Many studies have been conducted to explain the phase transition of the Josephson junction in the resistive shunted Josephson circuit. However, most of the studies were limited to the case where the temperature condition was 0 K. In particular, it is necessary to introduce a strong correlation study technique when the coupling size of the Josephson junction and the external environment consisting of the RC resonator is large. In this paper, the resistively shunted Josephson circuit was analyzed using a strong coupling expansion technique that considers temperature based on Matsubara’s framework. First, the Hamiltonian of the given circuit was expressed in the form of a matrix using the macroscopic basis state, and the obtained result was substituted into the perturbation-theoretic diagram expansion technique of the partition function. Through the above process, the change of order parameter and spin-spin correlation function was calculated to investigate the state change of the Josephson junction according to the temperature.|저항 분로 조셉션 회로에 대하여 조셉션 접합이 나타내는 상전이를 설명하기 위하여 다양한 연구가 진행되었으나, 대다수의 연구는 온도 조건이 0K 인 경우에 국한되어 있었으며 특히 RC 공진기로 이루어진 외부 환경과 조셉션 접합의 결합의 크기가 큰 경우 강상관계 연구기법을 도입할 필요가 있다. 이 논문에서는 마츠바라 이론을 기반으로 하는 강상관계 풀이기법을 이용해 저항 분로 조셉션 회로를 분석하였다. 먼저 주어진 회로의 해밀토니안을 거시적인 기저상태를 이용하여 행렬의 형태로 나타내었고 얻어진 결과를 분배함수의 섭동론적 다이어그램 전개 기법에 대입하였다. 위 과정을 통해 온도에 따른 조셉션 접합의 상태변화를 알아보기 위하여 질서변수의 기댓값과 소산-변동정리를 기반으로 한 스핀-스핀 상관관계 함수를 계산하였다.
Table Of Contents
1. Introduction 1 2. Methodology 3 2.1 Field operator and Green’s function 3 2.2 Diagrammatic method 4 Diagrammatic expansion of Green’s function 4 Diagrammatic expansion in strong coupling case 5 Evaluation of correlation function 6 3. Model & Application 7 3.1 Circuit model 7 Hamiltonian of resistivity shunted Josephson junction 7 in Josephson phase basis 8 Matrix form of $\hat{N}$ in local eigen basis – 3-level case 9 Matrix form of $\hat{N}$ in local eigen basis – Multilevel case 10 Matrix form of cosΦ in local eigen basis – 3-level case 11 Matrix form of cosΦ in local eigen basis – Multilevel case 11 3.2 Diagrammatic method in the circuit Hamiltonian model 13 Calculation of bosonic action 13 Calculation of Propagator 16 Calculating correlation function 16 4. Numerical calculation for integro-differential equation 19 4.1 Structure of C++ code 19 4.2 Hybridzation function – Simpson’s Rule 20 4.3 Trapezoidal method 21 4.4 Iteration truncation – Relative entropy 23 4.5 Implementation of Correlation function 25 5. Result 28 5.1 Benchmark Setup and Simulation Environment 28 Benchmark: Exact diagonalization in single mode 28 Benchmark: Multi-mode case 29 Benchmark: α = 0 condition· 29 Simulation condition: Saturation test 30 Simulation condition: Size dependence 31 5.2 Orderparameter evaluation 31 Temperature independent result 31 Temperature dependent transition 34 Small alpha 38 5.3 41 Table. 4 condition 41 α in excess range 43 6. Conclusion 46 Conclusion 46 Future work 46 Appendix: Circuit Hamiltonian 47
