# Quantum thermalization of a single mode radiation field by atom reservoir

###### Abstract

This study concerns with the thermalization process of a single mode radiation field by an atomic reservoir. In particular, the temporal evolution of quantum thermalization of a single mode radiation field weakly coupled to an atomic reservoir analysed in the framework of a quantum micromaser model both analytically and numerically. The findings show that the system thermalizes by both multi-atom and multi-level atom reservoir. It’s shown that different reservoirs constitute different scaling laws with respect to their typical parameters on thermalization time. Moreover, a quantum reservoir parameter dependent thermalization time which can be used for faster heating or cooling a single radiation mode field derived analytically and verified numerically. We believe that obtained results will contribute convenient parameter selection options for expanding or reducing the open system thermalization time on demand. These results are of importance for boosting the efficiency of quantum photonic heat engines using radiation fields as working substances operating at finite time.

###### pacs:

03.65.Yz, 05.70.-a, 03.67.-a## I Introduction

The capability of developing quantitative formulations of dynamical behaviors of physical systems such as quantum statistics based finite temperature dependent predictions is of importance to quantum optics scully_quantum_1997 (), cold atoms bloch_quantum_2012 (); gardiner_quantum_2014 () , and electrical engineering and materials science callister_materials_2010 (). Particularly by the reduction of temporal and spatial scales of electronic and optoelectronic devices by the rapid technological progress, these systems would be referred to as open quantum systems in contact with quantum reservoirs dolcini_interplay_2013 (). In this picture, classical Boltzmann formulation jacoboni_monte_2002 () replaced by density matrix rossi_theory_2011 () or Wigner-function formulation.

Redefinition of work and heat in the quantum domain enhanced the interest to quantum thermodynamics and quantum heat engines applying thermodynamic principles to quantum matter as working substances quan_quantum_2007 (); kieu_second_2004 (); quan_quantum_2005 (); johal_quantum_2009 (); allahverdyan_work_2008 (); wang_efficiency_2012 (); dorfman_photosynthetic_2013 (); altintas_quantum_2014 (); altintas_rabi_2015 (); turkpence_quantum_2016 (); turkpence_photonic_2017 (). Recently the first experimental realization of a single atom heat engine has been demonstrated by laser cooling and electric field noise as cold and hot heat baths respectively rosnagel_single-atom_2016 (). In this context, ideas on hybridizing the engineering and quantum physics seems begin to emerge schleier-smith_editorial:_2016 (). Engineering the states of the photonic microwave fields appears to be a requirement for quantum communication and quantum information technologies. Photons are convenient quantum information carriers with the ability of encoding qubit information to the polarization states and can be transmitted via optical communication channels northup_quantum_2014 (). Microwave fields also have an active part for quantum information processing since they govern the mutual interaction between superconducting qubits as promising building blocks for quantum computers majer_coupling_2007 (); sillanpaa_coherent_2007 (). However, microwaves are not very efficient for long distance information transmission. But technically a microwave signal has the possibility of being transmitted to the optical field based on a mechanical oscillator interaction. barzanjeh_reversible_2012 (); andrews_bidirectional_2014 (). To this end, cooling the mechanical resonator to the ground state is essential in order to be protected against the classical noise chan_laser_2011 (); teufel_sideband_2011 ().

This manuscript investigates the temporal evolution of quantum thermalization of a single mode radiation field in contact with a thermal atomic reservoir. Thermalization of quantum systems is a basic concept of quantum statistical physics kubo_statistical_2003 () and has attracted renewed interest. Conventional quantum thermalization states that an open quantum system connected with a thermal reservoir will evolve towards an equilibrium state with the same reservoir temperature as an irreversible process breuer_theory_2007 (). If the system is a part of an isolated quantum system, then the process can be defined by the eigenstate thermalization hypothesis deutsch_quantum_1991 (); srednicki_chaos_1994 (). This study adopts a conventional quantum thermalization scheme by a thermal atom reservoir in the framework of a micromaser model in order to focus on the possible applications to conventional quantum heat engines filipowicz_theory_1986 (). Beyond the context of quantum optics, micromaser has been very popular for examining the thermalization dynamics representing the single atom reservoir liao_single-particle_2010 (), coherent or correlated two-atom reservoir li_quantum_2014 (); dillenschneider_energetics_2009 () or coherent non-thermal multi-atom reservoirs dag_multiatom_2016 (). Moreover, a micromaser setup was also introduced as photonic heat engines in which a the single radiation field is the working substance and atom reservoir is the heat bath scully_extracting_2003 (); turkpence_photonic_2017 (); liao_single-particle_2010 (); li_quantum_2014 (); dillenschneider_energetics_2009 (); dag_multiatom_2016 (). Since the thermalization process is a part of Quantum Otto engine cycle quan_quantum_2007 (), the ability to choose the reservoir parameters for faster heating or cooling the single mode field is also precious to boost the power output of the engine for finite-time operations geva_quantummechanical_1992 ().

Concisely, this study investigates the dynamical evolution of conventional quantum thermalization process of a single mode cavity field in contact with thermal atom reservoir for both multi-atom and multi-level cases. By the obtained numerical and analytical results, parameter dependence of thermalization dynamics qualitatively compared. Since the selected atom reservoirs are the thermal ones the effect of quantum coherence or correlations is beyond the scope of this study.

## Ii Theory and system dynamics

Novel approaches draw a general framework on repeated (random or regular) quantum interactions to define a quantum heat, work or information reservoir strasberg_quantum_2017 (); turkpence_neural_2017 (). Cavity QED systems have mature quantum control methods and a micromaser constitutes a good example of repeated random interactions for a single mode field and injected atoms which stand for the atomic reservoir. Since the objective of the study is to examine the thermalization dynamics of the single mode field inside the cavity, the focus is of the temperature of the cavity field at a steady state. When the steady state temperature of the cavity field is equal to the injected atom temperature, then the cavity field can be considered as thermalized by injected atoms (atom reservoir).

The original definition of a micromaser consists a single mode radiation field in a high-Q cavity and a beam of excited two-level atoms injecting into the cavity. Velocity selected atomic beams consist uncorrelated atoms sent from an oven. Each atomic interaction with the field can be represented by a Jaynes-Cummings Hamiltonian scully_quantum_1997 ()

(1) |

in the rotating wave approximation (RWA) where is the atomic energy level spacing, is the single cavity mode frequency and is the atom-field coupling. Here, coupling strength is small enough to neglect the counter-rotating terms in the Hamiltonian. Operators , , are the Pauli-z, Pauli-rising, Pauli-lowering operators and , are respectively, the bosonic annihilation and creation operators satisfies the bosonic commutation relation . In the study, Planck constant divided by was taken throughout the paper. According to the standard micromaser theory, dynamics of the system described by a coarse grained master equation iii_laser_1978 () by tracing out the atomic degrees of freedom. System related parameters are atom decay rate , cavity field decay , atom-field interaction time and injection rate . Since photon lifetime , cavity loss is negligible for each interaction. The limitation of atom-field interaction time such as guarantees that only a single atom is inside the cavity each time. It’s supposed that injected two-level atoms have a well defined temperature

(2) |

where is the Boltzmann constant and and stands for ground and excited level populations of the atom respectively.

Time evolution of cavity field can be defined by a coarse-grained equation of motion

(3) |

where is the time independent interaction Hamiltonian in the interaction picture and Trat is the partial trace operation over atomic degrees of freedom.

Here, is the density matrix representing the overall system such that where is the atomic and is the field density matrix. For the case of thermal reservoir of two-level atoms resonant interaction Hamiltonian and corresponding matrix representation reads

(4) |

and the initial atomic density matrix is

(5) |

including no coherence. The injected atoms define a positive temperature for and . By the calculations in Eq.(3) treated for second order in , becomes

(6) |

where and . In general, field density matrix can be considered in number state representation

(7) |

Since the solutions of Eq.(II) were searched in the steady state, the equation to be obtained is

(8) |

Here, is the rate of change of average photon number where and is the Fock number state representation for the cavity photons. By using the properties of photon statistics, one obtains

(9) |

where the solution is

(10) |

Here, can be reduced as

(11) |

where the last term is the thermal photon number according to Boltzmann distribution at steady state. The mean photon number decays from its initial value to a stationary value with a decay rate

(12) |

Thus, Eq.(II) states the continuous dynamics of the mean photon number of the cavity field through a steady state. A temperature definition can be attributed to the steady state of the cavity field in terms of thermal photon number such as

(13) |

In this study values were chosen small enough to keep Eq.(13) valid liao_single-particle_2010 (). During the injection of atoms approaches to a stationary value. When this value converges to the temperature of the reservoir defined by Eq.(2), the system is considered to be thermalized.

## Iii Results and discussion

In order to analyse the thermalization dynamics of the system with respect to reservoir dependent parameters, multi-atom clusters; non-interacting number of two-level atoms and single atoms with number of levels investigated as atom reservoir for cavity field.

### iii.1 Analytical results

First, Eqs. (II-12) would be used to write

(14) |

where is the initial photon number and is the thermalization time. Regarding this evolution, thermalization time inferred as

(15) |

By this result it becomes possible to analytically evaluate the thermalization time of the field in terms of relevant parameters. First, to derive the analytical results derivation of thermalization time for multi atom reservoir was considered.

Interaction of number of non-interacting atoms can be defined by Tavis-Cummings Hamiltonian tavis_exact_1968 (); tavis_approximate_1969 ()

(16) |

where and . As depicted in the left panel of Fig.1, two-level atom clusters are being injected into the cavity. Since the clusters are identical and involve thermal, non-correlated and non-interacting two-level atoms, they constitute a thermal atomic reservoir with a well defined temperature. The matrix representation of can be defined by tensor products of dimensional individual atomic density matrices

(17) |

Also the matrix representation of interaction Hamiltonian obtained by straightforward calculations

(18) |

where

(19) |

is the block diagonal part and

(20) |

is the inverse diagonal part of . Inserting these results into Eq.(3) and by following the steps described above, the master equation

(21) |

describing the dynamics of cavity field obtained where and is the number of two-level atoms of the injected multi-atom cluster. Here, is a Liouvillian superoperator in Lindblad form. Since due to the number conservation, Eq.(III.1) reduces to

(22) |

The dependence of here is apparent and its affect to the decay dynamics will be discussed below. At this point also the results of the multi-level reservoir to the systems dynamics evaluated by the same arguments. Interaction of a single atom with one excited and number of lower levels with the radiation field can be described by iii_laser_1978 ()

(23) |

where and . Here, coupling of atomic levels to the field was taken equal and the lower levels were taken as degenerate ground states and equally populated for simplicity. Then the conservation of total probability of the multi-level atom reads where is the population of one of the degenerate ground states. The injected each multi-level atoms which stand for the multi-level atom reservoir can be defined by a diagonal atomic density matrix

(24) |

representing a thermal reservoir with temperature .

In this case the matrix form of the interaction Hamiltonian is

(25) |

for ground levels. Following the same recipe, equation of motion of the system

(26) |

obtained.

A direct conclusion of Eqs. (III.1) and (26) can be drawn upon Eq.(15) by comparing the parameters in terms of their effect to the thermalization dynamics. Considering the multi-atom reservoir, thermalization time is

(27) |

Analysing this result in terms of excited level populations (by inserting ) and taking the low temperature condition into account, thermalization time becomes . By this expression, one concludes that large number of two-level atoms in the cluster reduce the thermalization time. On the other hand, in the presence of multi-level atom reservoir, thermalization time is

(28) |

Since one obtains in the low temperature limit. Conversely, the multi-level atom reservoir alters linearly with respect to . These results show that multi-atom or multi-level atom reservoirs would preferably be used to expand or reduce the thermalization time of a single mode field on demand.

### iii.2 Numerical results

Numerical examination of the analytical results were performed with an initial state where the field state updated after each interaction with the atoms. The initial cavity field which is the system of interest, defined by a temperature dependent thermal state

(29) |

Here, , and is the partition function. The field density matrix was mathematically represented by a truncated Hilbert space in the calculations. As stated in the previous sections multi-atom and multi-level atom reservoirs analysed in which the single mode field is in contact with. First, multi-atom reservoir numerically investigated. All the two-level atoms were assumed to be resonant with the field with identical frequencies . In generic micromaser procedure injection rate defined by where is the atom field interaction time and is the time elapsed while there is no atom present in the cavity. The calculations were performed with the assumption of a regular atomic injection with in accordance with the coarse-grained master equation used in the analytical calculations. A microscopic master equation in Markov approximation

(30) |

was used in order to implement the numerical calculations. Here, is the Hamiltonian defined in Eq.(16), is the atomic decay and is the cavity field decay. According to key assumptions of micromaser theory filipowicz_theory_1986 () the effect of cavity decay is negligible during . Since the time elapsed during the absence of the atom in the cavity was taken , analytical coarse-grained master equation conditions recovered numerically. In order to numerically analyse the time evolution of the mean photon number and the field temperature defined in Eq.(13), cavity field dynamics was obtained by tracing out the atomic degrees of freedom with the corresponding master equation . In the calculations, the cavity field state is evolved iteratively through a steady state during the thermal atom clusters defined by Eq.(17) injected into the cavity.

Similarly, for the case of multi-level atom injection into the cavity field, a convenient microscopic master equation is

(31) |

where for . The system dynamics obtained with the same arguments and parameter regimes defined above as in the multi-atom case.

Fig. 2 presents the evolution of thermalization dynamics of a single cavity field in contact with a multi- atom heat reservoir. The initial and prominent result is the reduction of thermalizaton time depending on the increment of the number of two-level atoms . Figs. 2(b), (e) express that the thermalization time is independent of the initial thermal field temperature both for heating and cooling processes. On the other hand, in Figs. 2(c), (f) the dependence of thermalization time to the reservoir temperature is visible. These results are in accordance with the analytical result in Eq.(27). Since the initial field temperature and also the reservoir temperature is low, () reservoir temperature dependence of thermalization time is slightly evident. In Fig 3, the cavity field contacted to the hot and cold multi-level atom type reservoirs respectively. This time number of ground levels is the typical parameter of the reservoir and the simulations stated that larger extends the thermalization time in accordance with Eq.(28). Fig. 3 (a), (d) confirms this result by depicting the thermalization in contact with the hot and cold heat bath depending on the number of ground states of the multi-level atom reservoir. Figs. 3(b), (e) and 3(c), (f) express the evolution of second order correlation function in the time domain for multi-level atom and multi-atom reservoirs respectively. Second order correlation functions define the characteristics of light sources and distinguish the non-classical light from the classical or thermal ones scully_quantum_1997 (); iii_laser_1978 ().

A normalized zero time delay second order correlation function defined as . For a thermal photon state always while is larger than the thermal one as a signature of squeezed thermal state kim_properties_1989 (). The numerical results confirm that the field state is a thermal state since converges to 2 at the end of the contact duration to the atomic thermal reservoirs. The delay to reach the thermal state in Figs. 3(b), (e) for larger for multi-level reservoir and shortening the thermalization time for larger for multi-atom reservoir in Figs. 3(c), (f) are consistent with the explored thermalization characteristics of the atom reservoirs discussed above.

### iii.3 Discussion

Atomic reservoirs come out by different ways in physical systems. For instance, cold atoms are convenient examples as atom reservoirs in contact with sites of finite one dimensional Bose atom lattice kordas_non-equilibrium_2015 (). Moreover cold-atom setups are recently referred to as atomtronics as components of electronic devices kolovsky_microscopic_2017 (). On the other hand, the atom reservoirs are also referred to as the heat baths of quantum photonic heat engines with light-matter interaction as discussed in this study. Photonic quantum heat engines turkpence_quantum_2016 (); turkpence_photonic_2017 (); scully_extracting_2003 () operates with a single radiation mode field as working substance and the radiation field drives the engine piston which is a highly reflective cavity mirror. Quantum engines operate with classical thermodynamic cycles with a quantum working agent quan_quantum_2007 ().

For instance, a Quantum-Otto cycle consists four paths with isentropic compression, hot isochoric stage, isentropic expansion, and cold isochoric stage. During the isochoric stages the system is in contact with a reservoir, heat exchange occurs and no work is done during the process. These parts are the quantum thermalization parts of the cycle and end up with a thermal quantum state of the working substance. Isentropic stages are the quantum adiabatic paths in which the unitary evolution (system disconnected with the reservoir) takes place with a time dependent system parameter. Work is done in these stages while there is no heat transfer between the system and environment. Efficiency and power of quantum heat engines has conventional definitions as the ratio of work output and the work input and the work output per unit time respectively. Since the quantum adiabatic paths are infinitely long, power can be a quantifier for finite-time quantum thermodynamic cycles geva_quantummechanical_1992 (); feldmann_performance_2000 (). As the name implies, in a finite time thermodynamic quantum cycle adiabatic paths also operate in finite-time. It’s a straightforward inference to expect high power output with a shorter cycle time. However, in finite-time thermodynamics, increase in efficiency generally yields a power decrease and vice versa feldmann_performance_2000 (). Moreover, by the rapid adiabatic drive unwanted entropy production; an internal friction occurs during the finite-time adiabatic paths due to the non-commuting Hamiltonians at different instances feldmann_quantum_2006 (); plastina_irreversible_2014 (); cakmak_irreversibility_2016 (). Since these effects limits the performance of quantum thermal machines, shortening the isochoric paths (Quantum thermalization) of the cycle becomes important in order to enhance their performance.

Therefore the results of this study can be used to boost the performance of thermal machines through reducing the thermalization time by determining the reservoir and its parameters. Though, isochoric paths of the cycles is much shorter than the adiabatic paths, any shortening of thermalization time will directly be translated to the improvement of the performance of the cycle without effecting the work output since no work is done on these paths. The results of this study would also be beneficial for open quantum systems demanding longer thermalization times.

## Iv Conclusions

Summarizing, this study investigated the quantum thermalization dynamics of bosonic fields in the presence of atom reservoirs by a micromaser model. Particularly multi-atom reservoir consisting two-level atoms and multi-level atom reservoir consisting ground state levels separately considered as a quantum reservoir to a single mode cavity field. Reservoir parameter dependent thermalization time during the contact of these reservoirs derived analytically. It’s found that the thermalization time is inverse proportional with the the increase of the number of two level atoms in the presence of multi-atom reservoir. On the other hand, it is direct proportional with the number of ground state levels in the presence of multi-atom reservoir. These results verified qualitatively by numerical calculations adopting a microscopic master equation approach. Also possible applications of these results for open quantum devices or the possible beneficial effects to the performance of quantum thermal machines were discussed.

###### Acknowledgements.

I acknowledge support from İstanbul Technical University. I specially thank to Ferdi Altintas for guiding discussions. I also thank T. Çetin Akıncı and Serhat Şeker for their motivational support.## References

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