Abstract
Topological superconductors represent a fruitful playing ground for fundamental research as well as for potential applications in faulttolerant quantum computing. Especially Josephson junctions based on topological superconductors remain intensely studied, both theoretically and experimentally. The characteristic property of these junctions is their 4πperiodic groundstate fermion parity in the superconducting phase difference. Using such topological Josephson junctions, we introduce the concept of a topological Josephson heat engine. We discuss how this engine can be implemented as a Josephson–Stirling cycle in topological superconductors, thereby illustrating the potential of the intriguing and fruitful marriage between topology and coherent thermodynamics. It is shown that the Josephson–Stirling cycle constitutes a highly versatile thermodynamic machine with different modes of operation controlled by the cycle temperatures. Finally, the thermodynamic cycle reflects the hallmark 4πperiodicity of topological Josephson junctions and could therefore be envisioned as a complementary approach to test topological superconductivity.
Introduction
The promise of faulttolerant quantum computing has made topological superconductors the focus of intense research during the past decade^{1,2}. In this context, topological Josephson junctions based on nanowires^{3,4} or on topological insulators^{5,6,7,8,9,10,11} provide an alternative route for probing topological superconductivity. Their topological nature is reflected in a groundstate fermion parity that is 4πperiodic in the superconducting phase difference ϕ. Finding unambiguous experimental evidence for this 4πperiodicity still proves a difficult task, however^{12,13,14,15,16}, and it is therefore desirable to have several different and complementary approaches. Here we propose a topological Josephson heat engine implemented by a Josephson–Stirling cycle and discuss its thermodynamic^{17,18,19,20,21,22,23} properties. Using a Josephson junction based on a quantum spin Hall (QSH) insulator as an example, we show how topological Josephson junctions represent versatile thermodynamic machines with various operating modes. Moreover, the thermodynamic cycle properties reflect the 4πperiodicity of the topological ground state, distinguishing between parityconserving and nonparityconserving engines. Interestingly, we find that parity conservation generally boosts both the efficiency and power of the topological heat engine with respect to its nontopological counterpart. Our results are applicable not only to QSHbased junctions but also to any topological Josephson junction and establish topological Josephson heat engines as a novel testbed for the 4πperiodicity of the groundstate fermion parity by its entropic signature.
Results and discussion
Basic ideas
In our proposed setup (Fig. 1a), an external magnetic flux controls the superconducting phase bias ϕ across the junction. The temperature T of the QSH system is assumed to be externally modulated compared to the bath temperature T_{b}. For example, this could be done with radiative heating of the system^{24,25,26} or by having the superconductors acting as reservoirs whose temperature is controlled via resistors or superconductor/insulator/superconductor tunnel junctions^{17}.
A Josephson–Stirling cycle^{23} is composed by a sequence of (i) an isothermal phase change of ϕ = 0 → ϕ_{f} at an externally set temperature T = T_{e}, followed by (ii) an isophasic temperature change T = T_{e} → T_{b} at constant ϕ = ϕ_{f}, (iii) an isothermal phase change of ϕ = ϕ_{f} → 0 at T = T_{b}, and (iv) an isophasic temperature change T = T_{b} → T_{e} at ϕ = 0 to complete the cycle (Fig. 1b). If the reference phase ϕ_{f} is chosen as (an integer multiple of) ϕ_{f} = 2π, the work released by the engine crucially differs between a setup without fermionparity constraints and a setup with constant fermion parity. In the former case, the free energy and other thermodynamic quantities are 2πperiodic. This requires that no work or heat is generated or absorbed during each of the isothermal phase changes ϕ = 0 → 2π and ϕ = 2π → 0. If we assume, on the other hand, that the fermion parity can be kept constant throughout all processes, the thermodynamic quantities are 4πperiodic. Work and heat are then exchanged with the reservoirs during the isothermal phase changes ϕ = 0 → 2π and ϕ = 2π → 0. Hence, for ϕ_{f} = 2π a topological heat engine releases work only when parity can be conserved (Fig. 1c, d).
Model and thermodynamic properties
While the concepts outlined above are expected for any topological Josephson junction, we will discuss them explicitly for the example of a short, topological Josephson junction based on a QSH insulator. Here the pairing in the superconducting (S) regions is induced from nearby swave superconductors (see Fig. 1a, also for the coordinate system). Assuming two independent edges of the QSH system, the corresponding Bogoliubovde Gennes Hamiltonian for the QSH edge states then reads
where s = ↑/↓ ≡ ±1 describes the natural (outofplane) spin projection, σ = t/b ≡ ±1 the top and bottom edges, and τ_{j} (with j = x, y, z) denote Pauli matrices of the particlehole degrees of freedom (see Supplementary Notes I).
We study a short junction with a normal QSH region of width L_{N}, approximated by a δlike profile. The proximityinduced pairing amplitude is Δ and we use the phase convention Φ(x) = Θ(x)ϕ to describe the superconducting phase difference ϕ between the two S regions. Furthermore, \({\hat{p}}_{x}\) denotes the momentum operator, and V_{0} is the potential difference between the normal QSH and proximitized S regions. We employ a scattering approach to determine the Andreev bound states and the continuum spectrum of Eq. (1), from which we obtain the free energy—up to some additive ϕindependent contributions—as
with the Boltzmann constant k_{B} and the temperature T of the QSH states (see Supplementary Notes II and III). Here Eq. (2) arises solely from the Andreev bound state energies and the prefactor 2 takes into account contributions from the top and bottom edges.
Equation (2) describes a situation where the states of the system have equilibrium occupations without any external constraints. If fermionparity conservation is enforced, the free energy acquires an additional term and becomes
where we use the convention that p = ±1 corresponds to even and odd groundstate parity, respectively. In Eq. (3), we again omit additive ϕindependent contributions to F_{p}, which are also parity independent. The contribution
originates from the superconducting electrodes, where the energy scale E_{S} = ℏv_{F}/L_{S} is related to the total length L_{S} of the superconducting QSH edge^{7}. Following refs. ^{7,27}, we have assumed rigid boundary conditions in writing down Eqs. (2) and (3) and do therefore not take into account the inverse proximity effect since L_{S} ≫ L_{N}^{21}.
The total free energy F of the QSH junction, given by Eq. (2) or (3), allows us to calculate the total Josephson current via^{27}
where e is the elementary charge, and the entropy via
From S, one can subsequently obtain the heat capacity of the junction
Importantly, Eq. (2) is 2πperiodic in ϕ, while Eq. (3) is 4πperiodic. Consequently, the quantities derived from Eq. (2) or (3) inherit the respective periodicities. This is illustrated by Fig. 2a, b, which shows F and S for junctions without and with parity constraints. For simplicity, we assume a temperatureindependent proximity gap, Δ(T) ≈ Δ(T = 0) and ∂Δ/∂T ≈ 0, during our calculations, which is reasonably valid for the setup considered here (see Supplementary Notes III).
Thermodynamic processes
For the Josephson–Stirling cycle, we need to describe different thermodynamic processes. We study quasistatic processes, during which the system passes through quasiequilibrium states. Then the work done and heat released during a process i → f are W_{i→f} = −ℏ/(2e)∫dϕ I(ϕ, T) and Q_{i→f} = ∫dS T, respectively. The sign convention is such that W_{i→f} is positive when the system releases work to the environment, while Q_{i→f} is positive when the system absorbs heat from the environment.
For an isothermal process where ϕ is changed from ϕ_{i} → ϕ_{f} at constant T, W_{i→f} = −[F(ϕ_{f}, T) − F(ϕ_{i}, T)] and Q_{i→f} = T[S(ϕ_{f}, T) − S(ϕ_{i}, T)] can be directly obtained from Eqs. (2) and (3) and their temperature derivatives. During an isophasic process, T is changed from T_{i} → T_{f} at constant ϕ. In this case, W_{i→f} = 0, while
can be calculated from the total heat capacity. The ϕdependent contribution δC(ϕ, T) = C(ϕ, T) − C_{0}(T) can be directly calculated from Eq. (2) or (3) and its derivatives and is measured with respect to ϕ = 0. In principle, we also need to account for the ϕindependent contribution C_{0}(T) arising from the terms omitted in Eqs. (2) and (3). For additional details, we refer to Supplementary Notes IV and V, where C_{0}(T) is calculated using the Bardeen–Cooper–Schrieffer density of states.
Josephson–Stirling cycle
We are now in a position to explicitly compute the total work and heat produced during each of the processes of the Josephson–Stirling cycle introduced above (Fig. 1b). As mentioned above, W_{1→2} and W_{3→4} correspond to integrals over the current–phase relation but can also be computed directly from F. The total work W = W_{1→2} + W_{3→4} of each cycle thus coincides with the difference between the integrated areas over the current–phase relation (Fig. 3a, b). The heat exchanged with the hot (T = T_{e}) and cold reservoirs (T = T_{b}) is Q_{e} = Q_{1→2} + Q_{4→1} and Q_{b} = Q_{2→3} + Q_{3→4}, respectively. Conservation of energy dictates W = Q, where Q = Q_{e} + Q_{b} is the total heat exchange during the cycle. Note that, in our setup, there are no separate hot and cold reservoirs, but the environment acts successively as hot and cold reservoir.
In Fig. 3c, we show W as a function of the reference phase ϕ_{f} and compare the case without and with parity constraints. Without parity conservation, W is maximal for ϕ_{f} = π, whereas W = 0 for ϕ_{f} = 2π. The latter is a consequence of the 2πperiodicity of F_{0}(ϕ, T) = F_{0}(ϕ + 2π, T), causing W_{1→2} = −[F_{0}(ϕ_{f}, T_{e}) − F_{0}(0, T_{e})] and W_{3→4} = −F_{0}(0, T_{b}) − F_{0}(ϕ_{f}, T_{b})] to each vanish for ϕ_{f} = 2π. If fermion parity is kept constant, on the other hand, F_{p}(ϕ, T) = F_{p}(ϕ + 4π, T) and W is maximal for ϕ_{f} = 2π. A topological heat engine with ϕ_{f} = 2π thus releases work only if parity is conserved and can thus serve as a test for the 4πperiodicity of the groundstate fermion parity.
Different operating modes
Until now, we have focused only on an engine. Depending on the relative values of T_{e} and T_{b}, the Josephson–Stirling cycle can, however, exhibit also other operating modes. This is illustrated by Fig. 4, which shows W and cycle efficiencies for different combinations of T_{e} and T_{b} without (Fig. 4a, b) and with (Fig. 4c, d) parity constraints. Here ϕ_{f} is chosen to yield the maximal work, that is, ϕ_{f} = π without parity constraints (Fig. 4a) and ϕ_{f} = 2π if parity is conserved (Fig. 4c).
For T_{e} > T_{b}, the Josephson–Stirling cycle/machine acts as an engine: The machine absorbs Q_{e} > 0 from the hot reservoir and releases ∣Q_{b}∣ < Q_{e} to the cold reservoir. Hence, W > 0 is done on the environment and the engine efficiency is given by η = W/Q_{e}. A comparison of the engine efficiency and maximal power shows that a parityconserving engine is on average more efficient and more powerful than its nonparityconserving implementation (Fig. 4b, d). We interpret the stronger power as due to an increased phase space available: to obtain a finite work, the work integral can be integrated over a 0–2π range if parity is preserved, whereas one needs to remain within the 0–π range without parity conservation. Second, the lower efficiency of the nonparityconserving engine can be understood as due to the competition between mutually exclusive processes with opposite parities. Indeed, Eq. (3) shows that the additional parityrelated terms contribute with different signs to F_{p}, implying opposite contributions to the heat exchange. Consequently, the nonparitypreserving engine can be interpreted as a thermal machine composed of two mutually exclusive engines working in an opposite manner, thereby reducing the total efficiency.
If T_{e} < T_{b}, the systems with and without parity conservation act as refrigerators with a coefficient of performance COP = Q_{e}/∣W∣^{23} or as Joule or cold pumps. Controlling T_{e} vs T_{b} thus enables multiple operating modes of the Josephson–Stirling cycle. If the cycle is set up as in Fig. 1b, refrigerators as well as Joule and cold pumps require that T_{e} < T_{b}. While it is possible by superconductor/insulator/superconductor cooling to bring T_{e} below T_{b}^{17}, a more promising way to realize refrigerators, Joule, or cold pumps is to shift the cycle by interchanging the initial and final phases, ϕ = 0 and ϕ = ϕ_{f}. Such a setup implies the same phase diagrams as in Fig. 4 but with T_{e} and T_{b} interchanged (see Supplementary Notes VI). Hence the “shifted” Josephson–Stirling cycle can be used to realize operating modes other than engines, making it a highly versatile thermodynamic machine.
Potential routes to experimental realization
Importantly, a topological Josephson heat engine implemented as a Josephson–Stirling cycle can be used to test the hallmark 4πperiodicity of the phasedependent groundstate fermion parity. In this implementation, a major challenge is to quickly modulate the temperature of the proximitized QSH junction while preserving its fermion parity. While this condition precludes galvanic channels of heat transfer to the QSH system, others such as phononic^{28}, photonic^{29,30}, or radiative^{24} channels could be used. In such setups, the timescale of the temperature modulation could be as low as 0.1 ns (see Supplementary Notes VII), much smaller than typical quasiparticle poisoning timescales, which are of the order of 1 μs^{31,32,33}. Hence, implementing a parity conserving Josephson–Stirling cycle appears feasible. The work per cycle can be experimentally determined by measuring the current–phase relation during each isothermal process to compute the work integrals. Such current–phase measurements have, for example, been successfully performed in topological junctions with scanning superconducting quantum interference device microscopes^{34,35} (see also Supplementary Notes VIII).
Conclusions
In this manuscript, we have explicitly discussed topological Josephson heat engines for the example of a short QSHbased Josephson junction. Since the concept is only based on the 4πperiodicity of the groundstate fermion parity, it is also applicable to long as well as nanowirebased topological Josephson junctions. Schemes to detect signatures of the 4πperiodic groundstate parity in topological Josephson junctions often require careful interpretation of the measurements^{36,37}. It is therefore desirable to have several different, complementary approaches. As their properties reflect the 4πperiodic groundstate of topological Josephson junctions, topological Josephson heat engines could be used as such a complementary setup to test the topological superconductivity as long as fermion parity can be conserved. However, even without conserving the groundstate fermion parity, topological Josephson junctions represent versatile machines with various operating modes.
Data availability
The data that support the findings of this study are available within the paper and its Supplementary Information. Additional data are available from the corresponding author upon request.
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Acknowledgements
B.S. and E.M.H. acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through SFB 1170, ProjectID 258499086, through Grant No. HA 5893/41 within SPP 1666 and through the WürzburgDresden Cluster of Excellence on Complexity and Topology in Quantum Matter – ct.qmat (EXC 2147, ProjectID 390858490) as well as by the ENB Graduate School on Topological Insulators. E.S., A.B., and F.G. acknowledge partial financial support from the EU’s Horizon 2020 research and innovation program under Grant Agreement No. 800923 (SUPERTED), the CNRCONICET cooperation program “Energy conversion in quantum nanoscale hybrid devices,” the SNSWIS jointlab QUANTRA funded by the Italian Ministry of Foreign Affairs and International Cooperation, and the Royal Society through the International Exchanges between the UK and Italy (Grant Nos. IES R3 170054 and IEC R2192166). This publication was supported by the Open Access Publication Fund of the University of Würzburg.
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E.H. and F.G. conceived the idea of applying the concepts of heat engines to topological Josephson junctions. These concepts were then further developed and refined into the theoretical formulation presented in this work after intense discussions with B.S., A.B., and E.S. B.S. performed the calculations. All authors contributed to the writing of the manuscript.
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Scharf, B., Braggio, A., Strambini, E. et al. Topological Josephson heat engine. Commun Phys 3, 198 (2020). https://doi.org/10.1038/s42005020004636
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