## Amplitude and phase modulation of time-energy entangled two-photon states

Optics Express, Vol. 16, Issue 21, pp. 16452-16458 (2008)

http://dx.doi.org/10.1364/OE.16.016452

Acrobat PDF (338 KB)

### Abstract

We experimentally demonstrate amplitude and phase modulation of a time-energy entangled two-photon wave function. The entangled photons are produced by spontaneous parametric down-conversion, spectrally dispersed in an prism compressor, modulated in amplitude and/or phase, and detected in coincidence by sum-frequency generation. First, we present a Fourier optical analysis of the optical setup yielding an analytic expression for the resulting field distribution at the exit plane of the shaping apparatus. We then introduce amplitude and/or phase shaping and present results which can only be obtained through a combination of the two. Specifically, we use a shaper-based interferometer to measure the two-photon interference of an almost bandwidth-limited two-photon wave function.

© 2008 Optical Society of America

## 1. Introduction

1. E. Schrödinger, “Die gegenwärtige Situation in der Quantenmechanik,” Naturwissenschaften **23**, 807–812, 823–828, 844–849 (1935). [CrossRef]

2. S. Popescu and D. Rohrlich, “The joy of entanglement,” in *Introduction to quantum computation and information*, H.-K. Lo, S. Popescu, and T. Spiller eds. (World Scientific, 1998), pp. 29–48. [CrossRef]

3. N. Gisin and R. Thew, “Quantum communication,” Nature Phot. **1**, 165–171 (2007). [CrossRef]

5. D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. **25**, 84–87 (1970). [CrossRef]

6. Z. Y. Ou and L. Mandel, “Violation of Bell’s inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. **61**, 50–53 (1988). [CrossRef] [PubMed]

7. Y. H. Shih and C. O. Alley, “New type of Einstein-Podolsky-Rosen-Bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett **61**, 2921–2924 (1988). [CrossRef] [PubMed]

9. P. G. Kwiat, A. M. Steinberg, R. Y. Chiao, P. H. Eberhard, and M. D. Petroff, “High-efficiency single-photon detectors,” Phys. Rev. A **48**, 867–870 (1993). [CrossRef]

10. R. Ghosh and L. Mandel, “Observation of nonclassical effects in the interference of two photons,” Phys. Rev. Lett. **59**, 1903–1905 (1987). [CrossRef] [PubMed]

11. P. G. Kwiat, W. A. Vareka, C. K. Hong, H. Nathel, and R. Y. Chiao, “Correlated two-photon interference in a dual-beam Michelson interferometer,” Phys. Rev. A **41**, 2910–2913 (1990). [CrossRef] [PubMed]

12. D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of Two-Photon “Ghost” Interference and Diffraction,” Phys. Rev. Lett. **74**, 3600–3603 (1995). [CrossRef] [PubMed]

13. T. B. Pittman, Y. H. Shih, D.V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A **52**, 3429–3432 (1995). [CrossRef]

14. J. Jacobson, G. Björk, I. Chuang, and Y. Yamamoto, “Photonic de Broglie waves,” Phys. Rev. Lett. **74**, 4835–4838 (1995). [CrossRef] [PubMed]

15. 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**, 2733–2736 (2000). [CrossRef] [PubMed]

5. D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. **25**, 84–87 (1970). [CrossRef]

16. 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]

17. J. D. Franson, “Bell inequality for position and time,” Phys. Rev. Lett. **62**, 2205–2208 (1989). [CrossRef] [PubMed]

18. Z. Y. Ou and Y. J. Lu, “Cavity enhanced spontaneous parametric down-conversion for the prolongation of correlation time between conjugate photons,” Phys. Rev. Lett. **83**, 2556–2559 (1999). [CrossRef]

19. M. Bellini, F. Marin, S. Viciani, A. Zavatta, and F. T. Arecchi, “Nonlocal Pulse Shaping with Entangled Photon Pairs,” Phys. Rev. Lett. **90**, 043602 (2003). [CrossRef] [PubMed]

20. S. Viciani, A. Zavatta, and M. Bellini, “Nonlocal modulations of the temporal and spectral profiles of an entangled photon pair,” Phys. Rev. A **69**, 053801 (2004). [CrossRef]

21. B. Dayan, A. Pe’er, A. A. Friesem, and Y. Silberberg, “Nonlinear Interactions with an Ultrahigh Flux of Broadband Entangled Photons,” Phys. Rev. Lett. **94**, 043602 (2005). [CrossRef] [PubMed]

22. B. Dayan, A. Pe’er, A. A. Friesem, and Y. Silberberg, “Temporal Shaping of Entangled Photons,” Phys. Rev. Lett. **94**, 073601 (2005). [CrossRef] [PubMed]

21. B. Dayan, A. Pe’er, A. A. Friesem, and Y. Silberberg, “Nonlinear Interactions with an Ultrahigh Flux of Broadband Entangled Photons,” Phys. Rev. Lett. **94**, 043602 (2005). [CrossRef] [PubMed]

22. B. Dayan, A. Pe’er, A. A. Friesem, and Y. Silberberg, “Temporal Shaping of Entangled Photons,” Phys. Rev. Lett. **94**, 073601 (2005). [CrossRef] [PubMed]

21. B. Dayan, A. Pe’er, A. A. Friesem, and Y. Silberberg, “Nonlinear Interactions with an Ultrahigh Flux of Broadband Entangled Photons,” Phys. Rev. Lett. **94**, 043602 (2005). [CrossRef] [PubMed]

## 2. Experimental realization

**94**, 043602 (2005). [CrossRef] [PubMed]

_{4}(PPKTP) crystal. Both photons had the same polarization and the entanglement was with respect to time-energy. The spectral bandwidth of the pump laser was approximately 5MHz and the maximum pump power 5W. Given a poling periodicity of

*G*=9

*µ*m, phase matching allowed for generating an approximately 50nm broad spectrum centered at twice the pump wavelength. The exact shape and bandwidth of the spectrum was dominated by the crystal temperature. Here, the temperature was set to 29.5 °C maximizing the spectral bandwidth as well as the conversion efficiency, which was on the order of 10

^{-7}. The focusing lens for the pump laser was selected according to the optimum focusing condition [23

23. G. D. Boyd and D. A. Kleinman, “Parametric Interaction of Focused Gaussian Light Beams,” J. Appl. Phys. **39**, 3597–3639 (1968). [CrossRef]

^{-9}. A spectral filter (4mm BG18) suppressed the remaining photon pairs and the sum-frequency photons were collected by a multi-mode fiber connected to a single-photon counter (PerkinElmer SPCM-AQR-15). Its efficiency at 532nm is about 33 times higher than at 1064 nm. Detecting the sum-frequency photons was essentially similar to a coincidence detection scheme with an extremely high temporal resolution [21

**94**, 043602 (2005). [CrossRef] [PubMed]

24. A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. **71**, 1929–1960 (2000). [CrossRef]

*z*axis. It is sufficient to treat the problem in one dimension,

*x*, as the other dimension, y, remains unaffected by the prism pairs. We use the transfer functions for free-space propagation

*T*(

_{P}*x*,

*ω*)=exp[

*iγ*(ω-

*ω*)

_{c}*x*], with the center frequency of the optical wavepacket

*ω*, its center wavelength

_{c}*λ*, and

_{c}*f*and the distances crystal–lens and lens–SLM are

*a*and

*b*, respectively, yielding a magnification of

*m*=

*b*/

*a*. Without loss of generality, we assume that the first prism follows immediately after the imaging lens and the second prism is located right before the intermediateSLM plane. Then, we find for a classical light field in the last image plane, i.e. at the position of the up-conversion crystal

## 3. Quantum optical description of the measurement

*ℳ*(

*ω*). While the pump field is treated classically, the signal and the idler fields are quantized. Signal and idler photons have the same polarization and, thus, experience the same index of refraction. The two-photon wave function generated in SPDC has been derived in reference [26

26. T. E. Keller and M. H. Rubin, “Theory of two-photon entanglement for spontaneous parametric down-conversion driven by a narrow pump pulse,” Phys. Rev. A **56**, 1534–1541 (1997). [CrossRef]

*E*̃

*(*

_{p}*ω*)=

*E*̃

_{p}*δ*(

*ω*−

*ω*) with a frequency

_{pc}*ω*and the two-photon wave function is

_{pc}*α*, Δ

*k*≐

*k*(

_{p}*ω*)−

_{p}*k*(

_{s}*ω*)−

_{s}*k*(

_{i}*ω*)−2

_{i}*π*/

*G*is the phase mismatch, and

*L*the length and

*G*the periodicity of the periodically poled crystal. The sum-frequency signal measured after the second crystal is proportional to the second order coherence function assuming that for a perfectly aligned setup there is no delay between the signal and the idler photon. That is, the coincidence rate is given by

*M*(

_{s}*ω*)

_{s}*M*(

_{i}*ω*−

_{pc}*ω*)=

_{s}*ℳ*(

*ω*).

_{s}## 4. Experimental results

*M*

_{s,i}(ω)]=0. Figure 2(a) shows the signal versus the position of a spectral edge filter. By moving the edge filter across the spectrum more and more frequencies are blocked and the signal drops to zero. Note that zero signal is reached exactly when one half of the spectrum is blocked; in other words, removing all idler photons from all photon pairs is sufficient to destroy all coincidences measured at the second crystal. When only a spectral slice is blocked and scanned across the spectrum, the signal drops to a minimal value, exhibits a small peak around the center frequency, and then increases back to the initial value, as shown in Fig. 2(b). The signal is minimal when either most of the idler or most of the signal photons are blocked. The height and shape of the central peak depend on the width of the spectral block compared to the spectrum. Here, the spectral block is almost half as wide as the spectrum and, consequently, the peak is barely visible. Figure 2(c) shows the signal as a function of the position of an amplitude grating. The signal exhibits the same periodicity as the grating, is zero when the grating is asymmetric and maximal when the grating is symmetric with respect to the center frequency. In the asymmetric case the idler photons of one half of all photon pairs and the signal photons of the other half of all photon pairs are blocked causing the overall coincidence signal to disappear. Although the overall number of photons has been reduced by only one half, the signal is zero because not a single photon pair is left intact. In the symmetric case the amplitude grating blocks all idler and all signal photons of the same photon pairs, but passes all other photon pairs unaffected.

*M*

_{s,i}(

*ω*)|=1. Various phase-only modulation examples have already been published by the Silberberg group [21

**94**, 043602 (2005). [CrossRef] [PubMed]

22. B. Dayan, A. Pe’er, A. A. Friesem, and Y. Silberberg, “Temporal Shaping of Entangled Photons,” Phys. Rev. Lett. **94**, 073601 (2005). [CrossRef] [PubMed]

27. B. Dayan, Y. Bromberg, I. Afek, and Y. Silberberg, “Spectral polarization and spectral phase control of time-energy entangled photons,” Phys. Rev. A **75**, 043804 (2007). [CrossRef]

*M*

_{s,i}(

*ω*)=exp[

*i*Φ

_{2}/2(

*ω*−

*ω*/2)

_{pc}^{2}], the two-photon wave function is smeared out in time. In such a broadened two-photon wave function the chances for a coincidence are reduced resulting in a decreasing signal with increasing |Φ

_{2}|, as seen in Fig. 3(a). The result is similar for the classical light field of a coherent short pulse. Observing the maximum signal at Φ

_{2}=0 confirms that the four-prism arrangement has been aligned such that it compensates for all positive quadratic dispersion. Next, we apply a V-shaped phase modulation. Such a phase modulation was already used in reference [21

**94**, 043602 (2005). [CrossRef] [PubMed]

*ξ*(

*ω*). From the result presented in Fig. 3(a) we can deduce a coherence time of the two-photon wave function of approximately 150 fs. In both experiments the theoretical predictions agree well with the experimental results.

*r*and the transmission

*t*of the beamsplitter and the time delay

*τ*. A pulse shaper allows mimicking a much more flexible transfer function, such as

*γ*=1 and

*ϕ*=0 eq. 8 resembles eq. 7 and an unbalanced Michelson interferometer may be simulated. The measured coincidence rate versus time delay, i.e. the two-photon interference, for

*γ*=1 and

*ϕ*=0 is shown Fig. 4(a). The signal oscillates with a periodicity that is determined by the frequency

*ω*/2. The inset indicates that by selecting

_{pc}*ϕ*=0 or

*ϕ*=

*π*allows to switch between the two output ports of the simulated interferometer; the two signals are exactly half a period out of phase. In order to measure these two signals with a real Michelson interferometer would require to move the coincidence detection apparatus from one exit port of the beam splitter to the other. If we select

*γ*=0, only the slowly varying amplitudes of the signal and idler photons are delayed in time leaving their carrier frequencies unaffected. The results change quite dramatically, i.e. the oscillations completely disappear, as seen in Fig. 4(b). The results in Fig. 4(b) are especially interesting, because the two curves can readily be used to extract the fringe visibility as a function of time delay. From both measurements we can extract the coherence properties of the two-photon wave function, which must be close to bandwidth-limited because all second order dispersion has been compensated for.

## 5. Conclusion

## Acknowledgments

## References and links

1. | E. Schrödinger, “Die gegenwärtige Situation in der Quantenmechanik,” Naturwissenschaften |

2. | S. Popescu and D. Rohrlich, “The joy of entanglement,” in |

3. | N. Gisin and R. Thew, “Quantum communication,” Nature Phot. |

4. | D. Bouwmeester, A. Ekert, and A. Zeilinger |

5. | D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. |

6. | Z. Y. Ou and L. Mandel, “Violation of Bell’s inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. |

7. | Y. H. Shih and C. O. Alley, “New type of Einstein-Podolsky-Rosen-Bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett |

8. | A. A. Malygin, A. N. Penin, and A. V. Sergienko, “Absolute calibration of the sensitivity of photodetectors using a biphotonic field,” Sov. Phys. JETP Lett. |

9. | P. G. Kwiat, A. M. Steinberg, R. Y. Chiao, P. H. Eberhard, and M. D. Petroff, “High-efficiency single-photon detectors,” Phys. Rev. A |

10. | R. Ghosh and L. Mandel, “Observation of nonclassical effects in the interference of two photons,” Phys. Rev. Lett. |

11. | P. G. Kwiat, W. A. Vareka, C. K. Hong, H. Nathel, and R. Y. Chiao, “Correlated two-photon interference in a dual-beam Michelson interferometer,” Phys. Rev. A |

12. | D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of Two-Photon “Ghost” Interference and Diffraction,” Phys. Rev. Lett. |

13. | T. B. Pittman, Y. H. Shih, D.V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A |

14. | J. Jacobson, G. Björk, I. Chuang, and Y. Yamamoto, “Photonic de Broglie waves,” Phys. Rev. Lett. |

15. | 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. |

16. | C. K. Hong, Z. Y. Ou, and L. Mandel “Measurement of Subpicosecond Time Intervals between Two Photons by Interference,” Phys. Rev. Lett. |

17. | J. D. Franson, “Bell inequality for position and time,” Phys. Rev. Lett. |

18. | Z. Y. Ou and Y. J. Lu, “Cavity enhanced spontaneous parametric down-conversion for the prolongation of correlation time between conjugate photons,” Phys. Rev. Lett. |

19. | M. Bellini, F. Marin, S. Viciani, A. Zavatta, and F. T. Arecchi, “Nonlocal Pulse Shaping with Entangled Photon Pairs,” Phys. Rev. Lett. |

20. | S. Viciani, A. Zavatta, and M. Bellini, “Nonlocal modulations of the temporal and spectral profiles of an entangled photon pair,” Phys. Rev. A |

21. | B. Dayan, A. Pe’er, A. A. Friesem, and Y. Silberberg, “Nonlinear Interactions with an Ultrahigh Flux of Broadband Entangled Photons,” Phys. Rev. Lett. |

22. | B. Dayan, A. Pe’er, A. A. Friesem, and Y. Silberberg, “Temporal Shaping of Entangled Photons,” Phys. Rev. Lett. |

23. | G. D. Boyd and D. A. Kleinman, “Parametric Interaction of Focused Gaussian Light Beams,” J. Appl. Phys. |

24. | A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. |

25. | J. C. Diels and W. Rudolph, |

26. | T. E. Keller and M. H. Rubin, “Theory of two-photon entanglement for spontaneous parametric down-conversion driven by a narrow pump pulse,” Phys. Rev. A |

27. | B. Dayan, Y. Bromberg, I. Afek, and Y. Silberberg, “Spectral polarization and spectral phase control of time-energy entangled photons,” Phys. Rev. A |

**OCIS Codes**

(270.5570) Quantum optics : Quantum detectors

(320.5540) Ultrafast optics : Pulse shaping

(190.4223) Nonlinear optics : Nonlinear wave mixing

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: June 18, 2008

Revised Manuscript: August 21, 2008

Manuscript Accepted: September 24, 2008

Published: October 1, 2008

**Citation**

F. Zäh, M. Halder, and T. Feurer, "Amplitude and phase modulation of time-energy entangled two-photon states," Opt. Express **16**, 16452-16458 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-21-16452

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### References

- E. Schrödinger, "Die gegenwärtige Situation in der Quantenmechanik," Naturwissenschaften 23, 807-812, 823-828, 844-849 (1935). [CrossRef]
- S. Popescu and D. Rohrlich, "The joy of entanglement," in Introduction to quantum computation and information, H.-K. Lo, S. Popescu, T. Spiller, eds. (World Scientific, 1998), pp. 29-48. [CrossRef]
- N. Gisin and R. Thew, "Quantum communication," Nature Phot. 1, 165-171 (2007). [CrossRef]
- D. Bouwmeester, A. Ekert, and A. Zeilinger The Physics of Quantum Information (Springer, 2000).
- D. C. Burnham and D. L. Weinberg, "Observation of simultaneity in parametric production of optical photon pairs," Phys. Rev. Lett. 25, 84-87 (1970). [CrossRef]
- Z. Y. Ou and L. Mandel, "Violation of Bell�??s inequality and classical probability in a two-photon correlation experiment," Phys. Rev. Lett. 61, 50-53 (1988). [CrossRef] [PubMed]
- Y. H. Shih and C. O. Alley, "New type of Einstein-Podolsky-Rosen-Bohm experiment using pairs of light quanta produced by optical parametric down conversion," Phys. Rev. Lett 61, 2921-2924 (1988). [CrossRef] [PubMed]
- A. A. Malygin, A. N. Penin, and A. V. Sergienko, "Absolute calibration of the sensitivity of photodetectors using a biphotonic field," Sov. Phys. JETP Lett. 33, 477-480 (1981).
- P. G. Kwiat, A. M. Steinberg, R. Y. Chiao, P. H. Eberhard, and M. D. Petroff, "High-efficiency single-photon detectors," Phys. Rev. A 48, 867-870 (1993). [CrossRef]
- R. Ghosh and L. Mandel, " Observation of nonclassical effects in the interference of two photons," Phys. Rev. Lett. 59, 1903-1905 (1987). [CrossRef] [PubMed]
- P. G. Kwiat, W. A. Vareka, C. K. Hong, H. Nathel, and R. Y. Chiao, "Correlated two-photon interference in a dual-beam Michelson interferometer," Phys. Rev. A 41, 2910-2913 (1990). [CrossRef] [PubMed]
- D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, "Observation of Two-Photon "Ghost" Interference and Diffraction," Phys. Rev. Lett. 74, 3600-3603 (1995). [CrossRef] [PubMed]
- T. B. Pittman, Y. H. Shih, D.V. Strekalov, and A. V. Sergienko, "Optical imaging by means of two-photon quantum entanglement," Phys. Rev. A 52, 3429-3432 (1995). [CrossRef]
- J. Jacobson, G. Björk, I. Chuang, and Y. Yamamoto, "Photonic de Broglie waves," Phys. Rev. Lett. 74, 4835-4838 (1995). [CrossRef] [PubMed]
- 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, 2733-2736 (2000). [CrossRef] [PubMed]
- 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]
- J. D. Franson, "Bell inequality for position and time," Phys. Rev. Lett. 62, 2205-2208 (1989). [CrossRef] [PubMed]
- Z. Y. Ou and Y. J. Lu, "Cavity enhanced spontaneous parametric down-conversion for the prolongation of correlation time between conjugate photons," Phys. Rev. Lett. 83, 2556-2559 (1999). [CrossRef]
- M. Bellini, F. Marin, S. Viciani, A. Zavatta, and F. T. Arecchi, "Nonlocal Pulse Shaping with Entangled Photon Pairs," Phys. Rev. Lett. 90, 043602 (2003). [CrossRef] [PubMed]
- S. Viciani, A. Zavatta, and M. Bellini, "Nonlocal modulations of the temporal and spectral profiles of an entangled photon pair," Phys. Rev. A 69, 053801 (2004). [CrossRef]
- B. Dayan, A. Pe�??er, A. A. Friesem, and Y. Silberberg, "Nonlinear Interactions with an Ultrahigh Flux of Broadband Entangled Photons," Phys. Rev. Lett. 94, 043602 (2005). [CrossRef] [PubMed]
- B. Dayan, A. Pe�??er, A. A. Friesem, and Y. Silberberg, "Temporal Shaping of Entangled Photons," Phys. Rev. Lett. 94, 073601 (2005). [CrossRef] [PubMed]
- G. D. Boyd and D. A. Kleinman, "Parametric Interaction of Focused Gaussian Light Beams," J. Appl. Phys. 39, 3597-3639 (1968). [CrossRef]
- A. M. Weiner, "Femtosecond pulse shaping using spatial light modulators," Rev. Sci. Instrum. 71, 1929-1960 (2000). [CrossRef]
- J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic Press, 1996).
- T. E. Keller and M. H. Rubin, "Theory of two-photon entanglement for spontaneous parametric down-conversion driven by a narrow pump pulse," Phys. Rev. A 56, 1534-1541 (1997). [CrossRef]
- B. Dayan, Y. Bromberg, I. Afek, and Y. Silberberg, "Spectral polarization and spectral phase control of timeenergy entangled photons," Phys. Rev. A 75, 043804 (2007). [CrossRef]

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