Unconventional Quantum Electrodynamics with Hofstadter-Ladder Waveguide

The interactions between quantum emitters and the unavoidable baths with large degrees of freedom are the central topic of quantum optics Cohen-Tannoudji et al. (1998); Clerk et al. (2010). For example, in the present of a vacuum bath, the emitter will spontaneously decay to its ground state as well as its frequency being renormalized due to Lamb shifts Lamb and Retherford (1947); Scully and Zubairy (1997). By shaping the size of the environment or narrowing its spectrum bandwidth, many intriguing phenomena, such as isotropic propagation of photons and non-Markovian evolution arise John and Quang (1994); Lambropoulos et al. (2000); Giraldi and Petruccione (2011); Lodahl et al. (2015); Stewart et al. (2020); Ferreira et al. (2021). As discussed in Refs. Petersen et al. (2014); Bliokh and Nori (2015); Bliokh et al. (2015); Lodahl et al. (2017); Lang et al. (2022), chiral emission can be observed via the subwavelength confinement in nanophotonic systems, which opens the possibilities to realize cascaded quantum networks. Moreover, when considering an emitter coupling to the bandgaps of a bath John and Wang (1990); Goban et al. (2014); González-Tudela et al. (2015); Douglas et al. (2016); Liu and Houck (2017); Chang et al. (2018); Wang et al. (2021a), photonic bound states (in the form of an evanescent field) emerge Douglas et al. (2015). In this scenario, between atoms there are long-range dipole-dipole interactions by exchanging the virtual photons in the waveguide Shahmoon and Kurizki (2013); Ying et al. (2019).

In recent years, exploring quantum electrodynamics (QED) with emitters coupling to structured lattice environments, has attracted a lot of interests Ramos et al. (2016); Calajo et al. (2016); González-Tudela and Cirac (2018); González-Tudela et al. (2019); Leonforte et al. (2021). Those artificial lattice reservoirs are widely studied in condensed matter physics, and usually have unconventional spectra, or topological properties with nontrivial vacuum modes. In Refs. González-Tudela and Cirac (2017, 2017), by considering emitters interacting with a 2D tight-binding lattice environment, the authors showed that both superradiance and subradiance of collective atoms emerge in the nonperturbative regime. The unusual chiral bound states and directional dipole-dipole interaction were also demonstrated in a topological waveguide QED system Bello et al. (2019); Kim et al. (2021). In Refs. Roccati et al. (2022); Gong et al. (2022), the authors discussed how to realize bound states and dipole-dipole interactions in non-Hermitian photonic lattices. All these studies indicate those structured lattice reservoirs with reduced dimensionality are versatile toolboxes for exploring novel quantum optical phenomena, as well as the potential applications in quantum information processing.

In artificial baths with the spin-orbit coupling, the motion and spin freedoms of a particle are linked, and many anomalous phenomena such as spin-Hall effect and topological insulators can be observed Zhang (2000); Murakami et al. (2003); Sinova et al. (2004); Wunderlich et al. (2005); Kane and Mele (2005); Galitski and Spielman (2013); Zhou et al. (2013); Wu et al. (2016); Kartashov et al. (2017); Livi et al. (2016); Liu et al. (2011). Since neither the spin nor the momentum is the well-defined quantum number to describe the dispersion relation, the spin-orbit coupling will produce nontrivial energy bands and photonic modes Sala et al. (2015); Salerno et al. (2017). The quantum optics with emitters interacting with baths of the spin-orbit coupling, is rarely studied. It is a simple but interesting toy model in condensed matter physics (see Fig. 1) Creutz (2001); Narozhny et al. (2005); Jaefari and Fradkin (2012); Atala et al. (2014); Tai et al. (2017); Yuan et al. (2019); Guan et al. (2020). As discussed in Ref. Hügel and Paredes (2014), the ladder contains two legs which play the roles of two freedoms in an effective spin. In the present of synthetic gauge fields, the effective spin will be locked to momentum freedom.

In this work, we discuss QED phenomena in a setup composed by quantum emitters and a Hofstadter-ladder waveguide. Different from previous studies based on lattice environment with synthetic gauge fields Sánchez-Burillo et al. (2020); Wang et al. (2020); De Bernardis et al. (2021); Dong et al. (2021), here we mainly focus on unconventional QED phenomena induced by the spin-orbit coupling. First, we assume that the emitter is of small atom form which frequency is resonant with the lower energy band. Due to spin-momentum locking, the emitter chirally dissipates almost all its energy into one direction of the waveguide, which is different from the directional emission along the edge states of a 2D topologically non-trivial lattice Mittal et al. (2014). In our study the chiral emission into the 1D Hofstadter-ladder waveguide stems from the effective spin-orbit coupling, and does not require any topological protection. Our proposal is possible to demonstrate chiral quantum optics, which has been extensively studied in Refs. Calajó et al. (2019); Gheeraert et al. (2020); Wang et al. (2022); Solano et al. (2021); Kannan et al. (2022). Second, the emitter is considered as giant atom form, and couples to the waveguide at multiple sites Kockum et al. (2014); Guo et al. (2017); Kockum et al. (2018); Kockum (2020); Kannan et al. (2020); Zhao and Wang (2020); Du et al. (2021); Zhang et al. (2021); Wang et al. (2021b); Du et al. (2022). Given that emitters’ frequency is below two degenerate minima points induced by the spin-orbit coupling, there will be bound state in which the photonic energy will be trapped. The bound state induced by time-delay effects of giant atom has been investigated in Ref. Guo et al. (2020). In this work, we focus on another giant atom effect, i.e., the quantum interference between different coupling points. We find that, due to quantum interference and unconventional spectrum of the Hofstadter-ladder waveguide, the bound state will be periodically modulated by the giant atom’s size. The periodical length is tunable by controlling the parameters of the Hofstadter ladder waveguide. Based on this mechanism, we show that by tuning the interference as constructive/destructive, the dipole-dipole interaction between two giant emitters will be enhanced/suppressed.

The model of the QED setup we study is depicted in Fig. 1, where a quantum emitter interacts with an artificial one-dimensional waveguide along the $x$ direction, which behaves as a photonic analog of the Hofstadter-ladder model. The Hofstadter ladder can be viewed as the two-leg edge of the Harper-Hofstadter model Ozawa et al. (2019), where a synthetic gauge field $\Phi=2\phi$ is applied through each plaquette (see Fig. 1). Here we consider it working as a 1D artificial waveguide which allows photons traveling along it. In this situation, two legs in of the ladder waveguide serve as channel A and B of the waveguide. For convenience, we set the length of one unit site as $d_{0}=1$. The ladder waveguide is composed by two legs, which can be viewed as two quantum channels for the emitter. Two sites in each rung are coupled with strength $t^{\prime}$, which is set as $t^{\prime}=1$. By adopting a Landau gauge along the $x$ direction, the phase connections only appear in each leg. Therefore, the hopping amplitude between two nearest neighbor sites is $te^{i\phi}$ ($te^{-i\phi}$) for channel A (B). Consequently, by setting $\hbar=1$, the tight-binding Hamiltonian of the waveguide is Guan et al. (2020)

where $a_{x},b_{x}$ ($a_{x}^{\dagger},b_{x}^{\dagger}$) are the annihilation (creation) operators of the sites $a,b$ at position $x$, and $\omega_{0}$ is the identical frequency of those bosonic modes. In the following we work in the rotating frame of the constant part $\sum_{x}\omega_{0}(a_{x}^{\dagger}a_{x}+b_{x}^{\dagger}b_{x})$.

Under the periodic boundary condition and in the momentum space with

we can diagonalize the waveguide Hamiltonian as

where $\eta=t^{\prime}/2t$, $g\left(k\right)$ and $f\left(k\right)$ are respectively expressed as

As shown in Eq. (6), the Hamiltonian $H_{B}$ is expressed in the Pauli operators, indicating that the upper-lower leg degree of freedom behaves as an effective spin. Due to the synthetic gauge field, $H_{B}$ contains the effective spin-orbit coupling term $(\sin\phi\sin k)\sigma_{z}$, which will lead to spin-momentum locking Hügel and Paredes (2014). Note that in condensed matter physics the concept of “spin” is extensively used for models consisting of“A” and “B” sublattices. Similarly, the spin-orbit coupling is a generalized concept from atomic physics, which describes a two-component internal freedom coupling to the momentum of a particle Ozawa et al. (2019). For example, in Refs. Sala et al. (2015); Hafezi et al. (2011), the spin-orbit coupling and spin Hall insulators for photons have been successfully demonstrated in experiments.

The energy spectrum can be derived by simply diagonalizing $\mathcal{H}_{B}$. Consequently, the energy bands and eigenmodes are derived as

where $\theta_{k}=\arctan[\eta/f(k)]$. Now we define the average spin as

Given that $\langle\sigma_{z}\rangle_{k}>0$ ($\langle\sigma_{z}\rangle_{k}<0$), the mode $k$ asymmetrically distributes on channel $A$ ($B$) with more probabilities. In Fig. 2(a), we plot $E_{\pm}\left(k\right)$ and $\langle\sigma_{z}\rangle_{k}$ of two bands versus $k$. It is found that, due to spin-momentum locking, the chiral current $\langle\sigma_{z}\rangle_{k}$ of the lower band is opposite to the upper band for certain momentum $k$.

Moreover, the spin-orbit coupling significantly modifies the dispersion relation of the waveguide. For a free particle without spin-orbit coupling, the energy minimum point is usually with zero momentum $k=0$ (or $k=\pm\pi$). When spin-orbit coupling appears, the spin-up and spin-down modes minimize their energies by carrying non-zero opposite momentum $k$ Galitski and Spielman (2013). As depicted in Fig. 2, the photonic dispersion relation of the ladder waveguide becomes spin-dependent, and the energy minima are degenerate at two points with non-zero momentum $k\neq 0$. Moreover, there is Kramers degeneracy for a pair of modes $\pm k$ due to spin-momentum interaction, and the field distribution is mostly localized in channel A (B) for $k>0$ ($k<0$) Hügel and Paredes (2014). Those properties allow us to realize unconventional phenomena of quantum optics, which will be addressed in the following discussions.

We first consider that the two-energy-level emitter is of small atom form, i.e., couples to the Hofstadter-ladder waveguide at one site $x=0$ (see Fig. 1). Its frequency lies resonantly within the lower energy band. In the rotating frame of bosonic frequency $\omega_{0}$, the system Hamiltonian is written as

where $\Delta_{q}=\omega_{q}-\omega_{0}$ with $\omega_{q}$ being the emitter’s transition frequency, $\sigma_{z}=|e\rangle\langle e|-|g\rangle\langle g|$ and $\sigma_{+}=(\sigma_{-})^{\dagger}=|e\rangle\langle g|$, with $|e\rangle$ ($|g\rangle$) being the exited (ground) state of the emitter. Applying inverse Fourier transform, one obtains $a^{\dagger}_{0}=\sum_{k}a_{k}^{\dagger}/\sqrt{N}$. According to Eqs. (9, 10), $a_{k}^{\dagger}$ can be decomposed as the superposition of $C_{k\pm}^{\dagger}$. Finally, the interaction Hamiltonian is written as

As shown in Fig. 2(a), $\omega_{q}$ is set in the cyan regime, and only the lower band $E_{-}(k)$ is resonant with the emitter. To avoid the non-Markovian effects led by the band tops Calajo et al. (2016), we require $\Delta_{q}$ far away from two band edges, i.e., $|\delta_{\pm k0}|\gg 0$. By dropping the off-resonant terms with upper band modes $C_{k+}^{\dagger}$, the interacting Hamiltonian is reduced as

After substituting Eq. (9) into Eq. (15), we can divide $H_{\mathrm{int}}$ into two parts which describe interactions with channel A and B respectively:

From Eq. (16) and as depicted in Fig. 2(a), we find four dissipation terms by assuming the resonant position at $\Delta_{q}=E_{-}(\pm k_{r})$, i.e., the left/right direction of channel A (B). After applying the unitary transformation $U_{0}(t)=\exp(-iH_{0}t)$, the interaction operator with channel A becomes

where $\Delta_{k}=E_{-}\left(k\right)-\Delta_{q}$. Similar to Eq. (19), the interaction operator with channel B can also be written in an integral form. We consider the spontaneous decay process with an excitation initially localized in the emitter. In the single-excitation subspace, the state of the whole system is expressed as $|\psi(t)\rangle=\sum_{k}[c_{ka}(t)|g,1_{ka}\rangle+c_{kb}(t)|g,1_{kb}\rangle]+c_{e}(t)|e,0\rangle$, and the evolution of the whole system governed by $H_{\mathrm{int}}$ is derived from the following differential equations

By substituting the internal form of Eqs. (21, 22) into Eq. (20), the evolution of $c_{e}(t)$ is derived as

As depicted in Fig. 2(a), we approximate the dispersion relation around $\pm k_{r}$ to be linear, i.e.,

where $v_{g}$ is the group velocity at $k_{r}$ ($k_{r}>0$). By setting $\delta k=k-k_{r}$, the detuning is written as $\Delta_{k}\simeq v_{g}\delta k$. In the Born-Markovian regime, the decay rate is required to be much smaller than the band width $\{\delta_{+k0},\delta_{-k0}\}$, and we can extend the integral bound $\pm\pi$ to be infinite. Consequently, Eq. (23) is reduced as

where $\Gamma_{i\pm}$ correspond to the emission rates into the right/left direction of channel $i$, which are derived as

which show that $\Gamma_{A(B)\pm}$ are determined by $\theta_{\pm kr}$. In this part we only focus on the Markovian decay regime (cyan area in Fig. 2) where both band edges and the upper energy band do not take apparent effects. We plot $\Gamma_{A,B}$ in Fig. 2(b) [in units of $g^{2}/(2v_{g})$], and find that the emission to channel A (B) is spatially asymmetric (symmetric), i.e.,

Specially, under the following condition

the emission field mostly distributes on the right side of channel A. Therefore, the spontaneous emission field will chirally propagate along the Hofstadter-ladder waveguide. As discussed in Sec. II, the chirality is led by the effective spin-orbit coupling mechanism.

We assume that the coupling position is at $x=0$, and therefore, the field intensities on the right (left) side of channel A and B are defined as

where $c_{A(B),x}$ is the field amplitude of site $a(b)$ of the rung at $x$ of the ladder. For example, if the right side of channel A is the desired direction, both the dissipation into channel B and into the left hand side of channel A will lead to photonic leakage. Consequently, the chiral factor $\mathcal{C}$ is defined as Lodahl et al. (2017)

By adopting expressions of $\Gamma_{A(B)\pm}$ in Eqs. (26,27), the analytical chiral factor is derived as

where we employ the relation $\tan\theta_{k}=\eta/f(k)$. In Fig. 2(b), we plot chiral factor $\mathcal{C}$ changing with resonant wave number $k_{r}$. Given that $-\pi<\phi<0$, $f\left(k_{r}\right)=\sin\phi\sin k_{r}<0$ (Note that we have restricted $k_{r}$ as positive, i.e., $k_{r}>0$). In this case, $\mathcal{C}=0$, indicating that the field hardly dissipates into the desired channel. When $0<\phi<\pi$, the chiral factor is simplified as

Therefore, under the condition

most excitation energy will dissipate into the right side of channel A. Those discussions indicate that both the waveguide’s parameters and the resonant position $k_{r}$ will directly determine the chiral factor.

By adopting the parameters in Fig. 2, we obtain

which is a high chiral factor to realize cascaded quantum networks with multiple nodes. To verify our above analysis, in the following we numerically simulate the system’s evolution by adopting the system’s Hamiltonian in Eq. (12). Note that the ladder’s Hamiltonian is expressed in real space [see Eq. (1)], and the ladder length is set as $N=1000$, which is long enough to avoid field reflection by the bounds. Figure 3(a) shows the evolution of the emitter with $g=0.4$ and $g=3$, respectively. Given that $g=0.4$, the analytical decay rate is calculated as $\sum_{i=A,B}\left(\Gamma_{i+}+\Gamma_{i-}\right)\simeq 0.0014$ [according to Eqs. (26, 27)], which is much smaller than the band width $\delta_{\pm k0}$, and the Markovian approximation is valid. The emitter’s evolution $|c_{e}(t)|^{2}$ is shown in Fig. 3(a), which decays with time and matches well with the analytical exponential form. In Fig. 3(b), we plot the field distributions at $t=100$ for both channel A and B, where the photonic field mostly distributes on the right side of channel A. By adopting Eq. (29) the numerical chirality is about $\mathcal{C}\simeq 0.943$, which is very close to the analytical result derived in Eq. (34).

Note that all the above results for the chiral decay are derived within the Born-Markovian approximation. When the emitter-waveguide interaction strength is comparable to the band width $\{\delta_{+k0},\delta_{-k0}\}$, both modes around $k=0$ and two energy minima points $k_{\pm\text{min}}$ (see Fig. 2 and Fig. 5) with zero group velocity will prevent the emitter from decaying. Partial excitation energy will be trapped around the coupling position in the form of bound states Calajo et al. (2016). By increasing the emitter-waveguide interaction beyond the Markovian regime with $g=3$, we plot the emitter’s evolution in Fig. 3(a), which is no longer of exponential decaying form. After dissipating partial energy into the waveguide, the rest is trapped within the emitter. Therefore, to work as a well-performed cascaded quantum system, the emitter in each node should couple to the waveguide within the Markovian regime. Next, we will discuss the behavior of bound states of both small and giant emitters due to band edge effects.

As discussed in experimental work in Refs. Underwood et al. (2012); Houck et al. (2012), the bosonic modes in the Hofstadter-ladder waveguide can be made by cavities or LC resonators, which will experience decoherence led by the noisy environment. The decay of each site is assumed in the Lindblad form, i.e.,

where $\kappa$ is the photonic decay rate of each cite. By setting $\kappa=0$ and $0.01$ respectively, we plot the corresponding field distributions along the waveguide at $t=100$. When the photonic wavepacket propagates along the dissipative waveguide, it will decay the energy into the environment. We find that the amplitude of the photonic field becomes much lower than the non-dissipative case. However, the field distribution is still chiral, which is not affected by this local coherence.

Additionally, in experiments the waveguide’s length cannot be infinite, and therefore, the chiral field will be reflected by the hard-wall boundary of the waveguide after a long-time propagation. In Fig. 4(c), we plot the field distribution at $t=180$, when the wavepacket already touches the waveguide’s boundary. The energy flow direction is shown in the insets. We find that, due to the hopping rates $t^{\prime}$ between two channels at the boundary, most of the energy will be reflected into channel B. Since the photon decay of each rate is also considered, the amplitude of the reflected wavepacket is much lower than that in Fig. 4(a).

In this work, we explore the unconventional quantum optics by considering a Hofstadter-ladder waveguide interacting with both small and giant emitters. Due to the effective spin-orbit coupling, both the waveguide’s vacuum modes and spectrum show nontrivial properties. In the first part, we consider a small emitter which frequency is resonant with one energy band. Two legs of the ladder, which can be viewed as the freedoms in an effective spin locked with the momemtum freedom, provide two dissipating channels. The spontaneous emission is chiral with most photonic field decaying unidirectionally. Both numerical and analytical results show that the Hofstadter-ladder waveguide can work as a well-performed quantum bus of a chiral network.

In the second part, the emitters are assumed of giant atom form with frequencies below the lower energy band. In this scenario, only the modes around two energy minima points induced by the spin-orbit coupling contribute significantly to the system’s dynamics, which will lead to emitter-waveguide bound states. Since the energy minima modes carry non-zero momentum, the coupling strengths are periodically modulated by the giant emitter’s size due to quantum interference. Specially the giant emitter will decouple with the waveguide when maximum destructive interference happens. This mechanism provides a novel approach to control long-range dipole-dipole interactions between emitters by modulating their sizes.