Submicrometer-scale temperature sensing using quantum coherence of a superconducting qubit

Interest is growing in the development of quantum sensing based on the principles of quantum mechanics, such as discrete energy levels, quantum superposition, and quantum entanglement. Superconducting flux qubits are quantum two-level systems whose energy is sensitive to a magnetic field. Therefore, they can be used as high-sensitivity magnetic field sensors that detect the magnetization of a spin ensemble. Since the magnetization depends on temperature and the magnetic field, the temperature can be determined by measuring the magnetization using the flux qubit. In this study, we demonstrated highly sensitive temperature sensing with high spatial resolution as an application of a magnetic field sensor using the quantum coherence of a superconducting flux qubit. By using a superconducting flux qubit to detect the temperature dependence of the polarization ratio of electron spins in nano-diamond particles, we succeeded in measuring the temperature with a sensitivity of 1.3 µKµ Hz−1 at T = 9.1 mK in the submicrometer range.


Introduction
Temperature, an important parameter in science and technology, can be measured in a variety of ways. In the vicinity of room temperature, highly precise thermometry is possible using the thermal expansion of materials. Such thermometers work, for examples, by detecting the change in the resonance frequency of an optical resonator [1,2]. In addition, thermometers based on nitrogen-vacancy (NV) centers in a diamond crystal have been proposed and realized [3][4][5][6][7]. The temperature in this case is measured using the change in zero magnetic field splitting due to thermal expansion of the crystal. However, thermal expansion cannot be used as a principle of thermometry at low temperatures, since it does not occur at temperatures much lower than the Debye temperature. The thermometers most widely used at low temperatures are based on the temperature dependence of the electric resistance of semiconductors. This allows for easy temperature measurements but requires batch-to-batch calibration in all temperature ranges using a primary thermometer. Furthermore, self-heating during measurement causes errors at low temperatures, making accurate temperature measurement difficult. Here, primary thermometers using different principles in the low-temperature range, such as those based on shot noise measurement [8], magnetic noise measurement [9], and nuclear orientation measurement [10,11] have been realized. In particular, ultra-high-sensitivity temperature measurements have been reported that exploit the polarizability of paramagnetic materials [12,13]. Moreover, recent progress in nanotechnology has led to the development of new small thermometer devices that can make local temperature measurements by using, for example, NV centers in a diamond [3,5], quantum dots [14,15], single-electron transistors [16], Coulomb blockade [17], superconducting junctions [18][19][20], and quantum two-level systems in superconducting microresonators [21].
A micrometer-scale thermometer has high spatial resolution and can be used for imaging the temperature distribution in a sample. Such a capability is especially important, for example, in measuring the activity of organelles in cells [5] and imaging current distributions [22] in condensed matter physics. To obtain an accurate thermal image with high spatial resolution, the thermometer should not affect the target object's temperature when they are in contact. To this end, the thermometer should be small to reduce its heat capacity. In addition, a small heat capacity has the advantage of reducing the time constant of the measurement [23]. However, because the temperature is obtained by measuring a physical quantity of the thermometer, the smaller the thermometer is, the lower the measurement accuracy becomes. More quantitatively, the measurement accuracy is proportional to the square root of the volume V of the thermometer. Therefore, it is challenging to measure the temperature accurately on the micrometer scale.
Quantum sensing is one possible way to avoid this trade-off [24]. The simplest method in quantum sensing detects the changes in the energy of a quantum system, while another method uses quantum coherence like Ramsey interference. By using these methods, we can approach the sensitivity bounded by the standard quantum limit (SQL). In addition, the use of quantum entanglement such as Greenberger-Horne-Zeilinger (GHZ) states can improve the sensitivity of the thermometer beyond the SQL [25]. Therefore, quantum sensing holds out the possibility of realizing a thermometer with both high precision and high spatial resolution.
In this study, we demonstrated highly precise thermometry with a spatial resolution below a micrometer by measuring the polarization of paramagnetic electron spins in nano-particles, which have quite small heat capacities. The polarization was detected as a phase change in the quantum superposition state of a superconducting flux qubit. This is one of the applications of highly sensitive magnetic field detection using superconducting flux qubits [26][27][28][29][30] utilizing developing superconducting quantum technologies. The methodology provides a new thermometry tool for condensed matter physics and thermodynamics in microscale areas.

Methods
To demonstrate submicrometer thermometry, a superconducting flux qubit was fabricated on a silicon substrate, on which nano-diamond powder was spread ( figure 1(a)). To form the qubit, aluminum was evaporated on the substrate with a thickness of 90 nm. The qubit chip had a microwave line for exciting the qubit and a heater line for heating the nano-diamonds. We applied a perpendicular small magnetic field B ⊥,bias (order of 10 µT) to control the qubit frequency and an in-plane magnetic field B ∥ = 2.5 mT to polarize spin ensembles in the nano-diamonds ( figure 1(a)). If the alignment of the magnetic field B ∥ is not perfect, the perpendicular component appears and the operating point of the flux qubit changes. We can cancel this component by adjusting B ⊥,bias to keep the flux qubit at the operating point. In this configuration, when the magnetic field generated from the magnetization of the nano-diamonds (B M in figure 1(a)) passes through the loop of the qubit, the qubit energy is shifted. The width of the aluminum wire of the superconducting flux qubit is 300 nm. Especially, the magnetization of the nano-diamond on this line can be best detected due to the large coupling between the nano-diamond and the superconducting line.
The flux qubit was magnetically coupled to a superconducting quantum interference device (SQUID) loop. In combination with coplanar waveguide, the SQUID constituted a 6.95 GHz transmission line resonator to readout the qubit state. We used the Josephson bifurcation readout method [31,32] to shorten the readout interval of the qubit [33] to 1 µs. The phase and amplitude of the readout microwave pulse passing through the resonator changes according to the state of the qubit. After amplification by the HEMT amplifier, a readout pulse having the information of the qubit is detected, and the information of the qubit readout. The qubit and nano-diamonds were set in a dilution refrigerator and cooled to below 100 mK. From the measured spectra, the energy gap and the persistent loop current of the qubit were determined to be ∆/h = 5.16 GHz and I p = 297 nA, respectively.
The nano-diamond particles are placed in thermal contact with the target object to measure its temperature. In this experiment, the target objects were the flux qubit and the substrate. Because of the thermal contact, the spin temperature of the nano-diamond particles reached the temperature of the target. The polarization of the spin ensemble of nano-diamond particles depends on the external magnetic field and the temperature. Therefore, a change in the temperature of the nano-diamond could be detected when the superconducting flux qubit was under a fixed magnetic field.
As a spin ensemble to convert the change in temperature to a magnetic field, we chose commercially available nano-diamonds (NDNV140nmHi10ml, Adàmas Nanotechnologies). They were synthesized in a high-pressure high-temperature (HPHT) method and had a typical size of 140 nm. Many kinds of nitrogen defects may have formed in the nano-diamonds, but single substitutional nitrogen centers (called P1 centers) are more likely than either NV centers (3 ppm) or defects composed of a plurality of nitrogen atoms.
The electron spin of the P1 center in the nano-diamond is a two-level quantum system (S = 1 2 ) with an isotropic g-factor, g = 2.00216 [34], and an external magnetic field B ∥ causes Zeeman splitting. Moreover, the level of the P1 center splits further into three due to the hyperfine interaction of 14 N with the nuclear spin The heater line excites one of the parasitic resonant modes of the circuit and heats the nano-diamond by its energy. Resonant microwaves of the qubit are emitted from the qubit excitation line and control the quantum state of the qubit. (b) Schematic diagram of measurement principle in quantum hybrid system. The magnetization of the nano-diamond is changed by the heat from the target object. The change in the qubit state due to the magnetization can be detected through the SQUID by using the Josephson bifurcation readout method. The temperature of the target can be estimated from the qubit state. (c) Magnetic field dependence of the measured qubit energy spectrum. ∆B ⊥ is the difference between the applied magnetic field and the magnetic field at the degeneracy point of the qubit. The energy of the qubit is changed by the magnetic field shift caused by the change in the spin polarization. I = 1. In general, both the electron spins and the nuclear spins contribute to the magnetization. The expected value of the magnetization can be described using a density matrix ρ: where M S0 and M I0 are the saturation magnetizations of the electron spins and the nuclear spins. In the thermal equilibrium state, the diagonal elements of the density matrix are expressed by a Boltzmann distribution: Here, E i is the ith eigenenergy, k B is the Boltzmann constant, and T is the temperature. In terms of eigenenergies and eigenstates |ϕ i ⟩, the expectation value of the magnetization of the thermal equilibrium state is described as Since the ratio of the gyromagnetic ratio of the electron spin to that of 14 N is γe γ14 N = 4.5 × 10 3 , the second term can be ignored. Under the condition |E i | ≪ k B T, the magnetization is Here, the magnetization does not contain a temperature-independent term because Tr(Ŝ) = 0. This formula clearly shows that the magnetization is proportional to the inverse temperature under our experimental conditions. In general, diamond crystals have various defects with different hyperfine parameters [35,36]. However, such differences would not have a significant effect on the magnetization for the magnetic field and temperature ranges in our experiments. For example, at T = 50 mK, the deviation from equation (4) is about 0.4% of the magnitude of magnetization (see the appendix).
In the experiment, we first measured the temperature dependence of the spectra of the qubit with nano-diamonds from 100 mK down to the base temperature of the refrigerator (about 10 mK). Here, the electron spin in the P1 center is polarized when a constant external magnetic field is applied to the nano-diamonds. Since the polarization of electron spin systems are determined by thermal excitation, the lower the temperature is, the larger the polarization becomes. Therefore, the magnetic field B M generated by the magnetization of the nano-diamonds grows as the temperature decreases. The perpendicular magnetic field B ⊥,M felt by the flux qubit depends on B M . Since the flux qubit spectrum shift is related to B ⊥,M , the flux qubit measurement can be used to determine the temperature. The experimental results are presented in section 3.
Furthermore, we performed thermometry by using the quantum coherence of a flux qubit. A quantum superposition state composed of the ground state |g⟩ and the excited state |e⟩ was prepared on a flux qubit by applying a resonant microwave pulse. In the quantum superposition state |ψ(t)⟩, the phase difference between |g⟩ and |e⟩ evolves over time depending on the energy difference E: This formula shows that the magnetic field B ⊥ can be measured by detecting the phase, since the qubit energy has the magnetic field dependence E(B ⊥ ). In the experiment, we waited for a time t = τ and measured the qubit state with the X basis to see this time evolution. To prevent unwanted temperature fluctuations at the base temperature, the temperature of the refrigerator was stabilized at T = 50 mK. Under this condition, the parasitic resonance mode at 2.023 GHz of the qubit chip was used to control the temperature of the nano-diamonds locally. The nano-diamonds were heated by the large microwave current flow by applying the resonant microwave signal to the parasitic resonator via the heater line. The experimental results are presented in section 4.

Temperature dependence of magnetic field shift
We measured the temperature dependence of the magnetic field shift by measuring the flux qubit spectrum. The qubit spectrum was measured by sweeping the bias magnetic field B ⊥,bias in a range of about 300 nT near the degenerate point. The resulting shift in the magnetic field of the qubit ∆Φ/S q = B ⊥,M is plotted as a function of inverse temperature 1/T in figure 2. Here, B ⊥,M is the magnetic field generated by the magnetization of the nano-diamonds at the position of the flux qubit and S q is the qubit loop area. We can estimate B ⊥,M from the measured value of ∆Φ by assuming that S q is constant. Without the external magnetic field (B ∥ = 0 mT), the magnetic field shift of the qubit shows almost no temperature dependence. On the other hand, there is a clear temperature-dependent shift when an external magnetic field (B ∥ = 2.5 mT) is applied. This dependence is linear in the range from 50 to 20 mK, with slope dB ⊥,M dT −1 = 6.158 nT·K. A deviation from the theoretical curve (equation (3)) is visible below 20 mK, but it is within the range where the two calibrated temperatures show different values (black and red triangles in figure 2). In general, a measurement made by a resistance thermometer is affected by self-heating in the region below several 10 mK. The deviation below 20 mK is considered to be caused by the inaccuracy of the calibration of the RuO thermometer. There is also a slight deviation from the theoretical curve at high temperatures above 50 mK. It could be speculated that this deviation originated from the NV center with spin S = 1, but our analysis (see the appendix) shows that its effect is negligible. Another possibility is a change in the effective loop area of the flux qubit due to the temperature dependence of the magnetic penetration depth. The effective loop area has the following temperature dependence: Here, λ ⊥ (T) is the magnetic penetration depth of thin superconducting film under a perpendicular magnetic field at temperature T. In general, the penetration depth of superconductors is constant at temperatures well below T C , but increases as the temperature approaches T C . Even if the bias magnetic field B ⊥,bias applied to the qubit is constant, when the effective loop area of the flux qubit changes with temperature, the bias magnetic flux of the qubit also changes, resulting in qubit energy shifts. Therefore, the relationship between the magnetic field B ′ ⊥ , which is estimated from the qubit energy under the assumption of constant loop area, and the true magnetic field B ⊥ = B ⊥,M + B ⊥,bias is expressed as follows: To validate this effect, we measured the spectra around the following negative and positive bias fields of the flux qubit: B ⊥,bias = −10 µT (− 1 2 Φ 0 ) or B ⊥,bias = +10 µT (+ 1 2 Φ 0 ). Here, the polarity in the temperature-dependent magnetic field shift depends on the sign of B ⊥,bias . We measured this temperature dependence at B ∥ = 0 mT. A slight shift in the magnetic flux with the same polarity to the bias field was observed in the high-temperature range over 50 mK (figure 2). Measurement results at B ∥ = 0 mT in figure 2 are corresponded to B ′ ⊥ − B ⊥ in equation (6). We know the loop area, so we can estimate λ ⊥ (T) − λ ⊥ (0) from these measurement data using equation (6). It was also found from this estimation that the penetration depth of the aluminum thin film of the superconducting qubit loop changes by a few nanometers in the high temperature region above 50 mK (figure 2 inset). Adding the effect of the temperature change on the experimentally obtained penetration depth to the theoretical curve gives the orange dotted line in figure 2. The orange line well matches the experimental data at B ∥ = 2.5 mT above T = 50 mK. We can use the measurement of the temperature dependence to relate the magnetic field shift and the inverse temperature. Therefore, we can determine the temperature below T = 50 mK from the measured shift in energy of the qubit by using the extrapolated theoretical curve.

Thermometry using Ramsey oscillation
To carry out highly sensitive thermometry using quantum coherence, the Ramsey oscillation of the qubit was measured by setting the temperature of the refrigerator at T = 50 mK. Figure 3(a) shows the dependence of the Ramsey oscillation on the heater power. Here, the heater power is defined as the microwave input power to the heater line on the chip by taking into account the attenuation of attenuators at each temperature stage. Since the qubit energy is modulated by the temperature-dependent magnetic field generated by the nano-diamonds, the heating changes the frequency (inverse of the period) of the Ramsey oscillation. In addition to the heat, the magnetic effect of microwaves can change the energy of the qubit, but in this case, the effect disappears in a very short time after the microwaves are turned off. In the case of the heat, on the other hand, because of the heat capacity of the sample, the effect remains until the sample reaches thermal equilibrium after the microwaves are turned off. We confirmed that the energy shift of the qubit, which appears when a microwave is added to the chip, remains for a while even after the microwave is turned off. This indicates that the energy shift of the qubit is caused by a thermal effect, not a magnetic effect. A clear decrease in the frequency of the Ramsey oscillation is observed as the heater power increases, at least up to 2.5 nW. The dependence of the temperature increase on the heater power can be derived using the relationship between qubit energy and temperature (figures 1(c) and 2) and is shown in figure 3(b). The applied heater power is absorbed by the parasitic resonator and heats the chip. The heat capacity of the nano-diamonds is small enough compared to that of the chip that the nano-diamond reaches the same temperature as the chip without affecting the chip's temperature. The change in the temperature of the chip due to the heater was measured by detecting the temperature change of the nano-diamond with the qubit. As shown in figure 3, the sub-millikelvin change in temperature of the nano-diamonds caused by the below-nanowatt incident power is detected by the quantum coherence of the flux qubit. Now let us estimate the sensitivity of the thermometer. The Ramsey oscillation was measured by reading the probability of the qubit in |+⟩ = 1 √ 2 (|e⟩ + |g⟩) state. The measured probability can be expressed as where P 0 is the offset, V is the visibility of the measurement, and T 2 is the phase relaxation time of the qubit. δω 2π is the frequency detuning defined as Here, the first term on the right-hand side is the resonance frequency of the qubit expressed in terms of the energy gap ∆ and the energy detuning ε, and ω 0 is the angular frequency of the applied microwave pulse. The sensitivity of the magnetic field as determined from the Ramsey measurement can thus be written as Here, ∆P is the standard deviation of the readout probability P. By choosing the most sensitive conditions, i.e. τ = T 2 and δωT 2 = 1 2 + n π, the sensitivity becomes We can relate the derivative of the energy detuning of the qubit with respect to the magnetic field to the qubit parameters: dε dB ⊥ = 2I p S q . Here, I p is the persistent current of the qubit and we have used the relationship ε = 2I p Φ − 2n+1 2 Φ 0 ; Φ = B ⊥ S q is the magnetic flux through the qubit loop, and ε = 0 corresponds to the qubit degeneracy point. Note that the value 1 h dε dB ⊥ = 95.06 GHz µT −1 is entirely determined from qubit parameters that were obtained from figure 1(c). The phase relaxation time of the qubit was measured to be T 2 = 23.5 ns. The other parameters were derived from the qubit spectrum: V = 7.88 × 10 −2 , √ ∆ 2 + ε 2 /h = 5.227 GHz, and ε/h = 0.834 GHz. The parameter ∆P is derived from the readout condition of the qubit. Qubits are read out in the |+⟩ or |−⟩ state; thus, the readout value follows a binomial distribution. Therefore, the dependence of the standard deviation of the readout probability on the number of repetitions of the measurement N is ∆P = P(1−P) N . We chose P = 1 2 to minimize the standard deviation and the qubit readout interval of 1 µs, which corresponds to N = 10 6 times per second. Thus, the standard deviation of the readout probability is ∆P = 5.0 × 10 −4 for the measurement duration of one second. From these parameters, the sensitivity to the magnetic field was estimated to be 112 pT √ Hz −1 . We can convert the magnetic field sensitivity to temperature sensitivity as follows: The temperature sensitivity was determined from the experimental relationship between temperature and the magnetic field generated by the magnetization dB ⊥ dT −1 (figure 2) to be 45.5 µK √ Hz −1 at T = 50 mK.
As shown in equation (11), the sensitivity of thermometry is expected to increase at lower temperatures. We estimated its ultimate sensitivity at the base temperature of the dilution refrigerator. First, we evaluated the base temperature from the magnetic field shift B ⊥,M due to the magnetization of the nano-diamonds because it is in a range where the RuO thermometer calibration is unreliable. The base temperature was determined to be T = 9.1 mK from the theoretical curve in figure 2. At this temperature, the sensitivity reached 1.3 µK √ Hz −1 with the parameters of T 2 = 18.7 ns, √ ∆ 2 + ε 2 /h = 5.261 GHz, and ε/h = 1.027 GHz, and V = 8.24 × 10 −2 . If the system temperature can be cooled down without limitation, energy splitting due to hyperfine interactions in diamond defects constrains sensitivity. To prevent this, spin systems with no or small hyperfine interactions are desirable. Even with the current setup, the temperature range reachable by the dilution refrigerator can be measured with sufficient sensitivity.
We found that the frequency of the Ramsey oscillation fluctuated at the base temperature. There were two Ramsey oscillations at intervals of about 5 min, as shown in figure 4. We can estimate the frequencies of the Ramsey oscillations by fitting each measurement using equation (7). As shown in figure 3, the frequency of the Ramsey oscillations are stable at T = 50 mK, so the frequency change of these Ramsey oscillations detect the change in the base temperature rather than the magnetic field change due to vibration. The temperature difference between the two measurements was determined from the frequencies of the Ramsey oscillation to be T diff = 7.2 µK. This difference might have been caused by the instability of the temperature of the dilution refrigerator, as we will discuss later.
Equation (11) assumes that the Curie law is obeyed. Accordingly, the magnetization of the nano-diamond M is proportional to the external magnetic field B ∥ , so the sensitivity is also expected to be proportional to the external magnetic field. However, at high magnetic fields, the magnetization is saturated and the Curie law cannot be applied; in such case, the temperature dependence of the magnetization disappears and the sensitivity of the thermometer decreases. For our thermometry, we can show that there is a specific magnetic field at which sensitivity is maximized. Specifically, the sensitivity of the thermometer is maximized by a magnetic field that satisfies the following condition: In this particular experiment, the optimal magnetic field was calculated to be B opt ∥ = 10.1 mT. At the optimum magnetic field, the sensitivity is expected to be 2.4 times higher than that at B ∥ = 2.5 mT.
Although the standard derivation of the readout probability can be estimated by assuming a perfect binomial distribution, noise arising from sources that were not included in our considerations could have dominated the experiment in reality. To investigate the properties of the noise in detail, we measured the Allan deviation [37] of the magnetic field B ⊥ (figure 5). As shown in figure 1(b), in the measurement system, the quantum state of the flux qubit changes with the magnetization of the diamond, and the change in loop current of the flux qubit due to the change in the quantum state is read by the Josephson bifurcation readout. When the qubit is in a quantum superposition state, we can obtain the stability of the all the system of the magnetic field measurement using quantum coherence because the phase of the qubit is changed by the magnetization of the diamond. On the other hand, when the qubit is in the ground state, the phase of the qubit does not change with the magnetic field, so it is expected that the stability are obtained only for the qubit readout system. Measurements were carried out when the qubit was in the quantum superposition state used for thermometry and in the ground state as a control experiment. The slope of the Allan deviation A ) indicate that white noise is dominant up to about 100 ms. Around 1 s, the effect of 1/f noise (σ ∝ τ 0 A ) dominates the system with a minimum value at around 10 s. Over 100 s, the Allan deviation increases as a result of the effect of a random walk (σ ∝ τ 1 A ). By comparing these two curves, we can conclude that the noise of the thermometer is not dominated by the qubit measurement system, since the Allan deviation for the ground state is always smaller than that of the superposition state. A possible source of the excess noise could be the fluctuation of the magnetic field and temperature which the nano-diamond spin and flux qubit sense.

Conclusion
We demonstrated high-sensitivity thermometry with submicrometer-scale spatial resolution by measuring the polarization ratio of electron spins trapped at defects in nano-diamond particles by using the quantum coherence of a superconducting flux qubit. We succeeded in measuring the temperature with high sensitivity, 1.3 µK √ Hz −1 . Because the magnitude of the spin magnetization obeys a simple physical law, the temperature can be measured at low temperatures without having to fully calibrate the thermometer, provided that the required parameters are known. That is, if we can determine the magnetic field shift generated by the magnetization of the spins at several temperatures, we can extend the measurement range of the thermometer to lower temperatures while keeping its high accuracy. Therefore, we believe that our thermometer is not only an important application of quantum sensing, but that it is also useful for measuring physical properties at low temperatures.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).
At low temperatures of Dβ ≫ 1, there is no temperature dependence, and only a constant magnetization 4 3 g 2 µ 2 B NB VD appears. In the case of powders, the magnetization is averaged out and does not appear except in the direction of the magnetic field.
On the other hand, there is no magnetic anisotropy in the case of a system with a spin angular momentum of S = 1 2 , such as a P1 center. Thus, the Zeeman energy and hyperfine interaction are the main contributions to the spin Hamiltonian The g-factor g and hyperfine parameters A ∥ and A ⊥ vary depending on the type of defect. Using the Hamiltonian, the magnetization can be expressed as  interaction is at most 100 MHz (∼5 mK). These energy scales are lower than the thermal energy of the experiment. Therefore, equation (17) can be expanded in the high temperature limit as As in the case of NV centers in diamond, since the occupation numbers of the four orientations of the crystal axes are the same, the angular dependence of the magnetization parallel to the magnetic field disappears. Therefore, the magnetization parallel to the magnetic field direction can be described as The hyperfine interaction appears in terms of order β 3 or above. Therefore, the difference in effect of the defect type on the magnetization appears only in the temperature range lower than the energy of the hyperfine interaction. At an external magnetic field B = 2.5 mT, the Zeeman energy gµ B B is converted to a temperature of 3.4 mK. The hyperfine energies of diamond P1 defect A ∥ and A ⊥ are converted to a temperature of 5.5 mK and 2.8 mK, respectively. For example, at T = 50 mK, the ratio of the second and first terms in equation (19) is calculated to be 0.4%. The temperature dependence of the magnetization for major defects is calculated up to β 7 in figure 6(a). The difference in the hyperfine interaction appears only at low temperatures. Since the nano-diamond used here was synthesized with the HPHT method, most of the defects would have been composed of nitrogen. Among the various nitrogen defects, the P1 center is most likely to be formed stochastically, because it is substituted by only one nitrogen. In addition, since the difference in magnetization originating from the effect of hyperfine interactions is small in our temperature range, it is reasonable to treat the magnetization from the defects with S = 1 2 as being due to P1 centers. Magnetization measurements detect the sum of magnetic moments from all defects involved. Thus, the temperature dependence of the magnetization varies for different ratios of NV and P1 centers ( figure 6(b)). At low temperatures, the NV center has a constant magnetization and its effect is negligible. However, at high temperatures, the temperature dependence is affected by the ratio. In standard diamond crystals, the effect of NV centers is negligible even at high temperatures since the P1 centers are the dominant defects.