Polarization-entangled mid-infrared photon …

1 Polarization-entangled mid- infrared photon generation in p-doped semiconductor quantum wellsRAZALI, Razif, IKONIC, Zoran, INDJIN, Dragan and HARRISON, Paul < >Available from Sheffield Hallam University Research Archive (SHURA) at: document is the author deposited version. You are advised to consult the publisher's version if you wish to cite from versionRAZALI, Razif, IKONIC, Zoran, INDJIN, Dragan and HARRISON, Paul (2016). Polarization-entangled mid- infrared photon generation in p-doped semiconductor quantum wells.

2 Semiconductor Science and Technology, 31 (11), p. 115011. Copyright and re-use policySee Hallam University Research mid- infrared photongeneration inp-doped semiconductor quantum wellsR Razali 1,3, Z. Ikoni c1, D. Indjin1and P. Harrison21 School of Electronic and Electrical Engineering, University of Leeds,Woodhouse Lane, Leeds LS2 9JT, United Kingdom2 Materials and Engineering Research Institute, Sheffield Hallam University,Howard Street, Sheffield, South Yorkshire S1 1WB, United Kingdom3 Physics Department, Faculty of Science, Universiti Teknologi Malaysia, 81310,Skudai, Johor, optimal design of double quantum well structures for generationof Polarization-entangled photons in the mid- infrared range.

3 Based on the valenceintersubband transitions spontaneous parametric downconversion, is efficiency and frequency selectivity of the process are also numbers: 1315, 9440 TKeywords: optimised quantum well, SPDC, nonlinear optics, entangled twin photonsSubmitted to:Semicond. Sci. mid- infrared photon generation inp-doped semiconductor quantum wells21. IntroductionQuantum correlated photon pairs, also known as entan-gled photon pairs, are the main ingredient in quantumcommunications [1], quantum computing [1 3], quan-tum key distribution (QKD) [4], quantum teleporta-tion [5], superdense coding [6] and many other appli-cations of quantum information theory.

4 These pho-ton pairs can be generated in spontaneous parametricdown conversion (SPDC), or in spontaneous four wavemixing (SFWM) [7]. SPDC, which is also known asparametric fluorescence [8], is based on second ordernonlinearity ( (2)), while SFWM is based on third or-der nonlinearity ( (3)). Raman scattering noise, whichis hard to suppress in SFWM [9], has made the SPDC scheme more attractive. The SPDC and SWFM pro-cesses, producing entangled photon pairs, can be in-duced in different media, like bulk crystals (possiblywith tailored inhomogeneous nonlinearity [10]), quan-tum well heterostructures, in quantum dots [11, 12], orNV centres in diamond [13].

5 This work focuses on SPDC process basedon intersubband transitions in the valence bandof quantum well incentiveto consider this case comes from the fact thatSPDC based on intersubband transitions within theconduction band -valley (the most frequent case)does not provide polarization entanglement, sincethese transitions are active only for Z-polarizationof light (perpendicular to the well layer)[14].Incontrast, valence band intersubband transitions areactive for various polarizations, which comes fromtheirp-like, rather thans-like character, enablingpolarization entanglement in the SPDC process (inparticular, the type II SPDC).

6 Unlike the case ofconduction band intersubband transitions, the opticalparameters of valence band intersubband transitionscannot be calculated by the effective mass method, butrather byk pmethod (in particular, for structuresbased on wider band gap materials, assumed in thiswork, the 6-bandk pmethod is sufficiently accurate).As for the structure design and optimisation, themethodology employer here is similar to what we haveused previously [14]As pointed in [15], which considered the optimiza-tion of the second harmonic generation (SHG) in p-type GaAs-AlAs step quantum well structures, thesymmetry of hole state wavefunctions enables only 5non-zero components of the second-order polarizabil-ity.

7 ZXY, ZXX, XYZ, XXZ and ZZZ, where the firstcomponent denotes the generated SHG photon polar-ization, the other two being the pump photons. SPDCis similar to SHG, but reversed in time, so these se-lection rules apply to SPDC as well. Therefore, gener-ation of Polarization-entangled twin photons can relyon ZXY, XYZ or XXZ configuration. Practical consid-erations, suggesting that SPDC will be used in waveg-uide layout (co-propagating waves), imply that polar-izations of the three waves cannot be all different, henceonly the XXZ polarization will be considered SPDC based on valence intersubbandtransitionsSPDC is a second order optical process, with thenonlinear polarizationP= 0 (2)E2(1)determined by the second-order nonlinear susceptibil-ity (2)as given in [16, p.]

8 174]. For intersubbandtransitions involving hole states in quantum wells thestate energiesEl(kx,ky) depend on their quantum in-dexland the in-plane wave vector (kx,ky), and thetransition matrix elements also depend on (kx,ky),so the summation over all states, discretised in the(kx,ky) plane, is written as (2)( p+ q, p, q) =e3 kx ky8 2Lz 0 h2 kx,ky lmn[fFD(El(kx,ky),EF) fFD(Em(kx,ky),EF)]{dxlndxnmdzml[( nl p q) i nl] [( ml p) i ml]+dxlndznmdxml[( nl p q) i nl] [( ml q) i ml]+dxlndxnmdzml[( nm+ p+ q) +i nm] [( ml p) i ml]+dzlndxnmdxml[( nm+ p+ q) +i nm] [( ml q) i ml]},(2)whereeis the electron charge, kxand kyare themesh steps inx- andy-directions,Lzis the total lengthof the structure inz-direction, 0is the free spacepermittivity, p, qand p+ qare the frequenciesof three photons interacting in the nonlinear system,and h nm=En(kx,ky) Em(kx,ky) is the subbandspacing at a particular (kx,ky).

9 In the SPDC case, pand qmay denote the signal and idler photons,while p+ qis the pump photon that will be down-converted in the SPDC process. ijis the linewidth ofi jtransition, anddxijis thex-component of dipolematrix element for this transition. The state energiesEm(kx,ky) and the matrix elementsdxijwere calculatedusing 6 6 method [17, p. 407].The state populations in (2) are given by theFermi-Dirac functionfFD:fFD(Ea(kx,ky),EF) =[1 + e(Ea(kx,ky) EFkT)] 1.(3)whereEFis the Fermi energy in the structure,kis theBoltzman s constant andTis the mid- infrared photon generation inp-doped semiconductor quantum wells33.

10 Degenerate Twin photon GenerationQuantitative analysis of twin- photon generation in thepresence of optical losses [18] shows that the degeneratetwin photon generation is given byPTwin=4| |2P3L3/23 2 |g|3e | 123|L2| 12 3L|3/2 | 12 3L|0sinh(| 12 3L| x2)dx .(4)where 123= 1+ 2+ 3, 12 3= 1+ 2 3,g= [ 2 / 2], and = 2 n1/ 1. The iin (4) is theabsorption coefficient at photon frequency i,Lis thelength of the device,P3is the pump power, and isrelated to (2)via = 0deff 2 1 2n1n2n3 Seff( 0 0)32(5)wheredeff= (2),Seffis the pump beam crosssection.