We develop an approach to study the entanglement in two coupled harmonic oscillators. We start by introducing an unitary transformation to end up with the solutions of the energy spectrum. These are used to construct the corresponding coherent states through the standard way. To evaluate the degree of the entanglement between the obtained states, we calculate the purity function in terms of the coherent and number states, separately. The result is yielded to two parameters dependance of the purity, which can be controlled easily. Interesting results are derived by fixing the mixing angle of such transformation as π/2. We compare our results with already published work and point out the relevance of these findings to a systematic formulation of the entanglement effect in two coupled harmonic oscillators.
We develop an approach to study the entanglement in two coupled harmonic oscillators. We start by introducing an unitary transformation to end up with the solutions of the energy spectrum. These are used to construct the corresponding coherent states through the standard way. To evaluate the degree of the entanglement between the obtained states, we calculate the purity function in terms of the coherent and number states, separately. The result is yielded to two parameters dependance of the purity, which can be controlled easily. Interesting results are derived by fixing the mixing angle of such transformation as π/2. We compare our results with already published work and point out the relevance of these findings to a systematic formulation of the entanglement effect in two coupled harmonic oscillators.
Quantum entanglement serves as a key phenomenon in understanding correlations in many-body systems, but analytical results remain scarce for coupled three-body oscillators. In this work, we address this gap by introducing a geometrical diagonalization approach that constrains Euler angles, thereby reducing the degrees of freedom in the entanglement analysis. It consists of deriving analytical expressions for linear entropy and purity under the bipartitions (x∣yz), (y∣xz), and (xy∣z) using the Wigner function framework. Our results indicate that excitations in any oscillator basically enhance the redistribution of correlations across the system. The mixing angle θ governs entanglement intensity, ranging from separability to maximal correlation. Moreover, we reveal the symmetry relations, notably SLy[(n,m,l),θ]=SLz[(n,m,l),−θ] and an intrinsic symmetry within (x∣yz). Hence, we clarify how excitation levels and mixing angles create and enhance entanglement in the three coupled harmonic oscillators.
We investigate the quantum entanglement dynamics of two coupled harmonic oscillators with a time-dependent interaction. Using the Lewis-Riesenfeld invariant method, we derive the exact analytical wave functions without any perturbative or adiabatic approximations and combine this with a phase-space analysis using the Wigner function to provide a complete description of the system's quantum state evolution. We obtain general expression form for the purity and the linear entropy SL=1−P for arbitrary excitation numbers (n,m), allows a systematic study of entanglement for a large class of quantum states. We show that the entanglement dynamics is very sensitive to the interplay between the detuning parameters θ and ϑ2, the frequency parameter β0 and the coupling strength ε: the increase of detuning takes the system from slow irregular oscillations to fast and regular periodic behavior, and the stronger coupling systematically enhances both the amplitude and the average value of the linear entropy. Most importantly, for the resonance case ω1=ω2=1 and strong couplings ε≈0.99, the system shows robust undamped synchronized periodic oscillations of the linear entropy for all quantum states considered, indicating preserved quantum coherence without saturation. Our findings demonstrate that linear entropy is a sensitive and practical entanglement witness, and we establish explicit analytical relations between the coupling parameters of the system and its entanglement properties, which are directly relevant to quantum information processing and the control of quantum correlations in continuous-variable systems.
Abstract We develop an approach using coherent states and path integral to investigate the dynamics of entanglement in a simple two-dimensional non-commutative harmonic oscillator. We start by employing a Bopp shift to convert the Hamiltonian describing the system into a commutative equivalent one. This allows us to construct coherent states and calculate the propagator in standard way. By deriving the explicit expression of the time-dependent coherent states and considering its connection with the number states, we provide exact results for evaluating the degree of entanglement between the ground state and any excited state through the purity function. The interesting emerging result is that, as long as the non-commutativity parameter is non-zero, our system exhibits the phenomenon of collapse and revival of entanglement.
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.
Abstract From the definition of the standard Perelomov coherent states we introduce the Perelomov number coherent states for any su(2) Lie algebra. With the displacement operator we apply a similarity transformation to the su(2) generators and construct a new set of operators which also close the su(2) Lie algebra, being the Perelomov number coherent states the new basis for its unitary irreducible representation. We apply our results to obtain the energy spectrum, the eigenstates and the partition function of two coupled oscillators. We show that the eigenstates of two coupled oscillators are the SU(2) Perelomov number coherent states of the two-dimensional harmonic oscillator with an appropriate choice of the coherent state parameters.
Harmonic chains are used as a model in quantum information theory for ion traps and simple solid state systems. Entanglement is a quantum effect in which observations made on one object instantaneously affect measurement outcomes on the other. This study investigates interparticle entanglement in finite chains of coupled harmonic oscillators as a function of the vibrational modes, mode excitations, and bipartition of oscillators. The methods used and results extend previous work for the Gaussian ground state to excited states. Entanglement is analyzed by calculating the purity of the reduced density matrix of the combined wavefunctions of the oscillators in the chain, tracing over subensembles. Analytic and numerical results for a varying effective spring constants between the particles and number of particles in the chain are presented. Interesting correlations between the symmetries of the vibrational modes and their excitations, and the symmetries of the partitions were discovered. Results show that higher excitation numbers result in higher entanglement. Adding energy to the modes will never decrease entanglement, however, the relationship between energy and entanglement is not simple. Different combinations of mode energy excitations will produce the same entanglement, even if the total energy of the system differs. Patterns indicate that the combination of symmetry in mode shapes and partition determines excitation number purity equivalence classes.