## Weak interference in the high-signal regime |

Optics Express, Vol. 20, Issue 17, pp. 18869-18875 (2012)

http://dx.doi.org/10.1364/OE.20.018869

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### Abstract

Weak amplification is a signal enhancement technique which is used to measure tiny changes that otherwise cannot be determined because of technical limitations. It is based on: a) the existence of a weak interaction which couples a property of a system (*the system*) with a separate degree of freedom (*the pointer*), and b) the measurement of an anomalously large mean value of the pointer state (*weak mean value*), after appropriate pre-and post-selection of the state of the system. Unfortunately, the weak amplification process is generally accompanied by severe losses of the detected signal, which limits its applicability. However, we will show here that since weak amplification is essentially the result of an interference phenomena, it should be possible to use the degree of interference (*weak interference*) to get relevant information about the physical system under study in a more general scenario, where the signal is not severely depleted (high-signal regime).

© 2012 OSA

## 1. Introduction

1. Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. **60**, 1351–1354 (1988). [CrossRef] [PubMed]

*measuring device*, whose aim is to unveil the value of a property that characterizes the other interacting subsystem (

*the system*).

*weak mean value*) can yield an unexpectedly large value (

*weak amplification*). Surprisingly, this value can lay outside the range of small displacements of the measurement pointer caused by each one of the possible states of the system. Unfortunately, this is also accompanied by a severe depletion of the intensity of the signal detected, due to the quasi-orthogonality of the input and final states of the system. This prevents the applicability of the weak measurement concept to experiments which are already limited by a low signal-to-noise ratio [2

2. O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science **319**, 787–790 (2008). [CrossRef] [PubMed]

3. N. W. Ritchie, J. G. Story, and R. G. Hulet, “Realization of a measuremernt of a *weak value*,”Phys. Rev. Lett. **66**, 1107–1110 (1991). [CrossRef] [PubMed]

*〉 and final |Ψ*

_{in}*〉 states of the polarization of the system, so that 〈Ψ*

_{out}*|Ψ*

_{out}*〉 ∼*

_{in}*ε*(

*ε*is small), the mean value of the shift of the position of the light beam is 〈

*x*〉/Δ = (〈

*A*〉

*), where the weak value 〈*

_{w}*A*〉

*is defined as In this example, the power of the output signal (*

_{w}*P*) is severely reduced, i.e.,

_{out}*P*/

_{out}*P*∼

_{in}*ε*

^{2}. The signal enhancement of the position of the beam due to weak amplification can thus be observed only if the input signal intensity (

*P*) can be enhanced.

_{in}2. O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science **319**, 787–790 (2008). [CrossRef] [PubMed]

3. N. W. Ritchie, J. G. Story, and R. G. Hulet, “Realization of a measuremernt of a *weak value*,”Phys. Rev. Lett. **66**, 1107–1110 (1991). [CrossRef] [PubMed]

4. P. Ben Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. **102**, 173601 (2009). [CrossRef] [PubMed]

5. D. J. Starling, P. Ben Dixon, A. N. Jordan, and J. C. Howell, “Precision frequency measurements with interferometric weak values,” Phys. Rev. A **82**, 063822 (2010). [CrossRef]

6. N. Brunner, A. Acin, D. Collins, N. Gisin, and V. Scarini, “Optical telecom networks as weak quantum measurements with postselection,” Phys. Rev. Lett. **91**, 180402 (2003). [CrossRef] [PubMed]

7. D. R. Solli, C. F. McCormick, R. Y. Chiao, S. Popescu, and J. H. Hickmann, “Fast light, slow light, and phase singularities: a connection to generalized weak values,” Phys. Rev. Lett. **92**, 043601 (2004). [CrossRef] [PubMed]

8. I. M. Duck, P. M. Stevenson, and E. C. G. Sudarhshan, “The sense in which a ’weak measurement’ of a spin-1/2 particles’s spin component yields a value of 100,” Phys. Rev. D **40**, 2112–2117 (1989). [CrossRef]

9. J. C. Howell, D. J. Starling, P. B. Dixon, K. P. Vudyasetu, and A. N. Jordan, “Interferometric weak value deflections: quantum and classical treatments,” Phys. Rev. A **81**, 033813 (2010). [CrossRef]

10. G. J. Pryde, J. L. O’Brien, A. G. White, T. C. Ralph, and H. M. Wiseman, “Measurement of quantum weak values of photon polarization,” Phys. Rev. Lett. **94**, 220405 (2005). [CrossRef] [PubMed]

11. M. E. Goggin, M. P. Almeida, M. Barbieri, B. P. Lanyon, J. L. O’Brien, A. G. White, and G. J. Pryde, “Violation of the Leggett-Garg inequality with weak measurements of photons,” Proc. Natl. Acad. Sci. U.S.A. **108**, 1256–1261 (2011). [CrossRef] [PubMed]

*weak mean value*might not convey any relevant information about the system or the measuring device. This can open new applications based on weak interference, especially for interactions where it is not possible to enhance the signal-to-noise ratio of the measurement. For example, when the intensity of the input signal cannot be increased.

## 2. Weak interference: general scenario

*x*) propagates in a medium that couples the polarization and spatial degrees of freedom, thereby generating a polarization-dependent displacement of the photons, Δ

_{1}and Δ

_{2}. The input state of the system (polarization) and the measuring device (position of the beam) can be written, up to a normalization constant, as The global state of the system and measuring device after the interaction writes [3

3. N. W. Ritchie, J. G. Story, and R. G. Hulet, “Realization of a measuremernt of a *weak value*,”Phys. Rev. Lett. **66**, 1107–1110 (1991). [CrossRef] [PubMed]

*φ*stands for any polarization-dependent phase difference that can occur during the interaction. Finally, before being detected, the photons are projected into the polarization state |Ψ

*〉 = cos*

_{out}*α*|

*H*〉 + exp(

*iξ*)sin

*α*|

*V*〉. The intensity distribution of the output light beam can now be written as with

*θ*=

*φ*−

*ξ*.

*w*

_{0}. Making use of the product theorem for Gaussian integrals [12

12. E. Wolf, J. T. Foley, and F. Gori, “Frequency shifts of frequency lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A **6**, 1142–1149 (1989). [CrossRef]

*x*〉 = ∫

*dxxI*(

*x*)/∫

*dxI*(

*x*) reads where Δ

_{+}= Δ

_{1}+ Δ

_{2}and Δ

_{−}= Δ

_{1}− Δ

_{2}. In all cases, Δ

_{1}, Δ

_{2}≪

*w*

_{0}, so

*γ*∼ 1.

_{+}can not be amplified with the present scheme. In many experiments [2

2. O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science **319**, 787–790 (2008). [CrossRef] [PubMed]

*weak value*,”Phys. Rev. Lett. **66**, 1107–1110 (1991). [CrossRef] [PubMed]

_{1}= −Δ

_{2}, so Δ

_{+}= 0, and as a consequence the weak measurement amplifies all the relevant information. Therefore, the weak amplification happens for Δ

_{−}, with an amplification factor

*𝒜*that reads Figure 1 shows the amplification factor for

*θ*= 0.01°. The maximum amplification takes place for the angle

*α*

_{0}= −1/2 sin

^{−1}(

*γ*cos

*θ*), where the factor reaches the value of

*𝒜*= (1 −

_{max}*γ*

^{2}cos

^{2}

*θ*)

^{−1/2}.

*P*is the total input and output power of the optical beam, when integrated over all space. Figure 2 shows the loss in the measurement expressed in dB, as 10 log(

_{in,out}*P*/

_{out}*P*). Even though the enhancement of Δ can be a large staggering value for angles close to −45° (Fig. 1 shows an enhancement close to 10

_{in}^{3}), this is unfortunately accompanied by a severe loss penalty close to 70 dB (Fig. 2). For instance, if the goal of the experiment is to attain a signal-to-noise ratio of 10 dB at the measurement stage, the input signal has to be correspondingly increased 70 dB above this level.

*θ*, which should be chosen close to zero in order to achieve the maximum amplification. This angle can be modified by choosing the appropriate value

*ξ*of the polarization of the output state of the system. The power of the output signal also depends on this angle. Figure 3 shows the signal loss for a few selected angles:

*α*= −45°, −30°, 0° and 45°. In all cases, the loss goes from log[(1 + sin2

*α*)/2] for

*θ*= 0, to −3 dB for

*θ*= ±

*π*/2, which corresponds to post-selecting polarizations cos

*α*|

*H*〉 +sin

*α*|

*V*〉 and cos

*α*|

*H*〉 +

*i*sin

*α*|

*V*〉, respectively. Importantly, the dependence of the losses on the polarization selected at the output for

*α*≠ 0°, 90° is a consequence of the interference effect which is the essence of the weak measurement concept [8

8. I. M. Duck, P. M. Stevenson, and E. C. G. Sudarhshan, “The sense in which a ’weak measurement’ of a spin-1/2 particles’s spin component yields a value of 100,” Phys. Rev. D **40**, 2112–2117 (1989). [CrossRef]

## 3. Weak measurement in a high-signal regime

*P/P*, where Δ

_{in}*P*=

*P*−

_{out}*P*. One obtains that Figure 4 shows the value of Δ

_{in}*P/P*as a function of the spatial shift Δ for

*α*= 45° and

*θ*= 0, 0.1°, 0.2° and 0.3°. The dependence of the fractional loss on Δ comes from the relationship between

*γ*and

*δ*, as given in Eq. (6). Notice that in all cases, the total losses of the system are below 3 dB, which is significantly below the loss found in the usual regime of weak amplification, where losses can easily reach tens of dB for large amplifications. Moreover, in all cases shown in Fig. 4, the mean value of the beam position is 〈

*x*〉 = 0, so the weak value concept does not convey any relevant information here.

*α*= 45° and

*θ*= 0°. By choosing other values for these angles, one can decrease the fractional loss, making its detection easier. However, this would decrease the sensitivity, making the distinction between different spatial shifts Δ more difficult. Note that here we are assuming that there are not other sources of polarization-dependent losses.

13. C. W. Freudiger, W. Min, B. G. Saar, S. Lu, G. R. Holtom, C. He, J. C. Tsai, J. X. Kang, and X. S. Xie, “Label-free biomedical imaging with high sensitivity by stimulated Raman scattering microscopy,” Science **322**, 1857–1861 (2008). [CrossRef] [PubMed]

14. B. G. Saar, C. W. Freudiger, J. Reichman, C. M. Stanley, G. R. Holtom, and X. S. Xie, “Video-rate molecular imaging in vivo with stimulated Raman scattering,” Science **330**, 1368–1370 (2010). [CrossRef] [PubMed]

*P/P*< 10

^{−7}. The key point is to modulate the input signal at MHz rates to remove the low-frequency laser noise, implementing a detection scheme that is effectively shot-noise limited. In this way, one can increase the signal-to-noise ratio in a weak interference configuration, where the sensitivity is now limited by the presence of shot noise [15

15. D. J. Starling, P. Ben Dixon, A. N. Jordan, and J. C. Howell, “Optimizing the signal-to-noise ratio of a beam-deflection measurement with interferometric weak values,” Phys. Rev. A **80**, 041803 (2009). [CrossRef]

16. A. Nishizawa, K. Nakamura, and M. Fujimoto, “Weak-value amplification in a shot-noise-limited interferometer,” Phys. Rev. A **85**, 062108 (2012). [CrossRef]

## 4. Conclusions

10. G. J. Pryde, J. L. O’Brien, A. G. White, T. C. Ralph, and H. M. Wiseman, “Measurement of quantum weak values of photon polarization,” Phys. Rev. Lett. **94**, 220405 (2005). [CrossRef] [PubMed]

11. M. E. Goggin, M. P. Almeida, M. Barbieri, B. P. Lanyon, J. L. O’Brien, A. G. White, and G. J. Pryde, “Violation of the Leggett-Garg inequality with weak measurements of photons,” Proc. Natl. Acad. Sci. U.S.A. **108**, 1256–1261 (2011). [CrossRef] [PubMed]

10. G. J. Pryde, J. L. O’Brien, A. G. White, T. C. Ralph, and H. M. Wiseman, “Measurement of quantum weak values of photon polarization,” Phys. Rev. Lett. **94**, 220405 (2005). [CrossRef] [PubMed]

*weak value*,”Phys. Rev. Lett. **66**, 1107–1110 (1991). [CrossRef] [PubMed]

5. D. J. Starling, P. Ben Dixon, A. N. Jordan, and J. C. Howell, “Precision frequency measurements with interferometric weak values,” Phys. Rev. A **82**, 063822 (2010). [CrossRef]

*usefulness*of the approach considered in this type of experiments will depend on the flux rate of photons generated, and the capability to generate changes of power above the noise level of the experiment.

*u*〉 is a base of the N-dimensional space.

_{i}17. J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. **80**, 3217–3219 (1998). [CrossRef]

18. M. V. Vasnetsov, J. P. Torres, D. V. Petrov, and L. Torner, “Measurement of orbital angular momentum spectrum of a light beam,” Opt. Lett. **28**, 2285–2287 (2003). [CrossRef] [PubMed]

*= 2*

_{m}*m*Ω, where

*m*is the OAM mode index.

## Acknowledgments

## References and links

1. | Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. |

2. | O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science |

3. | N. W. Ritchie, J. G. Story, and R. G. Hulet, “Realization of a measuremernt of a |

4. | P. Ben Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. |

5. | D. J. Starling, P. Ben Dixon, A. N. Jordan, and J. C. Howell, “Precision frequency measurements with interferometric weak values,” Phys. Rev. A |

6. | N. Brunner, A. Acin, D. Collins, N. Gisin, and V. Scarini, “Optical telecom networks as weak quantum measurements with postselection,” Phys. Rev. Lett. |

7. | D. R. Solli, C. F. McCormick, R. Y. Chiao, S. Popescu, and J. H. Hickmann, “Fast light, slow light, and phase singularities: a connection to generalized weak values,” Phys. Rev. Lett. |

8. | I. M. Duck, P. M. Stevenson, and E. C. G. Sudarhshan, “The sense in which a ’weak measurement’ of a spin-1/2 particles’s spin component yields a value of 100,” Phys. Rev. D |

9. | J. C. Howell, D. J. Starling, P. B. Dixon, K. P. Vudyasetu, and A. N. Jordan, “Interferometric weak value deflections: quantum and classical treatments,” Phys. Rev. A |

10. | G. J. Pryde, J. L. O’Brien, A. G. White, T. C. Ralph, and H. M. Wiseman, “Measurement of quantum weak values of photon polarization,” Phys. Rev. Lett. |

11. | M. E. Goggin, M. P. Almeida, M. Barbieri, B. P. Lanyon, J. L. O’Brien, A. G. White, and G. J. Pryde, “Violation of the Leggett-Garg inequality with weak measurements of photons,” Proc. Natl. Acad. Sci. U.S.A. |

12. | E. Wolf, J. T. Foley, and F. Gori, “Frequency shifts of frequency lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A |

13. | C. W. Freudiger, W. Min, B. G. Saar, S. Lu, G. R. Holtom, C. He, J. C. Tsai, J. X. Kang, and X. S. Xie, “Label-free biomedical imaging with high sensitivity by stimulated Raman scattering microscopy,” Science |

14. | B. G. Saar, C. W. Freudiger, J. Reichman, C. M. Stanley, G. R. Holtom, and X. S. Xie, “Video-rate molecular imaging in vivo with stimulated Raman scattering,” Science |

15. | D. J. Starling, P. Ben Dixon, A. N. Jordan, and J. C. Howell, “Optimizing the signal-to-noise ratio of a beam-deflection measurement with interferometric weak values,” Phys. Rev. A |

16. | A. Nishizawa, K. Nakamura, and M. Fujimoto, “Weak-value amplification in a shot-noise-limited interferometer,” Phys. Rev. A |

17. | J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. |

18. | M. V. Vasnetsov, J. P. Torres, D. V. Petrov, and L. Torner, “Measurement of orbital angular momentum spectrum of a light beam,” Opt. Lett. |

**OCIS Codes**

(030.1670) Coherence and statistical optics : Coherent optical effects

(270.0270) Quantum optics : Quantum optics

(350.7420) Other areas of optics : Waves

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: May 1, 2012

Revised Manuscript: June 17, 2012

Manuscript Accepted: June 18, 2012

Published: August 2, 2012

**Citation**

Juan P. Torres, Graciana Puentes, Nathaniel Hermosa, and Luis Jose Salazar-Serrano, "Weak interference in the high-signal regime," Opt. Express **20**, 18869-18875 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-17-18869

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### References

- Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett.60, 1351–1354 (1988). [CrossRef] [PubMed]
- O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science319, 787–790 (2008). [CrossRef] [PubMed]
- N. W. Ritchie, J. G. Story, and R. G. Hulet, “Realization of a measuremernt of a weak value,”Phys. Rev. Lett.66, 1107–1110 (1991). [CrossRef] [PubMed]
- P. Ben Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett.102, 173601 (2009). [CrossRef] [PubMed]
- D. J. Starling, P. Ben Dixon, A. N. Jordan, and J. C. Howell, “Precision frequency measurements with interferometric weak values,” Phys. Rev. A82, 063822 (2010). [CrossRef]
- N. Brunner, A. Acin, D. Collins, N. Gisin, and V. Scarini, “Optical telecom networks as weak quantum measurements with postselection,” Phys. Rev. Lett.91, 180402 (2003). [CrossRef] [PubMed]
- D. R. Solli, C. F. McCormick, R. Y. Chiao, S. Popescu, and J. H. Hickmann, “Fast light, slow light, and phase singularities: a connection to generalized weak values,” Phys. Rev. Lett.92, 043601 (2004). [CrossRef] [PubMed]
- I. M. Duck, P. M. Stevenson, and E. C. G. Sudarhshan, “The sense in which a ’weak measurement’ of a spin-1/2 particles’s spin component yields a value of 100,” Phys. Rev. D40, 2112–2117 (1989). [CrossRef]
- J. C. Howell, D. J. Starling, P. B. Dixon, K. P. Vudyasetu, and A. N. Jordan, “Interferometric weak value deflections: quantum and classical treatments,” Phys. Rev. A81, 033813 (2010). [CrossRef]
- G. J. Pryde, J. L. O’Brien, A. G. White, T. C. Ralph, and H. M. Wiseman, “Measurement of quantum weak values of photon polarization,” Phys. Rev. Lett.94, 220405 (2005). [CrossRef] [PubMed]
- M. E. Goggin, M. P. Almeida, M. Barbieri, B. P. Lanyon, J. L. O’Brien, A. G. White, and G. J. Pryde, “Violation of the Leggett-Garg inequality with weak measurements of photons,” Proc. Natl. Acad. Sci. U.S.A.108, 1256–1261 (2011). [CrossRef] [PubMed]
- E. Wolf, J. T. Foley, and F. Gori, “Frequency shifts of frequency lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A6, 1142–1149 (1989). [CrossRef]
- C. W. Freudiger, W. Min, B. G. Saar, S. Lu, G. R. Holtom, C. He, J. C. Tsai, J. X. Kang, and X. S. Xie, “Label-free biomedical imaging with high sensitivity by stimulated Raman scattering microscopy,” Science322, 1857–1861 (2008). [CrossRef] [PubMed]
- B. G. Saar, C. W. Freudiger, J. Reichman, C. M. Stanley, G. R. Holtom, and X. S. Xie, “Video-rate molecular imaging in vivo with stimulated Raman scattering,” Science330, 1368–1370 (2010). [CrossRef] [PubMed]
- D. J. Starling, P. Ben Dixon, A. N. Jordan, and J. C. Howell, “Optimizing the signal-to-noise ratio of a beam-deflection measurement with interferometric weak values,” Phys. Rev. A80, 041803 (2009). [CrossRef]
- A. Nishizawa, K. Nakamura, and M. Fujimoto, “Weak-value amplification in a shot-noise-limited interferometer,” Phys. Rev. A85, 062108 (2012). [CrossRef]
- J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett.80, 3217–3219 (1998). [CrossRef]
- M. V. Vasnetsov, J. P. Torres, D. V. Petrov, and L. Torner, “Measurement of orbital angular momentum spectrum of a light beam,” Opt. Lett.28, 2285–2287 (2003). [CrossRef] [PubMed]

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