# Massive particle interferometry with lattice solitons: robustness against
ionization

Piero Naldesi  Université Grenoble-Alpes, LPMMC, F-38000 Grenoble, France and
CNRS, LPMMC, F-38000 Grenoble, France  Institute for Quantum Optics and
Quantum Information of the Austrian Academy of Sciences, Innsbruck, Austria
Peter D Drummond  Centre for Quantum Science and Technology Theory, Swinburne
University of Technology, Melbourne 3122, Australia  Vanja Dunjko  Department
of Physics, University of Massachusetts Boston, Boston Massachusetts 02125,
USA  Anna Minguzzi  Université Grenoble-Alpes, LPMMC, F-38000 Grenoble, France
and CNRS, LPMMC, F-38000 Grenoble, France  Maxim Olshanii
<EMAIL_ADDRESS>Department of Physics, University of Massachusetts
Boston, Boston Massachusetts 02125, USA

###### Abstract

We revisit the proposal of Castin and Weiss [Phys. Rev. Lett. vol. 102, 010403
(2009)] for using the scattering of a quantum matter-wave soliton on a barrier
in order to create a coherent superposition state of the soliton being
entirely to the left of the barrier and being entirely to the right of the
barrier. In that proposal, is was assumed that the scattering is perfectly
elastic, i.e. that the center-of-mass kinetic energy of the soliton is lower
than the chemical potential of the soliton. Here we relax this assumption.
Also, we introduce an interferometric scheme, which uses interference of
soltions, that can be used to detect the degree of coherence between the
reflected and transmitted part of the soliton. Using exact diagonalization, we
numerically simulate a complete interferometric cycle for a soliton consisting
of six atoms. We find that the interferometric fringes persist even when the
center-of-mass kinetic energy of the soliton is above the energy needed for
complete dissociation of the soliton into constituent atoms.

###### Contents

  1. I Introduction
  2. II Description of the model system
    1. II.1 The lattice Hamiltonian used in the numerical calculations
    2. II.2 The auxiliary continuum Hamiltonian
    3. II.3 Discrete-to-continuum map
  3. III The regime of interest: no lattice renormalization and the continuum model applies
    1. III.1 Preparation
    2. III.2 Beam-splitting
    3. III.3 Mirrors and recombination
    4. III.4 Inelastic effects in beam-splitting
    5. III.5 Calculations for $N\gg 1$
  4. IV A complete interferometric cycle with a uniform field as a phase object
    1. IV.1 The prediction for the fringes, assuming an elastic scattering of the CoM off the barrier.
    2. IV.2 A derivation of Eq. (14) for the signal absent inelastic effects
      1. IV.2.1 The signal for a given phase difference $\phi_{\text{right}}-\phi_{\text{left}}$
      2. IV.2.2 The vanishing of the unperturbed phase shift, $\phi_{\text{\rm right}}^{(0)}-\phi_{\text{\rm left}}^{(0)}$
      3. IV.2.3 An explicit calculation of the fringe shift due to a uniform field
  5. V Numerical simulations
    1. V.1 Results
  6. VI Conclusions
  7. VII Acknowledgments

##  I Introduction

There is considerable interest in developing matter-wave interferometry with
bright solitons, because they have a potental to improve sensitivity by as
much as a factor of 100 [1], as well as enable high-precision force sensing
[1] and measurement of small magnetic-field gradients [2]. A related possible
application is the creation of Schrödinger-cat states [3, 4, 5]. Atomtronic
devices with featuring soliton interferometry were considered in Refs. [6, 7,
8]. How to realize very narrow barriers to be used as beam splitters in
soliton interfermoetry was explored in Ref. [9]. What effect harmonic
confinement can have on the internal degrees of freedom of a quantum soliton
was investigated in [10].

The major inspiration for the present paper is the work of Castin and Weiss
[5, 11]. In that proposal, a bright soliton is scattered off a broad Gaussian
barrier in such a way that after the scattering, the atomic gas is in a
coherent superposition of being entirely to the left of the barrier and being
entirely to the right of the barrier. Here is how that happens.

First of all, the parameter regime is such that if the soliton were a point
particle, there would be a threshold for incoming velocity such that, if the
incoming velocity is below the threshold, the particle is completely
reflected, and if above, it is perfectly transmitted. The center-of-mass (CoM)
velocity of the incoming soliton is tuned to be exactly at this threshold.

At the same time, the CoM soliton speed is kept low enough that the CoM
kinetic energy of the soliton is lower than the chemical potential of the
soliton. This ensures that the scattering is perfectly elastic: the soliton is
energetically forbidden from ‘ionizing’ (shedding particles).

Now we note that the soliton wavepacket is not perfectly monochromatic, and
thus it contains Fourier components both slightly below and slightly above the
threshold needed to cross the barrier. So the scattering process behaves as
filtering in Fourier space, where the components that are below the threshold
are completely reflected and those above it are completely transmitted. But,
as we saw above, the soliton cannot fragment. Therefore, the only option for
the soliton is to coherently split into a reflected and a transmitted part.

We should note that scattering of bright solitons on barriers such that the
soliton is typically either wholly transmited or wholly reflected was
experimentally realized and studied in [12]. The authors say that their
solitons are too large to form mesoscopic quantum superpositions in the
process, but note that such superpositions should be obervable for smaller
numbers of atoms.

The purpose of the present work is to investigate what happens if the
assumption of perfectly ellastic scattering is relaxed, and the CoM kinetic
energy is allowed to be above the ionization threshold. Moreover, our barrier,
rather than being very broad, will be very narrow.

To do this, we need to be able to detect the degree of coherence between the
reflected and transmitted part of the soliton (or, rather, of what remains of
the soliton after it possibly sheds some particles). For this purpose, we use
the coherent splitting described above to construct an interferometer that is
sensitive to the presence of an external constant field. The quality of the
interferometric fringes is our measure of coherence.

Our main result is that, remarkably, the interferometric fringes persist even
if the CoM kinetic energy of our six-particle soliton is high enough that it
would be energetically allowed for the soliton to completely disintegrate upon
impact with the barrier.

##  II Description of the model system

###  II.1 The lattice Hamiltonian used in the numerical calculations

We are considering the Hamiltonian

$\hat{\mathcal{H}}=-J\sum_{j=1}^{L}(\hat{a}^{\dagger}_{j}\hat{a}_{j+1}+\text{h.c.})+\frac{U}{2}\sum_{j=1}^{L}\hat{n}_{j}(\hat{n}_{j}-1)\\\
+W\hat{n}_{j_{\text{center}}}+\sum_{j=1}^{L}\kappa_{\text{mirror}}(j-j_{\text{center}})^{2}d^{2}\,\hat{n}_{j}\\\
+\sum_{j=1}^{L}(j-j_{\text{center}})\,F\,d\,\hat{n}_{j}\,.$ (1)

Here $\hat{n}_{j}\equiv\hat{a}^{\dagger}_{j}\hat{a}_{j}$, $J$ is the hopping
constant, $U$ is the onsite coupling constant, $W$ is the barrier strength,
$\kappa_{\text{mirror}}$ is the spring constant for a harmonic potential used
to form the interferometer arms, and $F$ is a constant force acting to the
left—the “phase object”.. We impose periodic boundary conditions,
$\hat{a}_{L+1}\equiv\hat{a}_{1}$. We will be working in a sector with a fixed
number of particles $\hat{N}\equiv\sum_{j=1}^{L}\hat{n}_{j}$. We assume that
number of sites $L$ is odd. The center point of the lattice is given by
$j_{\text{center}}\equiv\frac{L+1}{2}\,\,.$ The length $d$ is the distance
between neighboring lattice sites.

The initial state is the ground state of the following Hamiltonian:

$\hat{\mathcal{H}}_{0-}=-J\sum_{j=1}^{L}(\hat{a}^{\dagger}_{j}\hat{a}_{j+1}+\text{h.c.})+\frac{U}{2}\sum_{j=1}^{L}\hat{n}_{j}(\hat{n}_{j}-1)\\\
+\sum_{j=1}^{L}\kappa_{\text{preparation}}(j-(j_{\text{center}}-L_{0}))^{2}d^{2}\,\hat{n}_{j}\,.$

The Hamiltonian $\hat{\mathcal{H}}_{0-}$ is similar to $\hat{\mathcal{H}}$ in
Eq. (1). However, $\hat{\mathcal{H}}_{0-}$ includes no barrier and no force.
Furthermore, its harmonic potential has a different curvature and is offset to
the left from the center by $L_{0}$ lattice sites.

###  II.2 The auxiliary continuum Hamiltonian

We further introduce the Hamiltonian

$\hat{\mathcal{H}}=\int_{-\infty}^{\infty}\\!dx\,\left\\{\frac{\hbar^{2}}{2m}\hat{\Psi}^{\dagger}_{x}\hat{\Psi}_{x}\right.\\\
+\frac{g}{2}\hat{\Psi}^{\dagger}\hat{\Psi}^{\dagger}\hat{\Psi}\hat{\Psi}+\hat{\Psi}^{\dagger}\hat{\Psi}V_{\text{barrier}}(x)\\\
\left.+\hat{\Psi}^{\dagger}\hat{\Psi}\frac{m\omega_{\text{mirror}}^{2}x^{2}}{2}+\hat{\Psi}^{\dagger}\hat{\Psi}Fx\right\\}\,,$
(2)

where $V_{\text{barrier}}(x)=\tilde{g}\,\delta(x)$ and
$\omega_{\text{mirror}}$ is the frequency of the harmonic potential that is
used to form an interferometer. The number-of-particles operator is
$\hat{N}\equiv\int\\!dx\,\hat{\Psi}^{\dagger}_{x}\hat{\Psi}_{x}$.

The initial state is the ground state of

$\hat{\mathcal{H}}_{0-}=\int\\!dx\,\left\\{\frac{\hbar^{2}}{2m}\hat{\Psi}^{\dagger}_{x}\hat{\Psi}_{x}+\frac{g}{2}\hat{\Psi}^{\dagger}\hat{\Psi}^{\dagger}\hat{\Psi}\hat{\Psi}\right.\\\
\left.+\hat{\Psi}^{\dagger}\hat{\Psi}\frac{m\omega_{\text{preparation}}^{2}(x-(-L_{0}d))^{2}}{2}\right\\}\,.$
(3)

###  II.3 Discrete-to-continuum map

Effective one-body mass is given by

$\displaystyle m=\frac{1}{2}\frac{\hbar^{2}}{Jd^{2}}\,\,.$

Next, let us introduce a two-body scattering length $a$, which is the same in
both the continuum and the lattice cases:

$\displaystyle
a=-\frac{2dJ^{\text{rel.}}_{K=0}}{U}\left(1-\frac{1}{\pi^{2}}\frac{U}{J^{\text{rel.}}_{K=0}}\right)\,\,.$

The corresponding coupling constant is given in terms of the scattering length
in the usual way, from which we may deduce its dependence on the parameters of
the lattice model:

$\displaystyle g=-\frac{\hbar^{2}}{(m/2)a}\approx Ud\,\,.$

Here, $J^{\text{rel.}}_{K}=2J\cos(Kd/2)$ is the hopping constant of the
lattice on which the relative motion [13]. For an explanation of the lattice
renormalization factor
$\left(1-\frac{1}{\pi^{2}}\frac{U}{J^{\text{rel.}}_{K=0}}\right)$, see Ref.
[14].

Note that it is not an accident that the lattice and continuum models share
the same scattering length. In fact, the effective continuum coupling $g$ is
introduced in such a way that it reproduces the lattice scattering length
exactly.

In a similar manner, we introduce a scattering length for the particle-barrier
interaction,

$\displaystyle\tilde{a}=-\frac{2dJ}{W}\left(1-\frac{1}{\pi^{2}}\frac{W}{J}\right)\,,$

and the corresponding particle-barrier coupling constant:

$\displaystyle\tilde{g}=-\frac{\hbar^{2}}{m\tilde{a}}\approx Wd\,\,.$

The frequency of the “mirror” satisfies
$\frac{1}{2}m\omega_{\text{mirror}}^{2}=\kappa_{\text{mirror}}$, so that

$\omega_{\text{mirror}}=\frac{2d}{\hbar}\sqrt{\kappa_{\text{mirror}}J}\,.$

Analogously, the “preparation” frequency will be given by
$\omega_{\text{preparation}}=\frac{2d}{\hbar}\sqrt{\kappa_{\text{preparation}}J}$.

##  III The regime of interest: no lattice renormalization and the continuum
model applies

We assume that there is no need for lattice renormalization, either for the
interactions or for the barrier. This happens in the regime of weak coupling,

$|U|\ll 2\pi^{2}J\quad\text{and}\quad W\ll\pi^{2}J\,.$

Further, we assume that the continuum model applies, which will be the case
when the healing length

$\displaystyle\ell=2\,\frac{\hbar^{2}}{m|g|N}=\frac{a}{N}$ (4)

satisfies $\ell\gg d$. This is equivalent to

$UN\ll 4J\,.$

###  III.1 Preparation

The initial soliton will be prepared so that its center of mass (CoM) is in
the ground state of a “preparation” harmonic trap, of frequency
$\omega_{\text{preparation}}$. Consider the mean kinetic energy of the
soliton,

$E_{\text{kinetic, CoM}}=\frac{1}{2}M\bar{\mathcal{V}}^{2},$ (5)

where $M=Nm$ is the soliton mass and $\bar{\mathcal{V}}$ is the mean CoM
velocity. Also consider the uncertainty in this kinetic energy

$\displaystyle\delta E_{\text{kinetic, CoM}}\approx
M\bar{\mathcal{V}}\delta\mathcal{V}\,\,,$

where

$\displaystyle\delta\mathcal{V}=\sqrt{\frac{\hbar\omega_{\text{preparation}}}{2M}}$

is the preparation r.m.s. velocity. We will be assuming that the uncertainty
in the kinetic energy is finite, but small relative to the mean kinetic
energy. It then follows that the preparation r.m.s. velocity is small relative
to the mean CoM velocity:

$\displaystyle\delta E_{\text{kinetic, CoM}}\ll E_{\text{kinetic,
CoM}}\Rightarrow\delta\mathcal{V}\ll\bar{\mathcal{V}}\,.$

###  III.2 Beam-splitting

The condition for a 50%–50% classical filtering as quantum beam-splitting
reads [5, 11]

$\displaystyle E_{\text{kinetic, CoM}}=\max_{X}V_{\text{soliton-on-
barrier}}(X)\,,$

where $X$ is the CoM position. Note that when the number of atoms $N$ is
large, the right-hand side is given by Eq. (10), below. See Ref. [15] for a
clear derivation of the one-body density matrix for a localized CoM position.
The original result is in Ref. [16].

###  III.3 Mirrors and recombination

. We will be using another harmonic trap, of frequency
$\omega_{\text{mirror}}$, as a “mirror” on each end, to ensure the return of
the wavepackets for recombination. The trap frequency and the initial position
of the CoM wavepacket, $-L_{0}$, will conspire to produce the incident energy
we need:

$\displaystyle\frac{M\omega_{\text{mirror}}^{2}(L_{0})^{2}}{2}=\max_{X}V_{\text{soliton-
on-barrier}}(X)\,,$ (6)

where, again, if the number of atoms $N$ is large, the right-hand side is
given by Eq. (10).

###  III.4 Inelastic effects in beam-splitting

. The full internal energy of the soliton, in the continuum limit (C. L.), is
given by

$\displaystyle E^{(N)}_{\text{S, interaction}}\stackrel{{\scriptstyle\text{C.\
L.}}}{{=}}-\,\frac{(N+1)N(N-1)}{24}\frac{mg^{2}}{\hbar^{2}}\,.$ (7)

The gap to the first excitation is

$\displaystyle E^{(N-1)}_{\text{S, interaction}}-E^{(N)}_{\text{S,
interaction}}\stackrel{{\scriptstyle\text{C.\
L.}}}{{=}}\frac{N(N-1)}{8}\frac{mg^{2}}{\hbar^{2}}\,;$

the gap to the first two-atom excitation is

$\displaystyle E^{(N-2)}_{\text{S, interaction}}-E^{(N)}_{\text{S,
interaction}}\stackrel{{\scriptstyle\text{C.\
L.}}}{{=}}\frac{(N-1)^{2}}{4}\frac{mg^{2}}{\hbar^{2}}\,.;$

the gap to the first three-atom excitation is

$\displaystyle E^{(N-3)}_{\text{S, interaction}}-E^{(N)}_{\text{S,
interaction}}\stackrel{{\scriptstyle\text{C.\
L.}}}{{=}}\frac{3N^{2}-9N+8}{8}\frac{mg^{2}}{\hbar^{2}}\,;$

and the gap to a complete ‘ionization’ is

$\displaystyle 0-E^{(N)}_{\text{S,
interaction}}\stackrel{{\scriptstyle\text{C.\
L.}}}{{=}}\frac{(N+1)N(N-1)}{24}\frac{mg^{2}}{\hbar^{2}}\,.$

We will be studying how the stability of the CoM interferometric signal
depends on the potential for ionization 111It may seem that breathing
excitations constitute another potential inelastic channel. However,
curiously, the soliton does not possess true localized excitations: the mean-
field breathers are, in reality, unbound. Interestingly, the absence of bound
excitations is confirmed in the Bogoliubov approximation. Such a restoration
may seem accidental, if Bogoliubov is to be considered as an approximation of
mean-filed equations. However, as a linearization of the Heisenberg equations
of motion for the quantum field, Bogoliubov can give predictions that are more
accurate than the mean-field ones, albeit limited to (perhaps multiple)
monomer excitations.. Note the following: (a) Refs. [5] and [11] conjecture
that being below the single-atom ionization threshold is a prerequisite for a
coherence between the transmitted and reflected parts of the CoM wavepacket;
(b) Ref. [3] indicates that such coherence is preserved even above the six-
atom ionization threshold, in a 100-atom soliton.

Thus, the ionization threshold is

$\displaystyle E_{\text{kinetic, CoM}}>E^{(N-1)}_{\text{S,
interaction}}-E^{(N)}_{\text{S, interaction}}\,.$ This works out to
$\displaystyle E_{\text{kinetic,
CoM}}>\frac{N(N-1)}{8}\frac{mg^{2}}{\hbar^{2}}\,.$ (8)

Meanwhile, the following gives the energy window in which a one-atom
ionization is allowed but already a two-atom one is forbidden:

$E^{(N-2)}_{\text{S, interaction}}-E^{(N)}_{\text{S, interaction}}\geq\\\
E_{\text{kinetic, CoM}}>E^{(N-1)}_{\text{S, interaction}}-E^{(N)}_{\text{S,
interaction}}\,,$

which works out to

$\displaystyle\frac{(N-1)^{2}}{4}\frac{mg^{2}}{\hbar^{2}}\geq
E_{\text{kinetic, CoM}}>\frac{N(N-1)}{8}\frac{mg^{2}}{\hbar^{2}}\,.$ (9)

###  III.5 Calculations for $\bm{N\gg 1}$

If the number of atoms is large, the mean-field approximation simplifies the
calculations.

The number-density distribution in a soliton is

$\displaystyle n(x)=\frac{1}{2}\frac{N}{\ell}\text{sech}^{2}(x/\ell)\,\,,$

where $\ell$ is the healing length, Eq. (4). The potential seen by the CoM of
the soliton scattered off a $\delta$-function barrier

$\displaystyle v_{\text{barrier}}(x)=\tilde{g}\delta(x)$

can be computed as

$\displaystyle V_{\text{soliton-on-barrier}}\stackrel{{\scriptstyle N\gg
1}}{{\approx}}\int\\!dx^{\prime}\,n(x^{\prime})v_{\text{barrier}}(x^{\prime}+X)=\tilde{g}n(X)\,;$

see Refs. [5, 11]. This gives

$\displaystyle\max_{X}V_{\text{soliton-on-barrier}}(X)\stackrel{{\scriptstyle
N\gg 1}}{{\approx}}\frac{1}{4}\frac{m|g|\tilde{g}N^{2}}{\hbar^{2}}\,.$ (10)

The ionization threshold (8) and the single-atom ionization window (9)
respectively become

$\displaystyle E_{\text{kinetic, CoM}}>\mu_{N}\,\,,$ (11)

and

$\displaystyle 2\mu_{N}\geq E_{\text{kinetic, CoM}}>\mu_{N}\,\,,$ (12)

where

$\displaystyle\mu_{N}=\frac{N^{2}}{8}\frac{mg^{2}}{\hbar^{2}}$ (13)

is the chemical potential of an $N$-atom soliton, in the mean-field limit. For
the 6-atom solitons used in this work, exact formulas must be used starting
from about $3$-atom excitations.

##  IV A complete interferometric cycle with a uniform field as a phase object

Figure 1 shows a complete interferometric cycle. The atoms are prepared in the
solitonic ground state of a harmonic oscillator of frequency
$\omega_{\text{preparation}}$. At $t=0$, the soliton is released from its
initial CoM position at $-L_{0}$, and at $t=T/4$, it hits the barrier
($T\equiv 2\pi/\omega_{\text{mirror}}$). The barrier is tuned to a 50%–50%
splitting. At $t=3T/4$, both wavepackets return to the barrier, where they
interfere. We will be interested in the probability of finding the soliton to
(say) the left of the barrier. This probability will be sensitive to the
presence of a phase object, which will be represented by a uniform force field
of a (one-body) intensity $F$.

Figure 1:  A complete interferometric cycle.

###  IV.1 The prediction for the fringes, assuming an elastic scattering of
the CoM off the barrier.

In the absence of inelastic effects, for a spatially even beamsplitter, the
signal will behave as

$\displaystyle\text{Prob}_{\text{left}}(F)=\sin^{2}\left(\frac{2NFL_{0}}{\hbar\omega_{\text{mirror}}}\right)\,;$
(14)

we will derive this in the next subsection. Notice the “quantum advantage
factor” $N$ appearing in the argument of the sine function.

Due to velocity filtering, there appears a difference between the kinetic
energies of the reflected (left) and transmitted (right) wavepackets. However,
this does _not_ introduce any phase shift on recombination.

###  IV.2 A derivation of Eq. (14) for the signal absent inelastic effects

Let us first define the phase $\phi_{\text{right}}$ as the _total_ phase
accumulated by the right wavepacket of the CoM wavefunction between the beam-
splttinng and recombination. It will not include the phase acquired in course
of the beamsplitting process itself. The phase $\phi_{\text{right}}$ can be
decomposed as

$\displaystyle\phi_{\text{right}}=\phi_{\text{right}}^{(0)}+\delta\phi_{\text{right}}\,.$
(15)

Here $\phi_{\text{right}}^{(0)}$ is the phase that would be accumulated if the
“phase object” were not present, and $\delta\phi_{\text{right}}$ is the
contribution from the “phase object.” Anologously, we introduce

$\displaystyle\phi_{\text{left}}=\phi_{\text{left}}^{(0)}+\delta\phi_{\text{left}}\,.$
(16)

####  IV.2.1 The signal for a given phase difference
$\phi_{\text{right}}-\phi_{\text{left}}$

For our interferometer, the signal will be defined as the probability of
finding the soliton to the left from the barrier after the recombination.

We are assuming elastic scattering and no external potential besides the
barrier. Then, asymptotically, the scattering solution
$\psi_{\text{CoM}}(\bar{X})$ for the CoM wave function assumes the form

$\displaystyle\psi_{\text{CoM}}(\bar{X})=\left\\{\begin{array}[]{ccc}e^{+i\bar{K}\bar{X}}+re^{-ik\bar{X}}&\text{
for }&\bar{X}\to-\infty\\\ te^{+i\bar{K}\bar{X}}&\text{ for
}&\bar{X}\to+\infty\end{array}\right.\,\,,$

where $r$ and $t$ are respectively the reflection and transmission
coefficients.

It is easy to show that, after the beam-splitting and recombination, the
signal has the form

$\displaystyle\text{Prob}_{\text{left}}=|r^{2}e^{i\phi_{\text{left}}}+t^{2}e^{i\phi_{\text{right}}}|^{2}\,.$

From the conservation of matter, we have $|r|^{2}+|t|^{2}=1$, so that, a
priori, the family of possible values for $r$ and $t$ is parametrized by three
real numbers. But if the scatterer used for both beam-splitting and
recombination is spatially even, then $r$ and $t$ are more constrained and
their possible values form a family parametrized by two real numbers,
$\eta_{\text{e}}$ and $\eta_{\text{o}}$ (see, e.g. [18]). Here ‘e’ stands for
even and ‘o’ for odd waves. This parametrization is as follows:

$\displaystyle r=f_{\text{e}}-f_{\text{o}}$ $\displaystyle
t=1+f_{\text{e}}+f_{\text{o}}\,\,,$

where

$\displaystyle f_{\text{e,o}}=-\frac{1}{1+i\eta_{\text{e,o}}}$

are the scattering amplitudes for the even and odd waves. The signal now reads

$\text{Prob}_{\text{left}}=\frac{1}{\left(\eta_{\text{e}}^{2}+1\right)^{2}\left(\eta_{\text{o}}^{2}+1\right)^{2}}\\\
\times\Big{\\{}-2(\eta_{\text{e}}\eta_{\text{o}}+1)^{2}(\eta_{\text{e}}-\eta_{\text{o}})^{2}\cos\left(\phi_{\text{right}}-\phi_{\text{left}}\right)\\\
+(\eta_{\text{e}}-\eta_{\text{o}})^{4}+(\eta_{\text{e}}\eta_{\text{o}}+1)^{4}\Big{\\}}\,\,.$

We next require that the scatterer be a 50%–50% beam-splitter:

$\displaystyle|t|^{2}=|r|^{2}=\frac{1}{2}\,\,.$

There are two disjoint one-parametric families of $r$ and $t$ that have this
property, and both can be parametrized by $\eta_{\text{e}}$:

$\eta_{\text{o}}=-\frac{\eta_{\text{e}}+1}{\eta_{\text{e}}-1}\quad\text{and}\quad\eta_{\text{o}}=+\frac{\eta_{\text{e}}-1}{\eta_{\text{e}}+1}\,.$

For both families, $\eta_{\text{e}}$ can be any real number except $1$ for the
first family and $-1$ for the second. While the magnitudes of $r$ and $t$ are
now fixed to 1/2, their phases will depend on the choice of family and the
choice of the value of $\eta_{\text{e}}$. Nonetheless, unexpectedly and
inexplicably (in the sense that we don’t know of any a priori reason why the
mathematics had to work out this way), in an interferometer featuring a
50%–50% beam-splitter and a 50%–50% recombiner, the produced signal obeys a
universal formula that depends _only_ on the phases accumulated between beam-
splitting and recombination (and not on the phases of $r$ and $t$, i.e.
neither on the choice of the family nor on the choice of $\eta_{\text{e}}$):

$\displaystyle\text{Prob}_{\text{left}}=\sin^{2}\left(\frac{\phi_{\text{right}}-\phi_{\text{left}}}{2}\right)\,.$
(17)

To reiterate, this formula applies to any spatially _even_ scatterer, which
must be _the same_ for both beam-splitting and recombination.

####  IV.2.2 The vanishing of the unperturbed phase shift, $\phi_{\text{\rm
right}}^{(0)}-\phi_{\text{\rm left}}^{(0)}$

First, let us discuss the effect of the velocity filtering. There will be a
difference in kinetic energies between the slow part of the incident
wavepacket that gets reflected (left interferometer arm) by the barrier and
the fast component that is transmitted (right interferometer arm). A priori,
this difference is expected to introduce a phase shift between the arms on
recombination. This phase shift will depend on both the width and the shape of
the velocity distribution of the incident wavepacket. Moreover, since the
“slow” and the “fast” trajectories arriving at the recombiner at the same time
would have left leave the beamspitter at two different instances of time, our
interferometer would require a degree of spatio-temporal coherence. Notice,
however, that for an interferometer formed by a _harmonic potential_ , the
latter effect disappears. This is not a coincidence, but an indication that in
a harmonic interferometer, velocity filtering does _not_ introduce any
additional left-right arm phase shift at all. This is indeed the case and can
be proven in three ways: (a) quantum-mechanically, (b) semi-classically, using
an explicit calculation, and (c) semi-classically, using a variational
principle, for small energy differences between the arms only. Moreover, each
of the two phases, $\phi_{\text{\rm left}}^{(0)}$ and $\phi_{\text{\rm
right}}^{(0)}$, vanish separately.

  * (a)

In a harmonic potential, any initial state $\psi(x,\,t\\!=\\!0)$ gets
transformed, after a half-period, to a state
$\psi\left(x,\,t\\!=\\!\pi/\omega_{\text{mirror}}\right)=-i\psi(-x,\,t\\!=\\!0)$,
i.e. to a mirror image of the initial state. No energy-dependent effects are
present.

  * (b)

Semi-classically, the phase acquired by the right wavepacket between beam-
splitting and recombination will be given by the classical action:

$\displaystyle\phi_{\text{\rm right}}^{(0)}$ $\displaystyle=S_{\text{\rm
right}}^{(0)}/\hbar$
$\displaystyle=\frac{1}{\hbar}\int_{t^{\text{BS}}}^{t^{\text{REC}}=t^{\text{BS}}+T/2}\\!dt\,\mathcal{L}(\bar{X}_{\text{\rm
right}}^{(0)}(t),\,\bar{\mathcal{V}}_{\text{\rm right}}^{(0)}(t))$
$\displaystyle=\frac{1}{\hbar}\int_{t^{\text{BS}}}^{t^{\text{REC}}=t^{\text{BS}}+T/2}\\!dt\,\left\\{\frac{1}{2}M\bar{\mathcal{V}}_{0}^{2}\cos^{2}(\omega_{\text{mirror}}t)\right.$
$\displaystyle\hskip
50.00008pt\left.-\frac{1}{2}m\omega_{\text{mirror}}^{2}(\bar{\mathcal{V}}_{0}/\omega_{\text{mirror}})^{2}\sin^{2}(\omega_{\text{mirror}}t)\right\\}$
$\displaystyle=\frac{1}{\hbar}\frac{E_{\text{\rm
right}}}{\omega_{\text{mirror}}}\int_{t^{\text{BS}}/\omega_{\text{mirror}}}^{t^{\text{BS}}/\omega_{\text{mirror}}+\pi}\\!d\xi\,\left\\{\cos^{2}(\xi)-\sin^{2}(\xi)\right\\}$
$\displaystyle=0\,\,,$

where $\mathcal{L}(\bar{X},\,\bar{\mathcal{V}})\equiv
M\bar{\mathcal{V}}^{2}/2-M\omega_{\text{mirror}}^{2}\bar{X}^{2}/2$ is the
classical Lagrangian for the CoM motion,

$\bar{X}_{\text{\rm
right}}^{(0)}(t)=(\bar{\mathcal{V}}_{0}/\omega_{\text{mirror}})\sin(\omega_{\text{mirror}}t)$

and

$\bar{\mathcal{V}}_{\text{\rm
right}}^{(0)}(t)=\bar{\mathcal{V}}_{0}\cos(\omega_{\text{mirror}}t)$

are the unperturbed CoM trajectory and the corresponding velocity dependence
in the right arm, and $M\equiv mN$ is the soliton mass. Further,

$\bar{\mathcal{V}}_{0}=L_{0}\omega_{\text{mirror}}$

is the velocity of the wavepacket on the beamsplitter, $E_{\text{\rm
right}}=M\bar{\mathcal{V}}_{0}^{2}/2$ is its energy, and $t^{\text{BS}}$ and
$t^{\text{REC}}$ are the beamsplitting and recombination time instances. As
one can see, the action _vanishes identically_ for any energy of the
wavepacket. This derivation can be repeated verbatim for the left
interferometer arm. This proof shows that $\phi_{\text{\rm left}}^{(0)}=0$ and
$\phi_{\text{\rm right}}^{(0)}=0$ separately.

  * (c)

Finally, the absence of the energy dependence of the phase accumulated between
the beamsplitting and recombination can be proven variationally, within the
semicalssical approximation, for small energy variations. Notice, again, that
the slow (left) and the fast (right) wavepackets share the initial and the
final points of their trajectories, in _both_ space and time. If the energy
difference between the trajectories is small, then one of them can be
considered a small variation of the other, with fixed space-time end-points.
Since the latter trajectory obeys laws of classical mechanics, it must obey
principle of least action. Thus, the difference between the two actions (hence
between the two quantum phases, in the semi-classical approximation) must
vanish to linear order in the amplitude of trajectory variation.

To sum up, we have just showed that

$\displaystyle\phi_{\text{right}}^{(0)}-\phi_{\text{left}}^{(0)}=0\,\,.$ (18)

This property is specific for an interferometric cycle driven by a harmonic
potential. Since the right-left energy disparity is conjectured to be
unavoidable in interferometry with massive objects [5, 11], a harmonic control
of the interferometer arms may provide a remedy for a possible dependence of
the fringe position on the energy and shape of the wavepacket.

####  IV.2.3 An explicit calculation of the fringe shift due to a uniform
field

Finally, we turn to the phase shift accumulated due to the “phase object.” Let
us compute the phase shift induced by the uniform field $Fx$ on, for example,
the right arm of the interferometer. Note that the potential energy correction
$Fx$ refers to a single atom. For the CoM, the energy correction _and the
resulting quantum phase correction accumulated_ must both be multiplied by the
number of atoms $N$. The resulting correction to the CoM Lagrangian becomes

$\delta\mathcal{L}(\bar{X},\,\bar{\mathcal{V}})=-NF\bar{X}\,.$

This amplification, combined with a suppression (which is much harder to
achieve) of decoherence of the CoM motion to other degrees of freedom, paves
the way to _quantum advantage in particle interferometry_.

Using the principle of the least action, one can easily show (see, e.g. Ref.
[19]) that a correction to the arm trajectory introduced by a correction to
the Lagrangian does not contribute to a correction to the acton, in the first
order in $\delta\mathcal{L}$. Thus,

$\displaystyle\delta\phi_{\text{\rm right}}$ $\displaystyle=\delta
S_{\text{\rm right}}/\hbar$
$\displaystyle=\frac{1}{\hbar}\int_{t^{\text{BS}}}^{t^{\text{REC}}=t^{\text{BS}}+T/2}\\!dt\,\delta\mathcal{L}(\bar{X}_{\text{\rm
right}}(t),\,\bar{\mathcal{V}}_{\text{\rm right}}(t))$
$\displaystyle=\frac{1}{\hbar}\int_{t^{\text{BS}}}^{t^{\text{REC}}=t^{\text{BS}}+T/2}\\!dt\,(-NFL_{0})\sin(\omega_{\text{mirror}}t)$
$\displaystyle=-\frac{NFL_{0}}{\hbar\omega_{\text{mirror}}}\int_{t^{\text{BS}}/\omega_{\text{mirror}}}^{t^{\text{BS}}/\omega_{\text{mirror}}+\pi}\\!d\xi\,\sin(\xi)$
$\displaystyle=-\frac{2NFL_{0}}{\hbar\omega_{\text{mirror}}}\,\,.$

An analogous computation for the left arm gives

$\delta\phi_{\text{\rm
left}}=+\frac{2NFL_{0}}{\hbar\omega_{\text{mirror}}}\,.$

Putting the two contributions together, we get

$\displaystyle\delta\phi_{\text{right}}-\delta\phi_{\text{left}}=-\frac{4NFL_{0}}{\hbar\omega_{\text{mirror}}}\,\,.$
(19)

Combining our results in Eqs. 15, 16, 17, 18 and 19, we finally obtain Eq.
(14).

##  V Numerical simulations

Our principal interest was to investigate how the degree to which soliton
‘ionization’ is energetically allowed (i.e. how large is the CoM kinetic
energy of the soliton, Eq. (5), as compared to the various thresholds in Sec.
III.4) affects the degradation of interference fringes from their idealized
behavior in Eq. (14).

The ratio of the CoM kinetic energy to the ionization thresholds can be tuned
by adjusting one or more of $\kappa_{\text{mirror}}$, $U$, and $W$. Once the
parameters that determine this ratio are set, we compute
$\text{Prob}_{\text{left}}$ (in fact, simply the number of atoms
$N_{\text{L}}$ detected to the left of the barrier, since
$\text{Prob}_{\text{left}}=N_{\text{L}}/N$) for various values of $F$. This
produces a curve like that in Fig. 3, featuring interference fringes which are
a function of $F$. We will present two such curves, corresponding to two
different ratios of the CoM kinetic energy to the ionization threshold.

We work in a system of units in which $d=J=\hbar=1\,.$ In all the runs that we
will present, the number of atoms is $N=6$, the number of lattice sites is
$L=29$, and the initial offset is $L_{0}=7$. In principle, the condition in
Eq. (6) links the values of $\kappa_{\text{preparation}}$, $U$, $W$, so that
only two of them can be chosen independently. However, after some
experimentation, we found that best fringes are obtained if this condition is
slightly violated. We ended up varying just the value of $U$ while keeping
$\kappa_{\text{preparation}}=0.00002813$ and $W=0.24$. Furthermore, we also
kept $\kappa_{\text{mirror}}=0.001800$.

Under these conditions, we can vary the ratio of the CoM kinetic energy to the
ionization thresholds by varying $U$. For each choice of $U$, need to produce
a curve of $\text{Prob}_{\text{left}}$ versus $F$. We do this by picking a
range of values of $F$; for each such value, we numerically simulate, using
exact diagonalization, the complete interferometric cycle from Fig. 1, and
compute the number of atoms $N_{\text{L}}$ detected to the left of the barrier
at the end of the cycle.

We should note that we ended up slightly modifying the interferometric cycle
in Fig. 1, in the folllowing way: instead of having the field $F$ be present
for the entire cycle, we only ‘turn it on’ slightly before the soliton first
hits the barrier (at $t=17.9$ time units), and then ‘turn it off’ slightly
after the recombination (at $t=46.5$ time units). The reason we did this is
that the influence of the $F$ field on the trajectory of the solitons is not
completely negligible, and, at the recombining stage, the two wavepackets
arrive at the beamsplitter at somewhat diferent times. This effect is not
accounted for at all in our idealized theory that leads to Eq. (14). Indeed,
for a sufficiently strong $F$, these times of arrival will be so different
that the incoming wavepackets will never overlap atop the beamsplitter: they
will completely miss each other and there will be no recombination. In order
to minimize this effect, we keep $F$ ‘turned off’ for as long as possible at
the beginning and at the end of the cycle.

Having obtaned a numerical curve of $N_{\text{L}}$ versus $F$, we compare it
to the idealized result in Eq. (14) by fitting the numerical curve to the
following function:

$y(F)=a_{1}+a_{2}F+a_{3}F^{2}\\\
+\left(b_{1}+b_{2}F+b_{3}F^{2}\right)\left[\cos^{2}\left(\omega_{\text{f\/it}}F+\phi_{0}\right)-\frac{1}{2}\right]\,.$

There are 8 fitting parameters
($a_{1},\,a_{2},\,a_{3},\,b_{1},\,b_{2},\,b_{3},\,\omega_{\text{f\/it}},\,$
and $\phi_{0}$), but in reality we are only interested in
$\omega_{\text{f\/it}}$. According to Eq. (14), $\omega_{\text{f\/it}}$ should
be correspond to $\frac{2NL_{0}}{\hbar\omega_{\text{mirror}}}$, which works
out to $990$ for our chosen values of parameters. All the other fitting
parameters are merely ‘empirical’, introduced to account for deviations from
the idealized behavior in Eq. (14).

###  V.1 Results

Our results are presented in Figs. 3 and 3, corresponding respectively to the
choices $U=-0.3$ and $U=-0.4$.

A priori, our expectation is that $U=-0.3$ allows for a complete
disintegration of the soliton onto six individual atoms, so a significant
suppression of fringes is expected. On the other hand, $U=-0.4$ brings us
slightly above the double ionization: we have enough energy to extract two
atoms from the total of six, but no more. Thus, some suppression of fringes is
expected, but less than for $U=-0.3$.

It is not clear if the numerical results bare out these expectations. From the
figures, it seems that the $U=-0.3$ plot is noisier than the $U=-0.4$ one, but
it is not obvious that the fringes are more suppressed for $U=-0.3$. Moreover,
the fitted value of the frequency is closer to the idealized one (990) for
$U=-0.3$ than for $U=-0.4$.

Our main result is that the interferometric signal still exists even when a
complete disintegration of the soliton onto six individual atoms is
energetically allowed.

Figure 2: Interference fringes for $U=-0.3$. The CoM kinetic energy is
sufficiently high that total ionization is energetically allowed. The fitted
value of the frequency of oscillations is $\omega_{\text{f\/it}}=846$.  Figure
3: Interference fringes for $U=-0.4$. The CoM kinetic energy is just above the
threshold necessary in order for shedding of two atoms (two-particle
ionization) to be energetically allowed. The fitted value of the frequency of
oscillations is $\omega_{\text{f\/it}}=667$.

##  VI Conclusions

We simulated the matter-wave interferometer of the kind originally proposed by
Weiss and Castin [5, 11]. Unlike in the original proposal, we considered
situations where either partial or complete disintegration of the soliton is
energetically allowed. Our expectation was that the suppression of the
interferometric fringes will depend in some simple way on the number of atoms
that the soliton is energetically allowed to lose. The numerical simulations
did not bare this out: the fringe distorsion sequence is more complicated, and
further study is required to make sense of it. Our main result is that,
surprisingly, the interferometric signal survives even when the soliton has
enough energy to completely disintegrate.

##  VII Acknowledgments

We benefited from useful discssions with Luigi Amico.

This work was supported by the NSF grants PHY-1912542 and PHY-1607221.

## References

  * Cuevas  _et al._ [2013] J. Cuevas, P. G. Kevrekidis, B. A. Malomed, P. Dyke, and R. G. Hulet, Interactions of solitons with a Gaussian barrier: splitting and recombination in quasi-one-dimensional and three-dimensional settings, New J. Phys. 15, 063006 (2013).
  * Veretenov  _et al._ [2007] N. Veretenov, Y. Rozhdestvensky, N. Rosanov, V. Smirnov, and S. Fedorov, Interferometric precision measurements with Bose-Einstein condensate solitons formed by an optical lattice, Eur. Phys. J. D 42, 455 (2007).
  * Streltsov  _et al._ [2009a] A. I. Streltsov, O. E. Alon, and L. S. Cederbaum, Scattering of an attractive Bose-Einstein condensate from a barrier: Formation of quantum superposition states, Phys. Rev. A 80, 043616 (2009a).
  * Streltsov  _et al._ [2009b] A. I. Streltsov, O. E. Alon, and L. S. Cederbaum, Efficient generation and properties of mesoscopic quantum superposition states in an attractive Bose-Einstein condensate threaded by a potential barrier, J. Phys. Bs 42, 091004 (2009b).
  * Weiss and Castin [2009] C. Weiss and Y. Castin, Creation and detection of a mesoscopic gas in a nonlocal quantum superposition, Phys. Rev. Lett. 102, 010403 (2009).
  * Naldesi  _et al._ [2020] P. Naldesi, J. P. Gomez, V. Dunjko, H. Perrin, M. Olshanii, L. Amico, and A. Minguzzi, Enhancing sensitivity to rotations with quantum solitonic currents (2020), arXiv:1901.09398 [cond-mat.quant-gas] .
  * Polo  _et al._ [2021] J. Polo, P. Naldesi, A. Minguzzi, and L. Amico, The quantum solitons atomtronic interference device, Quantum Sci. Technol. 7, 015015 (2021).
  * Naldesi  _et al._ [2019] P. Naldesi, J. P. Gomez, B. Malomed, M. Olshanii, A. Minguzzi, and L. Amico, Rise and fall of a bright soliton in an optical lattice, Phys. Rev. Lett. 122, 053001 (2019).
  * Grimshaw  _et al._ [2021] C. L. Grimshaw, T. P. Billam, and S. A. Gardiner, Soliton interferometry with very narrow barriers obtained from spatially dependent dressed states (2021), arXiv:2104.11511 [cond-mat.quant-gas] .
  * Holdaway  _et al._ [2012] D. I. H. Holdaway, C. Weiss, and S. A. Gardiner, Quantum theory of bright matter-wave solitons in harmonic confinement, Phys. Rev. A 85, 053618 (2012).
  * Weiss and Castin [2012] C. Weiss and Y. Castin, Elastic scattering of a quantum matter-wave bright soliton on a barrier, J. Phys. A 45, 455306 (2012).
  * Boissé  _et al._ [2017] A. Boissé, G. Berthet, L. Fouché, G. Salomon, A. Aspect, S. Lepoutre, and T. Bourdel, Nonlinear scattering of atomic bright solitons in disorder, Eur. Phys. Lett. 117, 10007 (2017).
  * Valiente and Petrosyan [2008] M. Valiente and D. Petrosyan, Two-particle states in the Hubbard model, J. Phys. B 41, 161002 (2008).
  * Castin [2004] Y. Castin, Simple theoretical tools for low dimension Bose gases, J. Phys. IV France 116, 89 (2004).
  * Castin [2009] Y. Castin, Internal structure of a quantum soliton and classical excitations due to trap opening, Eur. Phys. J. B 68, 317 (2009).
  * Calogero and Degasperis [1975] F. Calogero and A. Degasperis, Comparison between the exact and Hartree solutions of a one-dimensional many-body problem, Phus. Rev. A 11, 265 (1975).
  * Note [1] It may seem that breathing excitations constitute another potential inelastic channel. However, curiously, the soliton does not possess true localized excitations: the mean-field breathers are, in reality, unbound. Interestingly, the absence of bound excitations is confirmed in the Bogoliubov approximation. Such a restoration may seem accidental, if Bogoliubov is to be considered as an approximation of mean-filed equations. However, as a linearization of the Heisenberg equations of motion for the quantum field, Bogoliubov can give predictions that are more accurate than the mean-field ones, albeit limited to (perhaps multiple) monomer excitations.
  * Olshanii [1998] M. Olshanii, Atomic scattering in the presence of an external confinement and a gas of impenetrable bosons, Phys. Rev. Lett. 81, 938 (1998).
  * Ol’shanii [1994] M. Ol’shanii, Atomic interferometry, Laser Physics 4, 995 (1994).