The Quantum Josephson Hamiltonian In The Phase Representation

Keywords: Mathieu Equation

Derivation

We want to derive the Hamiltonian $\hat{H}=\frac{E_c}{4}\left({\hat{a}}_1^{+}{\hat{a}}_1^{+}{\hat{a}}_1{\hat{a}}_1+{\hat{a}}_2^{+}{\hat{a}}_2^{+}{\hat{a}}_2{\hat{a}}_2\right)-\frac{E_J}{N}\left({\hat{a}}_1^{+}{\hat{a}}_2+{\hat{a}}_2^{+}{\hat{a}}_1\right)$ from $\hat{H}=\int {\Phi }^{†}\left[-\frac{\hslash }{2m}{\nabla }^2+V_{ext}+\frac{g}{2}{\Phi }^{†}\Phi \right]\Phi dr$

We start from

$$\begin{array}{rll}\hat{H}& =\int {\Psi }^{†}\left[-\frac{\hslash }{2m}{\nabla }^2+V_{ext}+\frac{g}{2}{\Psi }^{†}\Psi \right]\Psi dr& \Psi \approx {\Phi }_1(r)a_1^{†}+{\Phi }_2(r)a_2^{†}\\ & =\int ({\Phi }_1(r)a_1+{\Phi }_2(r)a_2{)}^{†}\left[-\frac{\hslash }{2m}{\nabla }^2+V_{ext}+\frac{g}{2}({\Phi }_1(r)a_1+{\Phi }_2(r)a_2{)}^{†}({\Phi }_1(r)a_1+{\Phi }_2(r)a_2)\right]({\Phi }_1(r)a_1+{\Phi }_2(r)a_2)dr& \\ & =\int ({\Phi }_1^{\ast }(r)a_1^{†}+{\Phi }_2^{\ast }(r)a_2^{†})\left[-\frac{\hslash }{2m}{\nabla }^2+V_{ext}+\frac{g}{2}({\Phi }_1^{\ast }(r)a_1^{†}+{\Phi }_2^{\ast }(r)a_2^{†})({\Phi }_1(r)a_1+{\Phi }_2(r)a_2)\right]\left({\Phi }_1(r)a_1+{\Phi }_2(r)a_2\right)dr& \text{Splitting the integrals}\\ & & \\ & =a_1^{†}\int {\Phi }_1^{\ast }(r)\left[-\frac{\hslash }{2m}{\nabla }^2+V_{ext}+\frac{g}{2}({\Phi }_1^{\ast }(r)a_1^{†}+{\Phi }_2^{\ast }(r)a_2^{†})({\Phi }_1(r)a_1+{\Phi }_2(r)a_2)\right]{\Phi }_1(r)dr{a}_1& \\ & +a_1^{†}\int {\Phi }_1^{\ast }(r)\left[-\frac{\hslash }{2m}{\nabla }^2+V_{ext}+\frac{g}{2}({\Phi }_1^{\ast }(r)a_1^{†}+{\Phi }_2^{\ast }(r)a_2^{†})({\Phi }_1(r)a_1+{\Phi }_2(r)a_2)\right]{\Phi }_2(r)dr{a}_2& \\ & +a_2^{†}\int {\Phi }_2^{\ast }(r)\left[-\frac{\hslash }{2m}{\nabla }^2+V_{ext}+\frac{g}{2}({\Phi }_1^{\ast }(r)a_1^{†}+{\Phi }_2^{\ast }(r)a_2^{†})({\Phi }_1(r)a_1+{\Phi }_2(r)a_2)\right]{\Phi }_1(r)dr{a}_1& \\ & +a_2^{†}\int {\Phi }_2^{\ast }(r)\left[-\frac{\hslash }{2m}{\nabla }^2+V_{ext}+\frac{g}{2}({\Phi }_1^{\ast }(r)a_1^{†}+{\Phi }_2^{\ast }(r)a_2^{†})({\Phi }_1(r)a_1+{\Phi }_2(r)a_2)\right]{\Phi }_2(r)dr{a}_2& \text{Splitting the sum }\\ & & \\ & =a_1^{†}\int {\Phi }_1^{\ast }(r)\left[-\frac{\hslash }{2m}{\nabla }^2+V_{ext}\right]{\Phi }_1(r)dr{a}_1+a_1^{†}\frac{g}{2}\int {\Phi }_1^{\ast }(r)\left[({\Phi }_1^{\ast }(r)a_1^{†}+{\Phi }_2^{\ast }(r)a_2^{†})({\Phi }_1(r)a_1+{\Phi }_2(r)a_2)\right]{\Phi }_1(r)dr{a}_1& \\ & +a_1^{†}\int {\Phi }_1^{\ast }(r)\left[-\frac{\hslash }{2m}{\nabla }^2+V_{ext}\right]{\Phi }_2(r)dr{a}_2+a_1^{†}\frac{g}{2}\int {\Phi }_1^{\ast }(r)\left[({\Phi }_1^{\ast }(r)a_1^{†}+{\Phi }_2^{\ast }(r)a_2^{†})({\Phi }_1(r)a_1+{\Phi }_2(r)a_2)\right]{\Phi }_2(r)dr{a}_2& \\ & +a_2^{†}\int {\Phi }_2^{\ast }(r)\left[-\frac{\hslash }{2m}{\nabla }^2+V_{ext}\right]{\Phi }_1(r)dr{a}_1+a_2^{†}\frac{g}{2}\int {\Phi }_2^{\ast }(r)\left[({\Phi }_1^{\ast }(r)a_1^{†}+{\Phi }_2^{\ast }(r)a_2^{†})({\Phi }_1(r)a_1+{\Phi }_2(r)a_2)\right]{\Phi }_1(r)dr{a}_1& \\ & +a_2^{†}\int {\Phi }_2^{\ast }(r)\left[-\frac{\hslash }{2m}{\nabla }^2+V_{ext}\right]{\Phi }_2(r)dr{a}_2+a_2^{†}\frac{g}{2}\int {\Phi }_2^{\ast }(r)\left[({\Phi }_1^{\ast }(r)a_1^{†}+{\Phi }_2^{\ast }(r)a_2^{†})({\Phi }_1(r)a_1+{\Phi }_2(r)a_2)\right]{\Phi }_2(r)dr{a}_2& {\int }_{\mathbb{R}}{\Phi }_1(r{)}^2{\Phi }_2(r{)}^{2}dr\approx 0\\ & & \\ & =\int {\Phi }_1^{\ast }(r)\left[-\frac{\hslash }{2m}{\nabla }^2+V_{ext}\right]{\Phi }_1(r)dr{a}_1^{†}{a}_1+a_1^{†}\frac{g}{2}\int {\Phi }_1^{\ast }(r)\left[{\Phi }_1^{\ast }(r)a_1^{†}{\Phi }_1(r)a_1\right]{\Phi }_1(r)dr{a}_1& \\ & +\int {\Phi }_1^{\ast }(r)\left[-\frac{\hslash }{2m}{\nabla }^2+V_{ext}\right]{\Phi }_2(r)dr{a}_1^{†}{a}_2& \\ & +\int {\Phi }_2^{\ast }(r)\left[-\frac{\hslash }{2m}{\nabla }^2+V_{ext}\right]{\Phi }_1(r)dr{a}_2^{†}{a}_1& \\ & +\int {\Phi }_2^{\ast }(r)\left[-\frac{\hslash }{2m}{\nabla }^2+V_{ext}\right]{\Phi }_2(r)dr{a}_2^{†}{a}_2+a_2^{†}\frac{g}{2}\int {\Phi }_2^{\ast }(r)\left[{\Phi }_2^{\ast }(r)a_2^{†}{\Phi }_2(r)a_2\right]{\Phi }_2(r)dr{a}_2& \text{Simplifying}\\ & & \\ & =\int {\Phi }_1^{\ast }(r)\left[-\frac{\hslash }{2m}{\nabla }^2+V_{ext}\right]{\Phi }_1(r)dr{a}_1^{†}{a}_1& \\ & +\int {\Phi }_1^{\ast }(r)\left[-\frac{\hslash }{2m}{\nabla }^2+V_{ext}\right]{\Phi }_2(r)dr{a}_1^{†}{a}_2& \\ & +\int {\Phi }_2^{\ast }(r)\left[-\frac{\hslash }{2m}{\nabla }^2+V_{ext}\right]{\Phi }_1(r)dr{a}_2^{†}{a}_1& \\ & +\int {\Phi }_2^{\ast }(r)\left[-\frac{\hslash }{2m}{\nabla }^2+V_{ext}\right]{\Phi }_2(r)dr{a}_2^{†}{a}_2& \\ & +\frac{g}{2}\int {\Phi }_1^{\ast }(r){\Phi }_1^{\ast }(r){\Phi }_1(r){\Phi }_1(r)dr{a}_1^{†}{a}_1^{†}{a}_1{a}_1& \\ & +\frac{g}{2}\int {\Phi }_2^{\ast }(r){\Phi }_2^{\ast }(r){\Phi }_2(r){\Phi }_2(r)dr{a}_2^{†}{a}_2^{†}{a}_2{a}_2& E_c:=2g{\int }_{\mathbb{R}}|{\Phi }_1(r){|}^{4}dr=2g{\int }_{\mathbb{R}}|{\Phi }_1(r){|}^{4}dr\\ & & \\ & =\int {\Phi }_1^{\ast }(r)\left[-\frac{\hslash }{2m}{\nabla }^2+V_{ext}\right]{\Phi }_1(r)dr{a}_1^{†}{a}_1& \\ & +\int {\Phi }_1^{\ast }(r)\left[-\frac{\hslash }{2m}{\nabla }^2+V_{ext}\right]{\Phi }_2(r)dr{a}_1^{†}{a}_2& \\ & +\int {\Phi }_2^{\ast }(r)\left[-\frac{\hslash }{2m}{\nabla }^2+V_{ext}\right]{\Phi }_1(r)dr{a}_2^{†}{a}_1& \\ & +\int {\Phi }_2^{\ast }(r)\left[-\frac{\hslash }{2m}{\nabla }^2+V_{ext}\right]{\Phi }_2(r)dr{a}_2^{†}{a}_2& \\ & +\frac{E_c}{4}(a_1^{†}{a}_1^{†}{a}_1{a}_1+a_2^{†}{a}_2^{†}{a}_2{a}_2)& E_J:=-N{\int }_{\mathbb{R}}{\Phi }_1(r)\left[-\frac{{\hslash }^2}{2m}{\nabla }^2+V_{ext}+\frac{gN}{2}[{\Phi }_2(r{)}^2+{\Phi }_2(r{)}^2]\right]{\Phi }_2(r)dr\\ & & \\ & =\int {\Phi }_1^{\ast }(r)\left[-\frac{\hslash }{2m}{\nabla }^2+V_{ext}\right]{\Phi }_1(r)dr{a}_1^{†}{a}_1& \\ & +\int {\Phi }_1^{\ast }(r)\left[-\frac{\hslash }{2m}{\nabla }^2+V_{ext}\right]{\Phi }_2(r)dr{a}_1^{†}{a}_2& \\ & +\int {\Phi }_2^{\ast }(r)\left[-\frac{\hslash }{2m}{\nabla }^2+V_{ext}\right]{\Phi }_1(r)dr{a}_2^{†}{a}_1& \\ & +\int {\Phi }_2^{\ast }(r)\left[-\frac{\hslash }{2m}{\nabla }^2+V_{ext}\right]{\Phi }_2(r)dr{a}_2^{†}{a}_2& \\ & +\frac{E_c}{4}(a_1^{†}{a}_1^{†}{a}_1{a}_1+a_2^{†}{a}_2^{†}{a}_2{a}_2)& E_J:=-N{\int }_{\mathbb{R}}\Phi (r)\left[-\frac{{\hslash }^2}{2m}{\nabla }^2+V_{ext}+\frac{gN}{2}[{\Phi }_2(r{)}^2+{\Phi }_2(r{)}^2]\right]{\Phi }_2(r)dr\\ & & \\ & =\dots & \end{array}$$

Assumptions made:

1. $\Psi \approx {\Phi }_1(r)a_1+{\Phi }_2(r)a_2$, i.e. our Wave function can be approximated as a combination of creation operators.
2. $⟨{\Phi }_1,{\Phi }_2⟩=0$, i.e. ${\int }_{\mathbb{R}}{\Phi }_1^{\ast }(r){\Phi }_2(r)dr=0$
3. $\Vert {\Phi }_i{\Vert }_2=1$
4. Terms proportional to ${\int }_{\mathbb{R}}{\Phi }_1^2(r){\Phi }_2^2(r)dr$ are neglected.
5. Terms proportional to ${\int }_{\mathbb{R}}{\Phi }_1(r){\Phi }_2^3(r)dr$ are neglected.
6. Terms proportional to ${\int }_{\mathbb{R}}{\Phi }_1^3(r){\Phi }_2(r)dr$ are neglected.
7. ${\int }_{\mathbb{R}}|{\Phi }_1(r){|}^{4}dr={\int }_{\mathbb{R}}|{\Phi }_2(r){|}^{4}dr$
8. Charging energy $E_c:=2g{\int }_{\mathbb{R}}|{\Phi }_1(r){|}^{4}dr=2g{\int }_{\mathbb{R}}|{\Phi }_1(r){|}^{4}dr$
9. Josephson coupling energy $E_J:=-N{\int }_{\mathbb{R}}{\Phi }_1(r)\left[-\frac{{\hslash }^2}{2m}{\nabla }^2+V_{ext}+\frac{gN}{2}[{\Phi }_2(r{)}^2+{\Phi }_2(r{)}^2]\right]{\Phi }_2(r)dr$.
10. $N:=a_1^{†}{a}_1+a_2^{†}{a}_2$ .