Interacting Electrodynamics of Short Coherent Conductors in Quantum Circuits

When combining lumped mesoscopic electronic components to form a circuit, quantum fluctuations of electrical quantities lead to a nonlinear electromagnetic interaction between the components, which is generally not understood. The Landauer-Büttiker formalism that is frequently used to describe noninteracting coherent mesoscopic components is not directly suited to describe such circuits since it assumes perfect voltage bias, i.e., the absence of fluctuations. Here, we show that for short coherent conductors of arbitrary transmission, the Landauer-Büttiker formalism can be extended to take into account quantum voltage fluctuations similarly to what is done for tunnel junctions. The electrodynamics of the whole circuit is then formally worked out disregarding the non-Gaussianity of fluctuations. This reveals how the aforementioned nonlinear interaction operates in short coherent conductors: Voltage fluctuations induce a reduction of conductance through the phenomenon of dynamical Coulomb blockade, but they also modify their internal density of states, leading to an additional electrostatic modification of the transmission. Using this approach, we can quantitatively account for conductance measurements performed on quantum point contacts in series with impedances of the order of $R_K=h/e^2$. Our work should enable a better engineering of quantum circuits with targeted properties.

Popular Summary

Electronic circuits exploiting the laws of quantum mechanics have emerged in recent years as a promising platform for performing tasks that will forever remain beyond the reach of classical circuits. It is known that individual quantum electronic components should be regarded as scatterers of electrons. However, when combined in simple circuits with other components (e.g., even a single commonplace resistor), the classical impedance combinations rules break down because quantum components interact in a nonlocal and nonlinear way. Here, we develop replacement rules that apply to short coherent conductors.

We first extend the Landauer-Büttiker formalism in order to take into account quantum voltage fluctuations arising from electromagnetic degrees of freedom. We accordingly provide a consistent description of a quantum component and its surrounding circuit that fulfills, at the minimum level, Kirchhoff’s circuit laws in the presence of quantum fluctuations. Our approach exposes a key mechanism of interaction in quantum circuits: Voltage fluctuations modify the transport properties through a nonuniversal electrostatic effect. Our theory accounts well for existing experimental results displaying strong interactions, for which no fully predictive theory exists.

By providing a tractable way of quantitatively predicting the electrodynamics of quite general quantum circuits, we expect that our findings will pave the way for improved engineering of their properties.

Figure 1

(a) Schematics of a hypothetical quantum circuit where electrical quantities are operators with quantum fluctuations. We single out a mesoscopic two-terminal electronic component $Q_1$ in the circuit. The Landauer-Büttiker formalism is extended to describe transport in the component, taking into account voltage fluctuations. This electron scattering approach predicts that the component generates current fluctuations, which the whole circuit transforms into matching voltage fluctuations. (b) The energy dependence of the scattering matrix implies that the component has an implicit internal node with an associated degree of freedom (${\widehat{V}}_3$) whose dynamics needs to be taken into account. The small ac-signal electrodynamics of the component is described by an admittance matrix, which depends not only on the scattering matrix of $Q_1$ but also on the voltage fluctuations across $Q_1$ that are determined by the whole circuit. The drawing shows a possible lumped-element equivalent circuit of the admittance matrix [1]. (c) Disregarding non-Gaussianity, current and voltage fluctuations at a given bias point are related simply by the small ac-signal (locally linear) electrodynamics of the full circuit. As seen from $Q_1$, the rest of the circuit is also described by an admittance matrix $Y_{\mathrm{ext}}$ that includes, in particular, all geometrical capacitances. The voltage fluctuations ${\tilde{V}}_1$, ${\tilde{V}}_2$, ${\tilde{V}}_3$ across $Q_1$ result from the noise sources in the whole circuit (drawn here as current sources), all treated on the same footing. Depending on the circuit details, the lumped elements depicted here may depend on the dc bias condition, resulting in a globally nonlinear response of the system.

Figure 2

(a) QPC in series with an $RC$ circuit. The circuit can be biased by a dc voltage source, and an additional small ac excitation enables measuring its differential conductance using a lock-in technique. In the experimental implementation, the $RC$ circuit could be short-circuited on chip in order to measure the effect of its presence. When the switch is closed, the QPC is voltage biased: Voltage fluctuations are suppressed, and the usual LB description applies. (b) Lumped-element model of the QPC. Assuming the QPC is symmetric with respect to exchange of nodes 1 and 2, the lumped-element model of Fig. 1 has only two independent elements and can be represented as shown here, where $C_3(\omega )=Y_3(\omega )/i\omega$ is the quantum capacitance of the conductor and $Y_0(\omega )$ its two-point admittance. In this panel, we also symbolically represent the internal node of the device and, in grey, the geometrical capacitances that may affect it and which are not (and cannot be) included in the admittance matrix obtained from the electronic scattering matrix. (c) Small-signal equivalent of the full circuit used to determine voltage fluctuations across the QPC. In the simple case we consider here, the general analysis [see Eq. (12) and Fig. 1] reduces to a simple parallel combination of two-terminal components. $Y_3$ is greyed out because it has a negligible contribution in the fluctuations across the QPC.

Figure 3

Left panel: Relative DCB reduction of conductance of a QPC as a function of a reference transmission ${\mathcal{T}}_{\text{ref}}$ of the QPC obtained using the Landauer formula in a situation where DCB is suppressed. Symbols are data from Fig. 3 of Ref. [19], and lines are adjustments using our theory. The curves are obtained combining the two right graphs. Top right panel: Relative reduction of conductance due to generalized the dynamical Coulomb blockade for energy-independent transmission given by Eqs. (24) and (27) and the parameters of Table I. Because of the self-consistency of the phase fluctuations, these curves very slightly deviate from the straight line (shown as dashes). These straight lines would generalize the $(1-{\mathcal{T}}_0)$ Fano factor reduction of the tunnel limit of DCB previously known to be valid only in the case of weak reduction of conductance. Bottom right panel: Modified transmission of the QPC due to a change of the quantum capacitance, calculated using the parameters of Table I. In this panel and the left panel, the transmission ${\mathcal{T}}_{\text{ref}}$ on the horizontal axis represents either the bare channel transmission ${\mathcal{T}}_0$ (top curve) or the “high voltage” transmission ${\mathcal{T}}_{\infty }$ (bottom curves, see text).