論文 - 大橋　洋士

Spin susceptibility and effects of a harmonic trap in the BCS-BEC crossover regime of an ultracold Fermi gas

Tajima H., Hanai R., Ohashi Y.

Physical Review A （Physical Review A）  96 （ 3 ） 033614(1-11) 2017年09月

研究論文（学術雑誌）, 共著, 最終著者, 査読有り,  ISSN  24699926

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© 2017 American Physical Society. We theoretically investigate magnetic properties of a trapped ultracold Fermi gas. Including pairing fluctuations within the framework of an extended T-matrix approximation, as well as effects of a harmonic trap in the local density approximation, we calculate the local spin susceptibility χt(r,T) in the Bardeen-Cooper-Schrieffer-Bose-Einstein condensation crossover region. We show that pairing fluctuations cause nonmonotonic temperature dependence of χt(r,T). Although this behavior looks similar to the spin-gap phenomenon associated with pairing fluctuations in a uniform Fermi gas, the trapped case is found to also be influenced by the temperature-dependent density profile, in addition to pairing fluctuations. We demonstrate how to remove this extrinsic effect from χt(r,T), to study the interesting spin-gap phenomenon purely originating from pairing fluctuations. Since experiments in cold-atom physics are always done in a trap, our results would be useful for the assessment of preformed pair scenario, from the viewpoint of spin-gap phenomenon.

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Single-particle excitations and effects of hetero-pairing fluctuations in a Bose–Fermi mixture with a Feshbach resonance

Kharga D., Inotani D., Hanai R., Ohashi Y.

Journal of the Physical Society of Japan （Journal of the Physical Society of Japan）  86 （ 8 ） 084301 (1-8) 2017年08月

ISSN  00319015

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©2017 The Physical Society of Japan. We theoretically investigate normal-state properties of a gas mixture of single-component bosons and fermions with a hetero-nuclear Feshbach resonance. Including strong hetero-pairing fluctuations associated with the Feshbach resonance, we calculate single-particle density of states, as well as the spectral weight at various interaction strengths. For this purpose, we employ an improved T-matrix approximation (TMA), where the bare Bose Green’s function in the non-selfconsistent TMA self-energy is modified so as to satisfy the Hugenholtz–Pines relation at the Bose–Einstein condensation (BEC) temperature T BEC . In the unitary regime at T BEC , we show that hetero-pairing fluctuations couple Fermi atomic excitations with Fermi molecular excitations, as well as with Bose atomic excitations. Although a similar coupling phenomenon by pairing fluctuations is known to give a pseudo-gapped density of states in the unitary regime of a two-component Fermi gas, such a dip structure is found to not appear even in the unitary limit of a Bose–Fermi mixture. It only appears in the strong-coupling regime. Instead, a spectral peak along the molecular dispersion appears in the spectral weight. We also clarify how this coupling phenomenon is seen in the Bose channel. Since a hetero-nuclear Feshbach resonance, as well as the formation of Bose–Fermi molecules, have been realized, our results would be useful for the study of strong-coupling properties of this unique quantum gas.

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Normal-state properties of a unitary Bose–Fermi mixture: A combined strong-coupling approach with universal thermodynamics

Kharga D., Tajima H., Van Wyk P., Inotani D., Ohashi Y.

Journal of the Physical Society of Japan （Journal of the Physical Society of Japan）  86 （ 7 ） 074302(1-7) 2017年07月

最終著者, 査読有り,  ISSN  00319015

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©2017 The Physical Society of Japan. We theoretically investigate normal-state properties of a unitary Bose–Fermi mixture. Including strong hetero-pairing fluctuations, we evaluate the Bose and Fermi chemical potential, internal energy, pressure, entropy, as well as specific heat at constant volume C V , within the framework of a combined strong-coupling theory with exact thermodynamic identities. We show that hetero-pairing fluctuations at the unitarity cause non-monotonic temperature dependence of C V , being qualitatively different from the monotonic behavior of this quantity in the weak- and strong-coupling limit. On the other hand, such an anomalous behavior is not seen in the other quantities. Our results indicate that the specific heat C V , which has recently become observable in cold atom physics, is a useful quantity for understanding strong-coupling aspects of this quantum system.

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Zero-Temperature Properties of a Strongly Interacting Superfluid Fermi Gas in the BCS–BEC Crossover Region

Tajima H., van Wyk P., Hanai R., Kagamihara D., Inotani D., Horikoshi M., Ohashi Y.

Journal of Low Temperature Physics （Journal of Low Temperature Physics）  187 （ 5-6 ） 677 - 684 2017年06月

研究論文（国際会議プロシーディングス）, 共著, 最終著者, 査読有り,  ISSN  00222291

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© 2016, Springer Science+Business Media New York. We investigate thermodynamic properties and effects of quantum fluctuations in the Bardeen–Cooper–Schrieffer (BCS)–Bose–Einstein condensation (BEC) crossover region of a superfluid Fermi gas in the low-temperature limit. Including strong-coupling corrections within the framework of an extended T-matrix approximation, we numerically compute the isothermal compressibility χn. While quantum fluctuation effects on χn in the strong-coupling BEC regime are explained by the quantum depletion due to a repulsive interaction between tightly bound molecules, effects of self-energy shift on the Fermi chemical potential are found to enhance χn in the weak-coupling BCS region. We also show that the calculated χn agrees well with the recent experiment on a 6Li Fermi gas done from the weak-coupling region to the unitarity limit. Our result would be useful for the study of many-body quantum corrections in the BCS–BEC crossover region of a strongly interacting Fermi superfluid.

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Single-Particle Properties of a Strongly Interacting Bose–Fermi Mixture Above the BEC Phase Transition Temperature

Kharga D., Inotani D., Hanai R., Ohashi Y.

Journal of Low Temperature Physics （Journal of Low Temperature Physics）  187 （ 5-6 ） 661 - 667 2017年06月

研究論文（国際会議プロシーディングス）, 共著, 最終著者, 査読有り,  ISSN  00222291

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© 2017, Springer Science+Business Media New York. We theoretically investigate the normal state properties of a Bose–Fermi mixture with a strong attractive interaction between Fermi and Bose atoms. We extend the ordinary T-matrix approximation (TMA) with respect to Bose–Fermi pairing fluctuations, to include the Hugenholtz–Pines’ relation for all Bose Green’s functions appearing in TMA self-energy diagrams. This extension is shown to be essentially important to correctly describe the physical properties of the Bose–Fermi mixture, especially near the Bose–Einstein condensation instability. Using this improved TMA, we clarify how the formation of composite fermions affects Bose and Fermi single-particle excitation spectra, over the entire interaction strength.

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Specific Heat of Ultracold Fermi Gas with a Uniaxially Anisotropic p-Wave Interaction at the Superfluid Transition Temperature

Inotani D., van Wyk P., Ohashi Y.

Journal of Low Temperature Physics （Journal of Low Temperature Physics）  187 （ 5-6 ） 685 - 691 2017年06月

研究論文（国際会議プロシーディングス）, 共著, 査読有り,  ISSN  00222291

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© 2016, Springer Science+Business Media New York. We theoretically investigate the specific heat at constant volume CV and strong-coupling effects in a Fermi gas with p-wave interaction. In a 40K Fermi gas, a uniaxial anisotropy of a p-wave interaction associated with a p-wave Feshbach resonance is expected as a result of the split of the p-wave Feshbach resonance by a dipole interaction. Including this, as well as pairing fluctuations, we show that CV is significantly affected by this anisotropy in the strong-coupling regime. We also discuss the physical origin of this effect. Our results would contribute to the further understanding of an ultracold p-wave Fermi gas.

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Zero-Temperature Properties of a Strongly Interacting Superfluid Fermi Gas in the BCS-BEC Crossover Region

H. Tajima, P. van Wyk, R. Hanai, D. Kagamihara, D. Inotani, M. Horikoshi, Y. Ohashi

J. Low Temp. Phys. 187   677 - 684 2017年06月

研究論文（国際会議プロシーディングス）, 共著, 最終著者, 査読有り

Strong-Coupling Effects and Shear Viscosity in an Ultracold Fermi Gas

Kagamihara D., Ohashi Y.

Journal of Low Temperature Physics （Journal of Low Temperature Physics）  187 （ 5-6 ） 692 - 698 2017年06月

研究論文（国際会議プロシーディングス）, 共著, 最終著者, 査読有り,  ISSN  00222291

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© 2017, Springer Science+Business Media New York. We theoretically investigate the shear viscosity η, as well as the entropy density s, in the normal state of an ultracold Fermi gas. Including pairing fluctuations within the framework of a T-matrix approximation, we calculate these quantities in the Bardeen–Cooper–Schrieffer (BCS)–Bose–Einstein condensation (BEC) crossover region. We also evaluate η/ s, to compare it with the lower bound of this ratio, conjectured by Kovtun, Son, and Starinets (KSS bound). In the weak-coupling BCS side, we show that the shear viscosity η is remarkably suppressed near the superfluid phase transition temperature T c , due to the so-called pseudogap phenomenon. In the strong-coupling BEC side, we find that, within the neglect of the vertex corrections, one cannot correctly describe η. We also show that η/ s decreases with increasing the interaction strength, to become very close to the KSS bound, ħ/ 4 πk B , on the BEC side.

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Pseudogap Phenomena Near the BKT Transition of a Two-Dimensional Ultracold Fermi Gas in the Crossover Region

Matsumoto M., Hanai R., Inotani D., Ohashi Y.

Journal of Low Temperature Physics （Journal of Low Temperature Physics）  187 （ 5-6 ） 668 - 676 2017年06月

研究論文（国際会議プロシーディングス）, 共著, 査読有り,  ISSN  00222291

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© 2016, Springer Science+Business Media New York. We investigate strong-coupling properties of a two-dimensional ultracold Fermi gas in the normal phase. In the three-dimensional case, it has been shown that the so-called pseudogap phenomena can be well described by a (non-self-consistent) T-matrix approximation (TMA). In the two-dimensional case, while this strong-coupling theory can explain the pseudogap phenomenon in the strong-coupling regime, it unphysically gives large pseudogap size in the crossover region, as well as in the weak-coupling regime. We show that this difficulty can be overcome when one improves TMA to include higher-order pairing fluctuations within the framework of a self-consistent T-matrix approximation (SCTMA). The essence of this improvement is also explained. Since the observation of the BKT transition has recently been reported in a two-dimensional 6Li Fermi gas, our results would be useful for the study of strong-coupling physics associated with this quasi-long-range order.

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Local photoemission spectra and effects of spatial inhomogeneity in the BCS-BEC-crossover regime of a trapped ultracold Fermi gas

Ota M., Tajima H., Hanai R., Inotani D., Ohashi Y.

Physical Review A （Physical Review A）  95 （ 5 ） 053623(1-9) 2017年05月

研究論文（学術雑誌）, 共著, 最終著者, 査読有り,  ISSN  24699926

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© 2017 American Physical Society. We theoretically investigate single-particle excitations in the BCS-BEC-crossover regime of an ultracold Fermi gas. Including strong pairing fluctuations within a T-matrix approximation, as well as effects of a harmonic trap potential in the local-density approximation, we calculate the local photoemission spectrum in the normal state. Recently, the JILA group measured this quantity in a K40 Fermi gas in order to examine homogeneous single-particle properties of this system. Comparing our results with this experiment, we show that this attempt indeed succeeds under JILA's experimental condition. However, we also find that the current local photoemission spectroscopy still has room for improvement in order to examine the pseudogap phenomenon predicted in the BCS-BEC-crossover region. Since ultracold Fermi gases are always in a trap, our results would be useful in applying this system to various homogeneous Fermi systems as a quantum simulator.

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Appropriate conditions to realize a p -wave superfluid state starting from a spin-orbit-coupled s -wave superfluid Fermi gas

Yamaguchi T., Inotani D., Ohashi Y.

Physical Review A （Physical Review A）  95 （ 5 ） 053609(1-12) 2017年05月

研究論文（学術雑誌）, 共著, 最終著者, 査読有り,  ISSN  24699926

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© 2017 American Physical Society. We theoretically investigate a spin-orbit-coupled s-wave superfluid Fermi gas, to examine the time evolution of the system, after an s-wave pairing interaction is replaced by a p-wave one at t=0. In our recent paper [T. Yamaguchi, D. Inotani, and Y. Ohashi, J. Phys. Soc. Jpn. 86, 013001 (2017)JUPSAU0031-901510.7566/JPSJ.86.013001], we proposed that this manipulation may realize a p-wave superfluid Fermi gas because the p-wave pair amplitude that is induced in the s-wave superfluid state by a parity-broken antisymmetric spin-orbit interaction gives a nonvanishing p-wave superfluid order parameter, immediately after the p-wave interaction is turned on. In this paper, using a time-dependent Bogoliubov-de Gennes theory, we assess this idea under various conditions with respect to the s-wave and p-wave interaction strengths, as well as the spin-orbit coupling strength. From these, we clarify that the momentum distribution of Fermi atoms in the initial s-wave state (t<0) is a key to produce a large p-wave superfluid order parameter. Since the realization of a p-wave superfluid state is one of the most exciting and difficult challenges in cold Fermi gas physics, our results may provide a possible way to accomplish this.

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Specific heat and effects of uniaxial anisotropy of a p-wave pairing interaction in a strongly interacting ultracold fermi gas

Inotani D., Wyk P., Ohashi Y.

Journal of the Physical Society of Japan （Journal of the Physical Society of Japan）  86 （ 4 ）  2017年04月

研究論文（学術雑誌）, 共著, 最終著者, 査読有り,  ISSN  00319015

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We investigate the specific heat C at constant volume and effects of uniaxial anisotropy of a p-wave attractive interaction in the normal state of an ultracold Fermi gas. Within the framework of the strong-coupling theory developed by Nozières and Schmitt-Rink, we evaluate this thermodynamic quantity as a function of temperature, in the whole interaction regime. While the uniaxial anisotropy is not crucial for C in the weak-coupling regime, C is found to be sensitive to the uniaxial anisotropy in the strong-coupling regime. This originates from the population imbalance among pi-wave molecules (i = x; y; z), indicating that the specific heat is a useful observable to see which kinds of p-wave molecules dominantly exist in the strong-coupling regime when the p-wave interaction has uniaxial anisotropy. Using this strong point, we classify the strong-coupling regime into some characteristic regions. Since a p-wave pairing interaction with uniaxial anisotropy has been discovered in a K Fermi gas, our results would be useful in considering strong-coupling properties of a p-wave interacting Fermi gas, when the interaction is uniaxially anisotropic. V V V 40

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Strong-coupling corrections to ground-state properties of a superfluid Fermi gas

Tajima H., Van Wyk P., Hanai R., Kagamihara D., Inotani D., Horikoshi M., Ohashi Y.

Physical Review A （Physical Review A）  95 （ 4 ） 043625(1-5) 2017年04月

研究論文（学術雑誌）, 共著, 最終著者, 査読有り,  ISSN  24699926

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© 2017 American Physical Society. We theoretically present an economical and convenient way to study ground-state properties of a strongly interacting superfluid Fermi gas. Our strategy is that complicated strong-coupling calculations are used only to evaluate quantum fluctuation corrections to the chemical potential μ. Then, without any further strong-coupling calculations, we calculate the compressibility, sound velocity, internal energy, pressure, and Tan's contact, from the calculated μ without loss of accuracy, by using exact thermodynamic identities. Using a recent precise measurement of μ in a superfluid Li6 Fermi gas, we show that an extended T-matrix approximation (ETMA) is suitable for our purpose, especially in the BCS-unitary regime, where our results indicate that many-body corrections are dominated by superfluid fluctuations. Since precise determinations of physical quantities are not always easy in cold Fermi gas physics, our approach would greatly reduce experimental and theoretical efforts toward the understanding of ground-state properties of this strongly interacting Fermi system.

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Strong-coupling properties of a p-wave interacting fermi gas on the viewpoint of specific heat at constant volume

Inotani D., Van Wyk P., Ohashi Y.

Journal of the Physical Society of Japan （Journal of the Physical Society of Japan）  86 （ 2 ） 024302(1-6) 2017年02月

研究論文（学術雑誌）, 共著, 最終著者, 査読有り,  ISSN  00319015

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©2017 The Physical Society of Japan. We theoretically investigate the specific heat CV at constant volume in the normal state of a p-wave interacting Fermi gas. Including fluctuations in the p-wave Cooper channel within the framework of the strong-coupling theory developed by Nozières and Schmitt-Rink, we clarify how CV as a function of temperature varies, as one moves from the weak-coupling regime to the strong-coupling limit. In the weak-coupling regime, CV is shown to be enhanced by p-wave pairing fluctuations, near the superfluid phase transition temperature Tc. Similar enhancement of CV(T ≃ Tc) is also obtained in the strong-coupling regime, which, however, reflects that system is close an ideal Bose gas of p-wave two-body bound molecules. Using these results, we classify the normal state into (1) the normal Fermi gas regime, (2) the p-wave molecular Bose gas regime, and (3) the region between the two, where p-wave pairing fluctuations are dominant. Since the current experiments can only access the normal phase of a p-wave interacting Fermi gas, our results would be useful for experiments to understand strong-coupling properties of this Fermi system above Tc.

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A novel route to reach a p-wave Superfluid Fermi gas

Yamaguchi T., Inotani D., Ohashi Y.

Journal of the Physical Society of Japan （Journal of the Physical Society of Japan）  86 （ 1 ） 013001(1-5) 2017年01月

研究論文（学術雑誌）, 共著, 最終著者, 査読有り,  ISSN  00319015

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© 2017 The Physical Society of Japan. We theoretically propose an idea to realize a p-wave superfluid Fermi gas. To overcome the experimental difficulty that a p-wave pairing interaction to form p-wave Cooper pairs damages the system before the condensation growth, we first prepare a p-wave pair amplitude (Φ p ) in a spin-orbit coupled s-wave superfluid Fermi gas, without any p-wave interaction. Then, by suddenly changing the s-wave interaction with a p-wave one (U p ) by using a Feshbach resonance, we reach the p-wave superfluid phase with the p-wave superfluid order parameter being symbolically written as Δ p ∼ U p Φ p . In this letter, we assess this scenario within the framework of a time-dependent Bogoliubov-de Gennes theory. Our results would contribute to the study toward the realization of unconventional pairing states in an ultracold Fermi gas.

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Dip-hump temperature dependence of specific heat and effects of pairing fluctuations in the weak-coupling side of a P-wave interacting fermi gas

Inotani D., Van Wyk P., Ohashi Y.

Journal of the Physical Society of Japan （Journal of the Physical Society of Japan）  85 （ 12 ） 123301(1-5) 2016年12月

研究論文（学術雑誌）, 共著, 最終著者, 査読有り,  ISSN  00319015

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©2016 The Physical Society of Japan. We investigate the specific heat CV at constant volume in the normal state of a p-wave interacting Fermi gas. Including p-wave pairing fluctuations within the strong-coupling theory developed by Nozières and Schmitt-Rink, we show that, in the weak-coupling side, CV exhibits a dip-hump behavior as a function of the temperature. While the dip is associated with the pseudogap phenomenon near Tc, the hump structure is found to come from the suppression of Fermi quasiparticle scattering into a p-wave molecular state in the Fermi degenerate regime. Since the latter phenomenon does not occur in the ordinary s-wave interacting Fermi gas, it may be viewed as a characteristic phenomenon associated with a p-wave pairing interaction.

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Triplet pair amplitude in a trapped s-wave superfluid Fermi gas with broken spin rotation symmetry. II. Three-dimensional continuum case

Inotani D., Hanai R., Ohashi Y.

Physical Review A （Physical Review A）  94 （ 4 ） 043632(1-9) 2016年10月

研究論文（学術雑誌）, 共著, 最終著者, 査読有り,  ISSN  24699926

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© 2016 American Physical Society. We extend our recent work [Y. Endo, Phys. Rev. A 92, 023610 (2015)]PLRAAN1050-294710.1103/PhysRevA.92.023610 for a parity-mixing effect in a model of two-dimensional lattice fermions to a realistic three-dimensional ultracold Fermi gas. Including effects of broken local spatial inversion symmetry by a trap potential within the framework of the real-space Bogoliubov-de Gennes theory at T=0, we point out that an odd-parity p-wave Cooper-pair amplitude is expected to have already been realized in previous experiments on an (even-parity) s-wave superfluid Fermi gas with spin imbalance. This indicates that when one suddenly changes the s-wave pairing interaction to an appropriate p-wave one by using a Feshbach technique in this case, a nonvanishing p-wave superfluid order parameter is immediately obtained, which is given by the product of the p-wave interaction and the p-wave pair amplitude that has already been induced in the spin-imbalanced s-wave superfluid Fermi gas. Thus, by definition, the system is in the p-wave superfluid state, at least just after this manipulation. Since the achievement of a p-wave superfluid state is one of the most exciting challenges in cold Fermi gas physics, our results may provide an alternative approach to this unconventional pairing state. In addition, since the parity-mixing effect cannot be explained as far as one deals with a trap potential in the local density approximation (LDA), it is considered as a crucial example which requires us to go beyond the LDA.

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Rashbon Bound States Associated with a Spherical Spin–Orbit Coupling in an Ultracold Fermi Gas with an s-Wave Interaction

Yamaguchi T., Inotani D., Ohashi Y.

Journal of Low Temperature Physics （Journal of Low Temperature Physics）  183 （ 3-4 ） 161 - 168 2016年05月

研究論文（国際会議プロシーディングス）, 共著, 査読有り,  ISSN  00222291

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© 2016, Springer Science+Business Media New York. We investigate the formation of rashbon bound states and strong-coupling effects in an ultracold Fermi gas with a spherical spin–orbit interaction, (Formula presented.) (where (Formula presented.) are Pauli matrices). Extending the strong-coupling theory developed by Nozières and Schmitt-Rink (NSR) to include this spin–orbit coupling, we determine the superfluid phase transition temperature (Formula presented.) , as functions of the strength of a pairing interaction (Formula presented.) , as well as the spin–orbit coupling strength (Formula presented.). Evaluating poles of the NSR particle–particle scattering matrix describing fluctuations in the Cooper channel, we clarify the region where rashbon bound states dominate the superfluid phase transition in the (Formula presented.) – (Formula presented.) phase diagram. Since the antisymmetric spin–orbit interaction (Formula presented.) breaks the inversion symmetry of the system, rashbon bound states naturally have not only a spin-singlet and even-parity symmetry, but also a spin-triplet and odd-parity symmetry. Thus, our results would be also useful for the study of this parity-mixing effect in the BCS–BEC crossover regime of a spin–orbit coupled Fermi gas.

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Momentum Distribution of Cooper Pairs and Strong-Coupling Effects in a Two-Dimensional Fermi Gas Near the Berezinskii–Kosterlitz–Thouless Transition

Matsumoto M., Inotani D., Ohashi Y.

Journal of Low Temperature Physics （Journal of Low Temperature Physics）  183 （ 3-4 ） 199 - 207 2016年05月

研究論文（国際会議プロシーディングス）, 共著, 最終著者, 査読有り,  ISSN  00222291

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© 2015, Springer Science+Business Media New York. We investigate strong-coupling properties of a two-dimensional ultracold Fermi gas in the normal state. Including pairing fluctuations within the framework of a T-matrix approximation, we calculate the distribution function (Formula presented.) of Cooper pairs in terms of the center of mass momentum (Formula presented.). In the strong-coupling regime, (Formula presented.) is shown to exhibit a remarkable increase with decreasing the temperature in the low temperature region, which agrees well with the recent experiment on a two-dimensional (Formula presented.) Li Fermi gas (Ries et al. in Phys Rev Lett 114:230401, 2015). Our result indicates that the observed remarkable increase of the number of Cooper pairs with zero center of mass momentum can be explained without assuming the Berezinskii–Kosterlitz–Thouless (BKT) transition, when one properly includes pairing fluctuations that are enhanced by the low-dimensionality of the system. Since the BKT transition is a crucial topic in two-dimensional Fermi systems, our results would be useful for the study toward the realization of this quasi-long-range order in an ultracold Fermi gas.

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Non-equilibrium Properties of a Pumped-Decaying Bose-Condensed Electron–Hole Gas in the BCS–BEC Crossover Region

Hanai R., Littlewood P., Ohashi Y.

Journal of Low Temperature Physics （Journal of Low Temperature Physics）  183 （ 3-4 ） 127 - 135 2016年05月

研究論文（国際会議プロシーディングス）, 共著, 最終著者, 査読有り,  ISSN  00222291

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© 2016, Springer Science+Business Media New York. We theoretically investigate a Bose-condensed exciton gas out of equilibrium. Within the framework of the combined BCS-Leggett strong-coupling theory with the non-equilibrium Keldysh formalism, we show how the Bose–Einstein condensation (BEC) of excitons is suppressed to eventually disappear, when the system is in the non-equilibrium steady state. The supply of electrons and holes from the bath is shown to induce quasi-particle excitations, leading to the partial occupation of the upper branch of Bogoliubov single-particle excitation spectrum. We also discuss how this quasi-particle induction is related to the suppression of exciton BEC, as well as the stability of the steady state.

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