How does charging energy determine whether a qubit is a charge qubit or a transmon?

The ratio E_J/E_C defines the qubit operating regime. When E_C is large relative to E_J (E_J/E_C near 1), the qubit is a charge qubit with strong charge dispersion and large anharmonicity. When E_C is reduced by adding a large shunt capacitor (E_J/E_C of 50 to 100), the qubit becomes a transmon with exponentially suppressed charge sensitivity and moderate anharmonicity of approximately negative E_C (~200 to 300 MHz). Engineers select E_C by choosing the shunt capacitor geometry, typically an interdigital or parallel-plate capacitor with C_shunt of 50 to 100 fF.

What is a typical charging energy value for a modern transmon qubit?

Modern transmon qubits target E_C/h of 200 to 350 MHz, corresponding to a total island capacitance C_Σ of roughly 65 to 100 fF. For the widely used fixed-frequency transmon with a qubit frequency near 5 GHz, typical design values are E_C/h = 250 MHz and E_J/h = 12.5 GHz, giving E_J/E_C = 50. The junction capacitance itself contributes only 2 to 5 fF; the dominant contribution comes from the lithographically defined shunt capacitor pads.

Quantum Computing RF

Charging Energy

Q: How is charging energy measured experimentally?

Charging energy is extracted from two-tone spectroscopy. The qubit is driven at f_01 (the 0-to-1 transition) while a second tone probes f_12 (the 1-to-2 transition). The anharmonicity alpha = f_12 minus f_01 equals negative E_C for a transmon in the weakly anharmonic limit. A measurement of alpha = negative 250 MHz directly yields E_C/h = 250 MHz. Alternatively, charge dispersion measurements (sweeping gate voltage and observing frequency oscillations) can extract E_C in the charge qubit regime, though this is impractical for transmons where dispersion is sub-kilohertz.

/chahr-jing en-er-jee/

The electrostatic energy cost of adding one electron to a superconducting island, defined as E_C = e² / (2C_Σ), where C_Σ is the total island capacitance including junction, gate, and shunt contributions. Charging energy is the primary design parameter that determines whether a charge qubit or transmon regime is reached: large E_C produces strong anharmonicity but high charge noise sensitivity, while small E_C (achieved by adding a shunt capacitor) suppresses charge dispersion exponentially, enabling the long coherence times of modern transmon qubits.

Category: Quantum Computing RF

Transmon E_C/h: 200 to 350 MHz

CPB E_C/h: 0.5 to 5 GHz

Understanding Charging Energy

In a superconducting qubit circuit, the island is a mesoscopic conductor small enough that adding even a single electron measurably changes its electrostatic potential. The charging energy E_C quantifies this effect. For a bare Josephson junction with a capacitance of 2 to 5 fF, E_C/h can exceed 5 GHz, placing the device firmly in the charge qubit regime. By lithographically adding a large interdigital or parallel-plate shunt capacitor (50 to 100 fF), engineers reduce E_C/h to 200 to 350 MHz, crossing into the transmon regime where charge dispersion drops below 10 kHz and coherence times improve by orders of magnitude.

The ratio E_J/E_C is the single most important dimensionless parameter in superconducting qubit design. It controls anharmonicity (α ≈ −E_C for a transmon), charge dispersion (exponentially suppressed as E_J/E_C grows), and the number of bound states in the cosine potential well. Designing a qubit therefore starts with choosing E_C through capacitor geometry, then selecting junction critical current I_C to set E_J for the desired transition frequency f₀₁ ≈ (8E_JE_C)^½/h − E_C/h.

Formulas and Design Equations

Charging Energy:
E_C = e² / (2C_Σ)

Total Island Capacitance:
C_Σ = C_J + C_g + C_shunt + C_coupling

Transmon Frequency (E_J/E_C ≫ 20):
f₀₁ ≈ (√(8E_JE_C) − E_C) / h

Anharmonicity:
α = f₁₂ − f₀₁ ≈ −E_C/h

Where e = 1.602 × 10^-19 C, C_J = junction capacitance (2 to 5 fF typical), C_shunt = designed shunt (50 to 100 fF for transmon), h = Planck's constant.

Charging Energy Across Qubit Types

Qubit Type	E_C/h	C_Σ	E_J/E_C	Anharmonicity
Cooper Pair Box	1 to 5 GHz	5 to 20 fF	0.1 to 5	1 to 5 GHz
Quantronium	0.5 to 1 GHz	20 to 40 fF	5 to 20	0.5 to 1 GHz
Transmon	200 to 350 MHz	65 to 100 fF	50 to 100	200 to 350 MHz
Heavy Fluxonium	0.5 to 2 GHz	5 to 20 fF	2 to 10	GHz-scale

Common Questions

Frequently Asked Questions

How does charging energy determine the qubit operating regime?

The ratio E_J/E_C defines the regime. When E_C is large relative to E_J (ratio near 1), the qubit is a charge qubit with strong charge dispersion and large anharmonicity. Reducing E_C by adding a shunt capacitor pushes E_J/E_C to 50 to 100, creating a transmon with exponentially suppressed charge sensitivity and moderate anharmonicity of approximately −E_C (~200 to 300 MHz). Engineers select E_C by choosing the shunt capacitor geometry, typically interdigital or parallel-plate with C_shunt of 50 to 100 fF.

What is a typical charging energy for a modern transmon?

Modern transmons target E_C/h of 200 to 350 MHz, corresponding to C_Σ of roughly 65 to 100 fF. For a fixed-frequency transmon near 5 GHz, typical values are E_C/h = 250 MHz and E_J/h = 12.5 GHz, giving E_J/E_C = 50. The junction capacitance contributes only 2 to 5 fF; the lithographically defined shunt capacitor pads provide the dominant contribution.

How is charging energy measured experimentally?

E_C is extracted from two-tone spectroscopy. The qubit is driven at f₀₁ while a second tone probes f₁₂. The anharmonicity α = f₁₂ − f₀₁ equals −E_C for a transmon in the weakly anharmonic limit. Measuring α = −250 MHz directly yields E_C/h = 250 MHz. In the charge qubit regime, sweeping gate voltage and observing frequency oscillations (charge dispersion) can also extract E_C.

Related Terms

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