Coherence in Spontaneous Emission

Creston Herold

July 8, 2013JQI Summer School (1st annual!)

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• 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

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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

So we can choose simultaneous energy eigenstates:

commutes with all the (motion, collisions don’t change internal state)

(operates on CoM coords. only)

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internal energy

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Formalism: Atomic States

degeneracy:

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Formalism: Atom-Light Interaction

Field interaction with jth atom:(here, dipole approx. but results general!)

momentum conjugate to

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 BasisEach of the states is connected to many others throughspontaneous emission/absorption (any “spin” could flip).

As with angular momentum, and commute; therefore we can reorganize into eigenstates of :

“cooperation” number

degeneracy:

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Formalism: Better BasisDetermine all the eigenstates by starting with the largest :

and applying the lowering operator,

lowering operatornormalization

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 RatesThrough 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 is the single atom spontaneous emission rate (set ).

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Level Diagram

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collective states,single photon transitions!

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Examples: Collective Coherence2-atom Rydberg blockade demonstration:

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Gaëtan, A. et al. Nature Physics 5, 115 (2009) [Browaeys & Grangier]See also E. Urban et al. Nature Physics 5, 110 (2009) [Walker &

Saffman]

single atom2-atom, 1.38(3)x faster!

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Examples: Collective Coherence“many-body Rabi oscillations … in regime of Rydberg excitation blockade by just one atom.”

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Dudin, Y. et al. Nature Physics 8, 790 (2012) [Kuzmich]

Shared DAMOP 2013 thesis prize!

Neff = 148

Neff = 243

Neff = 397

Neff = 456

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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

• Directionality to coherence, emission• Same general approach applies– Eigenstates for particular (incomplete)– Include rest of to complete basis

(decoherence, can change “cooperation number” )

constant vectors

Have to keep spatial extent of field:

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Extended Cloud

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Extended Cloud

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Incorporate spatial phase into raising/lowering operators:

Rate per solid angle:

Generate eigenfunctions of

For specific, fixed

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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.”

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i.e. sets of with their own

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DLCZ protocol

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Speedup!

Strong pump (s e) recalls single e g photon

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DLCZ, storage times

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• 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]

H. J. Kimball. Nature 453, 1029 (2008)

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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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