Probabilistic magnetometry with two-spin system in diamond
Raúl Coto Hossein T. Dinani Ariel Norambuena Mo Chen Jerónimo R. Maze
quant-ph cond-mat.mes-hall




raul.coto@umayor.cl
Centro de Investigación DAiTA Lab, Facultad de Estudios Interdisciplinarios, Universidad Mayor, Chile

Centro de Investigación DAiTA Lab, Facultad de Estudios Interdisciplinarios, Universidad Mayor, Chile
Instituto de Física, Pontificia Universidad Católica de Chile,
Casilla 306, Santiago, Chile

Centro de Investigación DAiTA Lab, Facultad de Estudios Interdisciplinarios, Universidad Mayor, Chile


Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA


Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA


Instituto de Física, Pontificia Universidad Católica de Chile,
Casilla 306, Santiago, Chile



Solid-state magnetometers like the Nitrogen-Vacancy center in diamond have been of paramount importance for the development of quantum sensing with nanoscale spatial resolution. The underlying protocol is a Ramsey sequence, that imprints an external static magnetic field into the phase of the quantum sensor, which is subsequently read out. In this theoretical work we propose a sensing scheme that harnesses the hyperfine coupling between the Nitrogen-Vacancy center and a nearby nuclear spin to set a post-selection protocol. We show that concentrating valuable sensing information into a single successful measurement yields an improvement in sensitivity over Ramsey in the range of short transverse relaxation times. By considering realistic experimental conditions, we found that the detection of weak magnetic fields in the μT range can be achieved with a sensitivity of few tens of nTHz^-1/2 at cryogenic temperature (4 K), and μTHz^-1/2 at room temperature. 


Probabilistic magnetometry with two-spin system in diamond
    Jerónimo R. Maze
    December 30, 2023
==========================================================





§ INTRODUCTION


Quantum metrology takes advantage of the quantum properties of the system to achieve better precisions than allowed by its classical counterpart <cit.>. In particular, probabilistic quantum metrology aims to further increase the retrievable information of a parameter by means of a selective measurement. Inspired by the findings of Aharonov, Albert and Vaidman <cit.>, where a measurement sequence consisting of pre-selection, weak measurement and  post-selection leads to an anomalous amplification of the measurement result, several theoretical works <cit.> and experiments <cit.> have followed to explore such anomalous amplification. Nevertheless, there remains a longstanding controversy regarding whether probabilistic quantum metrology has practical advantages over standard techniques for parameter 
estimation <cit.>. In this work, we investigate the important case of spin magnetometry with color centers in diamond, and provide experimental parameter regimes where probabilistic quantum metrology is expected to succeed.

Quantum sensors, in particular solid-state magnetometers like the Nitrogen-Vacancy (NV) center in diamond have attracted widespread attention as a powerful tool at the nanoscale <cit.>. Development of sensing protocols and experimental techniques facilitates detection of weak magnetic fields, featuring applications including sensing of single protein  <cit.>, small molecules  <cit.>, single spins  <cit.>, and more recently 3D reconstruction of a nuclear spin cluster  <cit.>. 

In the following, we focus on DC magnetometry using a single NV center. The conventional measurement is realized by a Ramsey sequence with only the electronic spin of the NV. Here, we consider in addition a coupled nuclear spin located nearby the NV center. We show that by following a particular sequence involving post-selection one can achieve magnetic field sensitivity, the minimum detectable magnetic field normalized by the total sequence time, that is comparable with Ramsey sensitivity and outperforms the latter in suboptimal conditions. 



§ THEORY


The original idea proposed by Aharonov, Albert and Vaidman <cit.> contributed to the field of probabilistic metrology, promoting the development of protocols that amplify the effect of the system-meter coupling strength to further estimate this coupling constant <cit.>. Moreover, some of these ideas have been implemented in color centers in diamond for estimation of the hyperfine coupling <cit.> and weak measurements <cit.>. 

In this work, we propose a protocol consisting of pre-selection, system-meter interaction and post-selection. However, instead of amplifying the system-meter coupling strength itself, we take advantage of this interaction and use it to enhance the sensitivity for sensing an external magnetic field. We remark that a weak interaction is not required. We look for retrieving more information in a single successful estimation of the accumulated phase during the system-meter interaction. For convenience, we represent the initial state preparation and final state post-selection by three unitary rotations. The composite system then evolves according to U=R_1(θ_f)U_τR_1(θ_i)R_2(α), where the operator R_1(ϕ) (R_2(ϕ)) represents rotations of the system (meter) for an angle ϕ and U_τ is the free evolution of the system for time τ under the external magnetic field to sense. Thus, after post-selection upon the system in a target state |ψ_f⟩, the meter state will be given by ρ_post=ψ̱_̱f̱Uρ(0) U^†ψ̨_̨f̨, and the expectation value of the meter observable is

    ⟨σ_2^z⟩ =_2[ρ_postσ_2^z]/_2[ρ_post],

with σ_2^z the Pauli z operator acting on the meter. 

As a first step, we are interested in the shape of the signal ⟨σ_2^z⟩ that can be tuned by parameters {α, θ_i, θ_f } and the free evolution time τ. This particular feature will be our starting point to enhance the magnetic field measurement, since it allows us to set the optimal interrogation time.

In the next section, we will show how this protocol can be implemented efficiently with a two-spin system in diamond at cryogenic temperature. The low temperature allows us to perform single-shot readout, which improves the signal-to-noise ratio. We will also discuss the scenario where the protocol is performed at room temperature.




§ THE MODEL

For a more versatile sensing protocol that works both under room temperature and cryogenic temperature, we consider without loss of generality a concrete bipartite system model given by an electronic spin-1 (S=1) of a negatively charged Nitrogen-Vacancy center (NV^-) and a nearby nuclear spin-1/2 (I=1/2) of a Carbon-13 (), as illustrated in Fig. <ref> (a). Note that we have a native nuclear spin-1/2 (spin-1) from ^15N (^14N) as well, which enables single-shot readout of the NV^- for the post-selection at room temperature.

  

The ground state of the NV^- is a spin triplet labelled by the spin quantum number m_s = 0, ± 1 (S = 1). An external magnetic field B_z along the N-V axis (z-axis) induces Zeeman energy splitting between the spin sublevels m_s=+1 and m_s=-1 and lifts their degeneracy. The is hyperfine coupled to the center, yielding the system Hamiltonian (ħ=1)

    H_0 = D S_z^2 + γ_e S_z (B_z + B) +  γ_c I_z (B_z + B) +  S_z A_zz I_z,

where D/2π = 2.87 GHz is the zero-field splitting of the NV^- center, γ_e/2π≈ 2.8 MHz/G,  and γ_c/2π≈ 1.07 kHz/G are the gyromagnetic ratios of the electron and nuclear spins, respectively. B is the small magnetic field to detect and A_zz is the hyperfine coupling strength. We choose a weakly coupled nuclear spin aligned close to the N-V axis, such that its anisotropic hyperfine coupling terms such as A_zx are small and thus neglected here.


To further simplify the analysis, we focus on the 2-level submanifold {0̨ (m_s=0),1̨ (m_s=-1) } for the electronic spin of the NV^-, while for the we consider the complete basis  (m_I^c=+1/2) and  (m_I^c=-1/2). In Fig. <ref> (b) we show the energy levels of our configuration, indicating the relevant transitions  with blue solid lines. 

In order to manipulate the energy sublevels depicted in Fig. <ref> (b), we apply a series of microwave (MW) and radiofrequency (RF) pulse sequences (square pulses) that result in the total Hamiltonian in a multi-rotating frame 


    H̃=
    1/2[  γ_c B  Ω_0^c    Ω^e      0;  Ω_0^c -γ_c B      0    Ω^e;    Ω^e      0 2δ_1^↑      0;      0    Ω^e      0 2δ_1^↓ ],

where Ω^e and Ω^c_0  are the Rabi frequencies of the MW and RF fields acting on the electron and nuclear spins, respectively. δ_1^↑=-γ_eB-A_zz/2 +γ_c B/2 and δ_1^↓=-γ_eB+A_zz/2 -γ_c B/2. For more details of the Hamiltonian and the rotating frame see Appendix <ref>. 

In what follows, we describe our protocol that is represented in Fig. <ref> (c), taking the electronic spin of the as the system and the nuclear spin as the meter. 
First, the bipartite system is initialized to the state Ψ̨_̨į=0̨⊗|↓⟩. Efficient nuclear spin initialization has been demonstrated in Refs. <cit.>. Second, we prepare the nuclear spin in a coherent superposition state 0̨⊗(cos(α/2) + sin(α/2)) via the RF field (Ω_0^c). Third, a strong MW pulse rotates the electronic spin by an angle θ_i which is independent of the nuclear spin state, yielding


    Ψ̨_̨p̨r̨ę   =    (cos(θ_i/2)1̨ + sin(θ_i/2)0̨)
       ⊗   (cos(α/2) + sin(α/2)).


We will refer to Eq. (<ref>) as the pre-selected state and it serves two goals in the sensing protocol: firstly, it acquires a phase proportional to B, directly contributing to the magnetometry. Additionally, it enables the interaction between the nuclear spin and the  that is fundamental for probabilistic quantum metrology and the enhancement of the sensitivity. Next step, we let the system evolve for an interrogation time τ, leading to   



    Ψ̨_̨1̨   =   cos(θ_i/2)cos(α/2)e^-iδ_1^↑τ1̨⊗
       +   cos(θ_i/2)sin(α/2)e^-iδ_1^↓τ1̨⊗
       +   sin(θ_i/2)cos(α/2)e^-iγ_cBτ/20̨⊗
       +   sin(θ_i/2)sin(α/2)e^iγ_cBτ/20̨⊗.



Finally the system is post-selected upon NV^- in a target state ψ̨_̨f̨=cos(θ_f/2)1̨ + sin(θ_f/2) 0̨. This process is illustrated in Fig.<ref>-(c). In sensing a weak magnetic field B, we have γ_c Bτ≪ 1. Therefore the post-selection leaves the nuclear spin in the state 


    ϕ̨̨̅_̨p̨ǫs̨t̨   =   cos(α/2)cos(θ_f/2)cos(θ_i/2)e^-iδ_1^↑τ
       +   cos(α/2)sin(θ_f/2)sin(θ_i/2)
       +   sin(α/2)cos(θ_f/2)cos(θ_i/2)e^-iδ_1^↓τ
       +   sin(α/2)sin(θ_f/2)sin(θ_i/2).

The above state needs to be normalized, such that ϕ̨_̨p̨ǫs̨t̨=ϕ̨̨̅_̨p̨ǫs̨t̨/√(P_s), where P_s is the probability of having a successful post-selection,

    P_s    =   1/2[1+cos(θ_f)cos(θ_i)] + sin(θ_f)sin(θ_i)/2
       ×   [cos^2 (α/2) cos(δ_1^↑τ)+sin^2(α/2)cos(δ_1^↓τ)].


The final signal is proportional to

    ⟨ I_z⟩   =   1/4P_s[1+cos(θ_f)cos(θ_i)] cos(α) 
       +   sin(θ_f)sin(θ_i)/4P_s
       ×   [ cos^2 (α/2) cos(δ_1^↑τ) - sin^2(α/2)cos(δ_1^↓τ)].
 

We now compare the signal obtained in Eq. (<ref>) to the simple case of Ramsey spectroscopy (π/2)_x-τ-(π/2)_x considering a single spin, the electronic spin. The Ramsey signal follows ⟨ S_z⟩_R = 1/2-cos(γ_eBτ)/2. Notice that our protocol shares a common ground with the Ramsey technique when sensing DC magnetic field as phase estimation. Furthermore, when the nuclear spin related part is removed, our protocol converges to the conventional Ramsey sequence with θ_i=θ_f=π/2. Nevertheless, our protocol employs a more elaborate procedure involving the additional nuclear spin, allowing further gain in standard deviation via the post-selection process at shorter times (Fig. <ref> (b)). To emphasize the role of post-selection, we analyze the case where no post-selection is carried out. 
Since ⟨ I_z⟩=cos(α)/2 carries no information about the magnetic field in this case, we calculate the expectation value of I_x instead:

    ⟨ I_x⟩=sin(α)( cos^2(θ_i/2)cos((A_zz-γ_c B)τ)+ sin^2(θ_i/2) ).


This brings no benefits as compared to the simpler case achieved with a single spin by the Ramsey sequence because the nuclear spin itself is not a sensitive magnetometer. It is the post-selection that allows us to imprint the phase information into the nuclear spin, in such a way that variations of the post-selected angle θ_f calibrates the amount of information extracted from the measurement process <cit.>. 



§ RESULTS



The magnetic field sensitivity is defined as the minimum detectable 
magnetic field normalized by the total sequence time <cit.>. 
The minimum detectable magnetic field is found through the standard deviation Δ B <cit.>,

    Δ B = Δ I_z/|∂⟨ I_z⟩/∂ B |,

where Δ I_z= √(⟨I_z^2⟩ - ⟨ I_z⟩^2) is the standard deviation of the signal from the nuclear spin. 


Clearly, in order to increase the sensitivity, we need a sharp response of the signal ⟨ I_z⟩ to B through the post-selection process. To begin with, we fix B=10^-2 G and for convenience, we choose both spins initially prepared in a superposition state with α=θ_i=π/2 and the post-selected state to be parallel to the pre-selected electronic spin state θ_f=π/2. Without loss of generality, we consider a ^13C nuclear spin with A_zz = 500 kHz. We can simplify the signal in Eq. (<ref>) with these parameters


    ⟨ I_z⟩=	-sin(A_zzτ/2) sin (γ_e Bτ)/2 (cos(A_zzτ/2) cos ( γ_e Bτ)+1).


A_zz contributes in two ways. Explicitly, it conduces to oscillations in the signal (<ref>). Implicitly, it sets the upper limit of the nuclear spin control speed due to power broadening. As a result, stronger hyperfine coupling reduces the required nuclear spin gate time, and decreases the total sensing time of the protocol. 

In Fig. <ref> (a) we show the behavior of ⟨ I_z⟩ as a function of the interrogation time τ. Notice the region around τ=2 μs where the signal is sharp. We numerically found that the optimal interrogation time is close to the extreme values of ⟨ I_z⟩. The price to pay is a lower probability of successful post-selection (P_s). This trade-off between the signal gain and the probability of success is common in protocols that rely on post-selection, as stated in the field of weak value amplification <cit.>. Consequently, many trials are required for the successful post-selection. Interestingly, even when outcomes are discarded, there is valuable information in the post-selection's statistics that can be used <cit.>, given that P_s in Eq. (<ref>) is  a function of B. For more details see Appendix <ref>.

  

In Fig. <ref> (b), we show the minimum detectable field allowed in our protocol obtained from Eq. (<ref>). One observes from the plot that our protocol constitutes a better route than the Ramsey sequence towards minimizing Δ B at short interrogation time (τ), and this time is close to the coherence time of most NV centers. 

Therefore, concentrating valuable sensing information into a single successful measurement by post-selection provides a competitive approach for improving the minimum detectable magnetic field in a parameter range of interest.

The standard deviation Δ B does not take into account the number of failed post-selections, which increases the total time for the experiment. Hence, we now characterize our sensing protocol using the sensitivity (η) <cit.>, 

    η=Δ B √(t_m),

where t_m is the total sequence time consisting of initialization, interrogation and measurement, and includes those in failed post-selections. t_m = N(t_i + τ + t_p) + t_r. The factor N=1/P_s accounts for the average trials of the experiment for one successful post-selection. The initialization (t_i) and measurement times are usually fixed in a sequence, making the sensitivity dependent on the interrogation time. We split the measurement time for a single run into the post-selection time  t_p (NV^- readout) and nuclear spin readout time t_r. 

Firstly, we focus on the cryogenic temperature regime, 4 K, where we can perform single-shot readout of the NV^- electronic spin with high fidelity <cit.>. In combination with a nuclear spin controlled CNOT gate, one can readout the nuclear spin efficiently <cit.>. We take t_p=3.7μ s for as reported in ref. <cit.>, and t_r=5.7μ s for ^13C assuming a nuclear spin controlled CNOT gate limited by the hyperfine interaction strength in speed. 

The interrogation time τ is limited by transverse relaxation of the magnetometer, given by the characteristic time T_2^∗ of the electronic spin. This relaxation process (T_2^∗) for naturally occuring electronic spins in a natural abundance diamond sample is typically around a few microseconds <cit.>. We model this pure dephasing process with the following Markovian master equation,



    dρ/dt = -i[H̃,ρ] +  Γ(2S_zρ S_z - S_z^2ρ -ρ S_z^2),
 
that introduces an exponential decay exp(-Γ t) on the off-diagonal elements of ρ, with Γ=1/T_2^∗. More general non-Markovian magnetic noise can be modelled using a stochastic interaction ruled by the Ornstein-Uhlenbeck (OU) statistics <cit.>. For instance, this type of noise has been observed in samples with high density of paramagnetic nitrogen centers (P1 centers) <cit.>. We discuss this case in Appendix <ref>.


For further comparison, once again we use the Ramsey sequence for reference, where the total sequence time reduces to t_m=t_i+τ+t_p, with t_i=1 μs and t_p=3.7 μs. 


In Fig. <ref> (a), we show the sensitivity for both Ramsey and post-selection protocols as a function of transverse relaxation time T_2^∗. We consider a weak magnetic field B =10^-2 G and α=θ_i=θ_f=π/2. Each value of sensitivity is taken at the optimal interrogation time (τ). For long relaxation times T_2^∗, the sensitivity decreases (improves), and Ramsey performs better. As T_2^∗ decreases, the sensitivity of both magnetometers deteriorates, and for a suboptimal scenario T_2^∗≲ 2 μs (for ) post-selection yields better sensitivity. It is worth noticing that Ramsey's optimal interrogation time is always greater than the one for post-selection. 

 


So far we have considered a small magnetic field B =10^-2 G. In Appendix <ref> we show the sensitivity as a function of a magnetic field up to B=1 G in the absence of losses. 

Hereafter, we consider the T_2^∗ of naturally occuring electronic spins, T_2^∗ = 2 μs <cit.>. In addition, we take into account the readout inefficiency by introducing a factor C≤ 1 and defining the sensitivity as η_C=η/C <cit.>. Assuming ideal single-shot readout, this factor approaches to unity. Therefore, we can roughly estimate the sensitivity of our magnetometer for an optimal interrogation time τ=1.3 μs to be η=43.5 nTHz^-1/2 (C=1). For a non-ideal scenario (C=0.707 <cit.>) we get η_C=61.5 nTHz^-1/2.

We remark that the protocol can be improved if we limit ourselves to cryogenic temperature applications only (4 K). We can replace the meter by the native Nitrogen (assuming Nitrogen-15 () here) <cit.>. The  nuclear spin-1/2 is always present for the particular defect (), which allow us to make the protocol universal. Moreover, it only has isotropic hyperfine coupling A_zz=3.03 MHz, that is stronger than the coupling (A_zz = 500 kHz), and thus enables faster nuclear spin gates. In Fig. <ref> (a) we show the sensitivity using the . Due to the shorter initialization and gate times, the sensitivity with  is improved from that with and it performs better than Ramsey in the suboptimal regime (short T_2^∗). For further illustration, we calculate the ratio η_ post/η_ Ramsey (for the ) as a function of the relaxation time (T_2^∗) and magnetic field (B), see Fig. <ref> (b). One can observe now the region η_ post/η_ Ramsey<1, where our post-selection's based protocol exhibits an enhancement over the Ramsey sequence.

Next, we evaluate the performance of the protocol at room temperature. In this setting, we require the native Nitrogen nuclear spin as an ancilla and a high bias magnetic field (B_z>2000 G) to realize single shot readout for the post-selection through a repetitive readout protocol <cit.>. With this protocol, single shot readout of the  is achieved by firstly mapping the  state to the Nitrogen and then read out the nuclear spin repetitively. Considering 2000 repetitions we obtain t_p= 5 ms <cit.> and t_r= 8 ms <cit.>, leading to η_C=1.5 μTHz^-1/2. Based on numerical simulations we found the larger overhead for post-selection and readout does not change our conclusion fundamentally.

We further remark that the setting of our protocol, that is the NV center coupled to a nearby nuclear spin, is very versatile and has been explored extensively in both theories and experiments. For instance, the nuclear spin can provide different functionalities such as the quantum memory <cit.>, the ancilla in quantum error correction <cit.>, or the computational qubit itself <cit.>. In particular, recent works have taken advantages of the long coherence time of nuclear spin and use it as a quantum memory to extend the interrogation time or refocus static noise <cit.>. 
All these functionalities, together with the post-selection protocol presented here, make the composite NV and nuclear spin system an ideal playground for exploring quantum information applications in a small scale. 

Finally, we introduce the Fisher information (FI) to quantify the amount of information that can be retrieved about the magnetic field B for each protocol and show the results are consistent with the sensitivity analysis. FI can be calculated as <cit.>

    F=∑_i1/P_i(B)(∂ P_i(B)/∂ B)^2,

where P_i(B) is the probability of the measurement outcome i and the sum is over all posible outcomes. For the case of post-selection, we multiply the FI by the probability of having a successful measurement (P_s in Eq. (<ref>)) <cit.>. In Fig. <ref> (c) we show that the FI for post-selection is higher than the one for Ramsey, and maximum at short sensing times (same region where quantum Fisher information is maximum, see Appendix <ref>). We can conclude that, in the presence of high decoherence, the post-selection protocol focuses more valuable information on a single successful measurement, which is available at a shorter sensing time, resulting in an advantage over the Ramsey protocol.





§ CONCLUSIONS


In summary, we have proposed a new experimentally feasible protocol based on post-selection to estimate a weak magnetic field. The information of the field is stored in the relative phase acquired by the electronic spin of a NV^- center that is coupled to a nearby  nuclear spin. Using this protocol, the information regarding the magnetic field is focused on a single successful measurement upon the nuclear spin. Taking into account realistic conditions of losses and readout inefficiencies, we found that post-selection protocol is comparable with Ramsey in sensitivity in a wide range of transverse relaxation time T_2^∗, and outperforms Ramsey for a range of B<10^-2 G and T_2^∗≲ 3 μs. Furthermore, for this range our protocol results in a higher Fisher information that is achieved at shorter sensing times. We found that decreasing further initialization and gate times improves sensitivity, as shown in the case of native  nuclear spin. At cryogenic temperature (4 K), B=10^-2 G and T_2^∗=2 μs the expected reduction (improvement) in sensitivity for post-selection with the  is around 28% over Ramsey. At room temperature the most limiting factor is the number of repetitions for the readout of the nuclear spin. This could be improved by using Bayesian estimation <cit.>. Moreover, nuclear spin introduces functionalities such as the quantum memory or the ancillae to implement error correction. 




§ ACKNOWLEDGMENTS
  RC acknowledges support from Fondecyt Iniciación No. 11180143. H.T.D. acknowledges support from Fondecyt-postdoctorado Grant No. 3170922. H.T.D and AN acknowledges support from Universidad Mayor through the Postdoctoral fellowship. JRM acknowledges support from Fondecyt Regular No. 1180673 and AFOSR FA9550-18-1-0513.





§ MULTI-ROTATING FRAME


The total Hamiltonian is given by

    H =    DS_z^2 + γ_e S_z(B_z + B) + γ_c I_z(B_z + B) + S_zA_zzI_z 
       + √(2)Ω^e S_x cos(ω_e t) + 2 Ω_0^c I_x cos(ω_c t)
    
    =    DS_z^2 + γ_e S_z(B_z + B) + γ_cI_z(B_z + B) + S_zA_zzI_z 
       + √(2)Ω^e/2S^+ e^-iω_e t +2 Ω_0^c/2I^+ e^-iω_c t + h.c.,

where Ω^e and Ω_0^c are the Rabi frequencies of the electronic and nuclear spin transitions, respectively. S^+=S_x+i S_y, I^+=I_x+i I_y and S_i (I_i) are the electron (nuclear) spin-1 (-1/2) operators. In a rotating frame defined by the unitary operator V=exp[-i (ω_e S_z + ω_c I_z)t], the transformed Hamiltonian H̃=V^† H V +i(dV^†/dt) V reads

    H̃=    (DS_z^2+D S_z) + γ_e B S_z + γ_c B I_z + S_zA_zzI_z 
       + √(2)Ω^e/2S_x +2 Ω_0^c/2I_x .


Here we have taken ω_e=-D+γ_e B_z and ω_c=γ_c B_z. After taking the rotating wave approximation, the transformed Hamiltonian can be explicitly written in a matrix form as



    H̃=1/2[  A_zz + 4D + (2γ_e+γ_c) B                         0                         0                         0                         0                         0;                         0 -A_zz + 4D + (2γ_e-γ_c) B                         0                         0                         0                         0;                         0                         0                     γ_c B                     Ω_0^c                       Ω^e                         0;                         0                         0                     Ω_0^c                    -γ_c B                         0                       Ω^e;                         0                         0                       Ω^e                         0     -A_zz + (γ_c -2γ_e) B                         0;                         0                         0                         0                       Ω^e                         0       A_zz - (γ_c+2γ_e) B ].



We now obtain the desired Hamiltonian in the m_s=0,-1 manifold by truncating this matrix. This leaves us

    H̃=
    1/2[  γ_c B  Ω_0^c    Ω^e      0;  Ω_0^c -γ_c B      0    Ω^e;    Ω^e      0 2δ_1^↑      0;      0    Ω^e      0 2δ_1^↓ ],


where δ_1^↑=-γ_e B - A_zz/2 + γ_cB/2 and δ_1^↓=-γ_e B +A_zz/2 - γ_cB/2.










§ FISHER INFORMATION



The Fisher information (FI) quantifies the amount of information that can be extracted about an unknown parameter from a measurement using a probe state. The quantum Fisher information (QFI) is the maximization of the FI over all the possible measurements <cit.>. According to the quantum Cramér-Rao bound <cit.> the variance in estimating a parameter, labeled by B here, using an unbiased estimative of B, is lower bounded by the inverse of the QFI, var(B)≥ 1/(N F_Q), for N measurements.

The FI is defined as 

    F=∑_i1/P_i(B)(∂ P_i(B)/∂ B)^2,

where P_i(B) is the probability of the measurement outcome i and the sum is over all the outcomes. The QFI is given by

    F_Q= Tr(ρ L^2),

with L the symmetric logarithmic derivative defined as 

    ∂ρ/∂ B=1/2(Lρ+ρ L).

The matrix L can be obtained by finding the eigenvalues p_s and eigenvectors |ψ_s⟩ of the density matrix, by which we can write  ρ=∑_sp_s|ψ_s⟩⟨ψ_s|. In this eigenbasis we have <cit.>

    ⟨ψ_i|∂ρ|ψ_j⟩ =1/2(p_i+p_j)⟨ψ_i| L|ψ_j⟩.

The elements L_ij=⟨ψ_i|L|ψ_j⟩ for which p_i=p_j=0 can be set to zero <cit.>.
For a pure state |ψ⟩ the QFI can be simply written as <cit.>

    F_Q=4⟨∂_Bψ|∂_B ψ⟩-4|⟨∂_Bψ|ψ⟩|^2.


For an electron spin of a center which is prepared in the state |ψ⟩=(|0⟩+|1⟩)/√(2) and evolves under the Hamiltonian H=γ_e B S_z for time τ, the QFI is (γ_e τ)^2. In the absence of decoherence, the FI retrieved from a Ramsey sequence saturates this QFI, i.e., F_ Ramsey=F_Q=(γ_e τ)^2, making Ramsey the optimal protocol. 

We now investigate the amount of information that can be extracted from the post-selection protocol with the . Firstly, we focus on the lossless condition, i.e. 1/T^∗_2∼0 (in our numerical simulation we set T^∗_2=1 s),  and we calculate the QFI for the state |Ψ_1⟩ given in Eq. (5) (F_ Q, |Ψ_1⟩), which sets the upper bound. Then, we calculate the QFI resulting from the post-selection (F_ Q, post) (for the normalized post-selected state |ϕ_post⟩), that is weighted by the probability of success P_s in Eq. (<ref>). In addition, we calculate the FI contained in the post-selection statistics (F_ ps). We show the results in Fig. <ref> (a), and one can observe that there is a significant amount of information in the post-selection statistics F_ ps, but also in the favorable outcome F_ Q, post at short sensing time. We remark that since the nuclear spin is not a good sensor itself, the QFI F_ Q, |Ψ_1⟩ agrees with F_ Ramsey in the absence of losses.




Secondly, we compare the FI for the post-selection protocol with the Ramsey sequence in the presence of a transverse relaxation process with T^∗_2=2 μs, see Fig. <ref> (b). It can be seen that in the region of time τ where more information can be retrieved, as given by the QFI F_ Q, |Ψ_1⟩, post-selection (F_ Q, post) significantly outperforms Ramsey. We remark that the FI for post-selection is the same as its QFI F_ Q, post, as we have assumed that the post-selected state does not undergoes decoherence. The oscillations in F_ Q, post originates from the large value of A_zz of the . 

To conclude, these results confirm that even by only performing measurements on the post-selected state (and discarding the post-selection statistics) we can achieve an enhancement over the Ramsey sequence for ranges of B and T^∗_2 given by  B<10 mG and T^∗_2 ≲ 3 μs. 




§ STOCHASTIC NOISE


Non-markovian magnetic noise can be described using the following stochastic Hamiltonian ruled by the Ornstein-Uhlenbeck (OU) statistics (ħ = 1)


    H_ noise(t) = γ_e B(t) S_z,     ⟨ B(t) ⟩ = 0, 
       ⟨ B(t)B(t') ⟩ = B_s^2 e^-|t-t'|/τ_c,


where γ_e is the electron gyromagnetic ratio, B(t) is the stochastic magnetic field, S_z is the spin operator for S = 1,  B_s = √(⟨ B(t)^2 ⟩) is the magnetic field intensity, and τ_c is the correlation time of the OU noise. It is clear that H_ noise(t) commutes with the system Hamiltonian (and ) H_s = γ_e B S_z + γ_n B I_z + S_z A_zz I_z. Therefore, the time evolution of the system can be obtained through the application of the time evolution operator U(t)=U_ noise(t)U_s(t). Here, U_ noise(t) = (-i ∫_0^t H_ noise(τ)   dτ) and  U_ s(t) = (-i H_s t ). The magnetic noise can be generated using the recursive formula <cit.>


    B(t+dt) = B(t)e^-dt/τ_c + [c τ_c  2(1-e^-2 dt/τ_c) ]^1/2n,


where dt >0 is the time step, n is a normal random variable with mean value 0 and variance 1, and c can be written in terms of the transverse relaxation time T^*_2 as c = 4/(T_2^∗)^2 τ_c <cit.>. We note that the spin-bath interaction is intrinsically non-Markovian with a correlation time τ_c. However, for evolution times t≫τ_c, the Markov approximation is valid, and the Lindblad super-operator associated with the stochastic Hamiltonian is given by

    ρ̇ = ℒ_ markov[ρ] = γ[S_z ρ S_z^† - 1  2{S_z S_z^†, ρ}],

where ℒ_ markov[ρ] is a pure-dephasing dissipation channel and γ = 4  τ_c/(T_2^∗)^2 is the dephasing rate. In Fig. <ref> (a)-(c), we show the signal ⟨ I_z ⟩ in the presence of an OU noise using both a stochastic approach and Markovian approximation. We have used a range of correlation times for a system with T_2^∗ = 20  μs. We note that systems with T_2^∗≫τ_c are reasonably well described by the Markovian approximation, which is the case for the present work. Systems with a large memory time are beyond the scope of the proposed protocol. In such a case, a more complex envelope effect disturbs the signal ⟨ I_z ⟩. For instance, samples with high density of paramagnetic nitrogen centers, termed as P1 centers <cit.>, exhibits a correlation time τ_c=13 μs and transverse relaxation time of few microseconds. In Fig. <ref> (d) we show the sensitivity as a function of the interrogation time (τ) and observe a similar behavior as compared to the case modeled in the main text.  






§ VARYING MAGNETIC FIELD




In this section we show the variation of the sensitivity (η) to detect the magnetic field as a function of the magnitude of the field, for fixed interrogation times. In Fig. <ref>, the sensitivity remains constant (η=9.7 nTHz^-1/2) for τ=3.0 μs, while for τ=3.2 μs it exhibits oscillations. The magnetic field have been tuned in the range 10^-2-10^0 G. Then, we can conclude that our protocol is suitable in a wide range of weak magnetic field sensing. 





unsrt
10

Escher11
B. M. Escher, R. L. de Matos Filho, and L. Davidovich.
General framework for estimating the ultimate precision limit in
  noisy quantum-enhanced metrology.
Nature Physics 7, 406 (2011).


Aharonov88
Y. Aharonov, D. Z. Albert, and L. Vaidman.
How the result of a measurement of a component of the spin of a
  spin-1/2 particle can turn out to be 100.
Phys. Rev. Lett. 60, 1351 (1988).

Dressel14
J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W.
  Boyd.
Colloquium: Understanding quantum weak values: Basics and
  applications.
Rev. Mod. Phys. 86, 307 (2014).

Torres16
J. P. Torres and L. J. Salazar-Serrano.
Weak value amplification: a view from quantum estimation theory that
  highlights what it is and what isn't.
Sc. Rep. 6, 19702 (2016).

Coto17
R. Coto, V. Montenegro, V. Eremeev, D. Mundarain, and M. Orszag.
The power of a control qubit in weak measurements.
Sc. Rep. 7, 6351 (2017).

Magana16
O. S. Magaña-Loaiza, J. Harris, J. S. Lundeen, and R. W. Boyd.
Weak-value measurements can outperform conventional measurements.
Phys. Scr., 92, 023001 (2016).

Arvidsson-Shukur19
D. R. M. Arvidsson-Shukur, N. Y. Halpern, H. V. Lepage,
  A. A. Lasek, C. H. W. Barnes, and S. Lloyd.
Quantum advantage in postselected metrology.
Nat. Comm. 11, 3775 (2020).

Wu09
S. Wu and K. Mølmer.
Weak measurements with a qubit meter.
Phys. Lett. A 374, 34 (2009).

Brunner10
N. Brunner and C. Simon.
Measuring small longitudinal phase shifts: Weak measurements or
  standard interferometry?
Phys. Rev. Lett. 105, 010405 (2010).

Feizpour11
A. Feizpour, X. Xing, and A. M. Steinberg.
Amplifying single-photon nonlinearity using weak measurements.
Phys. Rev. Lett. 107, 133603 (2011).

Zhang15
L. Zhang, A. Datta, and I. A. Walmsley.
Precision metrology using weak measurements.
Phys. Rev. Lett. 114, 210801 (2015).

Vitalie
V. Eremeev and M. Orszag.
Phonon maser stimulated by spin postselection.
Phys. Rev. A  101, 063815 (2020).
  
Ritchie91
N. W. M. Ritchie, J. G. Story, and R. G. Hulet.
Realization of a measurement of a “weak value”.
Phys. Rev. Lett. 66, 1107 (1991).

Pryde05
G. J. Pryde, J. L. O'Brien, A. G. White, T. C. Ralph, and H. M. Wiseman.
Measurement of quantum weak values of photon polarization.
Phys. Rev. Lett. 94, 220405 (2005).

Dixon09
P. Ben Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell.
Ultrasensitive beam deflection measurement via interferometric weak
  value amplification.
Phys. Rev. Lett. 102, 173601 (2009).

Starling09
D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell.
Optimizing the signal-to-noise ratio of a beam-deflection measurement
  with interferometric weak values.
Phys. Rev. A 80, 041803 (2009).

Kocsis11
S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P.
  Mirin, L. K. Shalm, and A. M. Steinberg.
Observing the average trajectories of single photons in a two-slit
  interferometer.
Science 332, 1170 (2011).

Hosten08 O. Hosten and P. Kwiat. 
Observation of the Spin Hall Effect of Light via Weak Measurements.
Science 319,  787 (2008). 

Blok14
M. S. Blok, C. Bonato, M. L. Markham, D. J. Twitchen, V. V. Dobrovitski, and
  R. Hanson.
Manipulating a qubit through the backaction of sequential partial
  measurements and real-time feedback.
Nat. Phys. 10, 189 (2014).

Alves17
G. Bié Alves, A. Pimentel, M. Hor-Meyll, S. P. Walborn, L. Davidovich, and
  R. L. de Matos Filho.
Achieving metrological precision limits through postselection.
Phys. Rev. A 95, 012104 (2017).

Combes14
J. Combes, C. Ferrie, Z. Jiang, and C. M. Caves.
Quantum limits on postselected, probabilistic quantum metrology.
Phys. Rev. A 89, 052117 (2014).

Knee13
G. C. Knee, G. A. D. Briggs, S. C. Benjamin, and E. M. Gauger.
Quantum sensors based on weak-value amplification cannot overcome
  decoherence.
Phys. Rev. A 87, 012115 (2013).

Knee14
G. C. Knee and E. M. Gauger.
When amplification with weak values fails to suppress technical
  noise.
Phys. Rev. X 4, 011032 (2014).

Ferrie14
C. Ferrie and J. Combes.
Weak value amplification is suboptimal for estimation and detection.
Phys. Rev. Lett. 112, 040406 (2014).

Tanaka13
S. Tanaka and N. Yamamoto.
Information amplification via postselection: A parameter-estimation
  perspective.
Phys. Rev. A 88, 042116 (2013).

Alves15
G. Bié Alves, B. M. Escher, R. L. de Matos Filho, N. Zagury, and
  L. Davidovich.
Weak-value amplification as an optimal metrological protocol.
Phys. Rev. A 91, 062107 (2015).

Schloss18
J. M. Schloss, J. F. Barry, M. J. Turner, and R. L. Walsworth.
Simultaneous broadband vector magnetometry using solid-state spins.
Phys. Rev. Applied 10, 034044 (2018).

Arai18
K. Arai, J. Lee, C. Belthangady, D. R. Glenn, H. Zhang, and R. L. Walsworth.
Geometric phase magnetometry using a solid-state spin.
Nat. Comm. 9, 4996 (2018).

Degen17
C. L. Degen, F. Reinhard, and P. Cappellaro.
Quantum sensing.
Rev. Mod. Phys. 89, 035002 (2017).

Taylor08
J. M. Taylor, P. Cappellaro, L. Childress, L. Jiang, D. Budker, P. R. Hemmer,
  A. Yacoby, R. Walsworth, and M. D. Lukin.
High-sensitivity diamond magnetometer with nanoscale resolution.
Nat. Phys. 4, 810 (2008).

Maze08
J. R. Maze, P. L. Stanwix, J. S. Hodges, S. Hong, J. M. Taylor, P. Cappellaro,
  L. Jiang, A. S. Zibrov, A. Yacoby, R. Walsworth, and M. D. Lukin.
Nanoscale magnetic sensing with an individual electronic spin qubit
  in diamond.
Nature 455, 644 (2008).

Zaiser16
S. Zaiser, T. Rendler, I. Jakobi, T. Wolf, S.-Y. Lee,
  S. Wagner, V. Bergholm, T. Schulte-Herbruggen, P. Neumann,
  and J. Wrachtrup.
Enhancing quantum sensing sensitivity by a quantum memory.
Nat. Comm. 7, 12279 (2016).

Aiello13
C. D. Aiello, M. Hirose, and P. Cappellaro.
Composite-pulse magnetometry with a solid-state quantum sensor.
Nat. Comm. 4, 1419 (2013).

Balasubramanian09
G. Balasubramanian, P. Neumann, D. Twitchen, M.
  Markham, R. Kolesov, N. Mizuochi, J. Isoya, J. Achard,
  J. Beck, J. Tissler, V. Jacques, P. R. Hemmer, F.
  Jelezko, and J. Wrachtrup.
Ultralong spin coherence time in isotopically engineered diamond.
Nat. Mater. 8, 383 (2009).

Hirose12
M. Hirose, C. D. Aiello, and P. Cappellaro.
Continuous dynamical decoupling magnetometry.
Phys. Rev. A 86, 062320 (2012).

Bonato
C. Bonato, M. S. Blok, H. T. Dinani, D.W. Berry, L. Markham,
D. J. Twichen, and R. Hanson.
Optimized quantum sensing with a single electron spin using real-time adaptive measurements.
Nat. Nanotechnol. 11, 247 (2016).  

Lovchinsky16
I. Lovchinsky, A. O. Sushkov, E. Urbach, N. P. de Leon, S. Choi, K. De Greve,
  R. Evans, R. Gertner, E. Bersin, C. Müller, L. McGuinness, F. Jelezko,
  R. L. Walsworth, H. Park, and M. D. Lukin.
Nuclear magnetic resonance detection and spectroscopy of single
  proteins using quantum logic.
Science, 351, 836 (2016).

Glenn18
D.R. Glenn, D.B. Bucher, J. Lee, M.D. Lukin, H. Park, and R.L. Walsworth.
High-resolution magnetic resonance spectroscopy using a solid-state
  spin sensor.
Nature 555, 351 (2018).

Sushkov14l
 A. O. Sushkov, I. Lovchinsky, N. Chisholm, R. L. Walsworth, H. Park, and M. D. Lukin. 
Magnetic resonance detection of individual proton spins using quantum
  reporters.
Phys. Rev. Lett. 113, 197601 (2014).

Shi15
F. Shi, Q. Zhang, P. Wang, H. Sun, J. Wang, X. Rong, M.
  Chen, C. Ju, F. Reinhard, H. Chen, J. Wrachtrup,
  J. Wang, and J. Du.
Single-protein spin resonance spectroscopy under ambient conditions.
Science 347, 1135 (2015).

Abobeih19
M. H. Abobeih, J. Randall, C. E. Bradley, H. P. Bartling, M. A. Bakker, M. J.
  Degen, M. Markham, D. J. Twitchen, and T. H. Taminiau.
Atomic-scale imaging of a 27-nuclear-spin cluster using a quantum
  sensor.
Nature 576, 411 (2019).

Shikano
Y. Shikano and S. Tanaka.
Estimation of Spin-Spin Interaction by Weak Measurement Scheme.
EPL 96, 40002 (2011).

Pfender
M. Pfender, P. Wang, H. Sumiya, S. Onoda, W. Yang, D. B. R. Dasari, P. Neumann, X.-Y. Pan, J. Isoya, R.-B. Liu, and J. Wrachtrup. 
High-Resolution Spectroscopy of Single Nuclear Spins via Sequential Weak Measurements.
Nat. Comm 10, 594 (2019).

Cramer16
J. Cramer, N. Kalb, M. A. Rol, B. Hensen, M. S. Blok, M. Markham, D. J. Twitchen, R. Hanson, and T. H. Taminiau
Repeated quantum error correction on a continuously encoded qubit by real-time feedback.
Nature Communication 7, 11526 (2016).

Taminiau14
H.T. Taminiau, J. Cramer, T. van der Sar, V. V. Dobrovitski, and R. Hanson.
Universal control and error correction in multi-qubit spin registers
  in diamond.
Nat Nano 9, 171 (2014).

Jamonneau16
P. Jamonneau, G. Hétet, A. Dréau, J.-F. Roch, and V. Jacques.
Coherent population trapping of a single nuclear spin under ambient
  conditions.
Phys. Rev. Lett. 116, 043603 (2016).

Coto17b
R. Coto, V. Jacques, G. Hétet, and J. R. Maze.
Stimulated raman adiabatic control of a nuclear spin in diamond.
Phys. Rev. B 96, 085420 (2017).

Yun19
J. Yun, K. Kim, and D. Kim.
Strong polarization of individual nuclear spins weakly coupled to
  nitrogen-vacancy color centers in diamond.
New J. Phys. 21, 093065 (2019).

Bradley19
C. E. Bradley, J. Randall, M. H. Abobeih, R. C. Berrevoets, M. J. Degen, M. A.
  Bakker, M. Markham, D. J. Twitchen, and T. H. Taminiau.
A ten-qubit solid-state spin register with quantum memory up to one
  minute.
Phys. Rev. X 9, 031045 (2019).

Hensen15
B. Hensen, H. Bernien, A. E. Dreau, A. Reiserer, N. Kalb, M. S. Blok,
  J. Ruitenberg, R. F. L. Vermeulen, R. N. Schouten, C. Abellan, W. Amaya,
  V. Pruneri, M. W. Mitchell, M. Markham, D. J. Twitchen, D. Elkouss,
  S. Wehner, T. H. Taminiau, and R. Hanson.
Loophole-free bell inequality violation using electron spins
  separated by 1.3 kilometres.
Nature 526, 682 (2015).

Maurer12
P. C. Maurer, G. Kucsko, C. Latta, L. Jiang, N. Y. Yao, S. D. Bennett,
  F. Pastawski, D. Hunger, N. Chisholm, M. Markham, D. J. Twitchen, J. I.
  Cirac, and M. D. Lukin.
Room-temperature quantum bit memory exceeding one second.
Science 336, 1283 (2012).

Dreau13
A. Dréau, P. Spinicelli, J. R. Maze, J.-F. Roch, and V. Jacques.
Single-shot readout of multiple nuclear spin qubits in diamond under
  ambient conditions.
Phys. Rev. Lett. 110, 060502 (2013).

deLange_science
G. de Lange, Z. H. Wang, D. Risté, V. Dobrovitski, and R. Hanson
Universal Dynamical Decoupling of a Single Solid-State Spin from a Spin Bath
Science 330, 60 (2012).

deLange_Scirep
G. de Lange, T. van der Sar, M. Blok, Z-H. Wang, V. Dobrovitski, and R. Hanson
Controlling the quantum dynamics of a mesoscopic spin bath in diamond
  ambient conditions.
Sc. Rep. 2, 382 (2012).

Albrecht2014
A. Albrecht, G. Koplovitz, A. Retzker, F. Jelezko, S. Yochelis, D. Porath, Y. Nevo, 
O. Shoseyov, Y. Paltiel, and M. B. Plenio. Self-assembling hybrid diamond–biological quantum devices
New. J. Phys. 16, 093002 (2014).

Lei2017 C. Lei, S. Peng, C. Ju, M-H Yung, and J. Du.
Decoherence Control of Nitrogen-Vacancy Centers
Sc. Rep. 7, 11937 (2017).

Aslam17
N. Aslam, M. Pfender, P. Neumann, R. Reuter, A. Zappe, F. Fávaro de Oliveira, A. Denisenko, H. Sumiya, S. Onoda, J. Isoya, J. Wrachtrup.
Nanoscale nuclear magnetic resonance with chemical resolution
Science 357, 64 (2017).

Gurudev07 
M. V. G. Dutt et al., 
Quantum register based on individual electronic and nuclear spin qubits in diamond.
Science 316, 1312 (2007).


Zhou18
S. Zhou, M. Zhang, J. Preskill, and L. Jiang.
Achieving the heisenberg limit in quantum metrology using quantum
  error correction.
Nat. Comm. 9, 78 (2018).






Matsuzaki16
Y. Matsuzaki, T. Shimo-Oka, H. Tanaka, Y. Tokura,
  K. Semba, and N. Mizuochi.
Hybrid quantum magnetic-field sensor with an electron spin and a
  nuclear spin in diamond.
Phys. Rev. A 94, 052330 (2016).

Neumann10b
P. Neumann, J. Beck, M. Steiner, F. Rempp, H. Fedder,
  P. R. Hemmer, J. Wrachtrup, and F. Jelezko.
Single-shot readout of a single nuclear spin.
Science 329, 542 (2010).

Jacques09 
V. Jacques, P. Neumann, J. Beck, M. Markham, D. Twitchen, J. Meijer, F. Kaiser, G. Balasubramanian, F. Jelezko, and J. Wrachtrup.
Dynamic Polarization of Single Nuclear Spins by Optical Pumping of Nitrogen-Vacancy Color Centers in Diamond at Room Temperature.
Phys. Rev. Lett. 102, 057403 (2009).

Dinani19
 H. T. Dinani, D. W. Berry, R. Gonzalez, J. R. Maze, and C. Bonato.
Bayesian estimation for quantum sensing in the absence of single-shot detection.
Phys. Rev. B 99, 125413 (2019).

Braunstein
S. L. Braunstein, and C. M. Caves.
Statistical distance and the geometry of quantum states.
Phys. Rev. Lett. 72, 3439 (1994).



Liu
J. Liu, H. Yuan, X-M. Lu, and X. Wang.
Quantum Fisher information matrix and multiparameter estimation.
J. Phys. A: Math. Theo. 53, 023001 (2020).

Liu14
J. Liu, X-X. Jing, W. Zhong, and X-G. Wang.
Quantum Fisher Information for Density Matrices with Arbitrary Ranks.
Comm. Theor. Phys. 61, 45 (2014).

Gillespie
Daniel T. Gillespie
Exact numerical simulation of the Ornstein-Uhlenbeck process and its integral
Phys. Rev. E 54, 2084 (1996).







