Heat flow from a measurement apparatus monitoring a dissipative qubit

We investigate the heat flow of a qubit coupled to heat baths under continuous quantum measurement. In the steady-state limit, we show that heat always flows from the measurement apparatus into the qubit regardless of the measured qubit state and derive lower and upper bounds for the heat current between the qubit and the measurement apparatus. Furthermore, we study the transient dynamics of the heat current and the excess heat during the transient regime.

Figure 1

Setup of continuous quantum measurement of a qubit coupled to heat baths with temperatures $T_r$ ($r=1,2,\cdots ,N$). Continuous quantum measurement of the dissipative qubit can induce heat exchange between the qubit and the measurement apparatus.

Figure 2

(a) Bloch sphere. (b) Steady-state heat current as a function of $\theta$ for ${\mathrm{\Gamma }}_{+}/\mathrm{\Delta }=0.01$ and $\gamma /\mathrm{\Delta }=0.001$, 0.01, 0.05.

Figure 3

Elastic thermal photon processes. In the absorption process (left), a thermal photon with energy $\mathrm{\Delta }$ travels from the measurement apparatus to the qubit and then to one of the heat baths. In the emission process (right), the opposite occurs.

Figure 4

Dynamics of the heat current for ${\mathrm{\Gamma }}_{+}=0.02\mathrm{\Delta }, {\mathrm{\Gamma }}_{-}=\gamma =0.01\mathrm{\Delta }, \varphi =0$, and $\theta =0$ (blue), $\pi /6$ (light blue), $\pi /4$ (green), $\pi /3$ (orange), $\pi /2$ (red). The initial state is prepared with (a) ${\langle {\sigma }_x\rangle }_0=1, {\langle {\sigma }_y\rangle }_0={\langle {\sigma }_z\rangle }_0=0$ and (b) ${\langle {\sigma }_x\rangle }_0={\langle {\sigma }_x\rangle }_{\mathrm{ss}}=0, {\langle {\sigma }_y\rangle }_0={\langle {\sigma }_y\rangle }_{\mathrm{ss}}=0, {\langle {\sigma }_z\rangle }_0={\langle {\sigma }_z\rangle }_{\mathrm{ss}}=-{\mathrm{\Gamma }}_{-}/{\mathrm{\Gamma }}_{+}$.

Figure 5

Excess heat flowing from the measurement apparatus into the qubit as a function of $\theta$ for ${\mathrm{\Gamma }}_{+}=0.02\mathrm{\Delta }, {\mathrm{\Gamma }}_{-}=0.01\mathrm{\Delta }$, and $\varphi =0$.

Figure 6

(a) Schematic diagram of the $\mathrm{\Lambda }$ model with energy differences $\mathrm{\Delta }$ and $\delta$ ($<\mathrm{\Delta }$). The three-level system is coupled to the two bosonic heat baths ($T_{\mathrm{h}}>T_{\mathrm{c}}$). The hot bath allows transitions between $|0\rangle$ and $|2\rangle$, while the cold bath allows them between $|1\rangle$ and $|2\rangle$. (b) The steady-state heat current $J_{\mathrm{M}}$ as a function of $\gamma$ under the measurement on $(|0\rangle +|1\rangle )/\sqrt{2}$ for $\delta /\mathrm{\Delta }=0.5, k_{\mathrm{B}}{T}_{\mathrm{h}}/\mathrm{\Delta }=5, k_{\mathrm{B}}{T}_{\mathrm{c}}/\mathrm{\Delta }=2$, and ${\kappa }_{\mathrm{h}}={\kappa }_{\mathrm{c}}=0.01$.