Subradiance-protected excitation transport

TL;DR

This paper demonstrates that in a one-dimensional atomic chain, excitation transport can be made highly subradiant, enabling long-lived, dispersionless wave packet transport controllable by magnetic fields, with potential applications in energy transfer and light storage.

Contribution

It reveals the existence of subradiant states with linear dispersion in a 1D atomic chain, enabling long-distance, low-loss excitation transport with controllable direction and velocity.

Findings

Transport is subradiant with significantly extended lifetimes.

Subradiant states exhibit linear dispersion enabling dispersionless transport.

Transport properties can be controlled via external magnetic fields.

Abstract

We explore excitation transport within a one-dimensional chain of atoms where the atomic transition dipoles are coupled to the free radiation field. When the atoms are separated by distances smaller or comparable to the wavelength of the transition, the exchange of virtual photons leads to the transport of the excitation through the lattice. Even though this is a strongly dissipative system, we find that the transport is subradiant, that is, the excitation lifetime is orders of magnitude longer than the one of an individual atom. In particular, we show that a subspace of the spectrum is formed by subradiant states with a linear dispersion relation, which allows for the dispersionless transport of wave packets over long distances with virtually zero decay rate. Moreover, the group velocity and direction of the transport can be controlled via an external uniform magnetic field while preserving its subradiant character. The simplicity and versatility of this system, together with the robustness of subradiance against disorder, makes it relevant for a range of applications such as lossless energy transport and long-time light storage.

Figures

Equations

$$\dot{\rho }(t)=-\frac{\mathrm{i}}{\hslash }[H,\rho ]+\mathcal{D}(\rho ).$$

$$H=\hslash \sum _{\alpha \ne \beta }{\mathbf{d}}_{\alpha }^{\prime }\cdot {\bar{V}}_{\alpha \beta }{\mathbf{d}}_{\beta },$$

$${\bar{V}}_{\alpha \beta }=\frac{3\gamma }{8}\left(\begin{array}{ccc}{V}_{11}& -V_{10}^{\ast }& V_{+-}^{\ast }\\ -V_{10}& V_{00}& V_{10}^{\ast }\\ V_{+-}& V_{10}& V_{11}\end{array}\right),$$

$$V_{11}=\left(2-3{\sin }^2\theta \right)\left(\frac{\sin \kappa }{{\kappa }^2}+\frac{\cos \kappa }{{\kappa }^3}\right)-(2-{\sin }^2\theta )\frac{\cos \kappa }{\kappa }$$

$$V_{10}=\frac{1}{\sqrt{2}}\sin 2\theta e^{i\phi }\left[\frac{\cos \kappa }{\kappa }-3\left(\frac{\sin \kappa }{{\kappa }^2}+\frac{\cos \kappa }{{\kappa }^3}\right)\right]$$

$$V_{00}=2\left[\left(1-3{\cos }^2\theta \right)\left(\frac{\sin \kappa }{{\kappa }^2}+\frac{\cos \kappa }{{\kappa }^3}\right)-{\sin }^2\theta \frac{\cos \kappa }{\kappa }\right]$$

$$V_{+-}={\sin }^2\theta e^{2i\phi }\left[3\left(\frac{\sin \kappa }{{\kappa }^2}+\frac{\cos \kappa }{{\kappa }^3}\right)-\frac{\cos \kappa }{\kappa }\right].$$

$$H_{\Delta }=\sum _{\alpha =1}^N{\mathbf{d}}_{\alpha }^{\prime }\cdot {\bar{\Delta }}_{\alpha \alpha }{\mathbf{d}}_{\alpha }.$$

$${\bar{\Delta }}_{\alpha \alpha }={\mu }_{\mathrm{B}}g\left(\begin{array}{ccc}{B}_z& \frac{B_x-i{B}_y}{\sqrt{2}}& 0\\ \frac{B_x+i{B}_y}{\sqrt{2}}& 0& \frac{B_x-i{B}_y}{\sqrt{2}}\\ 0& \frac{B_x+i{B}_y}{\sqrt{2}}& -B_z\end{array}\right),$$

$$\mathcal{D}(\rho )=\sum _{\alpha ,\beta }\left({\mathbf{d}}_{\alpha }\cdot {\bar{\Gamma }}_{\alpha \beta }\rho {\mathbf{d}}_{\beta }^{\prime }-\frac{1}{2}\left\{{\mathbf{d}}_{\alpha }^{\prime }\cdot {\bar{\Gamma }}_{\alpha \beta }{\mathbf{d}}_{\beta },\rho \right\}\right).$$

$${\bar{\Gamma }}_{\alpha \beta }=\frac{3\gamma }{4}\left(\begin{array}{ccc}{\Gamma }_{11}& -{\Gamma }_{10}^{\ast }& {\Gamma }_{+-}^{\ast }\\ -{\Gamma }_{10}& {\Gamma }_{00}& {\Gamma }_{10}^{\ast }\\ {\Gamma }_{+-}& {\Gamma }_{10}& {\Gamma }_{11}\end{array}\right),$$

$${\Gamma }_{11}=(2-{\sin }^2\theta )\frac{\sin \kappa }{\kappa }+\left(2-3{\sin }^2\theta \right)\left(\frac{\cos \kappa }{{\kappa }^2}-\frac{\sin \kappa }{{\kappa }^3}\right)$$

$${\Gamma }_{10}=\frac{1}{\sqrt{2}}\sin 2\theta e^{i\phi }\left[-\frac{\sin \kappa }{\kappa }-3\left(\frac{\cos \kappa }{{\kappa }^2}-\frac{\sin \kappa }{{\kappa }^3}\right)\right]$$

$${\Gamma }_{00}=2\left[{\sin }^2\theta \frac{\sin \kappa }{\kappa }+\left(1-3{\cos }^2\theta \right)\left(\frac{\cos \kappa }{{\kappa }^2}-\frac{\sin \kappa }{{\kappa }^3}\right)\right]$$

$${\Gamma }_{+-}={\sin }^2\theta e^{2i\phi }\left[\frac{\sin \kappa }{\kappa }+3\left(\frac{\cos \kappa }{{\kappa }^2}-\frac{\sin \kappa }{{\kappa }^3}\right)\right].$$

$$\rho =\left(\begin{array}{cc}{\rho }_{GG}& {\mathbf{\rho }}_{Ge}\\ {\mathbf{\rho }}_{eG}& {\bar{\rho }}_{ee}\end{array}\right),$$

$${\dot{\bar{\rho }}}_{ee}=-\frac{\mathrm{i}}{\hslash }\left[H_{\text{eff}},{\bar{\rho }}_{ee}\right],$$

$$H_{\mathrm{eff}}=\hslash \left(\bar{V}-\mathrm{i}\frac{\bar{\Gamma }}{2}\right)+\bar{\Delta }.$$

$$|\psi (t)⟩=\sum _{m=\pm 1,0}\sum _{\alpha =1}^N{c}_{\alpha }^m(t)|e^m{⟩}_{\alpha },$$

$$P_{\mathrm{sur}}(t)=\sum _{\alpha =1}^N\sum _{m=\pm 1,0}|c_{\alpha }^m(t){|}^2.$$

$$⟨K(t)⟩=\sum _{\alpha ,\beta }⟨{\mathbf{d}}_{\alpha }^{\prime }\cdot {\bar{\Gamma }}_{\alpha \beta }{\mathbf{d}}_{\beta }⟩.$$

$$|k_m⟩=\frac{1}{\sqrt{N}}\sum _{\alpha =1}^N{e}^{i\frac{2\pi }{N}(\alpha -1)(k-q)}|e^m{⟩}_{\alpha },$$

$$|\psi (0)⟩=\sum _{\alpha }{c}_{\alpha }^0|e^0{⟩}_{\alpha }=\frac{1}{\sqrt{N}}\sum _k{c}_k|k_0⟩,$$

$$|\psi (t)⟩=\frac{1}{\sqrt{N}}\sum _k{c}_k{e}^{-i\left(V_k-i\frac{{\Gamma }_k}{2}\right)t}|k_0⟩,$$

$$|\psi (0)⟩=\frac{1}{\sqrt{N}}\sum _k|k_0⟩,$$

$$|\psi (t)⟩=\frac{1}{\sqrt{N}}\sum _k{e}^{-i\left(V_k-i\frac{{\Gamma }_k}{2}\right)t}|k_0⟩.$$

$$|\psi (0)⟩=\frac{1}{\sqrt{\sqrt{2\pi }{\sigma }_k}}\sum _k{e}^{-\frac{{\left[p(k)-p(k_s)\right]}^2}{4{\sigma }_k^2}}|k_0⟩.$$

$$|\psi (0)⟩=\sqrt{\frac{{\sigma }_k}{\sqrt{2\pi }}}\sum _{\alpha }{e}^{-ip(k_s)a(\alpha -1)}{e}^{-\frac{{\left[\alpha -1\right]}^2{a}^2}{4/{\sigma }_k^2}}|e^0{⟩}_{\alpha },$$

$${\dot{\rho }}_{GG}=$$

$${\dot{\rho }}_{G{e}_{\alpha }^m}=$$

$${\dot{\rho }}_{e_{\alpha }^{m}G}=$$

$${\dot{\rho }}_{e_{\alpha }^m{e}_{\beta }^l}=$$

$$-{\rho }_{e_{\alpha }^m{e}_{\gamma }^n}\left(\hslash Z_{\gamma \beta }^{nl\ast }+{\Delta }_{\gamma \gamma }^{nl}{\delta }_{\gamma \beta }\right)],$$

Full text

I Introduction

An ensemble of emitters couples collectively to a common electromagnetic bath, as was already investigated theoretically in the seminal papers of Dicke, Lehmberg and Agarwal in the 1950s and 70s Dicke (1954); Lehmberg (1970); Agarwal (1971). Here, the exchange of virtual photons results in induced dipole-dipole interactions Olmos et al. (2013); Sutherland and Robicheaux (2016); Bettles et al. (2016) and collective Lamb and Lorentz-Lorenz shifts Friedberg et al. (1973); Scully (2009); Keaveney et al. (2012); Pellegrino et al. (2014); Jenkins et al. (2016); Bromley et al. (2016); Peyrot et al. (2018). Moreover, the emission of photons into the bath takes place at a rate much faster or slower (so-called super- and subradiance, respectively) than the single atom decay rate Bienaimé et al. (2012); Ott et al. (2013); de Oliveira et al. (2014); Guerin et al. (2016); Araújo et al. (2016); Roof et al. (2016); Araújo et al. (2018); Cottier et al. (2018). This cooperative behavior is featured both in dense ensembles, where the interparticle separations are comparable to the wavelength of the scattered light, and in dilute ones with a very large number of emitters. Super- and subradiance have been observed experimentally not only in atomic gases, but also in QED circuits Filipp et al. (2011); Lalumière et al. (2013), metamaterial arrays Jenkins et al. (2017), and quantum dots Lodahl et al. (2004); Tighineanu et al. (2016); Kim et al. (2018). This collective atom-light coupling has found a variety of applications such as storage of light via the preparation of subradiant states through phase-imprinting protocols Scully et al. (2006); Plankensteiner et al. (2015); Scully (2015); Jen et al. (2016); Facchinetti et al. (2016); Asenjo-Garcia et al. (2017); Jen (2017, 2018); Jen et al. (2018); Moreno-Cardoner et al. (2019); Guimond et al. (2019), topologically protected transport of excitations Bettles et al. (2017); Perczel et al. (2017), or efficient long-range energy transport Willingham and Link (2011); Giusteri et al. (2015); Leggio et al. (2015); Doyeux et al. (2017).

In this paper, we show that it is possible to realize subradiance-protected transport of a wave packet through a dense atomic chain with lifetimes many orders of magnitude longer than the one of an individual atom. This is achieved by maximizing the overlap of the wave packet with a subradiant manifold of states that possess a linear dispersion relation. Further control over the transport can be attained by effectively changing the orientation of the transition dipoles via an external uniform magnetic field. In particular, we show that the group velocity of the wave packet can be brought close to zero while preserving its long lifetime. Finally, we analyse the effect of disorder, which arises from the width of the external wavefunction of the atoms in each lattice trap and is inevitable in a realistic experimental scenario. Even though this can lead to the suppression of the transport of the wave packet due to localization Deng et al. (2018); Botzung et al. (2018), we find that the subradiant character of the dynamics is robust against the presence of disorder Akkermans and Gero (2013); Biella et al. (2013).

II Interaction between the atoms and the radiation field

II.1 Master Equation

We consider an ensemble of $N$ atoms at positions $\mathbf{r}_{\alpha}$ with $\alpha=1,\dots,N$, each one tightly trapped in the sites of a one-dimensional lattice with lattice constant $a$ [see Figs. 1(a) and (b)]. The internal degrees of freedom of each atom are considered as a generic $J=0\to 1$ transition, with a single ground state $\left|g\right>$ and three degenerate excited states $\left|-1\right>$, $\left|0\right>$ and $\left|+1\right>$. The energy difference between the ground and excited states is denoted by $\hbar\omega=hc/\lambda$, where $\lambda$ is the wavelength of the transition. We choose the transition dipole moment $\mathbf{d}^{0}$ of the $\left|g\right>\to\left|0\right>$ transition to be aligned with the quantization axis ($z$-axis) [Fig. 1(c)].

The atoms are in contact with the radiation field, which we model as a thermal bath at zero temperature, whose degrees of freedom are traced out. Within the Born-Markov and secular approximations Dicke (1954); Lehmberg (1970); Agarwal (1971), the master equation for the dynamics of the internal degrees of freedom encoded in the reduced density matrix $\rho$ yields

[TABLE]

The Hamiltonian $H$ describes the coherent long-ranged exchange of virtual photons among the atoms and is given by

[TABLE]

where $\mathbf{d}_{\alpha}=\begin{pmatrix}d_{\alpha}^{+1}&d_{\alpha}^{0}&d_{\alpha}^{-1}\\ \end{pmatrix}^{T}$ and $\mathbf{d}_{\alpha}^{{}^{\prime}}=\begin{pmatrix}d_{\alpha}^{+1\dagger}&d_{\alpha}^{0\dagger}&d_{\alpha}^{-1\dagger}\\ \end{pmatrix}^{T}$ with the atomic lowering and raising operators defined as $d_{\alpha}^{m}=|{g}\rangle_{\alpha}\langle{m}|$ and ${d^{m{\dagger}}_{\alpha}}=|{m}\rangle_{\alpha}\langle{g}|$, respectively, for $m=-1,0,+1$ and $\alpha=1,\dots,N$. The coherent exchange rate between two atoms $\alpha$ and $\beta$ is represented by the coefficient matrix

[TABLE]

with

[TABLE]

All exchange rates between the internal states are proportional to the single-atom decay rate $\gamma$ and depend on the reduced distance between the two atoms $\kappa=2\pi r_{\alpha\beta}/\lambda$. Here, ${r}_{\alpha\beta}=\left|\mathbf{r}_{\alpha\beta}\right|$ is the modulus of the separation between the two atoms $\mathbf{r}_{\alpha\beta}=\mathbf{r}_{\alpha}-\mathbf{r}_{\beta}$, and $\theta$ and $\varphi$ are the angles between $\mathbf{r}_{\alpha\beta}$ and the transition dipole moment $\mathbf{d}^{0}$ and the $x$-axis, respectively [see Fig. 1(d)].

For small values of $\kappa$ (near-field) the exchange interactions (3) decay approximately as $1/\kappa^{3}$. Here, for a fixed value of $\kappa$, both the strength and sign of the interactions can be tuned by changing the angle $\theta$ [e.g. $V_{00}\approx 2\left(1-3\cos^{2}{\theta}\right)/\kappa^{3}$]. Control over this angle and, hence, the interactions, is obtained by applying a uniform magnetic field $\mathbf{B}=(B_{x},B_{y},B_{z})$, represented in the master equation (1) by substituting $H\to H+H_{\Delta}$, with

[TABLE]

Here, the matrix $\bar{\Delta}_{\alpha\alpha}$ reads

[TABLE]

with $\mu_{\mathrm{B}}$ being the Bohr magneton and $g$ the Landé g-factor.

The second term of the master equation (1) represents the dissipation via incoherent emission of photons into the radiation field and it is given by

[TABLE]

The coefficient matrix $\bar{\Gamma}_{\alpha\beta}$ encodes the dissipative couplings between the atoms and has a similar structure to the coherent interaction matrix:

[TABLE]

with

[TABLE]

The atoms couple to the radiation field as a collective and not as individuals. As a consequence, the decay rates in the system differ significantly from those of single emitters Dicke (1954); Lehmberg (1970); Agarwal (1971), with some being much larger and others much smaller than the single-atom decay rate $\gamma$ (corresponding to so-called superradiant and subradiant emission modes, respectively). This collective character becomes more pronounced for small reduced distances $\kappa$, i.e. small ratios $a/\lambda$, reaching regimes where some of the radiation modes are almost completely dark (with virtually zero decay rate). The population of these subradiant modes is the mechanism that allows for the prolongued storage of light in the atomic system.

II.2 Dynamics in the single excitation sector

Throughout, we will assume that the initial state contains a single excitation localized over a few lattice sites of the chain [Figs. 1(a) and (b)]. This single excitation (in one of the three states $|{-1}\rangle,|{0}\rangle$, or $|{+1}\rangle$) is transported via the exchange interactions given by $H$ (which conserve the number of excitations), while the action of dissipation can only decrease the number of excitations to zero. Thus, the dynamics can be described in a truncated space formed by the many-body ground state $|{G}\rangle\equiv|{g}\rangle_{1}\otimes|{g}\rangle_{2}\dots\otimes|{g}\rangle_{N}$ and the single-excitation states $|{e^{m}}\rangle_{\alpha}\equiv|{g}\rangle_{1}\otimes|{g}\rangle_{2}\dots|{m}\rangle_{\alpha}\dots\otimes|{g}\rangle_{N}$, for all $\alpha=1\dots N$ and $m=-1,0,+1$. Here, the density matrix takes the form

[TABLE]

where $\rho_{GG}=\langle{G}|\rho|{G}\rangle$, $\bm{\rho}_{Ge}=\langle{G}|\rho|{\mathbf{e}}\rangle$, $\bm{\rho}_{eG}=\langle{\mathbf{e}}|\rho|{G}\rangle$, and $\bar{\rho}_{ee}=\langle{\mathbf{e}}|\rho|{\mathbf{e}}\rangle$, with $|{\mathbf{e}}\rangle$ being a row vector containing all single-excitation states $|{e^{m}}\rangle_{\alpha}$. The time-evolution of the elements of $\bar{\rho}_{ee}$ (a $3N\times 3N$ matrix) is decoupled from the dynamics of the remaining elements (see Appendix A), obeying the equation

[TABLE]

with

[TABLE]

Here, $\bar{V}$, $\bar{\Gamma}$, and $\bar{\Delta}$, are $3N\times 3N$ matrices whose components for $\alpha,\beta=1,\dots,N$ are given by Eqs. (3), (7) and (5), respectively 111Note that $\bar{V}_{\alpha\alpha}=0$ and $\bar{\Delta}_{\alpha\beta}=\bar{\Delta}_{\alpha\alpha}\delta_{\alpha\beta}$.

We will consider in all cases a pure state as the initial state, and hence $\bar{\rho}_{ee}=|{\psi(t)}\rangle\langle{\psi(t)}|$ with

[TABLE]

where $c_{\alpha}^{m}(t)$ is the probability amplitude of the $\alpha$-th atom being excited to the $|{m}\rangle$ state. The state (11) evolves under the non-hermitian Hamiltonian $H_{\mathrm{eff}}$. The survival probability, i.e. the probability for not emitting a photon into the radiation field, is given by the norm of the wave function

[TABLE]

The instantaneous photon emission rate, also called activity, is given by

[TABLE]

A value of the activity larger (smaller) than the single atom decay rate $\gamma$ for a state with large excitation density is indicative collective superradiant (subradiant) behavior of the photon emission.

III Subradiant transport on a ring lattice

First, we focus on a ring lattice, as illustrated in Fig. 1(a) Jen (2018); Jen et al. (2018); Moreno-Cardoner et al. (2019). Here, the matrices $\bar{V}$ and $\bar{\Gamma}$ are symmetric circulant due to the periodic boundary conditions such that, for each orientation of the transition dipole $m$, the simultaneous eigenstates for both matrices are given by the plane waves

[TABLE]

with $k=-\left\lfloor{N/2}\right\rfloor,\dots\left\lfloor{(N-1)/2}\right\rfloor$ and $q=\left\lfloor{N/2}\right\rfloor$. For illustration purposes, we will consider in the following the case where only the $|{0}\rangle$ state is excited initially, and where the ring plane is perpendicular to the dipole moment $\mathbf{d}^{0}$ (i.e. $\theta=\pi/2$). Here, both the coherent and incoherent couplings between $|{0}\rangle$ and $|{\pm 1}\rangle$ vanish [$V_{10}=0$ and $\Gamma_{10}=0$ in Eqs. (3) and (7)], and hence the dynamics is determined solely by $V_{00}$ and $\Gamma_{00}$. The initial state can be written as

[TABLE]

where the coefficients $c_{\alpha}^{0}$ and $c_{k}$ represent the probability amplitude distribution of the initial state in real and momentum space, respectively. Conversely, the time-evolved state takes the form

[TABLE]

where $V_{k}$ and $\Gamma_{k}$ are the eigenvalues of the matrices $\bar{V}$ and $\bar{\Gamma}$, respectively. Note that the collective character of the dissipation is reflected here in the decay rates $\Gamma_{k}$, which are either larger or smaller than the single atom decay rate $\gamma$, corresponding to $|{k_{0}}\rangle$ having superradiant or subradiant character, as illustrated in Fig. 2(a), while $V_{k}$ represents the energy of the corresponding mode.

Let us start by considering the initial state to be $|{e^{0}}\rangle_{1}$, i.e. an excitation localized on a single site of the lattice such that $c_{\alpha}^{0}=\delta_{\alpha 1}$. This state can be written as a symmetric superposition of all plane waves (14), i.e.,

[TABLE]

and its time-evolution is then given by

[TABLE]

This dynamics is depicted in Fig. 2(b), where we observe that the initial wave packet splits into two that travel in opposite directions. These wave packets disperse quickly due to the non-linearity of $V_{k}$ as a function of $p(k)=2\pi k/(Na)$ [see Fig. 2(a)]. More importantly, the superradiant components (with large $\Gamma_{k}$) decay very fast and only the subradiant ones remain populated. This is seen in Fig. 2(c), which shows a plateau in the survival probability $P_{\mathrm{sur}}$, and near-zero emission rate $\left<K\right>$ after a rapid initial decay. The height of the plateau of $P_{\mathrm{sur}}$ is approximately given by the number of subradiant eigenstates [much larger than the number of superradiant ones, as can be seen in Fig. 2(a)] divided by the total number of modes $N$ 222Note that this approximation only holds for small enough values of $a/\lambda$. As this ratio is increased, the decay rate $\Gamma_{k}$ of the subradiant states get closer to $\gamma$ and hence the plateau only holds for a small period of time $\tau$, given by the difference between $1/\Gamma_{k}$ and $1/\Gamma_{k+1}$.. For a fixed value of $a/\lambda$, this ratio remains almost constant when increasing the number of atoms $N$. For a fixed size of the system $N$, on the other hand, reducing $a/\lambda$ increases the relative number of subradiant eigenstates and hence the survival probability, as can be seen in Fig. 2(d). In all cases considered, the lifetime of the excitation is dramatically longer than in the case of a single atom.

As can be observed in Figs. 2(a) and 3(a), the dispersion relation in the subradiant part of the spectrum is approximately linear [excluding the states with momentum $p(k)$ close to $\pm\pi/a$ and near the superradiant region]. Therefore, one can expect to have lossless-propagating wave packets with a constant group velocity (given by the gradient of $V_{k}$) without dispersing. We illustrate this by initialising the system with a Gaussian wave packet centered in momentum space at $p(k_{s})$ (center of the linear dispersion manifold) and width $\sigma_{k}$ small enough to ensure that most components of the wave packet are located in the linear dispersion regime [see blue solid line in Fig. 3(a)]:

[TABLE]

In real space this is also a Gaussian wave packet

[TABLE]

whose probability distribution is sketched on the left hand side of Fig. 3(b). Here, it is shown that such wave packet travels indeed without appreciable dispersion around the ring. Moreover, the lifetime of the excitation is extremely long: its effective decay rate $\Gamma_{\mathrm{eff}}$ is six orders of magnitude smaller than the single atom decay rate $\gamma$. A similar reduction of the decay rate $\Gamma_{\mathrm{eff}}$ is also achieved for different system sizes $N$ and ratios $a/\lambda$, as it can be observed in Fig. 3(c).

IV Finite linear chain: storage and transport control via magnetic field switching

In this Section we will focus on the control of the subradiant excitation transport and storage on a linear one-dimensional chain with open boundary conditions, as depicted in Fig. 4(a) Filipp et al. (2011); Jen et al. (2016); Jen (2017); Asenjo-Garcia et al. (2017). We consider the initial state to be $|{e^{0}}\rangle_{1}$, representing one excitation at the leftmost site with the rest of the atoms in the ground state. We further assume that a uniform magnetic field $\mathbf{B}$ is applied perpendicularly to the chain (which lies on the $y$-axis) and parallel to $\mathbf{d}^{0}$, such that $\theta=\pi/2$.

The initial excitation is transported to the right of the chain via the dipole-dipole interactions [see Fig. 4(b)] until it reaches the other edge of the lattice and bounces back. As in the case of the ring, the excitation quickly disperses, and acquires a subradiant character when reaching the bulk of the chain [see Fig. 4(c)]. However, as the excitation reaches the other edge, the survival probability decays faster, accompanied by an increase of the activity. Analogously with the case of the ring, the height of the plateau in $P_{\mathrm{sur}}$ can be increased by reducing the ratio $a/\lambda$, as illustrated in Fig. 4(d). Here, in order to facilitate the comparison, the time is scaled by $t_{\mathrm{pl}}$, which is approximately the time that the excitation takes to reach the middle of the chain (inversely proportional to the nearest neighbor exchange rate).

Since the excitation has almost zero decay rate within the bulk of the chain, its lifetime is ultimately limited by the time it takes for it to reach the other boundary, i.e. by size of the system and the value of the exchange interactions. One can ask, thus, whether it is possible to freeze the transport of the wave packet and confine it in the subradiant states of the bulk. This can indeed be done by adiabatically changing the direction of the external magnetic field, exploiting that the strength and sign of the exchange interactions depends on the angle between the transition dipole moment $\mathbf{d}^{0}$ and the direction of the separation between the atoms $\theta$.

Let us illustrate this protocol via an example depicted in Fig. 5 for the same parameters used in Fig. 4. At $t=0$, $\theta=\pi/2$, such that the nearest neighbor interactions are larger than the single atom decay rate $\gamma$ [as shown in Fig. 5(c), orange line], and make the excitation propagate into the bulk [see Figs. 5(a) and (b)]. When the activity reaches a minimum at $t=t_{\mathrm{min}}$, the direction of the magnetic field is changed within the $yz$-plane. This switch is done adiabatically, such that it is followed by the transition dipole moment of the excitation (i.e. the switching time $\tau\gg 1/\Delta$), but quickly enough to keep the excitation from leaving the bulk of the lattice. The change of the magnetic field direction is mathematically equivalent to a rotation of the angle $\theta$ between the quantization axis and the chain from its initial value $\theta_{\mathrm{in}}=\pi/2$ to a final value $\theta_{\mathrm{f}}$, which leads in turn to modified interactions. In particular, in order to slow down the excitation transport in the bulk, we fix the final value such that the nearest-neighbor interaction coefficient is zero, $V_{00}(a,\theta_{\mathrm{f}})=0$ [see Fig. 5(c), blue line]. While this change does not freeze the excitation transport entirely due to the non-zero values of the exchange rates beyond nearest neighbors, it does slow it down notably, as one can see in Fig. 5(a). Most importantly, the subradiant character of the propagation is preserved, reflected in a constant survival probability $P_{\mathrm{sur}}$ and vanishing activity, as shown in Fig. 5(b).

The versatility of the system using the change in the magnetic field direction is further illustrated in Fig. 5(d). Here, we show an example where several changes in the direction of the magnetic field allow to switch the direction of travel of the excitation. Most importantly, the activity remains close to zero throughout all of these changes, as long as the excitation stays in the bulk of the chain.

V Disorder

Finally, we briefly consider the effect of disorder on the subradiant transport discussed in the previous sections. In particular, we consider the disorder introduced due to the finite width of the external wavefunction of each atom, which we model as a three dimensional Gaussian with width $\sigma$ centered in the respective lattice sites.

Since the long-ranged exchange interactions $\bar{V}$ [given by (3)] are functions of the separation between the atoms, the uncertainty in the atomic positions translates into disorder in the hopping rates in the exchange Hamiltonian $H$ given by Eq. (2). This kind of positional disorder in Hamiltonians with long-ranged hopping has been recently studied and found to give rise to localization Deng et al. (2018). Consistently with this, we find that as we increase $\sigma$ the eigenstates of the Hamiltonian become localized, inhibiting transport. This can be seen in the top panels of Figs. 6(a) (b) and (c), where we show the excitation probability $\left|c_{\alpha}^{0}(t)\right|^{2}$ as a function of time for increasing disorder (ratio $\sigma/a$) from left to right.

Subradiance is, however, not a fine-tuned property but rather known to be robust in the presence of disorder Akkermans and Gero (2013); Biella et al. (2013). Indeed, in each lower panel in Figs. 6(a), (b) and (c) one can observe that, while increasing disorder has a detrimental effect on the subradiant state manifold, in all cases considered the excitation features lifetimes dramatically longer than the ones of an individual atom.

VI Conclusions and outlook

We have investigated the transport of an excitation in a one-dimensional atomic lattice that occurs due to the coupling of the atoms to the radiation field. In particular, we have shown that there is a high dimensional subradiant manifold that allows for dispersionless transport, control and storage of wave packets.

However, there are a number of experimental challenges to overcome when considering the realization of such long lived excitation storage in a dissipative system. One is to achieve a sufficiently small ratio $a/\lambda$ such that highly subradiant states emerge. An example of such a system, using a transition in the triplet series of alkaline earth metal atoms with a particularly long wavelength ($2.6$ $\mu$m in strontium), was introduced in Olmos et al. (2013). The trapping of these alkaline earth metal atoms is currently realized experimentally both in optical lattices Yamamoto et al. (2016); Miranda et al. (2017) and tweezer arrays Norcia et al. (2018). Even smaller ratios $a/\lambda$ can be achieved by using Rydberg states, where the transition wavelengths are much longer than in low-lying states. An alternative approach that allows subradiant states to emerge for large ratios $a/\lambda$ is changing the radiation field’s boundary conditions by placing, e.g., a surface Jones et al. (2018) or a waveguide Asenjo-Garcia et al. (2017); Le Kien and Rauschenbeutel (2017); Lodahl et al. (2017); Buonaiuto et al. (2019); Albrecht et al. (2019) near the atoms, which in turn modifies the exchange interaction and dissipation. Another experimental challenge is the preparation of the subradiant wave packets. In particular, preparing states with one excitation localized on one or a few sites will require single-site resolution and addressability, which has been achieved experimentally in optical lattices and tweezer arrays Weitenberg et al. (2011); Kim et al. (2016); Endres et al. (2016); Barredo et al. (2016); Cooper et al. (2018). Moreover, creating a wave packet with a linear dispersion relation will require a phase imprinting mechanism, which may be challenging to implement experimentally.

A future direction connecting to this work will be to move away from the linear optics regime (single excitation sector of the dynamics) Asenjo-Garcia et al. (2017); Jen (2018); Moreno-Cardoner et al. (2019); Guimond et al. (2019); Zhang and Mølmer (2019) and consider situations where two or more wave packets interfere with each other effectively realizing photon-photon interactions in a subradiant decoherence-free manifold. Such platform can find applications ranging from the creation of non-classical states of light to the realization of photon-photon quantum gates Gorshkov et al. (2011); Tiarks et al. (2019).

acknowledgements

The authors acknowledge fruitful discussions with Katarzyna Macieszczak, Matteo Marcuzzi, and Thomas Fernholz. BO was supported by the Royal Society and EPSRC [Grant No. DH130145]. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) [ERC Grant Agreement No. 335266 (ESCQUMA)] and from the European Union’s H2020 research and innovation programme [Grant Agreement No. 800942 (ErBeStA)]. Funding was also received from the EPSRC [Grant No. EP/M014266/1 and No. EP/R04340X/1]. IL gratefully acknowledges funding through the Royal Society Wolfson Research Merit Award.

Appendix A Dynamics in the truncated Hilbert space

In this Appendix we give the expressions for the equations of motion of each component of the truncated density matrix (8). We obtain these equations by simply substituting the expression (8) into the master equation (1), such that we can split the density matrix into the four components $\rho_{GG}=\langle{G}|\rho|{G}\rangle$, $\rho_{Ge^{m}_{\alpha}}=\langle{G}|\rho|{e^{m}}\rangle_{\alpha}$, $\rho_{e^{m}_{\alpha}G}={}_{\alpha}\langle{e^{m}}|\rho|{G}\rangle$ and $\rho_{e^{m}_{\alpha}e^{l}_{\beta}}={}_{\alpha}\langle{e^{m}}|\rho|{e^{l}}\rangle_{\beta}$. The time evolution of each component is given by

[TABLE]

and

[TABLE]

where $Z_{\alpha\beta}^{ml}=V_{\alpha\beta}^{ml}-\frac{\mathrm{i}}{2}\Gamma_{\alpha\beta}^{ml}$ [Eqs. (3) and (7)].