Experimental Verification of Dissipation-Time Uncertainty Relation

Dissipation is vital to any cyclic process in realistic systems. Recent research focus on nonequilibrium processes in stochastic systems has revealed a fundamental trade-off, called dissipation-time uncertainty relation, that entropy production rate associated with dissipation bounds the evolution pace of physical processes [Phys. Rev. Lett. 125, 120604 (2020)]. Following the dissipative two-level model exemplified in the same Letter, we experimentally verify this fundamental trade-off in a single trapped ultracold ${}^{40}{\mathrm{Ca}}^{+}$ ion using elaborately designed dissipative channels, along with a postprocessing method developed in the data analysis, to build the effective nonequilibrium stochastic evolutions for the energy transfer between two heat baths mediated by a qubit. Since the dissipation-time uncertainty relation imposes a constraint on the quantum speed regarding entropy flux, our observation provides the first experimental evidence confirming such a speed restriction from thermodynamics on quantum operations due to dissipation, which helps us further understand the role of thermodynamical characteristics played in quantum information processing.

Figure 1

Level schemes constructed for an effective two-level system in contact with two heat baths in the trapped ${}^{40}{\mathrm{Ca}}^{+}$ ion. (a) Levels constructed for the dissipative channels regarding cold heat bath. (b) Levels constructed for the dissipative channels regarding hot heat bath. Each dissipative channel is constructed by two heterochromatic lasers with the wavelengths of 729 and 854 nm, respectively, where the Rabi frequency ${\mathrm{\Omega }}_{n1}$ (${\mathrm{\Omega }}_{n2}$), with $n=1$, 2, 3, and 4, represents the resonant coupling of a sublevel of the ground state (metastable state) to a sublevel of the metastable state (excited state), and the decay of the excited state to the ground state accomplishes the effective procedure of the dissipative channel. Here, we set ${\mathrm{\Omega }}_{n2}\gg {\mathrm{\Omega }}_{n1}$ to guarantee the steady dissipative process, and the lifetime of the excited state is 6.9 ns; i.e., the decay rate $\mathrm{\Gamma }/2\pi =23.1\mathrm{MHz}$, while the metastable state has a negligible decay rate due to 1.2 s lifetime.

Figure 2

Decay rates in different dissipative channels constructed in the trapped ${}^{40}{\mathrm{Ca}}^{+}$ ion, where the black solid (orange dashed) line indicates the analytical solutions of the channels with the single(double)-decay processes in Eq. (4), and the experimental data are represented by blue dots, red circles, magenta diamonds, and green squares corresponding to the dissipative channels ①, ②, ③, and ④, respectively. Here the experimental data are measured with 4000 repetitions and the parameter in the horizontal axis is defined as $\mathcal{R}/{10}^4={\mathrm{\Omega }}_{k1}^2/{\mathrm{\Omega }}_{k2}^2$ with $k$ denoting the $k$th channel.

Figure 3

(a) Two exemplified stochastic trajectories of the dissipative processes, started from $|g⟩$ (blue line) or $|e⟩$ (red line), obtained by the discretization-repetition ensemble average method. (b) Population in state $|e⟩$ in the equilibrium process obtained by the ensemble average of 500 stochastic trajectories, where the dashed line indicates the analytical solution. (c) Energy flow in a single stochastic trajectory of the process, where a duration of 4 ms is amplified for scrutinizing the details. The experimental parameters are selected from group I of Table 1. The ensemble case of the energy flow with evident characteristic that the time reversed process is rare can be found in Fig. S12 in the Supplemental Material [51].

Figure 4

(a) Time evolution of the survival probabilities $p^s(t)$ of the process trajectories associated with different activation energies. Inset: the energy transfer in a stochastic trajectory for four different types of Markovian jumps induced by the cold and hot heat baths. (b) Time-averaged rate $\overline{r}(t)$ of the energy transfer process for different activation energies, where the green dashed line denotes the entropy flux of the stationary dynamics. Inset: the instantaneous rate $r(t)$ for an activation energy $E=\epsilon$, obtained by the survival probability, shows the stochastic entropy flux, and the small imbalance between the positive and negative directions at the initial period also illustrates the nonequilibrium property of the process with the predominant positive process. (a),(b) The black, red, and blue curves correspond to $E=\epsilon ,2\epsilon$, and $3\epsilon$, respectively, where the shaded area indicates the uncertainty of the date due to errors. Here, we choose the experimental values of the dissipative channels from group I of Table 1.

Figure 5

Dissipation-time uncertainty relation under activation energy $E$, where the green dashed line indicates the lower bound (LB), i.e., $k_B$. The error bars represent statistical standard deviation of our experimental observation by 4000 measurements for each data point.