[Paper Review] A diagrammatic method to compute the effective Hamiltonian of driven nonlinear oscillators

Xu Xiao, Jayameenakshi Venkatraman+3 more|arXiv (Cornell University)|Apr 26, 2023

Quantum Information and Cryptography4 citations

TL;DR

This paper introduces a novel diagrammatic method for computing the effective Hamiltonian of driven nonlinear oscillators using a self-consistent phase-space perturbation expansion. Each diagram corresponds to a Hamiltonian term with a prefactor derived from topological counting, enabling automated computation to arbitrary order and achieving consistency with established methods like Schrieffer-Wolff and classical harmonic balance in the $̂\hbar\rightarrow0$ limit.

ABSTRACT

In this work, we present a new diagrammatic method for computing the effective Hamiltonian of driven nonlinear oscillators. At the heart of our method is a self-consistent perturbation expansion developed in phase space, which establishes a direct correspondence between the diagram and algebra. Each diagram corresponds to a Hamiltonian term, the prefactor of which, like those in Feynman diagrams, involves a simple counting of topologically equivalent diagrams. Leveraging the algorithmic simplicity of our diagrammatic method, we provide a readily available computer program that generates the effective Hamiltonian to arbitrary order. We show the consistency of our schemes with existing perturbation methods such as the Schrieffer-Wolff method. Furthermore, we recover the classical harmonic balance scheme from our result in the limit of $\hbar ightarrow0$. Our method contributes to the understanding of dynamic control within quantum systems and achieves precision essential for advancing future quantum information processors. To demonstrate its value and versatility, we analyze five examples from the field of superconducting circuits. These include an experimental proposal for the Hamiltonian stabilization of a three-legged Schrödinger cat, modeling of energy renormalization phenomena in superconducting circuits experiments, a comprehensive characterization of multiphoton resonances in a driven transmon, a proposal for an inductively shunted transmon circuit, and a characterization of classical ultra-subharmonic bifurcation in driven oscillators. Lastly, we benchmark the performance of our method by comparing it with experimental data and exact Floquet numerical diagonalization.

Motivation & Objective

• To develop a systematic, diagrammatic approach for computing the effective Hamiltonian of driven nonlinear oscillators in quantum systems.
• To establish a direct correspondence between diagram topology and algebraic terms in the Hamiltonian, simplifying perturbative calculations.
• To provide a computationally efficient, extensible algorithm that generates the effective Hamiltonian to arbitrary order.
• To ensure consistency with existing methods such as the Schrieffer-Wolff transformation and classical harmonic balance in the $̂\hbar\rightarrow0$ limit.
• To validate the method through benchmarking against experimental data and exact Floquet diagonalization in superconducting circuit models.

Proposed method

• The method employs a self-consistent perturbation expansion formulated directly in phase space, avoiding traditional time-ordered or Dyson series approaches.
• Each diagram represents a specific term in the effective Hamiltonian, with its prefactor determined by counting topologically equivalent diagrams.
• The diagrammatic rules are derived from a generating functional formalism in phase space, enabling systematic enumeration of all contributing terms.
• The approach is implemented as a computer program that automatically generates the effective Hamiltonian up to any desired order.
• The method naturally recovers the Schrieffer-Wolff transformation in the weak-coupling limit and reduces to the classical harmonic balance method as $̂\hbar\rightarrow0$.
• The formalism is applied to five superconducting circuit examples, including multiphoton resonances and quantum cat state stabilization.

Experimental results

Research questions

• RQ1 How can a diagrammatic method be systematically constructed to compute the effective Hamiltonian of driven nonlinear oscillators in phase space?
• RQ2 What is the precise algebraic correspondence between diagram topology and the prefactors of Hamiltonian terms in such systems?
• RQ3 How does the method ensure consistency with established perturbative frameworks like Schrieffer-Wolff and classical harmonic balance?
• RQ4 To what extent can the method be automated and scaled to arbitrary orders of perturbation?
• RQ5 How does the method perform when benchmarked against exact numerical diagonalization and experimental data in superconducting circuits?

Key findings

• The diagrammatic method successfully reproduces the Schrieffer-Wolff transformation in the weak-coupling limit, confirming consistency with established quantum perturbation theory.
• In the classical limit ($\hbar\rightarrow0$), the method recovers the harmonic balance scheme, validating its classical correspondence.
• The method enables automated, high-order computation of the effective Hamiltonian through a dedicated computer program, significantly reducing manual derivation effort.
• Benchmarking against exact Floquet diagonalization shows excellent agreement in energy level shifts and resonance structures across multiple superconducting circuit models.
• The method accurately models complex phenomena such as energy renormalization, multiphoton resonances, and ultra-subharmonic bifurcations in driven oscillators.
• The approach successfully supports experimental proposals, including the stabilization of a three-legged Schrödinger cat state via engineered Hamiltonian control.