Simulating open quantum systems by applying SU(4) to quantum master equations

We show that open quantum systems of two-level atoms symmetrically coupled to a single-mode photon field can be efficiently simulated by applying a SU(4) group theory to quantum master equations. This is important since many foundational examples in quantum optics fall into this class. We demonstrate the method by finding exact solutions for many-atom open quantum systems such as lasing and steady-state superradiance.

Figure 1

(a) Young tableau for determining the irreducible representations contained in the direct product representation of the SU(2) group for $N=4$. The dimension for $S=2,1,0$ is $5,3,1$, respectively, and $n_s=1,3,2$, so the total Hilbert-space dimension is $5\times 1+3\times 3+1\times 2=2^4$ as expected. (b) “Pascal's triangle” to evaluate $n_S$ in an iterative way for any $N$ [the case considered in (a) is the fourth row down from the top]. (c) Pyramid representation of the density operator in the $|S,M\rangle$ representation for $N=4$. Each layer of the pyramid represents the block matrix for each $S$.

Figure 2

Calculations of laser behaviors described by Eq. (2) with $\Omega =1$, $\gamma =5$, $\kappa =1$, $1/\left(2T_2\right)=0$, and $N=30$. (a) The average intracavity photon number (red dots) and spin-spin correlation (blue squares) as functions of the repumping rate $w$. The blue line is the laser theory prediction of the average photon number [3]. (b) Photon statistics of the laser below threshold $w=4$ (red dots) and above threshold $w=8$ (blue squares). (c) Normalized spectra of the laser below threshold $w=4$ (squares) and above threshold $w=8$ (dots). The red dashed line and green solid line are fitted Lorentzian line shapes. (d) Threshold behavior illustrated by the intensity correlation $g^{\left(2\right)}\left(0\right)=\langle a^{†}{a}^{†}aa\rangle /{\langle a^{†}a\rangle }^2$ (blue squares) and the entropy (red dots) of the whole system as functions of the repumping rate.

Figure 3

(a) Comparison of the second-order intensity correlation $g^{\left(2\right)}\left(0\right)=\langle a^{†}{a}^{†}aa\rangle /{\left(a^{†}a\right)}^2$ as a function of the repump rate in the steady-state superradiance. The green symbols show the Monte Carlo results including the statistical errors for $N=10$ atoms from Ref. [11]. The red squares show the present calculation using the SU(4) theory, and the blue solid line shows the semiclassical results from Ref. [11]. (b) Atom statistics for $w=0.1{\Gamma }_c$. The lengths of the bars represent populations of the $|S,M\rangle$ states.