OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 19 — Sep. 15, 2008
  • pp: 14629–14634

Optics InfoBase > Optics Express > Volume 16 > Issue 19 > Page 14629

« Show journal navigation

Sorting photons of different rotational Doppler shifts (RDS) by orbital angular momentum of single-photon with spin-orbit-RDS entanglement

Lixiang Chen and Weilong She  »View Author Affiliations


Optics Express, Vol. 16, Issue 19, pp. 14629-14634 (2008)
http://dx.doi.org/10.1364/OE.16.014629


View Full Text Article

Acrobat PDF (317 KB)


Advanced Search

Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We demonstrate that single photons from a rotating q-plate exhibit an entanglement in three degrees of freedom of spin, orbital angular momentum, and the rotational Doppler shift (RDS) due to the nonconservation of total spin and orbital angular momenta. We find that the rotational Doppler shift Δω=Ω(Δs+Δl), where s, l and Ω are quantum numbers of spin, orbital angular momentum, and rotating velocity of the q-plate, respectively. Of interest is that the rotational Doppler shift directly reflects the rotational symmetry of q-plates and can be also expressed as Δω=Ωn, where n=2(q-1) denotes the fold number of rotational symmetry. Besides, based on this single-photon spin-orbit-RDS entanglement, we propose an experimental scheme to sort photons of different frequency shifts according to individual orbital angular momentum.

© 2008 Optical Society of America

1. Introduction

Recently, optical angular momentum is receiving a great deal of interest from both the classical and quantum points of view [1

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999). [CrossRef]

–5

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

]. One of those in focus is the rotational Doppler shift (RDS) [6

M. J. Padgett, “Electromagnetism: Like a speeding watch,” Nature 443, 924–925 (2006). [CrossRef] [PubMed]

]. The RDS effect was first described by Garetz and Arnold, who used a rotating half-wave plate of angular velocity Ω to imprint a frequency shift 2Ω to the circularly polarized light [7

B. A. Garetz and S. Arnold, “Variable frequency shifting of circularly polarized laser radiation via a rotating half-wave retardation plate,” Opt. Commun. 31, 1–3 (1979). [CrossRef]

]. By use of spin Poincaré-sphere, Simon et al. [8

R. Simon, H. J. Kimble, and E. C. G. Sudarshan, “Evolving geometric phase and its dynamical manifestation as a frequency shift: an optical experiment,” Phys. Rev. Lett. 61, 19–22 (1988). [CrossRef] [PubMed]

] demonstrated that the dynamically evolving geometric phase (Berry phase) arising from a rotating quarter-wave plate can offset the frequency of a laser beam. Later, Bretenaker and Le Floch [9

F. Bretenaker and A. Le Floch, “Energy exchanges between a rotating retardation plate and a laser beam,” Phys. Rev. Lett. 65, 2316 (1990). [CrossRef] [PubMed]

] suggested that the frequency shift is a consequence of the conservation of energy and associated with the spin angular momentum exchange. Unlike the intrinsic photon spin associated with circular polarization, the orbital angular momentum (OAM) was related to helical wavefront or optical vortex with phase singularity, however, much attention is still focused on seeking the analogy between them [10

L. Allen and M. Padgett, “Equivalent geometric transformations for spin and orbital angular momentum of light,” J. Mod. Opt. 54, 487–491 (2007). [CrossRef]

].

After recognizing that a twisted photon of l-intertwined helical phase front exp(ilϕ) carries a well-defined OAM of lħ [11

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). [CrossRef] [PubMed]

], Allen et al. also suggested using twisted photons to achieve the azimuthal Doppler shift [12

L. Allen, M. Babiker, and W. L. Power, “Azimuthal Doppler shift in light beams with orbital angular momentum,” Opt. Commun. 112, 141–144 (1994). [CrossRef]

]. Later experimental observation by Courtial et al. showed that when a mm-wave beam bearing OAM is rotating with a velocity Ω, it is imparted by a frequency shift of Ωl [13

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beams possessing orbital angular momentum,” Phys. Rev. Lett. 80, 3217–3219 (1998). [CrossRef]

]. Of striking interest is that the spin and orbital angular momentum were indistinguishable for the RDS effect [14

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. 81, 4828–4830 (1998). [CrossRef]

, 15

M. J. Padgett and J. Courtial, “Poincaré-sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. 24, 430–432 (1999). [CrossRef]

]. It was found that a number of experiments on the RDS can be treated as the special cases of the atomic systems lacking rotational invariance but having stationary state in a rotating frame, as predicted by Bialynicki-Birula and Bialynicka-Birula [16

I. Bialynicki-Birula and Z. Bialynicka-Birula, “Rotational Frequency shift,” Phys. Rev. Lett. 78, 2539–2542 (1997). [CrossRef]

]. Basistiy et al. reported their first all-optical detection of the RDS effect using an off-axis optical vortex beam [17

I. V. Basistiy, V. V. Slyusar, M. S. Soskin, M. V. Vasnetsov, and A. Ya. Bekshaev, “Manifestation of the rotational Doppler effect by use of an off-axis optical vortex beam,” Opt. Lett. 28, 1185–1187 (2003). [CrossRef] [PubMed]

]. Barreiro et al. demonstrated the first spectroscopic observation of the RDS effect due to OAM via the coherent interaction between light beams and an ensemble of rubidium atoms at room temperature [18

S. Barreiro, J. W. R. Tabosa, H. Failache, and A. Lezama, “Spectroscopic observation of the rotational Doppler effect,” Phys. Rev. Lett. 97, 113601 (2006). [CrossRef] [PubMed]

]. Recently, a quantum theory of rotating photons with RDS effect was proposed by van Enk and Nienhuis [19

S. J. van Enk and G. Nienhuis, “Photons in polychromatic rotating modes,” Phys. Rev. A 76, 053825 (2007). [CrossRef]

]. Here we demonstrate an experiment scheme to show that single photons from a rotating q-plate exhibit an entanglement in spin, orbital angular momentum, and the rotational Doppler shift, reflecting the rotational symmetry of the q-plate and, photons with different frequency shifts can be sorted according to individual orbital angular momentum. It has been recognized that the utility of several degrees of freedom of a single-photon to encode multiple qubits opens the possibility for building a deterministic quantum information processor [20

N. J. Cerf, C. Adami, and P. G. Kwiat, “Optical simulation of quantum logic,” Phys. Rev. A 57, 1477–1480 (1998). [CrossRef]

–22

Y.-H. Kim, “Single-photon two-qubit entangled states: Preparation and measurement,” Phys. Rev. A 67, 040301 (2003). [CrossRef]

]. It was also shown that single-photon two-qubit states might be useful for deterministic cryptographic schemes [23

A. Beige, B. G. Englert, C. Kurtsiefer, and H. Weinfurter, “Secure communication with single-photon two-qubit states,” J. Phys. A: Math. Gen. 35, L407–L413 (2002). [CrossRef]

]. Recently, beating the channel capacity limit for linear photonic supperdence coding was reported, in which single-photon spin-orbital Bell-state analysis played a key role [24

J. T. Barreiro, T. -C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nature Phys. 4, 282–286 (2008). [CrossRef]

]. So, we anticipate that the single-photon spin-orbit-RDS entanglement would have potential applications for quantum computation, quantum communication or optical tweezers system.

2. Theory and Application

Fig. 1. Illustration of some types of q-plates, in which the orientation of optical axis is the locally tangent to the line: (a) is rotational invariance; while (b), (c), (d), and (e) are of one-, two-, three-, and four-fold rotational symmetry, respectively.

Recently, Marrucci and colleagues [25

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006). [CrossRef] [PubMed]

] built a novel device called q-plate (or “Pancharatnam-Berry phase optical element”) that reverses the spin of photons while transferring the change of spin angular momentum into the orbital kind. A general q-plate is a planar slab of a uniaxial birefringent medium, with an inhomogeneous orientation of the optical axis lying in x-y plane and a homogeneous phase retardation of χ along z-axis. Suppose the orientation of optical axis has this form in a polar coordinate: α(r,ϕ)=qϕ+α 0, where q and α 0 are two constant. Some types of q-plates are illustrated in Fig. 1. On the circular bases, the transmitted matrix for the q-plate can be expressed as

Tq= 1+exp (iχ)2 [ 1 0 0 1]+ 1−exp (iχ)]2 [ 0 exp [i2α (r,φ)] exp [ −i2α (r,φ)] 0].
(1)

We find that these q-plates can be used for the generation of single photons exhibiting the spin-orbit-RDS entanglement, as sketched in Fig. 2. Assume the input are single photons with horizontal linear polarization, namely, |in〉=|L〉/√2+R〉/√2, where |L〉 and |R〉 denote left- and right-handed circular polarizations, respectively. A suitable computer-generated phase hologram is employed to serve as an OAM adder, which imprints the m-intertwined helical phasefront to incoming photons while remains the linear polarization unchanged [26

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995). [CrossRef] [PubMed]

], thus, yielding the state |2〉=in〉⊗|m〉=|L,m〉/√2+|R,m〉/√2. Subsequently, the photons are directed onto the q-plate. As the q-plate is installed in a rotating device, the local orientation of optical axis, with respect to a fixed reference coordinates, is varying along with the time t, namely, α′(r,ϕ,t)=qϕ+(q-1)Ωt+α 0. Unlike the use of Dove prism for rotating the light beam [13

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beams possessing orbital angular momentum,” Phys. Rev. Lett. 80, 3217–3219 (1998). [CrossRef]

, 14

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. 81, 4828–4830 (1998). [CrossRef]

], in present setup the q-plate is rotating, from which the emerging state becomes

|out〉= α1|in,m〉 + α2exp [i2 ( q−1)Ωt] | L,m+2q 〉+ α3exp [ −i2] ( q−1)Ωt | R,m−2q〉.
(2)

where α 1=[1+exp(iX)]/2,α 2=[1-exp(iX)]exp(i2α 0)/2√2, and α 3=[1-exp(iX)]exp(-i2α 0)/2√2. Of particular interest is that Eq. (2) shows the correlations among three degrees of freedom of spin, orbital angular momentum, and rotational Doppler shift. Actually, this state consists of three orthogonal OAM states, i.e., |l=m〉, |l=m-2q〉, and |l=m+2q〉, which correspond respectively to three different Pancharatnam-Berry phases, 0, 2(q-1)Ωt, and -2(q-1)Ωt, therefore, three different frequency shifts, Δω=0, 2(q-1)Ω, and -2(q-1)Ω. The two nonzero frequency shifts originate from the nonconservation of total spin and orbital angular momenta. Detailedly, for the state |l=m〉, as the angular momentum change Δj=0, no RDS is expected; while for |l=m+2q〉, Δj=Δs+Δl=2(q-1) and the frequency shift Δω=2(q-1)Ω; and for |l=m-2q〉, Δj=-2(q-1) and Δω=-2(q-1)Ω. Table 1 lists some example input states and q against resulting frequency shifts.

Table 1.  The example input states and q against resulting frequency shifts.
Input states
q-plates		 s=+1 (SAM)
m=-1 (OAM)		 s=+1 m=+1 s=-1 m=-1 s=-1 m=+1
q=0Δω=-2ΩΔω=-2ΩΔω=2ΩΔω=2Ω
q=1/2Δω=-ΩΔω=-ΩΔω=ΩΔω=Ω
q=1Δω=0Δω=0Δω=0Δω=0

Obviously, the rotational Doppler shift is only determined by the change of total spin and orbital angular momenta, irrelative of the absolute orbital angular momentum (m) bore by the incoming photons. The single-photon spin-orbit-RDS entanglement can be understood as follows: exp(i0t), exp(inΩt), and exp(-inΩt) are orthogonal to each other in the space [0,T] [n=2(q-1),T=2π/(nΩ)], so they can be taken as a set of bases. By denoting exp (i0t) 2π = |Δω=0〉, exp (inΩt) 2π= |Δω=nΩ〉, and exp (−inΩt) 2π= |Δω=−nΩ〉, Eq. (2) can be written as

∣ out〉 = α1∣in , m〉∣Δω= 0〉 + α2 ∣L ,m+2 q〉 ∣Δω=n Ω〉 + α3∣R ,m−2q〉∣Δω= −nΩ〉.
(3)

The state described by Eq. (3) is an entangled state, which is achieved among three different degrees of freedom of spin, orbital angular momentum, and the rotational Doppler shift in single photons. Also, Equation (3) could be considered as a standard Schmidt decomposition, whose Schmidt number [27

E. Schmidt, “Zur Theorie der linearen und nicht linearen Integralgleichungen,” Math. Ann. 63, 433–466 (1907). [CrossRef]

] κ=(|α 1|4+|α 2|4+|α 3|4)-1=8/[3(cosχ+1/3)2+8/3]∈ (1,3](χ≠2π) and, when cosχ=-1/3 (χ=109.5°), it reaches its maximum value κ=3. One can describe the single-photon state in terms of a tensor product Hilbert space, namely H=H 1⊗H 2⊗H 3, where H 1, H 2, and H 3 are respectively disjoint Hilbert spaces corresponding to three different degrees of freedom of spin, orbital angular momentum, and rotational Doppler shift. As well known, the entanglement is not limited to different particles, but generally applicable to different degrees of freedom in single particles [28

Y. Hasegawa, R. Loidl, G. Badurek, M. Baron, and H. Rauch, “Violation of a Bell-like inequality in single-neutron interferometry,” Nature 425, 45–48 (2003). [CrossRef] [PubMed]

–30

S. J. van Enk, “Entanglement of electromagnetic fields,” Phys. Rev. A 67, 022303 (2003). [CrossRef]

]. In fact, single-photon spin-orbit entangled state (two-qubit) has been recently reported [31

L. Marrucci, “Rotating light with light: Generation of helical modes of light by spin-to-orbital angular momentum conversion in inhomogeneous liquid crystals,” Proc. of SPIE 6587, 658708 (2007). [CrossRef]

–33

L. Chen and W. She, “Electro-optically forbidden or enhanced spin-to-orbital angular momentum conversion in a focused light beam,” Opt. Lett. 33, 696–698 (2008). [CrossRef] [PubMed]

] and has been utilized to implement a simple quantum Deutsch algorithm [34

A. N. de Oliveira, S. P. Walborn, and C. H. Monken, “Implementing the Deutsch algorithm with polarization and transverse spatical modes,” J. Opt. B: Quantum Semiclass. Opt. 7, 288–292 (2005). [CrossRef]

]. However, it would be more proper that the single-photon spin-orbit or spin-orbit-RDS entanglement described by Eq. (3) is called classical entanglement [35

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998). [CrossRef]

].

We know from Eq. (3) that, the single-photon RDS is proportional to the change of its total angular momentum, namely, Δω=ΩΔj=Ω(Δs+Δl). The rotational Doppler shift can be understood in terms of ray optics and the torque action on the rotating optical components [36

M. J. Padgett, “The mechanism for energy transfer in the rotational frequency shift of a light beam,” J. Opt. A: Pure Appl. Opt. 6, S263–S265 (2004). [CrossRef]

] or the accompanying energy exchange [37

I. V. Basistiy, A. Ya. Bakshaev, M. V. Vasnetsov, V. V. Slyusar, and M. S. Soskin, “Observation of the rotational doppler effect for optical beams with helical wave front using spiral zone plate,” JETP Lett. 76, 486–489 (2002). [CrossRef]

]. It is also illuminating for us to attribute this effect to the rotational symmetry of the system. It is known that if the rotational invariance around the propagation direction of light holds, then the angular momentum along this direction is expected to be conserved [38

S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B: Quantum Semiclass. Opt. 4, S7–S16 (2002). [CrossRef]

]. This is the case for q=1. For q≠1, the nonconservation of angular momentum as well as the frequency shift will happen synchronously. As shown in Fig. 1, it is evident that the q-plate with parameter q has 2(q-1)-fold rotational symmetry, just equal to the angular momentum increment. So we argue that the RDS can be considered as a nature consequence of the rotational symmetry of the q-plate and the RDS can be also simply expressed by Δω=nΩ, where n=2(q-1) denotes the fold number of rotational symmetry. For the special case of q=0, the q-plate will reduce to the half-wave plate, which is of twofold rotational symmetry [see Fig. 1(c)] and Δω=±2Ω, due to pure spin angular momentum [7

B. A. Garetz and S. Arnold, “Variable frequency shifting of circularly polarized laser radiation via a rotating half-wave retardation plate,” Opt. Commun. 31, 1–3 (1979). [CrossRef]

–9

F. Bretenaker and A. Le Floch, “Energy exchanges between a rotating retardation plate and a laser beam,” Phys. Rev. Lett. 65, 2316 (1990). [CrossRef] [PubMed]

], where only the spin-RDS entanglement can exist.

Fig. 2. The experimental scheme using q-plate to generate single-photon spin-orbit-RDS entanglement; and sort photons of different RDS according to individual OAM: BS, 50:50 non-polarizing beam splitter; DP, Dove prism; CBS, circularly polarizing beam splitter; M, mirror.

Of particular interest is that the single-photon spin-orbit-RDS entanglement provides us an approach to sort photons of different rotational Doppler shifts according to individual orbital angular momentum. Assume q=1/2 and α 0=0 is considered [39

S. C. McEldowney, D. M. Shemo, and R. A. Chipman, “Vortex retarders produced from photo-aligned liquid crystal polymers,” Opt. Express 16, 7295–7308 (2008). [CrossRef] [PubMed]

] and the orbital angular momentum adder is trivially absent (m=0), then the orbital angular momentum to be sorted are |l=-1〉, |l=0〉, and |l=1〉. The Mach Zehnder interferometer with two Dove prisms having a relative rotation angle performs the OAM sorting, which can work at a single photon level [40

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004). [CrossRef] [PubMed]

]. In general, to sort single photons with three different OAM states, two stages of cascaded Mach Zehnder interferometers are required. However, due to the fact that single photons exhibit the spin-orbit entanglement in present case, only one stage of Mach Zehnder interferometer together with a circularly polarizing beam splitter (CBS) is enough. With the relative angle of two Dove prisms set as π/2 (resulting in a π rotation to the beam), a relative phase difference π between two arms is introduced for the state |l=±1〉, as shown in Fig. 3, therefore, the resulted destructive interference will make photons of l=±1〉 emerge from port A1 (Here, phase retardation χ=109.5° is assumed to reach κ=3); while for photons of |l=0〉, the condition of constructively interference always holds, therefore they come out from port A2. Subsequently, as photons from Port A1 have the spin-orbit-RDS entanglement, namely |out〉A1=(|L〉|l=1〉|Δω=-Ω〉+|R〉|l=-1〉|Δω=Ω〉)/√2, the CBS, serving as a photon spin sorter, is capable to separate photons of |l=+1〉 and |l=-1〉 according to individual circular polarizations. Then photons of |l=+1〉 and |l=-1〉 come out from B1 and B2 port, respectively. For the state |l=0〉, we know Δω=0, without RDS; while for |l=±1〉, Δω=∓Ω, with down- and up-shift, respectively. Besides, if the input linearly polarized light is replaced by an arbitrarily polarized one, then the yield of photons with different RDS could be manipulated.

Fig. 3. Phase diagrams of photons with OAM (a) l=-1, (b) l=0, and (c) l=+1, respectively.

3. Conclusion

We have demonstrated that the single-photon entanglement in three degrees of freedom of spin, orbital angular momentum, and rotational Doppler shift can be prepared from a rotating q-plate and based on this type of entanglement the photons with different rotational Doppler shifts can be sorted according to individual orbital angular momentum. We anticipate the single-photon spin-orbit-RDS entanglement would have potential applications for quantum computation, quantum communication or optical tweezers system.

Acknowledgments

The authors thank Prof. L Marrucci for the helpful discussions and Prof. S J van Enk for his kind help, and gratefully acknowledge the financial support from the National Natural Science Foundation of China (Grant No. 10574167).

References and links

1.

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999). [CrossRef]

2.

L. Allen, “Induction to the atoms and angular momentum of light special issue,” J. Opt. B: Quantum Semiclass. Opt. 4, S1–S6 (2002). [CrossRef]

3.

M. J. Padgett, J. Courtial, and L. Allen, “Light’s orbital angular momentum,” Phys. Today 57, 35–40 (2004). [CrossRef]

4.

J. Courtial and K. O’Holleran, “Experiments with twisted light. Some of the mechanical and quantummechanical properties of optical vorteicest,” Eur. Phys. J. Special Topics 145, 35–47 (2007). [CrossRef]

5.

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

6.

M. J. Padgett, “Electromagnetism: Like a speeding watch,” Nature 443, 924–925 (2006). [CrossRef] [PubMed]

7.

B. A. Garetz and S. Arnold, “Variable frequency shifting of circularly polarized laser radiation via a rotating half-wave retardation plate,” Opt. Commun. 31, 1–3 (1979). [CrossRef]

8.

R. Simon, H. J. Kimble, and E. C. G. Sudarshan, “Evolving geometric phase and its dynamical manifestation as a frequency shift: an optical experiment,” Phys. Rev. Lett. 61, 19–22 (1988). [CrossRef] [PubMed]

9.

F. Bretenaker and A. Le Floch, “Energy exchanges between a rotating retardation plate and a laser beam,” Phys. Rev. Lett. 65, 2316 (1990). [CrossRef] [PubMed]

10.

L. Allen and M. Padgett, “Equivalent geometric transformations for spin and orbital angular momentum of light,” J. Mod. Opt. 54, 487–491 (2007). [CrossRef]

11.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). [CrossRef] [PubMed]

12.

L. Allen, M. Babiker, and W. L. Power, “Azimuthal Doppler shift in light beams with orbital angular momentum,” Opt. Commun. 112, 141–144 (1994). [CrossRef]

13.

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beams possessing orbital angular momentum,” Phys. Rev. Lett. 80, 3217–3219 (1998). [CrossRef]

14.

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. 81, 4828–4830 (1998). [CrossRef]

15.

M. J. Padgett and J. Courtial, “Poincaré-sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. 24, 430–432 (1999). [CrossRef]

16.

I. Bialynicki-Birula and Z. Bialynicka-Birula, “Rotational Frequency shift,” Phys. Rev. Lett. 78, 2539–2542 (1997). [CrossRef]

17.

I. V. Basistiy, V. V. Slyusar, M. S. Soskin, M. V. Vasnetsov, and A. Ya. Bekshaev, “Manifestation of the rotational Doppler effect by use of an off-axis optical vortex beam,” Opt. Lett. 28, 1185–1187 (2003). [CrossRef] [PubMed]

18.

S. Barreiro, J. W. R. Tabosa, H. Failache, and A. Lezama, “Spectroscopic observation of the rotational Doppler effect,” Phys. Rev. Lett. 97, 113601 (2006). [CrossRef] [PubMed]

19.

S. J. van Enk and G. Nienhuis, “Photons in polychromatic rotating modes,” Phys. Rev. A 76, 053825 (2007). [CrossRef]

20.

N. J. Cerf, C. Adami, and P. G. Kwiat, “Optical simulation of quantum logic,” Phys. Rev. A 57, 1477–1480 (1998). [CrossRef]

21.

B. G. Englert, C. Kurtsiefer, and H. Weinfurter, “Universal unitary gate for single-photon two-qubit states,” Phys. Rev. A 63, 032303 (2001). [CrossRef]

22.

Y.-H. Kim, “Single-photon two-qubit entangled states: Preparation and measurement,” Phys. Rev. A 67, 040301 (2003). [CrossRef]

23.

A. Beige, B. G. Englert, C. Kurtsiefer, and H. Weinfurter, “Secure communication with single-photon two-qubit states,” J. Phys. A: Math. Gen. 35, L407–L413 (2002). [CrossRef]

24.

J. T. Barreiro, T. -C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nature Phys. 4, 282–286 (2008). [CrossRef]

25.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006). [CrossRef] [PubMed]

26.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995). [CrossRef] [PubMed]

27.

E. Schmidt, “Zur Theorie der linearen und nicht linearen Integralgleichungen,” Math. Ann. 63, 433–466 (1907). [CrossRef]

28.

Y. Hasegawa, R. Loidl, G. Badurek, M. Baron, and H. Rauch, “Violation of a Bell-like inequality in single-neutron interferometry,” Nature 425, 45–48 (2003). [CrossRef] [PubMed]

29.

S. J. van Enk, “Single-particle entanglement,” Phys. Rev. A 72, 064306 (2005). [CrossRef]

30.

S. J. van Enk, “Entanglement of electromagnetic fields,” Phys. Rev. A 67, 022303 (2003). [CrossRef]

31.

L. Marrucci, “Rotating light with light: Generation of helical modes of light by spin-to-orbital angular momentum conversion in inhomogeneous liquid crystals,” Proc. of SPIE 6587, 658708 (2007). [CrossRef]

32.

L. Deng, H. Wang, and K. Wang, “Quantum CNOT gates with orbital angular momentum and polarization of single-photon quantum logic,” J. Opt. Soc. Am. B 24, 2517–2520 (2007). [CrossRef]

33.

L. Chen and W. She, “Electro-optically forbidden or enhanced spin-to-orbital angular momentum conversion in a focused light beam,” Opt. Lett. 33, 696–698 (2008). [CrossRef] [PubMed]

34.

A. N. de Oliveira, S. P. Walborn, and C. H. Monken, “Implementing the Deutsch algorithm with polarization and transverse spatical modes,” J. Opt. B: Quantum Semiclass. Opt. 7, 288–292 (2005). [CrossRef]

35.

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998). [CrossRef]

36.

M. J. Padgett, “The mechanism for energy transfer in the rotational frequency shift of a light beam,” J. Opt. A: Pure Appl. Opt. 6, S263–S265 (2004). [CrossRef]

37.

I. V. Basistiy, A. Ya. Bakshaev, M. V. Vasnetsov, V. V. Slyusar, and M. S. Soskin, “Observation of the rotational doppler effect for optical beams with helical wave front using spiral zone plate,” JETP Lett. 76, 486–489 (2002). [CrossRef]

38.

S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B: Quantum Semiclass. Opt. 4, S7–S16 (2002). [CrossRef]

39.

S. C. McEldowney, D. M. Shemo, and R. A. Chipman, “Vortex retarders produced from photo-aligned liquid crystal polymers,” Opt. Express 16, 7295–7308 (2008). [CrossRef] [PubMed]

40.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004). [CrossRef] [PubMed]

OCIS Codes
(230.5440) Optical devices : Polarization-selective devices
(270.5585) Quantum optics : Quantum information and processing

ToC Category:
Quantum Optics

History
Original Manuscript: June 3, 2008
Revised Manuscript: August 13, 2008
Manuscript Accepted: August 13, 2008
Published: September 3, 2008

Citation
Lixiang Chen and Weilong She, "Sorting photons of different rotational Doppler shifts (RDS) by orbital angular momentum of single-photon with spin-orbit-RDS entanglement," Opt. Express 16, 14629-14634 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-19-14629


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," Prog. Opt. 39, 291-372 (1999). [CrossRef]
  2. L. Allen, "Induction to the atoms and angular momentum of light special issue," J. Opt. B: Quantum Semiclass. Opt. 4, S1-S6 (2002). [CrossRef]
  3. M. J. Padgett, J. Courtial, and L. Allen, "Light??s orbital angular momentum," Phys. Today 57, 35-40 (2004). [CrossRef]
  4. J. Courtial and K. O'Holleran, "Experiments with twisted light. Some of the mechanical and quantum-mechanical properties of optical vorteicest," Eur. Phys. J. Special Topics 145, 35-47 (2007). [CrossRef]
  5. G. Molina-Terriza, J. P. Torres, and L. Torner, "Twisted photons," Nature Phys. 3, 305-310 (2007). [CrossRef]
  6. M. J. Padgett, "Electromagnetism: Like a speeding watch," Nature 443, 924-925 (2006). [CrossRef] [PubMed]
  7. B. A. Garetz and S. Arnold, "Variable frequency shifting of circularly polarized laser radiation via a rotating half-wave retardation plate," Opt. Commun. 31, 1-3 (1979). [CrossRef]
  8. R. Simon, H. J. Kimble, and E. C. G. Sudarshan, "Evolving geometric phase and its dynamical manifestation as a frequency shift: an optical experiment," Phys. Rev. Lett. 61, 19-22 (1988). [CrossRef] [PubMed]
  9. F. Bretenaker and A. Le Floch, "Energy exchanges between a rotating retardation plate and a laser beam," Phys. Rev. Lett. 65, 2316 (1990). [CrossRef] [PubMed]
  10. L. Allen and M. Padgett, "Equivalent geometric transformations for spin and orbital angular momentum of light," J. Mod. Opt. 54, 487-491 (2007). [CrossRef]
  11. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992). [CrossRef] [PubMed]
  12. L. Allen, M. Babiker, and W. L. Power, "Azimuthal Doppler shift in light beams with orbital angular momentum," Opt. Commun. 112, 141-144 (1994). [CrossRef]
  13. J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, "Measurement of the rotational frequency shift imparted to a rotating light beams possessing orbital angular momentum," Phys. Rev. Lett. 80, 3217-3219 (1998). [CrossRef]
  14. J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational frequency shift of a light beam," Phys. Rev. Lett. 81, 4828-4830 (1998). [CrossRef]
  15. M. J. Padgett and J. Courtial, "Poincaré-sphere equivalent for light beams containing orbital angular momentum," Opt. Lett. 24, 430-432 (1999). [CrossRef]
  16. I. Bialynicki-Birula and Z. Bialynicka-Birula, "Rotational Frequency shift," Phys. Rev. Lett. 78, 2539-2542 (1997). [CrossRef]
  17. I. V. Basistiy, V. V. Slyusar, M. S. Soskin, M. V. Vasnetsov, and A. Ya. Bekshaev, "Manifestation of the rotational Doppler effect by use of an off-axis optical vortex beam," Opt. Lett. 28, 1185-1187 (2003). [CrossRef] [PubMed]
  18. S. Barreiro, J. W. R. Tabosa, H. Failache, and A. Lezama, "Spectroscopic observation of the rotational Doppler effect," Phys. Rev. Lett. 97, 113601 (2006). [CrossRef] [PubMed]
  19. S. J. van Enk and G. Nienhuis, "Photons in polychromatic rotating modes," Phys. Rev. A 76, 053825 (2007). [CrossRef]
  20. N. J. Cerf, C. Adami, and P. G. Kwiat, "Optical simulation of quantum logic," Phys. Rev. A 57, 1477-1480 (1998). [CrossRef]
  21. B. G. Englert, C. Kurtsiefer, and H. Weinfurter, "Universal unitary gate for single-photon two-qubit states," Phys. Rev. A 63, 032303 (2001). [CrossRef]
  22. Y.-H. Kim, "Single-photon two-qubit entangled states: Preparation and measurement," Phys. Rev. A 67, 040301 (2003). [CrossRef]
  23. A. Beige, B. G. Englert, C. Kurtsiefer, and H. Weinfurter, "Secure communication with single-photon two-qubit states," J. Phys. A: Math. Gen. 35, L407-L413 (2002). [CrossRef]
  24. J. T. Barreiro, T. -C. Wei, and P. G. Kwiat, "Beating the channel capacity limit for linear photonic superdense coding," Nature Phys. 4, 282-286 (2008). [CrossRef]
  25. L. Marrucci, C. Manzo, and D. Paparo, "Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media," Phys. Rev. Lett. 96, 163905 (2006). [CrossRef] [PubMed]
  26. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995). [CrossRef] [PubMed]
  27. E. Schmidt, "Zur Theorie der linearen und nicht linearen Integralgleichungen," Math. Ann. 63, 433-466 (1907). [CrossRef]
  28. Y. Hasegawa, R. Loidl, G. Badurek, M. Baron, and H. Rauch, "Violation of a Bell-like inequality in single-neutron interferometry," Nature 425, 45-48 (2003). [CrossRef] [PubMed]
  29. S. J. van Enk, "Single-particle entanglement," Phys. Rev. A 72, 064306 (2005). [CrossRef]
  30. S. J. van Enk, "Entanglement of electromagnetic fields," Phys. Rev. A 67, 022303 (2003). [CrossRef]
  31. L. Marrucci, "Rotating light with light: Generation of helical modes of light by spin-to-orbital angular momentum conversion in inhomogeneous liquid crystals," Proc. of SPIE 6587, 658708 (2007). [CrossRef]
  32. L. Deng, H. Wang, and K. Wang, "Quantum CNOT gates with orbital angular momentum and polarization of single-photon quantum logic," J. Opt. Soc. Am. B 24, 2517-2520 (2007). [CrossRef]
  33. L. Chen and W. She, "Electro-optically forbidden or enhanced spin-to-orbital angular momentum conversion in a focused light beam," Opt. Lett. 33, 696-698 (2008). [CrossRef] [PubMed]
  34. A. N. de Oliveira, S. P. Walborn, and C. H. Monken, "Implementing the Deutsch algorithm with polarization and transverse spatical modes," J. Opt. B: Quantum Semiclass. Opt. 7, 288-292 (2005). [CrossRef]
  35. R. J. C. Spreeuw, "A classical analogy of entanglement," Found. Phys. 28, 361-374 (1998). [CrossRef]
  36. M. J. Padgett, "The mechanism for energy transfer in the rotational frequency shift of a light beam," J. Opt. A: Pure Appl. Opt. 6, S263-S265 (2004). [CrossRef]
  37. I. V. Basistiy, A. Ya. Bakshaev, M. V. Vasnetsov, V. V. Slyusar, and M. S. Soskin, "Observation of the rotational doppler effect for optical beams with helical wave front using spiral zone plate," JETP Lett. 76, 486-489 (2002). [CrossRef]
  38. S. M. Barnett, "Optical angular-momentum flux," J. Opt. B: Quantum Semiclass. Opt. 4, S7-S16 (2002). [CrossRef]
  39. S. C. McEldowney, D. M. Shemo, and R. A. Chipman, "Vortex retarders produced from photo-aligned liquid crystal polymers," Opt. Express 16, 7295-7308 (2008). [CrossRef] [PubMed]
  40. J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, "Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon," Phys. Rev. Lett. 92, 013601 (2004). [CrossRef] [PubMed]

Cited By

 Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1. Fig. 2. Fig. 3.
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited