# Spin Coherent Manipulation in Josephson Weak Links

Javier Cerrillo<EMAIL_ADDRESS>Área de Física Aplicada, Universidad
Politécnica de Cartagena, E-30202 Cartagena, Spain  Departamento de Física
Teórica de la Materia Condensada C-V, Universidad Autónoma de Madrid, E-28049
Madrid, Spain  M. Hays  Department of Applied Physics, Yale University, New
Haven, CT 06520, USA  V. Fatemi  Department of Applied Physics, Yale
University, New Haven, CT 06520, USA  Alfredo Levy Yeyati<EMAIL_ADDRESS>Departamento de Física Teórica de la Materia Condensada C-V, Universidad
Autónoma de Madrid, E-28049 Madrid, Spain  Condensed Matter Physics Center
(IFIMAC) and Instituto Nicolás Cabrera, Universidad Autónoma de Madrid,
E-28049 Madrid, Spain

###### Abstract

Novel designs of Josephson weak links based on semiconducting nanowires
combined with circuit QED techniques have enabled the resolution of their fine
structure due to spin-orbit interactions, opening a path towards Andreev spin
qubits. Nevertheless, direct manipulation of the spin within a given Andreev
state is in general suppressed compared to inter-doublet manipulation in the
absence of Zeeman effects. In addition, noisy spin-flip mechanisms limit any
coherent manipulation protocol to spin post-selection. We propose a
combination of a spin polarization protocol analogous to sideband cooling with
stimulated Raman adiabatic passage specifically tailored for these systems. We
show this approach is robust for a large range of design parameters, including
the currently rather stringent coherence times.

While conventional superconducting qubits are becoming a mature technology
Arute2019 , other radically new approaches involving superconducting
nanostructures remain to be explored Karzig2017 . In particular some qubit
proposals rely on the Andreev bound states (ABS) which appear in highly
transmissive few-channel Josephson weak links Desposito2001 ; Zazunov2003 ;
Janvier2015 ; Chtchelkatchev2003 ; Padurariu2010 ; Park2017 . Within these
proposals one can distinguish those operating when the ABS are populated with
an even number of particles (the even sector), which are called Andreev level
qubits (ALQ) Desposito2001 ; Zazunov2003 ; Janvier2015 ; and the ones in which
an extra unpaired quasiparticle is trapped in the weak link, which corresponds
to the so-called Andreev spin qubits (ASQ) Chtchelkatchev2003 ; Padurariu2010
; Park2017 . This type of sytems combines interesting aspects of microscopic
degrees of freedom (such as their isolation from the environment) with the
potential scalability of solid state devices. Another attractive aspect of ASQ
is their inherently fermionic character and their connection to Majorana
physics Prada2019 . In fact, under sufficiently large magnetic fields the
proximitized nanowires used in ASQ experiments could host Majorana states at
their edges Lutchyn2010 ; Oreg2010 .

$\left|A\right\rangle$$\left|B\right\rangle$$\left|C\right\rangle$$\Omega_{A}$$\Omega_{B}$$\Delta$a)b)c)d)$\epsilon_{ks}/\Delta_{S}$0.00.51.0$\phi/\pi$01.02.0$\Phi$$v_{f}$$v_{g}$$\left|1\right\rangle$$\left|2\right\rangle$$\left|\downarrow\right\rangle$$\left|\uparrow\right\rangle$$\Omega_{g}$$\Omega_{f}$$\alpha\Omega_{g}$$g$

Figure 1: a) Traditional STIRAP setup with a three-level system in
$\Lambda$-configuration and a Raman coupling generated by two fields
characterized by Rabi frequencies $\Omega_{A}$ and $\Omega_{B}$. b) Schematic
diagram of the experimental setup: a superconducting loop (blue) threaded by a
magnetic flux $\Phi$ is weakly linked by a semiconducting nanowire (orange).
The weak link can be controlled by AC gate voltage $v_{g}$ and inductively by
an AC flux voltage $v_{f}$. c) Level dispersion relation as a function of
$\phi$ for a device showing spin-split ABSs as reported in Tosi2019 . The thin
lines correspond to the ballistic case and the thick lines to the presence of
backscattering ($r\neq 0$). d) The four level system at a fixed phase value
$\phi$ as a coupled two-spin system, the first (mode) qubit represented by
Pauli matrices $\tau_{i}$ and the second (spin) qubit represented by Pauli
matrices $\sigma_{i}$. Gate and flux drivings induce Rabi terms
$\Omega_{g}\tau_{x}$ and $\Omega_{f}\tau_{x}$ respectively. Additionally, gate
driving induces a $\alpha\Omega_{g}\tau_{x}\sigma_{x}$ interaction and a
ferromagnetic term $g\tau_{z}\sigma_{z}$ exists between both qubits.

Recent experiments on weak links formed by semiconducting nanowires coupled to
a superconducting resonator in a circuit QED geometry have enabled detection
of their ABS spectrum Woerkom2017 ; Hays2018 , revealed their fine structure
caused by spin-orbit interactions Tosi2019 , and offer the possibility to
perform time resolved detection of their population dynamics Hays2019 . While
these results constitute a significant step forward, these systems still have
limited controllability and, more specifically, direct spin flip transitions
can be hardly implemented in the absence of magnetic fields Park2017 . Thus,
the coherent manipulation of the spin of a single quasiparticle trapped within
these junctions requires special techniques which are still to be developed in
the context of superconducting nanodevices.

In the present work we propose to address these problems by adapting
techniques which are common in the field of quantum optics. A useful protocol
for the control of population transfer between two states that lack a direct
coupling is the stimulated Raman adiabatic passage (STIRAP) Gaubatz1990 ;
Bergmann2019 . It consists in the application of two delayed overlapping
pulses in Raman configuration with respect to a third state, as illustrated in
Fig.1a). STIRAP is known to be robust against pulse shape modifications. This
is in stark contrast with the simultaneous application of two fields in Raman
configuration, which requires very accurate adjustment of the pulse area to a
value of $\pi$, as well as careful adjustment of the Rabi frequency $\Omega$
of both fields to similar values. In addition, a large detuning
$\Delta\gg\Omega$ is needed in order to suppress population transfer to the
excited levels. This reduces the effective Rabi frequency, which translates
into a transfer time that is proportional to $\frac{\Delta}{\Omega^{2}}$. This
is in general a factor $\frac{\Delta}{\Omega}$ slower than STIRAP.

In the context of superconducting circuits, the STIRAP protocol was first
proposed for Cooper-pair boxes Siewert2006  and experimentally demonstrated on
transmons Kumar2016 . Unfortunately, STIRAP techniques cannot be applied
directly to the Josephson weak link case. In the devices of Refs.Tosi2019 ;
Hays2019  with a typical junction length of the order of 400 nm, two set of
spin-split ABSs appear within the superconducting gap. Their characteristic
dispersion as a function of the superconducting phase difference $\phi$ is
illustrated in Fig.1c). Thus, when $\phi$ is fixed to a value different from
$0$ or $\pi$, there are four states within the odd sector which can be
connected by single particle transitions, instead of the three usually
considered for STIRAP. These can be viewed as two coupled qubits, one with
transition frequencies of the order of 10GHz, which is associated with
different Andreev longitudinal modes and one with frequencies of the order of
1GHz which is associated with the spin degree of freedom. Ambient flux or
charge noise fully mixes the spin degree of freedom Hays2019  and thus
additional spin-polarization techniques are necessary to prepare an initially
pure state which is required to apply the STIRAP protocol successfuly. We also
notice that intra-doublet spin-flip transitions are in general suppressed
compared to inter-doublet transitions Park2017  while spin-flip transitions
between manifolds are allowed with an efficiency $\alpha(\phi)$ with respect
to the spin-conserving ones. This efficiency has been experimentally observed
to take values in a range between 0 and 1 both for gate driving Tosi2019 ;
Metzger2020  and for flux driving Hays2019 ; Hays2020 . Although the parameter
is kept in all subsequent derivations, for simplicity in the simulations we
shall assume a combination of a purely spin-conserving drive ($\alpha=0$) as
well as an $\alpha>0$ drive.

We first discuss a simplified modeling of the ABSs in these systems and then
propose an implementation of a protocol analogous to STIRAP. We analyze the
most appropriate preparation technique and produce an analytical model for its
optimization. We show that the protocol is robust for a large range of driving
strengths and spin decohering rates, thus providing an avenue for the control
of ASQ regardless of device specifications.

$\left|1\downarrow\right\rangle$$\left|1\uparrow\right\rangle$$\left|2\downarrow\right\rangle$$\left|2\uparrow\right\rangle$$\alpha\Omega_{g}$$\Omega_{g}$$\alpha\Omega_{g}$$\Omega_{g}$$\Omega_{f}$$\Omega_{f}$$\Delta$$\Delta^{\prime}$a)

Figure 2: a) Schematic level diagram with available couplings generated by
gate AC driving (blue) and magnetic flux AC driving (orange). The frequencies
$\omega_{g}$ and $\omega_{f}$ are chosen such that the Raman resonance
condition Eq.(5) is fulfilled. b) Rabi frequencies $\Omega_{g}$ and
$\Omega_{f}$ as a function of time for $\alpha=1$. Otherwise, the intensity of
the gate pulse has to be enhanced by $\alpha^{-1}$ to achieve transfer. c)
Value of $\theta=\arctan(\alpha\Omega_{g}/\Omega_{f})$ as a function of time
for the pulses in b). d) Dynamics of the occupation probabilities of states
$\left|1\downarrow\right\rangle$, $\left|1\uparrow\right\rangle$,
$\left|2\downarrow\right\rangle$ and $\left|2\uparrow\right\rangle$ under the
effect of the pulses in b) and for $\alpha=1$, $\Delta=0$ and
$\Delta^{\prime}=2\pi\times 2{\rm GHz}$.

System Modeling — In order to obtain a simple model for the ABSs in the
nanowire Josephson junction we consider the ballistic case with a single
occupied subband, characterized by two modes $\left|m_{1,2}\right\rangle$ with
different Fermi velocities $v_{1,2}$ Tosi2019 ; SM . The low energy effective
Hamiltonian can be written in this basis as

$\displaystyle H_{eff}$ $\displaystyle=$
$\displaystyle\sum_{j=1,2}\frac{\Delta_{S}}{1+\lambda_{j}}\frac{\pi-(-1)^{j}\phi\sigma_{z}}{2}\left|m_{j}\right\rangle\left\langle
m_{j}\right|$ $\displaystyle+$ $\displaystyle
r\Delta_{S}\left(\left|m_{1}\right\rangle\left\langle
m_{2}\right|+H.c.\right),$

where $\Delta_{S}$ is the superconducting gap, $\lambda_{j}=\Delta_{S}L/\hbar
v_{j}$ where $L$ is the junction length, $r$ the reflection coefficient of the
weak link and $\sigma_{z}$ the Pauli matrix for the pseudospin degree of
freedom in the basis spanned by states $\left|\uparrow\right\rangle$ and
$\left|\downarrow\right\rangle$. More details are provided in the
supplementary information SM .

After diagonalization, it can be written as SM

$\displaystyle H_{eff}$ $\displaystyle=$
$\displaystyle\frac{\omega}{2}\tau_{z}-\frac{g}{2}\tau_{z}\sigma_{z}+\frac{\nu}{2}\sigma_{z}+E_{0}$
$\displaystyle=$ $\displaystyle\sum_{\begin{subarray}{c}k=1,2\\\
s=\uparrow,\downarrow\end{subarray}}\epsilon_{ks}\left|ks\right\rangle\left\langle
ks\right|$

where we now define $\tau_{z}$ as the Pauli matrix for the Andreev
longitudinal mode degrees of freedom in the basis spanned by states
$\left|1\right\rangle$ and $\left|2\right\rangle$ with an attractive
interaction term of strength $g$ that inverts the spin orientation in the
upper level and a global energy offset $E_{0}$. Note that the degrees of
freedom have become rotated with respect to Eq.(Spin Coherent Manipulation in
Josephson Weak Links).

We consider a gate AC-driving at frequency $\omega_{g}$ that produces
pseudospin-conserving transitions of strength $\Omega_{g}$ and pseudospin-
flipping transitions of weaker strength $\alpha\Omega_{g}$

$H_{g}(t)=\Omega_{g}\cos(\omega_{g}t)\left(1+\alpha\sigma_{x}\right)\tau_{x},$
(3)

and an additional flux AC-driving at frequency $\omega_{f}$ that produces
pseudospin-conserving transitions of strength $\Omega_{f}$ and no pseudospin-
flipping transitions ($\alpha=0$)

$H_{f}(t)=\Omega_{f}\cos(\omega_{f}t)\tau_{x}.$ (4)

A layout of the available degrees of freedom and couplings is shown in
Fig.2a).

STIRAP Implementation — As shown in Fig.2a), we consider a four level system
unlike the typical STIRAP setting of three levels (see Fig.1a)). In order to
implement a STIRAP protocol between the initial state
$\left|1\downarrow\right\rangle$ and the final state
$\left|1\uparrow\right\rangle$, the following Raman resonance condition is
imposed

$\Delta\equiv\left(\epsilon_{2\uparrow}-\epsilon_{1\downarrow}\right)-\omega_{g}=\left(\epsilon_{2\uparrow}-\epsilon_{1\uparrow}\right)-\omega_{f}$
(5)

such that a rotating wave approximation (RWA) in the relevant interaction
picture reveals the transitions

$\displaystyle H^{\prime}_{g}+H^{\prime}_{f}$ $\displaystyle\simeq$
$\displaystyle\frac{\Omega_{f}}{2}\left|1\uparrow\right\rangle\left\langle
2\uparrow\right|+\frac{\alpha\Omega_{g}}{2}\left|1\downarrow\right\rangle\left\langle
2\uparrow\right|$ (6)
$\displaystyle+\frac{\Omega_{g}}{2}\left|1\downarrow\right\rangle\left\langle
2\downarrow\right|+\mathrm{H.c.}$

including the wanted $\Lambda$-coupling that satisfies the Raman condition and
an unwanted vertical coupling detuned by
$\Delta^{\prime}=\left(\epsilon_{2\downarrow}-\epsilon_{1\downarrow}\right)-\omega_{g}$.
It is important to notice that, beyond these three couplings, the RWA neglects
three additional unwanted non-resonant couplings as shown in Fig.2a)). In
particular, the coupling between $\left|1\uparrow\right\rangle$ and
$\left|2\uparrow\right\rangle$ of strength $\Omega_{g}$ may be even more
detrimental due to its proximity to resonance. These couplings can only be
captured in a full time-dependent numerical simulation of the sort performed
in this work. Their effect is in general suppressed with the choice
$\Delta=0$, which is the default choice in the remainder of the discussion
unless otherwise stated.

$\alpha\Omega_{g}$$\Omega_{g}$$\alpha\Omega_{g}$$\Omega_{g}$$-\Delta$a)

Figure 3: a) Resonance condition
$\omega_{g}=\epsilon_{2\downarrow}-\epsilon_{1\uparrow}$ for the preparation
of state $\left|1\downarrow\right\rangle$ with a single gate pulse (Note the
difference with the arrangement of Fig.2a). b) Population dynamics under the
effect of a continuous gate pulse of strength $\Omega_{g}=2\pi\times
3\,\mathrm{MHz}$ for $\gamma=20\,\mathrm{rad/ms}$ and
$\gamma_{d}=100\,\mathrm{rad/\mu s}$. c) Purity of the steady state of the
spin degrees of freedom as a function of the driving Rabi frequency
$\Omega_{g}$ and the spin mixing rate $\gamma$ for
$\gamma_{d}=100\,\mathrm{rad/\mu s}$. d) Purity of the steady state of the
spin degrees of freedom as a function of the driving Rabi frequency
$\Omega_{g}$ and the decoherence rate $\gamma_{d}$ for
$\gamma=10\,\mathrm{rad/ms}$. e) Purity of the steady state of the spin
degrees of freedom as a function of the driving Rabi frequency $\Omega_{g}$
and the detuning $\Delta$ for $\gamma=5\,\mathrm{rad/ms}$ and
$\gamma_{d}=100\,\mathrm{rad/\mu s}$. Purity values may range between $1$
(pure state) and $0.5$ (fully mixed 2-level system).

The STIRAP protocol consists of the introduction of a slow modulation of the
parameters $\Omega_{f}$ and $\Omega_{g}$

$\Omega_{f}(t)=\alpha
Ae^{-\left(\frac{t-t_{f}}{T}\right)^{2}},\quad\Omega_{g}(t)=Ae^{-\left(\frac{t-t_{g}}{T}\right)^{2}},$
(7)

at times $t_{f}<t_{g}$ with pulse amplitudes $\alpha A$, $A$ and widths $T$.
Under the time-dependent transformation of the lower states into a bright
$\left|B\right\rangle$ and a dark $\left|D\right\rangle$ states defined by

$\displaystyle\left|B\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\Omega}\left|1\right\rangle\left(\alpha\Omega_{g}\left|\downarrow\right\rangle+\Omega_{f}\left|\uparrow\right\rangle\right),$
$\displaystyle\left|D\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\Omega}\left|1\right\rangle\left(\Omega_{f}\left|\downarrow\right\rangle-\alpha\Omega_{g}\left|\uparrow\right\rangle\right),$
(8)

with $\Omega=\sqrt{\Omega_{f}^{2}+\alpha^{2}\Omega_{g}^{2}}$ the root mean
square Rabi frequency, the effective Hamiltonian simplifies to the form

$\displaystyle H(t)$ $\displaystyle=$ $\displaystyle
H^{\prime}_{eff}+H^{\prime}_{g}+H^{\prime}_{f}$ $\displaystyle\simeq$
$\displaystyle\left(\Delta\left|\uparrow\right\rangle\left\langle\uparrow\right|+\Delta^{\prime}\left|\downarrow\right\rangle\left\langle\downarrow\right|\right)\left|2\right\rangle\left\langle
2\right|$
$\displaystyle+\frac{\Omega}{2}\left(\left|2\uparrow\right\rangle\left\langle
B\right|+H.c.\right)+i\dot{U}U^{\dagger},$

where the time-dependent transformation $U=\left|D\right\rangle\left\langle
1\downarrow\right|+\left|B\right\rangle\left\langle
1\uparrow\right|+\sum_{\sigma=\downarrow,\uparrow}\left|2\sigma\right\rangle\left\langle
2\sigma\right|$ introduces the term

$i\dot{U}U^{\dagger}=i\dot{\theta}\left|B\right\rangle\left\langle
D\right|+H.c.$ (10)

with $\tan\theta=\alpha\frac{\Omega_{g}}{\Omega_{f}}$ so that
$\dot{\theta}=\alpha\frac{\dot{\Omega}_{g}\Omega_{f}-\dot{\Omega}_{f}\Omega_{g}}{\Omega^{2}}$.
For very slow (adiabatic) variations of the Rabi frequency, the dark state
$\left|D\right\rangle$ remains decoupled. The mechanism of STIRAP is
straightforward from here. A system is initially prepared in
$\left|1\downarrow\right\rangle$, which becomes the dark state under the
effect of the first pulse, and can be adiabatically transformed into any
superposition with $\left|1\uparrow\right\rangle$ by an appropiate choice of
the timing of $\theta$. The condition of adiabaticity can be expressed as

$\frac{\dot{\theta}}{\Omega}\ll 1$ (11)

at all times $t$.

We present an implementation of the protocol in Fig.2. As an example,
following Hays2019  we may assume a spectrum with non-degenerate transitions
such that

$\displaystyle\epsilon_{1\uparrow}-\epsilon_{1\downarrow}$ $\displaystyle=$
$\displaystyle 2\pi\times 1\,\mathrm{GHz},$
$\displaystyle\epsilon_{2\downarrow}-\epsilon_{1\downarrow}$ $\displaystyle=$
$\displaystyle 2\pi\times 13\,\mathrm{GHz},$
$\displaystyle\epsilon_{2\uparrow}-\epsilon_{1\downarrow}$ $\displaystyle=$
$\displaystyle 2\pi\times 11\,\mathrm{GHz},$ (12)

the choice $\Delta=0$ imposes $\omega_{g}=2\pi\times 11\,\mathrm{GHz}$,
$\omega_{f}=2\pi\times 10\,\mathrm{GHz}$, and $\Delta^{\prime}=2\pi\times
2\,\mathrm{GHz}$. Assuming no suppression of spin-flipping transitions
$\alpha=1$, a choice of timings $t_{g}=2T=-t_{f}=2.5\,\mathrm{ns}$ imposes a
minimum pulse strength $A=2\pi\times 400\,\mathrm{MHz}$ in order for Eq.(11)
to be satisfied. The population transfer corresponding to such pulses
(Fig.2b)-c)) is shown in Fig.2d). Note a rather small transfer of population
to level $\left|2\uparrow\right\rangle$ due to the presence of the strong non-
resonant coupling between $\left|1\uparrow\right\rangle$ and
$\left|2\uparrow\right\rangle$ generated by the gate driving. Our parametric
choice prevents this effect from affecting the success of STIRAP.

Dissipation and state preparation — As investigated in Hays2019 ; Hays2020 ,
the states have both a limited lifetime and coherence time. In their sample,
two pseudospin-conserving decay channels between excited and ground levels are
found of rates $\Gamma$ ranging between $250\,\mathrm{rad/ms}$ and
$333\,\mathrm{rad/ms}$. Additionally, a pseudospin mixing term of rate
$\gamma$ drives the system into a maximal mixture of states
$\left|\downarrow\right\rangle$ and $\left|\uparrow\right\rangle$. The numeric
value has been investigated in Hays2019  with an inconclusive range between
$\simeq 24.4\,\mathrm{rad/ms}$ and $\simeq 0.4\,\mathrm{rad/ms}$, as discussed
in SM . Additionally, the coherence of the spin degrees of freedom decays at a
rate $\gamma_{d}$ in the order of $100\,\mathrm{rad/\mu s}$ Hays2020 . We
model these decoherence effects by means of the master equation

$\displaystyle\frac{d}{dt}\rho$ $\displaystyle=$
$\displaystyle\left[H(t),\rho\right]+\frac{\gamma_{d}}{2}\mathcal{D}\left[\tau_{z}\right](\rho)+\frac{\gamma_{d}}{2}\mathcal{D}\left[\sigma_{z}\right](\rho)$
$\displaystyle+$
$\displaystyle\frac{\Gamma}{2}\mathcal{D}\left[\tau_{-}\right](\rho)+\frac{\gamma}{2}\mathcal{D}\left[\sigma_{-}\right](\rho)+\frac{\gamma}{2}\mathcal{D}\left[\sigma_{+}\right](\rho),$

where $\tau_{-}=\left|1\right\rangle\left\langle 2\right|$,
$\sigma_{-}=\left|\downarrow\right\rangle\left\langle\uparrow\right|$,
$\sigma_{+}=\left(\sigma_{-}\right)^{\dagger}$ and with the Lindbladian
superoperator $\mathcal{D}\left[s\right](\rho)=2s\rho
s^{\dagger}-s^{\dagger}s\rho-\rho s^{\dagger}s$. Note that we assume a similar
coherence lifetime of superpositions of $\left|1\right\rangle$ and
$\left|2\right\rangle$ states in line with the findings in Hays2020 .

Due to these processes, the system is initially in the fully mixed state
$\rho(0)=1/2\left|1\right\rangle\left\langle
1\right|\sum_{s=\downarrow,\uparrow}\left|s\right\rangle\left\langle
s\right|$, which imposes the need to prepare the sample in a state of low
entropy for any control protocol. Fortunately, the gate pulse in the STIRAP
arrangement (Fig.2a)) automatically produces an optical pumping effect that
depletes the state $\left|1\downarrow\right\rangle$, similar to the case shown
in Hays2020 . For an appropriate choice of $\Delta$ (see Fig.3a)), the state
$\left|1\uparrow\right\rangle$ may be depleted instead (Fig.3b)). This is due
to the resonance with the
$\left|1\uparrow\right\rangle-\left|2\downarrow\right\rangle$ transition,
which pumps population from $\left|\uparrow\right\rangle$ to
$\left|\downarrow\right\rangle$ that then preferentially decays to
$\left|1\downarrow\right\rangle$.

Figure 4: a) Trace distance between the spin state before and after the STIRAP
protocol as a function of the Rabi frequency $\Omega_{g}$ of the preparation
pulse and the decoherence rate $\gamma_{d}$ for $\gamma=10\,\mathrm{rad/ms}$.
STIRAP pulses as in Fig.2b). b) Population transferred from state
$\left|1\downarrow\right\rangle$ to state $\left|1\uparrow\right\rangle$ after
application of the STIRAP protocol as a function of the Rabi frequency $A$ and
the pulse separation $2(t_{g}-t_{f})$ for $T=5\,\mathrm{ns}$, a spin mixing
rate $\gamma=10\,\mathrm{rad/ms}$ and a decoherence rate
$\gamma_{d}=100\,\mathrm{rad/\mu s}$.

The effect of the polarizing pulse may be described in terms of the rate
equation SM

$\frac{d}{dt}\begin{pmatrix}p_{\downarrow}\\\
p_{\uparrow}\end{pmatrix}=\begin{pmatrix}-\gamma_{\uparrow}-\gamma&\gamma_{\downarrow}+\gamma\\\
\gamma_{\uparrow}+\gamma&-\gamma_{\downarrow}-\gamma\end{pmatrix}\begin{pmatrix}p_{\downarrow}\\\
p_{\uparrow}\end{pmatrix},$ (14)

For an optimal choice of parameters, $\gamma_{\uparrow}\simeq 0$ and
$\gamma_{\downarrow}\simeq\Gamma$, so that the population inversion
$\Gamma/(2\gamma+\Gamma)$ is reached. This is only going to approach 1 for
$\gamma\ll\Gamma$ (see Fig.3b), and this becomes a crucial criterion to select
the parametric region for the implementation of STIRAP. The choice of
$\Omega_{g}$ is additionally affected by the decoherence rate $\gamma_{d}$, as
shown in Fig.3c). Finally, it is equally possible to address the detuning
value $\Delta=0$ in order to prepare the state $\left|1\uparrow\right\rangle$
with similar efficiency.

Global efficiency of the protocol — The final success of state transfer from
$\left|1\downarrow\right\rangle$ into $\left|1\uparrow\right\rangle$ after the
polarization pulse into $\left|1\downarrow\right\rangle$ and the STIRAP
protocol can be measured in terms of the trace distance
$Tr[\sqrt{(\rho_{0}-\rho_{t})^{2}}]$ between the spin density matrices before
$\rho_{0}$ and after $\rho_{t}$ the STIRAP protocol. Its maximum is 1 for
orthogonal states, while it vanishes for equal states. For a 2-level system,
it represents an appropriate measure of the success of the transfer for mixed
states. This is shown in Fig.4a). The transfer is successful for a large range
of preparation pulse strengths and decoherence rates, but it is mostly
dependent on the purity of the initial state, as one can deduce from comparing
Figs. 4b) and 3d). At the same time, variations in the overall Rabi frequency
$A$ and the separation betweeen pulses $t_{g}-t_{f}$ do not substantially
affect the final transfer efficiency, as shown in Fig.4b). On the contrary,
application of continuous Raman pulses (corresponding to the left edge of the
same figure) is highly sensitive to pulse strength, whereas STIRAP is robust
over a large region. This makes this proposal suitable for varying design
parameters.

Conclusions — We have proposed a means to the manipulation of the pseudospin
degrees of freedom of Andreev bound states that include a polarization pulse
and subsequent coherent manipulation with the STIRAP technique. We have shown
that the protocol could be efficient even for the stringent conditions of
actual present experiments in which coherent times are rather short and is in
line with current efforts to develop an ASQ by several experimental groups.

Acknowledgments — A.L.Y., J.C. acknowledge European Union’s Horizon 2020
research and innovation programme for financial support grant no. 828948
(AndQC) and QuantERA project no. 127900 (SuperTOP). J.C. acknowldeges the
support from Ministerio de Ciencia, Innovación y Universidades (Spain)
(“Beatriz Galindo” Fellowship BEAGAL18/00081). A.L.Y. acknowledges support by
Spanish MICINN through grants FIS2017-84860-R and through the “María de
Maeztu” Programme for Units of Excellence in R&D (Grant No. MDM-2014-0377).
M.H. and V.F. acknowldege support by the US Office of Naval Research
(N00014-16-1-2270) and by the US Army Research Office (W911NF-18-1-0020,
W911NF-18-1-0212 and W911NF-16-1-0349).

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  * (13) M. Hays, G. de Lange, K. Serniak, D.J. van Woerkom, D. Bouman, P. Krogstrup, J. Nygård, A. Geresdi, and M.H. Devoret, “Direct Microwave Measurement of Andreev-Bound-State Dynamics in a Semiconductor-Nanowire Josephson Junction”, Phys. Rev. Lett. 121, 047001 (2018).
  * (14) L. Tosi, C. Metzger, M.F. Goffman, C. Urbina, H. Pothier, S. Park, A. Levy Yeyati, J. Nygård, and P. Krogstrup, “Spin-Orbit Splitting of Andreev States Revealed by Microwave Spectroscopy”, Phys. Rev. X 9, 011010 (2019).
  * (15) M. Hays, V. Fatemi, K. Serniak, D. Bouman, S. Diamond, G. de Lange, P. Krogstrup, J. Nygård, A. Geresdi, and M. H. Devoret, “Continuous monitoring of a trapped, superconducting spin”, Nature Phys. (2020).
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  * (21) M. Hays, V. Fatemi, K. Serniak, D. Bouman, S. Diamond, G. de Lange, P. Krogstrup, J. Nygård, A. Geresdi, and M. H. Devoret, “Coherent spin manipulation of an Andreev spin qubit”, in preparation (2020).
  * (22) See supplementary material.
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##  I Supplemental Information

##  II Effective Model derivation

Ballistic case: As shown in Ref. sTosi2019  a $\sim$ 400 nm long nanowire
Josephson junction typically exhibits two sets of spin-split ABSs for a single
channel situation. We take the ballistic case analyzed in Refs. sPark2017 ;
sTosi2019  as a reference to describe the type of phase dependence and the
spin splitting exhibited by these states. For energies small compared to the
superconducting gap, $\Delta_{S}$, the lowest energy states can be
approximated as sTosi2019

$\displaystyle\epsilon_{m_{1},\uparrow}(\phi)$ $\displaystyle=$
$\displaystyle\frac{\Delta_{S}}{1+\lambda_{1}}\frac{\pi+\phi}{2}$
$\displaystyle\epsilon_{m_{1},\downarrow}(\phi)$ $\displaystyle=$
$\displaystyle\frac{\Delta_{S}}{1+\lambda_{1}}\frac{\pi-\phi}{2}$
$\displaystyle\epsilon_{m_{2},\uparrow}(\phi)$ $\displaystyle=$
$\displaystyle\frac{\Delta_{S}}{1+\lambda_{2}}\frac{\pi-\phi}{2}$
$\displaystyle\epsilon_{m_{2},\downarrow}(\phi)$ $\displaystyle=$
$\displaystyle\frac{\Delta_{S}}{1+\lambda_{2}}\frac{\pi+\phi}{2}\;,$

where $m_{1,2}$ indicate the modes with different Fermi velocities and
$\uparrow,\downarrow$ the corresponding pseudospin index. On the other hand
$\lambda_{j=1,2}=L\Delta_{S}/(\hbar v_{j})$, where $L$ is the junction length
and

$v_{j}=\frac{\hbar
k_{Fj}}{m^{*}}+(-1)^{j}\frac{\alpha_{SO}\left(E^{\perp}_{1}/2-(-1)^{j}\alpha_{SO}k_{Fj}\right)}{\hbar\sqrt{\left(E^{\perp}_{1}/2-(-1)^{j}\alpha_{SO}k_{Fj}\right)^{2}+\eta^{2}}},$
(15)

where $E_{1}^{\perp}$ is the ground state energy of the transverse modes,
$\alpha_{SO}$ the strength of the spin-orbit coupling, $\eta$ the spin-
flipping coupling strength to the first excited transverse mode and $k_{Fj}$
are the Fermi wave vectors satisfying $E_{s}(k_{Fj})=\mu$, with

$E_{s}(k_{x})=\frac{\hbar^{2}k^{2}_{x}}{2m^{*}}+\frac{E^{\perp}_{1}+E^{\perp}_{2}}{2}-\sqrt{\left(\frac{E^{\perp}_{1}-E^{\perp}_{2}}{2}-s\alpha_{SO}k_{x}\right)^{2}+\eta^{2}},$
(16)

where $E_{2}^{\perp}$ is the energy of the first excited transverse mode and
$s=\mp 1$ corresponds to the fast and the slow modes respectively. Introducing
the Pauli matrix
$\sigma_{z}=\left|\uparrow\right\rangle\left\langle\uparrow\right|-\left|\downarrow\right\rangle\left\langle\downarrow\right|$
and states $\left|m_{1}\right\rangle$ and $\left|m_{2}\right\rangle$, the
effective Hamiltonian in the ballistic case can then be written as

$H_{0}=\frac{\Delta_{S}}{1+\lambda_{1}}\frac{\pi+\phi\sigma_{z}}{2}|m_{1}\rangle\langle
m_{1}|+\frac{\Delta_{S}}{1+\lambda_{2}}\frac{\pi-\phi\sigma_{z}}{2}|m_{2}\rangle\langle
m_{2}|\;.$ (17)

Beyond ballistic case: The main effect of elastic scattering is to introduce
coupling between ballistic states and thus can be modelled as an extra term in
the Hamiltonian coupling the modes without mixing the pseudospin, which yield
the model Hamiltonian given in Eq. (1) of the main text

$H_{eff}=H_{0}+\Delta_{S}r\left(\left|m_{1}\right\rangle\left\langle
m_{2}\right|+H.c.\right).$ (18)

Notice that this model is valid for $\phi\ll\pi$ (around $\pi$ the coupling
with other modes should be taken into account). This Hamiltonian may be
analytically diagonalized since

$\left\langle\uparrow\left|H_{eff}\right|\uparrow\right\rangle=\Delta_{S}\begin{pmatrix}\frac{1}{1+\lambda_{1}}\frac{\pi+\phi}{2}&r\\\
r&\frac{1}{1+\lambda_{2}}\frac{\pi-\phi}{2}\end{pmatrix}$ (19)

and

$\left\langle\downarrow\left|H_{eff}\right|\downarrow\right\rangle=\Delta_{S}\begin{pmatrix}\frac{1}{1+\lambda_{1}}\frac{\pi-\phi}{2}&r\\\
r&\frac{1}{1+\lambda_{2}}\frac{\pi+\phi}{2}\end{pmatrix}.$ (20)

This produces eigenstates

$\displaystyle\left|1\uparrow\right\rangle$ $\displaystyle=$
$\displaystyle\left(\cos\theta_{\uparrow}\left|m_{1}\right\rangle+\sin\theta_{\uparrow}\left|m_{2}\right\rangle\right)\left|\uparrow\right\rangle$
$\displaystyle\left|2\uparrow\right\rangle$ $\displaystyle=$
$\displaystyle\left(-\sin\theta_{\uparrow}\left|m_{1}\right\rangle+\cos\theta_{\uparrow}\left|m_{2}\right\rangle\right)\left|\uparrow\right\rangle$
$\displaystyle\left|1\downarrow\right\rangle$ $\displaystyle=$
$\displaystyle\left(\cos\theta_{\downarrow}\left|m_{1}\right\rangle+\sin\theta_{\downarrow}\left|m_{2}\right\rangle\right)\left|\downarrow\right\rangle$
$\displaystyle\left|2\downarrow\right\rangle$ $\displaystyle=$
$\displaystyle\left(-\sin\theta_{\downarrow}\left|m_{1}\right\rangle+\cos\theta_{\downarrow}\left|m_{2}\right\rangle\right)\left|\downarrow\right\rangle\;,$

with

$\displaystyle\tan 2\theta_{\uparrow}$ $\displaystyle=$ $\displaystyle
4r\frac{\left(1+\lambda_{1}\right)\left(1+\lambda_{2}\right)}{\lambda_{2}\left(\pi+\phi\right)-\lambda_{1}\left(\pi-\phi\right)+2\phi}$
$\displaystyle\tan 2\theta_{\downarrow}$ $\displaystyle=$ $\displaystyle
4r\frac{\left(1+\lambda_{1}\right)\left(1+\lambda_{2}\right)}{\lambda_{2}\left(\pi-\phi\right)-\lambda_{1}\left(\pi+\phi\right)-2\phi}$

and eigenvalues

$\displaystyle\epsilon_{1\uparrow}(\phi)$ $\displaystyle=$
$\displaystyle\frac{\Delta_{S}}{4}\left[\frac{\pi+\phi}{1+\lambda_{1}}+\frac{\pi-\phi}{1+\lambda_{2}}-\sqrt{\left(\frac{\pi+\phi}{1+\lambda_{1}}-\frac{\pi-\phi}{1+\lambda_{2}}\right)^{2}+16r^{2}}\right]$
$\displaystyle\epsilon_{2\uparrow}(\phi)$ $\displaystyle=$
$\displaystyle\frac{\Delta_{S}}{4}\left[\frac{\pi+\phi}{1+\lambda_{1}}+\frac{\pi-\phi}{1+\lambda_{2}}+\sqrt{\left(\frac{\pi+\phi}{1+\lambda_{1}}-\frac{\pi-\phi}{1+\lambda_{2}}\right)^{2}+16r^{2}}\right]$
$\displaystyle\epsilon_{1\downarrow}(\phi)$ $\displaystyle=$
$\displaystyle\frac{\Delta_{S}}{4}\left[\frac{\pi-\phi}{1+\lambda_{1}}+\frac{\pi+\phi}{1+\lambda_{2}}-\sqrt{\left(\frac{\pi-\phi}{1+\lambda_{1}}-\frac{\pi+\phi}{1+\lambda_{2}}\right)^{2}+16r^{2}}\right]$
$\displaystyle\epsilon_{2\downarrow}(\phi)$ $\displaystyle=$
$\displaystyle\frac{\Delta_{S}}{4}\left[\frac{\pi-\phi}{1+\lambda_{1}}+\frac{\pi+\phi}{1+\lambda_{2}}+\sqrt{\left(\frac{\pi-\phi}{1+\lambda_{1}}-\frac{\pi+\phi}{1+\lambda_{2}}\right)^{2}+16r^{2}}\right]\;,$

so that

$H_{eff}=\sum_{\begin{subarray}{c}k=1,2\\\
s=\uparrow,\downarrow\end{subarray}}\epsilon_{ks}\left|ks\right\rangle\left\langle
ks\right|.$ (21)

This four level system can be understood as a pair of two level systems with
bases $\left\\{\left|1\right\rangle,\left|2\right\rangle\right\\}$ and
$\left\\{\left|\uparrow\right\rangle,\left|\downarrow\right\rangle\right\\}$
and Pauli matrices $\tau_{z}=\left|2\right\rangle\left\langle
2\right|-\left|1\right\rangle\left\langle 1\right|$ and
$\sigma_{z}=\left|\uparrow\right\rangle\left\langle\uparrow\right|-\left|\downarrow\right\rangle\left\langle\downarrow\right|$
such that

$H_{eff}=\frac{\omega}{2}\tau_{z}-\frac{g}{2}\tau_{z}\sigma_{z}+\frac{\nu}{2}\sigma_{z}+E_{0},$
(22)

with

$\displaystyle E_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\sum_{\begin{subarray}{c}k=1,2\\\
s=\uparrow,\downarrow\end{subarray}}\epsilon_{ks}=\frac{\pi\Delta_{S}}{4}\left(\frac{1}{1+\lambda_{1}}+\frac{1}{1+\lambda_{2}}\right)\;,$
$\displaystyle\nu=\epsilon_{2,\uparrow}+\epsilon_{1,\uparrow}-2E_{0}$
$\displaystyle=$
$\displaystyle\frac{\phi\Delta_{S}}{2}\frac{\lambda_{2}-\lambda_{1}}{\left(1+\lambda_{1}\right)\left(1+\lambda_{2}\right)},$
$\displaystyle\omega=\epsilon_{2\uparrow}+\epsilon_{2,\downarrow}-2E_{0}$
$\displaystyle=$
$\displaystyle\frac{\Delta_{S}}{4}\sqrt{\left(\frac{\pi-\phi}{1+\lambda_{1}}-\frac{\pi+\phi}{1+\lambda_{2}}\right)^{2}+16r^{2}}+\frac{\Delta_{S}}{4}\sqrt{\left(\frac{\pi+\phi}{1+\lambda_{1}}-\frac{\pi-\phi}{1+\lambda_{2}}\right)^{2}+16r^{2}},$
$\displaystyle g=\epsilon_{2,\downarrow}+\epsilon_{1,\uparrow}-2E_{0}$
$\displaystyle=$
$\displaystyle\frac{\Delta_{S}}{4}\sqrt{\left(\frac{\pi-\phi}{1+\lambda_{1}}-\frac{\pi+\phi}{1+\lambda_{2}}\right)^{2}+16r^{2}}-\frac{\Delta_{S}}{4}\sqrt{\left(\frac{\pi+\phi}{1+\lambda_{1}}-\frac{\pi-\phi}{1+\lambda_{2}}\right)^{2}+16r^{2}}.$

Figure 5: Probability of detected states as a function of time for two initial
preparations (a) and (b) as in Fig.3 (d) and (g) of Hays2019 . Each data
series is fit to an independent exponential decay, each providing disparate
$\tau$ values. The probability of detecting a state (black dots) increases
with time as the dynamics of the system (other colors) diminish.

##  III Spin-Mixing Rate

The values of the parameters in the main text are taken from real experimental
devices such as sTosi2019  and sHays2019 . Nevertheless, the value for the
spin-mixing rate $\gamma$ when there exists simultaneous driving is ambiguous
and leaves some space for discussion.

The method of datapoint assignation described in Hays2019  leaves out a set
such that we can distinguish between detected and undetected states. We
enlarge the probability space accordingly: beyond the four relevant physical
states $s\in\\{1\downarrow,1\uparrow,2\downarrow,2\uparrow\\}$, the
probability distribution $p(s,d)$ is defined over an additional
detected/undetected space $d\in\\{1,0\\}$. The data in Fig.3 of Hays2019
corresponds to the distribution $p(s,1)$, which appears unnormalized since
there are a number of undetected cases and
$\sum_{i}p(i,1)=1-\sum_{i}p(i,0)<1$. We define $p_{d}(1)=\sum_{i}p(i,1)$ and
plot it along the data of Fig.3(d) and Fig.3(g) of sHays2019  in the fits of
Fig.5 (a) and (b), which evidence dynamics in the $d$ subspace.

We want to extract the rate equation that best reproduces the evolution of the
probability distribution $p_{s}(s)=\sum_{i}p(s,i)$

$\frac{d}{dt}\vec{p}_{s}(t)=R\vec{p}_{s}(t)$ (23)

where $R$ is the associated rate matrix and
$\vec{p}_{s}(t)=\left[p_{s}(1\downarrow),p_{s}(1\uparrow),p_{s}(2\downarrow),p_{s}(2\uparrow)\right]^{T}$.
We want to include spin-conserving decay $\Gamma_{\downarrow}$ and
$\Gamma_{\uparrow}$ from 2 into 1 and spin mixing $\gamma$ in both channels 1
and 2, so that

$R=\begin{pmatrix}-\gamma&\gamma&\Gamma_{\downarrow}&0\\\
\gamma&-\gamma&0&\Gamma_{\uparrow}\\\
0&0&-\Gamma_{\downarrow}-\gamma&\gamma\\\
0&0&\gamma&-\Gamma_{\uparrow}-\gamma\end{pmatrix}.$ (24)

The fit with a rate equation is only reliable with a normalized distribution.
As evidenced by Fig.5, this is not the case for this data and a strategy for
normalization is required.

We considered different procedures to obtain a normalized $\vec{p}_{s}(t)$.
Our first intention was to assume independent distributions $p(s,d)\simeq
p_{s}(s)p_{d}(d)$, such that $p_{s}(s)\simeq p(s,1)/p_{d}(1)$. Nevertheless,
it seems dynamics in the $s$ subspace increase the likelihood of $d=0$, so
that distributions are somewhat correlated. Since $p(1\downarrow)$ is largely
static in Fig.5(a), we only renormalize $p(1\uparrow)$ and $p(2\uparrow)$. We
use full normalization for Fig.5(b).

We obtain values for $\gamma$, $\Gamma_{\downarrow}$ and $\Gamma_{\uparrow}$,
reproduced in Fig.6. Whereas it seems confirmed that the mode 2 lifetimes are
in the range of $2-3\mu$s, the spin lifetime produces a much larger value than
the ones obtained in sHays2019 . We could refine our rate matrix to allow for
non-full spin mixing, or discuss other normalization procedures. Nevertheless,
all attempts at refitting the data have provided lifetimes well above
$40\mu$s. It seems confirmed that this data indicates a longer spin lifetime
than what is measured in Fig.3 of sHays2019 . In addition, the similarity of
the values $\Gamma_{\downarrow}$ and $\Gamma_{\uparrow}$ justifies the
consideration of a spin independent mode 2 decay rate $\Gamma$ for simplicity.

Figure 6: Fits of the datapoints with the rate equation method described in
the text.

##  IV Rate Equation for Polarization

We can take advantage of the faster dissipation rate of the mode levels in
order to polarize the pseudospin degree of freedom by means of the application
of a constant gate AC driving. Concepts of laser cooling can be useful for the
optimization of this initial pulse. In order to do so, we recognize that our
polarization mechanism is analogous to a sideband cooling technique in the
context of laser cooling of trapped ions and follow derivations similar to
those presented in sCirac1992 . An interaction picture with respect to
$\omega_{g}\tau_{z}/2$ and a RWA delivers the effective Hamiltonian

$H^{\prime}=\frac{\Delta}{2}\tau_{z}+\frac{\Omega_{g}}{2}\tau_{x}+\bm{\tau}\cdot\bm{\sigma}+\frac{\nu}{2}\sigma_{z}$
(25)

with the vectors $\bm{\tau}=\left(\tau_{x},\tau_{y},\tau_{z}\right)$ and
$\bm{\sigma}=\left(\frac{\alpha\Omega_{g}}{2}\sigma_{x},0,-\frac{g}{2}\sigma_{z}\right)$
and the detuning $\Delta=\omega-\omega_{g}$. The resonance choice of the main
text corresponds to $\Delta=-\nu$, so that this is a protocol analogous to a
sideband cooling scheme.

Now we derive a rate equation for the pseudospin populations. For this, we
work in the interaction picture with respect to

$H_{0}=\frac{\Delta}{2}\tau_{z}+\frac{\Omega_{g}}{2}\tau_{x}-\frac{g}{2}\tau_{z}\sigma_{z}+\frac{\nu}{2}\sigma_{z},$
(26)

and analyze the perturbation generated by
$H_{1}=\frac{\alpha\Omega_{g}}{2}\tau_{x}\sigma_{x}$. We assume the adiabatic
state of the system is

$\rho(t)=\sum_{s=\downarrow,\uparrow}p_{s}(t)\left|s\right\rangle\left\langle
s\right|\rho_{s},$ (27)

where $\rho_{s}$ is the steady state of the mode degrees of freedom given
pseudospin polarization $s$. Pseudospin flip processes are only possible as
second order effects encoded in the Born-Markov approximation

$\frac{d}{dt}\rho(t)=-\left[H_{1}(t),\int_{0}^{t}\left[H_{1}(t^{\prime}),\rho(t)\right]dt^{\prime}\right],$
(28)

so that the rates in equation

$\frac{d}{dt}\begin{pmatrix}p_{\downarrow}\\\
p_{\uparrow}\end{pmatrix}=\begin{pmatrix}-\gamma_{\uparrow}&\gamma_{\downarrow}\\\
\gamma_{\uparrow}&-\gamma_{\downarrow}\end{pmatrix}\begin{pmatrix}p_{\downarrow}\\\
p_{\uparrow}\end{pmatrix},$ (29)

correspond to

$\displaystyle\gamma_{s}$ $\displaystyle=$ $\displaystyle
2Re\left[\int_{0}^{\infty}e^{\pm i\nu t}C_{s}(t)dt\right],$ (30)
$\displaystyle C_{s}(t)$ $\displaystyle=$ $\displaystyle
Tr\left\\{\tau_{x}\left\langle
s\right|U\left|s\right\rangle\tau_{x}\left\langle\bar{s}\right|U^{\dagger}\left|\bar{s}\right\rangle\rho_{\bar{s}}\right\\},$
(31)

where $U=e^{-iH_{0}t}$. Note the lack of unitarity of the evolution and
$\left\langle\bar{s}\right|U\left|s\right\rangle=0$. We approximate Eq.(31) by
means of the regression theorem and an associated modified master equation for
the mode degrees of freedom alone represented by the Bloch equations

$\frac{d}{dt}\langle\bm{\tau}\rangle^{T}=\begin{pmatrix}-\frac{\Gamma}{2}-\gamma_{d}+b&\frac{\Delta}{2}+a&0\\\
-\frac{\Delta}{2}-a&-\frac{\Gamma}{2}-\gamma_{d}+b&-\frac{\Omega_{g}}{2}\\\
0&\frac{\Omega_{g}}{2}&-\Gamma\end{pmatrix}\langle\bm{\tau}\rangle^{T}-\begin{pmatrix}0\\\
0\\\ \Gamma\end{pmatrix}$ (32)

The steady state $\rho_{\downarrow,\uparrow}$ is computed with $b=0$ and
$a=\pm\frac{g}{2}$ respectively, whereas the regression theorem for each
correlation function $C_{\downarrow,\uparrow}$ assumes Bloch equations with
$a=0$ and $b=\mp i\frac{g}{2}$ respectively, which account for the lack of
unitarity of the evolution. With this, it is possible to compute the real part
of the half-sided Fourier transform of the correlation function Eq.(31).

If we add the spin mixing processes, the dynamics of the populations are
finally governed by the rate equation

$\frac{d}{dt}\begin{pmatrix}p_{\downarrow}\\\
p_{\uparrow}\end{pmatrix}=\begin{pmatrix}-\gamma_{\uparrow}-\gamma&\gamma_{\downarrow}+\gamma\\\
\gamma_{\uparrow}+\gamma&-\gamma_{\downarrow}-\gamma\end{pmatrix}\begin{pmatrix}p_{\downarrow}\\\
p_{\uparrow}\end{pmatrix}.$ (33)

## References

  * (1) L. Tosi, C. Metzger, M.F. Goffman, C. Urbina, H. Pothier, S. Park, A. Levy Yeyati, J. Nygård, and P. Krogstrup, “Spin-Orbit Splitting of Andreev States Revealed by Microwave Spectroscopy”, Phys. Rev. X 9, 011010 (2019).
  * (2) S. Park and A. Levy Yeyati, “Andreev spin qubits in multichannel Rashba nanowires”, Phys. Rev. B 96, 125416 (2017).
  * (3) M. Hays, V. Fatemi, K. Serniak, D. Bouman, S. Diamond, G. de Lange, P. Krogstrup, J. Nygård, A. Geresdi, and M. H. Devoret, “Continuous monitoring of a trapped, superconducting spin”, arXiv:1908:02800.
  * (4) J. I. Cirac, R. Blatt, P. Zoller, and W. D. Phillips, “Laser cooling of trapped ions in a standing wave”, Phys. Rev. A 46, 2668 (1992).


[1,2]Anastassia M.Makarieva [1]Andrei V.Nefiodov [3]Antonio DonatoNobre
[4]UgoBardi [5,6,7]DouglasSheil [8]MaraBaudena [9]Scott R.Saleska
[10]AnjaRammig

1]Theoretical Physics Division, Petersburg Nuclear Physics Institute, 188300
Gatchina, St. Petersburg, Russia 2]Institute for Advanced Study, Technical
University of Munich, Lichtenbergstrasse 2 a, 85748 Garching, Germany 3]Centro
de Ciência do Sistema Terrestre INPE, São José dos Campos, 12227-010 São
Paulo, Brazil 4]Department of Chemistry, University of Florence, Italy
5]Forest Ecology and Forest Management Group, Wageningen University &
Research, PO Box 47, 6700 AA, Wageningen, The Netherlands 6]Center for
International Forestry Research (CIFOR), Kota Bogor, Jawa Barat, 16115,
Indonesia 7]Faculty of Environmental Sciences and Natural Resource Management,
Norwegian University of Life Sciences, Ås, Norway 8]National Research Council
of Italy, Institute of Atmospheric Sciences and Climate (CNR-ISAC), Turin,
Italy 9]Department of Ecology and Evolutionary Biology, University of Arizona,
Tucson, 85721, Arizona, USA 10]Technical University of Munich, School of Life
Sciences, Hans-Carl-von-Carlowitz-Platz 2, 85354 Freising, Germany

A. D. Nobre (anobre27@gmail.com), A. M. Makarieva<EMAIL_ADDRESS>
# How transpiration by forests and other vegetation determines alternate
moisture regimes

###### Abstract

The terrestrial water cycle links the soil and atmosphere moisture reservoirs
through four fluxes: precipitation, evaporation, runoff and atmospheric
moisture convergence. Each of these fluxes is essential for human and
ecosystem well-being. However, predicting how the water cycle responds to
changes in vegetation cover, remains a challenge (Lawrence and Vandecar, 2015;
Ellison et al., 2017; te Wierik et al., 2021). Recently, rainfall was shown to
decrease disproportionally with reduced forest transpiration following
deforestation (Baudena et al., 2021). Here, combining these findings with the
law of matter conservation, we show that in a sufficiently wet atmosphere
forest transpiration can control atmospheric moisture convergence such that
increased transpiration enhances atmospheric moisture import. Conversely, in a
drier atmosphere increased transpiration reduces atmospheric moisture
convergence and runoff. This previously unrecognized dichotomy can explain the
seemingly random observations of runoff and soil moisture sometimes increasing
and sometimes reducing in response to re-greening (e.g., Zheng et al., 2021).
Evaluating the transition between the two regimes is crucial both for
characterizing the risk posed by deforestation as well as for motivating and
guiding global ecosystem restoration.

[Precipitation, column moisture and mass conservation]

Forests play an important role for atmospheric moisture generation and
transport through transpiration and atmospheric moisture convergence.
Deforestation and reforestation can therefore strongly alter the hydrological
cycle depending on the prevailing climate regime in the region. Based on data
for a tropical island, Holloway and Neelin (2010) showed that the rainfall
probability rises sharply with increasing column water vapor $W$ (see also
Yano and Manzato, 2022). Using the radiosonde data for several meteostations
in Brazil (Fig. 1a), Makarieva et al. (2014) concluded that, in the Amazon
forest, a small relative increment of $W$ should lead to a larger relative
increment in rainfall probability. Baudena et al. (2021) found that in the
relatively flat part of the Amazon including the Amazon-Cerrado transition
zone (0$-$18oS and 65$-$50oW), the mean hourly precipitation $P$ (ERA5 data
for 2002-2013, binned for each 1 mm of $W$) depends non-linearily on $W$ (Fig.
1a)

$\frac{dP}{P}=k_{P}(W)\frac{dW}{W},$ (1)

where $k_{P}\gg 1$ for $W>30$ mm, while $k_{P}\simeq 0$ for
$10\leavevmode\nobreak\ {\rm mm}<W<20$ mm (Fig. 1b). (We excluded observations
at $W\leq 8$ mm and $W\geq 67$ mm from our consideration (all bins with less
than 20,000 data points, or about $0.1\%$ of all observations, see inset in
Fig. 1b), since $k_{P}$ displayed an erratic behavior apparently due to the
relatively small number of rainfall events at these extreme $W$ values.)

Interestingly, relationship (1) established for local hourly rainfall for a
specific study area in the Amazon region (Baudena et al., 2021) encompasses
characteristic precipitation rates over a broad range of spatial and temporal
scales, from annual precipitation in deserts, temperate and tropical forests,
to hurricane and flood-causing rainfall (Fig. 1a, data from Makarieva et al.,
2013, 2017; Smirnova et al., 2017; Beudert et al., 2018; Almazroui, 2020;
Kreienkamp et al., 2021).

Figure 1: Atmospheric moisture budget terms $P$, $E$ and $C$ as related to
column water vapor content $W$. Hourly precipitation data (from Baudena et
al., 2021, re-averaged arithmetically) give the blue curve in (a), which we
then integrate according to Eq. (2) to obtain $E$ (black curves in (a)) and
$k_{P}(W)$ (the black curve in (b)). The inset in (b) shows the number of data
points $N(W)$ for each 1 mm bin (Baudena et al., 2021). Constrained by the
mass balance Eq. (3) illustrated in (c), we derive moisture convergence $C$
(d) from $P(W)$ and $E(W)$. Circles in (a) represent daily averaged $P$ and
$W$ for several meteostations in Brazil (Makarieva et al., 2014, Table 2,
columns 4 and 11); arrows indicate characteristic precipitation rates in
different locations in the world (see text). In (c), the steady-state water
balance is shown for a forest location that receives moisture solely from the
ocean (thin blue arrows indicate moisture inflow $C_{+}$ and outflow $C_{-}$,
cf. Eltahir and Bras, 1994, their Fig. 12). A certain part of the recycled
transpired moisture (red dashed lines) flows to the forest further inland, the
rest precipitates locally. Thick yellowish arrows indicate the ascending and
descending air motions that generate precipitation and are responsible for
local moisture convergence $C>0$ (and steady-state runoff $R=C$). In (a) and
(d), red squares indicate the point where the moisture convergence (and
runoff) are minimal, but begin to grow with increasing $E$ at larger $W$.

Evapotranspiration $E$ adds moisture to the atmosphere, so we expect that $W$
grows with $E$. We can write such a dependence as follows:

$\frac{dE}{P}=k_{E}(W)\frac{dW}{W},$ (2)

where $k_{E}>0$. In the Amazon basin, with mean transpiration $T=45$ mm
month-1 (Staal et al., 2018) and mean annual rainfall of $P=2200$ mm year-1
(Marengo, 2006), a loss of evapotranspiration due to the loss of transpiration
from the entire basin is equal to $\Delta E/P=-T/P=-0.25$, i.e. a decrease of
25% (in proportion to precipitation). According to Baudena et al. (2021),
zeroing transpiration in a region reduces $W$ in a given location by the
fraction of water vapor originating from transpiration in the considered
region. In the Amazon basin, the fraction of water vapor originating from the
Amazon transpiration is equal to $0.32$ (Staal et al., 2018). With $\Delta
W/W=-0.32$ and $\Delta E/P=-T/P=-0.25$, we have $k_{E}=0.8$.

For a steady-state atmospheric column, the vertically integrated continuity
equation representing the water vapor convergence over the whole atmospheric
column as equal to the difference of precipitation and evaporation fluxes (see
scheme in Fig. 1c), reads:

$C{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\equiv-\int\limits_{0}^{\infty}\mathrm{div}(\rho_{v}\mathbf{u})dz}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-\int\limits_{0}^{\infty}\dot{\rho}_{v}dz\equiv}P-E,$
(3)

where $\mathbf{u}$ is the air velocity vector, $\rho_{v}$ is water vapor
denstity, $\dot{\rho}_{v}$ is the mass source/sink and $C$ is water vapor
convergence (net flux of water vapor through the column surface). The mass
balance (3) can be applied both locally and on a regional scale. It is valid
when the rate of change in moisture content is small compared to precipitation
($dW/dt\ll P$ ), i.e., on a time scale $\tau\lesssim\tau_{a}$, where
$\tau_{a}\equiv W/P$ (a few days) is the time scale of atmospheric moisture
turnover via precipitation. In a system of units where liquid water density
$\rho_{l}=10^{3}$ kg m-3 is set to unity, column water vapor
$W=\int\limits_{0}^{\infty}\rho_{v}dz$ has the units of length (mm), while
$C$, $P$ and $E$ are measured in units of length per unit of time (for
example, mm day-1).

##  1 Ecosystem’s two moisture regimes

Using the dependence $P(W)$, Eq. (1) we can integrate Eq. (2) assuming
$E(W_{\rm min})=0$ for minimal water vapor content $W_{\rm min}=9$ mm. Then we
can find $C(W)$ combining Eqs. (1) and (2)

$dC=dP-dE=(k_{P}-k_{E})dE.$ (4)

This reveals two regimes, for high and low $W$. In the drier regime with
$k_{P}<k_{E}$, moisture convergence declines with increasing
evapotranspiration and moisture content, while precipitation remains
relatively constant. In the wetter regime with $k_{P}>k_{E}$, moisture
convergence increases together with evapotranspiration, moisture content and
precipitation (Fig. 1d). (If the soil moisture content is steady, runoff $R$
is equal to moisture convergence $C$ and thus behaves similarly.)

The quantitative details of the $P(W)$, $E(W)$ and $C(W)$ dependencies will be
specific to each region, season and ecosystem type and depend on the spatial
and temporal scale of observations (Fig. 1a, see also Peters and Neelin,
2006). However, the established pattern is qualitatively robust with respect
to different values of $E(W_{\rm min})$ (not shown) and $k_{E}$ (Fig. 1d). At
higher $k_{E}$ (i.e., a slower accumulation of atmospheric moisture with
growing $E$), there appears an interval of $W$ with negative moisture
convergence. This corresponds to dry conditions when the ecosystem transpires
at the expense of previously accumulated soil moisture or at the expense of
irrigation.

Equations (1)–(3) make it clear that both regimes exist irrespective of
specific $k_{E}$ and $k_{P}$ values, provided that $k_{P}<k_{E}$ at lower $W$
and $k_{P}>k_{E}$ at higher $W$. The robustness of the pattern has a profound
physical meaning related to the saturated state of the atmosphere. Since
precipitation requires saturation, at low $W$ far from the saturation, $k_{P}$
is low, i.e., an increase in local $W$ does not markedly increase the
probability precipitation that may then be determined by non-local weather
systems (Makarieva et al., 2014). Evaporation, on the contrary, is significant
at high water vapor deficits and enriches the atmosphere with water vapor, so
if $E$ grows, so will $W$. This dry regime can be characterized as “abiotic”,
because the ecosystem exploits the geophysical moisture flows and, at $C<0$,
the previously accumulated water stores (Fig. 1d).

As the air column approaches saturation at high $W$, precipitation begins to
increase markedly with $W$. Evaporation, on the other hand, depends on the
moisture deficit near the surface atmosphere, which is largely decoupled from
total water vapor content $W$ (Holloway and Neelin, 2009, their Fig. 3e,f).
(Indeed, evapotranspiration and moisture convergence have distinct physics.
Moisture convergence occurs when the air rises and water vapor condenses. In
contrast, evapotranspiration is not explicitly linked to directional air
motions; it adds water vapor directly to the atmospheric column facilitated by
turbulent diffusion.) Under nearly saturated conditions, evaporation can only
proceed if precipitation depletes moisture from the atmosphere creating a
water vapor deficit (Murakami, 2006, 2021; Jiménez-Rodríguez et al., 2021).
Therefore, at high $W$ precipitation $P$ and evapotranspiration $E$ should
grow approximately proportionally to each other. The recycling ratio $E/P$ and
the runoff-to-precipitation ratio $R/P=C/P=1-E/P$ stabilize at high values of
$W$ and then remain approximately constant (Fig. 1d). In this wet regime all
the components of the water cycle should be under biotic control.

The wetter the atmosphere, the stronger the water cycle control and the more
resilient the forest: by slightly changing evapotranspiration, it can
compensate for unfavorable disturbances of the water cycle (Makarieva et al.,
2014). Conversely, in a drier atmosphere with less rainfall, the forest is
more vulnerable to external perturbations. This elucidates why the Amazon
forest appears to be losing resilience mostly in the regions where the
rainfall is relatively low (rather than where rainfall is decreasing) (Boulton
et al., 2022).

##  2 Conclusions

Our findings, that forest transpiration can control and switch atmospheric
moisture convergence, corroborate the biotic pump concept (Makarieva and
Gorshkov, 2007; Makarieva et al., 2014). Independent studies have shown that
the Amazon forest transpiration during the late dry season moistens the
atmosphere and triggers the wet season and associated ocean-to-land moisture
inflow (Wright et al., 2017). This dry-season transpiration has a phenological
and, hence, evolutionary component that encodes the Amazon dry-season greening
(Saleska et al., 2016). Enhanced forest transpiration preceding the wet season
was also observed in Northeast India and Congo basin (Pradhan et al., 2019;
Worden et al., 2021). Regulation of moisture convergence could explain how
forests buffer precipitation extremes across continents (O’Connor et al.,
2021; Silva de Oliveira et al., 2021). While over a broad range of
$30\leavevmode\nobreak\ {\rm mm}\lesssim W\lesssim 60$ mm the value of $k_{P}$
is relatively constant (pink area in Fig. 1b), it increases sharply at larger
$W$. The interval of $W>60$ mm harbors very high precipitation rates observed
under extreme weather conditions (Fig. 1a). It remains to be investigated
whether/how forest disturbances influence the probability of such extremes. If
natural forest ecosystems have evolved to stabilize and sustain the
continental water cycle, their destruction contributes to destabilization and
impoverishment of regional water cycles. This contribution is underestimated
(Sheil et al., 2019). Future studies of vegetation cover impacts on
atmospheric moisture flows must emphasize the role of natural forests (Zemp et
al., 2017; Makarieva et al., 2020; Leite-Filho et al., 2021; Hua et al.,
2022).

The existence of the two regimes with a possibly non-steady transition between
them have important implications for large-scale afforestation efforts e.g.,
in China where increasing atmospheric moisture content with progressive re-
greening can be in drier areas associated with a decline of runoff and soil
moisture content (e.g., Jiang and Liang, 2013; Zhang et al., 2021). This
decline has been interpreted as limiting further ecosystem restoration (Feng
et al., 2016). However, Fig. 1d suggests that if re-greening is continued, the
ecosystem can pass a tipping point and enter a wetter state when further re-
greening will enhance both rainfall, moisture convergence and runoff. Indeed,
in the wetter areas in China re-greening did cause an increase in runoff
(e.g., Wang et al., 2018). On the Loess Plateau, soil moisture content
decreased in the drier, but increased in the wetter, parts of the region
(Zheng et al., 2021, Fig. 10a). Establishing key parameters of the two regimes
and assessing the potential transition from the drier to the wetter state
(during which the recovering ecosystem might require extra water inputs) can
inform and guide afforestation and reforestation strategies, including
possible recovery of ecosystem productivity in the driest regions.

## Acknowledgments

We thank Dr. Arie Staal for his useful comments on an early draft of this
work. Work of A.M. Makarieva is funded by the Federal Ministry of Education
and Research (BMBF) and the Free State of Bavaria under the Excellence
Strategy of the Federal Government and the Länder, as well as by the Technical
University of Munich – Institute for Advanced Study.

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