Heating and cooling processes via phaseonium-driven dynamics of cascade systems

The search for strategies to harness the temperature of quantum systems is one of the main goals of quantum thermodynamics. Here we study the dynamics of a system made of a pair of quantum harmonic oscillators, represented by single-mode cavity fields, interacting with a thermally excited beam of phaseonium atoms, which act as ancillas. The two cavities are arranged in a cascade configuration, so that the second cavity interacts with phaseonium atoms only after their interaction with the first one. We provide exact closed dynamics of the first cavity for arbitrarily long interaction times. We highlight the role played by the characteristic coherence phase of phaseonium atoms in determining the steady states of the cavity fields as well as that of the ancillas. Also, we show how the second cavity follows a non-Markovian evolution due to interactions with the “used” ancillary atoms, which enable information exchange with the first cavity. Adjusting the parameters of the phaseonium atoms, we can determine the final stable temperature reached by the cavities. In this way, the cavities can be heated up as well as cooled down. These results provide useful insights into the use of different types of ancillas for thermodynamic cycles in cavity QED scenarios.

Figure 1

Standard collision model for a phaseonium bath interacting with a multipartite system. The system of interest is a cascade of two single-mode cavity fields $S_1, S_2$, which is described by a density operator $\rho$. The environment is made up of $N$ three-level atoms in $\lambda$ configuration, called phaseonium atoms, all prepared equally. These atoms play the role of ancillas and are described by a density operator ${\eta }_k$ ($k=0,...,N$). Ancillas travel at speed $v$ and enter the cavities at a rate $r$. They interact with each cavity field for a time $\mathrm{\Delta }t$. The speed and rate of the phaseonium atoms are selected such that there is, at most, one ancilla in each cavity at a time.

Figure 2

Range of steady-state temperatures of one cavity in contact with a phaseonium bath, as a function of excited-state population $\alpha$ and ground-state coherences $\varphi$ of phaseonium atoms. Temperature is symmetric in $\varphi$, so it is shown in the range $0\le \alpha \le 1, 0\le \varphi \le \pi$. Temperature is given in units of $K_B/\left(\hslash \omega \right)$.

Figure 3

Behavior of temperature of the first cavity (left column) and second cavity (right column) taking different values for the collision time. The initial state of both subsystems is taken to be a thermal state at $T=1$. Each plot shows the thermalization process obtained involving ancillas with a different amount of coherences (but the same $\alpha =0.25$). More precisely, for the hot curves, $\varphi =1.5855$, and for the cold curves, $\varphi =2.4043$, which correspond to stationary temperatures for the cavities $T=1.5$ and $T=0.5$, respectively. We observe that as the collision time increases, the thermalization process becomes faster for small $\mathrm{\Delta }t$, does not change for $\mathrm{\Delta }t\sim 0.6$ and becomes slower for $\mathrm{\Delta }t>1.2$. Interaction times are given in units of $1/\mathrm{\Omega }$, while temperature is shown in units of $K_B/\left(\hslash \omega \right)$.

Figure 4

Behavior of the temperature of the first cavity $S_1$ (left panel) and second cavity $S_2$ (right panel) using interaction times extracted at every step $k$ from a Gaussian distribution, $\mathrm{\Delta }{t}_k=(0.4\pm 0.2)$, where the standard deviation is used as the error on the mean. Here we selected ${\varphi }^{*}=\pi /2$, and for the hot curves we use ${\alpha }^{*}=0.2517517$, corresponding to a stationary temperature $T^{*}=1.5$, while for the cold curves we have ${\alpha }^{*}=0.451963$, corresponding to a stationary temperature $T^{*}=0.5$. The mean value across 10 simulations of the evolving temperature is plotted as a continuous line, surrounded by its standard deviation. For reference, the constant interaction time case with $\mathrm{\Delta }{t}_k=0.4$ is plotted with a dotted line. It is evident that both heating and cooling the cavities with noise in the interaction times slows down the thermalization process, although the final stable temperature is reached at some point. Interaction times are given in units of $1/\mathrm{\Omega }$, while temperature is shown in units of $K_B/\left(\hslash \omega \right)$.

Figure 5

Behavior of the temperature of the first cavity $S_1$ (left panel) and second cavity $S_2$ (right panel) using coherence phases ${\varphi }_k$ extracted at every step $k$ from a Gaussian distribution with standard deviation $\sigma$. For the cooling process, we select ${\alpha }^{*}=0.25$ and ${\varphi }_k^{*}=1.585589386$, with $\sigma =0.2$, corresponding to a stationary temperature $T^{*}=0.5$. For the heating process, we have two simulations with increasing standard deviation, using ${\alpha }^{*}=0.5$ and ${\varphi }_k^{*}=1.26768686$, with $\sigma =0.2$ for the red curve and $\sigma =0.35$ for the brown curve, both corresponding to a stationary temperature $T^{*}=1.5$. Interaction time $\mathrm{\Delta }{t}_k=0.2$ is fixed. The mean value across 10 simulations of the evolving temperature is plotted as a continuous line, surrounded by its standard deviation across different simulations. For reference, the constant coherence case with $\sigma =0$ is plotted with a dotted line. The cooling process is not much affected by random variations in the coherence phase: the first cavity reaches a noisy stable state at $T_1=(0.498\pm 0.006)$, and the second thermalizes at $T_2=(0.498\pm 0.004)$. Noise is higher in the heating scenario, with $T_1=(1.50\pm 0.03)$ and $T_2=(1.50\pm 0.02)$ for the red curve with $\sigma =0.2$ and $T_1=(1.52\pm 0.05)$ and $T_2=(1.52\pm 0.03)$ for the brown curve with $\sigma =0.35$. Interaction times are given in units of $1/\mathrm{\Omega }$, while temperature is shown in units of $K_B/\left(\hslash \omega \right)$.

Figure 6

Behavior of the temperature of the first cavity $S_1$ (right panel) and second cavity $S_2$ (left panel) using coherence phases ${\varphi }_k$ extracted at every step $k$ from a Gaussian distribution with standard deviation $\sigma =0.2$. Here we selected the same set of parameters ${\alpha }^{*}=0.25$ and ${\varphi }_k^{*}=(2.404315987\pm 0.2)$, corresponding to a stationary temperature $T^{*}=1.5$, but with two different initial conditions for the cavities: starting at temperatures $T_0=2.0$ or $T_0=1.0$. Interaction time $\mathrm{\Delta }{t}_k=0.2$ is fixed. The mean value across 10 simulations of the evolving temperature is plotted as a continuous line, surrounded by its standard deviation. For reference, the constant coherence case with ${\varphi }_k=2.404315987$ and ${\varphi }_k=1.585589386$ is plotted with a dotted line. A clear difference arises in the reference thermalization at $T^{*}=1.5$ and the two stochastic processes. In fact, in both cases, the first cavity thermalizes to a noisy temperature $T_1=(1.37\pm 0.04)$ and the second to $T_2=(1.37\pm 0.03)$. Interaction times are given in units of $1/\mathrm{\Omega }$, while temperature is shown in units of $K_B/\left(\hslash \omega \right)$.