Experimental simulation of non-equilibrium quantum piston on a programmable photonic quantum computer

Abstract

Quantum fluctuation relations provide a microscopic formulation of thermodynamics beyond equilibrium, but experimentally accessing many-body quantum work statistics remains an outstanding challenge. The quantum piston constitutes a canonical model of boundary-driven nonequilibrium dynamics, where finite-time deformation of a confining potential generates non-adiabatic transitions, dissipation and irreversibility. Here we experimentally simulate the nonequilibrium dynamics of a two-boson quantum piston on a programmable photonic quantum computer. Using two indistinguishable photons, we encode a truncated piston propagator through a quasi-unitary embedding, with an ancilla mode representing leakage into higher-energy states outside the resolved manifold. This architecture enables direct reconstruction of thermodynamic transition statistics for both expansion and compression protocols as functions of driving speed and final trap length. We observe the crossover from quasi-adiabatic to strongly non-adiabatic evolution and show that bosonic interference restructures the resulting two-particle Fock-state populations and work distributions. The measured statistics are in close agreement with theoretical predictions and satisfy the Jarzynski equality across expansion and compression protocols for cyclic driving we further quantify irreversibility through dissipated work and state overlap. Our work identifies programmable photonic quantum hardware as a powerful platform for simulating nonequilibrium quantum thermodynamics and for experimentally resolving how indistinguishability and many-body interference shape quantum work, dissipation and entropy production.

Figures (6)

• Figure 1: Quantum piston expansion and compression protocol. Identical bosons of mass $M$ are confined in a one-dimensional box of time-dependent length $\lambda(t)$. At $t=0$, the system is prepared in a thermal quantum state at temperature $T$, corresponding to a Gibbs mixture of energy eigenstates $\{|i^{\lambda_0}\rangle\}$ with Boltzmann weights. The single-particle energy levels are $E_n(\lambda)=\frac{\hbar^2\pi^2 n^2}{2m\lambda^2}, \quad n=1,2,\dots$. Then the system is detached from the bath and one boundary moves from $\lambda_0$ to $\lambda_\tau$ during time $\tau$ with constant velocity $v=(\lambda_\tau-\lambda_0)/\tau$. Expansion ($\lambda_\tau>\lambda_0$) decreases the level spacing, while compression ($\lambda_\tau<\lambda_0$) increases it. The resulting unitary evolution induces transitions between initial eigenstates $\{|i^{\lambda_0}\rangle\}$ and final eigenstates $\{|f^{\lambda_\tau}\rangle\}$. Within the two-projective-measurement scheme, quantum work is defined as $W=E_f(\lambda_\tau)-E_i(\lambda_0)$, and its distribution is determined by the thermal occupation probabilities and the transition amplitudes generated by the piston evolution.
• Figure 2: Programmable photonic implementation of the quantum piston transformation. (a) Schematic of the universal $12\times12$ Clements interferometer used to implement the quantum piston transformation. The chip is partitioned into a state-preparation unit (left), a programmable quantum piston transformation unit realizing an effective $U_{5\times5}(v,\lambda_0,\lambda_\tau)$, and a photon-number-resolving (PNR) unit (right). The highlighted five-mode region implements a quasi-unitary extension of the truncated $4\times4$ piston matrix. Each spatial mode in this region corresponds to a piston energy eigenstate: the top four physical modes represent the lowest energy levels $\{E_1,E_2,E_3,E_4\}$ (top to bottom), while the fifth mode serves as an ancilla representing all higher energy levels together ($E>E_4$). Remaining modes of the $12$-mode mesh are configured in passive routing cross ($\theta=0,2\pi$)/bar ($\theta=1\pi$) states. (b) Representative Mach--Zehnder interferometer (MZI). Left: the simplified two-mode element used symbolically in panel (a) for circuit clarity. Right: the physical MZI structure comprising two fixed 50:50 beam splitters and tunable internal ($\theta$) and external ($\phi$) phase shifters, enabling arbitrary SU(2) transformations required by the Clements decomposition. (c) Preparation of bunched two-photon Fock states (e.g., $\ket{2000}$, $\ket{0200}$, $\ket{0020}$, $\ket{0002}$) by configuring the preparation MZI as a balanced 50:50 beam splitter, corresponding to Hong-Ou-Mandel interference. (d) Preparation of antibunched two-photon Fock states (e.g., $\ket{1100}$, $\ket{1010}$, $\ket{1001}$, $\ket{0110}$, $\ket{0101}$, $\ket{0011}$) by routing photons through passive waveguide configurations prior to the piston unit.
• Figure 3: Experimental setup for the programmable quantum piston experiment implemented on the photonic quantum processor Noor-Q. Correlated photon pairs are generated via type-II spontaneous parametric down-conversion (SPDC) in a periodically poled KTP (PPKTP) crystal pumped by a 785 nm continuous-wave laser. The degenerate photons (centered near 1570 nm) are collected into single-mode fibers and injected into a 12$\times$12 universal interferometer based on the Clements architecture. Fiber polarization controllers before the chip ensure efficient coupling to the TE mode, and polarization control after the chip maximizes detector efficiency. The thermo-optic phase shifters within the interferometer are driven by low-noise current sources with active temperature stabilization. All output modes are connected to superconducting nanowire single-photon detectors (SNSPDs) housed in a 2.6 K cryostat, and coincidence events are recorded using a multi-channel Swabian Time Tagger with a 2 ns coincidence window. The inset shows the Hong–Ou–Mandel interference measured at the state-preparation stage. The coincidence data are fitted with a triangular function $y = a - b|x - x_0|$, yielding a visibility of 0.991 and a coherence width of 1.738 mm in optical delay. The delay is tuned by translating one photon path prior to chip injection. The high visibility confirms strong photon indistinguishability and temporal overlap, ensuring that the observed two-photon dynamics are governed by bosonic interference in the programmable piston transformation.
• Figure 4: Thermal two-photon state evolution and thermodynamic signatures under velocity-swept expansion and compression driving. Panel (a) shows expansion from $\lambda_0=1.0$ to $\lambda_\tau=3.0$ at $T=5.0$, and panel (b) shows compression from $\lambda_0=3.0$ to $\lambda_\tau=1.0$ at $T=0.5$. In each panel, the top-left and top-middle maps present the experimentally reconstructed and theoretically predicted two-photon output probability distributions in the Fock basis as the piston velocity is swept. Fock states are ordered along the vertical axis in ascending order of total energy, with the lowest-energy state at the bottom. The top-right plot reports the state overlap (Bhattacharyya coefficient), with the dashed reference line indicating unity. The inset shows the unitary error of the implemented $5\times5$ quasi-unitary matrices, quantifying residual non-unitarity. The lower-left block displays velocity-resolved thermal work distributions. The lower-right plot summarizes free-energy/work metrics: mean work $\langle W\rangle=\sum_W p(W)\,W$, experimental free-energy estimate $\Delta F_{\mathrm{exp}}=-T\ln\langle e^{-W/T}\rangle$ (Jarzynski estimator), and theoretical free-energy change $\Delta F_{\mathrm{th}}=-T\ln\![Z(\lambda_\tau)/Z(\lambda_0)]$ from equilibrium partition functions. In the velocity-sweep analysis, $\Delta F_{\mathrm{exp}}$ is additionally shown with a horizontal fit, while $\langle W\rangle$ is fitted by a quadratic function of $|v|$ (labeled Quadratic fit), capturing the increasing work free-energy gap with driving speed. $\Delta F_{\mathrm{th}}$ is shown as a dashed connected trend. For each branch, the temperature is chosen so that at $\lambda_0$ the probability that both bosons occupy the first four energy levels exceeds $95\%$, consistent with the four-level truncation used in the analysis. All quantities are reported in natural units with $k_B=1$, boson mass $M=1$, and $\hbar=1$.
• Figure 5: Thermal two-photon state evolution and thermodynamic metrics versus final trap parameter. Panel (a) shows the $\lambda$-sweep expansion branch with fixed velocity $v=1.1~\mathrm{mm/s}$, $\lambda_0=1.0$, and $T=5.0$, while panel (b) shows the $\lambda$-sweep compression branch with fixed velocity $v=-0.7~\mathrm{mm/s}$, $\lambda_0=5.0$, and $T=0.3$. In each panel, the top-left and top-middle heatmaps are the experimental and theoretical two-photon output probabilities in the Fock basis as $\lambda_\tau$ is varied. Fock states are ordered along the vertical axis in ascending order of total energy, with the lowest-energy state at the bottom. The top-right plot gives the experimental--theoretical state overlap (Bhattacharyya coefficient), with the dashed line indicating perfect fidelity. Insets report the unitary error of the implemented $5\times5$ quasi-unitary matrices, quantifying residual non-unitarity associated with representing leakage from the truncated four-level subspace with a single ancilla mode. The lower-left block shows the corresponding work distributions at representative $\lambda_\tau$ points, and the lower-right plot summarizes $\langle W\rangle=\sum_W p(W)\,W$, $\Delta F_{\mathrm{exp}}=-T\ln\!\langle e^{-W/T}\rangle$ (Jarzynski estimator), and $\Delta F_{\mathrm{th}}=-T\ln\![Z(\lambda_\tau)/Z(\lambda_0)]$. Temperatures are chosen so that, at $\lambda_0$, the probability that both bosons occupy the first four energy levels exceeds $95\%$, ensuring consistency with the four-level truncation. All quantities are in natural units with $k_B=1$, boson mass $M=1$, and $\hbar=1$.