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+1 734 763 8041, Email: Hero@Eecs.Umich.Edu TWO STATE MARKOV MODELLING AND DETECTION OF SINGLE ELECTRON SPIN SIGNALS Alfred O. Hero , Michael Ting, Jeffrey A. Fessler Department of Electrical Engineering and Computer Science, University of Michigan Ann Arbor, MI 48109-2108, USA ¡ phone: +1 734 763 0564, fax: +1 734 763 8041, email: [email protected] ABSTRACT suring the perturbation of the cantilever position from its nor- The focus of this paper is the optimal detection of piecewise mal oscillatory behaviour. The iOSCAR method considered constant binary valued continuous-time (C-T) signals with here uses an externally modulated radio frequency (RF) field Markovian state transitions. One example is the classic ran- to manipulate the electron spins in such a way as to produce dom telegraph signal for which the number of states is two periodic forces on the oscillating cantilever [1, 2]. This re- sults in small changes in the cantilever’s natural frequency and the transitions follow a Poisson process. This signal de- ω tection problem arises in many different areas of engineer- 0. A laser interferometer measures the cantilever displace- ing and science, notably modulation classification in digital ment; detection of these frequency shifts in the cantilever dis- communications and detection of digital content in continu- placement signal identifies the presence of electron spins. ous signals or images with high noise contamination. Our One model for the cantilever-electron spin interaction is motivation is single electron-spin detection in Magnetic Res- suggested by the Stern-Gerlach experiment [3]. The resultant onance Force Microscopy (MRFM), which is an emerging signal model takes the form of the classic C-T random tele- molecular scale imaging modality. We will obtain an optimal graph process. A finite-dimensional optimal detector does discrete time iterative detection algorithm and an associated not exist; however, a hybrid Bayes/Generalized Likelihood upper bound on the Receiver Operating Characteristic (ROC) Ratio (GLR) detector was developed [4, 5]. Unfortunately, curve. These results will be used to show that the optimal it has a running time that makes a real-time implementation detector reduces to a simple filtered energy detector when unfeasible at this point. Consequently, we have formulated the state transition probabilities are symmetric, the signal- a (generalized) discrete-time (D-T) analog of the C-T ran- to-noise ratio (SNR) is low, and the time-bandwidth product dom telegraph process. The optimal Likelihood Ratio Test (LRT) for the D-T random telegraph can be derived, with a is high. In the general case when the state transition proba- ¢¤£ bilities can be either symmetric or asymmetric, the optimal complexity of N ¥ . Here, N is the number of samples per observation. Surprisingly, it can be shown that there exist detector reduces to a hybrid filtered energy/amplitude/energy ¢¦£ detector when the SNR is low and there are a large number simpler detectors, all with N ¥ complexity, that approxi- of samples per observation. mate the LRT. This paper describes two results. Firstly, the filtered en- ergy detector is approximately asymptotically optimal for 1. INTRODUCTION the D-T random telegraph model under the conditions of The detection of piecewise constant binary valued C-T sig- low SNR, high time-bandwidth product, and symmetric tran- nals with Markovian state transitions occurs often in differ- sition probabilities. Secondly, in the general D-T ran- ent areas of science and engineering. One of the most com- dom telegraph model (which includes both symmetric and mon examples of such Markovian C-T signals is the random asymmetric transition probabilities), a hybrid filtered en- telegraph signal whose transitions are governed by a Pois- ergy/amplitude/energy detector is approximately asymptot- son process. In this paper, we propose optimal detectors for ically optimal under the conditions of low SNR and long ob- signals that can be approximated by two-state signals with servation time. The MRFM single electron spin detection re- Markov transitions. The prime motivation for our work is sults described in this paper are more fully developed in [6], the detection of a single electron spin in a very sensitive as are additional finite (non-binary) state Markov models and physics (MRFM) experiment called interrupted Oscillating detectors. Cantilever-driven Adiabatic Reversal (iOSCAR) described The outline of this paper is as follows. In Section 2, we below. Accordingly, while our optimal filter structures and briefly review the iOSCAR experiment. Then, in Section 3, bounds are more generally applicable, we present all of our we describe the C-T and D-T random telegraph process. Sec- results in this specific application domain. tion 4 develops the optimal LRT detector for the D-T random A general MRFM experiment involves the detection of telegraph model and its approximately optimal forms. Simu- perturbations of a thin micrometer-scale cantilever whose tip lation results are presented in Section 5. incorporates a submicron ferromagnet. When no electron spins are present, the cantilever acts as a harmonic oscillator. 2. DESCRIPTION OF THE IOSCAR EXPERIMENT Unpaired electron spins in the sample behave like magnetic dipoles, exerting perturbing forces on the cantilever. Thus, In the experiment, a submicron ferromagnet is placed on the presence of electron spins can be detected based on mea- the tip of a cantilever that sits approximately 50 nanome- ters above a sample. In the presence of an applied RF field, This work was supported in part by the DARPA Mosaic program under the electron in the sample undergoes magnetic resonance if ARO contract DAAD19-02-C-0055. the RF field frequency matches the Larmor frequency. Only those spins that are within a thin resonant slice will satisfy Additive White Gaussian Noise (AWGN) with variance σ 2, £ w the condition for magnetic resonance and interact with the and s t ¥ is a random telegraph signal containing only the ran- cantilever. If the cantilever is forced into mechanical oscilla- dom transitions. The detection problem for this model is to tion by positive feedback, the tip motion will cause the po- design a test between the two hypotheses: sition of the resonant slice to oscillate. As the slice passes £ £ £ ¥ back and forth through an electron spin in the sample, the H0 (spin absent) : y t ¥ w t £ £ spin direction will be cyclically inverted due to an effect £ £ ¥ ¥ H1 (spin present) : y t ¥ s t w t (1) called adiabatic rapid passage [7]. The cyclic inversion is synchronous with the cantilever motion and affects the can- © for t ¨ 0 T . tilever dynamics by changing the effective stiffness of the cantilever. Therefore, the spin-cantilever interaction can be 3.2 Generalized D-T Random Telegraph detected by measuring small shifts in the period of the can- ¥¥¥ © tilever oscillation using a laser interferometer. This method- ¨ Here, we treat y y0 yN 1 as samples of the baseband ology has been used successfully to detect small ensembles output of the frequency demodulator§ and correlator. The D-T of electron spins [1, 2]. random telegraph signal is a D-T Markov chain, and will be £ ζ ζ ¢ ! ! ¢ ζ ¥ Denote by B1 t the amplitude of the RF magnetic field, denoted by i, where i A A 0 i N 1, and 0 £ " B0 t ¥ the amplitude of the tip magnetic field at the spin lo- is equally likely to be either A, where A can be computed £ £ ¡ ¥£¢ cation, ωRF the RF field frequency, and ∆B0 t ¥ B0 t from experimental parameters. The transition probabilities ¤ ζ # ωRF γ the off-resonance field amplitude. The constant γ of i i 1 are as follows: 10 1 1 £ π § ∆ ¦ § ¥ 5 ¥ 6 10 s T is the gyromagnetic ratio. If B t ' 0 & ζ ζ varies sufficiently slowly such that the adiabatic criterion p i i 1 A ¤ ( § £ £ 2 £ ζ ζ ¢ *¢ ¥ © ¥ ¨ ∆ γ 1 p i A i 1 A ) d B0 t dt B t is satisfied, the spin can be assumed ' § % 1 ζ ζ ¥ $ £ £ P i i 1 ζ ζ (2) +¢ § q i i 1 A ¥ ¢ ¥ to remain aligned with either Beff t or Beff t , where Beff ζ § ζ *¢ 1 ¢ q i A i 1 A is the effective magnetic field. These are the spin-lock and § anti-spin-lock conditions, respectively. , We restrict 0 , p q 1. If p q, we say that the tran- sition probabilities are symmetric, and when p - q, we shall 3. MRFM SIGNAL MODELS say that they are asymmetric. For the symmetric case, we 3.1 C-T random telegraph can match the C-T model to the D-T model by equating the expected number of transitions of the Poisson process λ ¢ Assume that the cyclic adiabatic criterion holds. In the to that of the Markov chain. This results in p 1 T , £ s iOSCAR protocol, B1 t ¥ is turned off after every Nskip cycles where Ts is the sampling time interval and λ is the expected over a half-cycle duration: this induces periodic transitions number of transitions per second. Define the signal vector £ between the spin-lock and anti-spin-lock states. Let z˜ t ¥ be £ ζ ¨ ζ ¥¥¥ ζ © +¨ ¥¥¥ ©. 0 N 1 and the noise vector w w0 wN 1 . § § the result of frequency demodulating z t ¥ to baseband. Using £ We shall model the wi’s as independent and identically dis- the perturbation analysis in [8], it can be shown that z˜ t ¥ is a £ tributed (i.i.d.) Gaussian random variables (r.v.s) with mean square wave whose transitions coincide with those of B1 t ¥ 0 and variance σ 2. The detection problem is then to decide and is approximately periodic. Thus, spin detection can be £ between: accomplished by correlating z˜ t ¥ with a square wave refer- £ ence derived from B1 t ¥ .
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