## Quantum holography

Optics Express, Vol. 9, Issue 10, pp. 498-505 (2001)

http://dx.doi.org/10.1364/OE.9.000498

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### Abstract

We propose to make use of quantum entanglement for extracting holographic information about a remote 3-D object in a confined space which light enters, but from which it cannot escape. Light scattered from the object is detected in this confined space entirely without the benefit of spatial resolution. Quantum holography offers this possibility by virtue of the fourth-order quantum coherence inherent in entangled beams.

© Optical Society of America

## 1. Introduction

1. E. Schrödinger, “Die gegenwartige Situation in der Quantenmechanik,” Naturwissenchaften23, 807–812, 823–828, 844–849 (1935). [CrossRef]

3. D. Gabor, “A new microscopic principle,” Nature **161**, 777–778 (1948). [CrossRef] [PubMed]

5. J. F. Clauser and A. Shimony, “Bell’s Theorem. Experimental tests and implications,” Rep. Prog. Phys. **41**, 1881–1927 (1978). [CrossRef]

*any*point on its surface, whether scattered or not, but is totally incapable of discerning the location at which the photon arrives.

## 2. Method

_{k}〉 of the mode with wave vector

**k**,

*δ*is the Dirac delta function, and the state probability amplitude

*φ*(

**x**) is normalized such that

**∫**

_{S}

*d*

**x**|

*φ*(

**x**)|

^{2}=1. Here |

*φ*(

**x**)|

^{2}represents the probability density that a photon pair is emitted from point

**x**in the source plane. As a consequence of the state in Eq. (1), each photon is individually in a mixed state (described by the density operator

**=**ρ ^

**∫**

_{S}

*d*

**x**|

*φ*(

**x**)|

^{2}|1

_{x}〉〈1

_{x}|) that exhibits no second-order coherence [9

9. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. **87**, 123602 (2001). [CrossRef] [PubMed]

*φ*(

**x**) represents the spatial distribution of the pump field [8].

**S**, the one directed through the opening of the chamber may (or may not) be scattered from the object and impinges on the chamber wall at position

**x**

_{1}∊C, where C represents the set of points on the chamber wall. The optical system between the source and the chamber, idealized as a simple lens in Fig. 1, as well as everything inside the chamber including the object, is assumed to be linear and is characterized by an impulse response function

*h*

_{1}(

**x**

_{1},

**x**). The other photon is transmitted through a linear optical system characterized by an impulse response function

*h*

_{2}(

**x**

_{2},

**x**), where

**x**

_{2}∊D the single-photon-sensitive scanning (or array) detector.

**x**

_{1}and

**x**

_{2}is described by a probability density

*p*(

**x**

_{1},

**x**

_{2}) given by

^{7}

*p*(

**x**

_{1},

**x**

_{2}) may be regarded as the coherent image of a point

**x**

_{1}∊C formed through an optical system represented by the following cascade (see Fig. 1): propagation through

*h*

_{1}in the reverse direction toward the source (from

**x**

_{1}to

**x**), modulation by

*φ*(

**x**) at the source, and subsequent transmission from the source through the system

*h*

_{2}to the point

**x**

_{2}. Equation (2) may therefore be written symbolically as follows:

*p*(

**x**

_{1},

**x**

_{2})∝|

*h*

_{2}∗φ·

*h*

_{1}|

^{2}, where ∗ represents transmission through a linear system (convolution in the shift-invariant case) and · represents multiplication or modulation. The expression is to be read in reverse order, from right to left, as is the custom in operator algebra.

**x**

_{1}∊C on the chamber wall (C is a bucket detector) we must integrate over C, whereupon the coincidence rate in Eq. (2) becomes

*g*

_{1}(

**x**,

**x**′)=

**∫**

_{C}

*d*

**x**

_{1}

*h*

_{1}(

**x**

_{1},

**x**)

*h**

_{1}(

**x**

_{1},

**x′**). The function

*p̄*

_{2}(

**x**

_{2}) is therefore the marginal probability density of detecting one photon at

**x**

_{2}and another at any point

**x**

_{1}∊C. In spite of this integration, it is clear from Eq. (3) that

*p̄*

_{2}(

**x**

_{2}) contains information about the system

*h*

_{1}, and therefore about the object, via the function

*g*

_{1}. The function

*p̄*

_{2}(

**x**

_{2}) is the incoherent superposition of many coherent images of the form given in Eq. (2), originating from all points of C. This is therefore a modal expansion of a partially coherent system [12]. It is important to note the distinction between

*p̄*

_{2}(

**x**

_{2}) (which traces over C) and the singles rate measured by the detector array D, which results from tracing over the other photon (i.e., the photon incident on the chamber) in the two-photon state in Eq. (1). This distinction is highlighted in Ref. [9

9. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. **87**, 123602 (2001). [CrossRef] [PubMed]

## 3. Example: Scattering objects

**x**

^{(1)}inside C as depicted in Fig. 2. The system

*h*

_{1}comprises two contributions. The first is a direct path to the chamber wall, represented by the system

*h*

^{(0)}(see Fig. 2). The second is a scattering path to the chamber wall, represented by the illumination system

**x**

^{(1)}), and the system

*h*

^{(1)}that carries light from the scatterer to the chamber wall. These two paths are mutually coherent, so that the probability amplitudes of the two paths are added, thereby leading to

**x**

_{2}) is the marginal coincidence rate in absence of the scatterer. It is identical to that in Eq. (3) with

*h*

_{1}replaced by

*h*

^{(0)}. This term represents the direct path in Fig. 2. (2) The second term

**x**

_{2}) is the marginal coincidence rate arising from the scatterer alone, and is represented by the scattering path in Fig. 2. (3) The third term represents interference between these two paths, and is therefore the term of interest for quantum holography. It is the fourth-order analog of second-order interference in Gabor’s original conception of holography [3

3. D. Gabor, “A new microscopic principle,” Nature **161**, 777–778 (1948). [CrossRef] [PubMed]

*r*(

**x**

_{2},

**x**

^{(1)}) and

*q*(

**x**

_{2},

**x**

^{(1)}), which are defined in Eqs. (8) and (9), respectively, by the symbolic relations:

*r*=

*h**

_{2}∗

*φ**·

*h*

^{(0)}

^{*}∗

*h*

^{(1)}and

*q*=

*h*

_{2}∗φ·

*r*(

**x**

_{2},

**x**

^{(1)}) is the image formed by a point at the location of the scatterer

**x**

^{(1)}through a cascade of the systems

*h*

^{(1)}(traveling forward) and

*h*

^{(0)}(traveling backward), followed by modulation by

*φ*, and finally traveling forward through the system

*h*

_{2}to the point

**x**

_{2}. This is the term that includes the holographic information. The quantity

*q*(

**x**

_{2},

**x**

_{(1)}), by which

*r*is multiplied in Eq. (5), is the image of a point at

**x**

^{(1)}traveling backward through

*φ*and then forward propagation through

*h*

_{2}. If the optical system is designed such that

*q*is independent of

**x**

^{(1)}and becomes unimportant. Note that integration over the area of the chamber is essential for achieving quantum holography. Thus a point detector, for example [8], cannot be used for this purpose by virtue of Eqs. (8)–(10). Furthermore, ray tracing techniques, such as those used in used in Ref. [13

13. T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Twophoton geometric optics,” Phys. Rev. A **53**, 2804–2815 (1996). [CrossRef] [PubMed]

*N*static scatterers are located at positions

**x**

^{(j)},

*j*=1..

*N*, inside C, whereupon the impulse response function

*h*

_{1}becomes

**x**

_{2}), that due to the scatterers alone, includes the sum of the contributions of each scatterer independently, and terms resulting from interference amongst the scatterers. The third term in Eq. (12) includes the holographic information. The results can then be generalized to any object by replacing the discrete summation in Eq. (11) by an integral. The results also apply to objects that do not scatter.

*p̄*

_{2}(

**x**

_{2}) is holographic by virtue of Eq. (12). This equation has the structure of a conventional hologram obtained by illuminating the scatterers with coherent light through a composite system involving the optics of both beams, with the state probability amplitude

*φ*(

**x**) serving as an effective coherent aperture. This result is consistent with the duality between entanglement and coherence [8]. The function

*p̄*

_{2}(

**x**

_{2}), recorded in a computer via the observation of coincidences over an adequate period of time, is the hologram of the object in the chamber. The process of reconstruction can be subsequently implemented by recording the hologram on a photographic film or electronically on a micromirror array [14

14. T. Kreis, P. Aswendt, and R. Höfling, “Hologram reconstruction using a digital micromirror device,” Opt. Eng. **40**, 926–933 (2001). [CrossRef]

*t*is proportional to the marginal coincidence rate,

*t*(

**x**

_{2})∝

*p̄*

_{2}(

**x**

_{2}), where

*p̄*

_{2}(

**x**

_{2}) is given by Eq. (5). To reconstruct the object, the transparency is illuminated by a coherent plane wave. The wave transmitted through the hologram will then contain four components [15

15. B. E. A. Saleh and M. C. Teich, *Fundamentals of Photonics* (Wiley, New York, 1991), *Ch. 4.* [CrossRef]

**x**

_{2}) and

**x**

_{2}) in Eq. (5) are approximately plane waves since these terms are nearly independent of

**x**

_{2}. The third and fourth terms in Eq. (5) give rise to the other two waves, representing the desired reconstructed object and its conjugate [15

15. B. E. A. Saleh and M. C. Teich, *Fundamentals of Photonics* (Wiley, New York, 1991), *Ch. 4.* [CrossRef]

*h*

^{(1)},

*h*

^{(0)}, and

*h*

^{2}.

## 4. Conclusion

*The main conclusion of our analysis is that*

*p̄*

_{2}(

**x**

_{2}) (

*the marginal probability density of detecting one photon at*

**x**

_{2}

*and another at any point*

**x**

_{1}∊C)

*is a hologram of the 3-D object concealed in the chamber. It may then be recorded on a 2-D photographic plate and viewed with ordinary light in the usual fashion of holographic reconstruction*.

9. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. **87**, 123602 (2001). [CrossRef] [PubMed]

## 5. Acknowledgements

## References and links

1. | E. Schrödinger, “Die gegenwartige Situation in der Quantenmechanik,” Naturwissenchaften23, 807–812, 823–828, 844–849 (1935). [CrossRef] |

2. | A. Einstein in |

3. | D. Gabor, “A new microscopic principle,” Nature |

4. | D. Gabor, “Microscopy by reconstructed wavefronts, I,” Proc. Roy. Soc. (London) |

5. | J. F. Clauser and A. Shimony, “Bell’s Theorem. Experimental tests and implications,” Rep. Prog. Phys. |

6. | D. N. Klyshko, |

7. | B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A |

8. | A. V. Belinskii and D. N. Klyshko, “Two-photon optics: diffraction, holography, and transformation of two-dimensional signals,” Zh. Eksp. Teor. Fiz.105, 487–493 (1994) [Sov. Phys. JETP78, 259–262 (1994)]. |

9. | A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. |

10. | W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes. I,” Phys. Rev. |

11. | D. N. Klyshko, “Coherent decay of photons in a nonlinear medium,” Pis’ma Zh. Eksp. Teor. Fiz.6, 490–492 (1967) [Sov. Phys. JETP Lett.6, 23–25 (1967)]. |

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

13. | T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Twophoton geometric optics,” Phys. Rev. A |

14. | T. Kreis, P. Aswendt, and R. Höfling, “Hologram reconstruction using a digital micromirror device,” Opt. Eng. |

15. | B. E. A. Saleh and M. C. Teich, |

**OCIS Codes**

(090.0090) Holography : Holography

(190.0190) Nonlinear optics : Nonlinear optics

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: October 1, 2001

Published: November 5, 2001

**Citation**

Ayman Abouraddy, Bahaa Saleh, Alexander Sergienko, and Malvin Teich, "Quantum holography," Opt. Express **9**, 498-505 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-10-498

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### References

- E. Schr�dinger, "Die gegenwartige Situation in der Quantenmechanik," Naturwissenchaften 23, 807-812, 823-828, 844-849 (1935). [CrossRef]
- A. Einstein, in The Born-Einstein Letters (Walker, New York, 1971), p. 158, translated by I. Born.
- D. Gabor, "A new microscopic principle," Nature 161, 777-778 (1948). [CrossRef] [PubMed]
- D. Gabor, "Microscopy by reconstructed wavefronts, I," Proc. Roy. Soc. (London) A197, 454 (1949).
- J. F. Clauser, A. Shimony, "Bell's Theorem. Experimental tests and implications," Rep. Prog. Phys. 41, 1881-1927 (1978). [CrossRef]
- D. N. Klyshko, Photons and Nonlinear Optics (Nauka, Moscow, 1980) [translation: Gordon and Breach, New York, 1988].
- B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, "Duality between partial coherence and partial entanglement," Phys. Rev. A 62, 043816 (2000). [CrossRef]
- A. V. Belinskii and D. N. Klyshko, "Two-photon optics: diffraction, holography, and transformation of two-dimensional signals," Zh. Eksp. Teor. Fiz. 105, 487-493 (1994) [Sov. Phys. JETP 78, 259-262 (1994)].
- A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Role of entanglement in two-photon imaging," Phys. Rev. Lett. 87, 123602 (2001). [CrossRef] [PubMed]
- W. H. Louisell, A. Yariv, and A. E. Siegman, "Quantum fluctuations and noise in parametric processes. I," Phys. Rev. 124, 1646-1654 (1961). [CrossRef]
- D. N. Klyshko, "Coherent decay of photons in a nonlinear medium," Pis'ma Zh. Eksp. Teor. Fiz. 6, 490-492 (1967) [Sov. Phys. JETP Lett. 6, 23-25 (1967)].
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, New York, 1995).
- T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, "Two-photon geometric optics," Phys. Rev. A 53, 2804-2815 (1996). [CrossRef] [PubMed]
- T. Kreis, P. Aswendt, and R. H�fling, "Hologram reconstruction using a digital micromirror device," Opt. Eng. 40, 926-933 (2001). [CrossRef]
- B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Ch. 4. [CrossRef]

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