Thermalization as an invisibility cloak for fragile quantum superpositions

We propose a method for protecting fragile quantum superpositions in many-particle systems from dephasing by external classical noise. We call superpositions “fragile” if dephasing occurs particularly fast, because the noise couples very differently to the superposed states. The method consists of letting a quantum superposition evolve under the internal thermalization dynamics of the system, followed by a time-reversal manipulation known as Loschmidt echo. The thermalization dynamics makes the superposed states almost indistinguishable during most of the above procedure. We validate the method by applying it to a cluster of spins ½.

Figure 1

Magnetizations ${\langle M_z\rangle }_1\left(t\right)$ and ${\langle M_z\rangle }_2\left(t\right)$ for initial quantum states $|{\psi }_1\rangle =|\uparrow \uparrow \uparrow \cdots \rangle$ and $|{\psi }_2\rangle =|\downarrow \downarrow \downarrow \cdots \rangle$ of a periodic chain of 18 spins ½ governed by Hamiltonian (4). Symbols represent numerical calculations, lines are guides to the eye. The interaction Hamiltonian is reversed at ${\tau }_0=15$ indicated by the dotted line. This reversal leads to a nearly perfect revival at $t=2{\tau }_0$.

Figure 2

Coherence measures $C_{\text{I}}\left(2{\tau }_0\right)$ and $C_{\text{N}}\left(2{\tau }_0\right)$ computed respectively for interacting and noninteracting periodic chains of 18 spins ½ in the presence of external noise. Symbols represent numerical calculations, lines are guides to the eye.

Figure 3

Decay of coherence measures as a function of the number of spins $N_s$: (a) noninteracting case, (b) interacting case. In (a) and (b), symbols represent numerically computed $C_{\text{N}}\left(2{\tau }_0\right)$ and $C_{\text{I}}\left(2{\tau }_0\right)$, respectively, corresponding to $N_s$ indicated in plot legends; thin lines connecting symbols are guides to the eye; thick lines represent Eq. (9) for (a) and linear function $C_{\text{I}}\left(2{\tau }_0\right)\approx b(1-2{\mathrm{\Gamma }}_{\text{I}}{\tau }_0)$ with fitted parameters $b$ and ${\mathrm{\Gamma }}_{\text{I}}$ for (b). [Note the different scale of the vertical axis in (a) and (b).] (c),(d) Asymptotic decay rates ${\mathrm{\Gamma }}_{\text{N}}$ and ${\mathrm{\Gamma }}_{\text{I}}$ for the noninteracting and interacting cases, respectively, as a function of $N_s$. Symbols represent the values obtained by fitting the tails of $C_{\text{N}}\left(2{\tau }_0\right)$ and $C_{\text{I}}\left(2{\tau }_0\right)$; solid blue line represents Eq. (10); dashed orange line represents Eq. (11) with a fitted prefactor 0.96. Plots (c) and (d) differ only by the scale of the vertical axis.