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arXiv:1110.6311 · Physical Review B

Macroscopic quantum tunneling in quartic and sextic potentials: Application to a phase qubit

Nicolas Didier, F. W. J. Hekking

Published Mar 27, 2012• 10 citations
arXivHTMLPDF

Abstract

Macroscopic quantum tunneling of the phase is a fundamental phenomenon in the quantum dynamics of superconducting nanocircuits. The tunneling rate can be controlled in such circuits, where the potential landscape for the phase can be tuned with different external bias parameters. Precise theoretical knowledge of the macroscopic quantum tunneling rate is required in order to simulate and understand the experiments. We present a derivation, based on the instanton technique, of an analytical expression of the escape rate in general quartic and symmetric sextic potentials comprising two escape paths. These new potentials were recently realized when creating a noise-insensitive phase qubit in the camelback potential of a dc SQUID.

DOI
10.1103/physrevb.85.104522

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The two-Josephson-junction flux qubit with large tunneling amplitude

V. I. Shnyrkov, A. A. Soroka, S. I. Melnyk

In this paper we discuss solid-state nanoelectronic realizations of Josephson flux qubits with large tunneling amplitude between the two macroscopic states. The latter can be controlled via the height and form of the potential barrier, which is determined by quantum-state engineering of the flux qubit circuit. The simplest circuit of the flux qubit is a superconducting loop interrupted by a Josephson nanoscale tunnel junction. The tunneling amplitude between two macroscopically different states can be increased substantially by engineering of the qubit circuit if the tunnel junction is replaced by a ScS contact. However, only Josephson tunnel junctions are particularly suitable for large-scale integration circuits and quantum detectors with present-day technology. To overcome this difficulty we consider here a flux qubit with high energy-level separation between the “ground” and “excited” states, consisting of a superconducting loop with two low-capacitance Josephson tunnel junctions in series. We demonstrate that for real parameters of resonant superposition between the two macroscopic states the tunneling amplitude can reach values greater than 1K. Analytical results for the tunneling amplitude obtained within the semiclassical approximation by the instanton technique show good correlation with a numerical solution.

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Macroscopic quantum tunneling of the phase is a fundamental phenomenon in the quantum dynamics of superconducting nanocircuits. The tunneling rate can be controlled in such circuits, where the potential landscape for the phase can be tuned with different external bias parameters. Precise theoretical knowledge of the macroscopic quantum tunneling rate is required in order to simulate and understand the experiments. We present a derivation, based on the instanton technique, of an analytical expression of the escape rate in general quartic and symmetric sextic potentials comprising two escape paths. These new potentials were recently realized when creating a noise-insensitive phase qubit in the camelback potential of a dc SQUID.
We investigate a quadratic-quartic anharmonic oscillator formed by a potential well between two potential barriers. We realize this novel potential with a dc SQUID at near-zero current bias and flux bias near half a flux quantum. Escape out of the central well can occur via tunneling through either of the two barriers. We find good agreement with a generalized double-path macroscopic quantum tunneling theory. We also demonstrate an "optimal line" in current and flux bias along which the oscillator, which can be operated as a phase qubit, is insensitive to decoherence due to low-frequency current fluctuations.
Quantum tunneling remains unexplored in many regimes of many-body quantum physics, including the effect of quantum phase transitions on tunneling dynamics. In general, the quantum phase is a statement about the ground state and has no relation to far-from-equilibrium dynamics. Although tunneling is a highly dynamical process involving many excited states, we find that the quantum phase of the Bose-Hubbard model determines phase-dependent tunneling outcomes for the quantum tunneling escape, or quasi-bound problem. Superfluid and Mott insulator correlations lead to a new quantum tunneling rate, the quantum fluctuation rate. This rate shows surprising and highly dynamical features, such as oscillatory interference between trapped and escaped atoms and a completely different macroscopic quantum tunneling behavior for superfluid and Mott insulator phases. In the superfluid phase we find that escape dynamics are wave-like and coherent, leading to interference patterns in the density with a rapid decay process which is non-exponential. Quantum entropy production peaks when about half the atoms have escaped. In the Mott phase, despite stronger repulsive interactions, tunneling is significantly slowed by the presence of a Mott gap, creating an effective extra barrier to overcome. Only one atom can tunnel at a time, yet the decay process is nearly linear, completely defying the single-particle exponential model. Moreover, quantum entropy peaks when only about one quarter of the atoms have escaped. These and many other such effects go beyond the usual notions of single-particle quantum tunneling, quantum statistical effects on tunneling, and well-known semi-classical approaches from WKB to instanton theory. These results thus open up a new regime of exploration of far-from-equilibrium dynamics for quantum simulators and quantum dynamics.
Abstract: This study investigates the feasibility of exploiting macroscopic quantum tunnelling (MQT) in superconducting circuits as a constructive resource for fault-tolerant quantum computing. While MQT has historically been treated as a decoherence pathway associated with metastable escape and switching events, recent advances in circuit engineering suggest that controlled access to tunnelling dynamics can be integrated into protected qubit modalities and error-mitigating architectures. The present framework synthesizes quantum escape theory, circuit Hamiltonian models, and contemporary fault-tolerance criteria to propose an experimentally realistic pathway for harnessing MQT-driven transitions as calibrated, monitorable events that can support syndrome extraction and bias-preserving logical operations. A methods blueprint is developed that combines cryogenic device characterization, time-resolved switching statistics, and noise spectroscopy with scalable surface-code-inspired performance projections. Using literature-aligned representative device parameters, the results illustrate how barrier engineering and bias-point optimization can reshape the MQT rate landscape while maintaining coherence metrics compatible with microsecond-scale error-correction cycles. The analysis indicates that decoupling the tunnelling coordinate from dominant low-frequency noise sources and implementing fast, quantum-limited readout may enable an error budget in which MQT transitions become part of a controlled operational envelope rather than a catastrophic failure mode. The study concludes with design rules, limitations, and a roadmap for validating MQT-assisted fault-tolerant primitives in next-generation superconducting quantum processors. Keywords: macroscopic quantum tunnelling; superconducting circuits; fluxonium; fault tolerance; surface code; noise spectroscopy; Josephson junctions; quantum error correction.
FINDING: Macroscopic quantum tunneling (MQT) in superconducting qubits is the 2025 Nobel Prize physics focus; the key mathematical structure is the escape rate from a metastable potential well, governed by an instanton action that scales with the decay length. | MATH: The MQT rate is Γ=Aexp⁡(−Sinst/ℏ) \Gamma = A \exp(-S_{\text{inst}}/\hbar) Γ=Aexp(−Sinst​/ℏ), where Sinst S_{\text{inst}} Sinst​ is the Euclidean action. For a cubic potential V(q)=12mω02q2−λq3 V(q) = \frac{1}{2}m\omega_0^2 q^2 - \lambda q^3 V(q)=21​mω02​q2−λq3, the decay length ℓ=ℏ/2mΔU \ell = \hbar / \sqrt{2m\Delta U} ℓ=ℏ/2mΔU​ (where ΔU \Delta U ΔU is barrier height) enters as Sinst=365ΔUω0 S_{\text{inst}} = \frac{36}{5} \frac{\Delta U}{\omega_0} Sinst​=536​ω0​ΔU​. The qubit's Josephson energy ratio EJ/EC E_J/E_C EJ​/EC​ sets the effective mass; the crossover from quantum to thermal regime occurs at T∗=ℏω0/(2πkB) T^* = \hbar \omega_0 / (2\pi k_B) T∗=ℏω0​/(2πkB​). | CONNECTION: The decay length ratio ℓ/Δqzero-point \ell / \Delta q_{\text{zero-point}} ℓ/Δqzero-point​ (zero-point width Δq0=ℏ/(2mω0) \Delta q_0 = \sqrt{\hbar/(2m\omega_0)} Δq0​=ℏ/(2mω0​)​) yields \( \ell/\Delta q_0 = \sqrt{\Delta U / (\hbar \o Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
In this paper we discuss solid-state nanoelectronic realizations of Josephson flux qubits with large tunneling amplitude between the two macroscopic states. The latter can be controlled via the height and form of the potential barrier, which is determined by quantum-state engineering of the flux qubit circuit. The simplest circuit of the flux qubit is a superconducting loop interrupted by a Josephson nanoscale tunnel junction. The tunneling amplitude between two macroscopically different states can be increased substantially by engineering of the qubit circuit if the tunnel junction is replaced by a ScS contact. However, only Josephson tunnel junctions are particularly suitable for large-scale integration circuits and quantum detectors with present-day technology. To overcome this difficulty we consider here a flux qubit with high energy-level separation between the “ground” and “excited” states, consisting of a superconducting loop with two low-capacitance Josephson tunnel junctions in series. We demonstrate that for real parameters of resonant superposition between the two macroscopic states the tunneling amplitude can reach values greater than 1K. Analytical results for the tunneling amplitude obtained within the semiclassical approximation by the instanton technique show good correlation with a numerical solution.