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The Decoherence of the Electron Spin and Meta-Stability of ^{13}C Nuclear Spins in Diamond

## Abstract

**:**

## 1. Introduction

## 2. Central-Spin Model

## 3. The Generalized Master Equation

## 4. Discrete Fourier Transform

## 5. Transverse Single Electron Spin Decoherence

**Figure 1.**Time dependence of the transverse component of the single electron spin, ${\langle {\mathit{S}}^{+}\rangle}_{t}$, for spin baths of $N=8$ and $N=6$ ${}^{13}C$ atoms in Diamond. For $N=8$ the applied field is varied in 0–100 G, whereas for $N=6$ the applied field direction is varied relative to the z-axis at $100G$. The renormalization scale parameters for these expectations are given via Figure 2.

**Figure 2.**Applied field strength dependence of the logarithm of the argument of the Caputo integral in (13), for spin baths of $N=2-8\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}$${}^{13}C$ atoms in Diamond.

## 6. Summary

## References

- Childress, L.; Gurudev Dutt, M.V.; Taylor1, J.M.; Zibrov1, A.S.; Jelezko, F.; Wrachtrup, J.; Hemmer, P.R.; Lukin, M.D. Coherent dynamics of coupled electron and nuclear spin qubits in diamond. Science
**2006**, 314, 281–285. [Google Scholar] [CrossRef] [PubMed] - Gurudev Dutt1, M.V.; Childress1, L.; Jiang, L.; Togan, E.; Maze, J.; Jelezko, F.; Zibrov, A.S.; Hemmer, P.R.; Lukin, M.D. Quantum register based on individual electronic and nuclear spin qubits in diamond. Science
**2007**, 316, 1312–1316. [Google Scholar] - Neumann, P.; Mizuochi, N.; Rempp, F.; Hemmer, P.; Watanabe, H.; Yamasaki, S.; Jacques, V.; Gaebel, T.; Jelezko, F.; Wrachtrup, J. Multipartite entanglement among single spins in diamond. Science
**2008**, 320, 1326–1329. [Google Scholar] [CrossRef] [PubMed] - Takahashi, S.; Hanson, R.; van Tol, J.; Sherwin, M.S.; Awschalom, D.D. Quenching spin decoherence in diamond through spin bath polarization. Phys. Rev. Lett.
**2008**, 101, 047601. [Google Scholar] [CrossRef] [PubMed] - Lukin, M.D.; Imamoglu, A. Nonlinear optics and quantum entanglement of ultraslow single photons. Phys. Rev. Lett.
**2000**, 84, 1419–1422. [Google Scholar] [CrossRef] [PubMed] - He, X.F.; Manson, N.B.; Fisk, P.T.H. Paramagnetic resonance of photoexcited N-V defects in diamond. I. Level anticrossing in the 3A ground state. Phys. Rev. B
**2001**, 47, 8809–8815. [Google Scholar] [CrossRef] - Cini, M. Topics and Methods in Condensed Matter Theory: From Basic Quantum Mechanics to the Forefront of Research; Springer: Berlin, Germany, 2007; pp. 177–178. [Google Scholar]
- Felton, S.; Felton, S.; Edmonds, A.M.; Newton, M.E.; Martineau, P.M.; Fisher, D.; Twitchen, D.J. Electron paramagnetic resonance studies of the neutral nitrogen vacancy in diamond. Phys. Rev. B
**2008**, 77, 081201(R). [Google Scholar] [CrossRef] - Gali, A. Theory of the neutral nitrogen-vacancy center in diamond and its application to the realization of a qubit. Phys. Rev. B
**2009**, 79, 235210. [Google Scholar] [CrossRef] - Tamarat, Ph.; Gaebel1, T.; Rabeau, J.R.; Khan, M.; Greentree, A.D.; Wilson, H.; Hollenberg, L.C.L.; Prawer, S.; Hemmer, P.; Jelezko1, F.; et al. Stark shift control of single optical centers in diamond. Phys. Rev. Lett.
**2006**, 97, 083002. [Google Scholar] [CrossRef] [PubMed] - Jelezko, F.; Wrachtrup, J. Single defect centres in diamond: A review. Phys. Status Solidi A-Appl. Res.
**2006**, 203, 3207–3225. [Google Scholar] [CrossRef] - Delaney, P.; Greer, J.C.; Larsson, J.A. Spin-polarization mechanisms of the Nitrogen-vacancy center in Diamond. Nano Lett.
**2010**, 10, 610–614. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Manson, N.B.; Harrison, J.P.; Sellars, M.J. The nitrogen-vacancy center in Diamond re-visited. arXiv, 2006; arXiv:cond-mat/0601360. [Google Scholar]
- Rogers, L.J.; Armstrong, S.; Sellars, M.J.; Manson, N.B. Infrared emission of the NV center in diamond: Zeeman and uniaxial stress studies. New J. Phys.
**2008**, 10, 103204. [Google Scholar] [CrossRef] - Stoneham, A.M.; Harker, A.H.; Morley, G.W. Could one make a diamond-based quantum computer? J. Phys. Condens. Matter
**2009**, 21, 364222. [Google Scholar] [CrossRef] [PubMed] - Jelezko, F.; Gaebel, T.; Popa, I.; Domhan, M.; Gruber, A.; Wrachtrup, J. Observation of Coherent oscillation of a single nuclear spin and realization of a two-qubit conditional quantum gate. Phys. Rev. Lett.
**2004**, 93, 130501. [Google Scholar] [CrossRef] [PubMed] - Wrachtrup, J. Defect center room-temperature quantum processors. Proc. Nat. Acad. Sci.
**2010**, 107(21), 9479–9480. [Google Scholar] [CrossRef] [PubMed] - Balasubramanian, J.; Neumann, P.; Twitchen, D.; Markham, M.; Kolesov, R.; Mizuochi1, N.; Isoya, J.; Achard, J.; Beck, J.; Tissler, J.; et al. Ultralong spin coherence time in isotopically engineered diamond. Nat. Mater.
**2009**, 8, 383–387. [Google Scholar] [CrossRef] [PubMed] - Witzel, W.M.; Das Sarma, S. Quantum theory for electron spin decoherence induced by nuclear spin dynamics in semiconductor quantum computer architectures: Spectral diffusion of localized electron spins in the nuclear solid-state environment. Phys. Rev. B
**2006**, 74, 035322. [Google Scholar] [CrossRef] - Coish, W.A.; Loss, D. Hyperfine interaction in a quantum dot: Non-Markovian electron spin dynamics. Phys. Rev. B
**2004**, 70, 195340. [Google Scholar] [CrossRef] - Epstein, R.J.; Mendoza, F.M.; Kato, Y.K.; Awschalomv, D.D. Anisotropic interactions of a single spin and dark-spin spectroscopy in diamond. Nat. Phys.
**2005**, 1, 94–98. [Google Scholar] [CrossRef] - Jelezko, F.; Wrachtrup, J. Single defect centers in diamond: A review. Phys. Status Solidi A-Appl. Res.
**2006**, 203, 3207–3225. [Google Scholar] [CrossRef] - Goss, J.P.; Jones, R.; Breuer, S.J.; Briddon, P.R.; Öberg, S. The Twelve-Line 1.682eV Luminescence Center in Diamond and the Vacancy-Silicon Complex. Phys. Rev. Lett.
**1996**, 77, 3041–3044. [Google Scholar] [CrossRef] [PubMed] - Davies, G. Dynamic Jahn-Teller distortions at trigonal optical centers in Diamond. J. Phys. C
**1979**, 12, 2551–2566. [Google Scholar] [CrossRef] - Laidlaw, M.G.G.; DeWitt, C.M. Feyman Functional Integrals for Systems of Indistinguishable Particles. Phys. Rev. D
**1971**, 3, 1375–1378. [Google Scholar] [CrossRef] - McLachlan, A.D. Spin-spin coupling hamiltonian in spin multiplets. Mol. Phys.
**1963**, 6, 441–444. [Google Scholar] [CrossRef] - Greiter, M.; Schuricht, D. Many-spinon states and the secret significance of young tableaux. Phys. Rev. Lett.
**2007**, 98, 237202. [Google Scholar] [CrossRef] [PubMed] - Gómez, C.; Ruiz-Altaba, M.; Sierra, G. Quantum Groups in Two-Dimensional Physics; Cambridge University Press: Cambridge, MA, USA, 1996; p. 300. [Google Scholar]
- Bernstein, J.; Feinberg, G.; Lee, T.D.; Possible, C.T. Noninvariance in the Electromagnetic Interaction. Phys. Rev.
**1965**, 139, 1650–1659. [Google Scholar] [CrossRef] - Ham, F.S. Effect of linear Jahn-Teller coupling on paramagnetic resonance in a
^{2}E state. Phys. Rev.**1968**, 166, 307–321. [Google Scholar] [CrossRef] - Marinari, E.; Parisi, G.; Ricci-Tersenghi, F.; Ruiz-Lorenzo, J.J.; Zuliani, F. Replica symmetry breaking in short range spin glasses: A review of the theoretical foundations and of the numerical evidence. J. Stat. Phys.
**2000**, 98, 973–1019. [Google Scholar] [CrossRef] - Crompton, P.R. The quantum noise of ferromagnetic π-Bloch domain walls. Entropy
**2009**, 11, 548–559. [Google Scholar] [CrossRef] - Crompton, P.R. The partition function zeroes of quantum critical points. Nucl. Phys. B
**2009**, 810, 542–562. [Google Scholar] [CrossRef] - Hastings, M.B.; Sondhi, S.L. Breakdown of conformal invariance at strongly random critical points. Phys. Rev. B
**2001**, 64, 094204. [Google Scholar] [CrossRef] - Cappellaro, P.; Jiang, L.; Hodges, J.S.; Lukin, M.D. Coherence and control of quantum registers based on electronic spin in a nuclear spin bath. Phys. Rev. Lett.
**2009**, 102, 210502. [Google Scholar] [CrossRef] [PubMed] - Maze, J.R.; Taylor, J.M.; Lukin, M.D. Electron spin decoherence of single nitrogen-vacancy defects in diamond. Phys. Rev. B
**2008**, 78, 094303. [Google Scholar] [CrossRef] - Coish, W.A.; Loss, D.; Yuzbashyan, E.A.; Altshuler, B.L. Quantum versus classical hyperfine-induced dynamics in a quantum dot. J. Appl. Phys.
**2007**, 101, 081715. [Google Scholar] [CrossRef] - Bortz, M.; Stolze, J. Exact dynamics in the inhomogeneous central-spin model. Phys. Rev. B
**2007**, 76, 014304. [Google Scholar] [CrossRef] - Mueller, C.A. Diffusive spin transport. Lect. Notes Phys.
**2009**, 768, 277–314. [Google Scholar] - Wang, Z.; Fisher, M.P.A.; Girvin, S.M.; Chalker, J.T. Short-range interactions and scaling near integer quantum Hall transitions. Phys. Rev. B
**2000**, 61, 8326–8333. [Google Scholar] [CrossRef] - Wang, Z.; Xiong, S. Electron-electron interactions, quantum Coulomb gap, and dynamical scaling near integer quantum Hall transitions. Phys. Rev. B
**2002**, 65, 195316. [Google Scholar] [CrossRef] - Dong, J.; Xu, M. Applications of continuity and discontinuity of a fractional derivative of the wave functions to fractional quantum mechanics. J. Math. Phys.
**2008**, 49, 052105. [Google Scholar] [CrossRef] - Lim, S.C.; Teo, L.P. Topological symmetry breaking of self-interacting fractional Klein Gordon field theories on toroidal spacetime. J. Phys. A Math. Theor.
**2008**, 41, 145403. [Google Scholar] [CrossRef] - Laskin, N. Fractional Schrödinger equation. Phys. Rev. E
**2002**, 66, 056108. [Google Scholar] [CrossRef] - Naber, M. Time fractional Schrödinger equation. J. Math. Phys.
**2004**, 45, 3339. [Google Scholar] [CrossRef] - Goychuk, I. Anomalous relaxation and dielectric response. Phys. Rev. E
**2007**, 76, 040102(R). [Google Scholar] [CrossRef] - Wang, S.J.; Zhao, D.; Luo, H.G.; Cen, L.X.; Jia, C.L. Exact solution to the von Neumann equation of the quantum characteristic function of the two-level Jaynes-Cummings model. Phys. Rev. A
**2001**, 64, 052102. [Google Scholar] [CrossRef] - Luchko, Y.; Martinez, H.; Trujillo, J.J. Fractional Fourier transform and some of its applications. Fract. Calc. Appl. Anal.
**2008**, 11, 457–469. [Google Scholar] - Crompton, P.R. Exact nonperturbative renormalization. Phys. Rev. D
**2006**, 74, 096001. [Google Scholar] [CrossRef] - Zhang, W.; Konstantinidis, N.; Al-Hassanieh, K.A.; Dobrovitski, V.V. Modelling decoherence in quantum spin systems. J. Phys. Condens. Matter
**2007**, 19, 083202. [Google Scholar] [CrossRef] - Shenvi, N.; de Sousa, R.; Whaley, K.B. Universal scaling of hyperfine-induced electron spin echo decay. Phys. Rev. B
**2005**, 71, 224411. [Google Scholar] [CrossRef] - de Sousa, R.; Shenvi, N.; Whaley, K.B. Qubit coherence control in a nuclear spin bath. Phys. Rev. B
**2005**, 72, 045330. [Google Scholar] [CrossRef] - Popescu, S.; Short, A.J.; Winter, A. Entanglement and the foundations of statistical mechanics. Nat. Phys.
**2006**, 2, 754–758. [Google Scholar] [CrossRef] - Gurevicha, S.; Hadanib, R. On the diagonalization of the discrete Fourier transform. App. Comp. Harm. Anal.
**2009**, 27(1), 87–99. [Google Scholar] [CrossRef] - Bailey, D.H. FFTs in external or hierarchical memory. J. Supercomput.
**1990**, 4, 23–35. [Google Scholar] [CrossRef] - Leinaas, J.M.; Myrheim, J. On the theory of identical particles. Il Nuovo Cimento
**1977**, 37, 1–23. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Crompton, P.
The Decoherence of the Electron Spin and Meta-Stability of * ^{13}C *Nuclear Spins in Diamond.

*Entropy*

**2011**,

*13*, 949-965. https://doi.org/10.3390/e13050949

**AMA Style**

Crompton P.
The Decoherence of the Electron Spin and Meta-Stability of * ^{13}C *Nuclear Spins in Diamond.

*Entropy*. 2011; 13(5):949-965. https://doi.org/10.3390/e13050949

**Chicago/Turabian Style**

Crompton, Peter.
2011. "The Decoherence of the Electron Spin and Meta-Stability of * ^{13}C *Nuclear Spins in Diamond"

*Entropy*13, no. 5: 949-965. https://doi.org/10.3390/e13050949