1. 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

2. 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 ?

3. 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,
(2002)
Raman scattering (K ≈ 1000)
Chan, Law, and Eberly, PRA 68,
(2003)
Photoionization (K = ??)
Th. Weber, et al., PRL 84, 443 (2000)

4. 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

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

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

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

8. Frequency Entanglement in SPDC
where

10. Results (400nm pump, 0.8mm BBO)w

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

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

13. 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

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

15. 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

16. Transvere frequency entanglement on various orbital angular momentum
= 0.3

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

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

19. ( anti-parallel k and q )
Control parameter

21. Spatial density matrix of the spontaneous single photon-1 y

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

23. 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, (2003)

24. 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