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turned off. This can be achieved by increasing the po- tential between the two condensates. The coarse-grained Hamiltonian describing the relative phase φ$_{i}$ of the two superfluids corresponds to an XY model, to which we add a hopping term to describe the phase-locking in the initial state: 

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where Ω$_{0}$ is an overall (Josephson) energy scale, κ de- scribes the ratio of kinetic and potential energies. We can formally replace these parameters by Ω$_{0}$ κ/π = 2 Jn , π Ω$_{0}$ /κ = U (so that Ω$_{0}$ = √$_{2}$$_{JnU}$$_{,}$ κ = π √ 2 Jn/U ) and V ( t ) = 2 J$_{⊥}$ ( t ) n/ Ω$_{0}$, which gives a representation of two coupled Bose-Hubbard systems in the quantum rotor limit $^{33}$. In this limit the Bose operators are replaced by the phase-density representation and the fluctuations of density are assumed to be small. In the Bose-Hubbard model J is the in-plane hopping amplitude, U is the on- site interaction energy, n is the filling number, i.e. the number of particles per site, and J$_{⊥}$ is the inter-layer hopping amplitude J$_{⊥}$ . This representation gives at best a qualitative idea of how the model parameters relate to the parameters in experiment, but gives a more in- tuitive picture. We note that one can think about the continuum limit as discrete, where the lattice constant is approximately given by the zero-temperature healing 

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length in the system, i.e. the length over which density fluctuations are suppressed. 

We emphasize that despite the BKT transition being classical in origin, i.e. driven by thermal fluctuations, the mechanism of vortex or phonon creation in the process we consider comes from quantum fluctuations. Indeed when the superfluids are strongly coupled together the density (which plays the role of momentum conjugate of the phase) strongly fluctuates because of the zero point motion. The heating mechanism of this system can be thought of as enhancement of this zero point motion fol- lowing the quench. 

It is convenient to introduce the rescaled quantities ˜ t = Ω$_{0}$ t/ ℏ , ˜ φ = √ κ $_{π}$φ , and ˜ n = √ π $_{κ}$n . In terms of these, the classical equations of motion (EOMs) are 

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where we defined β = √ 2 π/κ . The indices j$_{i}$ describe the four neighboring sites of site i . 

