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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 24 — Nov. 21, 2011
  • pp: 24228–24240
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Quantum imaging with N-photon states in position space

E. Brainis  »View Author Affiliations


Optics Express, Vol. 19, Issue 24, pp. 24228-24240 (2011)
http://dx.doi.org/10.1364/OE.19.024228


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Abstract

We investigate the physics of quantum imaging with N > 2 entangled photons in position space. It is shown that, in paraxial approximation, the space-time propagation of the quantum state can be described by a generalized Huygens-Fresnel principle for the N-photon wave function. The formalism allows the initial conditions to be set on multiple reference planes, which is very convenient to describe the generation of multiple photon pairs in separate thin crystals. Applications involving state shaping and spatial entanglement swapping are developed.

© 2011 OSA

1. Introduction

In the recent years, there has been a growing interest in producing quantum states of light in which more than two photons are entangled. So far, up to six time-entangled photons have been produced through entanglement swapping [1

1. C. Lu, X. Zhou, O. Guhne, W. Gao, J. Zhang, Z. Yuan, A. Goebel, T. Yang, and J. Pan, “Experimental entanglement of six photons in graph states,” Nat. Phys. 3, 91–95 (2007). [CrossRef]

] and photon triplets start being produced through cascaded nonlinear processes [2

2. H. Hübel, D. R. Hamel, A. Fedrizzi, S. Ramelow, K. J. Resch, and T. Jennewein, “Direct generation of photon triplets using cascaded photon-pair sources,” Nature 466, 601–603 (2010). [CrossRef] [PubMed]

] or photon number post-selection in a single parametric process [3

3. E. Waks, E. Diamanti, and Y. Yamamoto, “Generation of photon number states,” New J. Phys. 8, 4–8 (2006). [CrossRef]

]. Multi-photon interferometry techniques have been developed to demonstrate the hyper-entanglement of a large number of qubits [4

4. W.-B. Gao, C.-Y. Lu, X.-C. Yao, P. Xu, O. Gühne, A. Goebel, Y.-A. Chen, C.-Z. Peng, Z.-B. Chen, and J.-W. Pan, “Experimental demonstration of a hyper-entangled ten-qubit Schrödinger cat state,” Nat. Phys. 6, 331–335 (2010). [CrossRef]

] as well as phase super-sensitivity [5

5. T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the Standard Quantum Limit with Four-Entangled Photons,” Science 316, 726–729 (2007). [CrossRef] [PubMed]

, 6

6. R. Okamoto, H. F. Hofmann, T. Nagata, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit: phase super-sensitivity of N-photon interferometers,” New J. Phys. 10, 073033 (2008). [CrossRef]

] (for a recent revue on N-photon entanglement and interferometry see [7

7. J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multi-photon entanglement and interferometry,” to appear in Rev. Mod. Phys. (2011).

]). Spatial entanglement [8

8. S. P. Walborn, C. H. Monken, S. Pádua, and P. H. Souto Ribeiro, “Spatial correlations in parametric down-conversion,” Phys. Rep. 495, 87–139 (2010). [CrossRef]

] is particularly interesting for quantum information processing and communication because large Hilbert spaces can be easily manipulated using passive masks or active spatial light modulator [9

9. L. Neves, S. Pádua, and C. Saavedra, “Controlled generation of maximally entangled qudits using twin photons,” Phys. Rev. A 69, 042305 (2004). [CrossRef]

14

14. C. Bonato, S. Bonora, A. Chiuri, P. Mataloni, G. Milani, G. Vallone, and P. Villoresi, “Phase control of a path-entangled photon state by a deformable membrane mirror,” J. Opt. Soc. Am. B 27, A175–A180 (2010). [CrossRef]

]. It can also be used to improve imaging resolution (quantum super-resolution imaging, quantum lithography) [15

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]

, 16

16. P. Kok, A. N. Boto, D. S. Abrams, C. P. Williams, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: Towards arbitrary two-dimensional patterns,” Phys. Rev. A 63, 063407 (2001). [CrossRef]

] and produce distributed or “ghost” images [17

17. 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]

19

19. 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]

]. Schemes to realise spatial N-photon entanglement (with N > 2) have been proposed [20

20. T. E. Keller, M. H. Rubin, Y. Shih, and L.-A. Wu, “Theory of the three-photon entangled state,” Phys. Rev. A 57, 2076–2079 (1998). [CrossRef]

, 21

21. J. Wen, E. Oh, and S. Du, “Tripartite entanglement generation via four-wave mixings: narrowband triphoton W state,” J. Opt. Soc. Am. B 27, A11–A20 (2010). [CrossRef]

]. One can foresee that the manipulation of spatial entanglement of few-photon states of light [22

22. J. Wen, M. H. Rubin, and Y. Shih, “Transverse correlations in multiphoton entanglement,” Phys. Rev. A 76, 045802 (2007). [CrossRef]

25

25. J. Wen, S. Du, and M. Xiao, “Improving spatial resolution in quantum imaging beyond the Rayleigh diffraction limit using multiphoton W entangled states,” Phys. Lett. A 374, 3908 – 3911 (2010). [CrossRef]

] will become an important research field in the forthcoming decade.

In this work, we investigate the physics of quantum imaging with N > 2 entangled photons. We describe the quantum state of the electromagnetic field using a N-particle wave function in position representation. Position representation is preferred to momentum representation because photon-position is what is actually detected by the photon-counters in a quantum optics experiment. However, in many experiments on spatial entanglement that have been performed so far, photons are generated by spontaneous parametric down-conversion in a nonlinear crystal and detected in the crystal far-field; in this particular case, people usually prefer working in momentum representation of the photon state since the position correlations in the far-field mimic the momentum correlations in the nonlinear crystal plane. In the case of an arbitrary optical system (made of lenses, mirror, masks, beam splitters, ...) between the crystal plane and the detection region, the use of the momentum representation for the field in the crystal plane will involve mixed propagators g(k, rj), where rj are the positions of the photon-counters and k wave vectors [22

22. J. Wen, M. H. Rubin, and Y. Shih, “Transverse correlations in multiphoton entanglement,” Phys. Rev. A 76, 045802 (2007). [CrossRef]

, 23

23. J. Wen, P. Xu, M. H. Rubin, and Y. Shih, “Transverse correlations in triphoton entanglement: Geometrical and physical optics,” Phys. Rev. A 76, 023828 (2007). [CrossRef]

]. Such propagators arise because the photons generated through nonlinear interactions are first regarded as populating an ensemble of plane-wave modes (momentum representation). Then, when it comes to calculate the joint detection probabilities at some selected positions, the local interference of all these modes (that is the position representation) is calculated. Instead, if position representation is chosen from the beginning, the propagators have the usual and intuitive point-to-point form h(r, rj). For applications of the photon position representation to quantum imaging problems with bi-photons (N = 2), see [11

11. R. Shimizu, K. Edamatsu, and T. Itoh, “Quantum diffraction and interference of spatially correlated photon pairs and its Fourier-optical analysis,” Phys. Rev. A 74, 013801 (2006). [CrossRef]

, 12

12. W. H. Peeters, J. J. Renema, and M. P. van Exter, “Engineering of two-photon spatial quantum correlations behind a double slit,” Phys. Rev. A 79, 043817 (2009). [CrossRef]

].

Application of standard Fourier optics techniques to the position representation wave function shows that the propagation of N photons through an optical system can be described by a generalized Huygens-Fresnel principle. This has been first noticed for biphoton states produced by parametric down-conversion in nonlinear crystals [26

26. 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]

28

28. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Entangled-photon Fourier optics,” J. Opt. Soc. Am. B 19, 1174–1184 (2002). [CrossRef]

]. Our generalization to N-photon states brings new interesting features: (i) it introduces time in the diffraction integral (giving a space-time picture of entanglement propagation), (ii) it shows how to deal with interferometers in the photon path, and most importantly (iii) the formalism applies to quantum systems made of an arbitrary number of photons, possibly emitted by sources that are located in different transverse planes Σj. To our best knowledge, this last situation has never been considered before in the context of quantum imaging despite its practical importance (see the generation scheme in [1

1. C. Lu, X. Zhou, O. Guhne, W. Gao, J. Zhang, Z. Yuan, A. Goebel, T. Yang, and J. Pan, “Experimental entanglement of six photons in graph states,” Nat. Phys. 3, 91–95 (2007). [CrossRef]

], for instance).

The article is organized as follows. In Sec. 2, we review the foundations of the position wave function representation of light. Then, we turn to paraxial approximation, derive the generalized Huygens-Fresnel principle (Sec. 3) and discuss the connection between the photon wave function and photodetection probabilities (Sec. 4). Finally, in Sec. 5 and 6, we apply the general formalism to specific examples. The example of Sec. 5 shows how an initial 3-photon position entanglement can be used to shape a 2-photon wave function through the detection of the third photon. In Sec. 6, we analyse an entanglement-swapping scheme acting on a pair of bi-photons produced in two different nonlinear crystals: combining the detection of two photons with wave front shaping of remaining two photons, entangled images can be created.

2. Photon wave functions in position representation

Replacing the complex coefficients f±(k) by annihilation operators â±(k) in Eq. (1), the fundamental photon field is found to be proportional to the positive frequency part of the electric field:
Ψ^(r,t)=h=±d3kh¯kceh(k)a^h(k)ei(krkct)(2π)3/2=i2ɛ0E^(+)(r,t).
Note that Ψ̂ ∝ Ê(+) holds information about both electric and magnetic field. Therefore, it provides a complete information about electromagnetic configuration. To show this explicitly, one decomposes Ψ̂ again into its helicity components ψ̂+ and ψ̂ and subtract them: this yields B^(+)=μ0/2(ψ^+ψ^), the positive frequency part of the magnetic field. In the second quantization formalism, the state of a single-photon wave packet writes: |Ψ〉 = Σh ∫d3k fh(k) |1k,h〉, where |1k,h=ah(k)|0 and f±(k) are the same spectral amplitudes that appear in Eq. (1). The connection between the first and second quantization formalism is given by the relation Ψ(r,t)=0|Ψ^(r,t)|Ψ=i2ɛ00|E^(+)(r,t)|Ψ. Since 〈Φ|Ψ̂(r,t)|Ψ〉 = 0 for all |Φ〉 ≠ |0〉, we have Ψi*(q)Ψi(q)=Ψ|Ψ^i(q)Ψ^i(q)|Ψ=2ɛ0E^i()(q)E^i(+)(q) for any pair of points q = (r,t) and q′ = (r′,t′), where the indexes (i,i′) ∈ {x,y,z}2 represent Cartesian components. This relates the Bialynicki-Birula-Sipe wave-function to the usual first-order correlation functions of coherence theory. Therefore, in the context of first-order perturbation theory and dipole moment interaction with matter, |Ψ(r,t)|2 is proportional to the photon absorption (detection) probability at point r at time t (see Sec. 4). In addition, ∫|Ψ(r,t)|2d3r = 〈Ψ|Ĥ|Ψ〉 is the expectation value of the photon energy [32

32. I. Bialynicki-Birula, “On the wave function of the photon,” Acta Phys. Pol. A 86, 97–116 (1994).

34

34. J. E. Sipe, “Photon wave functions,” Phys. Rev. A 52, 1875–1883 (1995). [CrossRef] [PubMed]

]. As a consequence, |Ψ(r,t)|2 can also be interpreted as a spatial density of electromagnetic energy at time t. Strictly speaking, one cannot interpret |Ψ(r,t)|2 in terms of photon probability density in position space despite it is a measure of photon energy localization [32

32. I. Bialynicki-Birula, “On the wave function of the photon,” Acta Phys. Pol. A 86, 97–116 (1994).

35

35. Note that the quantum mechanical scalar product Ψ(2)|Ψ(1)=h=±d3k[f±(2)(k)]*f±(1)(k)=d3r1d3r2[Ψ(2)(r2)]*Ψ(1)(r1)𝒲(r1r2) is evaluate using a double integral in the position representation with a non local kernel 𝒲 (ρ) = (h̄c2π2|ρ|2)−1.

].

The generalization to N-photon states
|Ψ=h1,,hNd3k1d3kNfh1,,hN(k1,,kN)|1k1,h1,,1kN,hN
is straightforward. The connection between wave functions and fields is given by
Ψi1iN(q1,,qN)=(i)N(2ɛ0)N/20|E^iN(+)(qN)E^i1(+)(q1)|Ψ
(3)
and
Ψi1iN*(q1,,qN)Ψi1iN(q1,,qN)=(2ɛ0)NE^i1()(q1)E^iN()(qN)E^iN(+)(qN)E^i1(+)(q1).
(4)
Eq. (4) shows that any field correlation function of a N-photon system can be computed as a product of two tensor elements of the N-photon wave function. As in the one particle case, the N-photon wave-function can be interpreted in terms of photodetection. Σi1,...,iNi1,...,iN (q1,..., qN)|2 is proportional to the joint probability of detecting the photons at space-time points (q1,..., qN).

It should be noted that the multi-particle wave functions constructed in this section are different from those that are constructed from the bi-vector form ψ̄(r,t) = [ψ+(r,t) ψ(r,t)] of the Bialynicki-Birula-Sipe wave function [36

36. B. J. Smith and M. G. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006). [CrossRef]

,37

37. B. J. Smith and M. G. Raymer, “Photon wave functions, wave-packet quantization of light, and coherence theory,” New J. Phys. 9, 414 (2007). [CrossRef]

]. The later cannot be directly interpreted in terms of N-photon detection amplitude because they contain a magnetic part to which electric dipole detectors are insensitive. However, since in paraxial approximation the magnetic field carries the same energy as the electric field, the square of both wave functions are proportional. Therefore, joint photo-detection probabilities calculated from vector and bi-vector wave functions are identical.

3. Generalized Huygens-Fresnel principle

If propagation from ρi to ri is through an optical system, the free space propagator
hfs(ri,ρi)=iλiexp(i2πλi|riρi|)|riρi|
(8)
must be replaced by the appropriate one hi(ri, ρi), which can be computed using standard Fourier optics techniques [38

38. J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, Englewood, 2005), 3rd ed.

]. With this generalization, Eq. (7) becomes
a(r1,t1,,rN,tN)=1d2ρ1Nd2ρNh1(r1,ρ1)hN(rN,ρN)a(ρ1,t1l(r1,ρ1)c,,ρN,tNl(rN,ρN)c),
(9)
where l(ri,ρi) is the optical path length from ρi to ri. Formula (9) assumes that there is only one optical path from ρi to ri. However, interferometers with arms having different path lengths can be placed between ρi and ri. To take this into account, we generalize (9) in the following way:
a(r1,t1,,rN,tN)=1d2ρ1Nd2ρNk1,,kNa(ρ1,t1lk1(r1,ρ1)c,,ρN,tNlkN(rN,ρN)c)h1(k1)(r1,ρ1)hN(kN)(rN,ρN).
(10)
The indexes ki label the different paths from ρi to ri.

That Fourier optics techniques can be applied to multi-photon wave functions has been first noticed in works [26

26. 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]

28

28. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Entangled-photon Fourier optics,” J. Opt. Soc. Am. B 19, 1174–1184 (2002). [CrossRef]

]. Compared to these works, the present formulation brings new interesting features: (i) It introduces time in the diffraction integral. This gives a space-time picture of entanglement and enables a time-resolved analysis of N-photon wave packet propagation and detection [39

39. In [26,28], time is only introduced to account for the bandwidth of the continuous biphoton stream and compute coincidence rates in the slow detector limit.

]. Time-resolved detection is already practical with very monochromatic photons emitted by atomic sources [40

40. T. Legero, T. Wilk, M. Hennrich, G. Rempe, and A. Kuhn, “Quantum beat of two single photons,” Phys. Rev. Lett. 93, 070503 (2004). [CrossRef] [PubMed]

]. (ii) Eq. (10) shows how to deal with interferometers in quantum imaging set-ups. As seen from Eq. (10), one cannot account for unbalanced interferometers in the propagators only. Placing independent interferometers in the paths of entangled photons makes it possible to explore the spatial and/or multi-particle counterparts of Franson-like interferometry [41

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

]. (iii) The present formalism applies to quantum systems made of an arbitrary number of photons, possibly emitted by sources that are located in different transverse planes Σj. In schemes that generate N-photon entanglement through multiple nonlinear interactions (up- and down-conversions), the surfaces Σj may represent the (thin) nonlinear crystals in which the interaction takes place. The propagation of photons coherently created in these planes, possibly at different times, can be calculated using Eq. (9). An example of such a situation is worked out in Sec. 6.

4. Detection process and wave function reduction

As pointed out in Sec. 2, |Ψ(q1,..., qN)|2 is proportional to the joint detection probability to detect the N photons at points q1 = (r1, t1) to qN = (rN, tN). What happens when a photon is indeed detected at point qN*? In contrast to standard detection theory, the quantum state of the Nth is not projected on a localized state |qN* but rather on vacuum |0〉 since the photon disappears. Therefore photodetection process should be modeled by the projection operator |0qN*| acting on the N photon quantum state |Ψ〉. In terms of position wave functions, this simply means that the N-particle wave function Ψ(q1,..., qN) is instantaneously reduced to a (N – 1)-particle wave function Ψ(q1,,qN1|qN*) in which qN* is not a variable anymore. It is important to realize that the quantum state is still a pure state after the detection of one of its photons. Therefore, starting with N + M photons, it is possible to shape a desired N-photon wave function by designing an appropriate detection scheme for the M ancillary photons. An example illustrating this point will be developed in Sec. 5.

The situation is different if “bucket” detectors are used. Such detector record the arrival time but not the position of the detected photon. If the Nth photon of a N-photon system is detected in a bucket detector at time tN*, the remaining N – 1 photons are projected on a mixed quantum state. The joint detection probability of the remaining photons at points q1 = (r1,t1) to qN–1 = (rN–1, tN–1) is given by 𝒟|Ψ(q1,qN1|rN*,tN*)|2drN*, where 𝒟 is the region in which the Nth photon could have been detected. As pointed out before for biphotons [27

27. 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]

], this probability usually differs from the one obtained by simply tracing over the degrees of freedom of the Nth photon.

5. Shaping the wave function through the detection process: application to super-resolution imaging

To illustrate how the reduction of a three photon state can be used to shape the wave function of the remaining two photons, consider the set-up of Fig. 1. A multi-step nonlinear process as in [20

20. T. E. Keller, M. H. Rubin, Y. Shih, and L.-A. Wu, “Theory of the three-photon entangled state,” Phys. Rev. A 57, 2076–2079 (1998). [CrossRef]

] produces photon triplets in a thin nonlinear crystal. We assume perfect energy and momentum conservation, so that the photon triplets are entangled in space and time. In the plane of the device:
a(ρ1,t1,ρ2,t2,ρ3,t3)=E(t3)F(ρ3)δ(ρ1ρ3)δ(ρ2ρ3)δ(t1t3)δ(t2t3),
where E and F define the temporal and transverse width of the 3-photon wave function. A dichroic beam splitter (DBS) separates two degenerate photons (wavelength λ1) from the third one (wavelength λ2). In the λ1 output, two thin lenses (focal f1 and f3) image the output of the crystal to the object plane O first (magnification M = s2/(s0 + s1)), then to the Da-detector plane (Da is a two-photon detector). The single-photon detector Db is placed on the optical axis. Using formula (9), one can calculate the 3-photon amplitude in the O- and Db-planes. If a photon is detected by Db at time t*, the wave function of the λ1 photons is projected on (see Sec. 4)
a(r,t,r,t)=δ(t(t*+τ))δ(tt)δ(rr)F(r/M)exp(i2π|r|2(1+1/M)/(λ1s2))h2(0,r/M).
where τ = (s1 + s2s3s4)/c and h2(rb,ρ3) is the propagator from the nonlinear device to the detector Db. This state is a linear superposition of localized two-photon Fock states of light. It has been shown [42

42. V. Giovannetti, S. Lloyd, L. Maccone, and J. H. Shapiro, “Sub-Rayleigh-diffraction-bound quantum imaging,” Phys. Rev. A 79, 013827 (2009). [CrossRef]

] that illuminating an object with such a state enables coherent super-resolution imaging of the object with a diffraction-limited lens f3 and a point-like detector Da. To get a good quality image, controlling the phase-curvature of the illuminating beam is also crucial [43

43. E. Brainis, C. Muldoon, L. Brandt, and A. Kuhn, “Coherent imaging of extended objects,” Opt. Commun. 282, 465–472 (2009). [CrossRef]

]. The scheme makes it possible to control the wavefront curvature of the λ1 2-photon state by tailoring the detection of the λ2 photon. For instance, one can make the wave front of λ1 photons flat in the object plane by placing a lens (focal f2) in the path to D2 (see Fig. 1) and choosing s3, s4 and f2 so that 2(λ2/λ1)M(M + 1) = s2/((1/f2 – 1/s4)−1 – (s0 + s3)). This solution exists if the inequalities s4(s0 + s3)/(s0 + s3 + s4) < f2 < s4 are satisfied. In this way, a phase accumulated by the λ1 photons is cancelled by a phase accumulated by the λ2 photon.

Fig. 1 Generation of heralded linear superposition of localized two-photon states of light.

By moving the detector Db along the z-axis, other wave-front curvatures can be produced. In addition, by moving this detector transversally one creates a tilt in the direction of propagation of the λ1 photons. If the plane O and the detector Db are in distant laboratories (O in Bob’s laboratory and Db in Alice’s laboratory), Alice can remotely generate a given 2-photon state at Bob’s side, conditionally to the detection of the λ2 photons at the appropriate position. Such a procedure is an example of a remote state preparation protocol (RSP). RSP is similar to quantum teleportation (QT) in the sense that in both cases Alice wants to send a quantum state to Bob. The difference with QT is that in RSP Alice knows the state that she want to transmit to Bob. Recently, an efficient method for remotely preparing spatial qubits has been designed [44

44. G. Taguchi, T. Dougakiuchi, N. Yoshimoto, K. Kasai, M. Iinuma, H. F. Hofmann, and Y. Kadoya, “Measurement and control of spatial qubits generated by passing photons through double slits,” Phys. Rev. A 78, 012307 (2008). [CrossRef]

, 45

45. M. A. Solis-Prosser and L. Neves, “Remote state preparation of spatial qubits,” Phys. Rev. A 84, 012330 (2011). [CrossRef]

]. In another publication, a teleportation of the entire angular spectrum of a single photon has been reported [46

46. S. P. Walborn, D. S. Ether, R. L. de Matos Filho, and N. Zagury, “Quantum teleportation of the angular spectrum of a single-photon field,” Phys. Rev. A 76, 033801 (2007). [CrossRef]

]. In the setup of Fig. 1, the spatial profile of the bi-photon at λ1 will be quasi-Gaussian. Therefore, the tilt controls the propagation axis of the beam and the wavefront curvature controls its beam waist. The waist position is the only parameter that Alice cannot control independently. If she communicates to Bob one real number – the offset Bob needs to apply to his reference frame in order to have the beam waist at the right z-coordinate – she could create at Bob’s site any Gaussian beam. This RSP requires an entangled state and the communication of a real number. This simple protocol can probably be improved to reduce the amount of the required classical communication.

6. Entangled images

To illustrate how to apply the GHF principle when the photons are generated in two different nonlinear crystals, consider the scheme of Fig. 2. A pump pulse is split in two parts using a 50/50 beam splitter (BS). The two pump parts are coherent and produce non degenerated collinear photon pairs (at λ1 and λ2) in two different non linear crystals (NLC) Γ1 and Γ2. Two dichroic mirrors are used to separate λ1 and λ2 photons. Finally, photons at λ1 are made to interfere on a second 50/50 BS and are detected using two point-like single-photon detectors, Da and Db, placed on the optical axis. Note that we call z the coordinate along the optical axis independently of its direction changes due to the BSs and DBSs.

Fig. 2 Scheme for quantum imaging with two independent photon pairs.

Assuming that the crystals are thin and that the pump pulse is a quasi-plane wave, the 4-photon position wave function generated in the NLCs reads
a(ρ1,t1,ρ2,t2,ρ1,t1,ρ2,t2)=E(t2)E(t2+Δ/c)(δ(t1t2)δ(t1t2)δ(ρ1ρ2)δ(ρ1ρ2)+δ(t1t2)δ(t1t2)δ(ρ1ρ2)δ(ρ1ρ2))
(11)
where E(t) is the temporal profile function of the pump wave. Eq. (11) assumes a perfect momentum conservation between the pump, the signal and the idler photons during the nonlinear generation process: |Ψ〉 = Σi,s δ(ωs + ωiωp)δ(ks + kikp)|ks〉|ki〉 (see [18

18. 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, R3429–R3432 (1995). [CrossRef] [PubMed]

]). Changing from the momentum to the position basis, it turns out that the signal and idler photons must originate from the same scattering point in the crystal. Another way of arriving to that conclusion in explained in [47

47. P. L. Saldanha and C. H. Monken, “Interaction between light and matter: a photon wave function approach,” New J. Phys. 13, 073015 (2011). [CrossRef]

], where the theory of spontaneous parametric down-conversion is explained in the wave function formalism. The wave function has been symmetrized with respect to the coordinates ρ1 and ρ1 of the λ1 photons. This symmetrization is necessary because, if a λ1 photon is detected at ra (or rb) there no way to know if it came from one NLC or the other one (see Fig. 2). No symmetrization with respect to the λ2 photons is needed because their is no ambiguity about their origin when they are detected at rc and rd. We assumed that the NLCs are placed at different positions (Δ) in the two arms of the setup in Fig. 2 to stress that geometrical symmetry is not required to made the λ1 photon indistinguishable.

The propagation to the points ra, rb, rc, and rd can be computed using the GHF intergral (9):
a(ra,ta,rb,tb,rc,tc,rd,td)=Γ1Γ2d2ρ1Γ1d2ρ2Γ1Γ2dρ1Γ2dρ2ha(ra,ρ1)hb(rb,ρ1)hc(rc,ρ2)hd(rd,ρ2)a(ρ1,tal(ra,ρ1)c,ρ2,tcl(rc,ρ2)c,ρ1,tbl(rb,ρ1)c,ρ2,tdl(rd,ρ2)c).
(12)
Note that according to Fig. 2, the integration surfaces for coordinates ρ2 and ρ2 are the non linear crystal planes Γ1 and Γ2. However for coordinates ρ1 and ρ1 the integration surface is Γ1 ∪ Γ2 since those coordinates represent photons that cannot be distinguished by the detection scheme. Inserting the two photon amplitude (11) into Eq. (12), one gets:
a(ra,ta,rb,tb,rc,tc,rd,td)=E(tcτ1)E(tdτ1)[δ(tatcτ2)δ(tbtdτ2)Γ1d2ρΓ2d2ρha(ra,ρ)hb(rb,ρ)hc(rc,ρ)hd(rd,ρ)]+δ(tbtcτ2)δ(tatdτ2)Γ1d2ρΓ2d2ρha(ra,ρ)hb(rb,ρ)hc(rc,ρ)hd(rd,ρ)],
(13)
where τ1 = (L + sc – (l + Δ))/c is the propagation delay from the NLC Γ1 to the detector rc, while τ2 = (L + q – sc)/c and τ2 = (L + qsd)/c are the propagation delay difference between the λ1 and the λ2 photons to their respective detectors. As expected, the detections of the λ2 photons are usually not synchronous although they happen during a time bin equal to the pump duration. However, a detection of the λ2 photons at times tc and td heralds the arrival of the λ1 photons at times tc + τ2 and td + τ2. Unless the λ2 photons are detected at times tc and td such that td = tc + τ2τ2, the λ1 photons are distinguishable by their arrival time (measuring the arrival time of the photon detected at ra reveals NLC it originates from). By post-selecting only coincidence detection events at space-time point (rc,t*) and (rd,t* + τ2τ2) the two-photon wave function of the λ1 photons is projected on
a(ra,ta,rb,tb)=E2(t*τ1)δ(tat*τ2)δ(tatb)[ϕac(ra)ϕbd(rb)+ϕad(ra)ϕbc(rb)],
(14)
where
ϕac(ra)=Γ1ha(ra,ρ)hc(rc,ρ)d2ρ,
(15a)
ϕad(ra)=Γ2ha(ra,ρ)hd(rd,ρ)d2ρ,
(15b)
ϕbd(rb)=Γ2hb(rb,ρ)hd(rd,ρ)d2ρ,
(15c)
ϕbc(rb)=Γ1hb(rb,ρ)hc(rc,ρ)d2ρ.
(15d)
Such a state is a pure entangled two-particle state, although not a maximally entangled Bell-state since photon position wave functions are not restricted to a two-dimensional Hilbert space. Furthermore, by designing the propagators hα (α ∈ {a,b,c,d}), a large variety of entangle states can be produced. In particular, if the propagation from the last BS to the detectors ra and rb is identical, ha(ra, ρ) = i hb(rb, ρ) and ha(ra, ρ′) = −i hb(rb, ρ′). As a result, ϕac(r) = i ϕbc(r) ≡ ϕ1(r), ϕbd(r) = −i ϕad(r) ≡ ϕ2(r), and the quantum state (14) becomes similar to the |Ψ+〉 Bell-state:
a(ra,ta,rb,tb)=E2(t*τ1)δ(tat*τ2)δ(tatb)[ϕ1(ra)ϕ2(rb)+ϕ2(ra)ϕ1(rb)].
(16)
As seen from Eq. (16), the scheme of Fig. 2 realises an entanglement swapping in the spatial domain. An heralded entangled photon pairs is produced from a 4-particle state of light. However in contrast with standard qubit entanglement swapping [48

48. J.-W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental entanglement swapping: Entangling photons that never interacted,” Phys. Rev. Lett. 80, 3891–3894 (1998). [CrossRef]

], entanglement is produced between continuous-variable states that live in infinite-dimensional Hilbert space. Such states have applications in continuous variable quantum information processing. As shown earlier, protocols initially designed for multi-photon field quadrature variables (for instance [49

49. N. J. Cerf, M. Lévy, and G. V. Assche, “Quantum distribution of gaussian keys using squeezed states,” Phys. Rev. A 63, 052311 (2001). [CrossRef]

]) can be efficiently implemented using (x,k) variables of single-photon fields [50

50. M. P. Almeida, S. P. Walborn, and P. H. Souto Ribeiro, “Experimental investigation of quantum key distribution with position and momentum of photon pairs,” Phys. Rev. A 72, 022313 (2005). [CrossRef]

52

52. D. S. Tasca, R. M. Gomes, F. Toscano, P. H. Souto Ribeiro, and S. P. Walborn, “Continuous-variable quantum computation with spatial degrees of freedom of photons,” Phys. Rev. A 83, 052325 (2011). [CrossRef]

]. Entangled qudits can also be produced by restricting the photon state to a d-dimensional linear subspace of the full Hilbert space, as has been recently investigated by many authors [9

9. L. Neves, S. Pádua, and C. Saavedra, “Controlled generation of maximally entangled qudits using twin photons,” Phys. Rev. A 69, 042305 (2004). [CrossRef]

, 53

53. G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007). [CrossRef]

, 54

54. B.-J. Pors, F. Miatto, G. W. ’t Hooft, E. R. Eliel, and J. P. Woerdman, “High-dimensional entanglement with orbital-angular-momentum states of light,” J. Opt. 13, 064008 (2011). [CrossRef]

].

The specific arrangement of Fig 2 is made of two positive lenses (focal length f) and two complex transmission masks (M1 and M2). The arrangement is such that the detectors ra and rb are in the Fourier planes of the lenses relatively to the two-photon imaging conditions. In other words, we assume that the following two-photon lens law is satisfied:
1+1p+q=1f,
(17)
with
=(2LlΔp)+λ2λ1(LlΔ+sc))=(2Llp)+λ2λ1(Ll+sd)).
(18)
Note that the second equality in Eq. (18) can only be satisfied simultaneously with the first one if
scsd=λ1+λ2λ2Δ.
(19)
The length ℒ is the propagation length from rc to the crystal Γ1 (and from rd to the crystal Γ2), then from the crystal to the lens in the geometrical optics interpretation of two photon-imaging [18

18. 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, R3429–R3432 (1995). [CrossRef] [PubMed]

]. We also assume that the detection points rc and rd lay on the optical axis.

The two-photon detection amplitude at detectors ra and rb can be deduced from Eqs. (15) and (16), after calculating the propagators hα (α ∈ {a,b,c,d}) using standard Fourier optics techniques [38

38. J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, Englewood, 2005), 3rd ed.

]. The result of the calculation is
ϕ1(r)=1λ12(p+q)exp[i2πλ1(2LlΔ+q)]exp[i2πλ2(LlΔ+sc)]exp[iπλ1x2+y2p+q]M˜1(xλ1(p+q),yλ1(p+q))
(20)
and
ϕ2(r)=1λ12(p+q)exp[i2πλ1(2Ll+q)]exp[i2πλ2(Ll+sd)]exp[iπλ1x2+y2p+q]M˜2(xλ1(p+q),yλ1(p+q)),
(21)
where
M˜α(ξ,η)=dxdyMα(x,y)exp[i2π(xξ+yη)]
(22)
is the spatial Fourier transform of the mask Mα (α ∈ {1,2}). Therefore
a(ra,ta,rb,tb)=E2(t*τ1)δ(tat*τ2)δ(tatb)[M˜1(ra)M˜2(rb)+M˜2(ra)M˜1(rb)]
(23)
up to constant or irrelevant phase factors. By designing the complex masks M1 and M2, one can prepare any two-photon position-entangled state of light. Moreover, since this scheme relies on entanglement swapping, the photon pair is heralded. We call the state (23) an entangled-image state because each photon that is detected is in a quantum superposition of two different images, but joint detections are correlated.

7. Conclusion

Acknowledgments

This research was supported by the Interuniversity Attraction Poles program, Belgium Science Policy, under grant P6-10 and the Fonds de la Recherche Scientifique - FNRS (F.R.S.-FNRS, Belgium) under the FRFC grant 2.4.638.09F.

References and links

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2.

H. Hübel, D. R. Hamel, A. Fedrizzi, S. Ramelow, K. J. Resch, and T. Jennewein, “Direct generation of photon triplets using cascaded photon-pair sources,” Nature 466, 601–603 (2010). [CrossRef] [PubMed]

3.

E. Waks, E. Diamanti, and Y. Yamamoto, “Generation of photon number states,” New J. Phys. 8, 4–8 (2006). [CrossRef]

4.

W.-B. Gao, C.-Y. Lu, X.-C. Yao, P. Xu, O. Gühne, A. Goebel, Y.-A. Chen, C.-Z. Peng, Z.-B. Chen, and J.-W. Pan, “Experimental demonstration of a hyper-entangled ten-qubit Schrödinger cat state,” Nat. Phys. 6, 331–335 (2010). [CrossRef]

5.

T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the Standard Quantum Limit with Four-Entangled Photons,” Science 316, 726–729 (2007). [CrossRef] [PubMed]

6.

R. Okamoto, H. F. Hofmann, T. Nagata, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit: phase super-sensitivity of N-photon interferometers,” New J. Phys. 10, 073033 (2008). [CrossRef]

7.

J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multi-photon entanglement and interferometry,” to appear in Rev. Mod. Phys. (2011).

8.

S. P. Walborn, C. H. Monken, S. Pádua, and P. H. Souto Ribeiro, “Spatial correlations in parametric down-conversion,” Phys. Rep. 495, 87–139 (2010). [CrossRef]

9.

L. Neves, S. Pádua, and C. Saavedra, “Controlled generation of maximally entangled qudits using twin photons,” Phys. Rev. A 69, 042305 (2004). [CrossRef]

10.

L. Neves, G. Lima, J. G. Aguirre Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett. 94, 100501 (2005). [CrossRef] [PubMed]

11.

R. Shimizu, K. Edamatsu, and T. Itoh, “Quantum diffraction and interference of spatially correlated photon pairs and its Fourier-optical analysis,” Phys. Rev. A 74, 013801 (2006). [CrossRef]

12.

W. H. Peeters, J. J. Renema, and M. P. van Exter, “Engineering of two-photon spatial quantum correlations behind a double slit,” Phys. Rev. A 79, 043817 (2009). [CrossRef]

13.

G. Lima, A. Vargas, L. Neves, R. Guzmán, and C. Saavedra, “Manipulating spatial qudit states with programmable optical devices,” Opt. Express 17, 10688–10696 (2009). [CrossRef] [PubMed]

14.

C. Bonato, S. Bonora, A. Chiuri, P. Mataloni, G. Milani, G. Vallone, and P. Villoresi, “Phase control of a path-entangled photon state by a deformable membrane mirror,” J. Opt. Soc. Am. B 27, A175–A180 (2010). [CrossRef]

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]

16.

P. Kok, A. N. Boto, D. S. Abrams, C. P. Williams, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: Towards arbitrary two-dimensional patterns,” Phys. Rev. A 63, 063407 (2001). [CrossRef]

17.

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]

18.

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, R3429–R3432 (1995). [CrossRef] [PubMed]

19.

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]

20.

T. E. Keller, M. H. Rubin, Y. Shih, and L.-A. Wu, “Theory of the three-photon entangled state,” Phys. Rev. A 57, 2076–2079 (1998). [CrossRef]

21.

J. Wen, E. Oh, and S. Du, “Tripartite entanglement generation via four-wave mixings: narrowband triphoton W state,” J. Opt. Soc. Am. B 27, A11–A20 (2010). [CrossRef]

22.

J. Wen, M. H. Rubin, and Y. Shih, “Transverse correlations in multiphoton entanglement,” Phys. Rev. A 76, 045802 (2007). [CrossRef]

23.

J. Wen, P. Xu, M. H. Rubin, and Y. Shih, “Transverse correlations in triphoton entanglement: Geometrical and physical optics,” Phys. Rev. A 76, 023828 (2007). [CrossRef]

24.

J. Wen and M. H. Rubin, “Distinction of tripartite Greenberger-Horne-Zeilinger and W states entangled in time (or energy) and space,” Phys. Rev. A 79, 025802 (2009). [CrossRef]

25.

J. Wen, S. Du, and M. Xiao, “Improving spatial resolution in quantum imaging beyond the Rayleigh diffraction limit using multiphoton W entangled states,” Phys. Lett. A 374, 3908 – 3911 (2010). [CrossRef]

26.

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]

27.

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28.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Entangled-photon Fourier optics,” J. Opt. Soc. Am. B 19, 1174–1184 (2002). [CrossRef]

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31.

M. Hawton, “Photon wave functions in a localized coordinate space basis,” Phys. Rev. A 59, 3223–3227 (1999). [CrossRef]

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33.

I. Bialynicki-Birula, “Photon wave function,” in Progress in Optics, vol. 36, E. Wolf, ed. (North-Holland, Elsevier, Amsterdam, 1996), chap. 5, pp. 248–294.

34.

J. E. Sipe, “Photon wave functions,” Phys. Rev. A 52, 1875–1883 (1995). [CrossRef] [PubMed]

35.

Note that the quantum mechanical scalar product Ψ(2)|Ψ(1)=h=±d3k[f±(2)(k)]*f±(1)(k)=d3r1d3r2[Ψ(2)(r2)]*Ψ(1)(r1)𝒲(r1r2) is evaluate using a double integral in the position representation with a non local kernel 𝒲 (ρ) = (h̄c2π2|ρ|2)−1.

36.

B. J. Smith and M. G. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006). [CrossRef]

37.

B. J. Smith and M. G. Raymer, “Photon wave functions, wave-packet quantization of light, and coherence theory,” New J. Phys. 9, 414 (2007). [CrossRef]

38.

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, Englewood, 2005), 3rd ed.

39.

In [26,28], time is only introduced to account for the bandwidth of the continuous biphoton stream and compute coincidence rates in the slow detector limit.

40.

T. Legero, T. Wilk, M. Hennrich, G. Rempe, and A. Kuhn, “Quantum beat of two single photons,” Phys. Rev. Lett. 93, 070503 (2004). [CrossRef] [PubMed]

41.

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

42.

V. Giovannetti, S. Lloyd, L. Maccone, and J. H. Shapiro, “Sub-Rayleigh-diffraction-bound quantum imaging,” Phys. Rev. A 79, 013827 (2009). [CrossRef]

43.

E. Brainis, C. Muldoon, L. Brandt, and A. Kuhn, “Coherent imaging of extended objects,” Opt. Commun. 282, 465–472 (2009). [CrossRef]

44.

G. Taguchi, T. Dougakiuchi, N. Yoshimoto, K. Kasai, M. Iinuma, H. F. Hofmann, and Y. Kadoya, “Measurement and control of spatial qubits generated by passing photons through double slits,” Phys. Rev. A 78, 012307 (2008). [CrossRef]

45.

M. A. Solis-Prosser and L. Neves, “Remote state preparation of spatial qubits,” Phys. Rev. A 84, 012330 (2011). [CrossRef]

46.

S. P. Walborn, D. S. Ether, R. L. de Matos Filho, and N. Zagury, “Quantum teleportation of the angular spectrum of a single-photon field,” Phys. Rev. A 76, 033801 (2007). [CrossRef]

47.

P. L. Saldanha and C. H. Monken, “Interaction between light and matter: a photon wave function approach,” New J. Phys. 13, 073015 (2011). [CrossRef]

48.

J.-W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental entanglement swapping: Entangling photons that never interacted,” Phys. Rev. Lett. 80, 3891–3894 (1998). [CrossRef]

49.

N. J. Cerf, M. Lévy, and G. V. Assche, “Quantum distribution of gaussian keys using squeezed states,” Phys. Rev. A 63, 052311 (2001). [CrossRef]

50.

M. P. Almeida, S. P. Walborn, and P. H. Souto Ribeiro, “Experimental investigation of quantum key distribution with position and momentum of photon pairs,” Phys. Rev. A 72, 022313 (2005). [CrossRef]

51.

L. Zhang, C. Silberhorn, and I. A. Walmsley, “Secure quantum key distribution using continuous variables of single photons,” Phys. Rev. Lett. 100, 110504 (2008). [CrossRef] [PubMed]

52.

D. S. Tasca, R. M. Gomes, F. Toscano, P. H. Souto Ribeiro, and S. P. Walborn, “Continuous-variable quantum computation with spatial degrees of freedom of photons,” Phys. Rev. A 83, 052325 (2011). [CrossRef]

53.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007). [CrossRef]

54.

B.-J. Pors, F. Miatto, G. W. ’t Hooft, E. R. Eliel, and J. P. Woerdman, “High-dimensional entanglement with orbital-angular-momentum states of light,” J. Opt. 13, 064008 (2011). [CrossRef]

OCIS Codes
(070.2580) Fourier optics and signal processing : Paraxial wave optics
(270.0270) Quantum optics : Quantum optics

ToC Category:
Quantum Optics

History
Original Manuscript: September 21, 2011
Revised Manuscript: October 29, 2011
Manuscript Accepted: November 1, 2011
Published: November 14, 2011

Citation
E. Brainis, "Quantum imaging with N-photon states in position space," Opt. Express 19, 24228-24240 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-24-24228


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References

  1. C. Lu, X. Zhou, O. Guhne, W. Gao, J. Zhang, Z. Yuan, A. Goebel, T. Yang, and J. Pan, “Experimental entanglement of six photons in graph states,” Nat. Phys.3, 91–95 (2007). [CrossRef]
  2. H. Hübel, D. R. Hamel, A. Fedrizzi, S. Ramelow, K. J. Resch, and T. Jennewein, “Direct generation of photon triplets using cascaded photon-pair sources,” Nature466, 601–603 (2010). [CrossRef] [PubMed]
  3. E. Waks, E. Diamanti, and Y. Yamamoto, “Generation of photon number states,” New J. Phys.8, 4–8 (2006). [CrossRef]
  4. W.-B. Gao, C.-Y. Lu, X.-C. Yao, P. Xu, O. Gühne, A. Goebel, Y.-A. Chen, C.-Z. Peng, Z.-B. Chen, and J.-W. Pan, “Experimental demonstration of a hyper-entangled ten-qubit Schrödinger cat state,” Nat. Phys.6, 331–335 (2010). [CrossRef]
  5. T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the Standard Quantum Limit with Four-Entangled Photons,” Science316, 726–729 (2007). [CrossRef] [PubMed]
  6. R. Okamoto, H. F. Hofmann, T. Nagata, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit: phase super-sensitivity of N-photon interferometers,” New J. Phys.10, 073033 (2008). [CrossRef]
  7. J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multi-photon entanglement and interferometry,” to appear in Rev. Mod. Phys. (2011).
  8. S. P. Walborn, C. H. Monken, S. Pádua, and P. H. Souto Ribeiro, “Spatial correlations in parametric down-conversion,” Phys. Rep.495, 87–139 (2010). [CrossRef]
  9. L. Neves, S. Pádua, and C. Saavedra, “Controlled generation of maximally entangled qudits using twin photons,” Phys. Rev. A69, 042305 (2004). [CrossRef]
  10. L. Neves, G. Lima, J. G. Aguirre Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett.94, 100501 (2005). [CrossRef] [PubMed]
  11. R. Shimizu, K. Edamatsu, and T. Itoh, “Quantum diffraction and interference of spatially correlated photon pairs and its Fourier-optical analysis,” Phys. Rev. A74, 013801 (2006). [CrossRef]
  12. W. H. Peeters, J. J. Renema, and M. P. van Exter, “Engineering of two-photon spatial quantum correlations behind a double slit,” Phys. Rev. A79, 043817 (2009). [CrossRef]
  13. G. Lima, A. Vargas, L. Neves, R. Guzmán, and C. Saavedra, “Manipulating spatial qudit states with programmable optical devices,” Opt. Express17, 10688–10696 (2009). [CrossRef] [PubMed]
  14. C. Bonato, S. Bonora, A. Chiuri, P. Mataloni, G. Milani, G. Vallone, and P. Villoresi, “Phase control of a path-entangled photon state by a deformable membrane mirror,” J. Opt. Soc. Am. B27, A175–A180 (2010). [CrossRef]
  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]
  16. P. Kok, A. N. Boto, D. S. Abrams, C. P. Williams, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: Towards arbitrary two-dimensional patterns,” Phys. Rev. A63, 063407 (2001). [CrossRef]
  17. 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]
  18. T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A52, R3429–R3432 (1995). [CrossRef] [PubMed]
  19. 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. A53, 2804–2815 (1996). [CrossRef] [PubMed]
  20. T. E. Keller, M. H. Rubin, Y. Shih, and L.-A. Wu, “Theory of the three-photon entangled state,” Phys. Rev. A57, 2076–2079 (1998). [CrossRef]
  21. J. Wen, E. Oh, and S. Du, “Tripartite entanglement generation via four-wave mixings: narrowband triphoton W state,” J. Opt. Soc. Am. B27, A11–A20 (2010). [CrossRef]
  22. J. Wen, M. H. Rubin, and Y. Shih, “Transverse correlations in multiphoton entanglement,” Phys. Rev. A76, 045802 (2007). [CrossRef]
  23. J. Wen, P. Xu, M. H. Rubin, and Y. Shih, “Transverse correlations in triphoton entanglement: Geometrical and physical optics,” Phys. Rev. A76, 023828 (2007). [CrossRef]
  24. J. Wen and M. H. Rubin, “Distinction of tripartite Greenberger-Horne-Zeilinger and W states entangled in time (or energy) and space,” Phys. Rev. A79, 025802 (2009). [CrossRef]
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