Waveguide quantum electrodynamics in squeezed vacuum

We study the dynamics of a general multiemitter system coupled to the squeezed vacuum reservoir and derive a master equation for this system based on the Weisskopf-Wigner approximation. In this theory, we include the effect of positions of the squeezing sources which is usually neglected in the previous studies. We apply this theory to a quasi-one-dimensional waveguide case where the squeezing in one dimension is experimentally achievable. We show that while dipole-dipole interaction induced by ordinary vacuum depends on the emitter separation, the two-photon process due to the squeezed vacuum depends on the positions of the emitters with respect to the squeezing sources. The dephasing rate, decay rate, and the resonance fluorescence of the waveguide-QED in the squeezed vacuum are controllable by changing the positions of emitters. Furthermore, we demonstrate that the stationary maximum entangled NOON state for identical emitters can be reached with arbitrary initial state when the center-of-mass position of the emitters satisfies certain conditions.

Figure 1

(a) Schematic setup for waveguide-QED in a 1D squeezed vacuum where the vacuum is squeezed from both directions. (b) The dispersion relations inside the waveguide. Here the atomic transition frequency is $\frac{1.2c\pi }{a}$, which is below the cutoff frequency of ${\text{TE}}_{11}$ mode. Considering the fact that the atomic dipole moment is along $y$ axis and $E_y\ne 0$ only for ${\text{TE}}_{10}$, we only need to consider ${\text{TE}}_{10}$ mode in our calculation.

Figure 2

(a) Dephasing dynamics of a single emitter in the squeezed vacuum. The black and red solid curves are the results of ${\sigma }_x$ and ${\sigma }_y$, respectively. The blue dotted line is the result when there is no squeezing (thermal reservoir). (b) The dephasing rates of ${\sigma }_x$ and ${\sigma }_y$ as a function of the emitter position. For (a) and (b), the squeezing parameters are chosen to be $r=0.5$.

Figure 3

Two-emitter case: transverse polarization decay of the first emitter as a function of time. (a) $r_{12}=0.5{\lambda }_{0z}$ for superscript (1) and $r_{12}=1.0{\lambda }_{0z}$ for superscript (2), $r=0.5$, and $r_c=0$; (b) population decay as a function of time when $r_{12}=0.5{\lambda }_{0z}, r=0.5$, and $r_c=0$. Solid lines are the results in squeezed vacuum and the dotted lines are the results in the thermal reservoir with $N={sinh}^2\left(r\right)$. Here the dynamics of ${\rho }_{++}$ and ${\rho }_{--}$ are highly identical. (c) Dephasing rate as a function of atom separation with the center of mass fixed at $r_c=0$. (d) Dephasing rate as a function of center-of-mass position with atom separation fixed at $r_{ij}={\lambda }_{0z}$, where the two-atom case is plotted in solid lines and the five-atom case is plotted in dashed lines.

Figure 4

(a) Concurrence evolution of different initial states in squeezed vacuum, where $r=1, r_c=0$, and $r_{12}=0.25{\lambda }_{0z}$. (b) Fidelity evolution of different initial states in the same environment.

Figure 5

(a) Concurrence of the steady state as a function of average photon number $N=sinh{\left(r\right)}^2$ and the position of the center mass $r_c=\frac{r_1+r_2}{2}$. (b) The impact of $r_c$'s fluctuations on concurrence for different average photon number $N$. $\mathrm{\Delta }{r}_c$ is the distance from $\frac{n}{4}{\lambda }_{0z}$ to the position where the entanglement vanishes.

Figure 6

Resonance fluorescence spectrum of the two-emitter system inside a 1D waveguide. For better comparison, the spectra are normalized to the intensity at $\omega ={\omega }_0$ with the coherent elastic scattering singularity removed. Coherent driving Rabi frequency is ${\mathrm{\Omega }}_R=4\gamma$. In (a) and (b), the solid curves are the spectra for the coupled emitters, while the dashed curves are the spectra without emitter-emitter coupling. Parameters: (a) $r_1=0,r_2=0.01{\lambda }_{0z}$, and squeezing parameter $r=0.5$. (b) $r_1=0,r_2=0.25{\lambda }_{0z}$, and $r=0.5$. (c) $r_1=0,r_2={\lambda }_{0z},\varphi =\pi /2, r=0.5$ for black line, and $r=1$ for red line. (d) $r_1=-0.125{\lambda }_{0z},r_2=0.125{\lambda }_{0z}$ for the red line; $r_1=-0.25{\lambda }_{0z},r_2=0.25{\lambda }_{0z}$ for the black line. $\varphi =0,r=0.5$.