Phase diagrams of tunable spin-orbit coupled Bose-Einstein condensates

We analytically calculate phase boundaries of tunable spin-orbit coupled BECs with effective two-body interactions by using variational method. Phase diagrams for periodically driving $^{87}\rm{Rb}$ and $^{23}\rm{Na}$ systems are presented, respectively, which display several characteristic features contrast with those of undriven systems. In the $^{87}\rm{Rb}$ BECs, the critical density $n_{c}$ (density at quantum tricritical point) can be dramatically reduced in some parameter regions, thus the prospect of observing this intriguing quantum tricritical point is greatly enlarged. Moreover, a series of quantum tricritical points emerge quasi-periodically with increasing the Raman coupling strength for fixed $^{87}\rm{Rb}$ density. In the $^{23}\rm{Na}$ BECs, two hyperfine states of $^{23}\rm{Na}$ atoms can be miscible due to driving under proper parameters. As a result, $^{23}\rm{Na}$ systems can stay in the stripe phase with small Raman frequency at typical density. This expands the region of stripe phase in the phase diagram. In addition, there is no quantum tricritical point in such $^{23}\rm{Na}$ system , which is different from $^{87}\rm{Rb}$ system.