Probing coherent quantum thermodynamics using a trapped ion

The study investigates how thermodynamic concepts must be refined in the quantum regime, where coherence, entanglement and quantum fluctuations are relevant. While microscopic thermal machines have been realized, conclusively verifying genuine quantum thermodynamic signatures beyond any classical explanation has remained challenging. The authors focus on the work fluctuation–dissipation relation (FDR), which in classical near-equilibrium settings links average dissipated work and equilibrium fluctuations. They aim to detect and quantify a coherent quantum correction to this relation—induced by noncommutativity of the Hamiltonian during driving (quantum friction)—using a single trapped-ion qubit. The core research question is whether experimentally measured work fluctuations and dissipation can be made statistically incompatible with any incoherent (classical) driving or with measurement (SPAM) errors, thus certifying a genuinely quantum thermodynamic effect beyond the slow-driving regime.

The paper situates the work within quantum thermodynamics and stochastic quantum thermodynamics. Prior studies have shown potential advantages due to coherence, addressed fluctuation theorems, and explored work statistics. Thermal machines at the single-quantum-system scale have been realized, with comparisons to classical counterparts, but unequivocal certification of genuinely quantum effects via fluctuations remains elusive. Theoretical works have predicted quantum corrections to work FDR near equilibrium and in slow processes (e.g., Miller et al., Scandi et al.). The two-point measurement (TPM) framework and its limitations have been discussed extensively in the literature, along with alternative approaches. Previous experimental verifications of fluctuation relations exist in classical or quasi-classical contexts, but a direct measurement of a quantum correction to the work FDR had not been demonstrated. This study builds on discrete-step protocols that emulate slow continuous driving while enabling practical experimental implementation, and leverages recent theory anticipating coherent corrections to FDR close to equilibrium.

Protocol and model:

• System: Single-qubit Hamiltonian H(θ) = (ħω_g/2)(sin θ σ_x − cos θ σ_z), with ħω_g set to 1 for units. H does not commute at different θ, enabling coherent effects (quantum friction). The parameter θ is varied from 0 to π/2.
• Discrete driving: Replace slow continuous driving by N equal steps Δθ = π/(2N). At each step j with θ_j = jΔθ, perform: (i) Thermalize to Gibbs state of H(θ_j): ρ_j = e^{−β H(θ_j)}/Z, Z = 1 + e^{−β}. For j=0 and computational basis, ρ_0 = |0⟩⟨0| + e^{−β}|1⟩⟨1|, with Tr(ρ_j σ_z) = 1 − 2p = tanh(β/2). (ii) Projective energy measurement in H(θ_j) basis, outcome e_j ∈ {0,1}. (iii) Coherent rotation R(Δθ) = exp(−i Δθ σ_x / 2). (iv) Projective energy measurement in H(θ_{j+1}) basis, outcome e_{j+1}. Steps (iii)-(iv) are equivalent to a quench H(θ_j) → H(θ_{j+1}) followed by an energy measurement.
• Work definition (TPM): Step work w_j = e_{j+1} − e_j (values −1, 0, +1 for a qubit). The distribution for the coherent protocol: Pr(w_j = +1) = (1 − p) sin^2(Δθ/2), Pr(w_j = −1) = p sin^2(Δθ/2), Pr(w_j = 0) = 1 − Pr(+1) − Pr(−1). Total work W = ∑_j w_j. Due to re-thermalization at each step, steps are i.i.d., effectively Markovian.
• FDR correction metric: Using the first two cumulants of W, define the deviation from the classical linear-response FDR as δ = Var(W) − ⟨W⟩ − ΔF. For these processes ΔF = 0 (basis change leaves free energy invariant). Analytical expressions and error analysis are in Supplementary Notes.

Experimental implementation:

• Platform: 40Ca+ ion in a segmented Paul trap. Qubit encoded in Zeeman sublevels of 4^2S_{1/2}: |0⟩ := |m_J = +1/2⟩, |1⟩ := |m_J = −1/2⟩; splitting ω_q = 2 × 10.5 MHz via static magnetic field.
• Initialization and thermal state prep: Optical pumping initializes |0⟩; a thermal Gibbs state in the logical basis is prepared by partial population transfer from |0⟩ to |1⟩ combined with the first TPM measurement to realize Boltzmann weights set by β through tanh(β/2) = 1 − 2p.
• Coherent control: Stimulated Raman transitions with co-propagating off-resonant beams, red-detuned by Δ_R = 2 × 250 GHz from 4^2S_{1/2} ↔ 4^2P_{1/2}. Resonant drive yields rotations R(Δθ) = exp(−i (Δθ/2) σ_x); Δθ set by beam intensity and exposure time.
• Readout: State-selective transfer from |0⟩ to 3^2D_{5/2}, then fluorescence detection at 397 nm: |1⟩ bright, |0⟩ dark. As readout is destructive, projection-valued measurement is emulated via re-initialization and conditional π pulses based on prior results.
• Data collection: For chosen N and β, perform N independent runs to sample step work values w_j: assign w_j = +1 if first readout |0⟩ and second |1⟩, w_j = −1 if first |1⟩ then |0⟩, otherwise w_j = 0. For N in {2,…,7} at β = 3.413 ± 0.025 (p = 0.032 ± 0.001), repeat protocol 8000 times per N; compute sample mean and variance of W.

Benchmarking against classical and SPAM scenarios:

• Driving speed: Define v = ||ΔH||/N, with ||ΔH|| the operator norm of the Hamiltonian change. Slow-driving regime v ≪ 1.
• Incoherent (classical) protocols: Simulate many protocols where H(t) always commutes (fixed σ_z eigenbasis) and only the energy gap changes: ||ΔH|| = (ω_f − ω)/2. For coherent protocol: ||ΔH|| = 1/√2. Compare measurements to the rescaled metric N·δ/||ΔH|| = δ/v to obtain a bound for incoherent processes over all speeds.
• SPAM errors: Account for readout errors in the second measurement of TPM with small dark/bright misclassification probability. Compute worst-case spurious correction N·δ_SPAM/||ΔH|| by performing measurements without intermediate rotations to bound SPAM-induced effects.

Temperature dependence:

• For fixed N = 5, measure N·δ/||ΔH|| versus inverse temperature β and compare with the theoretical prediction (Eq. 11) without free parameters. Error bars reflect counting statistics (8000 repetitions).

• Clear quantum correction to the classical work fluctuation–dissipation relation observed: the rescaled quantity N·δ/||ΔH|| for coherent driving tends to a constant with increasing N, in contrast to incoherent processes where N·δ_inco/||ΔH|| ∝ 1/N (i.e., δ_inco/v ∝ 1/N), consistent with validity of the classical FDR in the slow-driving regime.
• Statistical incompatibility: Experimental N·δ/||ΔH|| values lie outside the entire region attainable by any incoherent protocol (blue region in Fig. 2) by more than 10 standard deviations (>10σ, quantified as >10.9σ in Discussion) and are also more than 12σ away from SPAM-induced worst-case values (>12.1σ).
• Finite-size behavior: While ideal values are slightly reduced due to rotation pulse calibration and measurement imperfections, the measured points remain significantly above both incoherent and SPAM bounds.
• SPAM scaling: SPAM-induced correction N·δ_SPAM/||ΔH|| increases with N, implying that in the quasi-static limit (N → ∞) SPAM can mask genuine quantum corrections; the trapped-ion platform’s low readout error enables access to an intermediate-speed regime where quantum signatures are resolvable.
• Temperature dependence: For N = 5, N·δ/||ΔH|| versus β follows the theoretical prediction (Eq. 11) without free parameters across the explored range; the data remain below the SPAM-only bound.
• Experimental parameters: Demonstrations span N = 2–7, with β = 3.413 ± 0.025 (excited-state population p = 0.032 ± 0.001), with 8000 repetitions per N; ||ΔH|| = 1/√2 for the coherent protocol; for the incoherent simulations, ||ΔH|| = (ω_f − ω)/2.

The findings directly address whether observed work fluctuations in a driven qubit can be explained classically. By constructing a discrete step-wise protocol that induces coherent Hamiltonian changes (noncommuting H(θ)), and by carefully benchmarking against both incoherent driving scenarios and worst-case SPAM error models, the measured excess fluctuations are demonstrably incompatible with any classical (commuting) evolution and with measurement artifacts. This constitutes an unambiguous certification of a genuinely quantum thermodynamic effect: a coherent correction to the work FDR arising from quantum friction. The contrast in scaling behaviors—coherent N·δ/||ΔH|| approaching a constant versus incoherent scaling ∝ 1/N—highlights the fundamental role of coherence in determining near-equilibrium work statistics. The temperature-dependent measurements further corroborate theory without adjustable parameters, reinforcing the robustness of the effect beyond the strict slow-driving limit. These results validate the application of quantum stochastic thermodynamics in experiments to certify nonclassical behavior, establishing trapped ions as a precise platform for probing quantum thermodynamic relations. They open the path to exploring richer phenomena such as quantum many-body thermal machines, non-Markovian temporal coherences in work extraction, and connections to single-shot thermodynamics.

The study presents the first experimental measurement and certification of a genuine quantum correction to the classical work fluctuation–dissipation relation using a trapped-ion qubit. By implementing a discrete protocol with alternating thermalization, coherent rotations, and energy measurements, the authors measure an excess work fluctuation that is statistically incompatible with any incoherent protocol (>10.9σ) and with SPAM-induced errors (>12.1σ). The coherent correction exhibits a constant asymptote with increasing N, contrasting with the 1/N decay for incoherent processes, and matches temperature-dependent theoretical predictions without free parameters. This work demonstrates that quantum stochastic thermodynamics provides operational tools to benchmark and certify quantum effects beyond slow driving. Future research directions include probing quantum field thermal machines, exploiting temporal coherence and non-Markovian dynamics to mitigate SPAM and access slower protocols, and connecting with single-shot work extraction frameworks.

• SPAM/readout errors: Although small, misclassification in the second TPM measurement induces a spurious correction that grows with N, potentially obscuring genuine quantum effects in the quasi-static limit (N → ∞). Careful error characterization and mitigation are necessary.
• Calibration/systematics: Rotation pulse calibration and measurement imperfections cause measured excess fluctuations to fall short of ideal theoretical values, introducing systematic deviations.
• Destructive readout and re-initialization: TPM implementation requires re-initialization and conditional operations, which may introduce additional noise and complexity.
• Regime of operation: While the protocol duration is short compared to coherence times (allowing isolation assumption), accessing very slow protocols at larger N is limited by the growth of SPAM-induced contributions.
• Modeling assumptions: Comparison to incoherent protocols assumes commuting Hamiltonians with fixed eigenbasis; while broad, this bound does not encompass all conceivable classical noise models, though the statistical separation reported is substantial.