Abstract
A key challenge of magnetometry lies in the simultaneous optimization of magnetic field sensitivity and maximum field range. In interferometrybased magnetometry, a quantum twolevel system acquires a dynamic phase in response to an applied magnetic field. However, due to the 2π periodicity of the phase, increasing the coherent interrogation time to improve sensitivity reduces field range. Here we introduce a route towards both large magnetic field range and high sensitivity via measurements of the geometric phase acquired by a quantum twolevel system. We experimentally demonstrate geometricphase magnetometry using the electronic spin associated with the nitrogen vacancy (NV) color center in diamond. Our approach enables unwrapping of the 2π phase ambiguity, enhancing field range by 400 times. We also find additional sensitivity improvement in the nonadiabatic regime, and study how geometricphase decoherence depends on adiabaticity. Our results show that the geometric phase can be a versatile tool for quantum sensing applications.
Introduction
The geometric phase^{1,2} plays a fundamental role in a broad range of physical phenomena^{3,4,5}. Although it has been observed in many quantum platforms^{6,7,8,9} and is known to be robust against certain types of noise^{10,11}, geometric phase applications are somewhat limited, including certain protocols for quantum simulation^{12,13} and computation^{14,15,17}. However, when applied to quantum sensing, e.g., of magnetic fields, unique aspects of the geometric phase can be exploited to allow realization of both good magnetic field sensitivity and large field range in one measurement protocol. This capability is in contrast to conventional dynamicphase magnetometry, where there is a tradeoff between sensitivity and field range. In dynamicphase magnetometry using a twolevel system (e.g., two spin states), the amplitude of an unknown magnetic field B can be estimated by determining the relative shift between two energy levels induced by that field (Methods). A commonly used approach is to measure the dynamic phase accumulated in a Ramsey interferometry protocol. An initial resonant π/2 pulse prepares the system in a superposition of the two levels. In the presence of an external static magnetic field B along the quantization axis, the system evolves under the Hamiltonian H = ħγBσ_{z}/2, where γ denotes the gyromagnetic ratio and σ_{z} is the zcomponent of the Pauli spin vector. During the interaction time T (limited by the spin dephasing time T_{2}*), the Bloch vector s(t) depicted on the Bloch sphere precesses around the fixed Larmor vector R = (0, 0, γB), and acquires a dynamic phase ϕ_{d} = γBT. The next π/2 pulse maps this phase onto a population difference P = cos ϕ_{d}, which can be measured to determine ϕ_{d} and hence the magnetic field B (Supplementary Note 1).
Such dynamicphase magnetometry possesses two wellknown shortcomings. First, the sinusoidal variation of the population difference with magnetic field leads to a 2π phase ambiguity in interpretation of the measurement signal and hence determination of B. Specifically, since the dynamic phase is linearly proportional to the magnetic field, for any measured signal P_{meas} (throughout the text, this value corresponds to (ΔFL/FL) × k, where k is a constant that depends on NV readout contrast), there are infinite magnetic field ambiguities: B_{m} = (γT)^{−1} (cos^{−1}P_{meas} + 2πm), where m = 0, ±1, ±2 …±∞. Thus, the range of magnetic field amplitudes that one can determine without modulo 2π phase ambiguity is limited to one cycle of oscillation: B_{max} ∝ 1/T (Supplementary Note 2, Supplementary Figure 5). Second, there is a tradeoff between magnetic field sensitivity and field range, as the interaction time also restricts the shotnoiselimited magnetic field sensitivity: η ∝ 1/T^{1/2}. Consequently, an improvement in field range via shorter T comes at the cost of a degradation in sensitivity (Supplementary Note 3). Use of a closedloop lockin type measurement^{18}, quantum phase estimation algorithm^{19,20}, or nonclassical states^{21,22} can alleviate these disadvantages; however, such approaches require either a continuous measurement scheme with limited sensitivity, large resource overhead (additional experimental time) or realization of longlived entangled or squeezed states.
In the present work, we use the electronic spin associated with a single nitrogen vacancy (NV) color center in diamond to demonstrate key advantages of geometricphase magnetometry: (i) it resolves the 2π phase ambiguity limiting dynamicphase magnetometry; and (ii) it decouples magnetic field range and sensitivity, leading to a 400fold enhancement in field range at constant sensitivity in our experiment. We also show additional improvement of magnetic field sensitivity in the nonadiabatic regime of mixed geometric and dynamicphase evolution. By employing a power spectral density analysis^{23}, we find that adiabaticity plays an important role in controlling the degree of coupling to environmental noise and hence the spin coherence timescale.
Results
Geometricphase magnetometry protocol
To implement geometricphase magnetometry, we use a modified version of an experimental protocol (“Berry sequence”) previously applied to a superconducting qubit^{9}. In our realization, the NV spin sensor is placed in a superposition state by a π/2 pulse, where the driving frequency of the π/2 pulse is chosen to be resonant with the NV m_{s} = 0 ↔ m_{s} = + 1 transition at constant bias field B_{bias} (≈9.6 mT in our experiment) aligned with the NV axis. A small signal field B (~100 µT in our experiment) is then applied parallel to B_{bias}, and the NV spin acquires a geometric phase due to offresonant microwave driving with control parameters cycled along a closed path as illustrated in Fig. 1b (Methods). Under the rotating wave approximation, the effective twolevel Hamiltonian is given by:
Here, Ω is the NV spin Rabi frequency for the microwave driving field, ρ is the phase of the driving field, and σ = (σ_{x}, σ_{y}, σ_{z}) is the Pauli spin vector. By sweeping the phase, the Larmor vector R(t) = R*(sinθ cosρ, sinθ cosρ, cosθ), where cosθ = γB/(Ω^{2} + (γB)^{2})^{1/2}, R = (Ω^{2} +(γB)^{2})^{1/2}, rotates around the zaxis. The Bloch vector s(t) then undergoes precession around this rotating Larmor vector (for detailed picture of the measurement protocol, see Supplementary Fig 2). If the rotation is adiabatic (i.e., adiabaticity parameter \(A \equiv \dot \rho \sin \theta /2R \ll 1\)), then the system acquires a geometric phase proportional to the product of (i) the solid angle Θ = 2π(1 − cosθ) subtended by the Bloch vector trajectory and (ii) the number of complete rotations N of the Bloch vector around the Larmor vector in the rotating frame defined by the frequency of the initial π/2 pulse. We apply this Bloch vector rotation twice during the interaction time T, with alternating direction separated by a π pulse, which cancels the accumulated dynamic phase and doubles the geometric phase: ϕ_{g} = 2NΘ (Supplementary Note 1). A final π/2 pulse allows this geometric phase to be determined from standard fluorescence readout of the NV spinstate population difference:
This normalized geometricphase signal (Supplementary Note 1) exhibits chirped oscillation as a function of magnetic field. There are typically only a small number of field ambiguities that give the same signal P_{meas}; these can be resolved uniquely by measuring the slope dP_{meas}/dB (Supplementary Note 2, Supplementary Fig. 5). From the form of Eq. (2) it is evident that at large B, cosine signal approaches to zero like B^{−2}, and the slope goes to zero. Hence, we define the field range as the largest magnetic field value (B_{max}) that gives the last oscillation minimum in the signal: B_{max} ∝ Ω N^{1/2}. Importantly, the field range of geometricphase magnetometry has no dependence on the interaction time T. If the magnetic field is below B_{max}, then one can make a geometricphase magnetometry measurement with optimal sensitivity η ∝ Ω N^{−1} T^{1/2} (Supplementary Note 3).
Comparison between dynamic and geometricphase magnetometry
We implemented both dynamic and geometricphase magnetometry using the optically addressable electronic spin of a single NV color center in diamond (Fig. 2a) (Supplementary Figs. 13). NVdiamond magnetometers provide high spatial resolution under ambient conditions^{24,25,26}, and have therefore found wideranging applications, including in condensed matter physics^{27,28}, the life sciences^{29,30}, and geoscience^{31}. At an applied bias magnetic field of 9.6 mT, the degeneracy of the NV m_{s} = ± 1 levels is lifted. The twolevel system used in this work consists of the ground state magnetic sublevels m_{s} = 0 and m_{s} = +1, which can be coherently addressed by applied microwave fields. The hyperfine interaction between the NV electronic spin and the host ^{14}N nuclear spin further splits the levels into three states, each separated by 2.16 MHz. Upon green laser illumination, the NV center exhibits spinstatedependent fluorescence and optical pumping into m_{s} = 0 after a few microseconds. Thus, one can prepare the spin states and determine the population by measuring the relative fluorescence (see Methods for more details).
First, we performed dynamicphase magnetometry using a Ramsey sequence to illustrate the 2π phase ambiguity and show how the dependence on interaction time gives rise to a tradeoff between field range and magnetic field sensitivity. We recorded the NV fluorescence signal as a function of the interaction time T between the two microwave π/2 pulses (Fig. 1a). Signal contributions from the three hyperfine transitions of the NV spin result in the observed beating behavior seen in Fig. 2b. We fixed the interaction time at T = 0.2, 0.5, 1.0 μs, varied the external magnetic field for each value of T, and observed a periodic fluorescence signal with a 2π phase ambiguity (Fig. 2c). The oscillation period decreased as the interaction time was increased, indicating a reduction in the magnetic field range (i.e., smaller B_{max}). In contrast, the magnetic field sensitivity, which depends on the maximum slope of the signal, improved as the interaction time increased. For each value of T, we fit the fluorescence signal to a sinusoid dependent on the applied magnetic field and extracted the oscillation period and slope, which we used to determine the experimental sensitivity and field range. From this procedure, we obtained η ∝ T^{−0.49(6)} and B_{max} ∝ T^{−0.96(2)}, consistent with expectations for dynamicphase magnetometry and illustrative of the tradeoff inherent in optimizing both η and B_{max} as a function of interaction time (Supplementary Fig. 7).
Next, we used a Berry sequence to demonstrate two key advantages of geometricphase magnetometry: i.e., there is neither a 2π phase ambiguity nor a sensitivity/fieldrange tradeoff with respect to interaction time. For fixed adiabatic control parameters of Ω/2π = 5 MHz and N = 3, the observed geometricphase magnetometry signal P_{meas} has no dependence on interaction time T (Fig. 2d). Varying the external magnetic field with fixed interaction times T = 4.0, 6.0, 8.0 μs, P_{meas} exhibits identical chirped oscillations for all T values (Fig. 2e), as expected from Eq. (2). From the P_{meas} data we extract dP_{meas}/dB, which allows us to determine the magnetic field uniquely for values within the oscillatory range (Supplementary Note 2), and also to quantify B_{max} from the last minimum point of the chirped oscillation (Fig. 2e). Additional measurements of the dependence of P_{meas} on the adiabatic control parameters Ω, N, and T (Supplementary Figs. 4, 6) yield the scaling of sensitivity and field range: η ∝ Ω^{1.2(5)}N^{−0.92(1)}T^{0.46(1)} and B_{max} ∝ Ω^{0.9(1)}N^{0.52(5)}T^{0.02(1)}, which is consistent with expectations and shows that geometricphase magnetometry allows η and B_{max} to be independently optimized as a function of interaction time (Supplementary Fig. 7).
In Fig. 3 we compare the measured sensitivity and field range for geometricphase and dynamicphase magnetometry. For each point displayed, the sensitivity is measured directly at small B (0.01 ~ 0.1 mT), whereas the field range is calculated from the measured values of N and Ω (for geometricphase magnetometry) and T (for dynamicphase magnetometry, with T limited by the dephasing time T_{2}*), following the scaling laws give above. Since geometricphase magnetometry has three independent control parameters (T, N, and Ω), B_{max} can be increased without changing sensitivity by increasing N and Ω while keeping the ratio N/Ω fixed. Such “smart control” allows a tenfold improvement in geometricphase sensitivity (compared to dynamicphase measurements) for B_{max} ~ 1 mT, and a 400fold enhancement B_{max} at a sensitivity of ~2 μT Hz^{−1/2}. Similarly, the sensitivity can be improved without changing B_{max} by decreasing the interaction time, with a limit set by the adiabaticity condition (\(A \equiv \dot \rho \sin \theta /2R \approx N/\Omega T \ll 1\)).
Geometricphase magnetometry in nonadiabatic regime
Finally, we explored geometricphase magnetometry outside the adiabatic limit by performing Berry sequence experiments and varying the adiabaticity parameter by more than two orders of magnitude (from A ≈ 0.01−5). We find good agreement between our measurements and simulations, with an onset of nonadiabatic behavior for A \(\gtrsim\) 0.2 (Supplementary Figure 8). At each value of the adiabaticity parameter A, we determine the magnetic field sensitivity from the largest slope of the measured magnetometry curve. (The magnetometry curve is the plot of P_{meas} obtained as a function of applied magnetic field B.) To compare with the best sensitivity provided by dynamicphase magnetometry, we fix the interaction time at T \(\approx\) T_{2}*/2 in the nonadiabatic geometricphase measurements. We find that the sensitivity of geometricphase magnetometry improves in the nonadiabatic regime, and becomes smaller than the sensitivity from dynamicphase measurements for A \(\gtrsim\) 1.0 (Fig. 4a).
To understand this behavior, we recast the sensitivity scaling in terms of the adiabaticity parameter and interaction time, η ∝ A^{−1}T^{−1/2} and investigated the tradeoff between these parameters. (Note that in the nonadiabatic regime the Bloch vector no longer strictly follows the Larmor vector, and thus the sensitivity scaling is not exact.) We performed a spectral density analysis to assess how environmental noise leads to both dynamic and geometricphase decoherence, with the relative contribution set by the adiabaticity parameter A, thereby limiting the interaction time T. We take the exponential decay of the NV spin coherence W(T) ~ exp(−χ(T)), characterized by the decoherence function χ(T) given by
Here, S(ω) is a spectral density function that describes magnetic noise from the environment; F_{0}(ωT) = 2sin^{2}(ωT/2) is the filter function for geometricphase evolution in the Berry sequence, which is spectrally similar to a Ramsey sequence, with maximum sensitivity to static and low frequency (\(\lesssim\)1/T) magnetic fields; and F_{1}(ωT) = 8sin^{4}(ωT/4) is the filter function for dynamicphase evolution in the Berry sequence, which is spectrally similar to a Hahnecho sequence, with maximum sensitivity to higher frequency (\(\gtrsim\)1/T) magnetic fields (Supplementary Note 4).
Geometricphase coherence time
Figure 4b shows examples of the measured decay of the geometricphase signal (P_{meas}) as a function of interaction time T and adiabaticity parameter A. From such data we extract the geometricphase coherence time T_{2g} by fitting P_{meas} ~ exp[−(T/T_{2g})^{2}]. We observe four regimes of decoherence behavior (Fig. 4c), which can be understood from Eq. (3) and its schematic spectral representation in Fig. 4d. For A < 0.1 (adiabatic regime), dynamicphase evolution (i.e., Hahnecholike behavior) dominates the decoherence function χ(T) and thus T_{2g} ~ T_{2} ≈ 500 μs. For 0.1 ≤ A < 1.0 (intermediate regime), the coherence time is inversely proportional to the adiabaticity parameter (T_{2g} ~ T_{2}*/A) as geometricphase evolution (with Ramseylike dephasing) becomes increasingly significant. For A \(\approx\) 1.0 (nonadiabatic regime), geometricphase evolution dominates χ(T) at long times and thus T_{2g} ~ T_{2}* ≈ 50 μs. For \(A \gg 1.0\) (strongly nonadiabatic limit), the driven rotation of the Larmor vector is expected to average out during a Berry sequence (Fig. 1b) and only the zcomponent of the Larmor vector remains. Thus, the Berry sequence converges to a Hahnecholike sequence and the coherence time is expected to increase to T_{2} for very large A.
Discussion
In summary, we demonstrated an approach to NVdiamond magnetometry using geometricphase measurements, which avoids the tradeoff between magnetic field sensitivity and maximum field range that limits traditional dynamicphase magnetometry. For an example experiment with a single NV, we realize a 400fold enhancement in static (DC) magnetic field range at constant sensitivity. We also explored geometricphase magnetometry as a function of adiabaticity, with good agreement between measurements and model simulations. We find that adiabaticity controls the coupling between the NV spin and environmental noise during geometric manipulation, thereby determining the geometricphase coherence time. Furthermore, we showed that operation in the nonadiabatic regime, where there is mixed geometric and dynamicphase evolution, allows magnetic field sensitivity to be better than that of dynamicphase magnetometry. We expect that geometricphase AC field sensing will provide similar advantages to dynamicphase magnetometry, although the experimental protocol (Berry sequence) will need to be adjusted to allow only accumulation of geometric phase due to the AC field. The generality of our geometricphase technique should make it broadly applicable to precision measurements in many quantum systems, such as trapped ions, ultracold atoms, and other solidstate spins.
Methods
NV diamond sample
The diamond chip used in this experiment is an electronicgrade singlecrystal cut along the [110] direction into a volume of 4 × 4 × 0.5 mm^{3} (Element 6 Corporation). A highpurity chemical vapor deposition layer with 99.99% ^{12}C near the surface contains preferentially oriented NV centers. The estimated N and NV densities are 1×10^{15} and 3×10^{12} cm^{−3}, respectively. The ground state of an NV center consists of an electronic spin triplet with the m_{s} = 0 and ±1 Zeeman sublevels split by 2π × 2.87 GHz due to spin−spin interactions. Excitation with green (532 nm) laser light induces spinpreserving optical cycles between the electronic ground and excited states, entailing red fluorescence emission (637−800 nm). There is also a nonradiative decay channel from the m_{s} = ±1 excited states to the m_{s} = 0 ground state via metastable singlet states with a branching ratio of ~30%. Thus, the amount of red fluorescence from the NV center is a marker for the zcomponent of the spinstate, and continuous laser excitation prepares the spin into the m_{s} = 0 state over a few microseconds. The spin qubit used in this work consists of the m_{s} = +1 and 0 ground states. Nearresonant microwave irradiation allows coherent manipulation of the ground spin states. The NV spin resonance lifetimes are T_{1} ~ 3 ms, T_{2} ~ 500 µs, and T_{2}* ~ 50 µs.
Confocal scanning laser microscope
Geometricphase magnetometry using single NV centers is conducted using a homebuilt confocal scanning laser microscope (Supplementary Fig. 1). A threeaxis motorized stage (Micos GmbH) moves the diamond sample in three dimensions. An acoustooptic modulator (Isomet Corporation) operated at 80 MHz allows timegating of a 400 mW, 532 nm diodepumped solidstate laser (Changchun New Industries). An oilimmersion objective (×100, 1.3 NA, Nikon CFI Plan Fluor) focuses the green laser pulses onto an NV center. NV red fluorescence passes through the same objective, through a singlemode fiber cable with a modefielddiameter of ~5 μm (Thorlabs), and then onto a silicon avalanche photodetector (Perkin Elmer SPCMARQH12). The NV spin initialization and readout pulses are 3 µs and 0.5 µs, respectively. The change of fluorescence signal is calculated from ΔFL = FL^{+} − FL^{−}, where FL^{±} are the fluorescence counts obtained after spin projection using a microwave π/2pulse along the ±xaxis, respectively. For each measurement, the fluorescence count FL when the spin is in the m_{s} = 0 state is also measured as a reference. The temperature of the confocal scanning laser microscope is monitored by a 10k thermistor (Thorlabs) and stabilized to within 0.05 ^{o}C using a 15 W heater controlled with a PID algorithm.
Hamiltonian parameter control system
The Rabi frequency (Ω) and phase (ρ) of the microwave drive field, as well as the applied magnetic field to be sensed (B), are key variables of this work. It is thus crucial to calibrate the microwave driving system and magnetic field control system beforehand. Microwave pulses for NV geometric phase magnetometry are generated by mixing a high frequency (~3 GHz) local oscillator signal and a low frequency (~50 MHz) arbitrary waveform signal using an IQ mixer (Supplementary Fig. 1). The Rabi frequency and microwave phase are controlled by the output voltage of an arbitrary waveform generator (Tektronix AWG5014C) (Supplementary Fig. 2). The microwave pulses are amplified (Minicircuits ZHL16W43S+) and sent through a gold coplanar waveguide (10 µm gap, 1 µm height) fabricated on a glass coverslip by photolithography. An external magnetic field for magnetometry demonstration is created by sending an electric current through a copper electromagnetic coil (4 mm diameter, 0.2 mm thick, n = 40 turns, R = 0.25 Ω) placed h = 0.5 mm above the diamond surface. The electric current is provided by a highstability DC voltage controller (Agilent E3640A). To enable fine scan of the electric current with limited voltage resolution, another resistor with 150 Ω is added in series. Thus, a DC power supply voltage of 3 V approximately corresponds to I = 0.02 A, which creates an external field of B = μ_{0}nI/4πh ~ 16 G. One can determine the change of the external magnetic field as a function of DC power supply voltage ΔB(V) by measuring the shift of the resonance peak Δf in the NV electron spin resonance spectrum using Δf = γΔB. The result is ΔB/V = 0.50 ± 0.01 G V^{−1} (Supplementary Fig. 3). Joule heating produced by the coil is P = I^{2}R ~ 10^{−4} W. The mass and heat capacity of the coil are about 0.15 g and 0.06 J K^{−1}, respectively. Thus, the temperature rise is at most 2 mK s^{−1}. Since the temperature coefficient of the fractional resistivity change for copper is 0.00386 K^{−1}^{32}, the change of resistance due to Joule heating is negligible.
Numerical methods for geometric phase simulation
All simulations of NV spin evolution in this work are carried out by computing the timeordered time evolution operator at each time step.
where t_{i} and t_{f} are the initial and final time, respectively, \(\hat T\) is the timeordering operator, Δt is the time step size of the simulation, N=(t_{f} − t_{i})/Δt is the number of time step, and H(t) is the timedependent Hamiltonian (Eq. (1)). In the simulation, we used Δt=1 ns step size which is sufficiently small in the rotating frame. The algorithm is implemented with MATLAB^{®}.
Data and code availability
The data and numerical simulation code that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
This material is based upon work supported by, or in part by, the U.S. Army Research Laboratory and the U.S. Army Research Office under contract/grant numbers W911NF1510548 and W911NF1110400. This work was performed in part at the Center for Nanoscale Systems (CNS), a member of the National Nanotechnology Coordinated Infrastructure Network (NNCI), which is supported by the National Science Foundation under NSF award no. 1541959. J.L. was supported by the ILJU Graduate Fellowship. We thank John Barry, Jeff Thompson, Nathalie de Leon, Kristiaan de Greve, and Shimon Kolkowitz for helpful discussions.
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K.A., C.B., and R.L.W. conceived and K.A. and J.L. designed the experiments. K.A., J.L., and H.Z. performed the experiments and processed the data. All authors analyzed the results. K.A., J.L., D.R.G. and R.L.W. wrote the manuscript.
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Arai, K., Lee, J., Belthangady, C. et al. Geometric phase magnetometry using a solidstate spin. Nat Commun 9, 4996 (2018). https://doi.org/10.1038/s4146701807489z
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