Characteristic Function of Work

• Characteristic Function of Work is the Fourier transform of work distributions that quantifies energy differences in driven quantum processes.
• It generates work cumulants and underpins fluctuation relations such as the Jarzynski equality and Tasaki–Crooks relation in various quantum systems.
• The framework informs both theoretical analyses and experimental protocols, including TPM, interferometric measurements, and quasiprobability schemes.

The characteristic function of work is the Fourier transform of the work distribution associated with a driven quantum process. In the standard two-projective-measurement setting, it encodes the statistics of energy differences between initial and final energy measurements; in compact operator form, for an initial state diagonal in the initial energy basis, it is $$G(u)=\mathrm{Tr}[U^\dagger e^{iuH_f}U e^{-iuH_i}\rho_i]$$, while the work distribution is recovered from $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$. The logarithm $\ln G(u)$ generates the work cumulants, and the same object reappears, with different notation and operational meaning, in integrable Floquet systems, quantum-critical quenches, adiabatic open-system master equations, coherence-sensitive quasiprobability schemes, and interferometric measurement protocols (Russomanno et al., 2015, Mazzola et al., 2013).

1. TPM definition and cumulant structure

Within the two-projective-measurement framework, a nonequilibrium protocol starts from an initial Hamiltonian $H_0$, ends at $H_\tau$, and is generated by a unitary $$U(\tau)=\mathcal{T}\exp[-(i/\hbar)\int_0^\tau H(t)\,dt]$$. If the first energy measurement projects onto $\Pi_n^0$ and the second onto $\Pi_m^\tau$, then

$$P(W)=\sum_{m,n} p_n p_{m|n}\,\delta\!\left(W-[E_m^\tau-E_n^0]\right),$$

and the characteristic function is

$$G(u)=\int dW\,e^{iuW}P(W) =\mathrm{Tr}\!\left[U^\dagger(\tau)e^{iuH_\tau}U(\tau)e^{-iuH_0}\rho_0'\right],$$

with $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$0. If $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$1, then $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$2. A terminological point made explicitly in the Floquet-Ising analysis is that $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$3 is the characteristic function, whereas $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$4 is the cumulant generating function, with $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$5 (Mazzola et al., 2013, Russomanno et al., 2015).

For Gibbs initial states, the characteristic function directly yields fluctuation relations. Substituting $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$6 gives $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$7, i.e. the Jarzynski equality, and the corresponding forward and backward characteristic functions satisfy the Tasaki–Crooks relation. This TPM structure is also the reference point for later generalizations: phase-space methods, path-integral constructions, full-counting-statistics-like approaches, and quasiprobability schemes all retain the Fourier relation between $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$8 and $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$9, but modify either the operational definition of work or the treatment of initial coherence (Mazzola et al., 2013).

2. Exact structures in integrable and quadratic systems

In several exactly solvable many-body settings, the characteristic function factorizes over independent modes. For the periodically driven one-dimensional transverse-field Ising chain, Jordan–Wigner and Bogoliubov transformations reduce the Hamiltonian to decoupled $\ln G(u)$0-sectors, so that $\ln G(u)$1. At zero temperature and in the thermodynamic limit, the infinite-time cumulant generating function per site becomes

$\ln G(u)$2

which isolates the low-$\ln G(u)$3 physics controlling the asymptotic work statistics. In the same steady state, the mean work per site and the variance are explicit $\ln G(u)$4-integrals over $\ln G(u)$5 and $\ln G(u)$6 (Russomanno et al., 2015).

A distinct exact product form arises in the 2D Kitaev honeycomb model quenched across a critical line. Using the trace formula for quadratic fermionic Hamiltonians and the Landau–Zener solution for each $\ln G(u)$7 sector, the zero-temperature characteristic function is

$\ln G(u)$8

with

$\ln G(u)$9

Because each mode contributes a Bernoulli factor, the corresponding work distribution is a Poisson binomial distribution (Zhang et al., 2021).

Quadratic Hamiltonians also admit exact phase-space and path-integral representations. In the phase-space formulation, the Weyl symbol of the auxiliary operator obeys a Moyal evolution equation, and for quadratic systems the characteristic function reduces to Gaussian integrals along the classical linear flow; explicit closed forms are obtained for the forced harmonic oscillator, the oscillator with time-dependent mass and frequency, and coupled oscillators. In the path-integral construction, the characteristic function is rewritten as a double forward–backward path integral in which the counting field appears as two additional time strips of length $H_0$0; this reproduces the exact harmonic-oscillator and expanding-piston results obtained in Schrödinger form (Qian et al., 2019, Qiu et al., 2019).

3. Critical behavior, Floquet resonances, and universal asymptotics

In periodically driven integrable systems, the asymptotic structure of the characteristic function can encode non-equilibrium criticality. For the driven Ising chain, the $H_0$1 Floquet quasi-energies are $H_0$2, folded into the quasi-energy Brillouin zone. The resonance condition

$H_0$3

defines non-equilibrium critical points

$H_0$4

at which the Floquet spectrum is gapless at $H_0$5. For sinusoidal driving $H_0$6, coherent destruction of tunneling suppresses the resonance when $H_0$7 (Russomanno et al., 2015).

These Floquet resonances determine the small-$H_0$8 edge singularities of the asymptotic work distribution. For $H_0$9, the threshold is

$H_\tau$0

If $H_\tau$1, then

$H_\tau$2

which yields

$H_\tau$3

If $H_\tau$4, then

$H_\tau$5

and the edge becomes a derivative-of-delta singularity,

$H_\tau$6

For critical initialization $H_\tau$7, the large-$H_\tau$8 behavior changes from exponential to power-law, so $H_\tau$9 and $$U(\tau)=\mathcal{T}\exp[-(i/\hbar)\int_0^\tau H(t)\,dt]$$0 behaves as $$U(\tau)=\mathcal{T}\exp[-(i/\hbar)\int_0^\tau H(t)\,dt]$$1 at $$U(\tau)=\mathcal{T}\exp[-(i/\hbar)\int_0^\tau H(t)\,dt]$$2 or $$U(\tau)=\mathcal{T}\exp[-(i/\hbar)\int_0^\tau H(t)\,dt]$$3 at resonant $$U(\tau)=\mathcal{T}\exp[-(i/\hbar)\int_0^\tau H(t)\,dt]$$4, with correspondingly different small-$$U(\tau)=\mathcal{T}\exp[-(i/\hbar)\int_0^\tau H(t)\,dt]$$5 edges (Russomanno et al., 2015).

A second universality problem concerns work cumulants across a quantum critical surface. Adiabatic perturbation theory gives, for all cumulants,

$$U(\tau)=\mathcal{T}\exp[-(i/\hbar)\int_0^\tau H(t)\,dt]$$6

$$U(\tau)=\mathcal{T}\exp[-(i/\hbar)\int_0^\tau H(t)\,dt]$$7

$$U(\tau)=\mathcal{T}\exp[-(i/\hbar)\int_0^\tau H(t)\,dt]$$8

where $$U(\tau)=\mathcal{T}\exp[-(i/\hbar)\int_0^\tau H(t)\,dt]$$9 is the codimension of the critical surface. In the 2D Kitaev honeycomb model, the quench crosses a one-dimensional critical surface with $\Pi_n^0$0, so $\Pi_n^0$1 for all $\Pi_n^0$2. The exact characteristic function confirms this prediction, and numerical integration for a $\Pi_n^0$3 system gave $\Pi_n^0$4, $\Pi_n^0$5, and $\Pi_n^0$6 with slopes equal to $\Pi_n^0$7 on log–log plots (Zhang et al., 2021).

4. Open-system formulations and generalized measurement schemes

For open quantum systems described by adiabatic two-level quantum Markovian master equations, the characteristic function is constructed from the quantum-jump unraveling of the Lindblad dynamics. The inclusive work along a trajectory is

$\Pi_n^0$8

where $\Pi_n^0$9 and $\Pi_m^\tau$0 jumps exchange energies $\Pi_m^\tau$1 with the bath and $\Pi_m^\tau$2 corresponds to pure dephasing. The characteristic function $\Pi_m^\tau$3 is obtained from a $\Pi_m^\tau$4-deformed tilted Lindblad generator in which only the energy-exchange channels acquire counting factors $\Pi_m^\tau$5. Under instantaneous detailed balance and time-reversal symmetry, the forward and backward characteristic functions satisfy

$\Pi_m^\tau$6

which yields the Crooks relation and the Jarzynski equality in the adiabatic weak-coupling Markovian regime (Liu, 2014).

Measurement models based on Gaussian pointers modify the characteristic function by explicit apparatus parameters. In the scenario of two Gaussian energy measurements, the characteristic function is $\Pi_m^\tau$7 times an operator expression containing the energy-resolution scale $\Pi_m^\tau$8, the dephasing scale $\Pi_m^\tau$9, and the pointer correlation parameter $$P(W)=\sum_{m,n} p_n p_{m|n}\,\delta\!\left(W-[E_m^\tau-E_n^0]\right),$$0. In the Gaussian-pointer work-meter, the prefactor is $$P(W)=\sum_{m,n} p_n p_{m|n}\,\delta\!\left(W-[E_m^\tau-E_n^0]\right),$$1, with an otherwise analogous operator structure. In the strong-measurement limit both schemes recover the TPM characteristic function. In the weak, imprecise limit, the average work becomes the untouched mean work,

$$P(W)=\sum_{m,n} p_n p_{m|n}\,\delta\!\left(W-[E_m^\tau-E_n^0]\right),$$2

while the fluctuations are dominated by the measurement and diverge with $$P(W)=\sum_{m,n} p_n p_{m|n}\,\delta\!\left(W-[E_m^\tau-E_n^0]\right),$$3. For $$P(W)=\sum_{m,n} p_n p_{m|n}\,\delta\!\left(W-[E_m^\tau-E_n^0]\right),$$4-diagonal Gibbs states, the fluctuation-theorem value $$P(W)=\sum_{m,n} p_n p_{m|n}\,\delta\!\left(W-[E_m^\tau-E_n^0]\right),$$5 is replaced by $$P(W)=\sum_{m,n} p_n p_{m|n}\,\delta\!\left(W-[E_m^\tau-E_n^0]\right),$$6 for two Gaussian measurements or $$P(W)=\sum_{m,n} p_n p_{m|n}\,\delta\!\left(W-[E_m^\tau-E_n^0]\right),$$7 for the work-meter (Talkner et al., 2015).

5. Coherence, quasiprobabilities, and alternative work observables

When the initial state has coherence in the energy basis, the TPM characteristic function is insensitive to it because the first projective measurement implements the dephasing map $$P(W)=\sum_{m,n} p_n p_{m|n}\,\delta\!\left(W-[E_m^\tau-E_n^0]\right),$$8. One coherence-sensitive alternative is the class

$$P(W)=\sum_{m,n} p_n p_{m|n}\,\delta\!\left(W-[E_m^\tau-E_n^0]\right),$$9

associated with a quasiprobability distribution $$G(u)=\int dW\,e^{iuW}P(W) =\mathrm{Tr}\!\left[U^\dagger(\tau)e^{iuH_\tau}U(\tau)e^{-iuH_0}\rho_0'\right],$$0. It reduces exactly to TPM when $$G(u)=\int dW\,e^{iuW}P(W) =\mathrm{Tr}\!\left[U^\dagger(\tau)e^{iuH_\tau}U(\tau)e^{-iuH_0}\rho_0'\right],$$1. Its first two moments are

$$G(u)=\int dW\,e^{iuW}P(W) =\mathrm{Tr}\!\left[U^\dagger(\tau)e^{iuH_\tau}U(\tau)e^{-iuH_0}\rho_0'\right],$$2

while the dependence on $$G(u)=\int dW\,e^{iuW}P(W) =\mathrm{Tr}\!\left[U^\dagger(\tau)e^{iuH_\tau}U(\tau)e^{-iuH_0}\rho_0'\right],$$3 begins at the third moment. In the same framework, coherence modifies fluctuation relations through

$$G(u)=\int dW\,e^{iuW}P(W) =\mathrm{Tr}\!\left[U^\dagger(\tau)e^{iuH_\tau}U(\tau)e^{-iuH_0}\rho_0'\right],$$4

and the joint work–coherence quasidistribution obeys

$$G(u)=\int dW\,e^{iuW}P(W) =\mathrm{Tr}\!\left[U^\dagger(\tau)e^{iuH_\tau}U(\tau)e^{-iuH_0}\rho_0'\right],$$5

For initial thermal populations this yields $$G(u)=\int dW\,e^{iuW}P(W) =\mathrm{Tr}\!\left[U^\dagger(\tau)e^{iuH_\tau}U(\tau)e^{-iuH_0}\rho_0'\right],$$6 and

$$G(u)=\int dW\,e^{iuW}P(W) =\mathrm{Tr}\!\left[U^\dagger(\tau)e^{iuH_\tau}U(\tau)e^{-iuH_0}\rho_0'\right],$$7

Negative contributions are bounded by $$G(u)=\int dW\,e^{iuW}P(W) =\mathrm{Tr}\!\left[U^\dagger(\tau)e^{iuH_\tau}U(\tau)e^{-iuH_0}\rho_0'\right],$$8 for $$G(u)=\int dW\,e^{iuW}P(W) =\mathrm{Tr}\!\left[U^\dagger(\tau)e^{iuH_\tau}U(\tau)e^{-iuH_0}\rho_0'\right],$$9, linking negativity directly to off-diagonal coherence and noncommutativity; if $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$00, coherence does not affect any moment (Francica, 2021).

A Wigner-function-based quasiprobability introduces a continuous-variable ancilla with $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$01 and joint unitary

$$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$02

Its $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$03-resolved quasi-characteristic function is

$$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$04

and the $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$05-integrated $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$06 contains a Gaussian dephasing map $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$07. In this construction, the TPM characteristic function is recovered for $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$08, whereas finite $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$09 and initial energy-basis coherences generate negativity and interference fringes in the ancilla Wigner function (Cerisola et al., 2023).

A distinct line of work defines a Hermitian work operator $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$10 and sets

$$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$11

Here $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$12 is introduced by spectral calculus after constructing $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$13, so no time-ordering symbol is attached to the exponential itself. In this operator-based framework, Jarzynski is recovered only if $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$14. This suggests a substantive conceptual divide: some formulations retain the operational logic of measurements, some replace probabilities by quasiprobabilities to keep initial coherence, and some elevate work to an observable tied to a chosen internal-energy operator (Silva et al., 2021).

6. Experimental access and computational formalisms

The characteristic function can be measured interferometrically with an ancilla qubit. In a Ramsey protocol, the ancilla is prepared in $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$15, and controlled gates encode either a commuting-case unitary $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$16 or, in the general noncommuting case, the sequence

$$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$17

with

$$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$18

$$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$19

After a second Hadamard gate, the ancilla observables satisfy

$$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$20

Hybrid optomechanical and electromechanical implementations were proposed explicitly, with detailed gate decompositions based on controlled displacements and phase control (Mazzola et al., 2013).

Coherence-sensitive variants also admit ancilla-based readout. The detector construction for $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$21 extracts the characteristic function from two detector coherences after impulsive couplings to $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$22 and $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$23, while the Wigner-based protocol reconstructs $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$24 through tomography of the final ancilla state. Circuit QED and trapped-ion platforms were identified as suitable settings for the latter because they permit conditional displacements and Wigner tomography of the ancilla mode (Francica, 2021, Cerisola et al., 2023).

On the computational side, several non-equivalent but compatible formalisms have been developed. Phase-space methods use Weyl symbols and Moyal evolution, with an $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$25 truncation for general smooth potentials. Path-integral methods rewrite the characteristic function as a forward–backward double path integral with counting-field time strips appended to the contour. Nonequilibrium Green’s-function methods place the problem on a modified Schwinger–Keldysh contour $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$26, from which a second-order expansion yields $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$27 in terms of the drive spectral weight

$$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$28

and the equilibrium two-point cumulant $$P(W)=(1/2\pi)\int_{-\infty}^{\infty}du\,e^{-iuW}G(u)$$29. A plausible implication is that the characteristic function has become the common technical currency of quantum work statistics not because all definitions of work coincide, but because it remains the object that can be factorized, expanded, measured, or tomographically reconstructed across TPM, quasiprobability, phase-space, path-integral, and Keldysh formulations (Qian et al., 2019, Qiu et al., 2019, Fei et al., 2020).