## Measuring optical tunneling times using a Hong-Ou-Mandel interferometer

Optics Express, Vol. 16, Issue 20, pp. 16005-16012 (2008)

http://dx.doi.org/10.1364/OE.16.016005

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### Abstract

We report a prediction for the delay measured in an optical tunneling experiment using Hong-Ou-Mandel (HOM) interference, taking into account the Goos-Hänchen shift generalized to frustrated total internal reflection situations. We precisely state assumptions under which the tunneling delay measured by an HOM interferometer can be calculated. We show that, under these assumptions, the measured delay is the group delay, and that it is apparently ‘superluminal’ for sufficiently thick air gaps. We also show how an HOM signal with multiple minima can be obtained, and that the shape of such a signal is not appreciably affected by the presence of the optical tunneling zone, thus ruling out the explanation of the anomalously short tunneling delays in terms of a reshaping of the wavepacket as it goes through the tunneling zone. Finally, we compare the predicted tunneling delay to a relevant classical delay and conclude that our predictions involve no non-causal effect.

© 2008 Optical Society of America

1. E. H. Hauge and J. A. Støvneng, “Tunneling times: a critical review,” Rev. Mod. Phys. **61**, 917–936 (1989). [CrossRef]

2. Th. Martin and R. Landauer, “Time delay of evanescent electromagnetic waves and the analogy to particle tunneling,” Phys. Rev. A **45**, 2611–2617 (1992). [CrossRef] [PubMed]

3. A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. **71**, 708–711 (1993). [CrossRef] [PubMed]

*i*.

*e*. delays that appear to correspond to velocities greater than that of light in vacuum (see [4

4. H. G. Winful, “Tunneling time, the Hartman effect, and superluminality: A proposed resolution of an old paradox,” Phys. Rep. **436**, 1–69 (2006). [CrossRef]

*et al*. [5

5. R. Y. Chiao, P. G. Kwiat, and A. M. Steinberg, “Analogies between electron and photon tunneling: A proposed experiment to measure photon tunneling times,” Physica B **175**, 257–262 (1991). [CrossRef]

6. C. K. Hong, Z. Y. Ou, L. Mandel, and Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. **59**, 2044–2046 (1987). [CrossRef] [PubMed]

*n*≈1.5) facing each other, separated by a gap of air (index 1) of adjustable width

*d*on the

*µ*m scale.

*t*, which we have derived for an

_{G}*s*-polarized plane wave. This coefficient can be calculated for an evanescent wave with incidence angle

*θ*(

*θ*>

*θ*) in a similar way as for a Fabry-Pérot cavity with a propagating wave:

_{c}*λ*is the wavelength in vacuum of the incident plane wave.

*λ*is in the visible range and whose waist

_{i}*w*

_{0}is on the mm scale, impinging on the air gap with incidence angle

*θ*. The evanescent nature of the field inside the gap yields relevant effects only for values of the gap width

_{i}*d*which are greater than the characteristic attenuation length

*z*

_{0}of these evanescent waves. Additionally, we assume

*w*

_{0}≫

*d*/

*κ*, which ensures that the amplitude attenuation of a plane wave crossing the air gap with incidence angle

*θ*≈

*θ*

*is approximately constant over the angular dispersion*

_{i}*w*

_{0}, becomes apparently ‘superluminal’. We are then able to explain why such a short delay is not associated with any non-causal effect. We have not investigated the regime for which this assumption does not hold. Indeed, a transverse distortion of the incident beam as it crosses the tunneling zone would make it much harder to define or measure the tunneling delay.

*x*

*T*, along the second glass-to-air interface, of the centroid of the transmitted beam with respect to that of the incident beam [8

8. A. M. Steinberg and R. Y. Chiao, “Tunneling delay times in one and two dimensions,” Phys. Rev. A **49**, 3283–3295 (1994). [CrossRef] [PubMed]

*t*(

_{G}*) to each plane-wave component. The resulting integral can be evaluated, in the paraxial approximation, using the method of stationary phase, which yields the following analytical expression:*

**k***d*, this shift saturates

*λ*. on the

_{i}, i.e*µ*m scale. Realistic values for the width

*d*of the air gap, chosen such that intensity transmission through the tunneling zone is finite (albeit small), are of the same magnitude. Consequently, the GH shift has to be taken into account to predict the delay measured using the HOM interferometer.

*xF*is the position on the (second) air-glass interface corresponding to the position of the fiber-collimator on the output side of the second cube (

*cf*. Fig. 1), and

*cτ*=

*cτ*

_{1}-

*cτ*

_{2}is the difference in optical lengths between the ‘unknown’ and ‘reference’ arms of the interferometer, excluding the actual air gap. Considering the ideal case of a symmetrical beamsplitter, the field operators at exit ports 3 and 4 are given by

**E**^{(+)}

_{3,4}=

**E**^{(+)}

_{1}±

**E**^{(+)}

**. The photon-coincidence signal**

_{2}*S*

_{0}(

*t*

_{0})=

*∫dt*〈

*Ψ*|

**E**^{(-)}

_{3}_{(t0)}

**E**^{(-)}

_{4}_{(t0+t)}

**E**^{(+)}

_{4}_{(t0+t)}

**E**^{(+)}

_{3}_{(t0)}|

*Ψ*〉, integrated over the resolving time of the detectors, reduces to:

*cf*. the single HOM dips in Fig. 4). Such a numerical calculation is possible in the general case. Furthermore, if the spectral bandwidth

*σ*of the source is sufficiently narrow, the preceding integral can be calculated using the approximation of stationary phase. We thus obtain:

*µ*is the displacement, on the plot of

*S*

_{0}(

*τ*), of the minimum of the HOM dip with respect to its position when there is no air gap (

*t*=1). Therefore, it can be interpreted as a measurement of the delay of the photons crossing the gap. The expression for this delay depends on

_{G}*xF*, however it is only meaningful if the collecting fiber-collimator is positioned at the point where the transmitted beam is expected to exit the second cube, taking the generalized Goos-Hänchen shift into account:

*xF*=Δ

*xT*. One thus obtains the following prediction for the tunneling delay measured in this experiment:

*µ*involves the derivative of the gap transmission function with respect to frequency, and therefore corresponds to the group-delay. Our calculation thus provides theoretical grounds for Steinberg et al.’s experimental measurement, using an HOM interferometer, of the delay due to tunneling through a 1D photonic band-gap barrier, which they found to agree with the group-delay prediction [3

3. A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. **71**, 708–711 (1993). [CrossRef] [PubMed]

*µ*on the thickness

*d*of the air gap is represented in Fig. 3, where it is compared to

*d*/

*c*, the shortest possible delay required to cross the air gap at the speed of light. Our prediction for the delay is greater than

*d*/

*c*for small values of

*d*; however, it becomes smaller than

*d*/

*c*, and therefore apparently superluminal, for greater values of

*d*. It saturates to a finite value for large values of

*d*[8

8. A. M. Steinberg and R. Y. Chiao, “Tunneling delay times in one and two dimensions,” Phys. Rev. A **49**, 3283–3295 (1994). [CrossRef] [PubMed]

11. T. E. Hartman, “Tunneling of a Wave Packet,” J. Appl. Phys. **33**, 3427–3433 (1962). [CrossRef]

*σ*, and it allows the calculation of more elaborate two-photon coincidence signals than the single HOM dip. This allows us to investigate whether or not the shape of the coincidence signal is appreciably distorted by the tunneling zone. Were the coincidence signal to undergo a significant reshaping as it crossed the air gap, one might argue that the predicted superluminal delay of the wavepacket peak was simply an artifact of this reshaping.

4. H. G. Winful, “Tunneling time, the Hartman effect, and superluminality: A proposed resolution of an old paradox,” Phys. Rep. **436**, 1–69 (2006). [CrossRef]

4. H. G. Winful, “Tunneling time, the Hartman effect, and superluminality: A proposed resolution of an old paradox,” Phys. Rep. **436**, 1–69 (2006). [CrossRef]

**436**, 1–69 (2006). [CrossRef]

*d*, whereas the intensity that arrives at a time

*d*/

*c*before the peak of the pulse decreases as

*w*

_{0}≫

*d*/

*κ*. Therefore, the effective temporal width

*d*/

*c*for our calculation to be valid. The preceding condition replaces, in the case of 2D FTIR, the quasi-static condition for 1D tunneling [4

**436**, 1–69 (2006). [CrossRef]

*h*

_{0}~

*d*/

*κ*, following a virtual classical trajectory as represented on Fig. 5, contributes to the field at

*x*

*F*earlier than the calculated tunneling time. In the regime for which our calculations are valid (

*d*>

*z*

_{0}and

*w*

_{0}≫

*d*/

*κ*), the total contribution to the intensity of the

*h*≿

*h*

_{0}part of the incident beam is of the order of

## Acknowledgment

## References and links

1. | E. H. Hauge and J. A. Støvneng, “Tunneling times: a critical review,” Rev. Mod. Phys. |

2. | Th. Martin and R. Landauer, “Time delay of evanescent electromagnetic waves and the analogy to particle tunneling,” Phys. Rev. A |

3. | A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. |

4. | H. G. Winful, “Tunneling time, the Hartman effect, and superluminality: A proposed resolution of an old paradox,” Phys. Rep. |

5. | R. Y. Chiao, P. G. Kwiat, and A. M. Steinberg, “Analogies between electron and photon tunneling: A proposed experiment to measure photon tunneling times,” Physica B |

6. | C. K. Hong, Z. Y. Ou, L. Mandel, and Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. |

7. | D. Branning, A. L. Migdall, and A. V. Sergienko, “Simultaneous measurement of group and phase delay between two photons,” Phys. Rev. A |

8. | A. M. Steinberg and R. Y. Chiao, “Tunneling delay times in one and two dimensions,” Phys. Rev. A |

9. | K. Yasumoto and Y. Oishi, “A new evaluation of the Goos-Hänchen shift and associated time delay,” J. Appl. Phys. |

10. | L. Mandel and E. Wolf, |

11. | T. E. Hartman, “Tunneling of a Wave Packet,” J. Appl. Phys. |

**OCIS Codes**

(030.0030) Coherence and statistical optics : Coherence and statistical optics

(260.6970) Physical optics : Total internal reflection

**ToC Category:**

Physical Optics

**History**

Original Manuscript: August 12, 2008

Revised Manuscript: September 9, 2008

Manuscript Accepted: September 10, 2008

Published: September 24, 2008

**Citation**

D. J. Papoular, P. Clade, S. V. Polyakov, C. F. McCormick, A. L. Migdall, and P. D. Lett, "Measuring optical tunneling times using a Hong-Ou-Mandel interferometer," Opt. Express **16**, 16005-16012 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-16005

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### References

- E. H. Hauge and J. A. Støvneng, "Tunneling times: a critical review," Rev. Mod. Phys. 61, 917-936 (1989). [CrossRef]
- Th. Martin and R. Landauer, "Time delay of evanescent electromagnetic waves and the analogy to particle tunneling," Phys. Rev. A 45, 2611-2617 (1992). [CrossRef] [PubMed]
- A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, "Measurement of the single-photon tunneling time," Phys. Rev. Lett. 71, 708-711 (1993). [CrossRef] [PubMed]
- H. G. Winful, "Tunneling time, the Hartman effect, and superluminality: A proposed resolution of an old paradox," Phys. Rep. 436, 1-69 (2006). [CrossRef]
- R. Y. Chiao, P. G. Kwiat, and A. M. Steinberg, "Analogies between electron and photon tunneling: A proposed experiment to measure photon tunneling times," Physica B 175, 257-262 (1991). [CrossRef]
- C. K. Hong, Z. Y. Ou, and L. Mandel, "Measurement of subpicosecond time intervals between two photons by interference," Phys. Rev. Lett. 59, 2044-2046 (1987). [CrossRef] [PubMed]
- D. Branning, A. L. Migdall, and A. V. Sergienko, "Simultaneous measurement of group and phase delay between two photons," Phys. Rev. A 62, 063808 (2000). [CrossRef]
- A. M. Steinberg and R. Y. Chiao, "Tunneling delay times in one and two dimensions," Phys. Rev. A 49, 3283-3295 (1994). [CrossRef] [PubMed]
- K. Yasumoto and Y. Oishi, "A new evaluation of the Goos-Hänchen shift and associated time delay," J. Appl. Phys. 54, 2170-2176 (1983).
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press 1995).
- T. E. Hartman, "Tunneling of a Wave Packet," J. Appl. Phys. 33, 3427-3433 (1962). [CrossRef]

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