Leggett-Garg inequality for qubits coupled to thermal environment

We study violation of the Leggett-Garg inequality for the correlator $K_3$ in two cases: (i) for a qubit weakly coupled to a thermal decohering (dissipative and/or dephasing) environment and (ii) for a pair of qubits with only one of them coupled to the environment. In the case of a single qubit, we identify conditions depending both on initial state preparation and a measurement protocol when violation of the $K_3$ inequality becomes dependent only on dephasing and independent of dissipation and temperature. In the case of two qubits, we have found the temperature-independent regime as well as the regime where all three components of the decoherence can control violation of the Leggett-Garg inequality.

Figure 1

The Leggett-Garg correlator $K_3$ given by Eq. (17) versus the intermeasurement time interval $\tau$ for different values of the dephasing parameter $G$. The projector operators ${\Pi }_{\pm 1}$ in Eq. (14) are fixed for $\theta =\pi /2$ and $\phi =0$.

Figure 2

The Leggett-Garg correlator $K_3$ versus intermeasurement time interval $\tau$ for different measurements ${\Pi }_{\pm 1}$ in Eq. (14) parametrized by different values of $\theta$ and fixed $\phi =0$. (a) The case of unitary evolution $U$. (b) The system evolves under the Davies map with $A=G=1/10$ and $p=0$.

Figure 3

The Leggett-Garg correlator $K_3$ versus intermeasurement time interval $\tau$ for the measurement ${\Pi }_{\pm 1}$ in Eq. (14) with $\theta =\pi /3$ and $\phi =0$. (a) Different values of $G$ and fixed $A=1/10$ and $p=0$. (b) Different values of $A$ and fixed $G=1/10$ and $p=0$. (c) Different values of $p$ and fixed $A=G=1/10$. The insets show $K_3$ for small values of $\tau \le \pi /2$.

Figure 4

The Leggett-Garg correlator $K_3$ in Eq. (25) for a pair of qubits, one of which undergoes Davies evolution whereas the second evolves unitarly for the BSM measurement (23), versus intermeasurement time interval $\tau$. (a) For different $G$ and $A=1/10$. (b) For different $A$ and $G=1/10$.