Introduction to the NV center: Electronic structure and optical transitions

1. Introduction to the NV center: Electronic structure and optical transitions Lily Childress Bates College Sonderborg, Denmark, August 27 2010

2. The nitrogen-vacancy defect Why the NV- defect in diamond? NV0 5 electrons NV- 6 electrons • Diamond is a good host • Wide bandgap • 0 nuclear spin isotope • Stiff lattice • Ground state spin triplet • Optical transitions • Spin-dependent fluorescence • Optical pumping possible • => Very atom-like • Formation: natural or… • N implantation • irradiation creates vacancies • annealing removes barrier to vacancy diffusion Charging state depends on local Fermi energy From Weber et al. 2010 PNAS What is the electronic structure of the NV- center?

3. Atomic physics approach: NV~ an atom embedded in a crystal Electronic structure: new symmetries N V Spherical symmetry C3v symmetry s,p,d,f – irreducible representations of the group of rotations A1, A2, E – irreducible representations of the group of trigonal symmetries

4. e- e- e- e- e- e- Electronic structure: new symmetries N Excited state Strain 3E E V ~637 nm 637 nm Inhomogeneous broadening C3v symmetry Ground state ms = ±1 A1, A2, E – irreducible representations of the group of trigonal symmetries ms = 0 3A2 A2 “fine structure”

5. e- e- e- e- e- e- Electronic structure: Triplet states Singlet state(s)? N Excited state ISC metastable 3E 1A1 V ~637 nm Ground state Explains observed optical pumping into ms= 0 ms = ±1 ms = 0 3A2 Explains observed higher fluorescence intensity for ms= 0 Spin-conserving optical transitions Drabenstedt PRB 1999

6. e- e- e- e- e- e- Electronic structure: Triplet states Singlet state(s)? N Excited state 3E 1E V 1046 nm 1A1 ~637 nm 1E Ground state ms = ±1 ms = 0 3A2 Ma PRB 2010 Rogers NJP 2008

7. e- e- e- e- e- e- Electronic structure: Triplet states Singlets N Excited state 300K 3E ? V ~637 nm Phonon sidebands Ground state ms = ±1 ZPL ms = 0 3A2 Questions? Rogers NJP 2009

8. Operations SR form a group: “group of the Schrodinger equation” N V Symmetry and quantum mechanics The set of transformations associated with that symmetry leave the Hamiltonian invariant A system exhibits a symmetry  Operators SR commute with the Hamiltonian e.g. for atoms, the group of rotations For the NV center: C3v symmetry group elements: E = identity 2C3 = rotations by 2πn/3 3v = reflections 1 3 2

9. Symmetry and quantum mechanics The set of transformations associated with that symmetry leave the Hamiltonian invariant A system exhibits a symmetry  Operations SR form a group: “group of the Schrodinger equation” Operators SR commute with the Hamiltonian How do eigenstates of H transform under these symmetry operations? => Symmetry operations yield degenerate eigenstates Applying the set {SR}yields all (non-accidentally) degenerate states

10. This matrix encodes the action of SR in the basis { } Symmetry and quantum mechanics Suppose that energy level En has ln distinct degenerate states . Applying any symmetry operators to one state must yield some linear combination of the degenerate states The set of R (for all symmetry group elements R) forms an ln – dimensional representation of the symmetry group preserves the group multiplication rules It is also an irreducible representation because every state is connected to every other state by a symmetry operator, regardless of the basis chosen to describe the degenerate eigenstates.

11. Energy level En with degeneracy ln ln – dimensional IR of the group of symmetries  Classifying energy levels An energy level can be classified according to the irreducible representation that describes the action of the symmetry group on its eigenstates Example: “3d” states of hydrogen form a 5-fold degenerate level with states that transform under rotations according to a 5-dimensional matrix representation of the group SO3. • Knowing the IRs for the group of symmetries of a system yields • degeneracy of the levels • transformation properties of the “basis functions” (states) • selection rules for e.g. dipole transitions

12. N N V V C C C C C C Irreducible representations of C3v A1 A2 E Dimensionality = degeneracy 2 1 1 Basis functions transform into each other the way that the vectors x and y transform into each other under the C3v symmetry operations The identity representation – the basis function transforms into itself under all symmetry operations. The {1, -1} representation – the basis function picks up a minus sign under reflections. like s or pz like triplet ms = 0 like px and py Two a1 orbitals Symmetrized orbitals: N Two e orbitals V N C C C V C C C

13. N V From Weber et al. 2010 PNAS C C C Orbital electronic configurations of the NV center First orbital excited state: a1(1)2a1(2)e3 = one hole in a1(2) and one hole in e E Now add in spin: triplet or singlet Symmetrized orbitals Ground state: a1(1)2a1(2)2e2 a1, a1, ex, ey = two holes in the e orbitals A2 S = 1 transforms as A2 (Sz) and E (Sx, Sy)

14. Spin-spin interaction leads to zero-field ground state splitting Electronic states of the NV center from Jeronimo Maze, thesis 2010 Strain breaks symmetry, shifts energies, and mixes levels Batalov et al. 2009 PRL Optical transitions to excited state spin triplet Spin-orbit mediated crossing to spin singlet levels States with ms=0 Selection rules also from group theory Ground state spin triplet Optically induced spin polarization into ms= 0 spin projection

15. Optical transitions at low temperature Optical spectroscopy typically uses ZPL excitation and sideband detection (only few % into ZPL) Confocal geometry to isolate single defects Required to resolve excited state structure! Detect Excite Batalov et al. 2009 Early ensemble work: Davies, Collins, Manson, Rand

16. Optical transitions at low temperature Optical spectroscopy typically uses ZPL excitation and sideband detection (only few % into ZPL) high strain configuration (typical) In the absence of microwaves, only this transition would be visible And even then, it could disappear from a (~1% probability) spin-flip • Important questions: • Stability • Coherence • Tunability • Selection rules Optical spin polarization => mix spin states with MW to see all lines Batalov et al. 2009

17. Stability Photoionization In a few, select samples, stable lifetime-broadened lines possible Even when spin-flips aren’t a problem, continuous resonant illumination leads to photoionization Robledo et al. 2010, arXiv 532nm repump reverses ionization but causes spectral diffusion Spectral diffusion may be caused by ionization of nearby nitrogen impurities Tamarat et al. 2006 PRL Very sample dependent Fu et al. 2009, PRL

18. Coherence Batalov et al. 2008 PRL Optical Rabi oscillations Typical decay of oscillations yields a pure dephasing time similar to or longer than the lifetime ~12 ns Consistent with observed linewidths Promising for two-photon interference experiments lifetime = radiative lifetime? Robledo et al. 2010 arXiv

19. Optical properties depend strongly on strain Batalov et al. 2009 PRL High transverse strain (typical) Low strain: C3v symmetry Linearly polarized emission, spin conserving in the limit of high strain A2 A1 Selection rules determined by C3v symmetry Ex,y E1,2 Togan et al. 2010 Nature Significant mixing between spin states in lower branch creates lambda transitions at moderate strain Electric fields have the same effect as strain Tamarat et al. 2008 NJP

20. Stark shift control Ey transitions for several NV centers can be tuned several GHz with electric fields. Offset depends on local strain. Possible to bring different NV centers into resonance Or control selection rules, degree of mixing Tamarat et al. 2006 PRL

21. Selection rules: low strain A2 A1 Ex,y + E1,2 - -1 +1 X,Y Can be roughly understood from angular momentum conservation Difficult to realize for most NVs 0

22. Moderate strain: Lambda transitions Santori et al. PRL 2006 Coherent population trapping signatures (same mechanism as EIT) Mechanism for all-optical spin manipulation

23. Selection rules: high strain Off-resonant excitation Resonant excitation Sideband detection ZPL detection Fu et al. 2009, PRL Good linear selection rules in high strain NVs Phonon averaging between branches (worse at high temperatures) Kaiser et al. 2009, arXiv Degrades with temperature, indications of phonon-induced averaging between orbital states

24. High T Low T The transition from low to high temperature Linewidths broaden, selection rules disappear Electron spin resonance signal emerges in the excited state at ~100K Batalov et al. 2009 PRL Interpretation: Phonon processes average over the orbital degrees of freedom, leaving only the spin-spin interaction Strain independent Strain dependent Fu et al. 2009 PRL Rogers et al. 2009 NJP Degrades with temperature, indications of phonon-induced averaging between orbital states

25. Room temperature optical transitions: Optical spin polarization and detection Room temperature: Excited state Initial differential fluorescence => readout mechanism* ? Fluorescence Few s => polarization Ground state ms = ±1 Time ms = 0 Green on *This method isn’t single shot; typical experiments get 1/100 of a photon per shot Preparation and detection mechanism for spin resonance experiments

26. Deplete Excite Detect Room temperature optical transitions: Single defect imaging Confocal imaging of single defects Wrachtrup, Weinfurter, Grangier Photon anti-bunching • Applications: • single photon sources • fluorescent bio-markers Subwavelength imaging: Stimulated emission depletion (STED) Overlap beams; only see fluorescence at central point (size depends on depletion beam intensity) Rittweger et al. 2009 Nature Photonics Other methods for selective depletion using spin manipulation or shelving

27. Summary: Electronic structure and optical transitions Electronic structure and selection rules governed by C3v symmetry but highly affected by strain, which is almost always present Triplet-singlet intersystem crossings lead to spin polarization and spin-dependent fluorescence even at room temperature Lifetime broadened optical transitions at low temperature (ZPL) promising for interference experiments Strong coupling to phonons Significant phonon sidebands Orbital averaging at room temperature • Collection efficiency: • typically ~0.1% of expected counts • ZPL only a few % of that! Stability questions remain Spectral diffusion Photoionization Sample dependent properties

28. Increasing collection efficiency Coupling to cavities Directed emission Plasmon enhanced emission Babinec et al. 2010 Hadden et al. 2010 10x increase in counts Waveguides Santori et al. 2010, also Prawer group Degrades with temperature, indications of phonon-induced averaging between orbital states

29. Increasing collection efficiency Plasmon enhanced emission Directed emission Coupling to silver nanowires Theory: Chang 2006 Kolesov et al. 2009 Babinec et al. 2010 Hadden et al. 2010 Lifetime decrease 2.5 Or gold nanoparticles 10x increase in counts Barth et al. 2010 J. Lumin. Waveguides Santori et al. 2010, also Prawer group Degrades with temperature, indications of phonon-induced averaging between orbital states

30. Increasing collection efficiency Coupling to cavities Coupling nanocrystals to non-diamond optical resonators Santori et al. 2010 3-fold increase Q~1000 Silica microdisk Schlietinger, Benson 2009 Typical results: NV up to 10 times brighter at cavity resonance Nanopositioning techniques for PC cavities Single crystal diamond Babinec et al. 2010 Diamond fabrication Challenge: couple to stable (likely bulk) NVs Englund et al. 2010 Also polycrystalline Awschalom group Also Benson, Hanson groups