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 a single well of the tilted-washboard potential, which approximates a cubic potential near the bias point.

When biased near the critical current ($I_{\text{bias}}\lesssim I_c$), the washboard potential develops shallow wells with a finite number of bound states. The two lowest states serve as $|0⟩$ and $|1⟩$, with transition frequency ${\omega }_{01}$ tunable by adjusting $I_{\text{bias}}$. The anharmonicity comes from the cubic shape of the potential near the top of the barrier: higher levels are more closely spaced and eventually become unbound (tunneling into the continuum).

Readout exploits this: the $|1⟩$ state has a much higher tunneling rate out of the well than $|0⟩$, so a brief measurement pulse causes $|1⟩$ to tunnel (producing a voltage pulse across the junction) while $|0⟩$ remains trapped.

The phase qubit was historically important — the Martinis group (UCSB/Google) used it extensively from 2002–2013 — but has been largely superseded by the transmon, which offers superior coherence with simpler operation.

Hamiltonian

Current-biased Josephson junction:

$H=\frac{{\hat{Q}}^2}{2C}-E_J\cos \hat{\phi }-\frac{\hslash I_{\text{bias}}}{2e}\hat{\phi }$

Near the bottom of a well (expanding the tilted cosine to cubic order):

$H\approx \frac{{\hat{p}}^2}{2m}+\frac{1}{2}m{\omega }_p^2{\hat{x}}^2-\frac{m{\omega }_p^2}{6\Delta x}{\hat{x}}^3$

where ${\omega }_p=(2e{I}_c/\hslash C{)}^{1/2}(1-(I_{\text{bias}}/I_c{)}^2{)}^{1/4}$ is the plasma frequency and $\Delta x$ sets the barrier height.

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 demonstration — Martinis et al. (2002):

• Rabi oscillations observed in a large-area current-biased Josephson junction
• Demonstrated coherent control of quantum states in the phase regime
• Tunneling-based readout with ~85% fidelity

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

• Demonstrated entanglement between two capacitively coupled phase qubits
• Bell state fidelity sufficient to violate Bell inequality
• Established viability of multi-qubit superconducting processors

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

• Generated three-qubit GHZ states using phase qubits
• Process tomography demonstrated coherent multi-qubit operations
• Marked the largest entangled state in superconducting circuits at the time

Transition to transmon — post-2013:

• The Martinis group transitioned from phase qubits to transmon-based architectures (Xmon)
• Superior coherence times ($T_1>50\mu \text{s}$ vs. $\sim 1\mu \text{s}$) and non-destructive readout drove the switch

Key Metrics

Metric	Value	Notes	Fidelity reference
$T_1$	0.5–5 μs	Limited by dielectric loss	Martinis et al. 2002
$T_2$	0.1–2 μs	Bias noise dominated	—
Anharmonicity	1–5% of ${\omega }_{01}$	Cubic potential shape	—
Transition frequency	5–10 GHz	Tunable via bias current	—
Readout fidelity	85–96%	Tunneling-based, destructive	Martinis et al. 2002
Operating temperature	10–25 mK	Dilution refrigerator	—

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., “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)

Reviews

• J. Clarke and F. K. Wilhelm, “Superconducting quantum bits,” Nature 453, 1031 (2008)

Linked Papers

• martinis-2002-phase-qubit

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