Proposal for a Room-Temperature Diamond Maser

ARTICLE

Received 27 Nov 2014 | Accepted 3 Aug 2015 | Published 23 Sep 2015 DOI: 10.1038/ncomms9251 OPEN Proposal for a room-temperature diamond maser

Liang Jin1, Matthias Pfender2, Nabeel Aslam2, Philipp Neumann2, Sen Yang2,Jo¨rg Wrachtrup2 & Ren-Bao Liu1

The application of masers is limited by its demanding working conditions (high vacuum or low temperature). A room-temperature solid-state maser is highly desirable, but the lifetimes of emitters (electron spins) in solids at room temperature are usually too short (Bns) for population inversion. Masing from pentacene spins in p-terphenyl crystals, which have a long spin lifetime (B0.1 ms), has been demonstrated. This maser, however, operates only in the pulsed mode. Here we propose a room-temperature maser based on nitrogen-vacancy centres in diamond, which features the longest known solid-state spin lifetime (B5 ms) at room temperature, high optical pumping efﬁciency (B106 s 1) and material stability. Our numerical simulation demonstrates that a maser with a coherence time of approximately minutes is feasible under readily accessible conditions (cavity Q-factor B5 104, diamond size B3 3 0.5 mm3 and pump power o10 W). A room-temperature diamond maser may facilitate a broad range of microwave technologies.

1 Department of Physics and Centre for Quantum Coherence, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China. 2 3rd Institute of Physics, University of Stuttgart, 70569 Stuttgart, Germany. Correspondence and requests for materials should be addressed to R.-B.L. (email: [email protected]).

NATURE COMMUNICATIONS | 6:8251 | DOI: 10.1038/ncomms9251 | www.nature.com/naturecommunications 1 & 2015 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms9251

he maser1—with a frequency 0.3–300 GHz and a B105 s 1 in silicon carbide27). Therefore, the population wavelength between 1 m and 1 mm—is the microwave inversion can be easily achieved if a magnetic field is applied to 2–5 Tanalogue of lasers with several important applications shift the ms ¼ 0 ground state to above another spin state. such as in ultrasensitive magnetic resonance spectroscopy, Furthermore, the good thermal conductivity and material stability astronomy observation, space communication, radar and of diamond are also advantageous for masers. All of these features high-precision clocks. Such applications, however, are hindered suggest that the NV centres in diamond are a superb gain by the demanding operation conditions (high vacuum for a gas medium for a room-temperature solid-state maser. However, the maser6 and liquid-helium temperature for a solid-state maser7). diamond maser requires a magnetic field; this is a further Room-temperature solid-state masers are highly desirable. experimental complication that can, however, be overcome The key to a maser is the population inversion of the emitters because magnets that can provide stable and uniform magnetic and the macroscopic coherence among microwave photons. fields are commercially available28–30. Population inversion requires a spin relaxation rate lower than Here, we theoretically propose a room-temperature maser based the pump rate. This sets the bottleneck in room-temperature on NV centres in diamond. Our numerical simulation demon- solid-state masers, as the spin relaxation times in solids are strates that masing and microwave amplification are feasible under usually extremely short (approximately nanoseconds8) at room readily accessible conditions (cavity Q-factor B50,000, diamond temperature due to rapid phonon scattering7. The spin relaxation size B3 3 0.5 mm3 and NV centre concentration B2p.p.m.), induced by phonon scattering can be largely suppressed in with the 532-nm optical pump threshold B3W for microwave light-element materials (such as organic materials) where the amplifying and B4 W for masing. For a pump power o10 W, it is spin–orbit coupling is weak. Actually, the only room-temperature feasible to achieve masing with an output power of 410 nW, a solid-state maser demonstrated so far is based on a pentacene- coherence time of approximately minutes, an mHz linewidth and a doped p-terphenyl molecular crystal where the spin lifetime can sensitivity of o10 pT Hz 1/2 for magnetometry application. As a reach 135 ms at room temperature9. However, the active spins in room-temperature microwave amplifier, the noise temperature is pentacene-doped p-terphenyl are the intermediate metastable as low as B0.3 K under a few-watt pump. A room-temperature states instead of the ground states. Such an energy level structure diamond maser may facilitate a broad range of microwave greatly reduces the optical pumping efficiency and the pentacene technologies. maser requires high pump laser power. In addition, the organic material of p-terphenyl molecular crystal is unstable under high- Results power pump and at high temperature. The pentacence maser System and model works only at the pulsed mode with a quite low repetition rate . We consider an ensemble of NV centre spins B 9 in diamond resonantly coupled to a high-quality microwave ( 1 Hz) . Another candidate system under consideration is the 9,31 silicon vacancy (V ) centre in silicon carbide10. Very recently, it sapphire dielectric resonator cavity (Fig. 1a; Supplementary Si Note 1). This type of cavity has been experimentally has been clarified that the VSi centre has a spin-3/2 ground state, 5 10,11 demonstrated to have a Q-factor of 410 at room which allows population inversion by an optical pump . The 9,31 spin lifetime is quite long (B0.5 ms) at room temperature10,11. temperature . Note that many other types of microwave resonators10,32 may also be considered for implementing the Stimulated microwave emission from VSi centres has been 10 proposal in this paper. The spin sublevels jmsi (ms ¼ 0or observed at room temperature . Masing from the silicon ± carbide spins, however, is still elusive. The challenges include ms ¼ 1) of the NV triplet ground state have a zero-field j i j i the complexity of the defects in the compound material and the splitting about 2.87 GHz between 0 and 1 (ref. 12) (Fig. 1b). difficulty of engineering the V centres. The NV centres can be optically pumped to the state j0i (ref. 12). Si 4 Nitrogen-vacancy (NV) centre spins in diamond12 have been A moderate external magnetic field ( 1,000 G) splits the states extensively studied for quantum information processing12 and j1i and shifts the j1i state to below j0i so that the spins can quantum sensing13–15. This is due to their long coherence time at be inverted by an optical pump (Fig. 1b,c). The transition o room temperature and high efficiencies of initialization by optical frequency S between the spin ground state jgij1i and the j iji pumping and readout via photon detection. In particular, spin exited state e 0 is tuned resonant with the microwave o coupling between ensemble NV centre electron spins cavity frequency c. and superconducting resonators has been demonstrated for The maser is driven by coupling between the cavity mode and quantum information storage and retrieval in hybrid quantum the spins.P The Hamiltonian of the coupled spin–cavity system is 16,17 N þ y systems at cryogenic temperature . Enhanced quantum HI ¼ j¼1 gjðâ^sj þ a^ ^sj Þ, where â annihilates a microwave þ coherence of NV centre ensembles has been observed in the cavity photon, ^sj jeijjhgj is the raising operator of the j-th strong coupling regime at low temperature18. It should be noted ^ ^þ y that the superb spin coherence features of NV centres persist at spin, sj ¼ðsj Þ , and gj is the coupling constant. Without room temperature and even at temperatures up to at least 600 K changing the essential results, we assume that the spin–photon (ref. 19), and the requirement of low temperatures in refs 16–18 is coupling is uniform, that is, gj ¼ g and write the HamiltonianP as mostly for the resonators used. On the basis of these works, we ¼ ð^^ þ ^y^ Þ ^ N ^ HI g aS þ a S with the collective operators S j¼P1 sj , here propose a new class of quantum technologies based on NV which satisfy the commutation relation ½^S ; ^S ¼ N centres in diamond, namely, room-temperature solid-state masers þ j¼1 ^ and microwave amplifiers. ðjeijjhejjgijjhgjÞ Sz. When masing occurs, the spin polariza- NV centres in diamond possess all of the features needed for a tion (or population inversion) Sz h^Szi is a macroscopic room-temperature solid-state maser. NV centre spins have the number ½ OðNÞ, while the fluctuation d^Sz ^Sz Sz longest known lifetime (B5 ms) at room temperature20–22 pffiffiffiffi OðN1=2Þ is much smaller. Therefore, ^by ^S = S can be among all of the solid-state spins (B50 times longer than z interpreted as the creation operator of a collective mode with B0.1 ms in pentacene-doped p-terphenyl9 and B10 times longer ^ ^y than B0.5 ms in silicon carbide10,11). The spin is a triplet (spin-1) ½b; b ffi1. The creation operator generates coherent super- in the ground state and it can be optically pumped rapidly to the position states in the spin ensemble. For example, from a fully B 6 1 ms ¼ 0 ground state, with a pump rate as high as 10 s (refs polarized spin state, the collective mode state excited by one B 3 5 1 9 ^ 23–26) (versus 10 to 10 s in organic materials and cavity photon is a quantized spin wave bjgi1jgi2 ÁÁÁjgiN ¼

2 NATURE COMMUNICATIONS | 6:8251 | DOI: 10.1038/ncomms9251 | www.nature.com/naturecommunications & 2015 Macmillan Publishers Limited. All rights reserved. NATURE COMMUNICATIONS | DOI: 10.1038/ncomms9251 ARTICLE a amplification factor of the contribution of pump rate w to the C B collective mode decay rate kS (see Supplementary Note 1), due to C V N multiple pathways (Fig. 1c) of the pump process. The total decay Ã rate of the collective mode is thus kS ¼ geg þ 2=T2 þ qw. The C coupled quantum dynamics of the collective modes and the 33 c photons is described by the quantum Langevin equations for Pump ^ MW the photon and spin collectiveP mode operators â and S, the spin ^ N ^ populations Ne=g j¼1 jie=g jjhje=g , and polarization Sz (see +1 Methods and Supplementary Note 2). 3E 0 − For a specific system, we consider a single crystal bulk diamond 50 μs 1 − 3 1 of volume VNV ¼ 3 3 0.5 mm with a natural abundance (1.1%) of 13C nuclear spins, a P1 centre concentration of about 1 b A1 −1 20 p.p.m. and an NV centre concentration of 2 p.p.m. (for N-to- 70 μs 1 ns−1 1 NV conversion efficiency 10%). Such parameters are extracted ⏐+1〉 E from the diamond used in ref. 17. The ensemble spin decoherence Ã ⏐0〉 ⏐e〉 +1 time is T2 ¼ 0.4 ms (ref. 17) (see Supplementary Note 1).

Energy − Considering the four orientations of NV centres and three = 3A 1 μs 1 − 〉 〉 S c 2 0 14 ⏐ 1 ⏐g nuclear spin states of N, the number of NV centres coupled to S 14 −1 the cavity mode is estimated to be N ¼ rNVVNV/12 ¼ 1.32 10 . B E The external magnetic field 2,100 G results in oS/2p 3 GHz. The Figure 1 | Scheme of room-temperature diamond maser. (a) The diamond microwave dielectric resonator has its TE01d mode frequency maser system. A diamond sample is fixed inside a high-quality sapphire resonant with the spin collective mode, that is, oc ¼ oS. The microwave dielectric resonator loaded in a coaxial cylindrical cavity. cavity system (Fig. 1a) is composed of a cylindrical sapphire dielectric resonator (with radius r ¼ 15 mm and height h ¼ 16 Microwave signal outputs from the loop coupling to the TE01d mode magnetic field (dashed red circles). The Halbach magnet array (the outer mm), and a coaxial cylindrical cavity (with radius R ¼ 40 mm and cylindrical wall) provides a uniform external magnetic field along the NV height H ¼ 40 mm), placed inside a Halbach magnet array with a axis, which is set perpendicular to the cavity axial direction. The NV centres 50 (80)-mm inner (outer) radius, which provides the uniform magnetic field with inhomogeneity o0.01 G across the diamond are pumped by a 532-nm light (green arrow). (b) The energy levels of an 28,29 NV centre spin as functions of a magnetic field B. The zero-field splitting at size . The coupling between the microwave photons and the NV centre spins is calculated to be g/2pE0.02 Hz for the B ¼ 0 is about 2.87 GHz. The magnetic field is set such that the transition 9,34 E 3 frequency o between the states ji 1 ðÞjig andji 0 ðÞjie is resonant with effective cavity mode volume Veff 3cm . At room S temperature (T ¼ 300 K), the phonon scattering dominates the the cavity mode frequency oc.(c) The pump scheme. After the optical E 1 3 spin relaxation and geg 200 s (refs 20–22). The number of excitation by a 532-nm light (green arrows), the excited-state E can E 3 thermal photons inside the cavity is nth 2,100 at room directly return to the ground-state A2 via spin-conserving photon emission 1 temperature. at a rate of B70 ms , but the excited states jims ¼1 can also decay to 1 1 the singlet state A1 via inter-system crossing at a rate of B50 ms and quickly decay to the metastable state 1E, then relax back to the three Masing conditions. The quantum Langevin equations can be different ground states at a rate of B1 ms 1 in each pathway. solved at steady-state masing. When masing occurs, the quantum operators can be approximated as their expectation values, that is, ^ ^ ^ pffiffiffiffiffiffiffiffiffi P S S, â a, Ne=g Ne=g and Sz Sz. By dropping the N 1=N j¼1 jgi1 ÁÁÁjgij 1jeijjgij þ 1 ÁÁÁjgiN , which acts as a small quantum fluctuations, we reduce the quantum Langevin boson. In the masing state, both the photons and the spin equations to classical equations for the expectation values collective modes, coherently coupled to each other, have macro- (see Methods and Supplementary Note 2). Under the resonant scopic amplitudes. With the excitation number of the coherent condition (oc ¼ oS), the steady-state solution is ÀÁ ^y ^ ^ 2 spin collective mode nS hb bi¼hS þ S i=Sz OðNÞ, the Sz ¼ kSkc 4g ; ð1Þ spins are in a macroscopic quantum superposition state, which sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is maintained by the masing process. A prerequisite of masing is the spin population inversion. The w geg w þ geg S ¼ i Sz N Sz ; ð2Þ optical pumping rate w can be tuned by varying the pump light 2kS 2kS B 6 1 intensity, up to 10 s (refs 23–26). The cavity mode has a rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi decay rate determined by the cavity Q-factor, kc ¼ oc/Q, due to w geg w þ geg photon leakage and coupling to the input/output channels. The a ¼ N Sz: ð3Þ decay of the spin collective mode is caused by various 2kc 2kc 2 mechanisms. First, the spin relaxation (T1 process caused by When the spin polarization SzBO(N), we have S þ S BO(N ). 2 phonon scattering and resonant interaction between spins) FromÀÁ equation (3), the number of intracavity photons jja ¼ contributes a decay rate geg ¼ 1/T1. Second, the individual spins 2 2 2 4g kc S þ S / N and consequently the output power Pout ¼ experience local field fluctuations due to interaction with nuclear 2 2 2 ‘ oc Á kcjja / N , both scaling with the number of spins by N . spins, coupling to other NV and nitrogen (P1) centre electron 2 spins, and fluctuation of the zero-field splitting17. Such local field The requirement that photon number jja 40 leads to the fluctuations induce random phases to individual spins, making masing condition Ã 2 the bright collective mode decay to other modes at a rate 2=T2 , 4g w geg Ã kco N: ð4Þ where T2 is the dephasing time of the spin ensemble. Finally, the k w þ g optical pump of the NV centres, being incoherent, also induces S eg the decay of the collective mode. The collective mode decay rate On one hand, the pump rate needs to be greater than the spin E induced by the incoherent pump is qw, where q 16 is an relaxation rate for population inversion (w4geg). On the other

NATURE COMMUNICATIONS | 6:8251 | DOI: 10.1038/ncomms9251 | www.nature.com/naturecommunications 3 & 2015 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms9251 hand, the cavity Q-factor has to be above a threshold the correlation functions (see Methods and Supplementary w þ g k o Note 2), which apply to both masing and incoherent emission. Q ¼ eg S c ð5Þ C w g 4Ng2 The spin polarization Sz, the microwave output power Pout ¼ eg y ‘ oc Á kchâ âi and the spin–spin correlation h^S þ ^S i (shown in to have a sufficient number of photons for sustaining the Fig. 2a–c) are consistent with the results obtained from macroscopic quantum coherence. A stronger spin–photon equation (2) when the pump rate and the cavity Q-factor are coupling (g), a smaller spin collective mode decay rate (kS)ora above the masing threshold (white curve in the Fig. 2). It is clearly larger number of spins (N) can reduce this threshold cavity seen that the output power increases markedly when the pump Q-factor (see Supplementary Note 1 for more discussions). The rate is above the spin relaxation rate (population inverted) and cavity Q-factor threshold is equivalent to the requirement that when the cavity Q-factor is above the masing threshold (Q4QC) the spin collective mode decay rate kS should be kept below the (Fig. 2b). The fact that ^S ^S N unambiguously evidences 2 þ e maximal collective emission rate of photons 4Ng /kc. Otherwise the phase correlation among the large ensemble spins established overpumping would fully polarize the spins, making the - - by cavity photons in the masing region (Fig. 2c). The pump spin–spin correlation vanish (Sz N and S 0). condition optimal for spin–spin correlation is determined The emergence of macroscopic quantum coherence is by maximizing h^S ^S i¼ðS =2k Þ½ðw g ÞN ðw þ g ÞS : evidenced by macroscopic values of the spin–spin correlation, þ z S eg eg z Under the conditionÀÁ that the pump is well above the threshold, the photon number and the spin collective mode amplitude, and Ã the long coherence time in the masing region (Fig. 2b–d). We w geg; 1 qT2 , the spin–spin correlation reaches its 2 calculated these quantities by using the higher-order equations of maximum value h^S þ ^S iN =ðÞ8q at the optimal pump rate

abS 0 –2 z 6 10 6 10 N 10 Pout (W) 1.0 10–5 0.5 10–8 10–6 10–4 5 0.1 5

Q 10 Q 10 10–10 0.0 0.5 –15 0.9 –0.5 10 104 104 –1.0 102 104 106 102 104 106 w (s−1) w (s−1)

c ˆ ˆ d S+S− 9 6 6 10 10 10 Tcoh(s) 10–6 N 2 –5 10 10–3 6 –4 10 10 5 5 5 10

Q 10 Q 10 10–3 10–6 3 10 100 0 4 4 10 10–9 10 10–5

102 104 106 102 104 106 w (s−1) w (s−1) e f B√ T √ − 106 −1/2 106 (K Hz 1/2) (T Hz ) 10–10 –3 10–6 10 10–9 10–11 10–12 5 5 Q 10 Q 10 –6 10–8 10 10–9 10–7 10–9 104 104 10–12 102 104 106 102 104 106 w (s−1) w (s−1)

Figure 2 | Room-temperature masing of NV centre spins in diamond. Contour plots of (a) the spin polarization Sz,(b) the output power Pout, ^ ^ (c) the collective spin–spin correlation hS þ S i,(d) the macroscopic quantum coherence time Tcoh,(e) the sensitivity to external magnetic ﬁeld and (f) the sensitivity to temperature, as functions of the pump rate w and the cavity Q-factor. The masing threshold is indicated in the ﬁgures by the white curves. The blue dashed curve in d shows the pump rate optimal for maximum coherence time. The parameters are such that oc/2p ¼ oS/2p ¼ 3 GHz, Ã 14 1 g/2p ¼ 0.02 Hz, T2 ¼ 0.4 ms, N ¼ 1.32 10 and geg ¼ 200 s at T ¼ 300 K.

4 NATURE COMMUNICATIONS | 6:8251 | DOI: 10.1038/ncomms9251 | www.nature.com/naturecommunications & 2015 Macmillan Publishers Limited. All rights reserved. NATURE COMMUNICATIONS | DOI: 10.1038/ncomms9251 ARTICLE

maxcorr 2 E maxcorr 2 wopt 2Ng =ðÞqkc , where the spin polarization Sz N/2 and spin–spin correlation, that is, wopt 2Ng=ÀÁðÞqkc , and the ‘ o Á 2 2=ðÞk opt 2 2 3 the maser power Pout c N g 2q c (see Supplementary optimal coherence time reaches Tcoh 2N g qnthkc , which Note 3). scales with the spin number and the cavity Q-factor according to The maser linewidth is determined by the correlation of the opt 2 3 Tcoh / N Q (see Supplementary Note 3). Note that the quantum phase fluctuations of the photons or equivalently by that of the coherence sustained by active masing (with a pump) has a much spin collective modes. The coherence time is obtained (see longer lifetime than the spin coherence protected by passive Supplementary Note 4) as 18 ÀÁ coupling to the cavity (without a pump) . This is due to the 1 1 superradiant emission of photons from the spin collective modes Tcoh ¼ 4 kc þ kS ðÞnc þ nS nincoh; ð6Þ of the NV centres in the bulk diamond and the concomitant large y ^ ^ number of photons in the cavity. where nc ¼hâ âi is the photon number, nS ¼hS þ S i Sz is the spin collective mode number and nincoh ¼ nth þ Ne/Sz includes the thermal photon number (nth) andP the incoherent spin collective Diamond microwave amplifier. The coupled spin–cavity system ^ ^ N þ mode number (hS þ S i=Sz ¼ j¼1 h^sj ^sj i=Sz ¼ Ne=Sz if the can be configured as a room-temperature microwave amplifier correlation between different spins is forced to be zero). The when the spin population is inverted (Sz40) but when the cavity physical meaning of equation (6) is the following: the coherent Q-factor is below the masing threshold (QoQC). For the readily 4 excitations (photons and spin collectiveÀÁ excitations) have the accessible parameter Q ¼ 10 , the noise temperature is as low as same phase within the total lifetime k 1 þ k 1 of the photons B0.2 K (versus B1 K for the state-of-the-art ruby amplifier c S 3 and the spin collective modes; beyond the total lifetime, each working at liquid-helium temperature ). To study the incoherent excitation induces a random phase BO(p); the total amplification, we calculate the spin inversion, the microwave random phase is shared by all the coherent excitations. Thus the output power gain and the noise temperature with a weak random phase of a single photon or spin collectiveÀÁ excitation microwave resonant input (see Methods and Supplementary 1 1 Note 5). As shown in Fig. 3, the system linearly amplifies the accumulated during the total lifetime kc þ kS is O½p Á n =ðn þ n Þ . The coherence time is greatly microwave signal. The gain is about 6–10 dB under a 6–10-W incoh c S pump with noise temperature 340–280 mK. At high pump enhanced in the masing region (Fig. 2d). The spin collective 4 1 6 1 around 136 W (w ¼ 10 s ), the gain is as high as 20 dB with mode decay rate of the NV centres kS45 10 s , while for a good microwave cavity (QB105), the photon decay rate noise temperature as low as 200 mK. The low noise temperatures B 4 1 indicate the single-photon noise level. kc 6p 10 s , thus the photon number nc ¼ nSkS/kc is much greater than the spin collective mode number, and the macroscopic quantum coherence is mainly maintained by the Discussion photons in the cavity. For readily accessible cavities in The ultralong coherence time of the maser is useful for metrology. experiments, the fractional frequency instability of a room- The collective excitation of a large number of spins (B1014) and 12 1/2 temperature diamond maser is B10 t as listed in cavity photons enhances the sensitivity. The sensitivity to a slow- 4 pffiffiffi Tables 1 and 2 for a cavity with Q ¼ 5 10 . For comparison, varying magneticpffiffiffiffiffiffiffiffiffiffiffiffi field noise (with frequency rkS/2) dB t ¼ the fractional frequency instability of a hydrogen maser is g 1ðÞ1 þ k =k 2T 1 for measurement time t (see 13 1/2 35 15 1/2 NV S c coh B10 t at room temperature or B10 t at Supplementary Note 6), where g /2p ¼ 2.8 MHz G 1 is the 36 NV cryogenic temperature , and the state-of-the-art ytterbium NV centre gyromagnetic ratio. The magnetic field sensitivity is 37 atomic clock has a fractional frequency instability of estimated to be in the order of 10–1 pT Hz 1/2 at room 16 1/2 B10 t . The optimal pump condition for a long temperature (Tables 1 and 2). The temperature noise would coherence time can be obtained from equation (6). In the induce cavity frequency fluctuation via thermal expansion and 2=k Ã good-cavity or large ensemble limit where 2Ng c 1 T2 and dielectricpffiffiffi constant variation.pffiffiffiffiffiffiffiffiffiffiffiffi The temperature sensitivity, E 1 1 at room temperature where nincoh nth, the optimal pump rate dT t ¼ g0 ðÞ1 þ kc=kS 2Tcoh (see Supplementary Note 6), for maximum coherence time is close to that for maximum is estimated to be in the order of 100–10 nK Hz 1/2 at room

Table 1 | Performance of room-temperature diamond maser under low pump. ffiffiffi ffiffiffi ffiffiffi 1 p 1/2 p 1/2 p 12 Ppump (W) w (s ) Pout (nW) Tcoh (s) DvST (mHz) dB s (pT Hz ) dT s (nK Hz ) ry (s) s (10 ) 6.0 (1.0) 440.1 (440.1) 12.5 (1.0) 98 (5) 3.2 (61.1) 11.6 (34.3) 148.2 (666.2) 8.1 (36.6) 8.0 (1.4) 586.9 (586.9) 27.4 (8.9) 215 (49) 1.5 (6.5) 7.8 (11.2) 100.1 (218.1) 5.5 (12.0) 10.0 (1.7) 733.6 (733.6) 42.2 (16.9) 332 (92) 1.0 (3.5) 6.3 (8.2) 80.6 (158.6) 4.4 (8.7) pffiffiffi pffiffiffi Ppump, the pump power; w, the pump rate; Pout, the output power; Tcoh, the coherence time; DnST, the maser linewidth; dB t, the magnetic field sensitivity; dT t, the temperature sensitivity; sy ðtÞ,the 4 fractional frequency instability, where t is the measurement time. The cavity Q ¼ 5 10 . The numbers in the brackets in the first column show the absorbed power Pabsorb, and the numbers in the brackets in other columns are values with a 0.2 K temperature fluctuation taken into account.

Table 2 | Performance of room-temperature diamond maser under high pump. ffiffiffi ffiffiffi ffiffiffi 1 p 1/2 p 1/2 p 12 Ppump (W) w(s ) Pout (lW) Tcoh (s) DvST (lHz) dB s (pT Hz ) dT s (nK Hz ) ry (s) s(10 ) 13.6 (2.4) 103 (103) 0.07 (0.007) 543 (27) 585.8 (11,891.6) 4.9 (10.7) 62.9 (308.6) 3.5 (17.0) 136.3 (23.7) 104 (104) 1.0 (0.2) 7,609 (836) 41.8 (380.6) 1.4 (2.0) 16.8 (55.0) 0.9 (3.0) 1,363.2 (236.7) 105 (105) 9.2 (1.5) 69,580 (5,312) 4.6 (59.9) 0.6 (1.0) 5.5 (21.2) 0.3 (1.2) pffiffiffi pffiffiffi Ppump, the pump power; w, the pump rate; Pout, the output power; Tcoh, the coherence time; DnST, the maser linewidth; dB t, the magnetic field sensitivity; dT t, the temperature sensitivity; sy ðtÞ,the 4 fractional frequency instability, where t represents the measurement time. The cavity Q ¼ 5 10 . The numbers in the brackets in the first column show the absorbed power Pabsorb, and the numbers in the brackets in other columns are values with a 0.5 K temperature fluctuation taken into account.

NATURE COMMUNICATIONS | 6:8251 | DOI: 10.1038/ncomms9251 | www.nature.com/naturecommunications 5 & 2015 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms9251

a b 1.0 Sz 1010 6 6 10 N 10 GdB (dB) 0.5 1.0 105 100 0.0 Masing Masing 0 –0.5 0.5 10 50 5 Ampl. 5 Ampl.

10 Q 10 Absorb. Absorb. –1.0 10–5 2 6 0.0 2 6 0 Absorbing 10 10 10 10 Masing –0.5 –50 50 dB 104 104 Amplifying 10 dB –1.0 –100 102 104 106 102 104 106 −1 w (s ) w (s−1)

c d 4 100 Q = 2 × 10 10–6 106 w = Tn (K) 103 s−1 101 104 s−1 Masing 10–9 105 s−1 5 Ampl. (W)

QQ 10 Absorb. 4 –1 out = × 10 Q 1 10 0 P –12 102 106 10 10 w = 103 s−1 4 −1 104 144 mK 10 s 10–15 105 s−1 102 104 106 10–18 10–15 10–12 10–9 −1 w (s ) Pin (W)

Figure 3 | Room-temperature diamond microwave amplifier. Contour plots of (a) the spin polarization Sz,(b) the power gain G and (c) the noise 1 temperature Tn, as functions of the pump rate w and the cavity Q-factor for resonant input microwave power Pin ¼ 1fW.Forwogeg ¼ 200 s , the system is 1 in the absorbing region (population not inverted). For w4geg ¼ 200 s and the cavity Q-factor below/above masing threshold (the dashed curves), the system operates in the amplifying/masing mode. The insets show dependence on the pump rate for a fixed cavity Q ¼ 5 104 (the absorbing, amplifying and masing regions marked as the grey, green and white, respectively). Amplifying is unstable in the masing region (see Supplementary Fig. 1). (d) Output power as a function of input microwave power for various pump rates (w ¼ 103,104 or 105 s 1), with the cavity Q-factor Q ¼ 104 and 2 104 corresponding to the amplifying and masing regions, respectively. In the amplifying regime (Q ¼ 104), the amplification is linear in a large range of input power (see Supplementary Fig. 2 for more information). The parameters are the same as in Fig. 2.

E temperature (Tables 1 and 2), where we have taken g0 (a þ b/ 1 a 1.0 b 2)oc ¼ 165 kHz K with a and b being the temperature 10–6 coefﬁcients of thermal expansion and permittivity for sapphire, respectively (see Supplementary Note 1). For a higher cavity 10–9 Q-factor, the thermometry sensitivity is enhanced while the /N (W)

z 0.5 magnetometry sensitivity is reduced (see Fig. 2e,f) due to the

− out S w =103 s 1 P frequency dragging effect (the steady-state masing frequency is a − –12 w =104 s 1 10 weighted average of the spin and cavity frequency38, − w =105 s 1 o ¼ (kcoS þ kSoc)/(kc þ kS), see Supplementary Note 2). The 0.0 10–15 sensitivities to the magnetic field and the temperature noises set 012 012 ΔT (K) ΔT (K) the requirements on stability of the set-up for maintaining the long coherence time of the maser. c 106 d 106 The pump power Ppump ¼ ‘ opðÞS=s ðÞ4w is proportional to the pump light frequency (op), the pump rate (w) and the area of 103 103 the pump light spot (S) divided by the NV centre absorption cross-section (s) (see Supplementary Note 1). The threshold (s) (Hz) 100 100 pump power is low because the spin relaxation time is long in ST coh 4 T Δ diamond. For a readily accessible cavity with Q ¼ 5 10 ,we 10–3 10–3 show diamond maser performance in Tables 1 and 2. The threshold pump power for microwave amplifying (population 10–6 10–6 inversion) is estimated to be 2.7 W, above which the net photon 012 0 1 2 emission into the cavity amplifies the signal. The threshold pump ΔT (K) ΔT (K) power for masing is estimated to be 4.3 W for Q ¼ 5 104 cavity Figure 4 | Temperature fluctuation effects on room-temperature masing. (see Supplementary Note 1 for pump thresholds), above which (a) The spin polarization, (b) the output power, (c) the linewidth and the emitted photons show collective coherence and the photon 2 (d) the coherence time as functions of temperature fluctuation for cavity number scales with the spin number (N)byncpN . Q ¼ 5 104, for various pump rates (w ¼ 103,104 or 105 s 1). The sharp The optical pump may heat the system, inducing both changes indicate the transition between amplifying and masing. The temperature increase and fluctuation. The frequency shifts of parameters are the same as in Fig. 2. the spins and the cavity due to temperature increase are not an

6 NATURE COMMUNICATIONS | 6:8251 | DOI: 10.1038/ncomms9251 | www.nature.com/naturecommunications & 2015 Macmillan Publishers Limited. All rights reserved. NATURE COMMUNICATIONS | DOI: 10.1038/ncomms9251 ARTICLE issue since once the steady-state is reached, the spin transition The coherence time and the linewidth are calculated using the spectrum of the frequency can be tuned to resonance with the cavity mode by phase fluctuations. The Langevin equations are linearized for the fluctuations, which are much smaller than the expectation values at steady state. The linearized tuning the magnetic field. equations are It has been demonstrated that NV centre spins still have good ^ ÀÁ ddNe ^ ^ y Ã ^ ^ ^ coherence properties at least up to 600 K (ref. 19). According to ¼þwdNg gegdNe þ ig S dâ S þ da^ þ ig a dS adS þ þ Fe; ref. 19, the spin polarization, the pump rate and the TÃ dt 2 ð14Þ decoherence time are only slightly changed at a temperature as high as 650 K. The longitudinal spin relaxation time T1 is reduced ddN^ ÀÁ g ¼wdN^ þ g dN^ ig S dây S da^ ig aÃd^S ad^S þ F^ ; to 0.34 ms at 600 K, more than 10 times shorter than that at room dt g eg e þ þ g temperature, and the pump threshold for microwave amplifying ð15Þ (w41/T1) is fulfilled when the pump power is 440 W. Also, the dd^S k ÀÁ contribution of the longitudinalÀÁ spin relaxation to spin collective ¼ S d^S þ igS da^ þ iga dN^ dN^ þ F^ ; ð16Þ Ã dt 2 z e g S mode decay is negligible 1=T1 qw; 1 T2 . The temperature fluctuation leads to transition frequency shifts ddâ k ¼ c da^ igd^S þ F^ : ð17Þ of the spins (via lattice expansion) and the cavity mode (via dt 2 c dielectric constant variation and mode volume expansion), with By Fourier transform of these equations, the spectrum of the phase noise can be 1 1 1 (2p) DoS/DTE 74 kHz K (ref. 39) and (2p) Doc/ calculated and hence the maser coherence time and linewidth are determined. DTE 165 kHz K 1 (ref. 40) (see Supplementary Note 1). The To investigate the correlations in both the masing and the incoherent emission masing conditions considered in this paper correspond to the regimes, we derive the equations for the correlation functions and take the expectation values of the relevant operators. That leads to photon leakage rate being much slower than the decay rate of ^ DE the spin collective mode ðÞk k . Thus, the effects of d Ne y c S ¼þw N^g g Nê þ ig a^ ^S ^S þ â ; ð18Þ temperature fluctuation on the maser result mainly from the dt eg DE cavity mode frequency fluctuation. Figure 4 shows the spin d N^ g ¼w N^ þ g N^ ig ây^S ^S â ; ð19Þ polarization, the output power, the masing linewidth and the dt g eg e þ coherence time as functions of the temperature fluctuation DT for DE 4 3 4 y^ DE DE cavity Q ¼ 5 10 , and various pump rates (w ¼ 10 ,10, d â S k þ k 1 5 1 ¼ S c a^y^S þ ig 1 ^S ^S þ N^ þ âyâ ^S ; 10 s ). Note that the threshold cavity Q-factor is the lowest dt 2 N þ e z 4 1 4 1 near w ¼ 10 s (Fig. 2), thus the rate near w ¼ 10 s is the ð20Þ most robust to temperature fluctuation. The masing condition ^ ^ DE is still fulfilled for up to 1.0 K temperature fluctuation at d S þ S ^ ^ ^ y^ ^ 4 1 ¼kS S þ S ig Sz â S S þ a^ ; ð21Þ w ¼ 10 s . However, temperature fluctuation does affect the dt maser performance. In Table 1 (2), we compare the maser DE performance with and without the 0.2 K (0.5 K) temperature d âya^ DE DE ¼k âyâ ig ây^S ^S a^ þ k n : ð22Þ fluctuation for Q ¼ 5 104 cavity at low (high) pump power. For dt c þ c th a larger number of spins, a larger temperature fluctuation can be Here, to make the equations close, we have used the approximation ^y^^ ^y^ ^ ^y^ ^ ^ ^y^ ^ ^ ^ ^ ^ ^ tolerated (DTmaxpN) due to the reduced threshold cavity Q- ha aSziha aihSzi, ha SzS i Sz ha S i and hS þ SzaihSzihS þ ai, factor. neglecting the higher-order correlations, which is well justified for Gaussian fluctuations. Methods Microwave amplifier and noise temperature. To investigate the microwave Maser equations The theoretical study is based on the standard Langevin . amplifier, we solve the mean-field equations with a steady-state input s e ioint as equations33 in 0 ¼ wN g N þ igðÞ aÃS S a ; ð23Þ ^ g eg e þ dNe ^ ^ y^ ^ ^ ¼þwNg gegNe þ ig a^ S S þ a^ þ Fe; ð7Þ dt kS 0 ¼ iðÞo o S S þ igS a; ð24Þ in S 2 z ^ dNg y ¼wN^ þ g N^ ig a^ ^S ^S a^ þ F^ ; ð8Þ kc pffiffiffiffiffiffi dt g eg e þ g 0 ¼ iðÞo o a a igS þ k s ; ð25Þ in c 2 ex in ffiffiffiffiffiffi d^S k ÀÁ p ^ S ^ ^ ^ ^ sout ¼ sin kexa; ð26Þ ¼ioSS S þ ig Ne Ng a^ þ FS; ð9Þ dt 2 2 2 from which the power gain G ¼ jjsout jjsin is obtained. A power gain G 1is ^ k possible under the resonant condition oin ¼ oc ¼ oS, but it will be reduced to O(1) da c ^ ^ 7 ¼ioca^ â igS þ Fc; ð10Þ at off-resonance, that is, oin oS;c kS;c\1. The intrinsic noise temperature of dt 2 the diamond maser is given by ^ where Fc=S=e=g is the noise operator that causes the decay of the photons (c), the ÀÁ ‘ 1 LdB LdB Ne oc spin collective modes (S), the population in the excited state (e) or that in the Tn ¼ 1 G T þ 1 þ ; ð27Þ ground state (g). Note that the total spin number is written as an operator N^ to take GdB GdB Sz kB into account the fluctuation due to the population of the third spin state jiþ 1 and where ‘ is the Planck constant, kB is the Boltzmann constant, T is the environment other intermediate states. The population fluctuation, however, has no effect on the kc trt temperature, GdB ¼ 10log10 G and LdB ¼10 log e is the cavity power loss phase fluctuation of the maser (see Supplementary Note 4). 10 inÂÃ decibel duringÀÁffiffiffiffi the time of a microwave photon roundtrip time trt ¼ By replacing the operators with their expectation values, we obtain the mean- p E 2 R þ ðÞr r0 er 1 c (er 10 is the sapphire dielectric permittivity in the field equations for the maser at the steady state direction perpendicular to the c axis (resonator axis) of the sapphire crystal, R is the Ã radius of the cylindrical cavity and r (r ) is the external (internal) radius of the 0 ¼ wNg g Ne þ igðÞ a S S þ a ; ð11Þ 0 eg dielectric resonator). k 0 ¼io S S S þ igS a; ð12Þ S 2 z References 1. Gordon, J. P., Zeiger, H. J. & Townes, C. H. The maser—new type of microwave kc 0 ¼ioca a igS ; ð13Þ amplifier, frequency standard, and spectrometer. Phys. 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NATURE COMMUNICATIONS | 6:8251 | DOI: 10.1038/ncomms9251 | www.nature.com/naturecommunications 7 & 2015 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms9251

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Protecting a spin ensemble against decoherence in the strong- This work was supported by the Hong Kong Research Grants Council Project No. coupling regime of cavity QED. Nat. Phys. 10, 720–724 (2014). 14303014 and The Chinese University of Hong Kong Vice Chancellor’s One-off 19. Toyli, D. M. et al. Measurement and control of single nitrogen-vacancy center Discretionary Fund. spins above 600 K. Phys. Rev. X 2, 031001 (2012). 20. Redman, D. A., Brown, S., Sands, R. H. & Rand, S. C. Spin dynamics and Author contributions electronic states of N-V centers in diamond by EPR and four-wave-mixing R.-B.L. conceived the idea and supervised the project. R.-B.L. and L.J. formulated the spectroscopy. Phys. Rev. Lett. 67, 3420–3423 (1991). theory. R.-B.L., L.J., P.N., M.P., N.A., S.Y. and J.W. designed the physical system. L.J. 21. Takahashi, S., Hanson, R., van Tol, J., Sherwin, M. S. & Awschalom, D. D. carried out the calculations. R.-B.L. and L.J. wrote the paper. All authors commented on Quenching spin decoherence in diamond through spin bath polarization. Phys. the manuscript. Rev. Lett. 101, 047601 (2008). 22. Jarmola, A., Acosta, V. M., Jensen, K., Chemerisov, S. & Budker, D. Temperature- and magnetic-field-dependent longitudinal spin relaxation in Additional information nitrogen-vacancy ensembles in diamond. Phys. Rev. Lett. 108, 197601 (2012). Supplementary Information accompanies this paper at http://www.nature.com/ 23. Harrison, J., Sellars, M. J. & Manson, N. B. Optical spin polarization of the N-V naturecommunications centre in diamond. J. Lumin. 107, 245–248 (2004). 24. Manson, N. B., Harrison, J. P. & Sellars, M. J. Nitrogen-vacancy center in Competing financial interests: The authors declare no competing financial interests. diamond: model of the electronic structure and associated dynamics. Phys. Rev. B 74, 104303 (2006). Reprints and permission information is available online at http://npg.nature.com/ 25. Robledo, L., Bernien, H., van der Sar, T. & Hanson, R. Spin dynamics in the reprintsandpermissions/ optical cycle of single nitrogen-vacancy centres in diamond. N. J. Phys. 13, How to cite this article: Jin, L. et al. Proposal for a room-temperature diamond maser. 025013 (2011). Nat. Commun. 6:8251 doi: 10.1038/ncomms9251 (2015). 26. Tetienne, J. P. et al. Magnetic-field-dependent photodynamics of single NV defects in diamond: an application to qualitative all-optical magnetic imaging. N. J. Phys. 14, 103033 (2012). This work is licensed under a Creative Commons Attribution 4.0 27. Neu, E. et al. Electronic transitions of single silicon vacancy centers in the International License. The images or other third party material in this near-infrared spectral region. Phys. Rev. B 85, 245207 (2012). article are included in the article’s Creative Commons license, unless indicated otherwise 28. Raich, H. & Blu¨mler, P. Design and construction of a dipolar Halbach array in the credit line; if the material is not included under the Creative Commons license, with a homogeneous field from identical bar magnets: NMR Mandhalas. users will need to obtain permission from the license holder to reproduce the material. Concepts Magn. Reson. Part B Magn. Reson. Eng. 23B, 16–25 (2004). To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/