Analysis of measurement errors for a superconducting phase qubit

We analyze several mechanisms leading to errors in a course of measurement of a superconducting flux-biased phase qubit. Insufficiently long measurement pulse may lead to nonadiabatic transitions between qubit states ∣1⟩ and ∣0⟩, before tunneling through a reduced barrier is supposed to distinguish the qubit states. Finite (though large) ratio of tunneling rates for these states leads to incomplete discrimination between ∣1⟩ and ∣0⟩. Insufficiently fast energy relaxation after the tunneling of state ∣1⟩ may cause the repopulation of the quantum well in which only the state ∣0⟩ is supposed to remain. We analyze these types of measurement errors using analytical approaches as well as a numerical solution of the time-dependent Schrödinger equation.

Figure 1

(a) Circuit schematic of the flux-biased phase qubit, controlled by external magnetic flux $\varphi \left(t\right)$. (b) Sketch of the qubit potential energy $U\left(\delta \right)$, where $\delta$ is the superconducting phase difference across the Josephson junction. During the measurement pulse the energy barrier is lowered so that the state ∣1⟩ escapes from the “left” well into the neighboring “right” well; this escape is sensed by a nearby SQUID.

Figure 2

The modified qubit potential (8) (thick solid lines, in GHz) and the normalized wave functions ${\mid {\psi }_n\left(\delta \right)\mid }^2$ (thin solid lines), shifted vertically by the energy eigenvalues $E_n$ for (a) $N=5$ and (b) $N=1.355$; other qubit parameters are given by Eq. (7). Dashed lines correspond to the actual potential (2). The energy origin is chosen at the well minimum.

Figure 3

Time dependence of the instantaneous nonadiabatic error defined in three different ways: $P_{01}\left(t\right)$ (solid lines), $P_{10}\left(t\right)$ (dashed lines), and $1-P_{00}\left(t\right)$ (dotted lines), for two durations of the linear pulse: $\tau =0.2\mathrm{ns}$ and $\tau =2\mathrm{ns}$ (for $\tau =2\mathrm{ns}$ the data are multiplied by the factor 20). The solid, dashed, and dotted lines are practically indistinguishable, except near the pulse end for $\tau =0.2\mathrm{ns}$.

Figure 4

The nonadiabatic error $P_e$ as a function of the duration $\tau$ of the linear measurement pulse. (a) Numerical results (thick solid line) and results obtained using either Eq. (25) (thin solid line) or Eq. (27) (dashed line) for the pulse starting with the barrier height $N=5$ (corresponding to ${\varphi }_0=5.09$) and ending with $N=1.355$ $({\varphi }_1=5.308)$. (b) Numerical results for pulses starting with $N=5$ and ending with $N=1.355$ (thick solid line), 1.5 (dashed line), and 1.65 (thin solid line); the corresponding maximum fluxes are $\varphi =5.308$, 5.298, and 5.287.

Figure 5

Dependence of the measurement error $P_e$ on the pulse duration $\tau$ calculated either numerically (thick solid lines) or using the simple formula (28), in which the spectral component is taken at the frequency ${\omega }_{10}$ corresponding to either beginning (thin solid lines) or end (dashed lines) of the measurement pulse. The pulse is (a) linear, $g\left(\theta \right)=\theta$, (b) of the shape $g\left(\theta \right)=1-{\cos }^4(\pi \theta ∕2)$, or (c) of the Gaussian shape. For the Gaussian pulse $\tau$ is defined as the r.m.s. width of the pulse.

Figure 6

(a) Several shapes of the measurement pulse and (b) the corresponding dependences $P_e\left(\tau \right)$. The pulse shapes are $g\left(\theta \right)=\theta$ [thin solid lines in (a) and (b)], $\sin (\pi \theta ∕2)$ (thin dotted lines), $1-\cos (\pi \theta ∕2)$ (thin dashed lines), ${\sin }^2(\pi \theta ∕2)$ (thick solid lines), ${\sin }^4(\pi \theta ∕2)$ (thick dashed lines), and $1-{\cos }^4(\pi \theta ∕2)$ (thick dotted lines).

Figure 7

Structure of the energy levels $E_n$ (in GHz) as a function of external flux $\varphi$ (in units of critical flux ${\varphi }_c=5.43$). The levels with $n$ between 168 and 218 (counted from the right well bottom) are shown. The dashed line shows the barrier height $\Delta U$. The vectors $\mid k⟩$ on the left enumerate the states localized in the left well.

Figure 8

Solid lines: the WKB tunneling rates ${\Gamma }_1$ and ${\Gamma }_0$ for the states ∣1⟩ and ∣0⟩ as functions of the applied flux $\varphi$. Dashed line shows the ratio ${\Gamma }_1∕{\Gamma }_0$ (scale at the right).

Figure 9

The probabilities $P_{s,1}$ (thick lines) and $P_{s,0}$ (thin lines) of tunneling to the right well during the measurement pulse, starting from the states ∣1⟩ or ∣0⟩, correspondingly, as functions of the maximum flux ${\varphi }_1$ during the pulse. Calculations are based on the WKB approximation. Initial flux is ${\varphi }_0∕{\varphi }_c=0.94$ $(N=5)$. The pulse duration is $\tau =2\mathrm{ns}$ (solid lines), $10\mathrm{ns}$ (dashed lines), or $50\mathrm{ns}$ (dotted lines).

Figure 10

The switching probabilities $P_{s,0}$ (thin lines) and $P_{s,1}$ (thick lines) as functions of the maximum flux ${\varphi }_1$, calculated numerically (solid lines) and using WKB approximation (dashed lines). The measurement pulse is of parabolic shape with duration $\tau =2\mathrm{ns}$, starting and ending at flux ${\varphi }_0∕{\varphi }_c=0.94$ $(N=5)$. Oscillations of the solid lines reflect level discreteness in the right well. (At $N=5$ there are $N_r=174$ levels in the right well.)

Figure 11

The level populations $P_n$ $(169⩽n⩽204)$ as functions of time $t$ for the measurement pulse with duration $\tau =2\mathrm{ns}$, starting and ending at ${\varphi }_0∕{\varphi }_c=0.94$ $(N=5)$ with maximum flux in the middle ${\varphi }_1∕{\varphi }_c=1.01$. The curves for $169⩽n⩽197$ are thin solid lines, the curves for $198⩽n⩽204$ are denoted by the value of $n$. Inset: Energies $E_n(168⩽n⩽218)$ versus time $t$ for the rising part of the pulse (similar to Fig. 7).

Figure 12

The level populations $P_n$ at $t=1\mathrm{ns}$ and $t=2\mathrm{ns}$ for the process shown in Fig. 11 (${\varphi }_1∕{\varphi }_c=1.01$, $\tau =2\mathrm{ns}$). The data are shown by dots and squares, connected by lines as guides for the eye.

Figure 13

The switching probability $P_{s,0}$ for the initial state ∣0⟩ [see Eq. (41)] (a) versus the maximum flux ${\varphi }_1$ (normalized by the critical flux ${\varphi }_c$) for $\tau =2\mathrm{ns}$ and (b) versus the pulse duration $\tau$ for ${\varphi }_1={\varphi }_c$. Solid lines are for our usual set (7) of qubit parameters; dashed lines are for the qubit with parameters (42), having much smaller number $N_r$ of discrete levels in the right well.

Figure 14

Energies $E_n$ (in GHz) as a function of flux $\varphi$ (in units of ${\varphi }_c=3.55$) for the qubit with parameters (42) corresponding to $N_r=30$. The levels with $24⩽n⩽57$ are shown. The dashed line corresponds to the barrier height $\Delta U$.