Cooling and Autonomous Feedback in a Bose-Hubbard Chain with Attractive Interactions

We engineer a quantum bath that enables entropy and energy exchange with a one-dimensional Bose-Hubbard lattice with attractive on-site interactions. We implement this in an array of three superconducting transmon qubits coupled to a single cavity mode; the transmons represent lattice sites and their excitation quanta embody bosonic particles. Our cooling protocol preserves the particle number—realizing a canonical ensemble—and also affords the efficient preparation of dark states which, due to symmetry, cannot be prepared via coherent drives on the cavity. Furthermore, by applying continuous microwave radiation, we also realize autonomous feedback to indefinitely stabilize particular eigenstates of the array.

Figure 1

(a) A schematic (not to scale) of the experimental setup described in the text. (b) Spectroscopically measured eigenfrequencies of the one- and two-particle states of the array as a function of current through the external bias coil. For a given current, the flux through the two SQUIDs in the array differs by 2.5%; 17 mA roughly corresponds to a quarter of a flux quantum. Solid lines denote measured frequencies with fits to the 1D Bose-Hubbard Hamiltonian shown as overlaid dashed lines. Red lines correspond to two-particle states; green lines are one-particle states. The inset shows raw data near the $|E_1⟩$ frequency, from which the darkness of the $|G⟩\to |E_1⟩$ transition discussed in the text becomes apparent.

Figure 2

(a) and (b) Examples of spontaneous decays from two-particle states to the global ground state via the one-particle subspace. In (a), $|F_5⟩$ decays rapidly and almost entirely to $|E_2⟩$, which then decays to $|G⟩$, while in (b), $|F_2⟩$ decays with roughly equal rates to $|E_1⟩$ and $|E_2⟩$. Overlaid black lines are obtained from fitting the decay data for all nine excited states to a single rate-equation model, as described in the Supplemental Material [39]. (c) An illustration of the natural decay pathways of the system, from the two-particle subspace on the left, through the one-particle subspace in the middle, to the zero-particle state on the right. Each black arrow represents a decay channel, with the opacity of the line indicating the rate of the transition. Darker lines indicate faster rates, as the legend shows. Also shown are representations of the eigenstate wave functions in the qubit basis. The black circle radius is proportional to the mean particle number.

Figure 3

(a) An approximate representation of the eigenstates of the array-cavity system; states in the left-hand (right-hand) column contain zero (one) cavity photons. In this picture, our cooling process can be understood qualitatively as follows (taking $|E_2⟩\to |E_1⟩$) as an example: the cooling drive at frequency ${\omega }_{\mathrm{c}}-{\mathrm{\Delta }}_c$ induces a transition from $|E_2⟩|0⟩$ to $|E_1⟩|1⟩$, where the second ket indicates the cavity photon number. The cavity state $|1⟩$ decays via photon emission, leaving the system in the state $|E_1⟩|0⟩$, as desired. (b) Example of cooling from $|E_2⟩\to |E_1⟩$ at an incident power corresponding to 1.3 drive photons in the cavity. (c) The cooling rate versus drive frequency is Lorentzian, centered around ${\omega }_{\mathrm{c}}-(E_2-E_1)$, with a linewidth roughly $\kappa$. As the drive power is increased, the Lorentzian peak shifts due to a Stark shift of the relevant transition frequency. The inset shows that in the regime where the cooling rate is much lower than $\kappa$, the rate scales linearly with incident power. (d) Example of cascaded cooling, where the system is cooled from $|F_3⟩$ to $|F_1⟩$ via the intermediate state $|F_2⟩$. (e) Connected graphs representing the measured cooling rates per photon in the linear regime for the one- (left-) and two- (right-)particle subspaces, with the width of the line indicating the rate of the corresponding transition. The dispersive shifts for the $|F_3⟩$ and $|F_4⟩$ states are almost identical, so they cannot be distinguished by our measurement. We thus do not measure cooling from $|F_4⟩$ to $|F_3⟩$.

Figure 4

Steady state population at the different stages of the persistent stabilization scheme of the two particle ground state $|F_1⟩$. The top trace shows the thermal equilibrium population, where 78% of the population is in $|G⟩$. In each subsequent trace a further drive is added: first a coherent drive $|G⟩\to |E_3⟩$ (I), then cooling from $|E_3⟩\to |E_1⟩$ (II), coherent drive from $|E_1⟩\to |F_4⟩$ (III), and finally a cooling drive from $|F_4⟩\to |F_1⟩$ (IV). At the end, 67% of the population is in the desired state, $|F_1⟩$. $|F_4⟩$ contains the bulk, 13%, of the residual population. Single-particle ($|E_1⟩$) stabilization achieves a population of 80% (data not shown).