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ICOP 2009-International Conference on Optics and Photonics CSIO, Chandigarh, India, 30 Oct.-1 Nov. 2009 SIMULTANEOUS POLARIZATION SQUEEZING OF ALL THREE STOKES PARAMETERS IN PURE PHOTON NUMBER STATES Ranjana Prakash1,2* & Namrata Shukla1** 1 2 Physics Department, University of Allahabad, Allahabad 211002, INDIA M.N. Saha Centre of Space Studies, Institute of Inter-Disciplinary Studies, University of Allahabad, Allahabad 211002, INDIA [email protected] [email protected] Abstract: We study Polarization squeezing of a pure photon number state which is obviously polarized but if the basis of linear polarization is adopted, simultaneous polarization squeezing in all the stokes parameters is observed. We use our definition of Polarization squeezing, which is similar to the recent definition of Prakash and Kumar used for Atomic Squeezing and formulated by uncertainty relations taken into account. 1.INTRODUCTION Products of quantum fluctuations in two noncommuting observables satisfy the uncertainty relation but the individual fluctuations can be reduced and this gives the well known concept of squeezing. In classical optics the polarization state of light beam can be visualized as direction of a Stokes vector in Poincare sphere and is determined by the four Stokes parameters [1,2] S0 , S1 , S 2 , and S3 . Quantum nature of Polarization of light [3-8] has received more attention in a number of papers because of more chances in the same for Quantum information communication. The quantum Stokes parameter operators [9-14] and associated Poincare sphere describe the quantum optical polarization properties of light, that can also apply to non classical light [10,12,15-18]. Operators, S0 , S1 , S 2 , and S3 follow commutation relations and their variances are restricted by quantum mechanical uncertainty relations. Polarization squeezing is a non classical property which can be mathematically defined with a criterion developed using the uncertainty relations under consideration. Polarization squeezing was firstly discussed by Chirkin [9] and the quantum Stokes parameters and polarization squeezing is quite widely studied [9-11, 19-23]. Some definitions for polarization squeezing have been given and formation and conversion of polarization squeezed states has been done [24-26]. It has also been defined in such a way that uncertainty relations as well as MUS limit are satisfied [23, 27] but using this definition we can investigate squeezing only along the coordinate axes formed ∧ ∧ ∧ by stokes parameter operators S 1 , S 2 , S 3 . The main outline of this paper is as follows: We give a general definition of Polarization squeezing and using that definition, we investigate simultaneous squeezing in all the three components of stokes parameter operators.of a pure photon number state which is polarized, by adopting the linear basis of polarization. 2. CRITERION FOR POLARIZATION SQUEEZING The Hermitian Stokes operators can be defined as quantum versions of their classical counterparts [1, 2], by ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ S 0 = a x† a x + a y† a y , S1 = a x† a x − a y† a y , ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ (1) S 2 = a x † a y + a y † a x , S 3 = i a y † a x − a x † a y , ∧ ∧ ∧ ∧ where a x , a y , a x † , a y † refer to the photon annihilation and creation operators respectively of two orthogonal polarization modes x and y satisfying the commutation ∧ ∧ relations, a j , a k = δ jk for j , k = x, y . The operator ∧ S 0 is equal to the total photon number operator and corresponds to the beam intensity. Polarization state is ∧ ∧ described by the four operators S 0 and S (given by ∧ ∧ ∧ S 1 , S 2 and S 3 ). Commutation relations for these stokes parameters are, ∧ ∧ ∧ ∧ ∧ S 0 , S i = 0 , i = 1,2,3 , S 1 , S 2 = 2i S 3 , ∧ ∧ ∧ ∧ ∧ ∧ (2) S 2 , S 3 = 2i S 1 , S 3 , S 1 = 2i S 2 . These relations are parallel to the commutation relations for components of the angular momentum operator [28-31, see also 32]. These non zero 1 ICOP 2009-International Conference on Optics and Photonics CSIO, Chandigarh, India, 30 Oct.-1 Nov. 2009 commutators show that simultaneous exact measurement of the quantities represented by these Stokes operators are impossible and the following uncertainty relations hold, 2 ∧ 2 ∧ , V2V3 ≥ S 1 V1V2 ≥ S 3 The criterion for squeezing of the component.V1 is then ∧ , V3V1 ≥ S 2 ∧ . ∧ of ∧ 2 ∧ ∧ 2 ∧ i.e. variance of stokes operator S j should be less than the corresponding variance in the coherent mode [331] . Any polarization state which follows equation (6) will be a non-classical state. This definition of polarization squeezing actually does not contain minimum uncertainty state (MUS) limits of equation (3) and hence may show only quadrature squeezing observed by calculating stokes parameters. According to the improved definition of polarization squeezed states containing MUS limit, polarization squeezed state [23, 27] is one in which one of Stokes variances lies not only below the coherent limit but also below respective MUS limit. i.e.. light is said to be polarization squeezed [34-36] if, ∧ V j < S l < Vk , j, k , l = 1,2,3 , j ≠ k ≠ l ≠ j (6) .We now make more general considerations and define polarization squeezed states with squeezing any general direction in the coordinate axes system. If one considers a set of three mutually perpendicular unit vectors ( n̂ , nˆ ⊥1 , nˆ ⊥ 2 ), for of ^ ^ S n ⊥1 = n ⊥1 . S ∧ S and ^ along these, ^ ^ S n⊥2 = n ⊥2 . S = (n.S), V = (∆S nˆ ) ≤ 1 (Snˆ ⊥1 or/and 1 (Snˆ ⊥2 .. (7) 2 2 Since the maximum value of the right hand sides is 2 − S nˆ 2 . (10) D = 1- S (11) 3.PHOTON NUMBER STATE AND POLARIZATION SQUEEZING Pure state with N photons polarized along complex unit vector, ∧ ∧ ∧ (12) ε = e x cos (θ / 2 ) + e y sin (θ / 2 )e iφ can be written as, ψ = (N! ) −1/2 (ε x a x † + ε y a y † ) N 0,0 x,y = 0,0 , the most general criterion for polarization squeezing is [4-7], is V = 〈∆Sn2〉 < [〈S〉 2 − 〈Sn〉 2] ½. (8) (13) ε,ε⊥ in (ε , ε ⊥ ) basis with, ∧ ∧ ∧ ∧ ∧ ∧ * * a ε = ε x a x + ε y a y , a ε ⊥ = −ε y a x + ε x a y (14) If we consider the same state in ( x , y ) basis of polarization, creation and annihilation operators in both the modes can be written as, ∧ ∧ ∧ ∧ ∧ ∧ ∗ * a x = ε x aε − ε y aε ⊥ , a y = ε y aε + ε x aε ⊥ (15) Initially, the state of polarization is N ,0 and ε x = cos(θ / 2), ε y = e iφ sin(θ / 2) (16) Straight forward calculations for expectation values of the Stokes parameters and their variances for the state (13) give ∧ ∧ S 0 = N , S 1 = N cos θ = n x ∧ S 2 = N sin θ cos φ = n y , ^ 2 2 2 The Degree of Squeezing is often defined as S n = n. S , uncertainty relations give squeezing if S − Snˆ S ∧ 2 (4) S2 + S3 = S0 + 2S0 , and it is fuzzy and ill defined because of the inherent uncertainties. . According to the definition [9-10] a state is said to be polarization squeezed if, V j < V j (coh), j = 1,2,3 (5) component V S = the quantum stokes parameter S 1 and so on. The mean value of the radius of quantum Poincare sphere is given by square root of expectation value of either side of the equation, S 12 + (9) We also define squeezing factor, as the ratio of the left and right hand sides of Eq. (8) ∧ ∧ 2 ∧ + S3 S2 (3) 2 S 12 − S 1 Here V1 stands for the variance 2 ∧ V1 < ∧ (17a,b,c) S 3 = N sin θ sin φ = nz , and, ( ) = N (1 − sin θ cos φ ) = N (1 − n = N (1 − sin θ sin φ ) = N (1 − n V1 = N 1 − cos 2 θ = N (1 − n x 2 ) V2 2 2 2 2 V3 where n = nx e x + ny e y + nz e z y z 2 2 ) , (18a,b,c) ) = cos θ e x + sinθ (cosφ e y + sinφ e z ) (20) 2 ICOP 2009-International Conference on Optics and Photonics CSIO, Chandigarh, India, 30 Oct.-1 Nov. 2009 Is a unit vector in the direction specified by the angles θ and φ . ∧ As per the criterion (9), S 1 would be squeezed if 2 ∧ V1 < 2 ∧ . Eqs. (18a) and (17b,c) + S3 S2 then give, (1-nx2) < (1-nx2)1/2, which is always true except when nx is 0 or 1. Similary conditions for squeezing are V 2 < ∧ of ∧ S3 2 ∧ + S1 ∧ and S2 2 , V3 < ∧ S1 S3 2 ∧ 2 + S2 , which lead to (1-ny2) < (1-ny2)1/2 and (1-nz2) < (1nz2)1/2 , respectively, which are always true unless the component of n under consideration has the value 0 or 1. Thus we find that all Stokes parameters (all ∧ Components of S ) are squeezed in general. To study the exceptions, we may consider the squeezing factors, given by Eq. (10). Squeezing factors corresponding to all these operators are as, S1 = 1− cos 2 θ = , (1-nx2), S2 = 1 − sin 2 θ cos 2 φ = , (1-ny2) S3.= 1 − sin 2 θ sin 2 φ 2 = , (1-nz ) This shows that any Stokes parameter which has a nonzero mean value, i.e., is in a direction perpendicular to n̂ , is squeezed. 4. RESULT AND DISCUSSION Starting with a pure number state, polarization squeezed light can be achieved and that is possible just by changing the basis of polarization of pure photon number state. Squeezing factors show that all the stokes parameter operators visualizing polarization of state are found to be squeezed. ACKNOWLEDGEMENT We would like to thank Prof. H. Prakash and Prof. N. Chandra for their support, interest and critical comments. REFERENCES [1] G.G.Stokes, “On the Composition and Resolution of Streams of Polarized Light from different Sources”, Trans. Cambridge Philos.Soc.9, 399 (1852) [2] M.Born and E.Wolf , “Principles of Optics”, Cambridge University Press, Cambridge,England,1999. [3] H.Prakash, N.Chandra, “Density Operator of Unpolarized Radiation” Phys.Rev.A4, 796 (1971) [4] H.Prakash, N.Chandra, “Density Operator of Unpolarized Radiation”, Phys. Lett. 34A, 28 (1971). [5] H.Prakash, N.Chandra, “Density Operator of Unpolarized Radiation” Phys.Rev. A9, 1021 (1974). [6] D.N.Klyschko, “ Multiphoton Interference and Polarization Effects” Phys. Lett.A 163, 349 (1992) [7] A.P.Aldojants, S.M.Arkelian, A.S.Chirkin, “ Polarization quantum states of light in nonlinear distributed feedback systems: quantum non demolition measurements of stokes parameters of Light and atomic angular momentum” Appl. Phys. B. 66, 53-65 (1998) [8] M.Lassen, M.Sabunch, P.Buchhave, U.L.Anderson, “Generation of Polarization Squeezing with Periodically Poled KTP at 1064 nm”, Optics Express. 15, 5077 (2007) [9] A.S.Chirkin, A.A.Orlov and D.Yu.Paraschuk, “ Quantum Theory of two-mode Interactions in Optically anisotropic Media with Cubic Nonlinearities :Generation of Quadrature and polarization squeezed light” Kvant [Elektron. 20, 999(1993)] Quantum Electron. 23, 870 (1993) [10] N.Korolkova, G.Leuchs, R.Louden, T.C.Ralph, S.Silberhorn “Polarization Squeezing and Continuous Variable Polarization Entanglement”, Phys.Rev. A, 65, 052306 (2002) [11] P.Usachev, J.Soderholm, G.Bjork and A.Trifonov, “Experimental Verification of Differences between Classical and Quantum Polarization Properties” Opt. Commun. 193, 161 (2001) [12] L.K.Shalm, R.B.A.Adamson, A.M. Steinberg, “Squeezing and Over squeezing of Triphotons”, Nature 475, 07624(2009) [13] A.Luis, L.L.Sanchez-Soto. “Quantum Phase Difference, Phase Measurements and Quantum Stokes Parameters”, Prog. Opt. 41, 421 (2000) [14] W.P.Bowen, N.Treps, R.Schnabel, T.C.Ralph, P.K.Lam, “Continuous Variable Polarization Entanglement, Experiment and Analysis” J. Opt.B. 5, 467-478 (2003) [15] W. P. Bowen, R.Schanabel, H.A.Bachor and P.K.Lam, “Polarization Squeezing of Continuous Variable Stokes Parameters” Phys. Rev. Letters. 88, 093601 (2002) [16] J.Hald, J.N.Serensen, C.Schori, E.S. Polzik, “Entanglement Transfer from Light to Atom”, J.Mod. Optics. 47, 2599 (2001) [17] W..P.Bowen, N. Treps, R. Schnabel, P.K.Lam, “Experimental demonstration of Continuous Variable Polarization Entanglement”, Phys. Rev. Letters . 89, 253601 (2002) [18] A.P.Alodjants, S.M.Arakelian, “Quantum Phase Measurement and Non Classical Polarization States of Light”, J.Mod.Opt. 46, 475-507 (1999) [19] D.N. Klyshko, “Polarization of Light: Fourthorder Effects and Polarization- Squeezed States” JETP 84, 1065-1079 (1997) [20] A.P.Aldojants, S.M.Arakelian, A.S.Chirkin, “Interaction of Two Polarization Modes in a Spetioperiodic Non linear Medium: Generation of 3 ICOP 2009-International Conference on Optics and Photonics CSIO, Chandigarh, India, 30 Oct.-1 Nov. 2009 Polarization Squeezed Light and Quantum Nondemolition measurements of the Stokes Parameters”, Quantum Semiclass. Opt. 9, 311-329 (1997) [21] R.Tanas, S.Kielich, “Quantum Fluctuations in the stokes parameters of Light Propagating in a Kerr Medium”, J.Mod.Opt. 37, 1935-1945 (1990) [22] R.Tanas, T.S.Gantsog, “Quantum Fluctuations in the Stokes Parameters of Light Propagating in a Kerr Medium with Dissipation”, J.Mod.Opt. 39, 749-760 (1992) [23] J.Heersink, T.Gaber, S.Lorenz, O.Glock, N.Korolkova and G.Leuchs, “Polarization Squeezing of Intense Pulses with Fiber Optic Sagnac Interferometer” Phys.Rev.A.68, 013815 (2003) [24] N.V..Korolkova and A.S.Chirkin, “Formation and Conversion of the Polarization squeezed Light”, J. Mod. Optics 43, 869, (1996) [25] A.S.Chirkin, V.V.Volokhovsky, “Formation of a Nonclassical Polarization state in an Isotropic Gyrotropic Nonlinear Optical Medium”, J.Russ. Laser. Res. 16, 6 (1995) [26] J.F.Sherson, K.Mølmer, “Polarization Squeezing by Optical Faraday Rotation”, Phys.Rev.Lett. 97, 143602-143606 (2006) [27] J.Heersink, V.Josse, G.Leuchs, V.L.Anderson, “Efficient Polarization Squeezing in optical Fibers”, Optics Letters. 30, 1192-1194 (2005) [28] D.F. Walls and P. Zoller, “Reduced Fluctuation in Resonance Fluorescence”, Phys. Rev. Lett. 47, 709 (1981). [29] H. Prakash and R. Kumar, “Atomic Squeezing in Interaction with Single Mode Coherent Radiation of an Assembly of Two Two-Level Atoms Initially in the Superradiant State”, Proc. of COPE – 03, Editors: L. S. Tanwar and Anurag Sharma, pp. 92-95 (2003). [30] H.Prakash and R. Kumar, “Simultaneous Squeezing of Two Orthogonal Spin Components”, J. Opt. B: Quantum and Semiclass. Opt. 7, S757 (2005). [31] H. Prakash and R. Kumar, “Atomic Squeezing in Assembly of Two Two-Level Atoms Interacting with a Single Mode Coherent Radiation, Eur. Phys. J. D 42, 475 (2007). [32] Alfredo Luis, “Polarization Squeezing and Non classical Properties of Light”, Phys. Rev. A 74, 043817 (2006). [33] Yu.M.Golubev, T.Yu.Golubeva, M.I.Kolobov, E.Giacobino , “Polarization Squeezing in Vertical Cavity Surface Emitting Lasers”, Phys.Rev.A.70, 053817 (2004) [34] N.Korolkova, R.Louden, “Nonseperability and Squeezing of Continuous Polarization Variables”, Phys. Rev. A. 71, 032343 (2005) [35] U.L.Anderson, P.Buchhave, “Polarization Squeezing and Entanglement Produced by a Frequency Doubler”, J.Opt. B. 5, 486-491(2003) [36] O.GlÖckl, J.Heersink, N.Korolkova, G.Leuchs, S.Lorenz, “A Pulsed Source of Continuous Variable polarization Entanglement”, J.Opt. B. 5, 492-496 (2003) 4