Waveguide QED: Power spectra and correlations of two photons scattered off multiple distant qubits and a mirror

We study two-level systems (2LS) coupled at different points to a one-dimensional waveguide in which one end is open and the other is either open (infinite waveguide) or closed by a mirror (semi-infinite). Upon injection of two photons (corresponding to weak coherent driving), the resonance fluorescence and photon correlations are shaped by the effective qubit transition frequencies and decay rates, which are substantially modified by interference effects. In contrast to the well-known result in an infinite waveguide, photons reflected by a single 2LS coupled to a semi-infinite waveguide are initially bunched, a result that can be simply explained by stimulated emission. As the number of 2LS increases (up to 10 are considered here), rapid oscillations build up in the correlations that persist for a very long time. For instance, when the incoming photons are slightly detuned, the transmitted photons in the infinite waveguide are highly antibunched. On the other hand, upon resonant driving, incoherently reflected photons are mostly distributed within the photonic band gap and several sharp side peaks. These features can be explained by considering the poles of the single-particle Green function in the Markovian regime combined with the time delay. Our calculation is not restricted to the Markovian regime, and we obtain several fully non-Markovian results. We show that a single 2LS in a semi-infinite waveguide can not be decoupled by placing it at the node of the photonic field, in contrast to recent results in the Markovian regime. Our results illustrate the complexities that ensue when several qubits are strongly coupled to a bus (the waveguide) as might happen in quantum information processing.

Figure 1

Schematic of the waveguide-QED system, in which equally separated, identical 2LS are coupled to a semi-infinite waveguide with one open end and another closed at $x=0$. For an infinite waveguide with two open ends, the 2LS are instead placed symmetrically with respect to $x=0$ to simplify the calculation.

Figure 2

Normalized power spectra (resonance fluorescence) of multiple qubits (from top to bottom: $N=1,2,3,5,10$) coupled to the infinite waveguide with $k_{0}L=\pi /2$ (separation $L={\lambda }_0/4$). For the first column, the incoming photons are on resonance, $E/2={\omega }_0=100\mathrm{\Gamma }$; for the second column, the frequency is chosen such that the single-photon transmission is 50% in each case. The total fluorescence (black solid line) is broken down into the reflected (blue dashed) and transmitted (red dotted) components. The vertical lines indicate the real part of the poles. For $N=10$ in the off-resonant case, the height of the central peak goes up to $\sim 60$, which is not shown for better visibility. The frequencies used in the second column are $E/2\mathrm{\Gamma }=\{99.5,99.29,99.34,99.43,99.48\}$ (from top to bottom) [105].

Figure 3

Time delay $\tau$ (top), single-photon transmission spectrum $T={\left|t\left(k\right)\right|}^2$ (middle), and poles of the transmission amplitude $t\left(k\right)$ (bottom) as a function of frequency. The system consists of 10 qubits coupled to an infinite waveguide with $k_{0}L=\pi /4$ (left column) and $k_{0}L=\pi /2$ (right column). For the sake of clarity, we show only the red-detuned side; for $k_{0}L=\pi /2$, the poles are symmetric with respect to the qubit frequency ${\omega }_0=100\mathrm{\Gamma }$, while for $k_{0}L=\pi /4$ five poles are not shown. The vertical lines indicate the real parts of the poles. In panel (d), the black squares give the incident frequencies used in Fig. 8, the red dots give those used in Fig. 9, and the dashed line labels $T=50\%$. The dashed-dotted lines in panels (e) and (f) label the origin ($\tilde{\mathrm{\Gamma }}=0$).

Figure 4

Normalized power spectra (resonance fluorescence) of multiple qubits (from top to bottom: $N=2,3,5,10$) coupled to the infinite waveguide with separation $k_{0}L=\pi /4$ ($L={\lambda }_0/8$). The incoming photon frequency is $E/2={\omega }_0=100\mathrm{\Gamma }$ for the first column (on resonance) and is chosen such that $T=50\%$ for the second column. The total fluorescence (black solid line) is broken down into the reflected (blue dashed) and transmitted (red dotted) components. The vertical lines indicate the real part of the poles. The frequencies used in the second column are $E/2\mathrm{\Gamma }=\{99.66,99.73,99.77,99.78\}$ (from top to bottom) [105].

Figure 5

Reflection $g_2$ of multiple qubits (from left to right: $N=1,2,3,5,10$) coupled to an infinite waveguide with separation $k_{0}L=\pi /2$ (first row) and $\pi /4$ (second row). The two incoming photons are on resonance ($E/2={\omega }_0=100\mathrm{\Gamma }$). This comparison shows that the limitation $g_2\left(0\right)=0$ for single 2LS is removed by adding more 2LS, and that a chain of few 2LS can cause long-time beating.

Figure 6

In the off-resonant case, $g_2$ of multiple qubits (left: $N=5$; right: $N=10$) coupled to an infinite waveguide with separation $k_{0}L=\pi /2$. The first row is for two transmitted photons and the second for two reflected ones. The gray, dashed line is the $N=1$ result serving as a reference. The solid curve is calculated using the Markovian approximation while the full non-Markovian result is given by the dots. The frequency of the incoming photons is chosen such that $T=50\%$, and the qubit frequency is ${\omega }_0=100\mathrm{\Gamma }$.

Figure 7

In the off-resonant case, $g_2$ of multiple qubits (left: $N=5$; right: $N=10$) coupled to an infinite waveguide with separation $k_{0}L=\pi /4$. The first row is for two transmitted photons and the second for two reflected ones. The gray, dashed line is the $N=1$ result serving as a reference. The frequency of the incoming photons is chosen such that $T=50\%$, and the qubit frequency is ${\omega }_0=100\mathrm{\Gamma }$.

Figure 8

$g_2$ of 10 qubits coupled to an infinite waveguide with separation $k_{0}L=\pi /2$. The first (second) row is for two transmitted (reflected) photons. The driving frequencies are chosen such that $T=50\%$ and are labeled as black squares on the single-photon transmission spectrum in Fig. 3. The qubit frequency is ${\omega }_0=100\mathrm{\Gamma }$.

Figure 9

Transmission $g_2$ of 10 qubits in an infinite waveguide with separation $k_{0}L=\pi /2$. The driving frequencies are chosen such that the single-particle transmission is (a) 20%, (b) 50%, and (c) 80% [labeled by red dots in Fig. 3], close to the most subradiant pole. The qubit frequency is ${\omega }_0=100\mathrm{\Gamma }$.

Figure 10

Normalized power spectra of one or two qubits coupled to a semi-infinite waveguide with qubit-qubit separation $k_{0}L=\pi /2$ (for $N=2$). The qubit-mirror separation is $k_{0}a=\pi /2$ in (a) and (c), and $\pi /4$ in (b). The qubit frequency is ${\omega }_0=100\mathrm{\Gamma }$.

Figure 11

Normalized power spectra of two qubits coupled to a semi-infinite waveguide with qubit-qubit separation $k_{0}L=\pi /2$ and qubit-mirror separation $k_{0}a=\pi /4$. The driving frequencies for each plot are $E/2\mathrm{\Gamma }=96.5,99,100,101,\text{and}103.5$ (from bottom to top). The black ticks label the position of the two poles (and the qubit frequency ${\omega }_0=100\mathrm{\Gamma }$ is at the center), and the blue arrow indicates the incident frequency.

Figure 12

A representative case of (a) $S\left(\omega \right)$ and (b) $g_2$ for 10 qubits coupled to the semi-infinite waveguide with $k_{0}L=k_{0}a=\pi /2$. The system is driven resonantly ($E/2={\omega }_0=100\mathrm{\Gamma }$). The vertical lines indicate the real parts of the poles.

Figure 13

$g_2$ of a single qubit coupled to a semi-infinite waveguide. The qubit-mirror separation is (a), (c) $k_{0}a=\pi /2$ and (b), (d) $\pi /4$. The frequency of the incoming photons is (a), (b) resonant with the 2LS ($E/2={\omega }_0$) and (c), (d) detuned by $+1\mathrm{\Gamma }$. Due to the modulated effective qubit frequency [Eq. (13)], for $\pi /4$ the $g_2$ with detuning $-1\mathrm{\Gamma }$ is same as the resonant case; for $\pi /2$ the $g_2$ with detuning $-1\mathrm{\Gamma }$ is same as the $+1\mathrm{\Gamma }$ detuned case. The dots in panel (d) are the results of the full non-Markovian numerical calculation, and the solid curves are based on the Markovian approximation. The qubit frequency is ${\omega }_0=100\mathrm{\Gamma }.$

Figure 14

$g_2$ of a single qubit coupled to a semi-infinite waveguide with resonant incoming photons. The solid (red) curves are based on the Markovian approximation while the (blue) dots result from the full non-Markovian numerical calculation. (a) $k_{0}a=41\pi /2$. Note the breakdown of the Markovian approximation for this large value of $a$. (b) $k_{0}a=20\pi$, thus the qubit is at a node of the single-photon wave function ($a=10{\lambda }_0$). In the Markovian approximation, the qubit is decoupled from the waveguide and $g_2\left(t\right)=1$. Clearly, this is not the case in the full solution: there is both bunching and antibunching. The qubit frequency is ${\omega }_0=100\mathrm{\Gamma }.$

Figure 15

$g_2$ of two qubits coupled to a semi-infinite waveguide with qubit-qubit separation $k_{0}L=\pi /2$. The qubit-mirror separation in the first column is $k_{0}a=\pi /2$ and in the second $\pi /4$. The first row has resonant driving ($E/2={\omega }_0=100\mathrm{\Gamma }$) and the second is detuned by $-1\mathrm{\Gamma }$.