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Journal of Physics A Mathematical and General

On polariton instability and thermodynamic phase transition in a photon-matter system

M. Yamanoi

Published Sep 1, 1979• 8 citations
Publisher

Abstract

It is shown that the rotating wave approximation in the minimal coupling Hamiltonian with the A 2 term leads to an incorrect energy spectrum of the polariton mode: in particular, to a spurious polariton instability which should not occur for a photon-matter system consisting of radiation and atoms without the Coulomb interaction. Thus the counter-rotating term in the presence of the A 2 term plays a crucial role in the instability property of the system. The super-radiant phase transition in the Dicke model is shown to be accompanied by polariton instability, and therefore by instability of the ground state of the photon-matter system. The rotating wave approximation, as well as the neglect of the A 2 term, should therefore not be made for the study of the thermodynamic phase transition and instability properties of the ground state of the system.

DOI
10.1088/0305-4470/12/9/025

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It is shown that the rotating wave approximation in the minimal coupling Hamiltonian with the A 2 term leads to an incorrect energy spectrum of the polariton mode: in particular, to a spurious polariton instability which should not occur for a photon-matter system consisting of radiation and atoms without the Coulomb interaction. Thus the counter-rotating term in the presence of the A 2 term plays a crucial role in the instability property of the system. The super-radiant phase transition in the Dicke model is shown to be accompanied by polariton instability, and therefore by instability of the ground state of the photon-matter system. The rotating wave approximation, as well as the neglect of the A 2 term, should therefore not be made for the study of the thermodynamic phase transition and instability properties of the ground state of the system.
A non-relativistic quantum theory of matter-photon interaction is formulated, within the framework of the dipole approximation, by employing a canonical transformation to diago­ nalize the A' term plus the free photon Hamiltonian. The matter-photon interaction is there­ by expressed rigorously in a renormalized A·p form. Using this form for the Dicke model of two-level atoms interacting with a single-mode radiation field, we examine a possibility of the occurrence of stationary coherent states which could arise for a photon and atomic polarization as the ground state of the system, deducing a condition for such coherent states. The condition, expressed as an inequality which imposes a stable nontrivial Bloch angle of the uniform atomic polarization, is identical with that for the occurrence of a second order phase transition, and indicates that the non-existence of a second order phase transition pointed out by Rza2ewski et al. io; in fact a consequence of the present treatment of the A'­ term renormalization. However, a fulfillment of the condition is recovered, if exchange­ type atom-atom interaction is taken into account in the matter system. We also present a method to construct an effective Hamiltonian of atom-atom interaction which is equivalent to the above Dicke model in the thermodynamic limit.
A very simple treatment is presented to describe a phase transition present in Dicke's model for super-radiance, in the off-resonance case and for a single radiation mode. The results agree with previous authors, although the use of mod n( states provides a transparent physical interpretation. The results show that if the A 2 term is kept in the Hamiltonian, the phase transition persists, thus clarifying some recent controversy on the subject. The thermodynamic variables are calculated above and below T c . The phase transition is shown to be a consequence of an instability of the ground state. The behaviour of (a Dagger a) and (S Z ) is explored as a function of temperature. (a Dagger a) goes to zero as (T c -T) near the critical temperature.
The thermodynamic equilibrium properties of the Dicke Hamiltonian for a collection of spin- 1 / 2 particles interacting via magnetic dipole interaction with a single mode of the radiation field and with each other by spin-spin interaction are analysed and the conditions for the existence of a super-radiant phase transition are derived. It is shown, using the approximate long-range spin-spin interaction (MI), in the Dicke Hamiltonian, that the conditions for the existence of the super-radiant phase transition are invariant with respect to isotropic MI whether or not the A 2 term is included.
The Dicke model can exhibit quantum phase transition between the normal and the superradiant phases when the strength of the light-matter coupling exceeds the ultrastrong-coupling regime. However, it is challenging to observe this phase transition in practical systems due to limited coupling strength or finite two-photon A2{A}^{2}A2 terms. Here we show that by applying a periodic modulation to the frequency of the two-level systems in a standard Dicke model in the strong-coupling regime, an anisotropic Dicke model with tunable rotating and counter-rotating terms in the ultrastrong-coupling regime can be achieved. We calculate the ground state and the excitation spectrum of this model in terms of the modulation parameters. Our result shows that the superradiant phases can be observed in cavity- or circuit-quantum electrodynamics systems with strong coupling.
The unitary transformation method is used to study the properties of the polaritonic states and Bose-Einstein condensation in the Rabi lattice model, where the on-site two-level systems (TLSs) are coupled with the intersite hopping photons. It is shown that the counter-rotating coupling (CRC) between TLS and photon, which breaks down the conservation of the excitation numbers, may induce a long-range Ising-like interaction among intersite TLSs. We have shown that the coupled TLSs and photons are hybridized to form the polaritonic states, and the corresponding ground state and polaritonic excitation spectra are calculated by diagonalizing our transformed Hamiltonian. When the photon hopping JJJ is weak, the excitation spectrum is gapped. But the gap decreases with increasing JJJ, and at critical value J=JcJ={J}_{c}J=Jc​ the gap becomes zero at the $\mathrm{\ensuremath{\Gamma}}$ point and the ground state becomes instable. Thus, Jc{J}_{c}Jc​ is the phase transition point where the polaritons are condensed at the $\mathrm{\ensuremath{\Gamma}}$ point to form the delocalized superradiant phase. The results show that the larger detuning between TLS and photon favors the disorder insulator phase, which requires the stronger Jc{J}_{c}Jc​ to get the long-range order phase. The phase diagram of the delocalized superradiant phase transition has been obtained where there are no Mott lobes since the CRC breaks down the number conservation. The ground state and the excitation spectra of the delocalized superradiant phase are also calculated. It is shown that in the delocalized superradiant state the order parameter is the nonzero ground-state average of the photon annihilation operator. The polaritonic excitation has a finite excitation gap because the CRC breaks down the number conservation but leaves only a discrete symmetry, the parity conservation.
The Hamiltonian of the two-photon Dicke model is diagonalized for the normal phase and super-radiant phase beyond the mean-field method respectively, giving the critical coupling strength. Besides a spectral collapse, the super-radiant phase transition is presented by the vanishing of the excitation energy, which is lower than the mean-field results. Finite-size scaling exponents for the ground-state energy and the atomic angular momentum are analytically derived from scaling hypothesis, belonging to the same scaling universality of the one-photon Dicke model.