Decoherence in a System of Many Two-Level Atoms

I show that the decoherence in a system of $N$ degenerate two-level atoms interacting with a bosonic heat bath is for any number of atoms $N$ governed by a generalized Hamming distance (called “decoherence metric”) between the superposed quantum states, with a time-dependent metric tensor that is specific for the heat bath. The decoherence metric allows for the complete characterization of the decoherence of all possible superpositions of many-particle states, and can be applied to minimize the overall decoherence in a quantum memory. For qubits which are far apart, the decoherence is given by a function describing single-qubit decoherence times the standard Hamming distance. I apply the theory to cold atoms in an optical lattice interacting with blackbody radiation.

Figure 1

The function $\widehat{f}(t,r_{ij},\vartheta )$, for $\kappa =0.01$ and $\vartheta =\pi /2$ at $T=0$. The limit $r\to 0$ represents the single-qubit decoherence, $f_{ii}(t)$.

Figure 2

Decoherences $d_{\mathbf{s},{\mathbf{s}}^{\prime }}(t)$ of 9 qubits in a $3\times 3$ square optical lattice at $t=200.0$ as function of the Hamming distances $D^H(\mathbf{s},{\mathbf{s}}^{\prime })$. Small red squares are for $a=580d$, large blue circles for $a=1000d$. Same parameters as in Fig. 1.

Figure 3

Statistical analysis of all $2^{23}$ independent decoherences of $3\times 4$ qubits in a 2D square lattice for $a=100d\dots 1100d$ (parameters as in Fig. 2). The average decoherence, given by the trace of the DMT, is independent of the lattice spacings (central green curve, error bars represent $\pm$ 1 standard deviation). The minimum decoherence (bottom curve, blue) indicates a DFS at small spacings; the top curve (red) is the maximum decoherence. The inset shows a histogram of all decoherences for $a=580d$.