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arXiv:0810.1334 · Physical Review B

Saturation of two-level systems and charge noise in Josephson junction qubits

Magdalena Constantin, Clare C. Yu, John M. Martinis

Published Mar 20, 2009• 35 citations
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Abstract

We study the charge noise ${S}_{Q}$ in Josephson qubits produced by fluctuating two-level systems (TLSs) with electric-dipole moments in the substrate. The TLSs are driven by an alternating electric field of angular frequency $\ensuremath{\Omega}$ and electric field intensity $I$. It is not widely appreciated that TLS in small qubits can easily be strongly saturated if $I⪢{I}_{c}$, where ${I}_{c}$ is the critical electric field intensity. To investigate the effect of saturation on the charge noise, we express the noise spectral density in terms of density-matrix elements. To determine the dependence of the density-matrix elements on the ratio $I/{I}_{c}$, we find the steady-state solution for the density matrix using the Bloch-Redfield differential equations. We then obtain an expression for the spectral density of charge fluctuations as a function of frequency $f$ and the ratio $I/{I}_{c}$. We find $1/f$ charge noise at low frequencies, and that the charge noise is white (constant) at high frequencies. Using a flat density of states, we find that TLS saturation has no effect on the charge noise at either high or low frequencies.

DOI
10.1103/physrevb.79.094520

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We study the charge noise SQ{S}_{Q}SQ​ in Josephson qubits produced by fluctuating two-level systems (TLSs) with electric-dipole moments in the substrate. The TLSs are driven by an alternating electric field of angular frequency \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ and electric field intensity III. It is not widely appreciated that TLS in small qubits can easily be strongly saturated if I⪢IcI⪢{I}_{c}I⪢Ic​, where Ic{I}_{c}Ic​ is the critical electric field intensity. To investigate the effect of saturation on the charge noise, we express the noise spectral density in terms of density-matrix elements. To determine the dependence of the density-matrix elements on the ratio I/IcI/{I}_{c}I/Ic​, we find the steady-state solution for the density matrix using the Bloch-Redfield differential equations. We then obtain an expression for the spectral density of charge fluctuations as a function of frequency fff and the ratio I/IcI/{I}_{c}I/Ic​. We find 1/f1/f1/f charge noise at low frequencies, and that the charge noise is white (constant) at high frequencies. Using a flat density of states, we find that TLS saturation has no effect on the charge noise at either high or low frequencies.
We show that charge noise SQS_QSQ​ in Josephson qubits can be produced by fluctuating two level systems (TLS) with electric dipole moments in the substrate using a flat density of states. At high frequencies the frequency and temperature dependence of the charge noise depends on the ratio J/JcJ/J_cJ/Jc​ of the electromagnetic flux JJJ to the critical flux JcJ_cJc​. It is not widely appreciated that TLS in small qubits can easily be strongly saturated with J/Jc≫1J/J_c\gg 1J/Jc​≫1. Our results are consistent with experimental conclusions that SQ∼1/fS_Q\sim 1/fSQ​∼1/f at low frequencies and SQ∼fS_Q\sim fSQ​∼f at high frequencies.
Author(s): Yu, Clare C; Constantin, Magdalena; Martinis, John M | Abstract: We show that charge noise SQS_QSQ​ in Josephson qubits can be produced by fluctuating two level systems (TLS) with electric dipole moments in the substrate using a flat density of states. At high frequencies the frequency and temperature dependence of the charge noise depends on the ratio J/JcJ/J_cJ/Jc​ of the electromagnetic flux JJJ to the critical flux JcJ_cJc​. It is not widely appreciated that TLS in small qubits can easily be strongly saturated with J/Jc≫1J/J_c\gg 1J/Jc​≫1. Our results are consistent with experimental conclusions that SQ∼1/fS_Q\sim 1/fSQ​∼1/f at low frequencies and SQ∼fS_Q\sim fSQ​∼f at high frequencies.
We show that charge noise SQS_QSQ​ in Josephson qubits can be produced by\nfluctuating two level systems (TLS) with electric dipole moments in the\nsubstrate using a flat density of states. At high frequencies the frequency and\ntemperature dependence of the charge noise depends on the ratio J/JcJ/J_cJ/Jc​ of the\nelectromagnetic flux JJJ to the critical flux JcJ_cJc​. It is not widely\nappreciated that TLS in small qubits can easily be strongly saturated with\nJ/Jcgg1J/J_c\\gg 1J/Jc​gg1. Our results are consistent with experimental conclusions that\nSQsim1/fS_Q\\sim 1/fSQ​sim1/f at low frequencies and SQsimfS_Q\\sim fSQ​simf at high frequencies.
The electrical noise of mesoscopic devices can be strongly influenced by the quantum motion of electrons. To probe this effect, we have measured the current fluctuations at high frequency (5 to 90 gigahertz) using a superconductor-insulator-superconductor tunnel junction as an on-chip spectrum analyzer. By coupling this frequency-resolved noise detector to a quantum device, we can measure the high-frequency, nonsymmetrized noise as demonstrated for a Josephson junction. The same scheme is used to detect the current fluctuations arising from coherent charge oscillations in a two-level system, a superconducting charge qubit. A narrow band peak is observed in the spectral noise density at the frequency of the coherent charge oscillations.
Superconducting circuits are a leading platform for quantum computing. However, their coherence times are still limited and exhibit temporal fluctuations. Those phenomena are often attributed to the coupling between qubits and material defects that can be well described as an ensemble of two-level systems (TLSs). Among them, charge fluctuators inside amorphous oxide layers contribute to both low-frequency 1/f1/f1/f charge noise and high-frequency dielectric loss, causing fast qubit dephasing and relaxation. Moreover, spectral diffusion from mutual TLS interactions varies the noise amplitude over time, fluctuating the qubit lifetime. Here, we propose to mitigate those harmful effects by engineering the relevant TLS noise spectral densities. Specifically, our protocols smooth the high-frequency noise spectrum and suppress the low-frequency noise amplitude via depolarizing and dephasing the TLSs, respectively. As a result, we predict a drastic stabilization in qubit lifetime and an increase in qubit pure dephasing time. Our detailed analysis of feasible experimental implementations shows that the improvement is not compromised by spurious coupling from the applied noise to the qubit.
We study the effect of external driving on the two-level systems (TLSs) assumed to be a major obstacle in increasing the coherence time of solid-state Josephson junction qubits. We find, by use of a Bloch-Redfield approach, that external driving has two major effects on the TLS. The first is increased fluctuations between the two states of the TLS; the significance of this effect compared to thermal fluctuations depends on the energy splitting of the TLS compared to temperature. The second effect is a reduction in the intensity of the noise spectrum at low frequencies, and at the same time an increase in intensity around the renormalized Rabi frequency of the TLS and the driving frequency and at beatings between these two frequencies. Finally, we study the ensemble-averaged noise spectrum for a typical distribution of TLSs known to give origin to 1/f1/f1/f noise. We find that strong driving leads to reduced noise at low frequencies, and therefore to an increased dephasing time T2Q{T}_{2}^{Q}T2Q​ of the qubit. However, this effect is exponentially suppressed when the driving frequency is large compared to temperature, as we typically find for Josephson qubits. We suggest that external driving at frequencies much lower than the qubit frequency might be used in order to enhance the the qubit coherence time.