Nonreciprocal frequency conversion with chiral $\Lambda$-type atoms

Nonreciprocal photon transmissions play significant roles in fabricating various integrated photonic elements and thus have important applications ranging from quantum information processing to quantum network engineering NR1 ; NR2 ; NR3 . In particular, it is beneficial to break the Lorentz reciprocity at the single-photon level to avoid the harmful backflows between quantum devices and thereby to protect the signal sources. In general, nonreciprocal photon transmissions can be achieved via magneto-optical effects magop1 ; magop2 ; magop3 ; magop4 ; magop5 , optical nonlinearities opNL1 ; opNL2 ; opNL3 ; opNL4 , dynamic modulations dymod1 ; dymod2 ; dymod3 ; dymod4 ; dymod5 ; dymod6 ; dymod7 ; dymod8 ; dymod9 ; dymod10 ; dymod11 , synthetic magnetism syn1 ; syn2 ; syn3 ; syn4 ; syn5 ; syn6 ; syn7 , and reservoir engineerings clerkX ; clerkNP .

On the other hand, the interactions between light and matter (e.g., atoms) can also be direction dependent: atoms can interact differently with photons propagating along different directions. Such phenomena, known as “chiral quantum optics” in the literature chiralZoller ; chiralAW , provide a new paradigm for quantum network engineering. To date, significant progress has been made on the basis of chiral quantum optics, such as cascaded quantum systems cas1 ; cas2 ; cas3 ; cas4 , deterministic photon routing rout1 ; rout2 , and non-destructive photon detection detec . Moreover, non-Markovian retardation effects, e.g., time-delayed quantum feedbacks, have also been widely investigated in chiral quantum optical systems FB1 ; FB2 ; FB3 ; FB4 ; FB5 . In experiments, chiral atom-field interactions can be achieved via several approaches, such as the spin-momentum locking effect of light in one-dimensional optical fibers chiralfiber1 ; chiralfiber2 ; chiralfiber3 , inserting circulators in superconducting circuits SCchiral1 ; SCchiral2 ; SCchiral3 , topological engineering topo1 ; topo2 , and synthesizing artificial gauge fields sawtooth . In spite of the broken time-reversal symmetry, however, it has been shown that such chirality is not sufficient to support nonreciprocal scatterings solely in the absence of intrinsic atomic dissipations yanweibin . To achieve high-performance nonreciprocal transmissions, one should exploit, for instance, considerable intrinsic atomic dissipations or atoms with more transitions chirally coupled to the waveguide field rout1 ; shenchiral ; extract .

In this paper, we begin with the small-atom model considered in Ref. rout1 , where both the transitions of a $\Lambda$-type atom are chirally coupled to the waveguide at a single point. We briefly review the nonreciprocal single-photon scatterings of this model, which are allowed even if the intrinsic atomic dissipation is ignored. This is quite different from the case of a chiral two-level atom, where the intrinsic dissipations are indispensable for observing nonreciprocal scatterings. Afterwards, we extend this chiral $\Lambda$-type atom to its giant-atom version, where both the atomic transitions couple chirally to the waveguide at two separated points with the separation distance comparable to the wavelength of the waveguide field. In the past few years, nonchiral giant atoms have been well investigated fiveyears , which demonstrated a series of intriguing interference effects such as frequency-dependent Lamb shift and relaxation rate Lamb , decoherence-free interatomic interactions decofree1 ; decofree2 ; decofree3 , various bound states oscillate ; WZHbound ; GAYuan , and modified topological effects GACheng ; GATudela . Most recently, the concept of chiral quantum optics has also been brought to giant-atom structures WXprl ; AFKchiral ; WXarxiv ; DLsyn , which shows the possibility of combining the advantages of the two paradigms. Here, we study the nonreciprocal single-photon frequency conversion of the chiral $\Lambda$-type giant atom in both the Markovian and non-Markovian regimes, which are defined depending on whether the propagation time of photons between different atom-waveguide coupling points is negligible or not. In both regimes, the chiral giant-atom model exhibits intriguing interference effects such as ultra-narrow scattering windows. In particular, non-Markovicity induced nonreciprocal scatterings are demonstrated under specific conditions.

We begin by considering a $\Lambda$-type small atom, where both the atomic transitions ($|g\rangle\leftrightarrow|e\rangle$ and $|f\rangle\leftrightarrow|e\rangle$) interact chirally with the waveguide field at $x=0$, as shown in Fig. 1(a). Similar model has been considered in Ref. rout1 to demonstrate nonreciprocal few-photon scatterings. However, we briefly review this model here to pave the way for introducing the giant-atom model in the next section and facilitate the comparison between the two models. The Hamiltonian of such a small-atom model can be written as ($\hbar=1$ hereafter)

$$\begin{split}H&=H_{\text{a}}+H_{\text{w}}+H_{\text{int}},\\ H_{\text{a}}&=\omega_{f}|f\rangle\langle f|+\omega_{e}|e\rangle\langle e|,\\ H_{\text{w}}&=\int_{-\infty}^{+\infty}dx\Big{[}c_{L}^{{\dagger}}(x)\Big{(}\omega_{0}+iv_{g}\frac{\partial}{\partial x}\Big{)}c_{L}(x)\\ &\quad\,+c_{R}^{{\dagger}}(x)\Big{(}\omega_{0}-iv_{g}\frac{\partial}{\partial x}\Big{)}c_{R}(x)\Big{]},\\ H_{\text{int}}&=\int_{-\infty}^{\infty}dx\delta(x)\Big{\{}|g\rangle\langle e|[g_{R}c_{R}^{{\dagger}}(x)+g_{L}c_{L}^{{\dagger}}(x)]\\ &\quad+|f\rangle\langle e|[\xi_{R}c_{R}^{{\dagger}}(x)+\xi_{L}c_{L}^{{\dagger}}(x)]+\text{H.c.}\Big{\}},\end{split}$$

(1)

where $H_{\text{a}}$ is the free Hamiltonian of the atom, with $\omega_{f}$ and $\omega_{e}$ the energies of the states $|f\rangle$ and $|e\rangle$ with respect to the state $|g\rangle$, respectively; $H_{\text{w}}$ is the Hamiltonian of the waveguide, with $c_{R}^{{\dagger}}(x)$ [$c_{L}^{{\dagger}}(x)$] the creation operator of the right-moving (left-moving) waveguide mode at position $x$ and $v_{g}$ the group velocity; $\omega_{0}$ is a chosen frequency that is away from the cutoff frequency, around which the dispersion relation of the waveguide can be linearized as $E=\omega_{0}+kv_{g}$ ($k$ is the renormalized wave vector of photons in the waveguide) shen2005 ; shen2009 ; $H_{\text{int}}$ describes the interactions between the atomic transitions and the waveguide modes, with $g_{R}$ and $g_{L}$ ($\xi_{R}$ and $\xi_{L}$) the coupling strengths of the transition $|g\rangle\leftrightarrow|e\rangle$ ($|f\rangle\leftrightarrow|e\rangle$) with the right-moving and left-moving waveguide modes (assumed to be real), respectively. For simplicity, we assume $\xi_{R}=g_{R}$ and $\xi_{L}=g_{L}$ in this section. Moreover, we have assumed that the two lower states $|g\rangle$ and $|f\rangle$ are quasi-degenerate, such that both the transition frequencies $\omega_{e}$ and $\omega_{e}-\omega_{f}$ fall into the linearized region around $\omega_{0}$ NJPLam ; shenLam ; DLlambda ; FanLambda . The case of nondegenerate lower states will be discussed in Sec. III.4, where the group velocities corresponding to the two transition frequencies are different, yet our main results are shown to be insensitive to such a difference.

In the single-excitation space, the eigenstate of Hamiltonian (1) can be written as

$$\begin{split}|\psi\rangle&=\sum_{\alpha=g,f}\int_{-\infty}^{+\infty}dx\Big{[}R_{\alpha}(x)c_{R}^{{\dagger}}(x)\\ &\quad\,+L_{\alpha}(x)c_{L}^{{\dagger}}(x)\Big{]}|0,\alpha\rangle+u_{e}|0,e\rangle,\end{split}$$

(2)

where $R_{\alpha}(x)$ [$L_{\alpha}(x)$] is the probability amplitude of finding a right-moving (left-moving) photon in the waveguide at position $x$ and the atom finally in the state $|\alpha\rangle$; $u_{e}$ is the probability amplitude of the atom in the excited state $|e\rangle$. Solving the stationary Schrödinger equation $H|\psi\rangle=E|\psi\rangle$ with Eqs. (1) and (2), one can obtain

$$\begin{split}ER_{g}(x)&=\Big{(}\omega_{0}-iv_{g}\frac{\partial}{\partial x}\Big{)}R_{g}(x)+g_{R}\delta(x)u_{e},\\ EL_{g}(x)&=\Big{(}\omega_{0}+iv_{g}\frac{\partial}{\partial x}\Big{)}L_{g}(x)+g_{L}\delta(x)u_{e},\\ ER_{f}(x)&=\Big{(}\omega_{0}+\omega_{f}-iv_{g}\frac{\partial}{\partial x}\Big{)}R_{f}(x)+g_{R}\delta(x)u_{e},\\ EL_{f}(x)&=\Big{(}\omega_{0}+\omega_{f}+iv_{g}\frac{\partial}{\partial x}\Big{)}L_{f}(x)+g_{L}\delta(x)u_{e},\\ \Delta u_{e}&=g_{R}[R_{g}(0)+R_{f}(0)]+g_{L}[L_{g}(0)+L_{f}(0)],\end{split}$$

(3)

where $\Delta=E-\omega_{e}$ is the detuning between the propagating photons and the transition $|g\rangle\leftrightarrow|e\rangle$.

We first consider that a single photon is incident from the left side of the waveguide and the atom occupies the state $|g\rangle$ initially. If the atom is excited by the photon, then the atom can spontaneously decay back to $|g\rangle$ and reemit a photon with the same wave vector $k$, or it can decay to another lower state $|f\rangle$ and reemit a photon with wave vector $q=k-\omega_{f}/v_{g}$ NJPLam ; shenLam ; DLlambda ; FanLambda . We refer to these two processes as elastic and inelastic scatterings, respectively, depending on whether the frequency of the input photon is converted or not. For the left-incident photon here, one can assume

$$\begin{split}R_{g}(x)&=[\Theta(-x)+t_{1}\Theta(x)]e^{ikx},\\ L_{g}(x)&=r_{1}\Theta(-x)e^{-ikx},\\ R_{f}(x)&=t_{2}\Theta(x)e^{iqx},\\ L_{f}(x)&=r_{2}\Theta(-x)e^{-iqx},\end{split}$$

(4)

where $\Theta(x)$ is the Heaviside step function; $t_{1}$ and $r_{1}$ ($t_{2}$ and $r_{2}$) are the transmission and reflection coefficients of the elastic (inelastic) scattering processes, respectively. Substituting Eq. (4) into Eq. (3), one readily obtains

$$\begin{split}t_{1}&=\frac{\Delta+i\Gamma_{L}}{\Delta+i(\Gamma_{L}+\Gamma_{R})},\\ t_{2}&=\frac{-i\Gamma_{R}}{\Delta+i(\Gamma_{R}+\Gamma_{L})},\end{split}$$

(5)

where $\Gamma_{R}=g_{R}^{2}/v_{g}$ and $\Gamma_{L}=g_{L}^{2}/v_{g}$ are the radiative decay rates of $|e\rangle$ induced by the right-moving and left-moving waveguide modes, respectively. Note that we only focus on the two transmission coefficients in this paper, with which it is enough to understand the underlying physics of our models.

Similarly, for a single photon incident from the right side of the waveguide (the atom is still assumed in the state $|g\rangle$ initially), with the ansatz

$$\begin{split}R_{g}(x)&=\tilde{r}_{1}\Theta(x)e^{ikx},\\ L_{g}(x)&=[\Theta(x)+\tilde{t}_{1}\Theta(-x)]e^{-ikx},\\ R_{f}(x)&=\tilde{r}_{2}\Theta(x)e^{iqx},\\ L_{f}(x)&=\tilde{t}_{2}\Theta(-x)e^{-iqx},\end{split}$$

(6)

the elastic and inelastic transmission coefficients in this case can be solved as

$$\begin{split}\tilde{t}_{1}&=\frac{\Delta+i\Gamma_{R}}{\Delta+i(\Gamma_{R}+\Gamma_{L})},\\ \tilde{t}_{2}&=\frac{-i\Gamma_{L}}{\Delta+i(\Gamma_{R}+\Gamma_{L})}.\end{split}$$

(7)

It is clear from Eqs. (5) and (7) that nonreciprocal scatterings (i.e., $|t_{1}|\neq|\tilde{t}_{1}|$ and/or $|t_{2}|\neq|\tilde{t}_{2}|$) can be achieved if $g_{R}\neq g_{L}$ (i.e., $\Gamma_{R}\neq\Gamma_{L}$), yet this is impossible for the case of a chiral two-level atom in the absence of intrinsic atomic dissipations (see Appendix A for more details). Physically, the nonreciprocity here is because, for an input photon that can be absorbed by the transition $|g\rangle\leftrightarrow|e\rangle$, the other transition $|f\rangle\leftrightarrow|e\rangle$ effectively serves as a “dissipation channel”, which makes the elastic scattering process behaves like that of a two-level atom with intrinsic dissipations. The excitation probabilities of the atom are different for photons propagating along opposite directions due to the chiral atom-waveguide couplings, which thus result in nonreciprocal elastic and inelastic scatterings.

Figures 2(a) and 2(b) depict the transmission contrasts $I_{1}=T_{1}-\tilde{T}_{1}$ and $I_{2}=T_{2}-\tilde{T}_{2}$ versus the detuning $\Delta$ and the decay ratio $\Gamma_{L}/\Gamma_{R}$, where $T_{1}=|t_{1}|^{2}$ and $\tilde{T}_{1}=|\tilde{t}_{1}|^{2}$ ($T_{2}=|t_{2}|^{2}$ and $\tilde{T}_{2}=|\tilde{t}_{2}|^{2}$) are the elastic (inelastic) transmission rates for the left-incident and right-incident photons, respectively. It can be seen that the optimal nonreciprocal scatterings ($|I_{1}|=|I_{2}|=1$) occur if $\Gamma_{L}=0$ (or $\Gamma_{R}=0$, leading to reversed nonreciprocal scatterings), while the scatterings are totally reciprocal ($I_{1}=I_{2}=0$) if $\Gamma_{L}=\Gamma_{R}$. To show more details, we also plot in Figs. 2(c) and 2(d) the profiles of the transmission rates of two cases, i.e., the ideal chiral case with $\Gamma_{L}/\Gamma_{R}=0$ and the nonideal chiral case with $\Gamma_{L}/\Gamma_{R}=2$. In the ideal chiral case, the elastic (inelastic) transmission rate of the left-incident single photon shows a standard anti-Lorentzian (Lorentzian) line shape and efficient frequency conversion with $T_{2}=1$ can be achieved, whereas for a right-incident photon, the inelastic scattering is totally suppressed ($\tilde{T}_{2}\equiv 0$) because the atom is decoupled with the left-moving waveguide mode. In the nonideal chiral case, the atom can still exhibit nonreciprocal transmissions, yet both the transmission contrasts and the conversion efficiency are degraded in this case. In view of this, we focus on the ideal chiral case hereafter in this paper.

Now we consider a giant-atom version of the chiral $\Lambda$-type atom, where both the transitions $|g\rangle\leftrightarrow|e\rangle$ and $|f\rangle\leftrightarrow|e\rangle$ are coupled chirally with the waveguide modes at two separate points $x_{1}=0$ and $x_{2}=d$, as shown in Fig. 1(b). The case of a nonchiral $\Lambda$-type giant atom has been investigated in our previous work DLlambda , where the optimal frequency conversion efficiency is still at most one half in spite of the giant-atom interference effects. In this section, we would like to demonstrate that while the chiral couplings enable nonreciprocal and efficient frequency conversion as shown above, the giant-atom structure brings some intriguing interference effects, especially in the non-Markovian regime.

For the giant atom considered here, the interaction Hamiltonian becomes

$$\begin{split}H_{\text{int}}&=\sum_{\alpha=g,f}\int_{-\infty}^{\infty}dx|\alpha\rangle\langle e|\Big{\{}c_{R}^{{\dagger}}(x)[g_{1,R}\delta(x)\\ &\quad\,+g_{2,R}\delta(x-d)]+c_{L}^{{\dagger}}(x)[g_{1,L}\delta(x)\\ &\quad\,+g_{2,L}\delta(x-d)]\Big{\}}+\text{H.c.}.\end{split}$$

(8)

Here, $g_{1,R}$ and $g_{2,R}$ ($g_{1,L}$ and $g_{2,L}$) are the coupling strengths (assumed to be real again) between the atomic transitions and the right-moving (left-moving) waveguide modes at the points $x=x_{1}$ and $x=x_{2}$, respectively. Similar to the small-atom case, we have assumed that both the atomic transitions couple to the right-moving (left-moving) waveguide mode with the same strength $g_{j,R}$ ($g_{j,L}$) at each coupling point ($j=1,2$). In this case, the probability amplitudes of the waveguide modes can be written as

$$\begin{split}R_{g}(x)&=\{\Theta(-x)+A[\Theta(x)-\Theta(x-d)]\\ &\quad\,+t_{1}\Theta(x-d)\}e^{ikx},\\ L_{g}(x)&=\{r_{1}\Theta(-x)+B[\Theta(x)-\Theta(x-d)]\}e^{-ikx},\\ R_{f}(x)&=\{M[\Theta(x)-\Theta(x-d)]+{t}_{2}\Theta(x-d)\}e^{iqx},\\ L_{f}(x)&=\{r_{2}\Theta(-x)+N[\Theta(x)-\Theta(x-d)]\}e^{-iqx}\end{split}$$

(9)

for a left-incident photon and

$$\begin{split}R_{g}(x)&=\{\tilde{r}_{1}\Theta(x-d)+\tilde{A}[\Theta(x)-\Theta(x-d)]\}e^{ikx},\\ L_{g}(x)&=\{\Theta(x-d)+\tilde{B}[\Theta(x)-\Theta(x-d)]\\ &\quad\,+\tilde{t}_{1}\Theta(-x)\}e^{-ikx},\\ R_{f}(x)&=\{\tilde{r}_{2}\Theta(x-d)+\tilde{M}[\Theta(x)-\Theta(x-d)]\}e^{iqx},\\ L_{f}(x)&=\{\tilde{N}[\Theta(x)-\Theta(x-d)]+\tilde{t}_{2}\Theta(-x)\}e^{-iqx}\end{split}$$

(10)

for a right-incident photon. Here $A$ and $B$ ($M$ and $N$) are the probability amplitudes of finding right-moving and left-moving photons with wave vector $k$ ($q$) in the region of $x_{1}<x<x_{2}$, respectively, if the photon is incident from the left side, and those with tildes correspond to a right-incident photon. Then the transmission coefficients can be solved as

$$\begin{split}t_{1}&=\frac{\Delta+i(\Gamma_{1,L}+\Gamma_{2,L}+F_{-})}{\Delta+i(\Gamma_{1,R}+\Gamma_{2,R}+\Gamma_{1,L}+\Gamma_{2,L}+F_{+})},\\ t_{2}&=\frac{-i[\Gamma_{1,R}+\Gamma_{2,R}e^{i(\phi_{1}-\phi_{2})}+\Gamma_{12,R}(e^{i\phi_{1}}+e^{-i\phi_{2}})]}{\Delta+i(\Gamma_{1,R}+\Gamma_{2,R}+\Gamma_{1,L}+\Gamma_{2,L}+F_{+})},\\ \tilde{t}_{1}&=\frac{\Delta+i(\Gamma_{1,R}+\Gamma_{2,R}+\tilde{F}_{-})}{\Delta+i(\Gamma_{1,R}+\Gamma_{2,R}+\Gamma_{1,L}+\Gamma_{2,L}+F_{+})},\\ \tilde{t}_{2}&=\frac{-i[\Gamma_{1,L}+\Gamma_{2,L}e^{i(\phi_{2}-\phi_{1})}+\Gamma_{12,L}(e^{-i\phi_{1}}+e^{i\phi_{2}})]}{\Delta+i(\Gamma_{1,R}+\Gamma_{2,R}+\Gamma_{1,L}+\Gamma_{2,L}+F_{+})},\end{split}$$

(11)

with

$$\begin{split}F_{+}&=(\Gamma_{12,R}+\Gamma_{12,L})(e^{i\phi_{1}}+e^{i\phi_{2}}),\\ F_{-}&=\Gamma_{12,R}(e^{i\phi_{2}}-e^{-i\phi_{1}})+\Gamma_{12,L}(e^{i\phi_{1}}+e^{i\phi_{2}}),\\ \tilde{F}_{-}&=\Gamma_{12,R}(e^{i\phi_{1}}+e^{i\phi_{2}})+\Gamma_{12,L}(e^{i\phi_{2}}-e^{-i\phi_{1}}),\end{split}$$

(12)

where $\Gamma_{j,\beta}=g_{j,\beta}^{2}/v_{g}$ and $\Gamma_{12,\beta}=\sqrt{\Gamma_{1,\beta}\Gamma_{2,\beta}}$ ($\beta=L,R$); $\phi_{1}=kd=(\omega_{e}-\omega_{0}+\Delta)\tau$ and $\phi_{2}=qd=(\omega_{e}-\omega_{f}-\omega_{0}+\Delta)\tau$ are the phases of photons accumulated between the two atom-waveguide coupling points, with $\tau=d/v_{g}$ the corresponding propagation time. Clearly, both $\phi_{1}$ and $\phi_{2}$ consist of a constant part and a $\Delta$-dependent part, such that we define $\phi_{1}=\phi_{1,0}+\tau\Delta$ and $\phi_{2}=\phi_{2,0}+\tau\Delta$ for simplicity with $\phi_{1,0}=(\omega_{e}-\omega_{0})\tau$ and $\phi_{2,0}=(\omega_{e}-\omega_{f}-\omega_{0})\tau$. In this way, we can study single-photon scatterings in both the Markovian and non-Markovian regimes, depending on whether the propagation time $\tau$ is negligible or not, as discussed in detail below. Note that we focus on the ideal chiral case of $\Gamma_{1,L}=\Gamma_{2,L}=0$ in this section, which enables efficient frequency conversion as discussed above.

In summary, we have generalized a chiral quantum optical model, where both the transitions of a $\Lambda$-type atom couple chirally to the waveguide field, to its giant-atom version by assuming that the chiral $\Lambda$-type atom interacts twice with the waveguide at two separated points. We have studied their nonreciprocal single-photon scatterings that are impossible for the case of a chiral two-level atom in the absence of intrinsic atomic dissipations. Compared with the small-atom case, the chiral giant-atom model exhibits intriguing interference effects that are closely related to the propagation time of photons between the two atom-waveguide coupling points, such as ultra-narrow scattering windows. In particular, we have explored the scattering behaviors of the chiral giant-atom model in both the Markovian and non-Markovian regimes, which are defined depending on whether the propagation time is negligible or not. It has been shown that the scattering behaviors are quite different in the two regimes, especially when either of the two atomic transitions is completely suppressed due to the destructive interferences: in this case, the scatterings are always reciprocal in the Markovian regime, while the non-Markovicity induced nonreciprocity with multiple nonreciprocal scattering windows can be observed in the non-Markovian regime. The results in this paper provide a deeper sight into the combination of chiral quantum optics and giant-atom physics, and have potential applications in, e.g., integrated photonics and quantum network engineering.

We would like to point out that the investigations of multi-level giant atoms are still at the initial stage. More peculiar phenomena can be expected based on the marriage of multi-level giant atoms and various quantum effects. For instance, one can consider a $\Delta$-type giant atom, whose interference effect arising from its closed-loop level structure may bring more intriguing effects. Moreover, one can also extend some exotic effects for two-level giant atoms to their multi-level versions, such as oscillating bound states in the non-Markovian regime oscillate and decoherence-free interactions between giant atoms decofree1 ; decofree2 ; decofree3 .

This work is supported by the National Natural Science Foundation of China (under Grants No. 11774024, No. 12074030, and No. U1930402) and the Science Challenge Project (Grant No. TZ2018003).