Exchange interaction, entanglement, and quantum noise due to a thermal bosonic field

We analyze the indirect exchange interaction between two two-state systems, e.g., spins $1∕2$, subject to a common finite-temperature environment modeled by bosonic modes. The environmental modes, e.g., phonons or cavity photons, are also a source of quantum noise. We analyze the coherent vs noise-induced features of the two-spin dynamics and predict that for low enough temperatures the induced interaction is coherent over time scales sufficient to create entanglement. A nonperturbative approach is utilized to obtain an exact solution for the onset of the induced interaction, whereas for large times, a Markovian scheme is used. We identify the time scales for which the spins develop entanglement for various spatial separations. For large enough times, the initially created entanglement is erased by quantum noise. Estimates for the interaction and the level of quantum noise for localized impurity electron spins in Si-Ge type semiconductors are given.

Figure 1

Short-time correction to the induced exchange interaction for the Ohmic case. The arrow shows the order of the curves for increasing ${\omega }_c\mid \mathbf{d}\mid ∕c_s=0,1,\dots ,10$.

Figure 2

Development of the concurrence as a function of time, calculated with ${\alpha }_1^x=1∕20$ and $k_{B}T∕{\omega }_c=1∕20$. The curves correspond to various spin-spin separations, as can be read off the left inset, which shows the distribution of the concurrence in the $\mid \mathbf{d}\mid -t$ plane. The right inset presents the dynamics of the diagonal density matrix elements $P_{\uparrow \uparrow }\equiv ⟨\uparrow \uparrow \mid {\rho }_S\mid \uparrow \uparrow ⟩$, etc., on the same time scale.

Figure 3

The magnitude of the time-dependent Hamiltonian corresponding to the initial correction as a function of time and distance. The Ohmic $(n=1)$ case is shown. The inset demonstrates the onset of the cross-qubit interaction on the same time scale.

Figure 4

Dynamics of the occupation probabilities for the states ∣↑↑⟩, ∣↑↓⟩, ∣↓↑⟩, and ∣↓↓⟩. The parameters are the same as in Fig. 2. The inset shows the structure of the reduced density matrix (the nonshaded entries are all zero).

Figure 5

Dynamics of the inverse-diagonal matrix elements for the same system as in Fig. 4. Note that $\mathrm{Im}{\rho }_{\downarrow \uparrow ,\uparrow \downarrow }=0$.

Figure 6

The magnitude of the induced exchange interaction Hamiltonian for the Ohmic $n_m=1$ and super-Ohmic $n_m=2,3,4,5$ baths.

Figure 7

The magnitude of the noise for the Ohmic $n_m=1$ and super-Ohmic $n_m=2,3,4,5$ baths, for $2k_{B}T∕{\omega }_c=0.01$.

Figure 8

The magnitude of the induced spin-spin interaction for a 3D Si-Ge type structure: The dominant interaction amplitude, which is the same for the $x$ and $y$ spin components, is shown. The arrow indicates increasing $b$ values for the curves shown, with $b=0.03$, 0.09, 0.15, 0.21, 0.27, 0.33. The inset estimates the level of the noise (for $2k_{B}T{a}_B∕c_s=0.01$): The bottom curve is ${\eta }_c^m\left(0\right)$, with $m=x,y$. For $b\lesssim 0.4$, the amplitude ${\eta }_s^m\left(\mathbf{d}\right)$ can be comparable, and its values, calculated numerically for $0⩽r⩽50$, are shown as long as they exceed ${\eta }_c^m\left(0\right)$, with the top curve corresponding to the maximum value, at $r=0$.

Figure 9

The magnitudes, measured in units of ${\mathrm{s}}^{-1}$, of the induced spin-spin interaction, the EM coupling strength, and the level of noise for P-impurity electron spins in Ge. Here $H_z=3\times {10}^4\mathrm{G}$, and low temperature, $2k_{B}T∕\Delta ⪡1$, was assumed.