This document discusses using a superconducting parametric cavity to simulate quantum lattice models. It demonstrates a 4-mode lattice system that shows early hints of topological features, including zero states when the flux is 0 and soliton-like states when the flux is π, resembling an emergent Creutz ladder. The superconducting parametric cavity allows complex hopping terms between modes that can be controlled in both strength and phase by a pump signal, opening possibilities to simulate topological and other quantum models.

1. Quantum simulations of
lattice models using a
superconducting parametric
cavity
Jamal Busnaina, Jimmy Shih-Chun
Hung, M. V. Moghaddam, CWS Chang,
A. M. Vadiraj, C. M. Wilson
Engineered Quantum Systems Laboratory
Hadiseh Alaeian1, Enrique Rico2
1Universitat Stuttgart
2University of the Basque Country
Experiment Theory

2. Motivation
 There has been growing interest in building quantum simulators
for classically intractable quantum systems
 The scalability and flexibility of superconducting parametric
cavities allow us to develop a toolbox to simulate a wide variety of
quantum models
 Implementations of lattice models on the platform:- Hints of
topological features.

3. Photonic Lattices
Complex hopping terms with controllable phases opens the door to, e.g.,
 Lattice gauge field simulations
 Topological and chiral dynamics
𝑖
𝑗
𝐴 𝑒𝑓𝑓 𝑑𝑙 = 𝜙𝑖𝑗
Fang et al., Nature Photon 6, 782 (2012)

4. Topological states in
Creutz ladder
End States
Soliton state
𝒏 𝒏 + 𝟐

5.  Long cavity puts a number of modes into our measurement band
 Impedance stepping for unequal frequency spacing
 SQUID termination for tunability and parametric modulation of cavity
modes
Parametric Cavities
Simoen et al., JAP (2015)

6. Parametric 2nd order interactions mediated by SQUID:
Flux pump Cavity signals
Applying appropriate rotating-wave approximations:
Complex hopping term
Both coupling strength and phase controlled by the pump
signal
𝐻 𝐻𝑜𝑝𝑝𝑖𝑛𝑔 = ℏ 𝑔′
(𝑎𝑖 𝑎𝑗
†
𝑒 𝑖𝜃
+ 𝑎𝑖
†
𝑎𝑗 𝑒−𝑖𝜃
)
𝐻𝑆𝑄 ≈ ℏ𝑔0 (𝛼 𝑝 + 𝛼 𝑝
∗)( 𝑎𝑖 + 𝑎𝑖
†
𝑖
)2
𝐻𝑆𝑄 = 𝐸𝐽 𝑐𝑜𝑠(𝜋(𝛷𝑝 − 𝛷 𝑒𝑥𝑡 )/𝛷0) 𝑐𝑜𝑠(2𝜋𝛷𝑐/𝛷0)
NIST, Nature Physics 7, 599 (2011)
Realizing the hopping term

7. 4-node lattice
Experiment System

8. 4-mode System Spectrum
Mode Hybridization

9. 4-mode System Spectrum
Mode Hybridization
Moving Eigen-Energies
Φ

10. Transport measurement
Identifying system states

11. 4-mode System Spectrum vs Flux

12. Emerging Topological Features
Zero state at Ф = 0
Twisting

13. Emerging Topological Features
Degeneracy at Ф = 𝜋

14. Emerging Topological Features
Aharonov–Bohm caging
𝒏
𝜽 = 𝝅
𝒏 + 𝟏 𝒏 + 𝟐
𝜽 = 𝝅

15. Summary
 We demonstrated the capability of superconducting
parametric cavity as potential platform for simulating
quantum lattice models
 We observed hints of topological features of 4-mode
lattice which looks like an emergent Creutz ladder
states.
- Zero states at Ф = 0
- Solion-like states at Ф = 𝜋