Two electrons in harmonic confinement coupled to light in a cavity

The energy and wave function of a harmonically confined two-electron system coupled to light is calculated by separating the wave functions of the relative and center-of-mass (CM) motions. The relative motion wave function has a known quasianalytical solution. The light only couples to the CM variable and the coupled equation can be solved analytically. The approach works for any coupling strength. Examples of wave functions of light-matter hybrid states are presented.

Figure 1

(a) Frequencies of the eigensolutions of Eq. (2.32) as a function of photon frequency; ${\omega }_s$ (black curve starts at zero at $\omega =0$) and ${\omega }_t$ (red curve) for $x=1/10$. (b) The lowest eigenstates of $H_c$ with $n_s+n_t\le 3$ for $x=10$. The lowest transition is between states $(n_s=0,n_t=1)$ and $(n_s=1,n_t=0)$ at $\omega ={\omega }_0$ with energy gap $\mathrm{\Omega }$. Next, in the middle, there is a transition between the $(0,2)$ and $(2,0)$ states with energy gap $2\mathrm{\Omega }$. In the top, there is a transition between states $(0,3)$ and $(3,0)$ with $3\mathrm{\Omega }$ and between states $(2,1)$ and $(1,2)$ with energy gap $\mathrm{\Omega }$.

Figure 2

Two-dimensional electron densities of two electrons confined in a harmonic potential with ${\omega }_0=1, \omega =0.5$ a.u. and spin $S=0$ for different $j$ (CM quantum number) and $\lambda$ values. First row: $j$ = 0, (a) $\lambda =0.5$, (b) $\lambda =2$. Second row: $j$ = 1, (c) $\lambda =0.5$, (d) $\lambda =2$. Third row: $j=2$, (e) $\lambda =0.5$, (f) $\lambda =2$. Fourth row: $j=5$, (g) $\lambda =0.5$, (h) $\lambda =2$. The $x$ axis is the horizontal, the $y$ axis is the vertical direction. The color bar shows the probability.

Figure 3

The energy levels as a function of $\omega$ for different $\lambda$ values (the confinement strength is ${\omega }_0=0.5$ a.u.): (a) $\lambda =\sqrt{\omega }$ and (b) $\lambda =\sqrt{\omega }/10$. The energy of the relative motion is added to the energy in (a), but not in (b) so that (b) can be compared to Fig. 1.

Figure 4

Two-dimensional densities of two electrons confined by a harmonic potential with ${\omega }_0=1/3, \omega =0.5$ a.u. and spin $S=1$. for different $j$ (CM quantum number) and $\lambda$ values. First row: $j=0$, (a) $\lambda =0.01$, (b) $\lambda =0.5$, (c) $\lambda =2$. Second row: $j=1$, (d) $\lambda =0.01$, (e) $\lambda =0.5$, (f) $\lambda =2$. Third row: $j=2$, (g) $\lambda =0.01$, (h) $\lambda =0.5$, (i) $\lambda =2$. Fourth row: $j=5$, (j) $\lambda =0.01$, (k) $\lambda =0.5$, (l) $\lambda =2$. The $x$ axis is the horizontal, the $y$ axis is the vertical direction. The color bar shows the probability.

Figure 5

Two-dimensional electron densities for the $S=1$ case (${\omega }_0=0.18055$ a.u., $\omega =1$ a.u. and $\lambda =1$ a.u.) in different photon subspaces: (a) total density, (b) density in the $n=1$ space, (c) density in the $n=3$ space, (d) density in the $n=5$ space. The $x$ axis is the horizontal, the $y$ axis is the vertical direction. The color bar shows the probability.

Figure 6

The logarithm of the occupation numbers in case of two the photon modes. The photon frequencies are $\omega$ and $2\omega$. The coupling vectors are $\mathit{\lambda }$ and $-\mathit{\lambda }$. The vertical axis is the photon number for $\omega$, the horizontal axis is the photon number for $2\omega$ (${\omega }_0=1$ a.u., $\omega =1$ a.u., $\lambda =1$ a.u.).

Figure 7

Energy levels as a function of $\omega$ for the two-photon modes. The photon frequencies are $\omega$ and $2\omega$, the coupling vectors are $\mathit{\lambda }$ and $-\mathit{\lambda }$, with $\lambda =\sqrt{\omega }$ (${\omega }_0=1$ a.u.)

Figure 8

(a) Probability of the occupation of a $jn$ subspace ($\lambda =0.5$ a.u. and $\omega =0.5$ a.u.), $P_j\left(n\right)$. (b) $P_n={\sum }_j{P}_n\left(j\right)$. (c) $P_j={\sum }_n{P}_n\left(j\right)$. ${\omega }_0=1/3$ a.u. is used in the calculations.