Fully-Balanced Heat Interferometer

A tunable and balanced heat interferometer is proposed and analyzed. The device consists of two superconductors linked together to form a double-loop interrupted by three Josephson junctions coupled in parallel. Both superconductors are held at different temperatures allowing the heat currents flowing through the structure to interfere. As we show here, thermal transport is coherently modulated through the application of a magnetic flux. Furthermore, such modulation can be tailored at will through the application of an extra control flux. In addition we show that, provided a proper choice of the system parameters, a fully balanced interferometer is obtained. The latter means that the phase-coherent part of heat current can be controlled to the extent of being fully suppressed. Such a device allows for a versatile operation appearing, therefore, as an attractive key to the onset of low-temperature coherent caloritronic circuits.

A tunable and balanced heat interferometer is proposed and analyzed. The device consists of two superconductors linked together to form a double-loop interrupted by three parallel-coupled Josephson junctions. Both superconductors are held at different temperatures allowing the heat currents flowing through the structure to interfere. We demonstrate that thermal transport is coherently modulated through the application of a magnetic flux. Furthermore, such modulation can be tailored at will or even suppressed through the application of an extra control flux. Such a device allows for a versatile operation appearing as an attractive key to the onset of low-temperature coherent caloritronic circuits.
Manipulation of heat currents in mesoscopic circuits represents a youthful field of research. 1,2 The role of dissipative phenomena at the nanoscale and the benefits that can be derived from them are becoming to be understood. Starting from cooling applications and fine temperature tuning in cryogenic detectors, 1 or superconducting circuits for quantum computing, 3 and ending with the emergence of caloritronic circuits. [4][5][6][7][8][9][10][11][12][13] Towards this end, a magnetic flux-controllable superconducting heat interferometer was recently theoretically conceived 14 and subsequently realized experimentally. 15 This achievement served to show, on the one hand, how quantum coherence between two weak-linked superconducting condensates extends also to dissipative observables such as heat current. 16 On the other, this device might constitute the building block for the implementation of superconducting hybrid coherent caloritronic circuits like, for instance, thermal modulators, heat transistors and splitters, etc. In this Letter we propose a step forward this goal by envisioning and theoretically analyzing a fullybalanced heat interferometer. Compared to the previous realization, 14,15 our device provides an enhanced control over the flux-to-heat current transfer function therefore enabling the choice of convenient operation points for different applications. Additionally, this fully-balanced heat interferometer is much more robust against fabrication deficiencies such as differences between the normal-state resistances of the Josephson junctions. Such deficiencies can be now "corrected" during the operation therefore allowing the maximization of the phase-dependent component of heat current (J int ) or its complete annihilation.
Our thermal circuit consists of two superconductors S 1 and S 2 , weak linked forming a double-loop interrupted by three parallel Josephson junctions (see Fig. 1). Let us denote R i the normal-state resistance of junction i, and ϕ i the macroscopic phase difference across junction i. This structure behaves as a conventional superconducting quantum interference device (SQUID) pierced by a magnetic flux Φ 1 in which one of the junctions has been replaced by a DC SQUID. The characteristics of Our superconducting circuit consists of three parallel Josephson junctions (characterized by their Josephson currents ia, i b and ic, respectively) that define a double-loop heat interferometer. The temperature in S1 is risen up to TS 1 ≥ TS 2 which yields a steady-state heat current Jtot flowing from S1 to S2. Moreover, the voltage drop across the device is set to zero. The main loop is pierced by a magnetic flux Φ1 whereas a control flux Φ2 is applied through the second loop enabling enhanced performance. this second "junction" can be tuned thanks to the application of a control magnetic flux Φ 2 . The system is temperature biased by setting the temperature in S 1 to be T S1 ≥ T S2 , T S2 being the temperature in S 2 . Furthermore, the voltage drop across the whole structure is set to zero. Under these circumstances, a thermal gradient arises across the junctions and a stationary heat current J tot will flow from S 1 to S 2 , which are in steady-state thermal equilibrium. [16][17][18] As it was argued in Ref. 14, J tot results from the sum of two terms, Here, J qp is the heat current carried by quasiparticles that depends only on the temperatures of both superconductors. On the other hand, J int is the interference component of the heat current that depends on the temperature of both superconductors and on the macroscopic phase difference across each junction 16 as well. This phase-dependent term is peculiar of weakly-coupled superconductors, and arises as a consequence of the interplay between Cooper pairs and quasiparticles on tunneling events through Josephson junctions. The Cooper pair condensate carries no entropy therefore leading to a zero contribution to the heat current. 16,17 In the following we shall concentrate on the phasedependent heat current only. Taking into account that arXiv:1210.7187v3 [cond-mat.supr-con] 4 Apr 2013  we are dealing with a three-junction circuit, J int reads (1) We emphasize that, whereas charge current depends on the sine of the phase difference across a Josephson junction, 19 heat current depends on the cosine of ϕ i .
In Eq.
is the temperature-dependent energy gap of superconductor S j , Θ(x) is the Heaviside step function, k B is the Boltzmann constant and e is the electron charge. For the sake of completeness we also provide with the expression for the heat current carried by quasiparticles, 16 The explicit dependence of J int on the applied magnetic fluxes and the circuit symmetry parameters, i.e., on the ratio between the normal-state resistances of the junctions, can be calculated analytically as follows. On the one hand the fluxoid quantization on both loops imposes where Φ 0 = 2.067 × 10 −15 Wb is the flux quantum, and n and m are integers. In these expressions we have neglected the geometric inductance of the SQUID which means neglecting the self-induced flux in the loops. In a practical situation, these loops can be designed so to exhibit inductances L j of the order of a few tens of pH where j = 1, 2 refers to the loop pierced by Φ 1 and Φ 2 , respectively. Such geometry ensures a sufficiently good coupling between the SQUID loops and the additional control coils that couple Φ 1 and Φ 2 while providing reasonably low self-inductances. Assuming that each Josephson junction i attains a maximal critical current i i J of the order of 10 2 nA one obtains a total screening parameter β = 2L j i i J /Φ 0 ∼ 10 −3 << 1 allowing us to neglect the SQUID inductance. On the other hand, the consideration of finite L j makes the deduction of analytical expressions impossible and does not contribute to the understanding of the essential physics in our device. The conservation of the circulating supercurrent in both loops, on the other hand, imposes where, according to the generalized Ambegaokar-Baratoff model, 20,21 i i J is proportional to R −1 i . In writing Eq. (3) we have established a given current sign convention (see yellow arrows in Fig. 1). We define now the circuit symmetry parameters as can vary between 0 ≤ r 2 ≤ 1 since setting r 2 > 1 is equivalent to exchange the roles of R b and R c . Combining Eqs.
(1), yield finally the following expression for J int , We note that, if we set r2 = 0, i.e., Rc → ∞, one recovers the expression corresponding to the single-loop heat interferometer conceived in Ref. 14. Consequently, for the case in which r1 = 1 and r2 = 0, i.e., Ra = R b and Rc → ∞, Eq. (5) becomes equal to Jint = 2J b int | cos(πΦ1/Φ0)| [see the inset in Fig. 2(b)]. This function is maximized for Φ1 = kΦ0 and minimized for Φ1 = kΦ0/2, k being an integer. The amplitude of the oscillation δJint, defined as the difference between We now turn our attention towards the double-loop heat interferometer discussed here. The straightest choice is r1 = r2 = 1, i.e., Ra = R b = Rc. Although being the most simple configuration, it enables us to infer most of the characteristics of our thermal interferometer. We can distinguish between two regimes, one defined by 0 ≤ Φ2 < Φ0/3 [ Fig.  2(a)] and a second one covered by Φ0/3 ≤ Φ2 ≤ Φ0/2 [ Fig.  2(b)]. If no magnetic flux is applied to the control loop, i.e., Φ2 = 0, the phase-dependent term of heat current is given by Jint = J b int 1 + 8 cos 2 (πΦ1/Φ0). As shown in Fig. 2(a), this function exhibits the same expected Φ0-periodicity and amplitude of the oscillation as that of the limit case in which r1 = 1 and r2 = 0 [see the inset of Fig. 2(b)] but the appearance of the oscillation is drastically different. In addition, notice that the minimum of Jint does not go to zero anymore, i.e., there is no possibility of suppressing the phase-dependent component of heat current. As we increase the amplitude of the control flux Φ2 the mean value and the shape of the curves continue to evolve whereas δJint holds unchanged. Furthermore, the curves turn out to be shifted horizontally.
At Φ2 = Φ0/3 a noticeable phenomenon takes place. Under these circumstances Eq. (5) takes the form Jint = 2J b int | cos[π(Φ1/Φ0 − 1/6)]|. This is to say, apart from a small shift equal to Φ0/6, one recovers the same dependence on Φ1 obtained for the symmetric single-loop heat interferometer. If we continue increasing the control flux we enter in a new regime. Under these circumstances it can be shown that δJint decreases linearly with Φ2 whereas the mean value of Jint remains constant. Furthermore, at Φ2 = Φ0/2, the modulation disappears completely and Jint becomes independent of Φ1, i.e., Jint = J b int . The aforementioned characteristics are emphasized in Figs. 2(c) and 2(d) where we plot the transfer function, TJ = ∂Jint/∂Φ1, for both regimes.
It is worthwhile now to analyze what happens by reducing the symmetry of our device. In Fig. 3 we show the density plots of Jint vs. Φ1 and Φ2 for three representative cases, including the symmetric double-loop that we have analyzed previously. In general, the maximum of Jint is always reduced for the cases in which one resistance is different from the others. Let us analyze in more detail what happens with δJint. If r1 ≤ 1 , i.e., Ra ≤ R b , we find that the maximum amplitude of oscillation at fixed Φ2 is given by δJint = 2J b int r1 for whatever value of r2, i.e., for whatever value of Rc. This is to say, δJint is independent of the degree of asymmetry of the control loop and increases linearly by increasing the symmetry on the main loop. On the other hand, if r1 > 1 the dependence of δJint at fixed Φ2 is more complicated. For r1 − r2 ≤ 1, δJint is independent from the degree of symmetry in both loops reaching it maximum value, that, in this case, is given by δJint = 2J a int . Finally, for r1 − r2 > 1, δJint decreases with r2. These regimes are summarized in Fig. 4(a).
Let us ascertain now whether it is always possible to suppress completely Jint. The minimum of Jint can be easily determined from Eq. (5). By requesting ∂Jint/∂Φ1 = 0 and ∂Jint/∂Φ2 = 0, and by summing the resulting equations we get that r1 sin(2πΦ1/Φ0) + r2 sin(2πΦ2/Φ0) = 0. This equation gives us the family of values of Φ1 and Φ2 that maximize or minimize Jint for any given r1 and r2. For instance, if Φ1 = Φ0/2, there are only two solutions, Φ2 = 0 and Φ2 = Φ0/2, that satisfy this condition and, in addition, correspond to a minimum. Inserting these solutions into Eq. (5) and imposing Jint = 0 we obtain r1 − r2 = 1 and r1 + r2 = 1, respectively. These equations define two straight lines in the r1 − r2 space plotted in Fig. 4. Within this area there exists at least one value of Φ2 that enables us to write Jint as a function of | cos[π(Φ1/Φ0 − θ)]| where θ is a shift in Φ1. The aforementioned conditions take a more eloquent form when expressed in terms of the Josephson critical currents of each junction, giving Unlikely to a single-loop heat interferometer, 14 even a quite asymmetric double-loop structure for which inequality (6) holds, offers the possibility of suppressing completely Jint through an appropriate choice of Φ2. The three cases plotted in Fig. 3 satisfy the aforementioned conditions and shall therefore be useful to illustrate this behavior. In Fig. 4(b) we plot Jint(Φ1) for Φ2 = 0 (dashed lines) and for the corresponding value of Φ2 that provides the fully suppression of Jint (solid lines). The curves corresponding to the symmetric double-loop have already been plotted in Fig. 2(a) and 2(b), and are therefore not shown. Notably, when setting r1 = 0.5 and r2 = 0.5, i.e., Ra = Rc = 2R b , Jint cancels precisely when Φ2 = Φ0/2 at Φ1 = Φ0/2 ± kΦ0 since this case satisfies exactly the condition r1 + r2 = 1. Notice that the total amplitude of the oscillation is reduced by one half with respect to the previous example. Let us finally consider a last illustrative case belonging to the region defined by 1 ≤ r1 ≤ r2 + 1, i.e, the diagonally stripped region in Fig. 4(a). If r1 = 1.25 and r2 = 0.75, i.e., Ra = 0.6Rc and 2R b = 0.75Rc, although corresponding to a substantially asymmetric interferometer, it should be possible to suppress completely Jint while conserving the maximum amplitude of oscillation. As we can see in the bottom panel of Fig. 4(b) this is exactly the case. Setting Φ2 = Φ0/4, the phase-dependent component of heat transport cancels at Φ1 = 3Φ0/5 ± kΦ0 with δJint = 2J a int . We shall finally dedicate a few words to some potential applications and practical aspects related to the fabrication of the heat interferometer proposed here. Our structure can be integrated within, not only superconducting elements, but also hybrid mesosocopic circuits composed of, e.g., normal metals, two dimensional electron gases and semiconductor nanowires as well. A precise control of the amount of heat flowing through such circuits is of crucial importance. 1,2 Temperature determines, for instance, the phase transition in superconductors, 19 the amount of heat exchanged between electron and lattice phonons, 1 the energy level occupation in quantum systems, 3 or the critical current flowing through Josephson junctions. 1 Mastering the heat current in superconducting circuits would enable the in-situ fine tuning of radiation detectors. 1,22 Controlling the temperature of a twolevel quantum system can eventually have influence on its decoherence time or contribute to its initialization in quantum computing architectures. 3 Furthermore, fully tunable Josephson junctions of different kinds can be envisioned. In such devices, the direct relation between the electronic temperature and the critical current can be exploited to modulate the latter via the application of a magnetic flux. Unlike the usual voltage-controlled hot-electron Josephson transistors, 23-25 the principle of operation proposed here can lead to magnetic fluxcontrolled thermal Josephson transistors. Last but not least, our device is furnished with two external control knobs that correspond to the two separately coupled magnetic fluxes. This opens the way to perform closed cycles in its control space parameters, which can eventually lead to the realization of a heat pump. 26 Regarding the experimental realization of our double-loop heat interferometer we refer to the successful fabrication of analogous Josephson devices composed of two or more loops with independent magnetic flux controls operating as charge interferometers, 27,28 that prove the feasibility of this structure. On the other hand, a single-loop double-junction superconducting heat interferometer has indeed recently been realized experimentally. 15 Our double-loop scheme provides further advantages whereas it does not imply extra difficulties from the point of view of the fabrication. Such structure can be easily fabricated by standard electron-beam lithography and shadow mask evaporation of superconducting metals, e.g., aluminum. Aluminum oxide for the Josephson barriers and copper for the normal metal electrodes can be used. A temperature gradient can easily be established across the superconducting double loop by intentionally heating one SQUID branch while maintaining the other well thermalized at the minimum bath temperature. Temperature detection and manipulation can be performed through two normal metal leads tunnel-coupled to one superconducting electrode of the SQUID that allow for the implementation of normal metal-insulator-superconductor thermometers and heaters. 1 To conclude, we have provided with all the informations required for designing a fully-balanced heat interferometer. Such a device allows, on the one hand, to modify the form and phase shift of the phase-dependent heat current by tuning the control flux. This would enable the user to choose a convenient point of operation within the available flux-to-heat current transfer characteristic. On the other hand, we have demonstrated that if condition (6) holds, a fully-balanced interferometer is obtained, meaning that the phase-dependent part of the heat current can be completely annihilated.
We acknowledge C. Altimiras, M. Cuoco, T. T. Heikkilä and P. Solinas for comments, and the FP7 program No. 228464 "MICROKELVIN" for partial financial support.