In 1973 Swiss physicist Klaus Hepp and American physicist Eliot Lieb rigorously proved the existence of second-order phase transition in the celebrated Jeynes-Cummings model describing a resonance interaction of atoms with the quantized electromagnetic field. For sufficiently high density and large transition dipole moment, below a critical temperature, a state with spontaneously generated photons appears in thermal equilibrium. In 1875, with Krzysztof Wódkiewicz and Władysław Żakowicz, we have shown that the superradiant phase transition is spurious and does not exist in a gauge invariance Hamiltonian. Three years later, in my only joint paper with IBB, we have generalized the no-go theorem for the superradiant phase transition from a two-level atom to a full, realistic multi-level system. For several decades, despite many attempts, our negative results were holding. Surprisingly, recently Tilman Esslinger from ETH Zurich, found a way out. Without contradicting our no-go theorem, Esslinger found a dynamical system, mathematically equivalent to the original JC Hamiltonian, and demonstrated the analog of the superradiant phase transition. This important experiment is yet another demonstration of the value of the physics of quantum gases.