Entanglement dynamics of nonidentical oscillators under decohering environments

We study the evolution of entanglement for a pair of coupled nonidentical harmonic oscillators in contact with an environment. For both cases of a common bath and of two separate baths for each of the oscillators, a full master equation is provided without rotating-wave approximation. The entanglement dynamics is analyzed as a function of the diversity between the oscillators’ frequencies and their positive or negative mutual coupling and also the correlation between the occupation numbers. The singular effect of the resonance condition (identical oscillators) and its relationship with the possibility of preserving asymptotic entanglement are discussed. The importance of the bath’s memory properties is investigated by comparing Markovian and non-Markovian evolutions.

Figure 1

Time evolution of an entangled two-mode squeezed initial state Eq. (10) with $r=2$, in the case of (a) separate baths and (b) a common bath. The parameters are $k_{B}T=10{\omega }_1$, $\gamma =0.001{\omega }_1$, and cutoff frequency $\Lambda =50{\omega }_1$. The four curves correspond to the points in Figs. 2 and 4, namely, A (dotted black), B (solid black), C [dotted orange (gray)], and D [solid orange (gray)]. Two features are apparent: In the presence of a nonzero coupling $\lambda$ (continuous lines), entanglement oscillates strongly; and second, in the case of separate baths, any initial entanglement vanishes fast, while, in the case of a common bath, it can be made to survive by having identical frequencies.

Figure 2

Parameter scan of $t_F$ (see text) as a function of the oscillators’ coupling and frequency detuning, in the case of separate baths. We have restricted the scan within a limited detuning (${\omega }_2⩽2{\omega }_1$) as a result of the absence of any important features outside this region. In addition, the condition $\lambda <{\omega }_1{\omega }_2$ ensures reality of the eigenfrequencies in the problem. The behavior of $t_F$ is basically monotonic in ${\omega }_2/{\omega }_1$, with a superimposed arenalike shape, which comes from the oscillatory nature of entanglement. See text for more details. The dots A, B, C, and D are given in Fig. 1 as examples.

Figure 3

Three-dimensional representation of Fig. 2. Here, the arenalike shape can be better appreciated.

Figure 4

Parameter scan of $t_F$ (see text) as a function of the oscillators’ coupling and frequency detuning, in the case of a common bath. As seen here, $t_F$ is huge whenever the detuning is small and on the order of tens of periods for high detuning. As expected, at resonance, one of the eigenmodes decouples from the baths, which leads to an infinite $t_F$. We have truncated the plot at heights of ${\omega }_1{t}_F=210$, otherwise, this mountain riff is infinitely high (but only strictly infinite when ${\omega }_1={\omega }_2$).

Figure 5

Three-dimensional representation of Fig. 4. Apart from the riff at equal frequencies, the shape is very similar to the case of separate baths, with an arenalike bowl shape. We have drawn a mesh to guide the reader’s eye. Again, heights over $t_F=210/{\omega }_1$ have been truncated (yellow region).

Figure 6

Comparison between the time evolutions of entanglement (black) and the correlation measure $\max [0,-\langle :(n_1-n_2){}^2:\rangle$] [orange (gray)], for the case of (a) separate baths and (b) a common bath. We have used equal frequencies, coupling $\lambda =0.1{\omega }_1^2$, damping $\gamma =0.001{\omega }_1$, temperature $k_{B}T/{\omega }_1=10$, and initial squeezing parameter $r=0.5$. The envelope of the correlation measure seems to be closely related to the entanglement, except for a delay of half a period.

Figure 7

Comparison between Markovian [orange (gray)] and non-Markovian (black) entanglement dynamics. The system parameters are ${\omega }_1={\omega }_2$, $\lambda =0.2{\omega }_1^2$, $\gamma =0.01\times 2/\pi {\omega }_1$, $\Lambda =20{\omega }_1$, and $k_{B}T=10{\omega }_1$. The initial squeezings are (a) $r=2$ and (b) $r=0$.