Quantum detectors for the third cumulant of current fluctuations

We consider the measurement of the third cumulant of current fluctuations arising from a point contact, employing the transitions that they cause in a quantum detector connected to the contact. We detail two generic detectors: a quantum two-level system and a harmonic oscillator. In these systems, for an arbitrary relation between the voltage driving the point contact and the energy scales of the detectors, the results can be expressed in terms of an effective detector temperature $T_{\mathrm{eff}}$. The third cumulant can be found from the dependence of $T_{\mathrm{eff}}$ on the sign of the driving voltage. We find that proper ordering of the fluctuation operators is relevant in the analysis of the transition rates. This is reflected in the effective Fano factor for the third cumulant measured in such setups: it depends on the ratio of the voltage and an energy scale describing the circuit where the fluctuations are produced.

Figure 1

Schematic idea of the fluctuation measurements: Current fluctuations in the noise source are coupled to a quantum detector, where they induce transitions between the detector energy levels. Another source for the transition rates is the intrinsic Gaussian environment of the detector, which can be modeled via an ensemble of harmonic oscillators. Up to the third order in the coupling coefficient $g$, the transition rates in the detector depend on the second and third cumulant of the fluctuations. As the latter is odd in the driving average current through the noise source, its effect can be measured by detecting the change in the rates when the sign of the driving average current is reversed.

Figure 2

(Color online) Third-order contributions to the excitation (solid lines) and relaxation (dashed lines) rates of a quantum two-level system coupled to a point contact. The rates are plotted for four different types of contacts with equal conductance $G$: tunnel contact (blue circles, $F_2=F_3=1$), dirty interface (Ref. 27) (black squares, $F_2=1∕2$, $F_3=1∕4$), diffusive wire (Ref. 28) (green stars, $F_2=1∕3$, $F_3=1∕15$), and a chaotic cavity (Ref. 29) (red triangles, $F_2=1∕4$, $F_3=0$). The other parameters used in this plot are $\varphi =\pi ∕4$ and $\hslash {\omega }_c=20\Omega$. Inset shows the low-voltage region. The rates are proportional to the dimensionless constant $G{R}_K$ where $R_K=h∕e^2$ is the resistance quantum.

Figure 3

(Color online) Effective temperature of the two-level system as a function of the voltage through different types of point contacts coupled to it. The different curves correspond, from top to bottom, to a tunnel contact (solid blue line), dirty interface (dashed black line), diffusive wire (dashed-dotted green line), and a chaotic cavity (dotted red line). The behavior is mostly dictated by the Fano factor $F_2$. Other parameters in the plot are $g=0.05$, $\hslash {\omega }_c=20\Omega$, $\varphi =\pi ∕4$, $\Lambda =1$, $k_B{T}_e=\Omega$, and $G=1∕R_K$.

Figure 4

(Color online) Effective temperature of the two-level system as a function of the voltage through a diffusive point contact coupled to it via circuits characterized with different ${\omega }_c$. On the right, from top to bottom: $\hslash {\omega }_c∕\Omega =100$ (dashed-dotted green line), 10 (dotted red line), 0.1 (solid blue line), and 1 (dashed black line). Other parameters are as in Fig. 3.

Figure 5

(Color online) Asymmetric part of the temperature $\Delta T_{\mathrm{TLS}}$ as a function of the voltage over the point contact for different types of point contacts. On the right, from top to bottom: chaotic cavity (dotted red line), diffusive wire (dashed-dotted green line), dirty interface (dashed black line), and a tunnel junction (solid blue line). Other parameters are as in Fig. 3.

Figure 6

(Color online) Asymmetric part of the temperature $\Delta T_{\mathrm{TLS}}$ as a function of the voltage over a diffusive point contact with different frequencies characterizing the circuit: on the right, from top to bottom: $\hslash {\omega }_c∕\Omega =100$ (dashed-dotted green line), 10 (dotted red line), 0.1 (solid blue line), and 1 (dashed black line). Other parameters are as in Fig. 3. The curve for $\hslash {\omega }_c=100\Omega$ is in the logarithmic regime, Eq. (21), for all plotted voltages, and it is characterized by the Fano factor $F_2-F_3$. The curves for $\hslash {\omega }_c⩽1$ are in the linear regime, Eq. (22), and the case $\hslash {\omega }_c=10\Omega$ has a crossover between the two. Inset shows the low-voltage regime, reflecting the behavior described in Eq. (14).

Figure 7

Fluctuation measurement schemes using superconducting qubits: (a) persistent current qubit, (b) phase qubit, and (c) charge qubit.

Figure 8

Fluctuation measurements with oscillators whose mass or spring constant is driven with the fluctuations: (a) Cooper-pair box and (b) driven Josephson junction. In both cases, the Josephson energy is tuned with the fluctuations. In (a), the difference to a charge qubit is in the fact that the Cooper-pair box is placed as a part of a resonant circuit.