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

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References

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  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]
  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]
  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, and L. Mandel, "Measurement of subpicosecond time intervals between two photons by interference," Phys. Rev. Lett. 59, 2044-2046 (1987).
    [CrossRef] [PubMed]
  7. 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]
  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]
  9. K. Yasumoto and Y. Oishi, "A new evaluation of the Goos-Hänchen shift and associated time delay," J. Appl. Phys. 54, 2170-2176 (1983).
  10. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press 1995).
  11. T. E. Hartman, "Tunneling of a Wave Packet," J. Appl. Phys. 33, 3427-3433 (1962).
    [CrossRef]

2006 (1)

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

2000 (1)

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]

1994 (1)

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

1993 (1)

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]

1992 (1)

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]

1991 (1)

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]

1989 (1)

E. H. Hauge and J. A. Støvneng, "Tunneling times: a critical review," Rev. Mod. Phys. 61, 917-936 (1989).
[CrossRef]

1987 (1)

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]

1983 (1)

K. Yasumoto and Y. Oishi, "A new evaluation of the Goos-Hänchen shift and associated time delay," J. Appl. Phys. 54, 2170-2176 (1983).

1962 (1)

T. E. Hartman, "Tunneling of a Wave Packet," J. Appl. Phys. 33, 3427-3433 (1962).
[CrossRef]

Branning, D.

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]

Chiao, R. Y.

A. M. Steinberg and R. Y. Chiao, "Tunneling delay times in one and two dimensions," Phys. Rev. A 49, 3283-3295 (1994).
[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]

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]

Hartman, T. E.

T. E. Hartman, "Tunneling of a Wave Packet," J. Appl. Phys. 33, 3427-3433 (1962).
[CrossRef]

Hauge, E. H.

E. H. Hauge and J. A. Støvneng, "Tunneling times: a critical review," Rev. Mod. Phys. 61, 917-936 (1989).
[CrossRef]

Hong, C. K.

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]

Kwiat, P. G.

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]

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]

Landauer, R.

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]

Mandel, L.

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]

Martin, Th.

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]

Migdall, A. L.

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]

Oishi, Y.

K. Yasumoto and Y. Oishi, "A new evaluation of the Goos-Hänchen shift and associated time delay," J. Appl. Phys. 54, 2170-2176 (1983).

Ou, Z. Y.

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]

Sergienko, A. V.

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]

Steinberg, A. M.

A. M. Steinberg and R. Y. Chiao, "Tunneling delay times in one and two dimensions," Phys. Rev. A 49, 3283-3295 (1994).
[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]

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]

Støvneng, J. A.

E. H. Hauge and J. A. Støvneng, "Tunneling times: a critical review," Rev. Mod. Phys. 61, 917-936 (1989).
[CrossRef]

Winful, H. G.

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

Yasumoto, K.

K. Yasumoto and Y. Oishi, "A new evaluation of the Goos-Hänchen shift and associated time delay," J. Appl. Phys. 54, 2170-2176 (1983).

J. Appl. Phys. (2)

K. Yasumoto and Y. Oishi, "A new evaluation of the Goos-Hänchen shift and associated time delay," J. Appl. Phys. 54, 2170-2176 (1983).

T. E. Hartman, "Tunneling of a Wave Packet," J. Appl. Phys. 33, 3427-3433 (1962).
[CrossRef]

Phys. Rep. (1)

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

Phys. Rev. A (3)

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]

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]

Phys. Rev. Lett. (2)

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]

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]

Physica B (1)

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]

Rev. Mod. Phys. (1)

E. H. Hauge and J. A. Støvneng, "Tunneling times: a critical review," Rev. Mod. Phys. 61, 917-936 (1989).
[CrossRef]

Other (1)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press 1995).

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Figures (5)

Fig. 1.
Fig. 1.

Left: experimental setup. Inset: schematics of the tunneling zone and the incident, reflected and transmitted beams, taking the generalized Goos—Hänchen shift into account. Note that the width of the air gap, the size of the prisms, and the waist of the beam are not to scale.

Fig. 2.
Fig. 2.

Magnitude of the Goos-Hänchen shift on the transmitted beam ΔxT as a function of the gap width d, for incident wavelength λi =702nm in vacuum, incidence angle θi =41.9°, and prism glass index n=1.5.

Fig. 3.
Fig. 3.

Predicted delay µ as a function of the gap width d, calculated for incident wavelength λi =702nm in vacuum, incidence angle θ i =41.9°, and prism glass index n=1.5.

Fig. 4.
Fig. 4.

Left: Coincidence signals with one (red) and two (green) minima, calculated for d=10µm, λi =702nm, σ=20nm, tW(ωi +u)=0.4+0.6cos(τ 0 u)+0.7sin(τ 0 u/2), and τ0=2/σ, both without the tunneling zone (solid lines) —adding the delay d/c (shortest possible delay required to cross the gap) —and with the tunneling zone (dashed lines). The numbers of coincidence counts are normalized to 1 in the non-interfering region; for large values of τ, the ratio between the numbers of counts with and without the air gap is ≈|tG (ωi,d )|2=10-6. The signals presented here are the exact numerical results obtained from Eq. 3, hence the contrasts of the incident and transmitted signals are not exactly the same, as they would have been in the stationary phase approximation. Right: Combination of beamsplitters and delay lines that can be used to produce the transmission function tw . The transmission and reflection coefficients tk and rk characterising the beamsplitters, the phase shifts ϕ k , and the optical delays Lk are chosen such that the interference between the five paths of the optical circuit yields an overall transmission which is proportional to tw .

Fig. 5.
Fig. 5.

Ray optics trajectory between a reference wavefront before the first prism and another wavefront in the second prism, which is shorter than our prediction for the tunneling delay. If the waist w 0 of the incident beam is large enough, the distance h of this trajectory to the centroid is such that the corresponding intensity is comparable to that of the transmitted beam.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

t G = ( cosh ( d / z 0 ) i sinh ( d / z 0 ) 1 + n 2 cos ( 2 θ ) 2 n κ cos θ ) 1
{ E 1 ( + ) = ω a 1 ( ω ) e i ω ( t τ 1 ) E 2 ( + ) = ω t G ( ω ) a 2 ( ω ) e i ω ( t τ 2 ) e i k x x F
s = + d u e u 2 / 2 σ 2 [ t G ( ω i + u ) 2 Re ( t G ( ω i + u ) t G * ( ω i u ) e 2 i u τ ) ] .
S 0 t G ( ω i ) 2 ( 1 e 2 σ 2 ( τ μ ) 2 )
π = ( arg t G ) ω ω = ω i 1 ω i tan θ i arg t G θ i θ = θ i .

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