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I have a question about the dynamical passage from superfluid to Mott insulator state in the Bose-Hubbard model. Is it possible to go from superfluid region to the Mott insulator by changing the lattice depth adiabatically? Adiabatically means that the rate of change is slow enough that the state adjust itself to the ground state. Although there is experiment by Greiner et al. and theoretical analysis by Kashurnikov et al. or Zakrzewski, it doesn't talk to me. I would imagine that due to different phases, rate of change would grow to infinity close to the transition point.

Moreover, if you look at the BHM phase diagram you can choose many paths in the Mott insulator state because mean number of particles is everywhere constant.

enter image description here

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It is indeed possible to change between these phases adiabatically. Since, as you noted, the ground state changes between being a superfluid and a Mott insulator, starting in the ground state and making an adiabatic change means that you track that change in state by definition.

Note that this diagram is only formally true for the Grand Canonical ensemble, which is necessary to have a well-defined $\mu$. So the path that you take in a phase transition is uniquely set by what $\mu$ is, which is a physical parameter in your system. So there is no ambiguity about which path into the MI region the system takes.

One reason you might be confused is that experiments are often in a number-conserving setting, in which case this plot does not apply. In particular, if you have a uniform system with a non-integer occupancy per site and the number cannot change, you will clearly never get a Mott insulator. The experiments get around this by having a non-uniform system, in which case part of it can be a Mott insulator.

Edit (3/10/2018): This answer is not correct. When crossing the phase boundary between a superfluid and Mott insulator, there will always be some amount of excitation that is determined by the rate of change. This is the quantum version of the Kibble-Zurek mechanism.

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