Nonperturbative approach to circuit quantum electrodynamics

We outline a rigorous method which can be used to solve the many-body Schrödinger equation for a Coulomb interacting electronic system in an external classical magnetic field as well as a quantized electromagnetic field. Effects of the geometry of the electronic system as well as the polarization of the quantized electromagnetic field are explicitly taken into account. We accomplish this by performing repeated truncations of many-body spaces in order to keep the size of the many particle basis on a manageable level. The electron-electron and electron-photon interactions are treated in a nonperturbative manner using “exact numerical diagonalization.” Our results demonstrate that including the diamagnetic term in the photon-electron interaction Hamiltonian drastically improves numerical convergence. Additionally, convergence with respect to the number of photon states in the joint photon-electron Fock space basis is fast. However, the convergence with respect to the number of electronic states is slow and is the main bottleneck in calculations.

Figure 1

Energy spectra for the lowest $\sim$60 states with (a) one electron and $x$ polarization and (b) two electrons and $y$ polarization. The diamagnetic $A^2$ term in the $e$-EM interaction Hamiltonian is both included (blue solid) and omitted (red dashed). In (a), the system is on resonance between the one electron states ${\left|1\right)}_1$ and ${\left|2\right)}_1$ with a DGC strength of $|{\mathcal{G}}_{12}|=0.290$ and $\hslash {\omega }_p=0.185$ meV. In (b), the system is on resonance between the two electron states ${\left|1\right)}_2$ and ${\left|5\right)}_2$ with a DGC strength of $|{\mathcal{G}}_{15}|=0.987$ and $\hslash {\omega }_p=1.025$ meV. As can be seen from the figure, omitting the $A^2$ term does give accurate results for small ${\mathcal{E}}_c$, while for large ${\mathcal{E}}_c$ the energy spectrum takes a steep dive downwards. This dive also takes place in the two electron case; however it can't be seen in the chosen range of ${\mathcal{E}}_c$. There is no physical significance in these dives since the results are highly divergent in those areas.

Figure 2

Convergence calculations with respect to $N_{\mathrm{mesT}}$ for (a) $x$ polarization and (b) $y$ polarization. Note that the order of the curves from top to bottom is the same as on the legend. In (a), the system is on resonance between the one electron states ${\left|1\right)}_1$ and ${\left|2\right)}_1$ giving $\hslash {\omega }_p=0.185$ meV and $|{\mathcal{G}}_{12}|=0.290$. The results are convergent up to ${\mathcal{E}}_c\simeq 0.37$ meV or $|{\mathcal{G}}_{12}|{\mathcal{E}}_c/\hslash {\omega }_p\simeq 0.6$. In (b) the system is on resonance between the one electron states ${\left|1\right)}_1$ and ${\left|5\right)}_1$ giving $\hslash {\omega }_p=1.03$ meV and $|{\mathcal{G}}_{15}|=0.701$. The results are convergent up to ${\mathcal{E}}_c\simeq 1.05$ meV or $|{\mathcal{G}}_{15}|{\mathcal{E}}_c/\hslash {\omega }_p\simeq 0.7$. For this run we have $a=100$, $b=150$, $c=200$, and $d=250$ [see Eqs. (37) and (38) for definitions]. $N_{\mathrm{EM}}$ is kept constant at 20.

Figure 3

Convergence calculations with respect to $N_{\mathrm{mesT}}$ for (a) $x$ polarization and (b) $y$ polarization. Note that the order of the curves from top to bottom is the same as on the legend. In (a), the system is on resonance between the two electron states ${\left|1\right)}_2$ and ${\left|2\right)}_2$ giving $\hslash {\omega }_p=0.516$ meV and $|{\mathcal{G}}_{12}|=0.648$. The results are convergent up to ${\mathcal{E}}_c\simeq 0.25$ meV or $|{\mathcal{G}}_{12}|{\mathcal{E}}_c/\hslash {\omega }_p\simeq 0.3$. In (b) the system is on resonance between the two electron states ${\left|1\right)}_2$ and ${\left|5\right)}_2$ giving $\hslash {\omega }_p=1.03$ meV and $|{\mathcal{G}}_{15}|=0.987$. The results are convergent up to ${\mathcal{E}}_c\simeq 0.25$ meV or $|{\mathcal{G}}_{15}|{\mathcal{E}}_c/\hslash {\omega }_p\simeq 0.24$. For this run we have $a=100$, $b=150$, $c=200$, and $d=250$ [see Eqs. (37) and (38) for definitions]. Other accuracy parameters are $N_{\mathrm{ses}}=50$ and $N_{\mathrm{EM}}=20$.

Figure 4

Convergence calculations with respect to $N_{\mathrm{EM}}$ for (a) $x$ polarization and (b) $y$ polarization. Note that the order of dashed and solid curves from top to bottom is the same as on the legend. Values of $\hslash {\omega }_p$ and $|{\mathcal{G}}_{\lambda \kappa }|$ are the same as in Fig. 2 for both polarizations. We can see that for $N_{EM}=20$ (green, respectively, the third solid or dashed curve from top), the results are acceptable for the whole range of ${\mathcal{E}}_c$ considered. For this run we have $a=10$, $b=15$, $c=20$, and $d=25$ [see Eqs. (37) and (38) for definitions). The electron state number is kept constant at $N_{\mathrm{mesT}}=200$.

Figure 5

Convergence calculations for two electrons with respect to $N_{\mathrm{mesT}}$ for (a) $x$ polarization and (b) $y$ polarization. Note that the order of the curves from top to bottom is the same as on the legend. For both polarizations, the system is off resonance with $\hslash {\omega }_p=0.4$ meV. In both cases, the results are convergent up to ${\mathcal{E}}_c\simeq 0.26$ meV or ${\mathcal{E}}_c/\hslash {\omega }_p\simeq 0.65$. For this run we have $a=100$, $b=150$, $c=200$, and $d=250$ [see Eqs. (37) and (38) for definitions]. Other accuracy parameters are $N_{\mathrm{ses}}=50$ and $N_{\mathrm{EM}}=20$.