Restoring the Heisenberg limit via collective non-Markovian dephasing

In this work an exactly solvable model of $N$ two-level systems interacting with a single bosonic dephasing reservoir is considered to unravel the role played by collective non-Markovian dephasing. We show that phase estimation with entangled states for this model can exceed the standard quantum limit and demonstrate Heisenberg scaling with the number of atoms for an arbitrary temperature. For a certain class of reservoir densities of states decoherence can be suppressed in the limit of a large number of atoms and the Heisenberg limit can be restored for arbitrary interrogation times. We identify the second class of densities when the Heisenberg scaling can be restored for any finite interrogation time. We also find the third class of densities when the standard quantum limit can be exceeded only on the initial stage of dynamics in the Zeno regime.

Figure 1

The dephasing factor ${\gamma }_N(T,t)$ as a function of $w_{s}t$, [see Eq. (10)], for the density (11) with $s=4$, for different temperatures $T$ and for $N={10}^3, \overline{w}=w_4$, and ${\alpha }_4{w}_4^5\mathrm{\Gamma }(5/2)/2=0.12$. Solid, dashed, and dash-dotted lines correspond to $T=0,0.5,1$ (in units of $\hslash \overline{w}/k_B$), respectively. The inset shows the dependence of the stationary value of the dephasing factor on $T$, for $N={10}^2$ (dashed line) and $N={10}^3$ (solid line).

Figure 2

The Fisher information $F_N(0,t)$ at the optimal detuning $\varphi$ as a function of $w_{s}t$ [see Eq. (18)], for the spectral density (11) with $s=4$, at $T=0, \overline{w}=w_4$, and ${\alpha }_4{w}_4^5\mathrm{\Gamma }(5/2)/2=0.12$. Thin (thick) lines correspond to one-by-one measurements with the uncorrelated initial state (to the GHZ state with $N$ simultaneously measured qubits). Solid, dashed, dash-dotted, and dotted lines correspond to $N={10}^3,{10}^4,{10}^5,{10}^6$, respectively.

Figure 3

The ratio $F_N(0,t_{\mathrm{best}})/F_1(0,t_{\max })$ versus $N$ [see Eq. (18)]. The solid line corresponds to the density (11) with $s=4$ and to $\overline{w}=w_4, {\alpha }_4{w}_4^5\mathrm{\Gamma }(5/2)/2=0.12, t_{\max }=100w_4^{-1}$. The dashed line corresponds to the density (11) with $s=2$ and to $\overline{w}=w_2, {\alpha }_2{w}_2^3\mathrm{\Gamma }(3/2)/2=0.02, t_{\max }=100w_2^{-1}$. The inset shows the optimal measurement time $t_{\mathrm{best}}$ (in units of $w_4^{-1}$) versus $N$, for $s=4$. The switching from $t_{\mathrm{best}}=100$ to $t_{\mathrm{best}}=7.364$ occurs at $N=254$. Dotted and dashed lines show the exact numerical solution of Eq. (19) and the analytical approximation (20), respectively.

Figure 4

The Fisher information $F_N(0,t)$ at the optimal detuning $\varphi$ as a function of $w_{s}t$ [see Eq. (18)], for the spectral density (11) with $s=2$, at $T=0, \overline{w}=w_2$, and ${\alpha }_2{w}_2^3\mathrm{\Gamma }(3/2)/2=0.12$. Thin (thick) lines correspond to one-by-one measurements with the uncorrelated initial state (to the GHZ state with $N$ simultaneously measured qubits). Solid, dashed, dash-dotted, and dotted lines correspond to $N={10}^3,{10}^4,{10}^5,{10}^6$, respectively.

Figure 5

The dependence of the best time $t_{\mathrm{best}}$ on the temperature $T$ (in units of $\hslash \overline{w}/k_B)$ and on the number of qubits $N$ (in the log scale), for $s=4, t=20w_4^{-1}, \overline{w}=w_4$, and ${\alpha }_4{w}_4^5\mathrm{\Gamma }(5/2)/2=0.12$.