Dissipative state transfer and Maxwell's demon in single quantum trajectories: Excitation transfer between two noninteracting qubits via unbalanced dissipation rates

We introduce a protocol to transfer excitations between two noninteracting qubits via purely dissipative processes (i.e., in the Lindblad master equation there is no coherent interaction between the qubits). The fundamental ingredients are the presence of collective (i.e., nonlocal) dissipation and unbalanced local dissipation rates (the qubits dissipate at different rates). The resulting quantum trajectories show that the measurement back-action changes the system wave function and induces a passage of the excitation from one qubit to the other. While similar phenomena have been witnessed for a non-Markovian environment, here the dissipative quantum state transfer is induced by an update of the observer knowledge of the wave function in the presence of a Markovian (memoryless) environment—this is a single quantum trajectory effect. That is, a postselection of a jumpless trajectory allows a transfer even for a non-Markovian environment where no quantum jumps have taken place. Beyond single quantum trajectories and postselection, such an effect can be observed by histogramming the ratio of quantum jumps at different times along several realizations. By investigating the effect of the temperature in the presence of unbalanced local dissipation, we demonstrate that, if appropriately switched on and off, the collective dissipator can act as a Maxwell's demon. These effects are a generalized measure equivalent to the standard projective measure description of quantum teleportation and Maxwell's demon. They can be witnessed in state-of-the-art setups given the extreme experimental control in, e.g., superconducting qubits, Rydberg atoms, and nitrogen-vacancy (NV) centers.

Figure 1

(a) As a function of time, ${\widehat{\sigma }}_{+}^{\left(1\right)}{\widehat{\sigma }}_{-}^{\left(1\right)}$ (qubit 1, blue curve, starting from 1) and ${\widehat{\sigma }}_{+}^{\left(2\right)}{\widehat{\sigma }}_{-}^{\left(2\right)}$ (qubit 2, red curve, starting from 0) for an ideal counting quantum trajectory in the presence of collective dissipation and local dissipation identical for qubits 1 and 2. As time passes, the wave function tends towards the Bell state $|{\mathrm{\Psi }}^{-}\rangle$, until a quantum jump occurs at time ${\gamma }_{c}t\simeq 6$. The black dashed lines represent the analytical results of Eq. (13), while the blue (red) squares (circles) indicate the results of the postselection in Eq. (16) for qubit 1 (2). Notably, the creation of the entangled state $|{\mathrm{\Psi }}^{-}\rangle$ is mediated by both the environment and by a continuous update of an observer's knowledge (who knows that quantum jumps have not occurred) due to the continuous measurement. (b) As a function of time and for a single trajectory, the percentage of local jumps ${\widehat{\sigma }}_{-}^{\left(1\right)}$ happening for qubit 1 (blue bars, starting from $100\%$) and ${\widehat{\sigma }}_{-}^{\left(2\right)}$ for qubit 2 (red bars, starting from $0\%$) without keeping track of the collective quantum jumps due to $({\widehat{\sigma }}_{-}^{\left(1\right)}+{\widehat{\sigma }}_{-}^{\left(2\right)})/\sqrt{2}$. We simulated ${10}^6$ trajectories. Parameters: ${\omega }_1={\omega }_2=10{\gamma }_c$ (the relevant condition is ${\omega }_1={\omega }_2$), ${\gamma }_1/{\gamma }_c=0.2,$ and ${\gamma }_2/{\gamma }_c=0.2$. The system is always initialized in the state $|e,g\rangle$.

Figure 2

(a) As a function of time, the expectation value of ${\widehat{\sigma }}_{+}^{\left(1\right)}{\widehat{\sigma }}_{-}^{\left(1\right)}$ (qubit 1, blue curve, starting from 1) and ${\widehat{\sigma }}_{+}^{\left(2\right)}{\widehat{\sigma }}_{-}^{\left(2\right)}$ (qubit 2, red curve, starting from 0) for an ideal counting quantum trajectory in the presence of collective dissipation and unbalanced local dissipation. While collective dissipation would create a Bell state $|{\mathrm{\Psi }}^{-}\rangle$, the unbalance in local dissipation transforms $|{\mathrm{\Psi }}^{-}\rangle$ into $|g,e\rangle$. These two processes ensure the excitation swap, until a quantum jump occurs at ${\gamma }_{c}t\simeq 8$. The black dashed lines represent the analytical results of Eq. (13), while the blue (red) squares (circles) indicate the results of the postselection in Eq. (16) for qubit 1 (2). In this case, a state is transferred because each time no excitation is emitted, it becomes more probable that the excitation is in the less dissipative qubit. Again, this is the effect of the environment combined with the continuous measurement. (b) As a function of time, the percentage of local jumps ${\widehat{\sigma }}_{-}^{\left(1\right)}$ happening for qubit 1 (blue bars, starting from $100\%$) and ${\widehat{\sigma }}_{-}^{\left(2\right)}$ for qubit 2 (red bars, starting from $0\%$) without keeping track of the collective quantum jumps due to $({\widehat{\sigma }}_{-}^{\left(1\right)}+{\widehat{\sigma }}_{-}^{\left(2\right)})/\sqrt{2}$. Despite the remarkable difference in the emission rates, the statistics of jumps at long times clearly indicates the excitation swap. The last time intervals are noisy due to the lack of statistics. We simulated ${10}^6$ trajectories. Parameters: ${\omega }_1={\omega }_2=10{\gamma }_c$ (the relevant condition is ${\omega }_1={\omega }_2$), ${\gamma }_1/{\gamma }_c=2.2,$ and ${\gamma }_2/{\gamma }_c=0.2$. The system is always initialized in the state $|e,g\rangle$.

Figure 3

As a function of time, the expecation value of ${\widehat{\sigma }}_{+}^{\left(1\right)}{\widehat{\sigma }}_{-}^{\left(1\right)}$ (for qubit 1, blue curve, jumping to 1 at ${\gamma }_{1}t\simeq 0.1$) and ${\widehat{\sigma }}_{+}^{\left(2\right)}{\widehat{\sigma }}_{-}^{\left(2\right)}$ (for qubit 2, red curve, slowly raising from zero for small ${\gamma }_{1}t$) for an ideal quantum state transfer in the presence of thermal processes, $n_{\mathrm{th}}^{\left(1\right)}=0.05, n_{\mathrm{th}}^{\left(2\right)}=0.1$, and $n_{\mathrm{th}}^{\left(c\right)}=0$. Qubit 1 is at a low-temperature while qubit 2 has a bath at a higher temperature. The collective dissipation, i.e., the element which allows a state transfer, is only activated (${\gamma }_c\ne 0$) once the cold qubit absorbs one excitation, inducing the hot qubit to become excited as a consequence of the dissipative state transfer. Parameters: ${\omega }_1={\omega }_2=4.5{\gamma }_1, {\gamma }_2={\gamma }_1/11$, and when active ${\gamma }_c=5{\gamma }_1/11$ (i.e., when ${\gamma }_c\ne 0$, the rates are those in Fig. 2).

Figure 4

As a function of time, the expectation value of ${\widehat{\sigma }}_{+}^{\left(1\right)}{\widehat{\sigma }}_{-}^{\left(1\right)}$ (for qubit 1, blue curve, starting from 1) and ${\widehat{\sigma }}_{+}^{\left(2\right)}{\widehat{\sigma }}_{-}^{\left(2\right)}$ (for qubit 2, red curve, starting from 0) for an ideal homodyne trajectory ($\beta \to \infty$, see Ref. [3]). This time, the results of a single trajectory differ profoundly from those predicted in Eq. (13) (the black dashed lines). The system is initialized in the state $|e,g\rangle$. Parameters are as in Fig. 2.

Figure 5

As a function of time, the average over 1000 counting trajectories in Eq. (5) [i.e., approximately the result of the LME in Eq. (2)] of the expectation value of ${\widehat{\sigma }}_{+}^{\left(1\right)}{\widehat{\sigma }}_{-}^{\left(1\right)}$ (qubit 1, blue curve, starting from 1) and ${\widehat{\sigma }}_{+}^{\left(2\right)}{\widehat{\sigma }}_{-}^{\left(2\right)}$ (qubit 2, red curve, starting from 0). The black dashed lines represent the analytical results of Eq. (13), which not applicable in this case. Parameters are as in Fig. 2.

Figure 6

As a function of time, the expectation value of ${\widehat{\sigma }}_{+}^{\left(1\right)}{\widehat{\sigma }}_{-}^{\left(1\right)}$ (qubit 1, blue curve, starting from 1) and ${\widehat{\sigma }}_{+}^{\left(2\right)}{\widehat{\sigma }}_{-}^{\left(2\right)}$ (qubit 2, red curve, starting from 0) for an ideal counting quantum trajectory in the presence of collective dissipation and unbalanced local dissipation for the Hamiltonian in Eq. (C1) for coupling: (a) $g=0.01{\gamma }_c$, (b) $g=0.1{\gamma }_c$, and (c) $g=0.5{\gamma }_c$. This picture corresponds to Fig. 2, but in the presence of qubit-qubit interactions. The black dashed lines represent the analytical results of Eq. (13) of the main text, and allows one to gauge the difference in the presence of nonzero $g$. Parameters: ${\omega }_1={\omega }_2=10{\gamma }_c, {\gamma }_1/{\gamma }_c=2.2$, and ${\gamma }_2/{\gamma }_c=0.2$. The system is always initialized in the state $|e,g\rangle$.