### Photon gas with hyperbolic dispersion relations - INSPIRE-HEP

The hyperbolic dispersion results in an extremely large optical density of states that . When a periodic dielectric gap is placed within an HMM, the photonic band Therefore, the relationship between disorder-induced high wavevector mode . Next, using an Oxford P with SF6 and C4F8 gases, the exposed silicon was. In this paper, we will also visit the thermodynamics of photon gas with this deformed dispersion relation, but with maximum energy bound as. We study the effect of the resulting hyperbolic dispersion relations on the black body spectral density. We show that for both of the possible wavevector space.

The second method involves the tuning of the QED vacuum, achieved through cavities, waveguides and photonic bandgaps 141516 Up to now, these electrodynamic methods have relied on resonant effects that require large quality factors along with extensive nanofabrication steps.

## Super-Coulombic atom–atom interactions in hyperbolic media

It is an open question, however, whether there exists alternative non-resonant techniques for controlling DDIs that would be robust to broad spectral lineshapes of atoms or molecules with possible room temperature applications.

Here we present work related to this new avenue of research. In this study, we reveal a class of divergent excited-state atom—atom interactions that can occur in natural and artificial media with hyperbolic dispersion.

Unlike the above mentioned approaches, which engineer radiative coupling, we show that the homogeneous hyperbolic medium itself fundamentally alters the Coulombic near-field. We show that this interaction affects the entire landscape of real photon and virtual photon phenomena such as the cooperative Lamb shift CLSthe cooperative decay rate CDRresonance energy transfer rates and frequency shifts, as well as resonant interatomic forces.

### Super-Coulombic atom–atom interactions in hyperbolic media

Although we find that the singularity is curtailed by material absorption, it still allows for interactions with much larger magnitudes and longer ranges than those found in any conventional media. We also show that atoms in a hyperbolic medium will exhibit a strong orientational dependence that can effectively switch the dipolar interaction off or on, providing an additional degree of freedom to control DDI. Our investigation reveals a marked contrast between ground-state and excited-state interactions which can be used to distinguish the super-Coulombic effect in experiment.

Finally, we provide a unified perspective for controlling DDIs on multiple experimental platforms for hyperbolic media including plasmonic super-lattices, hyperbolic metasurfaces and natural hyperbolic media such as hexagonal boron nitride h-BN. We emphasize that the materials platform we introduce in this study, to enhance DDIs, is fundamentally different from the cavity QED 1819 or waveguide QED regimes 582021 see Supplementary Table 1.

We do not rely on atom confinement 256719cavity resonances or modal effects such as the quasi transverse electromagnetic TEM mode in circuit QED 8the band-edge slow light as in PhC waveguides 5622the low-mode volume of plasmonic waveguides 2123 or the infinite phase velocity at the cutoff frequency of epsilon-near-zero ENZ waveguides 14 A remarkable similarity is noticed between the results of this Lorentz violating model and another recently reported Lorentz violating model [ 35 ].

Partition Function The expression of the partition function in usual statistical mechanics for massless bosons in grand canonical ensemble can be obtained [ 36 ]: Now, changing the sum to integral, we find out In [ 1 ], Magueijo and Smolin proposed a DSR model where the dispersion relation for a massive particle takes the form For massless photon gas, 6 takes the usual form.

Take a look at [ 3 ] to see how the phase space volume remains unaltered in this model. One might also wonder whether phase space is invariant under canonical transformation. In case of usual special relativity SRthe Hamiltonian equation for massless particles is. Now, of course, any canonical transformation must keep the form of this equation invariant. And it is known that these canonical transformations also keep the usual phase space volume invariant.

In case of model 1, the dispersion relation changes due to the existence of the Planck scale [see 1 ]. Now, if we only consider the massless case of model 1, one can see then that the dispersion relation is nothing but 1 just like the usual SR scenario. Therefore, in the massless scenario, the dispersion relation of model 1 reduces to the usual special relativity.

And we already know that these canonical transformations not only keep the Hamiltonian equation invariant, but also keep the phase space volume invariant. Therefore, in this case, it is enough to take the usual phase space volume as phase space volume for the massless case of model 1.

But a point to note is that it is only true for the massless case. In case of massive particles, the dispersion relation Hamiltonian equation changes [see 1 ] and therefore the canonical transformation changes compared to the usual SR.

As a result, the phase space volume also changes. But as we are dealing only with the massless scenarios, the phase space volume mentioned in here is justified. Now, due to the presence of an upper bound in energy, in this model, the partition function becomes Here, is the total volume of the system.

Equation 7 can be written as where the function is defined in the Appendix. In order to calculate 7one can choose. As a result, the upper limit of integration should also be changed accordingly.