Entropy transport through a superfluid quantum point contact: A Keldysh field-theory approach

The gist: The study describes matter and entropy transport between two ultra-cold neutral Fermi-gas reservoirs linked by a quantum point contact under a chemical-potential gradient, confirming well-known particle current behavior while extending the computation to entropy current in the BCS regime.

Model and Formalism

The research investigates a model consisting of two superfluid reservoirs coupled via a quantum point contact (QPC), described within the framework of standard BCS mean-field theory. The Hamiltonian in the interaction picture is given by Eq. (1), which incorporates terms for particle dynamics and tunneling between the leads. The system is modeled using a Nambu-spinor representation, leading to a more compact Hamiltonian in Eq. (2). The particle and heat currents are extracted by applying the Heisenberg equation of motion to the particle number and heat operators, resulting in expressions for the change of rate of particle and heat on the left reservoir.

Current Calculation

The calculation involves evaluating two-body operators for the rate of heat transfer in frequency space, which are then averaged over time to yield DC components. The rates of particle and heat transfer are given by Eq. (10), where the central quantity to evaluate is the correlation between the left and right fields. The total current of interest is defined as the difference between the change of rate of particle and heat associated with the left reservoir, J(c)n, and those associated with the right reservoir, J(d)n. The entropy current is related to the heat current through thermodynamics considerations as explained in Appendix A.

Transport Regimes

The study analyzes results for a wide range of values of the junction’s transparency, focusing on high (in Sec. IV) and low (in Sec. V) transparency regimes, with a more detailed focus given to the ballistic case. In the high-transparency regime, the particle current is shown in Fig. 6, where a clear enhancement in the current even at small chemical potential differences is observed, consistent with ballistic channel behavior. In contrast, the entropy current shows an oscillatory behavior (see right panel of the same figure).

Oscillatory Behavior Mechanism

The oscillatory behavior observed deep within the superconducting gap is attributed to multiple Andreev reflections (MARs). The competition between two mechanisms—Andreev composite reflection, in which a particle is effectively reflected as a hole with opposite spin within the same reservoir, and Andreev composite tunneling, in which a particle ultimately tunnels into the opposite reservoir—is responsible for the predicted rich nonmonotonic behavior of the entropy current. The contribution from odd MAR (oMAR) processes provides a larger heat contribution than eMAR ones, leading to the oscillatory behavior when oMAR contribution is reduced until the next process is activated.

Comparison with Experiments

The BCS mean-field approach successfully captures the particle current as measured in experiments, but the entropy current exhibits clear quantitative deviations. The results indicate that such a relation between chemical potential and temperature gradient does not hold in the BCS regime, highlighting a distinct mechanism for transport in the BCS side. Furthermore, in the ballistic limit, the entropy current is found to be approximately two orders of magnitude smaller than experimentally observed values, suggesting that additional contributions—such as pairing fluctuations or beyond–mean-field effects—play an essential role.

Limitations and Extensions

The study goes beyond the linear dispersion approximation by considering the full quadratic dispersion relation to overcome the limitation of suppressing the heat current due to particle-hole symmetry. The analysis suggests that considering two different superconducting gaps, ∆c ≠ ∆d, with either linear or quadratic dispersion can produce a distinct and larger contribution compared to the oscillations observed in the quadratic dispersion case. The paper also notes that the cited experiments are performed in cold-atom setups in the unitary Fermi gas regime, which is intrinsically distinct from the weak-coupling BCS framework employed in this analysis.

Thermodynamic Derivations

The entropy current is related to the heat current through Eq. (12), which shows that JS = 1/4T squared - ∆T squared (4T JH + ∆µ∆T JN). The derivation of the quantum case for the entropy current involves combining Eq. (A6), Eq. (A7), and Eq. (A8) to derive the expression for the entropy current in the quantum case. The thermodynamic transport equations are derived from which we can extract the particle and entropy currents.

Transmission Function

The transmission function T 2(E) is defined in the metallic limit by Eq. (B3), which simplifies to T squared = 4(τ /W) squared / (1 + (τ /W) 2) squared. The general expression for the tunneling coupling τ is given by Eq. (B4). For the superconducting case, the LDoS is obtained in Eq. (30). The analysis of the truncated system with ωcutoff = 0 shows that the oscillatory behavior remains present even in the absence of a divergent LDoS at the gap edge. The results are consistent with Fig. 12, which shows entropy current behavior of the truncated SS junction with ωcutoff = 0, at transparency T = 0.99.

Conclusion

The competition between the two mechanisms of Andreev composite reflection and Andreev composite tunneling is responsible for the predicted rich nonmonotonic behavior of the entropy current. The paper concludes that focusing on the ballistic limit reveals pronounced oscillations arising from MARs, revealing a rich structure of subgap entropy transport that goes well beyond a simple monotonic response. The comparison with experimental results suggests that additional contributions—such as pairing fluctuations or beyond–mean-field effects—play an essential role. The work provides a clear direction for future theoretical extensions, particularly regarding the distinction between the BCS regime and the unitary Fermi system. The paper notes that even when including a higher number of MAR processes, the current will ultimately start to converge to zero around the same bias due to the non-unitary value of T. The final conclusion is that Andreev composite reflection provides a larger heat contribution than eMAR ones, which explains the oscillatory behavior. The results are consistent with Fig. 11, which shows the local density of state (LDoS) for the truncated system at ωcutoff = 0 as a function of the frequency with EF = 20∆. The paper concludes that focusing on the ballistic limit reveals pronounced oscillations arising from MARs, revealing a rich structure of subgap entropy transport that goes well beyond a simple monotonic response. The competition between the two mechanisms of Andreev composite reflection and Andreev composite tunneling is responsible for the predicted rich nonmonotonic behavior of the entropy current.

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The gist: The study describes matter and entropy transport between two ultra-cold neutral Fermi-gas reservoirs linked by a quantum point contact under a chemical-potential gradient, confirming well-known particle current behavior while extending the computation to entropy current in the BCS regime.

How it works

1. The system is modeled using a Hamiltonian incorporating both particle dynamics and tunneling terms between leads.

2. Particle and heat currents are extracted by applying the Heisenberg equation of motion to the respective operators.

3. The DC component of the current is obtained by taking the limit t → t′ in frequency space, yielding rates given by Eq. (10).

1. Improved AI system can accurately model out-of-equilibrium transport phenomena in quantum systems by implementing Keldysh field-theory approaches to derive current-bias characteristics for both particle and entropy transport.

2. Improved AI system can predict oscillatory behavior in the entropy current at low voltage in the ballistic junction limit by analyzing the competition between even and odd multiple Andreev reflection processes.

3. Improved AI system can distinguish between transport mechanisms contributing to the entropy current, specifically differentiating between Andreev composite reflection (oMAR) and tunneling (eMAR) based on their heat contributions.

4. Improved AI system can quantify how increasing transparency affects the entropy current by modeling the behavior of the transmission function T as it moves from low to high values.

5. Improved AI system can predict the nonmonotonic behavior of the entropy current within a superconducting gap by analyzing how multiple Andreev reflection processes are activated at specific chemical potential gradients.

We study the matter and entropy transport between two ultra-cold neutral Fermi-gas reservoirs linked by a quantum point contact under a chemical-potential gradient. We describe the two leads with a BCS mean-field model and derive the current-bias characteristics for both particle and entropy transport. We compute the out of equilibrium steady-state currents by using the Keldysh formalism. In accordance with previous works in the literature, we confirm the well-known behavior for the particle current and extend the computation to the entropy current in the BCS regime. The entropy current shows an oscillatory behavior at low voltage in the ballistic junction limit. We analyze the results for a wide range of values of the junction's transparency. We also compare our findings with experimental results in cold atomic gases in the unitary regime.