* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download polarized quantum states
Particle in a box wikipedia , lookup
Quantum field theory wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
Bell test experiments wikipedia , lookup
Quantum decoherence wikipedia , lookup
Wave–particle duality wikipedia , lookup
Hydrogen atom wikipedia , lookup
Quantum dot wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Scalar field theory wikipedia , lookup
Wheeler's delayed choice experiment wikipedia , lookup
Many-worlds interpretation wikipedia , lookup
Quantum fiction wikipedia , lookup
Path integral formulation wikipedia , lookup
Double-slit experiment wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
Quantum entanglement wikipedia , lookup
Probability amplitude wikipedia , lookup
Quantum computing wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
EPR paradox wikipedia , lookup
Delayed choice quantum eraser wikipedia , lookup
History of quantum field theory wikipedia , lookup
Quantum group wikipedia , lookup
Quantum teleportation wikipedia , lookup
Bell's theorem wikipedia , lookup
Quantum machine learning wikipedia , lookup
Quantum state wikipedia , lookup
Hidden variable theory wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
Canonical quantization wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Coherent states wikipedia , lookup
Quantum dot cellular automaton wikipedia , lookup
Polarization descriptions of quantized fields Anita Sehat, Jonas Söderholm, Gunnar Björk Royal Institute of Technology Stockholm, Sweden Pedro Espinoza, Andrei B. Klimov Universidad de Guadalajara, Jalisco, Mexico Luis L. Sánchez-Soto Universidad Complutense, Madrid, Spain Outline • Motivation • Stokes parameters and Stokes operators • Unpolarized light – hidden polarization • Quantification of polarization for quantized fields • Generalized visibility • Polarization of pure N-photon states • Orbits and generating states • Arbitrary pure states •Summary Motivation • • • • The polarization state of a propagating electromagnetic field is relatively robust The polarization state is relatively simple to transform Transformation of the polarization state introduces only marginal losses The polarization state can easily and relatively efficiently be measured The polarization is an often used property to encode quantum information Typically, photon counting detectors are used to measure the polarization => The post-selected polarization states are number states A (semi)classical description of polarization is insufficient. The Stokes parameters In 1852, G. G. Stokes introduced operational parameters to classify the polarization state of light If E a1ei(t 1 )eˆx a2ei(t 2 )eˆy where a1 , a2 ,1 , and 2 are real, then S x 2a1a2 cos tests x linear polarization S y 2a1a2 sin tests circular polarization S z a12 a22 tests + linear polarization S 0 a12 a22 , where 1 2 PC S 2 x S y2 S z2 S0 1/ 2 the degree of polarizati on If P=0, then the light is (classically) unpolarized The Stokes operators S x S 0 sˆx aˆ bˆ aˆbˆ S y S 0 sˆ y aˆ bˆ aˆbˆ i E. Collett, 1970: S z S 0 sˆz aˆ aˆ bˆ bˆ S Nˆ aˆ aˆ bˆ bˆ 0 PSC sˆ 2 x sˆ y Two-mode thermal state 2 Nˆ sˆz 2 1/ 2 a " quantum" degree of polarizati on 0 PSC 1 E. Collett, Am. J. Phys. 38, 563 (1970). Any two-mode coherent state A problem with PSC Any two - mode coherent state , has PSC 1. lim 0 PSC ,0 1 A two-mode coherent state, arbitrarily close to the vacuum state is fully polarized according to the semiclassical definition! SU(2) transformations – realized by geometrical rotations and differential-phase shifts A geometrica l rotation by the angle is realized by exp i sˆ y A differenti al - phase shift by is realized by exp i sˆz A unitary representa tion of the group SU(2) can be realized by the operator Uˆ , , exp i sˆ exp i sˆ exp i sˆ z y z Only waveplates, rotating optics holders, and polarizers needed for all SU(2) transformations and measurements. Another problem: Unpolarized light – hidden polarization PSC = 0 => Is the corresponding state is unpolarized? Consider t he state 1,1 Counter 1,1 Frequency BBO doubled pulsed Type II Ti:Sapphire laser =780 nm =390 nm HWP PBS Detector Experimental results sy HWP sx sz sy QWP Single counts per 10 sec 11000 10000 9000 8000 HWP@ QWP@ [email protected]+QWP@ light off 7000 6000 5000 4000 3000 2000 1000 sx sz 0 0 30 60 90 120 150 180 210 240 270 300 Phase plate rotation , deg The state is unpolarized according to the classical definition P. Usachev, J. Söderholm, G. Björk, and A. Trifonov, Opt. Commun. 193, 161 (2001). 330 360 Unpolarized light in the quantum world A quantum state which is invariant under any combination of geometrical rotations (around its axis of propagation) and differential phase-shifts is unpolarized. H. Prakash and N. Chandra, Phys. Rev. A 4, 796 (1971). G. S. Agarwal, Lett. Nuovo Cimento 1, 53 (1971). J. Lehner, U. Leonhardt, and H. Paul, Phys. Rev. A 53, 2727 (1996). 1,1 Coincidence counter Detector BBO HWP PBS Detector Type II Coincidence counts per 10 sec A coincidence count experiment 800 HWP@ 700 curve fit 600 500 400 300 200 100 0 0 15 30 45 60 75 90 105 120 135 150 165 180 195 Half-wave plate rotation , deg Since the state is not invariant under geometrical rotation, it is not unpolarized. The raw data coincidence count visibility is ~ 76%, so the state has a rather high degree of (quantum) polarization although by the classical definition the state is unpolarized. This is referred to as “hidden” polarization. D. M. Klyshko, Phys. Lett. A 163, 349 (1992). States invariant to differential phase shifts “Linearly” polarized quantum states Eigenstate s to sˆz : Classical polarization Quantum polarization 2,0 Vertical Vertical 0,2 Horizontal Horizontal Unpolarized! Neutral, but fully polarized? 1,1 The “linear” neutrally polarized state lacks polarization direction (it is symmetric with respect to permutation of the vertical and horizontal directions). It has no classical counterpart. For all even total photon numbers such states exist. Rotationally invariant states Circularly polarized quantum states 0,2 1 0,2 2 2,0 1 0,2 2 1,1 1 0,2 2,0 2 Classical polarization Quantum polarization i 2 1,1 2,0 Left handed Left handed i 2 1,1 2,0 Right handed Right handed Unpolarized Neutral, but fully polarized? The circular neutrally polarized state is rotationally invariant but lacks chirality. It has no classical counterpart. For all even total photon numbers such states exist. States with quantum resolution of geometric rotations Consider: 2 1 0,2 0 i 1,1 0 2,0 3 0 2 i 2 i 0,2 2,0 6 6 A geometrical rotation of this state by /3 (60 degrees) will yield the state: 1 i 2 / 3 3 e 0,2 0 i 1,1 0 ei 2 / 3 2,0 0 3 A rotation of 2 by 2 /3 (120 degrees) or by - /3 will yield the state: 1 i 2 / 3 1 e 0,2 0 i 1,1 0 ei 2 / 3 2,0 0 3 1 2 1 3 2 3 0 Complete set of orthogonal two-mode two photon states. There states are not the “linearly” polarized quantum states PSC = 0 for these states => Semiclassically unpolarized, “hidden” polarization Experimental demonstration Measured data (dots) and curve fit for the overlap e iSˆ y 2 2 2 Coincidence counts per 500 s 2500 2 2 2000 1500 1000 3 1 3 1 Background level 500 0 -180 -120 -60 0 60 120 180 Polarization rotation angle (deg) T. Tsegaye, J. Söderholm, M. Atatüre, A. Trifonov, G. Björk, A.V. Sergienko, B. E. A. Saleh, and M. C. Teich, Phys. Rev. Lett., vol. 85, pp. 5013 -5016, 2000. Existing proposals for quantum polarization quantification Stokes operator uncertaint y relation : Δsˆx2 Δsˆ y2 Δsˆz2 2 Nˆ N N m 2cos m 2eim m, N m SU(2) coherent states : N , , sin m 0 m N 1 SU(2) Q - function : Q , N , , ˆ N , , N 0 4 N Degree of polarizati on : PSU ( 2) 1 1 4 dQ , 2 The measures quantify to what extent the state’s SU(2) Q-function is spread out over the spherical coordinates. That is, how far is it from being a Stokes operator minimum uncertainty state? A. Luis, Phys. Rev. A 66, 013806 (2002). Examples SU(2) coherent state : PSU(2) N N 1 2 That is, the vacuum state is unpolarized and highly excited states are polarized Note that: N Max{PSU(2)} 1 1/4 2 4/9 4 16/25 Two - mode number state N , N m : PSU(2) 2N 1 2N 1 2 N 1 2m 2 N 1 N m Degree of polarization based on distance to unpolarized state Another proposal is to define the degree of polarization as the distance (the distinguishability) to a proximal unpolarized state. Will be covered in L. Sánchez-Soto’s talk. Proposal for quantification of polarization – Generalized visibility PQ 1 Min U Original state Uˆ 2 , where Uˆ is a unitary polarizati on transfo rmation Transformed state How orthogonal (distinguishable) can the original and a transformed state become under any polarization transformation? All pure, two-mode N-photon states are polarized One can show that all pure, two-mode N-photon states with N ≥ 1 have unit degree of polarization using this definition, even those states that are semiclassically unpolarized => No ”hidden” polarization. For the vacuum state PQ 0 For a two - mode coherent state , where α β N : PQ 1 exp 2 2 2 2 Orbits From the definition of PQ it follows that if Uˆ then the two states have the same degree of quantum polarizati on. The set of all such states define an orbit. If one state in an orbit can be generated, then we can experimentally generate all states in the orbit. Orbit generating states Orbits type 1 : Discrete orbits, generating states N , N n Number of orbits N 2 1 (rounded downwards to the nearest integer) cos Orbits type 2 : Generating states cos 2,0 sin 0,2 0 for N 2 sin ei cos sin sin cos for N 3, etc. sin sin ei cos sin Orbit generating states where the orbit spans the whole Hilbert space As we have seen, the state 2,0 0,2 2 can generate a whole basis set (3 states). Moreover, to generate the basis set we need only make geometrical rotations or differential phase shifts. Such orbits are of particular interest for experimentalists to implement 3dimensional quantum information protocols, and to demonstrate effects of twophoton interference. It can be shown that the state 3,0 0,3 2 can generate a whole basis set for N 4. Another basis generating state for N 4 is 2,1 1,2 2. In higher excitation manifolds it is not known if it is possible to find completebasis generating orbits, but it seems unlikely. Summary Polarization is a useful and often used characteristic for coding of quantum info. The classical, and semiclassical description of polarization is unsatisfactory for quantum states. Other proposed measures have been discussed and compared. We have proposed to use the generalized visibility under (linear) polarization transformations as a quantitative polarization measure. Polarization orbits naturally appears under this quantitative measure. Orbits spanning the complete N-photon space have special significance and interest for experiments and applications. Schematic experimental setup 2 Generated state: 2 i 2 i 0,2 2,0 6 6 1,1 BBO Type II 4 Phase shift arccos 23 4 2 HWP Coincidence Detector 4 Phase shift Projection onto the state .2 (This state causes coincidence counts.) PBS Detector