Constraining work fluctuations of non-Hermitian dynamics across the exceptional point of a superconducting qubit

Thermodynamics constrains changes to the energy of a system, both deliberate and random, via its first and second laws. When the system is not in equilibrium, fluctuation theorems such as the Jarzynski equality further restrict the distributions of deliberate work done. Such fluctuation theorems have been experimentally verified in small, non-equilibrium quantum systems undergoing unitary or decohering dynamics. Yet, their validity in systems governed by a non-Hermitian Hamiltonian has long been contentious, due to the false premise of the Hamiltonian's dual and equivalent roles in dynamics and energetics. Here we show that work fluctuations in a non-Hermitian qubit obey the Jarzynski equality even if its Hamiltonian has complex or purely imaginary eigenvalues. With post-selection on a dissipative superconducting circuit undergoing a cyclic parameter sweep, we experimentally quantify the work distribution using projective energy measurements and show that the fate of the Jarzynski equality is determined by the parity-time symmetry of, and the energetics that result from, the corresponding non-Hermitian, Floquet Hamiltonian. By distinguishing the energetics from non-Hermitian dynamics, our results provide the recipe for investigating the non-equilibrium quantum thermodynamics of such open systems.

Introduction.The concept of a small system coupled to a large reservoir is elemental to both thermodynamics and open quantum systems.In thermodynamics, a reservoir allows one to distinguish between two types of energetics: heat Q, the random energy transferred to the system from the reservoir, and work W , deliberately imparted to the system.The energy U of the system is additively changed by the two, thereby encoding the first law of quantum thermodynamics, ∆U = Q + W (Fig. 1 right inset) [1][2][3][4][5][6].Conversely, a closed quantum system is governed by a Hermitian Hamiltonian H(t), undergoes unitary evolution with zero heat exchange, and its energy is equal to the expectation value of the generator of its dynamics H(t).When coupled to a reservoir, one describes its evolution by averaging over possible, consistent micro-states of the reservoir.This averaging leads to (engineered) decoherence and dissipation [7]; the resulting dynamics are described by a Lindblad equation ∂ t ρ = Lρ for the reduced density matrix ρ(t) of the system [8].Here, the system-reservoir coupling results in trajectory-dependent heat and work that, when added together, gives trajectory-independent change in the energy of the system [9].In such cases, the internal energy operator H(t), which encodes the energy U (t) ≡ Tr[ρ(t)H(t)], is distinct from the generator L of its temporal dynamics.
In addition to the work-energy theorem, work fluctuations of a non-equilibrium system with internal energy operator H(t) are further constrained by the Jarzynski equality [10][11][12][13] Here ⟨•⟩ denotes trajectory-ensemble average, β −1 is the reservoir temperature, Z(t) ≡ Tr exp[−βH(t)] is the system partition function, and ∆F ≡ F (τ ) − F (0) is the Helmholtz free energy change in time τ .The equality (1) supersedes the Jensen inequality ∆F ≤ ⟨W ⟩ that constrains the amount of work done on a system and its free-energy change.In a quantum system with indefinite energy, a two-point-measurement (TPM) protocol quantifies changes in a system's energy ∆U [14].It entails performing a pair of projective measurements in the energy basis to quantify ∆U in terms of transition probabilities between instantaneous eigenstates of the internal energy operator H(t) (Fig. 1).These transition probabilities differ for unitary and Lindblad evolution and yet Jarzynski equality (1) holds [15] for unital quantum maps [16], as has been experimentally verified [17][18][19][20][21].
In recent years, a third model of quantum dynamics obtained by post-selecting on quantum trajectories with no quantum jumps has emerged [22,23].With a non-Hermitian generator H eff (t) = H(t) + iΓ(t) and a nonlinear, trace-preserving equation of motion [24], it maps pure states into pure states but changes the entropy of mixed states [25], thereby commingling salient features of unitary and Lindblad evolution.When the non-Hermitian H eff has a real spectrum, its role in dynamics has been conflated with energetics, leading to predicted violations of the Jarzynski equality and Crooks fluctuation theorem when the spectrum of H eff turns complex [26][27][28][29][30][31].  1. Thermodynamics of open quantum systems.The first law of thermodynamics (right inset) states that the internal energy of a system (S) coupled to a reservoir (R) is additively changed by the heat Q and the work W .The generalized second law or the Jarzynski equality (1) governs the trajectory-dependent work fluctuations.For a quantum system starting in an equilibrium density matrix ρeq, these work fluctuations are characterized by projective energy measurements leading to a discrete work distribution P (W ).We show that for cyclic parameter variations in H eff (t), the average exponentiated work is unity when the corresponding Floquet Hamiltonian H F eff has a parity-time symmetry, and its Floquet energy operator matches the system's initial energy operator.
in such dynamics?
Here, we demonstrate the Jarzynski equality in a non-Hermitian qubit undergoing cyclic parameter changes, including cases where H eff (t) has complex eigenvalues at all times.The qubit dynamics is characterized by the nonunitary , where H F is the Floquet internal energy operator.The Jarzynski equality ⟨e −βW ⟩ = 1 is satisfied when H F eff has an explicit or emergent parity-time symmetry that guarantees real or complex-conjugate eigenvalues, and H F ∝ H(0), i.e. the two internal energy operators have the same eigenbasis (see Methods).Measuring work distribution.Our experimental platform comprises a superconducting transmon circuit with energy eigenstates labeled {|g⟩, |e⟩, |f ⟩} dispersively coupled to a microwave cavity (see Methods).Bath engineering allows us to tune the radiative decay rates such that the decay rate γ e = 1.57µs −1 of state |e⟩ by quantum jumps |e⟩ → |g⟩ is much larger than the decay rate γ f = 0.21 µs −1 leading to a decay contrast γ ≡ γ e − γ f ≈ γ e .We post-select quantum trajectories with no quantum jumps to the |g⟩ state, thereby limiting the dynamics only to the excited-state subspace {|e⟩, |f ⟩}.With the addition of a drive that couples the states |e⟩ and |f ⟩ with detuning ∆(t) and rate J(t) (Fig. 2a), the evolution of this qubit subspace is described by an effective non-Hermitian Hamiltonian, where Experimental setup.(a) A non-Hermitian qubit is realized as the sub-manifold (dashed box) of the lowest three levels of a transmon circuit.The system exhibits decay from the |e⟩ state to |g⟩ at rate γe and is driven by a microwave drive with detuning ∆(t) and coupling rate J(t).(b) The experimental protocol involves preparing an initial eigenstate | + x⟩ or | − x⟩, dynamically tuning the energy operator H(t) for a certain time τ and returning it to its initial value H(τ ) = H(0), followed by post-selective quantum state tomography.(c) By sampling both initial states |j⟩ ∈ {| + x⟩, | − x⟩} and post-selecting cases where the system does not project onto the ground state, we determine the transition probabilities Pij within the excited-state manifold {|e⟩, |f ⟩}.∆(t) ≡ 0, the Hamiltonian H eff (t) commutes with the antilinear operator PT = σ x K, where K denotes complex conjugation, at all times.In the static case, this explicit PT -symmetry underlies the purely real or purely imaginary eigenvalues λ ± = ± J 2 − (γ/4) 2 of H eff with an exceptional-point (EP) degeneracy at J EP = γ/4.For sampling the work distribution, we implement three timeperiodic parameter paths, where τ is the protocol duration and J = (J max +J min )/2.When ∆(t) ̸ = 0, the instantaneous eigenvalues λ ± (t) of the non-Hermitian H eff (t) are complex and the Hamiltonian has no explicit PT -symmetry.The basis of the experiment is to determine the work distribution P (W ) after the system is driven by the cyclic internal energy operator H(t) (Fig. 2b).In our experiment, the TPM procedure consists of three steps: (i) With a sequence of resonant rotations to the transmon circuit, we initialize the system in the eigenstates 2 of the energy operator H(0) = J max σ x .A Gibbs state with inverse temperature β is then synthesized by preparing the two eigenstates with relative probabilities P ±x ∝ exp(∓βJ max ).Throughout this work we set β = 0.5 µs/rad, which corresponds to P +x = 0.98.(ii) We dynamically apply work to the qubit by tuning the parameters J(t), ∆(t) as in Eqs. ( 3)-( 5).(iii) We perform a final projective measurement in the basis {|g⟩, |+x⟩, |−x⟩} via a single shot, multi-state readout of the qutrit, which gives probabilities {p g,j , p +x,j , p −x,j } that add up to unity.The TPM protocol determines the total energy change ∆U = W + Q whose distribution is characterized by the transition probabilities [32] where i, j = ±x label the eigenstates of the internal energy operator H(0).The state-dependent denominator in (6) captures the norm-preserving nature of the post-selection process, i P ij (τ ) = 1 (Fig. 2b,c).The exponentiated-work expectation value (1) is obtained as where the statistical weights P ±x = {0.98,0.02} reflect the reservoir temperature and transition probabilities P ij (τ ) are experimentally measured for loop duration τ ranging from 0.1 µs ≤ τ ≤ 1 µs.
Figure 3a shows that for the first path with zero detuning, Eq. ( 3), the survival and transition probabilities are equal for the two energy eigenstates.On the contrary, for the second path, Eq. ( 4) with ∆ max = 10π rad/µs, the probabilities P ++ ̸ = P −− (or equivalently, P −+ ̸ = P +− ) are clearly asymmetrical (Fig. 3b).Both cases have J max = J min = 3.74 rad/µs.We observe stark differences between these two cases which correspond to Hamilto- nians H eff (t) with or without an explicit PT symmetry, respectively.Jarzynski equality and its violation.Figure 4 summarizes the experimental results for ⟨e −βW ⟩(τ ) for J(t) variations (a-d) and ∆ 1 (t) variations (e-f); each parameter path and the location of the EP is schematically shown in the corresponding panel inset.We see that ⟨e −βW ⟩ ≃ 1 for J(t) variations that range from the static case, J max = J min (a), to paths confined to the PT -symmetric region, J min = 0.5J max (b), to paths that traverse across the EP into the PT -broken region with J min = 0 (c).Panel (d) shows that starting from the PT -broken region and traversing across the EP into the PT -symmetric region also maintains the Jarzynski equality, with much smaller fluctuations arising from the smaller energy scale (upper inset).Thus, the Jarzynski equality is satisfied for arbitrary J(t) sweeps, independent of the real or imaginary nature of eigenvalues of H eff (t) as long as the Hamiltonian has an explicit PTsymmetry.For the first path, this explicit PT -symmetry also ensures that the corresponding Floquet energy operator H F has the same energetics as the system's initial energy operator H(0) = J max σ x .In sharp contrast, for one-sided sweeps ∆ 1 (t), the average exponentiatedwork ⟨e −βW ⟩(τ ) exceeds one for ∆ max < 0 (e) and is below unity for ∆ max > 0 (f), thereby indicating that the Jarzynski equality is violated when H eff (t) or its Floquet counterpart H F eff do not have an antilinear (parity-time) symmetry.
Lastly, we investigate a case where H eff (t) has no explicit PT symmetry, and yet its Floquet counterpart H F eff is parity-time symmetric.In this case, the Jarzynski equality is satisfied only at specific loop times: times where the Floquet energy operator H F aligns with the system's initial energy operator H(0) = J max σ x .We introduce a new parameter path, Eq. ( 5), which obeys ∆ 2 (t) = −∆ 2 (τ − t).As a consequence of the zero average detuning, the corresponding Floquet Hamiltonian has an emergent parity-time symmetry, i.e.H F eff eigenvalues are always real or complex conjugates.Figure 5a shows that the measured probability asymmetry ∆P (τ ) ≡ P ++ (τ ) − P −− (τ ) (brown), or equivalently, ∆P (τ ) = P +− (τ ) − P −+ (τ ) (orange), is generally nonzero.However, the symmetry under the eigenstatelabel exchange +x ↔ −x, indicated by ∆P (τ ) = 0, is recovered at loop times τ 1 = 0.455 µs and τ 2 = 0.572 µs.The corresponding simulated exponentiated-work average shows that although generally violated, the Jarzynski equality is satisfied along black dashed contours (Fig. 5b).These contours intersect with the experimentally investigated region at ∆ max = 10π rad/µs (red solid line).In general, the experimentally measured ⟨e −βW ⟩(τ ) is not equal to unity (Fig. 5c).However, at loop times τ 1 and τ 2 the two equalities ⟨e −βW ⟩ = 1 and ∆P = 0 are satisfied simultaneously.A higher-resolution measurement of exponentiated-work in a smaller loop-time window shows this effect clearly (Fig. 5d).Discussion and Outlook.Non-Hermitian Hamiltonians with real spectra [33], first realized in open classical systems [34][35][36], have recently materialized in the quantum domain [22,[37][38][39][40][41].On top of their role in dynamics, their real eigenvalues are often mistaken for allowed energies of a quantum system [26][27][28][29][30][31], with implications to thermodynamics.Although the two share conceptual roots, the thermodynamics of non-Hermitian systems remains an open challenge.A consistent formulation of its first law requires distinguishing the Hermitian part H that gives allowed energies [9] from the non-Hermitian Hamiltonian H eff = H + iΓ that governs the temporal dynamics.Using the same distinction we have verified a fluctuation theorem for exponentiated work, i.e.Jarzynski equality (1) for cyclic variations of H eff (t) that include parameter regions with complex eigenvalues.
The Jarzynski equality, rigorously tested in the classi- cal domain [42,43], trivially extends to isolated quantum systems (Q = 0) by equating the work distribution P (W ) with the TPM protocol that, technically, generates the distribution P (∆U ) of internal-energy changes [14].It also holds in decohering quantum systems [21], but the equality's validity in driven qubits with dissipation, continuous monitoring, or feedback, as tested with the TPM protocol, is disputed [44,45].In such settings, the tests of Jarzynski equality require modifications that reflect the energetic cost of information or, equivalently, new dynamics that encode the monitoring, feedback, and measurement processes [46][47][48][49][50][51].Our results show that the coherent qubit dynamics of non-Hermitian Hamiltonians is a new class where the Jarzynski equality for cyclic variations is preserved when two symmetry considerations are met.First, the presence of explicit or emergent parity-time symmetry ensures that the qubit has symmetrical amplification or decay rates.Second, energetic changes of the qubit (sometimes termed quantum heat [6]) vanish when the measurement basis of H(0) = H(τ ) (| ± x⟩, in the present case) align with the equivalent Floquet energy operator H F basis.With our symmetry-governed, consistent formulation of the second law of thermodynamics, we anticipate new opportunities in quantum, non-equilibrium thermodynamics through non-Hermitian models.
Methods: Analytical Results.For a qubit with internal energy operator H(0) = H(τ ) = J max σ x , the exponentiatedwork average ( 7) is given by ⟨e −βW ⟩ = e −2βJmax P +− (τ )P −x + e +2βJmax P −+ (τ )P +x It is easy to verify that the right-hand side is equal to unity when the initial density matrix is thermal, i.e.P ±x = exp(∓βJ max )/2 cosh(βJ max ), and the transition probabilities are symmetric under the eigenstate-label exchange +x ↔ −x (Fig. 3a).The exchange-symmetry constraint on the P ij (τ ) holds provided the elements of the time-evolution matrix G(τ ) satisfy By expressing the traceless non-Hermitian Floquet Hamiltonian as H F eff = h x σ x + h y σ y + h z σ z , the exchange-symmetry constraint can be written as where , and S = sin(τ |h|)/|h|.The terms C, S are real if and only if |h| is real or purely imaginary, which means the Floquet Hamiltonian H F eff , with eigenvalues ±|h|, has an explicit or emergent parity-time (antilinear) symmetry [52].Eq. ( 9) further requires that h x ∈ R and h y , h z are purely imaginary.Thus, the non-Hermitian, parity-time symmetric, Floquet Hamiltonian H F eff = H F + iΓ F is further constrained to an internal energy operator H F = h x σ x that is aligned with the system's initial energy operator H(0).Note that the mere requirement of parity-time symmetry allows for h x ∈ R and a complex h y = h * z ∈ C.However, in such cases, H F = h x σ x + ℜh y (σ y + σ z ) is not aligned with H(0), and the constraint that guarantees exchange symmetry for probabilities, Eq.( 9), is not fulfilled.Thus, Jarzynski equality requires a parity-time symmetric H F eff with its Floquet energy operator proportional to the system's initial energy operator.Experimental setup.The experimental setup comprises a superconducting circuit that was fabricated and provided by the Superconducting Qubits at Lincoln Laboratory (SQUILL) Foundry at MIT Lincoln Laboratory.The experiments utilize a sub-portion of a multi-qubit chip with relevant components consisting of a tunable transmon qubit with maximum frequency ω ge /2π = 4.373 GHz, dispersively coupled to a readout resonator at coupling rate g/2π = 33 MHz and linewidth κ/2π = 246 kHz, qubit drive line, and an off-chip coupling line.A solenoid coil fixed to the package allows control of the global flux through the transmon SQUID loop, with bias current filtered at the 4K stage with a low pass filter (QDevil Q015 QFilter).The qubit is operated at ω ge /2π = 4.25 GHz and resonator frequency ω r /2π = 6.88865GHz.To realize the non-Hermiticity, an off-chip coaxial filter is coupled to the qubit to enhance the |e⟩ decay rate to γ e = 1.57µs −1 .The readout signal probes the resonator via a common bus line and is amplified by a Josephson parametric amplifier (BBN-PS2-JPA-DEVICE-QEC) operating with ∼ 15 dB of gain.Simulations.The evolution of the three-level system can be solved using Lindblad equation  10) is solved in MATLAB using the Runge-Kutta method to obtain ρ 3 (τ ) with suitable initial conditions and thereby calculate each transition probability P ij (τ ).For γ f , γ 2e , γ 2f ≪ γ e , the Lindblad results for the {|e⟩, |f ⟩} manifold are identical to those obtained from the non-Hermitian Hamiltonian [32], Post-selection and error analysis.For the experimental data, we employ post-selection by normalizing the state readouts to the population within the {|e⟩, |f ⟩} manifold of states, the resulting evolution can be described by the non-Hermitian Hamiltonian (2).For each transition probability, we repeat the experiment a total of 8000 times, yet through post-selection up to ∼ 65% of the data is discarded.The statistical (trinomial) error associated with the state readout is typically less than 0.016 for the transition probabilities and less than 0.012 for the exponentiated work.Remnant, point-to-point fluctuations are likely due to residual low-frequency (1/f) fluctuations in the experimental setup.For Fig. 5a,c-d we utilized 24,000 experimental repetitions per point.
Fundamentally, the coherent, non-unitary, non-unital dynamics generated by H eff begs the question: What are the constraints on quantum work fluctuations arXiv:2309.12393v1[quant-ph] 21 Sep 2023