Conservation of Quantum Correlations in Multimode Systems with $U(1)$ Symmetry

We present a theoretical investigation of the properties of quantum correlation functions in a multimode system. We define a total $m$th order equal-time correlation function, summed over all modes, which is shown to be conserved if the Hamiltonian possesses U(1) symmetry. It is also conserved in the presence of dissipation, provided the loss rate is the same for all modes of the system. As examples, we demonstrate this conservation using numerical simulations of a coupled cavity system and the Jaynes-Cummings model.

Figure 1

Numerical solution for $H_{\mathrm{JC}}$, with a Rabi term $\eta =0.25$ and ${\omega }_a={\omega }_{\sigma }=0$, prepared in an initial state with $|{\alpha }_0{|}^2=0.18$ particles in the cavity mode and the atom in the excited state. (a) The number of photons, $n_a$, and excitation level of the atom, $n_{\sigma }$. $n_{\sigma }=0$ corresponds to the atom in its ground state, $n_{\sigma }=1$ to the excited state. (b) The correlation functions ${\mathcal{G}}_{i,j}^{(2)}=⟨a_i^{†}{a}_j^{†}{a}_i{a}_j⟩/N^2$ for the atomic, bosonic, and for the cross terms. The total correlations for the emitted light is a constant of motion; here it is always sub-Poissonian ($g_{\text{tot}}^{(2)}<1$).

Figure 2

Numerical solutions for the number of excitation for two coupled bosonic modes, with coupling $\tau =1.5\gamma$ and nonlinearity $g=0.25\gamma$, prepared into an initial state with $|{\alpha }_1{|}^2=1.7$ particles in mode 1 and $|{\alpha }_2{|}^2=0.22$ particles in mode 2 (${\alpha }_1$ and ${\alpha }_2$ both real and positive), and ${\omega }_1={\omega }_2=0$. (a) Populations of the two modes, $n_1$ and $n_2$. (b) The correlation functions ${\mathcal{G}}_{i,j}^{(2)}=⟨a_i^{†}{a}_j^{†}{a}_i{a}_j⟩/N^2$ for each mode and for the cross terms. The total correlation function for the emitted light is a constant of motion; in this case $g_{\text{tot}}^{(2)}=1$, as the initial state is coherent. (c) The time evolution of the normalized SOC functions for the individual modes; each shows a nontrivial behavior, even though the total correlation function is stationary.