arXiv:1111.7272 · Physical Review B

Pure dephasing in flux qubits due to flux noise with spectral density scaling as 1 / f α

Steven M. Anton, Clemens Müller, Jeffrey Birenbaum, Sean R. O’Kelley, Andrew D. Fefferman, Dmitry S. Golubev, Gene C Hilton, Hsiao-Mei Cho, K. D. Irwin, F. C. Wellstood, Gerd Schön, Alexander Shnirman, John L. Clarke

Published Jun 5, 2012• 44 citations

arXiv HTML PDF

Abstract

For many types of superconducting qubits, magnetic flux noise is a source of pure dephasing. Measurements on a representative dc superconducting quantum interference device (SQUID) over a range of temperatures show that ${S}_{\ensuremath{\Phi}}(f)={A}^{2}/{(f/1\phantom{\rule{0.28em}{0ex}}\text{Hz})}^{\ensuremath{\alpha}}$, where ${S}_{\ensuremath{\Phi}}$ is the flux noise spectral density, $A$ is of the order of 1 $\ensuremath{\mu}{\ensuremath{\Phi}}_{0}\phantom{\rule{0.28em}{0ex}}{\text{Hz}}^{\ensuremath{-}1/2}$, $0.61\ensuremath{\leqslant}\ensuremath{\alpha}\ensuremath{\leqslant}0.95$, and ${\ensuremath{\Phi}}_{0}$ is the flux quantum. For a qubit with an energy level splitting linearly coupled to the applied flux, calculations of the dependence of the pure dephasing time ${\ensuremath{\tau}}_{\ensuremath{\varphi}}$ of Ramsey and echo pulse sequences on $\ensuremath{\alpha}$ for fixed $A$ show that ${\ensuremath{\tau}}_{\ensuremath{\varphi}}$ decreases rapidly as $\ensuremath{\alpha}$ is reduced. We find that ${\ensuremath{\tau}}_{\ensuremath{\varphi}}$ is relatively insensitive to the noise bandwidth, ${f}_{1}\ensuremath{\leqslant}f\ensuremath{\leqslant}{f}_{2}$, for all $\ensuremath{\alpha}$ provided the ultraviolet cutoff frequency ${f}_{2}>1/{\ensuremath{\tau}}_{\ensuremath{\varphi}}$. We calculate the ratio ${\ensuremath{\tau}}_{\ensuremath{\varphi},E}/{\ensuremath{\tau}}_{\ensuremath{\varphi},R}$ of the echo ($E$) and Ramsey ($R$) sequences and the dependence of the decay function on $\ensuremath{\alpha}$ and ${f}_{2}$. We investigate the case in which ${S}_{\ensuremath{\Phi}}({f}_{0})$ is fixed at the ``pivot frequency'' ${f}_{0}\ensuremath{\ne}1$ Hz while $\ensuremath{\alpha}$ is varied and find that the choice of ${f}_{0}$ can greatly influence the sensitivity of ${\ensuremath{\tau}}_{\ensuremath{\varphi},E}$ and ${\ensuremath{\tau}}_{\ensuremath{\varphi},R}$ to the value of $\ensuremath{\alpha}$. Finally, we present calculated values of ${\ensuremath{\tau}}_{\ensuremath{\varphi}}$ in a qubit corresponding to the values of $A$ and $\ensuremath{\alpha}$ measured in our SQUID.

DOI: 10.1103/physrevb.85.224505

Related papers

Magnetic flux noise in SQUIDs and qubits

Steven M. Anton

Jan 1, 20130 citations

Evolution of 1/f Flux Noise in Superconducting Qubits with Weak Magnetic Fields

David A. Rower, Lamia Ateshian, Lauren H. Li +18 more

arXiv:2301.07804Physical Review LettersMay 31, 202352 citations

Decoherence in Josephson-junction qubits due to critical-current fluctuations

Dale J. Van Harlingen, Timothy L. Robertson, B. L. T. Plourde +3 more

arXiv:cond-mat/0404307Physical Review BAug 30, 2004171 citations

Origin and Reduction of 1=f Magnetic Flux Noise in Superconducting Devices

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Magnetic Flux Noise in dc SQUIDs: Temperature and Geometry Dependence

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Spectroscopy of low-frequency noise and its temperature dependence in a superconducting qubit

Fei Yan, Jonas Bylander, Simon Gustavsson +7 more

arXiv:1201.5665Physical Review BMay 22, 2012106 citations

arXiv:1111.7272 · Physical Review B

Pure dephasing in flux qubits due to flux noise with spectral density scaling as 1 / f α

Published Jun 5, 2012• 44 citations

arXiv HTML PDF

Abstract

{S}_{\ensuremath{\Phi}}(f)={A}^{2}/{(f/1\phantom{\rule{0.28em}{0ex}}\text{Hz})}^{\ensuremath{\alpha}}

, where

{S}_{\ensuremath{\Phi}}

is the flux noise spectral density,

A

is of the order of 1

\ensuremath{\mu}{\ensuremath{\Phi}}_{0}\phantom{\rule{0.28em}{0ex}}{\text{Hz}}^{\ensuremath{-}1/2}

0.61\ensuremath{\leqslant}\ensuremath{\alpha}\ensuremath{\leqslant}0.95

, and

{\ensuremath{\Phi}}_{0}

is the flux quantum. For a qubit with an energy level splitting linearly coupled to the applied flux, calculations of the dependence of the pure dephasing time

{\ensuremath{\tau}}_{\ensuremath{\varphi}}

of Ramsey and echo pulse sequences on

\ensuremath{\alpha}

for fixed

A

show that

{\ensuremath{\tau}}_{\ensuremath{\varphi}}

decreases rapidly as

\ensuremath{\alpha}

is reduced. We find that

{\ensuremath{\tau}}_{\ensuremath{\varphi}}

is relatively insensitive to the noise bandwidth,

{f}_{1}\ensuremath{\leqslant}f\ensuremath{\leqslant}{f}_{2}

, for all

\ensuremath{\alpha}

provided the ultraviolet cutoff frequency

{f}_{2}>1/{\ensuremath{\tau}}_{\ensuremath{\varphi}}

. We calculate the ratio

{\ensuremath{\tau}}_{\ensuremath{\varphi},E}/{\ensuremath{\tau}}_{\ensuremath{\varphi},R}

of the echo (

E

) and Ramsey (

R

) sequences and the dependence of the decay function on

\ensuremath{\alpha}

and

{f}_{2}

. We investigate the case in which

{S}_{\ensuremath{\Phi}}({f}_{0})

is fixed at the ``pivot frequency''

{f}_{0}\ensuremath{\ne}1

Hz while

\ensuremath{\alpha}

is varied and find that the choice of

{f}_{0}

can greatly influence the sensitivity of

{\ensuremath{\tau}}_{\ensuremath{\varphi},E}

and

{\ensuremath{\tau}}_{\ensuremath{\varphi},R}

to the value of

\ensuremath{\alpha}

. Finally, we present calculated values of

{\ensuremath{\tau}}_{\ensuremath{\varphi}}

in a qubit corresponding to the values of

A

and

\ensuremath{\alpha}

measured in our SQUID.

DOI: 10.1103/physrevb.85.224505

Related papers

Magnetic flux noise in SQUIDs and qubits

Steven M. Anton

For over three decades, the presence of magnetic flux noise with a power spectral density scaling roughly as $S_Phi(f) propto 1/f^alpha$ , where $alpha lesssim 1$ , 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 $alpha$ 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 $1/f$ 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 $Phisq$ from independently fluctuating spins on the surface of thin-film loops of arbitrary geometry. By reciprocity, $Phisq$ is proportional to $langleBbf(rbf)^2rangle$ , where $Bbf(rbf)$ is the magnetic field generated by a circulating current around the loop and $rbf$ 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 $Phisq$ 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 $Phisq propto R/W$ in the limit where the loop radius $R$ is much greater than the linewidth $W$ . We further show that the previously neglected contribution of edge spins to $Phisq$ is significant---even dominant---in narrow-linewidth loops.To calculate theoretical dephasing rates in qubits, we consider flux noise with a spectral density $S_Phi(f) = A^2/(f/1~hbox{Hz})^alpha$ , where $A$ is of the order of 1~ $muPhi_0 , hbox{Hz}^{-1/2}$ and $0.6 leq alpha leq 1.2$ ; $Phi_{0}$ 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 $tau_phi$ of Ramsey and echo pulse sequences on $alpha$ for fixed $A$ show that $tau_phi$ decreases rapidly as $alpha$ is reduced. We find that $tau_phi$ is relatively insensitive to the noise bandwidth, $f_1 leq f leq f_2$ , for all $alpha$ provided the ultraviolet cutoff frequency $f_2 > 1/tau_phi$ . We calculate the ratio $tau_{phi,E} / tau_{phi,R}$ of the echo ( $E$ ) and Ramsey ( $R$ ) sequences, and the dependence of the decay function on $alpha$ and $f_2$ . We investigate the case in which $S_Phi(f_0)$ is fixed at the ``pivot frequency'' $f_0 neq 1$ ~Hz while $alpha$ is varied, and find that the choice of $f_0$ can greatly influence the sensitivity of $tau_{phi,E}$ and $tau_{phi,R}$ to the value of $alpha$ .Finally, we conclude with a brief review of our principal results and conclusions. We also comment on promising avenues of future research.For over three decades, the presence of magnetic flux noise with a power spectral density scaling roughly as $S_Phi(f) propto 1/f^alpha$, where $alpha lesssim 1$, 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 $alpha$ 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 $1/f$ 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 $Phisq$ from independently fluctuating spins on the surface of thin-film loops of arbitrary geometry. By reciprocity, $Phisq$ is proportional to $langleBbf(rbf)^2rangle$, where $Bbf(rbf)$ is the magnetic field generated by a circulating current around the loop and $rbf$ 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 $Phisq$ 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 $Phisq propto R/W$ in the limit where the loop radius $R$ is much greater than the linewidth $W$. We further show that the previously neglected contribution of edge spins to $Phisq$ is significant---even dominant---in narrow-linewidth loops.To calculate theoretical dephasing rates in qubits, we consider flux noise with a spectral density $S_Phi(f) = A^2/(f/1~hbox{Hz})^alpha$, where $A$ is of the order of 1~$muPhi_0 , hbox{Hz}^{-1/2}$ and $0.6 leq alpha leq 1.2$; $Phi_{0}$ 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 $tau_phi$ of Ramsey and echo pulse sequences on $alpha$ for fixed $A$ show that $tau_phi$ decreases rapidly as $alpha$ is reduced. We find that $tau_phi$ is relatively insensitive to the noise bandwidth, $f_1 leq f leq f_2$, for all $alpha$ provided the ultraviolet cutoff frequency $f_2 > 1/tau_phi$. We calculate the ratio $tau_{phi,E} / tau_{phi,R}$ of the echo ($E$) and Ramsey ($R$) sequences, and the dependence of the decay function on $alpha$ and $f_2$. We investigate the case in which $S_Phi(f_0)$ is fixed at the ``pivot frequency'' $f_0 neq 1$~Hz while $alpha$ is varied, and find that the choice of $f_0$ can greatly influence the sensitivity of $tau_{phi,E}$ and $tau_{phi,R}$ to the value of $alpha$.Finally, we conclude with a brief review of our principal results and conclusions. We also comment on promising avenues of future research.

Jan 1, 20130 citations

Evolution of 1/f Flux Noise in Superconducting Qubits with Weak Magnetic Fields

David A. Rower, Lamia Ateshian, Lauren H. Li +18 more

The microscopic description of 1/f magnetic flux noise in superconducting circuits has remained an open question for several decades despite extensive experimental and theoretical investigation. Recent progress in superconducting devices for quantum information has highlighted the need to mitigate sources of qubit decoherence, driving a renewed interest in understanding the underlying noise mechanism(s). Though a consensus has emerged attributing flux noise to surface spins, their identity and interaction mechanisms remain unclear, prompting further study. Here, we apply weak in-plane magnetic fields to a capacitively shunted flux qubit (where the Zeeman splitting of surface spins lies below the device temperature) and study the flux-noise-limited qubit dephasing, revealing previously unexplored trends that may shed light on the dynamics behind the emergent 1/f noise. Notably, we observe an enhancement (suppression) of the spin-echo (Ramsey) pure-dephasing time in fields up to B=100 G. With direct noise spectroscopy, we further observe a transition from a 1/f to approximately Lorentzian frequency dependence below 10 Hz and a reduction of the noise above 1 MHz with increasing magnetic field. We suggest that these trends are qualitatively consistent with an increase of spin cluster sizes with magnetic field. These results should help to inform a complete microscopic theory of 1/f flux noise in superconducting circuits.The microscopic description of 1/f magnetic flux noise in superconducting circuits has remained an open question for several decades despite extensive experimental and theoretical investigation. Recent progress in superconducting devices for quantum information has highlighted the need to mitigate sources of qubit decoherence, driving a renewed interest in understanding the underlying noise mechanism(s). Though a consensus has emerged attributing flux noise to surface spins, their identity and interaction mechanisms remain unclear, prompting further study. Here, we apply weak in-plane magnetic fields to a capacitively shunted flux qubit (where the Zeeman splitting of surface spins lies below the device temperature) and study the flux-noise-limited qubit dephasing, revealing previously unexplored trends that may shed light on the dynamics behind the emergent 1/f noise. Notably, we observe an enhancement (suppression) of the spin-echo (Ramsey) pure-dephasing time in fields up to B=100 G. With direct noise spectroscopy, we further observe a transition from a 1/f to approximately Lorentzian frequency dependence below 10 Hz and a reduction of the noise above 1 MHz with increasing magnetic field. We suggest that these trends are qualitatively consistent with an increase of spin cluster sizes with magnetic field. These results should help to inform a complete microscopic theory of 1/f flux noise in superconducting circuits.

arXiv:2301.07804Physical Review LettersMay 31, 202352 citations

Decoherence in Josephson-junction qubits due to critical-current fluctuations

Dale J. Van Harlingen, Timothy L. Robertson, B. L. T. Plourde +3 more

We compute the decoherence caused by $1∕f$ fluctuations at low frequency $f$ in the critical current ${I}_{0}$ of Josephson junctions incorporated into flux, phase, charge, and hybrid flux-charge superconducting quantum bits (qubits). The dephasing time ${\ensuremath{\tau}}_{\ensuremath{\phi}}$ scales as ${I}_{0}∕\ensuremath{\Omega}\ensuremath{\Lambda}{S}_{{I}_{0}}^{1∕2}(1\phantom{\rule{0.3em}{0ex}}\mathrm{Hz})$, where $\ensuremath{\Omega}∕2\ensuremath{\pi}$ is the energy-level splitting frequency, ${S}_{{I}_{0}}(1\phantom{\rule{0.3em}{0ex}}\mathrm{Hz})$ is the spectral density of the critical-current noise at $1\phantom{\rule{0.3em}{0ex}}\mathrm{Hz}$ , and $\ensuremath{\Lambda}\ensuremath{\equiv}\ensuremath{\mid}{I}_{0}d\ensuremath{\Omega}∕\ensuremath{\Omega}d{I}_{0}\ensuremath{\mid}$ is a parameter computed for given parameters for each type of qubit that specifies the sensitivity of the level splitting to critical-current fluctuations. Computer simulations show that the envelope of the coherent oscillations of any qubit after time $t$ scales as $\mathrm{exp}(\ensuremath{-}{t}^{2}∕2{\ensuremath{\tau}}_{\ensuremath{\phi}}^{2})$ when the dephasing due to critical-current noise dominates the dephasing from all sources of dissipation. We compile published results for fluctuations in the critical current of Josephson tunnel junctions fabricated with different technologies and a wide range in ${I}_{0}$ and area $\mathcal{A}$ , and show that their values of ${S}_{{I}_{0}}(1\phantom{\rule{0.3em}{0ex}}\mathrm{Hz})$ scale to within a factor of 3 of $[144{({I}_{0}∕\ensuremath{\mu}\mathrm{A})}^{2}∕(\mathcal{A}∕\ensuremath{\mu}{\mathrm{m}}^{2})]{(\mathrm{pA})}^{2}∕\mathrm{Hz}$ at $4.2\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ . We empirically extrapolate ${S}_{{I}_{0}}^{1∕2}(1\phantom{\rule{0.3em}{0ex}}\mathrm{Hz})$ to lower temperatures using a scaling $T(\mathrm{K})∕4.2$ . Using this result, we find that the predicted values of ${\ensuremath{\tau}}_{\ensuremath{\phi}}$ at $100\phantom{\rule{0.3em}{0ex}}\mathrm{mK}$ range from $0.8\phantom{\rule{0.5em}{0ex}}\text{to}\phantom{\rule{0.5em}{0ex}}12\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{s}$, and are usually substantially longer than values measured experimentally at lower temperatures.We compute the decoherence caused by $1∕f$ fluctuations at low frequency $f$ in the critical current ${I}_{0}$ of Josephson junctions incorporated into flux, phase, charge, and hybrid flux-charge superconducting quantum bits (qubits). The dephasing time ${\ensuremath{\tau}}_{\ensuremath{\phi}}$ scales as ${I}_{0}∕\ensuremath{\Omega}\ensuremath{\Lambda}{S}_{{I}_{0}}^{1∕2}(1\phantom{\rule{0.3em}{0ex}}\mathrm{Hz})$, where $\ensuremath{\Omega}∕2\ensuremath{\pi}$ is the energy-level splitting frequency, ${S}_{{I}_{0}}(1\phantom{\rule{0.3em}{0ex}}\mathrm{Hz})$ is the spectral density of the critical-current noise at $1\phantom{\rule{0.3em}{0ex}}\mathrm{Hz}$, and $\ensuremath{\Lambda}\ensuremath{\equiv}\ensuremath{\mid}{I}_{0}d\ensuremath{\Omega}∕\ensuremath{\Omega}d{I}_{0}\ensuremath{\mid}$ is a parameter computed for given parameters for each type of qubit that specifies the sensitivity of the level splitting to critical-current fluctuations. Computer simulations show that the envelope of the coherent oscillations of any qubit after time $t$ scales as $\mathrm{exp}(\ensuremath{-}{t}^{2}∕2{\ensuremath{\tau}}_{\ensuremath{\phi}}^{2})$ when the dephasing due to critical-current noise dominates the dephasing from all sources of dissipation. We compile published results for fluctuations in the critical current of Josephson tunnel junctions fabricated with different technologies and a wide range in ${I}_{0}$ and area $\mathcal{A}$, and show that their values of ${S}_{{I}_{0}}(1\phantom{\rule{0.3em}{0ex}}\mathrm{Hz})$ scale to within a factor of 3 of $[144{({I}_{0}∕\ensuremath{\mu}\mathrm{A})}^{2}∕(\mathcal{A}∕\ensuremath{\mu}{\mathrm{m}}^{2})]{(\mathrm{pA})}^{2}∕\mathrm{Hz}$ at $4.2\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. We empirically extrapolate ${S}_{{I}_{0}}^{1∕2}(1\phantom{\rule{0.3em}{0ex}}\mathrm{Hz})$ to lower temperatures using a scaling $T(\mathrm{K})∕4.2$. Using this result, we find that the predicted values of ${\ensuremath{\tau}}_{\ensuremath{\phi}}$ at $100\phantom{\rule{0.3em}{0ex}}\mathrm{mK}$ range from $0.8\phantom{\rule{0.5em}{0ex}}\text{to}\phantom{\rule{0.5em}{0ex}}12\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{s}$, and are usually substantially longer than values measured experimentally at lower temperatures.

arXiv:cond-mat/0404307Physical Review BAug 30, 2004171 citations

Origin and Reduction of 1=f Magnetic Flux Noise in Superconducting Devices

P. Senthil Kumar, David P. Pappas, R. McDermott +5 more

Magnetic flux noise is a dominant source of dephasing and energy relaxation in superconducting qubits. The noise power spectral density varies with frequency as 1/fα, with α1, and spans 13 orders of magnitude. Recent work indicates that the noise is from unpaired magnetic defects on the surfaces of the superconducting devices. Here, we demonstrate that adsorbed molecular O2 is the dominant contributor to magnetism in superconducting thin films. We show that this magnetism can be reduced by appropriate surface treatment or improvement in the sample vacuum environment. We observe a suppression of static spin susceptibility by more than an order of magnitude and a suppression of 1/f magnetic flux noise power spectral density of up to a factor of 5. These advances open the door to the realization of superconducting qubits with improved quantum coherence.Magnetic flux noise is a dominant source of dephasing and energy relaxation in superconducting qubits. The noise power spectral density varies with frequency as 1/fα, with α1, and spans 13 orders of magnitude. Recent work indicates that the noise is from unpaired magnetic defects on the surfaces of the superconducting devices. Here, we demonstrate that adsorbed molecular O2 is the dominant contributor to magnetism in superconducting thin films. We show that this magnetism can be reduced by appropriate surface treatment or improvement in the sample vacuum environment. We observe a suppression of static spin susceptibility by more than an order of magnitude and a suppression of 1/f magnetic flux noise power spectral density of up to a factor of 5. These advances open the door to the realization of superconducting qubits with improved quantum coherence.

Journal of SuperconductivityOct 18, 20169 citations

Magnetic Flux Noise in dc SQUIDs: Temperature and Geometry Dependence

Steven M. Anton, Jeffrey Birenbaum, Sean R. O’Kelley +11 more

The spectral density S(Φ)(f) = A(2)/(f/1 Hz)(α) of magnetic flux noise in ten dc superconducting quantum interference devices (SQUIDs) with systematically varied geometries shows that α 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. These results are incompatible with a model based on the random reversal of independent, surface spins.The spectral density S(Φ)(f) = A(2)/(f/1 Hz)(α) of magnetic flux noise in ten dc superconducting quantum interference devices (SQUIDs) with systematically varied geometries shows that α 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. These results are incompatible with a model based on the random reversal of independent, surface spins.

Physical Review LettersApr 5, 2013110 citations

Spectroscopy of low-frequency noise and its temperature dependence in a superconducting qubit

Fei Yan, Jonas Bylander, Simon Gustavsson +7 more

We report a direct measurement of the low-frequency noise spectrum in a superconducting flux qubit. Our method uses the noise sensitivity of a free-induction Ramsey interference experiment, comprising free evolution in the presence of noise for a fixed period of time followed by single-shot qubit-state measurement. Repeating this procedure enables Fourier-transform noise spectroscopy with access to frequencies up to the achievable repetition rate, a regime relevant to dephasing in ensemble-averaged time-domain measurements such as Ramsey interferometry. Rotating the qubit's quantization axis allows us to measure two types of noise: effective flux noise and effective critical-current or charge noise. For both noise sources, we observe that the very same $1/f$ -type power laws measured at considerably higher frequencies ($0.2\ensuremath{-}20$ MHz) are consistent with the noise in the $0.01\ensuremath{-}100$-Hz range measured here. We find no evidence of temperature dependence of the noises over $65\ensuremath{-}200$ mK, and also no evidence of time-domain correlations between the two noises. These methods and results are pertinent to the dephasing of all superconducting qubits.We report a direct measurement of the low-frequency noise spectrum in a superconducting flux qubit. Our method uses the noise sensitivity of a free-induction Ramsey interference experiment, comprising free evolution in the presence of noise for a fixed period of time followed by single-shot qubit-state measurement. Repeating this procedure enables Fourier-transform noise spectroscopy with access to frequencies up to the achievable repetition rate, a regime relevant to dephasing in ensemble-averaged time-domain measurements such as Ramsey interferometry. Rotating the qubit's quantization axis allows us to measure two types of noise: effective flux noise and effective critical-current or charge noise. For both noise sources, we observe that the very same $1/f$-type power laws measured at considerably higher frequencies ($0.2\ensuremath{-}20$ MHz) are consistent with the noise in the $0.01\ensuremath{-}100$-Hz range measured here. We find no evidence of temperature dependence of the noises over $65\ensuremath{-}200$ mK, and also no evidence of time-domain correlations between the two noises. These methods and results are pertinent to the dephasing of all superconducting qubits.

arXiv:1201.5665Physical Review BMay 22, 2012106 citations

{S}_{\ensuremath{\Phi}}(f)={A}^{2}/{(f/1\phantom{\rule{0.28em}{0ex}}\text{Hz})}^{\ensuremath{\alpha}}

, where

{S}_{\ensuremath{\Phi}}

is the flux noise spectral density,

A

is of the order of 1

\ensuremath{\mu}{\ensuremath{\Phi}}_{0}\phantom{\rule{0.28em}{0ex}}{\text{Hz}}^{\ensuremath{-}1/2}

0.61\ensuremath{\leqslant}\ensuremath{\alpha}\ensuremath{\leqslant}0.95

, and

{\ensuremath{\Phi}}_{0}

is the flux quantum. For a qubit with an energy level splitting linearly coupled to the applied flux, calculations of the dependence of the pure dephasing time

{\ensuremath{\tau}}_{\ensuremath{\varphi}}

of Ramsey and echo pulse sequences on

\ensuremath{\alpha}

for fixed

A

show that

{\ensuremath{\tau}}_{\ensuremath{\varphi}}

decreases rapidly as

\ensuremath{\alpha}

is reduced. We find that

{\ensuremath{\tau}}_{\ensuremath{\varphi}}

is relatively insensitive to the noise bandwidth,

{f}_{1}\ensuremath{\leqslant}f\ensuremath{\leqslant}{f}_{2}

, for all

\ensuremath{\alpha}

provided the ultraviolet cutoff frequency

{f}_{2}>1/{\ensuremath{\tau}}_{\ensuremath{\varphi}}

. We calculate the ratio

{\ensuremath{\tau}}_{\ensuremath{\varphi},E}/{\ensuremath{\tau}}_{\ensuremath{\varphi},R}

of the echo (

E

) and Ramsey (

R

) sequences and the dependence of the decay function on

\ensuremath{\alpha}

and

{f}_{2}

. We investigate the case in which

{S}_{\ensuremath{\Phi}}({f}_{0})

is fixed at the ``pivot frequency''

{f}_{0}\ensuremath{\ne}1

Hz while

\ensuremath{\alpha}

is varied and find that the choice of

{f}_{0}

can greatly influence the sensitivity of

{\ensuremath{\tau}}_{\ensuremath{\varphi},E}

and

{\ensuremath{\tau}}_{\ensuremath{\varphi},R}

to the value of

\ensuremath{\alpha}

. Finally, we present calculated values of

{\ensuremath{\tau}}_{\ensuremath{\varphi}}

in a qubit corresponding to the values of

A

and

\ensuremath{\alpha}

measured in our SQUID.

For over three decades, the presence of magnetic flux noise with a power spectral density scaling roughly as

S_Phi(f) propto 1/f^alpha

, where

alpha lesssim 1

, 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

alpha

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/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

Phisq

from independently fluctuating spins on the surface of thin-film loops of arbitrary geometry. By reciprocity,

Phisq

is proportional to

langleBbf(rbf)^2rangle

, where

Bbf(rbf)

is the magnetic field generated by a circulating current around the loop and

rbf

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

Phisq

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

Phisq propto R/W

in the limit where the loop radius

R

is much greater than the linewidth

W

. We further show that the previously neglected contribution of edge spins to

Phisq

is significant---even dominant---in narrow-linewidth loops.To calculate theoretical dephasing rates in qubits, we consider flux noise with a spectral density

S_Phi(f) = A^2/(f/1~hbox{Hz})^alpha

, where

A

is of the order of 1~

muPhi_0 , hbox{Hz}^{-1/2}

and

0.6 leq alpha leq 1.2

;

Phi_{0}

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

tau_phi

of Ramsey and echo pulse sequences on

alpha

for fixed

A

show that

tau_phi

decreases rapidly as

alpha

is reduced. We find that

tau_phi

is relatively insensitive to the noise bandwidth,

f_1 leq f leq f_2

, for all

alpha

provided the ultraviolet cutoff frequency

f_2 > 1/tau_phi

. We calculate the ratio

tau_{phi,E} / tau_{phi,R}

of the echo (

E

) and Ramsey (

R

) sequences, and the dependence of the decay function on

alpha

and

f_2

. We investigate the case in which

S_Phi(f_0)

is fixed at the ``pivot frequency''

f_0 neq 1

~Hz while

alpha

is varied, and find that the choice of

f_0

can greatly influence the sensitivity of

tau_{phi,E}

and

tau_{phi,R}

to the value of

alpha

.Finally, we conclude with a brief review of our principal results and conclusions. We also comment on promising avenues of future research.

We compute the decoherence caused by

1∕f

fluctuations at low frequency

f

in the critical current

{I}_{0}

of Josephson junctions incorporated into flux, phase, charge, and hybrid flux-charge superconducting quantum bits (qubits). The dephasing time ${\ensuremath{\tau}}_{\ensuremath{\phi}}$ scales as ${I}_{0}∕\ensuremath{\Omega}\ensuremath{\Lambda}{S}_{{I}_{0}}^{1∕2}(1\phantom{\rule{0.3em}{0ex}}\mathrm{Hz})$, where $\ensuremath{\Omega}∕2\ensuremath{\pi}$ is the energy-level splitting frequency,

{S}_{{I}_{0}}(1\phantom{\rule{0.3em}{0ex}}\mathrm{Hz})

is the spectral density of the critical-current noise at

1\phantom{\rule{0.3em}{0ex}}\mathrm{Hz}

, and $\ensuremath{\Lambda}\ensuremath{\equiv}\ensuremath{\mid}{I}_{0}d\ensuremath{\Omega}∕\ensuremath{\Omega}d{I}_{0}\ensuremath{\mid}$ is a parameter computed for given parameters for each type of qubit that specifies the sensitivity of the level splitting to critical-current fluctuations. Computer simulations show that the envelope of the coherent oscillations of any qubit after time

t

scales as $\mathrm{exp}(\ensuremath{-}{t}^{2}∕2{\ensuremath{\tau}}_{\ensuremath{\phi}}^{2})$ when the dephasing due to critical-current noise dominates the dephasing from all sources of dissipation. We compile published results for fluctuations in the critical current of Josephson tunnel junctions fabricated with different technologies and a wide range in

{I}_{0}

and area

\mathcal{A}

, and show that their values of

{S}_{{I}_{0}}(1\phantom{\rule{0.3em}{0ex}}\mathrm{Hz})

scale to within a factor of 3 of $[144{({I}_{0}∕\ensuremath{\mu}\mathrm{A})}^{2}∕(\mathcal{A}∕\ensuremath{\mu}{\mathrm{m}}^{2})]{(\mathrm{pA})}^{2}∕\mathrm{Hz}$ at

4.2\phantom{\rule{0.3em}{0ex}}\mathrm{K}

. We empirically extrapolate

{S}_{{I}_{0}}^{1∕2}(1\phantom{\rule{0.3em}{0ex}}\mathrm{Hz})

to lower temperatures using a scaling

T(\mathrm{K})∕4.2

. Using this result, we find that the predicted values of ${\ensuremath{\tau}}_{\ensuremath{\phi}}$ at

100\phantom{\rule{0.3em}{0ex}}\mathrm{mK}

range from $0.8\phantom{\rule{0.5em}{0ex}}\text{to}\phantom{\rule{0.5em}{0ex}}12\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{s}$, and are usually substantially longer than values measured experimentally at lower temperatures.

1/f

-type power laws measured at considerably higher frequencies ($0.2\ensuremath{-}20$ MHz) are consistent with the noise in the $0.01\ensuremath{-}100$-Hz range measured here. We find no evidence of temperature dependence of the noises over $65\ensuremath{-}200$ mK, and also no evidence of time-domain correlations between the two noises. These methods and results are pertinent to the dephasing of all superconducting qubits.