U. Dorner, and I. Walmsley, “Real-world quantum sensors: evaluating resources for precision measurement,” Phys. Rev. Lett. **107**(11), 0113603 (2011). [CrossRef]

*N*00

*N*state generation efficiency. In order to achieve phase supersensitivity, we would need to obtain unit fringe visibility and increase our net system efficiency to 0.707 for an

*N*= 2

*N*00

*N*state. To achieve this criterion with an

*N*= 4

*N*00

*N*state by our current experimental apparatus, we would need unit fringe visibility and a unit net system efficiency. We are working towards improving our interference visibility and net system efficiency for future experiments.

## Acknowledgments

## References and links

1. | V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics |

2. | A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. |

3. | M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature |

4. | P. Walther, J. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “de Broglie wavelength of a nonlocal four-photon state,” Nature |

5. | T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science |

6. | D. V. Korobkin and E. Yablonovitch, “Two-fold spatial resolution enhancement by two-photon exposure of photographic film,” Opt. Eng. |

7. | N. Thomas-Peter, B. Smith, A. Datta, L. Zhang, U. Dorner, and I. Walmsley, “Real-world quantum sensors: evaluating resources for precision measurement,” Phys. Rev. Lett. |

8. | G. M. Gehring, H. Shin, R. Boyd, C. M. Kim, and B. S. Ham, “Tunable optical time delay of quantum signals using a prism pair,” Opt. Express |

9. | B. C. Barish and R. Weiss, “LIGO and the detection of gravitational waves,” Phys. Today |

10. | F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica |

11. | F. Marquardt and S. Girvin, “Optomechanics,” Physics |

12. | B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman, and G. J. Pryde, “Entanglement-free Heisenberg-limited phase estimation,” Nature |

13. | S. Braunstein and C. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. |

14. | R. A. Fisher, “Theory of statistical estimation,” Mathematical Proceedings of the Cambridge Philosophical Society |

15. | K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O’Brien, and A. G. White, “Time-reversal and super-resolving phase measurements,” Phys. Rev. Lett. |

16. | R. Campos, B. Saleh, and M. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A |

17. | H. Shin, K. W. C. Chan, H. J. Chang, and R. W. Boyd, “Quantum spatial superresolution by optical centroid measurements,” Phys. Rev. Lett. |

18. | O. Steuernagel, “de Broglie wavelength reduction for a multiphoton wave packet,” Phys. Rev. A |

19. | J. U. White, “Long optical paths of large aperture,” J. Opt. Soc. Am. |

**OCIS Codes**

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.3940) Instrumentation, measurement, and metrology : Metrology

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: November 21, 2012

Revised Manuscript: January 9, 2013

Manuscript Accepted: January 9, 2013

Published: January 29, 2013

**Citation**

Heedeuk Shin, Omar S. Magaña-Loaiza, Mehul Malik, Malcolm N. O’Sullivan, and Robert W. Boyd, "Enhancing entangled-state phase estimation by combining classical and quantum protocols," Opt. Express **21**, 2816-2822 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-3-2816

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### References

- V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics5(4), 222–229 (2011). [CrossRef]
- A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett.85(13), 2733–2736 (2000). [CrossRef] [PubMed]
- M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature429(6988), 161–164 (2004). [CrossRef] [PubMed]
- P. Walther, J. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “de Broglie wavelength of a nonlocal four-photon state,” Nature429(6988), 158–161 (2004). [CrossRef] [PubMed]
- T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science316(5825), 726–729 (2007). [CrossRef] [PubMed]
- D. V. Korobkin and E. Yablonovitch, “Two-fold spatial resolution enhancement by two-photon exposure of photographic film,” Opt. Eng.41, 1729–1732 (2002). [CrossRef]
- N. Thomas-Peter, B. Smith, A. Datta, L. Zhang, U. Dorner, and I. Walmsley, “Real-world quantum sensors: evaluating resources for precision measurement,” Phys. Rev. Lett.107(11), 0113603 (2011). [CrossRef]
- G. M. Gehring, H. Shin, R. Boyd, C. M. Kim, and B. S. Ham, “Tunable optical time delay of quantum signals using a prism pair,” Opt. Express18(18), 19156–19162 (2010). [CrossRef] [PubMed]
- B. C. Barish and R. Weiss, “LIGO and the detection of gravitational waves,” Phys. Today52(10), 44–50 (1999). [CrossRef]
- F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica9(7), 686–698 (1942). [CrossRef]
- F. Marquardt and S. Girvin, “Optomechanics,” Physics2, 40 (2009). [CrossRef]
- B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman, and G. J. Pryde, “Entanglement-free Heisenberg-limited phase estimation,” Nature450(7168), 393–396 (2007). [CrossRef] [PubMed]
- S. Braunstein and C. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett.72(22), 3439–3443 (1994). [CrossRef] [PubMed]
- R. A. Fisher, “Theory of statistical estimation,” Mathematical Proceedings of the Cambridge Philosophical Society22(05), 700–725 (1925). [CrossRef]
- K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O’Brien, and A. G. White, “Time-reversal and super-resolving phase measurements,” Phys. Rev. Lett.98(22), 0223601 (2007). [CrossRef]
- R. Campos, B. Saleh, and M. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A40(3), 1371–1384 (1989). [CrossRef] [PubMed]
- H. Shin, K. W. C. Chan, H. J. Chang, and R. W. Boyd, “Quantum spatial superresolution by optical centroid measurements,” Phys. Rev. Lett.107, 083603 (2011). [CrossRef] [PubMed]
- O. Steuernagel, “de Broglie wavelength reduction for a multiphoton wave packet,” Phys. Rev. A65, 033820 (2002). [CrossRef]
- J. U. White, “Long optical paths of large aperture,” J. Opt. Soc. Am.32(5), 285–288 (1942). [CrossRef]

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