# Nonsinusoidal current-phase relations in
semiconductor-superconductor-ferromagnetic insulator devices

Andrea Maiani  Center for Quantum Devices, Niels Bohr Institute, University of
Copenhagen, DK-2100 Copenhagen, Denmark  Karsten Flensberg  Center for Quantum
Devices, Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen,
Denmark  Martin Leijnse  Center for Quantum Devices, Niels Bohr Institute,
University of Copenhagen, DK-2100 Copenhagen, Denmark  Division of Solid State
Physics and NanoLund, Lund University, S-22100 Lund, Sweden  Constantin
Schrade  Center for Quantum Devices, Niels Bohr Institute, University of
Copenhagen, DK-2100 Copenhagen, Denmark  Saulius Vaitiekėnas  Center for
Quantum Devices, Niels Bohr Institute, University of Copenhagen, DK-2100
Copenhagen, Denmark  Rubén Seoane Souto  Center for Quantum Devices, Niels
Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark
Division of Solid State Physics and NanoLund, Lund University, S-22100 Lund,
Sweden

###### Abstract

Coherent tunneling processes of multiple Cooper pairs across a Josephson
junction give rise to higher harmonics in the current phase relation. In this
work, we propose and study Josephson junctions based on semiconductor-
superconductor-ferromagnetic insulator heterostructures to engineer
nonsinusoidal current-phase relations. The gate-tunability of charge carriers
density in the semiconductor, together with the adjustable magnetization of
the ferromagnetic insulator, provides control over the content of the
supercurrent harmonics. At finite exchange field, hybrid junctions can undergo
a 0 – $\pi$ phase transition, resulting in the supercurrent reversal. Close to
the transition, single-pair tunneling is suppressed and the current-phase
relation is dominated by the second-harmonic, indicating transport primarily
by pairs of Cooper pairs. Finally, we demonstrate that non-collinear
magnetization or spin-orbit coupling in the leads and the junction can lead to
a gate-tunable Josephson diode effect with efficiencies of up to $\sim 30\%$.

In Josephson junctions with insulating weak links, the flow of a
dissipationless supercurrent arises from individual Cooper-pair tunneling
events, which is typically characterized by a sinusoidal current-phase
relation (CPR) [1]. For junctions with high transparency, additional
contributions to the supercurrent appear thanks to the simultaneous coherent
tunneling of multiple Cooper pairs [2, 3, 4, 5, 6, 7, 8, 9, 10]. These
tunneling events give rise to higher harmonics in the CPR, leading to
deviations from the standard sinusoidal form.

In a simple weak link, the Josephson energy is minimized when the
superconducting phases on both sides of the junction are equal. However, when
time-reversal symmetry is broken, a phase transition can occur, resulting in
an equilibrium state with the relative phase difference of $\pi$. This shift
in the macroscopic degree of freedom leads to a phase transition between the
so-called $0$ and $\pi$ phases [11, 12, 13, 14, 15].

Josephson junctions with broken time-reversal symmetry can be tuned to a
regime where both the $0$ and the $\pi$ phases are metastable with the
Josephson potential showing two local minima [16, 17]. In this case, the
junction is said to be $0^{\prime}$ or $\pi^{\prime}$ regime if the global
minimum is at $0$ or $\pi$ phase difference. The fundamental harmonic changes
sign when moving from $0^{\prime}$ to $\pi^{\prime}$ and vanishes at the
crossover between the two. At this point, the current is dominated by higher
harmonics, causing a nonsinusoidal CPR [18]. The study of such nonsinusoidal
CPRs has recently gained significant interest due to potential applications in
developing protected superconducting qubits [19, 20, 21, 22, 23] and
supercurrent diodes [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37,
38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]. These novel
applications of Josephson junctions require not only a static nonsinusoidal
CPR, but also the ability to control its harmonic content.

Figure 1:  Josephson junction in a semiconductor-ferromagnet-superconductor
device. Sketch of a ferromagnetic hybrid junction (a) and the continuum model
considered in this paper (b). The nanowire divides into three regions: the
lateral left ($\mathrm{L}$) and right ($\mathrm{R}$) regions feature
proximity-induced superconductivity from the superconductor shells, while in
the central ($\mathrm{C}$) region the superconductor is etched. All the
regions are subjected to the ferromagnetic proximity effect. The density in
the central region and the barriers between the central and the lateral ones
can be controlled by gates.

In this work, we propose and study a platform for tunable nonsinusoidal CPRs
based on Josephson junctions comprised of semiconductor-superconductor-
ferromagnetic insulator materials, Fig. 1. Recently, such platforms have been
realized experimentally in InAs nanowires coated with epitaxial Al and EuS
shells [52], which exhibited signatures of spin-polarized subgap states [53,
54], and supercurrent reversal [55]. We show that these hybrid materials offer
a new way to control the harmonic content of the CPR by combining the gate-
tunable charge carriers density of the semiconductor and the adjustable
magnetization of ferromagnetic insulator insets. Additionally, in these
devices, both the superconducting pairing and the exchange field are induced
in the semiconductor through proximity effects. This allows for unique regimes
where the exchange field in the superconductor is below the Chandrasekhar-
Clogston limit [56, 57], while in the semiconductor it can exceed the induced
pairing potential. As a result, this platform is suited for studying
superconductivity under extreme exchange fields that can surpass the induced
pairing potential.

To illustrate the prospects of this platform, we discuss an approach for
realizing a pure second-harmonic CPR characterized by the coherent tunneling
of pairs of Cooper pairs. We find this condition to be present both in the
open and in the quantum dot regime. We show that non-collinear magnetization,
or, alternatively, non-collinear spin-orbit coupling fields in the leads and
the junction allow for a gate-tunable Josephson diode effect with efficiencies
up to $\sim 30\%$. Our work expands on the possible technological applications
of ferromagnetic hybrid devices beyond the realization of topological
superconductivity by exploring the tunability of nonsinusoidal CPR in hybrid
Josephson junctions.

##  I Model

The system we are considering can be conceptually split into three regions:
two lateral ($\mathrm{L}$ and $\mathrm{R}$), and a central region
($\mathrm{C}$), see Fig. 1(a). The superconductor primarily induces a
superconducting pairing potential $\Delta(x)$ in the semiconductor through the
proximity effect. It also contributes to the electrostatic potential landscape
$V(x)=-\mu(x)$. The ferromagnetic insulator induces an exchange field
$\mathbf{h}(x)$ in both the superconductor and the semiconductor. Given that
both the exchange field and the pairing potential are induced in a
semiconductor with controllable charge carrier density, there is not a fixed
hierarchy of energy scales, and, in principle, all regimes can be realized in
the system. We consider that the exchange field is sufficiently weak in the
superconductor such that superconductivity persists. This condition is relaxed
in the semiconductor, where the induced exchange field can overcome the
induced pairing potential. Therefore, ferromagnetic hybrid junctions allow
exploring the parameter space beyond the conventional regime
($\mu\gg\Delta>|\mathbf{h}|$). Note that we do not refer to a particular
arrangement of the layers in the lateral region, as different combinations of
interfaces, for instance, tripartite arrangement [58, 59] and tunneling
arrangement [60, 61], allows induction of both superconducting pairing and
exchange field in the semiconductor.

The system Hamiltonian is $H=\frac{1}{2}\int\psi^{\dagger}\mathcal{H}\psi$,
where the Bogoliubov-de Gennes (BdG) Hamiltonian $\mathcal{H}$ in the Nambu
spinor basis
$\psi^{\dagger}=\begin{pmatrix}\psi_{\uparrow}^{\dagger}&\psi_{\downarrow}^{\dagger}&-\psi_{\downarrow}&\psi_{\uparrow}\end{pmatrix}$
is

$\begin{split}\mathcal{H}=\quantity[\frac{\hbar^{2}k_{x}}{2m^{*}}-\mu]\tau_{z}+\mathbf{h}\cdot{\bf\it\sigma}+\Delta\tau_{+}+\Delta^{\dagger}\tau_{-}+\mathcal{H}_{\mathrm{SOC}}\,.\end{split}$
(1)

Here, $k_{x}=-i\partial_{x}$ is the momentum operator (we consider a single
mode in the junction), $m^{*}$ is the effective mass, and $\sigma_{j}$ and
$\tau_{j}$ are the Pauli matrices in the spin and particle-hole space,
respectively. The spin-orbit coupling Hamiltonian $\mathcal{H}_{\mathrm{SOC}}$
is given in Eq. (9) and discussed in Sec. II.4.

The proximity-induced exchange field $\mathbf{h}$ is due to the coupling to
the ferromagnetic insulator and, in principle, can be spatially inhomogeneous
due to the micromagnetic configuration. The magnitude of the $\mathbf{h}$
field can also vary due to a nonuniform coupling strength. Moreover, recent
theoretical investigations of ferromagnetic InAs-Al-EuS nanowires showed that
the electrostatic environment is crucial in modulating the effect of the EuS
on the InAs [58, 59, 61], suggesting that, in principle, it is possible to
tune the induced exchange field electrostatically.

In this work, we consider that the exchange field $\mathbf{h}$ takes a
constant value $\mathbf{h}_{j}$ in each of the three regions
$i\in\\{\mathrm{L},\mathrm{C},\mathrm{R}\\}$. We will call collectively
$\mathrm{L}$ and $\mathrm{R}$ lateral regions, with an exchange field
$\mathbf{h}_{\mathrm{L}}=\mathbf{h}_{\mathrm{R}}=\mathbf{h}_{\mathrm{lat}}$
while we will use the symbol $\mathbf{h}_{\mathrm{all}}$ when considering a
homogeneous value for the exchange field. We use the same notation for the
chemical potential $\mu$ of the three regions. In addition, we introduce
potential barriers with height $V_{\mathrm{B}}$ at the interfaces between the
central and the lateral regions, tuning the system from the open
($V_{\mathrm{B}}=0$) to the quantum dot regimes
($V_{\mathrm{B}}\gg-\mu_{\mathrm{lat}}$). The induced pairing potential is
$\Delta_{j}=\Delta_{0,j}e^{i\phi_{j}}$ with the modulus $\Delta_{0,j}$ taking
finite value only in the $\mathrm{L}$ and $\mathrm{R}$ regions while the
superconducting phase difference between the two leads is
$\phi=\phi_{\mathrm{R}}-\phi_{\mathrm{L}}$.

In all the simulations we use realistic parameters for InAs-Al-EuS
heterostructures, taking $\Delta_{0}=$0.250\text{\,}\mathrm{meV}$$ and
effective mass $m^{*}=0.026\,m_{e}$. We consider a nanowire of total length
including both the lateral and central regions of
$\ell_{\mathrm{W}}=3~{}\mu$m. To obtain numerical results, we discretize the
Hamiltonian using a finite-differences scheme with a lattice spacing of
$a=$2\text{\,}\mathrm{nm}$$ implemented using the KWANT package [62]. The code
and the results of the simulations are available at [63].

We focus on the short junction limit, and we consider a central region of
length $\ell_{\mathrm{C}}=$180\text{\,}\mathrm{nm}$$, such that the
quantization energy is comparable to the other energy scales. We fix the
chemical potential in the lateral regions to be
$\mu_{\mathrm{L}}=\mu_{\mathrm{R}}=16~{}\Delta=$4\text{\,}\mathrm{meV}$$.
Directly solving for the quasiparticle spectrum of the continuum model allows
treating on equal footing the current carried by Andreev bound states (ABSs)
and the quasi-continuum of states above the gap. Usually, the continuum
current is subdominant, except for strong exchange fields, where its
contribution is comparable to or even larger than the ABS one [64, 65, 66, 67,
55].

The supercurrent in a Josephson junction is an equilibrium phenomenon that can
be described by a function $E_{\mathrm{J}}(\phi)$, called Josephson potential
or phase dispersion relation. This can be evaluated from the quasiparticle
spectrum, assuming that the quasiparticles are in thermal equilibrium and
calculating the free energy at a fixed phase

$\begin{split}E_{\mathrm{J}}\quantity(\phi)&=-k_{\mathrm{B}}T_{\mathrm{p}}\ln\tr
e^{-\frac{H}{k_{\mathrm{B}}T_{\mathrm{p}}}}\\\
&=-k_{\mathrm{B}}T_{\mathrm{p}}\sum_{n}\ln\quantity[2\cosh\quantity(\frac{\omega_{n}\quantity(\phi)}{2k_{\mathrm{B}}T_{\mathrm{p}}})]\,,\end{split}$
(2)

where $\omega_{n}$ is the quasiparticle spectrum, $k_{\mathrm{B}}$ is the
Boltzmann constant and $T_{\mathrm{p}}$ is the quasiparticle temperature [68].
From $E_{\mathrm{J}}$, the CPR is calculated through the thermodynamic
relation

$\langle
I\rangle=\frac{2e}{\hbar}\partialderivative{E_{\mathrm{J}}(\phi)}{\phi}\,.$
(3)

The maximum current that can flow in the junction in equilibrium is called
critical current, $I_{\mathrm{c}}$, while we define the critical phase,
$\phi_{\mathrm{c}}$, as the phase difference where this is reached

$I_{\mathrm{c}}=|I(\phi_{\mathrm{c}})|,\qquad\phi_{\mathrm{c}}\equiv\operatorname*{argmax}_{\phi\in[0,\pi]}\absolutevalue{I\quantity(\phi)}\,,$
(4)

where we restricted the search domain to $[0,\pi]$ in the reciprocal case. We
define the quantity $I_{0}\equiv 2e\Delta/\hbar$ as the relevant current scale
that has the numerical value $I_{0}=$122\text{\,}\mathrm{nA}$$ for Al. The
sign of $I(\phi_{\mathrm{c}})$ defines the direction of the supercurrent at
the critical phase.

It is useful to decompose the phase dispersion in its harmonic components

$E_{\mathrm{J}}(\phi)=\sum_{k=1}^{\infty}\quantity[C_{k}\cos(k\phi)+S_{k}\sin(k\phi)]\,,$
(5)

since each harmonic corresponds to the tunneling of multiplets of Cooper pairs
between the two superconducting regions. To see this, it is necessary to
interpret the phase in Eq. (5) as an operator and rewrite the expression in
the charge basis by interpreting $\exp(ik\phi)$ as a translation operator.
Neglecting the constant term, the result is

$\mathcal{H}_{J}=\sum_{n}\sum_{k}\frac{C_{k}+iS_{k}}{2}\ket{n+k}\bra{n}+\mathrm{H.c.}$
(6)

where $\ket{n}$ is the state with a difference of $n$ Cooper pairs in the two
leads. In this way, it is easy to see how the terms $\cos(k\phi)$ mediate the
tunneling of $k$ Cooper pairs.

A purely $\cos(2\phi)$ Josephson junction can be used to create a qubit with
nearly-degenerate ground states that are a superposition of only states with
the same parity of Cooper pair number. [19, 21]. In this setup, $\cos(\phi)$
and $\sin(\phi)$ perturbations can be used to implement rotation in the Bloch
sphere [23].

If time-reversal and inversion symmetries are not simultaneously broken, the
Josephson junction is reciprocal and
$E_{\mathrm{J}}(\phi)=E_{\mathrm{J}}(-\phi)$. It subsequently results in the
absence of the ${S_{k}}$ components, called anomalous. These anomalous
components are necessary for $\phi_{0}$ junctions and the diode effects.

In the reciprocal case, assuming all the components $\\{C_{k}\\}$ with $k>2$
are negligible, the only minima of the Josephson potential are located at
$\phi=0$ and $\pi$ only if $|C_{1}|\geq 4C_{2}$. For $|C_{1}|<4C_{2}$, minima
can be found at $\phi=\pm\arctan(C_{1}/\sqrt{16C_{2}^{2}-C_{1}^{2}})$. This
case, dubbed $\pm\phi_{0}$-junction, does not require an inversion-symmetry
breaking mechanism [69].

###  I.1 Single level model

Figure 2:  Phase diagram in the quantum dot regime. (a) Phase diagram, and (b)
critical current for a system in the dot regime as a function of the chemical
potential in the central region, $\mu_{\mathrm{C}}$, and the exchange field,
$h_{\mathrm{all}}$, considered homogeneous in the heterostructure. The dashed
lines are an overlay of the analytical model in Eq. (7) where we selected
$\varepsilon=16~{}\Delta$ and $\gamma=0.2~{}\Delta$. (c) and (d) show the
Josephson potential and the CPR for some selected points in the parameter
space [crosses in (a) and (b)]. Parameters for the continuum BdG model:
$\mu_{\mathrm{lat}}=$4\text{\,}\mathrm{meV}$$,
$V_{\mathrm{B}}=$3\text{\,}\mathrm{meV}$$.

Before proceeding to the numerical results, we introduce a single-level model
in which a level with energy $\varepsilon$ couples to two high carrier density
superconducting lateral regions, see App. A and Ref. [55].

To get an expression for the critical lines separating the different phases,
we consider the condition where an ABS crosses the Fermi level, obtained from
Eq. 17 for $\omega=0$. When a spin-split ABS crosses the Fermi level, its
occupation changes and provides no contribution to the current. Therefore, the
residual current is entirely due to the continuum of states and has a
characteristic $\pi$ contribution [55].

In the case of equal exchange fields in the leads,
$h_{\mathrm{L}}=h_{\mathrm{R}}=h_{\mathrm{lat}}$, the Fermi level crossing
condition for ABS with spin $\sigma=\pm 1$ is given by the expression

$\begin{split}\frac{\sigma h_{\mathrm{C}}}{\gamma}+&\frac{\sigma
h_{\mathrm{lat}}}{\sqrt{\Delta^{2}-h_{\mathrm{lat}}^{2}}}=\\\
\pm&\sqrt{\frac{\varepsilon^{2}}{\gamma^{2}}+\frac{\Delta^{2}}{\Delta^{2}-h_{\mathrm{lat}}^{2}}[1-T\sin^{2}(\phi/2)]}\,,\end{split}$
(7)

where $\gamma=\gamma_{\mathrm{L}}+\gamma_{\mathrm{R}}$ is the tunnel rate to
the leads and
$T=4\gamma_{\mathrm{L}}\gamma_{\mathrm{R}}/(\gamma_{\mathrm{L}}+\gamma_{\mathrm{R}})^{2}$
is the transparency of the junction. When a spin-split ABS crosses the Fermi
level at $\phi=\pi$, a metastable $\pi$ phase appears, marking the transition
from $0$ to $0^{\prime}$. When such a crossing happens for $\phi=0$, the $0$
phase becomes completely unstable, marking the $\pi^{\prime}$ to $\pi$
transition. Finally, the $0^{\prime}$ to $\pi^{\prime}$ critical line can be
approximated by $\phi=\pi/2$.

The quantum point contact limit, in which the intermediate state in the
junction is strongly hybridized with the states in the leads, can be obtained
from Eq. 17 for $\gamma\to\infty$. This results in a generalization of
Beenakker’s formula [2] for spin-split leads

$\omega=\pm\Delta\sqrt{1-T\sin^{2}(\phi/2)}-\sigma h_{\mathrm{lat}}\,.$ (8)

##  II Controllable CPR

###  II.1 Quantum dot regime

The quantum dot regime is reached when large barriers at the edge of the
central region are introduced, see Fig. 1. In this regime, electrons are
confined in the central regionThe quantum dot regime is optimal for the
electrostatic controllability of the 0 – $\pi$ transition. When the quantum
dot levels align with the chemical potential in the leads, the $\pi$ phase
appears at low exchange fields, and the critical line takes the form
$h_{\mathrm{C}}\propto\varepsilon$, see Fig. 2(c). This work does not consider
the electrostatic repulsion in the central region (Coulomb blockade). We
expect the mean-field picture to be valid for large exchange fields, and the
Coulomb repulsion to enhance the exchange field in the central region.

In this regime, $I_{\mathrm{c}}$ is maximal when the dot levels align with the
chemical potential of the leads, as shown in Fig. 2(b). In the off-resonance
condition, the critical lines converge to the spectral gap closing point,
$h=\Delta$. The phase diagram can be understood using the single-level model,
which predicts the hyperbolic critical lines [see the dashed lines in Fig.
2(a)]. The Josephson potential and CPRs on resonance are shown in Figs. 2(c)
and (d). The fundamental harmonic dominates the junction properties in both
the 0 and the $\pi$ phases (top and bottom panels). In contrast, high-harmonic
contributions become important in the $0^{\prime}$ and $\pi^{\prime}$ phases
because of the double minima Josephson potential.

To better understand the harmonic composition of the Josephson energy, we
present the lowest harmonic components of the CPR in Fig. 3. For a small value
of the exchange field, in the $0$ and $0^{\prime}$ phases, the harmonic
component coefficients show a peak corresponding to an energy level in the
quantum dot aligning with the Fermi level of the lateral regions. These peaks
have widths that decrease for higher-order components, allowing the relative
strength of the first two harmonics, $\delta C_{21}=|C_{2}|-|C_{1}|$, to be
tuned by slightly changing the energy of the quantum dot levels
electrostatically. In contrast, the sensitivity of $E_{\mathrm{J}}$ to the
chemical is almost negligible in the $\pi$ phase and the suppression of
$C_{2}$ is significant, resulting in a sinusoidal CPR, as shown in Fig. 2(d).
When the system is tuned to the vicinity of the $0^{\prime}$–$\pi^{\prime}$
transition, the fundamental harmonic is suppressed, leading to a regime
dominated by the second harmonic and a double-well Josephson potential.

Figure 3:  Harmonic components in the dot regime. Fundamental (a) and second
(b) harmonic for a ferromagnetic hybrid junction in the dot regime. The
transition from 0 to $\pi$ appears close to the gap closing ($h/\Delta=1$) for
a detuned dot, while, near resonance, the $\pi$ phase can appear at lower
magnetic fields. In the regions of the metastable phases, a strong $C_{2}$
component can be observed while the $C_{1}$ component vanishes. In panels (c)
and (d), we show two cuts of the harmonic components [blue and red ticks in
(a) and (b)]. For constant $h_{\mathrm{all}}$ the $C_{k}$ components show a
peak when a dot level crosses the Fermi level of the leads, but the width is
increasingly smaller for higher harmonics. For constant $\mu_{\mathrm{C}}$,
the CPR show a sinusoidal behavior until the system reaches the 0–$\pi$
transition, where the second harmonic dominates. The parameters are equal to
those in Fig. 2.

###  II.2 Open regime

In contrast to the quantum dot regime, the open regime shows a weak dependence
on chemical potential, except close to the edge of the band ($\mu=0$), Fig. 4
(a). The open regime shows a $\pi$ phase for $h_{\rm all}>\Delta$, as
predicted by the analytic expression in Eq. (7). The system shows extended
metastable $0^{\prime}$ and $\pi^{\prime}$ regions compared to the quantum dot
regime. In the open regime, the transition happens for
$h_{\mathrm{all}}=\Delta/\sqrt{1-T/2}$. For transparent junctions, this
critical line coincides with the zero-temperature paramagnetic limit for
superconductors meaning that this regime cannot be achieved in materials with
intrinsic superconductivity. Instead, semiconductor-superconductor devices are
ideal for reaching the $\pi^{\prime}$ and $\pi$ regimes. At the
$0^{\prime}$–$\pi^{\prime}$ transition point, the CPR is dominated by the
$\sin(2\phi)$ term, leading to a Josephson potential with two equivalent
minima within the $\phi\in[0,\pi]$ range. We note that the robustness against
local fluctuations in $\mu_{\mathrm{C}}$ is a unique feature of the open
regime. The other two transitions lines, for $0$–$0^{\prime}$ and
$\pi$–$\pi^{\prime}$, are also almost independent of the chemical potential
once $\mu_{C}\gtrsim 10$, taking place close to $h_{\mathrm{all}}=0$ and
$h_{\mathrm{all}}=\Delta\sqrt{1-T}$.

In both open and dot cases, when $C_{1}$ reaches the crossover point and
becomes zero, $C_{2}$ has a negative sign. This satisfies the condition
$|C_{1}|>4C_{2}$ discussed before, resulting in minima at $0$ and $\pi$. For
what concerns temperature dependence, since the higher harmonic components are
dictated mainly by the lowest ABS, the main effect of increasing temperatures
is a reduction of the metastable $0^{\prime}$ and $\pi^{\prime}$ regions.

Figure 4: Open regime. (a) Phase diagram and (b) difference between the second
and first CPR harmonic component for a ferromagnetic hybrid junction in the
open regime. The overlaid dashed lines are the prediction of the analytic
model in the $\gamma\to\infty$ limit and $T=1$. The sensitivity of the phase
boundaries to $\mu_{\mathrm{C}}$ is strongly suppressed in this limit, while
the phase space occupied by $0^{\prime}$ and $\pi^{\prime}$ phases is
increased. Parameters: $\mu_{\mathrm{lat}}=$4\text{\,}\mathrm{meV}$$,
$\ell_{\mathrm{B}}=$0\text{\,}\mathrm{nm}$$.

The open and dot regimes have different advantages and disadvantages for
practical applications of ferromagnetic hybrid junctions as a $\cos(2\phi)$
Josephson element. The open regime is insensitive to noise in the gate voltage
but requires a relatively high exchange field
$h_{\mathrm{all}}\simeq\Delta/\sqrt{2}$ for the second harmonic to dominate.
Increasing the barriers, and thus moving toward the dot regime, lowers the
required exchange field toward the theoretical limit $h_{\mathrm{all}}\simeq
0$ at the price of a higher sensitivity to gate noise. It also allows
electrostatic control of the harmonic content. Indeed, when the $C_{2}$
component approaches its maximum location, $C_{1}$ vanishes linearly. This
opens up the possibility of introducing a gate-controllable $\cos(\phi)$
component. Additionally, the ability to change from a dot to an open regime is
controlled by the electrostatic environment, allowing to tune the system
between the two regimes.

###  II.3 Inhomogeneous exchange field

The exchange field in the lateral and central regions affects CPR differently.
To reveal this difference, we now analyze the case where the exchange field in
the central ($h_{\mathrm{C}}$) and lateral regions
($h_{\mathrm{R}}=h_{\mathrm{L}}=h_{\mathrm{lat}}$) have different values while
being still aligned in the same direction, see Fig. 5(a) and (b). In the case
of small magnetization in the lateral regions ($h_{\mathrm{lat}}\sim 0$), we
find that a strong polarization in the central one $h_{\mathrm{C}}\gg\Delta$
is needed to induce the transition to the $\pi$ state in the open regime.

The exchange field strength necessary to induce a $0$ – $\pi$ transition
crucially depends on other parameters of the system. In particular, longer
junctions and low density in the central region are associated with
transitions at lower fields. The length dependence can be understood using a
semiclassical model, where $h_{\mathrm{C}}$ adds an extra phase accumulated by
quasiparticles in a round-trip between the leads. This phase is proportional
to $h_{\mathrm{C}}\,\ell_{\mathrm{C}}$ product, explaining why longer
junctions exhibit switches from the 0 to the $\pi$ phase at lower exchange
field values. In addition, it leads to a periodic pattern of $0$ and $\pi$
phases along the $h_{\mathrm{C}}$ axis. A similar effect can be obtained by
reducing $\mu_{\mathrm{C}}$, which reduces the Fermi velocity. This is further
discussed in App. B.

A sharp transition from $0$ to $\pi$ can also be obtained near gap closing
($\absolutevalue{h_{\mathrm{lat}}}=\Delta)$. In this case, the transition is
associated with a strong reduction in the magnitude of $I_{\mathrm{c}}$. This
behavior can be understood using the simplified one-level model in Eq. (7),
which explains the transition to the $\pi$ phase as the disappearance of the
contribution of the lowest ABS, thus reducing the total supercurrent in the
junction. The fact that the 0 – $\pi$ transition is associated with a decrease
of critical current only in the case of gap closing can be potentially used to
infer the dominant mechanism in experiments.

Figure 5:  Effect of inhomogeneous exchange field and spin-orbit coupling.
Phase diagram (a) and critical current (b) for a ferromagnetic hybrid junction
with an inhomogeneous exchange field. The overlaid dashed lines are the
prediction of the single-level model, Eq. (7). The system shows the
alternation of $0$ and $\pi$ phases as a function of $h_{\mathrm{C}}$ with
sharp transitions. A transition from $0$ to $\pi$ can also be obtained by gap
closing ($h/\Delta=1$). Phase diagrams (c) and critical current (d) of the
ferromagnetic hybrid junction in the presence of spin-orbit coupling such that
${\bf\it\kappa}\cdot\mathbf{h}=0$. Spin-orbit coupling leads to general
suppression of the $\pi$ phase and the expansion of the metastable phases.
Parameters: $\mu_{\mathrm{C}}=$1\text{\,}\mathrm{meV}$$,
$\mu_{\mathrm{lat}}=$4\text{\,}\mathrm{meV}$$,
$\ell_{\mathrm{B}}=$0\text{\,}\mathrm{nm}$$.

###  II.4 Spin-orbit coupling in the semiconductor

So far, we have neglected the effect of spin-obit coupling. Here, we consider
a simple model for linear spin-orbit coupling that, for a quasi-1D system,
takes the form

$\mathcal{H}_{\mathrm{SOC}}=k_{x}\quantity[\alpha_{z}\sigma_{y}+\beta\sigma_{x}]\tau_{z}=k_{x}[{\bf\it\kappa}\cdot\sigma]\tau_{z}\,,$
(9)

where we defined a spin-orbit coupling vector
${\bf\it\kappa}=(\beta_{x},\alpha_{z},0)$. In the simplest setup, $\alpha_{z}$
arises from the Rashba field and $\beta$ from the Dresselhaus term. We note
that the distinction between the two terms is artificial in a one-dimensional
model, as the two terms can be mapped onto each other by a unitary
transformation

$U=\exp(-i\theta/2\sigma_{z}\tau_{0})\,,$ (10)

that is a rotation in spin space around the $z$ axis by an angle $\theta$,
potentially inhomogeneous in space. Using $\theta=\arctan(\beta/\alpha_{z})$,
we can always remove the term proportional to $\sigma_{x}$ and align the spin-
orbit vector in the $y$ direction. This transformation, however, also rotates
the exchange field. This result illustrates the equivalence between
inhomogeneous spin-orbit fields and exchange fields.

We first focus on the homogeneous spin-orbit situation, displayed in Fig. 5
(c) and (d). Although spin-orbit coupling splits the Fermi surface, Cooper
pairs do not acquire a finite momentum unless time-reversal symmetry is
broken. Thus, the oscillation between triplet and singlet components is
absent, and it is impossible to obtain a $\pi$ phase. At a finite magnetic
field, the Rashba term couples the two spin-split ABSs reopening the gap,
unless the field aligns with the spin-orbit vector ${\bf\it\kappa}$. The
effect on the CPR of a transverse Rashba field is a substantial reduction of
the $\pi$ regions and an enlargement of the metastable phases. When the
exchange field $\mathbf{h}$ is instead aligned with ${\bf\it\kappa}$, the
spin-rotation symmetry is unbroken. This allows for $\pi$ phases, but
simultaneously, the system remains gapless for $h>\Delta$.

Anomalous Josephson effects can occur when a spin-orbit coupling vector is
aligned with a magnetic field [70]. However, it is not observed in the
homogeneous case as the combination of various spin-rotation symmetry-breaking
effects is necessary for its manifestation [71]. This can occur, for instance,
due to finite spin-splitting with a non-zero component in both the junction
direction and the transverse one [72], or multiple modes that can hybridize
[73]. The presence of anomalous currents in similar systems has been
considered in Refs. [74, 75].

Figure 6: Non-reciprocal behavior. We introduce a spin-orbit coupling
$\kappa=$2\text{\,}\mathrm{nm}\text{\,}\mathrm{eV}$$, misaligned with an angle
$\theta_{\mathrm{C}}$ between the central and the lateral regions. Panel (a)
shows the current at the critical phase, while the diode efficiency is shown
in (b). (c) The Josephson potential and (e) CPR for a specific point of the
parameter space [black cross in (a)] are displayed for increasing temperature.
Non-reciprocal behavior shows a non-monotonic temperature dependence, evident
in panel (f). This can be explained by the ABS spectrum [panel (d)] that
comprises a reciprocal lowest state. Therefore, increasing the temperature
suppresses the reciprocal contribution increasing the efficiency. Parameters:
$\mu_{\mathrm{C}}=$1\text{\,}\mathrm{meV}$$,
$\mu_{\mathrm{lat}}=$4\text{\,}\mathrm{meV}$$,
$\ell_{\mathrm{B}}=$0\text{\,}\mathrm{nm}$$, $h_{\mathrm{lat}}=0.8\Delta$,
$\kappa=$2\text{\,}\mathrm{meV}\text{\,}\mathrm{nm}$$.

The spin-orbit field depends on the local electrostatic environment. For this
reason, the spin-orbit direction can have different magnitudes and directions
in different regions of the ferromagnetic hybrid junction. We consider this
situation in Fig. 6. The spin-orbit direction is misaligned in the central
region by an angle $\theta_{\mathrm{C}}$ with respect to the lateral ones,
while we consider a homogeneous exchange field. Since the Rashba field is
proportional to the electric field, this scenario might appear in Josephson
junctions due to a varying electrostatic environment. This is equivalent to a
homogeneous spin-orbit coupling field and a misaligned exchange field in the
three regions. In this case, the CPR shows a non-reciprocal behavior,
$I(\phi)\neq I(-\phi)$, due to the presence of anomalous $\sin(k\phi)$ terms
in the Josephson potential. The non-reciprocal supercurrent has been recently
reported in superconductor-semiconductor nanowires [31]. The critical phase in
the non-reciprocal case reads as
$\phi_{\mathrm{c}}\equiv\operatorname*{argmax}_{[0,2\pi)}\absolutevalue{I}$.
To measure the non-reciprocal behavior, we define the critical current in the
two directions, $I_{\mathrm{c}}^{+}\equiv\max I$ and
$I_{\mathrm{c}}^{-}\equiv-\min I$, and the corresponding diode efficiency as
$\eta\equiv(I_{\mathrm{c}}^{+}-I_{\mathrm{c}}^{-})/(I_{\mathrm{c}}^{+}+I_{\mathrm{c}}^{-})$.
For the parameters considered, the efficiency can be as high as $30\%$ in the
region close to the 0 – $\pi$ transition. For this point, the CPR shows a
characteristic form $I(\phi)\sim\sin(\phi-\phi_{0})+\cos(2\phi)$.

The rectification effect exhibits a non-monotonic temperature dependence
$\eta(T_{p})$ that shows a maximum at finite temperature. This can be
explained by the ABS spectrum, see Fig. 6 (d): the lowest state is dominated
by the $\cos(k\phi)$ contribution, with a weak non-reciprocal behavior. The
first excited state has a predominant $\sin(\phi)$ contribution. Therefore,
increasing the temperature increases the non-reciprocal supercurrent
contribution through an interference process of the different CPR harmonics
[30]. At higher temperatures, more states become populated, washing out the
contribution from the harmonics and leading to a sinusoidal CPR, Fig. 6(e).

##  III Conclusions

In this work, we have analyzed the higher harmonics in the current-phase
relation appearing in ferromagnetic hybrid junctions. The induced exchange
field in the semiconductor can overcome the induced pairing potential without
causing a transition to the normal state. This results in spin-polarized
Andreev bound states with opposite spin crossing the Fermi level, leading to
the 0 – $\pi$ phase transition and supercurrent reversal. Higher harmonics in
the current-phase relation become dominant close to the 0 – $\pi$ transition,
where the supercurrent changes sign. In the dot regime (weak coupling to the
leads), the $\pi$ depends on the relative position of the dot levels with
respect to the leads chemical potential. In case the junction is tuned into
the open regime (large coupling to the leads), the onset of the $\pi$ phase is
less sensitive to changes in chemical potential.

We find that the spin-orbit coupling increases the coexisting region between
$0$ and $\pi$ phases with considerable amplitudes of higher-harmonic
components. Finally, we find that non-collinear spin-orbit coupling in the
junction, due to a varying electrostatic environment, results in a
supercurrent rectification effect whose efficiency peaks around 0 – $\pi$
transition.

The tunability of the harmonic content of CPRs is relevant for a number of
applications, including superconducting diodes [24, 25, 26, 27, 28, 29, 30,
31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49,
50, 51], ferromagnetic transmon qubits [76], and parity-protected qubits [19,
20, 21, 22, 23]. In this context, ferromagnetic junctions in the open regime
are promising thanks to their robustness to fluctuations in the charge
environment.

##  IV Acknowledgments

This work was supported by the Danish National Research Foundation, the Danish
Council for Independent Research |Natural Sciences, Swedish Research Council
(VR), the European Research Council (ERC) under the European Union’s Horizon
2020 research and innovation program under Grant Agreement No. 856526, and
Nanolund. We also acknowledge support from Microsoft and a research grant
(Project 43951) from VILLUM FONDEN.

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##  Appendix A Analytical model for the short-junction limit

To describe the transport through the system, we use the Green function
formalism, summarized in Ref. [77]. The retarded/advanced (R/A) Green function
of the central region is given by

$\hat{G}^{R/A}_{0}(\omega)=\left[\hat{g}_{0}^{-1}(\omega)-\hat{\Sigma}^{R/A}_{\mathrm{L}}(\omega)-\hat{\Sigma}^{R/A}_{\mathrm{R}}(\omega)\right]^{-1},$
(11)

where $\hat{g}_{0}^{-1}(\omega)=(\omega+\sigma h_{\rm
C})\hat{\tau}_{0}+\varepsilon\hat{\tau}_{z}$ describes the isolated normal
region as a function of the electron energy $\omega$, and
$\hat{\Sigma}^{R/A}_{\nu}$ is the self-energy describing the coupling to the
lateral regions $\nu$.

In the wide bandwidth limit, the self-energy of these regions is given by

$\hat{\Sigma}^{R/A}_{\nu}(\omega)=\sum_{\nu}\Gamma_{\nu}\left[\mathfrak{g}^{R/A}_{\nu}(\omega)\hat{\tau}_{0}+\mathfrak{f}^{R/A}_{\nu}(\omega)\tau_{x}e^{\hat{\tau}_{y}\phi_{\nu}}\right]\,.$
(12)

Here

$\mathfrak{g}^{R/A}_{\nu}(\omega)=-\frac{\omega+\sigma h_{\nu}\pm
i\eta}{\sqrt{\Delta^{2}-(\omega+\sigma h_{\nu}\pm
i\eta)^{2}}}\,,\qquad\mathfrak{f}^{R/A}_{\nu}(\omega)=\frac{\Delta}{\sqrt{\Delta^{2}-(\omega+\sigma
h_{\nu}\pm i\eta)^{2}}}\,,$ (13)

and $\eta$ is the Dynes parameter, controlling the width of the
superconducting coherent peaks at $\omega=\pm\Delta$, which we take
infinitesimal in this case.

In the short junction case, we can use a minimal model where the central
region is described by a single electronic site with energy $\varepsilon$ and
exchange splitting $h_{\mathrm{C}}$. To determine the ABS spectrum, we search
for the poles of the retarded Green’s function by solving
$\det[(G^{R})^{-1}]=0$. This results in the complicated expression

$\quantity(\omega+\sigma h_{\mathrm{C}}+\sum\gamma_{\nu}\frac{\omega+\sigma
h_{\nu}}{\sqrt{\Delta^{2}-(\omega+\sigma
h_{\nu})^{2}}})^{2}=\varepsilon^{2}+\absolutevalue{\sum_{\nu}\frac{\Delta\gamma_{\nu}e^{i\phi_{\nu}}}{\sqrt{\Delta^{2}-(\omega+\sigma
h_{\nu})^{2}}}}^{2}$ (14)

The expression becomes more compact if we consider
$h_{\mathrm{R}}=h_{\mathrm{L}}=h_{\mathrm{lat}}$. In this case, the expression
simplifies to

$\quantity(\omega+\sigma
h_{\mathrm{C}}+\sum_{\nu}\gamma_{\nu}\frac{\omega+\sigma
h_{\mathrm{lat}}}{\sqrt{\Delta^{2}-(\omega+\sigma
h_{\mathrm{lat}})^{2}}})^{2}=\varepsilon^{2}+\frac{\Delta^{2}}{\Delta^{2}-(\omega+\sigma
h_{\mathrm{lat}})^{2}}b^{2}(\phi)$ (15)

where we defined the phase potential

$b(\phi)=\absolutevalue{\sum_{\nu}\gamma_{\nu}e^{i\phi_{\nu}}}\,.$ (16)

Note that with this minimal simplification, the dependence on the phase is
entirely condensed in the function $b(\phi)$.

By defining the total coupling
$\gamma=\gamma_{\mathrm{L}}+\gamma_{\mathrm{R}}$ and the transmission as
$T=4\gamma_{\mathrm{L}}\gamma_{\mathrm{R}}/(\gamma_{\mathrm{L}}+\gamma_{\mathrm{R}})^{2}$,
we get a simplified expression that reads as

$\frac{\omega+\sigma h_{\mathrm{C}}}{\gamma}+\frac{\omega+\sigma
h_{\mathrm{lat}}}{\sqrt{\Delta^{2}-(\omega+\sigma
h_{\mathrm{lat}})^{2}}}=\pm\sqrt{\frac{\varepsilon^{2}}{\gamma^{2}}+\frac{\Delta^{2}}{\Delta^{2}-(\omega+\sigma
h_{\mathrm{lat}})^{2}}[1-T\sin^{2}(\phi/2)]}\,.$ (17)

The quantum point contact limit is defined for $\gamma\to\infty$ and results
in a generalization of Beenakker’s formula for spin-split leads

$\omega=\pm\Delta\sqrt{1-T\sin^{2}(\phi/2)}-\sigma h_{\mathrm{lat}}\,,$ (18)

while the case for $\omega=0$ gives the Fermi level crossing formula used in
Eq. (7).

##  Appendix B Semiclassical analysis of a long junction

The more conventional regime is described by the energy scale hierarchy
$\mu\gtrsim\Delta\gtrsim h$ in the lateral regions, and $\mu\gtrsim h$ in the
central one. In this case, the device behaves substantially as an SFS
junction. For long junctions with a sufficiently high density and small
polarization, the behavior can be approximately described by a semiclassical
model. This assumption is not met in the system studied by the numerical
model. Nevertheless, this approach provides a clearer qualitative picture of
the physics of this system. We follow the steps of Refs. [2, 3] including a
spin splitting oriented in the wire direction with magnitudes $h_{\mathrm{L}}$
and $h_{\mathrm{R}}$. Later we include a spin-splitting field in the central
region captured by the magnetic phase $\Phi_{M}$.

The discrete spectrum is obtained by solving the equation

$\det[1-S_{\mathrm{C}}(\varepsilon)S_{A}(\varepsilon)]=0$ (19)

where $S_{A}$ is the scattering matrix at the interfaces while
$S_{\mathrm{C}}$ is the scattering matrix for the transmission through the
central region. We use the basis
$\psi^{\mathrm{in}}=\quantity(c^{e}_{+}(0)\,c^{e}_{-}(\ell_{\mathrm{C}})\,c^{h}_{-}(0)\,c^{h}_{+}(\ell_{\mathrm{C}}))$
and
$\psi^{\mathrm{out}}=\quantity(c^{e}_{-}(0)\,c^{e}_{+}(\ell_{\mathrm{C}})\,c^{h}_{+}(0)\,c^{h}_{-}(\ell_{\mathrm{C}}))$
such that $S_{\mathrm{C}}\psi^{\mathrm{in}}=\psi^{\mathrm{out}}$ and
$S_{A}\psi^{\mathrm{out}}=\psi^{\mathrm{in}}$.

For $\varepsilon<\Delta$, we can assume that no normal reflection happens at
the interface between the central and lateral regions if the interfaces are
clean. In this case, the Andreev reflection matrix takes the form

$S_{A}=\begin{pmatrix}0&s_{A}^{eh}\\\ s_{A}^{he}&0\end{pmatrix}\,.$ (20)

where the submatrices are

$\displaystyle s_{A}^{eh}$
$\displaystyle=\begin{pmatrix}e^{-i\arccos[(\varepsilon+\sigma
h_{\mathrm{L}})/\Delta]}e^{+i\phi/2}&0\\\ 0&e^{-i\arccos[(\varepsilon+\sigma
h_{\mathrm{R}})/\Delta]}e^{-i\phi/2}\end{pmatrix}\,,$ (21) $\displaystyle
s_{A}^{he}$ $\displaystyle=\begin{pmatrix}e^{-i\arccos[(\varepsilon+\sigma
h_{\mathrm{L}})/\Delta]}e^{-i\phi/2}&0\\\ 0&e^{-i\arccos[(\varepsilon+\sigma
h_{\mathrm{R}})/\Delta]}e^{+i\phi/2}\end{pmatrix}\,,$ (22)

For the central region, we assume a general scattering matrix that is block-
diagonal in the electron-hole subspaces:

$S_{\mathrm{C}}=\begin{pmatrix}s_{N}^{ee}&0\\\ 0&s_{N}^{hh}\end{pmatrix}\,,$
(23)

where the two scattering matrices are related.

Using the property $\det\begin{pmatrix}A&B\\\
C&D\end{pmatrix}=\det\quantity(AD-ACA^{-1}B)$ we eliminate the particle-hole
blocks to simplify Eq. (19) to

$\det\quantity(1-s_{A}^{he}s_{N}^{ee}s_{A}^{eh}s_{N}^{hh})=0\,.$ (24)

To proceed, we need to introduce some specific assumptions on the form of
$S_{\mathrm{C}}$. For a clean system, we can assume a free propagation that,
in Andreev approximation, results in a scattering matrix with the form

$s_{N}^{ee}=\exp\quantity(i\frac{\pi}{2}\frac{\varepsilon+\sigma
h_{\mathrm{C}}}{E_{\mathrm{T}}})\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$ (25)

where we defined the Thouless energy $E_{\mathrm{T}}=\frac{\pi}{2}\frac{\hbar
v_{F}}{L_{C}}$. This results in an equation for the bound states

$\pi\frac{\varepsilon+\sigma
h_{\mathrm{C}}}{E_{T}}\pm\phi-\sum_{\mathrm{lat}=L,R}\arccos\quantity(\frac{\varepsilon+\sigma
h_{\mathrm{lat}}}{\Delta})=2\pi n\,.$ (26)

If we take the exchange field in the lateral regions to be equal
$h_{\mathrm{L}}=h_{\mathrm{R}}=h_{\mathrm{lat}}$, this simplifies to

$\frac{\pi}{2}\frac{\varepsilon}{E_{T}}+\sigma\Phi_{M}\pm\frac{\phi}{2}-\arccos\quantity(\frac{\varepsilon+\sigma
h_{\mathrm{lat}}}{\Delta})=\pi n\,,$ (27)

where we defined the magnetic phase
$\Phi_{M}\equiv\frac{\pi}{2}\frac{h_{\mathrm{C}}}{E_{T}}=\frac{h_{\mathrm{C}}\ell_{\mathrm{C}}}{\hbar
v_{F}}$. In the case of a short central region, we can neglect the spin-
independent phase shift in the central region and get the simplified relation

$\varepsilon_{0,\sigma}=\pm\Delta\cos\quantity(\pm\frac{\phi}{2}+\sigma\Phi_{M})-\sigma
h_{\mathrm{lat}}$ (28)

In the case of a low-density regime, the conduction-band polarization
$h_{\mathrm{C}}/\mu_{\mathrm{C}}$ can reach high values and exceed one at the
transition to a half-metallic regime. The presence of inhomogeneities in the
chemical potential or exchange field causes the breakdown of quasiclassical
Andreev solutions that manifests in the hybridization of the solutions of Eq.
26 and the opening of sizable gaps in the spectrum [78, 79].

Some features of this simple model can be connected to the results of the non-
linearized BdG model displayed in Fig. 7. Even in the short SNS junction limit
($\ell_{\mathrm{C}}\ll\xi$), the junction length modulates the magnetic phase
acquired in the transport in the normal region. Systems with a longer normal
region show an alternation of $0$ and $\pi$ phases together with $0^{\prime}$
and $\pi^{\prime}$ regions. The phase diagram shows a series of $\pi$ regions
with a $\frac{\ell_{\mathrm{C}}}{\mu_{\mathrm{C}}^{a}}\sim b$ shapes, with $a$
and $b$ constants. This can be understood on the basis of the semiclassical
result where the effect of the ferromagnetic insulator in the central region
enters the ABSs spectrum through the magnetic phase
$\Phi_{M}=\frac{\pi}{2}h_{\mathrm{C}}\ell_{\mathrm{C}}\quantity(\frac{2\mu_{\mathrm{C}}}{m*})^{-1/2}$.
Therefore changing the density can have effects similar to changing the length
of the junction.

Figure 7: Dependence of the phase diagram on the length of the central region.
Parameters: $h_{\mathrm{lat}}=0$,
$h_{\mathrm{C}}=$0.1875\text{\,}\mathrm{meV}$$,
$\mu_{\mathrm{C}}=$3.125\text{\,}\mathrm{meV}$$
$\mu_{\mathrm{lat}}=$1.25\text{\,}\mathrm{meV}$$,
$V_{\mathrm{B}}=$0.5\text{\,}\mathrm{meV}$$.


# Uncertainty principles for the imaginary Ornstein-Uhlenbeck operator

Nicola Garofalo  Dipartimento d’Ingegneria Civile e Ambientale (DICEA)
Università di Padova
Via Marzolo, 9 - 35131 Padova, Italy<EMAIL_ADDRESS>
###### Abstract.

We prove two forms of uncertainty principle for the Schrödinger group
generated by the Ornstein-Uhlenbeck operator. As a consequence, we derive a
related (in fact, equivalent) result for the imaginary harmonic oscillator.

###### Key words and phrases:

Uncertainty principle. Schrödinger equation. Ornstein-Uhlenbeck operator

###### 1991 Mathematics Subject Classification:

35H20, 35B09, 35R03, 53C17, 58J60

The author is supported in part by a Progetto SID (Investimento Strategico di
Dipartimento): “Aspects of nonlocal operators via fine properties of heat
kernels”, University of Padova (2022); and by a PRIN (Progetto di Ricerca di
Rilevante Interesse Nazionale) (2022): “Variational and analytical aspects of
geometric PDEs”. He is also partially supported by a Visiting Professorship at
the Arizona State University

##  1\. Introduction

Given $m\in\mathbb{N}$, let $L$ indicate the Ornstein-Uhlenbeck operator in
$\mathbb{R}^{m}$ defined by

$L\varphi=\Delta\varphi-\langle x,\nabla\varphi\rangle,$

see [12]. As it is well-known, see [4], the invariant measure for $L$ is
$d\gamma(x)=e^{-\frac{|x|^{2}}{2}}dx$. The operator $L$ is symmetric with
respect to $d\gamma$, i.e., for any $\varphi,\psi\in
C^{\infty}_{0}(\mathbb{R}^{m})$,

$\int_{\mathbb{R}^{m}}\varphi L\psi d\gamma=\int_{\mathbb{R}^{m}}\psi L\varphi
d\gamma.$

Since $L$ is self-adjoint in $L^{2}(\mathbb{R}^{m},d\gamma)$, by Stone’s
theorem, see [15, Theorem 1, p. 345], there exists a strongly-continuous group
$e^{itL}$ of unitary operators in $L^{2}(\mathbb{R}^{m},d\gamma)$. Such group
provides the solution operator for the Cauchy problem in
$\mathbb{R}^{m}\times(0,\infty)$ for the Schrödinger equation

(1.1) $\begin{cases}\partial_{t}f-iLf=0,\\\ f(x,0)=\varphi(x).\end{cases}$

Notably, the differential equation in (1.1) is invariant with respect to the
(complex) left-translation

$\tau_{(x,s)}(y,t)=(x,s)\circ(y,t)=(y+e^{it}x,t+s),$

in the sense that
$(\partial_{t}-iL)(\tau_{(x,s)}f)=\tau_{(x,s)}[(\partial_{t}-iL)f]$. It is
worth emphasising here that (1.1) represents a basic model for a more general
class of (possibly) degenerate operators of interest in mathematics and
physics, introduced in the work [10].

The main objective of this note is the following form of uncertainty principle
for the group $e^{itL}$. When $\varphi\in L^{2}(\mathbb{R}^{m},d\gamma)$, we
will write $f(x,t)=e^{itL}\varphi(x)$.

######  Theorem 1.1.

Assume that for some $a,b>0$ one has

(1.2)
$||e^{a|\cdot|^{2}}f(\cdot,0)||_{L^{2}(\mathbb{R}^{m},d\gamma)}+||e^{b|\cdot|^{2}}f(\cdot,s)||_{L^{2}(\mathbb{R}^{m},d\gamma)}<\infty.$

If $ab\sin^{2}s\geq\frac{1}{16}$, then $f\equiv 0$ in
$\mathbb{R}^{m}\times\mathbb{R}$.

We note explicitly that the assumption $ab\sin^{2}s\geq\frac{1}{16}$
automatically excludes the possibility that $s=k\pi$, with $k\in\mathbb{Z}$.
This discrete set of points is where the covariance matrix $Q(t)I_{m}$, with
$Q(t)$ defined in (2.8) below, becomes singular and the representation (2.5)
of $e^{itL}$ ceases to be valid. The proof of Theorem 1.1 combines our formula
(2.20) in Proposition 2.4 with the interesting $L^{2}$ version of the
classical theorem of Hardy for the Fourier transform due to Cowling,
Escauriaza, Kenig, Ponce and Vega in [5].

Theorem 1.1 implies (and is in fact equivalent to) the following uncertainty
principle for the Schrödinger group $e^{itH}$ associated with the harmonic
oscillator $H=\Delta-\frac{|x|^{2}}{4}$, 111This operator is usually defined
as $H=\Delta-|x|^{2}$. We are using the $1/4$ normalisation in order not to
have to change in $\Delta-2\langle v,\nabla\rangle$ that of the Ornstein-
Uhlenbeck operator in the statement of Proposition 4.2 below. representing the
solution operator for the Cauchy problem

(1.3) $\begin{cases}\partial_{t}u-iHu=0,\\\ u(x,0)=u_{0}(x).\end{cases}$

Again by Stone’s theorem, there exists a strongly-continuous group $e^{itH}$
of unitary operators in $L^{2}(\mathbb{R}^{m})$. If for $u_{0}\in
L^{2}(\mathbb{R}^{m})$ we let $u(x,t)=e^{itH}u_{0}(x)$, then we have the
following result.

######  Corollary 1.2.

Assume that for some $a,b>0$ one has

(1.4)
$||e^{a|\cdot|^{2}}u(\cdot,0)||_{L^{2}(\mathbb{R}^{m})}+||e^{b|\cdot|^{2}}u(\cdot,s)||_{L^{2}(\mathbb{R}^{m})}<\infty.$

If $ab\sin^{2}s\geq\frac{1}{16}$, then $u\equiv 0$ in
$\mathbb{R}^{m}\times\mathbb{R}$.

The passage from Theorem 1.1 to Corollary 1.2 (and vice-versa) is based on
Proposition 4.2 below. We note that combining Proposition 2.4 with the
uncertainty principle for the Fourier transform due to Hardy, see [9] or also
[5, Theorem 1.2], we obtain corresponding $L^{\infty}$ versions of Theorem 1.1
and Corollary 1.2. For the Schrödinger group $e^{itL}$ we have the following.

######  Theorem 1.3.

Suppose that for some $C,a,b>0$ one has for any $x\in\mathbb{R}^{m}$

(1.5) $|e^{-\frac{|x|^{2}}{4}}f(x,0)|\leq Ce^{-a|x|^{2}},\ \ \ \ \ \
|e^{-\frac{|x|^{2}}{4}}f(x,s)|\leq Ce^{-b|x|^{2}}.$

If $ab\sin^{2}s\geq\frac{1}{16}$, then $f\equiv 0$ in
$\mathbb{R}^{m}\times\mathbb{R}$.

We mention that, for the harmonic oscillator $H$, an $L^{\infty}$ uncertainty
principle such as Theorem 1.3, in which the Gaussian function
$e^{-\frac{|x|^{2}}{4}}$ does not appear in the hypothesis (1.5), was found in
Theorem 6.2 of the interesting paper [2]. Uncertainty inequalities is a vast
subject and there exist many beautiful and important results scattered in the
literature which is impossible to quote here. We refer the reader to [6] and
[13] for an interesting account which covers up to 2004.

In closing, we state an optimal dispersive estimate that we obtain combining
Proposition 2.4 with Beckner’s deep improvement of the Hausdorff-Young
inequality, see [1]. For the definition of the class
$\mathscr{K}(\mathbb{R}^{m})$ see (2.3) below.

######  Proposition 1.4.

Let $\varphi\in\mathscr{K}(\mathbb{R}^{m})$, and $f(x,t)=e^{itL}\varphi(x)$
with $t\not=k\pi$, with $k\in\mathbb{Z}$. For any $1\leq p\leq 2$ one has

(1.6)
$||e^{-\frac{|\cdot|^{2}}{4}}f(\cdot,t)||_{L^{p^{\prime}}(\mathbb{R}^{m})}\leq\left(\frac{p^{1/p}}{{p^{\prime}}^{1/p^{\prime}}}\right)^{\frac{m}{2}}\
\frac{1}{(4\pi|\sin t|)^{m(\frac{1}{2}-\frac{1}{p^{\prime}})}}\
||e^{-\frac{|\cdot|^{2}}{4}}\varphi||_{L^{p}(\mathbb{R}^{m})},$

where $1/p+1/{p^{\prime}}=1$. The estimate (1.6) is optimal, in the sense that
it cannot possibly hold in the range $2<p\leq\infty$. Moreover, equality is
attained in (1.6) by initial data
$\varphi(x)=e^{-\alpha|x|^{2}+\frac{|x|^{2}}{4}}$, $\Re\alpha>0$.

Note that as $t\to 0$ the estimate (1.6) displays the same behaviour
$t^{-m(\frac{1}{2}-\frac{1}{p^{\prime}})}$ as that of the free Schrödinger
group $e^{it\Delta}$ (for the latter, see e.g. [8, Lemma 1.2]). It may be of
interest to compare (1.6) with the following dispersive estimate for the
Schrödinger equation with friction

(1.7) $\partial_{t}v-i\Delta v+\langle x,\nabla v\rangle=0.$

The PDE (1.7) is very different from (1.1), as one can readily surmise from
its invariance group of (real) left-translations

$\sigma_{(x,s)}(y,t)=(x,s)\circ(y,t)=(y+e^{t}x,t+s).$

As a special case of the result in [7, Theorem 4.1], one obtains for the
semigroup $\\{\mathcal{T}(t)\\}_{t\geq 0}$ associated with the Cauchy problem
for (1.7)

(1.8) $||\mathcal{T}(t)\varphi||_{L^{p^{\prime}}(\mathbb{R}^{m})}\leq C(m,p)\
\frac{e^{\frac{m}{p^{\prime}}t}}{(1-e^{-2t})^{m(\frac{1}{2}-\frac{1}{p^{\prime}})}}||\varphi||_{L^{p}(\mathbb{R}^{m})}.$

The behaviour as $t\to 0^{+}$ in (1.6) and (1.8) is the same, but because of
the presence of $e^{-\frac{|\cdot|^{2}}{4}}$ in the former, the two estimates
are incomparable.

##  2\. The imaginary Ornstein-Uhlenbeck

In this section we solve the Cauchy problem (1.1) by constructing a
representation of the solution operator. Our main result is (2.20) in
Proposition 2.4. It rests on Proposition 2.2, which could be derived from the
physicists’ Wick rotation, see [14, Section 3], in the Mehler formula for the
harmonic oscillator $H=-\Delta+\frac{|x|^{2}}{4}$, see e.g. p.154 in [3] or
p.55 in [6]. Such derivation however requires justifying some nontrivial
passages. Our elementary construction makes the solution operator (2.5)
immediately available independently from the harmonic oscillator and also
leads to directly unveil the basic identity (2.20).

In what follows, given a function $\varphi$ we indicate with $\psi$ the
function

(2.1) $\psi(x)=e^{-\frac{|x|^{2}}{4}}\varphi(x),\ \ \ \ \ \
x\in\mathbb{R}^{m}.$

Notice from (2.1) that $\psi\in
L^{2}(\mathbb{R}^{m})\Longleftrightarrow\varphi\in
L^{2}(\mathbb{R}^{m},d\gamma)$, and

(2.2)
$||\psi||_{L^{2}(\mathbb{R}^{m})}=||\varphi||_{L^{2}(\mathbb{R}^{m},d\gamma)}.$

We denote by

(2.3) $\mathscr{K}(\mathbb{R}^{m})=\\{\varphi\in
C^{\infty}(\mathbb{R}^{m})\mid\psi\in\mathscr{S}(\mathbb{R}^{m})\\}.$

It is clear that $\mathscr{K}(\mathbb{R}^{m})$ is dense in
$L^{2}(\mathbb{R}^{m},d\gamma)$.

######  Remark 2.1.

It follows from Proposition 2.4 below that

(2.4) $e^{itL}:\mathscr{K}(\mathbb{R}^{m})\ \longrightarrow\
\mathscr{K}(\mathbb{R}^{m}).$

Throughout this note we let $J^{+}=(0,\pi)$, $J^{-}=(\pi,2\pi)$, and denote

$J=J^{+}\cup J^{-}.$

The reader should keep in mind that the covariance matrix $Q(t)=e^{-it}\sin t\
I_{m}$ defined by (2.8) below is invertible for any $t\in J$. Since such
matrix can be extended by periodicity to the whole of
$\mathbb{R}\setminus\pi\mathbb{Z}$, we will confine the attention to the set
$J$.

######  Proposition 2.2.

Let $\varphi\in\mathscr{K}(\mathbb{R}^{m})$. For every $x\in\mathbb{R}^{m}$
the function

(2.5)
$f(x,t)=\begin{cases}\frac{(4\pi)^{-\frac{m}{2}}e^{\frac{imt}{2}}}{e^{\frac{i\pi
m}{4}}(\sin
t)^{\frac{m}{2}}}\int_{\mathbb{R}^{m}}e^{i\frac{|e^{it/2}y-e^{-it/2}x|^{2}}{4\sin
t}}\varphi(y)dy,\ \ \ \ \ \ \ t\in J^{+},\\\ \\\
\frac{(4\pi)^{-\frac{m}{2}}e^{\frac{imt}{2}}}{e^{\frac{i3\pi m}{4}}|\sin
t|^{\frac{m}{2}}}\int_{\mathbb{R}^{m}}e^{-i\frac{|e^{it/2}y-e^{-it/2}x|^{2}}{4|\sin
t|}}\varphi(y)dy,\ \ \ \ \ \ \ t\in J^{-},\end{cases}$

and $f(x,0)=\varphi(x)$, solves (1.1) in $\mathbb{R}^{m}\times J$.

###### Proof.

We begin with a simple, but critical observation. Suppose that $v$ and $f$ are
connected by the relation

(2.6) $v(x,t)=f(e^{it}x,t).$

Then, $f$ is a solution of the Cauchy problem (1.1) if and only if $v$ solves
the problem

(2.7) $\begin{cases}\partial_{t}v-iQ^{\prime}(t)\Delta v=0,\\\
v(x,0)=\varphi(x),\end{cases}$

where we have let

(2.8)
$Q(t)=\int_{0}^{t}e^{-2is}ds=\frac{1-e^{-2it}}{2i}=e^{-it}\frac{e^{it}-e^{-it}}{2i}=e^{-it}\sin
t.$

To prove that $v$ solves (2.7), we argue as follows. The chain rule gives from
(2.6)

$v_{t}(x,t)=ie^{it}\langle x,\nabla f(e^{it}x,t)\rangle+f_{t}(e^{it}x,t).$

On the other hand, the PDE in (1.1) gives

$f_{t}(e^{it}x,t)=i\Delta f(e^{it}x,t)-ie^{it}\langle x,\nabla
f(e^{it}x,t)\rangle.$

Combining the latter two equations, we infer that $v$ solves

$v_{t}(x,t)=i\Delta f(e^{it}x,t).$

Next, differentiating (2.6) we find

$\Delta v(x,t)=e^{2it}\Delta f(e^{it}x,t).$

We thus conclude that

$v_{t}(x,t)=ie^{-2it}\Delta v(x,t)=iQ^{\prime}(t)\Delta v(x,t),$

where in the second equality we have used (2.8). Summarising, the function $v$
solves the problem (2.7). To find a representation formula for the latter, we
use the Fourier transform. Supposing that $v$ be a solution, we define

(2.9) $\hat{v}(\xi,t)=\int_{\mathbb{R}^{m}}e^{-2\pi
i\langle\xi,x\rangle}v(x,t)dx.$

Then (2.7) is transformed into

(2.10)
$\begin{cases}\partial_{t}\hat{v}+4\pi^{2}iQ^{\prime}(t)|\xi|^{2}\hat{v}=0,\\\
\hat{v}(\xi,0)=\hat{\varphi}(\xi),\end{cases}$

whose solution is given by

(2.11) $\hat{v}(\xi,t)=\hat{\varphi}(\xi)e^{-4\pi^{2}iQ(t)|\xi|^{2}}.$

Note that, with $Q(t)$ as in (2.8), for every $t\in J$ the matrix $Q(t)I_{m}$
is invertible. Moreover, we have

(2.12) $iQ(t)I_{m}=i(\cos t-i\sin t)\sin tI_{m}=\sin^{2}t\
I_{m}+i\frac{\sin{2t}}{2}I_{m}.$

We now invoke [11, Theorem 7.6.1], which we formulate as follows: Let $A\in
G\ell(\mathbb{C},m)$ be such that $A^{\star}=A$ and $\Re A\geq 0$. Then

(2.13)
$\mathscr{F}\left(\frac{(4\pi)^{-\frac{m}{2}}}{\sqrt{\operatorname{det}A}}e^{-\frac{\langle
A^{-1}\cdot,\cdot\rangle}{4}}\right)(\xi)=e^{-4\pi^{2}\langle
A\xi,\xi\rangle},$

where $\sqrt{\operatorname{det}A}$ is the unique analytic branch such that
$\sqrt{\operatorname{det}A}>0$ when $A$ is real. If in (2.13) we take
$A=iQ(t)I_{m}=ie^{-it}\sin t\ I_{m}$ with $t\in J^{+}$, then

$A^{-1}=-i\frac{e^{it}}{\sin t}I_{m},$

and $\Re A=\sin^{2}t\ I_{m}\geq 0$. We thus find

(2.14)
$e^{-4\pi^{2}iQ(t)|\xi|^{2}}=\mathscr{F}\left(\frac{e^{\frac{imt}{2}}}{e^{\frac{i\pi
m}{4}}(\sin
t)^{\frac{m}{2}}}(4\pi)^{-\frac{m}{2}}e^{ie^{it}\frac{|\cdot|^{2}}{4\sin
t}}\right)(\xi).$

From (2.11) and (2.14) we conclude that for every $x\in\mathbb{R}^{m}$ and
$t\in J^{+}$

(2.15) $v(x,t)=\frac{e^{\frac{imt}{2}}}{e^{\frac{i\pi m}{4}}(\sin
t)^{\frac{m}{2}}}(4\pi)^{-\frac{m}{2}}\int_{\mathbb{R}^{m}}e^{ie^{it}\frac{|y-x|^{2}}{4\sin
t}}\varphi(y)dy.$

Finally, keeping (2.6) in mind, after some elementary algebraic manipulations,
we obtain the representation (2.5) when $t\in J^{+}$. The part corresponding
to $t\in J^{-}$ follows by elementary changes if one observes that now
$A=e^{i\frac{3\pi}{2}}e^{-it}|\sin t|I_{m}$.

∎

######  Remark 2.3.

It may be of interest to compare (2.5) with the well-known
representation222Formula (2.16) is classical. One way to easily obtain it is
by taking $A=I_{m},B=-2\sqrt{\omega}I_{m},c=0$ in (1.2) in the opening of
[10].

(2.16) $\displaystyle u(x,t)$
$\displaystyle=(4\pi)^{-\frac{m}{2}}e^{mt\sqrt{\omega}}\left(\frac{2\sqrt{\omega}}{\sinh(2t\sqrt{\omega})}\right)^{\frac{m}{2}}$
$\displaystyle\times\int_{\mathbb{R}^{m}}\exp\left(-\frac{\sqrt{\omega}}{2\sinh(2t\sqrt{\omega})}|e^{t\sqrt{\omega}}y-e^{-t\sqrt{\omega}}x|^{2}\right)\varphi(y)dy$

of the solution of the Cauchy problem for the Ornstein-Uhlenbeck operator

(2.17) $\begin{cases}u_{t}-\Delta u+2\sqrt{\omega}\langle x,\nabla
u\rangle=0,\ \ \ \ \ \omega>0,\\\ u(x,0)=\varphi(x).\end{cases}$

If one takes $\omega=\frac{1}{4}$, and keeping in mind that $\sinh it=i\sin
t$, then it is clear that _formally_ substituting $t\to it$ in (2.16), one
obtains the case $t\in J^{+}$ of (2.5).

In what follows we assume without restriction that $t\in J^{+}$. To further
unravel (2.5), and also to better clarify the role of the class
$\mathscr{K}(\mathbb{R}^{m})$ in (2.3), note that expanding

(2.18) $\frac{|e^{it/2}y-e^{-it/2}x|^{2}}{4\sin
t}=\frac{e^{it}|y|^{2}+e^{-it}|x|^{2}-2\langle x,y\rangle}{4\sin t},$

we find

(2.19) $f(x,t)=\frac{(4\pi)^{-\frac{m}{2}}e^{\frac{imt}{2}}}{e^{\frac{i\pi
m}{4}}(\sin
t)^{\frac{m}{2}}}\int_{\mathbb{R}^{m}}e^{i\frac{e^{it}|y|^{2}+e^{-it}|x|^{2}-2\langle
x,y\rangle}{4\sin t}}\varphi(y)dy.$

The change of variable $y=4\pi\sin t\ z$ in the integral in (2.19) gives

$\displaystyle\int_{\mathbb{R}^{m}}e^{-i\frac{\langle x,y\rangle}{2\sin
t}}e^{i\frac{e^{it}|y|^{2}+e^{-it}|x|^{2}}{4\sin t}}\varphi(y)dy=(4\pi\sin
t)^{m}e^{i\frac{e^{-it}|x|^{2}}{4\sin t}}\int_{\mathbb{R}^{m}}e^{-2\pi
i\langle x,z\rangle}e^{i\frac{e^{it}|4\pi\sin tz|^{2}}{4\sin
t}}\varphi(4\pi\sin tz)dz$ $\displaystyle=(4\pi\sin
t)^{m}e^{\frac{|x|^{2}}{4}}e^{i\frac{\cot
t|x|^{2}}{4}}\int_{\mathbb{R}^{m}}e^{-2\pi i\langle x,z\rangle}e^{i\frac{\cot
t|4\pi\sin tz|^{2}}{4}}e^{-\frac{|4\pi\sin tz|^{2}}{4}}\varphi(4\pi\sin
tz)dz.$

Keeping (2.1) in mind, we can rewrite the above integral as

$\displaystyle\int_{\mathbb{R}^{m}}e^{-i\frac{\langle x,y\rangle}{2\sin
t}}e^{i\frac{e^{it}|y|^{2}+e^{-it}|x|^{2}}{4\sin t}}\varphi(y)dy=(4\pi\sin
t)^{m}e^{\frac{|x|^{2}}{4}}e^{i\frac{\cot
t|x|^{2}}{4}}\mathscr{F}\left(\delta_{4\pi\sin t}\ e^{i\frac{\cot
t|\cdot|^{2}}{4}}\psi\right)(x)$
$\displaystyle=e^{\frac{|x|^{2}}{4}}e^{i\frac{\cot
t|x|^{2}}{4}}\mathscr{F}\left(e^{i\frac{\cot
t|\cdot|^{2}}{4}}\psi\right)(\frac{x}{4\pi\sin t}),$

where we have denoted by $\delta_{\lambda}f(x)=f(\lambda x)$ the action of the
dilation operator on a function $f$. Going back to (2.19), we have finally
established the following basic result.

######  Proposition 2.4.

Let $\varphi\in\mathscr{K}(\mathbb{R}^{m})$. Then for every $t\in J^{+}$ the
function $f(x,t)=e^{itL}\varphi(x)$ is given by the formula

(2.20)
$e^{-\frac{|x|^{2}}{4}}f(x,t)=(4\pi)^{-\frac{m}{2}}\frac{e^{\frac{imt}{2}}}{e^{\frac{i\pi
m}{4}}(\sin t)^{\frac{m}{2}}}e^{i\frac{\cot
t|x|^{2}}{4}}\mathscr{F}\left(e^{i\frac{\cot
t|\cdot|^{2}}{4}}\psi\right)(\frac{x}{4\pi\sin t}),$

where $\psi$ is defined by (2.1).

Note that (2.20) shows that if $\varphi\in\mathscr{K}(\mathbb{R}^{m})$, then
$f(\cdot,t)\in\mathscr{K}(\mathbb{R}^{m})$, see Remark 2.1, and also

$||f(\cdot,t)||_{L^{2}(\mathbb{R}^{m},d\gamma)}=||\varphi||_{L^{2}(\mathbb{R}^{m},d\gamma)}$

The equation (2.20) unveils the intertwining between the group $e^{itL}$ and
the Fourier transform. In the next section we exploit it to prove Theorem 1.1.

##  3\. Proof of Theorem 1.1

In this section we prove the uncertainty principle in Theorem 1.1. We will
also use Proposition 4.2 in Section 4 to derive Corollary 1.2. We will need
the following result, see [5, Theorem 1.1]. We note for the reader that our
normalisation of the Fourier transform

$\hat{\varphi}(\xi)=\mathscr{F}(\varphi)(\xi)=\int_{\mathbb{R}^{m}}e^{-2\pi
i\langle\xi,x\rangle}\varphi(x)dx,$

differs from theirs, and this accounts for the different constants in (3.1)
below.

######  Theorem 3.1.

Assume that $h:\mathbb{R}^{m}\to\mathbb{R}$ be a measurable function that
satisfies

(3.1)
$||e^{a|\cdot|^{2}}h||_{L^{2}(\mathbb{R}^{m})}+||e^{b|\cdot|^{2}}\hat{h}||_{L^{2}(\mathbb{R}^{m})}<\infty.$

If $ab\geq\pi^{2}$, then $h\equiv 0$.

We are ready to give the

###### Proof of Theorem 1.1.

Let $\varphi\in L^{2}(\mathbb{R}^{m},d\gamma)$ and $f(x,t)=e^{itL}\varphi(x)$.
Let $a,b>0$ be as in the statement of the theorem, so that $\sin s\not=0$.
With $\psi$ as in (2.1), consider the function defined by

(3.2) $h_{s}(x)=e^{i\frac{\cot s|x|^{2}}{4}}\psi(x).$

We have

(3.3) $\displaystyle\int_{\mathbb{R}^{m}}e^{2a|x|^{2}}|h_{s}(x)|^{2}dx$
$\displaystyle=\int_{\mathbb{R}^{m}}e^{2a|x|^{2}}|\psi(x)|^{2}dx=\int_{\mathbb{R}^{m}}e^{2a|x|^{2}}|\varphi(x)|^{2}e^{-\frac{|x|^{2}}{2}}dx$
$\displaystyle=||e^{a|x|^{2}}f(\cdot,0)||^{2}_{L^{2}(\mathbb{R}^{m},d\gamma)}<\infty,$

in view of (1.2). On the other hand, (2.20) gives

$\left|\hat{h}_{t}\left(\frac{x}{4\pi\sin
t}\right)\right|=(4\pi)^{\frac{m}{2}}(\sin
t)^{\frac{m}{2}}e^{-\frac{|x|^{2}}{4}}|f(x,t)|,$

and therefore for every $t\in J^{+}$ we have

(3.4)
$\displaystyle\left(\int_{\mathbb{R}^{m}}e^{2b|x|^{2}}\left|\hat{h}_{t}\left(\frac{x}{4\pi\sin
t}\right)\right|^{2}dx\right)^{1/2}=(4\pi)^{\frac{m}{2}}(\sin
t)^{\frac{m}{2}}\left(\int_{\mathbb{R}^{m}}e^{2b|x|^{2}}|f(x,t)|^{2}e^{-\frac{|x|^{2}}{2}}dx\right)^{1/2}$
$\displaystyle=(4\pi)^{\frac{m}{2}}(\sin
t)^{\frac{m}{2}}||e^{b|x|^{2}}f(\cdot,t)||_{L^{2}(\mathbb{R}^{m},d\gamma)}.$

If $s\in J^{+}$ is such that
$||e^{b|x|^{2}}f(\cdot,s)||_{L^{2}(\mathbb{R}^{m},d\gamma)}<\infty$, see
(1.2), then we infer from (3.4) that

(3.5)
$\int_{\mathbb{R}^{m}}e^{2(16b\pi^{2}\sin^{2}s)|y|^{2}}|\hat{h}_{s}(y)|^{2}dy<\infty.$

In view of (3.3) and (3.5), applying Theorem 3.1 to the function $h_{s}$ we
conclude that, if

$16\pi^{2}ab\sin^{2}s\geq\pi^{2}\ \Longleftrightarrow\
ab\sin^{2}s\geq\frac{1}{16},$

then $h_{s}(x)=0$ for every $x\in\mathbb{R}^{m}$. From (3.2), it is clear that
this implies $\psi\equiv 0$, and therefore $\varphi\equiv 0$, in
$\mathbb{R}^{m}$.

∎

Next, we present the

###### Proof of Corollary 1.2.

Let $u_{0}\in Dom(H)\subset L^{2}(\mathbb{R}^{m})$, and let
$u(x,t)=e^{itH}u_{0}(x)$. Suppose that $a,b>0$ are such that (1.4) be
satisfied. For every $t\in\mathbb{R}$ such that $\sin t\not=0$, define

$h_{t}(x)=e^{i\frac{\cot t|x|^{2}}{4}}u_{0}(x).$

It is clear that

$||e^{a|\cdot|^{2}}h_{t}||_{L^{2}(\mathbb{R}^{m})}=||e^{a|\cdot|^{2}}u_{0}||_{L^{2}(\mathbb{R}^{m})}<\infty.$

Arguing as in the proof of Theorem 1.1, but this time using (4.11) in
Corollary 4.4, we infer that

$\left(\int_{\mathbb{R}^{m}}e^{2b|x|^{2}}\left|\hat{h}_{s}\left(\frac{x}{4\pi\sin
s}\right)\right|^{2}dx\right)^{1/2}\cong||e^{b|\cdot|^{2}}u(\cdot,s)||_{L^{2}(\mathbb{R}^{m})}<\infty.$

Again by Theorem 1.1 we conclude that, under the hypothesis
$ab\sin^{2}s\geq\frac{1}{16}$, we must have $u_{0}\equiv 0$.

∎

We close this section by noting that, with Proposition 2.4 in hand, the proof
of Proposition 1.4 follows directly from (2.20) and from Beckner’s sharp
version of the the Hausdorff-Young theorem, see [1]. We leave the relevant
details to the interested reader. We will return to more general dispersive
estimates for the group $e^{itL}$ in a future study.

##  4\. Appendix: The imaginary harmonic oscillator

In what follows we consider the harmonic oscillator in $\mathbb{R}^{m}$

(4.1) $H=\Delta-\frac{|x|^{2}}{4}$

and the Cauchy problem in $\mathbb{R}^{m}\times(0,\infty)$ for the associated
Schrödinger operator

(4.2) $\begin{cases}\partial_{t}u-iHu=0\\\ u(x,0)=u_{0}(x),\end{cases}$

where the initial datum $u_{0}$ will be taken e.g. in
$\mathscr{S}(\mathbb{R}^{m})$. The following lemma establishes a general
principle, one interesting consequence of which is that it allows to connect
(4.2) to the problem (1.1), and in fact show that they are equivalent. The
functions $\Phi$ and $h$ in its statement are assumed complex-valued.

######  Lemma 4.1.

Let $\Phi\in C(\mathbb{R}^{m+1})$ and $h\in C^{2}(\mathbb{R}^{m+1})$ be
connected by the following nonlinear Schrödinger equation

(4.3) $ih_{t}+\Delta h-|\nabla h|^{2}=\Phi.$

Then $u$ solves the partial differential equation

(4.4) $Pu=i(\Delta u+\Phi u)-u_{t}=0$

if and only if $f$ defined by the transformation

(4.5) $u(x,t)=e^{-h(x,t)}f(x,t),$

solves the equation

(4.6) $i(\Delta f-2\langle\nabla h,\nabla f\rangle)-f_{t}=0.$

###### Proof.

With $u$ as in (4.5), we find

$\displaystyle Pu$ $\displaystyle=i\Delta(e^{-h}f)+i\Phi
e^{-h}f-(e^{-h}f)_{t}$ $\displaystyle=if\Delta(e^{-h})+ie^{-h}\Delta
f+2i\langle\nabla(e^{-h}),\nabla f\rangle+i\Phi
e^{-h}f-(e^{-h})_{t}f-e^{-h}f_{t}$ $\displaystyle=ie^{-h}f|\nabla
h|^{2}-ie^{-h}f\Delta h+ie^{-h}\Delta f-2ie^{-h}\langle\nabla h,\nabla
f\rangle+i\Phi e^{-h}f-(e^{-h})_{t}f-e^{-h}f_{t}$
$\displaystyle=e^{-h}\left\\{i\left[\Phi-ih_{t}-(\Delta h-|\nabla
h|^{2})\right]f+i\Delta f-2i\langle\nabla h,\nabla f\rangle-f_{t}\right\\}.$

This computation proves that if $h$ and $\Phi$ solve (4.3), then $u$ solves
(4.4) if and only if $f$ is a solution of (4.6).

∎

With Lemma 4.1 in hand, we now return to (4.1) and prove the following result.

######  Proposition 4.2.

A function $u$ solves the Cauchy problem (4.2) if and only if the function

(4.7) $f(x,t)=u(x,t)\ e^{\frac{|x|^{2}}{4}+i\frac{m}{2}t}$

solves (1.1) with $f(x,0)=\varphi(x)=u_{0}(x)\ e^{\frac{|x|^{2}}{4}}$.

###### Proof.

It is clear from (4.4) that, in order to obtain from it the PDE in (4.2), we
need $\Phi(x,t)=-\frac{|x|^{2}}{4}$. With this choice, we look for a function
$h(x,t)$ that is connected to such $\Phi$ by the equation (4.3). A natural
ansatz is $h(x,t)=A|x|^{2}+Bt$, with $A,B\in\mathbb{C}$ to be determined.
Since $\Delta h=2mA$, $h_{t}=B$ and $|\nabla h|^{2}=4A^{2}|x|^{2}$, to satisfy
(4.3) we want

$iB+2mA-4A^{2}|x|^{2}=-\frac{|x|^{2}}{4},$

which holds iff $A=\frac{1}{4},B=i\frac{m}{2}$, and thus

(4.8) $h(x,t)=\frac{|x|^{2}}{4}+i\frac{m}{2}t.$

With such choice of $h(x,t)$, the equation (4.5) in Lemma 4.1 shows that
$f(x,t)$ defined in (4.7) solves the Cauchy problem (1.1), with
$f(x,0)=\varphi(x)=u_{0}(x)\ e^{\frac{|x|^{2}}{4}}$. The “if and only if”
character of the statement is obvious.

∎

If $u_{0}\in\mathscr{S}(\mathbb{R}^{m})$, then it is clear that
$e^{\frac{|\cdot|^{2}}{4}}u_{0}\in\mathscr{K}(\mathbb{R}^{m})$. According to
Proposition 4.2, we can express the group $e^{itH}$ by the formula

(4.9)
$e^{itH}u_{0}(x)=e^{-\frac{|x|^{2}}{4}-i\frac{m}{2}t}e^{itL}(e^{\frac{|\cdot|^{2}}{4}}u_{0})(x).$

Applying (2.5), we infer from (4.9).

######  Corollary 4.3.

Given $u_{0}\in\mathscr{S}(\mathbb{R}^{m})$, for $t\in J^{+}$ one has

(4.10)
$e^{itH}u_{0}(x)=\frac{(4\pi)^{-\frac{m}{2}}e^{-\frac{|x|^{2}}{4}}}{e^{\frac{i\pi
m}{4}}(\sin
t)^{\frac{m}{2}}}\int_{\mathbb{R}^{m}}e^{i\frac{|e^{it/2}y-e^{-it/2}x|^{2}}{4\sin
t}}e^{\frac{|y|^{2}}{4}}u_{0}(y)dy.$

Using (2.18) in (4.10), we thus obtain the following counterpart of
Proposition 2.4.

######  Corollary 4.4.

Given $u_{0}\in\mathscr{S}(\mathbb{R}^{m})$, let $u(x,t)=e^{itH}u_{0}(x)$.
Then for every $t\in J^{+}$ one has

(4.11) $u(x,t)=\frac{(4\pi)^{-\frac{m}{2}}}{e^{\frac{i\pi m}{4}}(\sin
t)^{\frac{m}{2}}}e^{i\frac{\cot t|x|^{2}}{4}}\mathscr{F}\left(e^{i\frac{\cot
t|\cdot|^{2}}{4}}u_{0}\right)(\frac{x}{4\pi\sin t}).$

##  5\. Declarations

Data availability statement: This manuscript has no associated data.

Funding and/or Conflicts of interests/Competing interests statement: The
author declares that he does not have any conflict of interest for this work

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