Nonadiabatic approach to entanglement distribution over long distances

Entanglement distribution between trapped-atom quantum memories, viz. single atoms in optical cavities, is addressed. In most scenarios, the rate of entanglement distribution depends on the efficiency with which the state of traveling single photons can be transferred to trapped atoms. This loading efficiency is analytically studied for two-level, $V$-level, $\Lambda$-level, and double-$\Lambda$-level atomic configurations by means of a system-reservoir approach. An off-resonant nonadiabatic approach to loading $\Lambda$-level trapped-atom memories is proposed, and the ensuing trade-offs between the atom-light coupling rate and input photon bandwidth for achieving a high loading probability are identified. The nonadiabatic approach allows a broad class of optical sources to be used, and in some cases it provides a higher system throughput than what can be achieved by adiabatic loading mechanisms. The analysis is extended to the case of two double-$\Lambda$ trapped-atom memories illuminated by a polarization-entangled biphoton.

Figure 1

(Color online) Three architectures for entanglement distribution: (a) To-the-memory architecture, in which two entangled photons, generated at the source, carry the entangled state to the quantum memory units and load them with this state. (b) From-the-memory architecture, in which, by the use of a common source, each memory generates a flying qubit entangled with itself. Entanglement swapping is then accomplished by means of a Bell-state measurement on the photons, at the midpoint, which leaves the memories in an entangled state. (c) Memory-to-memory configuration, in which a flying qubit entangled with one of the memories is generated that then propagates to and loads the other memory, i.e., transfers its state to that memory.

Figure 2

(Color online) A single two-level atom trapped in a single-ended high-finesse optical cavity. A single photon illuminates the cavity at the proper frequency to excite the atom to its upper state.

Figure 3

(Color online) Loading probability versus normalized time for $\kappa T=2$ and a hyperbolic secant pulse shape at $\gamma =\Delta =0$. Upon arrival of the photon, there is a peak in the loading probability, which occurs at the time that the photon is most likely to have been absorbed by the atom. This maximum loading probability is a function of $g∕\kappa$, and there exists an optimum value for this parameter that maximizes the chance of loading.

Figure 4

(Color online) The optimum coupling rate, and the maximum loading probability achieved at this rate versus $\kappa T$ for loading a two-level atom by a single photon with hyperbolic secant pulse shape on resonance. Several spontaneous decay rates, $\gamma$, are considered.

Figure 5

(Color online) Left: loading probability at $\kappa T=2$, $g∕\kappa =1$, and $\gamma =\Delta =0$ for various pulse shapes. Right: the baseband pulse shapes. In order of their maximum loading probabilities they are the hyperbolic secant (with the highest peak), rectangular, exponentially rising, and exponentially decaying pulse shapes.

Figure 6

(Color online) A $\Lambda$-level trapped atom illuminated by a single photon. The cavity mode corresponding to the single photon drives the atom from its ground state $\mid g⟩$ to the auxiliary state $\mid r⟩$. A $z$-polarized beam then shelves the atom in the metastable level $\mid e⟩$. Here, $g_c$ is the coupling rate for the $\mid g⟩$-to-$\mid r⟩$ transition, and $\Omega$ is the Rabi frequency associated with the control field, which is proportional to $\mid E_z\mid$. These transitions are off-resonant by detunings ${\Delta }_1$ and ${\Delta }_2$, respectively, as defined in the text. The control field may also include a time-varying phase ${\varphi }_z\left(t\right)$.

Figure 7

(Color online) The loading probability for the adiabatic method as a function of $\kappa T$ versus $g_c^2∕\left(\kappa {\Delta }_1\right)$, at the two-photon resonance ${\Delta }_1={\Delta }_2$ with ${\varphi }_z\left(t\right)=0$, ${\gamma }_r=0$, and several values of $\kappa T$. These curves were obtained by numerically solving Eq. (18) using the control pulse shapes, shown in the inset, that correspond to hyperbolic secant input pulse shapes.

Figure 8

(Color online) The optimum coupling rate and the maximum loading probability achieved at this rate for both adiabatic and nonadiabatic approaches. (Nonadiabatic curves start from $\kappa T=0.5$; adiabatic curves start from $\kappa T=4.5$.) For all curves, we assume that $\Delta ={\gamma }_r=0$. A hyperbolic secant pulse shape has been used for the incoming photon.

Figure 9

(Color online) Loading probability versus normalized timing offset in the photon arrival. In the adiabatic case (the left curve), this timing error results in applying the control field earlier or later than the correct time. In the nonadiabatic case (the right curve), it results in stopping the control field earlier or later than $T_{\mathrm{Load}}$. Both curves assume $\Delta ={\gamma }_r=0$ for their respective optimum coupling rates for hyperbolic secant input pulse shapes.

Figure 10

(Color online) (a) Loading a double-$\Lambda$ atom driven by a polarization-state single photon. By adiabatically eliminating the upper states, this system becomes equivalent to the $V$–level system shown in (b). (c) Reduction of the $V$–level loading problem to two two-level loading problems by means of linear superposition.

Figure 11

(a) MIT-NU architecture for long-distance quantum communications consisting of a dual-OPA source that produces polarization-entangled photons, and two quantum memories, ${\mathrm{QM}}_1$ and ${\mathrm{QM}}_2$, separated by $2L_0\mathrm{km}$. (b) Dual-OPA source of polarization-entangled photons. OPAs 1 and 2 are coherently pumped ($\pi$-rad out of phase), continuous-wave, type-II phase matched, doubly resonant amplifiers operated at frequency degeneracy whose orthogonally polarized signal $\left(\right\{S_k\left\}\right)$ and idler $\left(\right\{I_k\left\}\right)$ outputs are combined, as shown, on the polarizing beam splitter (PBS). (c) Notional schematic for the relevant hyperfine levels of ${}^{87}\mathrm{Rb}$. Each quantum memory consists of a single trapped rubidium atom that can absorb arbitrarily polarized photons, storing their coherence in the long-lived $D$ levels. A nondestructive load verification is effected by means of the $A$-to-$C$ cycling transition.

Figure 12

(Color online) (a) Loading a pair of double-$\Lambda$ atoms illuminated by a polarization-entangled biphoton. This is a fair approximation to the MIT-NU loading problem. (b) Breaking the loading problem in (a) into two simpler loading problems for a pair of $\Lambda$-level atoms, each illuminated by a biphoton state.

Figure 13

(Color online) A two-level-atom model for the systems shown in Fig. 12b. Here, we have assumed that the Rabi frequency $\Omega$ is a constant and we have adiabatically eliminated the upper state in the $\Lambda$-level atoms. In the absence of spontaneous decay, the effective coupling rate is $g=g_c\Omega ∕{\Delta }_1$. The effective detuning $\Delta$ can be made zero by a proper choice of parameters, as shown in Eq. (18).

Figure 14

(Color online) (a) MIT-NU loading probability for different values of $g∕\kappa$ at $\kappa T=\kappa T_0=2$. The maximum probability in each case is a function of $g∕\kappa$, which implies the existence of an optimum value for $g∕\kappa$. (b) MIT-NU loading probability for different values of $\kappa T_0$ at $g∕\kappa =1$ and $\kappa T=2$. In all curves, we have used are a hyperbolic secant pulse, a rectangular pulse, an exponentially rising pulse, and an exponentially decaying pulse for the pump beam with ${\gamma }^{\prime }=0$.

Figure 15

(Color online) (a) The optimum value of $g∕\kappa$ versus $\kappa T$ and $\kappa T_0$ for the MIT-NU nonadiabatic loading problem. (b) The corresponding maximum loading probability at $(g∕\kappa {)}_{\mathrm{opt}}$. We have used a hyperbolic secant pulse shape for the pump beam with ${\gamma }^{\prime }=0$.