Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
  • Editors' Suggestion

Modeling temperature-dependent population dynamics in the excited state of the nitrogen-vacancy center in diamond

S. Ernst, P. J. Scheidegger, S. Diesch, and C. L. Degen
Phys. Rev. B 108, 085203 – Published 24 August 2023
PDFHTMLExport Citation
Abstract
Authors
See Also
Article Text
  • INTRODUCTION
  • RATE MODEL
  • SIMULATION OF SPIN RELAXATION
  • COMPARISON TO EXPERIMENTAL DATA
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    The nitrogen-vacancy (NV) center in diamond is well known in quantum metrology and quantum information for its favorable spin and optical properties, which span a wide temperature range from near zero to over 600 K. Despite its prominence, the NV center's photophysics is incompletely understood, especially at intermediate temperatures between 10–100 K where phonons become activated. In this paper, we present a rate model able to describe the crossover from the low-temperature to the high-temperature regime. Key to the model is a phonon-driven hopping between the two orbital branches in the excited state (ES), which accelerates spin relaxation via an interplay with the ES spin precession. We extend our model to include magnetic and electric fields as well as crystal strain, allowing us to simulate the population dynamics over a wide range of experimental conditions. Our model recovers existing descriptions for the low- and high-temperature limits and successfully explains various sets of literature data. Further, the model allows us to predict experimental observables, in particular the photoluminescence (PL) emission rate, spin contrast, and spin initialization fidelity relevant for quantum applications. Lastly, our model allows probing the electron-phonon interaction of the NV center and reveals a gap between the current understanding and recent experimental findings.

    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    3 More
    • Received 6 April 2023
    • Accepted 2 June 2023

    DOI:https://doi.org/10.1103/PhysRevB.108.085203

    ©2023 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Color centersElectron-phonon couplingMagneto-optical spectraPoint defectsSingle photon sourcesSpin-phonon couplingTemperature
    1. Physical Systems
    Nitrogen vacancy centers in diamond
    1. Techniques
    Lindblad equationMaster equationQuantum master equation
    Condensed Matter, Materials & Applied PhysicsQuantum Information, Science & TechnologyAtomic, Molecular & Optical

    Authors & Affiliations

    S. Ernst1,*, P. J. Scheidegger1,*, S. Diesch1, and C. L. Degen1,2,†

    • 1Department of Physics, ETH Zurich, Otto Stern Weg 1, 8093 Zurich, Switzerland
    • 2Quantum Center, ETH Zurich, 8093 Zurich, Switzerland

    • *These authors contributed equally to this work.
    • †degenc@ethz.ch

    See Also

    Temperature Dependence of Photoluminescence Intensity and Spin Contrast in Nitrogen-Vacancy Centers

    S. Ernst, P. J. Scheidegger, S. Diesch, L. Lorenzelli, and C. L. Degen
    Phys. Rev. Lett. 131, 086903 (2023)

    Temperature-Dependent Photophysics of Single NV Centers in Diamond

    Jodok Happacher, Juanita Bocquel, Hossein T. Dinani, Märta A. Tschudin, Patrick Reiser, David A. Broadway, Jeronimo R. Maze, and Patrick Maletinsky
    Phys. Rev. Lett. 131, 086904 (2023)

    Article Text (Subscription Required)

    Click to Expand

    References (Subscription Required)

    Click to Expand
    Issue

    Vol. 108, Iss. 8 — 15 August 2023

    Reuse & Permissions
    Access Options
    Author publication services for translation and copyediting assistance advertisement

    Authorization Required

    Log In
    Other Options
    ×

    Images

    • Figure 1
      Figure 1

      NV center coordinate system used in this paper. Carbon atoms are shown in black, the substitutional nitrogen atom in blue, and the vacancy in gray. The x axis is along one of the carbon bonds. B⃗ is the applied magnetic field (red), δ⃗ the strain or electric field (green), and δ⊥ its in-plane component. The remaining symbols are explained in the text.

      Reuse & Permissions
    • Figure 2
      Figure 2

      (a) Energy levels of the ES as a function of strain at zero magnetic field. In (a) and (c), level names are printed where they are approximately eigenstates, and level-anticrossings (LAC) are indicated by black markers. The dashed lines are the eigenlevels of Hspin, obtained by a partial trace over the orbital subspace. They constitute the observed ES energy levels at room temperature. (b) Under strain, two orbital branches Ex and Ey form, split by approximately 2δ⊥. The eigenstates of Horbit⊗Hspin are composite states of the orbital states |x〉 and |y〉 and superpositions of the spin states |0〉 and |±〉∝(|+1〉±|−1〉). Only two spin states mix significantly in this example, with ɛ=ε|y〉|0〉,|y〉|−〉. Level spacings ωx/y are given for the discussion in Fig. 4. (c) Energy levels of the ES as a function of axial magnetic field at δ⊥=20GHz. Under high strain and magnetic field, eigenstates of orbit and spin are formed, for example, |Ex,+1〉=|x〉|+1〉. (d) Rate model employed in this paper. The rates listed in Table 1 and the text are depicted by arrows. The dashed arrow indicates a low decay probability, allowing for the optical initialization and readout of the GS spin state. Excitation by green light and emission of red light are indicated by wavy arrows. The two bases zf [see left in (a)] and hf [right in (c)] are connected via basis transformations (dark gray arrows) to the ez basis of the ES Hamiltonian.

      Reuse & Permissions
    • Figure 3
      Figure 3

      (a) Sequence of the initialization and readout protocol used to simulate the time t resolved PL(t) for determining the SNR. (b) Simulation of the expectation value 〈mS=0〉 (sum of GS and ES, black) and PL(t) (orange). We start in a thermal state and obtain the spin state defined as the mS=0 initialized state (solid lines) by a laser pulse, followed by a waiting time. The state defined as the mS=±1 initialized state (dashed lines) is obtained by an additional π pulse. In a second laser pulse, the integration time tint for the spin-state readout is marked by the blue shaded area. The duration of the laser pulses and waiting time given here are typical experimental values [35]. In simulations, we evaluate the steady-state solution for each step.

      Reuse & Permissions
    • Figure 4
      Figure 4

      Illustration of the temperature-dependent spin relaxation process in the ES. We use the parameters from Table 1. In this setting, the ES levels are ordered as in Fig. 2. (a) Orbital hopping rates and their one- and two-phonon contributions as a function of temperature (adapted from Ref. [35]). The inverse lifetime TES−1≈1/10ns of the ES as well as the doubled Larmor frequencies 2ωx/y (units MHz) are indicated by horizontal lines. Spin relaxation is most efficient at temperatures where the hopping rates are resonant with 2ωx/y, as indicated by the gray shaded area (here 30K−40K). [(b)–(d)] Exemplary realizations of 3ns time evolution on the side and top views of the Bloch sphere in the |0〉−|−〉-spin manifold, plotted for three selected temperatures [marked in (a) and (h)]. The color of the time trace encodes the current orbital branch (Ex: Blue, Ey: Pink). Dashed lines show the quantization axes given by the energy eigenstates in spin space of the two orbital branches, about which Larmor precession occurs. [(e)–(g)] Values of mS after a time evolution of duration TES under random discrete jumps (average: Black arrow) with a corresponding prediction of the master equation with Lindblad jump operators (red arrow). (h) Signal-to-noise ratio as a function of temperature. The three cases from above mark three distinct regimes, as discussed in the text. Prominently, the SNR reflects the rate of the spin relaxation process.

      Reuse & Permissions
    • Figure 5
      Figure 5

      (a) Simulated spin initialization fidelity after a long laser pulse followed by a long waiting time (see Fig. 3) at different laser powers. β=0.3 corresponds to the laser power P=2.34mW used elsewhere in this paper. When spin relaxation is maximal (around 35K), increasing the laser power is beneficial, approaching a limit related to the SS branching ratio rS (horizontal grey line). (b) Simulated readout SNR (black), for which we assume an initialization of 〈mS=0〉=1. We optimize the integration time tint (see Fig. 3) of the readout for each temperature. The resulting optimal tint (orange) follows the same trend as the readout SNR.

      Reuse & Permissions
    • Figure 6
      Figure 6

      (a) Effect of strain δ⊥ or electric field and its angle ϕδ on the SNR. ES LACs as indicated in Fig. 2 lead to regions of reduced SNR. [(b), (c)] Effect of strain or electric field on the SNR as a function of temperature. The lines at 0K for the two angles ϕδ in (b) and (c) correspond to the lines indicated by markers in (a). A multifaceted landscape of regions with reduced SNR is predicted by these simulations.

      Reuse & Permissions
    • Figure 7
      Figure 7

      Comparison of our master equation model, evaluated at different temperatures, to the common rate model at 300K (“Classical Model”). (a) Simulated steady state PL as a function of magnetic field. The two LACs of the ES average to a single LAC at room temperature {see Fig. 2 and Ref. [35]}. Here, we use θB=1.9∘ and ϕB=194∘ (NV-2 in Ref. [35]), and other parameters from Table 1. (b) PL(t) dynamics during a readout laser pulse as in Fig. 3. In both panels, complete quantitative agreement with the classical rate model is recovered at room temperature.

      Reuse & Permissions
    • Figure 8
      Figure 8

      Simulation of effective ISC rates obtained from a single exponential decay fit of the PL after exciting into the various states of the ES in conditions where zf basis states are almost eigenstates (labels). Experimental data and parameters are taken from Goldman et al. [23].

      Reuse & Permissions
    • Figure 9
      Figure 9

      Simulated steady state PL at the ES LAC as a function of magnetic field and strain at 300K. (a) Room-temperature classical rate model with the additional temperature reduction factor in the Hamiltonian as derived by Plakhotnik et al. [19]. The LAC (minimum in PL) is marked by the dotted line. A magnetic field alignment of θB=1.9∘ and ϕB=194∘ (NV-2 in Ref. [35], other parameters as in Table 1) is used. (b) The master equation model developed in this paper. The same line plotted in (b) is shown to demonstrate the excellent overlap with the predicted position of the LAC. (c) For a given magnetic field alignment angle θB (here: 0.3∘), a sweep of the in-plane magnetic field angle ϕB is predicted to reveal the strain angle ϕδ and magnitude δ⊥. White lines are contours of constant PL. (d) Full width at half maximum (FWHM) of the LAC at the narrowest (ϕB=ϕδ+180∘, red line) and widest (ϕB=ϕδ, blue line) in-plane magnetic field angle ϕB as a function of δ⊥. Magnetic field sweeps to determine the FWHM are indicated in (c) for one value of δ⊥.

      Reuse & Permissions
    • Figure 10
      Figure 10

      (a) Steady-state PL as a function of temperature for several phonon cutoff energies Ω. The Ω values correspond to the discussion in Sec. 4d. The increase in PL above 100K is due to the decreasing SSL [Eq. (7)]. The effect of the SSL can be distinguished from the ES phonon-driven averaging by comparison with the classical room-temperature model (gray curve). (b) Phonon mode integral I(T,δ⊥) plotted for the different Ω values. Vertical gray lines indicate the onset of the cut-off at T≈0.1·Ω/kB. (c) k↓ plotted for the different Ω values.

      Reuse & Permissions
    ×

    Sign up to receive regular email alerts from Physical Review B

    Log In

    Cancel
    ×

    Search

    Article Lookup

    Paste a citation or DOI
    Enter a citation
    ×