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arXiv:2003.14244

Probing Environmental Spin Polarization with Superconducting Flux Qubits

Trevor M. Lanting, M. H. S. Amin, Baron, C., Michael C. Babcock, J. Boschee, Sergio Boixo, Vadim Smelyanskiy, M. Foygel, A. G. Petukhov

Published Mar 31, 2020• 10 citations
arXivHTMLPDF

Abstract

We present measurements of the dynamics of a polarized magnetic environment coupled to the flux degree of freedom of rf-SQUID flux qubits. The qubits are used as both sources of polarizing field and detectors of the environmental polarization. We probe dynamics at timescales from 5 $μ$s to 5 ms and at temperatures between 12.5 and 22 mK. The measured polarization versus temperature provides strong evidence for a phase transition at a temperature of $5.7\pm 0.3$ mK. Furthermore, the environmental polarization grows initially as $\sqrt{t}$, consistent with spin diffusion dynamics. However, spin diffusion model deviates from data at long timescales, suggesting that a different phenomenon is responsible for the low-frequency behavior. A simple $1/f$ model can fit the data at all time scales but it requires empirical low- and high-frequency cutoffs. We argue that these results are consistent with an environment comprised of random clusters of spins, with fast spin diffusion dynamics within the clusters and slow fluctuations of the total moments of the clusters.

DOI
10.48550/arxiv.2003.14244

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We present measurements of the dynamics of a polarized magnetic environment coupled to the flux degree of freedom of rf-SQUID flux qubits. The qubits are used as both sources of polarizing field and detectors of the environmental polarization. We probe dynamics at timescales from 5 μμμs to 5 ms and at temperatures between 12.5 and 22 mK. The measured polarization versus temperature provides strong evidence for a phase transition at a temperature of 5.7±0.35.7\pm 0.35.7±0.3 mK. Furthermore, the environmental polarization grows initially as t\sqrt{t}t​, consistent with spin diffusion dynamics. However, spin diffusion model deviates from data at long timescales, suggesting that a different phenomenon is responsible for the low-frequency behavior. A simple 1/f1/f1/f model can fit the data at all time scales but it requires empirical low- and high-frequency cutoffs. We argue that these results are consistent with an environment comprised of random clusters of spins, with fast spin diffusion dynamics within the clusters and slow fluctuations of the total moments of the clusters.
The intrinsic flux noise observed in superconducting quantum interference devices (SQUIDs) is thought to be due to the fluctuation of electron-spin impurities, but the frequency and temperature dependence observed in experiments do not agree with the usual 1/f1/f1/f models. We present theoretical calculations and experimental measurements of flux noise in rf SQUID flux qubits that show how these observations, and previous reported measurements, can be interpreted in terms of a spin-diffusion constant that increases with temperature. We fit measurements of flux noise in 16 devices, taken in the 20–8020\text{--}8020–80 mK temperature range, to the spin-diffusion model. This allows us to extract the spin-diffusion constant and its temperature dependence, suggesting that the spin system is close to a spin-glass phase transition.
We experimentally investigated the temperature dependence of magnetic flux threading four Nb dc SQUIDs of different loop sizes and an rf SQUID flux qubit from 10 mK to 1 K. All of the devices were cooled down in a residual magnetic field of ∼ 50 nT. The data shows a very weak temperature dependence of the trapped flux, indicating negligible spin polarization at low temperatures. This result agrees well with the recent theoretical model suggested by Faoro and Ioffe (2008 Phys. Rev. Lett. 100 227005).
It is well known that that superconducting qubits based on Josephson junctions have the advantages of scalability and the qubit states are easy to prepare and control. In addition, demonstration of Rabi oscillations in various superconducting circuits shows that superconducting qubits are promising for scalable quantum information processing. However, despite flexibility of design and fabrication, easier to scale up, and fast gate speed, superconducting qubits usually have much shorter decoherence time than trapped ions, NMR etc. due to the relatively strong interactions between qubits and environment. Recent experiments show that low frequency flux noise is the dominant mechanism of decoherence in superconducting flux qubits. However, despite extensive effort the origin of flux noise is still not well understood. The goal of this work is to identify the source and characterize the property of flux noisein rf SQUID flux qubit through various spectroscopy and time-resolved measurements. Our result show that one can determine all circuit parameters needed for reconstructing qubit Hamiltonian with high accuracy via measurement of macroscopic resonant tunneling (MRT) and photon assisted tunneling (PAT) and that the amount of flux noise in rf SQUID qubits scales linearly with self inductance of the qubits. In addition, we have investigated the dynamics of a three-level flux qubit in incoherent regime. The result demonstrates that treating a multi-level physical qubit, such as the superconducting flux qubit, as an ideal two-level quantum system may be inadequate under certain circumstances.
At low temperature and low damping, the magnetic flux of a rf Superconducting QUantum Interference Device (SQUID) acts as a macroscopic quantum variable whose dynamic is described by a double well potential. A rf SQUID consists of a superconducting loop interrupted by a thin insulating barrier, known as a Josephson junction. The modified rf SQUID design discussed in this thesis has independent, in situ, controls that allow varying the barrier height between the potential wells and their relative tilt. This high level of control of the potential together with the capability to implement an experimental setup optimized to reduce coupling to external sources of noise, made this system ideal not only for tests of quantum transition at a macroscopic level, but also as a scalable solid state device for quantum computation. A variety of measurements of quantum phenomena are reported in this thesis, including microwave level spectroscopy, coherent oscillations, and macroscopic resonant tunneling (MRT). These measurements are compared with theoretical models to study the possible sources of decoherence in a rf SQUID qubit. The analysis confirmed that the main source of decoherence is not due to coupling to the external environment, but it can be linked to intrinsic low frequency flux noise. Work from many groups over the last several years has shown that the performance of a superconductive qubit is limited by coupling to spurious defects in the materials used for the device fabrication. In this work we analyze the possibility of using the MRT as a tool for future studies of effects of materials and fabrication process in the realization of improved qubits, and we also explore the feasibility for measurements of quantum phenomena that involve fast pulsing of the potential barrier height.
For over three decades, the presence of magnetic flux noise with a power spectral density scaling roughly as SPhi(f)propto1/falphaS_Phi(f) propto 1/f^alphaSP​hi(f)propto1/falpha, where alphalesssim1alpha lesssim 1alphalesssim1, has been known to limit the low-frequency performance of dc superconducting quantum interference devices (SQUIDs). In recent years, experiments indicate that this same noise persists to frequencies up to 1~GHz and is a dominant source of dephasing in flux-sensitive superconducting quantum bits (qubits). Thus, the reduction of flux noise presents a major hurdle towards the successful realization of scalable quantum computers that are based on flux-based qubits. In this thesis, we present experimental measurements, theoretical analyses, and numerical simulations that support a more detailed understanding of both the microscopic and macroscopic properties of flux noise.Our experimental work begins with flux noise measurements of a large number of SQUIDs in the temperature range from 0.1~K to 4~K. We report on measurements of ten SQUIDs with systematically varied geometries and show that alphaalphaalpha increases as the temperature is lowered; in so doing, each spectrum pivots about a nearly constant frequency. The mean square flux noise, inferred by integrating the power spectra, grows rapidly with temperature and at a given temperature is approximately independent of the outer dimension of a given SQUID washer. We show that these results are incompatible with a model based on the random reversal of independent, spins that are located at the surface of the SQUID washer.In the course of our flux noise measurements, we became aware of a spurious contribution to low-frequency critical current noise in Josephson junctions---normally attributed to charge trapping in the barrier---arising from temperature instabilities inherent in cryogenic systems. These temperature fluctuations modify the critical current via its temperature dependence. By computing cross-correlations between measured temperature and critical current noise in Al-AlOx-Al junctions, we show that, despite excellent temperature stability, temperature fluctuations induce observable critical current fluctuations. Particularly, because 1/f1/f1/f critical current noise has decreased with improved fabrication techniques in recent years, it is important to understand and eliminate this additional noise source.Next, we introduce a numerical method of calculating the mean square flux noise PhisqPhisqPhisq from independently fluctuating spins on the surface of thin-film loops of arbitrary geometry. By reciprocity, PhisqPhisqPhisq is proportional to langleBbf(rbf)2ranglelangleBbf(rbf)^2ranglelangleBbf(rbf)2rangle, where Bbf(rbf)Bbf(rbf)Bbf(rbf) is the magnetic field generated by a circulating current around the loop and rbfrbfrbf varies over the loop surface. By discretizing the loop nonuniformly, we efficiently and accurately compute the current distribution and resulting magnetic field, which may vary rapidly across the loop. We use this method to compute PhisqPhisqPhisq in a number of scenarios in which we systematically vary physical parameters of the loop. We compare our simulations to an earlier analytic result predicting that PhisqproptoR/WPhisq propto R/WPhisqproptoR/W in the limit where the loop radius RRR is much greater than the linewidth WWW. We further show that the previously neglected contribution of edge spins to PhisqPhisqPhisq is significant---even dominant---in narrow-linewidth loops.To calculate theoretical dephasing rates in qubits, we consider flux noise with a spectral density SPhi(f)=A2/(f/1 hboxHz)alphaS_Phi(f) = A^2/(f/1~hbox{Hz})^alphaSP​hi(f)=A2/(f/1 hboxHz)alpha, where AAA is of the order of 1~muPhi0,hboxHz−1/2muPhi_0 , hbox{Hz}^{-1/2}muPhi0​,hboxHz−1/2 and 0.6leqalphaleq1.20.6 leq alpha leq 1.20.6leqalphaleq1.2; Phi0Phi_{0}Phi0​ is the flux quantum. For a qubit with an energy level splitting linearly coupled to the applied flux, our calculations of the dependence of the pure dephasing time tauphitau_phitaup​hi of Ramsey and echo pulse sequences on alphaalphaalpha for fixed AAA show that tauphitau_phitaup​hi decreases rapidly as alphaalphaalpha is reduced. We find that tauphitau_phitaup​hi is relatively insensitive to the noise bandwidth, f1leqfleqf2f_1 leq f leq f_2f1​leqfleqf2​, for all alphaalphaalpha provided the ultraviolet cutoff frequency f2>1/tauphif_2 > 1/tau_phif2​>1/taup​hi. We calculate the ratio tauphi,E/tauphi,Rtau_{phi,E} / tau_{phi,R}tauphi,E​/tauphi,R​ of the echo (EEE) and Ramsey (RRR) sequences, and the dependence of the decay function on alphaalphaalpha and f2f_2f2​. We investigate the case in which SPhi(f0)S_Phi(f_0)SP​hi(f0​) is fixed at the ``pivot frequency'' f0neq1f_0 neq 1f0​neq1~Hz while alphaalphaalpha is varied, and find that the choice of f0f_0f0​ can greatly influence the sensitivity of tauphi,Etau_{phi,E}tauphi,E​ and tauphi,Rtau_{phi,R}tauphi,R​ to the value of alphaalphaalpha.Finally, we conclude with a brief review of our principal results and conclusions. We also comment on promising avenues of future research.
Abstract 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.