Abstract:
A system for performing Walsh reconstruction of time dependent fields with a quantum probe that includes a diamond structure that includes nitrogen-vacancy (NV) centers for sensing magnetic and electric fields. An arbitrary waveform generator and microwave waveguides manipulate the quantum spins in the diamond structure using a plurality of coherent control sequences to measure the arbitrary profile of time-varying magnetic fields. A laser that optically addresses the quantum spins in the diamond structure for initialization and readout.

Description:
PRIORITY INFORMATION 
       [0001]    This application claims priority from provisional application Ser. No. 61/824,600 filed May 17, 2013, which is incorporated herein by reference in its entirety. 
     
    
     SPONSORSHIP INFORMATION 
       [0002]    This invention was made with government support under Contract No. HR0011-11-C-0073 awarded by the Defense Advanced Research Projects Agency. The government has certain rights in the invention. 
     
    
     BACKGROUND OF THE INVENTION 
       [0003]    The invention is related to the field of quantum sensors and quantum dynamic parameter estimation, and in particular to time-resolved magnetic sensing with electronic spins in diamond. 
         [0004]    Precision measurement is among the most important applications of quantum physics. For example, quantum control of atomic systems forms the physical basis of the world&#39;s best clocks. The purpose of quantum metrology is to detect a small external field, coupled to the sensor by a Hamiltonian: H b   S =b(t)κS z , where S z  is the spin operator of the quantum sensor. Here, b(t) can be an external magnetic field or the detuning of a laser from a clock transition while κ is the spin&#39;s coupling to the field. 
         [0005]    The working principle of almost any quantum metrology scheme can be reduced to a Ramsey experiment, where the field is measured via the induced phase difference between two states of the quantum sensor. The quantum probe is prepared in a superposition of two eigenstates of the operator S z , e.g. 
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         [0000]    and is then let evolve freely under the action of the external field. The eigenstates acquire a phase that is proportional to their eigenvalues (here 0 and 1) and to the parameter to be estimated: 
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         [0000]    with φ(τ)κ∫ 0   τ b(t)dt. This phase difference can be converted in a population difference and measured out. The figure of merit for quantum sensitivity is the smallest field Υb min  that can be read out during a total time T. For a single spin ½, if the sensing time is limited to τ (e.g. by environmental decoherence) then: 
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       SUMMARY OF THE INVENTION 
       [0006]    According to one aspect of the invention, there is provided a system for performing Walsh reconstruction of time dependent fields with a quantum probe. The system includes a diamond structure that includes nitrogen-vacancy (NV) centers for sensing magnetic and electric fields. An arbitrary waveform generator and microwave waveguides manipulate the quantum spins in the diamond structure using a plurality of coherent control sequences to measure the arbitrary profile of time-varying magnetic fields. A laser that optically addresses the quantum spins in the diamond structure for initialization and readout. 
         [0007]    According to another aspect of the invention, there is provided a method of performing Walsh reconstruction. The method includes positioning a diamond structure that includes nitrogen-vacancy (NV) centers for sensing magnetic and electric fields. Also, the method providing a plurality of coherent control sequences to measure the arbitrary profile of time-varying magnetic fields arbitrary using waveform generator and microwave waveguides that manipulate the quantum spins in the diamond structure. The method includes optically addressing the quantum spins in the diamond structure for initialization and readout using a laser. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0008]      FIG. 1  is a schematic diagram illustrating the Walsh reconstruction technique for measuring time-varying field with quantum probes; 
           [0009]      FIGS. 2A-2B  are graphs illustrating the coherent control sequences and the N-point functional approximation of the field is obtained by sampling the field; 
           [0010]      FIGS. 3A-3C  are graphs illustrating the invention approach to Walsh reconstruction; 
           [0011]      FIGS. 4A-4B  are graphs illustrating a 32-point reconstruction of a bichromatic magnetic field; and 
           [0012]      FIGS. 5A-5B  are graphs illustrating a 64-point reconstruction of a magnetic field radiated by a skew normal impulse flowing through the physical model of a neuron. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0013]    The invention introduces a novel approach to accurately reconstruct the arbitrary profile of time-varying fields using coherent control sequences associated with the Walsh functions. These sequences act as digital filters that efficiently extract information about the field dynamics while suppressing decoherence. One has experimentally demonstrated the inventive approach by performing proof-of-principle reconstruction of the magnetic field radiated by a physical model of a neuron using a single electronic spin in diamond. The method is useful for performing time-resolved magnetic sensing with quantum probes in a broad array of physical and biological systems at the nanometer scale. 
         [0014]    Quantum probes enable the sensitive detection of time-varying fields with nanometer spatial resolution. In particular, nitrogen-vacancy (NV) centers in diamond have recently emerged as promising sensors of magnetic and electric fields. Although coherent control techniques have measured the amplitude of constant or oscillating fields, these techniques are unable to measure fields with unknown dynamics. Quantum estimation techniques can be applied to measure time-varying fields by monitoring the shift in the resonance energy of a qubit sensor, e.g., via Ramsey interferometry measurements. The qubit sensor, first prepared in an equal superposition of its eigenstates, accumulates a phase 
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         [0000]    where γ is the strength of the interaction with the time-varying field b(t) during the acquisition period T. 
         [0015]    The field dynamics could be mapped by measuring its amplitude over N successive acquisition periods of duration δt=T/N; however, this sequential sensing method is often impractical due to short coherence times and it is inefficient. Instead, the invention proposes to reconstruct the temporal profile of time-varying fields by using a set of digital filters, implemented with coherent control sequences over the whole acquisition period T, which simultaneously extract information about the field dynamics and protect the qubit sensor from noise. The method can achieve the field dynamics reconstruction N-times faster than the sequential sensing method (or with a √N smaller error for the same fixed time). Specifically, one can use control protocols, as shown in  FIG. 1 , associated with the Walsh functions, which form a complete orthonormal basis of digital filters and are easily implementable experimentally. Because a π-pulse effectively reverses the evolution of the qubit sensor, sequences of π-pulses act as digital filters that sequentially switch the sign of the evolution between ±1. 
         [0016]    In particular,  FIG. 1  shows the Walsh reconstruction technique  2  for measuring time-varying field with quantum probes in accordance with an exemplary embodiment of the invention. A single NV center in diamond  6 , optically initialized and read out by confocal microscopy  4 , is manipulated with coherent control sequences  6  to measure the arbitrary profile of time-varying magnetic fields  12  radiated by a coplanar waveguide  8  under ambient conditions. 
         [0017]    If w m (t/T) is the digital filter created by applying m control π-pulses at the zero-crossings of the m-th Walsh function, the normalized phase acquired by the qubit sensor is 
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         [0000]    where {circumflex over (b)}(m) is the m-th Walsh coefficient, defined as the Walsh transform of b(t) evaluated at sequence number (sequency) m. Successive measurements with the first N Walsh sequences {w m (t/T)} m=0   N−1  give a set of N Walsh coefficients {{circumflex over (b)}(m)} m=0   N−1  that can be used to reconstruct an N-point functional approximation to the field 
         [0000]        b   N ( t )=Σ {circumflex over (b)} ( m ) w   m ( t/T )  Eq. 2
 
         [0000]    Eq. (2) is the inverse Walsh transform of order N, which gives the best least-squares Walsh approximation to b(t). With few assumptions or prior knowledge about the field dynamics, the reconstruction can be shown to be accurate with quantifiable truncation errors and convergence criteria. 
         [0018]    The Walsh reconstruction method was experimentally demonstrated by measuring increasingly complex time-varying magnetic fields with a single NV center in an isotopically purified diamond sample. NV centers in diamond have recently emerged as promising sensors for magnetic and electric fields, rotations and temperature. These sensors are ideal for nanoscale imaging of living biological systems due to their low cytotoxicity, surface functionalizations, optical trapping capability and long coherence time under ambient conditions. A single NV center is optically initialized and read out by confocal microscopy under ambient conditions. A coplanar waveguide  10  delivers both resonant microwave pulses and off-resonant magnetic fields produced by an arbitrary waveform generator. 
         [0019]      FIG. 2A  shows the coherent control sequences, acting as digital filters on the evolution of the qubit sensor to extract information about time-varying fields.  FIG. 2B  shows an N-point functional approximation of the field is obtained by sampling the field with a set of N digital filters taken from the Walsh basis, which contain some known set of decoupling sequences such as the even-parity CPMG sequences (w 2n ) and the odd-parity PDD sequences (w 2n−1 ) .    
         [0020]    First, the monochromatic sinusoidal fields, b(t)=b sin(2πvt+α), are reconstructed by measuring the Walsh spectrum up to fourth order (N=2 4 ). The m-th Walsh coefficient {circumflex over (f)}(m) of the normalized field f(t)=b(t)/b was obtained by sweeping the amplitude of the field and measuring the slope of the signal S m (b)=sin(γb{circumflex over (f)}(m)T) the origin ( FIG. 3A ).  FIG. 3B  shows the measured non-zero Walsh coefficients of the Walsh spectrum. As shown in  FIG. 3C , the 16-point reconstructed fields are in good agreement with the expected fields. The Walsh reconstruction method is thus phase selective as it discriminates between fields with the same frequency but different phases. 
         [0021]    One can further reconstruct a bichromatic field b(t)=b[a 1  sin(2πv 1 t+α 1 )+a 2  sin(2πv 2 t+a 2 )].  FIG. 4A  shows the measured Walsh spectrum up to fifth order (N=2 5 ). As shown in  FIG. 4B , the 32-point reconstructed field agrees with the expected field, which demonstrates the accuracy of the Walsh reconstruction method. In contrast, sampling the field with an incomplete set of digital filters, such as the CPMG and PDD sequences, extracts only partial information about the field dynamics. By linearity of the Walsh transform, the Walsh reconstruction method applies to any polychromatic field (and by extension to any time-varying field), whose frequency spectrum lies in the acquisition bandwidth [1/T, 1/τ] set by the coherence time T&lt;T 2  and the maximum sampling time τ=T/N, which is in turn limited by the finite duration of the control π-pulses. 
         [0022]    The performance of the Walsh reconstruction method is determined by the reconstruction error e N  and the measurement sensitivity η N . The least-squares reconstruction error e N =//b N (t)−b(t)// 2  due to truncation of the Walsh spectrum up to N==2 n  coefficients is bounded by e N ≦max tε[0,T] |∂ t b(t)|/2 n+1  and vanishes to zero as N tends to infinity (as needed for perfect reconstruction). This implies that although the resources grow exponentially with n, the error converges exponentially quickly to zero, and only a finite number of coefficients is needed to accurately reconstruct the field. 
         [0023]    The measurement sensitivity of the m-th Walsh sequence in M measurements, η m =δbm √{square root over (MT)}, gives the minimum field amplitude, δbm=ΔS m /∂S m /∂b m | √{square root over (M)}, that can be measured with fixed resources: 
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         [0000]    where γ e =2π·28 Hz/nT is the gyromagnetic ratio of the NV electronic spin, n NV  is the density of NV centers in the sensing volume, and C accounts for inefficient photon collection and finite contrast due to spinstate mixing during optical measurements [1, 23]. Here vm=(e −T/T   2   (m) ) p(m) ≦1 is the visibility decay, where T 2 (m) and p(m) characterize the decoherence of the qubit sensors during the m-th Walsh sequence in the presence of a specific noise environment. 
         [0024]    The sensitivity η m  is thus the ratio between a field independent factor ̂ηm and a field-dependent Walsh coefficient |{circumflex over (f)}(m)|. One can estimate {circumflex over (η)} m ˜100 pT √{square root over (μm 3 )}/√{square root over (Hz)} based on physical parameters and previous experiments with ensemble of NV centers. Note if the field dynamics is known, so that the Walsh coefficients can be precomputed, the best estimate of the field amplitude is obtained by choosing the Walsh sequence that gives the optimal measurement sensitivity. Because the Walsh sequences suppress dephasing noise and can extend coherence times by many order of magnitudes, the choice of the most sensitive Walsh sequence involves a trade-off between large Walsh coefficients and long coherence times. 
         [0025]    The amplitude resolution of the Walsh reconstruction method, δb N =√{square root over (Σ m δ{circumflex over (b)} m   2 )}, gives the smallest variation of the reconstructed field that can be measured from the Walsh spectrum of order N. If each Walsh coefficient is obtained from M measurements over the acquisition period T, the measurement sensitivity of the Walsh reconstruction method, η N =δb N √{square root over (MNT)}, is 
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         [0026]    The measurement sensitivity η N  combines with the reconstruction error e N  to determine the accuracy of the Walsh reconstruction method. If some small coefficients cannot be resolved due to low signal visibility, the increase in reconstruction error can be analytically quantified using data compression results. In the same way, the acquisition time can be reduced by sampling only the most significant coefficients and discarding other negligible coefficients. Furthermore, if the field is sparse in some known basis, which is often the case, a logarithmic scaling in resources can be achieved by using compressed sensing methods based on convex optimization algorithms. 
         [0027]    The Walsh reconstruction method is readily applicable to measuring time-varying fields in a variety of physical systems. Promising applications include magnetic resonance spectroscopy of spins extrinsic to the diamond lattice, measurements of the dynamics of magnetic nanostructures or magnetic vortices in nanodisk chains. Whereas spectroscopic methods based on periodic dynamical decoupling have been used to reconstruct the noise spectral density in a variety of solid-state quantum systems, the Walsh reconstruction method could be used to simplify the problem of spectral reconstruction of stochastic fields by removing the need for functional approximations or non-linear deconvolution algorithms. 
         [0028]    A paradigmatic application for quantum sensors is measuring magnetic fields radiated by single neurons, triggered on-demand by external excitation. The flow of ions in the neuronal axon radiates a magnetic field on the order of hundreds of pT that could be measured with state-of-the-art diamond magnetometers over ms acquisition periods. As a proof-of-principle implementation, the magnetic field radiated is measured by a physical model of a neuron undergoing an action potential Φ(t) approximated by a skew normal impulse. Due to its linear response in the kHz regime], the coplanar waveguide acts as the physical model of a neuron, with the radiated magnetic field given by the derivative of the electric field: b(t)=dΦ(t)/dt. 
         [0029]    The Walsh coefficients were measured by fixing the amplitude of the field and sweeping the phase of the last read-out pulse to reconstruct the absolute field b(t) rather than the normalized field f(t).  FIG. 5A  shows the measured Walsh spectrum up to fifth order (N=25). As shown in  FIG. 5B , the 32-point reconstructed field is in good agreement with the expected field. Although the acquisition time and the field amplitude were chosen to satisfy the limitations of the experimental apparatus, extending the reconstruction method to measuring in-vivo neuronal magnetic fields is within reach with improvements in collection efficiency, pulse duration, coherence times, proximity to the field source and by using ensemble of NV centers. 
         [0030]    The invention utilizes a novel approach where control sequences acting as digital filters on the evolution of a single NV electronic spin to efficiently sample and accurately reconstruct the arbitrary profile of time-varying fields. The Walsh reconstruction method generalizes current sensing methods and allows for accurate reconstruction of time-varying fields with quantifiable errors and formal convergence criteria. The method achieves faster dynamic parameter reconstruction than naïve, sequential methods or a sensitivity enhancement at fixed time. Possible extensions include exploiting other digital sampling techniques, such as the wavelet transform. The method is compatible with compressed sensing algorithms that achieve a reduction in both resources and reconstruction error. The Walsh reconstruction method is readily implementable with quantum probes in practical applications such as neuronal activity recording with nanodiamonds functionalized to target specific receptor sites on neuronal cells, spatiotemporal field imaging of neuronal. 
         [0031]    Although the present invention has been shown and described with respect to several preferred embodiments thereof, various changes, omissions and additions to the form and detail thereof, may be made therein, without departing from the spirit and scope of the invention.