Circuit QED and Cavity QED: Light-Matter Interaction Across Scales

Table of Contents

1. Introduction
2. Fundamentals of Cavity Quantum Electrodynamics (Cavity QED)
3. Jaynes-Cummings Model and Quantum Rabi Oscillations
4. Strong and Ultra-Strong Coupling Regimes
5. Atom-Cavity Systems in Cavity QED
6. Experimental Realizations of Cavity QED
7. From Cavity QED to Circuit QED
8. Superconducting Qubits as Artificial Atoms
9. Transmission Line Resonators in Circuit QED
10. Coupling Strength and Coherence in Circuit QED
11. Hamiltonian of Circuit QED
12. Dispersive and Resonant Regimes
13. Readout via Dispersive Shifts
14. Multi-Qubit Coupling and Quantum Buses
15. Quantum Gates in Circuit QED
16. Quantum Error Correction in cQED Architectures
17. Quantum Simulation and Many-Body Physics
18. Comparison Between Cavity and Circuit QED
19. Challenges and Scaling in Circuit QED
20. Conclusion

1. Introduction

Cavity Quantum Electrodynamics (QED) and Circuit QED (cQED) explore the interaction of light and matter at the quantum level. Cavity QED uses natural atoms and optical cavities, while cQED employs superconducting qubits and microwave resonators. Together, they underpin much of quantum optics and quantum information science.

2. Fundamentals of Cavity Quantum Electrodynamics (Cavity QED)

Cavity QED studies the interaction between a single atom and a quantized electromagnetic field confined in a high-Q cavity. It enables precise control over atom-photon coupling.

3. Jaynes-Cummings Model and Quantum Rabi Oscillations

The Jaynes–Cummings Hamiltonian describes the interaction:
[
H = \hbar \omega_c a^\dagger a + \hbar \omega_a \sigma_z + \hbar g(a^\dagger \sigma^- + a \sigma^+)
\]
It predicts coherent Rabi oscillations between atomic and photonic excitations.

4. Strong and Ultra-Strong Coupling Regimes

In the strong coupling regime \( g > \kappa, \gamma \), coherent energy exchange dominates over decay. In ultra-strong coupling, non-rotating terms (from the full Rabi model) become significant.

5. Atom-Cavity Systems in Cavity QED

• Rydberg atoms in microwave cavities
• Alkali atoms in Fabry–Pérot cavities
• Trapped ions or atoms in photonic crystal cavities
  These setups test fundamental quantum phenomena and serve as building blocks for quantum communication.

6. Experimental Realizations of Cavity QED

• High-finesse optical cavities
• Cryogenic microwave cavities
• Fiber-coupled systems
  Measurement of vacuum Rabi splitting and photon blockade are key milestones.

7. From Cavity QED to Circuit QED

Circuit QED adapts these ideas to the microwave domain using artificial atoms (superconducting qubits) embedded in on-chip microwave resonators. It offers stronger couplings and lithographic scalability.

8. Superconducting Qubits as Artificial Atoms

Transmon, flux, and phase qubits behave like discrete two-level systems with tunable energy levels. They replace natural atoms with engineered anharmonic oscillators.

9. Transmission Line Resonators in Circuit QED

1D microwave resonators confine electromagnetic fields on a chip, replacing optical cavities. Coplanar waveguides and 3D cavities are common in state-of-the-art experiments.

10. Coupling Strength and Coherence in Circuit QED

Coupling strengths \( g \) range from 10–100 MHz, often exceeding loss rates. Coherence times have improved from ~100 ns to >100 μs, enabling multiple gate operations.

11. Hamiltonian of Circuit QED

The effective cQED Hamiltonian resembles Jaynes–Cummings but includes nonlinearities:
\[
H = \hbar \omega_r a^\dagger a + \hbar \omega_q \sigma_z/2 + \hbar g(a^\dagger \sigma^- + a \sigma^+)
\]
Dispersive shifts appear when \( \Delta = \omega_r – \omega_q \) is large.

12. Dispersive and Resonant Regimes

• Resonant: photon and qubit exchange energy coherently
• Dispersive: interaction shifts qubit or cavity frequency; ideal for readout

13. Readout via Dispersive Shifts

In the dispersive regime, the cavity frequency depends on qubit state:
[
\omega_r o \omega_r \pm \chi
\]
This allows non-demolition readout using microwave transmission measurements.

14. Multi-Qubit Coupling and Quantum Buses

Multiple qubits couple via a shared resonator, enabling entanglement and multi-qubit gates. Parametric modulation or tunable couplers enhance flexibility.

15. Quantum Gates in Circuit QED

• iSWAP, CZ, and CPHASE gates
• Cross-resonance gates
• Tunable qubit interactions
  Gate fidelities now exceed 99%, enabling error correction protocols.

16. Quantum Error Correction in cQED Architectures

Surface codes, bosonic codes (e.g., cat codes), and repetition codes are implemented using high-Q cavities and ancilla qubits for syndrome detection.

17. Quantum Simulation and Many-Body Physics

Arrays of coupled cavities and qubits simulate Bose-Hubbard models, spin chains, and lattice gauge theories. cQED enables analog and digital quantum simulation.

18. Comparison Between Cavity and Circuit QED

Feature	Cavity QED	Circuit QED
Atom type	Natural (atoms)	Artificial (qubits)
Operating frequency	Optical	Microwave
Coupling strength	~kHz–MHz	~MHz–GHz
Scalability	Limited	Lithographically scalable
Integration	3D free-space	On-chip planar or 3D

19. Challenges and Scaling in Circuit QED

• Crosstalk and frequency crowding
• Thermal photon noise
• Coherent control of many qubits
• Cryogenic infrastructure and wiring complexity

20. Conclusion

Cavity QED and Circuit QED offer complementary platforms for exploring light–matter interactions. While cavity QED provides precision in atomic systems, circuit QED unlocks scalability and strong coupling for quantum computing and simulation.