Phonon cooling and lasing with nitrogen-vacancy centers in diamond

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Phonon cooling and lasing with nitrogen-vacancy centers in diamond

K. V. Kepesidis, S. D. Bennett, S. Portolan, M. D. Lukin, and P. Rabl

Phys. Rev. B 88, 064105 – Published 27 August 2013

Abstract

We investigate the strain-induced coupling between a nitrogen-vacancy impurity and a resonant vibrational mode of a diamond nanoresonator. We show that under near-resonant laser excitation of the electronic states of the impurity, this coupling can modify the state of the resonator and either cool the resonator close to the vibrational ground state or drive it into a large-amplitude coherent state. We derive a semiclassical model to describe both effects and evaluate the stationary state of the resonator mode under various driving conditions. In particular, we find that by exploiting resonant single- and multiphonon transitions between near-degenerate electronic states, the coupling to high-frequency vibrational modes can be significantly enhanced and dominate over the intrinsic mechanical dissipation. Our results show that a single nitrogen-vacancy impurity can provide a versatile tool to manipulate and probe individual phonon modes in nanoscale diamond structures.

Received 25 June 2013

DOI:https://doi.org/10.1103/PhysRevB.88.064105

Authors & Affiliations

K. V. Kepesidis¹, S. D. Bennett², S. Portolan¹, M. D. Lukin², and P. Rabl¹

¹Institute of Atomic and Subatomic Physics, TU Wien, Stadionallee 2, 1020 Wien, Austria
²Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA

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Vol. 88, Iss. 6 — 1 August 2013

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Images

Figure 1
(a) Setup. A single NV $^{-}$ defect center is embedded in an all-diamond doubly clamped beam. Vibrations of the beam with frequency $ω_{m}$ modulate the local strain and shift the energy levels of the electronically excited defect states. (b)–(d) Illustration of the phonon-assisted optical transitions for the case where the state $| y 〉$ is driven by a laser of frequency $ω_{L}$ and detuning $δ_{L} = ω_{L} - ω_{y}$ . The spacing between the two excited levels is defined as $Δ = ω_{x} - ω_{y}$ . (b) Phonons coupled to $Σ_{∥}$ only affect the driven state $| y 〉$ and lead to cooling (heating) effects for a laser detuning $δ_{L} \approx - ω_{m}$ ( $δ_{L} \approx + ω_{m}$ ). (c) For phonon-induced transitions between the excited states with coupling $\sim$ $Σ_{⊥}$ , a resonant cooling process occurs for $Δ = ω_{m}$ and with resonant excitation of the $| y 〉$ state. (d) For an opposite level ordering, i.e., $Δ = - ω_{m}$ , the same process leads to resonant phonon emission, leading to heating and lasing effects discussed in Sec. 4.Reuse & Permissions
Figure 2
(a) Schematic top view of the NV defect and its dangling bond representation. The shaded areas depict the (hybridized) $s p^{3}$ bonding orbitals ( $σ_{1, 2, 3}$ for the carbon and $σ_{N}$ for the nitrogen atoms). (b) Ground-state single-particle configuration in the electron (black) and in the hole (empty arrows) representations. (c) Energy level diagram of the NV center showing the spin sublevels of the ground and the first excited triplet state. The relevant three-level structure used in this work is highlighted in black. (d) Excited-state energy splitting induced by the nonaxial strain (Refs. 5, 42, and 43).Reuse & Permissions
Figure 3
Density plots of the Lamb-Dicke cooling rate (in units of $λ^{2} / Γ$ ) as a function of the detuning $δ_{L}$ ( $y$ axis) and the frequency difference $Δ$ of the excited levels ( $x$ axis) for four different values of the Rabi frequency: (a) $Ω / Γ = 0.5$ , (b) $Ω / Γ = 1$ , (c) $Ω / Γ = 2$ , and (d) $Ω / Γ = 5$ . For all plots it has been assumed that $λ_{⊥} = λ_{∥} = λ$ , $λ_{0} = 0$ , and $Γ_{ϕ} = 0$ .Reuse & Permissions
Figure 4
(a) The stationary P function $P (r, \infty)$ is plotted for different values of the Rabi frequency $Ω$ given in the inset. Each curve is rescaled by its maximal value $P_{\max}$ and the other parameters used for this plot are (in units of $ω_{m}$ ) $N_{t h} = 20$ , $γ = 10^{- 6}$ , $λ_{⊥} = 0.001$ , $Γ = 0.05$ , and $Γ_{ϕ} = 0$ . (b) The final phonon occupation number $n_{f}$ is plotted as a function of $Ω$ and other parameters as in (a). The dashed line indicates the approximate result derived from the Gaussian P function given in Eq. (31). (c) Under the same conditions the Fano factor $F$ (solid line) and the correlation function $g^{2} (0)$ (dashed line) are plotted as a function of the driving strength. In (b) and (c) the vertical dashed line indicates the position of the threshold given in Eq. (30).Reuse & Permissions
Figure 5
Numerically evaluated final phonon occupation number $n_{f}$ as function of $Δ$ and $δ_{L}$ and assuming an initial occupation of $N_{t h} = 80$ . The other parameters used for this plot are (in units of $ω_{m}$ ) $Ω = 0.05$ , $Γ = 0.05$ , $γ = 10^{- 6}$ , $λ_{⊥} = λ_{∥} = 0.005$ , and $Γ_{ϕ} = 0$ . The dashed lines indicate the resonance conditions for single- and multiphonon sidebands.Reuse & Permissions
Figure 6
Scattered photon flux $I_{η = x, y}$ as functions of the laser detuning $δ_{L}$ and normalized to the resonant scattering rate $I_{0}$ . (a) Photon flux from the $| y 〉$ state and assuming a dominant $Σ_{∥}$ coupling of strength $λ_{∥} = 0.05 ω_{m}$ and an equilibrium occupation number of $N_{t h} = 80$ . At low driving, $Ω = 0.001 ω_{m}$ (solid line), phonon sidebands at $δ_{L} = \pm ω_{m}$ are of approximately the same height. At larger probe strength, $Ω = 0.01 ω_{m}$ (dashed line), the probe laser induces cooling and heating effects, which result in a pronounced asymmetry between the sidebands. The other parameters for this plot are (in units of $ω_{m}$ ) $Γ = 0.1$ , $Γ_{ϕ} = 0$ , $γ = 10^{- 6}$ . In (b) and (c) the scattered photon flux from the $| x 〉$ state is plotted for $Δ = ω_{m}$ and $Δ = - ω_{m}$ , respectively. In (b) the height of the scattered intensity peak provides a direct measurement of the phonon number $〈 n 〉$ . In (c) the transition to the lasing regime at large $Ω$ results in a phonon-induced Rabi splitting of the signal proportional to $\sim$ $2 λ_{⊥} \sqrt{〈 n 〉}$ . For these two plots a $Σ_{⊥}$ -type coupling with strength $λ_{⊥} = 0.01 ω_{m}$ has been assumed and $Ω = 10^{- 2.5}$ (solid lines), $Ω = 10^{- 2}$ (dashed lines), and $Ω = 10^{- 1.5}$ (dotted lines). The other parameters are as in (a).Reuse & Permissions

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