Entropy of field interacting with two two-qubit atoms

Abstract

We use quantum field entropy to measure the degree of entanglement for a coherent state light field interacting with two atoms that are initially in an arbitrary two-qubit state. The influence of different mean photon number of the coherent field on the entropy of the field is discussed in detail when the two atoms are initially in one superposition state of the Bell states. The results show that the mean photon number of the light field can regulate the quantum entanglement between the atoms and light field.

Full text

Chin. Phys. B Vol. 27, No. 9 (2018) 090303
Entropy of field interacting with two two-qubit atoms∗
Tang-Kun Liu(刘堂昆)†, Yu Tao(陶宇), Chuan-Jia Shan(单传家), and Ji-Bing Liu(刘继兵)
College of Physics and Electronic Science, Hubei Normal University, Huangshi 435002, China
(Received 9 February 2018; revised manuscript received 25 June 2018; published online 28 August 2018)
We use quantum field entropy to measure the degree of entanglement for a coherent state light field interacting with
two atoms that are initially in an arbitrary two-qubit state. The influence of different mean photon number of the coherent
field on the entropy of the field is discussed in detail when the two atoms are initially in one superposition state of the Bell
states. The results show that the mean photon number of the light field can regulate the quantum entanglement between the
atoms and light field.
Keywords: quantum field entropy, coherent state light field, quantum entanglement
PACS: 03.65.Ud, 42.50.Pq DOI: 10.1088/1674-1056/27/9/090303
1. Introduction
In 1935, Einstein et al.[1]and Schr¨
odinger[2]proposed
the concept of entangled states respectively. Quantum en-
tanglement is a distinctive feature of quantum physics, and
it is very useful in quantum information processing, includ-
ing quantum communication and quantum computation.[3–6]
In order to quantitatively describe the degree of entanglement
between microsystems of the light field interacting with the
atom, Phoenix and Knight[7]studied the entanglement and dy-
namics between light fields and atoms using von Neumann
quantum entropy theory. Thereafter, the coupling entropy
characteristic between the light field and the atom has attracted
a great deal of attention, the entropy of quantum systems has
been studied, and many results have been obtained.[8–18]How-
ever, the above literatures are based on the eigenvalue method
for calculating the reduced density operator of the atom of
the subsystem. In the previous work,[19,20]we formulated the
quantum entropy of the interaction between two two-level en-
tangled atoms and a light field of the coherent state and the
Schr¨
odinger cat state by using the eigenvalue method to cal-
culate the reduced density operator of the light field of the
subsystem. In this paper, we investigate the quantum field en-
tropy in a system of arbitrary two qubit atoms interacting with
the coherent state light field. For our purpose, we choose the
initial atoms in four different states and examine the evolu-
tion characteristics of the quantum field entropy by means of
numerical calculations.
2. Theoretical model and its solution
We consider a system composed of two identical two-
level atoms resonantly interacting with a single-mode cavity
field simultaneously, the Hamiltonian of the system in the ro-
tating wave approximation can be written as (¯
h=1)[21]
H=Ωa+a+1
2ω
2
∑
i=1
σz
i+g
2
∑
i=1
(a+σ−
i+aσ+
i),(1)
where a+and aare the creation and annihilation operators
of the field mode, ωis the atomic transition frequency, and
σz
i=|eiihei|,σ+
i=|eiihgi|, and σ−
i=|giihei|are the spin op-
erators of the i-th atom (i=1,2). |eidenotes an excited state
of the atom, |gidenotes a ground state of the atom, and gis
the atom–field coupling constant.
The two atoms are initially prepared in an arbitrary two-
qubit state
|Ψa(0)i=a|e1,e2i+b|g1,g2i+c|e1,g2i+d|g1,e2i,(2)
where |a|2+|b|2+|c|2+|d|2=1. The radiation field is ini-
tially prepared in a single-mode coherent state
|Ψf(0)i=
∞
∑
n=0
Fn|ni=
∞
∑
n=0
e−1
2|α|2αn
√n!|ni,(3)
where α=|α|eiϕ,|α|2is the mean photon number of the
coherent light field, and ϕis the phase angle of the coherent
field.
When two atoms interact with the light field, the system
state vector at any time will evolve as
|Ψs(t)i=|αi|e1,e2i+|βi|e1,g2i+|γi|g1,e2i
+|δi|g1,g2i,(4)
where
|αi=
∞
∑
n=0
A(n,t)|ni,|βi=
∞
∑
n=0
C(n,t)|ni,
|γi=
∞
∑
n=0
D(n,t)|ni,|δi=
∞
∑
n=0
B(n,t)|ni.(5)
∗Project supported by the National Natural Science Foundation of China (Grant No. 11404108).
†Corresponding author. E-mail: tkliu@hbnu.edu.cn
© 2018 Chinese Physical Society and IOP Publishing Ltd http://iopscience.iop.org/cpb http://cpb.iphy.ac.cn
090303-1