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