Cavity-Free Optical Isolators and Circulators Using a Chiral Cross-Kerr Nonlinearity

Optical nonlinearity has been widely used to try to produce optical isolators. However, this is very difficult to achieve due to dynamical reciprocity. Here, we show the use of the chiral cross-Kerr nonlinearity of atoms at room temperature to realize optical isolation, circumventing dynamical reciprocity. In our approach, the chiral cross-Kerr nonlinearity is induced by the thermal motion of $N$-type atoms. The resulting cross phase shift and absorption of a weak probe field are dependent on its propagation direction. This proposed optical isolator can achieve more than 30 dB of isolation ratio, with a low loss of less than 1 dB. By inserting this atomic medium in a Mach-Zehnder interferometer, we further propose a four-port optical circulator with a fidelity larger than 0.9 and an average insertion loss less than 1.6 dB. Using atomic vapor embedded in an on-chip waveguide, our method may provide chip-compatible optical isolation at the single-photon level of a probe field.

Figure 1

(a) Schematic diagram of our setup for an optical isolator and circulator. For an optical isolator, we consider only the upper channel embedded with a cloud of $N$-type atoms. The photon passing through the atoms suffers a phase shift $\varphi$ and an amplitude transmission of $\xi$, dependent on its propagation direction. To perform an optical circulator, the lower channel is added to form a Mach-Zehnder interferometer, using beam splitters BS1 and BS2, with the upper one. In the lower branch, a phase shift $\vartheta$ is used to compensate the phase shift of the backward-moving photon in the upper branch. (b) Level diagram of $N$-type atoms. The switching (${\mathrm{\Omega }}_s$), coupling (${\mathrm{\Omega }}_c$), and probe (${\mathrm{\Omega }}_p$) fields couple to the transitions $|1⟩\leftrightarrow |2⟩$, $|3⟩\leftrightarrow |2⟩$, and $|3⟩\leftrightarrow |4⟩$, with detunings ${\mathrm{\Delta }}_s$, ${\mathrm{\Delta }}_c$, and ${\mathrm{\Delta }}_p$, respectively.

Figure 2

Transmission of an isolator for the right-moving (blue curves) and left-moving (red curves) probe fields and the isolation ratio (green curves associating with the right vertical axis) as a function of the probe detuning ${\mathrm{\Delta }}_p$. Solid (dashed) curves are for $L=2(4)\mathrm{cm}$. Other parameters are $N_a=5\times {10}^{12}{\mathrm{cm}}^{-3}$, ${\mathrm{\Gamma }}_3=0.1{\gamma }_0$, ${\mathrm{\Omega }}_c=20{\gamma }_0$, ${\mathrm{\Omega }}_s=4{\gamma }_0$, and $\delta =0$.

Figure 3

Circulator performance versus the detuning ${\mathrm{\Delta }}_p$. (a) Phase shifts (blue curves) and transmission amplitudes (red curves) for right- (solid curves) and left-moving (dashed lines) probe fields as a function of the probe detuning ${\mathrm{\Delta }}_p$. (b) Fidelity (green curves) and average insertion loss (blue dashed curves) of an optical circulator as a function of ${\mathrm{\Delta }}_p$. The vertical black dashed lines in the two figures show the optimal detuning ${\mathrm{\Delta }}_p^{\mathrm{opt}}/{\gamma }_0=7.77$ when ${\varphi }_f^{\mathrm{opt}}-{\varphi }_b^{\mathrm{opt}}=\pi$ and ${\xi }_f^{\mathrm{opt}}\approx {\xi }_b^{\mathrm{opt}}\approx 0.66$. The length of the atomic medium is 3.33 mm. Other parameters are as in Fig. 2.

Figure 4

Transmission matrix of an optical circulator at the optimal point ${\mathrm{\Delta }}_p^{\mathrm{opt}}/{\gamma }_0=7.77$. The numbers inside the color squares are the transmission between the two ports. The transmissions in white areas are zero. Other parameters are as in Fig. 3.