Dissipative quantum-light-field engineering

We put forward a dissipative preparation scheme for strongly correlated photon states. Our approach is based on a two-photon loss mechanism that is realized via a single four-level atom inside a bimodal optical cavity. Each elementary two-photon emission event removes one photon out of each of the two modes. The dark states of this loss mechanism are given by NOON states and arbitrary superpositions thereof. We find that the steady state of the two cavity modes exhibits entanglement and, for certain parameters, a mixture of two coherent entangled states is produced. We discuss how the quantum correlations in the cavity modes and the output fields can be measured.

Figure 1

A single four-level atom interacts with two cavity modes and a classical laser field. $A_{\text{in}}$ and $B_{\text{in}}$ are coherent input fields that drive the corresponding cavity resonantly, and $A_{\text{out}}$, $B_{\text{out}}$ are the output fields. The inset shows the atomic level scheme. The laser field with frequency ${\omega }_{\text{L}}$ and Rabi frequency ${\Omega }_{\text{L}}$ couples to the $|2\rangle \leftrightarrow |3\rangle$ transition, and the cavity mode with frequency ${\omega }_a$ interacts with the $|3\rangle \leftrightarrow |1\rangle$ transition. The second cavity mode with frequency ${\omega }_b$ interacts with the $|4\rangle \leftrightarrow |2\rangle$ transition. $g_a$ and $g_b$ are the single-photon Rabi frequencies corresponding to mode $a$ and $b$, respectively. The parameters ${\gamma }_{ij}$ are the decay rates of the various transitions, $\delta$ and $\Delta$ label the detuning of the cavity fields with transition $|3\rangle \leftrightarrow |1\rangle$ and $|4\rangle \leftrightarrow |2\rangle$, respectively, and $\varepsilon$ is the two-photon detuning.

Figure 2

Population of the cavity Fock states $|n_a,n_b\rangle$ in steady state for (a) ${\Omega }_a={\Omega }_b=1.32\times {10}^{-2}{\gamma }_{42}$, ${\kappa }_a={\kappa }_b={10}^{-2}{\gamma }_{42}$, (b) ${\Omega }_a={\Omega }_b=6.26\times {10}^{-3}{\gamma }_{42}$, ${\kappa }_a={\kappa }_b=0.5\times {10}^{-2}{\gamma }_{42}$, and (c) ${\Omega }_a={\Omega }_b=1.16\times {10}^{-3}{\gamma }_{42}$, ${\kappa }_a={\kappa }_b={10}^{-3}{\gamma }_{42}$. The common parameters in (a)–(c) are $\Gamma ={\gamma }_{42}/100$, $\Delta =U=0$, $\delta =0$, and $\varepsilon =0$. The mean photon number for all states in (a)–(c) is $\langle \widehat{N}\rangle =5$.

Figure 3

Numerical evaluation of (a) the negativity and (b) the inequality in Eq. (33) for the two cavity modes in steady state. The results are shown as a function of the two-particle loss rate $\Gamma$ and for different intensities of the coherent driving fields. The parameters are $\phi =\pi /2$, ${\kappa }_a={\kappa }_b={10}^{-2}{\gamma }_{42}$, $\Delta =U=0$, $\delta =0$, and $\varepsilon =0$. The red dashed line corresponds to ${\Omega }_a={\Omega }_b=0.2\times {10}^{-2}{\gamma }_{42}$, the blue dotted line stands for ${\Omega }_a={\Omega }_b=0.4\times {10}^{-2}{\gamma }_{42}$, the green dashed-dotted line corresponds to ${\Omega }_a={\Omega }_b=1\times {10}^{-2}{\gamma }_{42}$, and the black solid line stand for ${\Omega }_a={\Omega }_b=1.6\times {10}^{-2}{\gamma }_{42}$.

Figure 4

Squeezing spectra. The blue dashed line corresponds to $S_u\left(\omega \right)$, the red dotted line to $S_v\left(\omega \right)$, and the black solid line shows $S_u\left(\omega \right)+S_v\left(\omega \right)$. The parameters are $\phi =\pi /2$, $\kappa ={\kappa }_a={\kappa }_b={10}^{-2}{\gamma }_{42}$, $\Gamma ={\gamma }_{42}/100$, ${\Omega }_a={\Omega }_b=0.7\times {10}^{-2}{\gamma }_{42}$, $\Delta =U=0$, $\delta =0$, and $\varepsilon =0$.

Figure 5

Measurement of the two-mode squeezing spectrum $S_u\left(\omega \right)$ in Eq. (C8). The output field $A_{\text{out}}$ ($B_{\text{out}}$) is superimposed with a strong coherent field at a $50:50$ beam splitter, and the photocurrents of each detector are subtracted. The photocurrents of the two balanced homodyne detections are added and fed into a spectrum analyzer. The resulting spectrum is given by Eq. (C10) and directly proportional to $S_u\left(\omega \right)$.