Decoherence in a superconducting quantum bit circuit

Decoherence in quantum bit circuits is presently a major limitation to their use for quantum computing purposes. We present experiments, inspired from NMR, that characterize decoherence in a particular superconducting quantum bit circuit, the quantronium. We introduce a general framework for the analysis of decoherence, based on the spectral densities of the noise sources coupled to the qubit. Analysis of our measurements within this framework indicates a simple model for the noise sources acting on the qubit. We discuss various methods to fight decoherence.

Figure 1

(a) Circuit diagram of the quantronium. The Hamiltonian of this circuit is controlled by the gate charge $N_g\propto U$ on the island between the two small Josephson junctions and by the phase $\delta$ across their series combination. This phase is determined by the flux $\varphi$ imposed through the loop by an external coil, and by the bias current $I_b$. The two lowest-energy eigenstates form a quantum bit whose state is read out by inducing the switching of the larger readout junction to a finite voltage V with a bias-current pulse $I_b\left(t\right)$ approaching its critical current. (b) Qubit transition frequency ${\nu }_{01}$ as a function of $\delta$ and $N_g$. The saddle point $P_0$ indicated by the arrow is optimal for a coherent manipulation of the qubit. (c) Bloch sphere representation in the rotating frame. The quantum state is manipulated by applying resonant microwave gate pulses $N_g\left(t\right)$ with frequency ${\nu }_{\mu w}$ and phase $\chi$, and/or adiabatic trapezoidal $N_g$ or $I_b\left(t\right)$ pulses. Microwave pulses induce a rotation of the effective spin $S$ representing the qubit around an axis in the equatorial plane making an angle $\chi$ with $X$, whereas adiabatic pulses induce rotations around $Z$.

Figure 2

(Color online) (a) Schematics of the experimental setup used in this work with temperatures indicated on the left. Rectangles labeled in dB are $50\Omega$ attenuators whereas rectangle labeled :10 is a high-impedance voltage divider by 10. Squares labeled Fcp and $F\mu$ are copper powder filters and microfabricated distributed $\mathit{RC}$ filters, respectively. Single lines, double lines, and twisted pairs are $50\Omega$ coaxes, lossy coaxes made of a manganin wire in a stainless steel tube, and shielded lossy manganin twisted pairs, respectively. (b) Scanning electron microscope pictures of the sample. The whole aluminum loop (right) of about $5.6\mu {\mathrm{m}}^2$ is defined by $200\text{−}\mathrm{nm}$-wide lines and includes a $890\times 410{\mathrm{nm}}^2$ island (left) delimited by two $160\times 160{\mathrm{nm}}^2$ junctions, and a $1.6\mu \mathrm{m}\times 500\mathrm{nm}$ readout junction. Note also the presence of gold quasiparticle traps (bright pads).

Figure 3

Top: oscillogram of a typical microwave gate pulse, measured at the top of the cryostat. The arrow indicates the effective duration $\tau$ of the pulse. Bottom: the Rabi precession of the qubit state during a microwave pulse results in oscillations of the switching probability $p$ with the pulse length $\tau$, at a frequency ${\nu }_{R0}={\omega }_{R0}∕2\pi$ proportional to the reduced microwave amplitude $\Delta N_g$ (right). The two arrows in the left panel correspond to the so-called $\pi$ and $\pi ∕2$ pulses used throughout this work. The arrow in the right panel indicates the point that corresponds to the data shown in the left panel.

Figure 4

Readout of the quantronium. Top left: full $I_b\left(t\right)$ variation measured at the top of the cryostat, when the qubit is operated at the optimal point (see text) and read out with maximum sensitivity. The current is first kept at zero to avoid heating in the bias resistor, then prebiased at a negative value that corresponds to $\delta =0$, then increased in about $50\mathrm{ns}$ to a value close to the critical current of the readout junction, and maintained at this value during about $100\mathrm{ns}$, a time period over which the switching of the junction may occur. $I_b$ is then slightly lowered and maintained at this lower value to let the voltage develop along the measuring line if the junction has switched. Finally it is set back to zero. Top right: detail of readout pulse (but without negative prebias), measured at room temperature at the bottom of the bias line, before cooling the cryostat. Bottom left: persistent currents in the loop for the ∣0⟩ (solid line) and ∣1⟩ (dashed line) states, computed at $N_g=1∕2$ as a function of $\delta$, using the measured sample parameters. The vertical dot-dashed line indicates the readout point ${\delta }_M$ where the experimental difference $i_1-i_0$ was found to be maximum. Bottom right: variation of the switching probability $p$ with respect to the peak current $I_M$, measured without microwave (solid line), measured after a $\pi$ microwave pulse (dotted line), and calculated from the sample parameters for state ∣1⟩ (dashed line). The vertical dot-dashed line indicates a maximum fidelity of 0.4 (instead of the expected 0.95), obtained with the pulse shown in the top-left panel. The two arrows of the upper and lower left panels indicate the adiabatic displacement in $\delta$ between operation and readout of the qubit.

Figure 5

Equivalent schematic drawing of the noise sources responsible for decoherence in the quantronium. These sources are coupled to $E_J$, $N_g$, or $\delta$. In part they are of microscopic nature like the two-level fluctuators inside the junction that induce $E_j$ variations, like a charged TLF (represented as a minus sign in a small double arrow) coupled to $N_g$, or like moving vortices $\left({\Phi }_{\mathit{\text{micro}}}\right)$ in the vicinity of the loop. The macroscopic part of the decoherence sources is the circuitry, which is represented here as an equivalent circuit as seen from the qubit. The relevant resistances and temperatures of the dissipative elements are indicated. Capacitance with no label represent a shunt at the qubit frequency and an open circuit at frequencies below $200\mathrm{MHz}$.

Figure 6

Experimental $T_1$ values measured at $N_g=1∕2$ as a function of $\delta$ (left panel), and at $\delta =0$ as a function of $N_g$ (right panel). The vertical line separating the two panels corresponds to the optimal point $P_0=(N_g=1∕2,\delta =0)$. The dashed line joining the points is a guide for the eye. The correspondence between $\delta$, $N_g$, and ${\nu }_{01}$ is given by the upper horizontal axis. Inset: Example of $T_1$ measurement. The switching probability $p$ (dots) is measured as a a function of the delay $t$ between a $\pi$ pulse and the readout pulse. The fit by an exponential (full line) leads to $T_1$ ($0.5\mu \mathrm{s}$ at $P_0$ in this example).

Figure 7

Ramsey signals at the optimal point $P_0$ for ${\omega }_{R0}∕2\pi =106\mathrm{MHz}$ and $\Delta \nu \approx 50\mathrm{MHz}$, as a function of the delay $\Delta t$ between the two $\pi ∕2$ pulses. Top and middle panels: solid lines are two successive records showing the partial irreproducibility of the experiment. Dashed lines are a fit of the envelope of the oscillations in the middle panel (see text) leading to $T_2=300\mathrm{ns}$. The dotted line shows for comparison an exponential decay with the same $T_2$. Bottom panels: zoom windows of the middle panel. The dots represent now the experimental points whereas the solid line is a fit of the whole oscillation with $\Delta \omega ∕2\pi =50.8\mathrm{MHz}$. Arrows point out a few sudden jumps of the phase and amplitude of the oscillation, attributed to strongly coupled charged TLFs.

Figure 8

Ramsey oscillations as a function of the delay $\Delta t$ between the two $\pi ∕2$ pulses, for different working points located on the lines $N_g=1∕2$ (left column) and $\delta =0$ (right column). The Rabi frequency is ${\omega }_{R0}∕2\pi =162\mathrm{MHz}$ for all curves. The nominal detunings $\Delta \nu$ are 50, 53, 50, 50, 40, 100, and $80\mathrm{MHz}$ (left, top to bottom) and $35–39\mathrm{MHz}$ (right). Dots are experimental points whereas full lines are exponentially damped sinusoids fitting the experimental results and leading to the $T_2$ values reported on Fig. 15.

Figure 9

Phase detuning pulse technique for measuring $T_2$. Top: Ramsey signal at the optimal point $P_0$, with $\Delta \nu \simeq 50\mathrm{MHz}$, when no detuning dc pulse is applied. The dashed line corresponds to an exponential decay with $T_2\left(P_0\right)=200\mathrm{ns}$. Bottom: signal obtained with a delay $\Delta t=275$ ns between the two $\pi ∕2$ pulses (corresponding to the dashed vertical line of the upper panel) and with an adiabatic current pulse maintaining $\delta ∕2\pi =0.063$ during a time $\Delta t_2$. The oscillation of the signal with $\Delta t_2$ decays with a characteristic time of about $70\mathrm{ns}$ (note the different horizontal scales on the two graphs). The pictograms on the right illustrate the two $\pi ∕2$ microwave pulses and the $I_b\left(t\right)$ signal.

Figure 10

Charge detuning pulse technique used for measuring $T_2$ at four points $P=(0,1∕2+\Delta N_g)$. Dots are experimental points whereas full lines are fits using Gaussian damped sinusoids at frequencies $\Delta \omega \left(\Delta N_g\right)$. The extracted $T_2$ are indicated on each panel.

Figure 11

Top panels: line shape (thin lines) and central position (dots) of the resonance lines as a function of $\delta$ at $N_g=1∕2$ (left) and as a function of $N_g$ at $\delta =0$ (right). The optimal point $P_0$ corresponds to the double arrow in the center of the graph. Note the two different vertical scales and the occasional substructure of resonance lines pointed out by small arrows. Bold lines are fits of the peak positions leading to $E_J=0.87k_B\mathrm{K}$, $E_C=0.66k_B\mathrm{K}$, and $d<13\%$. Bottom panel: asymmetric line shape recorded (dots) at $P_0$ with a microwave power small enough to desaturate the line. The dashed line is the theory, with a $T_{\phi }$ that corresponds to that of Fig. 7. The solid line is the convolution of this theoretical line and of a Lorentzian corresponding to a decay time of $600\mathrm{ns}$ (see text).

Figure 12

Spin echoes (bold lines) obtained at the optimal point $P_0$ with a detuning $\Delta \nu \simeq 50\mathrm{MHz}$. Pictograms illustrate the experimental protocol: the delay $\Delta t_3$ between the $\pi$ pulse and the last $\pi ∕2$ pulse is kept constant while altering the timing of the first $\pi ∕2$ pulse. Each panel corresponds to a different $\Delta t_3$. Vertical arrows indicate the sequence duration $\Delta t=2\Delta t_3$ for which the echo amplitude is expected to be maximal and where $p=p_E$ is minimum. For the sake of comparison, the corresponding Ramsey signal (thin lines) is shown in all panels.

Figure 13

Echo signal $p_E$ (linked big dots) measured at the optimal point $P_0$ by keeping a $\pi$ pulse precisely in the middle of the sequence while sweeping the sequence duration $\Delta t$ (pictogram). The Rabi frequency is ${\omega }_{R0}∕2\pi =130\mathrm{MHz}$ and the detuning $\Delta \nu =20\mathrm{MHz}$. For comparison, the Ramsey signal (oscillating line) and its envelope (dashed line leading to $T_{\phi }=450\mathrm{ns}$) are also shown. The dotted line is a fit of $p_E$ that leads to the characteristic decay time of $f_{z,E}$, $T_{\phi ,E}=1.3\mu \mathrm{s}$, and that shows that the $\pi ∕2$ pulses were actually 15% too short whereas the $\pi$ pulse was correct. The resulting echo time is $T_E\sim 600\mathrm{ns}$. Inset: comparison between $p_E$ (linked dots) and the echo signal recorded with a fixed $\pi$ pulse (solid line), as presented in Fig. 12.

Figure 14

Echo signals $p_E\left(\Delta t\right)$ measured (dots) at different working points indicated in each panel, with a Rabi frequency ${\omega }_{R0}∕2\pi =140\mathrm{MHz}$ and $\Delta \nu \approx 50\mathrm{MHz}$. Full lines are exponential fits leading to $T_E$ values reported on Fig. 15. Note that the amplitude of the signal depends on the working point.

Figure 15

Echo times $T_E$ (open circles) and coherence times $T_2$ measured from the resonance linewidth (solid dots), from the decay of Ramsey signals (triangles), and from the detuning pulse method (squares), at $N_g=1∕2$ as a function of $\delta$ (left panel) and at $\delta =0$ as a function of $N_g$ (right panel). The full and dashed lines are best fits (see text) of $T_E$ and $T_2$ times, respectively, leading to the phase and charge noise spectral densities depicted at the bottom. The spectra are even functions of $\omega$.

Figure 16

Decay of the Rabi signals at the optimal point $P_0$ for different Rabi frequencies ${\nu }_{R0}={\omega }_{R0}∕2\pi$. The experimental data (oscillating solid lines) are fitted by exponentially damped sinusoids (dotted lines in the top panels), while their lower envelopes are fitted by exponentials (monotonous solid lines) leading to the ${\tilde{T}}_2$ values reported in Fig. 17.

Figure 17

Characteristic decay times ${\tilde{T}}_2$ of the Rabi oscillations at the optimal point $P_0$, as a function of the Rabi frequency ${\nu }_{R0}$ (left panel) at zero detuning $\Delta \nu$, and as a function of $\Delta \nu$ (right panel) at ${\nu }_{R0}=15.4\mathrm{MHz}$ (dotted vertical line). ${\tilde{T}}_2({\nu }_{R0},\Delta \nu =0)$ turns out to be a constant of order $0.48\mu \mathrm{s}$ (left solid line). The difference with $4∕3T_1$ leads to an estimate for $T_{\nu }=1∕{\Gamma }_{\nu }$. The right solid line corresponds to Eq. (45) plotted using the experimentally determined values of $T_{\phi }$, $T_1$, and $T_{\nu }$.

Figure 18

Spin-locking signals (oscillatory lines) obtained at the optimal point $P_0$, using a detuning $\Delta \nu =8\mathrm{MHz}$, a locking microwave power corresponding to $24\mathrm{MHz}$, and a final microwave pulse of $\pi ∕2$ (top) or $3\pi ∕2$ (bottom). The bold solid lines are exponential fits corresponding to ${\tilde{T}}_1\sim 580\mathrm{ns}$. For comparison, the Ramsey envelope (dotted line with $T_2\sim 250\mathrm{ns}$) and the longitudinal relaxation (dashed line with $T_1\sim 450\mathrm{ns}$) are shown.