## Abstract

To identify the decoherence origin, frequency spectra using multiple π-pulses have been extensively studied. However, little has been discussed on how to define the spectral intensities from multiple-echo decays and how to incorporate the Hahn-echo *T*_{2} in the noise spectra. Here, we show that experiments based on two theories solve these issues. As proved in the previous theory, the spectral intensity is given as the decay in the long-time limit. Unlike the initial process of decays, this definition is not only theoretically proven but also validated experimentally, since long-time behaviors are generally free from experimental artifacts. The other is the fluctuation–dissipation theory, with which the Hahn-echo *T*_{2} is utilized as the zero-frequency limit of the noise spectrum and as an answer to the divergent issue on the 1/*f*^{n} noises. As a result, arsenic nuclear spins are found to exhibit 1/*f*^{2} dependences over two orders of magnitude in all the substrates of un-doped, Cr-doped semi-insulating and Si-doped metallic GaAs at 297 K. The 1/*f*^{2} dependence indicates that the noise is dominated by a single source with characteristic frequency *f*_{c}^{un} = 170 ± 10 Hz, *f*_{c}^{Cr} = 210 ± 10 Hz and *f*_{c}^{Si} = 460 ± 30 Hz. These *f*_{c} values are explained by a model that the decoherence is caused by the fluctuations of next-nearest-neighboring nuclear spins.

To realize quantum computing, electron-, nuclear-, and pseudo-spins have been extensively studied as promising candidates of quantum bits. One of the greatest findings is to apply multiple π-pulses for not just enhancing the coherence time but presenting decoherence-noise spectra. In these studies, some report the usual 1/*f* and others 1/*f*^{3} spectra^{1}, where very little has been discussed on how to deduce the spectral intensities from their raw data: the multiple-echo decay. Moreover, the Hahn-echo *T*_{2}, on which the coherence enhancement is based, has not been incorporated in noise spectra.

In this paper, we demonstrate that experiments based on two theories solve these issues. In our previous theory, it is proved that the spectral intensity should be defined as the multiple-echo decay in the long-time limit^{2}. We take advantage of this definition, which is free from experimental artifacts usually entailed in the initial decays. Experimentally, we employ a multiple alternating-phase-Meiboom-Gill (APCP) π-pulse sequence instead of the usually employed Carr-Purcell-Meiboom-Gill (CPMG)^{1}, resulting in multiple-echo decays for shorter π-pulse intervals. We also take into account the fluctuation–dissipation theorem^{3} that predicts that the Hahn-echo *T*_{2} reflects the zero-frequency intensity. This indicates that the Hahn-echo *T*_{2} generally solves the divergent issue on the 1/*f*^{n} noises and validates the spectrum as the maximum value at zero frequency. As a result, ^{75}As nuclear spins are found to exhibit, for all the substrate of un-doped, Cr-doped semi-insulating and Si-doped metallic GaAs, not the usual 1/*f* noise but a Loerntzian 1/*f*^{2} spectrum at 297 K. The 1/*f*^{2} dependences are characterized by the frequency *f*_{c}^{un} = 170 ± 10 Hz and *f*_{c}^{Cr} = 210 ± 10 Hz. These *f*_{c} values are explained by a model that the decoherence is caused by the fluctuations of next-nearest-neighboring nuclear spins. For the metallic GaAs, the somewhat larger value *f*_{c}^{Si} = 460 ± 30 Hz can be understood that the spin fluctuation of doped carriers enhances the dipolar coupling between the nearby nuclear spins, and hence the increase in *f*_{c}.

## Theories

Before discussing the results, we briefly outline the theories^{2, 3} on which the present experiments are based. The fluctuation–dissipation theorem proves that the spin–lattice relaxation time *T*_{1} and the Hahn-echo *T*_{2} are given as^{3}

where γ_{n} is the gyromagnetic ratio of the nuclear spin, *J*_{⊥}(*ω*) [*J*_{||}(0)] is the spectrum intensity due to the fluctuating magnetic field δ*B*_{x, y}(*t*) [δ*B*_{z}(*t*)] acting on the nuclear spin where a static magnetic field *B*_{z} is applied in the *z*-direction. The *ω* in Eq. (1) is the resonance frequency given as γ_{n} *B*_{z} of the order of 10–100 MHz, whereas 1/*T*_{2} is given by the *zero*-frequency *J*_{||}(0). The inclusion of the Hahn-echo *T*_{2} solves the “divergent” issue^{4} of the 1/*f*^{n} noise in the zero-frequency limit.

To extend Eq. (2) into *non-zero* frequency, we consider a decay of multiple echoes of spin (nuclear- or electron- or pseudo-spin) as a result of applying a sequence of multiple π-pulses. In the long-time limit, it is clarified^{2} that the multiple-echo decay intensity *s*_{APCP}(*t*) exhibits an exponential function as

This relation is found to hold for any multiple π-pulse sequence so long as all the π-pulse intervals (2τ) are equal and for the cases that *T*_{1} »*T*_{2}. With the *T*_{2}^{L} defined as the decay in the long-time limit [Eq. (3)], Eq. (2) is extended into *non-zero* frequencies as

where the frequency *f* is given as *f* = 1/4τ. The factor 1/(2* m* + 1)^{2} in the *J*_{||}((2* m* + 1)/4τ) term allows us to approximate Eq. (4) by the fundamental (*m* = 0) harmonic *J*_{||}(1/4τ) to a good approximation as

From the mathematical formula ∑1/(2* m* + 1)^{2} = π^{2}/8, the zero-frequency limit of Eq. (4) is reduced to Eq. (2). Alternatively, the Hahn-echo *T*_{2} that reflects *J*_{||}(0) [Eq. (2)] can be extended into non-zero frequency [Eq. (4)] as the generalized coherence time *T*_{2}^{L}.

## Experiments

All the measurements are done for commercially available substrate of un-doped, Cr-doped semi-insulating and Si-doped metallic GaAs (P/N N-5181–001, VAT, Switzerland) at 297 K using a standard NMR spectrometer which we built up and customized to ourselves. The static magnetic field *B*_{z} of 6.166 T generated by a superconducting magnet (Oxford 300/89) is applied perpendicular to [0 0 1] surface of the substrate. Frequency spectra of ^{75}As-NMR show a single line without any quadrupole splitting^{SI-1}, with the full width at half maximum of 5.8 kHz and 6.4 kHz, and 6.8 kHz for un-doped, Cr-doped semi-insulating and Si-doped metallic GaAs substrate, respectively. To obtain multiple spin-echo decays or the Hahn-echo *T*_{2}, we irradiate sufficiently strong power roughly 300W or more, resulting in the π/2-pulse width of 2.2 μs for un-doped GaAs, and 8.0 μs for both Cr-doped semi-insulating and Si-doped metallic GaAs. From these pulse widths, the frequency range that the irradiated pulses can manipulate is estimated to be 114 kHz (> 5.8 kHz) for un-doped GaAs and 31.3 kHz (> 6.4 or 6.8 kHz) for both Cr-doped semi-insulating and Si-doped metallic GaAs. This proves that the irradiated pulses manipulate all the nuclear spins observed in the frequency spectra. The assumption on which Eqs. (2–5) are based is *T*_{1} » *T*_{2},^{2} which is experimentally proved to hold in the present cases^{SI-5}.

Instead of the widely-used CPMG pulse sequence, we employ alternating-phase Carr-Purcell (APCP) pulses that alternate the polarity of multiple echoes (Fig. 1a), enabling us to obtain *T*_{2}^{L} for shorter 2τ’s. Also employed are the phase-cycling technique to eliminate ring-down noises from spin-echo signals, and the quadrature detection to increase signal-to-noise ratio^{5}. For a multi-exponential decay where a single-exponential behavior is theoretically expected, the long-time limit behavior is generally considered intrinsic since the initial decay depends on experimental artifacts. In the present cases, the slopes of initial decays are found to depend on the homogeneity of alternating magnetic field while 1/*T*_{2}^{L} is intact (Fig. 1c).

## Results

Figure 2 shows multiple-echo decays for various 2τ’s for ^{75}As nuclear spin in un-doped (Fig. 2a), Cr-doped semi-insulating (Fig. 2b) and Si-doped GaAs (Fig. 2c) substrate. The shorter the 2τ, the longer the *s*_{APCP}(*t*) persists, which is well known as the dynamical decoupling^{1}. To deduce *T*_{2}^{L} in the long-time limit, triple-exponential functions are found^{SI-3, SI-4} to consistently reproduce all the decays for various 2τ’s. To be consistent with *T*_{2}^{L} at nonzero frequencies, we define the Hahn-echo *T*_{2} not as the usually employed initial decay but as the exponential function in the long-time limit ^{2, SI-2}.

Figure 3 shows the noise spectra *J*_{||}(*f*) represented by 1/*T*_{2}^{L} as a function of frequency* f* = 1/4τ for un-doped (Fig. 3a), Cr-doped semi-insulating (Fig. 3b) and Si-doped GaAs (Fig. 3c) substrate. To reproduce the data both at zero- and high-frequency, the fitting function should be described as

where *A* is given by the long-time limit of the Hahn-echo decay as *A*^{un} = 1.60 ± 0.05 kHz for un-doped GaAs, *A*^{Cr} = 1.70 ± 0.08 kHz for Cr-doped semi-insulting GaAs and *A*^{Si} = 0.85 ± 0.04 kHz for Si-doped metallic GaAs. 1/*T*_{2}^{L} values at high frequencies give the *B* value as *B*^{un} = 7.3 ± 0.4 Hz, *B*^{Cr} = 24.7 ± 0.9 Hz and *B*^{Si} = 15.5 ± 0.7 Hz. For further investigations, we subtract the *B* term from all the 1/*T*_{2}^{L} values. As the broken curves in Fig. 3 show, for all the substrate, the *f*-dependence in Eq. (6) are found to hold over two orders of magnitude and that *n* = 2.00 ± 0.01. As a result, the noise spectra are reproduced by Eq. (6) with the fitting parameter *f*_{c}^{un} = 170 ± 10 Hz, *f*_{c}^{Cr} = 210 ± 10 Hz and *f*_{c}^{Si} = 460 ± 30 Hz.

## Discussions

First, we discuss the noises dominant at lower frequencies. From the basics of Fourier transformation, the first term with *n* = 2 in Eq. (6) indicates that the noise is due to a single source^{4} with the characteristic frequency *f*_{c} as in < *δB*_{z}(*t* + *t*’)*δB*_{z}(*t*) > ∝ exp(-*f*_{c}*t’*). To be noted is the *f*_{c}^{un} and *f*_{c}^{Cr} values of ~ 200 Hz that are extremely smaller than the fluctuation frequencies of other noise spectra^{1}. For un-doped and semi-insulating GaAs, neither spins of doped carriers nor magnetic impurities are likely. Here we show that *δB*_{z}(*t*) is caused by the fluctuation of surrounding nuclear spins from a rough estimation on the *f*_{c}^{un} and *f*_{c}^{Cr} values. Dipolar field on ^{75}As due to the nearest neighbor ^{69} Ga nuclear spins is estimated that < ^{69}*B*_{dip} > = ^{69}μ/*r*^{3} = ± 0.68 Gauss, where *r* is the distance between ^{69} Ga and ^{75}As nuclear spins. Similarly, the dipolar field by the nearest neighbor ^{71} Ga is given as < ^{71}*B*_{dip} > = ± ^{71}μ/*r*^{3} = ± 0.87 Gauss. The dipolar field due to the nearest-neighbor Ga nuclear spins result in the de-phasing of ^{75}As nuclear spin with the frequency of ^{75}γ < ^{69}*B*_{dip} > ~ 1.00 kHz and ^{75}γ < ^{71}*B*_{dip} > ~ 1.27 kHz. The estimated frequencies are significantly greater than the *f*_{c}^{un} value of 170 ± 10 Hz. If we assume that the de-phasing is caused by the next-nearest neighbor dipolar field that < ^{75}*B*_{dip} > = ± ^{75}μ/*r*^{3} = ± 0.11 Gauss, we obtain the frequency that ^{75}γ < ^{75}*B*_{dip} > ~ 163 Hz, which is very close to the experimental result that *f*_{c}^{un} = 170 ± 10 Hz. In an analogy with the RKKY interaction^{6}, the doped-carrier spins are likely to enhance the dipolar coupling between the adjacent As nuclear like-spins, resulting in the increase in de-phasing frequency of the ^{75}As nuclear spins. This is consistent with the fact that *f*_{c}^{Si} = 460 ± 30 Hz is greater than *f*_{c}^{un} = 170 ± 10 Hz and *f*_{c}^{Cr} = 210 ± 10 Hz. Thus, the decoherence noise in the lower frequencies are likely to come from the next-nearest-neighboring nuclear spins.

Next we consider the origin of the higher frequency *B* terms. Generally, in the limit that *f*_{c} → ∞, the Lorentzian spectrum exhibits a frequency-independent function and the intensity approaches zero. Thus, the higher-frequency *B* term can be viewed as another single noise of the Lorentzian spectrum with a characteristic frequency much larger than *f*_{c}^{un}, *f*_{c}^{Cr} and *f*_{c}^{Si}. As a candidate for the higher frequency *B* terms, we consider the possibility of the Fermi-contact interaction^{7} where the spins of carriers contribute to the ^{75}As nuclear-spin decoherence. For un-doped or semi-insulating [Si-doped] GaAs, the average distance between adjacent carriers is of the order of 10^{–5} [10^{–6}] cm from the carrier concentration of 10^{15} [10^{18}] cm^{-3}. Since the free-electron model gives us the Fermi velocity of roughly10^{3} [10^{4}] m/s for un-doped or semi-insulating [Si-doped] GaAs, the frequency with which carriers are diffracted by ^{75}As is estimated to be of the order of 10^{5} [10^{7}] kHz, which is sufficiently larger than the low-frequency *f*_{c}^{un} or *f*_{c}^{Cr} [*f*_{c}^{Si}]. This indicates that the higher frequency *B* term is dominated by the Fermi-contact interaction and explains the fact that *B*^{Si} > *B*^{un}. The finding that *B*^{Cr} > *B*^{un} indicates that the *B* term is also contributed by inhomogeniety of doped atoms.

Finally, we show how the noise spectra are dependent on the definition of the spectral intensity and on the inclusion of the Hahn-echo *T*_{2}. Figure 4 shows noise spectra without the Hahn-echo *T*_{2}. In this case, the spectra are fit by *A*/(*f*/*f*_{c})^{n}+ *B.* If we take the initial slope *T*_{2}^{S} as the spectral intensity, the spectra exhibit the usual 1/*f*^{1±0.3} spectra. On the other hand, if we take the *T*_{2}^{L} as the spectral intensity, all the spectra approximately result in 1/*f* ^{2±0.3} dependence, even in the absence of the Hahn-echo *T*_{2}. This proves the validity of *T*_{2}^{L} in the definition of the spectral intensity. Figure 5 indicates the importance of the Hahn-echo *T*_{2} to obtain the spectra of decoherence noise. Even when *T*_{2}^{S} is employed to define the spectral intensity, all the spectra are found to exhibit 1/*f* ^{2±0.3} dependence so long as the Hahn-echo *T*_{2} is incorporated as the zero-frequency intensity.

## Conclusions

To obtain the decoherence noise spectrum from a multiple spin-echo decay, we have followed the previous paper that the intensity should be defined as 1/*T*_{2}^{L}, the slope of the long-time limit. This method is found to avert the experimental artifacts usually entailed in the spin-echo decays. Based on the fluctuation–dissipation theorem, we have utilized the Hahn-echo *T*_{2} to validate the noise spectrum as the zero-frequency intensity, and to solve the apparent divergent behavior of the 1/*f*^{n} noises. With the help of these two theories, we have found that ^{75}As nuclear spins exhibit 1/*f*^{2} dependences over two orders of magnitude in all the substrates of un-doped, Cr-doped semi-insulating and Si-doped metallic GaAs at 297 K. The 1/*f*^{2} dependence indicates single noise source that is characterized by the frequency that *f*_{c}^{un} = 170 ± 10 Hz, *f*_{c}^{Cr} = 210 ± 10 Hz and *f*_{c}^{Si} = 460 ± 30 Hz. We have shown from a rough estimation that the decoherence is caused by the fluctuations of next-nearest-neighboring nuclear spins. We have also proved the validity of *T*_{2}^{L} as the spectral intensity and the inclusion of the Hahn-echo *T*_{2}.

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## Acknowledgements

The authors acknowledge financial supports from Tohoku University CSRN and KAKENHI Grants (Nos. 26287059 and 15H05867). S.S. and Y.H. are grateful to stimulated discussions in the meetings of Cooperative Research Project of Research Institute of Electrical Communication, Tohoku University. S.S. is also supported partly by the joint research program of Research Institute of Electrical Communication (RIEC) Tohoku University, the collaboration program of Institute of Materials and Systems for Sustainability (IMaSS), Nagoya University, KAKENHI grants (No 19H02580), the Naito Scholarship Foundation and the Yamgaguchi Educational and Scholarship Foundation.

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S.S. presented ideas, supervised and performed all experiments and analyses with the help of previous students, and wrote the paper. With S.S.’s instructions, T.M., K.I. and M.S. carried out experimental and analytical works with equal contributions. Sekikawa and Saito checked all the estimated values. T.Y. performed theoretical calculations and proved S.S.’s prediction on the long-time behavior. All authors, particularly Y.H., took part in discussing results and analyses in detail, and in editing the manuscript.

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Sasaki, S., Miura, T., Ikeda, K. *et al.* 1/*f*^{2} spectra of decoherence noise on ^{75}As nuclear spins in bulk GaAs.
*Sci Rep* **10, **10674 (2020). https://doi.org/10.1038/s41598-020-67636-9

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