Bose condensation of squeezed light

Light with a chemical potential and no mass is shown to possess a general phase-transition curve to Bose-Einstein condensation. This limiting density and temperature range is found by the diverging in-medium potential range of effective interaction. The inverse expansion series of the effective interaction from the Bethe-Salpeter equation is employed, exceeding the ladder approximation. While usually the absorption and emission with dye molecules is considered, here it is proposed that squeezing can also create such a mean interaction leading to a chemical potential. The equivalence of squeezed light with a complex Bogoliubov transformation of an interacting Bose system with finite lifetime is established, with the help of which an effective gap is deduced where the squeezing parameter is related to an equivalent gap by $\left|\mathrm{\Delta }\right(\omega \left)\right|=\hslash \omega /(coth2\left|z\right(\omega \left)\right|-1)$. This gap phase creates a finite condensate in agreement with the general limiting density and temperature range. In this sense, it is shown that squeezing induces the same effect on light as an interaction, leading to possible condensation. The phase diagram for condensation is presented due to squeezing and the appearance of two gaps is discussed.

Figure 1

The critical density times interaction range versus critical temperature for 2d-14th order in the expansion $\sigma \left(y_0\right)$ (right side from below to above). The simple parametric plot $\{x\left(\sigma \right),y_0\left(\sigma \right)\}$ is shown as a continuously increasing thin black line to the upper right. Below $T_{c1}$ and $n_{c1}$, the linear dependence Eq. (2) holds. The scaling is Eq. (15).

Figure 2

The frequency dependence of the squeezing parameter Eq. (17) with scaling Eq. (15).

Figure 3

The frequency dependence of the equivalent gap Eq. (30) with ${\mathrm{\Delta }}_0=\mathrm{\Delta }\left(0\right)$ with scaling Eq. (15).

Figure 4

The quasiparticle energy Eq. (32) for different phases of the squeezing parameter Eq. (17) with scaling Eq. (15).

Figure 5

Above: The ratio of quasiparticle energy (solid line) according to Eq. (28) and its inverse lifetime ${\epsilon }_{0i}=\hslash /\tau$ (dashed), according to Eqs. (26) to the frequency versus frequency which is the free quasiparticle energy for various phases. The scale is Eq. (15). Below: The interaction energy and its inverse lifetime according to Eqs. (26).

Figure 6

The region where the total density and the condensate density are nonzero versus gap and temperature and scaling Eq. (15).

Figure 7

The gap versus condensate density for different temperatures in units of Eq. (15).

Figure 8

The gap versus total density for different temperatures in units of Eq. (15).

Figure 9

The gap parameter versus temperature for various densities. The thick dashed line gives the gap where the minimal density appears, see Fig. 8. Above the density $n_{c1}$, two gaps appear for one temperature and or densities. The thin dashed line gives the limiting curve above which we have a finite condensate density according to Fig. 6.

Figure 10

The phase diagram in the density versus temperature plot in units of Eq. (15). The region where the gap is nonzero (shaded) includes the region where the condensate density is nonzero (dark gray), which lies above the straight borderline. The curve between the two right dots are the minima of the density with respect to the gap of Fig. 8. In this range, the straight line slightly above the borderline would be the line of zero gap in Fig. 8. For temperatures higher than $T_{c1}$, two gaps appear. The critical curve from Fig. 1 is given by the thick line for comparison. It starts exactly at $\{T_{c1},n_{c1}\}$ and is observed to end at $\{0,n_{c2}\}$.