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The quantum oscillator model plays a significant role in quantum optics and quantum information and has been one of the hot topics in related research fields. Inspired by the single-mode linear harmonic oscillator and the two-mode entangled state representation, we construct a two-mode coupled harmonic oscillator in this work. Different from the quantum transformation method used in previous literature, the entangled state representation is directly used in this work to solve its energy eigenvalues and eigenfunctions easily. The energy eigenvalues and eigenfunctions of this two-mode coupled harmonic oscillator are continuous compared with those of the one-mode harmonic oscillator. Using the matrix theory of quantum operators, we derive the transformation and inverse transformation of the time evolution operator corresponding to the two-mode coupled harmonic oscillator. In addition, using the entangled state representation, the specific form of the time evolution of the two-mode vacuum state under the action of the oscillator is obtained. Through the analysis of quantum fidelity, it is found that the fidelity of the output quantum state decreases with the oscillator frequency increasing, and the fidelity eventually tends to zero with the increase of time. When analyzing the orthogonal squeezing properties of the output quantum state, this type of two-mode oscillator does not have the orthogonal squeezing effect, but it has a strong quantum dissipation effect instead. This conclusion is further verified by the quasi-probability distribution Q function of the quantum state phase space. Therefore, the two-mode coupled harmonic oscillator has a major reference value in quantum control such as quantum decoherence and quantum information transmission. Like the two-mode squeezed vacuum state, the photon distribution of the output quantum light field corresponding to the two-mode harmonic oscillator presents a super-Poisson distribution, and the photons exhibit a strong anti-bunching effect. Using the three-dimensional discrete plot of the photon number distribution, the super-Poisson distribution and quantum dissipation effect of the output quantum state are intuitively demonstrated. Finally, the SV, which is an entanglement criterion, is used to determine that the output quantum state has a high degree of entanglement. Further numerical analysis shows that the degree of entanglement increases with the action time and the oscillator frequency. In summary, the two-mode coupled harmonic oscillator constructed in this work can be used to prepare highly entangled quantum states through a complete quantum dissipation process. This provides theoretical support for experimental preparing quantum entangled states based on dissipative mechanisms. -
Keywords:
- quantum oscillator /
- quantum dissipation /
- super-poisson distribution /
- quantum entanglement
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图 1 双模真空态保真度随时间的变化曲线
Figure 1. Curves for the change of fidelity with two-mode vacuum state over time.
图 2 量子态$\left| \tau \right\rangle $的相空间分布Q函数 (a) $\omega = 0.5, t = 0$; (b) $\omega = 0.5, t = 0.5$; (c) $\omega = 0.5, t = 1.5$; (d) $\omega = 1.5, t = 1.5$
Figure 2. Q-distribution function of quantum state $\left| \tau \right\rangle $ in phase space: (a) $\omega = 0.5, t = 0$; (b) $\omega = 0.5, t = 0.5$; (c) $\omega = 0.5, t = 1.5$; (d) $\omega = 1.5, t = 1.5$.
图 3 量子态$\left| \tau \right\rangle $的Q参数随时间的变化曲线
Figure 3. Curves for the change of Q-parameter with quantum state $\left| \tau \right\rangle $over time.
图 4 量子态$\left| \tau \right\rangle $的${R_{ab}}$函数随时间的变化曲线
Figure 4. Curves for the change of ${R_{ab}}$ function with quantum state $\left| \tau \right\rangle $over time.
图 5 量子态$\left| \tau \right\rangle $中光子分布概率 (a) $\omega = 0.5, t = 0.5$; (b) $\omega = 0.5, t = 2.5$; (c) $t = 0.5, \omega = 0.1$;(d) $t = 0.5, \omega = 2.5$
Figure 5. Photon number distribution probability for the quantum state $\left| \tau \right\rangle $: (a) $\omega = 0.5, t = 0.5$; (b) $\omega = 0.5, t = 2.5$; (c) $t = 0.5, \omega = 0.1$; (d) $t = 0.5, \omega = 2.5$.
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