Nonreciprocal few-photon routing schemes based on chiral waveguide-emitter couplings

We demonstrate the possibility of designing efficient, nonreciprocal few-photon devices by exploiting the chiral coupling between two waveguide modes and a single quantum emitter. We show how this system can show nonreciprocal photon transport at the single-photon level by exploiting a single-photon routing mechanism. Afterwards, we also demonstrate how the fundamentally different two-photon response makes the system show a transistorlike behavior, where a first photon can open a transmission channel for a second incoming photon. The efficiency in both cases is shown to be large for feasible experimental implementations. Our results illustrate the potential of chiral waveguide-emitter couplings for applications in quantum circuitry.

Figure 1

(a) Scheme of the system under study. A three level system in $\mathrm{\Lambda }$ configuration interacts with two independent waveguides, labeled $u$ and $d$. The transition $|g\rangle \leftrightarrow |e\rangle$, depicted in blue, is chirally coupled to the right- and left-propagating photons of the bottom waveguide, with coupling rates ${\gamma }_{dR}$ and ${\gamma }_{dL}$ respectively. The second transition, $|s\rangle \leftrightarrow |e\rangle$ (in red) is in turn chirally coupled to the upper waveguide, with coupling rates ${\gamma }_{uR}$ and ${\gamma }_{uL}$. Finally, the excited state $|e\rangle$ may decay radiatively into free space modes at a rate ${\mathrm{\Gamma }}^{*}$. The usual transmission and reflection amplitudes are named $t$ and $r$ respectively, whereas the processes by which the photon is routed into the second waveguide have scattering coefficients $\tilde{t}$ and $\tilde{r}$, corresponding to right and left propagating photons respectively. (b) Inverted W system in which two optically excited states $|f_{d,u}\rangle$ are connected to $|e\rangle$ through an off-resonant classical field with amplitude ${\mathrm{\Omega }}_{d,u}$ and detuning ${\mathrm{\Delta }}_{d,u}$, and to $|g\rangle /|s\rangle$ through the lower or upper waveguide. When $|{\mathrm{\Delta }}_{d,u}|\gg {\mathrm{\Omega }}_{d,u}$, the system is equivalent to that of panel (a), with renormalized coupling strengths ${\gamma }_{i\nu }\to \left(|{\mathrm{\Omega }}_i{|}^2/{\mathrm{\Delta }}_i^2\right){\gamma }_{i\nu }$ [and spontaneous emission ${\mathrm{\Gamma }}^{*}\to {\sum }_i\left(|{\mathrm{\Omega }}_i{|}^2/{\mathrm{\Delta }}_i^2\right){\mathrm{\Gamma }}^{*}]$.

Figure 2

(a) The single-photon transmittance for a qubit nonchirally coupled to a waveguide vanishes due to a destructive interference. (b) When the coupling is maximally chiral, however, the reflection is canceled and the balance between the previously interfering amplitudes is broken, resulting in full transmission. (c) If an extra decay channel is added to the qubit, perfect interference can be achieved again, and both transmission and reflection are canceled. (d) Our scheme uses a second waveguide to collect the photon emitted through the extra channel, achieving full routing.

Figure 3

(a) Total routing probability into port 3 vs Purcell factor, for different directionalities $D_d=D_u$. (b) Reflection probability determining the efficiency of the diode vs directionality $D_d$. Inset: Rectifier acting as a single-photon diode with respect to the bottom waveguide. A photon introduced through port 1 is routed with probability 1, and cannot reach port 2. On the other hand, a photon in port 2 is transmitted to port 1 with maximum probability. In both panels we fix $t=0$.

Figure 4

Performance of the single-photon transistor. (a) Success probability $P_{23}$ as a function of the directionalities $D_d$ and $D_u$. Each subpanel corresponds to a different value for the Purcell factor, and its domain is constrained by the fundamental limit Eq. (17). (b) $P_{23}$ vs Purcell factor $P_F$, for different values of the directionalities $D_d=D_u$. In both panels the couplings are tuned to fulfill $t=0$.