Measurement and Control of Transverse Photonic Degrees of Freedom via Parity Sorting and Spin-Orbit Interaction

Transcription

1 Measurement and Control of Transverse Photonic Degrees of Freedom via Parity Sorting and Spin-Orbit Interaction Cody Leary University of Warsaw October 21, 2010 Institute of Experimental Physics Optics Division

2 Measurement and Control of Transverse Photonic Degrees of Freedom via Parity Sorting and Spin-Orbit Interaction Cody Leary University of Warsaw October 21, 2010 Institute of Experimental Physics Optics Division

3 Paraxial beams and transverse spatial Photons & electrons have 4 degrees of freedom (DOFs) For paraxial beams, they may be taken as: 1.) Energy (frequency) 2.) Spin/Polarization (SAM) 3.) Orbital angular momentum z-component (OAM) 4.) Radial quantum number y modes All of these have been previously studied individually x However, interactions between DOF s have received less attention z

4 Paraxial beams and transverse spatial Photons & electrons have 4 degrees of freedom (DOFs) For paraxial beams, they may be taken as: 1.) Energy (frequency) 2.) Spin/Polarization (SAM) 3.) Orbital angular momentum z-component (OAM) 4.) Radial quantum number y modes All of these have been previously studied individually x However, interactions between DOF s have received less attention This talk concerns: Various manifestations of the interaction between the SAM and OAM of electrons and photons in cylindrical, beamlike geometries z

5 Paraxial quantum numbers 1.) Energy (continuous): 2.) Spin/Polarization: 3.) Orbital angular momentum z-component (integer): ω σ = ±1 m l m m l l = 0, n = 0 : = 1, n = 0 : 4.) Radial (nonnegative integer): n m l = 1, n = 1:

6 Paraxial quantum numbers 1.) Energy (continuous): 2.) Spin/Polarization: 3.) Orbital angular momentum z-component (integer): ω σ = ±1 m l m m l l = 0, n = 0 : = 1, n = 0 : 4.) Radial (nonnegative integer): n m l = 1, n = 1: In a weakly inhomogeneous medium, a monoenergetic wave function has the form: m ( ) ( ) imlφ i 0z t ˆ, n n m σ = ψ ρ e e β ω e l l OAM σ SAM β : 0 l, eˆ σ : Propagation constant, fixed by m n Spinor for electrons, vector for photons, with helicity ± λħ

7 Spin (SAM) Bloch Sphere Orbital (OAM) Bloch Sphere R L LG 0 +1 LG 0-1 D A HG +45 HG -45 H V HG 01 HG 10 R σ = +1 LG 0 +1 m = +1 l H φ ɶ θ ɶ V HG 01 φ ɶ θ ɶ HG 10 D HG +45 L σ = 1 LG 0-1 m = 1 l

8 Outline 1.) Geometric phase of a wave packet with SAM, OAM, and a well-defined trajectory

9 Outline 1.) Geometric phase of a wave packet with SAM, OAM, and a well-defined trajectory 2.) Interferometric parity operations on transverse modes Experimental manipulation of transverse spatial DOFs HG to LG mode conversion First 2-D parity sorting of Hermite-Gauss modes

10 Outline 1.) Geometric phase of a wave packet with SAM, OAM, and a well-defined trajectory 2.) 3.) Interferometric parity operations on transverse modes Experimental manipulation of transverse spatial DOFs HG to LG mode conversion First 2-D parity sorting of Hermite-Gauss modes Spin-orbit interaction of a particle in a cylindrical waveguide OAM-controlled rotation of particle spin (polarization) SAM-controlled rotation of transverse spatial state Spin-orbit interaction as geometric phase Experiment to observe SOI

11 Outline: Part 1.) 1.) Geometric phase of a wave packet with SAM, OAM, and a well-defined trajectory

12 Particle with well-defined momentum k along a curvilinear path Trajectory is a helix with radius a, helix pitch h z, pitch angle Global X,Y,Z axes fixed, local z axis aligned with the path Local x,y axes may rotate about z as the particle propagates This frame transport is determined by the geometric phase θ ( ) ( ) ( )

13 Particle with well-defined momentum k along a curvilinear path Trajectory is a helix with radius a, helix pitch h z, pitch angle Global X,Y,Z axes fixed, local z axis aligned with the path Local x,y axes may rotate about z as the particle propagates This frame transport is determined by the geometric phase θ ( ) ( ) ( )

14 Particle with well-defined momentum k along a curvilinear path Trajectory is a helix with radius a, helix pitch h z, pitch angle Global X,Y,Z axes fixed, local z axis aligned with the path Local x,y axes may rotate about z as the particle propagates This frame transport is determined by the geometric phase θ ( ) ( ) ( )

15 ( ) ( ) ( ) ( )

16 a.) h z z θ kˆ c b.) Ω C kz θ kˆ c k y k x a y x Q: How to calculate the phase? φ Ω A: The phase equals, the solid angle subtended by particle trajectory in momentum space. It is thus determined by the path geometry. φ

17 a.) h z z θ kˆ c b.) Ω C kz θ kˆ c k y k x a y x Geo ( λσ m ) Φ = + Ω l λ σ m l Ω magnitude of particle helicity SAM quantum number OAM quantum number solid angle subtended by particle trajectory in momentum space

18 Geometric Phase Discussion Φ = σ Ω Geo The SAM dependence of this result for photons was predicted by Bortolotti (1926) and calculated by Rytov and Vladimirsky ( ) for waves without OAM Forty years later, prompted by the work of Berry on the geometric phase (1984), Chiao, Wu, and Tomita carried out an experiment to measure this effect. (1986) In 1987, Bialynicki-Birula and Bialynicka-Birula extended the result to any particle with spin propagating along a well-defined curvilinear trajectory (without OAM) That same year, Kitano, Yabuzaki, and Ogawa showed that the phase manifests itself as an image rotation for a light beam undergoing out-of-plane refections. This image rotation effect was subsequently studied more rigorously by Segev, Solomon, and Yariv (1989), and Galvez and Holmes (1998), who demonstrated that the both the polarization and image of the light beam are rotated through the same transverse angle by any sequence of out-of-plane mirror refections. Perhaps surprisingly, the connection between the geometric phase, OAM, and transverse image rotation for light was not explicitly pointed out until a few years ago by Bliokh (2006)

19 Geometric Phase Discussion Φ = λσ Ω Geo The SAM dependence of this result for photons was predicted by Bortolotti (1926) and calculated by Rytov and Vladimirsky ( ) for waves without OAM Forty years later, prompted by the work of Berry on the geometric phase (1984), Chiao, Wu, and Tomita carried out an experiment to measure this effect. (1986) In 1987, Bialynicki-Birula and Bialynicka-Birula extended the result to any particle with spin propagating along a well-defined curvilinear trajectory (without OAM) That same year, Kitano, Yabuzaki, and Ogawa showed that the phase manifests itself as an image rotation for a light beam undergoing out-of-plane refections. This image rotation effect was subsequently studied more rigorously by Segev, Solomon, and Yariv (1989), and Galvez and Holmes (1998), who demonstrated that the both the polarization and image of the light beam are rotated through the same transverse angle by any sequence of out-of-plane mirror refections. Perhaps surprisingly, the connection between the geometric phase, OAM, and transverse image rotation for light was not explicitly pointed out until a few years ago by Bliokh (2006)

20 Geometric Phase Discussion ( λσ m ) Φ = + Ω Geo The SAM dependence of this result for photons was predicted by Bortolotti (1926) and calculated by Rytov and Vladimirsky ( ) for waves without OAM Forty years later, prompted by the work of Berry on the geometric phase (1984), Chiao, Wu, and Tomita carried out an experiment to measure this effect. (1986) In 1987, Bialynicki-Birula and Bialynicka-Birula extended the result to any particle with spin propagating along a well-defined curvilinear trajectory (without OAM) That same year, Kitano, Yabuzaki, and Ogawa noted that the phase manifests itself as an image rotation for a light beam undergoing out-of-plane refections. This image rotation effect was subsequently studied more rigorously by Segev, Solomon, and Yariv (1989), and Galvez and Holmes (1998), who demonstrated that the both the polarization and image of the light beam are rotated through the same transverse angle by any sequence of out-of-plane mirror refections. Perhaps surprisingly, the connection between the above relation for the orbital geometric phase, the OAM, and transverse image rotation for light was not explicitly pointed out until a few years ago by Bliokh (2006) l

21 Geometric Phase Discussion Geo ( λσ m ) Φ = + Ω l Aspects of this result have been experimentally verified in various semiclassical contexts (cf. Bliokh, Chiao) It plays a central role in both the design of our parity sorting interferometer and in understanding the physics of the spin-orbit interaction.

22 Outline: Part 2.) 1.) Geometric phase of a wave packet with SAM, OAM, and a well-defined trajectory 2.) Interferometric parity operations on transverse modes Experimental manipulation of transverse spatial DOFs HG to LG mode conversion First 2-D parity sorting of Hermite-Gauss modes

23 Transverse spatial mode combinations = i = + LG = HG10 i HG HG + 10 = HG 10 + HG 01 LG 0 +1 Transverse spatial modes can be represented in terms of Hermite or Laguerre-Gauss modes; first-order modes are decomposed as shown HG 01 HG +45 φ ɶ θ ɶ HG 10 LG 0-1

24 Previously realized parity interferometers x y (a) y x = + i 1 2 y z E x y E 1-D Parity Sorter: H. Sasada and M. Okamoto, Phys. Rev. A 68, (2003). GP B A x y (b) x 2-D Parity Sorter: J. Leach et al. Phys. Rev. Lett. 88, (2002) = + i 1 2 y x GP B A + i

25 2-D parity operation as 180 degree rotation (a) y Πˆ x y Πˆ y y x x x E (b) y x Π = Rˆ ˆ xy 180 x y E x x y 2-D Parity Sorter: y GP B A + i y x = + i 1 2 y x

26 1-D parity sorting experiment Laser BS3 BS1 GP d 1 d 2 BS2 B A BS4 CCD Ref

27 Experimental results: 1-D parity We use this 1-D sorter in later experiments

28 Intermission The previously realized interferometers are of the Mach-Zehnder type They are subject to phase noise and drift This makes their application impractical for quantum information applications using entangled photon sources We have developed a phase-stable (common path!) Sagnac interferometer based on 2-D parity Its design is based on the geometric phase + i y x 2-D Parity Sorter: = + i 1 2 y x GP B A

29 2-D parity sorting apparatus M2 Y β X Z M3 θ y M1 BS z x = + i M4 B A + i

30 Poincare sphere construction: reflection (a) Z (b) X y y Y α ˆk 1 ˆk 2 Y α ˆk 1 ˆk 2 x (c) x

31 Poincare sphere construction: 2-D sorter (a) (b) β θ Z = + i + i B A h z a z θ y kˆ c x Ω kz C θ kˆ c k y k x Geo ( λσ m ) Φ = + Ω iml iml imlφgeo e e e e l ( φ ) φ φ im Φ = l Geo A geometric phase accumulation is equivalent to a rotation for states with well-defined OAM Polarization also rotates

32 2-D parity sorting experiment C. C. Leary, L. A. Baumgardner, and M. G. Raymer, Optics Express, Vol. 17, Issue 4, pp (2009) BC = Berek polarization compensator (for polarization matching)

33 Experimental results: 2-D parity sorting

34 Outline 1.) Geometric phase of a wave packet with SAM, OAM, and a well-defined trajectory 2.) 3.) Interferometric parity operations on transverse modes Experimental manipulation of transverse spatial DOFs HG to LG mode conversion First 2-D parity sorting of Hermite-Gauss modes Spin-orbit interaction of a particle in a cylindrical waveguide OAM-controlled rotation of particle spin (polarization) SAM-controlled rotation of transverse spatial state Spin-orbit interaction as geometric phase Experiment to observe SOI

35 Spin-Orbit Interaction (SOI) in Spherical Potentials: ELECTRON IN AN INHOMOGENOUS SPHERICAL ELECTRIC POTENTIAL (ATOM) H ' 2 e 1 V = 2 2 2m c r r S L S = SAM r p = L = O A M (atomic fine structure) Spin-Orbit Interaction (SOI) in Cylindrical Potentials? H ' = S ˆ z = S z C Leary, D Reeb, M Raymer, NJP, 10, (2008). 2 e 2 1 V 2 2 m c ρ ρ S L z z L z = L zˆ e e Schrodinger-Pauli equation traveling wave solution: ĤΨ = EΨ Ψ ( 0 E ) i z t e β ħ

36 (a) Laboratory frame: Field is entirely electrostatic (b) Electron frame: A magnetic field is present, which interacts with the electron spin magnetic moment

37 The electron spin-orbit interaction has an intuitive physical model Solve Schrodinger (Dirac) equation: ĤΨ = 2 β Ψ U ( ρ ) χ ( ρ ) e U a ρ ( ) ( ρ ) = U ( 0) + χ ( ρ ) Hˆ = e 2 1 χ SzL ρ ρ z

38 The electron spin-orbit interaction has an intuitive physical model Solve Schrodinger (Dirac) equation: e U U ( ρ ) 2 ĤΨ = β Ψ χ ( ρ ) a ρ ( ) ( ρ ) = U ( 0) + χ ( ρ ) However, photons do not have a spin magnetic moment to contribute to the spinorbit energy shift Hˆ = e 2 1 χ SzL ρ ρ z

39 The electron spin-orbit interaction has an intuitive physical model Solve Schrodinger (Dirac) equation: ĤΨ = 2 β Ψ Solve Helmholtz (Maxwell) equations: ĤE 2 = β E U ( ρ ) χ ( ρ ) ( ) = 2 ( ) ε ρ ε ρ 0 n χ ( ρ ) e U a ρ ( ) ( ρ ) = U ( 0) + χ ( ρ ) γ ρ a ε ρ = ε 0 1 χ ρ ( ) ( ) ( ) ( ) Hˆ = e 2 1 χ SzL ρ ρ z Hˆ = γ 2 1 χ SzL ρ ρ z But it is remarkable that the photon SOI is completely analogous!

40 The electron spin-orbit interaction has an intuitive physical model Solve Schrodinger (Dirac) equation: ĤΨ = 2 β Ψ Solve Helmholtz (Maxwell) equations: ĤE 2 = β E U ( ρ ) χ ( ρ ) ( ) = 2 ( ) ε ρ ε ρ 0 n χ ( ρ ) e U a ρ ( ) ( ρ ) = U ( 0) + χ ( ρ ) γ ρ a ε ρ = ε 0 1 χ ρ ( ) ( ) ( ) ( ) Hˆ = e 2 1 χ SzL ρ ρ z Hˆ = γ 2 1 χ SzL ρ ρ z But it is remarkable that the photon SOI is completely analogous! Main Point: the S part of the Hamiltonian means that will zl β z undergo a small positive/negative shift depending upon whether and are oriented parallel/antiparallel to each other. L z S z

41 Spin-orbit phase shift Wavefunction: ( ) ( ρ ) imlφ i 0z t ˆ, Ψ n m e e β ω σ = ψ e m n l l σ OAM SAM Hamiltonian: Perturbation theory: 1 χ H = S L S L SOI z z z z 4β0 ρ ρ δβ = Ψ H Ψ σ m = σ µ m n ml σ SOI n ml σ l l Spin-orbit Phase shift: Ψ Ψ e = Ψ iδβ z n m σ n m σ n m σ l l l e iσµ δβ z

42 OAM-controlled spin rotation Free Space: Positive OAM R Positive OAM Positive OAM + L = Right Circular (Positive Helicity) Left Circular (Negative Helicity) Linear Polarization/Spin Cylindrical Waveguide: i z R e δβ + L e + iδβ z = Negative Phase Shift Positive Phase Shift orbit -controlled spin rotation z Polarization/Spin: ( δβ z) xˆ sin ( δβ z) cos ± yˆ States with parallel SAM and OAM experience a negative phase shift, while anti-parallel states undergo a positive shift. This results in rotation of the polarization/spin with propagation in z. x y

43 Spin-controlled orbital rotation Free Space: Positive OAM R Negative OAM + R = R Right Circular (Positive Helicity) Right Circular (Positive Helicity) Petal Mode with Zero average OAM Cylindrical Waveguide: i z R e δβ + R e + iδβ z = Negative Phase Shift Positive Phase Shift R spin -controlled orbital rotation Wave Function: ( φ δβ ) cos m ± z ; m = OAM States with parallel SAM and OAM experience a negative phase shift, while anti-parallel states undergo a positive shift. This results in rotation of the transverse mode with propagation in z.

44 Complementary spin-orbit effects Zel dovich, PRA 91 (for photons) orbit -controlled spin rotation spin -controlled orbit rotation Both of these effects may occur either in space or time The effects occur for both electrons and photons Independent of mass, charge, magnetic moment, etc. This spin-orbit interaction is a universal geometric phase (see PRA 80, (2009), NJP (2008))

45 Spin-orbit interaction and geometric phase The polarization rotation effect is reminiscent of an experiment of Chaio, where the geometric phase was observed as polarization rotation in a coiled single-mode fiber. z a Cyl Chaio Experiment z x a Mode Cylinder Single mode optical fiber y The Chaio effect involves only the zero OAM fundamental Gaussian mode Predicts an accumulated geometric phase of zero for a straight fiber! Nevertheless, by indentifying the cylinder radius with the average mode radius and the helical path with the OAM of the mode, the semiclassical description of Chaio agrees quantitatively with our more rigorous mode-picture predictions.

46 Evolution of particle spin under SOI R The spin/polarization Bloch vector of a particle with welldefined OAM precesses about the polar axis of the Bloch sphere The direction of precession is controlled by the sign of the particle OAM H D φ ɶ θ ɶ V L

47 Evolution of particle OAM under SOI L The orbital angular momentum Bloch vector of a particle with well-defined spin precesses about the polar axis of the Bloch sphere The direction of precession is controlled by the sign of the particle spin H D φ ɶ θ ɶ V R

48 Spin-orbit interaction experiment 3 Mode Fiber SM Fiber Laser Glass plates To Polarization Analyzer Fiber Crusher 2-D Sorter Odd Port Even Port To 1-D Sorter Test Fiber

49 Conclusions Geometric phase of a spin-orbit wave packet

50 Conclusions Geometric phase of a spin-orbit wave packet Experimental manipulation of transverse spatial modes via 1-D and 2-D parity-based interferometers First 2-D parity sorting measurements of HG modes Applications: entangled Bell states and heralded single photons in arbitrary first-order transverse spatial modes

51 Conclusions Geometric phase of a spin-orbit wave packet Experimental manipulation of transverse spatial modes via 1-D and 2-D parity-based interferometers First 2-D parity sorting measurements of HG modes Applications: entangled Bell states and heralded single photons in arbitrary first order spatial modes The spin-orbit interaction dynamics of the electron and photon are identical in a cylindrical inhomogeneous medium They have a common geometric origin Application: The SOI allows for the construction of both spin and orbital gates, which enables reversible transfer of entanglement between SAM and OAM DOFs Experimental progress toward observing the SOI

52 Supplementary Material

53 Transverse spatial mode combinations = i = + LG = HG10 i HG HG + 10 = HG 10 + HG 01 = + i = LG 0 1 = HG 10 + i HG HG 10 = HG10 HG 01 Transverse spatial modes can be represented in terms of Hermite or Laguerre-Gauss modes; first-order modes are decomposed as shown

54 Polarization state combinations = i = + R = H i V D = H + V = + i = L = H + i V A = H V The transverse spatial mode decomposition is similar to the more wellknown decomposition of polarization states of light

55 Spin (SAM) Bloch Sphere: Orbital (OAM) Bloch Sphere: Arbitrary superpositions of SAM and OAM states may be represented on the associated two-level Bloch spheres R LG 0 +1 H φ ɶ θ ɶ V HG 01 φ ɶ θ ɶ HG 10 D HG +45 L LG 0-1

56 Electrons have spin ½: A rotation is equivalent to a phase λσφ a h z X Y Z X x y X x y x z z X y z φ θ θ θ Z Z Z A B C ψ ρ, φ, z ψ ρ, φ + φ, z ψ ρ, φ + 2 φ, z ( ) ( ) ( ) φ

57 Photons have spin 1: A rotation is equivalent to a phase λσφ a h z X Y Z X x y X x y x z z X y z φ θ θ θ Z Z Z A B C ψ ρ, φ, z ψ ρ, φ + φ, z ψ ρ, φ + 2 φ, z ( ) ( ) ( ) φ

58 Both particles have OAM: For OAM states, a rotation is equivalent to a phase m φ l a h z X Y Z X x y X x y x z z X y z φ θ θ θ Z Z Z A B C ψ ρ, φ, z ψ ρ, φ + φ, z ψ ρ, φ + 2 φ, z ( ) ( ) ( ) φ

59 (a) (c) (d) ( xˆ ± yˆ ) 1 Action of 1-D interferometer = = φ = B 0 ŷ A ˆx 1 1 (b) φ = B π 2 ( ˆ ± ˆ ) x y 1 ŷ ( ) ŷ ŷ + φ = B 0 ŷ A φ = B π 2 A ( xˆ ± iyˆ ) ( xˆ iyˆ ) ŷ ( + ) ŷ ŷ ( + i )ŷ = A = ŷ ( i )ŷ

60 Action of 2-D interferometer (a) = + ( + i ) 1 φ = 0 A B = ( + i ) (b) = + ( + i ) 1 φ = B π 2 ( ) = i A + i + = + i = i( + i ) = i

61 Action of 1-D and 2-D sorters on Hermite-Gauss modes

62 Poincare sphere construction: 2-D sorter (a) (b) β θ Z = + i + i B A (c) π Ω (d) π Ψ π 2 π 2 π 4 π 2 θ π 8 π 4 3π 8 π 2 θ

63 Poincare sphere construction: 2-D sorter (a) (b) β θ Z = + i + i B A For a 2-D sorter, θ = ψ = π Other angles correspond to various OAM sorting interferometers Multiple interferometers may be stably cascaded for applications in quantum information processing π 4 (d) π π 2 Ψ π 8 π 4 3π 8 π 2 θ

64 Applications for quantum info. processing

65 SAM to OAM entanglement transfer Begin in a purely SAMentangled Bell state: Apply quarter wave plates: H V + V H R L + L R Apply OAM rotation SOI gate: + R L L R Apply quarter wave plates + and mode converters: D A A D Apply SAM rotation SOI gate: (End in purely OAM entangled state) + H H H H

66 Horizontally polarized z Chaio Experiment Detectors a.) h z Polarized HeNe a z Coupling lens θ y ˆk Cylinder Ω = x Single mode optical fiber 2π N 1 cosθ ( ) Where N = # of coils b.) Ω Polarizing beam splitter The geometric phase equals the surface area enclosed by the curve C, which is traced out by the momentum vector k. This phase causes a beating effect: polarization rotation C kz θ ˆk k y k x

67 Spatial vs. Temporal Rotation β 0 β anti-parallel parallel The SOI may be thought of a a splitting of the dispersion curve for parallel vs. anti-parallel states. For a given frequency, there are two values, so that the SOI splits the propagation constant (spatial rotation) β ω 0 ω If the both the frequencies and propagation constants of the parallel and anti-parallel states are originally slightly different, than the SOI acts to restore the propagation constants to degeneracy (temporal rotation)

68 Geometric phase approach Specifically, one gets a particularly simple form for the i z accumulated phase e δβ in the special case of a stepindex/potential of large radius a, for OAM states which are near cutoff that is, their wavefunction peak is near the fiber radius: γ = δβ z = σ ml z 2 2β a In this case the two approaches agree exactly. 0 z z θ ˆk h z a x y a y x