Entanglement and nonlocality in a coupled-cavity system Download: 637次
1. INTRODUCTION
The Bell inequality had been raised in the field of quantum entanglement in 1964 [1]. Measuring some different but coupled parts of an isolated system, we definitely obtain some statistical correlations. Unlike classical correlations, which satisfy the Bell inequality and are in consistence with our perceptions, the quantum correlations calculated following Bell’s method that, taking into consideration the entangled states, just revealed some phenomena that are incompatible with classical correlations. From then until now, violations of the Bell inequality have been observed in a large number of experiments, which prove the existence of quantum entanglement [2
Investigating the entanglement of a system helps us to understand the complex phenomenon of nonlocality, which can be exhibited via violation of the Bell inequality. Quantum entanglement brings some new features to our common knowledge about the world that are necessary for nonlocality. These features make quantum entanglement essential to quantum information science.
Entanglement is so important that we need a careful study about its features. Coupled-cavity systems, allowing the manipulation of states of quantum systems, are a kind of perfect candidate for our study of quantum entanglement. Nowadays, various methods for preparing and controlling the states have been demonstrated, among which are two-level atoms or artificial atoms with a two-level energy structure [8
The probability distribution for the bare states of the ground state depends on different parameters. The intersect states, the pure atomic state, and all photon states play different roles according to different parameters. Thus, they will be connected with the entropy of the system. We explore in detail their correlations. These correlations are used in the following discussion on bipartite entanglement entropies of the system, based on the von Neumann entropy method. To reveal the entanglement between subparts of subsystems in the whole system, we investigate the nonlocality, in which the Clauser–Horne–Shimony–Holt (CHSH) inequality [21] is checked, implying that the incompatibility definitely indicates a high entanglement.
The paper is organized as follows. In Section
2. CALCULATING FOR THE HAMILTONIAN
First, we introduce the two-site coupled cavity array system with -type three-level atoms. As is shown in Fig.
Fig. 1. Schematic configuration of the two-site coupled-cavity system. The system is composed of two harmonic resonators and two -type three-level atoms. For the atom, represents the excitation state, and and are the degenerate ground states of the atom.
When the atom in the excited state jumps to one of two degenerate ground states, an - or -polarized photon is emitted. For the case of emission of an -polarized photon, it would interact with the atom located in the state, stimulate the atom up to the excited state, stay in the cavity, or hop to the adjacent cavity. The Hamiltonian for the two-cavity system, , consists of three parts, where is a cavity Hamiltonian representing the harmonic oscillation parts for both cavities, is the atomic Hamiltonian, and is the atom–cavity interaction part. The three parts of the Hamiltonian can be rewritten as follows (setting ):
To obtain the matrix expression of the Hamiltonian, we consider a subspace of the two-excitation space, where all the states can freely evolve into another one through one or more steps. We represent a general form of the state in the following way: , where denotes the number of photonic excitations with polarization in cavity , and thus denotes the photonic state for -polarized photons in cavity . Similarly, denotes the photonic state for -polarized photons in cavity . The atomic excitation is denoted as , where represent the two ground states and the excited state for the atom in cavity . Thus, the bare states of the system can be written as follows: Photonic states when no atom is excited: Pure atomic states when both of the two atoms are excited: Intersect states, where there is one atomic excitation and one photonic excitation:
By far, after getting the Hamiltonian and the bare states of the subspace, we are definitely going to get its matrix form. In practice, we are not so interested in the actual eigen energies of the Hamiltonian. It is the ordering of the eigen values that matter. Thus, the matrix form of the Hamiltonian, with Eqs. (
3. ENTROPY OF THE GROUND STATE
In this section, we discuss the entropy for the ground state of the system, referring to an idea that is familiar in information theory. As is well known, entropy is often referred to as the number of possible arrangements or the degree of disorder. Here, we use this property to discuss the evolution of bare states for the ground states.
At first, let us examine the distributions of the probabilities of the bare states in ground states for different parameters, and . We can calculate the entropy (), which is well known in information theory. Assuming the ground state of the system to have the form , the entropy of this state is thus .
A contour plot of is shown in Fig.
Fig. 2. Entropies in the ground states as a function of detuning and hopping strength. Parameter values are , where is set as the basic unit throughout this paper.
Next, we show the probability plot for the pure atomic states in the ground states, as shown in Fig.
We show in Figs.
Fig. 4. Probabilities for the intersect states, one atom being in the excited state in one cavity, while a photon is in its eigenstate in the other cavity, in the ground states.
Fig. 5. Probabilities for the intersect states, two excitations being in the same cavity, in the ground states.
When the hopping strength goes up to a certain level, around the line represented by , things begin to change. Even though the atomic energy level is still much smaller than the energy of the photon for the same , the interacting strength between atom and photon is much smaller than the hopping strength. As a result, the chances for us to detect the photonic states and the atomic states draw near. This creates chaos around the , which obtains the high entropy region. With continued increase of the hopping strength, the chaos gradually declines, implying that the chances for detecting the system located in the atomic state or in the intersect states becomes dimmer, while the all-photon states become the main part of the ground state, as shown in Fig.
As the detuning increases, the atomic energy levels become smaller, while the photon energy gets higher. For the ground states, the probability for us to find the photons keeps on going up while the probability for the atoms keeps on going down. Because of this, the probability of the intersect states gets bigger, which raises the entropy of the system. The peak of entropy appears around the region. When we keep increasing the detuning, the opportunity for detecting the atoms becomes even smaller, thus the probabilities for the intersect states grow smaller. The all-photon states begin to take up the main parts of the ground states. As a result, the entropy goes down.
During the evolution process discussed above, the different polarized photons play different roles. We take the intersect states as an example, as shown in Fig.
Fig. 7. Probabilities of the intersect states: (a) one atom excited in one cavity and one -polarized photon excited in the other cavity, (b) one atom excited in one cavity and one -polarized photon excited in the other cavity, (c) one atom and one -polarized photon excited in the same cavity, and (d) one atom and one -polarized photon excited in the same cavity.
In this section, we discussed mainly the atom–atom, atom–photon, and photon–photon evolution for different parameters of the system. In analyzing the physical processes, we calculated the probability distributions for different part of the bare states. Results show that entropy for the bare state reflects the evolution or phase transformation quite well.
4. BIPARTITE ENTANGLEMENT
We discuss the bipartite entanglement about the ground state of the system, in which the von Neumann entropy method is employed. We assume that a system is composed of two parts, and , and is represented as in density matrix form. Given the reduced density matrices, and , the entanglement between and can be expressed as or .
Let us first look at an example. For the subspace composed of the -polarized photons in one cavity, we may obtain three different results (0 photon, 1 photon, and 2 photons) when a measurement is performed. We write these states as , and , respectively. These bare states are also the eigenstates of the density matrix for this part that is reduced from the whole. We can explicitly measure the probabilities for this subsystem to be in these three states. We will use these results in the following for discussing their entropies. The measurement operators for this subspace can be defined as
Fig. 8. Probability distribution of the -polarized photons under the small hopping limit. Parameter values are and .
For large negative detuning, the photon excitation, which has higher energy than the atomic excitation, is less likely to be detected in the ground states. The probability for detecting any -polarized photon is almost zero. As the detuning increases, the intersect states and the later all-photon states gradually take up the main parts in the ground states. We can find the growth trend in the line for the probability of one -polarized photon. This line rises ahead of the line for two -polarized photons. This is because, for the small detuning, the probability for the photon is low and the small hopping strength more or less prevents the photons from hopping into the adjacent cavity, which may be described as a two-photon state in this case. Finally, for the large positive detuning, the photon excitations carry smaller energy and are more likely to be detected in the ground states. The probability for detecting the -polarized two-photon states increases, while the probability for the -polarized single-photon states decreases. Because of the existence of the -polarized photons, it is still possible to detect the non-photon state even for the very large detuning. This is what we have already learned from Section
Fig. 9. Bipartite entanglement under the small hopping limit. Parameter values are and .
For the subsystems, such as -polarized photons in one cavity, the -polarized photons in one cavity, and the one-atom system, their eigenstates are the same as the bare states. As discussed in Section
However, the one-site entropy, which measures the entanglement between site I (including the photons and the atom) and site II, remains at a quite low level (be zero in this figure at the point of ) and after that rise to a high level. This is quite strange, so we need a more detailed understanding. Calculations show that this subsystem is mainly composed of three eigenstates, , , and , where are coefficients and satisfy for each state. We can find from Eqs. (
Fig. 10. Probability distribution of the eigenstates in the one-site subsystem. The inset shows the total probability for the three eigenstates. Parameter values are and .
Figure
Fig. 11. Bipartite entanglement under the large hopping limit. Parameter values are and .
5. NONLOCALITY: CHSH INEQUALITY
In Section
First, following the method reported in Refs. [21,22], let us measure the number of -polarized photons in cavity I; the results would be 0, 1, and 2. If no photon is detected, our measuring apparatus would respond as , while if any photon is detected, the response would be . The -polarized photons can be measured in the same way. We assume that the polarizations of two different photons are orthogonal, as and directions. For cavity II, we measure the number of photons in some orthogonal directions, excluding and , labeled as and , corresponding to the direction, in which . In each experiment, we choose two directions , where and , and would obtain two results and , where , respectively. Carrying on the experiment a large number of times, for a specific pair of directions , we would obtain the expectation value of the product for given measurement choices , where indicates the probability measuring the system in directions and , respectively, and getting the responses of and . Thus we would obtain the famous CHSH inequality
We check this inequality in the ground states for our system in the classical and quantum manners. The quantum measurement is performed as follows.
For the - and -polarized photons in cavity I, the eigenstates are , , and , and the responses are , , and , respectively, and we define the measurement operators:
The measurement operators in cavity II are thus
Fig. 12. Dependence of on the detuning . The solid curves show the results calculated in the classical manner, while the dashed curves are the results calculated in the quantum manner. From top to bottom, and from left to right, panels (a)–(i) correspond to the results calculated for directions , and , respectively. The gray lines are the gridlines of . Parameters are and .
6. CONCLUSIONS
We have examined mainly bipartite entanglement entropy and nonlocality, which definitely indicate the existence of entanglement in the ground state. For the entropy of the ground states, we analyzed the probability distributions of the bare states in detail over a wide range of the hopping and detuning parameters, in particular for different polarized photons.
For the bipartite entanglement entropy, the eigenstates of a small subsystem, e.g., a single -polarized photon, are in agreement with their bare states. It is not difficult to show their behaviors for bipartite entanglement entropy just by referring to the entropy for ground states. However, for a larger subsystem, e.g., a single site, their behaviors become rather complicated.
For nonlocality, we also consider just the ground states of the system for different parameters. Because of entanglement, there exists nonlocality. As the detuning changes, the entanglement between the photons in different cavities is also changed. It may become so small that the classical effect plays an important or even a main role in the experiment, which results in agreement with the CHSH inequality. Therefore, we can say that experiments agreeing with the CHSH inequality do not exclude the existence of entanglement, but results incompatible with the CHSH inequality indicate the existence of entanglement.
7 Acknowledgment
Acknowledgment. This work is supported by the Collaborative Innovation Center of Extreme Optics.
[1] J. S. Bell. On the Einstein-Podolsky-Rosen paradox. Physics, 1964, 1: 195-200.
Article Outline
Zheyong Zhang, Jianping Ding, Hui-Tian Wang. Entanglement and nonlocality in a coupled-cavity system[J]. Photonics Research, 2017, 5(3): 03000224.