Energy, Heat, and Work

Energy at a time is represented by a Hamiltonian. Heat and work are different: they describe how energy is transferred during a physical process. In small quantum systems this distinction is not a semantic nicety. Measurements can disturb the state, reservoirs can be structured, interaction energy can be non-negligible, and a “work distribution” usually refers to a protocol rather than to a single Hermitian operator.

The formulas below describe a weakly coupled system driven by external controls and exchanging energy with thermal reservoirs. The distinction between those two mechanisms is essential: a change in the density operator alone does not tell us whether energy came from a bath, a drive, or a detector. For their application to extended systems, see Driven Many-Body Systems.

Energy Changes and External Driving

For a system described by a density operator $\rho(t)$ and a chosen system Hamiltonian $H(t)$, the internal energy is

$$E(t) = \operatorname{Tr}\!\left[ \rho(t)H(t) \right].$$

If $\rho(t)$ and $H(t)$ are differentiable, then

$$\frac{dE}{dt} = \operatorname{Tr}\!\left[ \dot\rho(t)H(t) \right] + \operatorname{Tr}\!\left[ \rho(t)\dot H(t) \right].$$

The standard weak-coupling sign convention in this volume is:

$$\dot Q = \operatorname{Tr}\!\left[ \dot\rho(t)H(t) \right], \qquad \dot W = \operatorname{Tr}\!\left[ \rho(t)\dot H(t) \right].$$

Thus

$$\frac{dE}{dt} = \dot Q+\dot W.$$

Here $\dot Q$ is heat current into the system and $\dot W$ is power supplied to the system by externally changing the Hamiltonian. With the opposite sign convention for work, some thermodynamics books write power delivered by the system as $-\dot W$.

Quantity	Meaning in This Convention	Typical Source
$E=\operatorname{Tr}(\rho H)$	internal energy assigned to the chosen reduced system	state and system Hamiltonian
$\dot Q=\operatorname{Tr}(\dot\rho H)$	energy change from state evolution at fixed Hamiltonian	thermal reservoirs in the setting assumed here
$\dot W=\operatorname{Tr}(\rho\dot H)$	energy change from explicit Hamiltonian control	external drive, changing fields, moving boundaries

These expressions separate two terms in an exact derivative. Calling the first term heat requires the additional reservoir interpretation. If a coherent drive changes the expectation of a bare Hamiltonian $H_0$, while the generator is $H_0+V(t)$, then $-i\operatorname{Tr}(H_0[V,\rho])/\hbar$ is energy supplied by that drive. It is not thermal heat merely because it appears in $\dot\rho$. Similarly, detector-induced energy changes require a model of the detector.

What Is Being Assumed

The formulas above are exact as an identity for the selected pair $(\rho,H)$, but their thermodynamic interpretation assumes more:

• The system Hamiltonian is the energy observable whose expectation is being tracked.
• The external drive appears as explicit time dependence in $H(t)$.
• Terms called heat describe exchange with specified thermal reservoirs; detector and engineered-source contributions are identified separately.
• Interaction energy with untracked degrees of freedom is either negligible, included consistently in $H(t)$, or accounted for separately.
• The zero of energy and the frame used for $H(t)$ have been fixed.

These assumptions are usually appropriate for weak system–bath coupling and a controlled Hamiltonian drive. They can fail for strong coupling, finite reservoirs, non-Markovian dynamics, time-dependent rotating frames, or ambiguous system–environment partitions. For the microscopic starting point, see System–Bath Hamiltonians.

Energy Is an Observable, Work Is a Process Quantity

At a fixed time, energy measurements are described by the spectral projectors of $H(t)$. If

$$H(t) = \sum_n E_n(t)\Pi_n(t),$$

then an ideal energy measurement has outcomes $E_n(t)$ with probabilities $\operatorname{Tr}[\Pi_n(t)\rho(t)]$.

Work is not usually represented by a single observable measured at one time. It is energy transferred by a process. For a closed driven protocol from $0$ to $\tau$, one often compares an initial energy measurement of $H(0)$ with a final energy measurement of $H(\tau)$. That two-point measurement scheme defines random values

$$W_{m n} = E_m^\tau-E_n^0,$$

but it also defines a measurement protocol with backaction. If the initial state has coherence between energy eigenspaces, the initial energy measurement changes the state used in the distribution. For this reason the slogan “work is not an observable” means:

• there is no universal Hermitian operator $W$ whose spectral measure gives work for arbitrary driven quantum processes;
• work distributions depend on the operational protocol used to define them;
• the average energy change can be meaningful even when no unique work distribution has been specified.

For fluctuation relations based on two energy measurements, see Fluctuation Theorems.

Closed Driven Systems

For a closed system with unitary dynamics,

$$\dot\rho = - \frac{i}{\hbar}[H(t),\rho],$$

the Hamiltonian part does not change the expectation value of $H(t)$ at fixed time:

$$\operatorname{Tr}\!\left[ H(t)\dot\rho(t) \right] = - \frac{i}{\hbar} \operatorname{Tr}\!\left[ H(t)[H(t),\rho(t)] \right] = 0.$$

Thus, in the weak-coupling convention,

$$\dot Q=0, \qquad \frac{dE}{dt}=\dot W.$$

A closed system can still absorb or release energy when $H(t)$ is explicitly changed. The energy transfer is work because it is caused by a controlled parameter, not by exchange with an untracked thermal reservoir.

If the Hamiltonian is time-independent, closed unitary dynamics conserves $E$. If $H(t)$ changes cyclically so that $H(\tau)=H(0)$, the net work need not vanish: the drive may leave the state in a different energy distribution.

Open Systems and Heat Currents

For a Markovian master equation

$$\dot\rho = - \frac{i}{\hbar}[H(t),\rho] + \sum_\alpha \mathcal L_\alpha(\rho),$$

the energy current into the system from reservoir $\alpha$ is

$$\dot Q_\alpha = \operatorname{Tr}\!\left[ H(t)\mathcal L_\alpha(\rho) \right],$$

so that

$$\frac{dE}{dt} = \operatorname{Tr}\!\left[ \rho\dot H \right] + \sum_\alpha \operatorname{Tr}\!\left[ H\mathcal L_\alpha(\rho) \right].$$

This is a heat current when $\mathcal L_\alpha$ represents a thermal reservoir in the stated weak-coupling model. For a detector or engineered source it is first an energy current; its division into heat and work needs the corresponding physical model. A Lindblad equation is mathematically trace preserving and completely positive, but it is not automatically a thermal master equation.

For a thermal reservoir at inverse temperature $\beta_\alpha$, transition rates should obey detailed-balance constraints with respect to the Hamiltonian whose energy is being counted. See Thermal Master Equations and Detailed Balance.

Qubit Example

Consider a two-level system with ground energy $0$ and excited energy $\hbar\omega(t)$:

$$H(t) = \hbar\omega(t) |e\rangle\langle e|.$$

Let

$$p_e(t) = \langle e|\rho(t)|e\rangle.$$

Then the internal energy is

$$E(t) = \hbar\omega(t)p_e(t).$$

Differentiating gives

$$\frac{dE}{dt} = \hbar\omega(t)\dot p_e(t) + \hbar\dot\omega(t)p_e(t).$$

In the standard convention,

$$\dot Q = \hbar\omega(t)\dot p_e(t), \qquad \dot W = \hbar\dot\omega(t)p_e(t).$$

The interpretation is direct:

• changing the excited-state population through a thermal reservoir at fixed gap exchanges heat with that reservoir;
• changing the level spacing at fixed population performs work through the external control of $\omega(t)$.

For a thermal amplitude-damping model,

$$\dot p_e = \gamma_\uparrow p_g - \gamma_\downarrow p_e, \qquad p_g=1-p_e,$$

the heat current is

$$\dot Q = \hbar\omega \left( \gamma_\uparrow p_g - \gamma_\downarrow p_e \right).$$

Absorption from the bath gives positive heat into the system. Spontaneous or stimulated emission into the bath gives negative heat into the system.

Strong Coupling and Interaction Energy

For a closed system plus bath,

$$H_{\mathrm{tot}} = H_S+H_B+H_I,$$

the total energy is unambiguous once $H_{\mathrm{tot}}$ is specified. The reduced system energy $E_S=\operatorname{Tr}_S(\rho_S H_S)$ is less automatic because part of the energy can sit in the interaction term $H_I$.

When $H_I$ is weak and the bath is large, it is often accurate to neglect interaction energy in the reduced energy balance. When $H_I$ is not negligible, several inequivalent conventions appear:

• assign $H_I$ to neither side and track it separately;
• assign part of the interaction energy to the system, retaining the joint-state correlations needed to evaluate it;
• use a Hamiltonian of mean force for equilibrium thermodynamics;
• enlarge the “system” to include strongly coupled environmental modes.

These conventions can give different heat and work assignments while agreeing on the total energy budget. The chosen definition must agree with the measured work source and bath energy changes. A Hamiltonian of mean force generally depends on temperature; its thermodynamic internal energy is not simply its mean expectation without the corresponding temperature derivative.

Time-Dependent Energy Offsets and Frames

Adding a scalar offset changes the energy accounting:

$$H'(t) = H(t)+c(t)I.$$

Then

$$E'(t) = E(t)+c(t), \qquad \dot W' = \dot W+\dot c(t),$$

while transition frequencies and closed-system state evolution are unchanged up to a global phase. This shows why the zero of energy must be fixed when quoting work. A time-dependent scalar offset can look like power in the energy accounting even when it has no observable effect on the isolated system.

Changing rotating frames can introduce similar hazards. The Hamiltonian in a rotating frame may be the generator of transformed dynamics, not the laboratory energy observable. Thermodynamic heat and work should be assigned using the energy operator appropriate to the physical energy budget, not merely whatever operator appears in the most convenient equation of motion.

Measurements, Information, and Energy

Measurements can change a system’s energy. Whether that energy change is called heat, work, or measurement backaction depends on the physical implementation.

Examples:

• An unread projective energy measurement preserves the energy distribution for every initial state, including coherent states and degenerate spectra. It removes coherence between different energy eigenspaces.
• A projective measurement in a non-energy basis can inject or remove energy because it changes the state at fixed Hamiltonian.
• Continuous monitoring can add diffusion or jumps to the conditional state; the detector and any feedback controller must be included in the energy budget.
• Feedback can extract work using a measurement record. To determine the net work of a repeatable cycle, also include the memory, controller, and their reset costs.

The safest language is operational: specify the apparatus, the record, and the external controller. Then assign energy flows according to that enlarged model. For monitored dynamics, see Quantum-Jump Trajectories and Stochastic Master Equations.

Heat Is Not Entropy

Heat and entropy are related but not identical. Heat is energy transfer. Entropy is a state function, such as the von Neumann entropy

$$S(\rho) = -k_B\operatorname{Tr}(\rho\ln\rho).$$

For an ideal thermal reservoir at temperature $T_\alpha$, heat current into the system contributes reservoir entropy change

$$\frac{dS_{\alpha}}{dt} = - \frac{\dot Q_\alpha}{T_\alpha}.$$

The system entropy production rate is then typically organized as

$$\dot\Sigma = \frac{dS}{dt} - \sum_\alpha \frac{\dot Q_\alpha}{T_\alpha},$$

with $\dot Q_\alpha$ defined as heat into the system. The inequality $\dot\Sigma\ge0$ requires a thermodynamically consistent model, not merely any equation that decreases coherence. See Entropy Production.

Exercises

Driven Level Spacing

A two-level system has $H(t)=\hbar\omega(t)|e\rangle\langle e|$ and excited-state population $p_e(t)$. Show that changing $\omega$ at fixed $p_e$ is work in the convention of this page, while reservoir-induced changes of $p_e$ at fixed $\omega$ are heat in a thermal amplitude-damping model.

Solution

The internal energy is

$$E(t)=\hbar\omega(t)p_e(t).$$

Differentiation gives

$$dE = \hbar p_e\,d\omega + \hbar\omega\,dp_e.$$

At fixed population, $dp_e=0$, so

$$dE=\hbar p_e\,d\omega.$$

This energy change comes from moving the level spacing through an external control parameter and is assigned to work. At fixed level spacing, $d\omega=0$, so

$$dE=\hbar\omega\,dp_e.$$

This energy change comes from changing the state populations at fixed Hamiltonian and is assigned to heat if the population change is caused by a reservoir.

Thermal Qubit Current

For the qubit master equation

$$\dot p_e = \gamma_\uparrow p_g-\gamma_\downarrow p_e, \qquad p_g=1-p_e,$$

with $H=\hbar\omega|e\rangle\langle e|$, compute the heat current into the system. What condition makes the stationary heat current vanish?

Solution

The heat current into the system is

$$\dot Q = \operatorname{Tr}(H\dot\rho) = \hbar\omega\dot p_e.$$

Using the rate equation,

$$\dot Q = \hbar\omega \left( \gamma_\uparrow p_g - \gamma_\downarrow p_e \right).$$

At stationarity, $\dot p_e=0$, so

$$\gamma_\uparrow p_g^{\mathrm{ss}} = \gamma_\downarrow p_e^{\mathrm{ss}}.$$

The stationary heat current then vanishes. For a thermal bath, detailed balance gives

$$\frac{\gamma_\uparrow}{\gamma_\downarrow} = e^{-\beta\hbar\omega},$$

so the stationary population ratio is Gibbsian:

$$\frac{p_e^{\mathrm{ss}}}{p_g^{\mathrm{ss}}} = e^{-\beta\hbar\omega}.$$

Energy Offset

Let $H'(t)=H(t)+c(t)I$. Show how $E$ and $\dot W$ change under this replacement. Why is this a warning about work energy accounting?

Solution

Since $\operatorname{Tr}\rho=1$,

$$E' = \operatorname{Tr}[\rho H'] = \operatorname{Tr}[\rho H]+c(t) = E+c(t).$$

The work rate changes as

$$\dot W' = \operatorname{Tr}[\rho\dot H'] = \operatorname{Tr}[\rho\dot H] + \dot c(t)\operatorname{Tr}\rho = \dot W+\dot c(t).$$

If $c(t)I$ has no observable effect except a global phase in an isolated system, then counting $\dot c(t)$ as physical work would be a convention-dependent artifact. A thermodynamic calculation must fix the energy zero and state which Hamiltonian represents the physical energy budget.

References

• R. Alicki, “The quantum open system as a model of the heat engine,” Journal of Physics A: Mathematical and General 12, L103–L107 (1979).
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002.
• M. Campisi, P. Hänggi, and P. Talkner, “Colloquium: Quantum fluctuation relations: Foundations and applications,” Reviews of Modern Physics 83, 771–791 (2011). arXiv:1012.2268.
• S. Deffner and S. Campbell, Quantum Thermodynamics: An Introduction to the Thermodynamics of Quantum Information, Morgan & Claypool, 2019.
• M. Esposito, U. Harbola, and S. Mukamel, “Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems,” Reviews of Modern Physics 81, 1665–1702 (2009).
• J. Gemmer, M. Michel, and G. Mahler, Quantum Thermodynamics, Springer, 2nd ed., 2009.
• P. Talkner, E. Lutz, and P. Hänggi, “Fluctuation theorems: Work is not an observable,” Physical Review E 75, 050102(R) (2007).
• S. Vinjanampathy and J. Anders, “Quantum thermodynamics,” Contemporary Physics 57, 545–579 (2016).