Photon-mediated qubit interactions in one-dimensional discrete and continuous models

In this work we study numerically and analytically the interaction of two qubits in a one-dimensional waveguide, as mediated by the photons that propagate through the guide. We develop strategies to assert the Markovianity of the problem, the effective qubit-qubit interactions, and their individual and collective spontaneous emission. We prove the existence of collective Lamb shifts that affect the qubit-qubit interactions and the dependency of coherent and incoherent interactions on the qubit separation. We also develop the scattering theory associated with these models and prove single-photon spectroscopy does probe the renormalized resonances of the single- and multiqubit models, in sharp contrast to earlier toy models in which individual and collective Lamb shifts cancel.

Figure 1

Parameters for the evolution of the two-level system coupled to the one-dimensional transmission line as a function of the cutoff, computed with Eq. (17) for model (8). In (a) we fix $\mathrm{\Delta }=1$ and try different cutoffs. In (b) we fix the cutoff ${\omega }_c=10\mathrm{\Delta }$ and change ${\mathrm{\Delta }}^{\prime }$. It is evident from this plot that since $|\mathrm{\Delta }-{\mathrm{\Delta }}^{\prime }|\simeq g^2\ll 1$, there is no big difference between using $\mathrm{\Delta }$ and ${\mathrm{\Delta }}^{\prime }$ in Eq. (19).

Figure 2

(a) Parameters for the evolution of the two qubits coupled to the one-dimensional transmission line as a function of the distance between qubits, measured in units of ${\lambda }_0$, the qubit wavelength. We plot $\gamma /{\gamma }_1$ (dash-dotted line), ${\gamma }_{12}/{\gamma }_1$ (dashed line), and $2g_{12}/{\gamma }_1$ (solid line), computed using (27), using a self-consistent formula for ${\mathrm{\Delta }}_{\pm }$. (b)–(d) Errors in the parameters $g_{12}$, ${\gamma }_{12}$, and $\gamma$ made by not using a Lamb-shift renormalization (dashed line) or by using the simplified renormalization scheme (22) (solid line). In all simulations ${\omega }_c=10\mathrm{\Delta }$ and $g=0.04\mathrm{\Delta }$.

Figure 3

Single-qubit spontaneous emission rate ${\gamma }_1$ and frequency renormalization $\delta$ as a function of the model cutoff, ${\omega }_c$ in (8). We plot the numerical result obtained by solving (11) with an implicit method (solid line), together with the self-consistent expression [(24), dashed line] and the usual prediction without Lamb shift [(23), dotted line].

Figure 4

Two-qubit parameters, $\gamma /{\gamma }_1$, ${\gamma }_{12}/{\gamma }_1$, and $g_{12}/{\gamma }_1$, as a function of the model cutoff ${\omega }_c$ in (8). We plot the numerical result obtained by solving (11) with an implicit method (solid black line), together with our prediction (27) using the self-consistent solution for ${\mathrm{\Delta }}_{\pm }$ (blue dashed line) and expressions for $\gamma ,{\gamma }_{12}$, and $g_{12}$ that do not include the Lamb shifts (red dotted line).

Figure 5

(a) Qubit variables in a single numerical experiment. We plot $|c_{+}{|}^2+{\left|c_{-}\right|}^2$ (black solid line), $|c_{-}{|}^2$ (red dashed line), and $|c_{+}{|}^2$ (blue dash-dotted line). Note how $c_{+}$ and $c_{-}$ depart at the time when the photon arrives at the second qubit. The simulation parameters are $L=20\pi$, $N=800$, $g=0.04$, and $d={\lambda }_0$. (b) We extend the same plot to all other qubit separations, representing $\left|\right|c_{-}{|}^2-|c_{+}{|}^2|$ as a function of time and distance. Outside of the light cone, there is no difference between $|c_{-}|$ and $|c_{+}|$, and the qubits are uncorrelated. As a guide to the eye we plot the dashed line $d=vt$.

Figure 6

(a) Single-qubit spontaneous emission, (b) collective decay, and (c) interaction for two qubits interacting with a one-dimensional gapless waveguide, $\omega \left(k\right)={\omega }_c\sqrt{1-cos\left(k\delta x\right)}$, as a function of the qubit separation $d$. As simulation parameters we use the same gap for both qubits, $\mathrm{\Delta }$, and the speed of light $v$ as units. The waveguide has a length $L=20{\lambda }_0$ and a discretization $\delta x=L/N$, with a varying number of modes: $N=200$ (dotted line), 400 (dot-dashed line), $N=1600$ (dashed line), and $N=3200$ (solid line).

Figure 7

Renormalized gap of the first qubit obtained from the approximate period of the functions $g_{12}$ and ${\gamma }_{12}$ as a function of the cutoff ${\omega }_c$ for $\mathrm{\Delta }=1$, $g=4\%\mathrm{\Delta }$.

Figure 8

(a) Single-qubit spontaneous emission, (b) collective decay, and (c) interaction for two qubits interacting with a one-dimensional photonic crystal, $\omega \left(k\right)={\omega }_0-Jcos\left(k\right)$. We use periodic boundary conditions for qubits with gap $\mathrm{\Delta }={\omega }_0$, $N=800$, a waveguide length $L=20{\lambda }_0$, and a varying bandwidth $J/{\omega }_0=0.5$ (dotted line), $0.6$ (dashed line), $0.8$ (dot-dashed line), and 1 (solid line). Note how the period of the curves is not significantly affected by the bandwidth, $J$.

Figure 9

(a) Comparison between the results of MPS simulations, $\gamma /{\gamma }_1$, ${\gamma }_{12}/{\gamma }_1$, and $2g_{12}/{\gamma }_1$ (solid line), and the RWA method (dashed line). (b) Difference $\left|\right|c_{+}{|}^2-|c_{-}{|}^2|$ as a function of time and qubit separation, together with the light cone (dashed line). The parameters of the simulation are $N=321$, $g=0.05$, $\mathrm{\Delta }=1$, and $v=1$.