Coherent control in a decoherence-free subspace of a collective multilevel system

Decoherence-free subspaces (DFS’s) in systems of dipole-dipole interacting multilevel atoms are investigated theoretically. It is shown that the collective state space of two dipole-dipole interacting four-level atoms contains a four-dimensional DFS. We describe a method that allows us to populate the antisymmetric states of the DFS by means of a laser field, without the need for a field gradient between the two atoms. We identify these antisymmetric states as long-lived entangled states. Further, we show that any single-qubit operation between two states of the DFS can be induced by means of a microwave field. Typical operation times of these qubit rotations can be significantly shorter than for a nuclear spin system.

Figure 1

(a) The system under consideration is comprised of two atoms that are located at ${\mathit{r}}_1$ and ${\mathit{r}}_2$, respectively. The relative position $\mathit{R}={\mathit{r}}_2-{\mathit{r}}_1$ of atom 2 with respect to atom 1 is expressed in terms of spherical coordinates. (b) Internal level structure of atom $\mu ∊\{1,2\}$. The ground state of each of the atoms is a $S_0$ state, and the three excited levels are Zeeman sublevels of a $P_1$ triplet. The states $\mid 1_{\mu }⟩$, $\mid 2_{\mu }⟩$, and $\mid 3_{\mu }⟩$ correspond to the magnetic quantum numbers $m_j=-1,0$, and 1, respectively. The frequency splitting of the upper levels is denoted by $\delta ={\omega }_3-{\omega }_2={\omega }_2-{\omega }_1$, where $\hslash {\omega }_i$ is the energy of state $\mid i_{\mu }⟩$.

Figure 2

(a) Plot of the vacuum-induced coupling terms ${\Omega }_{11}$ and ${\Omega }_{31}$ according to Eq. (15). ${\lambda }_0$ is the mean transition wavelength. If the interatomic distance $R$ approaches zero, the parameters ${\Omega }_{11}$ and ${\Omega }_{31}$ diverge. (b) Plot of the collective decay rates ${\Gamma }_{11}$ and ${\Gamma }_{31}$ according to Eq. (16). ${\Gamma }_{11}$ and ${\Gamma }_{31}$ remain finite in the limit $R\to 0$. The parameters in (a) and (b) are given by $\theta =\pi ∕2$ and $\varphi =0$.

Figure 3

Plot of the vacuum-induced energy shifts ${\Omega }_F$ and ${\Omega }_N$ as a function of the interatomic distance $R$ according to Eq. (38). These shifts enter the expressions for the eigenvalues of $H_{\Omega }$ in Eqs. (37, 41). Note that ${\Omega }_F$ decreases with $1∕R$ for large values of $R$, while ${\Omega }_N$ vanishes with $1∕R^2$.

Figure 4

Dependence of the parameters ${\Gamma }_a^i$ and ${\Gamma }_s^i$ on the interatomic distance $R$ according to Eq. (43). (a) In the limit $R\to 0$, the ${\Gamma }_a^i$ tend to zero and the antisymmetric states $\mid {\psi }_a^i⟩$ are subradiant. (b) The symmetric states $\mid {\psi }_s^i⟩$ decay twice as fast as compared to two independent atoms if $R$ approaches zero.

Figure 5

The atoms are aligned in a plane spanned by the unit vectors ${\mathit{e}}_z$ and ${\mathit{e}}_{\varphi }=(\cos \varphi ,\sin \varphi ,0)$. Within this plane, the relative position of the two atoms, $\mathit{R}=z{\mathit{e}}_z+l{\mathit{e}}_{\varphi }$, is described by the parameters $z$ and $l$. The energies of the eigenstates of $H_A+H_{\Omega }$ depend only on $z$ and $l$, but not on $\varphi$.

Figure 6

(Color online) Plot of the energy shifts that determine the energy levels of the antisymmetric states according to Eq. (45). In (a)–(c), the parameters ${\Lambda }_a^i$ are shown in a plane spanned by ${\mathit{e}}_z$ and ${\mathit{e}}_{\varphi }=(\cos \varphi ,\sin \varphi ,0)$. The relative position $\mathit{R}=z{\mathit{e}}_z+l{\mathit{e}}_{\varphi }$ of the atoms in this plane is parametrized by $z$ and $l$ (see also Fig. 5). Since the ${\Lambda }_a^i$ do not depend on $\varphi$, the energy surfaces shown in (a)–(c) do not change if ${\mathit{e}}_{\varphi }$ is rotated around the $z$ axis. While ${\Lambda }_a^1$ and ${\Lambda }_a^2$ tend to $-\infty$ in the limit $R\to 0$, ${\Lambda }_a^3$ tends to $+\infty$. The frequency splitting of the excited states is $\delta =\gamma$. In (d), the ${\Lambda }_a^i$ are shown as a function of the interatomic distance $R$; the parameters are $\theta =\pi ∕2$ and $\delta =\gamma$.

Figure 7

Complete level scheme of the nondegenerate system $(\delta \ne 0)$. For the special geometrical setup where the atoms are aligned in the $x-y$ plane $(\theta =\pi ∕2)$, the analytical expressions for the states $\mid {\phi }_a^i⟩$, $\mid {\phi }_s^i⟩$ and the frequency shifts ${\Lambda }_a^i$, ${\Lambda }_s^i$ are given in Eqs. (48, 53, 46, 52), respectively. The frequency shifts ${\Lambda }_a^i$ $\left({\Lambda }_s^i\right)$ of the antisymmetric (symmetric) states and the splitting of the excited states are not to scale. Note that the frequency shifts ${\Lambda }_a^i$ and ${\Lambda }_s^i$ depend on the relative position of the atoms.

Figure 8

(Color online) Laser-induced coupling of $\mid {\psi }_a^2⟩$ to the excited states $\mid i,j⟩$ $(i,j∊\{1,2,3\})$ in the case of the degenerate system. States that are not directly coupled to $\mid {\psi }_a^2⟩$ have been omitted (except for the ground state). The laser polarization that couples the antisymmetric state $\mid {\psi }_a^2⟩$ to a state $\mid i,j⟩$ $(i,j∊\{1,2,3\})$ is indicated next to the respective transition. $\mid {\psi }_a^2⟩$ is completely decoupled from a $y$-polarized laser field.

Figure 9

Time-dependent population of the states $\mid {\psi }_a^i⟩$ for different polarizations of the driving field. The initial state at $t=0$ is $\mid 4,4⟩$. The parameters are $\theta =\pi ∕2$, $\varphi =0$, $\delta =0$, and ${\Delta }_2=0$. (a) Population of $\mid {\psi }_a^2⟩$ for ${\Omega }_y\left({\mathit{r}}_1\right)={\Omega }_y\left({\mathit{r}}_2\right)=5\gamma$. The states $\mid {\psi }_a^1⟩$ and $\mid {\psi }_a^3⟩$ are not populated. (b) Population of $\mid {\psi }_a^3⟩$ for ${\Omega }_x\left({\mathit{r}}_1\right)={\Omega }_x\left({\mathit{r}}_2\right)=5\gamma$. The states $\mid {\psi }_a^1⟩$ and $\mid {\psi }_a^2⟩$ are not populated.

Figure 10

(Color online) Bloch sphere representation of the system dynamics in the subspace $\mathcal{Q}$ spanned by the states $\{\mid {\psi }_a^2⟩,\mid {\psi }_a^3⟩\}$. At $t=0$, the system is in the pure state $\mid {\psi }_a^2⟩$ and a static magnetic field is switched on. The Bloch vector is rotated around an axis in the $x\text{−}z$ plane, and the tilt of this axis in the $x$ direction increases with the magnetic field strength. The value of the parameter $\delta$ is (a) $\delta =3.15\gamma$, (b) $\delta =4.83\gamma$, and (c) $\delta =6.22\gamma$, and we chose $R=0.1{\lambda }_0$.

Figure 11

(Color online) Complete population transfer from $\mid {\psi }_a^2⟩$ to $\mid {\psi }_a^3⟩$ by means of a resonant rf field. At $t=0$, the Bloch vector ${\mathit{B}}_N$ points into the positive $z$ direction. At $t=\pi ∕{\Omega }_{\mathrm{rf}}$, the state of the system is $\mid {\psi }_a^3⟩$ and ${\mathit{B}}_N$ points into the negative $z$ direction. Note that the length of ${\mathit{B}}_N$ is slightly smaller than unity for $t>0$ due to the small probability of spontaneous emission to the ground state. The parameters are $R=0.05{\lambda }_0$, ${\delta }_0=\gamma$, ${\varphi }_{\mathrm{rf}}=\pi$, and ${\Delta }_{\mathrm{rf}}=0$.