Charger-Mediated Energy Transfer for Quantum Battery With Constant Time Dependent Step Function and Open System Approach

The energy charging of quantum battery is analysed by open quantum approach. The modelled of the charger and battery are described by harmonic oscillator model. We choose this model because the harmonic oscillator battery gives the largest maximum energy albeit having the longest maximum time. In this paper, the interaction as actual quantum system whose dynamic is determined by Lindblad Master Equation in terms of constant is a step function to set the time when energy flow and what time that energy stop to flow or the interaction between charger and battery has stopped. The energy equation was determined by solving the master equation with second order differential equation to find the first momenta from charger and battery. Based on the equation, the energy of charger will not be zero which mean after interaction in several times, the charger will not one hundred percent lose its energy when the energy storage in battery already done.


Introduction
Nowadays, quantum information has been developed which generated great expectation that quantum effect like entanglement could be exploited to perform more advantages over classical device [1].Energy processing in quantum system has already attracted attention for fundamental and practical reason.Quantum advantages for the energy processing was motivated research in microscopic thermal machine [2].One of the quantum advantages which is realized in physical system used the quantum effect is a quantum battery.
A battery is a physical system which capable to store energy supplied by an external force and making it available to other devices.Then, quantum battery is a energy storing system which is the principle of the battery use the quantum effect which the aim is to improve the performances with respect to conventional one [3].The model of charging process of quantum battery has been presented.A quantum battery B and its external energy supply A called "quantum charger" (also modelled as a quantum system).The quantum correlation between them has an important role to speed up the charging of the battery.Building up for this work, in the present work, has been introduced a generalization of a quantum battery and quantum charger model by explicitly embedding the whole system in terms of master equation [4].The master equation is based on the open quantum system approach.Open quantum system is a system which describe the interaction between the system and environment.This approach happen because the close quantum system is just a idealization of the real system.But, practically the interaction between environment and system cannot be avoided [5].A quantum battery is coupled to the charger which interact with external laser field as well as with environment.This arrangement on one hand allows the flow of energy for the field to battery while on the other partially isolates it form the effect of noise, improving quality of stored energy [2].
In this paper, we modelled the case of the charger and the battery are described by resonant of harmonic oscillator model.We choose this model because the harmonic oscillator battery gives the largest maximum energy and having the longest maximum time [6].In our treatment, we represent A and B as actual quantum system whose dynamic is determined by Lindblad Master Equation in terms of constant is a step function to set the time when energy flow and what time that energy stop to flow or the interaction between charger and battery has stopped.

Model of Equation
We consider that both the model and the battery are described by harmonic oscillator with following Hamiltonian [3]: Where,   is a Hamiltonian of charger,   is Hamiltonian of battery, △   () is a local modulation of the energy from charger which is driven by external classical field that may inject energy to a system,   is a interaction Hamiltonian between charger and battery,   is the fundamental frequency of the model (we set   = 0 in next equation), while g and F are coupling constants ganging the AB coupling and the driving field acting on A. For harmonic oscillator model, we defined the operator as annihilation and creation operator.In Equation 1,  † is a creation operator for charger, a is annihilation operator for charger,  † is a creation operator for battery and b is annihilation operator for battery.Therefore, the dynamic of QB can be described by Lindblad-Gorini-Kossakowski-Sudarshan master equation as [7]: Where the  † ,  are the Lindblad operator which describe the dephasing process of the battery and  is a coupling constant.We set  as () as a step function constant which dependent in time.It calls dependent of time because when the battery and charger start to interact, the value will be  and when there is no interaction between charger and battery the value of  will be zero.Then, the representation of master equation in this case reads: Accordingly, a complete characterization of Equation ( 3) can be obtained by simple determining first and second momenta of the field operator [3].In this paper, we determine the first momenta by the operator annihilation and creation based on Equation 3 to get the energy operator equation form for A and B , then we model the equation in graph to see the model of interaction.

Equation of motion
To examine the first momenta which is operator of annihilation and creation, we change   () into the operator.The Equation 3 becomes: From equation above, we can see that there is a coupling constant in first momenta of battery which means there is a dephasing in charger because the charger is interacted with environment.Then, we solve equation above by differentiate again Equation 8 until 11 by time.Then we get the second order differential equation.We set   will be a constant (  ) when there is an interaction between charger and battery and zero when not interacted.So that becomes:  +  2 )  † = 0 (17).
Use the formula to find the solution of characteristic function, so the solution for first momenta of battery becomes: √   2 4 − 4 2 .Then, equation 12 and 14 are nonhomogenous differential equation.So, the solution will be consist of homogenous part and particular part.For the homogenous part, use the same method as equation 16 and 17, and for particular part, first we find the integration factor: So, the solution becomes: We set   =  †  and   =  †  so the energy operator equation will be obtained by multiply Equation 18and 19 for energy operator of charger and multiply Equation 21 and 22 for energy operator of battery.The equation will be: The interaction between battery and charger based on the step function shown in Figure 1: Based on the Figure 1, before interaction the charger still have an energy but the energy of battery start at zero or null energy.We set the value of   is 2, g is between 0,01 until 1, F is 0 which mean there is no modulation, and range of their interaction between charger and battery is 60 seconds.When they start interact, the charger energy will decrease and the battery will fulfill with energy.Based on the equation 17, the energy of charger will not be zero which mean after interaction in several times, the charger will not one hundred percent lose its energy when the energy storage in battery already done.In this paper, we set the interaction time in the small range because of the constant step function.For the long interaction time, we use a step function which is dependent with time and its still conducting on future research.

Conclusion
The energy transfer between quantum charger and battery has been investigated.In this paper, we use harmonic oscillator model for modelling the charger and the battery because the harmonic oscillator battery gives the largest maximum energy albeit having the longest maximum time.The master equation is based on the open quantum system to find the energy equation of battery and charger.We used a interaction constant based on the step function to know the time interaction between them.This step function help to modelling the interaction based on the Figure 1.Further research can be conducted by changing the step function with the dependent time function for the long interaction time, or set the different interaction time so that we can get more precision interaction and energy flow between quantum charger and quantum battery.

Acknowledgement
The author would thank for Theoretical Physic Laboratory, Faculty of Mathematic and Natural Science, Bandung Institute of Technology and author supervisor for helping and give author a knowledge guidance and direction in making this paper.

Figure 1 .
Figure 1.The energy interaction between charger (blue line) and the battery (orange line).