# Tuning the Josephson diode response with an ac current

Rubén Seoane Souto  Instituto de Ciencia de Materiales de Madrid (ICMM),
Consejo Superior de Investigaciones Científicas (CSIC), Sor Juana Inés de la
Cruz 3, 28049 Madrid, Spain.  Division of Solid State Physics and NanoLund,
Lund University, S-22100 Lund, Sweden  Center for Quantum Devices, Niels Bohr
Institute, University of Copenhagen, 2100 Copenhagen, Denmark  Martin Leijnse
Division of Solid State Physics and NanoLund, Lund University, S-22100 Lund,
Sweden  Center for Quantum Devices, Niels Bohr Institute, University of
Copenhagen, 2100 Copenhagen, Denmark  Constantin Schrade  Center for Quantum
Devices, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen,
Denmark  Marco Valentini  Institute of Science and Technology Austria, Am
Campus 1, 3400 Klosterneuburg, Austria.  Georgios Katsaros  Institute of
Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria.
Jeroen Danon  Department of Physics, Norwegian University of Science and
Technology, NO-7491 Trondheim, Norway

###### Abstract

Josephson diodes are superconducting elements that show an asymmetry in the
critical current depending on the direction of the current. Here, we
theoretically explore how an alternating current bias can tune the response of
such a diode. We show that for slow driving there is always a regime where the
system can only carry zero-voltage dc current in one direction, thus
effectively behaving as an ideal Josephson diode. Under fast driving, the
diode efficiency is also tunable, although the ideal regime cannot be reached
in this case. We also investigate the residual dissipation due to the time-
dependent current bias and show that it remains small. All our conclusions are
solely based on the critical current asymmetry of the junction, and are thus
compatible with any Josephson diode.

Superconductivity offers the potential for a new generation of electronic
devices characterized by minimal or zero dissipation and rapid response times
[1]. Within this promising landscape, the non-reciprocal phenomenon in
superconducting systems known as the “superconducting diode effect” has
garnered substantial attention in recent times [2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 16, 27, 28,
29, 30, 17, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46,
47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65,
66, 67, 68, 69], for a recent review see Ref. [70].

In these systems, the critical currents in the two directions are different,
$|I_{c}^{+}|\neq|I_{c}^{-}|$. The conventional figure of merit for such
superconducting diodes is the diode efficiency, defined by
$\eta=|(I_{c}^{+}+I_{c}^{-})/(I_{c}^{+}-I_{c}^{-})|$. This metric quantifies
the asymmetry in critical currents, a pivotal aspect of diode functionality.
Therefore, maximizing $\eta$ is an important aspect for potential applications
of superconducting diodes. An ideal diode ($\eta=\pm 1$) is characterized by
supporting supercurrent only in one direction. So far, different directions
have been explored to approach unity efficiency, including multiple Andreev
reflections after applying a small bias voltage [55], concatenating several
junctions in parallel [19, 64], and three terminal superconducting devices
(triodes) [46]. Very recently, there was a proposal for an ideal diode with
dissipation based on the application of an electric field perpendicular to the
supercurrent propagation [71].

On the other hand, Ref. [47] demonstrated experimentally how a periodic
modulation of the bias current, $I_{\rm ac}$, can tune the effective diode
efficiency. Even though the system had a relatively low $\eta$ for $I_{\rm
ac}=0$, it could be tuned to the ideal regime at finite $I_{\rm ac}$, with a
measurable zero-resistance plateau that only extends for one direction of the
dc bias current. The possibility of modulating the diode response using time-
dependent voltage biasing has been studied theoretically in Ref. [72] in
topological junctions, demonstrating a regime where $\eta$ effectively
approaches one, but with very small critical current. The ideal regime was
also analyzed theoretically in quantum dot systems subject to two ac signals
out of phase [73, 74].

Motivated by the recent experimental observations of Ref. [47], we present
here the theoretical analysis of the current-driven superconducting diode. We
consider a superconducting diode subject to a current bias that has a dc and
an ac component, see Fig. 1(a). We focus on two limiting scenarios where the
frequency of the ac driving is either much smaller or much larger than the
inverse characteristic time of the junction, related to its critical current
and normal resistance, and we derive analytical insight in the diode response
in both cases. We explain why, independent of the origin of the diode effect,
a slowly varying $I_{\rm ac}$ can tune the diode to become effectively ideal.
In the opposite fast-driving regime, we show that the efficiency can be
modulated but will never reach one. Driven superconducting junctions can still
dissipate power due to fluctuations of the superconducting phase, even if the
resistance is zero. We therefore also calculate the dissipated power of the
driven diode, deriving analytic expressions for the slow and fast driving
limits. We show that the dissipation in the ideal diode regime can be several
orders of magnitude smaller than in the junction in the normal state for the
same dc current and normal resistance.

We exemplify our insights with a simple diode geometry based on a SQUID.
Higher harmonics in the current–phase relationship (CPR) can lead to an
asymmetric critical current when a magnetic flux penetrates the loop [19].
Several mechanisms can contribute to higher harmonics, including high-
transmission modes [75] and inductance effects [76, 6], studied before in the
context of quantum ratchets [77]. Here, we focus on the latter, which can be
experimentally realized using thin, disordered, or granular superconductors
[78, 79, 80]. We show that the system can be tuned to the ideal diode regime,
with a dissipation that is several orders smaller than in the metallic regime.

The circuit we consider is sketched in Fig. 1(a) and consists of a
superconducting junction (left) that is shunted by a resistor with resistance
$R$ (right). The circuit is connected to an external current source that can
be used to bias the circuit simultaneously with an ac and dc current, i.e.,
$I_{\rm bias}(t)=I_{\rm dc}+I_{\rm ac}\cos(\omega t)$. Assuming the resistance
$R$ to be small enough that the junction is in the overdamped limit, the
equation of motion for the superconducting phase difference over the junction
$\varphi$ becomes

$\dot{\varphi}+I(\varphi)=I_{\rm dc}+I_{\rm ac}\cos(\omega t),$ (1)

where all currents have been renormalized by $\hbar/2eR$, which gives them
units of $s^{-1}$. The CPR $I(\varphi)$ of the superconducting junction has to
be $2\pi$-periodic and can thus be expanded as

$I(\varphi)=\sum_{m\geq 1}I_{m}\sin(m\varphi+\gamma_{m}).$ (2)

In order to have a finite superconducting diode effect at zero driving, i.e.,
$I_{\rm ac}=0$, the junction must have an asymmetric CPR in such a way that
the system has a different maximal current in both directions,

$|I_{c}^{+}|=\left|\max_{\varphi}I(\varphi)\right|\neq|I_{c}^{-}|=\left|\min_{\varphi}I(\varphi)\right|.$
(3)

In this work, we will not speculate under what physical conditions such a
diode effect can arise, but we will treat the set of Fourier coefficients
$\\{I_{m},\gamma_{m}\\}$ as free parameters that can describe any
superconducting junction being part of the circuit shown in Fig. 1(a). A
simple example of a set of coefficients that gives rise to a diode effect is
$I_{1}=I_{2}\equiv i_{0}$, $\gamma_{2}=-\frac{\pi}{2}$, and all other
coefficients set equal to zero. This CPR yields $I_{c}^{+}=2i_{0}$ and
$I_{c}^{-}=-\frac{9}{8}i_{0}$ without current driving, and we will use it
below as “toy” example to illustrate the response of a superconducting diode
to finite ac current driving.

Figure 1: Diode response in the slow-driving regime. (a) Schematic of the
circuit we consider. (b) Average differential resistance as a function of the
two current biases. Red dashed lines show the boundaries of the zero-voltage
region as predicted by the adiabatic theory, given by $\tilde{I}^{\pm}_{\rm
dc}=I^{\pm}_{c}\mp I_{\rm ac}$. The black and blue dots denote the parameters
used for the lower panels of Fig. 2. (c) Average voltage drop across the
junction as a function of $I_{\rm dc}$, for $I_{\rm ac}=0$ (blue) and $I_{\rm
ac}=1.2i_{0}$ (red). (d) Effective diode efficiency as a function of $I_{\rm
ac}$, as extracted from (b). In all panels, we used
$I(\varphi)=i_{0}\sin(\varphi)+i_{0}\sin(2\varphi-\frac{\pi}{2})$ and
$\omega_{0}=0.1i_{0}$.

We can analyze the response of the circuit to driving by solving Eq. (1)
numerically, see for example Fig. 1(b), where we show the average differential
resistance across the system, $d\langle V\rangle/dI_{\rm dc}$, as a function
of the current biases $I_{\rm ac}$ and $I_{\rm dc}$, using $\omega=0.1i_{0}$,
where $\langle V\rangle$ is the averaged voltage drop over many periods after
the steady state is reached. This figure illustrates one of the main points we
explore in this work: the zero-voltage superconducting window is tunable via
the ac current bias $I_{\rm ac}$. Moreover, we note that there exists a regime
where all zero-voltage supercurrent is positive, and there is one ac bias in
particular where the maximum dc current supported at zero voltage is
$\tilde{I}_{\rm dc}^{+}>0$ whereas the minimum dc current is $\tilde{I}_{\rm
dc}^{-}=0$, where the tilde indicates that this is the critical dc current
under driving. Here, one could say that the system behaves effectively as an
ideal diode. In Fig. 1(c) we illustrate this regime by plotting the calculated
voltage over the junction $\langle V\rangle$ as a function of $I_{\rm dc}$
close to this special point (red trace). A zero-voltage plateau extends from
$I_{\rm dc}\approx 0$ to $I_{\rm dc}\approx 0.8i_{0}$. The blue trace shows
the $VI$ characteristic at $I_{\rm ac}=0$, for comparison, confirming the
limiting values given by the CPR, $I_{c}^{+}=2i_{0}$ and
$I_{c}^{-}=-\frac{9}{8}i_{0}$, corresponding to a diode efficiency of
$\eta=0.28$. Fig. 1(d) shows the effective efficiency
$\tilde{\eta}=|(\tilde{I}_{\rm dc}^{+}+\tilde{I}_{\rm dc}^{-})/(\tilde{I}_{\rm
dc}^{+}-\tilde{I}_{\rm dc}^{-})|$ as a function of $I_{\rm ac}$, as extracted
from the data presented in Fig. 1(b), showing an increase from
$\tilde{\eta}=0.28$ to $\tilde{\eta}=1$, as expected.

Below we will (i) present the simple picture that explains the behavior of the
junction in the slow-driving limit $\omega\ll|I_{c}^{\pm}|$, explored in Fig.
1, and (ii) derive analytic understanding of the opposite limit of fast
driving, $\omega\gg|I_{c}^{\pm}|$, where the extent of the zero-voltage
plateau is also tunable via $I_{\rm ac}$, although to a lesser extent.

In the limit of $\omega\ll|I_{c}^{\pm}|$, it is helpful to use the common
interpretation of Eq. (1) in terms of an equation of motion of a massless
particle with coordinate $\varphi$ in a time-dependent tilted “washboard”
potential. We thus write Eq. (1) in the form
$\dot{\varphi}+\partial_{\varphi}U(\varphi,t)=0$, where

$U(\varphi,t)=-\\!\\!\sum_{m\geq
1}\frac{I_{m}}{m}\cos(m\varphi+\gamma_{m})-[I_{\rm dc}+I_{\rm ac}\cos(\omega
t)]\varphi.$ (4)

Since there is no inertia, the particle always adjusts its velocity
$\dot{\varphi}$ instantaneously to the local gradient of the potential, such
that the friction force $-\dot{\varphi}$ and the force $-\partial_{\varphi}U$
cancel. A finite average voltage $\langle
V\rangle=(\hbar/2e)\langle\dot{\varphi}\rangle$ corresponds to a finite
average velocity of the particle, and in the slow-driving limit
$\langle\dot{\varphi}\rangle\neq 0$ can only arise when there are parts of the
driving period where $U$ has no local minima, i.e., it increases or decreases
monotonically. The particle will thus be stuck in the same minimum, resulting
in zero average voltage, when the equation
$\partial_{\varphi}U(\varphi,t)=I(\varphi)-I_{\rm dc}-I_{\rm ac}\cos(\omega
t)=0$ has solutions for all $t$. This leads to the simple condition

$I_{c}^{-}+I_{\rm ac}<I_{\rm dc}<I_{c}^{+}-I_{\rm ac},$ (5)

for the lowest zero-voltage plateau; the boundaries of this region are
indicated by the red dashed lines in Fig. 1(b). The effective diode efficiency
at finite $I_{\rm ac}$ follows as

$\tilde{\eta}=\frac{I_{c}^{+}+I_{c}^{-}}{I_{c}^{+}-I_{c}^{-}-2I_{\rm ac}},$
(6)

from which we find that the maximal efficiency $\tilde{\eta}=\pm 1$ occurs
simply when $I_{\rm ac}=\min\\{|I_{c}^{-}|,|I_{c}^{+}|\\}$. For larger $I_{\rm
ac}$, the zero-resistance window does not contain $I_{\rm dc}=0$ and the
interpretation of $\tilde{\eta}$ as an effective diode efficiency becomes
meaningless, gray area in Fig. 1(d).

Figure 2: Rocking washboard potential. Effective time-dependent potential
describing the Josephson diode under an ac current drive, $I=I_{\rm dc}+I_{\rm
ac}\cos(\omega t)$, where the dots illustrate the phase dynamics. Upper panels
show the situation where the diode is in the ideal regime with $I_{\rm
dc}=-0.3i_{0}$ (yellow), $0$ (orange), and $0.3i_{0}$ (brown). The phase stays
in the same potential minimum after a cycle for $I_{\rm dc}\geq 0$, while it
can drift for $I_{\rm dc}<0$, ending up in a different minimum after each
period. Lower panels correspond to the black and blue dots in Fig. 1(b),
$I_{\rm ac}=0.4i_{0}$ and $I_{\rm dc}=\pm i_{0}$. For $I_{\rm dc}=i_{0}$
(blue) the potential always has local minima and the phase is trapped. In
contrast, the phase can drift for $I_{\rm dc}=-i_{0}$. Depending on the
driving frequency, the phase can stay or drift between neighboring minima
(black and gray dots), leading to the doubling of the Shapiro steps shown in
Fig. 1(b).

Let us illustrate this picture for the example CPR considered in Fig. 1, where
the effective efficiency reaches $\tilde{\eta}=1$ for $I_{\rm
ac}=|I_{c}^{-}|$. For this particular driving strength, the behavior of the
time-dependent potential $U(\varphi,t)$ is sketched in the top row of Fig 2
for $I_{\rm dc}>0$, $I_{\rm dc}=0$, and $I_{\rm dc}<0$ (brown, orange, and
yellow curves, respectively). The driving provides a time-dependent tilt of
the potential; the potentials with maximal negative ($t=n\,T$) and positive
[$t=(n+\frac{1}{2})T$] tilt per period are shown in the first and third
columns of Fig. 2. For an adiabatic drive, the phase would be able to adapt to
the instantaneous potential, and a finite average voltage can thus develop as
soon as the potential becomes monotonous for part of the driving period. For
$I_{\rm dc}=0$ (orange curve), this happens exactly at $t=(n+\frac{1}{2})T$,
where the tilt is maximal. A small additional current bias $I_{\rm dc}\neq 0$
yields an extra time-independent tilt $-I_{\rm dc}\varphi$ in the potential.
If $I_{\rm dc}>0$ (brown curve), the extra dc tilt restores the barriers at
$t=(n+\frac{1}{2})T$ resulting in the particle being trapped in the same
minimum during the whole period and $\langle\dot{\varphi}\rangle=0$. In
contrast, any $I_{\rm dc}<0$ (yellow curve) would increase the slope of the
curve resulting in a part of the driving period around $t=(n+\frac{1}{2})T$
where the potential has no local minima and maxima, allowing a finite average
velocity $\langle\dot{\varphi}\rangle<0$ which makes the system resistive.
This illustrates the physics of the $\tilde{\eta}=1$ case, where a finite
voltage drop appears only for one direction of the dc current.

In reality, driving is never truly in the (infinitely) slow limit, and most
additional structure seen in Fig. 1(b) can be attributed to finite-frequency
effects. The Shapiro steps observed outside the zero-voltage region reflect
the fact that the number of local minima that the phase can drift per period
is discrete, as illustrated by the dots in the right panels of Fig. 2, and
this number increases with increasing $I_{\rm ac}$. The apparent doubling of
the number of Shapiro steps at $I_{\rm dc}<-I_{\rm ac}$ is a result of the
specific shape of our CPR [77]: The time-dependent potential for a point in
the region with additional Shapiro steps [black dot in Fig. 1(b)] is
illustrated by the black curve in the lower panels of Fig. 2. In this case,
the black potential with the smallest average slope has _two_ inequivalent
local minima. Considering the driving frequency to be finite, the first
dissipative processes yield a phase jump between the different potential
minima (gray dots). It then takes two periods to change the phase by an
integer of $2\pi$, whereas in Fig. 2(a) the first available process changes
the phase by a multiple of $2\pi$ each period. This explains the doubling of
the Shapiro steps shown in Fig. 1(b). For comparison, we show in blue the
situation with the opposite $I_{\rm dc}$ [blue dot in Fig. 1(b)], where the
phase does not drift, implying an asymmetry of the Josephson potential. All
this means that the detailed structure of the Shapiro steps in the $(I_{\rm
ac},I_{\rm dc})$-plane encodes information about the shape of the CPR.

Figure 3: Diode response in the fast-driving regime. (a) Differential
resistance as a function of the dc and ac bias currents. The red dashed lines
correspond to the critical currents predicted by Eq. (9). (b) Effective diode
efficiency as a function of $I_{\rm ac}$. In both panels we use the same CPR
as in Fig. 1 and we set $\omega=25i_{0}$.

In the case of fast driving, $\omega\gg|I_{c}^{\pm}|$, the reaction of the
phase to the driving is more complex. Still, the system can show a region with
an average zero voltage drop, as shown in Fig. 3(a), and we can derive
analytic expressions describing the boundaries of this region.

In the zero voltage drop situation, the phase is periodic in time and can be
written as

$\varphi(t)=\alpha_{0}+\sum_{n\geq 1}\alpha_{n}\cos(n\omega t+\beta_{n}).$ (7)

Inserting this Fourier expansion into the CPR (2), we find that we can write

$\displaystyle I(\varphi)={}$ $\displaystyle{}\sum_{m\geq 1}I_{m}{\rm
Im}\Bigg{[}e^{i\gamma_{m}}\prod_{n\geq
0}\sum_{p}i^{p}J_{p}(m\alpha_{n})e^{ipn\omega t}e^{ip\beta_{n}}\Bigg{]},$ (8)

with $J_{p}(x)$ being the $p$-th Bessel function of the first kind and using
$\beta_{0}=0$. We can then inspect Eq. (1) and equate all zero-frequency
terms, which yields the equation $I_{\rm dc}=\sum_{m\geq
1}I_{m}\sin(m\alpha_{0}+\gamma_{m})\prod_{n\geq 1}J_{0}(m\alpha_{n})$.
Equating all terms in Eq. (1) that oscillate with frequency $\omega$, and then
all terms that oscillate with $n\omega$ where $n>1$, we find that in the limit
$\omega\gg I_{m}$ we have $I_{\rm ac}\approx\omega\alpha_{1}$ and all
$\alpha_{n}$ with $n>1$ can be neglected. This finally yields

$I_{\rm dc}=\sum_{m\geq
1}I_{m}\sin(m\alpha_{0}+\gamma_{m})J_{0}\left(m\frac{I_{\rm
ac}}{\omega}\right),$ (9)

and the boundaries of the zero-voltage region are the maximal and minimal dc
current that can be supported according to this equation by adjusting
$\alpha_{0}$. We thus see that in the fast-driving limit the leading effect of
the driving is a “dressing” of all Fourier components of the CPR by a factor
$J_{0}(mI_{\rm ac}/\omega)$.

The red dashed lines in Fig. 3(a) show the maximal and minimal zero-voltage dc
currents $I_{\rm dc}^{\pm}$ found from Eq. (9) for the chosen parameters (see
captions of Figs. 1 and 3). For this particularly simple CPR, analytic
expressions for the critical currents can in fact be derived (see the
Appendix), but in general one needs to find the extrema of the dressed CPR (9)
numerically. The corresponding effective diode efficiency as a function of
$I_{\rm ac}$ is shown in Fig. 3(b), which is clearly tunable through $I_{\rm
ac}$ but to a lesser extent than in the slow-driving case. We note that the
directionality of the diode behavior can be changed by tuning $I_{\rm ac}$.
The efficiency does not exceed $|\tilde{\eta}|\approx 0.34$, but we still see
that it can be improved by approximately 10% as compared to the non-driven
case. Indeed, fast driving cannot tune the diode to become ideal
($\tilde{\eta}=1$) since also the dressed CPR is zero on average.

Figure 4: Inductance diode. (a) CPR. Inset: sketch of the system, consisting
of two conventional Josephson junctions in a SQUID geometry, where the arms of
the loop have a finite inductance. (b) Differential resistance. (c) Resistance
(blue) and dissipated Joule power (solid red) as a function of the dc current
for $\tilde{I}_{\rm ac}=1.32i_{0}$. Dashed line shows the dissipated power in
the normal state. (d) Dissipated power as a function of the ac and dc bias
current. The critical currents of the two junctions are taken as
$i_{1c}=i_{2c}/2=i_{0}$, $L=0.2\hbar/e\,i_{0}$, and the flux due to the
external magnetic field $\Phi_{E}/\Phi_{0}=0.35$, with $\Phi_{0}$ being the
flux quantum.

In all cases considered, even if the average voltage drop across the junction,
and therefore the calculated differential resistance, is zero, the driven
diode can still dissipate energy. This occurs because of shifts in the phase
$\varphi$ resulting from alterations in the location of the potential minima.
To quantify this effect, we calculate the average dissipated power as

$P=\frac{1}{T}\int_{0}^{T}dt\,V(t)I_{\rm bias}(t),$ (10)

where we integrate over one period, assuming that the system has reached
steady state. In the slow-driving regime, the residual dissipation is due to
changes of the effective potential that shifts the minima. Considering the
zero-voltage region, where the phase stays trapped in the same local minimum,
we can expand the potential around this minimum as $U(\varphi,t)\approx
U_{0}+\alpha(\varphi-\varphi_{\rm min})^{2}-I_{\rm bias}(t)\varphi$ and derive
from Eq. (1) (see the Appendix)

$P_{\rm slow}\approx\frac{\omega^{2}I^{2}_{\rm ac}R}{8\alpha^{2}}.$ (11)

From this expression, it is clear that decreasing the driving frequency or
increasing the sharpness of the confining potential, $\alpha$, will decrease
dissipation. In the fast-driving regime, one can use Eq. (7) to find

$P_{\rm fast}\approx\frac{I^{2}_{\rm ac}R}{2}.$ (12)

In Fig. S5 of the Appendix, we compare dissipation in the slow- and the fast-
driving regimes and show that the two approximate expressions (42,29) fit
numerical results well. We see that the system will dissipate much more energy
in the fast-driving than in the slow-driving regime, even in the regime where
differential resistance is zero. Indeed, the phase remains in the same
potential well (zero voltage drop in average), although it cannot adapt to the
minimum instantaneously, leading to a phase amplitude proportional to $I_{\rm
ac}$, and a dissipated power $\propto I^{2}_{\rm ac}$.

Finally, we discuss a physical implementation where our predictions can be
tested, although the described mechanism is general and compatible with all
diode proposals based on dc mechanisms. Here, we consider a SQUID with two
arms that have a finite inductance, as illustrated in Fig. 4(a), a well
understood device [76]. In that figure, we use an inductance for each arm that
has a value of $L=0.2\hbar/e\,i_{0}$ that would correspond to $\sim 1$nH for a
critical current of $\sim 100$nA. This condition can be easily reached in
superconductor-semiconductor junction where the critical current is tunable.
For simplicity, we consider that the two junctions have a sinusoidal CPR. The
inductance leads to the onset of higher harmonics in the CPR of each of the
junctions. We find an asymmetry in the SQUID’s critical current whenever the
critical currents or the inductances of the two junctions are different and
there is a finite magnetic flux penetrating the loop. The CPR is shown in Fig.
4(a), indeed showing an asymmetry between negative and positive directions.
The calculated differential resistance under driving is shown in Fig. 4(b) for
the slow-driving regime, showing signatures that are qualitative similar to
those of the simple diode example of Fig. 1(c). The blue curve in Fig. 4(c)
shows a line cut of the differential resistance at $I_{\rm ac}=1.32i_{0}$,
where $\tilde{\eta}=1$ for a realistic diode implementation. In Fig. 4(d) we
show the dissipated power for the inductance-based diode in the slow-driving
regime, as calculated from Eq. (10), using a logarithmic color scale. The
dissipated power is strongly suppressed in the zero differential resistance
region, see Fig. 4(c), where it is approximately given by Eq. (42). In Fig.
4(c) we also show a line cut of the dissipated power at $\tilde{\eta}=1$
(solid red line). For illustration, we show the dissipated power by the device
in the normal state with $I_{\rm ac}=0$ (dashed line). We note a reduction of
several orders of magnitude, demonstrating that ideal diodes are promising for
low-dissipation circuit elements.

To conclude, in this article we studied the possibility to control the
superconducting diode response using ac current driving. We have shown that a
slow drive can tune the system to a situation where the zero-resistance
plateau extends in only one direction, yielding an effective efficiency of 1,
independent of the diode origin. We illustrated this possibility in a
realistic implementation of a SQUID diode based on Josephson junctions with
significant inductance. Another example can be found in Ref. [47], where the
SQUID Josephson junctions themselves feature higher harmonics. In the opposite
regime of high-frequency driving, the drive can still tune the diode response,
but without ever reaching efficiency 1. In all cases, the system will
dissipate energy under driving due to phase fluctuations in time. We have
shown that in the slow-driving regime this dissipation can be small, opening
the door for new low-dissipation circuit elements. The proposal and
realization of an ideal superconducting diode is an open challenge in the
field.

Acknowledgements.- We acknowledge support from research grants Spanish CM
“Talento Program” (project No. 2022-T1/IND-24070), Spanish Ministry of
Science, innovation, and Universities through Grant PID2022-140552NA-I00,
Swedish Research Council under Grant Agreement No. 2020-03412, the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme under Grant Agreement No. 856526., Nanolund, FWF Project
F-8606, and Microsoft Corporation.

##  Appendix A Boundaries of the zero-voltage region under fast driving

The differential equation to solve is

$\dot{\varphi}+I(\varphi)=I_{\rm dc}+I_{\rm ac}\cos(\omega t),$ (13)

with all currents renormalized by $\hbar/2eR$. We assume the phase to be
periodic in steady state,

$\varphi(t)=\alpha_{0}+\sum_{n\geq 1}\alpha_{n}\cos(n\omega t+\beta_{n}).$
(14)

which yields

$\displaystyle\dot{\varphi}={}$ $\displaystyle{}-\sum_{n\geq
1}n\alpha_{n}\omega\sin(n\omega t+\beta_{n}),$ (15)

as well as

$\displaystyle I(\varphi)={}$ $\displaystyle{}\sum_{m\geq
1}I_{m}\cos\left[m\varphi+\gamma_{m}\right]$ $\displaystyle={}$
$\displaystyle{}\sum_{m\geq
1}\frac{I_{m}}{2}\Bigg{[}e^{i(m\alpha_{0}+\gamma_{m})}\prod_{n\geq
1}\sum_{p}i^{p}J_{p}(m\alpha_{n})e^{ipn\omega
t}e^{ip\beta_{n}}+e^{-i(m\alpha_{0}+\gamma_{m})}\prod_{n\geq
1}\sum_{p}i^{p}J_{p}(-m\alpha_{n})e^{ipn\omega t}e^{ip\beta_{n}}\Bigg{]}.$
(16)

This finally allows us to rewrite Eq. (13) as

$\displaystyle I_{\rm dc}+\frac{I_{\rm ac}}{2}\left(e^{i\omega t}+e^{-i\omega
t}\right)={}$ $\displaystyle{}\frac{i\omega}{2}\sum_{n\geq
1}n\alpha_{n}\left(e^{in\omega t}e^{i\beta_{n}}-e^{-in\omega
t}e^{-i\beta_{n}}\right)$ $\displaystyle{}+\sum_{m\geq
1}\frac{I_{m}}{2}\Bigg{[}e^{i(m\alpha_{0}+\gamma_{m})}\prod_{n\geq
1}\sum_{p}i^{p}J_{p}(m\alpha_{n})e^{ipn\omega t}e^{ip\beta_{n}}$
$\displaystyle{}\hskip 50.00008pt+e^{-i(m\alpha_{0}+\gamma_{m})}\prod_{n\geq
1}\sum_{p}i^{p}J_{p}(-m\alpha_{n})e^{ipn\omega t}e^{ip\beta_{n}}\Bigg{]}.$
(17)

We then start equating the Fourier components on the left and right hand side
of this equation. The constant part of the equation ($\propto 1$) becomes

$\displaystyle I_{\rm dc}=\sum_{m\geq
1}I_{m}\cos(m\alpha_{0}+\gamma_{m})\prod_{n\geq 1}J_{0}(m\alpha_{n}),$ (18)

which presents the modified effective current–phase relationship under
driving. The maximum and minimum dc currents follow from maximizing this
expression over all free parameters $\alpha_{n}$.

We then consider the first Fourier component of the equation ($\propto
e^{i\omega t}$), yielding

$\displaystyle\frac{I_{\rm ac}}{2}={}$
$\displaystyle{}\frac{i\omega}{2}\alpha_{1}e^{i\beta_{1}}+\sum_{m\geq
1}\frac{I_{m}}{2}\Bigg{[}e^{i(m\alpha_{0}+\gamma_{m})}\prod_{r\geq
1}\sum_{p}i^{p}J_{p}(m\alpha_{r})e^{ipr\omega t}e^{ip\beta_{r}}$
$\displaystyle{}\hskip 90.00014pt+e^{-i(m\alpha_{0}+\gamma_{m})}\prod_{r\geq
1}\sum_{p}i^{p}J_{p}(-m\alpha_{r})e^{ipr\omega
t}e^{ip\beta_{r}}\Bigg{]}_{\propto e^{i\omega t}},$ (19)

where the final subscript means that only the terms proportional to
$e^{i\omega t}$ should be considered. In the limit $\omega\gg I_{m}$ we can
reduce this equation to an equation for $\alpha_{1}$ only,

$I_{\rm ac}\approx i\omega\alpha_{1}e^{i\beta_{1}}.$ (20)

Working with $I_{\rm ac},\omega>0$, we see that $\beta=-\pi/2$ makes
$\alpha_{1}>0$, such that we can write $\alpha_{1}\approx I_{\rm ac}/\omega$.
The terms proportional to $e^{-i\omega t}$ in Eq. (17) give the same result
and all higher Fourier components in the equation $\propto e^{in\omega t}$
with $|n|>1$ only produce small corrections to the $\alpha_{n}$ with $n>1$.

We thus approximate $\alpha_{1}=I_{\rm ac}/\omega$ and $\beta_{1}=-\pi/2$ and
find

$\displaystyle I_{\rm dc}\approx{}$ $\displaystyle{}\sum_{m\geq
1}I_{m}\cos(m\alpha_{0}+\gamma_{m})J_{0}\left(m\frac{I_{\rm
ac}}{\omega}\right),$ (21)

where $\alpha_{0}$ is the only remaining free parameter, over which $I_{\rm
dc}$ can be maximized and minimized to find the boundaries of the zero-voltage
region,

$\displaystyle I_{\rm dc}^{+}={}$ $\displaystyle{}{\rm
max}_{\alpha_{0}}\left\\{\sum_{m\geq
1}I_{m}\cos(m\alpha_{0}+\gamma_{m})J_{0}\left(m\frac{I_{\rm
ac}}{\omega}\right)\right\\},$ (22) $\displaystyle I_{\rm dc}^{-}={}$
$\displaystyle{}{\rm min}_{\alpha_{0}}\left\\{\sum_{m\geq
1}I_{m}\cos(m\alpha_{0}+\gamma_{m})J_{0}\left(m\frac{I_{\rm
ac}}{\omega}\right)\right\\}.$ (23)

For a simple current–phase relationship, such as

$I(\varphi)=i_{0}\sin(\varphi)+i_{0}\sin(2\varphi-\tfrac{\pi}{2}),$ (24)

as we used as example in the main text, one can find analytic expressions for
the boundaries of the zero-voltage,

$\displaystyle I_{\rm dc}^{+}={}$
$\displaystyle{}\begin{cases}{\displaystyle-\frac{J_{0}(i_{\rm
ac})^{2}}{8J_{0}(2i_{\rm ac})}-J_{0}(2i_{\rm ac})}&\text{for $|J_{0}(i_{\rm
ac})|<-4J_{0}(2i_{\rm ac})$},\\\\[8.00003pt] {\rm max}\big{\\{}J_{0}(2i_{\rm
ac})\pm J_{0}(i_{\rm ac})\big{\\}}&\text{elsewhere},\end{cases}$ (25)
$\displaystyle I_{\rm dc}^{-}={}$
$\displaystyle{}\begin{cases}{\displaystyle-\frac{J_{0}(i_{\rm
ac})^{2}}{8J_{0}(2i_{\rm ac})}-J_{0}(2i_{\rm ac})}&\text{for $|J_{0}(i_{\rm
ac})|<4J_{0}(2i_{\rm ac})$},\\\\[8.00003pt] {\rm min}\big{\\{}J_{0}(2i_{\rm
ac})\pm J_{0}(i_{\rm ac})\big{\\}}&\text{elsewhere},\end{cases}$ (26)

where we used $i_{\rm ac}=I_{\rm ac}/\omega$.

##  Appendix B Calculation of the steady-state dissipation

In this Section, we will explain how we arrived at the two approximate
expressions for the average power dissipated in the fast- and slow-driving
limits. In all cases, we calculate the dissipation as

$\displaystyle P={}$ $\displaystyle{}\frac{1}{T}\int_{0}^{T}dt\,V(t)I_{\rm
bias}(t)=\frac{1}{T}\int_{0}^{T}dt\,R\dot{\varphi}[I_{\rm dc}+I_{\rm
ac}\cos(\omega t)],$ (27)

where we used that $V(t)=R\dot{\varphi}$ since $\dot{\varphi}$ was defined to
be renormalized by $\hbar/2eR$.

###  B.1 Fast-driving regime

In the fast-driving limit, we can use the results from the previous section.
We see from Eq. (15) that

$\dot{\varphi}\approx-I_{\rm ac}\sin(\omega t-\tfrac{\pi}{2}),$ (28)

where we made use of the result $\alpha_{1}=I_{\rm ac}/\omega$ and
$\beta_{1}=-\pi/2$. We then straightforwardly find from Eq. (27)

$\displaystyle P=\frac{I_{\rm ac}^{2}R}{2}.$ (29)

###  B.2 Slow-driving regime

In this case, we will assume that the driving is slow enough such that the
phase will more or less constantly follow the minimum of the time-dependent
potential $U(\varphi,t)$. The equation of motion reads as

$\dot{\varphi}=-\partial_{\varphi}U(\varphi,t),$ (30)

with

$U(\varphi,t)=-\sum_{m\geq 1}\frac{I_{m}}{m}\cos(m\varphi+\gamma_{m})-[I_{\rm
dc}+I_{\rm ac}\cos(\omega t)]\varphi.$ (31)

The first part of the potential (i.e., the effective Josephson energy of the
whole junction) can be expanded around its minimum,

$U(\varphi,t)\approx U_{0}+\alpha(\varphi-\varphi_{0})^{2}-[I_{\rm dc}+I_{\rm
ac}\cos(\omega t)]\varphi.$ (32)

As a zeroth approximation, we will assume that the phase follows the minimum
exactly and instantaneously (the minimum will be periodically shifted back and
forth by the rest of the rocking potential). We find the time-dependent
minimum of the potential from solving $\partial_{\varphi}U(\varphi,t)=0$,
yielding

$\varphi_{\rm min}(t)=\varphi_{0}+\frac{I_{\rm dc}+I_{\rm ac}\cos(\omega
t)}{2\alpha},$ (33)

and we thus assume that

$\dot{\varphi}(t)=\dot{\varphi}_{\rm min}(t)=-\frac{I_{\rm
ac}\omega}{2\alpha}\sin(\omega t).$ (34)

Inserting this approximation in Eq. (27), we see that it yields $P=0$. Indeed,
by insisting that the phase always exactly follows the potential minimum, it
never needs to “slide down” the potential, which is the process that yields
dissipation. The “particle” trapped in the rocking potential will only
periodically absorb and release energy as its potential energy is raised and
lowered by the rocking.

In order to describe dissipation we thus need to acknowledge that as soon as
$\omega>0$ the phase will always be slightly behind in following the potential
minimum, and is thus effectively constantly sliding down the potential while
trying to reach the instantaneous minimum. To describe this effect, we first
consider a potential with an initial minimum at $\varphi_{0}$ and a constant
increase of the tilt of the potential with rate $\eta$,

$U(\varphi,t)=U_{0}+\alpha(\varphi-\varphi_{0})^{2}+\eta t\varphi.$ (35)

The equation of motion now becomes

$\displaystyle\dot{\varphi}={}$
$\displaystyle{}-2\alpha(\varphi-\varphi_{0})+\eta t.$ (36)

which can be solved, yielding

$\varphi(t)=\varphi_{0}-\frac{\eta
t}{2\alpha}+\frac{\eta}{4\alpha^{2}}(1-e^{-2\alpha t}),$ (37)

where we used that $\varphi(0)=\varphi_{0}$. At long times, $t\gg 1/\alpha$,
we thus find

$\varphi(t)\approx\varphi_{0}-\frac{\eta
t}{2\alpha}+\frac{\eta}{4\alpha^{2}}.$ (38)

From Eq. (35) it is easy to find that the instantaneous position of the
minimum of the potential is $\varphi_{\rm min}(t)=\varphi_{0}-\eta t/2\alpha$
in this case. Comparing this with Eq. (38), we find

$\varphi(t)\approx\varphi_{\rm min}(t)+\frac{\eta}{4\alpha^{2}},$ (39)

i.e., the phase will follow the minimum of the potential at a distance given
by $\eta/4\alpha^{2}$.

Now we again focus on the rocking potential given in Eq. (32). We assume that
the modulation of the potential is slow enough such that we are always
effectively in the long-time limit where the phase has its dynamics adjusted
to the slowly changing instantaneous tilting speed $\eta(t)$. We thus identify

$\eta(t)=\omega I_{\rm ac}\sin(\omega t),$ (40)

yielding

$\varphi(t)\approx\varphi_{\rm min}(t)+\frac{\omega I_{\rm
ac}}{4\alpha^{2}}\sin(\omega t)=\varphi_{0}+\frac{I_{\rm dc}+I_{\rm
ac}\cos(\omega t)}{2\alpha}+\frac{\omega I_{\rm ac}}{4\alpha^{2}}\sin(\omega
t),$ (41)

where we also substituted for $\varphi_{\rm min}(t)$ the minimum of the
rocking potential. This allows for a straightforward calculation of the
dissipation,

$\displaystyle P=\frac{\omega^{2}I^{2}_{\rm ac}R}{8\alpha^{2}}.$ (42)

##  Appendix C Current–phase relationship for a SQUID with significant
inductance

Figure 5: Schematic of the model circuit we use to derive the current–phase
relationship of the inductance diode.

We consider the circuit model of a SQUID shown in Fig. 5, where both arms of
the SQUID have a significant inductance, denoted by $L_{1,2}$ and separated
from the junction for simplicity. The total current can be written as the sum
over the two currents $i_{1,2}$ through the Josephson junctions,

$I(\varphi)=i_{1}(\varphi_{1})+i_{2}(\varphi_{2}),$ (43)

where $\varphi_{j}$ is the phase difference over the given junction, as
indicated in the Figure. The phases can be related as

$\displaystyle\varphi$ $\displaystyle=$
$\displaystyle\varphi_{1}+L_{1}i_{1}(\varphi_{1}),$ (44)
$\displaystyle\varphi$ $\displaystyle=$
$\displaystyle\varphi_{2}+L_{2}i_{2}(\varphi_{2})-2\pi\frac{\Phi}{\Phi_{0}},$
(45)

where we have included the effect of the finite flux penetrating the loop into
the phase difference over junction $2$. For simplicity, we consider junctions
with purely sinusoidal current–phase relationships, given by

$i_{j}(\varphi_{j})=i^{c}_{j}\sin(\varphi_{j}).$ (46)

Eqs. (43–46) together implicitly define the current–phase relationship
$I(\varphi)$ of the whole circuit.

##  Appendix D Supporting results

Figure 6: From adiabatic to fast driving. Panels (a–d) correspond to
$\omega/i_{0}=0.2$, $0.5$, $1$, and $2$, respectively. We use the same
current–phase relationship as in Fig. 1 of the main text,
$I(\varphi)=i_{0}\sin(\varphi)+i_{0}\sin(2\varphi-\frac{\pi}{2})$.

In this Section, we present results that complement the ones presented in the
main text. In Fig. 6 we show the differential resistance, numerically
calculated as the derivative of the averaged voltage drop with respect to the
dc current, for the driving frequencies $\omega/i_{0}=0.2$, $0.5$, $1$, and
$2$, which span the crossover from slow to fast driving. In the slow-driving
regime, the differential resistance shows sharp steps, related to the phase
dynamics when the phase can overcome a certain number of potential barriers
during each driving period, see Fig. 6(a) (and Fig. 1 in the main text). In
the intermediate regime, the steps acquire additional structure, becoming
progressively broader when increasing the driving frequency $\omega$, see
Figs. 6(b,c). Finally, for $\omega\gtrsim i_{0}$ we approach the fast-driving
limit described above, the properties of the dc supercurrent then being
captured approximately by Eq. (21). In this limit, the critical currents in
the two directions oscillate as a function of the driving amplitude $I_{\rm
ac}$, see Fig. 7(a). We see that, even though the critical current can be
modulated using the ac current, the efficiency cannot be boosted
significantly, as illustrated in Fig. 7(b).

Figure 7: (a) Critical current in the fast-driving limit in the positive (red)
and negative (blue) directions. (b) The corresponding effective diode
efficiency. For both panels we used Eq. (21) to calculate the critical
currents.

Figure 8 shows the dissipated power in the slow and fast regimes as considered
in Figs. 1(b) and 3(a) in the main text, respectively. We used a logarithmic
scale to enhance features. In the slow-driving regime, the system shows a
region with significantly reduced dissipated power, see Fig. 8(a). In
contrast, the dissipation in the fast-driving regime increases rapidly as
$P=I^{2}_{\rm ac}R/2$ due to the stronger phase drift every cycle, making the
zero-resistance features not visible, see Fig. 8(b). Finally, in Figs. 8(c,d)
we show line cuts of the dissipated power for $I_{\rm dc}=0$, comparing the
numerically calculated dissipation (red solid) to the analytic expressions in
Eqs. (29) and (42) (blue dashed). Both approximate expressions describe
accurately the energy dissipated in these two limits. In the slow-driving
regime, the analytic expression starts to deviate from the analytic result
close to the boundary of the resistive regime, where corrections to the
harmonic approximation of the Josephson potential start to become important.

Figure 8: Dissipated power. We show numerically calculated results in the slow
(a) and fast (b) driving regimes. The corresponding parameters are the same as
those used in Figs. 1 and 3, of the main text. Panels (c) and (d) show line
cuts for the dissipated power at $I_{\rm dc}=0$ for slow and fast driving,
respectively [parameters are the same as in panels (a,b)]. Dashed lines show
the analytic results using Eq. (42), with the obtained $\alpha=1.29$ for the
used current–phase relationship, and Eq. (29).

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