Circuit quantization links a physical circuit to its corresponding quantum Hamiltonian. The standard quantization procedure generally assumes any external magnetic flux to be static. Time dependence naturally arises, however, when flux is modulated or when flux noise is considered. In this case, application of the existing quantization procedure can lead to inconsistencies. To resolve these, we generalize circuit quantization to incorporate time-dependent external flux.
Quantum circuit theory has emerged as an essential tool for the study of the dynamics of superconducting circuits. Recently, the problem of accounting for time-dependent driving via external magnetic fields was addressed by Riwar-DiVincenzo in their paper - 'Circuit quantization with time-dependent magnetic fields for realistic geometries' in which they proposed a technique to construct a low-energy Hamiltonian for a given circuit geometry, taking as input the external magnetic field interacting with the geometry. This result generalises previous efforts that dealt only with discrete circuits.…
The development of new superconducting circuits and the improvement of existing ones rely on the accurate modeling of spectral properties which are key to achieving the needed advances in qubit performance. Systematic circuit analysis at the lumped-element level, starting from a circuit network and culminating in a Hamiltonian appropriately describing the quantum properties of the circuit, is a well-established procedure, yet cumbersome to carry out manually for larger circuits. We present work utilizing symbolic computer algebra and numerical diagonalization routines versatile enough to tackl…
We propose a new quantization method for superconducting electronic circuits involving a Josephson junction device coupled to a linear microwave environment. The method is based on an exact impedance synthesis of the microwave environment considered as a blackbox with impedance function Z(s). The synthesized circuit captures dissipative dynamics of the system with resistors coupled to the reactive part of the circuit in a non-trivial way. We quantize the circuit and compute relaxation rates following previous formalisms for lumped element circuit quantization. Up to the errors in the fit our m…
We introduce CircuitQ, an open-source toolbox for the analysis of superconducting circuits implemented in Python. It features the automated construction of a symbolic Hamiltonian of the input circuit and a dynamic numerical representation of the Hamiltonian with a variable basis choice. Additional features include the estimation of the T1 lifetimes of the circuit states under various noise mechanisms. We review previously established circuit quantization methods and formulate them in a way that facilitates the software implementation. The toolbox is then showcased by applying it to practically…
We present a theoretical description for circuits consisting of weak anharmonic qubits coupled to cavity multimodes. We obtain a unitary transformation that diagonalizes harmonic sector of the circuit. Weak anharmonicity does not alter the normal mode basis, however it can modify energy levels. We study two examples of a transmon and two transmons coupled to bus resonator, and we determine dressed frequencies and Kerr nonlinearities in closed form formulas. Our results are valid for arbitrary frequency detuning and coupling within and beyond dispersive regime.
We present a method for calculating the energy levels of superconducting circuits that contain highly anharmonic, inductively-shunted modes with arbitrarily strong coupling. Our method starts by calculating the normal modes of the linearized circuit and proceeds with numerical diagonalization in this basis. As an example, we analyze the Hamiltonian of a fluxonium qubit inductively coupled to a readout resonator. While elementary, this simple example is nontrivial because it cannot be efficiently treated by the method known as "black-box quantization," numerical diagonalization in the bare harm…
We present a semi-classical method for determining the effective low-energy quantum Hamiltonian of weakly anharmonic superconducting circuits containing mesoscopic Josephson junctions coupled to electromagnetic environments made of an arbitrary combination of distributed and lumped elements. A convenient basis, capturing the multi-mode physics, is given by the quantized eigenmodes of the linearized circuit and is fully determined by a classical linear response function. The method is used to calculate numerically the low-energy spectrum of a 3D-transmon system, and quantitative agreement with …
By placing an atom into a cavity, the electromagnetic mode structure of the cavity is modified. In Cavity QED, one manifestation of this phenomenon is the appearance of a gauge-dependent diamagnetic term, known as the $A^2$ contribution. Although in atomic Cavity QED, the resulting modification in the eigenmodes is negligible, in recent superconducting circuit realizations, such corrections can be observable and may have qualitative implications. We revisit the canonical quantization procedure of a circuit QED system consisting of a single superconducting transmon qubit coupled to a multimode …
Improving the accuracy of circuit quantization using the electromagnetic properties of superconductors
Seoul National University（首尔大学）；Korea Research Institute of Standards and Science（韩国标准科学研究院）；Quantum Devices Labs, Korea Advanced Nano Fab Center（韩国先进纳米制造中心量子器件实验室）；Quantum Design Korea（量子设计韩国公司）；SKKU Adavanced Institute of Nanotechnology, Sungkyunkwan University（成均馆大学SKKU纳米技术高级研究所）
Alternating current RLC electric circuits form an accessible and highly tunable platform simulating Hermitian as well as non-Hermitian (nH) quantum systems. We propose here a circuit realization of nH Dirac and Weyl Hamiltonians subject to time-reversal invariant pseudo-magnetic field, enabling the exploration of novel nH physics. We identify the low-energy physics with a generic real energy spectrum from the nH Landau quantization of exceptional points and rings, which can avoid the nH skin effect and provides a physical example of a quasiparticle moving in the complex plane. Realistic detect…
Nonreciprocal devices effectively mimic the breaking of time-reversal symmetry for the subspace of dynamical variables that they couple, and can be used to create chiral information processing networks. We study the systematic inclusion of ideal gyrators and circulators into Lagrangian and Hamiltonian descriptions of lumped-element electrical networks. The proposed theory is of wide applicability in general nonreciprocal networks on the quantum regime. We apply it to pedagogical and pathological examples of circuits containing Josephson junctions and ideal nonreciprocal elements described by a…
In the context of quantum information theory, "quantization" of various mathematical and computational constructions is said to occur upon the replacement, at various points in the construction, of the classical randomization notion of probability distribution with higher order randomization notions from quantum mechanics such as quantum superposition with measurement. For this to be done "properly", a faithful copy of the original construction is required to exist within the new "quantum" one, just as is required when a function is extended to a larger domain. Here procedures for extending hi…
The electromagnetic field inside a cubic cavity filled up with a linear magnetodielectric medium and in the presence of external charges is quantized by modelling the magnetodielectric medium with two independent quantum fields. Electric and magnetic polarization densities of the medium are defined in terms of the ladder operators of the medium and eigenmodes of the cavity. Maxwell and constitutive equations of the medium together with the equation of motion of the charged particles have been obtained from the Heisenberg equations using a minimal coupling scheme. Spontaneous emission of a two …
Field quantization in three dimensional unstable optical systems is treated by expanding the vector potential in terms of non-Hermitean (Fox-Li) modes in both the cavity and external regions. The cavity non-Hermitean modes (NHM) are treated using the paraxial and monochromaticity approximations. The NHM bi-orthogonality relationships are used in a standard canonical quantization procedure based on introducing generalised coordinates and momenta for the electromagnetic (EM) field. The quantum EM field is equivalent to a set of quantum harmonic oscillators (QHO), associated with either the cavit…
Superconducting microwave circuits incorporating nonlinear devices, such as Josephson junctions, are one of the leading platforms for emerging quantum technologies. Increasing circuit complexity further requires efficient methods for the calculation and optimization of the spectrum, nonlinear interactions, and dissipation in multi-mode distributed quantum circuits. Here, we present a method based on the energy-participation ratio (EPR) of a dissipative or nonlinear element in an electromagnetic mode. The EPR, a number between zero and one, quantifies how much of the energy of a mode is stored …
The conventional approach to circuit quantization is based on node fluxes and traces the motion of node charges on the islands of the circuit. However, for some devices, the relevant physics can be best described by the motion of polarization charges over the branches of the circuit that are in general related to the node charges in a highly nonlocal way. Here, we present a method, dual to the conventional approach, for quantizing planar circuits in terms of loop charges. In this way, the polarization charges are directly obtained as the differences of the two loop charges on the neighboring l…
For a particle moving in a one-dimensional space an under a periodic external force, its quantization is study using the Hamiltonian (generalized linear momentum quantization) and constant of motion (velocity quantization) approaches. it is shown a great difference on the quantization of both approaches and the ambiguities arisen by using the quantization on the constants of motion.
Quantum-dot cellular automata (QCA) may provide low-power, general-purpose computing in the post-CMOS era. A molecular implementation of QCA features nanometer-scale devices and may support THz switching speeds at room-temperature. Here, we explore the ability of molecular QCA circuits to tolerate unwanted applied electric fields, which may come from a variety of sources. One likely source of strong unwanted electric fields may be electrodes recently proposed for the write-in of classical bits to molecular QCA input circuits. Previous models have shown that the input circuits are sensitive to …
We derive a standard quantum limit for probing mechanical energy quantization in a class of systems with mechanical modes parametrically coupled to external degrees of freedom. To resolve a single mechanical quantum, it requires a strong-coupling regime -- the decay rate of external degrees of freedom is smaller than the parametric coupling rate. In the case for cavity-assisted optomechanical systems, e.g. the one proposed by Thompson et al., zero-point motion of the mechanical oscillator needs to be comparable to linear dynamical range of the optical system which is characterized by the optic…
Immittance formulas for exact blackbox quantization and divergence-free effective models in circuit QED
Technical University of Munich（慕尼黑工业大学）；TUM School of Natural Sciences（TUM自然科学学院）；Physics Department（物理系）；Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften（瓦尔特·迈斯纳研究所，巴伐利亚科学院）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）
The quantization of a constant of motion for the harmonic oscillator with a time-explicitly depending external force is carried out. This quantization approach is compared with the normal Hamiltonian quantization approach. Numerical results show that there are qualitative and quantitative differences for both approaches, suggesting that the quantization of this constant of motion may be verified experimentally.
In the quest to produce quantum technology, superconducting networks, working at temperatures just above absolute zero, have arisen as one of the most promising physical implementations. The precise analysis and synthesis of such circuits have required merging the fields of physics, engineering, and mathematics. In this dissertation, we develop mathematically consistent and precise Hamiltonian models to describe ideal superconducting networks made of an arbitrary number of lumped elements, such as capacitors, inductors, Josephson and phase-slip junctions, gyrators, etc., and distributed ones l…
Based on a circuit QED qubit-cavity array a source of two-mode entangled microwave radiation is designed. Our scheme is rooted in the combination of external driving, collective phenomena and dissipation. On top of that the reflexion symmetry is broken via external driving permitting the appearance of chiral emission. Our findings go beyond the applications and are relevant for fundamental physics, since we show how to implement quantum lattice models exhibiting criticality driven by dissipation.
Quantum integrated photonics requires large-scale linear optical circuitry, and for many applications it is desirable to have a universally programmable circuit, able to implement an arbitrary unitary transformation on a number of modes. This has been achieved using the Reck scheme, consisting of a network of Mach Zehnder interferometers containing a variable phase shifter in one path, as well as an external phase shifter after each Mach Zehnder. It subsequently became apparent that with symmetric Mach Zehnders containing a phase shift in both paths, the external phase shifts are redundant, re…
With the increase of complexity and coherence of superconducting systems made using the principles of circuit quantum electrodynamics, more accurate methods are needed for the characterization, analysis and optimization of these quantum processors. Here we introduce a new method of modelling that can be applied to superconducting structures involving multiple Josephson junctions, high-Q superconducting cavities, external ports, and voltage sources. Our technique, an extension of our previous work on single-port structures [1], permits the derivation of system Hamiltonians that are capable of r…
We consider generation of an electrical pulse by an optical pulse in the ``virtual excitation'' regime. The electronic system, which is any electro-optic material including a quantum well structure biased by a dc electric field, is assumed to be coupled to an external circuit. It is found that the photon frequency is subject to an extra red shift in addition to the usual self-phase modulation, whereas the photon number is conserved. The Joule energy consumed in the external circuit is supplied only from the extra red shift.
Quantum computers that process information by harnessing the remarkable power of quantum mechanics are increasingly being put to practical use. In the future, their impact will be felt in numerous fields, including in online casino games. This is one of the reasons why quantum gambling theory has garnered considerable attention. Studies have shown that the quantum gambling theory often yields nontrivial consequences that classical theory cannot interpret. We devised a quantum circuit reproducing classical blackjack and found possible quantum entanglement between strategies. This circuit can be…
Quantum circuit complexity has played a central role in recent advances in holography and many-body physics. Within quantum field theory, it has typically been studied in a Lorentzian (real-time) framework. In a departure from standard treatments, we aim to quantify the complexity of the Euclidean path integral. In this setting, there is no clear separation between space and time, and the notion of unitary evolution on a fixed Hilbert space no longer applies. As a proof of concept, we argue that the pants decomposition provides a natural notion of circuit complexity within the category of 2-di…
We study the reparametrization invariant system of a classical relativistic particle moving in (5+1) dimensions, of which two internal ones are compactified to form a torus. A discrete physical time is constructed based on a quasi-local invariant observable. Due to ergodicity, it is simply related to the proper time on average. The external motion in Minkowski space can then be described as a unitary quantum mechanical evolution.
A general problem of quantum field theories is the fact that the free vacuum and the vacuum for an interacting theory belong to different, non-equivalent representations of the canonical (anti-)commutation relations. In the external field problem of QED, we encounter this problem in the form that the Dirac time evolution for an external field with non-vanishing magnetic components will not satisfy the Shale-Stinespring condition, known to be necessary and sufficient for the existence of an implementation on the fermionic Fock space. Therefore, a second quantization of the time evolution in the…
