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Acta Physica Sinica

Polaron-soliton squeezed states , ground state and quantum fluctuation in one-dimensional molecular crystals

Chaofan Yu, Liang Guo-Dong, Cao Xi-Jin, (1)广东教育学院物理系，广州 510303; (2)华南师范大学物理系，广州 510006; (3)暨南大学光电工程系，广州 510632

Published Jan 1, 2008• 4 citations
Publisher

Abstract

On the basis of the Hamiltonian of the Holstein one_dimensional molecular crystals, using the squeezed_coherent state expansion method, the influence of the electron_two phonon interaction on the properties of the ground state and quantum fluctuation for the polaron_soliton system were investigated by including the quantum correlation between the polarons and the squeezed phonons, and the renormalized displacement correction.The nonlinear Schringer equation for one_dimensional polaron_soliton state has been found in a closed form.By the use of the nonlinear expansion, we have given the analytical solution of the corresponding nonlinear equation so as to obtain the ground state energy,the quantum fluctuation and the polaron energy of the polaron_soliton system in analytical form. We have found that, when the electron_two phonon coupling strength g1g12n〉 and 〈Δ2p〉〈Δ2q〉 for the two-phonon effect are larger than the one_phonon one and the polaron energy for the two_phonons effect is more negative compared with the one_phonon one.

DOI
10.7498/aps.57.4402

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On the basis of the Hamiltonian of the Holstein one_dimensional molecular crystals, using the squeezed_coherent state expansion method, the influence of the electron_two phonon interaction on the properties of the ground state and quantum fluctuation for the polaron_soliton system were investigated by including the quantum correlation between the polarons and the squeezed phonons, and the renormalized displacement correction.The nonlinear Schringer equation for one_dimensional polaron_soliton state has been found in a closed form.By the use of the nonlinear expansion, we have given the analytical solution of the corresponding nonlinear equation so as to obtain the ground state energy,the quantum fluctuation and the polaron energy of the polaron_soliton system in analytical form. We have found that, when the electron_two phonon coupling strength g1g12n〉 and 〈Δ2p〉〈Δ2q〉 for the two-phonon effect are larger than the one_phonon one and the polaron energy for the two_phonons effect is more negative compared with the one_phonon one.
Based on the Holstein model Hamiltonian of one-dimensional molecular crystals, by making use of the expansion approach of the correlated squeezed-coherent states of phonon instead of the two-phonon coherent state expansion scheme, the properties of the ground state and the anomalous quantum fluctuations are investigated in a strongly coupled electron—phonon system with special consideration of the electron-two-phonon interaction. The effective renormalization ( i ) of the displacement of the squeezed phonons with the effect of the squeezed-coherent states of phonon and both the electron-displaced phonon and the polaron-squeezed phonon correlations have been combined to obtain the anomalous quantum fluctuations for the corrections of the coherent state. Due to these non-adiabatic correlations, the effective displacement parameter i is larger than the ordinary parameter α i (0) . In comparison with the electron-one-phonon interaction ( g ) corrected as i g , we have found the electron-two-phonon interaction ( g 1 ) corrected as i 2 g 1 is enhanced significantly. For this reason, the ground state energy ( E 0 (2) ) contributed by the electron-two-phonon interaction is more negative than the single-phonon case ( E 0 (1) ) and the soliton solution is more stable. At the same time, the effects of the electron-two-phonon interaction greatly increase the polaron energy and the quantum fluctuations. Furthermore, in a deeper level, we have considered the effect of the polaron-squeezed phonon correlation ( f -correlation). Since this correlation parameter f > 1, this effect will strengthen the electron-one and two-phonon interactions by f i g and f 2 i 2 g 1 , respectively. The final results show that the ground state energy and the polaron energy will appear more negative further and the quantum fluctuations will gain further improvement.
The influences of the squeezing–antisqueezing effect arising from variations of the electron density and motion of polarons and nonadiabatic phonon fluctuations on the properties of the ground state and the binding energy of the polaron in the Holstein model of coupled one-dimensional electron–phonon systems with spin 1/2 in the cases of high and low electron densities have been studied by using the Bogolyubov transformation and a new ansatz for states that includes correlation effects among the one-phonon coherent and two-phonon squeezed and polaronic states. The results obtained show that the squeezing–antisqueezing (correlation) effect results in that the ground state energy of the systems is significantly lowered, and the binding energy of the polaron is considerably increased. Thus, the stability of the systems and of the polaron are obviously enhanced in such a case when compared with the uncorrelated case. This shows that the ground state determined by the new state ansatz is most stable, and the new ansatz introduced here is very relevant for the coupled electron–phonon systems, especially in strongly coupled and largely squeezed cases.
Based on the coherent interaction and action–counteraction principles, we investigate the ground state properties for small polaron systems, the coherent-squeezed fluctuation correction, and the anomalous lattice quantum fluctuation, with the new variational generator containing correlated squeezed-coherent coupling and quantum entanglement. Noting that −2t is the T.B.A. energy, for the coherent interaction effect, we find the ground-state energy E 0 to be −2.428t, in which the coherent squeezed fluctuation correction −A 0 t is −0.463t (where t is the hopping integral, ω is the phonon frequency), with the electron–one-phonon coupling constant g = 1 and the electron–two-phonon coupling constant g 1 = −0.1. However, as a result of the action–counteraction effect, <?CDATA E∼0{\mathop{E}\limits^{\sim }}_{0}E∼​0​?> E ∼ 0 is −2.788t, but <?CDATA −A∼0t-{\mathop{A}\limits^{\sim }}_{0}t−A∼​0​t?> − A ∼ 0 t is −0.735t. As to the polaron binding energy (E P), for the coherent interaction effect, E P is −1.38ω, but for the action–counteraction effect, <?CDATA E∼P{\mathop{E}\limits^{\sim }}_{{\rm{P}}}E∼​P​?> E ∼ P is −1.88ω. In particular, the electron–two-phonon interaction noticeably enlarges the coherent interaction and the coherent squeezed quantum fluctuation correction. By intervening with the quantum entanglement, the evolutions of the squeezed coherent state and the lattice quantum fluctuation begin to take control. At that time, we encounter a new quantum phase coherence phenomenon — the collapse and revival of inversion repeatedly for the coherent state in the entangled evolution.
The effects of the nonadiabatic phonon behaviours on the ground state of the system and the quantum modifications of the polaron characteristics and the modification form of the uncertainty relation between the phonon coordinate and momentum due to fluctuations of the electron density in the coupled electron-phonon systems have been studied by using a new variational ansatz for coherent phonons with correlated displacement and squeezing including internations with electrons. The correlated effect results in the following effect: (1) the energy of the ground state of the system is found to be lower than that for the phonons with uncorrelated displacement and squeezing; (2) the binding energy of the polaron was found to be increased; (3) the nonadiabatic coupling with electrons enhances the quantum uncertainty for the phonon coordinate and momentum when compared with the adiabatic and uncorrelated case.
In this paper we have applied the correlated squeezed-state approach to investigate the ground-state properties of the one-dimensional Hubbard model with a local Holstein-type electron-phonon interaction. We have introduced a variational ground state in which the phonon subsystem is in a correlated squeezed state. The correlated squeezed state has built in the correlation between different phonon modes induced by the linear coupling with the electrons. With optimal values of the variational parameters the correlated squeezed state will by construction yield an energy lower than those obtained in earlier treatments. This means that our variational ansatz is more stable as the ground state of the system. Furthermore, our variational study shows that in the correlated squeezed state the polaronic reduction effect of phonons is much more alleviated, and thus the mass enhancement inherent to the polaron effect is considerably weakened. This weakening of the reduction effect should, in turn, affect other physical properties of the system.
Effects of the nonadiabatic phonon fluctuations on the ground state of coupled one-dimensional electron-phonon systems with high and low electron-densities were investigated by a new ansatz including correlated interaction of the displacement and squeezing. The correlated effect resulted in noticeable reduction of ground state energy of the systems, and obvious increases of binding energy of the polarons occurred. Thus, the stabilities of the polaron and of the systems were significantly enhanced, exactly steady ground state of the Holstein model, then, was obtained by the new ansatz for the coupled electron-phonon systems. In general, the new ansatz were found to be very relevant for the strong coupling and large squeezing cases in nonadiabatically coupled electron-photon systems.