C. K. Law

Department of Physics,

The Chinese University of Hong Kong

Collaborators:

Rochester group – K. W. Chan and J. H. EberlyCUHK group – T. W. Chen and P. T. Leung Moscow group – M. V. Fedorov

Analysis of quantum entanglement of spontaneous single photons

A B

Formation of entangled particles via breakup processes

momentum conservation

What are the physical features of entanglement ?

How do we control quantum entanglement ?

Can quantum entanglement be useful ?

Non-separable (in general)

energy conservation

Examples of two-particle breakup

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)

Spontaneous PDC (K ≈ 4.5)Law, Walmsley and Eberly, PRL 84, 5304 (2000)

Based on the Schmidt decomposition method, we will quantifyand characterize quantum entanglement of two basic processes :

In this talk

• Frequency entanglement• Transverse wave vector entanglement • recoil momentum entanglement

Representation of entangled states of continuous variables

Continuous-mode basis Discrete Schmidt-mode basis

Orthogonalmode pairing

Correlated observables

Local transformation

Characterization of (pure-state) entanglement via Schmidt decomposition

Degree of entanglement Pairing mode structure

Average numberof Schmidt modes

entropy

1

Example: Schmidt decomposition of gaussian states

where

Eigenstate of aharmonic oscillator

Two-mode squeezed state

Frequency Entanglement in SPDC

where

Results (400nm pump, 0.8mm BBO)

Phase-adjusted symmetrization:

Branning et al. (1999)

0

• 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)

Transverse Wave Vector Entanglement

A model of transverse two-photon amplitudes

Assumptions: (1) Paraxial approximation (2) Monochromatic limit with (3) Ignore refraction and dispersion effects

Let angular spread of the pump (assumed Gaussian)

Monkenet al.

Transverse momentum conservation

Longitudinal phase mismatchsubjected to the energy conservation constraint

Examples of Schmidt modes in transverse wave vector space

= 0.3m – orbital angular momentum quantum numbern – radial quantum number

Control parameter of the transverse entanglement in SPDC

= angular spread of the pump

Dash line corresponds to the K value of a gaussian approximation

Shorter crystal length L Higher entanglement

exact

Transvere frequency entanglement on various orbital angular momentum

= 0.3

Enhancement of entanglement: Selection of higher transverse wave vectors

Higher transverse wave vectors are “more entangled”

( )

70 % higherEntanglement !

= 0.3

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

y’

Very high entanglement via Raman scattering

Line widthcan be very small

motional linewidthradiative linewidth

Example: Cesium D-line transition. With = 15 GHz and = 300 MHz,

velocity spread ~ 1 m/s, can be as large as 5000, giving K ~ 1400.

K

Chan, Law, and Eberly, PRA 68, 02211 (2003)

=

We apply Schmidt decomposition to analyze the structure ofentanglement generated in two basic single photon emission processes involving continuous variables:

Summary

• Frequency entanglement• Transverse wave vector entanglement

• recoil momentum entanglement• Very high K possible