Figure 1

(a) Circuit diagram of the single-electron box capacitively coupled to an rf-SET electrometer. The tunnel junctions are represented by boxes divided by a horizontal line. The junction capacitance $C_J$ is the dominant component of the total box capacitance, $C_{\Sigma }^0=C_J+C_g+C_C.$ Additional circuit elements (not shown) apply an rf signal between the SET’s drain and source and detect the amount of rf power reflected $P_r$ from the rf-SET [15]. (b) Calibration of the Coulomb sawtooth is accomplished by varying the SET’s control gate voltage $V_{\mathrm{g}\mathrm{e}}=n_{\mathrm{g}\mathrm{e}}(e/C_{\mathrm{g}\mathrm{e}})$ about a fixed operating voltage $V_{\mathrm{o}\mathrm{p}}=n_{\mathrm{o}\mathrm{p}}(e/C_{\mathrm{g}\mathrm{e}})$ while the box gate is held at $n_g=0$. This applies a known charge signal $Q_{\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}}=e(n_{\mathrm{g}\mathrm{e}}-n_{\mathrm{o}\mathrm{p}})$ to the SET. The plot $P_r$ versus $n_{\mathrm{g}\mathrm{e}}-n_{\mathrm{o}\mathrm{p}}=Q_{\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}}/e$ (solid line, top axis) is a nonlinear map (implied by dotted lines) that converts $P_r$ versus $n_g$ (dashed lines, bottom axis) into $Q_{\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}}$ versus $n_g$. The electrometer’s operating point $n_{\mathrm{o}\mathrm{p}}=0.44$ and an alternative $n_{\mathrm{o}\mathrm{p}}=0.56$ and are indicated (two dots). (c) The Coulomb sawtooth, $Q_{\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}}$ versus $n_g$, (dashed line) on the right axis, and the Coulomb staircase, $Q_{\mathrm{b}\mathrm{o}\mathrm{x}}$ vs $n_g$, (solid line) on the left axis at $T=30\mathrm{m}\mathrm{K}$.Reuse & Permissions

Figure 2

(a) Coulomb staircases at $T=500\mathrm{m}\mathrm{K}$ (dotted line) and $T=100\mathrm{m}\mathrm{K}$ (solid line). The charging energy $E_C^N$ is extracted by fitting to the 500 mK data a theoretical staircase (not shown) broadened only by thermal fluctuations. The 100 mK staircase is compared to the thermal fluctuation theory with no adjustable parameters (dashed line). (b) The residuals of the 500 mK fit (dotted line) and of the 100 mK comparison (solid line). (c) Plotted versus $T$ on logarithmic scales are the measured value of $(1/\chi )$ (points), the expression $1/\chi =2k_{B}T/E_C^N$ (line) showing the expected behavior in the absence of quantum fluctuations, and the expression $1/\chi =2k_B\sqrt{(T^2+T_{\mathrm{s}\mathrm{p}}^2)}/E_C^N$ showing the expected behavior in the absence of quantum fluctuations but in the presence of a spurious broadening characterized by a phenomenological effective temperature $T_{\mathrm{s}\mathrm{p}}=25\mathrm{m}\mathrm{K}$ (dotted line). A model of temperature-independent spurious broadening does not contain the observed behavior of $\chi$ versus $T$. (d) The quantity $E_C^{*}(T)=2k_{B}T\chi$ (points) and the prediction of 9 (line) versus $T$ with no adjustable parameters. Dashed lines indicate the range of theory consistent with the uncertainties in $g$ and $E_C^N$. The error bars of two lowest $T$ points account for a systematic rounding introduced by the SET’s backaction [16].Reuse & Permissions

Figure 3

Coulomb staircases with the box in its normal state (line) and in its superconducting state, $B_{\mathrm{a}\mathrm{p}\mathrm{p}}=100\mathrm{m}\mathrm{T}$ (triangles). For both, the Coulomb sawtooths have been converted to staircases using ${\kappa }^0=C_C/C_{\Sigma }^0=3.9\times {10}^{-2}$, which would be the slope of the Coulomb sawtooth around $n_g=0$ in the absence of tunneling. The renormalization of $C_{\Sigma }^0$ is visible as the nonzero slope $(1/e)d{Q}_{\mathrm{b}\mathrm{o}\mathrm{x}}/d{n}_g$ of these plots at $n_g=0$. The inset shows ${\kappa }^S(B_{\mathrm{a}\mathrm{p}\mathrm{p}})$ versus $B_{\mathrm{a}\mathrm{p}\mathrm{p}}$ and the value of $E_C^S(B_{\mathrm{a}\mathrm{p}\mathrm{p}})$ inferred from ${\kappa }^S$, as the aluminum is driven from its superconducting to its normal state. In the superconducting state, a single out-of-equilibrium quasiparticle on the box’s island keeps the Coulomb staircase $e$ periodic.Reuse & Permissions

Figure 4

Coulomb staircases with the box at two different electrometer operating points, $n_{\mathrm{o}\mathrm{p}}=0.44$ (triangles) and $n_{\mathrm{o}\mathrm{p}}=0.56$ (circles); $V_{ds}=0$ for both. The SET’s backaction causes these two curves to deviate from each other around $n_g=0.5$. The theory plots are the charge on the box predicted for thermal fluctuations but no quantum fluctuations at $T=125\mathrm{m}\mathrm{K}$ (dashed dotted line) and $T=29\mathrm{m}\mathrm{K}$ (dashed line), and for the quantum fluctuations calculated to first order in $g$ (dotted line) and second order in $g$ (solid line) [6, 7]. The second order calculation is fit to the data with the adjustable parameter $\kappa =(3.375\pm 0.001)\times {10}^{-2}$, which is better constrained by this fit than by extracting the slope of the Coulomb sawtooth. Note we have plotted both data and theory in the form with $d{Q}_{\mathrm{b}\mathrm{o}\mathrm{x}}/d{n}_g=0$ at $n_g=0$ unlike Refs. 6, 7. Inset: the same data over a range $0<n_g<1$. The dashed box indicates the region plotted in the main figure.Reuse & Permissions