Experimental dissipative quantum sensing
Yijin Xie Jianpei Geng Huiyao Yu Xing Rong Ya Wang Jiangfeng Du
quant-ph






Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China
CAS Key Laboratory of Microscale Magnetic Resonance, University of Science and Technology of China, Hefei 230026, China
Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China


3. Physikalisches Institut, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany




Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China
CAS Key Laboratory of Microscale Magnetic Resonance, University of Science and Technology of China, Hefei 230026, China
Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China



xrong@ustc.edu.cn
Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China
CAS Key Laboratory of Microscale Magnetic Resonance, University of Science and Technology of China, Hefei 230026, China
Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China


Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China
CAS Key Laboratory of Microscale Magnetic Resonance, University of Science and Technology of China, Hefei 230026, China
Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China


djf@ustc.edu.cn
Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China
CAS Key Laboratory of Microscale Magnetic Resonance, University of Science and Technology of China, Hefei 230026, China
Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China






Quantum sensing utilizes quantum systems as sensors to capture weak signal, and provides new opportunities in nowadays science and technology.
 
 The strongest adversary in quantum sensing is decoherence due to the coupling between the sensor and the environment. The dissipation will destroy the quantum coherence and reduce the performance of quantum sensing. Here we show that quantum sensing can be realized by engineering the steady-state of the quantum sensor under dissipation. We demonstrate this protocol with a magnetometer based on ensemble Nitrogen-Vacancy centers in diamond, while neither high-quality initialization/readout of the sensor nor sophisticated dynamical decoupling sequences is required.
 Thus our method provides a concise and decoherence-resistant fashion of quantum sensing. The frequency resolution and precision of our magnetometer are far beyond the coherence time of the sensor. Furthermore, we show that the dissipation can be engineered to improve the performance of our quantum sensing.
 By increasing the laser pumping, magnetic signal in a broad audio-frequency band from DC up to 140kHz can be tackled by our method.
 
 
 Besides the potential application in magnetic sensing and imaging within microscopic scale, our results may provide new insight for improvement of a variety of high-precision spectroscopies based on other quantum sensors.



Experimental dissipative quantum sensing
    Jiangfeng Du
    December 21, 2021
========================================



Quantum sensing utilizes a quantum system to perform a measurement of a physical quantity, and it allows one to gain advantages over its classical counterpart <cit.>.  There are variety of quantum systems for quantum sensing, such as neutral atoms <cit.>, trapped ions<cit.>, solid-state spins <cit.>, and superconducting circuits <cit.>.
The regular procedure of quantum sensing includes three elementary steps:
(1) Initialize the state of the sensor to a superposition state.
(2) Let the sensor interact with the target field.
(3) Readout the final state of the  sensor.
For the step (1) and (3), high-quality and efficient  initialization and readout are required.
The step (2) is usually very fragile, due to the inevitable interactions between the sensor and the environment. To minimize the effect of the environmental noise, exquisite dynamical decoupling technologies have been developed to suppress the noise <cit.> and enhance the performance of quantum sensing <cit.>.

However, the decoherence sets an upper bound on the time delay between the pulses in the dynamical decoupling sequence, corresponding to a lower bound of the detectable frequency of signal. As a result, the detection of low frequency magnetic field, which is important in magnetic navigation <cit.>, magnetic-anomaly detection <cit.>  and bio-magnetic field detection <cit.>, can hardly benefit from the dynamical decoupling protocols.




Inspired by resent progress in dissipative quantum computation and quantum metrology <cit.>,
we propose and experimentally demonstrate a dissipative quantum sensing protocol for low frequency signal detection in an important type of solid-state quantum sensors, Nitrogen-Vacancy (NV) center in diamond <cit.>. Due to the atomic scale of NV centers, quantum sensing based on NV center provides exciting quantum technologies, such as nuclear magnetic resonance and magnetic resonance imaging with nanoscale.

An ensemble electron spins of NV center is taken as a quantum sensor for magnetic field measurement.
The steady state of the sensor under dissipation can be engineered to be sensitive to the detected magnetic field.
We show that the frequency resolution and precision go far beyond the spin coherence time.
Furthermore, the laser pumping procedure, during which the sensor can be initialized and read out, can be utilized to introduce an additional dissipation to the sensor.
We also show that such a laser-controlled dissipation can improve detection bandwidth for quantum sensing.
Our method is capable to sense magnetic signal over a broad audio-frequency band ranging from DC to 140kHz.




Figure <ref>(a) shows the schematic of the NV center, which is an atomic defect consisting of a substitutional nitrogen and a vacancy adjacent to it.
It is negatively charged since the center comprises of six electrons, two of which are unpaired.
The energy level diagram is shown in the right panel of Figure <ref>(a).
The electronic ground state is a spin triplet state ^3A_2 consisting of three spin sublevels |m_s=0⟩ and |m_s=±1⟩.
The NV center can be excited from ^3A_2 to the excited state ^3E by a laser pulse with 532nm, and decays back to ^3A_2 emitting photoluminescence.
The photoluminescence intensity is dependent on the spin state of the NV center, which can be utilized for readout of the spin state.
Due to the intersystem crossing process, there is a larger probability to decay into |m_s=0⟩ of ^3A_2 than |m_s=±1⟩.
Therefore, there is approximately a net decay rate Γ_1 from |m_s=+1⟩ (or |m_s=-1⟩) to |m_s=0⟩ of ^3A_2 after a round of the optical transition.
The decay rate Γ_1 can be harnessed with controllable power of the laser pulse.
The optical transitions can be utilized for initialization of the spin state.

The spin states |m_s=0⟩ and |m_s= +1⟩ of ^3A_2 are encoded as a quantum sensor.
This sensor can be manipulated by microwave field with angular frequency being ω_e = D - γ_e B_0, where D =2π× 2.87GHz is the ground state zero splitting, γ_e = -2π×28GHz/T is the gyromagnetic ratio of the electron spin and B_0 is the external magnetic field.





Left panel of figure <ref>(b) presents the regular quantum sensing based on NV center.
In part (1), a laser pulse together with a π/2 microwave pulse are applied on the NV center to prepare the quantum sensor to the superposition state (|0⟩-i|1⟩)/√(2).
In the procedure labeled by part (2), we let the quantum sensor interact with the magnetic signal.
The magnetic signal contributes to a relative phase shift ϕ of the quantum sensor state.
A spin echo technique is utilized to prolong the coherence time of the sensor.
A π pulse is applied to decouple the interaction between the quantum sensor and the environmental noise.
If the magnetic signal is in phase with the spin echo pulse sequence, the relative phase shift ϕ due to the magnetic signal is accumulated rather than cancelled as shown in the bottom of left panel of figure <ref>(b).
More sophisticated dynamical decoupling (DD) sequences can be applied instead of spin echo sequence to enhance the coherence time of the NV center.
Once the coherence time is prolonged, the accumulated phase shift ϕ increases and the sensitivity of quantum sensing will be improved <cit.>.
The information of ϕ can be extracted by procedure (3) consisting of a π/2 microwave pulse and a laser pulse.
The lowest frequency of detected signal is bounded by 1/2τ <cit.>, where τ stands for the delay time between microwave pulses.
However, this delay time is limited by spin docoherence, so that low frequency signal detection by this method is challenging. Furthermore, once multiple-pulse DD sequences are applied, imperfection of DD pulses contributes to the reduction of sensitivity.
The non-ideal initialization and readout of the NV center also contribute to the reduction of observed signal.

Right panel of figure <ref>(b) shows the basic idea of our dissipative quantum sensing protocol.
Laser and microwave fields are always turned on simultaneously during the quantum sensing.
The laser pumping introduces an additional dissipation with decay rate Γ_1 from |m_s=+1⟩ to |m_s=0⟩.
The dissipation can be described by an amplitude damping process.
The microwave field drives the electron spin continuously, corresponding to the evolution of the sensor governed by the Hamiltonian H=Δ S_z - γ_e b(t)S_z - γ_e B_1 S_x.
Here the detuning Δ≡ω_e - ω_mw is the difference between the transition angular frequency of the electron spin ω_e and the angular frequency of the microwave field ω_mw, B_1 corresponds to the strength of the microwave field, b(t) stands for the ac magnetic field to be detected along the NV axis, and S_x, y, z=σ_x, y, z/2 are the spin operators with σ_x, y, z being the Pauli operators.
The ac magnetic field can be written as b(t ) = b_accos(ω_ac t + ϕ_ac), where b_ac, ω_ac and ϕ_ac are the amplitude, angular frequency, and phase, respectively.
Besides the dissipation introduced by the laser pumping, the sensor undergoes dephasing due to the interaction with environment (e.g. the nuclear spin bath).
The dephasing, with a dephasing rate Γ_2, can be described by a phase damping process.




The master equation, which describes the dynamics of the state ρ of the sensor in a rotating frame, can be written as


    dρ/d t = -i [H, ρ] + ∑_j =1,2 (2 L_jρ L_j^† - L_j^† L_jρ -ρ L_j^† L_j).

The first term on the right-hand side of equation <ref> stands for the evolution under the Hamiltonian H.

The second term on the right-hand side of equation <ref> describes the dissipation process of the system.
The operator L_1 = √(Γ_1/2)σ_- corresponds to the amplitude damping, and L_2 = √(Γ_2)/2σ_z corresponds to the phase damping, where σ_- = (σ_x - iσ_y)/2.
Here, the intrinsic longitudinal relaxation of the sensor is neglected, since the intrinsic relaxation time (about 3.9ms) is much longer than the decay time under the laser pumping (1/Γ_1, about several microseconds).
The dissipation process leads the sensor to a steady state.
The characteristic time to reach the steady state depends on two parameters: the dephasing time under the laser, T_2=1/(Γ_2+Γ_1/2), which is measured to be about 200ns, and the decay time of amplitude damping, T_1 = 1/ Γ_1.

For low frequency signal detection, the ac magnetic field to detect varies in a way that is much slower than that the sensor reaches a steady state.
In this case, the ac magnetic field can be considered quasi-static, and the state of the sensor can be approximated by the steady state under the quasi-static magnetic field.













For weak signal detection, i.e., if |γ_e b(t)T_2|≪1, the steady state ρ can be approximated, up to the first order, as

    ρ=ρ_0+Kb(t),

where ρ_0 and Kb(t) stand for the time independent and dependengt part of the steady state, respectively.
The detailed information of ρ_0 and K are provided in the Supplemental Material <cit.>.






























The ac magnetic field to detect is encoded into the state of the quantum sensor in a linear fashion according to equation <ref>.
Since the laser is always applied on the NV centers, the photoluminescence intensity which reflects the probability of the state in |m_s = 0⟩ can be continuously monitored.
The probability in |m_s = 0⟩ is







    P_|m_s=0⟩=2+s+2Δ^2T_2^2/2(1 + s +Δ^2T_2^2) -γ_eΔ T_2^2s/(1+s+Δ^2T_2^2)^2b(t),

where s = γ_e^2B_1^2 T_1 T_2. The dynamics of the probability P_|m_s = 0⟩  is plotted schematically in the bottom of right panel of figure <ref>(b).
The oscillation of  P_|m_s = 0⟩ reflects the amplitude and frequency of the ac magnetic field b(t).
When the detuning is set to Δ = √(1+ s)/√(3)T_2, the optimal sensitivity for signal detection is reached.
Since ρ is a steady state which can last for arbitrary long time, the detection can be, in principle, in an arbitrary precision.








The experimental setup has been developed for ensemble NV based magnetometer. Always-on laser and microwave fields are applied on NV centers. The microwave field is fed on the NV centers by a double split-ring resonator. A coil is used to send the magnetic field b(t) on NV centers. The waveform of b(t)  can be generated by the wave generator (33500B, Keysight). Two photodiodes (PDs) are used to transform intensities of green laser and red fluorescence to voltages, which can be monitored by a two channel lock-in amplifier (HF2LI, Zurich Instruments). The time constant of the lockin-amplifier is set to zero when it is utilized as an oscilloscope. The voltage, v_1, from PD1, which receives the red fluorescence, is proportional to the probability P_|m_s=0⟩ of state in |m_s = 0⟩.  The voltage, v_2, from PD2, which receives the green laser, is used to cancel the long time drift due to the laser power instability.



Figure <ref>a shows the experimental data of v_1, which reflects P_|m_s=0⟩ of the steady state, as a function of time when the frequency of b(t) is set to 9 Hz.
The voltage v_1 is recorded by the lock-in amplifier which is utilized as an oscilloscope with the frequency of its reference signal set to zero.
It can be observed that there is a coherent oscillation which lasts for about four hours without decay.
The non-decay behavior shows that the quantum state of the sensor is a steady state as expected.



The ac magnetic field b(t) is encoded into v_1 according to equation <ref>.
The frequency of b(t) is the same as that of v_1, and the amplitude of b(t) is proportional to that of v_1.
The fast Fourier transformation of v_1 provides the spectrum of b(t) in the frequency domain.




The peak position of the spectrum corresponds to the frequency of b(t), and the linewidth of the spectrum is defined as the frequency resolution.
The frequency and frequency resolution are obtained by fitting the spectrum, with the fitting uncertainty of the frequency defined as the frequency precision <cit.>.
Figure <ref>b shows the obtained frequency resolution as a function of the measurement time T.
The frequency resolution improves with the measurement time as T^-1.
When the data of v_1 is measured for T=14000s, a frequency resolution of 138 μHz is obtained.

Figure <ref>c shows the frequency precision as a function of the measurement time T.
The frequency precision improves with the measurement time as T^-3/2.






Figure <ref> shows the experimental measurement of the detection bandwidth of our dissipative quantum sensing protocol.
Since the protocol is based on the steady state under a quasi-static magnetic field b(t), the detection bandwidth of b(t) depends on the rate that the sensor reaches the steady state.
The rate to reach the steady state can be engineered by varying the laser power, since the laser power influences the decay time T_1 of the amplitude damping.
At a certain laser power, as the frequency of b(t) increases, the rate that b(t) varies will be more and more comparable with the rate that the sensor reaches the steady state.
If the frequency of b(t) reaches a value large enough, the assumption of quasi-static b(t) will be broken down and the quantum state of the sensor will be no longer a steady state.
Therefore, the amplitude of v_1 is expected to decrease with increasing frequency of b(t).
In the experiment, the amplitude of v_1 is measured by the lock-in amplifier with the frequency of its reference signal set to that of b(t).
Figure <ref>a shows the measured amplitudes as functions of the frequency of b(t) when the laser power is set to a series of values.
For clarity, the amplitudes are normalized so that their values at the first point equal 1.
As expected, the normalized amplitude decreases as the frequency of b(t) increases for any certain laser power.

The frequency at which the normalized amplitude decreases to 1/√(2) is defined as the detection bandwidth.
Figure <ref>b shows the detection bandwidth as a function of the laser power.
It clearly shows that the detection bandwidth increases with the laser power increasing.
When the laser power is set to 1.8W, the detection bandwidth of 146kHz is reached.
Thus the improvement of the detection bandwidth by engineering the laser power has been demonstrated.



In conclusion, we have proposed and experimentally demonstrated a dissipative quantum sensing protocol based on the steady state of the sensor of ensemble NV centers in diamond.
The frequency resolution and precision is far beyond the limit of the sensor's coherence time.
We experimentally show that the frequency resolution and precision can be improved with the measurement time T as T^-1 and T^-3/2, respectively.
The detection bandwidth in our protocol can be engineered by controlling dissipation.
As has been demonstrated in the experiment, the detection bandwidth increases with the laser power increasing.
When the laser power is set to 1.8W, the detection of ac magnetic field over a broad band ranging from DC to about 140kHz can be achieved. Our method is essentially different from magnetic field detection by continuous-wave method based on NV centers<cit.>, whose bandwidth is limited by the time constant of the lockin-amplifier.
The sensitivity of our setup is measured to be about 1nT/√(Hz) when the frequency is higher than 1 kHz <cit.>.
The sensitivity can be further improved with optimization of the experimental parameters of our apparatus (e.g. the material properties of the diamond).
The dynamic range of our protocol is measured to be better than 80 dB when the frequency is larger than 1 kHz <cit.>.
The high dynamic range can be achieved with our method, because we can continuously monitor the quantum probe and get rid of the disadvantage of  regular procedure of quantum sensing <cit.>.
Our protocol provides new possibility and insight in quantum sensing, imaging, and spectroscopies based on quantum sensors.


This work was supported by the National Key R&D Program of China (Grants No. 2018YFA0306600 and No. 2016YFB0501603), the CAS (Grants No. GJJSTD20170001, No.QYZDY-SSW-SLH004 and No.QYZDB-SSW-SLH005), the NNSFC (Grants No. 81788101, No. 11761131011), and Anhui Initiative in Quantum Information Technologies (Grant No. AHY050000). X.R. thanks the Youth Innovation Promotion Association of Chinese Academy of Sciences for the support.

Y. X. and  J. G.  contributed equally to this work.

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Supplementary material for "Experimental dissipative quantum sensing"













§ HAMILTONIAN OF THE QUANTUM PROBE

The ensemble electron spins of nitrogen-vacancy centers in diamond are utilized as the quantum probe for detection of low frequency ac magnetic field.
The Hamiltonian of the NV center is described as

    H_NV=H_ZF+H_ZM+H_MW+H_d+H_hyp.

The first term, H_ZF=DS_z,3^2, corresponds to the zero-field splitting of the NV center electron spin, with D=2π×2.87GHz and S_j,3 (j=x, y, z) being the electron spin operator.
The subscript "3" denotes that the NV center electron spin is a 3-level (spin-1) system.
The second term, H_ZM=-γ_eB_0S_z,3, is the Zeeman splitting of the electron spin under a static magnetic field B_0 along the NV axis, where γ_e=-2π×28GHz/T is the gyromagnetic ratio.
The third term, H_MW=-γ_e×√(2)B_1cosω_mw tS_x,3, is the control Hamiltonian introduced by a microwave field with amplitude √(2)B_1 and angular frequency ω_mw.
The fourth term, H_d=-γ_eb(t)S_z,3, is the term for detection with b(t)=b_accos(ω_act+ϕ_ac) being the ac magnetic field to be detected along the NV axis.
The fifth term, H_hyp=AS_z,3I_z,3, is the hyperfine interaction with the ^14N nuclear spin, with A=-2π×2.16MHz being the hyperfine coupling strength and I_z,3 being the ^14N nuclear spin operator.
In the experiment, the ^14N nuclear spin is not polarized.
Dependent on the state |m_I⟩ of the nuclear spin, the Hamiltonian of the electron spin can be simplified into

    H_m_I=DS_z,3^2+(-γ_eB_0+m_IA)S_z,3-√(2)γ_eB_1cosω_mw tS_x,3-γ_eb(t)S_z,3,

where m_I=-1, 0, 1.

The frequency of the microwave field, ω_mw/2π, is set close to the transition frequency ω_e/2π between |m_s=0⟩ and |m_s=1⟩, where ω_e=D-γ_eB_0+m_IA.
The transition probability between |m_s=0⟩ and |m_s=-1⟩ is negligible due to the large detuning.
Considering the microwave field together with the always-on laser field which initializes the NV center electron spin into |m_s=0⟩, there is negligible population on state |m_s=-1⟩ during the experiment.
Therefore, the subspace spanned by |m_s=1⟩ and |m_s=0⟩ is considered and the Hamiltonian of the quantum probe is written as

    H_probe=ω_eS_z-2γ_eB_1cosω_mw tS_x-γ_eb(t)S_z,

where S_j=σ_j/2 (j=x,y,z) is the spin operator of the equivalent two-level system with Pauli operator σ_z=|m_s=1⟩⟨ m_s=1|-|m_s=0⟩⟨ m_s=0|, σ_x=|m_s=1⟩⟨ m_s=0|+|m_s=0⟩⟨ m_s=1|, and σ_y=-i|m_s=1⟩⟨ m_s=0|+i|m_s=0⟩⟨ m_s=1|.

By turning into a rotating frame which rotates along z-axis with angular frequency ω_mw relative to the laboratory frame, and by considering the rotating-wave approximation, we present the Hamiltonian in the rotating frame as

    H=Δ S_z-γ_eb(t)S_z-γ_eB_1S_x,

with Δ=ω_e-ω_mw being the detuning.



§ STEADY STATE OF THE QUANTUM PROBE

The state evolution of the quantum probe can be described by the master equation

    dρ/d t = -i [H, ρ] + ∑_j =1,2 (2 L_jρ L_j^† - L_j^† L_jρ -ρ L_j^† L_j),

where ρ is the state of the quantum probe, L_1=√(Γ_1/2)σ_- (with σ_-=(σ_x-iσ_y)/2) and L_2=√(Γ_2)/2σ_z correspond to the amplitude damping and phase damping process.
If the ac magnetic field b(t) to be detected varies sufficiently slowly, H can be considered as a quasi-static Hamiltonian at each time, and the state of the quantum probe can be considered as a steady state slowly varying in step with b(t).
By setting the right hand of Eq. <ref> to zero, the steady state can be derived as

    ρ = 1/2(
             [                                            s/1 + s +[Δ-γ_e b(t)]^2T_2^2 γ_e B_1 [ Δ- γ_e b(t) ]T_2^2 + iγ_e B_1 T_2/1 + s +[Δ- γ_e b(t)]^2T_2^2; γ_e B_1 [ Δ- γ_e b(t)]T_2^2 - iγ_e B_1 T_2 /1 + s +[Δ- γ_e b(t)]^2T_2^2               2 + s + 2[ Δ-γ_e b(t)]^2T_2^2 /1 + s +[Δ-γ_e b(t)]^2T_2^2;                                                                         ]),

where T_1=1/Γ_1, T_2=1/(Γ_2+Γ_1/2), and s=γ_e^2B_1^2 T_1 T_2.

For weak signal detection, i.e., if |γ_e b(t)T_2|≪1, the steady state ρ can be approximated to the first order of γ_e b(t)T_2, which is

    ρ=ρ_0+Kb(t),

with

    ρ_0 = 1/2(
             [                               s/1 + s +Δ^2T_2^2  γ_e B_1 Δ T_2^2 + iγ_e B_1 T_2/1 + s +Δ^2T_2^2; γ_e B_1 Δ T_2^2 - iγ_e B_1 T_2 /1 + s +Δ^2T_2^2               2 +s + 2Δ^2T_2^2 /1 + s +Δ^2T_2^2;                                                 ]), 
       K = -(
          [                                   -γ_eΔ T_2^2s/(1+s+Δ^2T_2^2)^2 γ_e^2B_1T_2^2(1+s-Δ^2T_2^2)-i2γ_e^2B_1Δ T_2^3/2(1+s+Δ^2T_2^2)^2; γ_e^2B_1T_2^2(1+s-Δ^2T_2^2)+i2γ_e^2B_1Δ T_2^3/2(1+s+Δ^2T_2^2)^2                                    γ_eΔ T_2^2s/(1+s+Δ^2T_2^2)^2;                                                                 ]).

The probability of the state ρ in |m_s=0⟩ is

    P_|m_s=0⟩=2+s+2Δ^2T_2^2/2(1 + s +Δ^2T_2^2) -γ_eΔ T_2^2s/(1+s+Δ^2T_2^2)^2b(t),







§ SCHEMATIC OF THE OPTICALLY DETECTED MAGNETIC RESONANCE SETUP 

The experimental setup has been developed for ensemble NV based magnetometer Fig. <ref>. Always-on laser and microwave fields are applied on NV centers. The microwave field is fed on the NV centers by a double split-ring resonator. A coil is used to send the magnetic field b(t) on NV centers. The waveform of b(t)  can be generated by the wave generator (33500B, Keysight). Two photodiodes (PDs) are used to transform intensities of green laser and red fluorescence to voltages, which can be monitored by a two channel lock-in amplifier (HF2LI, Zurich Instruments). The time constant of the lockin-amplifier is set to zero when it is utilized as an oscilloscope. The voltage, v_1, from PD1, which receives the red fluorescence, is proportional to the probability P_|m_s=0⟩ of state in |m_s = 0⟩.  The voltage, v_2, from PD2, which receives the green laser, is used to cancel the long time drift due to the laser power instability.





§ SENSITIVITY AND DYNAMICAL RANGE

For estimation of the sensitivity and dynamical range of our quantum probe in the dissipative sensing protocol, the voltage v_1 from the photodiode, which reflects the intensity of the red fluorescence and thus the probability P_|m_s=0⟩ of the state in |m_s=0⟩, is measured for various amplitudes and frequencies of the ac magnetic field to detect.
The solid lines in Fig. <ref> show the measured amplitude v_1m of the voltage v_1 as a function of the amplitude b_ac of the ac magnetic field when the frequency of the ac magnetic field is set to 100Hz, 2kHz, 11kHz, 30kHz, and 128kHz, respectively.
As is shown, the voltage amplitude v_1m increases linearly with b_ac for small values of b_ac, which is in agreement with expectation according to Eq. <ref>.
The voltage amplitude v_1m is fit with the linear function v_1m=kb_ac in the region where b_ac is small.
The fit result is shown with the dash lines in Fig. <ref>.
The fit value of the coefficient k decreases as the frequency of the ac magnetic field increases, indicating that the approximation of quasi-static magnetic field gradually breaks down.

The sensitivity is defined as the minimum detectable amplitude of the ac magnetic field with unit time of measurement, which depends on the noise level at the same frequency of the ac magnetic field.
The dash dot lines in Fig. <ref> show the noise levels in a bandwidth of 1Hz at frequencies 100Hz, 2kHz, 11kHz, 30kHz, and 128kHz.
The minimum detectable amplitude b_ac, min of the ac magnetic field, at which the signal-to-noise ratio of the measured voltage is generally considered to be 1, can be estimated as the abscissa of the intersection point of corresponding dash and dash dot lines.
The sensitivity estimated with this method is shown in Fig. <ref>.
When the frequency of the ac magnetic field to detect is 100Hz, the sensitivity is about 30nT/√(Hz).
As the noise level at the other measured frequencies is much smaller, the sensitivity at these frequencies improves with more than an order compared to that at 100Hz and approaches to 1nT/√(Hz).



As the amplitude b_ac of the ac magnetic field increases to a sufficiently large value, the increase of the voltage amplitude v_1m will be saturated.
Due to the hyperfine interaction of the quantum probe with the unpolarized ^14N nuclear spin, v_1m presents a 3-peak feature in the region of large b_ac, which is shown in Fig. <ref>.
The maximum detectable amplitude b_ac, max of the ac magnetic field is estimated as the abscissa of the first peak.
The short dash lines in Fig. <ref> show the positions of b_ac, max when the frequency of the magnetic field is 100Hz, 2kHz, 11kHz, 30kHz, and 128kHz.
The dynamical range is defined as 20(b_ac, max/b_ac, min).
Figure <ref> shows the dynamical range as a function of the frequency of the ac magnetic field.
The dynamical range reaches 90dB when the frequency of the ac magnetic field is, e.g., 2kHz.









§ SENSITIVITY OF OUR DISSIPATIVE QUANTUM SENSING PROTOCOL

The signal, which is the voltage from the photon diode, is measured when the amplitude of the AC magnetic field b(t) is set to different values for estimation of the sensitivity of our dissipative quantum sensing protocol.
Figure <ref>A shows the measured signal amplitude as a function of the amplitude of b(t) when the frequency of b(t) is set to 100Hz.
As is expected, the signal amplitude is proportional to the amplitude of b(t) when its value is small (i.e. less than 10 μT in Fig. <ref>A).
The linear fit of the signal amplitude is denoted by the orange dashed line.
To estimate the sensitivity, the noise level at 100Hz with a 1Hz bandwidth is also measured, which is shown as the black dotted line.
The intersection of the two lines indicates the sensitivity of our dissipative quantum sensing protocol for detection of a 100Hz magnetic field.
The sensitivity for sensing the AC magnetic field with other frequencies can be estimated similarly.



Figure <ref>B shows the estimated sensitivity as a function of the frequency of b(t).
It can be shown that the sensitivity is approaching 1nT/Hz^1/2 when the frequency of b(t) is larger than 1kHz.
Because the noise level at a frequency larger than 1kHz is much lower than that at 100Hz, the sensitivity for sensing b(t) with a frequency larger than 1kHz is more than one order of magnitude better than that when the frequency of b(t) is 100Hz.




