Cooling a mechanical resonator with nitrogen-vacancy centres using a room temperature excited state spin-strain interaction

Abstract

Cooling a mechanical resonator mode to a sub-thermal state has been a long-standing challenge in physics. This pursuit has recently found traction in the field of optomechanics in which a mechanical mode is coupled to an optical cavity. An alternate method is to couple the resonator to a well-controlled two-level system. Here we propose a protocol to dissipatively cool a room temperature mechanical resonator using a nitrogen-vacancy centre ensemble. The spin ensemble is coupled to the resonator through its orbitally-averaged excited state, which has a spin-strain interaction that has not been previously studied. We experimentally demonstrate that the spin-strain coupling in the excited state is 13.5±0.5 times stronger than the ground state spin-strain coupling. We then theoretically show that this interaction, combined with a high-density spin ensemble, enables the cooling of a mechanical resonator from room temperature to a fraction of its thermal phonon occupancy.

ARTICLE

Received 10 Aug 2016 | Accepted 19 Dec 2016 | Published 6 Feb 2017

Cooling a mechanical resonator mode to a sub-thermal state has been a long-standing challenge in physics. This pursuit has recently found traction in the eld of optomechanics in which a mechanical mode is coupled to an optical cavity. An alternate method is to couple the resonator to a well-controlled two-level system. Here we propose a protocol to dissipatively cool a room temperature mechanical resonator using a nitrogen-vacancy centre ensemble. The spin ensemble is coupled to the resonator through its orbitally-averaged excited state, which has a spinstrain interaction that has not been previously studied. We experimentally demonstrate that the spinstrain coupling in the excited state is 13.50.5 times stronger than the ground state spinstrain coupling. We then theoretically show that this interaction, combined with a high-density spin ensemble, enables the cooling of a mechanical resonator from room temperature to a fraction of its thermal phonon occupancy.

DOI: 10.1038/ncomms14358 OPEN

Cooling a mechanical resonator with nitrogen-vacancy centres using a room temperature excited state spinstrain interaction

E.R. MacQuarrie1, M. Otten1, S.K. Gray2 & G.D. Fuchs3

1 Department of Physics, Cornell University, Ithaca, New York 14853, USA. 2 Center for Nanoscale Materials, Argonne National Laboratory, Argonne, Illinois 60439, USA. 3 School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853, USA. Correspondence and requests for materials should be addressed to G.D.F. (email: mailto:[email protected]

Web End [email protected] ).

NATURE COMMUNICATIONS | 8:14358 | DOI: 10.1038/ncomms14358 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications 1

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14358

Cooling a mechanical resonator to a sub-thermal phonon occupation can enhance sensing by lowering the resonators thermal noise oor and extending a sensors

linear dynamic range14. Taken to the extreme, cooling a mechanical mode to the ground state of its motion enables the exploration of quantum effects at the mesoscopic scale57. These goals have motivated researchers in the eld of optomechanics to invent methods for cooling mechanical resonators through their interactions with light. Such techniques have been able to achieve cooling to the ground state from cryogenic starting temperatures6,7 and to near the ground state from room temperature812.

A well-controlled quantum system coupled to the motion of a resonator can also be used to cool a mechanical mode13,14. Recently, nitrogen-vacancy (NV) centres in diamond have been coupled to mechanical resonators through coherent interactions with lattice strain1522. The opportunity to use these interactions has stimulated the development of single-crystal diamond mechanical resonators2326 and motivated several theoretical proposals for cooling such resonators with a single NV centre13,14,27,28. In principle, replacing the single NV centre with a many-NV ensemble can provide a collective enhancement to the strain coupling, which could increase the cooling power of these protocols. In practice, however, ensembles can shorten spin coherence times and introduce inhomogeneities that may make collective enhancement impractical, depending on the proposed mechanism. To make ensemble coupling a useful resource, it thus becomes crucial to design a cooling protocol that is insensitive to these side effects.

In this work, we study the hybrid quantum system composed of an NV centre spin ensemble collectively coupled to a mechanical resonator with the goal of developing a method for cooling the resonator from ambient temperature. Experimentally, we characterize the previously unstudied spinstrain coupling within the room temperature NV centre excited state (ES), and we nd that it is 13.50.5 times stronger than the ground state (GS) spinstrain interaction. We then propose a dissipative cooling protocol that uses this ES spinstrain interaction and theoretically show that a dense NV centre ensemble can cool a high-Q mechanical resonator from room temperature to a fraction of its thermal phonon population. The proposed protocol requires neither long spin coherence times nor strong spin-phonon coupling, and the cooling power scales directly with the NV centre density. These properties make our proposed protocol a practical approach to cooling a room temperature resonator.

ResultsNV centrestrain interactions. To achieve substantial cooling from ambient conditions, we require a room temperature NV centrestrain interaction that can be enhanced by an ensemble. We rst consider the orbital-strain coupling that exists within the NV centre ES at cryogenic temperatures. This 850130 THz per strain interaction offers a promising route towards single NV centre-mechanical resonator hybrid quantum systems21,22. For ensemble coupling, however, inevitable static strain inhomogeneities will strongly broaden the orbital transition and prohibit collective enhancement. Moreover, the orbital coherence begins to dephase above 10 K because of phonon interactions29, limiting applications of orbital-strain coupling to cryogenic operation.

A weaker (21.51.2 GHz per strain) spinstrain coupling exists at room temperature within the NV centre GS (ref. 17). The resonance condition for this interaction is determined by a static magnetic bias eld which can be very uniform across an ensemble. This GS spinstrain interaction thus offers a path towards coupling an ensemble to a mechanical resonator. As the

NV centre density grows, however, the GS spin coherence will decrease30,31, limiting the utility of the collective enhancement.

Finally, we consider spinstrain interactions in the room temperature ES, which have not been thoroughly investigated but might provide the desired compatibility with dense ensembles. For temperatures above B150 K, orbital-averaging from the dynamic JahnTeller effect erases the orbital degree of freedom from the NV centre ES Hamiltonian, resulting in an effective orbital singlet ES at room temperature29,3234. Previously, magnetic spectroscopy measured an unidentied spin splitting within the room temperature ES that is on the order of 10 times stronger than the GS spinstrain interaction. These measurements hinted that this splitting might be a spinstrain interaction in the ES (refs 35,36). Like the GS spinstrain coupling, the resonance condition for such an interaction would be determined by a static magnetic bias eld, enabling collective enhancement with an ensemble. Furthermore, the NV centre density is not expected to affect the ES coherence time, which is limited by the ES motional narrowing rate34,37. Such an ES spinstrain interaction could thus offer a promising path towards coupling a dense NV centre ensemble to a mechanical resonator. Our rst goal then becomes to understand and precisely quantify this coupling.

Assuming this ES coupling is the result of a spinstrain interaction, we can write the spin Hamiltonian for an NV centre in the presence of a magnetic eld B and non-axial strain Ex. Both the GS and room temperature ES Hamiltonians then take the form (: 1)36,38

H D0S2z gNVS B d?Ex S2x S2y

AjjSzIz 1

where De0/2p 1.42 GHz and Dg0/2p 2.87 GHz are the ES

and GS zero-eld splittings, gNV/2p 2.8 MHz/G is the NV

centre gyromagnetic ratio, Aejj=2p 40 MHz (ref. 39) and

Agjj=2p 2:166 MHz are the ES and GS hyperne couplings to

the 14N nuclear spin, S (I) is the electronic (nuclear) spin-1 Pauli vector, and the z-axis runs along the NV centre symmetry axis. Perpendicular strain Ex couples the j ms

1i and j 1i spin

states with a strength de? in the ES and dg?/2p 21.51.2 GHz

per strain in the GS (ref. 17). As shown in Fig. 1a, this interaction enables direct control of the magnetically-forbidden j 1i

2j 1i spin transition within each orbital through resonant

strain.

Device details. The combination of a large hyperne splitting in the ES and a short ES lifetime broadens the spectral features of the ES spinstrain interaction. Measuring such a spectrum then requires large magnetic eld sweeps DBz 150 G

, which in turn

require a mechanical driving eld with a high carrier frequency om=2p\420 MHz

. To this end, we fabricate a high-overtone

bulk acoustic resonator (HBAR) capable of generating large amplitude strain at gigahertz-scale frequencies. The resonator used in this work was driven at a om/2p 529 MHz mechanical

mode that has a quality factor of Q 1,79020. An antenna

fabricated on the opposite diamond face provides high-frequency magnetic elds for magnetic spin control. The nal device is pictured in Fig. 1b.

Spinstrain spectroscopy. To measure mechanical spin driving within the ES, we execute the pulse sequences shown in Fig. 2a, as a function of the magnetic bias eld Bz. In the rst sequence, a 532 nm laser initializes the NV centre ensemble into the GS level g; ms

0

j i. A magnetic adiabatic passage (AP) then moves

the spin population to g; 1

j i. At this point, we pulse the

mechanical resonator at its resonance frequency om for 3 ms.

2 NATURE COMMUNICATIONS | 8:14358 | DOI: 10.1038/ncomms14358 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14358 ARTICLE

a b

ZnO transducer

Diamond

ZnO

Energy

AI

D

D e0

Microwave antenna

Confocal microscope objective

Pt

Magnetic bias field

Figure 1 | Energy levels and device schematic. (a) NV center ground state and excited state energy levels as a function of the magnetic bias eld. Energies have been plotted relative to the ms 0 state in each orbital, and a mechanical mode of frequency om has been drawn connecting the mI 1

hyperne sublevels. (b) Schematic of the device used in these measurements along with optical micrographs of the ZnO transducer used to generate the strain standing wave (150 mm scale bar) and the microwave antenna used to generate magnetic control elds (200 mm scale bar).

a

b

Measure S1= (Re-intialization + spin driving)

opt

t

t

1.0

0.8

0.6

0.4

0.2

0.00.0 0.1 0.2 0.3 0.4 0.5 0.6

Measure S2= Re-intialization

532 nm laser Magnetic driving

P g ,0

opt

S1 = (Re-initialization + spin driving)

Mechanical driving

Optical pulse length (s)

c

d

0.08

P +1

0.06

0.04

0.02

0.0020 93.0 93.5 94.0 94.5 95.0 95.5

40 60 80 100Bz (G) Bz (G)

120 140 160

GS transitions

0.08

0.06

0.04

0.02

0.00

ml = 1 ml = 0 ml = +1

Figure 2 | Spinstrain spectroscopy. (a) Pulse sequences used to measure excited state (ES) spin driving. (b) Population in g; 0

j i at the end of the pulse

sequences in a plotted against topt. (c) Spectrum of the spin population driven mechanically into 1

j i by the ES and ground state (GS) spinstrain

interactions. The red curve is a least squares t to the sum of six Lorentzians. (d) Zoomed in view of the GS spin transitions in c. The data in c,d were measured on one device with an NV centre ensemble, and error bars are from the s.d. in photon counting.

Just before the end of the mechanical pulse, we apply a topt

125 ns optical pulse with the 532 nm laser. This excites the ensemble to e; 1

j i and allows the spins to interact with the

mechanical driving eld in the ES. If the driving eld is resonant with the e; 1

j i2 e; 1

j i splitting, population will be driven

into e; 1

j i. The spins then follow either a spin-conserving

relaxation down to g; 1

j i or a relaxation to the singlet state S1

j i

through an intersystem crossing. The former preserves the spin state information, while relaxing to S1

j i re-initializes the state,

erases the stored signal and reduces the overall contrast of the measurement. After allowing the ensemble to relax, we apply the second magnetic AP to return the spin population in g; 1

j i to

g; 0

j i and measure the g; 0

j i population via uorescence read out.

We dene this signal as S1 and plot it as a function of topt in Fig. 2b.

In the second pulse sequence, the mechanical pulse occurs between the second AP and uorescence read out. Applying the mechanical pulse with the ensemble in g; 0

j i maintains the same

power load on the device but does not drive spin population. This sequence measures S2, the re-initialization of the ensemble from the topt optical pulse (Fig. 2b). Subtracting S2 S1 gives the

probability of nding the ensemble in 1

j i at the end of the rst

sequence. A third sequence with a single AP and a fourth with

two APs (both with topt 0) normalize the spin contrast at

each Bz.

Figure 2c shows the resulting experimental signal. The three broad, low peaks correspond to the hyperne-split e; 1

j i-

e; 1

j i transition, providing denitive evidence of a spinstrain

interaction within the room temperature ES. Population is also driven into g; 1

j i by the GS spinstrain interaction when the

NATURE COMMUNICATIONS | 8:14358 | DOI: 10.1038/ncomms14358 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications 3

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14358

mechanical driving eld is resonant with the g; 1

j i2 g; 1

j i

splitting. Figure 2c thus contains both ES and GS spectra. We t the data to the sum of three low, broad Lorentzians describing the ES spinstrain interaction and three taller, narrower Lorentzians describing the GS interaction (see Methods). Figure 2d highlights the GS driving with a zoomed in view of Fig. 2c about the GS resonances. The GS peaks have larger amplitudes than the ES peaks because the GS interaction acts for the entire duration of the 3 ms mechanical pulse, whereas the ES interaction only acts during the B125 ns that the spin population is in the ES. Also, the reversed sign of Aejj relative to Agjj is

consistent with ab initio calculations40 and was conrmed by measurements presented in Supplementary Note 1 that were conditional on the nuclear spin state.

Quantication of de?. To quantify the strength of the ES

spinphonon interaction, we rst calibrate the strain amplitude generated by the HBAR by mechanically driving Rabi oscillations within the GS and extracting the GS mechanical Rabi eld Og as a function of the applied power (Fig. 3a). Next, we spectrally isolate the e; 1

j i2 e; 1

j i transition by xing Bz 80 G. At this eld,

the applied strain is on resonance in the ES but off resonance in the GS (Fig. 2c). We then execute a modied version of the pulse sequence described above. Here, we use B20 ns magnetic p-pulses to address the g; 0

j i2 g; 1

j i transition and measure

both S1 and S2 as a function of topt for each power level applied to the HBAR.

As Fig. 3b shows, taking S2 S1 reveals a competition between

mechanical driving into e; 1

j i and re-initialization into 0

j i via

optical pumping. For nonzero topt, the ES mechanical driving eld Oe drives spin population from e; 1

j i to e; 1

j i,

increasing P e; 1

j i, but as topt grows, optical pumping re-initializes

the ensemble into 0

j i, vacating the ms { 1, 1} subspace and

reducing P e; 1

j i to zero. A seven-level master equation model

recreates this competition and provides good ts to the data. From these ts, we extract the value of Oe. The Methods section includes a detailed description of this model, which was designed to account for inhomogeneities within the NV centre ensemble and for the polarization of the nuclear spin sublevels, among other effects. Plotting Oe against Og (Fig. 3c) shows that the transverse spinstrain coupling in the ES is 13.50.5 times stronger than the GS coupling, or de?/2p 29020 GHz

per strain.

Resonator cooling protocol. With de? quantied, we now present

a dissipative protocol for cooling a mechanical resonator with an NV centre spin ensemble. In our proposed protocol, a 532 nm

laser continuously pumps the phonon sidebands of the ensembles optical transition, and a gigahertz frequency magnetic eld continuously drives the g; 0

j i2 g; 1

j i spin transition.

This generates a steady state population surplus in e; 1

j i as

compared with e; 1

j i, enabling the net absorption of phonons

by the ensemble. Spontaneous relaxation and subsequent optical pumping continually re-initialize the system, allowing the phonon absorption cycle to continue. Figure 4a summarizes this process.

The dissipative nature of this protocol enables resonator cooling without requiring strong spin-phonon coupling. Here, we dene strong coupling as a single-spin cooperativity Z l2T 2=gnth41, where l is the single spin-single

phonon coupling strength, T 2 is the inhomogeneous spin dephasing time, g om/Q is the mechanical dissipation rate

and nth kBT= om is the thermal phonon occupancy of the

resonator mode41. A cooperativity of Z41 marks the threshold for coherent interactions between the spin and the mechanical mode. Non-idealities in spin coherence and resonator fabrication have thus far prevented the experimental realization of NV centre cooperativities approaching unity, especially at room temperature. This makes the proposed dissipative protocol a practical and attractive approach because it does not require coherent interactions for resonator cooling to occur.

To analyse the performance of the protocol we start by considering a single two-state spin system coupled to a mechanical resonator. The resulting dynamics obey the master equation (: 1) _

r i H; r

LGr Lgr 2 where H describes the coherent coupling between the spin and the resonator, LG describes the incoherent spin processes, and Lg

describes the resonator rethermalization. For resonant coupling, the quantized Hamiltonian in the JaynesCummings form is42

H om aya S S

l S S

ay a

3

where aw (a) is the creation (annihilation) operator for the mechanical mode and S are the ladder operators for the spin state. The spin relaxation term in equation (2) takes the form LGr 2T1

1D S

r 2T 2

1 SzrSz r

, where

D S

r 2S rS S S r rS S

is the Lindblad

superoperator and T1 T 2

is the transverse (longitudinal) spin coherence time. The resonator rethermalization is described by

Lgr g2 nth 1

D a

r g2 nthDaw r.

Within this two-state model, an analytic expression for the steady state phonon number nf can be found by using the matrix of second order moments (Supplementary Note 3)43.

a b c

0.04

35

25

30

25 20 15

5

0.6

Ground state Rabi oscillations Exicted state spin driving

/2 = 26 3 MHz

e/2 , (MHz)

P g ,+1 , (Arbitrary units)

0.50.40.30.20.10.0

0.0 0.5 1.0 1.5Effective Rabi interval (s) Optical pulse length (s)

P e ,+1

0.03

0.02

0.01

0.00

2.0 2.5 3.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5

g/2 (MHz)

Figure 3 | Quantifying the excited state spinstrain coupling. (a) Mechanically driven Rabi oscillations within the ground state (GS) that have been t using the procedure described in Supplementary Note 2. (b) Population in e; 1

j i plotted as a function of topt. The red curves are least squares ts to a

seven-level master equation model of the measurement. The data in a,b were measured on a single device with an NV centre ensemble, and error bars are from the s.d. in photon counting. (c) The excited state mechanical driving eld plotted against the GS mechanical driving eld and t with a linear scaling. Each point corresponds to a single measurement, and error bars are standard error from the ts.

4 NATURE COMMUNICATIONS | 8:14358 | DOI: 10.1038/ncomms14358 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14358 ARTICLE

a

opt

Orbital

Spin

c d

6,000

Final phonon number

5,000

4,000

3,000

2,000

1,000

0

1017 1018 1019 1020 1021

1.0

0.8

0.6

0.4

0.2

0.0

Fractional cooling

Aligned NV density (cm3)

Figure 4 | A dissipative protocol for cooling a mechanical resonator. (a) The seven NV centre orbital and spin states at room temperature. Fast (slow) transitions are indicated by solid (dashed) one-way arrows. Coherent couplings are indicated by two-way arrows. (b) The toy model depiction of the proposed cooling protocol. (c) Schematic of a doubly-clamped beam. (d) Final phonon number achieved by the cooling protocol as a function of the density of properly aligned NV centers. Vertical lines indicate densities that have been realized in single-crystal diamonds (7.0 1017 cm 3 (ref. 31),

1.1 1018 cm 3 (ref. 58), 2.0 1018 cm 3 (ref. 56)) and nanodiamonds (4 1020 cm 3 (ref. 57)).

a

n: 4

For an ensemble of N spins coupled to the resonator but not to one another, each spin will add an additional damping term to the resonator dynamical equation. This allows us to rewrite the last term in equation (4) as

p leff. To determine a, we solve for the 7 7 density matrix describing the steady state

of the ensemble in the absence of the mechanical resonator, calculate the population difference between e; 1

j i and e; 1

j i,

and obtain a 0.017 for optimized control elds (Supplementary

Note 4).

Elasticity theory provides a means of calculating the remaining device parameters. For a doubly-clamped beam of length l, thickness t and width w, we compute leff from the strain because of the zero-point motion of the resonator E0(y, z) with coordinates as dened in Fig. 4c (see Methods). For a uniform distribution of properly aligned NV centres at a density n, we obtain41,48

leff

Under the secular approximation and working in the limit g; l 1=T1; 1=T 2, the dynamical equation for the

phonon occupancy n hawai can be simplied to dndt g nth n

4l2 2=T 2 1=T1

1=T1;i n. If each spin within the ensemble has the same T1 and T 2, we can factorize this expression and replace the individual li with an effective

ensemble-resonator coupling leff

PNi1l2i s

PNi14l2i 2=T 2;i

. For the case of

uniform coupling, this simplies to leff

N

de?

p

Z

Z

t=2

l t=2 E20 y; z

p l, which is equivalent to the effective coupling in the TavisCummings model44,45. Solving for the steady state of the system then gives

nf

gnth

g

dydz

s : 6

Evaluating equation (6), we nd that leff is independent of w and scales as leff G0

anw

0

p =l where G0 de?

nt

p

ak0=E

,

: 5

The problem now becomes mapping the seven-level NV centre structure pictured in Fig. 4a onto this two-state spin system. We do this by distilling the seven-level landscape to a toy model that contains only the two-states that couple to the mechanical resonator, e; 1

j i and e; 1

j i, as shown in Fig. 4b. Within this

simplied landscape, we assign T 2 to be the ES coherence time (T 2e 6.0 ns (ref. 46)) and T1 to be the ES lifetime of e; 1

j i

(T1e 6.89 ns (ref. 47)). At any one moment, only a fraction a

of the spins within the ensemble will be in the proper e; 1

j i; e; 1

j i

f g subspace to participate in the cooling.

We account for this by modifying leff !

120 GHz mm, and E 1,200 GPa is the Youngs modulus

of diamond. The frequency of the resonators fundamental mode scales as om k0t/l2. As described in Supplementary Note 5,

higher order mechanical modes are spectrally isolated from the NV centre spin dynamics in the devices considered here17,41. For any thin-beam resonator in the resolved-sideband regime

om=2p41=T 2e 170 MHz, the fractional cooling nf/nth is

insensitive to the physical dimensions of the resonator because the size of the ensemble scales with the size of the resonator. This can be seen by rewriting equation (5) as nf/nth (1 w) 1, where w

4 Qna de?

k0

4l2eff 2=T

2

1=T1

2

is independent of the resonator dimensions.

For illustrative purposes, we choose to examine a resonator with a

E 2=T

2e

1=T1e

NATURE COMMUNICATIONS | 8:14358 | DOI: 10.1038/ncomms14358 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications 5

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14358

a

Optically detected magnetic resonance Optically detected magnetic resonance

Bx frequency (GHz) Bx frequency (GHz)

b

1.000

Normalized P.L.

0.995

0.990

0.985

Normalized P.L.

1.000

0.995

0.990

0.985

Bz = 20.2 G

2.805

2.810 2.815 2.820 2.825

2.380 2.385 2.390 2.395 2.400

c

Normalized amplitude

0.6

0.5

0.4

0.3

0.2

0.1 20 40 60 80 100 120 140 160

Bz (G)

Figure 5 | Calibrating Bz-dependent nuclear polarization. (a,b) Normalized photoluminescence (P.L.) plotted as a function of the magnetic driving eld carrier frequency for (a) Bz 20.2 G and (b) Bz 171.5 G. The solid line in each plot is a least squares t to the sum of three Lorentzians. The data in a,b was

measured on a single device with an NV centre ensemble, and error bars are from the s.d. in photon counting. (c) Normalized amplitude of each mI hyperne sublevel as a function of Bz. The solid lines are least squares ts to a linear model, each point corresponds to a single measurement, and error bars are standard error from the ts.

om/2p 1 GHz fundamental mode and assume fully polarized

nuclear spins49. Potential device dimensions then become (l, t) (1.9, 0.19) mm. Finally, phononphonon interactions

limit the Q of an ideal diamond mechanical resonator at room temperature. For modes satisfying om/2p 41/T 2e, the maximum

Q 2 106 is independent of om (ref. 50), and we now have all

the parameters needed to study the performance of the protocol.

At this point, we note that distilling a seven-state model to the toy model we employ certainly requires validation. To justify our simplied model, we calculate the cooling predicted within the toy model and compare this both to an analytical LambDicke treatment of the seven-level model14,51 and to numerical simulations of a small number of seven-level NV centres coupled to a resonator. Because of the exponential growth of the Hilbert space, full seven-level numerical simulations were performed on the Titan supercomputer at Oak Ridge National Laboratory, with the most intensive simulations taking B104 core-hours. Comparing the toy model and LambDicke results to the numerical simulations, we determined that the two-level distillation outperforms the LambDicke approach in all test cases and provides an upper bound on nf (Supplementary

Note 6). This indicates that the proposed protocol cools a resonator more efciently than our toy model predicts5255.

Cooling performance. The lowest phonon occupancy that can be reached depends strongly on the density of properly aligned NV centres n. For instance, Choi, et al. reported measurements of an

NV centre ensemble with n 2.0 1018 cm 3 in single-crystal

diamond56. For this density and Q 2 106, we nd that the

proposed protocol cools a room temperature resonator to nf 0.86nth. Using the same Q and the density of n 4 1020

cm 3 reported by Baranov, et al. in nanodiamonds57, however, the protocol can cool to nf 0.03nth.

Increasing the size of the ensemble can thus dramatically increase the protocols cooling power. The magnetic eld noise from paramagnetic impurities will also grow with n, degrading the

GS coherence time. However, for large magnetic driving elds, this cooling protocol does not require a lengthy GS coherence time (Supplementary Note 4). The only coherence time that effects the protocol is T 2e, which is not expected to change with the defect density34,37. This means that large NV centre densities could in principle be used to cool a resonator with the ES spinstrain interaction. To study how increasing n affects the protocol, we plot nf against n in Fig. 4d for several different Q-values. For reference, we have included lines marking values of n that have been realized in single-crystal diamonds31,56,58 and in nanodiamonds57. The limiting density of NV centres in a single-crystal diamond nanostructure is currently unknown. Furthermore, while high defect densities have been shown to degrade the Q of om=2p 10 kHz frequency resonators59,

it remains to be seen how the gigahertz frequency resonators of interest here will be affected by the incorporation of a dense defect ensemble. These questions motivate future experimental work.

DiscussionThe insensitivity of the proposed protocol to the GS coherence time makes an ES cooling protocol an attractive and practical route to cooling a room temperature mechanical resonator with NV centres. Alternative approaches that use the GS spinstrain coupling18,19,25 are incompatible with the collective enhancement from a dense ensemble that makes the proposed protocol viable. Although the GS inhomogeneous dephasing time T 2g can be Bms long in high purity diamonds, T 2g scales roughly as 1=n in bulk

diamond and can be o100 ns for dense ensembles30,31. Within a nanostructure, effects such as exchange narrowing and the

6 NATURE COMMUNICATIONS | 8:14358 | DOI: 10.1038/ncomms14358 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14358 ARTICLE

truncation of the spin bath mitigate this reduction in T 2g (refs 57,60) and make it difcult to predict the decrease in T 2g inside a doubly-clamped beam. Nevertheless, we can roughly compare the ES and GS spinstrain interactions by calculating Z using coherence times measured in bulk diamond (Supplementary Note 7). For a moderate NV centre density of n 2.8 1013 cm 3 (ref. 18), the single-spin cooperativity for the

ES spinstrain interaction is 2.4-times larger than in the GS, and for n 7.0 1017 cm 3 (ref. 31), Z is 19-times larger in the ES.

In both cases, the ES offers the more efcient route to cooling, and as the collective enhancement grows, the ES interaction becomes increasingly more efcient than the GS interaction. A dense ensemble coupled via the ES spinstrain interaction thus becomes the more promising route to cooling a room temperature mechanical resonator with NV centres.

It is important to note that this analysis of the proposed protocol only applies for operation at room temperature. Reducing the bath temperature will lower nth and would thus ideally lower nf. However, the ES coherence time is limited by the ES motional narrowing rate, which increases as the bath temperature decreases34,37. This is expected to lead to a reduction in T 2e at lower temperatures, followed by a complete loss of

ES spin coherence below B150 K (ref. 32). As seen from equation (5), a reduction in T 2e will lead to a loss of cooling power. For cryogenic starting temperatures, it thus becomes necessary to use either the GS spin-phonon interaction or the orbital-strain interaction to cool a mechanical resonator with NV centres.

Finally, nf could be lowered further by simultaneously implementing an optomechanical cooling protocol812 alongside the proposed protocol. Optomechanical cooling has been demonstrated to cool gigahertz frequency resonators to

nf 0:01nth (refs 6,61). The cooling rate from an optimized

realization of the proposed protocol would combine additively with the optomechanical cooling rate, allowing the two complementary techniques to operate in conjunction and enhance the total cooling.

In conclusion, we have proposed a dissipative protocol for cooling a room temperature mechanical resonator that utilizes an ensemble of NV centre spins to realize a collective enhancement in the spin-phonon coupling. After experimentally determining that the spinstrain coupling in the room temperature ES is13.50.5 times stronger than the GS spinstrain coupling, we analysed the performance of the cooling protocol. For very dense NV centre ensembles, the proposed protocol can cool a room temperature resonator to a fraction of its thermal phonon occupancy. These results shed further light on the orbitally-averaged room temperature ES of the NV centre and demonstrate a practical path towards cooling a room temperature mechanical resonator with NV centres.

Methods

Sample details. Our HBAR consists of a 2.5 mm-thick ZnO piezoelectric lm sandwiched between a 25/200 nm Ti/Pt ground plane and a 250 nm thick apodized Al top electrode, all sputtered on one face of a h100i single-crystal diamond

substrate. Applying a high-frequency voltage to this transducer launches acoustic waves into the diamond, which then serves as a FabryProt cavity to generate a comb of standing wave resonances. By apodizing the shape of the Al electrode, we mitigate the loss of power into lateral modes formed across the diameter of the HBAR. The antenna fabricated on the opposite diamond face was patterned from 25/225 nm Ti/Pt with a lift-off process.

The CVD-grown diamond used in these measurements contained NV centres at a density of B4 1014 cm 3 as purchased. Our measurements thus address an

ensemble of B70 NV centres oriented with their symmetry axis parallel to Bz. NV centres of different orientations are spectrally isolated and contribute only a constant background to the measurements.

Spectrum tting. The spectrum pictured in Fig. 2 was t to the function

P 1

j i

ce

a Bz

Table 1 | Relaxation rates used in our seven-level master equation model47.

Parameter Value (MHz) Relaxation from k42 65.31.6 ES 1

j i to GS 1

j i

k31 64.91.5 ES 0

j i to GS 0

j i

k45 79.81.6 ES 1

j i to S1

j i

k35 10.61.5 ES 0

j i to S1

j i

k52 2.610.06 S1

j i to GS 1

j i

k51 3.000.06 S1

j i to GS 0

j i

e, +1

g, +1

g, 1

g, 0

0 B

@

a0 Bz

4 G2e Bz B0 Aejj=gNV

1 2

4 G2e Bz B0

1 2

a Bz

4 G2e Bz B0 Aejj=gNV

1 2

1 C

A

0 B

@

4 G2g Bz B0 Agjj=gNV

1 2

4 G2g Bz B0

1 2

a Bz

4 G2g Bz B0 Agjj=gNV

1 2

1 C

A

7

cg a Bz

a0 Bz

where ce and cg are constant amplitudes that quantify the driven spin contrast for the ES and GS resonances, ai[Bz] are eld-dependent scaling factors that account for the dynamic nuclear polarization of the hyperne sublevels49, Ge (Gg) is the

a

b

e/2 (MHz)

35

30

25

20

15

10

5

e, 1

e, 0

31

1.0

1.5 2.0 2.5

g/2 (MHz)

Figure 6 | Transitions and rates used in the seven-level model. (a) States and transitions included in our seven-level master equation model. The kij rates are listed in Table 1. (b) Excited state mechanical driving eld Oe plotted as a function of the ground state mechanical driving eld Og with the data labelled by the optical pumping rate G0 during each measurement. Each point corresponds to a single measurement, and error bars are standard error from the ts.

NATURE COMMUNICATIONS | 8:14358 | DOI: 10.1038/ncomms14358 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications 7

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14358

FWHM of the ES (GS) resonances, B0 is the resonant eld for the mI 0 hyperne

sublevel, and the other parameters are as dened above. Of these variables, ci, Gi, and B0 are free parameters in our tting procedure.

We calibrate ai[Bz] by performing hyperne-resolved magnetically-driven electron spin resonance (ESR) measurements within the NV centre GS at different values of the magnetic bias eld Bz. This is done by xing Bz and monitoring the

NV centre photoluminescence as the carrier frequency of a magnetic driving eld oriented along Bx is swept through the g; 0

j i2 g; 1

j i spin resonance. We t the

resulting curves to the function

P C

A

0

B

@

4 G2g Bz B0 Agjj

1 2

4 G2g Bz B0 Agjj

1 2

1

C

A

P0

8

A0

4 G2g Bz B0

1 2

A

j i Oe; Aejj

a 1P 1

j i Oe; 2Aejj

15

where the normalized amplitudes (

where P is the measured photoluminescence, C accounts for the driven spin contrast, Ai is the relative amplitude of each hyperne sublevel, P0 is the background photoluminescence, and we x

P

P

Ai 1. Figure 5a,b shows ESR

curves measured at Bz 20.2 G and 171 G. We have used the values of P0 returned

from the ts to normalize the photoluminescence.

Figure 5c shows the normalized amplitude of each nuclear sublevel plotted as a function of Bz. As expected, the nuclear polarization increases in the direction of the ES level anti-crossing at BLACz 507 G. In this gure, we have t each curve to a

straight line with a xed y-intercept of 13 to obtain the linear scaling functions ai[Bz] in equation (7). The sum of these scaling functions satises

P

ai 1) account for nuclear spin polarization

and have been measured separately via magnetic ESR. A least squares t of equation (15) to the data then provides Oe. Here, Oe is the only free parameter in the tting procedure.

When tting the relation between Oe and Og (Fig. 3c), we x the y-intercept of the linear tting function to be zero.

Elasticity theory. To analyse how strain couples to NV centres within a resonator, we start by assuming that the NV centres are aligned with the direction of beam deection such that the strain in an oscillating beam is entirely perpendicular to the NV centre symmetry axis. We then use elasticity theory to derive the scaling laws quoted above17,41.

The wave equation for doubly-clamped beams is

rdA @2

@t2 f t; z

EI

@4@z4 f t; z

ai[Bz] 1.

Seven-level master equation model. The master equation used to model our ES spin driving measurements is derived in the room temperature NV centre basis dened by the states g1; g0; g 1; e1; e0; e 1; S1

f g where, within the e and g

subspaces, a subscript denotes the ms value. The 7 7 density matrix evolves

according to (: 1)_

r i H; r

LGr: 9 In the rotating frame, the Hamiltonian is given by

H O2 e1

j i e 1

h j e 1

j i e1

h j

j i e1

h j 10 where Dm is the mechanical detuning. The incoherent NV centre processes are described by the superoperator

LGr GoptXi 1;0Lg ;e k42 Xi 1Le ;g k45 Xi 1Le ;S k52 Xi 1LS ;g

k31Le ;g k35Le;S k51LS ;g

1 T 2e

16

where f(t, z) is the transverse displacement in the y-direction, ^z points along the beam as indicated in Fig. 4c, A wt is the cross-sectional area of the resonator,

E 1,200 GPa is the Youngs modulus of diamond, rd 3.515 g cm 3 is the mass

density of diamond, and I wt3/12 is the resonators moment of inertia. Solutions

are of the form f(z, t) u(z)e iot where

un z an cos knz cosh knz

bn sin knz sinh knz

; 17 and the allowed k-vectors satisfy cos(knz) cosh(knz) 1. The wave vector and

amplitudes of the fundamental mode satisfy k0l 4.73 and a0/b0 1.0178.

We normalize un(z) by setting the free energy of the beam equal to the zero point energy of the mode:

W

1

2 EI

Dm e1

Xi 1;0Le ;e

11

1

2 on 18

where the eigenfrequencies of the resonator are given by on k2n

EI=rdA

where we dene

Li;f r f

j i i

h jr i

j i f

h j

1

2 i

j i i

h jr r i

j i i

h j

Z

L @2un

@z2

2

dz

0

p

: 12 Here, Gopt is the optical pumping rate of our 532 nm laser, T 2e 6.00.8 ns is the

ES coherence time46, and the relaxation rates kij are listed in Table 1. Figure 6a summarizes this landscape47.

Because optical initialization does not generate a pure state, we rst simulate the optical pumping process to obtain an initialized density matrix. To do so, we start with a thermal state rNV 13

Pi 1;0gij i gih j and apply equation (9) with Oe,

Dm 0 and Gopta0 for 10 ms. We take the resulting density matrix and apply

equation (9) for 5 ms with Oe, Dm, Gopt 0 to simulate the relaxation to g

j i.

A simulated p-pulse then swaps r22 and r33, providing the appropriate starting density matrix r0 for a given Gopt. From r0, we also extract the minimum and maximum spin contrast (smin g0

h jr0 g0

j i and smax g 1

h jr0 g 1

j i), which allow

us to properly normalize our simulations.

Next, we model the measurement of S2, the spin re-initialization. To do so, we apply equation (9) to r0 with Oe, Dm 0 and Gopta0 for a length of time topt.

Allowing the spin to relax as before gives us the measured density matrix r2. We normalize g 1

h jr2 g 1

j i using smin and smax, and repeat this simulation as a

function of topt to obtain a simulated S2 curve.

To account for spatial inhomogeneities in the optical power within the NV centre ensemble, we perform a weighted average of this simulation over the point spread function (PSF) of our microscope. We approximate the PSF by the function

Gopt z G0

sin k z0

z z0

f g

k z0

z z0

. This

expression for on can be simplied to on knt/l2, where kn knl

2

p

E=12rd

.

For the fundamental mode, k0 120 GHz mm as quoted above.

The spin-phonon coupling for a single NV centre located at (y, z) is given by l d>E0(y, z) where E0 y; z

y @@z un z

is the strain from the zero point

motion of the resonator mode. Here, the y-axis is zeroed at the neutral axis of the resonator. To compute the effective ensemble-resonator coupling, we assume a uniform distribution of properly aligned NV centres within the resonator and sum

the individual couplings in quadrature according to leff

a

PNi1 l2i s

, which gives

2

13

where G0 is the peak optical pumping rate, k[z0] denes the depth-dependent PSF width15, z is the distance below the diamond surface, and z0 7.90.9 mm is the

focus depth of the PSF. An ensemble measurement is then given by

Sens2 topt

equation (6).

Data availability. The data that support the ndings of this study are available from the corresponding author on reasonable request.

References

1. Albrecht, T. R., Grtter, P., Horne, D. & Rugar, D. Frequency modulation detection using high-Q cantilevers for enhanced force microscope sensitivity.J. Appl. Phys. 69, 668673 (1991).2. Cleland, A. N. & Roukes, M. L. A nanometre-scale mechanical electrometer. Nature 392, 160162 (1998).

3. Rugar, D., Budakian, R., Mamin, H. J. & Chui, B. W. Single spin detection by magnetic resonance force microscopy. Nature 430, 329332 (2004).

4. Poggio, M., Degen, C. L., Mamin, H. J. & Rugar, D. Feedback cooling of a cantilevers fundamental mode below 5 mk. Phys. Rev. Lett. 99, 017201 (2007).

5. OConnell, A. D. et al. Quantum ground state and single-phonon control of a mechanical resonator. Nature 464, 697703 (2010).

6. Chan, J. et al. Laser cooling of a nanomechanical oscillator into its quantum ground state. Nature 478, 8992 (2011).

7. Teufel, J. D. et al. Sideband cooling of micromechanical motion to the quantum ground state. Nature 475, 359363 (2011).

R

: 14

We discretize this integral to make it computationally tractable and perform a least squares t of Sens2 topt

to the measured data. G0 is the only free parameter in the tting procedure.

With the exception of the datum indicated in Fig. 6b, all of the measurements were taken at the same optical power. We simultaneously t each of these S2 curves and nd G0 19.11.6 MHz. For the measurement at a different optical power,

we nd G0 16.51.7 MHz.

To extract Oe, we then x G0 and repeat this procedure with Oea0 to simulate the S2 S1 measurement pictured in Fig. 3b. To account for inhomogeneities in the

mechanical driving eld, we must also include the spatial prole of the strain standing wave inside the weighted average. This function takes the formOe z O0 sin 2pz=l

j j where l 314 mm is the wavelength of the strain wave.

Dening the results of such a simulation as P 1

j i(Oe, Dm), we account for the

hyperne sublevels by computing the sum

P 1

j i Oe

a1P 1

j i Oe; 0

a0P 1

0 S2 topt; Gopt z

dz

R

0 Gopt z

1 1 dz

8 NATURE COMMUNICATIONS | 8:14358 | DOI: 10.1038/ncomms14358 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14358 ARTICLE

8. Metzger, C. H. & Karrai, K. Cavity cooling of a microlever. Nature 432, 10021005 (2004).

9. Gigan, S. et al. Self-cooling of a micromirror by radiation pressure. Nature 444, 6770 (2006).

10. Arcizet, O., Cohadon, P.-F., Briant, T., Pinard, M. & Heidmann, A. Radiation-pressure cooling and optomechanical instability of a micromirror. Nature 444, 7174 (2006).

11. Kleckner, D. & Bouwmeester, D. Sub-Kelvin optical cooling of a micromechanical resonator. Nature 444, 7578 (2006).

12. Schliesser, A., DelHaye, P., Nooshi, N., Vahala, K. J. & Kippenberg, T. J. Radiation pressure cooling of a micromechanical oscillator using dynamical backaction. Phys. Rev. Lett. 97, 243905 (2006).

13. Rabl, P. et al. Strong magnetic coupling between an electronic spin qubit and a mechanical resonator. Phys. Rev. B 79, 041302 (2009).

14. Kepesidis, K. V., Bennett, S. D., Portolan, S., Lukin, M. D. & Rabl, P. Phonon cooling and lasing with nitrogen-vacancy centers in diamond. Phys. Rev. B 88, 064105 (2013).

15. MacQuarrie, E. R., Gosavi, T. A., Jungwirth, N. R., Bhave, S. A. & Fuchs, G. D. Mechanical spin control of nitrogen-vacancy centers in diamond. Phys. Rev. Lett. 111, 227602 (2013).

16. Teissier, J., Barfuss, A., Appel, P., Neu, E. & Maletinsky, P. Strain coupling of a nitrogen-vacancy center spin to a diamond mechanical oscillator. Phys. Rev. Lett. 113, 020503 (2014).

17. Ovartchaiyapong, P., Lee, K. W., Myers, B. A. & Jayich, A. C. B. Dynamic strain-mediated coupling of a single diamond spin to a mechanical resonator. Nat. Commun. 5, 4429 (2014).

18. MacQuarrie, E. R. et al. Coherent control of a nitrogen-vacancy center spin ensemble with a diamond mechanical resonator. Optica 2, 233238 (2015).

19. Barfuss, A., Teissier, J., Neu, E., Nunnenkamp, A. & Maletinsky, P.

Strong mechanical driving of a single electron spin. Nat. Phys. 11, 820824 (2015).

20. MacQuarrie, E. R., Gosavi, T. A., Bhave, S. A. & Fuchs, G. D. Continuous dynamical decoupling of a single diamond nitrogen-vacancy center spin with a mechanical resonator. Phys. Rev. B 92, 224419 (2015).

21. Golter, D. A., Oo, T., Amezcua, M., Stewart, K. A. & Wang, H. Optomechanical quantum control of a nitrogen-vacancy center in diamond. Phys. Rev. Lett. 116, 143602 (2016).

22. Lee, K. W. et al. Strain coupling of a mechanical resonator to a single quantum emitter in diamond. Phys. Rev. Appl. 6, 034005 (2016).

23. Burek, M. J. et al. Diamond optomechanical crystals. Optica 3, 14041411 (2016).

24. Khanaliloo, B. et al. Single-crystal diamond nanobeam waveguide optomechanics. Phys. Rev. X 5, 041051 (2015).

25. Meesala, S. et al. Enhanced strain coupling of nitrogen-vacancy spins to nanoscale diamond cantilevers. Phys. Rev. Appl. 5, 034010 (2016).

26. Mitchell, M. et al. Single-crystal diamond low-dissipation cavity optomechanics. Optica 3, 963970 (2016).

27. Ma, Y., Yin, Z., Huang, P., Yang, W. L. & Du, J. Cooling a mechanical resonator to the quantum regime by heating it. Phys. Rev. A. 94, 053836 (2016).

28. Lau, H.-K. & Plenio, M. B. Laser cooling of a high-temperature oscillator by a three-level system. Phys. Rev. B 94, 054305 (2016).

29. Fu, K.-M. C. et al. Observation of the dynamic Jahn-Teller effect in the excited states of nitrogen-vacancy centers in diamond. Phys. Rev. Lett. 103, 256404 (2009).

30. Kittel, C. & Abrahams, E. Dipolar broadening of magnetic resonance lines in magnetically diluted crystals. Phys. Rev. 90, 238239 (1953).

31. Acosta, V. M. et al. Diamonds with a high density of nitrogen-vacancy centers for magnetometry applications. Phys. Rev. B 80, 115202 (2009).

32. Batalov, A. et al. Low temperature studies of the excited-state structure of negatively charged nitrogen-vacancy color centers in diamond. Phys. Rev. Lett. 102, 195506 (2009).

33. Rogers, L. J., McMurtrie, R. L., Sellars, M. J. & Manson, N. B. Time-averaging within the excited state of the nitrogen-vacancy centre in diamond. New J. Phys. 11, 063007 (2009).

34. Plakhotnik, T., Doherty, M. W. & Manson, N. B. Electron-phonon processes of the nitrogen-vacancy center in diamond. Phys. Rev. B 92, 081203 (2015).

35. Fuchs, G. D. et al. Excited-state spectroscopy using single spin manipulation in diamond. Phys. Rev. Lett. 101, 117601 (2008).

36. Doherty, M. W. et al. The nitrogen-vacancy colour centre in diamond. Phys. Rep 528, 1 (2013).

37. Fuchs, G. D. et al. Excited-state spin coherence of a single nitrogen-vacancy centre in diamond. Nat. Phys. 6, 668672 (2010).

38. Slichter, C. P. in Principles of Magnetic Resonance 3rd edn (Springer, 1990).39. Steiner, M., Neumann, P., Beck, J., Jelezko, F. & Wrachtrup, J. Universal enhancement of the optical readout delity of single electron spins at nitrogen-vacancy centers in diamond. Phys. Rev. B 81, 035205 (2010).

40. Gali, A., Fyta, M. & Kaxiras, E. Ab initio supercell calculations on nitrogen-vacancy center in diamond: electronic structure and hyperne tensors. Phys. Rev. B 77, 155206 (2008).

41. Bennett, S. D. et al. Phonon-induced spin-spin interactions in diamond nanostructures: application to spin squeezing. Phys. Rev. Lett. 110, 156402 (2013).

42. Jaynes, E. T. & Cummings, F. W. Comparison of quantum and semiclassical radiation theories with application to the beam maser. Proc. IEEE 51, 89109 (1963).

43. Wilson-Rae, I., Nooshi, N., Dobrindt, J., Kippenberg, T. J. & Zwerger, W. Cavity-assisted backaction cooling of mechanical resonators. New J. Phys. 10, 095007 (2008).

44. Tavis, M. & Cummings, F. W. Exact solution for an N-molecule-radiation-eld Hamiltonian. Phys. Rev 170, 379384 (1968).

45. Garraway, B. M. The Dicke model in quantum optics: Dicke model revisited. Philos. Trans. R. Soc. Lond. A: Math. Phys. Eng. Sci. 369, 11371155 (2011).

46. Fuchs, G. D., Falk, A. L., Dobrovitski, V. V. & Awschalom, D. D. Spin coherence during optical excitation of a single nitrogen-vacancy center in diamond. Phys. Rev. Lett. 108, 157602 (2012).

47. Robledo, L., Bernien, H., van der Sar, T. & Hanson, R. Spin dynamics in the optical cycle of single nitrogen-vacancy centres in diamond. New J. Phys. 13, 025013 (2011).

48. Schuster, D. I. et al. High-cooperativity coupling of electron-spin ensembles to superconducting cavities. Phys. Rev. Lett. 105, 140501 (2010).

49. Jacques, V. et al. Dynamic polarization of single nuclear spins by optical pumping of nitrogen-vacancy color centers in diamond at room temperature. Phys. Rev. Lett. 102, 057403 (2009).

50. Tabrizian, R., Rais-Zadeh, M. & Ayazi, F. Effect of phonon interactions on limiting the f.Q product of micromechanical resonators. in Solid-State Sensors, Actuators and Microsystems Conference. TRANSDUCERS 2009, 21312134 (IEEE, 2009).

51. Jaehne, K., Hammerer, K. & Wallquist, M. Ground-state cooling of a nanomechanical resonator via a Cooper-pair box qubit. N. J. Phys. 10, 095019 (2008).

52. Otten, M. et al. Entanglement of two, three, or four plasmonically coupled quantum dots. Phys. Rev. B 92, 125432 (2015).

53. Saad, Y. & Schultz, M. GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856869 (1986).

54. Balay, S. et al. PETSc users manual. Tech. Rep. ANL-95/11 - Revision 3.7, (Argonne National Laboratory, 2016).

55. Balay, S., Gropp, W. D., McInnes, L. C. & Smith, B. F. in Modern Software Tools in Scientic Computing (eds Arge, E., Bruaset, A. M. & Langtangen, H. P.), 163202 (Birkhauser Press, 1997).

56. Choi, S. et al. Depolarization dynamics in a strongly interacting solid-state spin ensemble. Preprint at http://arxiv.org/abs/1608.05471

Web End =http://arxiv.org/abs/1608.05471 (2016).

57. Baranov, P. G. et al. Enormously high concentrations of uorescent nitrogen-vacancy centers fabricated by sintering of detonation nanodiamonds. Small 7, 15331537 (2011).

58. Wee, T.-L. et al. Two-photon excited uorescence of nitrogen-vacancy centers in proton-irradiated type Ib diamond. J. Phys. Chem. A 111, 93799386 (2007).

59. Tao, Y., Boss, J. M., Moores, B. A. & Degen, C. L. Single-crystal diamond nanomechanical resonators with quality factors exceeding one million. Nat. Commun. 5, 3638 (2014).

60. Smith, M. J. A., Angel, B. R. & Emmons, R. G. Distribution of substitutional nitrogen donors in synthetic diamonds. Nature 210, 692694 (1966).

61. Safavi-Naeini, A. H. et al. Observation of quantum motion of a nanomechanical resonator. Phys. Rev. Lett. 108, 033602 (2012).

Acknowledgements

Research support was provided by the Ofce of Naval Research (ONR) (Grant N000141410812). Device fabrication was performed in part at the Cornell NanoScale Science and Technology Facility, a member of the National Nanotechnology Coordinated Infrastructure, which is supported by the National Science Foundation (Grant ECCS-15420819), and at the Cornell Center for Materials Research SharedFacilities which are supported through the NSF MRSEC program (DMR-1120296). Numerical simulations were performed in part at the Centre for Nanoscale Materials,a U.S. Department of Energy Ofce of Science User Facility under contractno. DE-AC02-06CH11357. This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supportedby the Ofce of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725.

Author contributions

E.R.M. and G.D.F. developed the concept and procedure for the experiment and the proposed cooling protocol. E.R.M. performed the experiments and analysed the data. E.R.M., M.O., S.K.G. and G.D.F. developed the toy model treatment of the proposed cooling protocol. M.O. and S.K.G. developed and performed the numerical simulations that validate the toy model treatment of the cooling protocol. E.R.M., M.O, S.K.G. and G.D.F. prepared the manuscript.

NATURE COMMUNICATIONS | 8:14358 | DOI: 10.1038/ncomms14358 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications 9

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14358

Additional information

Supplementary Information accompanies this paper at http://www.nature.com/naturecommunications

Web End =http://www.nature.com/ http://www.nature.com/naturecommunications

Web End =naturecommunications

Competing nancial interests: The authors declare no competing nancial interests.

Reprints and permission information is available online at http://npg.nature.com/reprintsandpermissions/

Web End =http://npg.nature.com/ http://npg.nature.com/reprintsandpermissions/

Web End =reprintsandpermissions/

How to cite this article: MacQuarrie, E. R. et al. Cooling a mechanical resonator with nitrogen-vacancy centres using a room temperature excited state spinstrain interaction. Nat. Commun. 8, 14358 doi: 10.1038/ncomms14358 (2017).

Publishers note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional afliations.

This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the articles Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

Web End =http://creativecommons.org/licenses/by/4.0/

r The Author(s) 2017

10 NATURE COMMUNICATIONS | 8:14358 | DOI: 10.1038/ncomms14358 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications