q-plates: some classical and quantum applications

Transcription

1 Workshop on Singular Op1cs and its Applica1ons to Modern Physics q-plates: some classical and quantum applications Lorenzo Marrucci Università di Napoli Federico II

2 Outline: q A brief reminder on the q-plate concept and operation principles q Let s go quantum : single photons with OAM q q-plate effect on single photons q A quantum interface : Quantum information transfer SAM OAM q Coherent unitary mapping SAM OAM q Generating a 2-photon quantum state with OAM correlations q Quantum information transfer: some examples of applications q Moving further up in the photon space dimensionality

3 A brief reminder on the q-plate concept and operation principles [L. Marrucci, C. Manzo, D. Paparo, PRL 96, (2006); APL 88, (2006)]

4 q-plate structure: patterned half-wave plates y The op1cal axis orienta1on in the plate is pa4erned Cell thickness and birefringence chosen so as to have uniform half- wave retarda0on x (x, y) = (r,) α = angle between the local op1cal axis n and a reference axis y r φ n α x

5 q-plate structure: patterned half-wave plates General α( xy, ) = α( r, ϕ) = qϕ+ α pa4ern: 0 with q integer or half- integer Three examples: Topological defect of charge q in the center q = ½ (α 0 = 0) q = 1 (α 0 = 0) q = 1 (α 0 = π/2)

6 q-plate optical effect: Jones calculus Jones matrix of an α- oriented half- wave plate: M = $ # cos2 sin 2 sin 2 cos2 % ' & Let us apply it to an input le@- circular polarized plane wave: M $ # 1 i % ' E 0 = & $ # cos2 + isin 2 (icos2 + sin 2 % ' E 0 = & $ # 1 (i % 'e i2 E 0 & The output polariza1on is uniform right- handed circular Pancharatnam- Berry geometrical phase (unrelated with op1cal path length) The wave has acquired a phase retarda0on = 2

7 q-plate optical effect For a non- uniform op0cal axis orienta0on: The wavefront gets reshaped For the specific q- plate pa4ern: α(, r ϕ) = qϕ+ α 0 ( ) ΔΦ (, xy) = ± 2α = ± 2qϕ+ ± 2α = mϕ+ cost. Helical phase with m = ±2q 0

8 q-plate optical effect Examples: LeZ circular polariza1on q = 1/2 OAM m = 1 Right circular polariza1on OAM m = 1 Polariza0on controlled OAM handedness

9 q-plate optical effect Examples: q = 1/2 OAM m = ±1 q = 1 OAM m = ± 2

10 Photon angular momentum balance: case q = 1 Le@- circular input: Spin: S z = +ħ Orbital: L z = 0 Total: J z = +ħ q- plate Spin: S z = ħ Orbital: L z = 2ħ Total: J z = +ħ Right- circular input: Spin: S z = ħ Orbital: L z = 0 Total: J z = ħ Spin: S z = +ħ Orbital: L z = 2ħ Total: J z = ħ Spin- to- orbital conversion of op0cal angular momentum

11 Cascading q-plates By mul1ple polariza1on control, one can access any value of OAM In principle, the switching can be as fast as GHz rates (MHz are fairly easy) [L. Marrucci, C. Manzo, D. Paparo, APL 88, (2006)]

12 Let s go quantum : single photons with OAM No1ce: we will actually be using the quantum language and nota1on also for describing op1cal processes which are fully within the scope of classical electromagne1sm

13 Single photons with OAM Nota0on: a photon having a given polariza0on (SAM) and OAM state ψ = SAM OAM = h π m o SAM (π ): a 2D space h = H, V (linear polarizations) h = L, R (circular polarizations) OAM (o ): an D space m = 0, ± 1, ± 2, ± 3,... (Interes0ng for quantum informa0on: lots of room in just one photon)

14 Quantum OAM superposi0ons Polariza1on superposi1ons: ψ = α L + β R = α H + β V π π π π A polariza0on qubit OAM superposi1ons: ψ = α β 2 o o An OAM qubit Higher- dimensional superposi0ons are also possible with OAM ( qudits ) In the following we will consider only OAM qubits with m = ±2

15 Quantum superposi0ons: Poincaré (or Bloch) sphere The (well known) case of polariza0on: V = 1 ( L R ) L i 2 ψ = α L + β R D = 1 2 ( H V ) 1 2 ( ) H = L + R R A = 1 2 ( H + V )

16 Quantum superposi0ons: Poincaré- like sphere The case of an OAM subspace ( m =2): v = 1 i 2 ( l r ) +2 = l ψ = α β 2 d = 1 2 ( h v ) h = 1 2 ( l + r ) 2 = r a = 1 2 ( h + v ) [M. J. Padged & J. Cour1al, Opt. Led. 24, 430 (1999)]

17 What is the behavior of a q- plate in the quantum domain? q- plate effect on single photons [ L. Marrucci, Proc. SPIE 6587, (2007)] [E. Nagali, F. Sciarrino, F. De Mar1ni, L. Marrucci, B. Piccirillo, E. Karimi, E. Santamato, PRL 103, (2009)]

18 q- plate quantum effect on single photons For SAM- OAM eigenstates, nothing new: L π 0 o R π + 2 o R π 0 o L π 2 o What happens with quantum superposi0ons?

19 q- plate quantum effect on single photons The q- plate is also expected to preserve the superposi0ons (it is coherent): ψ = α L 0 + β R 0 π o π o α R π β L 2 o π o In par0cular for a linearly polarized input (H or V): 1 = ( + ) 1 ( R L 2 ) H 0 L R 0 π o 2 π π o 1 = ( ) 1 ( R + 2 L 2 ) V 0 L R 0 π o i 2 π π o i 2 2 π π o o π π o o Entangled state of spin and orbital angular momentum of the same photon

20 q- plate quantum effect on single photons No1ce: this single- photon entanglement is not a non- local property and can be also understood classically 1 2 ( R L 2 ) π o π o = A non- separable polariza1on spa1al mode distribu1on S1ll interes1ng for quantum informa1on protocols and for making some fundamental tests on quantum mechanics

21 q- plate effect on single photons: experiment [E. Nagali, F. Sciarrino, F. De Mar1ni, L. Marrucci, B. Piccirillo, E. Karimi, E. Santamato, PRL 103, (2009)]

22 q- plate effect on single photons: experiment Quantum tomography of polariza0on- OAM entangled states Input H photons Input V photons real real imaginary imaginary ψ = 1 ( R L 2 ) ψ = 1 ( R + 2 L 2 ) 2 π o π o 2 π o π o Single- photon entanglement confirmed

23 q- plate quantum effect: what can we do with it? A quantum interface : Quantum informa0on transfer SAM OAM [E. Nagali, F. Sciarrino, F. De Mar1ni, L. Marrucci, et al., Phys. Rev. Le3. 103, (2009)] [E. Nagali, F. Sciarrino, F. De Mar1ni, L. Marrucci, et al., Opt. Express 17, (2009)]

24 Quantum informa0on transfer SAM OAM 1) SAM OAM Arbitrary polariza1on qubit: ψ = ψ = ( α L + β R ) 0 0 π π o π π o q- plate effect: α R π β L 2 o π o H polarizer: H ( 2 2 ) α + + β = H ψ π o o 2 π o Transfer completed But successful only 50% of 1mes

25 Quantum informa0on transfer SAM OAM 2) OAM SAM Arbitrary OAM qubit: ψ = H ψ = H ( α β 2 ) o π o π o o q- plate effect: α ( R L 0 ) + ( R 0 + L 4 ) β 2 2 π o π o π o π o Coupling to single mode fiber: ( L R ) α + β 0 = ψ π π o 2 π o Transfer completed Again successful only 50% of 1mes

26 Quantum informa0on transfer SAM OAM: the experiment #$%&'(# )*+(,' ?@ /-) *7 /454 %# 6*7 <# /454###; 0 /-) /454 9/ /: /454,# #$% = 6*7 ># %#&' 9/: 9/8 /454 [E. Nagali, F. Sciarrino, F. De Mar1ni, L. Marrucci, et al., Opt. Express 17, (2009)]

27 Quantum informa0on transfer SAM OAM: the experiment Poincaré sphere state reconstruc0ons SAM OAM OAM SAM

28 Quantum informa0on transfer SAM OAM: the experiment Typical quantum tomography results (SAM OAM): Typical experimental fideli0es = 98%

29 Quantum informa0on transfer SAM OAM: back and forth #$%&'(# )*+(,' ?@ /-) *7 /454 %# 6*7 <# /454###; 0 /-) /454 9/ /: /454,# #$% = 6*7 ># %#&' 9/: 9/8 /454 [E. Nagali, F. Sciarrino, F. De Mar1ni, L. Marrucci, et al., Opt. Express 17, (2009)]

30 Quantum informa0on transfer SAM OAM: back and forth $%&'()#$*+,)),) -. #$%&'()#$*+,)),) -. #$ #$ % $% &% # ' $& # #$% #$&% #$&% # #$% $ & % #$(% #$% #$&% # #$% #$&% #$&% # #$% #$&% #$&% # #$% #$% #$&% #$&% # #$% #$% #$&% #$&% # #$% #$% #$&% #$&% # #$%

31 Quantum informa0on transfer SAM OAM: cascaded transfer #$%&'(# )*+(,' ?@ /-) *7 /454 %# 6*7 <# /454###; 0 /-) /454 9/ /: /454,# #$% = 6*7 ># %#&' 9/: 9/8 /454 [E. Nagali, F. Sciarrino, F. De Mar1ni, L. Marrucci, et al., Opt. Express 17, (2009)]

32 Quantum informa0on transfer SAM OAM: cascaded transfer SAM OAM (subspace ±4) ' #&$ #$ #%$ #$ #%$ #%$ #$ #$%&'()&)*+#,-. /$'%-0'1#2/2 #$ % $ #$ #%$ #%$ #$ $ # #$ #%$ #%$ #$ #$ #%$ #%$ #$ #$ #%$ #%$ #$ ' #&$ #$ #%$ #$ #%$ #%$ #$ #$ #%$ #%$ #$ $ $ % #$ #%$ #%$ #$ #$ #%$ #%$ #$ #$ #%$ #%$ #$

33 Thus far only probabilis1c (lossy) transfer, with 50% success probability. Can we do beder? Coherent unitary mapping SAM OAM (or determis0c reversible quantum informa0on transfer) [E. Nagali, F. Sciarrino, F. De Mar1ni, L. Marrucci, et al., Opt. Express 17, (2009)] [E. Karimi, S. Slussarenko, B. Piccirillo, L. Marrucci, E. Santamato, PRA 81, (2010)]

34 Coherent unitary mapping SAM OAM Yes: reversible and determinis1c transfer is possible (ideally 100% success probability) : Sagnac interferometer with PBS input/output and Dove prism (DP) This scheme has not been tested yet in the single photon regime but we did it in the (equivalent) classical regime

35 Coherent unitary mapping SAM OAM A 3D version of the setup: [E. Karimi, S. Slussarenko, B. Piccirillo, L. Marrucci, E. Santamato, PRA 81, (2010)]

36 Coherent unitary mapping SAM OAM Experimental results (output mode images and interference paderns): All OAM states on the OAM Poincaré- like sphere can be reproduced using polariza1on control only.

37 Coherent unitary mapping SAM OAM A different (closed) path on the Poincaré sphere:

38 Yet another path: Coherent unitary mapping SAM OAM

39 Coherent unitary mapping SAM OAM Pancharatnam geometric phase resul1ng in the closed paths also transferred: 1.0 (a) 1.0 (b) Intensity Intensity x a.u x a.u.

40 Single photon: not uniquely quantum effects (just like classical op1cs, but at lower intensity) We need to test the case of two (or more) photons for having truly quantum correla0on effects

41 Genera0ng a 2- photon quantum state with OAM correla0ons [E. Nagali, F. Sciarrino, F. De Mar1ni, L. Marrucci, B. Piccirillo, E. Karimi, E. Santamato, PRL 103, (2009)]

42 2- photon quantum correla0ons in OAM ψ = Consider 2 photons with orthogonal linear polariza1ons H, V: 1 2 H V Same state in the circular- polariza1on basis: 1 ψ = + = + 2i ( L R ) ( L R ) ( L L R L L R R R ) i Coalescence enhancement For iden1cal photons: ψ = i 1 ( ) 2 L L R R 2- photon quantum interference When iden0cal, the two photons must always have the same polariza0on handedness: quantum correla0ons In polariza0on this has been already demonstrated, but now

43 2- photon quantum correla0ons in OAM SAM OAM Obtained a 2- photon state with OAM quantum correla0ons 1 ψ = + + i 2 ( ) How to verify? VH, Two- photons source Coincidence counter Photons separator (introducing a delay) There should be no coincidences when the photons are iden1cal

44 2- photon quantum correla0ons in OAM Our experimental results: 600 Coincidence counts in 900 s Temporal delay (fs) But, where are the photons going? Coalescence enhancement

45 2- photon quantum correla0ons in OAM Verifying coalescence enhancement: VH, Two- photons source Coincidence counter Coincidences Enhancement factor = Time delay (fs)

46 2- photon quantum correla0ons in OAM Coherence check: = 1 i 2 ( ) = h v = 1 ( 2 a a d d ) VH, Two- photons source Coincidence counter h a v d

47 But what can we do with these quantum informa1on transfer devices? Quantum informa0on transfer: some examples of applica0ons

48 Hybrid OAM SAM entanglement and quantum contextuality tests E. Karimi, J. Leach, S. Slussarenko, B. Piccirillo, L. Marrucci, L. Chen, W. She, S. Franke- Arnold, M. J. Padged, E. Santamato, PRA 82, (2010)

49 Hybrid OAM SAM entanglement and quantum contextuality tests Coincidence counts s a 0 π π 0 3 π π 0 π π 0 3 π π -ORBIT HYBRID ENTANGLEMENT OF PHOTONS8 AND PHYSICAL 8 REVIEW 4 A8 82, (2010) 2 Angle of hologram χ rad Angle of hologram χ rad S χ Experimental results: Power µw c π π 0 3 π π π π 0 3 π π Similar results (with C. Classicala slightly different experiment Angle of hologram χ rad Angle of hologram χ rad technique) In our final also experiment, reported we in [E. move Nagali to a classical and regime of F. Sciarrino, nonseparable Opt. optical Express modes 18, occupied (2010)] by many photons, corresponding see Fabio s to coherent talk quantum states. A 100-mW frequency- IG. 3. The CHSH S value in a region where FIG. it is2. larger (Color than theonline) The experimental coincidence counts as sical limit 2. The choice of the variables appearing in Eq. (4) is a function of orientation of the sector hologram for different values ollowing: θ = 0, θ = π/4, χ is the plot abscissa, χ doubled linearly polarized continuous wave Nd:YVO = χ + π/8. 4 laser light gray, gray, and black dots correspond of polarization to the experimental direction, beam for is (a) sent heralded in an optical single linephotons, equal arm (b) Aphoton of our quantum in the E. Karimi, case of single-photon J. Leach, S. Slussarenko, (a), photon-pairs pairs, B. Piccirillo, and (b), and (c) L. classicale (c) SAM-OAM experiments, respectively. The dashed line is SAM-OAM nonseparable mode coherent-states. Marrucci, L. apparatus Chen, W. to Black She, obtain, dots S. Franke- Arnold, after the q plate, a coherent state in the respresentm. θ J. = 0, dark gray dots θ = π/4, gray dots θ = 2π/4, and light gray dots + Padged, E. Santamato, PRA 82, (2010) given by Eq. (2) [26]. θ = 3π/4. quantum mechanical ideal prediction. In the two cases (a) and The calculated structure of this mode is shown in Fig. 2(d) Coincidence counts s b measurements for different angles θ and χ, thequantitys was evaluated from Eqs. (d) (4) and(5) and the violation of the Bell- kind CHSH(Clauser- Horne- Shimony- Holt) inequality was verified, as shown in Fig. 3 (light gray dots). This violation provides a demonstration of SAM-OAM inequality hybrid tested entanglement for demonstra1ng and nonlocality quantum for separated photon contextuality pairs. in different regimes

50 Quantum cloning of OAM qubits and SAM OAM qudits LETTERS PUBLISHED ONLINE: 22 NOVEMBER 2009 DOI: /NPHOTON Optimal quantum cloning of orbital angular momentum photon qubits through Hong Ou Mandel coalescence Eleonora Nagali 1,LindaSansoni 1, Fabio Sciarrino 1,2 *, Francesco De Martini 1,3,LorenzoMarrucci 4,5 *, Bruno Piccirillo 4,6, Ebrahim Karimi 4 and Enrico Santamato 4,6 PRL 105, (2010) P H Y S I C A L R E V I E W L E T T E R S week ending 13 AUGUST 2010 Experimental Optimal Cloning of Four-Dimensional Quantum States of Photons E. Nagali, 1 D. Giovannini, 1 L. Marrucci, 2,3 S. Slussarenko, 2 E. Santamato, 2 and F. Sciarrino 1,4, * 1 See Fabio s talk

51 The next step: Moving further up in the photon space dimensionality

52 A single- beam universal quantum gate in SAM OAM space: Main idea: to exploit OAM radial profile correla1ons arising in free propaga1on A q- box : semi birefringent plate m 0 I a.u m r w 0 The complete device: q- box λ/4 λ/2 [S. Slussarenko, E. Karimi, B. Piccirillo, L. Marrucci, E. Santamato, PRA 80, (2009)]

53 Controlling a higher- dimensional OAM subspace with a single q- plate ¼ j j (Color online) Alice apparatus is similar to what is sh [S. Slussarenko, E. Karimi, B. Piccirillo, L. Marrucci, E. Santamato, JOSA A 28, (2011)]

54 Acknowledgments Naples group: Rome group: (OAM line) Current sponsor: