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Examples of quantum circuits with Lagrangians using one or two types of generalized coordinates. (a) An inductor shunted by two capacitors: the Lagrangian of the circuit contains a single type of variable, the flux across the inductor, such that L(φ, ˙ φ). The charge-flux conjugate pairs are defined at the Lagrangian level. (b) A Josephson junction and a quantum phaseslip junction forming a loop. This quantum circuit cannot be described by a Lagrangian using a single type of variable but only with a Lagrangian that contains both charge and flux variables, L(φ, ˙ φ, q, ˙ q). (c) A more complex circuit of multiple Josephson junctions and quantum phase slips, where writing down a Hamiltonian requires geometrical arguments.
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Circuit quantization is an extraordinarily successful theory that describes the behavior of quantum circuits with high precision. The most widely used approach of circuit quantization relies on introducing a classical Lagrangian whose degrees of freedom are either magnetic fluxes or electric charges in the circuit. By combining nonlinear circuit el...
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Context 1
... a simple example, consider an inductor, L, with two capacitors, C 1 and C 2 , all in parallel [see Fig. 1(a)]. There is one degree of freedom, which can be identified as the flux φ across the inductor. Since ˙ φ is the voltage drop across the capacitor, one writes down a ...
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... q across the element as E ∼ cos(2π q/2e), one can prove that no Lagrangian of the circuit with a single type of variable (flux or charge) exists in general [29]. Here, φ 0 = h/2e is the fundamental flux quantum, h is the Planck constant, and e is the electron charge. Although for the minimal circuit of one JJ and one QPS included in a loop [see Fig. 1(b)], one can immediately write down a Hamiltonian [30] ...
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... involving even a few of these elements can involve nontrivial constraints [see Fig. 1(c)]. For example, the number of degrees of freedom is not equal to the number of JJs or QPSs; worse, the fluxes and charges across the different elements may not be conjugate pairs. Understanding how to even identify the dynamical degrees of freedom, let alone quantize the circuit, is an open problem. Moreover, modeling circuits that ...
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... (for example, branch fluxes), and then the conjugate momenta is found at the Lagrangian level. Any excess in degrees of freedom is dealt with by integrating out nondynamical variables. However, writing down a Lagrangian as a first step is not always possible. A simple example is the circuit that we discussed above, the dualmon qubit [30] shown in Fig. 1(b). The Hamiltonian in ...
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Citations
... Before such a task can be achieved, it is desirable to first develop a thorough understanding of the classical dynamics of electrical circuits interacting with a thermal bath, within a framework that is amenable to quantization. For dissipationless LC circuits, this has recently been achieved in an intriniscally Hamiltonian formulation of circuit mechanics [15,16], building on earlier work [17][18][19]. This Hamiltonian formulation contrasts with the more conventional Lagrangian approach used in much of the superconducting circuit literature [1, 4-6, 12, 20-27]. ...
... We are now ready to write down a locally valid dissipative Lagrangian (Definition 2.4) for general RLC circuits, as defined in the previous subsection. As has been discussed previously [15,16,44], it is possible that there are global singularities which require care to resolve, and we do not focus on resolving all such possible singularities in the present paper. We now introduce a summary of the notation we will use in what follows: ...
... In our framework, which will closely follow [15,43], one motivation for the importance of removing constraints can be more concretely seen as follows. Lagrangians of the form (3.15) are not in the MSR form (2.10). ...
We revisit the theory of dissipative mechanics in RLC circuits, allowing for circuit elements to have nonlinear constitutive relations, and for the circuit to have arbitrary topology. We systematically generalize the dissipationless Hamiltonian mechanics of an LC circuit to account for resistors and incorporate the physical postulate that the resulting RLC circuit thermalizes with its environment at a constant positive temperature. Our theory explains stochastic fluctuations, or Johnson noise, which are mandated by the fluctuation-dissipation theorem. Assuming Gaussian Markovian noise, we obtain exact expressions for multiplicative Johnson noise through nonlinear resistors in circuits with convenient (parasitic) capacitors and/or inductors. With linear resistors, our formalism is describable using a Kubo-Martin-Schwinger-invariant Lagrangian formalism for dissipative thermal systems. Generalizing our technique to quantum circuits could lead to an alternative way to study decoherence in nonlinear superconducting circuits without the Caldeira-Leggett formalism.
... The actual drop has to be determined by the electromagnetic field established in the circuit, which depends on its structure and geometry [81][82][83][84][85] and on the currents' spatial distribution [86][87][88][89][90][91][92]. In addition, the circuit variables bear a different nature depending on the configuration of the circuit, leading to continuous or discrete charge variables [93][94][95][96][97][98][99][100][101][102][103]. In the present work we , the middle region is relevant to describe how electrons move between leads. ...
We explore superconducting quantum circuits where several leads are simultaneously connected beyond the tunneling regime, such that the fermionic structure of Andreev bound states in the resulting multiterminal Josephson junction influences the states of the full circuit. Using a simple model of single-channel contacts and a single level in the middle region, we discuss different circuit configurations where the leads are islands with finite capacitance and/or form loops with finite inductance. We find situations of practical interest where the circuits can be used to define noise-protected qubits, which map to the bifluxon and 0 − π qubits in the tunneling regime. We also point out the subtleties of the gauge choice for a proper description of these quantum circuit dynamics.
Published by the American Physical Society 2024
The quantum mechanics of superconducting circuits is derived by starting from a classical Hamiltonian dynamical system describing a dissipationless circuit, usually made of capacitive and inductive elements. However, standard approaches to circuit quantization treat fluxes and charges, which end up as the canonically conjugate degrees of freedom on phase space, asymmetrically. By combining intuition from topological graph theory with a recent symplectic geometry approach to circuit quantization, we present a theory of circuit quantization that treats charges and fluxes on a manifestly symmetric footing. For planar circuits, known circuit dualities are a natural canonical transformation on the classical phase space. We discuss the extent to which such circuit dualities generalize to nonplanar circuits.
