[Paper Review] Origin and Implications of $A^2$-like Contribution in the Quantization of Circuit-QED Systems

Mohammad Moein Malekakhlagh, Hakan E. Türeci|arXiv (Cornell University)|Jun 9, 2015

Semiconductor Lasers and Optical Devices4 citations

TL;DR

This paper provides a rigorous derivation of the quantum Hamiltonian for a superconducting transmon qubit coupled to a multimode transmission line cavity, identifying the origin of the gauge-dependent $A^2$-like term as a physical consequence of qubit-induced modification to cavity eigenmodes. The authors show that this term arises from current conservation constraints and leads to measurable shifts in eigenfrequencies and coupling strengths, with implications for ultra-strong coupling regimes and superradiant phase transitions in circuit-QED systems.

ABSTRACT

By placing an atom into a cavity, the electromagnetic mode structure of the cavity is modified. In Cavity QED, one manifestation of this phenomenon is the appearance of a gauge-dependent diamagnetic term, known as the $A^2$ contribution. Although in atomic Cavity QED, the resulting modification in the eigenmodes is negligible, in recent superconducting circuit realizations, such corrections can be observable and may have qualitative implications. We revisit the canonical quantization procedure of a circuit QED system consisting of a single superconducting transmon qubit coupled to a multimode superconducting microwave resonator. A complete derivation of the quantum Hamiltonian of an open circuit QED system consisting of a transmon qubit coupled to a leaky transmission line cavity is presented. We introduce a complete set of modes that properly conserves the current in the entire structure and present a sum rule for the dipole transition matrix elements of a multi-level transmon qubit coupled to a multi-mode cavity. Finally, an effective multi-mode Rabi model is derived with coefficients that are given in terms of circuit parameters.

Motivation & Objective

• To rigorously derive the quantum Hamiltonian for a transmon qubit coupled to a multimode superconducting cavity, accounting for mode distortions due to the qubit.
• To identify the physical origin of the $A^2$-like term in circuit-QED, distinguishing it from gauge artifacts by linking it to current conservation and mode renormalization.
• To generalize the quantization procedure to open, leaky cavities connected to external waveguides, enabling realistic modeling of dissipation.
• To derive an effective multi-mode Rabi model with coupling coefficients fully determined by circuit parameters, including $A^2$-like corrections.
• To assess the observability of $A^2$-like effects in modern cQED platforms, particularly in ultra-strong coupling and long-waveguide systems.

Proposed method

• Derives the classical Hamiltonian for a transmon qubit coupled to a transmission line cavity, incorporating local capacitance perturbations at the qubit position.
• Introduces a complete set of eigenmodes that satisfy current conservation at the qubit location, leading to a modified mode structure with frequency shifts.
• Solves the eigenvalue problem for the modified cavity modes using a boundary condition that includes the qubit’s capacitive coupling, yielding a transcendental equation for eigenfrequencies.
• Establishes orthogonality and normalization conditions for the modified modes, including a non-local term involving the qubit position.
• Performs canonical quantization by expanding the vector potential and conjugate momentum in terms of the new eigenmodes, ensuring correct commutation relations.
• Derives an effective multi-mode Rabi Hamiltonian with coupling strengths $g_{mnl}$ expressed in terms of qubit matrix elements, mode functions, and circuit parameters.

Experimental results

Research questions

• RQ1 What is the physical origin of the $A^2$-like term in circuit-QED, and why is it not merely a gauge artifact?
• RQ2 How does the presence of a transmon qubit modify the eigenmodes and eigenfrequencies of a superconducting transmission line cavity?
• RQ3 To what extent are $A^2$-like corrections observable in modern cQED systems, particularly in ultra-strong coupling or long-cavity regimes?
• RQ4 How can the quantization procedure be consistently extended to open, leaky cavities with external waveguide coupling?
• RQ5 What are the implications of these corrections for the effective coupling in multi-mode Rabi models used in quantum information applications?

Key findings

• The $A^2$-like term arises from the qubit’s local modification of the cavity’s current distribution and eigenmode structure, not from gauge choice, making it physically observable.
• The eigenmodes of the modified cavity are piecewise sinusoidal, with discontinuities in the derivative at the qubit position, and satisfy a modified orthogonality condition involving a delta-function-like term at $x_0$.
• The eigenfrequency equation includes a parameter $\chi_c \propto \alpha^2 a_0 L / S$, showing that mode shifts are significant only when $S/L \sim \alpha^2 a_0$, a condition realizable in ultra-compact, high-impedance circuits.
• The modified cavity Hamiltonian becomes diagonal in the new mode basis, with eigenfrequencies determined by a transcendental equation involving $k_n L$ and the qubit’s coupling strength.
• The effective coupling strength $g_{mnl}$ in the Rabi model depends explicitly on the modified mode amplitude $\tilde{A}_l(x_0)$ at the qubit position, which includes the $A^2$-like correction.
• The derivation shows that $A^2$-like effects can be consistently incorporated into the Hamiltonian without breaking current conservation, providing a foundation for modeling realistic cQED systems with high accuracy.