Stochastic Variational Approach to Small Atoms and Molecules Coupled to Quantum Field Modes in Cavity QED

In this work, we present a stochastic variational calculation (SVM) of energies and wave functions of few particle systems coupled to quantum fields in cavity QED. The spatial wave function and the photon spaces are optimized by a random selection process. Using correlated basis functions, the SVM approach solves the problem accurately and opens the way to the same precision that is reached the nonlight coupled quantum systems. Examples for a two-dimensional trion and confined electrons as well as for the He atom and the ${\mathrm{H}}_2$ molecule are presented showing that the light-matter coupling drastically changes the electronic states.

Figure 1

(a) Energy convergence of the SVM calculation. (b) Photon space occupations for the ground state with $\overrightarrow{\lambda }=(0.5,0.5)\mathrm{a}.\mathrm{u}.$, solid line: SVM, dashed line: exact diagonalization.

Figure 2

Electron density of a harmonically confined 2D three-electron system ($\omega =2$ and $\lambda =4$) (a) $S=1/24$, (b) $S=3/2$. Photon distribution (logarithm of the occupation probability) for two photons modes in the 2D three-electron system, $S=1/2$, ${\omega }_1=1/2{\omega }_2=1$; (c) ${\overrightarrow{\lambda }}_1=(5,5)$, ${\overrightarrow{\lambda }}_2=(-5,-5)$, (d) ${\overrightarrow{\lambda }}_1=(5,5)$, ${\overrightarrow{\lambda }}_2=(-10,-10)$. $n$ is the number of photons in the ${\omega }_1$ mode and $m$ is the number of photons in the ${\omega }_2$ mode.

Figure 3

Two electrons and a hole in 2D coupled to one photon mode [${\omega }_1=2$, ${\overrightarrow{\lambda }}_1=(4,4)\mathrm{a}.\mathrm{u}.$]. The left column [(a), (d), (g), (j)] shows the total single particle density; the electron density is in the middle [(b), (e), (h), (k)]; and the right column [(c), (f), (i), (l)] shows the density of the hole. The first row belongs to the photon space $|0⟩$, the second to $|1⟩$, the third to $|2⟩$, and the fourth to $|3⟩$. The $x$ axis is the horizontal, the $y$ axis is the vertical direction.

Figure 4

Two electrons and a hole in 2D coupled to two photon modes [${\omega }_1=2$, ${\omega }_2=4$, ${\overrightarrow{\lambda }}_1=(4,4)$, ${\overrightarrow{\lambda }}_2=(-4,-4)\mathrm{a}.\mathrm{u}.$] The total single particle densities in photon space $|0⟩|0⟩$ (a), $|1⟩|0⟩$ (b), $|1⟩|1⟩$ (c), $|2⟩|2⟩$ (d). The electron density in $|2⟩|2⟩$ (e) and the hole density in $|2⟩|2⟩$ (f) are also shown. The $x$ axis is the horizontal, the $y$ axis is the vertical direction.

Figure 5

(a) Ground state energy of ${\mathrm{H}}_2$ as a function of bond length (solid line) and energies of the ${\mathrm{H}}_2$ molecule coupled to light. ${\overrightarrow{\lambda }}_i={\lambda }_i(1,1,0)$. (b) Energy of the lowest lying states of He as a function of $\lambda$ for $\omega =0.8\mathrm{a}.\mathrm{u}.$ (c) Energy of the $2{}^1{S}^e$ (starts at $-2.14\mathrm{a}.\mathrm{u}.$) and $2{}^1{P}^o$ (starts at $-2.12\mathrm{a}.\mathrm{u}.$) as a function of $\lambda$, for $\omega =0.8\mathrm{a}.\mathrm{u}.$ (solid lines), $\omega =0.4\mathrm{a}.\mathrm{u}.$ (dashed lines) and $\omega =0.2\mathrm{a}.\mathrm{u}.$ (dotted lines). (d) Energy of states lying around the $2{}^1{S}^e$ and $2{}^1{P}^o$ as a function of $\omega$ ($\sqrt{\omega }/\lambda$ is kept constant).