Diamonds for quantum nano sensing

Current Opinion in Solid State and Materials Science xxx (2016) xxx–xxx

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Diamonds for quantum nano sensing Taras Plakhotnik School of Mathematics and Physics, The University of Queensland, 4072 St Lucia, Australia

a r t i c l e

i n f o

Article history: Received 3 May 2016 Revised 18 July 2016 Accepted 7 August 2016 Available online xxxx Keywords: Diamond Nitrogen-vacancy center Electron paramagnetic resonance Relaxation processes Zero-phonon line

a b s t r a c t This paper reviews applications of diamonds for sensing of magnetic and electrical fields, pressure and temperature. Considerable attention is focussed on the interaction of spins with static and oscillating magnetic fields as well as applications of such fields to spin control. A particular focus is on the spin of nitrogen-vacancy centers. Electron-spin microwave resonances play a central role in the ultra-sensitive metrology on the nanoscale, but pure optical methods are also considered in this review. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction The purpose of this article is to review recent advances in the application of diamonds to quantum sensing on the nanoscale. To start, we need a few definitions clarifying the subject. First, the word quantum needs some explanation. It becomes quite fashionable to include quantum in a title or at least an abstract or the first paragraph of a paper to make it sound more appealing to readers outside the field. It is believed that the laws of quantum physics (quantum mechanics) form the foundation of all phenomena and therefore as a matter of fact quantum can be used almost in any context. But here we will call a measurement quantum if the sensitivity of the measurement is limited by quantum fluctuations. A similar motivation exists for the excessively wide usage of the prefix nano especially in combination with technology. Although any distance can be measured in nanometers, we will use the term nano when the characteristic dimensions describing the region of interest for sensing are smaller than approximately 50 nm. The border line is set by the limitations of the resolution in optical imaging due to diffraction. Application of diamonds for nano-sensing are almost exclusively based on the properties of so-called Nitrogen-Vacancy (NV) centers. Therefore we start with a brief review of the photophysics of these defects in diamond crystals. Such defects have been first described more than 40 years ago [1] but the boom of research began about 20 years later [2]. Then we will briefly review the interaction of electronic and nuclear magnetic moments with micro-wave (MW) radiation and static magnetic fields. The following sections will address applications of nano-diamonds to sensing

of magnetic and electrical fields, temperature and pressure. A number of relatively recent review papers are available on various applications of NV-centers [3,4]. These applications are at the proof-of-principle stage and therefore here we will focus more on these principles rather then on the diversity of experimental demonstrations. Examples of applications are included as illustrations in the appropriate sections and in the conclusion.

2. Nitrogen-vacancy centers and spin control Nitrogen-vacancy (NV) centers in diamond are defects of the crystal lattice (their geometry with C3v symmetry is shown in Fig. 1a) where one out of two neighboring carbon atoms is replaced with nitrogen while the other one is removed without substitution [5]. A single crystal of diamond can host NV centers with four different orientations of their axes. We will refer to such centers as different sets. These sets are not different from each other fundamentally, but the presence of more than one set allows us to define the 3D-vector of a magnetic field relative to these axes. Some of the sets may remain unoccupied in a crystal especially if the total number of centers is small. The wavefunction of the center is localized over a few lattice sites (the reason for calling them point defects) and this enables spatial resolution at the nano level. Among a large number of known defects in diamond [6] the NV centers are probably most famous. Even though some other crystal defects have been recently proposed to take over the leading role of the NV-centers, the position of the NV-centers has not been shaken. There are two types of such centers – electrically neutral and

http://dx.doi.org/10.1016/j.cossms.2016.08.001 1359-0286/Ó 2016 Elsevier Ltd. All rights reserved.

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T. Plakhotnik / Current Opinion in Solid State and Materials Science xxx (2016) xxx–xxx

(a)

(c)

(b)

(d)

Fig. 1. NV-center in diamond. Panel (a) shows center geometry and the standard orientation of the coordinate system. N and C atoms are in red and black. The vacancy is yellow. The x- and y-axes are shown as seen from the top of the z-axis. Panel (b) depicts the main features of the electronic states at room temperature. Due to double degeneracy of the molecular orbital of the excited spin-triplet state, there are 2 orthogonal transition dipole-moments. Spin-preserving luminescence and optical excitation are shown by the straight arrows downwards and upwards respectively. Panel (c) is an example of a luminescence spectrum (excitation at 590 nm). The ZPL is indicated by an asterisk. The wavy lines indicate non radiative transitions. Panel (d) presents an ODMR spectrum measured (dots) at about 5 mT external field. The 8 Lorentzian-shaped lines (shown by the solid curve) make 4 pairs (indicated by arrows) corresponding to 4 sets of NV centers in the crystal.

negatively charged. We will focus on NV
. Their unique properties which are most important for sensing applications are strong and resistant to bleaching photo luminescence with a zero-phonon line (ZPL) that is prominent even at room temperature, electron spintriplet nature of the electronic ground state (see Fig. 1b), and strong dependence of the luminescence intensity on the value of the spin projection on the symmetry axis of the center. An example of the luminescence spectrum of NV-centers is shown in Fig. 1c. A small feature at about 638 nm is the ZPL. The ZPL results from optical transitions which do not change the phonon population in the crystal. The rest of the spectrum is called phonon wing and is made by photons which were generated during the electronic energy relaxation accompanied by creation or annihilation of one or several phonons. Due to the dependence of the intersystem crossing on the magnetic quantum number m, (m ¼ 0; 1), the luminescence is strong when the center is in the m ¼ 0 state and is weak when the populated state is either of m ¼ 1. When the center is unperturbed, the energies of the m ¼ 1 states are nearly equal and about 2.87 GHz (the units of frequency are obtained when the energy is divided by the Planck constant and will be frequently used here instead of joule) higher than the energy of the m ¼ 0 state. At room temperature, the energy difference between the spin states is much smaller than the thermal energy kB T (the product of the Boltzmann constant and the temperature) and therefore the probabilities that a particular NV center occupies one of the three spin sub-levels are practically equal. But optical pumping in an about 100-nm broad band centered near 575 nm wavelength transfers most of the population to the m ¼ 0 level. This phenomenon is called spin polarization by optical pumping or simply optical spin-polarization. The reason for such polarization is again a much higher probability of the intersystem crossing from the triplet m ¼ 1 levels to singlet states sitting between the triplet ground and the first excited triplet states followed by the relaxation to the electronic ground m ¼ 0 state. Such a relaxation path

effectively converts population of the m ¼ 1 states into population of the m ¼ 0 state. Resonance MW-radiation at 2.87 GHz repopulates the m ¼ 1 states and this can be observed optically as a decrease in the luminescence intensity. Such a phenomenon enables optical detection of magnetic resonance, abbreviated as ODMR. An example of an ODMR spectrum is shown in Fig. 1d. The Hamiltonian of an NV-center in the presence of external magnetic and electrical fields reads

b ¼ Db Sy þ b S x Þ þ ce B b H S 2z þ E x ðb S 2x
b S 2y Þ þ E y ðb Sx b Sy b S

ð1Þ

In this equation, the coordinate system is chosen so that its z-direction coincides with the direction of the NV axis and the x-axis is positioned in the plane going through N, V and one of the nearest C (see Fig. 1a). Dimensionless (
h not included) spin 1 operators b S x;y;z are used for briefness. If B ¼ 0 and E x;y ¼ 0, the value of D  2:87 GHz determines the zero-field splitting between m ¼ 0 and the degenerate m ¼ 1 levels. The values of E x ¼ E ? cos /E and E y ¼ E ? sin /E characterize the stress in the crystal perpendicular to the symmetry axis of the NV center (/E is the polar angle in the xy plane). Such stress mixes the m ¼ 1 states into   E ji  p1ffiffi2 exp i/2E j1i  exp
i/ j
1i which have energy split of 2 2E ? . Note that even in nano-crystals where the stress is typically much larger than in a nearly perfect bulk diamond, the numerical value of E ? is about three orders of magnitude smaller than the value of D. The coordinate system can be rotated around z axis so S2
b S 2 Þ þ E y ðb Sx b Sy b that the expression E x ðb Sy þ b S x Þ is reduced to x

y

S 2x
b S 2y Þ. Such a choice simplifies the Hamiltonian (1) but may E ? ðb be inconvenient for treatment of, for example, spin-lattice coupling. It is apparent from Eq. (1) that the position of the EPR resonances will depend on the strength of the external magnetic filed B which will affect mainly the energies of the m ¼ 1 levels. This sensitivity can be used to measure the strength and direction of

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B. In a linear approximation, the first order perturbation theory the energy of the m ¼ 0 state will be unaffected and the energies of the m ¼ 1 states will become D  ce Bz , where Bz ¼ B cos hB (hB is the angle between the center axis and the direction of the magnetic field, the polar angle in the spherical coordinate system), and the electron gyromagnetic ratio ce  28:024 GHz/T. Note that in this approximations the energies of the mixed states ji are not affected by the magnetic field. Therefore if E ? – 0, the sensitivity to the field B is small but can be increased by adding a bias field Þ  E?. so that ce ðBz þ BðbiasÞ z External electrical fields mostly affect E while the parameter D depends primarily on the external hydrostatic pressure and the crystal temperature. These dependencies are used for sensing. It has been determined experimentally that at room temperatures the sensitivities of the D and E values to approximately hydrostatic pressure [7], crystal temperature [8], and electrical fields [9] can be characterized by the following non-zero partial derivatives

@D ¼ 14:6  1:4 MHz GPa
1 @P @D ¼
74  14 kHz K
1 sT ¼ @T @D sjj ¼ ¼ 0:35  2 Hz cm V
1 @Ez @E x @E y s? ¼ ¼ ¼ 17  3 Hz cm V
1 @Ex @Ey sP ¼

ð2Þ

Given the values of D; E, and the magnitude and direction of B, the energy levels of the Hamiltonian can be found exactly but this requires finding roots of a cubic polynomial which is quite messy in the general case and we will not explicitly obtain these roots here. For sensing applications, however, a reverse problem should be solved. That is the problem of finding the parameters D; E, and the three components of the vector B given two transition frequencies, f þ and f
. These frequencies correspond to two ESR frequencies (at small B this will be the transitions 0 ! jþi and 0 ! j
i). Obviously the five numbers cannot be determined using just two measured quantities. But if any three parameters are known or measured in one way or another, then the other two values can be found. Usually it is done by using the parameters in the Hamiltonian as fitting parameters which are tuned to maximize the likelihood of the experimental observations. The dependence of these parameters on the external fields, temperature and pressure is then used to infer E; B; T, and P. It is important to notice that the number of external factors affecting the resonances is also arger than the number of parameters in the Hamiltonian. Therefore some care is required in the analysis. For example, changes of the pressure and the temperature may interfere with each other. This point will be discussed in some details later. But in any case the required primary information is the transition frequencies f þ and f
. Despite the apparent simplicity of finding these frequencies from an ODMR spectrum similar to the one shown in Fig. 1, there are several serious obstacles affecting the accuracy of such determination. The intensity of the luminescence signal is obtained by counting detected photons and the number of counts is subject to Poisson statistics where the standard deviation of the counted number equals a square root of its value. The radiative lifetime srad of an NV-center and hence the maximum photon emission rate depend on the size of the crystal. For crystals much smaller than the radiated wavelength the luminescence lifetime (which is shorter or equal to the radiative time) varies from 25 ns [10] to 50 ns [11]. This is several times longer than 13 ns [12], the estimated radiative time in bulk crystals. The difference is due to the high refractive index of diamond (n  2:4) which affects the density of photon modes if the crystal is large and makes the emission

3

rate approximately proportional n [10]. All this means that in the case of an ideal detection scheme (all emitted photons are detected), the detection count rate will be at most 80 MHz (in bulk) but down to 20 MHz in nano crystals. In practice, the detection efficiency is up to 15% (NA = 0.9 for optics) in the case of nanocrystals [13] when the reflection and refraction by the surface of the diamond can be neglected. If the NV-center sits deep inside a bulk crystal, the collection efficiency is between 4% and 2% depending on the orientation of the NV axis (0.1% cited [39] but probably absorbs some other factors). These numbers take into account decrease in the numerical aperture of the collecting optics by 2.4 due to refraction and 20% reflection loss on the crystal boundary. The maximal detection rate will be further reduced by incomplete spin polarization. The residual occupation of the m – 0 levels is in avarage arround 20% of the total triplet population (but reported in the range of 4–60%) [5] and results in significant accumulation of the centers in the metastable non-radiating singlet state. Less than unity luminescence quantum yield (which is hard to measure) also reduces the emission rate. Summarising all of the above, R0 , the maximum count rate is expected at 200–500 kHz for a single center in nanodiamonds but a good quantitative agreement between the experimental values and the theoretical expectations has been never shown. The collection efficiency can be increased and the radiative lifetime decreased if the NV-center is strongly coupled to a micro-cavity [14,15]. A special grating etched on the surface of an approximately 300 nm thin diamond membrane drastically (approximately tenfold) lifts the photon collection efficiency [16]. This results in a detected count rate of about 2.7 MHz. All these methods are very promising for ultra-sensitive metrology but in current practice the collection efficiency is mainly defined by the numerical aperture of the microscope objective in use [13]. The accuracy of the frequency estimation is also affected by the contrast of the ODMR spectrum, that is the relative depth of the decrease of the luminescence intensity when the center initially at m ¼ 0 is brought in resonance with MW radiation. Smaller contrast makes the ODMR line less distinguishable from the random fluctuations of the base line. Fundamentally, the maximum contrast C m is limited by the ratio ðQ 0
Q  Þ=Q 0 , where Q 0 and Q  are the luminescence yields from m ¼ 0 and m ¼ 1 states. A relatively simple analysis reveals that the standard deviation of the random errors in the frequency estimates using direct ODMR measurements reads [17,18]

rf  0:8

C CðRtm Þ1=2

ð3Þ

where C is the contrast, R is the detected count rate at the base level (away from the MW resonance), C is the width of the ODMR line, and tm is the measurement time. Because the inverse proportionality of the measurement errors to the square root of the measurement time is typical for quantum limited measurement, a value of g  rt1=2 m is commonly used instead of the standard deviation. It is named the noise floor. The width of the ODMR line is ultimately determined by spinlattice coupling in the diamond crystals which leads to relaxation of the populations to their thermal equilibrium values. The relaxation time, called T 1 or longitudinal relaxation time, is temperature dependent and is relatively easy to measure. If more than two states are involved T 1 will depend on the state considered. For example, one can prepare an NV center in the state j0i by optical pumping and then observe recovery of the luminescence to its thermal equilibrium value [12,19]. The value of T 1 measured in [12] was 500 ms at 100 K. Conventional EPR methods have been used for systematic measurements of T 1 in a wide range of temperatures [19,20] and have showed temperature dependence of the longitudinal relaxation in agreement with a model where the spin

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is coupled to other NV-centers, a localized vibration [19,20], and lattice phonons. That is

1 A2 ¼ A1 C NV þ D=k T þ A3 T 5 ; T1 e b
1

ð4Þ

where A1 ¼ 0:8ð2Þ Hz=ppm; C NV is the concentration of NV centers in the crystal, A2 ¼ 2:1ð6Þ kHz; D ¼ 73ð4Þ meV (62:2  0:1 meV in [20]) is the local vibration energy, and A3 ¼ 2:2ð5Þ  10
11 Hz K
5. The value of T 1 reaches a plateau at temperatures below 50 K at a level inversely proportional to the concentrations of NV centers in the sample. At room temperature the second and third terms in Eq. (4) are approximately equal (120 Hz and 50 Hz respectively). In a super-pure crystal the results were in agreement with Eq. (4): T 1 ¼ 6 ms at 300 K and 600 ms at 77 K [21]. It is surprising that the values of T 1 above 200 K for crystals of different origin and purity are close to each other but are very different from a theoretical estimate of T 1 based on the material properties of diamond. Such an estimate gives a very long relaxation times, on the order of 100 s even at room temperature [22]. Perhaps the interactions of NVcenters with phonons were not properly accounted for in [22] as their model predicts T 7 temperature dependence for latticephonon contribution, different from T 5 as reported in [19,20]. The most current theory [23] is based on phenomenological parameters and a quantitative theoretical first-principle model is still to come. On a fundamental level, 1=T 2 defines the relaxation rate of offdiagonal elements of the density matrix. If the number of states is larger than two, there can be several values of T 2 . It can be derived that T 2 6 2T 1 by considering coupling of a small two-state system to a very large reservoir and using very general arguments [24]. The equality T 2 ¼ 2T 1 holds at zero temperature. Therefore the authors of [21] have been surprised to observe in the wide temperature range of 77–300 K the relation T 2 ¼ jT 1 with j  0:53ð2Þ instead of 2. The dephasing rate 1=T 2 is obtained theoretically in [23] using second order perturbation theory and the three states of spin projection. For the case of ultra-pure crystals and no coupling to other spins, the rate reads

1 ðÞ T2

¼

1

2

2T 1

T1

þ ð1Þ

þ ð2Þ



1

v

 sin hE cos 3/E þ v2 2



1 ð2Þ

2T 1

ð5Þ

where the superscripts (1) and (2) refer to the first and the second order contributions. The superscript ðÞ indicates the two transitions jþi $ j0i and j
i $ j0i for which the dephasing time is determined. The parameter v defines the ratio of hjVvib j0i to hjVvib ji, the matrix elements coupling the spin and vibrations of the crystal (phonons). This dephasing cannot be eliminated by so-called dynamical decoupling because it results from spinphonon interactions which have a very short correlation time, approximately inverse of the Debye frequency in diamond. The first order term is significant only at very low temperatures (where the second order term vanishes). The second term in Eq. (5) gives the observed T 2 ¼ 0:5T 1 . The rather weak dependence on the direction of the stress and the corresponding difference in the values of T 2 for the two transitions as defined by the last term in Eq. (5) are not yet confirmed experimentally. But one can easily prove that the minið2Þ

mal value of the third term is 1=2T 1 and therefore the relation (ignoring the first term) should be T 2 6 0:4T 1 . Apparently more research both experimental and theoretical is needed on this subject. The observed width of a MW resonance is defined by the socalled inhomogeneous dephasing time T 2 , that is C ¼ 1=ðpT 2 Þ. The name ‘‘inhomogeneous” originates in the early days of NMR research when inhomogeneity of the external magnetic field within the sample volume was responsible for the dominant contribution to the observed linewidth. In modern experiments and especially in the experiments with a single spin, this inhomogene-

ity has little or no effect on the width but the concept of T 2 is still in use and characterizes the observed linewidth. Temporal fluctuations of external magnetic fields (instability of the current in the magnetic coil, etc.) can contribute to the inhomogeneous broadening. The interaction between the spin of a particular NV-center and other spins present in the same crystal may have a similar effect. All these spins flip due to thermal excitation and this creates a fluctuating magnetic field at the location of each NV-center. Such fluctuations affect even a single NV-center. If many NV-centers are measured simultaneously, then there is also a contribution to the observed C due to static inhomogeneity within a single crystal. For example, different magnitudes and directions of internal stress (more significant in smaller crystals) result in different values of D and E which may be described as static inhomogeneity. The spins contributing to dephasing can be, for example, nuclear spins of 13 C, an isotope which is naturally present in diamond at concentrations of 1.1%. Note that the nuclear spin of 12 C is zero. Nitrogen is a necessary ingredient of NV-centers but substitution nitrogen atoms in diamond are usually ionized to N þ because they are good donors of electrons. These ions possess electronic spins which are much larger than nuclear spins and therefore they have a much larger effect on the linewidth. For example, the presence of 14 N at the concentrations around 100 ppm can cause the dephasing time to be as short as 0.1 ls, which corresponds to a linewidth of about 3 MHz. In nanocrystals, additional spins can be present on or near the crystal surface. The difference between the static inhomogeneity and a dynamic change in the local field is somewhat illusory. It all depends on the relevant time scale. For example, the fluctuations of the magnetic fields associated with nuclei can have characteristic times in a range between seconds and many minutes and therefore may be treated as static in most cases. Fluctuations of electronic spins are much faster and may be as fast as microseconds. The inhomogeneity of the crystal stress can last practically indefinitely unless treated, for example, by temperature cycling (annealing). When the measurement time shortens, some of these fluctuations may start to appear static while being dynamic at longer times. By shortening the measuring time more and more, the dephasing time can be made longer [21]. This phenomenon is called dynamical decoupling and it makes the relaxation time dependent on the experimental conditions. The experimentally observed dephasing time (with the dynamical decoupling in place) is called pure dephasing time of the spin (also labeled as T 2 ) which is somewhat confusing as it is not a completely intrinsic property of the spin and its physical neighborhood but also depends on the method of observation. It should be also noticed that generally speaking the resonance line is not Lorentzan-shaped in fluctuating environment and therefore the concept of a relaxation time is also not universal [25]. A record long T 2  3:3 ms at room temperatures has been observed [21] in isotopically and chemically pure bulk diamond (0.01% of 13 C , 0.01 ppm of N). This value is slightly longer than T 2  1:8 ms demonstrated experimentally at 300 K [22] in a different batch of ultra-pure crystals (0.3% of 13 C and only 0.01 ppb of N). Note that both numbers are just slightly shorter than the corresponding experimentally measured T 1 . The dynamical decoupling is usually achieved by applying a sequence of MW pulses to the sample. A pulse sequence which increases T 2 can be viewed as a band filter on the noise due to the fluctuating thermal bath. If the noise spectrum is known, the filter can be optimized to work more efficiently [26]. The effect of such optimization is rather marginal and the CPMG sequence (will be discussed later) may be the best practical solution [26].

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The dephasing can be reduced also by physically freezing the fluctuating spin bath. Because ce B  kB T at B  8 T and T  11 K, about 99.5% of the electronic spins will be in their ground state at 2 K. Such polarization of the spin bath will increase the value of T 2 from 8 ls at 20 K to 250 ls at 2 K [27]. Fluctuations of nuclear spins are much harder to freeze because nuclear magnetic moments are about 1000 times smaller than their electronic counterparts and their freezing will require proportionally smaller temperatures. As an application of Eq. (3), we estimate the noise floor for a 100-Hz linewidth, a detected phonon rate of about 300 kHz, and a contrast of 0.3. The noise floor for the frequency measurement would then be about 0.5 Hz s1=2 . But the problem of measuring such narrow lines is not at all simple. A high count rate used in the above estimate requires strong optical pumping which results in repopulation of the m ¼ 0 spin state (depopulated by the applied MW) and thus decreases the contrast. It takes absorption of only about 4 photons to restore the spin polarization [11]. As a matter of fact, the observed value of the contrast is a result of competition between repopulation of the m ¼ 1 states by the resonance MW and their depopulation by the optical pumping. If one attempts to overcome the problem by increasing the power of the MW, the unintended consequence will be power broadening of the ODMR resonance, that is increasing the value of C. To make the matter even more complicated, one has to mention also that the optical pumping to the electronically excited state effectively turns off the interaction between the spin state and the MW because the D value in the excited state [28] is only about 1.42 GHz and therefore the resonance conditions will not hold after the center makes its transition to the excited state. This will reduce the power broadening by MW fields at low and intermediate values of the optical pumping. But the same process will increase the ODMR linewidth at high rates of optical pumping by reducing the interaction time between the MW field and the spin states. The optical pumping does not broaden the MW resonance significantly only if 1=T 2 is smaller than the pumping rate. The conditions for optimal detection have been thoroughly analyzed in several publications [18,29]. Theoretical analysis with practically sufficient accuracy approximates the NV-center by considering at least 5 quantum states (this includes the electronically excited triplet state and the metastable singlet electronic state and the three spin states in the ground electronic state). It depends on a number of phenomenological relaxation constants and requires numerical simulation on a computer. Such numerical analysis demonstrates that the direct measurement of ODMR spectra does not use the full potential of the system especially if the MW resonances are narrow. Smaller measurement errors can be achieved by using pulsed ODMR schemes. A majority of such schemes have been developed for NMR experiments and then successfully implemented on NV-centers. The most useful sequences of these pulses such as Hahn echo [30], Carr and Purcell sequence (CP) [31] modified later by Meiboom and Gill (CPMG sequence) [32], and Ramsey fringes [33] have been named after their inventors. An apparent difference between typical NMR experiments and experiments with NVcenters is that NMR operates with billions of nuclei while only a single NV-center can be in operation when ODMR is recorded in a diamond crystal. The difference turns out to be not essential [34,35]. Detection of echoes from a single center requires multiple repetitive measurements. In each such measurement the local field acting on the center can be slightly different and effectively a single center acts as an ensemble of centers (it represents a new center at every repetition). The pulsed NMR spectroscopy is based on application of short pulses of the resonance MW field separated by certain waiting

times sw . The MW field at the location of the NV-center oscillates with frequency x and is linearly polarized (most usual case). Its magnitude is Bmw and its polar and azimuthal angles are hmw and /mw respectively. The optical pumping is switched off during the MW pulses and therefore the effect of the broadening due to the optical pumping is absent in all such schemes. Then the optical excitation is turned on for a short period of time to probe the population of m ¼ 0 and m ¼ 1 states by measuring the intensity of luminescence. In these measurements the optical excitation can be as high as practical (close to complete saturation of the optical transition) thus increasing the photon emission rate to its maximum. Because the spin of an NV-center has three eigenstates, a two-state analysis is not always applicable but frequently can be reduced to because one of the upper states can be excluded temporarily from the analysis due to off-resonance with the MW field. In the following equations, jui represents an ‘‘upper” state and may vary depending on the experimental conditions. For example, it can be j1i; j
1i, or their superpositions jþi and j
i. The coupling Hamiltonian in the rotating-frame approximation reads b mw ¼ 1 c Bmw sin hmw ðb b mw j  1i  H S x cos / þ b S y sin / Þ and h0j H mw

e

2

mw

b mw j  1ij. A resonance MW-pulse of duraXei/mw , where X  jh0j H

tion sp acting on a two-level system changes initial state jxi to b / ðXÞ in the basis j0i;j1i reads [36] b / ðXÞjxi. The matrix R R mw

mw

b / ðXÞ ¼ R mw

cos
i sin

Xsp 2

Xsp 2


i sin

ei/mw

Xsp 2

cos

e
i/mw

!

ð6Þ

Xsp 2

Other pairs j0i $ j
1i , j0i $ jþi, and j0i $ j
i can be treated using the above expressions and the linearity of the involved operators. Pulses ‘‘p=2”, ‘‘p”, and ‘‘2p” (named after the value of Xsp ) result in the following transformations
i/

i/

p p 1 ie mw 1 ie mw 2 2 j0i ! pffiffiffi j0i
pffiffiffi j1i and j1i ! pffiffiffi j1i
pffiffiffi j0i 2 2 2 2

p


i/mw

j0i !
ie

p

j1i and j1i !
ie

2p

i/mw

j0i

2p

j0i !
j0i and j1i !
j1i

ð7Þ

If the initial state of the spin is j0i or j1i, then a p=2-pulse puts the spin in a superposition of j0i and j1i with equal populations, a p-pulse flips the populations of the j0i and j1i but a 2p pulse returns the initial state multiplied by a factor
1. The overall phase of the final state vector may be important if the third state of the quantum system is involved in the spin manipulation. To understand how it works in the applications, we add a phase dependent factor e
iu in front of the j1i in the superposition state obtained after the action of a p=2-pulse. The accumulated phase u depends on the external fields or other factors which affect the energies of the spin sub-levels. Note that the j1i state accumulates h after free evolution for a waiting time sw , where a phase of DEsw =
DE is the energy difference between j1i and j0i states and
h is the Planck constant. The result of a second p=2-pulse acting on the
i/mw state p1ffiffi j0i
ie pffiffi e
iu j1i will be a superposition of the j0i and 2

2

j1i with coefficients depending on the accumulated phase. pþe
iu þp 2 2

j0i ! ¼
ie


iu=2

 u u  sin j0i þ e
i/mw cos j1i 2 2

ð8Þ

This protocol is called a Ramsey-type sequence and can be written p
sm
p2 as for briefness. In the Hahn echo sequence 2 p p , a p=2-pulse is followed by a p-pulse which
s
p
s
w w 2 2 is applied after a waiting time sw . The p-pulse turns jui into j0i and vice versa. Then the relative phase of the j0i and jui components will return to zero after the second time interval sw if the local field does not change during the 2sw -long time interval. Therefore

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when the final p=2-pulse acts on the spin, it will make together with the first p=2-pulse effectively a single p-pulse. The CP px sw
2
px
sw . . .
px
s2w
p2x sequence or for briefness 2 px sw n sw px

½ p



p =2-pulse followed by , is a sequence of a x 2 2 2 2 a large number (hundreds is not unusual) of the re-phasing ppulses separated by waiting times sw . The interval sw should be as short as possible. In the brief notation, the sequence in the square brackets is meant to be repeated by the number of times shown as a superscript (the sw -intervals are omitted). The subscript x shows the direction of the linearly polarized MW in the fixed laboratory reference frame. Each p-pulse creates an echo and reverses any phase difference which is caused by the fluctuations on the time scale longer than the time between the two p-pulses. The MGCP sequence p2x
s2w
½py n
s2w
p2x differs from the CP in two respects. First, all the pulses are coherent (they can be obtained, for example, by on/off modulation of a stable harmonically oscillating MW). Second, the phase of the initial p=2 pulse, the value of /mw in Eq. (7) is shifted by p=2 relative to the phase of the p pulses. The subscripts x and y indicate this phase shift. The CPMG sequence has the advantage of not accumulating the error in the area of the re-phasing pulses if they are slightly different from the exactly required value of p. Two consecutive py -pulses will compensate each other’s errors but only for the projection of magnetization on they-direction. Improved sequences correct errors for all projections and therefore can compensate, for example, deviations of the central frequency of the p-pulses from the exact resonance. Such sequences have been proposed in [37]. An example is a sequence called XY-8 which reads n o n px
s2w
½½py ; px 2 ; ½px ; py 2
s2w
p2x . 2 Note that application of the echo techniques to ODMR spectroscopy on NV-centers requires an additional p=2-pulse at the end of the sequence. This is required because the ODMR is sensitive to the difference in populations of the m ¼ 0 and m ¼ 1 states unlike in classical NMR where the precession of the spin is measured. The final p=2-pulse creates the population difference between j0i and jui as shown in Eq. (8) which can be monitored by measuring the luminescence intensity. 3. Magnetic field measurements If the values of D and E ? are known or determined by measuring the ODMR spectra at zero magnetic field, then the magnitude of B 2

and the quantity D  D cosð2hB Þ þ 2E ? cosð2/B Þ sin hB can be found explicitly. The angle /B is the azimuthal angle in the spherical coordinate system which defines the direction of the magnetic field vector in the xy-plane. The magnitude of the field can be determined using the following expression [38]

B¼

1

ce

2

2

f þ þ f

f þ f

D2
E 2? 3

!1=2 ð9Þ

In the definition of D the second term is very small (because D  E ? ) and if it is neglected then the angle hB can be determined in a unique way. Thus information about the two transition frequencies defines the position of the magnetic field vector on the cone surface which makes the angle hB with the NV axis subject to a small uncertainty due to the undetermined contribution from the term depending on the value of /B . The value of /B cannot be found from the measurement of f þ and f
. If the EPR frequencies are determined for two sets of NV centers, then the magnetic field direction will be determined by intersection of two conical surfaces which generally gives two possible directions. Three or more sets will define a single intersection of the cones. The axes of the available NV sets can be determined, for example, by measuring the

directions of known fields before the crystal is used in a magnetometer. There is however a fundamental uncertainty related to the symmetry of the physical system. For example, the measurements are not sensitive to the overall sign of the magnetic field vector (fields of the same magnitude but opposite directions are indistinguishable). In the direct ODMR, the random error in the field gf =ce (see Eq. (3)) may be large due to degradation of the contrast and C at a high level of optical pumping when R  R0 . This can be improved using a pulse sequence. The simplest pulse sequence [18] contains only one MW p-pulse and one optical pulse. The power of the MW pulse should be moderate not to broaden too much the MW resonance. It has been shown that the optimal length of the WM pulse is close to T 2 . Then a short but powerful optical pulse (length sl ) at 532-nm is applied. As has been mentioned before, it takes only about 4 optical excitations to get the spin state back to j0i and therefore the approximate relation R0 sl  4D holds, where D is the detection efficiency of the photon counting system. The pulse sequence is repeated many times and the resulting noise floor reads

gB 

1

2:3

ce CðR0 sl T 2 Þ1=2



1

1

ce CðDT 2 Þ1=2

ð10Þ

More sensitive NV-magnetometry proposed in [39] exploits the Ramsay-type sequence shown in Eq. (8). This sequence has a minor advantage over the above single pulse approach in the case of a DC magnetic field. The noise floor can be estimated using Eq. (10). But an interesting modification of the Ramsey-sequence allows ultrasensitive detection of AC magnetic fields, that is the amplitude in BAC cosðxAC tÞ. If the time interval 2sw is made equal to the period of the AC field oscillation, then the phase uB will be accumulated during the entire period. Moreover, a properly adjusted DC magnetic field applied to the sample will cause the nuclear spins (e.g. 13 C) to precess with double frequency 2xAC so that their contributions will be averaged out [40]. The noise floor optimized for detection of the AC amplitude reads [39]

gB 

1

1

ce CðDT 2 Þ1=2

ð11Þ

The record practical noise floor for AC fields has been reported at the level of gB ¼ 4 nT s1=2 [22] or about gf ¼ 110 Hz s1=2 if converted to the frequency scale by multiplying gB and ce , the field-sensitivity factor. Using NV-centers with the record long T 2 achieved in [21] would lower the noise floor by a factor of ð3:3=1:8Þ1=2 to the value of gB  3 nT s1=2 or correspondingly gf  80 Hz s1=2 . But such levels of the noise floor have not yet been demonstrated experimentally. With a small modification, AC magnetometry can be used to facilitate detection of the magnetic field produced by a single nucleus, a first step towards building an NMR spectrometer capable of working with one nucleus. A DC magnetic field will cause the magnetic moment of the nucleus to process with its specific Larmor frequency (which depends on the nuclear magnetic moment) and this will create an AC magnetic field which may be detected with NV-centers and a synchronized periodic pulse sequence as described above. Preliminary experiments with single protons have already been reported [41–43]. A variation of AC magnetometry is used in [44] where an electronic spin of nitroxide (used as a spin label on a single protein molecule) has been flipped by applying a resonance radio frequency field simultaneously with the MW p-pulses. A drawback of the pulsed measurement described above is their potential sensitivity to a small temperature change because such a change also affects the energies of the j1i and j
1i states. In direct ODMR spectroscopy the temperature effect can be eliminated by considering only the splitting between j1i and j
1i states which

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T. Plakhotnik / Current Opinion in Solid State and Materials Science xxx (2016) xxx–xxx

is not sensitive to the temperature change. In the pulsed measurements, this can be mitigated [45] by using jþi ¼ p1ffiffi2 ðj1i þ j
1iÞ and j
i ¼ p1ffiffi2 ðj1i
j
1iÞ as the base states of the upper levels of

the center. This requires very short MW-pulses, so short that their spectral width is much larger than the splitting between j1i and j
1i. Only the symmetrical state will be coupled to the state j0i and a p-pulse will turn the initially polarized j0i state into jþi. Evolution of this state in the presence of a static field results in different phase shifts of the j1i and j
1i components which effectively creates a superposition of jþi and j
i because

1 1 pffiffiffi ðe
iuB j1i þ eiuB j
1iÞ ¼ pffiffiffi ðcos uB jþi
i sin uB j
iÞ 2 2

ð12Þ

where uB ¼ 2pBz ce s. A second p-pulse applied after the time turns jþi into j0i and the spin state becomes

1 pffiffiffi ðcos uB j0i
i sin uB j
iÞ 2

s

ð13Þ

which can be probed by measuring the intensity of 532-nm excited luminescence. Note that ji have the same luminescence quantum yield as either of the two states j1i or j
1i. So far, we have been focussed on the noise floor estimate. This value tells us the smallest change in the field which can be distinguished from the noise in the signal. The accuracy is a measure of the difference between the measurement of the field and its actual value. How accurate NV magnetometers are has been investigated experimentally and theoretically in [46] using a sample of approximately 35-nm diamond crystals which contained about 20 NV-centers each. It has been shown that the fields measured by different crystals deviate from each other (we call this deviation a systematic error) an order of magnitude more than the uncertainty in the measurements caused by the fluctuations of the signal. In particular, the standard deviation of the systematic error is about 20 lT for the fields averaged at 20 mT and 10 mT. This error has been explained by the coupling of NV-centers with other spins in the sample. Theoretical estimates show that this error becomes proportional to the measured field for fields smaller than 5 mT. Thus the relative error in the sample containing other spins can be as high as 0.2–0.3%. It remains unclear what is the accuracy of the field measurements in ultra pure crystals with a low concentration of NV-centers.

lication [47] but is estimated elsewhere [17]. The reported noise floor in the pressure measurements is gP  0:6 MPa s1=2 or gf  8 kHz s1=2 if converted to the frequency scale. It is achieved using direct ODMR detection [7]. The relatively high levels of the noise floor for the frequency measurements here in comparison to the magnetic fields sensing (see the previous section) are either due to different crystal purity and/or the advantage of ACmeasurements and a Ramsey-type sequence over DC direct ODRM. When a Ramsey-type sequence is used to measure a temperature, one faces a problem of possible contribution to the resonance shifts from small changes in the magnetic field which may be hard to avoid in the laboratory unless it is magnetically shielded (the proper shielding may be quite expensive). A complementary problem has already been discussed in the previous section. In this case, to decouple the temperature and magnetic field effects, the measurement should be made sensitive to the value of D but not to the splitting between the j0i ! j  1i resonances. The idea of j0i n o ðþÞ ðþÞ and jþi being coupled by p2
sm
2pðþÞ
sm
p2 [47] is somewhat similar to the one explained in the section about the magnetic field measurement. After the initial p=2-pulse and a free evolution for a time interval sm (affected by the temperature and possibly unwanted magnetic field), the system ends in the superposition

1 e
iuT
iuB pffiffiffi j0i þ j1i þ eiuB j
1iÞ ðe 2 2 1 e
iuT ¼ pffiffiffi j0i þ pffiffiffi ðcos uB jþi
i sin uB j
iÞ 2 2

ð14Þ

where uT ¼ 2psT DT sm and uB as defined before. A 2p-pulse changes the signs in front of j0i; jþi and turns this state into

1 e
iuT
pffiffiffi j0i þ pffiffiffi ð
cos uB jþi
i sin uB j
iÞ 2 2 1 e
iuT iuB ¼
pffiffiffi j0i
ðe j1i þ e
iuB j
1iÞ 2 2

ð15Þ

Compare Eqs. (14) and (15) and note that the sign of the phase uB has changed due to the exchange of the amplitudes between j1i and j
1i. After the 2p-pulse, the state will evolve for the same time interval sm and will become
p1ffiffi2 j0i
p1ffiffi2 e
i2uT jþi. This final state it unaffected by the magnetic field. The terminal

p=2-pulse creates

a state where the population of j0i is proportional to sin uT . A difn o ð
1Þ ð
1Þ has ferent pulse sequence, p 2
sm
pð
1Þ pðþ1Þ pð
1Þ
sm
p 2 2

4. Temperature, pressure and electrical field measurements There are two different approaches to temperature/pressure measurements with NV centers. The most frequently used method exploits the sensitivity of the MW resonances to the external conditions. This approach does not require a lot of discussion because the basic principles of the ODMR and the corresponding estimates of the quantum fluctuations in the signal have been already explained in the previous sections. The only difference will be in using the factors sT and sP instead of ce for converting the resonance frequencies to the values of T and P. For example, the temperature noise floors of gT  9 mK s1=2 [47], gT  25 mK s1=2 [48], and gT  5 mK s1=2 [50] have been reported. These values are obtained with a Ramsey-type sequence and a single NV-center imbedded into a nearly perfect bulk crystal. By multiplying the smallest number with the sensitivity factor sT , one gets the corresponding frequency noise floor near 400 Hz s1=2 . Note that this level is achieved without using AC techniques. The noise floor increases to about 0.3 K s1=2 in direct intracellular ODMR measurements even with a nano-crystal containing about 100 centers [47] where the photon rate is 100 times larger than in the the case of a single NV. Note that this noise floor is not given in the original pub-

been proposed in [48,50] which works with spectrally narrow MW pulses interacting selectively with j1i and j
1i states. The three pulses in the middle do the same trick as the 2p-pulse in Eq. (15), they swap the amplitudes of j1i and j
1i. In the second approach, the external conditions are measured by detecting small changes in the width, the area (relative to the area under the entire spectrum) and the peak wavelength of the ZPL [17,49]. This approach is simpler to implement as it does not require anything but a standard optical microscope, a spectrometer and a sensitive CCD. This equipment is standard, for example, in Raman microscopy. As an example, Fig. 2 shows the temperature effect on the ZPL (narrowing the line and increasing its area at the lower temperature) as measured in [17] where it also has been shown that the ZPL amplitude A, that is the area of the ZPL divided by its width gives the smallest noise floor for the temperature sensing. The ratio of A to AR , the amplitude at a reference temperature T R reads

! A T 2R 2p2 SðT 2
T 2R Þ ; ¼ exp
AR T 2 3T 2D

ð16Þ

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T. Plakhotnik / Current Opinion in Solid State and Materials Science xxx (2016) xxx–xxx

2

measurement and a Hahn-type sequence. The observed shift of the MW resonance was about 28 kHz for the field of 3 kV/cm. Therefore a noise floor in the frequency domain is estimated near 2 kHz s1=2 in these experiments.

1

5. Conclusion, unsolved problems and outlook

Relative intensity

3

0 600

610

620

630

640

650

660

Wavelength (nm) Fig. 2. Luminescence spectra on NV-center measured at 383 K (upper curve) and at 295 K show clear increase in the ZPL amplitude and decrease in ZPL width at higher temperature.

where T D is the Debye temperature of diamond and S is the parameter determining the strength of coupling between the electronic degrees of freedom in the NV-center and the phonon bath of the crystal. The method has been tested on a batch of crystals of about pffiffiffi 35-nm across. The value of T D = S, the only parameter in Eq. (16) has been determined for these crystals to be approximately 1:0ð1Þ  103 K. The 10% variation is similar to the variation of the sT factor (see Eq. (2)). The theoretical sensitivity of the ZPL amplitude near room temperature reads [17]

360 K

gT  pffiffiffiffiffiffiffi AR

ð17Þ

where A is the relative area of the ZPL at room temperatures. The predictions for gT obtained from Eqs. (3) and (17) are identical for the following values: C ¼ 15 MHz; C ¼ 0:1, and A  0:05, the parameters routinely observed in nano-crystals. But the value of R in Eq. (17) can be close to R0 which makes the ZPL a more sensitive method in this example. Moreover, the ZPL of a large number of NVcenters is easy to measure simultaneously (this will increase the count rate proportionally to the number of centers involved) while the pulsed-ODMR under similar conditions will have limitations due to different orientation of the centers of different sets and therefore different coupling to the MW-field (this problem can be partially overcome by applying a DC bias magnetic field which will split the resonance of different sets so that the MW can be tuned in resonance with only one set). The reported noise floor achieved with nano-crystals hosting about 50 NV-centers is 0.3 K s1=2 [17]. The spin magnetometry wins 2 orders of magnitude even with a single NV center but requires a super-pure bulk crystal. There are two more temperature-dependent factors in the spectra of luminescence which one can see in Fig. 2, a change in the shape of the phonon wing around the ZPL region and a slight shift of the ZPL maximum at the higher temperature to a longer wavelength. The latter has the sensitivity of about 0.013 nm/K [51,52]. Inclusion of these factors in the data analysis should theoretically reduce the noise floor by a factor of 2. But a small thermal deformation or other mechanical instability may significantly alter the estimated temperature. For example, the position of the peak in the spectrum will change if the crystal moves. In the setup used in [17] (magnification of the microscope is 100 and a linear dispersion of the spectrometer is 19 nm/mm), a change in the position by 5 nm (well below the diffraction limited optical resolution) results in displacement of the ZPL, which is equivalent to a temperature change of 0.7 K. Therefore one has to mitigate such mechanical instabilities to use the position of the ZPL peak for the temperature measurements. The electrical fields can be measured with NV centers using the sensitivity factors sjj and s? . The reported noise floor for the electric field ODMR measurements is gE  202  6 V=cm s1=2 [9] for AC-

Measuring magnetic properties of materials at ultimate limits of sensitivity as discussed in the section on magnetometry has opened a way, for example, to NMR chemical analysis on a single molecule. Probing fluctuating magnetic fields of thermally excited Johnson currents in a metallic electrical conductors [53] or the fields of diamond-magnet composite nanoparticles [54] are other examples where extraordinary properties of NV-centers are in use. It has even been suggested to operate the NV-centers as frequency standards [55]. There are still quite a few obstacles on the way. In particular, the noise floor still requires considerable improvement. Note that the magnetic field noise floor of 30 pT s1=2 theoretically estimated using Eq. (3), C ¼ 0:3; R ¼ 300 kHz, and C ¼ 100 Hz is much smaller than the experimentally achieved one. One important factor affecting the noise floor is the linewidth. It is very desirable to increase the dephasing time T 2 to its ultimate level set by the spin-lattice coupling and the longitudinal relaxation. It is also important to improve understanding of the dephasing processes and, in particular, verify Eq. (5) as well as to obtain estimates (independent of the dephasing rates) for the phenomenological parameters of the theory such as the spin-lattice coupling constants. If the observed values of T 2  5 ms at room temperature are determined by the second order scattering processes similar to those resulted in Eq. (5), then improvement is hardly possible without cooling the crystal to cryogenic temperature where the competition with SQUIDs and their astonishing 3.6 fT s1=2 noise floor [56] (6 orders of magnitude lower than the best results recorded with NV centers) will be tough (although the SQUID loop of about 3 mm across makes NV centers more competitive than it first looks). Thermometry with nanometer spatial resolution has potential applications in nanotechnology, for example, as a way to study hot spots in highly integrated circuits [57]. It has also been suggested to use nano-sensors for intracellular measurements and thermogenesis in single cells have been reported recently. However, simple arguments based on energy conservation and kinetics of the heat transfer through cell boundaries [58] show that the possible temperature effects are several orders smaller than reported (see references in [58]). It is obvious that more accurate and reliable methods of nano thermometry are needed in this field. Better control of the composition and the quality of the crystals (chemical purity, no stress and crystal imperfections, and minimum spin possessing isotopes) may be a key to further progress. The problem is complicated if the centers should be positioned close to the surface or embedded in nano-crystals. In both cases the proximity of the surface contributes to the shortening of the dephasing time. The effect is very strong in ultra small crystals (so-called single digit nano-crystals) where the ODMR is not even observed although accounts on the efforts to produce and investigate such crystals have been published on a regular basis [59–63]. It has been noticed in several publications that the noise floor can be reduced by exploiting many NV centers instead of a single entity. The theoretical decrease of the noise floor may be large but the approach has several drawbacks. One of the most fundamental is the loss of spatial resolution and the coherence. According to Eq. (4), the coupling between NV centers provides a dominant contribution to the decoherence at room temperature at concentrations of NV centers above  20  1018 cm
3 . This concentration will also limit special resolution as 1000 NV centers

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T. Plakhotnik / Current Opinion in Solid State and Materials Science xxx (2016) xxx–xxx

occupy a region of about 30 nm across, the size of a large protein molecule [64], and will make single-molecule NMR significantly more difficult. The next paragraph points to other limits on the concentration. We note also that the NV centers from different sets will have different orientations and therefore different coupling with MW and DC magnetic fields. This also has to be mitigated in the case of MW pulses simultaneously operating on many centers. Achieving high concentrations of NV-centers especially in small nanocrystals is another important material science problem. This would be beneficial for sensing because luminescence intensity increases proportionally to the concentration. The highest experimentally observed concentrations of NV-centers are below 20 ppm (3:4  1018 cm
3 ) [65–67]. There are mounting evidences that the quantum yield of luminescence in NV-centers tends to decrease with the size of the crystals and also varies within an ensemble of apparently identical centers [68,69]. The reasons and therefore the solutions of this problem are not yet completely clear but it has been shown that oxidation of the surface increases the quantum yield by reducing the graphitisation of the surface layer [61]. More work needs to be done in this area. One should be also opened to searching for new defects in the diamond, which would eventually replace the NV-centers at least in some applications. For example, NE8 centers, a nickel atom surrounded by 4 nitrogens [70,71], Si-Vacancy (SiV) centers [72], and Cr-related centers [73] have attracted some attention. These centers have an advantage of much smaller phonon wing and much narrower ZPL. Other advantages are, for example, a much higher concentration of SiV than NV centers [74] and a high emission rate [75,73]. Magnetic sensitivity has been optically detected at low temperatures [76] but efficient optical spin polarization at room temperature in these centers has not been observed. The search for new centers should continue despite a limited success so far. The last comment is on the theoretical aspects of NV physics. As has been mentioned earlier, there is a discrepancy between theoretically estimated values of T 1 ; T 2 , and the experimentally observed values. Better understanding of spin-phonon coupling may be the key to a better agreement and potentially to longer dephasing times. Interestingly, but another problem related to phonons in diamond and their coupling to electronic degrees of freedom in NV-centers has been reported recently [77]. The dependent on the phonon frequency electron-phonon coupling in the electronically excited state appears to have an unexplained cutoff on the high frequency side of the acoustic phonon spectrum. Research on NV-centers has achieved amazing progress in the past 20 years. Implementation of different spin manipulation techniques, improvement in the optical detection and material properties have brought the applications of NV-centers in diamond to a stage which opens some quite fascinating possibilities. But when such applications become practical and routine is a bit premature to say. Acknowledgement This research has been supported by Australian Research Council (ARC) grant DP0771676. References [1] G. Davies, M.F. Hamer, Optical studies of the 1.945 eV vibronic band in diamond, Proc. R. Soc. London A 348 (1976) 285–298. [2] A. Gruber, A. Dräbenstedt, C. Tietz, L. Fleury, J. Wrachtrup, C. von Borczyskowski, Scanning confocal optical microscopy and magnetic resonance on single defect centers, Science 276 (1997) 2012–2014. [3] R. Schirhagl, K. Chang, M. Loretz, C.L. Degen, Nitrogen-vacancy centers in diamond: nanoscale sensors for physics and biology, Annu. Rev. Phys. Chem. 65 (2014) 83–105.

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