# Multiatom Quantum Coherences in Micromasers as Fuel for Thermal and Nonthermal Machines

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## Abstract

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## 1. Introduction

- Machines of the first kind are those fueled by a nonthermal bath, such as a squeezed-thermal or coherently-displaced thermal bath, that render the WF steady-state non-passive [30,31,32,33,34,35,36]. Such baths change the machine into a thermo-mechanical engine that, unlike a heat engine, is fueled by both mechanical work and heat imparted by the bath to the WF. The Carnot bound may be transgressed in such machines at the expense of work supplied by the bath. However, their efficiency bound cannot be properly compared to the Carnot bound, since the latter is a restriction imposed by the second law on heat [37], but not on work imparted by the bath.
- Machines of the second kind are those where the WF is thermalized by the nonthermal bath, as is the case of an engine fueled by a phaseonium bath. Such a machine is a proper heat engine, but the ability of the phaseonium bath to thermalize the WF to a temperature ${T}_{\phi}>T$ elevates its Carnot bound above that associated with an incoherent bath at temperature T.

## 2. Model and Effective Master Equation

## 3. Classification of Coherences as Different Types of Fuel

- The blocks adjacent to the main diagonal of the ${\rho}_{\mathrm{a}}$ matrix in the standard basis of Figure 2 contain coherences that can only induce absorption and emission processes in the field (WF), as they are associated with $\mathbb{L}\rho $ in the master Equation (7). We shall refer to these elements as heat-exchange coherences. They have a caloric (“flammable”) value, i.e., they may contribute to the thermalization of the cavity field. Heat-exchange coherences do not arise in the single-atom case, as they correlate states of the same energy, e.g., $\left(\right)open="|"\; close="\rangle ">eg$ in two-atom clusters and $\left(\right)open="|"\; close="\rangle ">eeg$ in three-atom clusters.
- Displacement coherences associated with the $-i[{H}_{\mathrm{eff}},\rho ]$ term in the master Equation (7) arise for all cluster sizes as they correlate states differing by one excitation, i.e., $\left|e\right.\u232a\phantom{\rule{-0.166667em}{0ex}}\left.\u2329g\right|$ and $\left|g\right.\u232a\phantom{\rule{-0.166667em}{0ex}}\left.\u2329e\right|$ in single atoms, $\left(\right)open="|"\; close="\rangle ">eg$ and its Hermitian conjugate in two-atom clusters and, say, $\left(\right)open="|"\; close="\rangle ">eeg$ in three-atom clusters.
- Squeezing coherences correspond to an exchange of two excitations and may exist in two-atom clusters in the form of $\left(\right)open="|"\; close="\rangle ">ee$ and its Hermitian conjugate or in three-atom clusters in, say, the form $\left(\right)open="|"\; close="\rangle ">eeg$.

## 4. Correlated Atomic Clusters as Fuel for Machines of the First and Second Kind

- If displacing or squeezing coherences are present in the bath, the cavity state becomes non-passive (displaced or squeezed, respectively), which implies that not only heat, but also work has been transferred from the bath to the cavity mode. Consequently, a machine fueled by such a bath is a machine of the first kind that operates thermo-mechanically.
- If the atomic state only contains heat-exchange coherences, the mode is thermalized by the bath, and only heat is exchanged. Such a setup is thus a viable implementation of a heat engine powered by a nonthermal bath, which has been dubbed a machine of the second kind.

#### 4.1. Conditions for Fueling Machines of the First Kind

#### 4.2. Conditions for Fueling Machines of the Second Kind

#### 4.2.1. Cavity Thermalization via Singly-Excited Entangled Three-Atom States

#### 4.2.2. GHZ States: Towards Infinite Effective Temperature

#### 4.2.3. States Leading to Ultrahigh Temperatures of the Cavity Field

## 5. Discussion

- An important insight that we have obtained is that two- and three-atom clusters are capable of acting as fuel for both kinds of machines in a highly effective fashion, so that there is no need to involve larger clusters. Still, a larger number of coherences as the cluster grows in size may further enhance the work output.
- For machines of the first kind, our analysis has revealed a particularly promising, simple, fuel in the form of two-atom clusters whose state is a nearly equal superposition of doubly-excited and doubly-ground states. Such a state is expected to give rise to very large squeezing of the cavity field. It may thus present a far superior alternative to existing squeezing schemes of cavity fields [46,47,48]. Such a strong squeezing may have fascinating applications [48] also outside of quantum thermodynamics. Our interest here is that this strong squeezing source may fuel a cavity field in a hybrid thermo-mechanical machine [29] with nearly 100% efficiency, at the expense of mechanical work supplied by the two-atom clusters.
- For machines of the second kind, we have found W-states of three-atom clusters to act as conventional heat-bath fuel at a positive finite temperature that is controllable by the W-state. By contrast, three-atom GHZ- and E-states have been found to correspond to effective baths at infinite or negative temperatures, respectively, that do not allow for a thermal steady-state solution for the cavity field. On the other hand, nearly-equal mixtures of W- and E-states have been identified as fuel capable of thermalizing the cavity field to an ultrahigh temperature.

- The arsenal of quantum gate operations [51] can in principle prepare two or three trapped atoms in an entangled state on demand, but such a preparation may require single-atom addressability.
- Another alternative is an optimized probabilistic scheme for multiatom entangled-state preparation in a cavity [58].

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Time-Evolution Operator for a One-Atom Micromaser

## Appendix B. Time-Evolution Operator for a Two-Atom Micromaser

## Appendix C. Time-Evolution Operator for a Three-Atom Micromaser

## Appendix D. Maser Threshold

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**Figure 1.**A schematic of the two- and three-atom micromaser model, where clusters of two-level atoms are injected into a single-mode cavity repeatedly in a Poissonian random sequence. The transition time of the atoms through the cavity is much shorter than the cavity lifetime, atomic relaxation and dephasing times or the mean free-time between the interactions, so that there can be at most one cluster present in the cavity at a time. The cavity-mode steady-state crucially depends on the state of the cluster, as shown here.

**Figure 2.**Energy levels of clusters of two- and three two-level atoms. The numbers next to the levels correspond to the indices used in the text to denote their corresponding position in the natural basis.

**Figure 3.**Density matrix of the atomic cluster for one (left), two (middle) and three (right) atoms, respectively, with color- and pattern-filled squares representing the different roles of the coherences with respect to the cavity-field evolution described by the master Equation (7). Red plain dark squares are populations, and light blue squares are ineffective coherences. Yellow diagonal striped squares are zeroth order coherences that can contribute to thermalization. Dark blue vertical striped squares are first order coherences that can contribute to the coherent displacement of the cavity field. Green horizontal striped squares are second order coherences contributing to the squeezing of the cavity field.

**Figure 4.**Trees of coherence for one- (left), two- (middle) and three- (right) atom clusters, respectively. The circles are the basis states, where the same color indicates the same number of excitations. Blue solid lines indicate heat-exchange coherences that may contribute to thermalization of the cavity field. These coherences between states with the same number of excitations only appear in the multipartite case. Displacement coherences (red dotted lines) between states differing by one excitation arise for all particle numbers. Squeezing coherences (green dashed lines) are between states differing by two excitations.

**Figure 5.**A doubly-excited state (

**top**) gives rise to a squeezed state of the cavity field. By contrast, a triply-excited state (

**bottom**) thermalizes the cavity to an ultrahigh temperature.

**Figure 6.**Squeezing parameter (19c) as a function of the coefficient $\vartheta <\pi /4$ of the two-atom state (15).

**Figure 7.**Steady-state mean number of photons in a cavity pumped randomly with three atom clusters in W class states, parameterized with angular variables θ and ψ, when $\delta =0$ and $\varphi =0$. The symmetric W-state yields the largest mean photon number in equilibrium and, hence, can be imagined as the “hottest” effective three-atom reservoir among the W class states.

**Figure 8.**Mean photon number (29) in a cavity pumped by atom clusters in the generalized GHZ state (28).

**Table 1.**Coefficients of the master Equation (7) for different cluster sizes. For later convenience, we have defined for three-atom clusters the abbreviations ${D}_{E}={b}_{22}+{b}_{33}+{b}_{44}$, ${D}_{W}={b}_{55}+{b}_{66}+{b}_{77}$, ${C}_{E}={b}_{23}+{b}_{24}+{b}_{32}+{b}_{34}+{b}_{42}+{b}_{43}$ and ${C}_{W}={b}_{56}+{b}_{65}+{b}_{57}+{b}_{75}+{b}_{67}+{b}_{76}$.

1 Atom | 2 Atoms | 3 Atoms | |
---|---|---|---|

${r}_{e}$ | ${b}_{11}$ | $2{b}_{11}+{b}_{22}+{b}_{33}+{b}_{23}+{b}_{32}$ | $3{b}_{11}+2{D}_{E}+{D}_{W}+{C}_{E}+{C}_{W}$ |

${r}_{g}$ | ${b}_{22}$ | $2{b}_{44}+{b}_{22}+{b}_{33}+{b}_{23}+{b}_{32}$ | $3{b}_{88}+2{D}_{W}+{D}_{E}+{C}_{E}+{C}_{W}$ |

λ | ${b}_{12}$ | ${b}_{12}+{b}_{13}+{b}_{24}+{b}_{34}$ | ${b}_{25}+{b}_{35}+{b}_{46}+{b}_{47}+{b}_{26}+{b}_{37}+{b}_{12}+{b}_{13}+{b}_{14}+{b}_{58}+{b}_{68}+{b}_{78}$ |

ξ | 0 | ${b}_{14}$ | ${b}_{28}+{b}_{38}+{b}_{48}+{b}_{15}+{b}_{16}+{b}_{17}$ |

**Table 2.**Steady-state properties following from the thermal master Equation (20) for different cluster sizes. Here, we have defined $C={C}_{E}+{C}_{W}$.

1 Atom | 2 Atoms | 3 Atoms | |
---|---|---|---|

$\frac{{k}_{\mathrm{B}}T}{\hslash {\omega}_{\mathrm{c}}}=ln{\left[\left(\right),\frac{{r}_{g}}{{r}_{e}}\right]}^{}-1$ | ${\left(\right)}^{ln}-1$ | ${\left(\right)}^{ln}-1$ | ${\left(\right)}^{ln}-1$ |

${\langle n\rangle}_{\mathrm{ss}}=\frac{{r}_{g}}{{r}_{g}-{r}_{e}}$ | $\frac{{b}_{22}}{{b}_{22}-{b}_{11}}$ | $\frac{2{b}_{44}+{b}_{22}+{b}_{33}+{b}_{23}+{b}_{32}}{2({b}_{44}-{b}_{11})}$ | $\frac{3{b}_{11}+2{D}_{E}+{D}_{W}+C}{3({b}_{88}-{b}_{11})+{D}_{W}-{D}_{E}}$ |

valid for (${r}_{g}>{r}_{e}$) | ${b}_{22}>{b}_{11}$ | ${b}_{44}>{b}_{11}$ | $3{b}_{88}+{D}_{W}>3{b}_{11}+{D}_{E}$ |

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**MDPI and ACS Style**

Dağ, C.B.; Niedenzu, W.; Müstecaplıoğlu, Ö.E.; Kurizki, G.
Multiatom Quantum Coherences in Micromasers as Fuel for Thermal and Nonthermal Machines. *Entropy* **2016**, *18*, 244.
https://doi.org/10.3390/e18070244

**AMA Style**

Dağ CB, Niedenzu W, Müstecaplıoğlu ÖE, Kurizki G.
Multiatom Quantum Coherences in Micromasers as Fuel for Thermal and Nonthermal Machines. *Entropy*. 2016; 18(7):244.
https://doi.org/10.3390/e18070244

**Chicago/Turabian Style**

Dağ, Ceren B., Wolfgang Niedenzu, Özgür E. Müstecaplıoğlu, and Gershon Kurizki.
2016. "Multiatom Quantum Coherences in Micromasers as Fuel for Thermal and Nonthermal Machines" *Entropy* 18, no. 7: 244.
https://doi.org/10.3390/e18070244