Characterization of electromagnetically-induced-transparency-based continuous-variable quantum memories

We present a quantum multimodal treatment describing electromagnetically induced transparency (EIT) as a mechanism for storing continuous-variable quantum information in light fields. Taking into account the atomic noise and decoherences of realistic experiments, we numerically model the propagation, storage, and readout of signals contained in the sideband amplitude and phase quadratures of a light pulse using phase space methods. An analytical treatment of the effects predicted by this model is then presented. Finally, we use quantum information benchmarks to examine the properties of the EIT-based memory and show the parameters needed to operate beyond the quantum limit.

Figure 1

(Color online) EIT level structure. $\hat{\mathcal{E}}\left(z,t\right)$ is the envelope operator of the probe field and ${\Omega }_c\left(t\right)$ is the coupling beam Rabi frequency. Almost all the atoms are pumped into state $|1⟩$ initially. $\gamma$ is the spontaneous emission rate from the upper state and ${\gamma }_0$, ${\gamma }_c$ are mean decoherence rates between the two ground states for pure dephasing and population exchange, respectively. These two quantities are usually referred to as $1/T_2$ and $1/T_1$ in the field of magnetic resonance.

Figure 2

(Color online) Phase space numerical simulations of quantum information storage using EIT. Amplitude and phase modulations at 190 kHz are applied to the pulse. The decoherence rates are ${\gamma }_0=250\text{Hz}$, ${\gamma }_c=100\text{Hz}$. (a) 3D graph showing the storage of the probe amplitude quadrature on a time-space grid. (b) Variances of the input and output fields for the amplitude and phase quadratures, with 1 corresponding to the quantum noise limit. (i)–(iii) is the power spectrum of the input and output states and (ii)–(iv) are the noise floor of the input and output states. The dashed lines corresponds to statistical standard deviations. These simulations are the average of 2000 trajectories.

Figure 3

Signal to noise ratios and noise results for (a) and (c) the numerical solutions and (b) and (d) the analytical solutions. (i) and (i') correspond to ${\gamma }_c=0.005\gamma$ and ${\gamma }_0=0$; (ii) and (ii') to ${\gamma }_c=0.005\gamma$ and ${\gamma }_0=0.005\gamma$; and (iii) and (iii') to ${\gamma }_0=0.005\gamma$ and ${\gamma }_c=0$. The same parameters as for the phase space simulations of light storage of Sec. 1A were chosen.

Figure 4

(Color online) (a) General schematics for characterizing an optical memory. A pair of EPR entangled beams are encoded with amplitude and phase quadrature information. One of these beams is injected into, stored, and readout from the optical memory while the other is being propagated in free space. A joint measurement with appropriate delay is then used to measure the quantum correlations between the quadratures of the two beams. (b) A classical teleporter scheme used as an optical memory. The input state is measured jointly on both quadratures using two homodyne detection schemes. Analogous to classical teleportation the measured information is stored for time $\tau$ before fed-forward onto an independent laser beam with a feed-forward gain, $g$. The feed-forward gain is analogous to a transmission of $\sqrt{\eta \left(\omega \right)}$ for EIT-based memories. (c) Quantum memory using EIT. The input state is stored in the long lived ground state coherence of three level atoms in a $\Lambda$ configuration.

Figure 5

(Color online) Fidelity as a function of memory loss for EIT for ${\alpha }_{\text{in}}\left(\omega \right)=10,5,2,1$. The nonclassical and no-cloning regimes are reached when $\mathcal{F}\ge 1/2$ and $\mathcal{F}\ge 2/3$, respectively, for large input state amplitudes.

Figure 6

(Color online) Diagram for total signal transfer coefficient $T$ versus conditional variance product $V$. The classical limit line shows the optimal performance of a classical teleporter. The linear loss limit defines the performance of an EIT-based memory that does not produce excess noise. The unity gain curve is obtained with increasing excess noise in an EIT system with no loss. Regions A, B, C, and D correspond to the quantum regime; regions B and C represent the regime where EPR entanglement is preserved; region D is the lossless amplification region; region C denotes the no-cloning limit.

Figure 7

(Color online) Classical, EPR and no-cloning regimes plotted as a function of EIT linear loss and excess noise in (a). A is the nonclassical regime, B is the EPR regime, and C is the no-cloning regime. These limits are drawn in (b) on a loss-gain plot and have been derived from Eq. (33). Contrary to the case of (a), with a large enough gain and sufficiently low losses in the memory, region (d) can be reached.

Figure 8

(Color online) $TV$ diagrams showing the performance of the EIT memory. (a) The evolution of its efficiency for three different ${\gamma }_0$ values, the dotted lines representing the loci for a constant ${\gamma }_0$ and varying ${\gamma }_c$. (b) The evolution the EIT memory for three different ${\gamma }_c$ values, the dotted lines representing the loci for a constant ${\gamma }_c$ and varying ${\gamma }_0$.