Analysis of quantum entanglement of spontaneous single photons C. K. Law Department of Physics, The Chinese University of Hong Kong Collaborators: Rochester group – K. W. Chan and J. H. Eberly CUHK group – T. W. Chen and P. T. Leung Moscow group – M. V. Fedorov

momentum conservation Formation of entangled particles via breakup processes A B Non-separable (in general) energy conservation momentum conservation What are the physical features of entanglement ? How do we control quantum entanglement ? Can quantum entanglement be useful ?

Examples of two-particle breakup Spontaneous PDC (K ≈ 4.5) Law, Walmsley and Eberly, PRL 84, 5304 (2000) Spontaneous emission (K ≈ 1) Chan, Law and Eberly, PRL 88, 100402 (2002) Raman scattering (K ≈ 1000) Chan, Law, and Eberly, PRA 68, 022110 (2003) Photoionization (K = ??) Th. Weber, et al., PRL 84, 443 (2000)

In this talk Based on the Schmidt decomposition method, we will quantify and characterize quantum entanglement of two basic processes : Frequency entanglement Transverse wave vector entanglement recoil momentum entanglement

Representation of entangled states of continuous variables Orthogonal mode pairing Continuous-mode basis Discrete Schmidt-mode basis

Characterization of (pure-state) entanglement via Schmidt decomposition Pairing mode structure Degree of entanglement Average number of Schmidt modes Correlated observables Local transformation entropy

Example: Schmidt decomposition of gaussian states where Eigenstate of a harmonic oscillator Two-mode squeezed state

Frequency Entanglement in SPDC where

Results (400nm pump, 0.8mm BBO) w

Phase-adjusted symmetrization: Branning et al. (1999) q = p q = 0

Transverse Wave Vector Entanglement Higher dimensional entanglement for quantum communication (making use of the orbital angular momentum) Vaziri, Weihs, Zeilinger PRL 89, 240401 (2002) Strong EPR correlation Howell, Bennink, Bentley, Boyd quant-ph/0309122  Applications in quantum imaging Gatti, Brambilla, Lugiato, PRL 90, 133603 (2003) Abouraddy et al., PRL 87, 123602 (2001)

A model of transverse two-photon amplitudes Assumptions: (1) Paraxial approximation (2) Monochromatic limit with (3) Ignore refraction and dispersion effects Monken et al. Longitudinal phase mismatch subjected to the energy conservation constraint Transverse momentum conservation

Examples of Schmidt modes in transverse wave vector space = 0.3 m – orbital angular momentum quantum number n – radial quantum number

Control parameter of the transverse entanglement in SPDC = angular spread of the pump Shorter crystal length L Higher entanglement Dash line corresponds to the K value of a gaussian approximation exact

Transvere frequency entanglement on various orbital angular momentum = 0.3

Enhancement of entanglement: Selection of higher transverse wave vectors 70 % higher Entanglement ! ( ) = 0.3 Higher transverse wave vectors are “more entangled”

Photon-Atom Entanglement in Spontaneous Emission How “pure” is the single photon state? What are the natural modes functions of the photon?

( anti-parallel k and q ) Control parameter

Spatial density matrix of the spontaneous single photon -1 y

Very high entanglement via Raman scattering Line width can be very small

K motional linewidth = radiative linewidth h Example: Cesium D-line transition. With D = 15 GHz and W = 300 MHz, velocity spread ~ 1 m/s, h can be as large as 5000, giving K ~ 1400. Chan, Law, and Eberly, PRA 68, 02211 (2003)

Summary We apply Schmidt decomposition to analyze the structure of entanglement generated in two basic single photon emission processes involving continuous variables: Frequency entanglement Transverse wave vector entanglement recoil momentum entanglement Very high K possible