Experimental Realization of Thermal Reservoirs with Tunable Temperature in a Trapped-Ion Spin-Boson Simulator

Abstract

We propose and demonstrate an experimental scheme to engineer thermal baths with independently tunable temperatures and dissipation rates for the motional modes of a trapped-ion system. This approach enables robust thermal-state preparation and quantum simulations of open-system dynamics in bosonic and spin-boson models at well-controlled finite temperatures. We benchmark our protocol by experimentally realizing out-of-equilibrium dynamics of a charge-transfer model at different temperatures. We observe that, when the process occurs at a higher temperature, the transfer rate spectrum broadens, with reduced rates at small donor-acceptor energy gaps and enhanced rates at large gaps. We then employ our scheme to study local-temperature effects in a two-mode vibrationally assisted exciton transfer system, where we observe thermally activated interference pathways for excitation transfer.

Figures (13)

• Figure 1: Proposed experimental setup for engineering a thermal reservoir with trapped ions. Independent control of the bath temperature and dissipation rate is achieved by simultaneously applying motional cooling and heating to the relevant mode. Trapped ions (green spheres) are addressed with cooling beams that remove phonon excitations from the collective vibrational mode (connecting springs). It is not necessary to apply cooling beams to all ions, as long as the addressed ions participate in the targeted motional mode. Concurrently, an antenna broadcasts electric-field noise at the motional-mode frequency from outside the vacuum chamber that hosts the ions to induce motional heating.
• Figure 2: Thermal bath controls. (A) Phonon-number dynamics from interacting with the engineered thermal bath. Red triangles and curves are associated with $\gamma_h=5.34\pm0.56$/ms, while blue circles and curves correspond to $\gamma_h=0.43\pm0.07$/ms. In both cases, $\gamma_c=4.03\pm 0.31$/ms. The curves are exponential functions with time constants $1/\gamma_c$ and the steady states $\braket{n(t\rightarrow\infty)}=\gamma_h/\gamma_c$. Data points with black outlines and dashed curves correspond to when the system starts from the Doppler-cooled temperature ($\braket{n(t=0)} \approx 7$), while data points with no outline and solid curves describe the realizations when the system is initialized from $\braket{n(t=0)}\approx0.1$. Light red and blue bands starting from $t=5/\gamma_c$ are the estimated $n_{\rm ss}=\gamma_h/\gamma_c$ for the two cases. Darkened data points are also used in (B) (and Fig. \ref{['Fig_steadystate_20250818']} of End Matter). (B) $n_{\rm ss}$ versus $\gamma_h$. Square data points correspond to the measured average phonon numbers after interacting with the engineered thermal bath, given by $\gamma_c = 1.00\pm0.10$/ms and $\gamma_h$, for 6 ms, while data points of other shapes are associated with the measured average phonon numbers after interacting with the engineered thermal bath, given by $\gamma_c=4.03\pm 0.31$/ms and $\gamma_h$, for 3 ms. The grey and purple bands are bounded by {$\gamma_h/(\gamma_c=1.00-0.10$/ms), $\gamma_h/(\gamma_c=1.00+0.10$/ms)} and {$\gamma_h/(\gamma_c=4.03-0.31$/ms), $\gamma_h/(\gamma_c=4.03+0.31$/ms)}, respectively. Colored data points are used in Fig. \ref{['Fig_steadystate_20250818']} of End Matter.
• Figure 3: Finite-temperature charge transfer. (A) Transfer rate versus donor-acceptor energy gap. Blue and red solid lines show the theoretical predictions of the transfer rate with $(V,\;g)=(0.2,\;1.1)\omega$ in contact with thermal reservoirs characterized by $\bar{n}=0.15$ and 0.80, respectively, at a dissipation rate of $\gamma = 0.0360\omega$. These solid lines also consider experimental imperfections with $(\gamma_z,\;\gamma_m) = (0.0014,\;0.0160)\omega$ (see Supplemental Material so2025fT_supp). Blue-filled circles and red-filled triangles represent the corresponding experimental results with error bars obtained from resampling so2024electrontransfer. (B) Donor population dynamics. Blue curves and circles show theoretical and experimental results, respectively, for the system interacting with the $\bar{n}=0.15$ reservoir, while red curves and triangles correspond to the $\bar{n}=0.80$ reservoir. Dashed curves and filled data points are for $\Delta E = 2.8\omega$, and solid curves with open data points are for $\Delta E = 6.0\omega$.
• Figure 4: Two-mode, vibrationally assisted exciton transfer at different local temperatures. (A) Transfer rate versus donor-acceptor energy gap. Blue and red solid lines show the theoretical predictions of the transfer rate with $(V,\;\omega_2,\;g_1,\;g_2)=(0.13,\;0.60,\;0.33,\;0.20)\omega_1$ in contact with local thermal reservoirs characterized by $(\bar{n}_1,\bar{n}_2)=(0.10,\;0.02)$ and $(0.10,\;0.80)$, respectively, at dissipation rates $\gamma_1 = 0.013\omega_1$ and $\gamma_2 = 0.010\omega_2$. These solid lines also consider experimental imperfections with $(\gamma_z,\;\gamma_m) = (0.0004,\;0.0040)\omega_1$ (see Supplemental Material so2025fT_supp). Blue-filled circles and red-filled triangles represent the corresponding experimental results. (B) Donor population dynamics for $\Delta E = 0.35\omega_1$. Blue curves and circles show the theoretical and experimental results, respectively, for the system interacting with the $\bar{n}_2=0.02$ bath, while red curves and triangles correspond to the $\bar{n}_2=0.80$ bath.
• Figure 5: Steady-state phonon-number distribution from the engineered thermal reservoir. (A) $\gamma_c = 4.03\pm0.31$/ms and $\gamma_h = 0.43\pm0.07$/ms for 3 ms (dark blue in Fig. \ref{['Fig_dynamics_20250818']} of the main text), (B) $\gamma_c = 4.03\pm0.31$/ms and $\gamma_h=5.34\pm0.56$/ms for 3 ms (dark red in Fig. \ref{['Fig_dynamics_20250818']} of the main text), and (C) $\gamma_c = 1.00\pm0.10$/ms and $\gamma_h=2.31\pm0.28$/ms for 6 ms (green in Fig. \ref{['Fig_dynamics_20250818']}B of the main text). Each inset corresponds to the spin dynamics of the steady-state system undergoing the probe blue-sideband drive for the phonon-number distribution measurement (black data points in the main figure). Bar charts represent the best-estimated thermal state bounded by $\gamma_h/\gamma_c$: (A) $0.09\pm0.02$, (B) $1.10\pm0.08$, and (C) $2.74\pm0.27$. Dashed red and solid blue curves in the insets describe the dynamics of the fitted phonon-number distribution and the estimated thermal state, respectively.