Control of entanglement dynamics in a system of three coupled quantum oscillators

Dynamical control of entanglement and its connection with the classical concept of instability is an intriguing matter which deserves accurate investigation for its important role in information processing, cryptography and quantum computing. Here we consider a tripartite quantum system made of three coupled quantum parametric oscillators in equilibrium with a common heat bath. The introduced parametrization consists of a pulse train with adjustable amplitude and duty cycle representing a more general case for the perturbation. From the experimental observation of the instability in the classical system we are able to predict the parameter values for which the entangled states exist. A different amount of entanglement and different onset times emerge when comparing two and three quantum oscillators. The system and the parametrization considered here open new perspectives for manipulating quantum features at high temperatures.


The Quantum System
We consider a system of two and three coupled quantum parametric oscillators in equilibrium with a common heat bath at temperature T. The system is described by the following Hamiltonian The total Hamiltonian H T consists of H S that represents a system of oscillators with N = 2, 3 (with equal mass m i = m 0 ) and their interactions and H SR that represents the interaction between oscillators and thermal reservoir. ω(t) is the angular frequency and {X i , P i } the position and momentum operators for the oscillators with i = 1, 2, 3; similarly ω k and {x k , p k } characterize the environment oscillators. The oscillators 1, 2 and 3 are coupled to each other by the functions c(t) and connected to the environment through the constants c k .
In the case N = 3 of three coupled oscillators, starting from [X] = (X 1 , X 2 , X 3 ) T and [P] = (P 1 , P 2 , P 3 ) T we can define two new vectors [X] = R. [  . In the case N = 2, the R matrix is: (9) and the Hamiltonian H T is decoupled as , in this case the commutator between the primed Hamiltonians is zero due to the orthogonality of the R transformation.
Here we consider, as initial condition, the coherent state ρ is a tensorial product of coherent states associated with parametric oscillators (see Supplementary). The coherent character of the initial Gaussian state is preserved by orthonormal transformations originated from R and it is maintained during the time evolution thanks to the bilinearity in the coordinates of the Hamiltonians (5) and (10). This implies that every quantum correlations will be obtained from the Covariance Matrix (CM) whose elements are given by The elements of the Covariance Matrix are easily obtained by using the primed operators and since the primed Hamiltonians commute between themselves an independent analysis leads to 0 Supplementary). In the calculation of the CM for the Hamiltonian (6) and (11) we have used the path integral formulation of Feynman where we calculate the propagator of the system J (see Supplementary and refs 9, 10), the temporal evolution of the density matrix isX To calculate the CM elements for the Hamiltonians (7), (8) and (12) we have used the Heisenberg representation (see Supplementary).

Experimental
In previous works 7, 8 , it has been shown that the onset of the entanglement is connected to the diverging solutions of the CM element equations of the operators X′ and P′. We indicated with  + t ( ) and t ( )  − the time dependence of the CM elements, in both cases of N = 2 and N = 3, for the oscillators with pulsation Ω + and Ω − respectively. +  t ( ) and −  t ( ) evolve in time according to the following differential equations where γ is the dissipation rate of the reservoir. We found that only the dynamical behaviour of the oscillators characterized by Ω − determines the survival of the entanglement at high temperature. Then we focused our attention on equation (15) to describe, also in experimental way, the space parameter regions where we expect entanglement arises.
We specialised the discussion to the case c t cm ( ) , with f(ω p , t) a suitable function and ω d the external driving pulsation. In that way we can define the dimensionless The experimental tests have been performed implementing an analog electronic version of the equation (15) with the Ω − introduced above, which corresponds to a perturbed Mathieu oscillator (see section Methods and Fig. 6). In terms of first order differential equations the system is described by represents the analytic form of the external stimulus and consists of a pulse train with adjustable amplitude and duty cycle (see blue curve in Fig. 1).
From a mathematical point of view the function f can be obtained as superposition of Fourier harmonics of angular frequency is the dimensionless perturbation period. It can be shown that the stability maps are almost invariant if more than five harmonics are used. This allows us to experimentally use a step function (red curve in Fig. 1) as the external stimulus without affecting the comparison between simulations and experimental data.
By observing the signal saturation, due to the integrated electronic component limits, we can experimentally reconstruct, in the parameter space is shown in Fig. 2(b). The experimental points (red dots in Fig. 2) were overlaid to the numerical simulations of instability regions characterized by positive values of the real part μ of the so called Floquet coefficient 11 . From a general point of view, Eq. (17) is a periodical and linear differential equation. The Floquet theorem guarantees that, under these conditions, the solution is of the form (x(t), y(t)) T = exp(Kt)F(t)C. The two eigenvalues λ ± of K are known as Floquet coefficients. Defining μ ± = Re{λ ± }, the system is unstable when μ + = μ > 0 as in our case μ − = −μ + .

Entanglement
We show the difference between the bipartite quantum entanglement in systems of 2 and 3 oscillators. Since the final system state is Gaussian, the theorem of positive partial transpose (PPT) 12, 13 is used as a criterion for entanglement. Furthermore, the logarithmic negativity 12 is employed to evaluate the entanglement level in the system of two oscillators: where v − is given by: v  I  I  I  I  I  I  I  2  2 ( 2 ) 2 , The two oscillators are entangled or separable for E N ≠ 0 and E N = 0 respectively. In the case of 3 oscillators we used again the PPT theorem to analise the entanglement bipartite between the oscillator 1 with the oscillators 2 and 3. As the hamiltonian (1) is invariant to coordinate changes the CM is fully symmetric 14 , then we can use the bi-partite logarithmic negativity given by where − n  is a eigenvalue of the CM transpose (only one of the 6 eigenvalues of the σ transpose can be negative) of the system with 3 oscillators 14 (see Supplementary).

Results
In all calculations the function f(ω, τ) has been approximated by the first five terms of its Fourier series and we used the following values A = 0.215, / 17   and dissipation rate γ = 0.01ω 0 . The initial condition of the system with 2 oscillators is the coherent state 2 0 α Ψ = that in the primed coordinates becomes αα Ψ = . The initial condition of the system with 3 oscillators is the coherent state In the primed coordinates this state is expressed by ααα Ψ = with 1/ 2 α = . In Fig. 2(c) we report, for N = 2, the logarithmic negativity against the dimensionless time τ, for different values of the duty cycle b  . For values of  b for which μ = 0 (see Fig. 2(b)) the system does not present entanglement. Otherwise, for . ≤ ≤ . b 0 515 1 000  , where μ > 0 and after a certain time the system presents entanglement. Once the system has acquired entanglement this grows approximately linearly with small superimposed oscillations.  Fig. 2(d), we observe that for the same values of  b and μ > 0 where entanglement is detected for two oscillators, a similar behaviour emerges for three oscillators with the . This reduction of entanglement, due to the third oscillator, is consistent with the monogamy inequality conjectured by Coffman, Kundu and Wootters 15 and extended by Osborne and Verstraete 16 . The same level of entanglement in the two cases (system with two oscillators and system with three oscillators) is reached if and only if for the three oscillators, the entanglement between the oscillators 2 and 3 is zero, i.e. 2 and 3 are separable oscillators.
In Fig. 3 ) between E N and | E N 1 23

(a) and (b) the surfaces of the bipartite logarithmic negativity
. Figure 3(c) reports the initial entanglement time against b  for 2 (blue dots) and 3 (red dots) oscillators showing that entanglement occurs first in the latter case.
In Fig. 3(d), the mean entanglement rates ζ for 2 and 3 oscillators, corresponding to the surfaces of Fig. 3(a) and (b), and the real part μ of the Floquet coefficient (green dots) are plotted as a function of  b. As already observed in Fig. 2(b-d), the entanglement is different from zero where the classical oscillator is unstable. Furthermore, ζ is higher for 2 oscillators, in fact the three oscillators system is faster to develop entanglement but with a lower value.
In Fig. 4 we show the average entanglement rate ζ as function of μ for the two systems. The parameter ζ has been numerically obtained by fitting the curves of Fig. 2

(c) and (d) using the nonlinear function
. From this figure it can be noted that the behaviour of ζ versus μ is linear and that the rate for 2 oscillators is larger than for 3 oscillators. In addition, the relation between μ and ζ is not bijective since in the system of 2 oscillators there are regions with two values of ζ (bifurcation of the entanglement rate) for the same value of μ. As it is possible to see in Fig. 3(d), the real part of the Floquet coefficient displays a local minimum as function of  b originating the bifurcation in the ζ vs μ curve.
The temperature dependence of the bipartite entanglement has been numerically investigated, for the case of three oscillators, and reported in Fig. 5 where we plot | E N 1 23 as function of τ for different values of  T for a fixed value of b 0 6 = .
 . From this figure we can state that a changing temperature does not affect the entanglement rate but it only slightly modify the initial entanglement time as in the case of two oscillators. We also observe that for a fixed value of τ the entanglement decreases as temperature increases.

Conclusions
In this work we have considered the bipartite quantum entanglement for a system of three oscillators in contact with a thermal reservoir at high temperature. The analysis of the classical counterpart by means of a suitable control implemented on a electronic single parametric oscillator allows to know the regions where entanglement will be originated. Peculiar differences emerge when the systems with two and three oscillators are compared, as the reduction of entanglement related with monogamy and the opposite behaviour in their initial times. We also note that the average bipartite entanglement rate is approximately a linear function of μ and it presents a bifurcation, related to the coupling between the two oscillators. Although, our experimental measurements are limited to the classical system we believe that future experiments confirming the quantum features could be done by using coupled optomechanical cavities as recently pointed out in refs 17-19.