1. Introduction

In the relativistic quantum theory, information cannot propagate faster than the speed of light, therefore a light cone exists to completely limit the propagation of information in the causal region. Although no such rigorous limitation in non-relativistic case, Lieb and Robinson proved that in exponentially decaying interaction systems, an effective linear light cone exists such that information propagation exponentially decreases with distance outside of the cone region [1]. Thus, the notion of "light cone" and causality can be extended into the fields of nonrelativistic quantum many-body systems such as condensed-matter physics [2], quantum chemistry, and quantum information science [3].

After the seminal work of Lieb and Robinson, many studies are devoted to extending the Lieb-Robinson bound to long-range interacting systems [4–7]. Most of these researches have focused on power-law decaying interactions [8–10] (i.e., the atom-atom interaction is proportional to ($r^{-\alpha }$) where $r$ is the atom distance and the power $\alpha$ is a positive real number) and are committed to answer the following questions: Is there still a speed limit in these power-law interacting systems [11]? How quickly does the information propagation velocity grow [12,13]? Under what condition does a causal region i.e. light cone exists [14,15], and what does it look like [16–18]? The first work that generalized Lieb-Robinson bound to power-law interacting systems was done by Hasting and Koma [19]. They proposed a logarithmic light cone ($t\sim \log r$) for $\alpha >D$ (D is the dimension of the system). This result was improved by Gong et al. [20] where they gave a new bound consisting of two parts: linear and logarithmic. An immediate consequence of the logarithmic boundary is that the maximum group velocity, defined by the slope of the light cone, grows exponentially with time. However, in Ref. [21] they showed the impossibility of this scenario and proposed a new light cone that bounded by a polynomial/power function ($t\sim r^\xi$ where $0<\xi <1$, approaches 1 for increasing $\alpha$) for $\alpha >2D$. Recently, Zhou et al. [22] demonstrated that with the increase of the exponent of power-law interaction, the light cone can continuously change from linear to power law and then to exponential shape. In addition, researchers have rigorously proved that the general criterion for the existence of a linear light cone in the power-law interaction systems is $\alpha >2D+1$ [23]. We emphasize that the distinction among these theories are partly due to the different definitions of light cone and the different systems they investigated. Besides, some experimental studies demonstrated that for sufficiently long-range interactions, the notion of locality is expected to break down completely and thus no light cone of any shape exists [24,25].

Recently, waveguide quantum electrodynamics (waveguide-QED) system attracts extensive attentions because it is a promising platform for realizing quantum network and scalable quantum computation [26–37]. In this system, the atom-atom interaction induced by the guided photon modes can be nearly ideal long-range which is a unique system for studying quantum many-body physics [38–47]. In this paper, we investigate the excitation and correlation propagations in an atom chain coupled to a 1D waveguide without Markovian approximation [48] under three different atom-atom interaction cases: exponentially-decaying, ideal long-range, and power-law decaying interactions. The extracted excitation propagation velocities and correlation propagation velocities for the three systems clearly show an obvious speedup of information propagation when the interaction changes from exponentially decaying to power-law decaying and to ideal long-range. In addition, we investigate the shapes of light cone for the three different interaction systems. A linear light cone appears in both the time evolutions of excitation and correlation for the exponentially decaying interaction system, and there is a blurry nonlinear light cone for the power-law decaying interaction system. However, no obvious light cone of any shapes emerges in the ideal long-range interaction, suggesting the rapid information transmission can be achieved in the waveguide-QED system with ideal long-range interaction. The results found here indicate that the waveguide-QED system can significantly speed up the quantum information flow which is very important for building faster quantum network and scalable quantum computation.

The paper is organized as follows. In Sec. II, we describe the system we study and illustrate the dynamical equations for three different interaction models. In Sec. III, we numerically calculate and compare the excitation and correlation propagations for the three different interaction models. Finally, we summarize the results.

2. Model and theory

The schematic model we consider is shown in Fig. 1 where an atom chain consisting of $N_{a}$ atoms is coupled to both the waveguide and nonwaveguide photon modes and we assume that all the atoms are identical and their transition frequencies are denoted as $\omega _{a}$. The total Hamiltonian of the system written in the rotating frame with respect to the atom transition frequency $\omega _{a}$ is given by [49–51].

The first two terms are the energies of the waveguide photons and the nonguided photons in the rotating frame, respectively. $\Delta \omega _{k}=\omega _{k}-\omega _{a}$ with $\omega _{k}$ being the frequency of the waveguide photons, and $a_{k}(a_{k}^{\dagger })$ is the annihilation (creation) operator of the waveguide photon mode with wavevector $k$. $a_{\vec {q}_{\lambda }}(a_{\vec {q}_{\lambda }}^{\dagger })$ is the annihilation (creation) operator of the nonguided photon mode with frequency $\omega _{\vec {q}_{\lambda }}$ and $\Delta \omega _{\vec {q}_{\lambda }}=\omega _{\vec {q}_{\lambda }}-\omega _{a}$ where $\vec {q}_{\lambda }$ is the three-dimensional wavevector. The third term is the atom-waveguide interaction Hamiltonian with coupling strength $g_{k}$ and the last term is the atom-vacuum interaction Hamiltonian with coupling strength $g_{\vec {q}_{\lambda }}$. $\sigma _{j}^{+}=|e\rangle _{j}\langle g|$ $(\sigma _{j}^{-}=|g\rangle _{j}\langle e|)$ is the raising (lowering) Pauli operators of the jth emitter with position $\vec {r}_{j}$ with its $z$ component denoted as $z_{j}$.

 

Fig. 1. Excitation and correlation propagations in an atom chain coupled to a one-dimensional waveguide. Initially, all atoms are in the ground states except the middle one which is in the excited state.

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Here, we mainly consider the single-excitation case in which only one atom is excited initially. In this case, the state of the system at arbitrary time can be expressed as

Substituting Eqs. (1) and (2) into the Schr$\ddot {o}$dinger equation $i\hbar \partial _{t}\vert \Psi (t)\rangle = H\vert \Psi (t)\rangle$, we can obtain the dynamics of the atom excitation probability amplitudes given by [50].

In the following, we would compare the excitation and correlation propagation patterns of the three different interaction regimes: (1) Short-range interaction, i.e., the atom-atom interaction is exponentially decaying with distance which occurs when the atom transition frequency is below the cutoff frequency of the waveguide; (2) Ideal long-range interaction, i.e., the magnitude of the atom-atom interaction does not change with distance which occurs when the atom transition frequency is above the cutoff frequency of the waveguide; (3) Multi-power-law interaction, i.e., the atom-atom interactions are multi-power-law functions of the distance which occurs when the waveguide is absent or the atoms are far away from the waveguide. The dynamical equations of the system for the above three interaction cases are given by

(i) Exponentially-decay interaction:

From Eqs. (4–6), we may either analytically or numerically calculate the excitation probability of each atom at arbitrary time from which we can also calculate the correlation propagation properties. The correlation function between a pair of atoms is given by $C_{ij}(t)=\langle \sigma _{i}^{z}(t)\sigma _{j}^{z}(t)\rangle -\langle \sigma _{i}^{z}(t)\rangle \langle \sigma _{j}^{z}(t)\rangle$ which indicates how one atom’s excitation affect the other atom’s excitation, and for single-excitation case in our system it can be calculated as

From Eq. (7), we can see that the correlation in this system is usually negative which means that they are anticorrelated to each other. This is understanable because when one atom’s excitation increases the other atoms’ excitations tend to decrease due to energy conservation. In the following, we mainly consider its amplitude (i.e., the absolute value) which reflects the correaltion strength. It is also noted from Eqs. (4–6) that time-retarded effect is included in the dynamics which is usually ignored in the previous studies of correlation propagation. Thus, the non-Markovian effects can be included in our theory. In the next section, we numerically calculate the excitation and correlation propagation along an atom chain with three different interaction models including the time-retarded effect.

3. Numerical simulations

In this section, we numerically calculate the excitation and correlation propagations of an atom chain. Here we choose the number of atoms $N_{a}=41$ without loss of generality. At $t=0$, all atoms are assumed to be in the ground state except the middle atom which is in the excited state i.e. $\alpha _{21}(0)=1$ and $\alpha _{i\neq 21}(0)=0$. Due to the interaction between the atoms, the energy of the middle atom can be released and excite other atoms at later times. In Sec. 3.1 and 3.2, we compare the excitation and correlation propagation velocities in the three different interaction models discussed in the previous section. For this purpose, we set the distance between two adjacent atoms be the same (e.g., $d=0.25\lambda$ where $\lambda$ is the radiation wavelength of the atoms) for all three cases and meanwhile assume that the coherent part of the coupling strengths between two nearest-neighbor atoms be equal for all three cases. To ensure the coherent part of the interaction strengths are the same when $d=0.25\lambda$ for all three cases, we set $\Gamma '_{1D}=2.72\Gamma _{1D}$ and $\gamma =1.65\Gamma _{1D}$ in the first two subsections. In Sec. 3.3 we discuss the effects of atom distance on the excitation and correlation propagation velocities of the three different interaction models.

3.1 Excitation propagation

In this subsection, we compare the excitation propagation velocities of the three different interaction models. The excitation probabilities for different atoms as a function of time are shown in the first column of Fig. 2. For the exponentially-decaying interaction, there is an obvious light-cone-like structure in the two-dimensional (2D) excitation propagation pattern (Fig. 2(a)) which is precisely the characteristic of the short-range interaction systems. However, when the interaction between the atoms is ideal long-range, surprisingly an inverse light-cone-like structure appears in the 2D excitation propagation pattern (Fig. 2(b)) which is very different from the usual short-range interaction case. For the multi-power-law-decaying interaction, we can barely see an obscure light-cone structure (Fig. 2(c)). In all three cases, interference patterns can be observed due to coherent feedback of the excitations. In these figures, we can see that there is a boundary effect, but it is not a hard boundary effect. The excitation can be reflected back by the boundary atom with certain probability while at the same time some energies are leaking out. Actually, the total atom excitation is decaying with time due to the leakage.

 

Fig. 2. The first column is the time evolution of atom excitation probabilities for three different interaction models. The second column is the propagation velocity extracted at a fixed amplitude of excitation probability (i.e., $|\alpha _{i}|^{2}=0.0003$). The third column is the propagation velocity extracted from the time at which the first maximum excitation is reached as a function of the distance. (a, d, g): Exponentially-decaying interaction with $\Gamma '_{1D}=2.72\Gamma _{1D}$ and $\kappa =4/\lambda$. (b, e, h): Ideal long-range interaction $\Gamma _{1D}=\omega _{a}/200\pi$. (c, f, i): Multi-power-law decaying interaction with $\gamma =1.65\Gamma _{1D}$. The atom distance $d=0.25\lambda$ for all three cases.

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To quantitatively compare the excitation propagation velocities of the three different cases, we extract the time at which each atom reaches a certain fixed excitation probability [24] (e.g., $|\alpha _{i}|^{2}=0.0003$ ) and the results are shown in the second column of Fig. 2. In all three cases, the time to reach the fixed excitation probability monotonously increases with the distance which indicates that the excitation sequentially propagates outward from the center atom. By linear fitting, we can extract the approximate propagation velocity. In the short-range interaction case (Fig. 2(d)), we can see that the excitation propagates faster when the atoms are close to the center atoms ($v_{1}=0.0217\lambda \Gamma _{1D}$) and slower when the atoms are relatively far away ($v_{2}=0.0051\lambda \Gamma _{1D}$) The overall average propagating velocity is about $\bar {v}=0.0076\lambda \Gamma _{1D}$. Different from the short-range interaction case, in the ideal long-range interaction case the excitation propagation velocity is about linear function of distance for all atoms and the extracted velocity is about $v=0.848\lambda \Gamma _{1D}$ which is about two orders of magnitudes faster than that in the short-range interaction case (Fig. 2(e)). For the multi-power-law decaying interaction in the free space, the excitation propagation speed is also slightly faster at the beginning ($v_1=0.068\lambda \Gamma _{1D}$) and slower at latter time ($v_2=0.026\lambda \Gamma _{1D}$) (Fig. 2(f)) with an overall average propagating velocity being about $\bar {v}=0.047\lambda \Gamma _{1D}$. Hence, the excitation propagation speed in the ideal long-range interaction case is also larger than that in the power-law interaction case shown here.

In addition to extracting the speed from a fixed excitation probability, in some studies [54–56], the propagation speed is extracted through the time at which the excitation reaches its first maximum value. Here, we also extract the time at which each atom reaches its first maximum excitation value and the results are shown in the third column of Fig. 2. For the short-range interaction case (Fig. 2(g)), we can see that the time for the first maximum excitation is about a linear function of the distance from which we can extract the propagation velocity $\nu =0.0039\lambda \Gamma _{1D}$. In contrast, the time for the first maximum excitation is surprisingly inverse proportional to the distance in the ideal long-range interaction case, i.e., the farthest atom reaches its first maximum excitation at the earliest time (Fig. 2(h)). This is quite different from the result of the short-range interaction case and it is the reason why an inverse light-cone-like structure appears in the 2D excitation pattern shown in Fig. 2(b). However, we should emphasize that it does not mean that the excitation propagates faster than the speed of light because the first maximum of the farthest atom is reached at a time limited by the speed of light. If we set a constant amplitude as benchmark (e.g., $|\alpha _{i}|^{2}=0.0003$) to compare the information arriving time, we can see that the closer atom can reach the benchmark value at an earlier time and the extracted speed is shown in Fig. 2(e) which is a finite value less than the speed of light. The result for the multi-power-law-decaying interaction case is shown in Fig. 2(i) from which we can see that the time for the first maximum excitation is irregular. This is due to the competition between the relatively long-range interaction (i.e, $1/r$) and the relatively short-range interaction (i.e., $1/r^3$) terms in the multi-power-law interaction Hamiltonian. In the relatively short-range interaction case, the first excitation maximum is arriving at the earliest time for the closest atom. However, in the relatively long-range interaction case, the first excitation maximum is arriving at the earliest time for the furthest atom. These two processes can compete to each other and therefore the time for reaching the first maximum excitation does not increase monotonously with the distance.

3.2 Correlation propagation

In this subsection, we numerically calculate the correlation propagations in the three different interaction cases. Here, we also assume that the atom separation and the nearest-neighbor coupling strength are the same for all three different cases. Initially, only the central atom is excited and other atoms are in the ground state. As the system evolves, the excitation propagates outwards and we calculate the correlation between the central atom and other atoms by Eq. (7). The numerical results are shown in Fig. 2. Similar to the excitation propagation, there is a clear light-cone-like structure in the correlation propagation with exponentially-decaying interaction (Fig. 2(a)) while there is an inverse light-cone-like structure in the ideal long-range interaction (Fig. 2(b)) and an obscure light-cone-like structure appears in the multi-power-law interaction (Fig. 2(c)). In Fig. 2(c), we can see that the correlation is mainly concentrated around the middle atom. This is because in this case the atoms can dissipate energy into the free space relatively quickly and therefore the excitation of the middle atom decay very fast. Therefore, only the atoms near the middle atom can have significant correlation with the middle atom.

Similar to the excitation case, we can also extract the correlation propagation velocities from the time at which the correlation reaches a fixed value (e.g. $C_{21,i}=0.0001$). The results are shown in the second column of Fig. 3 from which we can see that the overall features are similar to those shown in Fig. 2. For the exponentially-decaying interaction (Fig. 3(d)), the correlation propagates faster at the beginning ($v_{1}=0.068\lambda \Gamma _{1D}$) and slower later ($v_{2}=0.005\lambda \Gamma _{1D}$) with an overall average velocity being about $\nu \approx 0.0142\lambda \Gamma _{1D}$. For the ideal long-range interaction (Fig. 3(e)), the correlation propagates with almost a linear velocity ($v=0.983\lambda \Gamma _{1D}$) which is again about two orders of magnitude faster than that in the exponentially-decaying case. For the multi-power-law interaction (Fig. 3(f)), the correlation also propagates slightly faster at the beginning ($v_{1}=0.235\lambda \Gamma _{1D}$) than those at the later time ($v_{2}=0.0144\lambda \Gamma _{1D}$) with an overall average velocity being about $\bar {v}=0.199\lambda \Gamma _{1D}$ which is faster than that of the exponentially decaying case but slower than that of the ideal long-range case. Similar to the excitation, when we extract the time at which the first correlation maximum is reached, the correlations reach the first peak at the earliest time for the nearest atom for the exponentially-decaying interaction case (Fig. 3(g)) while the correlations reach the first peak at the earliest time for the farthest atom for the long-range interaction case (Fig. 3(h)). For the multi-power-law interaction case, overall the correlation peak arrives earlier for the atoms with shorter distance but there are some exceptions due to the competition between the interaction terms with different powers.

 

Fig. 3. The first column is the time evolution of correlations for 41 atoms. The second column is the propagation velocity extracted at a fixed amplitude of correlations ($C_{21,i}=0.0001$). The third column is the propagation velocity extracted from the time of the first maximum correlation as a function of the distance. (a, d, g) for exponentially-decaying interaction, (b, e, h) for long-range interaction, (c, f, i) for power-law decaying interaction. The parameters are the same as those used in Fig. 2.

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From the excitation propagation shown in the previous subsection and the correlation propagation shown in this subsection, we can clearly see that the information propagation can be significantly sped up with ideal long-range interaction. This provides a clear evidence showing that the waveguide-QED system is good for building fast quantum information processor.

3.3 Effect of the atomic distance on the propagation velocity

In the previous two subsections, we study the excitation and correlation propagation properties of the three different interaction models in a fixed atom separation. In this subsection, we investigate the effects of the atom separation on both the excitation and correlation propagation velocities. The numerical results of the time at which a fixed amplitude of excitation or correlation are reached versus the index of atom sites for different atom separations ($d=0.1\lambda \sim 0.5\lambda$) are shown in Fig. 4 where the coupling parameters are fixed and the values are the same as those used in the previous subsections, i.e., $\Gamma '_{1D}=2.72\Gamma _{1D}$ and $\gamma =1.65\Gamma _{1D}$. From Fig. 4, we can see that in all cases, both the excitation and the correlation propagate faster in terms of atom sites when the atom separation is smaller. In another word, the information can propagate to the nth atom faster when $d$ is smaller. However, we should emphasize that it does not mean that the information propagates faster because the actual propagation velocities are equal to the slopes of the fitted lines shown in Fig. 4 multiplied by the atom separation $d$ and the results are shown in Table 1. For the exponentially decaying interaction, the propagation velocity decreases with increasing atom separation d because the coupling strength decays exponentially with distance. However, for the long-range interaction and the multi-power-law decaying interaction, the propagation velocity is not necessary faster when the atom separation is smaller. In the current chosen parameters, the maximum excitation and correlation propagation velocity in the long-range interaction occur at $d=0.4\lambda$ and $d=0.1\lambda$, respectively.

 

Fig. 4. Excitation (a,c,e) and correlation (b,d,f) propagation velocities extracted at a fixed amplitude (tagged on each subfigures) of excitation or correlations for the three interaction systems with different atomic separation.

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Table 1. Excitation and correlation velocities extracted at a fixed amplitude of excitations or correlations for the three interaction systems with different atomic distances.

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4. Conclusions

We investigate the excitation and correlation propagation in an atom chain coupled to a 1D waveguide with exponentially decaying, ideal long-range, and multi-power-law decaying interactions. For the exponentially decaying interaction, a clearly linear light-cone-like structure exists in both the excitation and correlation propagation patterns which is consistent with the well-known Lieb-Robison bound. However, for the long-range interaction, an inverse light-cone-like structure appears in both the excitation and correlation propagation patterns and the extracted propagation velocities are about two orders of magnitudes faster than those in the short-range interaction case. For the multi-power-law decaying interaction, an obscure non-linear light cone appears in both the excitation and correlation diagrams which is due to the competition between long-range and short-range interaction and the extracted propagation velocities are about one order of magnitude larger than those in the short-range interaction but about one order of magnitude less than those in the ideal long-range interaction. Thus, the results here clearly show that the waveguide-QED system with long-range interaction is beneficial for building fast quantum network.