Zeno Regime of Collective Emission: Non-Markovianity beyond Retardation

The waveguide QED setup consists of $N$ two-level atoms with the ground state $\ket{g}$ and the excited state $\ket{e}$, and a 1D continuum of bosonic modes. The annihilation and generation operator of the waveguide mode with wavenumber $k$ are denoted by $a_{k}$ and $a_{k}^{\dagger}$, respectively. They satisfy the bosonic commutation relation $[a_{k},a_{k^{\prime}}^{\dagger}]=2\pi\delta(k-k^{\prime})$. Hamiltonian of the system is conventionally written in analogy with the multipolar gauge Hamiltonian of quantum optics, colloquially the “$\bm{d}\cdot\bm{E}$” Hamiltonian [28, 29, 30], though the bosons field may not be photonic, e.g. it could be surface acoustic waves [31, 32] and matter waves [33, 34], etc.,

where $\sigma_{i}=\ket{g}_{i}\bra{e}$, $\sigma_{i,X}=\sigma_{i}+\sigma_{i}^{\dagger}$, $x_{i}$ denotes the coordinate of atom $i$, $\Lambda$ is the cutoff of wavenumber. The coupling strength $g_{k}$ will be specified below. Here we assume a linear dispersion relation for the waveguide, $\omega_{k}=v_{g}\mathinner{\!\left\lvert k\right\rvert}$, where $v_{g}$ is the group velocity of the guided modes. (The Zeno time of a setup with non-linear dispersion relation is found qualitatively the same [35].) The atomic transition frequency $\omega_{0}$ defines a resonant wavenumber $k_{0}=\omega_{0}/v_{g}$. We assume $k_{0}\ll\Lambda$ so that the non-Markovianity induced by reservoir band edges [36] is irrelevant.

The counter-rotating terms of Hamiltonian (1) cannot be ignored for short-time dynamics [10]. Moreover, they are found to lead to non-linearity at the single-photon level [37]. Fortunately, theoretical difficulties brought by them can be avoided by turning to the gauge introduced by Drummond [38], which is also called the Jaynes-Cummings (JC) gauge [39, 40]. The transformation from the multipolar gauge to the JC gauge reads $H_{\text{JC}}=e^{-iS}H_{M}e^{iS}$, where

At the first order of $g_{k}$, we obtain

where $g_{k}^{\text{JC}}=2g_{k}\omega_{0}/(\omega_{k}+\omega_{0})$. The neglected terms at the order of $O(g_{k}^{2})$ give corrections to the atom and waveguide self-energies, while corrections to their couplings occur at the order of $O(g_{k}^{3})$ [10, 41, 42].

The absence of counter-rotating terms grants $H_{\text{JC}}$ (3) a nice property that its ground state is identical to that of its free part $\ket{G}=\ket{g_{1},g_{2}\cdots g_{N},\emptyset}$, an atom-field product state, where $\emptyset$ denotes the field vacuum. Then, the physical state of exciting an atom from the overall ground state, $\sigma_{i}^{\dagger}\ket{G}$, is also a product state. This kind of physical states are exactly the initial states interested by us. With respect to the multipolar gauge $H_{M}$, the same physical state is written as $e^{iS}\sigma_{i}^{\dagger}\ket{G}$, which is, however, an entangled state between the atoms and the field. Recall that initial states in the product form can greatly simply the theoretical analysis and are essential for theories of open quantum system [43, 44, 45]. Thus, we choose to work with $H_{\text{JC}}$ instead of $H_{M}$.

Next, let us specify the coupling strength $g_{k}$. In the literature of waveguide QED, a localized atom-field interaction is often assumed [46] so that $g_{k}$ is a $k$-independent constant [46, 28, 29, 30]. It can be spelled by the Markovian decay rate $\Gamma_{0}$ as $g^{2}_{k}=\Gamma_{0}v_{g}/2$. Another option for $g_{k}$ is $g_{k}\propto\mathinner{\!\left\lvert k\right\rvert}^{1/2}$, the same as the multipolar Hamiltonian of atoms in free space [30] (recall that an atom does not couple to the full displacement field but only its transverse component, which is a nonlocal field [47]). In this case, we have $g^{2}_{k}=\Gamma_{0}v_{g}\mathinner{\!\left\lvert k\right\rvert}/(2k_{0})$. The above two choices for $g_{k}$ correspond to a constant and a linear spectral density, hence will be denoted by “const-wQED” and “lin-wQED”, respectively.

We are now in a position to calculate the Zeno time. Given an initial state $\ket{\Psi_{0}}$ and a Hamiltonian $H$, the Zeno time $\tau_{Z}$ is defined from the short-time expansion of the non-decay probability

The Zeno time is a characterization to the duration of non-exponential decay, but not an exact measure. Nevertheless, we substitute $\ket{\Psi_{0}}=\ket{e,\emptyset}$ and $H_{\text{JC}}$ with $N=1$ into Eq. (4) and obtain

Remarkably, $\tau_{Z}$ of const-wQED is independent of the cutoff $\Lambda$, which is introduced in Eq. (1). This is not seen elsewhere. The above result implies that $\tau_{Z}\gg 1/\omega_{0}$, hence $\tau_{Z}\gg\tau_{\text{retard}}$, is valid if $\omega_{0}/\Gamma_{0}\gg 1$ (for const-wQED) or $\omega_{0}/\Gamma_{0}\gg\ln(\Lambda/k_{0})$ (for lin-wQED). Such weak atom-field couplings are satisfied commonly. Zeno time for $N>1$ is discussed in Ref. [35].

Suppose that the system is initialized with only one atomic excitation. Note that $H_{\text{JC}}$ preserves the number of excitations, thanks to the absence of counter-rotating terms. Thus, the evolution is captured by the singly-excited ansatz

where $\alpha_{i}(t)$ and $\beta_{k}(t)$ are superposition coefficients to be determined. The Schrödinger equation in the interaction picture implies the integro-differential equation

This equation is further transformed into an integral equation and solved numerically [35].

The above equation will be compared with the following one embodying only the non-Markovianity caused by retardation [20, 21]

where $r_{ij}=\mathinner{\!\left\lvert x_{i}-x_{j}\right\rvert}$ and $\Theta(t)=1$ for $t>0$ and vanishes otherwise. It can be derived from Eq. (7) via an approximation introduced in Ref. [18], see also Ref. [35]. Note that while the right-hand-side of Eq. (7) incorporates the entire history $\tau\in[0,t]$, the right-hand-side of Eq. (8) includes only a distance-dependent delay. Ignoring this delay immediately aligns it with the Markovian effective non-Hermitian Hamiltonian of waveguide QED [30]. Hereafter, data produced by Eq. (8) will be labeled by “retard.”

We shall characterize the non-exponential decay by instantaneous decay rate $\Gamma_{\text{inst}}$ and population in excited state $P_{e}(t)$ of the whole chain:

These two quantities can also be defined for every individual atom in an apparent way.

Let us start from the decay of a single atom. We plot $\Gamma_{\text{inst}}(t)$ in units of $\Gamma_{0}$ in Figs. 1(b,c) for both const-wQED (solid curves) and lin-wQED (dashed curves). Either one is calculated with three values of $\Gamma_{0}/\omega_{0}$, $10^{-2}$ (red), $10^{-3}$ (pink) and $10^{-6}$ (blue). The cutoff is set at $\Lambda/k_{0}=10^{4}$. For either const-wQED or lin-wQED, curves belonging to the three $\Gamma_{0}/\omega_{0}$ almost overlap; for each value of $\Gamma_{0}/\omega_{0}$, $\Gamma_{\text{inst}}(t)$ of lin-wQED increases faster at first ($t\lesssim 1/\omega_{0}$) but soon becomes more gradual than that of const-wQED. The latter increases to roughly $1.4\Gamma_{0}$ and turns to oscillating around $\Gamma_{0}$ with a waning amplitude. The non-Markovianity of const-wQED is more pronounced, as what we learn from its Zeno time (5). It is shown in Fig. 1(c) that the oscillation of the curves of const-wQED is still visible at $t=50/\omega_{0}$, equivalent to a distance of eight wavelengths for photon propagation.

Although the curves of $\Gamma_{\text{inst}}(t)$ clearly demonstrate the non-exponential decay, it is defined as the derivative with respect to time so that producing the curves requires a high temporal resolution ($\ll 1/\omega_{0}$) of measuring $P_{e}(t)$. This is of course experimentally challenging. The requirement of temporal resolution might be relaxed if the non-Markovianity can be manifested by $P_{e}(t)$ itself, or equivalently, $\Delta P_{e}(t)=P_{e}(t)-P_{e}(0)$. Unfortunately, we will see in Fig. 2(e) that it is not the case for $N=1$: $\Delta P_{e}(t)$ is averaged out to the memory-less Markovian result quickly. But fortunately, it would be possible for subwavelength atom arrays ($N>1$).

We consider a chain of atoms initialized in the timed-Dicke state

This kind of states are experimentally accessible [48, 49]. State (10) with $k=\pm k_{0}$ are the single-photon superradiant state [50]. We substitute $\ket{\Psi_{k_{0}}}$ with $N=20$, $\Gamma_{0}/\omega_{0}=10^{-4}$, and atom-atom separation $d=0.1\pi/k_{0}$ (see results of $d=0.5\pi/k_{0}$ in [35]) into into Eqs. (7) and (8) and show the results of $\Gamma_{\text{inst}}(t)$ in Fig. 2(a) for $t\leq 10/\omega_{0}$. Curves of Eq. (7) (red for lin-wQED and blue for const-wQED) show continuous growth while that of Eq. (8) (grey) gives a step-like increase. They agree well after $t\approx 6/\omega_{0}$.

Next, we pick five atoms, No. 1, 2, 5, 10, 20 ($x_{i}<x_{j}$ if $i<j$), and plot the individual instantaneous decay rate, $\Gamma_{j,\text{inst}}(t)=-d(\ln\mathinner{\!\left\lvert\alpha_{j}\right\rvert}^{2})/dt$ in Fig. 2(b). It shows that the atoms decay at different rates. Atom 1 decays slowly while Atom 20 accelerates to $20\Gamma_{0}$ (the opposite is obtained if we choose $\ket{\Psi_{k}}$ with $k=-k_{0}$). The three curves (const-wQED, lin-wQED and retard.) agree better for atoms in the middle, i.e., atom 10. In particular, for atom 1, significant derivations between three curves of $\Gamma_{1,\text{const}}$ are visible: the grey curve (retard.) shows two cycles of emission and absorption, the red curve (const-wQED) shows only emission while absorption is dominant for the blue curve (lin-wQED).

Such discrepancy inspires us to look at the change of individual population $\Delta P_{e}=\mathinner{\!\left\lvert\alpha_{j}(t)\right\rvert}^{2}-\mathinner{\!\left\lvert\alpha_{j}(0)\right\rvert}^{2}$, where $\mathinner{\!\left\lvert\alpha_{j}(0)\right\rvert}^{2}=1/N$ for state (10). In Fig. 2(c), we plot it for atom 1 of a shorter chain ($N=10$) within a longer time window $t\leq 50/\omega_{0}$ . To compare with Fig. 1, we apply the same three values of $\Gamma_{0}/\omega_{0}$ and the same coloring as in Fig. 1. Figure 2(c) shows that for $\Gamma_{0}/\omega_{0}=10^{-3}$ (pink) and $10^{-6}$ (blue) there is a gap of $\lesssim 10\Gamma_{0}/\omega_{0}$ between const-wQED (solid curves) and lin-wQED (dashed curves), while the predictions of Eq. (8) (dotted curves) are roughly in the middle. For stronger atom-waveguide coupling $\Gamma_{0}/\omega_{0}=10^{-2}$, the curves of const-wQED (red solid) and that of Eq. (8) (red dotted) have a tendency toward getting closer. We also plot $\Delta P_{e}(t)$ for the last atom (No. 10) in Fig. 2(d). It shows gaps between const-wQED (solid) and lin-wQED (dashed) of roughly the same scale as atom 1, except for the case of $\Gamma_{0}/\omega_{0}=10^{-2}$. A zoom-in view for $\omega_{0}t\in[45,50]$ is shown in the left-most panel of Fig. 2(e).

To compare, we plot $\Delta P_{e}(t)$ for the case of $N=1$ in the right three panels of Fig. 2(e), each for one choice of $\Gamma_{0}/\omega_{0}$. We find that predictions of the three models (solid, dashed and dotted curves) are almost indistinguishable for $\omega_{0}t\in[45,50]$. Thus, in terms of $\Delta P_{e}(t)$, the non-Markovian effect in the Zeno regime is much more prominent in collective emissions than in the decay of a single atom. And it is reasonable to conclude that the requirement of temporal resolution is significantly relaxed to the level of $\lesssim 10/\omega_{0}$.

The effective Hamiltonian of waveguide QED defines a subradiant eigenstate approximated by $\ket{\Psi_{\text{sub}}}=(\ket{\Psi_{k}}-\ket{\Psi_{-k}})/\sqrt{2}$ with $kd=\pi N/(N+1)$ [14, 15, 16]. We suppose the same parameters as in Fig. 2(a,b) and plot $\Gamma_{\text{inst}}(t)$ for a chain initialized in $\ket{\Psi_{\text{sub}}}$ in Fig. 3(a). In all cases, the subradiance is built through quick oscillations between emissions and absorptions. But the amplitudes are different: The piecewise curve predicted by Eq. (8) has the largest amplitudes. In Fig. 3(b), we plot $\Delta P_{e}(t)$ of the whole chain. It also shows apparent relative discrepancies between the three predictions.

It is of fundamental interest to extend the theory to atoms in free space. However, a controversial issue is that on which Hamiltonian all calculations should be based. Most works chose the Coulomb gauge ($\bm{A}\cdot\bm{p}$ interaction) disregarding the $\bm{A}^{2}$ term, see, e.g., Refs. [51, 52, 53, 54, 55]. Taking the counter-rotating terms into account, it has been found that $\tau_{Z}^{-2}\sim\ln\Lambda$ [10], the same as lin-wQED. Thus, we expect the same non-Markovianity, or perhaps even more pronounced, because the resonant dipole-dipole interaction in free space diverges as $1/r^{3}$ for short distances, resulting in strong photon blockade [56]. But recently Hamiltonians of quantum optics is revisited by causal perturbation theory [57]. In this sense, non-Markovianity beyond retardation might be viewed as a probe to determine which theory better captures the true physics.

For experimental tests, our plots show that temporal resolution at the scale of $\lesssim 10/\omega_{0}$ is favorable. Among various platforms of waveguide QED, superconducting circuits have the highest coupling efficiency [30] and fabricating the transmon qubits into a subwavelength chain is straightforward [58]. The transition frequency $\omega_{0}$ is in the GHz regime so that temporal resolution at nanosecond is sufficient. Subwavelength atom arrays can also be realized by trapping Sr atoms in optical lattices [59]. The wavelength of $\prescript{3}{}{P}_{0}-\prescript{3}{}{D}_{1}$ transition is $2.6\mathrm{\mu m}$ so the temporal resolution should be $\lesssim 10\mathrm{fs}$. Scenarios where the boson fields are surface acoustic waves [31, 32] and matter waves [33, 34] need further studies.

We have studied the Zeno regime of the decay of subwavelength atom arrays coupled to a 1D waveguide. Non-markovianity beyond retardation, characterized by instantaneous decay rates and population in the excited states, is addressed by comparing the full quantum treatment Eq. (7) with Eq. (8), which includes only the retardation effect. Specifically, the evolution of excited state population (in the single-photon superradiant state) manifests reservoir memory effect with a significantly relaxed temporal resolution. Our results might be useful for protecting the quantum information stored in compact atom ensembles [14, 60] via dissipation engineering [61], and studying the correlated noise in quantum computing processors [62], etc. For future works, one may explore such possibilities using the theoretical tools of non-Markovian open systems [43, 44, 45].