Sensing of Fluctuating Nanoscale Magnetic Fields Using NV Centres in Diamond
Liam T. Hall, Charles D. Hill, Jared H. Cole, Lloyd C.L. Hollenberg
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J u l Sensing of Fluctuating Nanoscale Magnetic Fields Using NV Centres in Diamond
Liam T. Hall, ∗ Jared H. Cole,
1, 2
Charles D. Hill, and Lloyd C.L. Hollenberg Centre for Quantum Computing Technology, School of Physics,University of Melbourne, Victoria 3010, Australia Institute f¨ur Theoretische Festk¨orperphysik and DFG-Centre for Functional Nanostructures (CFN),Universit¨at Karlsruhe, 76128 Karlsruhe, Germany
New magnetometry techniques based on Nitrogen-Vacancy (NV) defects in diamond allow forthe imaging of static (DC) and oscillatory (AC) nanoscopic magnetic systems. However, thesetechniques require accurate knowledge and control of the sample dynamics, and are thus limitedin their ability to image fields arising from rapidly fluctuating (FC) environments. We show herethat FC fields place restrictions on the DC field sensitivity of an NV qubit magnetometer, and thatby probing the dephasing rate of the qubit in a magnetic FC environment, we are able to measurefluctuation rates and RMS field strengths that would be otherwise inaccessible with the use of DCand AC magnetometry techniques. FC sensitivities are shown to be comparable to those of ACfields, whilst requiring no additional experimental overheads or control over the sample.
PACS numbers: 03.65.Yz, 07.55.Ge, 07.79.-v
The exploitation of controlled quantum systems asultra-sensitive nanoscale detectors has tremendous po-tential to advance our understanding of complex pro-cesses occurring in biological and condensed-matter sys-tems at molecular and atomic scales [1, 2, 3]. The strin-gent requirements for high sensitivity and spatial reso-lution has led to suggestions of using spin-based quan-tum systems as nanoscale magnetometers [4], or of imag-ing through detection of sample induced decoherence [5].One particularly attractive physical platform to imple-ment these ideas is the Nitrogen-Vacancy (NV) centre indiamond [Fig. 1(a)], chosen for its long coherence timesat room temperature and convenient optical readout ofthe spin state [6] [Fig. 1(b)]. As such, NV centres havebeen the focus of recent proposals to image static (DC)and oscillating (AC) magnetic fields [7, 8], which havesince been demonstrated experimentally [9, 10, 11].However, many important biological and condensedmatter systems exhibit non-sinusoidal fluctuating mag-netic fields with extremely low or zero mean values[Fig. 1(d)]. An important question is therefore to whatextent these quantum based magnetometry techniquesare applicable to such situations. In this paper we ad-dress this by quantifying the detection sensitivities forthese modes for samples with fluctuations characterizedby the RMS field and dominant spectral frequency. Theresults indicate that by probing the dephasing rate ofa spin qubit placed in such environments one can char-acterize the underlying fluctuation rates and RMS fieldstrengths that would be otherwise inaccessible with theuse of DC and AC magnetometry techniques, therebyopening the way for non-invasive nanoscale imaging of arange of biological and condensed matter systems.The theory behind the detection of magnetic fields us-ing quantum systems is heavily reliant on the phase esti-mation program of quantum metrology, particularly thedetermination of coupling parameters that are constant
FIG. 1: (colour online). Schematic of a scanning NVqubit magnetometer/decoherence probe for the detection ofnanoscale field fluctuations. (a) NV-centre diamond latticedefect. (b) NV spin detection through optical excitation andemission cycle. (c) Microwave control of the spin state ofthe NV centre and 531 nm optical pulse for read-out. (d)Magnetic field signals B ( t ) at the NV probe corresponding toregions I-IV of an inhomogeneous test sample with differentfluctuation amplitudes and frequency spectra. (e) The corre-sponding qubit excited state populations P ( t ) show that theregions can be distinguished by the dephasing information:I: Strong, rapid fluctuations → fast exponential dephasing.II: Strong, slow fluctuations → fast Gaussian dephasing. III:Weak, rapid fluctuations → slow exponential dephasing. IV:Weak, slow fluctuations → slow Gaussian dephasing. in time. In the context of DC magnetometry, this corre-sponds to measurement of the first moment (the mean)of the magnetic field strength. For zero mean fields, com-plex microwave control pulse sequences are necessary.For fields exhibiting oscillatory (AC) time dependencewith which either a spin-echo or Carr-Purcell-Meiboom-Gill [12] sequence may be synchronised, sensitivities arepredicted to be as low as 3 nTHz − / [8], based on thestandard quantum limit. Excellent agreement betweentheory and experiment has been demonstrated in [11].Such techniques require accurate knowledge of the fielddynamics which may not be available, or more commonly,the field strength may exhibit a stochastic time depen-dence. Examples include nuclear dipole fields of ion chan-nels [13] [Fig. 2(a)] and lipid bi-layers in biological cellmembranes [14], Overhauser fields in Ga-As quantumdots [15], and even self-diffusing water molecules [16, 17][Fig. 2(b)]. In what follows, we investigate the effects of amore general fluctuating (FC) field on the dephasing of aspin qubit as the primary detection mechanism, and theimplications for the characterisation of the magnetic fieldfrom the surrounding environment. In this sense, we areestimating the second moment of the environmental fieldstrength, and the corresponding temporal dynamics.A spin qubit placed in a randomly fluctuating mag-netic environment will experience a complex sequence ofphase kicks, leading to an eventual dephasing of the pop-ulation spectrum. For an NV centre, this will be in ad-dition to the intrinsic sources of dephasing, which aredue to paramagnetic impurities in the diamond lattice[18]. The dephasing rate can be quantified via repeatedprojective measurements of the qubit state, and the cor-responding dephasing envelope, D ( τ ), can be determinedvia a suitably chosen quantum state reconstruction tech-nique. We use the technique of Hamiltonian characterisa-tion [19] rather than quantum tomography techniques, asit requires only a single measurement basis yet is robustin the presence of dephasing [20].The motivation for the environment model used herecomes from consideration of magnetic dipoles in motion.Other models in which a two level system is coupled to abath of bistable fluctuators have been previously consid-ered [21, 22, 23, 24, 25, 26]. These models, however, donot capture the dephasing effects due to gradual transi-tions between environmental states in slowly fluctuatingfields. Later we will show this to be of particular impor-tance in the case of spin-echo based experiments. Addi-tionally, these models require a large number of fluctua-tors to model a continuous signal. In contrast, we wish toconsider the dephasing effects of small numbers of spinsin motion.Consider a qubit with gyromagnetic ratio γ p undergo-ing a π − τ − π Ramsey sequence [27] in the presence ofa classically fluctuating magnetic field, B ( t ). An exam-ple of a fluctuating magnetic field due to a uni-directionalspin current [Fig. 2(a)], and that of a bath of self-diffusingspins [Fig. 2(b)]. The field has mean h B i ≡ B , standarddeviation p h B i − h B i ≡ B ′ , and typical fluctuationrate f e ≡ /τ e , where τ e is the characteristic correla-tion time of the external field [Fig. 2(c)]. This gives riseto two natural frequency scales, given by ω = γ p B and ω ′ = γ p B ′ . The average precession frequency of the qubitis set by ω , and is found to be decoupled from all de-phasing effects for cases where ω ′ , f e ≪ ω . Additionalrelaxation processes may dominate the qubit evolution B ( t ) / B ’ t/ τ e (a) (b) (t−t’)/ τ e < B ( t ) B ( t ’ ) > / < B > (c) Parallel transportDiffusive transport FIG. 2: (colour online). Typical magnetic field, B ( t ) /B ′ , for(a) a channel of dipoles in unidirectional motion, and (b) aself diffusing dipole bath. (c) Temporal correlation function h B ( t ) B ( t ′ ) i / h B i . Time axes are rescaled by τ e . when this condition is violated, however such cases arenot considered here since we are interested in the char-acterisation of weak magnetic fields. The nature of thedephasing felt by the qubit will depend on the fluctu-ation rate of the environment, f e , or more specificallythe magnitude of the quantity defined by Θ ≡ f e /ω ′ . Inthe case of Θ ≫
1, or fast-fluctuation limit (FFL), thequbit will experience many environmental fluctuationsduring its natural timescale. Whilst B ′ need not nec-essarily be normally distributed, the accumulated phaseerror of the qubit at some time t ≫ /f e will be nor-mally distributed by way of the Central Limit Theorem.As such, the variance of the phase error at time t ≫ /ω ′ will be h ∆ φ i ∼ tγ p B ′ /f e , giving rise to an FFL dephas-ing rate of Γ fast ( B ′ , f e ) = γ p B ′ f e . (1)This is akin to the motional narrowing result from NMR[28] and reproduces the ubiquitous exponential dephasingenvelope given by D fast ( t ) = exp ( − Γ fast t ).In the slow-fluctuation limit (SFL), where Θ ≪ t about some initial time t : B ( t ) = P Nk =0 1 k ! d k Bdt k (cid:12)(cid:12)(cid:12) t ( t − t ) k ≡ P Nk =0 a k ( t − t ) k ,where each of the a k has a specific statistical distributioncontaining information about the k th order derivative of B ( t ), and thus gives rise to a different dephasing channel.For the special case where the a k are normally dis-tributed with mean µ k and variance σ k (as consistentwith random dipole motion), the resulting density ma-trix following the free evolution time τ , but prior to thesecond π/ ρ = ρ = 1 /
2, and ρ = ρ ∗ = Q ∞ k =0 D ( k )slow ( τ )Ω ( k )slow ( τ ); where D ( k )slow ( t ) = exp (cid:20) − (cid:16) Γ ( k )slow t (cid:17) k +2 (cid:21) , and (2)Ω ( k )slow ( t ) = exp (cid:20) − i (cid:16) ω ( k )slow t (cid:17) k +1 (cid:21) . (3)Thus we see the emergence of a hierarchy of dephasingand beating channels, with the dephasing rates and beatfrequencies of the k th channel given byΓ ( k )slow = (cid:18) √ σ k γ p k + 1 (cid:19) / ( k +1) (4) ω ( k )slow = (cid:18) µ k γ p k + 1 (cid:19) / ( k +1) (5)respectively. In the case of the zeroth order channel thiscorresponds to the rigid lattice result from NMR [28], andwe have σ = h B i − h B i . This effect will be suppressedby a spin echo pulse sequence. For the first order channel,we may approximate σ ∼ (cid:0) h B i − h B i (cid:1) f e .The relative contributions of each channel to the over-all dephasing rate of the qubit depend explicitly onthe dynamics of the field, however, it should be notedthat dominance of the zeroth order channel (ie Γ (0)slow > Γ ( j )slow , ∀ j ≥
1) is a necessary and sufficient conditionfor the system to exist in the slow fluctuation regime,Θ ≪
1. This justifies the use of the Taylor expansion,since the resulting polynomial may be well approximatedby a low-order truncation.The intermediate regime of Θ ∼ τ e , pure expo-nential dephasing behaviour is observed in all cases (withdephasing rate Γ fast ), however fast fluctuating environ-ments still exhibit slow (Gaussian) dephasing behaviouron timescales τ where ω ′ τ < √ / Θ. If Θ is large, con-tributions to D from the Γ ( k )slow will decay rapidly. Theabrupt transition from D slow → D fast is shown moreclearly in the corresponding insert.For the purpose of comparison with existing spin-based magnetometer proposals, we take the NV centreas our example qubit. The Hamiltonian used to de-scribe the time evolution of an NV-centre is given by H = S · D · S + ~ γ p B · S + H other , where H other describeshigher order effects such as hyper-fine splitting, interac-tion with optical fields, etc. which can be ignored in thepresent context. We consider weak external fields suchthat O ( ~ γ p B · S ) ≪ O ( S · D · S ), thereby ensuring thecrystal-field splitting tensor, D , sets the quantisation axisof the NV centre, and that ω ′ ≪ ω (even in the FFL).The shot-noise-limited DC magnetic field sensitivityfor an NV-based magnetometer subject to a Ramsey-style pulse sequence is given by [8] η dc ≡ B min √ T ≈ γ p C √ τ , (6)where √ T and C represent the combined effects of spinprojection and photon shot noise for N s measurements( C → τ is the free evolution timeof the qubit in a given experiment, and T = N s τ is thetotal averaging time for N s such experiments. Dephas-ing times due to the interaction of the NV centre with −8 −7 f e [Hz] η d c [ T H z − / ] (b) B’ = 1 µ T B’ = 10 µ T B’ = 100 µ T B’ = 32 µ T FIG. 3: (colour online). (a) Plot of simulated dephasing en-velopes for N s = 10 runs, showing agreement with Eqs. 1& 2. Time is in units of ( γ p B ′ ) − . (insert) Zoomed plotshowing that fast fluctuating environments still exhibit non-exponential dephasing for short timescales τ : ω ′ τ < √ / Θ.(b) log-log plot of DC magnetic field sensitivity, η dc , as afunction of f e for different contours of B ′ . Assumed parame-ter values are T ∗ = 1 µ s , and C = 0 . nearby paramagnetic lattice impurities will in general bedifferent for different centres and will thus require indi-vidual characterisation. For comparison with [8], we usethe commonly accepted value of τ = T ∗ ∼ µ s.We emphasise here that Eqn. 6 applies solely to theimaging of DC magnetic fields where the dephasing ofthe qubit is exclusively due to intrinsic crystal effects.If the sample being imaged produces a fluctuating fieldof sufficient amplitude, the dephasing time (1 / Γ) maybe shorter than T ∗ , resulting in poorer static field sensi-tivity. In this context, η dc refers to the sensitivity withwhich the mean field, h B i , may be measured as the fieldfluctuates over the course of the experiment. To gaininsight into the effect of fluctuating magnetic fields onthe DC field sensitivity, we consider again a π − τ − π sequence. The DC sensitivity as a function of B ′ and f e is shown in Fig. 3 (b). From this, we see that fluctuatingenvironments can have a dramatic effect on the DC fieldsensitivity of an NV based magnetometer, depending onboth field strength and fluctuation frequency.We now turn our attention to the the sensing of mag-netic field fluctuations themselves, including the case ofzero-mean magnetic fields. Using coherent control tech-niques (spin-echo for example), we may extend the de-phasing time of the NV-centre to T ∼ µ s, as dictatedby the 1.1% carbon-13 content in the lattice. The caseof perfectly oscillatory magnetic fields has been consid-ered in detail in [8], in which AC sensitivities may be aslow as 3 nT Hz − / [Fig. 4(b)]. This technique requiresthe π pulse to coincide with the first zero-crossing of themagnetic field, which requires an accurate knowledge ofthe oscillation frequency. This may prove difficult unlessthe frequency is externally controlled. Furthermore, ac-curate control will become difficult at high frequencies,and this may lead to further dephasing.Rather than considering an AC field, we now studythe magnetometer’s sensitivity to a more general fluc-tuating field via consideration of the induced dephasingrate [5]. For a π − τ − π − τ − π pulse sequence, theprobe will show decreased sensitivity to environments ex-hibiting fluctuation frequencies, f e less than 1 /τ . Forfast fluctuating fields, the effect will be negligible. Forthe imaging of slowly fluctuating fields, this may appearproblematic, however complete insensitivity only comeswith f e →
0. A spin echo sequence will modify the D ( k )slow via Γ ( k )slow (cid:0) − − k (cid:1) Γ ( k )slow , thus only the effects of thezeroth order dephasing channel will vanish.Perturbations on the dephasing rate may be measuredfrom (1 − D ) min = exp[( τ/T ) n ] C √ N s , where n describes theshape of the spin-echo dephasing envelope as dictatedby the presence of carbon-13 nuclei in the lattice, whichfor present purposes may be taken as n = 3 [18]. Thisimplies an optimal free-evolution time of τ ∼ T / √ /T dephas-ing rate as slow as 200 Hz for exponential dephasing and800 Hz for Gaussian dephasing may be detected by thismethod after 1 s of averaging time. By performing mea-surements of the total dephasing rate, Γ, both the fieldvariance and average fluctuation rate may be inferredfrom equations 1 & 4. Of course, the question remains ofwhich fluctuation regime a given sample system residesin. In the absence of any prior knowledge of the environ-ment being measured, this question may be answered viadetermination of the shape of the dephasing envelope, atask to which the Hamiltonian Characterisation methodis well suited [5].The optimal fluctuating magnetic field sensitivity willoccur when the Θ ∼ η fc = e / Cγ p √ T ,which for C = 0 . η fc = 1 . − / . In practicehowever, such sensitivity may be difficult to realise dueto memory effects in the fluctuating environment. Forsystems that satisfy Θ ≫
1, thus exhibiting long-timeexponential dephasing behaviour, Gaussian dephasing isstill exhibited for τ < /f e [Fig. 3(a)]. For spin-echo ex-periments, the effect is worsened as the dominant contri-bution to D slow comes from k = 1. Taking this into con-sideration, the minimum resolvable field obtained after T = 1 s averaging time is plotted in Fig. 4(a) as a functionof environment fluctuation frequency. We see that fluctu-ating field strengths as low as 4.5 nT may be achievableafter T = 1 s averaging time, or some N s ∼ /T . This is in direct contrast with the AC case,which shows poor sensitivity to fields oscillating with pe-riods less than T [Fig. 4(b)].We have theoretically investigated the effects of a fluc-tuating magnetic field on an NV centre spin qubit. Thisanalysis was used to place new limits on the sensitivitywith which the mean field strength may be measured. −8 −6 −4 (a) Fluctuating field sourcef e [Hz] B ’ m i n [ T ] −8 −6 −4 (b) Sinusoidal field source ν e [Hz] B a c m i n [ T ] FIG. 4: (colour online). (a) Minimum resolvable FC fieldstrength, B min , vs environmental fluctuation rate, f e , for T = 1 s averaging time. In contrast to the AC case, an FC de-tection requires no prior knowledge of fluctuation timescales.(b) Minimum resolvable AC field amplitude, B acmin , vs fieldoscillation frequency, ν e , for T = 1 s averaging time. Herewe have assumed that the AC field is initialised in phase withthe probe qubit, and that the oscillation frequency is knownaccurately enough that the π pulse of a spin-echo sequencecoincides with the first zero-crossing of the field. This willbecome increasingly difficult with increasing ν e , leading toadditional sources of dephasing. Furthermore, we have built upon the idea of decoherencemicroscopy [5] to theoretically demonstrate the abilityof an NV centre to measure field strengths and fluctua-tion rates of randomly fluctuating magnetic fields. Thisanalysis shows that the methods presented here requireno experimental resources beyond those of existing tech-niques, no prior control or knowledge of the external field,and thus may be implemented with current technology.We gratefully acknowledge discussions of the subjectmatter with A. Greentree, M. Testolin, F. Jelezko and J.Wrachtrup. This work was supported by the AustralianResearch Council (ARC) and the Alexander von Hum-boldt Foundation. ∗ Electronic address: [email protected][1] K. Hasselbach et al., Physica C , 140 (2000).[2] J. R. Kirtley et al., Appl. Phys. Lett. , 1138 (1995).[3] R. C. Black et al., Appl. Phys. Lett. , 2128 (1993).[4] B. M. Chernobrod and G. P. Berman, J. Appl. Phys. ,014903 (2004).[5] J. H. Cole and L. C. L. Hollenberg, arXiv:0811.1913v1[quant-ph] (2008).[6] F. Jelezko and J. Wrachtrup, Phys. Stat. Sol. , 3207(2006).[7] C. L. Degen, Appl. Phys. Lett. , 243111 (2008).[8] J. M. Taylor et al., Nature Phys. , 810 (2008).[9] G. Balasubramanian et al., Nature , 648 (2008).[10] G. Balasubramanian et al., Nature Mat. , 383 (2009).[11] J. R. Maze et al., Nature , 644 (2008).[12] S. Meiboom and D. Gill, Rev. Sci. Instrum. , 688691(1958).[13] B. Hillle, Ionic Channels of Excitable Membranes (Sin-auer Associates, Sunderland, MA, 2001), 3rd ed.[14] M. Patra et al., Biophys. J. , 3636 (2003). [15] D. J. Reilly et al., Phys. Rev. Lett. , 236803 (2008).[16] V. I. Tikhonov and A. A. Volkov, Science , 2363(2002).[17] A. Rahman and F. H. Stillincera, J. Chem. Phys. ,3336 (1971).[18] L. Childress et al., Science , 281 (2006).[19] J. H. Cole et al., Phys. Rev. A. , 062312 (2005).[20] J. H. Cole et al., Phys. Rev. A. , 062333 (2006).[21] A. Shnirman et al., Phys. Rev. Lett. , 127002 (2005).[22] J. Schriefl et al., New J. Phys. , 1 (2006). [23] E. Paladino et al., Phys. Rev. Lett. , 228304 (2002).[24] Y. M. Galperin et al., Phys. Rev. Lett. , 097009 (2006).[25] H. Gutmann et al., Phys. Rev. A , 020302 (2005).[26] M. Mottonen et al., Phys. Rev. A , 022332 (2006).[27] R. R. Ernst, G. Bodenhausen, and A. Wokaun, Prin-ciples of Nuclear Magnetic Resonance in One and TwoDimensions (Clarendon Press, Oxford, 1990).[28] C. Kittel,