Noiseless Subsystems and Synchronization

Abstract

In the previous chapter we have seen that common dissipation leads to the emergence of mutual synchronization between two oscillators, and we have also shown its relation with the slow decay of quantum correlations. Furthermore, this phenomenon is stronger the closer the natural frequencies of the oscillators are, as in the limiting case of equal frequencies only the normal mode corresponding to the center of mass position, $\hat{X}_+ = (\hat{x}_1 + \hat{x}_2)/2$, couples to the environment, while the other normal mode, the relative position $\hat{X}_{-} = (\hat{x}_1 - \hat{x}_2)/2$, is effectively uncoupled from any environmental action, leading to asymptotic entanglement between the two oscillators.

Appendices

Appendix

A.1 Analytical Derivation of Asymptotic Entanglement

As we pointed in Sect. 1.4.1, all the information about bipartite quantum correlations for a Gaussian continuous-variable state is condensed in its covariance matrix defined through the ten second-order moments of $\hat{q}_{(A, B)}$ and $\hat{p}_{(A, B)}$ (in our case first-order moments are initially zero). This bipartite covariance matrix defined for a system of two oscillators A and B, can be written as

$$\begin{aligned} V_{A B}= \left( {\begin{array}{cc} \alpha &{} \gamma \\ \gamma ^{t} &{} \beta \end{array} } \right) , \end{aligned}$$

(A.1)

where $\alpha , \beta$ and $\gamma$ are $(2\times 2)$ blocks: $\alpha (\beta )$ contains the second-order moments of oscillator subsystem A (B), and $\gamma$ contains correlations of both subsystems. The minimum symplectic eigenvalue (of the covariance matrix corresponding to the partially transposed density matrix), necessary to calculate the logarithmic negativity, is given by

$$\begin{aligned} \nu _{-} = \sqrt{\frac{1}{2}(a + b - 2g - \sqrt{(a + b- 2g)^2 - 4s)}}, \end{aligned}$$

(A.2)

with $a= 4 \det (\alpha )/\hslash ^2$, $b= 4 \det (\beta )/\hslash ^2$, $g= 4 \det (\gamma )/\hslash ^2$ and $s=16 \det V_{AB}/\hslash ^4$. Normal modes coupled to the environment will reach in the asymptotic limit a thermal state, given by the Gibbs distribution. For a normal mode (k), that is

$$\begin{aligned} \rho _{\mathrm {th}}^{(k)} = \frac{e^{-\frac{\hat{H}_k}{k_B T}}}{\mathrm {Tr}[e^{-\frac{\hat{H}_k}{k_B T}}]}, \end{aligned}$$

(A.3)

with $\hat{H}_k = \frac{1}{2} \left( \hat{P}_k^2 + \Omega _k^2 \hat{Q}_k^2 \right)$, yielding the second-order moments

$$\begin{aligned} \langle \hat{Q}_k^2 \rangle _{\mathrm {th}}&= \frac{\hslash }{2 \Omega _k} \coth \left( \frac{\hslash \Omega _k}{2 k_B T}\right) , \nonumber \\ \langle \hat{P}_k^2 \rangle _{\mathrm {th}}&= \frac{\hslash \Omega _k}{2} \coth \left( \frac{\hslash \Omega _k}{2 k_B T} \right) , \end{aligned}$$

(A.4)

where $\Omega _k$ is the corresponding frequency of the normal mode, and T the reservoir temperature. On the other hand, the uncoupled modes evolve freely. This means that the asymptotic covariance matrix can be calculated by expressing all second-order moments of natural oscillators in terms of the normal modes, and then substituting the asymptotic expressions corresponding to free modes or thermalized ones.

The covariance matrix in the asymptotic limit can be separated into three parts corresponding to the contributions of each normal mode, which we call $V_i$, for $i= 1, 2, 3$. In terms of the blocks we have

$$\begin{aligned} \alpha = \sum _{i=1}^3 \mathcal {F}_{A i}^2 V_i, ~~~~ \beta = \sum _{i=1}^2 \mathcal {F}_{B i}^2 V_i, ~~~~ \gamma = \gamma ^T = \sum _{i=1}^2 {\mathcal {F}}_{A i} {\mathcal {F}}_{B i} V_i, \end{aligned}$$

(A.5)

where $V_i$ can correspond either to a non-dissipative, or to a dissipative mode. For a non-dissipative normal mode, say n, we have:

$$\begin{aligned} V_{\mathrm {no-diss}}= \left( {\begin{array}{cc} \langle \hat{Q}_n^2 \rangle &{} \frac{\langle \{ \hat{Q}_n , \hat{P}_n \} \rangle }{2} \\ \frac{\langle \{ \hat{Q}_n , \hat{P}_n \} \rangle }{2} &{} \langle \hat{P}_n^2 \rangle \end{array} } \right) , \end{aligned}$$

(A.6)

and for a dissipative one, say k, we get

$$\begin{aligned} V_{\mathrm {diss}}= \left( {\begin{array}{cc} \langle \hat{Q}_k^2 \rangle _{\mathrm {th}} &{} 0 \\ 0 &{} \langle \hat{P}_k^2 \rangle _{\mathrm {th}} \end{array} } \right) . \end{aligned}$$

(A.7)

While elements in $V_{\mathrm {diss}}$ are given by the expressions (A.4), those of $V_{\mathrm {no-diss}}$ are the ones corresponding to a free evolution of an harmonic oscillator. This analysis gives all the necessary elements in order to calculate the asymptotic entanglement for pairs of oscillators in every particular situation, in which one or two of the normal modes are uncoupled from the environmental action.

1.1 A.1.1 One-Mode NS

Consider the specific case of an open chain $(\lambda _{1 3}=0)$ in which we have two equal frequencies $(\omega _1 = \omega _3 \equiv \omega )$ and two equal couplings $(\lambda _{1 2} = \lambda _{2 3} \equiv \lambda \ne 0)$ [Fig. 5.1a in Sect. 5.3]. In this case we get only one normal mode decoupled from the bath. In order to calculate the expression of the minimum symplectic eigenvalue, we have to first calculate the elements of the three normal modes, that are shown here as vector columns

$$\begin{aligned} {\mathbf C}_{\delta } = \frac{1}{\sqrt{2}} \left( {\begin{array}{cc} 1\\ 0 \\ -1 \end{array}} \right) ,~~~~~~~~ {\mathbf C}_{\pm } = c_{\pm } \left( {\begin{array}{cc} \lambda \\ \Omega _{\pm }^2 - \omega ^2 \\ \lambda \end{array}} \right) . \end{aligned}$$

Here we have labeled the non-dissipative mode as $\delta$ and the other two modes as $\{\pm \}$. Their corresponding frequencies are $\Omega _\delta = \omega$ and $\Omega _{\pm } = \sqrt{(\omega _2^2 + \omega ^2)/2 \pm \sqrt{\Delta }}$, defining $\Delta \equiv (\frac{\omega _2^2 - \omega ^2}{2})^2 + 2\lambda ^2$, and $c_{\pm }$ being nothing but a normalization constant. We can now obtain all the terms appearing in $V_{\pm }$.

The initial condition given in Eq. (5.27), can be now rewritten in terms of the non-dissipative normal mode as

$$\begin{aligned} \langle \hat{Q}_\delta ^2 (0) \rangle = \frac{\hslash }{2 \omega } e^{-2 r}, ~~~ \langle \hat{P}_\delta ^2 (0) \rangle = \frac{\hslash \omega }{2} e^{2 r}, ~~~ \langle \{ \hat{Q}_\delta , \hat{P}_\delta \}(0) \rangle = 0, \end{aligned}$$

and then their free evolution is given by

$$\begin{aligned}&\langle Q_{\delta }^{2}\rangle = \frac{\hslash }{2 \omega } (e^{2r}\sin ^{2}(\omega t) + e^{-2r}\cos ^{2}(\omega t)), \nonumber \\&\langle P_{\delta }^{2}\rangle = \frac{\hslash \omega }{2} (e^{-2r}\sin ^{2}(\omega t) + e^{2r}\cos ^{2}(\omega t)), \nonumber \\&\langle \{Q_\delta ,P_\delta \} \rangle = 2 \hslash \sinh (2r) \cos (\omega t)\sin (\omega t), \end{aligned}$$

(A.8)

where we have already used that $\Omega _\delta = \omega$. By substituting the above expressions in $V_{\mathrm {no-diss}}$ [Eq. (A.6)] we can now obtain the expressions of the determinants a, b, g and s. This yields for the minimum symplectic eigenvalue [Eq. (A.2)]:

$$\begin{aligned} \frac{\nu _{-}(t)^2}{ 2 \lambda ^2} = \mathcal{{G}}_0 + \mathcal{{G}}_1\cos (2 \omega t) - \sqrt{\left( \mathcal{{G}}_0 + \mathcal{{G}}_1\cos (2 \omega t) \right) ^2 - 4\sigma _P \sigma _Q }, \end{aligned}$$

(A.9)

which is an oscillatory function with frequency $2\omega$. Here

$$\begin{aligned} \mathcal{{G}}_0 = (\sigma _Q + \sigma _P) \cosh (2 r), ~~~~~ \mathcal{{G}}_1 = (\sigma _Q - \sigma _P) \sinh (2 r), \end{aligned}$$

and the dependence on the bath temperature and on the shape of the dissipative normal modes is given by

$$\begin{aligned}&\sigma _P = \frac{\Omega _{+}}{2 \omega } c_{+}^2 \coth \left( \frac{\Omega _{+}}{2 T} \right) + \frac{\Omega _{-}}{2 \omega } c_{-}^2 \coth \left( \frac{\Omega _{-}}{2 T} \right) , \nonumber \\&\sigma _Q = \frac{\omega }{2 \Omega _{+}} c_{+}^2 \coth \left( \frac{\Omega _{+}}{2 T} \right) + \frac{\omega }{2 \Omega _{-}} c_{-}^2 \coth \left( \frac{\Omega _{-}}{2 T} \right) . \end{aligned}$$

(A.10)

From Eq. (A.9), we can obtain the minimum entanglement (obtained for $t=(2n+1)\frac{\pi }{2 \omega };~ n=1,2,3,\ldots$) and the maximum one (for $t=(n+1)\frac{\pi }{\omega };~ n=1,2,3,\ldots$) in order to recover Eq. (5.29) with the proper definitions specified there.

1.2 A.1.2 Two-Mode NS

On the other hand, if we move to situation represented in Fig. 5.1e of Sect. 5.3 by fixing $\lambda = \tilde{\lambda }_0$ [see Eq. (5.25)], we have that the normal modes transform into

$$\begin{aligned} {\mathbf C}_{\delta } = \frac{1}{\sqrt{2}} \left( {\begin{array}{cc} 1\\ 0 \\ -1 \end{array}} \right) ,~~~ {\mathbf C}_{\varepsilon } = \frac{1}{\sqrt{6}} \left( {\begin{array}{cc} 1\\ -2 \\ 1 \end{array}} \right) ,~~~ {\mathbf C}_{\mathrm {c. m.}} = \frac{1}{\sqrt{3}} \left( {\begin{array}{cc} 1 \\ 1 \\ 1 \end{array}} \right) , \end{aligned}$$

being the center of mass, ${\mathbf C}_{\mathrm {c. m.}}$, the only dissipative mode. Their corresponding frequencies are respectively

$$\begin{aligned} \Omega _\delta = \omega , ~~~ \Omega _\varepsilon = \sqrt{2\omega _2^2 - \omega ^2}, ~~~ \Omega _{\mathrm {c. m.}} = \sqrt{2 \omega ^2 - \omega _2^2}. \end{aligned}$$

(A.11)

Naturally, we have to restrict ourselves to the regime $2\omega _3^2> \omega > \omega _3^2/2$ in order for these quantities to be real and positive.

Keeping the same initial condition as in the previous case, we have that nothing changes in the expression of the free evolution of mode $\delta$ [Eq. (A.8)], while the free evolution of mode $\varepsilon$ is given by

$$\begin{aligned}&\langle \hat{Q}_{\varepsilon }^{2}\rangle = \frac{2\omega _2 + \omega }{6 \Omega _\varepsilon ^2} \hslash e^{2r} \sin ^{2}(\Omega _\varepsilon t) + \frac{2\omega + \omega _2}{6 \omega \omega _2} \hslash e^{-2r}\cos ^{2}(\Omega _\varepsilon t), \\&\langle \hat{P}_{\varepsilon }^{2}\rangle = \frac{2\omega + \omega _2 \Omega _\varepsilon ^2}{6 \omega \omega _2} \hslash e^{-2r} \sin ^{2}(\Omega _\varepsilon t) + \frac{2 \omega _2 + \omega }{6} \hslash e^{2r}\cos ^{2}(\Omega _\varepsilon t), \\&\langle \{\hat{Q}_\varepsilon , \hat{P}_\varepsilon \}\rangle = \left( \frac{2\omega _2 + \omega }{3 \Omega _\varepsilon }e^{2r} - \frac{(2\omega + \omega _2)\Omega _\varepsilon }{3 \omega \omega _2} e^{-2r}\right) \times \\&~~~~~~~~~~~~~~~~~~~~ \times \hslash \cos (\Omega _\varepsilon t)\sin (\Omega _\varepsilon t). \end{aligned}$$

We have assumed the same squeezing parameter r in the central oscillator of the chain (notice that in the previous case the initial state of the central oscillator is not relevant and then we did not specify it). Following the same procedure as above, we calculate the expression for the minimum symplectic eigenvalue. It is worth noticing that in this case we have two contributions to the determinants of the free type $V_{\mathrm {no-diss}}$ [Eq.  (A.6)], corresponding to the two non dissipative modes, and a single dissipative one $V_{\mathrm {diss}}$ [Eq.  (A.7)], corresponding to the center of mass mode.

The minimum symplectic eigenvalue yields:

$$\begin{aligned} 2 \nu _{-}(t)^2 = \mathcal{{A}}_0 + \mathcal{{A}}_1(t) - \sqrt{\left( \mathcal{{A}}_0 + \mathcal{{A}}_1(t) \right) ^2 - \mathcal{{B}}_0 - \mathcal{{B}}_1(t)} \end{aligned}$$

(A.12)

where we have defined the following quantities in order to simplify the expression. The constant terms

$$\begin{aligned} \mathcal{{A}}_0&\equiv ~\cosh (2 r) \left( 4 (\sigma _Q + \sigma _P) + \mathcal {J}_{+}(\Omega _{\varepsilon }^2 + \omega ^2) \right) , \\ \mathcal{{B}}_0&\equiv ~64 \sigma _P \sigma _Q + \frac{4(\omega + \omega _2)^2}{81 \omega \omega _2} + \frac{32 \Omega _{\varepsilon } \omega \mathcal {J}_{+}}{3} \left( \frac{\omega \sigma _P}{\Omega _\varepsilon } + \frac{\Omega _\varepsilon \sigma _Q}{\omega } \right) , \end{aligned}$$

and the oscillating terms

$$\begin{aligned} \mathcal{{A}}_1(t) \equiv&~4 \cos (2 \omega t) \sinh (2 r) (\sigma _Q - \sigma _P) \\&+ \mathcal {J}_{+} \cos (2 \omega t) \sinh (2 r)(\Omega _\varepsilon ^2 + \omega ^2) \\&+ \mathcal {J}_{-} \cos (2 \Omega _{\varepsilon } t) \cosh (2r) (\Omega _\varepsilon ^2 - \omega ^2) \\&- \mathcal {J}_{-} \cos (2 (\Omega _\varepsilon - \omega ) t) \sinh (2r) \frac{(\Omega _\varepsilon + \omega )^2}{2} \\&- \mathcal {J}_{-} \cos (2 (\Omega _\varepsilon + \omega ) t) \sinh (2r) \frac{(\Omega _\varepsilon - \omega )^2}{2}, \\ \mathcal{{B}}_1(t) \equiv&~\cos (2 \Omega _\varepsilon t) \frac{32 \Omega _{\varepsilon } \omega \mathcal {J}_{+}}{3} \left( \frac{\Omega _\varepsilon \sigma _Q}{\omega } - \frac{\omega \sigma _P}{\Omega _\varepsilon } \right) , \end{aligned}$$

where different frequencies coming from the two non-dissipative modes are present. We have used $\mathcal {J}_{\pm } \equiv \frac{1}{12 \omega } \left( e^{2 r} \frac{2\omega _2 + \omega }{\Omega _\varepsilon ^2} \pm e^{-2 r}\frac{2\omega + \omega _2}{\omega \omega _2} \right)$ and the two bath-dependent functions are now given by the contribution of the c.m. mode:

$$\begin{aligned}&\sigma _P = \frac{\Omega _{\mathrm {c. m.}}}{6 \omega } \coth \left( \frac{\hslash \Omega _{\mathrm {c. m.}}}{2 k_B T} \right) , \nonumber \\&\sigma _Q = \frac{\omega }{6 \Omega _{\mathrm {c. m.}}} \coth \left( \frac{\hslash \Omega _{\mathrm {c. m.}}}{2 k_B T} \right) . \end{aligned}$$

(A.13)

A.2 Equations of Motion for the Second-Order Moments

As we are interested in classical and quantum correlations of the system oscillators, a description for the evolution of the first-order and second-order moments is necessary. The equations of motion for position, momenta, and the variances can be obtained from the Markovian master equation governing the dissipative dynamics, (5.39). In analogy to the case of two oscillators (Chap. 4) they can be indeed written in a simple form as $\dot{\mathbf {R}}= \mathcal {M}{\mathbf {R}}+{\mathbf N}$, where $\mathbf {R}$ is a column vector, now containing the $M= (2N + 1)N$ for $N=3$ independent second-order moments of the normal modes. The matrix $\mathcal {M}$ condenses all the information about their dynamical evolution and ${\mathbf N}$ determines the stationary values for long times (when $\dot{\mathbf {R}} = 0$). The dynamics of $\mathbf {R}$ can be solved in terms of the eigenvalues of $\mathcal {M}$:

$$\begin{aligned} \{{\mu _{i j}}\}=\{ -\frac{\Gamma _{i}+\Gamma _{j}}{2} \pm i\left| \Omega _i \pm \Omega _j\right| \},~~~ i \le j \end{aligned}$$

(A.14)

where the $i=j$ eigenvalues determine the evolution of $\langle \hat{Q}_i^2 \rangle$, $\langle \hat{P}_i^2 \rangle$ and $\langle \{\hat{Q}_i, \hat{P}_i\} \rangle$, while the ones with $i \ne j$ determine that of $\langle \hat{Q}_i \hat{Q}_j \rangle$, $\langle \hat{P}_i \hat{P}_j \rangle$ and $\langle \{\hat{Q}_i, \hat{P}_j\} \rangle$. Note that by virtue of Eqs. (5.40) and (A.14) the decay of the normal modes is entirely governed by the effective couplings mentioned above, thus differences in their magnitude produce disparate temporal scales for the dissipation and diffusion of normal modes.

We further stress that the stationary state of the dynamics is found to be $(\mathbf {R}_\infty = \mathcal {M}^{-1} {\mathbf N})$:

$$\begin{aligned}&\langle \hat{Q}_i^2 \rangle _\infty = \frac{D_i}{2 \Gamma _i \Omega _i^2}= \frac{\hslash }{2 \Omega _i} \coth \left( \frac{\Omega _i}{2 k_B T}\right) , \\&\langle \hat{P}_i^2 \rangle _\infty = \frac{D_i}{2 \Gamma _i} = \frac{\hslash \Omega _i}{2} \coth \left( \frac{\Omega _i}{2 k_B T} \right) , \end{aligned}$$

being all the other second-order moments equal to zero. Note that these expressions for the asymptotic limit recover the thermal state of the system at the bath temperature T given by the Gibbs distribution in Eq. (A.4).