Quantum thermodynamics studies heat, work and entropy at the scale of single quantum systems. Its devices — heat engines, refrigerators, batteries — are driven by time-dependent controls: a trap frequency that is compressed and expanded, couplings to hot and cold baths switched on and off, a field that charges a battery. Choosing these controls is an optimal control problem with thermodynamic objectives: extracted work, power, efficiency, entropy production, or the size of fluctuations [1, 2].

Engines as control problems

  • Origins. A three-level maser can be viewed as a heat engine, with an efficiency bounded by Carnot’s [3]. Open quantum systems driven by external fields and coupled to baths give a general model of heat engines [4].
  • The Otto cycle. In the quantum Otto cycle, a working medium (often a harmonic oscillator) is compressed and expanded by changing its Hamiltonian, and thermalized with two baths in between. When the Hamiltonian does not commute with itself at different times, fast strokes create excitations that cost work. This irreversible loss of power, called “quantum friction”, is of quantum origin [5]. At high temperature and in the quasistatic regime, the efficiency at maximum power coincides with the Curzon–Ahlborn value of endoreversible engines [5, 6]. The harmonic Otto cycle has been reviewed in [7].
  • Experiments. A single ion in a tapered Paul trap, coupled to engineered laser reservoirs, was proposed as a heat engine [8] and then realized [9]. With NV centres in diamond, two predicted effects of coherence in the working medium were observed: more power than an equivalent classical engine at small engine action, and the thermodynamic equivalence of different engine types [10].

Shortcuts against friction

Shortcuts to adiabaticity make the compression and expansion strokes fast without final excitations. This gives engine cycles that produce finite power with no friction [11]. It also raises power and efficiency and suppresses work fluctuations [12]. The shortcut fields are not free: the energetic cost of counterdiabatic driving grows as the duration shrinks [13]. Whether that cost is counted decides whether a shortcut pays off.

Optimal finite-time cycles

  • Thermodynamic length. For slow driving, the dissipated availability is governed by a metric on the space of equilibrium states, the thermodynamic length [14]. For Lindblad dynamics, minimal-dissipation protocols are geodesics of a corresponding metric [15]. Perturbation theory for slowly driven master equations gives the efficiency at maximum power of finite-time Carnot engines [16]. See Geometric & time-optimal control.
  • Optimal cycles. For low-dissipation Carnot engines, the optimal strategy is to run small cycles around a working point. The best point does not depend on how power and efficiency are weighted. Many-body working media can reach Carnot efficiency at finite power per constituent in the thermodynamic limit [17].
  • Several objectives. Power, efficiency and stability cannot all be maximized at once. Reinforcement learning has found the Pareto-optimal cycles of a quantum-dot engine, revealing abrupt changes in the optimal cycle shape [18]. See Reinforcement & machine learning for control.

Batteries

A quantum battery stores energy that can later be extracted as work. Entangling operations can extract more work from many copies than independent ones [19]. For charging in finite time, optimal protocols exist analytically for a qubit. Global operations on \(N\) qubits give an \(N\)-fold advantage in power per qubit, which numerical optimal control confirms [20]. This collective advantage is bounded when only \(k\) batteries interact at a time [21].

Feedback and fundamental limits

  • Feedback. With measurement and feedback, a Maxwell’s demon, the extractable work can exceed the free-energy difference by an amount set by the mutual information between system and controller. The second law is restored once the cost of information processing is included [22]. See Measurement-based feedback.
  • Cooling. The third law limits how fast a system can be cooled. The attainable temperature scales as an inverse power of the cooling time [23]. Cooling and reset are also State preparation & transfer and Control of open quantum systems problems.

Limitations

  • Definitions of work and heat are not unique when the system is strongly coupled to its baths or carries coherences, so results depend on the chosen definition.
  • Weak-coupling Markovian master equations, the usual models, can break down for fast strokes and fast switching of bath couplings.
  • The energy and entropy cost of the controller itself is often left out, although it can dominate at small scales.