Observation of the Bloch-Siegert Shift in a Qubit-Oscillator System in the Ultrastrong Coupling Regime (1005.1559v2)

Abstract: We measure the dispersive energy-level shift of an $LC$ resonator magnetically coupled to a superconducting qubit, which clearly shows that our system operates in the ultrastrong coupling regime. The large mutual kinetic inductance provides a coupling energy of $\approx0.82$~GHz, requiring the addition of counter-rotating-wave terms in the description of the Jaynes-Cummings model. We find a 50~MHz Bloch-Siegert shift when the qubit is in its symmetry point, fully consistent with our analytical model.

• The paper reveals a 50 MHz Bloch-Siegert shift in a qubit-oscillator system that surpasses predictions based on the rotating-wave approximation.
• It employs a superconducting circuit of a flux qubit and an LC resonator with magnetic coupling to precisely measure energy-level shifts.
• The findings broaden circuit QED models by validating counter-rotating effects, paving the way for advanced quantum computing applications.

Observation of the Bloch-Siegert Shift in a Qubit-Oscillator System in the Ultrastrong Coupling Regime

The research presented examines the Bloch-Siegert shift within a qubit-oscillator system operating under the ultrastrong coupling regime. The paper leverages a superconducting circuit composed of a flux qubit and an $LC$ resonator, utilizing magnetic coupling to probe the phenomenon. A critical element of this work is the investigation of a dispersive energy-level shift, demonstratively exceeding the limits often set by the rotating-wave approximation (RWA) typically applicable in quantum optical systems.

Theoretical Framework and Experimentation

The authors focus on a superconducting flux qubit coupled to a high-quality $LC$ resonator. In this setup, a significant coupling energy ($\sim 0.82$ GHz) necessitates a comprehensive understanding beyond RWA, making the counter-rotating terms of the Jaynes-Cummings model pivotal in analyzing the interaction dynamics.

In the circuit quantum electrodynamics (QED) setup utilized, the system's Hamiltonian is tuned to account for the ultrastrong coupling condition, $g/\omega_r \approx 0.1$. The effective Hamiltonian treats the counter-rotating terms with appropriate hierarchy, leading to the measurement of a 50 MHz Bloch-Siegert shift. This shift signifies a deviation influenced by the strength of interaction wherein the energy level splitting surpasses the regime where traditional RWA-based Jaynes-Cummings predictions hold true.

Experimental Setup and Results

The research involves a finite-frequency $LC$ resonator with a lumped element design and a flux qubit with engineered Josephson junctions. The mutual inductance, mediated via kinetic inductance of a narrow superconducting wire, plays a crucial role in achieving the significant coupling strength observed.

From the experimental data, the qubit and resonator system revealed stark deviations from RWA expectations, with avoided crossings and energy levels highlighting ultrastrong coupling characteristics. The comprehensive experimental methodology, including low-power driving and flux bias control, successfully distinguishes the Bloch-Siegert shift within the measurement resolution.

Implications and Future Directions

This investigation holds implications for quantum computation and simulation within superconducting circuits by delineating the regime of ultrastrong coupling, broadening the scope of feasible quantum phenomena beyond the typical RWA scenarios. Such experimental validation corroborates theoretical models predicting effects like Bloch-Siegert shifts in non-RWA applicable contexts.

Future developments may include enhanced measurement techniques to observe these non-perturbative effects more acutely and expand the ultrastrong coupling domain through refined qubit architecture and resonator design. These pathways could push the boundaries of circuit QED, offering richer platforms for simulating complex quantum systems that require accounting for both corotating and counter-rotating interactions.

By recognizing this ultrastrong coupling regime's unique attributes, the paper paves the way for more nuanced considerations of qubit-resonator interactions, particularly as they pertain to systems operating near the theoretical and practical limits of quantum coherent control. This research thereby contributes to the broader understanding of coupling dynamics in quantum circuits, instrumentally impacting the fields of quantum information processing and fundamental quantum mechanics investigations.