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

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 ${\tau }_{\varphi }$ scales as $I_0∕\Omega \Lambda S_{I_0}^{1∕2}\left(1\mathrm{Hz}\right)$, where $\Omega ∕2\pi$ is the energy-level splitting frequency, $S_{I_0}\left(1\mathrm{Hz}\right)$ is the spectral density of the critical-current noise at $1\mathrm{Hz}$, and $\Lambda \equiv \mid I_{0}d\Omega ∕\Omega d{I}_0\mid$ 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 $\exp (-t^2∕2{\tau }_{\varphi }^2)$ 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 $S_{I_0}\left(1\mathrm{Hz}\right)$ scale to within a factor of 3 of $[144{(I_0∕\mu \mathrm{A})}^2∕(\mathcal{A}∕\mu {\mathrm{m}}^2)]{\left(\mathrm{pA}\right)}^2∕\mathrm{Hz}$ at $4.2\mathrm{K}$. We empirically extrapolate $S_{I_0}^{1∕2}\left(1\mathrm{Hz}\right)$ to lower temperatures using a scaling $T\left(\mathrm{K}\right)∕4.2$. Using this result, we find that the predicted values of ${\tau }_{\varphi }$ at $100\mathrm{mK}$ range from $0.8\text{to}12\mu \mathrm{s}$, and are usually substantially longer than values measured experimentally at lower temperatures.

Figure 1

Effects of low-frequency flux and critical-current fluctuations in a superconducting qubit. (a) Flux modulation from vortices hopping into and out of a loop, and critical-current modulation from electrons $e^{-}$ temporarily trapped at defect sites in the junction barrier. (b) A single-charge trap blocks tunneling over an area $\Delta \mathcal{A}$, reducing the critical current. (c) Fluctuations modify the oscillation frequency, inducing phase noise which leads to decoherence in time-averaged ensembles of sequential measurements of the qubit observable $Z$.

Figure 2

One-junction flux qubit. (a) Schematic. (b) Symmetric double-well potential for flux bias ${\Phi }_x={\Phi }_0∕2$. (c) Flux fluctuation $\Delta \Phi$ couples to $\Omega$ only in second order. (d) Critical-current fluctuation $\Delta I_0$ produces an exponential change in $\Omega$.

Figure 3

Three quantities for the ground state of the one-junction flux qubit at the degeneracy point calculated using the standard WKB approximation (solid), WKB approximation corrected for the ground state (dashed), and numerical solution for the wave functions (points), plotted as a function of the dimensionless screening parameter ${\beta }_L$. (a) Splitting frequency between ground state and first excited state, (b) sensitivity parameter $\Lambda$, and (c) effect of critical current fluctuations on the tunneling rate for three values of $\delta I_0∕I_0$. Parameters are from Friedman et al. (Ref. 1): $L=240\mathrm{pH}$ and $C=104\mathrm{fF}$.

Figure 4

Eigenfunctions and energy levels for a one-junction flux qubit. Absolute values of eigenfunctions (thin lines), each offset so that it asymptotes to the corresponding energy level, are shown as a function of the junction phase $\delta$. The thick line is the asymmetric double-well potential. Eigenfunctions are labeled according to corresponding single-well harmonic-oscillator quantum numbers. Parameters are as in Fig. 3, with ${\beta }_L=1.5$; the flux ${\varphi }_x\approx 0.514\times 2\pi$ produces a resonance between the $3L$ (left) and $9R$ (right) states.

Figure 5

Numerical solution for the excited states of an asymmetric one-junction flux qubit. (a) Tunneling frequency between the third excited state in the shallow well and the ninth excited state in the deep well as a function of ${\beta }_L$ for a system on resonance at ${\beta }_L=1.5$. (b) Derived sensitivity to critical-current fluctuations. Device parameters are as in Fig. 3.

Figure 6

Three-junction flux qubit. (a) Schematic showing inductive loop, embracing ${\Phi }_0∕2$ interrupted by three Josephson junctions. (b) Tunneling frequency and (c) $\Lambda$ vs Josephson-to-charging energy ratio. Solid lines indicate dependence on large junction ratio ${\gamma }^{a,b}$ with ${\gamma }^c=28$, and dashed lines indicate dependence on small junction ratio ${\gamma }^c$ with ${\gamma }^a={\gamma }^b=35$. $E_C=7.4\mathrm{GHz}$ for all plots.

Figure 7

Single Josephson-junction qubit. (a) Schematic and (b) energy-level diagram. (c) Variation of energy separation with bias current. (d) $\Lambda$ as a function of bias current. Parameters are from Martinis et al. (Ref. 9): $C=6\mathrm{pF}$, corresponding to a junction area of $100\mu {\mathrm{m}}^2$, and $I_0=21.1\mu \mathrm{A}$.

Figure 8

The quantronium qubit. (a) Schematic showing phase difference $\delta$ across two small Josephson junctions with charge $N_g$ on island between them. Readout junction with critical current $I_0^r$ in superconducting loop allows for measurement of circulating current. (b) Level-splitting frequency $\Omega ∕2\pi$ and (c) critical-current sensitivity $\Lambda$ vs $N_g$. Curves are plotted for the parameters reported by Vion et al. (Ref. 7): $I_0=18\mathrm{nA}$, $C_j=2.7\mathrm{fF}$; at the optimal working point $N_g=1∕2$, $\delta =0$, $\Lambda =2^{-1∕2}$, and $\Omega ∕2\pi$ is calculated to be $17.9\mathrm{GHz}$, slightly different from the observed value of $16.5\mathrm{GHz}$.

Figure 9

Measurement sequences for mapping out coherent oscillations. (a) Method A: time-delay averaging. (b) Method B: time-sweep averaging. The interval between qubit state measurements is $t_Z$; the spacing of time-delay points is $t_d$.

Figure 10

(a) Simulated time sequence of critical-current changes for an experiment with $N={10}^4$ total qubit state measurements taken at intervals of $t_Z=1\mathrm{ms}$. (b) Corresponding $1∕f$ frequency spectrum.

Figure 11

Probability envelopes determined by simulations using measurement methods A and B for a qubit with $I_0=1\mu \mathrm{A}$, $S_{I_0}\left(1\mathrm{Hz}\right)=8.16\times {10}^{-24}{\mathrm{A}}^2{\mathrm{Hz}}^{-1}$, $\mathcal{A}=0.01\mu {\mathrm{m}}^2$, $\Lambda =100$, and $\Omega ∕2\pi =1\mathrm{GHz}$. The structure visible in the method B plot arises from periodic sampling of the oscillations and is evidence of the increased effective averaging relative to method A.

Figure 12

Simulated probability oscillations with large critical current fluctuations for measurement methods A and B. Qubit parameters as in Fig. 11, except $S_{I_0}\left(1\mathrm{Hz}\right)=1.39\times {10}^{-20}{\mathrm{A}}^2{\mathrm{Hz}}^{-1}$.

Figure 13

Distributions of dephasing times ${\tau }_{\varphi }$ calculated by method A (open symbols) and method B (closed symbols) for different number of flux measurement points $N=3\times {10}^4$ (squares), $3\times {10}^5$ (triangles), and $3\times {10}^6$ (circles). Each distribution includes 1000 simulations of the coherent oscillations accumulated into bins of width $2\mathrm{ns}$. Qubit parameters are as in Fig. 11.

Figure 14

Variation of the dephasing time ${\tau }_{\varphi }$ with the number of qubit-state measurements $N$ for methods A and B. Each point corresponds to the mean value of ${\tau }_{\varphi }$ from 50 simulations of the oscillation decay envelope. Qubit and noise parameters as in Fig. 11. The solid line is ${\tau }_{\varphi }^M$ obtained from Eq. (25).