Coherence in
Spontaneous Emission
Creston Herold
July 8, 2013
JQI Summer School (1st annual!)
• Emission from collective (many-body) dipole
• Super-radiance, sub-radiance
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Gross, M. and S. Heroche. Physics Reports 93, 301–396 (1982).
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•
•
•
•
Emission from collective (many-body) dipole
Super-radiance, sub-radiance
Nuclear magnetic resonance (NMR)
Duan, Lukin, Cirac, Zoller (DLCZ) protocol
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Classical: Dipole Antenna
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Simple Quantum Example
?
Spontaneous
emission rate
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Matrix Form: 2 atoms
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Matrix Form: 3 atoms
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Overview
• Write Hamiltonian for collection of atoms and
their interaction with EM field
• Build intuition for choice of basis
– Energy states (eigenspectrum)
– Simplify couplings by choosing better basis
• Effects of system size, atomic motion
• Experimental examples throughout!
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Formalism: Atomic States
Depends on CoM coords.
e.g. kinetic energy
commutes with all the
internal energy
(motion, collisions don’t change internal state)
So we can choose simultaneous energy eigenstates:
(operates on CoM coords. only)
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Formalism: Atomic States
degeneracy:
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Formalism: Atom-Light Interaction
momentum conjugate to
Field interaction with jth atom:
(here, dipole approx. but results general!)
is an odd operator, must be off-diagonal in representation with internal E diagonal:
constant vectors
For gas of small extent (compared to wavelength):
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Formalism: Better Basis
Each of the states
is connected to many others through
spontaneous emission/absorption (any “spin” could flip).
As with angular momentum,
reorganize into eigenstates of
and
:
commute; therefore we can
“cooperation” number
degeneracy:
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Formalism: Better Basis
Determine all the eigenstates
by starting with the largest
:
and applying the lowering operator,
normalization
lowering operator
Once done with
, construct states with
making them orthogonal to
; apply lowering operator.
Repeat (repeat, repeat, …); note
the rapidly increasing degeneracy!
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Spontaneous Emission Rates
Through judicious choice of basis, the field-atom interaction connects each
of the states
to two other states, with
.
Spontaneous emission rate is square of matrix element (lower sign):
where
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is the single atom spontaneous emission rate (set
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collective states,
single photon transitions!
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Level Diagram
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Examples: Collective Coherence
2-atom Rydberg blockade demonstration:
2-atom, 1.38(3)x faster!
single atom
Gaëtan, A. et al. Nature Physics 5, 115 (2009) [Browaeys & Grangier]
See also E. Urban et al. Nature Physics 5, 110 (2009) [Walker & Saffman]
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Examples: Collective Coherence
“many-body Rabi oscillations … in regime of Rydberg excitation blockade by just one atom.”
Neff = 148
Neff = 243
Neff = 397
Neff = 456
Shared DAMOP 2013 thesis prize!
Dudin, Y. et al. Nature Physics 8, 790 (2012) [Kuzmich]
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Example: Subradiance
• Takasu, Y. et al. “Controlled Production of Subradiant States
of a Diatomic Molecule in an Optical Lattice.” Phys. Rev.
Lett. 108, 173002 (2012). [Takahashi & Julienne]
• “The difficulty of creating and studying the subradiant state
comes from its reduced radiative interaction.”
• Observe controlled production of subradiant (1g) and
superradiant (0u) Yb2 molecules, starting from 2-atom Mott
insulator phase in 3-d optical lattice. (Yb is “ideal” for
observing pure subradiant state because it has no ground
state electronic structure).
• Control which states are excited by laser detuning.
Subradiant state has sub-kHz linewidth! Making is
potentially useful for many-body spectroscopy…
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Extended Cloud
Have to keep spatial extent of field:
constant vectors
• Directionality to coherence, emission
• Same general approach applies
– Eigenstates for particular
(incomplete)
– Include rest of
to complete basis (decoherence,
can change “cooperation number” )
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Extended Cloud
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Extended Cloud
Incorporate spatial phase into
raising/lowering operators:
Generate eigenfunctions of
For specific, fixed
Rate per solid angle:
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Extended Cloud
•
OK for fixed atoms, but I said we’d consider motion!
• We’ve incorporated CoM coordinates into
, the
“cooperation” operator; does not commute with !
• Thus, these are not stationary eigenstates of .
• Classically, relative motion of radiators causes decoherence,
but radiators with a common velocity will not decohere.
• Quantum mechanically, analogous simultaneous eigenstates
of and
are found with:
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Extended Cloud
• The states
are not complete.
• e.g. state after emitting/absorbing a photon with
is
not one of
.
• We can complete set of states “by adding all other orthogonal
plane wave states, each being characterized by a definite
momentum and internal energy for each molecule.”
i.e. sets of
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with their own
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DLCZ protocol
Speedup!
Strong pump (s  e) recalls
single e  g photon
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DLCZ, storage times
H. J. Kimball. Nature 453, 1029 (2008)
• 2-node entanglement realized by
Chou et al. Science 316, 1316
(2007). [Kimball]
• Ever longer storage times:
– 3 us: Black et al. Phys. Rev. Lett.
95, 133601 (2005). [Vuletic]
– 6 ms: Zhao et al. Nat. Phys. 5,
100 (2008). [Kuzmich]
– 13 s: Dudin et al. Phys. Rev. A 87,
031801 (2013). [Kuzmich]
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References
[1] Dicke, R. H. “Coherence in Spontaneous Radiation Processes.” Phys. Rev. 93, 99-110 (1954).
[2] Gross, M. and S. Haroche. “Superradiance: An essay on the theory of collective spontaneous
emission.” Physics Reports 93, 301–396 (1982).
[3] Gaëtan, A. et al. “Observation of collective excitation of two individual atoms in the Rydberg
blockade regime.” Nature Physics 5, 115-118 (2009);
also E. Urban et al. “Observation of Rydberg blockade between two atoms.” Nature Physics 5,
110-114 (2009).
[4] Dudin, Y. et al. “Observation of coherent many-body Rabi oscillations.” Nature Physics 8, 790
(2012).
[5] Takasu, Y. et al. “Controlled Production of Subradiant States of a Diatomic Molecule in an Optical
Lattice.” Phys. Rev. Lett. 108, 173002 (2012).
[6] Duan, L., M. Lukin, J. I. Cirac, P. Zoller. “Long-distance quantum communication with atomic
ensembles and linear optics.” Nature 414, 413-418 (2001).
[7] Chou, C. et al. “Functional quantum nodes for entanglement distribution over scalable quantum
networks.” Science 316, 1316-1320 (2007).
[8] Kimball, H. J. “The quantum internet.” Nature 453, 1023-1030 (2008).
[9] Black, A. et al. “On-Demand Superradiant Conversion of Atomic Spin Gratings into Single Photons
with High Efficiency.” Phys. Rev. Lett. 95 133601 (2005).
[10] Zhao, R., Y. Dudin, et al. “Long-lived quantum memory.” Nature Physics 5, 100 (2008).
[11] Dudin, Y. et al. “Light storage on the time scale of a minute.” Phys. Rev. A 87, 031801 (2013).
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Rydberg Blockade
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