Dynamics of multiple atoms in one-dimensional fields

We analyze the dynamics of a set of two-level atoms coupled to the electromagnetic environment within a waveguide. This problem is often tackled by assuming a weak coupling between the atoms and the environment as well as the associated Markov approximation. We show that the accuracy of such an approximation may be more limited than in the single-atom case and may also be strongly determined by the presence of collective effects produced by atom-atom interactions. To this aim, we solve the full problem with exact diagonalization and also the time-dependent density matrix renormalization group method, and we compare the result to that obtained within a weak-coupling master equation and with the Dicke approximation. Finally, we study the dynamics of the entanglement within the system when considering several interatomic distances and atomic frequencies.

Figure 1

Schematic representation of the ladderlike structure of our system in Eq. (2). The atoms, represented by the arrows, are locally coupled to the transformed oscillators of the environment, conforming the infinite chain below.

Figure 2

Evolution of the normalized total population of a different number $N$ of excited atoms, $P_{\text{tot}}={\sum }_{j=1}^N\langle {\sigma }_j^{+}\left(t\right){\sigma }_j\left(t\right)\rangle$, as given by the ME and t-DMRG. We consider all the atoms located at distances $L=1$, and the atomic frequencies within the gap $[\mathrm{\Delta }=-1$, top panel, where $N=1,2,3$ correspond to blue (upper), green (middle), and red (lower) lines, respectively] and within the band $[\mathrm{\Delta }=1$, bottom panel, where now $N=1,2,3$ corresponds to blue (lower), green (middle), and red (upper) lines, respectively].

Figure 3

Evolution of $N=3$ atoms as given by t-DMRG (solid lines) and ME (dashed lines) for the following cases: $|100\rangle$ blue (upper) lines and $|110\rangle$ green (middle) lines, and $|111\rangle$ red (lower) lines. As in Fig. 2, we consider all the atoms spread at a distance $L=1$.

Figure 4

Contour plot representing the difference between the population of atom 1 (which is initially excited) as given by the ME and a numerically exact solution based on exact diagonalization and quantified by $E(L,{\omega }_S)$ given by (21). The $y$-axis represents the frequency shift between the atomic and gap-edge frequencies, $\mathrm{\Delta }={\omega }_S-{\omega }_{\text{c}}$, where ${\omega }_{\text{c}}=A-B$. The inset represents the real part of the correlation function (10) for increasing distances $r_i-r_j=0,10,20$ in solid blue, dotted green, and dot-dashed red, respectively.

Figure 5

Evolution of the population for the initial condition $|{\psi }_1\rangle$ (two upper panels) and $|{\psi }_2\rangle$ (two lower panels). The curves, corresponding to different interatomic separations $L$, are displayed in a rainbow scale that goes from blue ($L=1$) to red ($L=10$), also indicated by the arrows. We also distinguish between even separations (solid lines) and odd separations (dashed lines). The presence of an even-odd difference for the initially entangled state is related to sub- and superradiant states, as explained in the main text.

Figure 6

Evolution of the population for $\mathrm{\Delta }=1$ considering the initial condition $|{\tilde{\psi }}_2\rangle$ (upper panel) and $|{\psi }_2\rangle$ (lower panel) and two different atomic distances, $L=0$ and 1, and as predicted by the exact result and the approximated ME.

Figure 7

Evolution of the concurrence starting from an initially entangled state $|{\psi }_2\rangle$. The upper and lower panels correspond to $\mathrm{\Delta }=-1$ and 1, respectively. As in Figs. 5 and 5, the solid blue, dashed green, dotted orange, and dot-dashed red correspond to interatomic distances $L=1$, 2, 9, and 10, respectively. In addition, the black squared curve represents the concurrence of atoms coupled to independent environments as given by (34).

Figure 8

Evolution of the von Neumann entropy, which accounts for the degree of entanglement between the atomic system and the environment. The different curves correspond to $N=1$ atom with $\mathrm{\Delta }=1$ (red dot-dashed curve) and $\mathrm{\Delta }=-1$ (green dotted), and $N=2$ atoms ($L=1$) with $\mathrm{\Delta }=1$ (blue dashed) and $\mathrm{\Delta }=-1$ (solid black).

Figure 9

Two panels representing the values of the two decaying scales for only two different conditions $|{\psi }_1\rangle$ and $|{\psi }_2\rangle$ with the interatomic distance. We have considered ${\omega }_S=49$ (top panel) and ${\omega }_S=51$ (bottom panel).

Figure 10

Two panels giving the total population of the steady state $P_{\mathrm{tot}}^{\text{ss}}$ corresponding to $s_1-s_2$ in (E1) for the three different initial conditions $|{\psi }_j\rangle$ for $j=1,2,3$ and corresponding to different interatomic distances. We have considered ${\omega }_S=49$ (top panel) and ${\omega }_S=51$ (bottom panel).

Figure 11

$E^{\mathrm{D}}(L,{\omega }_S)$ for atom 1. In the contour plot the color scale for the $z$-axis goes from dark blue for zero to bright orange for 0.5. The $x$-axis holds the values of the distance between the two spins and the $y$-axis the values of $\mathrm{\Delta }={\omega }_S-(A-B)$, the difference between the atomic distance and the band-gap edge. The inset represents the line $2\left|\xi \right|$ that represents the boundary between distances reached by atom-atom couplings (where collective effects exist) and larger distances where atoms behave as if they were coupled to independent reservoirs.

Figure 12

Two panels with the difference between the exact population for $L=10$ and the one calculated with the Dicke Hamiltonian (G1), also following t-DMRG. We consider single-atom excited and initial states $|10\rangle$ (dot-dashed red) and $|100\rangle$ (dotted green), and as many excitations as atoms, with initial state $|11\rangle$ (dashed blue) and $|111\rangle$ (solid black).