Determination of quantum phases via continuous measurements

We derive how the quantum phase of a complex system can be deduced from the record of a continuous and diffusive quantum measurement. We show that weak continuous probing of operators commuting with parts of the Hamiltonian defining different quantum phases lead to qualitatively distinct power spectra of the probe signals. This may be exploited to determine and define quantum phases of open systems based on the measurement record alone. This is in contrast to existing methods that use repeated destructive measurement of a single state, requiring very low entropy. We test the resulting phase criterion in a numerical simulation of the Bose-Hubbard model under two different, experimentally feasible measurements that satisfy complementary commutation relations. At low measurement strength, our criterion yields a critical point for the superfluid to Mott-Insulator transition in reasonable agreement with the quantum phase transition in the ground state of the closed system in the thermodynamic limit. At higher measurement strengths, the system's response enters a Zeno regime and becomes dominated by whichever phase is being measured.