Quantum Heat Engines and Refrigerators

A quantum heat engine is a thermodynamic machine whose working medium is described quantum mechanically. The working medium may be a two-level system, harmonic oscillator, spin ensemble, atom-cavity system, quantum dot, superconducting circuit, or another small system coupled to reservoirs and controls.

The word “quantum” does not suspend the ordinary thermodynamic questions. One still asks:

• which reservoirs provide heat;
• which degree of freedom stores or receives work;
• whether the process is cyclic or steady;
• whether the heat currents are defined by a thermodynamically consistent model;
• whether coherence, measurement, or finite-time driving changes power or irreversibility.

Quantum heat engines are most useful as precise models of microscopic energy conversion. They are not loopholes around Carnot’s bound. When the reservoirs are genuine thermal reservoirs and the accounting is complete, the usual second-law bounds reappear.

Heat Flows and Extracted Work

This chapter uses $Q_\alpha$ for heat transferred into the working medium from reservoir $\alpha$. For a cyclic engine between a hot bath $h$ and cold bath $c$,

$$\Delta E_{\mathrm{cycle}} = 0 .$$

Let

$$Q_h\gt0, \qquad Q_c\lt0,$$

where $Q_h$ is heat absorbed from the hot bath and $Q_c$ is heat dumped into the cold bath. Positive extracted work is

$$W_{\mathrm{ext}} = Q_h+Q_c .$$

The efficiency is

$$\eta = \frac{W_{\mathrm{ext}}}{Q_h} = 1+\frac{Q_c}{Q_h}.$$

The same process described using work supplied to the system would use $W_{\mathrm{sup}}=-W_{\mathrm{ext}}$. Many sign mistakes in heat-engine papers are just convention changes that were not stated clearly.

Carnot Bound from Entropy Production

For an ideal cycle coupled only to two thermal reservoirs at temperatures $T_h$ and $T_c$, with $T_h\gt T_c$, the working medium returns to its initial state. The total entropy production is therefore

$$\Sigma = - \frac{Q_h}{T_h} - \frac{Q_c}{T_c} \ge0 .$$

Since $Q_h\gt0$, this implies

$$Q_c \le - \frac{T_c}{T_h}Q_h .$$

Substituting into the efficiency gives

$$\eta \le 1-\frac{T_c}{T_h} \equiv \eta_{\mathrm C}.$$

This derivation is deliberately compact. It shows what must be true for the bound to apply: the reservoirs must be thermal, all entropy changes outside the working medium must be included, and the cycle must really close. If a squeezed reservoir, measurement record, chemical potential, or battery is present, it must be counted as an additional resource rather than hidden inside the words “hot bath.”

Refrigerators and Heat Pumps

A refrigerator uses supplied work to move heat from a cold reservoir to a hot reservoir. It is often clearer to use positive quantities

$$Q_c^{\mathrm{in}}\gt0, \qquad W_{\mathrm{in}}\gt0,$$

where $Q_c^{\mathrm{in}}$ is heat extracted from the cold bath and $W_{\mathrm{in}}$ is work supplied by the external source. The heat delivered to the hot bath is

$$Q_h^{\mathrm{out}} = Q_c^{\mathrm{in}}+W_{\mathrm{in}} .$$

The coefficient of performance for refrigeration is

$$\mathrm{COP}_{\mathrm R} = \frac{Q_c^{\mathrm{in}}} {W_{\mathrm{in}}}.$$

For two ideal thermal reservoirs,

$$\mathrm{COP}_{\mathrm R} \le \frac{T_c} {T_h-T_c}.$$

The same device can be called a heat pump when the useful output is heat delivered to the hot reservoir. Then

$$\mathrm{COP}_{\mathrm{HP}} = \frac{Q_h^{\mathrm{out}}} {W_{\mathrm{in}}} \le \frac{T_h} {T_h-T_c}.$$

Again, the bound assumes ideal thermal reservoirs and complete resource accounting.

Stroke Engines and Autonomous Engines

Quantum heat engines are commonly grouped into two architectural classes.

Stroke engines separate the tasks into time intervals:

• thermalization with a hot reservoir;
• isolated unitary driving;
• thermalization with a cold reservoir;
• another isolated unitary stroke.

The quantum Otto engine is the standard example. Its clean separation makes it useful pedagogically and experimentally, especially in trapped ions, spins, superconducting circuits, and oscillator platforms.

Autonomous engines operate continuously. Different transitions or subsystems are coupled to different reservoirs, and work is stored in a mode, load, field, or battery. The three-level maser is the classic autonomous model.

The two architectures can share the same thermodynamic efficiency in ideal limits, but they distribute irreversibility differently. Stroke engines emphasize finite-time driving and quantum friction. Autonomous engines emphasize steady-state currents, spectral filtering, and load modeling.

Master-Equation Description

In weak-coupling Markovian models, the working medium may obey

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

where each dissipator $\mathcal L_\alpha^t$ represents a reservoir. A heat current into the system is often defined as

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

The power supplied by explicit driving is

$$\dot W_{\mathrm{sup}} = \operatorname{Tr} \left[ \rho\frac{dH}{dt} \right].$$

The instantaneous first-law balance is then

$$\frac{d}{dt} \operatorname{Tr}[\rho H] = \dot W_{\mathrm{sup}} + \sum_\alpha \dot Q_\alpha .$$

For a periodically driven steady cycle, the average internal-energy change over one period vanishes, so

$$\overline{\dot W}_{\mathrm{ext}} = \sum_\alpha \overline{\dot Q}_\alpha, \qquad \overline{\dot W}_{\mathrm{ext}} = - \overline{\dot W}_{\mathrm{sup}} .$$

These energy currents have a thermal interpretation only if the dissipators are physically tied to the stated reservoirs. In the instantaneous thermal model written here, each reservoir generator must be consistent with the Hamiltonian used in the heat current. Rapid periodic driving can instead require a Floquet derivation with drive sidebands; assigning all jump energy to an instantaneous gap can miss drive work. See Thermal Master Equations and Detailed Balance.

Two-Level Quantum Otto Engine

Let the working medium be a two-level system with Hamiltonian

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

The Otto cycle uses two gaps:

$$\omega_h\gt\omega_c .$$

The four ideal strokes are:

• hot isochore at gap $\omega_h$;
• isolated expansion from $\omega_h$ to $\omega_c$;
• cold isochore at gap $\omega_c$;
• isolated compression from $\omega_c$ to $\omega_h$.

Let the excited-state populations after ideal hot and cold thermalization be

$$p_h = \frac{1} {1+e^{\beta_h\hbar\omega_h}}, \qquad p_c = \frac{1} {1+e^{\beta_c\hbar\omega_c}}.$$

The heat absorbed from the hot bath is

$$Q_h = \hbar\omega_h(p_h-p_c).$$

The heat dumped into the cold bath is

$$Q_c = \hbar\omega_c(p_c-p_h).$$

The net extracted work is

$$W_{\mathrm{ext}} = Q_h+Q_c = \hbar(\omega_h-\omega_c)(p_h-p_c).$$

The device acts as an engine when

$$p_h\gt p_c,$$

equivalently

$$\beta_h\omega_h \lt \beta_c\omega_c .$$

The ideal efficiency is

$$\eta_{\mathrm{Otto}} = \frac{W_{\mathrm{ext}}}{Q_h} = 1-\frac{\omega_c}{\omega_h}.$$

The engine condition ensures that this does not exceed the Carnot efficiency:

$$1-\frac{\omega_c}{\omega_h} \le 1-\frac{T_c}{T_h}.$$

For this particular Hamiltonian, all $H(t)$ commute: changing only the gap leaves the eigenvectors fixed. An isolated gap ramp therefore preserves populations at any speed and creates no coherence from a diagonal state. Nonadiabatic excitations arise in more general drives whose eigenvectors rotate, or in the physical oscillator frequency ramp below. Removing coherence by dephasing can itself dissipate useful energy or free energy; it is not a free reversal of that excitation.

Harmonic Oscillator Otto Engine

For a harmonic oscillator,

$$H(\omega) = \hbar\omega \left( a_\omega^\dagger a_\omega+\frac{1}{2} \right).$$

The ladder operator $a_\omega$ depends on frequency because the physical Hamiltonian is $p^2/(2m)+m\omega^2x^2/2$. Hamiltonians at different frequencies do not commute. The thermal mean occupation is

$$\bar n(\beta,\omega) = \frac{1} {e^{\beta\hbar\omega}-1}.$$

In an ideal adiabatic Otto cycle, the occupation number is unchanged during frequency ramps. If

$$\bar n_h = \bar n(\beta_h,\omega_h), \qquad \bar n_c = \bar n(\beta_c,\omega_c),$$

then

$$Q_h = \hbar\omega_h(\bar n_h-\bar n_c), \qquad Q_c = \hbar\omega_c(\bar n_c-\bar n_h).$$

The zero-point contribution cancels over the complete cycle. The extracted work is

$$W_{\mathrm{ext}} = \hbar(\omega_h-\omega_c) (\bar n_h-\bar n_c),$$

and the ideal efficiency is again

$$\eta_{\mathrm{Otto}} = 1-\frac{\omega_c}{\omega_h}.$$

The oscillator version is common in ion traps, optomechanics, nanomechanical systems, and circuit resonators. It also makes clear why “higher temperature” is not enough by itself; the relevant comparison is the occupation at the two different gaps.

Three-Level Maser Engine

The three-level maser is the classic autonomous quantum heat engine. Let the levels be

$$E_1=0, \qquad E_2=\hbar\omega_c, \qquad E_3=\hbar\omega_h,$$

with

$$\omega_h\gt\omega_c, \qquad \omega_w=\omega_h-\omega_c .$$

The hot bath couples the $1\leftrightarrow3$ transition, the cold bath couples the $1\leftrightarrow2$ transition, and the work output is associated with the $3\leftrightarrow2$ transition. Each hot quantum of energy $\hbar\omega_h$ can be converted into one work quantum $\hbar\omega_w$ plus one cold-bath quantum $\hbar\omega_c$.

The ideal efficiency is therefore

$$\eta = \frac{\hbar\omega_w} {\hbar\omega_h} = 1-\frac{\omega_c}{\omega_h}.$$

The engine regime requires population inversion on the work transition:

$$p_3\gt p_2 .$$

In the zero-load limit, the separate hot and cold transitions establish the ratios below. The resulting inversion gives the threshold for small-signal gain:

$$e^{-\beta_h\hbar\omega_h} \gt e^{-\beta_c\hbar\omega_c},$$

or

$$\beta_h\omega_h \lt \beta_c\omega_c .$$

The same inequality bounds the ideal tightly coupled efficiency by Carnot. Under finite loading, the populations need not retain the separate Gibbs ratios. Moreover, energy deposited in an output mode is not automatically all work: its entropy and extractable energy depend on the load dynamics. A power calculation must model the field or battery explicitly.

Power, Efficiency, and Tradeoffs

Efficiency is not the only figure of merit. A reversible engine can approach Carnot efficiency only with vanishing entropy production, which usually means vanishing power in ideal quasistatic limits. Practical microscopic engines are judged by several quantities:

• output power;
• efficiency or coefficient of performance;
• stability of the output load;
• work fluctuations;
• entropy production;
• control cost;
• robustness to parameter uncertainty;
• heat leaks and parasitic transitions.

For periodically driven engines, average power is

$$P_{\mathrm{out}} = \frac{W_{\mathrm{ext}}}{\tau_{\mathrm{cyc}}},$$

where $\tau_{\mathrm{cyc}}$ is the cycle time. For steady autonomous engines, power is an energy current into the work storage degree of freedom.

Quantum coherence can increase or decrease power depending on the model. It can enable coherent energy exchange with a load, but it can also create nonadiabatic excitations that reduce useful work. A trustworthy claim must specify the Hamiltonian, reservoirs, control protocol, and load.

Refrigeration Modes

The same quantum machine can often operate in several modes depending on parameters. For the two-level Otto model, changing frequency ratios and bath temperatures can switch among:

• engine operation, with $Q_h\gt0$ and $W_{\mathrm{ext}}\gt0$;
• refrigerator operation, with heat removed from the cold bath while work is supplied;
• heat pump operation, where useful output is heat delivered to the hot bath;
• dissipator operation, where supplied work is simply degraded into heat.

For absorption refrigerators, no externally driven work stroke is required. Instead, a third reservoir or transition plays the role of a work resource. Such machines are autonomous, and their performance must be bounded using entropy production for all reservoirs, not only a hot and cold pair.

Exercises

Carnot Bound with the Chapter Convention

Assume a cyclic engine absorbs $Q_h\gt0$ from a hot bath and has $Q_c\lt0$ for the cold bath, with no other reservoirs. Using

$$\Sigma = - \frac{Q_h}{T_h} - \frac{Q_c}{T_c} \ge0,$$

derive the Carnot efficiency bound.

Solution

The entropy-production inequality gives

$$- \frac{Q_c}{T_c} \ge \frac{Q_h}{T_h}.$$

Multiplying by $T_c$,

$$-Q_c \ge \frac{T_c}{T_h}Q_h .$$

Thus

$$Q_c \le - \frac{T_c}{T_h}Q_h .$$

The extracted work is $W_{\mathrm{ext}}=Q_h+Q_c$, so

$$\eta = \frac{W_{\mathrm{ext}}}{Q_h} = 1+\frac{Q_c}{Q_h} \le 1-\frac{T_c}{T_h}.$$

Two-Level Otto Engine Condition

For the two-level Otto engine, show that the engine condition $p_h\gt p_c$ is equivalent to

$$\beta_h\omega_h \lt \beta_c\omega_c .$$

Solution

The excited-state population of a thermal two-level system is

$$p(\beta,\omega) = \frac{1} {1+e^{\beta\hbar\omega}}.$$

This function decreases monotonically with $\beta\omega$. Therefore

$$p_h\gt p_c$$

is equivalent to

$$\beta_h\hbar\omega_h \lt \beta_c\hbar\omega_c .$$

Since $\hbar\gt0$, this is the same as

$$\beta_h\omega_h \lt \beta_c\omega_c .$$

A Numerical Otto Cycle

Set $\hbar=k_B=1$, $T_h=3$, $T_c=1$, $\omega_h=2$, and $\omega_c=1$. Compute the ideal qubit cycle’s heats, work, efficiency, and entropy production.

Solution

$p_h=1/(1+e^{2/3})\simeq0.33924$ and $p_c=1/(1+e)\simeq0.26894$. Hence

$$Q_h\simeq0.14060,\quad Q_c\simeq-0.07030,\quad W_{\mathrm{ext}}\simeq0.07030.$$

The efficiency is $1/2$, below the Carnot value $2/3$. The entropy production is $-Q_h/T_h-Q_c/T_c\simeq0.02343>0$. At the boundary $\omega_c/\omega_h=T_c/T_h$, the population difference and extracted work vanish; the limiting efficiency formula should not be mistaken for a finite-output Carnot engine.

Three-Level Maser Inversion

Assume the hot transition gives $p_3/p_1=e^{-\beta_h\hbar\omega_h}$ and the cold transition gives $p_2/p_1=e^{-\beta_c\hbar\omega_c}$. Derive the condition for population inversion on the work transition.

Solution

Population inversion on the work transition means

$$p_3\gt p_2 .$$

Using the two ratios,

$$\frac{p_3}{p_1} \gt \frac{p_2}{p_1}$$

becomes

$$e^{-\beta_h\hbar\omega_h} \gt e^{-\beta_c\hbar\omega_c}.$$

Taking logarithms and reversing the sign after multiplying by $-1$ gives

$$\beta_h\omega_h \lt \beta_c\omega_c .$$

This is the same condition that appears in the two-level Otto engine.

References

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