Quantum fuel with multilevel atomic coherence for ultrahigh specific work in a photonic Carnot engine

We investigate scaling of work and efficiency of a photonic Carnot engine with a number of quantum coherent resources. Specifically, we consider a generalization of the “phaseonium fuel” for the photonic Carnot engine, which was first introduced as a three-level atom with two lower states in a quantum coherent superposition by M. O. Scully, M. Suhail Zubairy, G. S. Agarwal, and H. Walther [Science 299, 862 (2003)], to the case of $N+1$ level atoms with $N$ coherent lower levels. We take into account atomic relaxation and dephasing as well as the cavity loss and derive a coarse-grained master equation to evaluate the work and efficiency analytically. Analytical results are verified by microscopic numerical examination of the thermalization dynamics. We find that efficiency and work scale quadratically with the number of quantum coherent levels. Quantum coherence boost to the specific energy (work output per unit mass of the resource) is a profound fundamental difference of quantum fuel from classical resources. We consider typical modern resonator set ups and conclude that multilevel phaseonium fuel can be utilized to overcome the decoherence in available systems. Preparation of the atomic coherences and the associated cost of coherence are analyzed and the engine operation within the bounds of the second law is verified. Our results bring the photonic Carnot engines much closer to the capabilities of current resonator technologies.

Figure 1

$N+1$ level atom phaseonium (NLAP) fuel. (a) Density matrix $\rho$ and complete graph representations of NLAP. $\rho$ is $N+1$ dimensional square matrix. Its coherent block can be represented by a complete graph with $N$ nodes and $S=N(N-1)/2$ links. Graphs were shown up to $N=5$ number of nodes. (b) NLAP for nondegenerate and degenerate atoms. The excited state is denoted by $a$ and the lower levels are denoted by $b_i$ with $i=1,\cdots ,N$. The upper level is well separated from the lower levels by an energy $\hslash \mathrm{\Omega }$ measured from the central lower level $b_{N/2}$.

Figure 2

Photonic Carnot engine with $N+1$-level atom phaseonium (NLAP) fuel. Photon gas in a high-quality cavity of frequency $\mathrm{\Omega }$ is the working substance and the mirrors of the cavity play the role of the piston. NLAP leaves the hohlraum at temperature $T_h$ and is subsequently prepared in a state with quantum coherence among its lower levels characterized by $N(N-1)/2$ phase parameters ${\varphi }_{ij}$ with $i,j=1,\cdots ,N$. Created NLAPs are repeatedly injected into the cavity at a rate $r$ in the quantum isothermal expansion process, where heat $Q_{\mathrm{in}}$ is transferred to the cavity. The cycle continues with quantum adiabatic expansion and quantum isothermal compression and is completed with a quantum adiabatic compression. An amount of heat $Q_{\mathrm{out}}$ is rejected into the entropy sink in the isothermal compression.

Figure 3

Extracted work ($W$) and efficiency ($\eta$) of photonic Carnot engine, with $N+1$ level atom phaseonium (NLAP) fuel, depending on the number of degenerate coherent ground-state levels $N$, for different decoherence factor models [(a) and (b)] $\xi =exp(-x)$ and [(c) and (d)] $\xi =exp(-Nx)$. [(e) and (f)] $\xi =exp(-N^{2}x)$, where $x={\gamma }_{\varphi }/\gamma$. Coherence parameter is $\lambda ={10}^{-6}$ and the initial thermal coherent atomic temperature is $T_h=4$ in units of $\hslash \mathrm{\Omega }/k_B$. $x$ values are 0.15, 0.1, 0.05, and 0.001 for (a) and (b); 0.14, 0.12, 0.1, 0.08 for (c) and (d); and 0.012, 0.01, 0.008, 0.006 for (e) and (f) for the dashed-dotted, dotted, dashed, and solid lines, respectively. The plots are given for the circuit QED parameters in Ref. [21]. The quantities $g=0.01$, $r=1\times {10}^{-4}$, $\kappa =6.25\times {10}^{-4}$, and $\gamma =5\times {10}^{-6}$, which are the coupling coefficient to the cavity field, atomic injection rate, cavity loss term, and atomic decay, respectively, are dimensionless and scaled with the resonance frequency $\mathrm{\Omega }\sim 10$ GHz. $\eta$ is dimensionless, and $W$ is dimensionless and scaled with $\mathrm{\Omega }$.

Figure 4

Extracted work ($W$) and efficiency ($\eta$) of microwave resonator photonic Carnot engine, with $N+1$ level atom phaseonium (NLAP) fuel, depending on the number of degenerate coherent ground-state levels $N$ for different decoherence factors [(a) and (b)] $\xi =exp(-x)$, [(c) and (d)] $\xi =exp(-Nx)$, and [(e) and (f)] $\xi =exp(-N^{2}x)$, where $x={\gamma }_{\varphi }/\gamma$. The coherence parameter is $\lambda ={10}^{-6}$ and the initial thermal coherent atomic temperature is $T_h=4$ in units of $\hslash \mathrm{\Omega }/k_B$. $x$ values are 0.1, 0.15, 0.05, 0.001 for (a) and (b); 0.12, 0.1, 0.08, 0.06 for (c); 0.14, 0.12, 0.1, 0.08 for (d); and 0.012, 0.01, 0.008, 0.006 for (e) and (f) for the dashed-dotted, dotted, dashed, and solid lines, respectively. The parameters $g=9.21\times {10}^{-7}$, $r=6.47\times {10}^{-5}$, $\kappa =1.96\times {10}^{-8}$, and $\gamma =9.54\times {10}^{-10}$ are the atom-field coupling coefficient, atomic injection rate, cavity loss term, and atomic decay, respectively. They are dimensionless and scaled with the typical resonance frequency is $\mathrm{\Omega }=51$ GHz [35]. $\eta$ is dimensionless, and $W$ is dimensionless and scaled with $\mathrm{\Omega }$.

Figure 5

Extracted work ($W$) and efficiency ($\eta$) of optical resonator photonic Carnot engine, with $N+1$ level atom phaseonium (NLAP) fuel, depending on the number of degenerate coherent ground-state levels $N$ for different decoherence factors [(a) and (b)] $\xi =exp(-x)$, [(c) and (d)] $\xi =exp(-Nx)$, and [(e) and (f)] $\xi =exp(-N^{2}x)$, where $x={\gamma }_{\varphi }/\gamma$. Coherence parameter is $\lambda ={10}^{-6}$ and the initial thermal coherent atomic temperature is $T_h=4$ in units of $\hslash \mathrm{\Omega }/k_B$. The $x$ values are 0.15, 0.1, 0.05, 0.001 for (a) and (b); 0.14, 0.12, 0.1, 0.08 for (c) and (d); and 0.01, 0.008, 0.006, 0.004 for (e) and (f) for the dashed-dotted, dotted, dashed, and solid lines, respectively. The parameters $g=6.28\times {10}^{-7}$, $r=8\times {10}^{-5}$, $\kappa =2.86\times {10}^{-7}$, and $\gamma =4.68\times {10}^{-8}$ are the coupling frequency to the cavity field, atomic injection rate, cavity loss term, and atomic decay, respectively. They are dimensionless and scaled with the typical resonance frequency is $\mathrm{\Omega }=350$ THz [36]. $\eta$ is dimensionless, and $W$ is dimensionless and scaled with $\mathrm{\Omega }$.

Figure 6

Time evolution of average photon number during thermalization process of PCE field while regular injection of 2LAP depending on different $N_{\mathrm{ex}}$ parameters. $N_{\mathrm{ex}}$ values are 4500, 1500, 150, 100, and 50 in decreasing order for the upper to lower curves, respectively. The coherence parameter is $\lambda ={10}^{-3}$, the initial field temperature $T_f=1$, and the temperature of the thermal coherent atoms is $T_h=2$ in units of $\hslash \mathrm{\Omega }/k_B$. The resonant field frequency is $\mathrm{\Omega }=51$ GHz, cavity quality factor is $Q=2\times {10}^{10}$, atom cavity field interaction time is $\tau =10\mu \mathrm{s}$, atom decay rate is $\gamma =33.3$ Hz, atom dephasing rate is ${\gamma }_{\varphi }=3.3$ Hz, and the atom cavity field coupling is $g=50$ kHz. Time is dimensionless and scaled with $\mathrm{\Omega }$.

Figure 7

Comparison of time evolution of average photon number in presence of thermal and coherent atom injection with $N=2$ (top figure) and $N=4$ (bottom figure). The horizontal dashed and dashed-dotted lines stands for the analytical ${\overline{n}}_{\varphi }$ and $\overline{n}$ values in absence of loss mechanisms. Solid and dotted lines stands for time evolution of ${\overline{n}}_{\varphi }$ and $\overline{n}$, respectively, in presence of dissipation channels. $N_{\mathrm{ex}}=4000$ for both subplots corresponding to ${\tau }_0=90\mu \mathrm{s}$. Insets magnifies the lines between $\mathrm{\Omega }t=600$ and $\mathrm{\Omega }t=800$. All the remaining parameters are the same with that of Fig. 6. Time is dimensionless and scaled with $\mathrm{\Omega }$.

Figure 8

Comparison of numerical results (black circles) with developed theory (solid line) of efficiency $\eta$ (top figure), harvested work $W$ (inset), and effective field temperature $T_{\text{eff}}$ (bottom figure), respectively, depending on the number of degenerate ground-state levels $N$. $N_{\mathrm{ex}}=12\times {10}^3$ and corresponding ${\tau }_0=2.01\mu \mathrm{s}$. All the remaining parameters are the same as in Fig. 6. $\eta$ is dimensionless, $T_{\text{eff}}$ and $W$ are dimensionless and scaled with $\mathrm{\Omega }$.