Phase Qubit

Figure

Description

The phase qubit is a superconducting qubit based on a current-biased Josephson junction operating in the phase regime ($E_J/E_C\gg 1$). The qubit states are the two lowest energy levels in one metastable well of the tilted-washboard potential, which becomes approximately cubic when the bias approaches the junction critical current.

When biased near the critical current ($I_{\text{bias}}\lesssim I_c$), the washboard potential has shallow metastable wells containing only a few quantized levels. The two lowest levels serve as $|0⟩$ and $|1⟩$, with transition frequency ${\omega }_{01}$ tunable by adjusting $I_{\text{bias}}$. The cubic asymmetry makes higher levels more closely spaced and gives excited states exponentially larger escape rates.

Readout exploits this escape-rate contrast. A fast bias pulse lowers the barrier so that the excited-state population escapes with much greater probability than $|0⟩$; the switched state is then amplified into a classical circulating-current or voltage signal. The original 2002 device first drove $|1⟩\to |2⟩$ to enhance escape, whereas later capacitively shunted devices used direct state-selective tunneling.

The phase qubit was historically important—the Martinis group used it extensively from 2002 through the early 2010s—but it was superseded in quantum processors by transmon-family circuits with much longer coherence and nondestructive dispersive readout. Current-biased Josephson junctions remain active as macroscopic quantum systems and threshold detectors, not as a competitive processor-qubit architecture.

Hamiltonian

For a current-biased junction, with $\hat{Q}=2e\hat{n}$, $[\hat{\phi },\hat{n}]=i$, and ${\Phi }_0/2\pi =\hslash /2e$, let $C$ denote the total capacitance seen by the junction phase mode, including the junction and intentional shunt capacitances,

$H=\frac{{\hat{Q}}^2}{2C}-E_J\cos \hat{\phi }-\frac{{\Phi }_0{I}_{\text{bias}}}{2\pi }\hat{\phi }.$

Writing $s=I_{\text{bias}}/I_c$, the local minimum is at ${\phi }_0=\arcsin s$. For $\hat{x}=\hat{\phi }-{\phi }_0$, expansion about that minimum gives

$U(\hat{x})=U(0)+\frac{E_J\cos {\phi }_0}{2}{\hat{x}}^2-\frac{E_J\sin {\phi }_0}{6}{\hat{x}}^3+O({\hat{x}}^4),$

with plasma frequency

${\omega }_p=\sqrt{\frac{2e{I}_c}{\hslash C}}(1-s^2{)}^{1/4}.$

The exact barrier from the local minimum to the adjacent maximum is

$\Delta U=2E_J\left[\sqrt{1-s^2}-s\arccos s\right]\approx \frac{4\sqrt{2}}{3}{E}_J(1-s{)}^{3/2},$

where the approximation holds for $s\to 1^{-}$. In the cubic-well limit the ground-state escape rate has the leading WKB dependence ${\Gamma }_0\propto \exp [-36\Delta U/(5\hslash {\omega }_p)]$; excited levels escape much faster.

Motivation

The phase qubit provided early demonstrations of quantum coherence and entanglement in superconducting circuits. Its straightforward readout mechanism (tunneling → voltage) was simpler than dispersive readout, making it an important stepping stone. However, its sensitivity to current-bias noise and the destructive nature of the tunneling measurement motivated the transition to transmon-based architectures.

Experimental Status

First coherent phase qubit — Martinis et al. (2002):

• Rabi oscillations observed in a large-area current-biased Josephson junction at ${\omega }_{01}/2\pi =6.9$ GHz
• Spectroscopic quality factor $Q\approx 1000$ and an inferred coherence time of about 10 ns in the reported device
• State-selective tunneling readout with 85% excited-state fidelity and greater than 99% ground-state preparation/readout fidelity

Capacitively shunted redesign — Steffen et al. (2006):

• Separated the shunt capacitor from the tunnel junction, reducing strongly coupled two-level defects by roughly an order of magnitude
• Demonstrated single-shot state tomography with about 90% measurement fidelity

Two-qubit entanglement — Steffen et al. (2006):

• Demonstrated entanglement between two capacitively coupled phase qubits
• Reconstructed a Bell state with 87% fidelity after correcting measurement error
• Reported $T_1=130$ ns and $T_2^{\ast }=80$ ns for the coupled devices

Three-qubit entanglement — Neeley et al. (2010):

• Generated both three-qubit W and GHZ states
• State tomography gave $F_W=0.78$ and $F_{\mathrm{GHZ}}=0.62$, both above their respective genuine-three-party-entanglement witness thresholds

Materials endpoint — Patel et al. (2013):

• A phase qubit with a single-crystal silicon shunt capacitor reached $T_1=1.6\mu \text{s}$ at 5.093 GHz
• The same device had $T_2^{\ast }=110$ ns because the architecture has no flux sweet spot

Recent status — 2024–2026:

• No phase-qubit processor result or coherence/gate record superseding the early-2010s endpoint was found
• Ouyang et al. (2024) experimentally revisited the bias-controlled crossover of a current-biased junction from a nearly harmonic oscillator to a nonlinear two-level artificial atom
• Badarne et al. (2025) theoretically proposed a memory-integrated phase qubit based on a superconducting-ferroelectric Josephson junction whose history-dependent barrier modulates the critical current; it is a device concept, not an experimental phase-qubit revival or processor benchmark
• Chai et al. (2025, 2026) developed current-biased Josephson-junction threshold-detector protocols from few-photon sensitivity toward simulated single-photon sensitivity using fast nonadiabatic bias sweeps; these are detector results, not a revival of the phase-qubit processor architecture

Key Metrics

Metric	Value	Notes	Fidelity reference
$T_1$	1.6 μs	Single-crystal-Si shunt; best value in the cited device	Patel et al. 2013
$T_2^{\ast }$	110 ns	No flux sweet spot; low-frequency flux noise dominated	Patel et al. 2013
Anharmonicity	${\omega }_{01}-{\omega }_{12}\approx 0.1{\omega }_{01}$	Representative shallow well with $\Delta U/\hslash {\omega }_p\sim 3$	Martinis et al. 2002
Transition frequency	5.093 GHz	Operating point of the 2013 crystalline-Si device	Patel et al. 2013
Readout fidelity	85% (2002); about 90% (2006)	Destructive, tunneling-based single-shot readout	Martinis et al. 2002; Steffen et al. 2006
Two-qubit Bell-state fidelity	87%	Tomographic fidelity after measurement-error correction	Steffen et al. 2006
Three-qubit state fidelity	$F_W=0.78$; $F_{\mathrm{GHZ}}=0.62$	Both exceeded the relevant entanglement-witness threshold	Neeley et al. 2010
Operating temperature	25 mK (2002); 35 mK (2013)	Dilution refrigerator	Martinis et al. 2002; Patel et al. 2013

References

Original proposal / demonstration

• J. M. Martinis et al., “Rabi Oscillations in a Large Josephson-Junction Qubit,” Phys. Rev. Lett. 89, 117901 (2002)

Experimental demonstrations

• M. Steffen et al., “State Tomography of Capacitively Shunted Phase Qubits with High Fidelity,” Phys. Rev. Lett. 97, 050502 (2006), arXiv:cond-mat/0602432
• M. Steffen et al., “Measurement of the Entanglement of Two Superconducting Qubits via State Tomography,” Science 313, 1423 (2006)
• M. Neeley et al., “Generation of Three-Qubit Entangled States using Superconducting Phase Qubits,” Nature 467, 570 (2010), arXiv:1004.4246
• U. Patel et al., “Coherent Josephson phase qubit with a single crystal silicon capacitor,” Appl. Phys. Lett. 102, 012602 (2013), arXiv:1210.1545

Recent current-biased-junction context

• P. H. Ouyang et al., “Experimental evidence for a current-biased Josephson junction acting as either a macroscopic boson or fermion,” Phys. Rev. Research 6, 013236 (2024)
• M. A. Badarne, E. G. Dalla Torre, and Y. Ivry, “Hybrid superconducting-ferroelectric quantum memristor,” Phys. Rev. Research 7, 043234 (2025)
• Y. Q. Chai et al., “Measuring weak microwave signals via current-biased Josephson junctions: Approaching the quantum limit of energy detection,” Phys. Rev. B 111, 024501 (2025)
• Y. Q. Chai et al., “Measuring weak microwave signals via a current-biased Josephson junction: Suppressing its thermal noise via fast bias current sweeps,” Phys. Rev. B 113, 224503 (2026), arXiv:2510.20570

Reviews

• J. Clarke and F. K. Wilhelm, “Superconducting quantum bits,” Nature 453, 1031 (2008)
• J. M. Martinis, “Superconducting phase qubits,” Quantum Inf. Process. 8, 81 (2009)

Linked Papers

• martinis-2002-phase-qubit
• steffen-2006-state-tomography-phase-qubits
• martinis-2009-superconducting-phase-qubits
• clarke-2008-superconducting-bits
• neeley-2010-generation-three-qubit
• steffen-2006-measurement-entanglement-superconducting
• patel-2013-coherent-phase-qubit-silicon-capacitor
• ouyang-2024-current-biased-josephson-boson-fermion
• badarne-2025-hybrid-superconducting-ferroelectric-memristor
• chai-2025-cbjj-quantum-limit-microwave-detection
• chai-2026-cbjj-single-photon-detection

Evergreen context

• josephson-junction-as-nonlinear-element — the tilted cosine potential and plasma-mode anharmonicity that the current-biased junction turns into a qubit
• coherence-time-hierarchy — how to interpret the short $T_1$ and $T_2$ numbers that ultimately made the phase qubit a transitional architecture
• dispersive-readout-mechanism — the non-destructive resonator readout paradigm that replaced tunneling-based measurement in mainstream superconducting processors

Related Entries

• transmon — successor architecture with superior coherence
• flux-qubit — alternative superconducting qubit in the phase regime
• cooper-pair-box-charge-qubit — charge-regime predecessor