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Introduction to Quantum Electromagnetic Circuits

Uri Vool, Michel H. Devoret

arXiv:1610.03438v2quant-phcond-mat.supr-con

TL;DR

The review addresses how to describe and quantize electromagnetic circuits, including dissipation and nonlinear Josephson elements, while connecting circuit variables to macroscopic quantum phenomena. It develops Hamiltonian and open-system treatments, surveys superconducting circuit building blocks and artificial atoms, and states the supported scope and exclusions of this overview.

  • Problem

    Understanding quantum circuits requires connecting classical circuit descriptions, Hamiltonian quantization, dissipation, and Josephson nonlinearity within a common framework.

  • Method

    The review presents systematic Hamiltonian construction, quantum treatment of linear dissipative circuits, quantum fluctuation-dissipation concepts, and Josephson-based superconducting circuit elements.

  • Results

    The review establishes a link between quantum electrodynamics and Josephson circuits through collective electrical variables, linear circuit media, and Josephson nonlinearity.

  • Takeaways & Limitations

    The review provides basic concepts for understanding the differing viewpoints used in specialized treatments of quantum effects in tunnel-junction circuits.

  • Takeaways & Limitations

    The review is intentionally non-comprehensive, with driven-dissipative circuits and several other topics excluded from its scope.

Abstract

from arXiv · show

The article is a short opinionated review of the quantum treatment of electromagnetic circuits, with no pretension to exhaustiveness. This review, which is an updated and modernized version of a previous set of Les Houches School lecture notes, has 3 main parts. The first part describes how to construct a Hamiltonian for a general circuit, which can include dissipative elements. The second part describes the quantization of the circuit, with an emphasis on the quantum treatment of dissipation. The final part focuses on the Josephson non-linear element and the main linear building blocks from which superconducting circuits are assembled. It also includes a brief review of the main types of superconducting artificial atoms, elementary multi-level quantum systems made from basic circuit elements.

1 What are quantum electromagnetic circuits?

Quantum electromagnetic circuits are macroscopic, designable systems whose collective electrical variables can behave quantum mechanically. Their circuit description uses conjugate flux and charge variables, while environmental coupling, dissipation, and Josephson nonlinearity determine which quantum effects are observable.

  • Macroscopic quantum mechanics: Mesoscopic circuits contain many atoms but possess collective degrees of freedom whose quantum behavior is tunable through design parameters.Their artificial construction distinguishes them from microscopic quantum particles and macroscopic classical objects.
  • Macroscopic quantum mechanics: An isolated LC oscillator is a lumped one-degree-of-freedom system described by conjugate flux Φ and capacitor charge Q.Typical values L = 1 nH and C = 10 pF give a resonant frequency of approximately 1.6 GHz.
  • Macroscopic quantum mechanics: The oscillator’s electromagnetic environment can be modeled by a frequency-dependent admittance Y(ω) connected in parallel, with coupling limiting the quanticity of Φ.A probing circuit with |Y(ω)|^-1 around 100 Ω corresponds to quality factor Q = 10 in the stated example.
  • Macroscopic quantum mechanics: A purely harmonic LC circuit displays only relatively simple quantum effects, with experimentally difficult signatures residing in temperature-dependent variances and higher moments.Directly observable non-trivial macroscopic quantum effects require at least one nonlinear component, for which Josephson junctions are the principal circuit element discussed.
  • Macroscopic quantum mechanics: A Josephson tunnel junction combines a nonlinear superconducting element with a parallel capacitance and is characterized by charge and tunneled Cooper-pair number degrees of freedom.The junction’s nonlinearity is associated with discrete charge tunneling, and its macroscopic parameter I0 depends on junction area and barrier transparency.
  • Macroscopic quantum mechanics: Josephson-junction fluctuations directly affect the circuit’s RF response, while dissipation combined with nonlinearity can produce localization above a resistance threshold of approximately 6.4 kΩ.This contrasts with the externally decoupled quantum fluctuations of a linear LC oscillator.

2.1 Non-dissipative circuits

The review builds Hamiltonians for non-dissipative circuits by reducing branch variables to node-based degrees of freedom, with fluxes as coordinates and charges as conjugate momenta. Capacitive and inductive energies then form kinetic and potential terms, while nonlinear inductive elements such as Josephson junctions fit the same framework under stated assumptions.

  • Circuit definitions: Circuits are networks of two-pole branches meeting at nodes, with loops formed by multiple paths between nodes.
  • Dynamical variables of the circuit: Branch voltage and current require an orientation and sign convention, while lumped-element assumptions make their field-based definitions largely path-independent.
  • Dynamical variables of the circuit: Branch fluxes and charges provide the variables needed for a Hamiltonian description, with static magnetic biases introduced adiabatically from an initially resting circuit.
  • Capacitive and inductive elements: Capacitive elements depend on charge and inductive elements on flux; linear capacitances and inductances have quadratic stored energies, while Josephson junctions provide nonlinear inductive behavior.
  • Finding the Hamiltonian of a circuit: The node construction requires a linear capacitive sub-network for nonlinear circuits; maximally nonlinear inductive and capacitive subnetworks remain an ongoing research problem.
  • Method of nodes: The node method eliminates superfluous variables by separating capacitive and inductive subnetworks, leaving at most N−1 true degrees of freedom associated with active nodes other than ground.
  • Finding the Hamiltonian of a circuit: In node variables, capacitive energy is kinetic and inductive energy is potential, producing a Hamiltonian whose first term depends on node charges and second term on node fluxes.

2.2 Circuits with linear dissipative elements

The review incorporates linear dissipation into Hamiltonian circuit theory through the Caldeira–Leggett model, replacing an admittance with infinitely many LC oscillators. This construction recovers irreversible behavior on physical timescales and supports derivations of classical and quantum fluctuation–dissipation relations.

  • The Caldeira-Leggett model: Hamiltonian dynamics can represent linear dissipation by extending the system with additional degrees of freedom rather than treating irreversible behavior directly.
  • The Caldeira-Leggett model: The Caldeira–Leggett model replaces a dissipative admittance Y(ω) with infinitely many series LC oscillators wired in parallel, with oscillator distributions determined by Y(ω).
  • The Caldeira-Leggett model: The infinite oscillator set reconciles formal time reversibility of Hamilton’s equations with irreversible behavior on physical timescales.
  • The Caldeira-Leggett model: An impedance can be represented by an infinite set of parallel LC circuits connected in series, providing the dual construction for Z(ω).
  • The Caldeira-Leggett model: The model determines the influence of a dissipative element on circuit collective variables, including through gauge-form coupling in the node representation.
  • Fluctuation-dissipation theorem: Thermal current and voltage fluctuation spectra can be derived by summing the independent oscillator correlations and expressed through admittance or impedance functions.

3.1 Non-dissipative quantum circuits

Quantum circuit quantization replaces classical circuit variables and Hamiltonians with operators while preserving the circuit’s correct conjugate degrees of freedom. For the LC oscillator, this yields temperature-dependent quantum fluctuations, including zero-point fluctuations and non-real correlation functions.

  • Quantization replaces classical variables with corresponding operators and the Hamiltonian function with an operator-valued function.
  • Correct quantization requires identifying true conjugate circuit degrees of freedom, because branch flux and charge operators are not generally conjugate Hamiltonian variables.
  • The LC oscillator is quantized using flux and capacitor charge as its canonical variables, with standard annihilation and creation operators describing its states.
  • The flux variance interpolates between temperature-dependent thermal fluctuations and the zero-point-fluctuation limit at low temperature.
  • Quantum fluctuations produce a non-real correlation function, so its Fourier transform cannot generally be interpreted as a classical measurable spectral density.

3.2.1 The quantum fluctuation-dissipation theorem

The quantum fluctuation-dissipation theorem derives circuit fluctuations from the dissipative part of a generalized impedance. Unlike classical spectra, quantum spectral densities distinguish positive and negative frequencies, which encode opposite energy-transfer processes.

  • The Caldeira-Leggett representation combines oscillator contributions to obtain branch-flux correlations for an arbitrary generalized impedance.
  • The quantum fluctuation-dissipation theorem relates the fluctuation spectrum to Re(Z[ω]) and the thermal factor coth(βℏω/2).
  • Quantum spectral densities are asymmetric in frequency, with Sφφ[−ω] ≠ Sφφ[ω].
  • In the voltage spectrum, the low-frequency limit gives 2kBT Re(Z[ω]), while the positive high-frequency limit gives 2ℏω Re(Z[ω]) and the negative limit vanishes.
  • For an Ohmic resistor, damping mainly rescales LC fluctuations, while quantum degrees of freedom inside the resistor make the uncertainty ellipse grow logarithmically with damping.

3.2.2 Input-Output theory

Input-output theory models dissipation as scattering between a circuit and traveling quantum fields on a transmission line. It places external driving and damping in one framework and connects field quantization to Langevin dynamics.

  • 3.2.2 Input-Output theory: Input-output theory treats the circuit as an elastic scatterer of environmental signals, putting external drive and dissipation on the same footing.
  • 3.2.2.1 Infinite transmission line: A resistance is modeled by a semi-infinite transmission line whose characteristic impedance equals the resistance, with incoming and outgoing wave amplitudes linked at the terminal.
  • 3.2.2.1 Infinite transmission line: Left- and right-moving wave amplitudes provide a common description of voltage, current, and power flow along the line.
  • 3.2.2.1 Infinite transmission line: Quantizing the transmission-line fields replaces Poisson brackets with commutators and introduces annihilation operators whose thermal occupations follow the Bose-Einstein expression.
  • 3.2.2.2 Semi-infinite transmission line: The resulting boundary conditions preserve the output fields’ commutation relations and lead to a quantum Langevin equation for the damped circuit.

3.3 Measurement operators

Measurement operators describe how traveling electromagnetic signals reveal different aspects of a quantum circuit. The stochastic master equation represents the resulting conditioned state evolution, including dynamics, dissipation, and measurement back-action.

  • The measurement framework replaces the environmental transmission line with a finite line terminated by an absorptive detector.
  • Homodyne measurement analyzes one phase-space quadrature, while heterodyne measurement analyzes two orthogonal components with added noise.
  • Photon measurement analyzes excitation number using Fock states indexed by photon number.
  • The stochastic master equation evolves the density conditioned on the sequence of measurement outcomes, or measurement record.
  • Its three parts are Hamiltonian evolution, Lindblad dissipative evolution, and stochastic measurement back-action.

4 Superconducting Artificial Atoms

The Josephson element is developed from Cooper-pair tunneling and represented equivalently in charge- and phase-based descriptions. Combining junctions in loops or arrays produces tunable nonlinear circuits and effective superinductances.

  • The Josephson element: The Josephson junction is modeled quantum mechanically through Cooper-pair tunneling between electrode states labeled by transferred-pair number N.The junction’s electrode difference N is the tunneling degree of freedom, while its quantum operator has macroscopic charge-transfer eigenstates.
  • The Josephson element: The charge and phase descriptions of the Josephson energy are two representations of the same physical Hamiltonian.Identifying the reduced phase modulo 2π with the unit-circle coordinate establishes the equivalence, while distinguishing electromagnetic flux from angular phase.
  • Loops and parallel junctions: External flux frustrates loop circuits by making the reduced branch flux generally nonzero and rendering the sum of offset fluxes around a loop observable.Zero frustration occurs when the applied flux equals the sum of branch offset fluxes.
  • Loops and parallel junctions: A DC SQUID behaves like a single junction with a Josephson energy tunable by external flux.The two junction phases are constrained by the loop relation, allowing the total energy to be rewritten as one cosine.
  • Junction arrays: A series array of M junctions realizes an effective linear inductance ML_J and enables superinductances unavailable from ordinary geometric inductors.Below the array self-resonance, these impedances can exceed the resistance quantum ℏ/4e^2.

4.2 Electromagnetic quantum circuit families

Superconducting artificial atoms occupy regimes set mainly by E_J/E_C and inductive energy, with circuit design trading sensitivity among charge noise, flux noise, nonlinearity, and readout performance. The review connects these regimes to representative qubits including the Cooper pair box, transmon, flux qubit, phase qubit, and fluxonium.

  • Circuit families: E_J/E_C acts as a mass parameter: low values favor charge as a good quantum number, whereas high values favor phase and a potential expansion.The second ratio, (E_J−E_L)/E_L, approximately counts phase-potential wells minus one at half a flux quantum.
  • Noise and circuit design: A gauge transformation converts charge noise into offset-flux noise, adding a term dependent on the time derivative of the charge noise.The transformed description makes the circuit sensitive to current rather than charge fluctuations.
  • Noise and circuit design: Current-noise sensitivity suppresses low-frequency charge noise by ω^2, but this suppression is weighted by E_L^2 and charge noise can dominate as the shunting inductance increases.When E_L=0, the circuit is insensitive to flux noise but remains sensitive to charge noise.
  • Noise and circuit design: Reducing E_C suppresses charge-noise sensitivity exponentially while reducing nonlinearity only linearly.This trade-off motivates large-capacitance designs in the high-E_J/E_C regime.
  • Representative artificial atoms: The transmon lowers electrostatic energy by adding a large shunt capacitance, screening offset-charge drift and removing its effect on the transition frequency.The resulting charge-noise insensitivity leads to higher coherence times.
  • Representative artificial atoms: At half a flux quantum, the flux qubit has two degenerate current states whose degeneracy is lifted by charging energy, while its transition depends sensitively on external flux and E_J.Moving slightly away from the optimal flux point significantly reduces coherence time.
  • Representative artificial atoms: A phase qubit uses a metastable potential well and DC-SQUID flux readout, achieving high readout signal-to-noise through macroscopic tunneling.Its circuit combines a large Josephson junction with a geometric inductance.
  • Representative artificial atoms: Fluxonium combines a small junction with a large inductive array, suppressing DC offset-noise components while requiring high characteristic impedance to avoid parasitic oscillator modes.A geometric inductance cannot meet the required impedance condition, motivating a Josephson-junction array implementation.

5 Conclusions and Perspectives

The review connects quantum electrodynamics with Josephson circuits through collective electromagnetic variables, treats dissipation quantum mechanically, and surveys circuit architectures and decoherence mechanisms. It also identifies Bloch oscillations and topological protection as topics outside its scope.

  • Conclusions: Josephson circuits provide a mesoscopic realization of quantum electrodynamics using collective electronic variables rather than microscopic particles.Linear circuit elements act as the electromagnetic medium, while the Josephson junction supplies a nonlinear interaction.
  • Conclusions: The review incorporates dissipative elements into circuit quantization and examines how noise contributes to qubit decoherence.Its examples include basic superconducting circuits and the noise mechanisms affecting their performance.
  • Perspectives: Bloch oscillations at the metrological level and proposals for topological protection from decoherence are explicitly left outside the review’s scope.These topics involve circuits in which flux and charge roles are interchanged relative to commonly used circuits such as the transmon.
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