Three-dimensional theory of quantum memories based on $\Lambda$-type atomic ensembles

We develop a three-dimensional theory for quantum memories based on light storage in ensembles of $\Lambda$-type atoms, where two long-lived atomic ground states are employed. We consider light storage in an ensemble of finite spatial extent and we show that within the paraxial approximation the Fresnel number of the atomic ensemble and the optical depth are the only important physical parameters determining the quality of the quantum memory. We analyze the influence of these parameters on the storage of light followed by either forward or backward read-out from the quantum memory. We show that for small Fresnel numbers the forward memory provides higher efficiencies, whereas for large Fresnel numbers the backward memory is advantageous. The optimal light modes to store in the memory are presented together with the corresponding spin waves and outcoming light modes. We show that for high optical depths such $\Lambda$-type atomic ensembles allow for highly efficient backward and forward memories even for small Fresnel numbers $F\gtrsim 0.1$.

Figure 1

Left: A schematic plot of the level scheme of the considered $\Lambda$-type atoms with two long-lived ground states $|0\rangle$ and $|1\rangle$. The quantum field carries the information to be stored and couples $|0\rangle$ and $|e\rangle$ with coupling strength $g$. States $|1\rangle$ and $|e\rangle$ are coupled by the strong classical control field with Rabi frequency $\Omega \left(t\right)$. $\Delta$ denotes the detuning of the light from the excited state $|e\rangle$, which can spontaneously decay at a rate $\gamma$. Right: Atomic ensemble of cylindrical symmetry with Gaussian distribution of atoms in the radial direction. The density along the $z$ direction is constant. The length of the sample is $L$ and the width is characterized by ${\sigma }_{\perp }$.

Figure 2

Maximum retrieval efficiency as a function of the Fresnel number of the atomic ensemble $F$ for different transverse azimuthal quantum numbers $m$, and for fixed peak optical depth $d_0=40$ (top) and $d_0=100$ (bottom). The dashed black lines denote the 1D limit of $F\to \infty$.

Figure 3

Top: The intensity of the optimal output light modes for retrieval at the end of the atomic ensemble, $\tilde{z}=1$ for (left) low $d_0=0.1$ and (right) high $d_0=100$ peak optical depths. Bottom: The corresponding density of the optimal spin-wave excitations stored in the atomic memory. In both cases, the Fresnel number of the ensemble is $F=0.5$ and retrieval is resonant $\tilde{\Delta }=0$. The radial position is scaled with the atomic ensemble width as $\tilde{\rho }=\rho /{\sigma }_{\perp }$ and time according to $\tilde{t}=t{\left|\Omega \right|}^2/\gamma$.

Figure 4

Schematic illustration of the two operation modes of a quantum memory: (top) forward, where the control fields $\Omega \left(t\right)$ during the storage and read-out are in the same direction, and (bottom) backward read-out with the opposite directions of the control fields.

Figure 5

The maximal efficiencies for storage followed by read-out in the forward (top) and backward (bottom) direction as a function of the Fresnel number $F$ for different values of the azimuthal quantum number $m$ and for a peak optical depth $d_0=100$. The dashed black lines denote the corresponding 1D limits with $F\to \infty$.

Figure 6

The maximal efficiencies for storage followed by backward (full circles) and forward (empty circles) read-out as functions of the Fresnel number $F$ for optical depth $d_0=40$ (bottom, dotted green lines), $d_0=100$ (middle, dashed-dotted red lines), and $d_0=200$ (top, solid blue lines). The dashed lines for large values of the Fresnel number $F$ denote the corresponding values from the one-dimensional theory.

Figure 7

The maximal efficiencies for the backward (black) and forward (light blue/gray) memory as a function of the peak optical depth $d_0$ for $F=0.2$ (empty circles) and $F=2$ (stars). The dashed lines correspond to the results of the 1D theory with $F\to \infty$.

Figure 8

Top: Purity of the optimal input light mode for backward (solid lines) and forward (dashed lines) memory as a function of the Fresnel number $F$ for the peak optical depth $d_0=100$ and different values of the azimuthal quantum number $m=0$ (circles), $m=1$ (squares), $m=2$ (triangles), and $m=3$ (stars). Bottom: The purity of the optimal input light mode for backward (solid lines) and forward (dashed lines) memory as a function of the Fresnel number $F$ for $m=0$ and different values of the peak optical depth $d_0=40$ (circles), $d_0=100$ (squares), and $d_0=200$ (triangles).

Figure 9

Top: The spatial profile of the intensity of the optimal incoming light pulse for $F=0.2$ (black lines) and $F=2$ (light blue/gray lines) for backward (dashed lines) and forward (solid lines) memory ($d_0=100$). Bottom: The corresponding spatial profile of the phases as in (top).

Figure 10

Top left: The dimensionless waist of the optimal input light multiplied by the square root of the Fresnel number $F$, ${\tilde{w}}_0\sqrt{F}=w_0/\left({\lambda }_{0}L\right)$, as a function of the Fresnel number for backward (solid lines with markers) and forward memory (dashed lines). Here we present the fits for the peak optical values of $d_0=40$ (blue circles), $d_0=100$ (red squares), and $d_0=200$ (green stars). Top right: The dimensionless waist of the optimal input light scaled as in (top left) as a function of the peak optical depth $d_0$ for forward read-out with $F=2$ (dashed blue line) and for backward read-out with $F=0.2$ (solid green line) and $F=2$ (dashed-dotted red line). Bottom left: The dimensionless distance between the beginning of the atomic ensemble and the focal plane in the free space ${\tilde{z}}_{\mathrm{f}}=z_{\mathrm{f}}/L$ of the optimal input light mode as a function of the Fresnel number $F$. Bottom right: The distance ${\tilde{z}}_{\mathrm{f}}$ as a function of the peak optical depth $d_0$.

Figure 11

Top: The spatial and temporal shape of the optimal outcoming light pulse. Middle: The optimal spin-wave excitation in the atomic ensemble. Bottom: The corresponding incoming light mode. Left: The backward memory with read-in and read-out at $\tilde{z}=0$. Right: The forward memory. All results are for $F=2$ and $d_0=100$. The radial position is scaled with the atomic ensemble width as $\tilde{\rho }=\rho /{\sigma }_{\perp }$ and time according to $\tilde{t}=t{\left|\Omega \right|}^2/\gamma$.