Coherence in Spontaneous Emission Creston Herold July 8, 2013 JQI Summer School (1 st annual!)

Emission from collective (many-body) dipole Super-radiance, sub-radiance 7/8/131JQI Summer School

7/8/13JQI Summer School2 Gross, M. and S. Heroche. Physics Reports 93, 301–396 (1982).

Emission from collective (many-body) dipole Super-radiance, sub-radiance Nuclear magnetic resonance (NMR) Duan, Lukin, Cirac, Zoller (DLCZ) protocol 7/8/133JQI Summer School

Classical: Dipole Antenna 7/8/134JQI Summer School

Simple Quantum Example ? Spontaneous emission rate 7/8/135JQI Summer School

Matrix Form: 2 atoms 7/8/136JQI Summer School

Matrix Form: 3 atoms 7/8/137JQI Summer School

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! 7/8/138JQI Summer School

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) 7/8/139JQI Summer School internal energy

Formalism: Atomic States degeneracy: 7/8/1310JQI Summer School

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): 7/8/1311JQI Summer School

Formalism: Better Basis Each of the states is connected to many others through spontaneous emission/absorption (any “spin” could flip). As with angular momentum, and commute; therefore we can reorganize into eigenstates of : “cooperation” number degeneracy: 7/8/1312JQI Summer School

Formalism: Better Basis Determine all the eigenstates by starting with the largest : and applying the lowering operator, lowering operator normalization Once done with, construct states with making them orthogonal to ; apply lowering operator. Repeat (repeat, repeat, …); note the rapidly increasing degeneracy! 7/8/1313JQI Summer School

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 is the single atom spontaneous emission rate (set ). 7/8/1314JQI Summer School

Level Diagram 7/8/1315JQI Summer School collective states, single photon transitions!

Examples: Collective Coherence 2-atom Rydberg blockade demonstration: 7/8/13JQI Summer School16 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 atom 2-atom, 1.38(3)x faster!

Examples: Collective Coherence “many-body Rabi oscillations … in regime of Rydberg excitation blockade by just one atom.” 7/8/13JQI Summer School17 Dudin, Y. et al. Nature Physics 8, 790 (2012) [Kuzmich] Shared DAMOP 2013 thesis prize! N eff = 148 N eff = 243 N eff = 397 N eff = 456

Example: Subradiance Takasu, Y. et al. “Controlled Production of Subradiant States of a Diatomic Molecule in an Optical Lattice.” Phys. Rev. Lett. 108, (2012). [Takahashi & Julienne] “The difficulty of creating and studying the subradiant state comes from its reduced radiative interaction.” Observe controlled production of subradiant (1 g ) and superradiant (0 u ) Yb 2 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… 7/8/13JQI Summer School18

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: 7/8/1319JQI Summer School

Extended Cloud 7/8/1320JQI Summer School

Extended Cloud 7/8/1321JQI Summer School Incorporate spatial phase into raising/lowering operators: Rate per solid angle: Generate eigenfunctions of For specific, fixed

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: 7/8/13JQI Summer School22

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.” 7/8/13JQI Summer School23 i.e. sets of with their own

DLCZ protocol 7/8/1324JQI Summer School Speedup ! Strong pump (s  e) recalls single e  g photon

DLCZ, storage times 7/8/13JQI Summer School25 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, (2005). [Vuletic] – 6 ms: Zhao et al. Nat. Phys. 5, 100 (2008). [Kuzmich] – 13 s: Dudin et al. Phys. Rev. A 87, (2013). [Kuzmich] H. J. Kimball. Nature 453, 1029 (2008)

References [1] Dicke, R. H. “Coherence in Spontaneous Radiation Processes.” Phys. Rev. 93, (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, (2009); also E. Urban et al. “Observation of Rydberg blockade between two atoms.” Nature Physics 5, (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, (2012). [6] Duan, L., M. Lukin, J. I. Cirac, P. Zoller. “Long-distance quantum communication with atomic ensembles and linear optics.” Nature 414, (2001). [7] Chou, C. et al. “Functional quantum nodes for entanglement distribution over scalable quantum networks.” Science 316, (2007). [8] Kimball, H. J. “The quantum internet.” Nature 453, (2008). [9] Black, A. et al. “On-Demand Superradiant Conversion of Atomic Spin Gratings into Single Photons with High Efficiency.” Phys. Rev. Lett (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, (2013). 7/8/13JQI Summer School26

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Rydberg Blockade 7/8/13JQI Summer School28