Energy cost and optimal entanglement production in harmonic chains

We investigate nonequilibrium entanglement generation in a chain of harmonic oscillators with time-dependent linear coupling. We use optimal control theory to determine the coupling modulation that leads to maximum logarithmic negativity for a pair of opposite oscillators, and show that it corresponds to a synchronization of the eigenmodes of the chain. We further analytically relate the maximum attainable entanglement to the irreversible work done to produce it. We find that the energy cost of optimal entanglement production scales exponentially with the amount of entanglement.

Figure 1

Logarithmic negativity, Eq. (8), for opposite oscillators in a chain of $N=8$ as a function of the even and odd normal-mode angles, ${\theta }_{\text{even}}={\theta }_{0,2,4,6}$ and ${\theta }_{\text{odd}}={\theta }_{1,3,5,7}$, for fixed squeezing $r_s=0.4$ $\left(s=1,\dots ,7\right)$ and coupling strength $c=0.05$. The logarithmic negativity has a constant maximum value when the condition ${\theta }_{\text{even}}-{\theta }_{\text{odd}}=\left(2k+1\right)\pi /2$, $k∊\mathbb{Z}$, is satisfied.

Figure 2

Logarithmic negativity, Eq. (8), for opposite oscillators in a chain of $N=8$ as a function of odd normal-mode angles, ${\theta }_1={\theta }_7$ and ${\theta }_3={\theta }_5$, for fixed squeezing $r_s=0.4$ $\left(s=1,\dots ,7\right)$ and coupling strength $c=0.05$ (due to the symmetry of the chain, angles are pairwise equal). The even normal angles were here set to $\pi /2$. The logarithmic negativity is maximum when ${\theta }_3-{\theta }_1=2k\pi /2$, $k∊\mathbb{Z}$. We note further that $E_N$ vanishes when the condition of Fig. 1 is maximally violated, that is, ${\theta }_{\text{even}}-{\theta }_{\text{odd}}=2k\pi /2$.

Figure 3

(Color online) Logarithmic negativity, Eq. (8), for opposite oscillators as a function of time for different modulations of the coupling strength $c\left(t\right)$ between $c=0$ and $c=0.05$ for $N=8$. The lower shaded curve corresponds to a sudden switch at $t=0$, while the curve in the center is generated by the optimal modulation shown in Fig. 5. The two upper curves are the maximum attainable entanglement for $T=0$ and $T=0.5$.

Figure 4

(Color online) Angles ${\theta }_s$ of the normal modes as a function of time for the two intervals A and B shown in Fig. 3. The synchronization of even and odd modes is clearly visible at the center of interval B and completely absent in interval A. Specifically, we note in interval B that eigenmodes with the same parity have equal values of $\text{sin}2\theta$ (the difference of their angles is thus an even multiple of $\pi /2$), while eigenmodes with different parity have opposite values of $\text{sin}2\theta$ (the difference of their angles is hence an odd multiple of $\pi /2$).

Figure 5

Optimal coupling modulation $c\left(t\right)$ that generates the logarithmic negativity shown in the center of Fig. 3.