Abdulla Rakhimov earned his M.Sc. in 1977 from Tashkent State University and completed his Ph.D. in Physics and Mathematics at the Joint Institute for Nuclear Research (JINR), Dubna, in 1983. He was awarded the Doctor of Sciences degree in 2005 by the National University of Uzbekistan. Prof. Rakhimov is currently Head Researcher of the Theoretical Physics Department at the Institute of Nuclear Physics. His research spans nonperturbative methods in quantum field theory, phase transitions in quantum magnets, superconductivity, Bose–Einstein condensates, and the application of path integral techniques in condensed matter physics. He has co-authored over 60 publications in peer-reviewed journals and has collaborated with numerous international institutions. Notable recent works include publications in Physical Review A, Journal of Physics: Condensed Matter, and Annals of Physics. Shukhrat Mardonov is Head of the Research Department at New Uzbekistan University. In 2012, he received a Ph.D. scholarship through the European Union–funded Erasmus Mundus program and completed his doctorate at the University of the Basque Country, Bilbao, in 2015. His research interests focus on the thermodynamics and dynamics of Bose–Einstein condensates, including triplon condensation and the optical properties of quantum gases. Prof. Mardonov has authored several peer-reviewed articles in journals indexed by Scopus and Web of Science, and contributed a chapter titled "Effects of Anomalous Velocity in Spin-Orbit Coupled Systems" to Spin Orbitronics and Topological Properties of Nanostructures (World Scientific Publishing, Lecture Notes of the 12th International School on Theoretical Physics).

• Preface.- Introduction.- Quantum fluids.- A system of noninteracting particles ideal gas.- Bec of interacting particles.- Experiments in a nutshell.- Path integrals for bosons.- Optimized perturbation theory.- Bec in optimal perturbation theory.- Tans contact in mean field theories.- The condensate phase and its relation to the anomalous density.- Two component bose mixture.- Quantum liquid droplets in two component bose mixtures.- Bose einstein condensation of triplons in quantum magnets.- Thermodynamics of dimerized quantum magnets.- Magnetocaloric effect and grueneisen parameter of quantum magnets with spin gap.- Weak anisotropy in spin gapped quantum magnets.- Optimized perturbation theory in δ2 order.- Appendices.

Abdulla Rakhimov earned his M.Sc. in 1977 from Tashkent State University and completed his Ph.D. in Physics and Mathematics at the Joint Institute for Nuclear Research (JINR), Dubna, in 1983. He was awarded the Doctor of Sciences degree in 2005 by the National University of Uzbekistan. Prof. Rakhimov is currently Head Researcher of the Theoretical Physics Department at the Institute of Nuclear Physics. His research spans nonperturbative methods in quantum field theory, phase transitions in quantum magnets, superconductivity, Bose–Einstein condensates, and the application of path integral techniques in condensed matter physics. He has co-authored over 60 publications in peer-reviewed journals and has collaborated with numerous international institutions. Notable recent works include publications in Physical Review A, Journal of Physics: Condensed Matter, and Annals of Physics. Shukhrat Mardonov is Head of the Research Department at New Uzbekistan University. In 2012, he received a Ph.D. scholarship through the European Union–funded Erasmus Mundus program and completed his doctorate at the University of the Basque Country, Bilbao, in 2015. His research interests focus on the thermodynamics and dynamics of Bose–Einstein condensates, including triplon condensation and the optical properties of quantum gases. Prof. Mardonov has authored several peer-reviewed articles in journals indexed by Scopus and Web of Science, and contributed a chapter titled "Effects of Anomalous Velocity in Spin-Orbit Coupled Systems" to Spin Orbitronics and Topological Properties of Nanostructures (World Scientific Publishing, Lecture Notes of the 12th International School on Theoretical Physics).

• Preface.- Introduction.- Quantum fluids.- A system of noninteracting particles ideal gas.- Bec of interacting particles.- Experiments in a nutshell.- Path integrals for bosons.- Optimized perturbation theory.- Bec in optimal perturbation theory.- Tans contact in mean field theories.- The condensate phase and its relation to the anomalous density.- Two component bose mixture.- Quantum liquid droplets in two component bose mixtures.- Bose einstein condensation of triplons in quantum magnets.- Thermodynamics of dimerized quantum magnets.- Magnetocaloric effect and grueneisen parameter of quantum magnets with spin gap.- Weak anisotropy in spin gapped quantum magnets.- Optimized perturbation theory in δ2 order.- Appendices.