††thanks: Corresponding author: J. D. W<EMAIL_ADDRESS>
# Beyond one-axis twisting: Simultaneous spin-momentum squeezing

John Drew Wilson  JILA and Department of Physics, University of Colorado, 440
UCB, Boulder, CO 80309, USA  Simon B. Jäger  JILA and Department of Physics,
University of Colorado, 440 UCB, Boulder, CO 80309, USA  Physics Department
and Research Center OPTIMAS, Technische Universität Kaiserslautern, D-67663,
Kaiserslautern, Germany  Jarrod T. Reilly  JILA and Department of Physics,
University of Colorado, 440 UCB, Boulder, CO 80309, USA  Athreya Shankar
Institute for Theoretical Physics, University of Innsbruck, Innsbruck, Austria
Institute for Quantum Optics and Quantum Information of the Austrian Academy
of Sciences, Innsbruck, Austria  Maria Luisa Chiofalo  Dipartimento di Fisica
“Enrico Fermi”, Università di Pisa, and INFN
Largo Bruno Pontecorvo, 3 I-56127 Pisa (Italy)  Murray J. Holland  JILA and
Department of Physics, University of Colorado, 440 UCB, Boulder, CO 80309, USA
Dipartimento di Fisica “Enrico Fermi”, Università di Pisa, and INFN
Largo Bruno Pontecorvo, 3 I-56127 Pisa (Italy)

###### Abstract

The creation and manipulation of quantum entanglement is central to improving
precision measurements. A principal method of generating entanglement for use
in atom interferometry is the process of spin squeezing whereupon the states
become more sensitive to $SU(2)$ rotations. One possibility to generate this
entanglement is provided by one-axis twisting (OAT), where a many-particle
entangled state of one degree of freedom is generated by a non-linear
Hamiltonian. We introduce a novel method which goes beyond OAT to create
squeezing and entanglement across two distinct degrees of freedom. We present
our work in the specific physical context of a system consisting of collective
atomic energy levels and discrete collective momentum states, but also
consider other possible realizations. Our system uses a nonlinear Hamiltonian
to generate dynamics in $SU(4)$, thereby creating the opportunity for dynamics
not possible in typical $SU(2)$ one-axis twisting. This leads to three axes
undergoing twisting due to the two degrees of freedom and their entanglement,
with the resulting potential for a more rich context of quantum entanglement.
The states prepared in this system are potentially more versatile for use in
multi-parameter or auxiliary measurement schemes than those prepared by
standard spin squeezing.

##  I Introduction

The creation and manipulation of quantum entanglement is central to developing
powerful quantum technologies [1, 2]. In particular, precision measurements
can greatly benefit from exploiting quantum entanglements [3, 4] because non-
classical states may be engineered for greater sensitivity to a parameter of
interest compared to their classical counterparts [5, 6]. This field has seen
rapid progress on several frontiers [7] including, but not limited to,
experimental demonstration of atomic clock timekeeping below the shot noise
limit [8], extensions of quantum error correction into quantum metrology
schemes [9], and machine learning and optimization for complex state
preparation [10, 11, 12] and measurement schemes [13]. Through this rapid
progress, there is the possibility that we will soon use quantum mechanical
devices to probe new fundamental physics via tabletop experiments [14, 15].

Many state-of-the-art atom interferometry schemes rely on the process of spin
squeezing [16, 17], where a set of quantum spins are correlated to prepare a
non-classical state that is sensitive to $SU(2)$ rotations at a precision
below the standard quantum limit (SQL) [18] of $\Delta\phi^{2}\propto 1/N$,
where $\Delta\phi^{2}$ is the mean square error and $N$ is the number of
particles used in the measurement. One candidate for generating this
entanglement is one-axis twisting (OAT), whereupon many particles become
entangled in a single degree of freedom under a non-linear Hamiltonian [19,
20]. Through entangling processes such as OAT, the SQL may be surpassed and a
limit in precision of $\Delta\phi^{2}\propto 1/N^{2}$ is achievable. This
limit is a result of a Heisenberg uncertainty-like principle between the
operator generating the unitary and the parameter one is measuring. This limit
is aptly named Heisenberg limited scaling (HLS) [21] and is the ultimate limit
for metrological systems [22].

Schemes using OAT provide below SQL improvements for single parameter
measurements, such as the angle a dipole sweeps under rotation generated by a
magnetic field. These improvements are realized by sacrificing the variance of
quantum fluctuations in one direction in exchange for a reduction in the
variance of fluctuations in the direction we wish to measure. This hints at a
natural extension of OAT; one where multiple degrees of freedom are entangled
and squeezed to provide below SQL improvements for multiple parameters
simultaneously.

In this paper, we introduce a novel method for squeezing and entangling two
distinct degrees of freedom: the internal energy levels of an atomic ensemble
and the collective atomic momentum. As a Gedanken experiment, we consider a
collimated packet of atoms passing through a cavity. The cavity mediated
emission and absorption of photons induces a twisting of the collective
internal and momentum degrees of freedom, while also rapidly creating
entanglement between these two degrees of freedom. The states prepared by this
system could have the potential for multiparameter sensing and estimation [23]
below the SQL, squeezed state Bragg interferometry [24], or single parameter
estimation benefiting from auxiliary measurements. By analyzing the Quantum
Fisher Information Matrix (QFIM) of the system, we find that the maximum
metrological gain in each individual degree of freedom is shown to scale
proportionally to HLS. Here, we focus on the squeezing and correlation of the
collective atomic internal energy state and momentum, but we emphasize that
the general process could be realized with any system having same structure in
its couplings and interactions. To this point, we discuss possible platforms
which might be made to generate similar forms of entanglement in the
conclusion of this paper.

The structure of this paper is as follows. In Section II, we cast the
Hamiltonian into a form that illustrates the entanglement generating process:
atomic emission and absorption of photons and the resulting momentum recoil.
From this form, we show that some features may be intuitively understood as a
generalization of the OAT Hamiltonian, while other important features have no
analog in OAT. In Section III, we explore the structure of the system and
Hamiltonian using an underlying Lie algebra, and use these to simplify the
subsequent analysis of the dynamics. In Section IV, we use the quantum Fisher
information matrix (QFIM) to discuss the results of a numerical simulation of
the time dynamics. Lastly, in Section V we show schematically two
interferometry protocols that benefit from the form of entanglement generated
by this scheme.

##  II Derivation of the Hamiltonian and System Dynamics

We consider the Gedanken experiment depicted in Fig. 1(a), where a collimated
packet of atoms passes through the center of the beam waist of a linear
optical cavity, similar to a pulsed version of the set up proposed in [25].
Each atom has a mass $m$, and two relevant internal energy levels labeled the
excited and ground states $\ket{e}$ and $\ket{g}$, respectively. These energy
levels are separated by the transition energy $\hbar\omega_{a}$. We assume
that the cavity supports a single optical mode with corresponding frequency
$\omega_{c}$, which is far detuned from the atomic transition by an amount
$\Delta=\omega_{a}-\omega_{c}$. The interaction strength between the cavity
photons and the $j$th atom is taken to be
$g(x_{j})=\frac{g}{2}\cos(k\hat{x}_{j})$. Furthermore, we assume $N$ atoms
enter the cavity with uniform velocity, and spend a time $t$ inside the light-
atom interaction volume. During this interaction time, the Hamiltonian is
then:

$\displaystyle\hat{H}=$
$\displaystyle\sum_{j=1}^{N}\left(\frac{\hat{p}_{j}^{2}}{2m}+\frac{\hbar\omega_{a}}{2}\hat{\sigma}_{j}^{z}\right)+\hbar\omega_{c}\hat{a}_{c}^{\dagger}\hat{a}_{c}$
(1) $\displaystyle+\frac{\hbar
g}{2}\sum_{j=1}^{N}\cos(k\hat{x}_{j})\left(\hat{a}_{c}\hat{\sigma}^{+}_{j}+\hat{a}_{c}^{\dagger}\hat{\sigma}^{-}_{j}\right),$

where $\hat{\sigma}^{z}_{j}=\ket{e}_{j}\bra{e}_{j}-\ket{g}_{j}\bra{g}_{j}$,
$\hat{\sigma}^{+}_{j}=(\hat{\sigma}^{-}_{j})^{\dagger}=\ket{e}_{j}\bra{g}_{j}$
are Pauli matrices for the $j^{\text{th}}$ atom, $\hat{p}_{j}$ ($\hat{x}_{j}$)
is the transverse momentum (position) operator for the $j^{\text{th}}$ atom
parallel to the cavity axis, and $\hat{a}_{c}^{\dagger}$ ($\hat{a}_{c})$ is
the photon creation (annihilation) operator of the cavity mode.

The two relevant processes at play are the exchange of photons between
different atoms and the atom’s recoil due to the emission and absorption of
photons. To simplify our study of these dynamics, we first take the
interaction picture with
$\hat{H}_{0}=\sum_{j=1}^{N}\hbar\omega_{a}\hat{\sigma}^{z}_{j}/2+\hbar\omega_{a}\hat{a}_{c}^{\dagger}\hat{a}_{c}$.
We assume the cavity is in the dispersive regime
$|\Delta|\gg\sqrt{N}g,\kappa$, where $\kappa$ is the cavity decay rate, such
that we can adiabatically eliminate the cavity degrees of freedom over a
coarse-grained timescale [26]. The resultant Hamiltonian becomes

${\hat{H}}=\sum_{j=1}^{N}\frac{\hat{p}_{j}^{2}}{2m}+\sum_{i,j=1}^{N}\frac{\hbar
g^{2}}{4\Delta}\cos(k\hat{x}_{i})\cos(k\hat{x}_{j})\hat{\sigma}^{+}_{i}\hat{\sigma}^{-}_{j}.$
(2)

The photon exchange has now been abstracted to an excitation exchange between
different atoms and a resultant change in momentum. We note that the operators
$\sum_{j=1}^{N}\cos(k\hat{x}_{j})\hat{\sigma}^{\pm}_{j}$ cause a change in an
atom’s momentum by $\pm\hbar k$ upon trading an excitation, as $\exp(\pm
ik\hat{x}_{j})$ are the momentum shift operators. Therefore, if the atomic
ensemble is prepared such that the atoms are in motional states differing in
their momentum by integer multiples of $\hbar k$, the atoms will never leave
this manifold under purely Hamiltonian evolution. We consider atoms in a
superposition of motional states of the form $\ket{n}_{j}\equiv\ket{n\hbar
k/2}_{j}$ for odd integers $n$. Preparation of such a state could be
accomplished with a diffraction grating [27] or via Kapitza-Dirac pulses and a
trapping potential [28].

Lastly, we assume that $\hbar Ng^{2}/(4\Delta)\ll(\hbar k)^{2}/m$, such that
the lowest two momentum states are far detuned from the rest of the quadratic
kinetic energy spectrum, as shown in Fig. 1(b). Therefore, if the atoms start
in the $\ket{\pm 1}_{j}$ states, they will in the subspace spanned by these
two states. Under these conditions, the total kinetic energy remains fixed at
$N(\hbar k)^{2}/(8m)$. As a result, we can ignore the constant kinetic energy.

In this regime, the momentum now has a spin-$1/2$ algebraic structure and so
the atom’s momentum is effectively mapped onto a two-level system. We define
$\hat{s}_{j}^{+}=(\hat{s}_{j}^{-})^{\dagger}=\ket{+1}_{j}\bra{-1}_{j},$ and
$\hat{s}_{j}^{z}=\ket{+1}_{j}\bra{+1}_{j}-\ket{-1}_{j}\bra{-1}_{j}$ such that
we can cast the translation operator
$\cos(k\hat{x}_{j})=[\exp(ik\hat{x}_{j})+\exp(-ik\hat{x}_{j})]/2$ in terms of
spin raising and lowering operators. We note that
$e^{+ik\hat{x}_{j}}=(e^{-ik\hat{x}_{j}})^{\dagger}=\hat{s}_{j}^{+}$ in this
regime and therefore
$2\cos(k\hat{x}_{j})=(\hat{s}_{j}^{+}+\hat{s}_{j}^{-})\equiv\hat{s}^{x}_{j}$,
thus we can rewrite our Hamiltonian in terms of these operators. Our
simplified Hamiltonian therefore becomes

$\displaystyle\hat{H}$
$\displaystyle=\chi\sum_{i,j=1}^{N}\hat{s}_{i}^{x}\hat{s}_{j}^{x}\hat{\sigma}_{i}^{+}\hat{\sigma}_{j}^{-},$
(3)

with $\chi=\hbar g^{2}/(16\Delta)$. This non-linear Hamiltonian dictates how
the atoms are to be entangled via cavity mediated interactions.

From Eq. 3, we see that if the atoms enter the cavity in the same momentum
state, with all atoms in the state $(\ket{+1}_{j}+\ket{-1}_{j})/\sqrt{2}$,
then the dynamics are generated by
$\hat{H}\approx\sum_{i,j=1}^{N}\hat{\sigma}_{i}^{+}\hat{\sigma}_{j}^{-}\propto(\hat{J}^{z})^{2}$,
where $\hat{J}^{z}=\sum_{j}^{N}\hat{\sigma}_{j}^{z}/2$, and one-axis twisting
is recovered. This is because the momentum flip operator, $\hat{s}^{x}_{j}$,
affects an atom in the state $(\ket{+1}_{j}+\ket{-1}_{j})/\sqrt{2}$ trivially.
Physically, this is the case that all the atoms are in the same equal
superposition of the $+\hbar k/2$ momentum states, so the recoil from emission
and absorption of light doesn’t affect the collective momentum, but the atom’s
internal degree of freedom remains free to evolve. With a starting state such
as $\ket{+}^{\otimes N}=(1/\sqrt{2})^{N}(\ket{e}+\ket{g})^{\otimes N}$ for the
internal atomic energies, the Hamiltonian induces standard OAT behavior,
leading to an effective spin squeezing. This starting state and behavior is
shown in Fig. 1(c), where the red arrows on the left Bloch sphere represent
the action of $(\hat{J}^{z})^{2}$.

We may also consider the case that the internal degrees of freedom don’t
affect the dynamics. This case is not physical, but rather provides an
important intuition for the behavior in the system. Here, we take
$\hat{H}\approx\chi\sum_{i,j=1}^{N}\hat{s}_{i}^{x}\hat{s}_{j}^{x}=4\chi(\hat{K}^{x})^{2}$,
where $\hat{K}^{x}=\sum_{j}^{N}\hat{s}_{j}^{x}/2$. While this is not
necessarily physical, it sheds light on the approximate behavior of the atomic
momentum: we expect the momentum states to experience OAT-like behavior
through the non-linear rotation under $(\hat{K}^{x})^{2}$. With a starting
state of $\ket{+1}^{\otimes N}$ for the momentum degrees of freedom, we would
expect to see operators orthogonal to $\hat{K}^{x}$, such as
$\hat{K}^{z}=\sum\hat{s}^{z}_{j}/2$, to undergo a twisting-like behavior. This
starting state and approximate behavior is shown in Fig. 1(c), where the red
arrow on the right Bloch sphere represents the action of $(\hat{K}^{x})^{2}$.

For the full Hamiltonian we expect the state $\ket{\psi}_{0}=\ket{+}^{\otimes
N}\otimes\ket{+1}^{\otimes N}$ to experience the corresponding spin twisting-
like behavior in both degrees of freedom, and to lead to interesting
entanglement between the two. In the subsequent sections, we demonstrate
mathematically that this state breaks an important symmetry typically found in
OAT, and then we numerically show this leads to entanglement that has
potential for metrological advantage.

Figure 1:  (a) Schematic of the proposed set up. Here, the momentum
perpendicular to the cavity controls the interaction time. The initial
momentum along the cavity axis selects the manifold of momentum states that
the cavity couples to. (b) The spectrum of the kinetic energy versus the
spectrum of momentum states. Here, we note the $\pm 3\hbar k/2$ states are far
from the $\pm\hbar k/2$ states, thus demonstrating that the lowest manifold of
$4$ states can be considered isolated from the rest of the quadratic spectrum.
(c) The two Bloch spheres for the collective two-level system. This picture is
only valid when there is no entanglement between the two degrees of freedom,
but it still provides a useful picture of the approximate behavior of the
system. The blue cloud is the starting state, while the green dashed line
represents the approximate distribution of the final state. The final state
may not be fully represented on these Bloch spheres due to entanglement
breaking the initial $SU(2)\otimes SU(2)$ symmetry needed to represent states
on two collective Bloch spheres. (d) The four-level system, and black arrows
representing each of the three unique $\ \mathfrak{su}(2)$ algebras acting on
the system.

##  III The Operator Algebras

In full, it is not immediately obvious how dynamics evolve under Eq. (3). The
$\hat{s}^{x}_{j}$ operators complicate the Hamiltonian compared to the usual
OAT Hamiltonian, preventing us from using methods typically used to solve OAT
models. However, we can use the symmetries of the system to recast the
Hamiltonian such that it is a member of an $\mathfrak{su}(4)$ algebra yielding
a clear picture of the full dynamics and allowing for efficient numerical
simulation.

The operators appearing in the Hamiltonian are all Pauli operators which
correspond to a single atom’s internal or momentum observable. For the
$j^{th}$ atom’s internal state, the operators
$\\{\hat{\sigma}^{x}_{j},\hat{\sigma}^{y}_{j},\hat{\sigma}^{z}_{j}\\}$ fully
describe any possible observable, where
$\hat{\sigma}^{x}_{j}=\sigma^{+}_{j}+\sigma^{-}_{j}$ and
$\hat{\sigma}^{y}_{j}=i(\sigma^{-}_{j}-\sigma^{+}_{j})$. Similarly, its
momentum state is fully described by
$\\{\hat{s}^{x}_{j},\hat{s}^{y}_{j},\hat{s}^{z}_{j}\\}$, where
$\hat{s}^{y}_{j}=i(\hat{s}_{j}^{-}-\hat{s}_{j}^{+})$ is needed for the
momentum operators to close under commutation. The total system is then
described, in part, by the collective atomic and momentum operators,
$\hat{J}^{i}=\sum_{j}^{N}\hat{\sigma}_{j}^{i}/2$ and
$\hat{K}^{i}=\sum_{j}^{N}\hat{s}_{j}^{i}/2$ for $i=x,y,z$ respectively. These
collective atomic and momentum operators each form an $\mathfrak{su}(2)$
algebra: $\mathfrak{J}=\\{\hat{J}^{z},\hat{J}^{\pm}\\}$ and
$\mathfrak{K}=\\{\hat{K}^{z},\hat{K}^{\pm}\\}$. These two algebras allow us to
fully describe any state which is seperable in the two degrees of freedom,
such as the state $\ket{\psi_{0}}$ which is represented on two composite Bloch
spheres in Fig. 1(c) in blue. Importantly, we note that the momentum operator
$\hat{K}^{z}$ corresponds to the observable for the center of mass momentum,
$\hat{P}_{\rm COM}=\hbar k\hat{K}^{z}$, which is intuitively the difference
between the number of atoms moving in the $+1$ and $-1$ eigenstates.

We can further simplify our analysis by mapping particles into the Schwinger
boson representation [29]. Here we use the simultaneous eigenstates of
$\hat{J}^{z}$ and $\hat{K}^{z}$ as the basis for the new representation, but
in general this could be done via the procedure shown in [30]. First, we
define

$\displaystyle\ket{\alpha,\beta,\gamma,\delta}=$ (4)
$\displaystyle\mathcal{S}\left(\ket{e,+1}^{\otimes\alpha}\ket{g,-1}^{\otimes\beta}\ket{e,-1}^{\otimes\gamma}\ket{g,+1}^{\otimes\delta}\right),$

where $\alpha+\beta+\gamma+\delta=N$ is the total number of atoms and
$\mathcal{S}$ is the symmeterization operator. Note that the symmetrizer is
defined with the normalization factor, shown explicitly in Appendix A.1, so
this representation is normalized. We can represent all the relevant operators
in this formalism as well by associating the annihilation (creation) operators
$\hat{a},\hat{b},\hat{c},\hat{d}$
($\hat{a}^{\dagger},\hat{b}^{\dagger},\hat{c}^{\dagger},\hat{d}^{\dagger}$) to
each of the four modes, such that
$\hat{a}\ket{\alpha,\beta,\gamma,\delta}=\sqrt{\alpha}\ket{\alpha-1,\beta,\gamma,\delta}$
and similarly for the other three modes as shown in Appendix A.2. Now, the
number of atoms in the excited state is simply $\alpha+\gamma$ for states of
the form in Eq. (LABEL:eq:SchwingState). Therefore, we define
$\hat{n}_{e}\ket{\alpha,\beta,\gamma,\delta}=(\hat{a}^{\dagger}\hat{a}+\hat{c}^{\dagger}\hat{c})\ket{\alpha,\beta,\gamma,\delta}$.
By the same process, we can recover the ground state number operator to be
$\hat{n}_{g}=\hat{b}^{\dagger}\hat{b}+\hat{d}^{\dagger}\hat{d}$, the $+1$
momentum state number operator to be
$\hat{n}_{+1}=\hat{a}^{\dagger}\hat{a}+\hat{d}^{\dagger}\hat{d}$, and the $-1$
momentum state number operator to be
$\hat{n}_{-1}=\hat{b}^{\dagger}\hat{b}+\hat{c}^{\dagger}\hat{c}$. Our
collective atomic and momentum operators are simple to represent in the form

$\displaystyle\hat{J}^{z}$
$\displaystyle=\frac{1}{2}(\hat{n}_{e}-\hat{n}_{g}),$ (5)
$\displaystyle\hat{K}^{z}$
$\displaystyle=\frac{1}{2}(\hat{n}_{+1}-\hat{n}_{-1}),$

and
$\hat{J}^{-}=\hat{a}\hat{d}^{\dagger}+\hat{c}\hat{b}^{\dagger}=(\hat{J}^{+})^{\dagger},\hat{K}^{-}=\hat{a}\hat{c}^{\dagger}+\hat{d}\hat{b}^{\dagger}=(\hat{K}^{+})^{\dagger}$.
Moreover, the Hamiltonian is also simply represented,

$\hat{H}=\chi(\hat{a}^{\dagger}\hat{b}+\hat{c}^{\dagger}\hat{d})(\hat{a}\hat{b}^{\dagger}+\hat{c}\hat{d}^{\dagger}).$
(6)

This is intuitively what should be expected because, for example,
$\hat{a}\hat{b}^{\dagger}$ is collective emission where a single atom goes
from the excited, +1 motional state to a ground, -1 motional state. The other
terms can be similarly understood.

Lastly, we introduce the raising and lowering operators
$\hat{E}^{+}=\hat{a}^{\dagger}\hat{b}+\hat{c}^{\dagger}\hat{d}=(\hat{E}^{-})^{\dagger}$,
and we notice that $[\hat{E}^{+},\hat{E}^{-}]=2\hat{J}^{z}$ and
$[\hat{J}^{z},\hat{E}^{\pm}]=\pm\hat{E}^{\pm}.$ Thus, we see that the set
$\mathfrak{E}=\\{\hat{J}^{z},\hat{E}^{\pm}\\}$ forms a third closed
$\mathfrak{su}(2)$ algebra on the system which succinctly represents the
entanglement generating processes due to absorption and emission. The three
sub-algebras $\mathfrak{J},\mathfrak{K}$ and $\mathfrak{E}$ taken together are
members of a complete $\mathfrak{su}(4)$ algebra, which generates an $SU(4)$
group that efficiently describes the dynamics of this system. The action of
three sub-algebras is represented schematically in Fig. 1(d) for a single
atom. In summary, within the full $\mathfrak{su}(4)$ describing our dynamics,
we find that there exists three $SU(2)$ subgroups each generated by
$\mathfrak{J},\mathfrak{K}$, or $\mathfrak{E}$, which matches the general
structure for $SU(4)$ [31]. Thus, the system can be considered as a collection
of hybridized angular momentum.

We can take advantage of the commutation structure in $\mathfrak{E}$ to
simplify the Hamiltonian even further,

$\displaystyle\hat{H}$ $\displaystyle=\chi\hat{E}^{+}\hat{E}^{-}$ (7)
$\displaystyle=\chi(\hat{E}^{2}-(\hat{J}^{z})^{2}+\hat{J}^{z}),$

where $\hat{E}^{2}=\hat{E}^{+}\hat{E}^{-}+(\hat{J}^{z})^{2}-\hat{J}^{z}$ is
the quadratic Casimir operator [32] for $\mathfrak{E}$. Now, Eq. (7) looks
like the familiar form of a OAT Hamiltonian, except for the important
difference that $\hat{K}^{y}$, $\hat{K}^{z}$ don’t commute with $\hat{E}^{2}$.
This means there exists states which are eigenstates of $\hat{K}^{z}$ that
evolve non-trivially under the operator $\hat{E}^{2}$, such as the starting
state discussed at the end of Section II. Furthermore, we can observe that the
operator $\hat{E}^{2}$ has shells corresponding to each of its eigenvalues,
similar to the shells typically defining eigenvalues for total angular
momentum observables. The starting state, $\ket{\psi_{0}}$ creates a
superposition over these shells and, with $\hat{E}^{2}$ contributing non-
trivially to the dynamics, each of the three pseudo-angular momentum subgroups
experience a twisting under this Hamiltonian.

##  IV Analysis of the Dynamics and Entanglement Generation

Now we use the Schwinger boson representation introduced in Section III to
numerically simulate the system and explore the dynamics. For these
simulations we assume the cavity decay at rate $\kappa$ and other dissipative
processes, such as spontaneous emission at rate $\gamma$, are negligible. This
assumption is valid in the limit that the timescale considered for unitary
dynamics, $t$, is much smaller than the relevant inverse decay rates. Further
analysis of the effects of decoherence is left to future work, but we attempt
to make explicit note of when one would expect decoherence to become non-
negligible, and the relevant bounds in these cases.

To simulate the system, we use the four annihilation/creation operators found
in the previous section, and model the atomic system as a system of four
harmonic oscillators. The Hilbert space of four harmonic oscillators has a
dimensionality of $(N+1)^{4}$ containing all states with atom numbers between
$0$ and $4N$ atoms. We may use either of the conditions that
$\hat{n}_{e}+\hat{n}_{g}=N$ or $\hat{n}_{+1}+\hat{n}_{-1}=N$ to project onto
only the states with $N$ atoms. This corresponds to restricting to only states
which are may be reached by $SU(4)$ action, and the typical argument of
putting $N$ atoms indistinguishably in four distinguishable states shows the
system scales at $(N+1)(N+2)(N+3)/6$ states for $N$ atoms. This now matches
the dimensionality of the basis states with an $SU(4)$ symmetry, given in Ref
[33], and is numerically more efficient than the initial $(N+1)^{4}$ scaling.

We use the starting state discussed in Section II,
$\ket{\psi}_{0}=\ket{+}^{\otimes N}\otimes\ket{+1}^{\otimes N}$. As noted in
the end of Section III, $\ket{\psi}_{0}$ is not an eigenstate of
$\hat{E}^{2}$. From the discussion of this state and the picture in Fig. 1(c),
we expect this initial state to lead to twisting-like behavior and
entanglement generation between the two degrees of freedom. The intuitive
picture to understand this behavior is the following. When an atom emits
light, its internal degree of freedom becomes entangled to that of the atom
which absorbs the emitted light. At the same time, both these atom’s momentum
states must switch, causing their external degrees of freedom to become
entangled similar to their internal ones.

To diagnose the amount of entanglement and potential metrological use, we
consider the case that one wants to prepare states which will be used to
estimate some phase, $\phi_{j}$, encoded by unitary evolution under some
operator, $\hat{G}^{j}$, so that the state evolves according to
$\exp(-i\phi_{j}\hat{G}^{j})$. Specifically we consider the cases that
$\hat{G}^{j}$ is in either $\mathfrak{J}$ or $\mathfrak{K}$, and choose the
indices $i,j$ so that if $i,j=1,2,3$ then
$\hat{G}^{i},\hat{G}^{j}=\hat{J}^{x},\hat{J}^{y},\hat{J}^{z}$ and if
$i,j=4,5,6$ then
$\hat{G}^{i},\hat{G}^{j}=\hat{K}^{x},\hat{K}^{y},\hat{K}^{z}$. In this
scenario the QFIM serves as both an entanglement measure [34] and a measure of
the potential metrological use of a state in quantum metrology [35]. We use
for the form of the QFIM given in Ref. [36] for pure states, since in the
present proof of concept we do not address decoherence. Under this condition,
the matrix elements are given by

$\displaystyle\mathcal{F}^{ij}=4\Big{(}\Big{\langle}\frac{\\{\hat{G}^{i},\hat{G}^{j}\\}}{2}\Big{\rangle}-\langle\hat{G}^{i}\rangle\langle\hat{G}^{j}\rangle\Big{)},$
(8)

where
$\\{\hat{G}^{i},\hat{G}^{j}\\}=\hat{G}^{i}\hat{G}^{j}+\hat{G}^{j}\hat{G}^{i}$
is the anti-commutator. For $i=j$, Eq. (8) returns the fourfold variance,
which captures the amount of squeezing and entanglement present. The condition
for an entanglement witness to squeezing is $\mathcal{F}^{ii}/N^{2}>1/N$,
which is equivalent to the condition given in Ref. [34]. If
$\mathcal{F}^{ii}/N^{2}$ approaches a constant as $N$ grows, then the
sufficient condition for entanglement is clearly met and the system’s
potential metrological use proportional to HLS is demonstrated. Meanwhile for
$i\neq j$, Eq. (8) returns the fourfold covariance, thereby capturing the
amount of quantum correlations between these observables. We observe that
$[\hat{J}^{i},\hat{K}^{j}]=0$ for all
$\hat{J}^{i}\in\mathfrak{J},\hat{K}^{j}\in\mathfrak{K}$. As a result the
covariance of two operators on the internal state and the momentum, such as
$\text{cov}(\hat{J}^{x},\hat{K}^{z})$, is non-zero only for pure states which
are entangled. The off-diagonal elements of the QFIM with $i\in\\{1,2,3\\}$
and $j\in\\{4,5,6\\}$ therefore represents the covariance between the atomic
and momentum operators, and acts as an entanglement witness of quantum
correlations between the two degrees of freedom. Thus, we use the sufficient
condition that $\mathcal{F}^{ij}\neq 0$ as an entanglement witness for the two
degrees of freedom as a pure state bipartite system. This is a modified
version of the condition given in Ref. [37].

In Fig. 2, we show the quantity $\mathcal{F}^{ii}/N^{2}$ for the four
operators of interest, and for four different numbers of atoms, $N$, each as a
function of interaction time with the cavity, $t$. We observe that
$\mathcal{F}^{ii}/N^{2}$ increases sharply before leveling off to a constant
value over time. Because $\mathcal{F}^{ii}/N^{2}>1/N$, the entanglement
witness condition is satisfied for each case. This condition is met in a short
interaction time, demonstrating that entanglement is quickly generated in both
the collective internal and momentum modes. Therefore we see that along with
the spin squeezing in the internal atomic degrees of freedom, this platform
also leads to an effective squeezing of the momentum degrees of freedom.

Figure 2:  Four of the six diagonal elements of the QFIM, for four different
atom numbers. The operators $\hat{J}^{z}$ and $\hat{K}^{x}$ are left out
because they commute with the Hamiltonian and are therefore conserved
quantities. We see that as the number of atoms grow, the behavior of the
diagonal QFIM elements converge. For atom numbers of $N\approx 50$ or more, a
plateau with respect to time appears, centered around $\chi t=\pi/4$. This is
similar to the behavior found in OAT where the QFIM for $\hat{J}^{x}$ and
$\hat{J}^{y}$ reach a plateau [5] centered around the same time. As $N$ grows,
the plateau exists almost everywhere in time. Here we only show even atom
numbers, $N$, but we note that for odd atom numbers the behavior is the same
except for at $\chi t=\pi/2$, where the concavity is opposite from what’s
shown here.

To quantify the potential metrological use of this system, we fix the time at
$\chi t=\pi/4$ and show how the diagonal element of QFIM for $\hat{J}^{x}$ and
$\hat{K}^{z}$ scales with atom number. The results are shown in Fig. 3.
Achieving an interaction time scale of $\chi t\sim 1$ would require a very big
cavity-atom coupling, such that $\chi\gg\gamma$. The same is true for any
other decoherence rate one might consider. As a result, this timescale may be
physically inaccessible with traditional cavities, but is theoretically
interesting nonetheless. These long timescales form the equivalent of the
“oversqueezed” timescales in standard OAT. We specifically choose the time
$\chi t=\pi/4$ because it is the center of the plateau in the QFIM’s diagonal
elements.

We see that both the atomic and momentum degrees of freedom scale
proportionally to $N^{2}$, i.e. with HLS. Similar behavior exists in OAT,
where one finds a plateau in the variance of the antisqueezed quadrature for
times between $1/\sqrt{N}\lesssim\chi^{\prime}t\lesssim\pi/2-1/\sqrt{N}$,
where $\chi^{\prime}$ is an appropriately defined frequency. However, in OAT
this plateau restricted to just the spin degree of freedom [5]. Our scheme
provides a squeezing mechanism for momentum degrees of freedom, creating the
possibility that spin-squeezing techniques used in Ramsey interferometry [38]
might be generalized to Bragg interferometry or that the two might be
performed simultaneously.

Figure 3:  The diagonal elements of the QFIM corresponding to $\hat{J}^{x}$
and $\hat{K}^{z}$ shown as a function of atom number, $N$. We fit
$4\Delta\hat{J}^{x}$ and $4\Delta\hat{K}^{z}$ with second order polynomials
$F_{J}(N)$ and $F_{K}(N)$ respectively. We fit for $N\geq 4$, because for
$N=2$ and $N=3$ the system has anomalous behavior for small atom numbers. We
find that $4\Delta\hat{J}^{x}$ is fit with the function $F_{J}(N)\approx
0.366N^{2}+0.793N-2.662$, and $4\Delta\hat{K}^{z}$ is fit with the function
$F_{K}(N)\approx 0.356N^{2}+0.599N+1.466$. Both of these demonstrate the HLS.

Now, we study the behavior of the entanglement between the degrees of freedom,
which has no analog in OAT. We study the entanglement via the fourfold
covariance between the two operators $\hat{J}^{x}$ and $\hat{K}^{z}$,
corresponding to an off-diagonal element of the QFIM. In Fig. 4(a), we see
that the system moves through a highly correlated state, with a high
covariance between the two degrees of freedom, before it approaches an
uncorrelated state for a moment in time at $\chi t=\pi/2$. At an interaction
time of $\chi t=\pi$, the system returns to its original state. In Fig. 4(b),
we see that for interaction times of $\chi t\approx\pi/4$ the correlations
only scale linearly with $N$. Therefore, interaction times which reach this
plateau prepare a system which is capable of quantum sensing for two
parameters at the Heisenberg limit, with relatively little error introduced by
the simultaneous measurement of the two parameters. This motivates the first
half of the next section, where a schematic representation of two parameter
interferometry is shown.

The time at which the system is maximally correlated is labeled
$t_{\text{max}}$, and we find $\chi t_{\text{max}}$ decreases with number of
atoms such that $\chi t_{\text{max}}\approx N^{\nu}$, where $\nu\approx-2/5$
is found from fitting. At this time, the maximum correlation scales
proportionally to $N^{2}$, which is on the order of the squeezing for the two
degrees of freedom.

To achieve an interaction time with these large correlations, one needs $\chi
t\sim N^{-2/5}$. Compared to the single particle emission, one has the
requirement $\chi\gg N^{-2/5}\gamma$, which can be achieved for sufficiently
large $N$. Therefore we expect single-particle decoherence to be negligible on
these timescales. In this regime, we instead expect that collective
decoherence processes, such as collective spontaneous emission mediated by the
cavity, would limit the amount of achievable entanglement. After adiabatic
elimination of the cavity, the collective decoherence rate is due to light
being incoherently scattered into the cavity and lost. This rate can be
estimated as $N\chi\kappa/\Delta\propto Ng^{2}\kappa/\Delta^{2}$. Therefore
one may reduce it by increasing the cavity-atom detuning, $\Delta$, at the
expense of reducing $\chi$. However, interaction times of $\chi t\sim
N^{-2/5}$ may still be possible in cavities with low photon loss rate
$\kappa$. We also note that a similar timescale of $\chi t\sim N^{-2/3}$ is
needed for production of optimally squeezed states in standard OAT [5, 39], so
it could be possible to achieve an interaction time on the order needed to see
strong correlations.

This short timescale with highly correlated degrees of freedom motivates the
second half of our next section, where a schematic representation of single
parameter interferometry is shown. The parameter is estimated via an
interaction with one degree of freedom, and an auxiliary measurement on the
other degree of freedom.

Figure 4:  Plots of $\mathcal{F}^{ij}=4\mathrm{cov}(\hat{J}^{x},\hat{K}^{z})$.
Left - The off diagonal of the QFIM, $\mathcal{F}^{ij}$, normalized by $N^{2}$
for four different values of $N$. We see the covariance between
$\hat{J}^{x},\hat{K}^{z}$ grows rapidly before decaying for longer time
scales, then in a collapse-revival like effect at $\chi t\approx\pi$ the
operators become correlated again before approaching the starting state. Right
- The same off diagonal element of the QFIM at two different times: $\chi
t=\pi/4$ when the correlations are decreasing, and $\chi t=N^{-2/5}$ when the
correlations are largest. We find that $\mathcal{F}^{ij}|_{\chi
t=\pi/4}\approx 4.103\cdot 10^{-3}N^{2}+0.926N$, and $\mathcal{F}^{ij}|_{\chi
t=N^{-2/5}}\approx 0.1782N^{2}-0.02721N$

.

##  V Interferometer Schemes

To demonstrate a possible metrological use, we numerically explore two
interferometry schemes. The first uses the system to detect two relative
phases: one encoded in the atom’s internal degree of freedom, and a second
encoded in the momentum degree of freedom. The second scheme uses this system
to detect a single parameter via auxiliary measurements. The version of the
auxiliary measurement scheme presented here is the case that the collective
internal degree of freedom accumulates phase and the momentum degree of
freedom is measured. However, this process would work similarly if the roles
were reversed.

For both schemes, we choose a new interaction picture for the Hamiltonian such
that $\hat{J}^{z}$ is removed from Eq. (7). This has no effect on the physics
described above, besides keeping the atomic ensemble polarized in
$\hat{J}^{x}$ instead of precessing about $\hat{J}^{z}$. This matches what is
often done in OAT, and the process is shown in more depth in Appendix B.

@C=1em @R=0.7em (a) &
—+⟩^⊗N 1e^-i ^H τ_2 e^-i θ_opt ^J^x e^-i ϕ_3 ^J^z ^J^x
—+1⟩^⊗N e^-i ^H τ_2 e^-i ϕ_5 ^K^y ^K^z

(b)
—+⟩^⊗N 1e^-i ^H τ_1 e^-i ϕ_1 ^J^x 1e^i ^H τ_1
—+1⟩^⊗N e^-i ^H τ_1 e^i ^H τ_1 ^K^z

Figure 5:  A quantum circuit schematic of the two schemes. The two tracks
represent the actions affecting either degree of freedom, with the top track
representing the internal states of the atoms, and the bottom track
representing the momentum. (a) The two parameter scheme. The interaction time
for this two parameter scheme, $\tau_{2}$, is fixed at $\chi\tau_{2}=\pi/4$ to
demonstrate metrological use on the platuea found in Section IV. (b) The
auxiliary measurement scheme. Here, $\chi\tau_{1}=N^{-2/5}$ is chosen such
that the ensembles are maximally correlated. The time-reversed unitary could
be achieved by changing the detuning on the cavity.

We start with the two parameter scheme. The relevant schematic representation
is shown in Fig. 5(a). Here, we first pass the atomic ensemble through the
cavity for an interaction time $\chi\tau_{2}=\pi/4$ to prepare the probe
state. We chose this time to show the metrological use for times near the
plateau, when correlations between the degrees of freedom are decreasing with
respect to interaction time. However, this multiparameter scheme could be used
for any interaction time, albeit with slight differences due to varying
correlation strengths. After the state preparation, a rotation generated by
$\hat{J}^{x}$ is performed so that the maximum fluctuation is in
$\hat{J}^{z}$, where the angle $\theta_{\text{opt}}$ is found numerically. For
the momentum degree of freedom, it was found that the state is already
prepared such that the maximal fluctuations are along $\hat{K}^{y}$ at this
time. The signal is encoded in the system by unitary

$\hat{V}=\exp(-i\phi_{3}\hat{J}^{z}-i\phi_{5}\hat{K}^{y}),$ (9)

where we assume for numerical convenience the phases $\phi_{3},\phi_{5}$ are
small, at $\phi_{3}=\phi_{5}=\pi/16$. However, we found that these results
hold for larger phases as well as for two phases which aren’t equal. After the
unitary, we measure the observables $\hat{J}^{x}$ and $\hat{K}^{z}$ and carry
out phase estimation for both phases simultaneously. To estimate the phase, we
simulate a Bayesian inferencing scheme [21] for two parameters and with a flat
prior, and to find the asymptotic behavior of this Bayesian inference, we
numerically calculate the Classical Fisher Information (CFI) as a function of
atom number. The exact algorithm for sampling and updating a probability
distribution, as well as the explicit form of the CFI are shown in Appendix C.
Using the CFI, we have a useful set of inequalities from the Cramér-Rao Bound
[40] (CRB):

$\sigma_{i}^{2}\geq\frac{1}{MI(\hat{G}^{i})}\geq\frac{1}{M\mathcal{F}^{ii}}$
(10)

where $i=3,5$ corresponds to either $\phi_{3},$ or $\phi_{5}$,
$\sigma_{i}^{2}$ is the variance of the probability distribution, $M$ is the
number of measurements, $I(\hat{G}^{i})$ is the CFI for a parameter encoded by
the operator $\hat{G}^{i}=\hat{J}^{z},\hat{K}^{y}$, and $\mathcal{F}^{ii}$ is
the diagonal element of the QFIM for the corresponding operator. The first
inequality is the classical CRB, and the second inequality is the quantum CRB.
By inverting this bound we find the following: $\mathcal{F}^{ii}\geq
I(\hat{G}^{i})\geq\frac{1}{M\sigma_{i}^{2}}$, so we can tell how close our
resultant probability distribution from Bayesian inferencing is to saturating
the CRB. In Fig. 6, we see the results of this analysis for $M=5000$
measurements. This measurement scheme saturates the classical CRB for both
parameters, and reaches a value of about $80\%$ of the quantum CRB. Moreover,
it does this simultaneously for both parameters.

We also note that, while not shown, as $\phi_{3},\phi_{5}$ tend towards zero
the CFI exactly saturates the quantum CRB, but Bayesian inferencing takes
asymptotically more measurements to saturate the classical CRB. This result
was found numerically, but it can be intuitively explained by the formation of
narrow, ring-like $Q$ functions on the collective Bloch spheres of the
internal and external degrees of freedom. Those rings form along the
$J^{x}$-$J^{z}$ plane and along the $K^{y}$-$K^{z}$ plane which makes them
sensitive to any rotation which results in leaving the corresponding planes.
For rotations of these planes around the $J^{z}$ and $K^{y}$ axes one can then
efficiently read out the applied phase by measuring $J^{x}$ and $K^{z}$,
respectively. With this picture in mind, we would expect the optimal
measurement for any value of $(\phi_{3},\phi_{5})$ is
$(\hat{J}^{x}\cos(\phi_{3})+\hat{J}^{y}\sin(\phi_{3}))\otimes(\hat{K}^{z}\cos(\phi_{5})+\hat{K}^{x}\sin(\phi_{5}))$,
such that the measurement will always be oriented the same relative to plane
this state is in. Numerically we find that this is in fact always saturates
the quantum CRB. However, using this measurement requires knowledge of
$(\phi_{3},\phi_{5})$.

Figure 6:  Left - A plot of the standard deviation corresponding to the final
result of Bayesian inferencing for estimating the phases $\phi_{3}$ and
$\phi_{5}$ with $M=5000$ measurements, and $\phi_{3}=\phi_{5}=\pi/16$. Right -
A plot of the quantities $\frac{1}{M\sigma_{i}^{2}}$ for
$\sigma_{i}=\sigma_{J},\sigma_{K}$, the CFI $I(\hat{G}^{i})$ for
$\hat{G}^{i}=\hat{J}^{z},\hat{K}^{y}$ corresponding to these measurements, and
the diagonal elements of the QFIM for these measurements. Note that because of
the rotation generated by $\hat{J}^{x}$ prior to the interferometry,
$\mathcal{F}^{33}=(\Delta\hat{J}^{z})^{2}$ now scales with HLS. We see that
the quantities $\frac{1}{M\sigma_{i}^{2}}=I(\hat{G}^{i})$ saturate the
classical CRB from the left half of Eq. (10), and nearly saturate the quantum
CRB. By fitting the diagonal QFIM elements and $\frac{1}{M\sigma_{i}^{2}}$ we
find the CFI scales as $I(\hat{J}^{z})\approx 0.3184N^{2}+0.9162N$,
$I(\hat{K}^{y})\approx 0.2022N^{2}+1.454N$, while
$\mathcal{F}^{33}=(\Delta\hat{J}^{z})^{2}\approx 0.3815N^{2}+0.1577N$,
$\mathcal{F}^{55}=(\Delta\hat{K}^{y})^{2}\approx 0.2512N^{2}+0.8727N$. This
indicates this measurement scheme scales at about $80\%$ the theoretical
maximum.

Now, we turn our attention to the auxiliary measurement scheme, shown in Fig
5(b). Here, the atomic ensemble first passes through the cavity for a time of
$\chi\tau_{1}=N^{-2/5}$, so that the observables $\hat{J}^{x}$ and
$\hat{K}^{z}$ are well correlated. Then, the phase is encoded on either the
internal degree of freedom or the momentum. By changing the detuning on the
cavity, the unitary may be reversed and a measurement on the non-interacting
degree of freedom may be used to determine the phase. We simulate this scheme
using a phase encoded on the atomic degree of freedom and a momentum
measurement. To diagnose the metrological use, we consider the fidelity
between the $\ket{+1}^{\otimes N}$ momentum state and the final momentum
state. This is the same as measuring if $\langle\hat{P}_{COM}\rangle$ is equal
to $+N\hbar k/2$ or not. We consider this measurement outcome because for
values of $\phi_{1}$ near zero, a $\hat{K}^{z}$ measurement outcome of $+N/2$
is the most likely outcome, and for $\phi_{1}=0$, it will be the only outcome.
As a result, this fidelity forms an effective probability distribution of
$\phi_{1}$ for just this one measurement outcome of $\hat{K}^{z}$, and groups
together the rest of the possible measurement outcomes. In Appendix D we show
that this effective probability distribution provides a lower bound for the
CFI. The standard deviation of this distribution may be used to calculate a
lower bound for the CFI of this measurement scheme. The standard deviation of
this fidelity is shown in Fig. 7 (a), while the inverted form of the standard
deviation from equation Eq. (10) is compared to the relevant QFIM diagonal
element and shown in Fig. 7 (b). Using the fidelity to represent only one of
the possible measurement outcomes, the uncertainty scales at
$1/\sigma_{Fid}^{2}\approx 0.1699N^{2}$ and from this we see that these
auxiliary measurements allow us to predict the real phase with an uncertainty
that scales with at least $59\%$ of the quantum CRB. This demonstrates that
the auxiliary measurement, while not optimal compared to a direct measurement,
still recovers a large amount of information about the degree of freedom not
being directly measured.

Figure 7:  Left - The standard deviation of the final state fidelity,
$\sigma_{Fid}$ with the $\ket{+1}^{\otimes N}$ momentum state. This is found
by fitting the central peak with a Gaussian and offset. Right - The quantities
$1/\sigma_{Fid}^{2}$ and the QFIM element corresponding to rotations about
$\hat{J}^{x}$. We see that $1/\sigma_{Fid}^{2}\approx 0.1699N^{2}+0.1069N$ and
$\mathcal{F}^{ii}|_{\chi t=N^{-2/5}}\approx 0.2874N^{2}-0.0577N$, showing that
this auxiliary measurement reaches about $0.6$ the quantum CRB.

##  VI Conclusion

In this work, we have introduced a novel method which individually squeezes
and entangles two degrees of freedom, and showed there exists a non-trivial
interplay between the atomic internal and momentum degrees of freedom. We have
demonstrated that these extra degrees of freedom might create the opportunity
for multi-parameter metrology at the Heisenberg limit in either degree of
freedom, or for novel metrology schemes which benefit from the entangled
degrees of freedom. The multiparamter and auxiliary schemes shown in the final
section have potential to be the basis for practical tools in matter wave
interferometry. This form of entanglement generation and manipulation
represents a possible new frontier for atom interferometry.

Future work could include adding decoherence in a numerical exploration [41],
and explorations of the existence of multipartite entanglement[42] that may be
realized by this system. We also note that the physical system explored here
might pose experimental challenges. Namely, the regime requiring
$\Delta\gg\sqrt{N}g$ leads to the parameter $\chi$ being small, thereby
requiring long interaction times which are hard to achieve in atomic beam-
cavity experiments. To explore the effects of the small $\chi$ and long
interaction times compared to the decoherence time, one could simulate this
system with full beam dynamics. It would also be interesting to explore the
use of a moving optical lattice [43] to select the atomic transverse momentum,
and trap the atoms in the cavity longer. We are also interested in the
possibility of using the auxiliary measurement scheme for much shorter
interaction times than shown here, $\chi t\ll 1$, such that the degrees of
freedom only become weakly correlated and measurements on one degree of
freedom only perturbatively affect the other degree of freedom. This could
allow for measurements which only extract a small amount of information, but
don’t destroy the quantum state of the other degree of freedom.

Lastly, we point out that the above discussion is centered on realizing Eq. 3,
however the principles discussed here may be relevant to different platforms.
Specifically, we believe coherently controlling a two-component Bose-Einstein
condensate [44, 45] in order to select for interactions, and engineering an
optical lattice to induce spin-momentum couplings in a Bose-Einstein [46]
might lead to similar Lie algebraic structure and allow for controlled
generation of metrologically useful entanglement. The use of a two component
BEC might have the added benefit of relaxing the condition on small $\chi$
that we have here [45].

## Acknowledgments

The authors thank John Cooper, Liang-Ying Chih, and Christopher Wilson for
stimulating discussions. This research was supported by the NSF PFC Grant No.
1734006 and the NSF Q-SEnSE Grant No. OMA 2016244. M.H. acknowledges support
from a Visiting Fellowship from the University of Pisa, Italy. M. L. C.
acknowledges support from the MIT-UNIPI program.

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##  Appendix A Schwinger Boson Representation

###  A.1 Normalization Coefficient

The symmeterizer in Eq. (LABEL:eq:SchwingState) is defined with the
normalization factor, $1/\mathcal{M}_{\alpha,\beta,\beta,\delta}$, such that

$\mathcal{M}_{q,q_{3},\sigma_{3}}=\sqrt{\frac{N!}{\alpha!\beta!\gamma!\delta!}},$
(11)

so that the bosonic state representation is normalized. In fact, we can see
that $\mathcal{M}_{\alpha,\beta,\beta,\delta}^{2}$ is just a multinomial
coefficient so this normalization makes our bosonic modes match a
straightforward second quantization of the system’s degrees of freedom.

###  A.2 Creation and Annihilation Operators

For posterity, we present the remaining three annihilation operators not shown
in the paper:

$\displaystyle\hat{b}\ket{\alpha,\beta,\gamma,\delta}$
$\displaystyle=\sqrt{\beta}\ket{\alpha,\beta-1,\gamma,\delta},$ (12)
$\displaystyle\hat{c}\ket{\alpha,\beta,\gamma,\delta}$
$\displaystyle=\sqrt{\gamma}\ket{\alpha,\beta,\gamma-1,\delta},$ (13)
$\displaystyle\hat{d}\ket{\alpha,\beta,\gamma,\delta}$
$\displaystyle=\sqrt{\delta}\ket{\alpha,\beta,\gamma,\delta-1}.$ (14)

##  Appendix B Interaction Picture for the Simplified Hamiltonian

Starting with the Hamiltonian Eq. (7), we can choose a different interaction
picture, such that

$\displaystyle\hat{H}-\chi\hat{J}^{z}$
$\displaystyle=\chi(\hat{E}^{2}-(\hat{J}^{z})^{2}+\hat{J}^{z})-\hat{J}^{z})$
$\displaystyle=\chi(\hat{E}^{2}-(\hat{J}^{z})^{2}).$ (15)

This is equivalent to choosing
$\hat{H}_{0}=\sum_{j=1}^{N}\hbar\omega_{a}\hat{\sigma}^{z}_{j}/2+\chi\sum_{j=1}^{N}\hat{\sigma}^{z}_{j}/2+\hbar\omega_{a}\hat{a}_{c}^{\dagger}\hat{a}_{c}$
for our transformation into the interaction picture. This leads to an extra
phase on the Pauli raising operator for the $j^{th}$ atom, so
$\hat{\sigma}^{+}_{j}(t)=e^{(-i\chi t)}\hat{\sigma}^{+}_{j}$ in the
interaction picture. However, this phase cancels after the adiabatic
elimination of the cavity mode. Thus, we may effectively ignore the
$\hbar\hat{J}^{z}$ appearing in the Hamiltonian. Our Hamiltonian is then

$\displaystyle\hat{H}$
$\displaystyle=\chi\sum_{i,j=1}^{N}\hat{s}_{i}^{x}\hat{s}_{j}^{x}\hat{\sigma}_{i}^{+}\hat{\sigma}_{j}^{-}-\chi\sum_{j=1}^{N}\hat{\sigma}^{z}_{j}/2$
(16)
$\displaystyle=\chi(\hat{a}^{\dagger}\hat{b}+\hat{c}^{\dagger}\hat{d})(\hat{a}\hat{b}^{\dagger}+\hat{c}\hat{d}^{\dagger})-\frac{\chi}{2}(\hat{a}^{\dagger}\hat{a}+\hat{c}^{\dagger}\hat{c}-\hat{b}^{\dagger}\hat{b}-\hat{d}^{\dagger}\hat{d})$
$\displaystyle=\chi(\hat{a}^{\dagger}\hat{a}\hat{b}^{\dagger}\hat{b}+\hat{c}^{\dagger}\hat{c}\hat{d}^{\dagger}\hat{d}+\hat{a}^{\dagger}\hat{b}\hat{c}\hat{d}^{\dagger}+\hat{a}\hat{b}^{\dagger}\hat{c}^{\dagger}\hat{d}+\hat{a}^{\dagger}\hat{a}+\hat{c}^{\dagger}\hat{c}-\frac{1}{2}\hat{n}_{e}+\frac{1}{2}\hat{n}_{g})$
$\displaystyle=\chi(\hat{a}^{\dagger}\hat{a}\hat{b}^{\dagger}\hat{b}+\hat{c}^{\dagger}\hat{c}\hat{d}^{\dagger}\hat{d}+\hat{a}^{\dagger}\hat{b}\hat{c}\hat{d}^{\dagger}+\hat{a}\hat{b}^{\dagger}\hat{c}^{\dagger}\hat{d}+\frac{1}{2}\hat{n}_{e}+\frac{1}{2}\hat{n}_{g})$
$\displaystyle=\chi(\hat{E}^{2}-(\hat{J}^{z})^{2}).$

In the last line we have used the fact that
$\frac{1}{2}\hat{n}_{e}+\frac{1}{2}\hat{n}_{g}=N/2$, and dropped this term due
to it only contributing a global phase.

##  Appendix C Bayesian Inferencing Algorithm

In Section V, we use Bayes theorem to carry out Bayesian inferencing. We aim
to construct a probability distribution $P(\vec{\phi}|\vec{\epsilon})$, where
$\vec{\phi}=(\phi_{3},\phi_{5})$ and $\vec{\epsilon}$ is a measurement log
derived from a weighted random sampling of possible measurement outcomes.
Here, we use the fact that
$\hat{J}^{x}=\sum_{i}\lambda^{x}_{i}\hat{\Pi}^{x}_{i}$ and
$\hat{K}^{z}=\sum_{j}\lambda^{z}_{j}\hat{\Pi}^{z}_{j}$, for eigenvalues
$\lambda^{x}_{i},\lambda^{z}_{j}$ and projective operators
$\hat{\Pi}^{x}_{i},\hat{\Pi}^{z}_{j}$. Both sets of projective operators form
a complete positive operator valued measure on the set of states.

We simulate a measurement by choosing an outcome, $\epsilon_{i,j}$,
corresponding to finding eigenvalue $\lambda^{x}_{i}$ for a $\hat{J}^{x}$
measurement and $\lambda^{z}_{j}$ for a $\hat{K}^{z}$ measurement. This
outcome is chosen at random by sampling a list of all possible outcomes with
weights
$P(\epsilon_{i,j})=\bra{\psi}\hat{V}^{\dagger}\hat{\Pi}^{x}_{i}\hat{\Pi}^{z}_{j}\hat{V}\ket{\psi}$,
where $\hat{V}$ is given in Eq. (9) and
$\ket{\psi}=\exp(-it\hat{H})\ket{\psi_{0}}$. Through this process we generate
the measurement log, $\vec{\epsilon}$.

We start with a flat prior distribution, $P(\vec{\phi})=(2\pi)^{-2}$, and
update our probability distribution with each measurement outcome according to

$P_{m+1}(\vec{\phi}|\epsilon_{i,j})=P(\epsilon_{i,j}|\vec{\phi})\frac{P_{m}(\vec{\phi})}{P(\epsilon_{i,j})},$
(17)

where
$P(\epsilon_{i,j}|\vec{\phi})=\bra{\psi}\hat{V}_{\text{est}}^{\dagger}(\vec{\phi})\hat{\Pi}^{x}_{i}\hat{\Pi}^{z}_{j}\hat{V}_{\text{est}}(\vec{\phi})\ket{\psi}$
with $\hat{V}_{\text{est}}(\vec{\phi})$ being a numerical reconstruction of
the unitary, $P(\epsilon_{i,j})$ is the probability of the measurement outcome
integrated over all values of $\phi_{3},\phi_{5}$, $P_{m}(\vec{\phi})$ is the
probability distribution from the first $m$ measurements, and
$P_{m+1}(\vec{\phi}|\vec{\epsilon})$ is the updated probability distribution.

We can predict the asymptotic behavior of Bayesian analysis from the classical
Fisher information (CFI). The CFI can be explicitly calculated:

$I(\hat{G}^{i})=\sum_{i}\left(\frac{d}{d\phi_{i}}\ln(P(\epsilon_{j}|\phi_{i}))\right)^{2}P(\epsilon_{j}|\phi_{i}),$
(18)

where $\epsilon_{j}$ represents the $j$th measurement outcome, and
$P(\epsilon_{j}|\phi_{i})$ is the same probability distribution, but
marginalized over any independent variables besides $\phi_{i}$. For example,
if $i=3$ such that $\hat{G}^{3}=\hat{J}^{z}$, we have that

$P(\epsilon_{j}|\phi_{3})=\Tr_{J}\left[\hat{\Pi}^{x}_{j}\
\Tr_{K}(\hat{V}_{\text{est}}(\vec{\phi})\ket{\psi}\bra{\psi}\hat{V}^{\dagger}(\vec{\phi}))\right],$
(19)

where $\Tr_{J}$ and $\Tr_{K}$ are the traces over the atomic internal degree
of freedom, and momentum degree of freedom respectively. The CFI we consider
is only dependent on a single degree of freedom because we only use it in a
comparison to a diagonal element of the QFIM.

##  Appendix D Fidelity as a Lower Bound of the CRB

The trace of one degree of freedom in this system is very hard to caclulate,
even just numerically, because correlations between the atomic energy level
and momentum states happen on an atom by atom basis–whereas large simulations
are only feasible using the 2nd quantization picture we show in this paper.
This provided challenges for calculating the scaling behavior of the axuiliary
scheme, where one wants to measure one degree of freedom but no the other.
Here we briefly show that the fidelity between an eigenstate of an observable
and the state one wishes to measure serves as a suitable lower bound on the
actual set of measurements, without needing to take the trace of a degree of
freedom.

One may analytically calculate the CFI with respect to a $\hat{J}^{x}$
rotation and a full $\hat{K}^{z}$ measurement as follows,

$I(\hat{J}^{x})=\sum_{m=-N/2}^{+N/2}p_{j}\left(\frac{\partial}{\partial\phi_{1}}\log(p_{j})\right)^{2}$
(20)

where $p_{j}$ represents the probability of the $j^{\text{th}}$ measurement
outcome of $\hat{K}^{z}$, for example
$p_{N/2}=\bra{+1}^{N}\text{Tr}_{J}(\ket{\psi}\bra{\psi})\ket{+1}^{N}$, where
$\text{Tr}_{J}$ means we first trace over the atomic degrees of freedom. We
also have that
$p_{j}\left(\frac{\partial}{\partial\phi_{1}}\log(p_{j})\right)^{2}\geq 0$ for
all outcomes $j$, so we can observe that

$p_{N/2}\left(\frac{\partial}{\partial\phi_{1}}\log(p_{N/2})\right)^{2}+p_{j\neq
N/2}\left(\frac{\partial}{\partial\phi_{1}}\log(p_{\neq
N/2})\right)^{2}\leq\sum_{m=-N/2}^{+N/2}p_{j}\left(\frac{\partial}{\partial\phi_{1}}\log(p_{j})\right)^{2},$
(21)

where $p_{j\neq N/2}=1-p_{N/2}$ is the probability of not measuring
$\hat{K}^{z}=+N/2$. These two probabilities, $p_{N/2}$ and $p_{j\neq N/2}$ can
be calculated without the use of a trace via the fidelity. This is because one
may observe that under the time evolution of the Hamiltonian, the only atomic
states entangled to $\ket{+1}^{N}$ momentum states are the ones in the initial
atomic configuration, $\ket{+}^{N}$. Otherwise, momentum flips occur in pairs
from the $\hat{s}^{x}_{i}\hat{s}^{x}_{j}$ term in the Hamiltonian. Therefore,
the CFI of this single probability distribution, $p_{N/2}$, serves as a lower
bound for the CFI by construction, because this would be the same as measuring
if $\hat{K}^{z}$ is $+N/2$ or not.


# Marginal Treatment Effects and Monotonicity

Henrik Sigstad BI Norwegian Business School, Department of Economics (e-mail:
henrik.sigstad@bi.no). Thanks to Magne Mogstad and Vitor Possebom.

###### Abstract

How robust are analyses based on marginal treatment effects (MTE) to
violations of Imbens & Angrist,  (1994) monotonicity? In this note, I present
weaker forms of monotonicity under which popular MTE-based estimands still
identify the parameters of interest.

##  1 Introduction

Marginal treatment effects (MTE), introduced by Björklund & Moffitt,  (1987)
and generalized by Heckman & Vytlacil,  (1999, 2005), provide a unified way of
estimating various treatment effects with continuous instruments. For
instance, MTE analysis can be used to identify the average treatment effect,
the average treatment effect on the treated and the untreated, and other
policy-relevant treatment effects. In contrast, with a continuous instrument,
two-stage least squares identifies a convex combination of treatment effects
that is not necessarily of policy interest (Heckman & Vytlacil, , 2007b). MTE
analysis, however, relies on Imbens & Angrist,  (1994) monotonicity—often a
strong assumption. For instance, in the context of judge IV designs, Imbens &
Angrist,  (1994) monotonicity requires that each judge is weakly stricter than
more lenient judges _in each case_. Thus, if Judge A is stricter than Judge B
in one case, Judge A can not be more lenient than Judge B in another case. As
shown in Sigstad,  (2024), this assumption is frequently violated among
judges.

It is thus important to understand how MTE-based treatment effect estimates
are affected by monotonicity violations. In this note, I derive necessary and
sufficient monotonicity conditions for MTE-based estimates of popular
treatment effects to identify the parameters of interest. Fortunately, it
turns out that even when Imbens-Angrist monotonicity is violated, MTE-based
estimates of these parameters might still be consistent. I first consider MTE-
based estimates of LATE—the average treatment effect for agents affected by
the instrument. The necessary and sufficient condition for MTE analysis to
identify LATE is that monotonicity holds between the two most extreme
instrument values. For instance, in the random-judge design, this condition
requires that the strictest judge is always stricter than the most lenient
judge. As shown in Sigstad,  (2024), this condition is much more plausible in
random-judge designs than Imbens & Angrist,  (1994) monotonicity. Thus, even
though conventional MTE analysis assumes Imbens-Angrist monotonicity, MTE-
based LATE estimates can be highly robust to plausible levels of monotonicity
violations.

Next, I consider estimates of the average treatment effect on the treated
(ATT) and the untreated (ATUT) for the complier population. MTE-based ATT
(ATUT) estimates are consistent as long as Imbens-Angirst monotonicity holds
for all pairs of instrument values involving the lowest (highest) instrument
value. For instance, in the random-judge design, these conditions require that
the most lenient (stringent) judge is most lenient (stringent) in all cases.
These conditions are more demanding than the condition required to estimate
LATE. Estimates of ATT and ATUT are thus more sensitive to monotonicity
violations.

I also consider MTE-based estimates of the average treatment effect (ATE),
which require extrapolation beyond the observed instrument values. As long as
this extrapolation is well specified, MTE-based ATE estimates are consistent
without any monotonicity assumption. Such estimates are equivalent to the
Arnold et al.,  (2021) approach to estimating average treatment effects.
Finally, I consider the use of MTEs to assess heterogeneous treatment effects
by treatment propensity. As long as attention is limited to aggregate
properties of the MTE curve, this practice also requires only mild
monotonicity assumptions.

While these analyses show that MTE-based estimators are relatively robust to
monotonicity violations, the intermediate step of estimating marginal
treatment effects is not a meaningful exercise when monotonicity is violated.
Instead, I propose to directly estimate the relevant treatment parameters
without first estimating an “MTE curve”. I show that whenever MTE analysis
identifies LATE, LATE is identified by a standard Wald estimand: the
difference in average outcomes between agents receiving the highest instrument
value and agents receiving the lowest instrument value divided by the
difference in treatment propensities. Similar results are obtained for the
average treatment effects on the treated and on the untreated for the complier
population. There are several reasons to prefer such a direct estimation of
treatment parameters over MTE-based estimation when monotonicity is violated.
First, the direct estimation is more honest and clarifies the necessary
identification assumptions. Second, the direct estimates can easily be
estimated non-parametrically and do not require a fully continuous instrument.
Finally, by targeting a specific parameter rather than the full MTE curve, the
parameter can be more precisely estimated.

##  2 Marginal Treatment Effects and Monotonicity

Fix a probability space with the outcome corresponding to a randomly drawn
agent. Define the following random variables: A binary treatment
$D\in\left\\{0,1\right\\}$, an outcome $Y\in\mathbb{R}$, and a continuous
instrument $Z\in\mathbb{R}$ with support $\left[\underline{z},\bar{z}\right]$.
To capture the idea that different agents might be induced into treatment in
different ways by the instrument, define a _response type_ as a mapping
$s:\left[\underline{z},\bar{z}\right]\rightarrow\left\\{0,1\right\\}$ from
instrument values to treatments.111Response types were introduced by Heckman &
Pinto,  (2018). Denote by the random variable $S$ the response type of the
randomly drawn agent. If $S=s$ for agent $i$, then $s\left(z\right)=1$
indicates that agent $i$ will receive the treatment if $Z$ is set to $z$.
Denote by $\mathcal{S}$ the set of all response types in the population.
Define $Y\left(0\right)$ and $Y\left(1\right)$ as the _potential outcomes_
when $D$ is set to $0$ and $1$, respectively. Denote by the random variable
$\beta\equiv Y\left(1\right)-Y\left(0\right)$ the treatment effect of agent
$i$. Let $p\left(z\right)\equiv\operatorname{E}\left[S\left(z\right)\right]$
be the share of agents receiving treatment at $Z=z$. I assume the following
throughout

######  Assumption 1.

(Exogeneity and Exclusion).
$\left\\{Y\left(0\right),Y\left(1\right),S\right\\}\perp Z$

######  Assumption 2.

(First stage). The propensity $p\left(z\right)$ is non-trivial function of
$z$.

To simplify the notation, assume (without loss of generality) that the
instrument values are labeled such that

######  Assumption 3.

$p\left(z\right)=z$.

Imbens & Angrist,  (1994) monotonicity is then defined by

######  Definition 1 (Imbens-Angrist Monotonicity).

For all $z,z^{\prime}\in\mathbb{R}$ and $s\in\mathcal{S}$

$z\geq z^{\prime}\Rightarrow s\left(z\right)\geq s\left(z^{\prime}\right)$

Marginal treatment effects were introduced by Björklund & Moffitt,  (1987) and
generalized by Heckman & Vytlacil,  (1999).222See Heckman & Vytlacil,
(2007b). In applied work (_e.g._ , Arnold et al.,  2018; Bhuller et al.,
2020), marginal treatment effect analysis relies on a generalized Roy,  (1951)
selection model

$D=\mathbf{1}\left[Z>U\right]$

where $U$ is a random variable.333See Heckman & Vytlacil,  (2007a, b). To
simplify the exposition, I disregard covariates. The agent receives treatment
if the instrument is above the agent’s unobserved _resistance to treatment_
$U$. This model implicitly assumes Imbens-Angrist monotonicity.444Consider two
instrument values $z_{1}\geq z_{2}$. Then $D\left(z_{2}\right)\geq
D\left(z_{1}\right)$ for all agents. A _marginal treatment effect_ is then
defined as the average treatment effect for agents with a given treatment
propensity:

$\operatorname{MTE}\left(u\right)=\operatorname{E}\left[\beta\mid U=u\right]$

Marginal treatment effects can then be identified using the local instrumental
variable approach (Heckman & Vytlacil, , 1999, 2005):

$\operatorname{LIV}\left(u\right)\equiv\frac{d\operatorname{E}\left[Y\mid
Z=u\right]}{du}$

Assume this derivative exists. Under Imbens-Angrist monotonicity, we have
$\operatorname{LIV}\left(u\right)=\operatorname{MTE}\left(u\right)$. But
$\operatorname{LIV}\left(u\right)$ is defined even when Imbens-Angrist
monotonicity does not hold.

The applied literature uses MTE analysis for two purposes. First, to estimate
meaningful treatment parameters such as LATE and ATE (_e.g._ , Arnold et al.,
2018; Bhuller et al.,  2020). Second, to assess heterogeneous treatment
effects based on the treatment propensity $U$ by directly inspecting
$\operatorname{LIV}\left(u\right)$ (_e.g._ , Doyle Jr,  2007; Maestas et al.,
2013; French & Song,  2014).

###  2.1 Using MTE to Identify Meaningful Treatment Parameters

Heckman & Vytlacil,  (1999, 2005) showed that many popular treatment
parameters—including the average treatment effect (ATE)—can be identified by a
weighted average of marginal treatment effects. MTE analysis can thus be used
to identify more meaningful treatment parameters than the weighted average of
individual treatment effects produced by 2SLS. Identifying ATEs using MTE
analysis, however, requires full support of $Z$ in $\left[0,1\right]$ or
extrapolation beyond the support of $Z$. Since $Z$ typically does not have
full support in practice, the literature instead normally seeks to estimate
the local average treatment effect (LATE) for agents receiving treatment when
$Z=\bar{z}$ and not when $Z=\underline{z}$—agents with
$D\left(\underline{z}\right)<D\left(\bar{z}\right)$. This parameter differs
from the 2SLS estimand by placing equal weight on all compliers. The
literature (_e.g._ , Bhuller et al.,  2020) has also considered the average
treatment effect on the treated and the average treatment effect on the
untreated for the same population. Since these treatment effects are
“local”—defined on the complier population—I label them LATT and LATUT,
respectively. Formally, define

$\displaystyle\operatorname{LATE}$ $\displaystyle\equiv$
$\displaystyle\operatorname{E}\left[\beta\mid
S\left(\underline{z}\right)=0,S\left(\bar{z}\right)=1\right]$
$\displaystyle\operatorname{LATT}$ $\displaystyle\equiv$
$\displaystyle\operatorname{E}\left[\beta\mid
S\left(\underline{z}\right)=0,D=1\right]$ $\displaystyle\operatorname{LATUT}$
$\displaystyle\equiv$ $\displaystyle\operatorname{E}\left[\beta\mid
S\left(\bar{z}\right)=1,D=0\right]$

The $\operatorname{LATE}$ parameter is the local average treatment effect for
agents receiving treatment under the highest instrument value but not under
the lowest instrument value. The $\operatorname{LATT}$ and
$\operatorname{LATUT}$ parameters are the (local) average treatment effects on
the treated and the untreated for a similar complier population.555The
$\operatorname{LATT}$ and $\operatorname{LATUT}$ complier population includes
all agents except never-takers and always-takers. The $\operatorname{LATE}$
complier population also ignores, for instance, response types receiving
treatment under some intermediate instrument values but not by the highest nor
the lowest instrument value. I do not see a way to identify a local average
treatment effect that covers also such compliers. As shown by Heckman &
Vytlacil,  (1999, 2005), these parameters are identified under Imbens-Angrist
monotonicity by:

$\displaystyle\tilde{\operatorname{LATE}}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{\bar{z}-\underline{z}}\int_{\underline{z}}^{\bar{z}}\operatorname{LIV}\left(u\right)du$
$\displaystyle\tilde{\operatorname{LATT}}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{\operatorname{E}\left[Z\right]-\underline{z}}\int_{\underline{z}}^{\bar{z}}\Pr\left[Z>u\right]\operatorname{LIV}\left(u\right)du$
$\displaystyle\tilde{\operatorname{LATUT}}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{\bar{z}-\operatorname{E}\left[Z\right]}\int_{\underline{z}}^{\bar{z}}\Pr\left[Z<u\right]\operatorname{LIV}\left(u\right)du$

When Imbens-Angrist monotonicity is violated, however, this method might lead
to wrong conclusions. How sensitive are these estimands to monotonicity
violations? Fortunately, it turns out that MTE analysis might still identify
$\operatorname{LATE}$, $\operatorname{LATT}$, and $\operatorname{LATUT}$ even
when Imbens-Angrist monotonicity is violated.

Formally, let $\mathcal{G}$ be the set of all possible joint distributions of
$\left(Y\left(1\right),Y\left(0\right),S\right)$. To allow for arbitrary
heterogeneous effects, we do not want to place any restrictions on this joint
distribution. The necessary and sufficient conditions for
$\tilde{\operatorname{LATE}}$, $\tilde{\operatorname{LATT}}$,
$\tilde{\operatorname{LATUT}}$ to be consistent under arbitrary heterogeneous
effects are the following much weaker monotonicity conditions:

######  Theorem 1.

(Identification results).

i)

$\tilde{\operatorname{LATE}}=\operatorname{LATE}$ for all $g\in\mathcal{G}$ if
and only if $s\left(\bar{z}\right)\geq s\left(\underline{z}\right)$ for all
$s\in\mathcal{S}$.

ii)

$\tilde{\operatorname{LATT}}=\operatorname{LATT}$ for all $g\in\mathcal{G}$ if
and only if $s\left(z\right)\geq s\left(\underline{z}\right)$ for all
$s\in\mathcal{S}$ and $z$.

iii)

$\tilde{\operatorname{LATUT}}=\operatorname{LATUT}$ for all $g\in\mathcal{G}$
if and only if $s\left(\bar{z}\right)\geq s\left(z\right)$ for all
$s\in\mathcal{S}$ and $z$.

Thus, for MTE analysis to identify $\operatorname{LATE}$, it is sufficient
that monotonicity holds between the lowest and the highest instrument value.
This condition is substantially weaker than Imbens-Angrist monotonicity,
especially when there are many possible instrument values. Similarly,
$\operatorname{LATT}$ is identified by MTE analysis whenever monotonicity
holds for all instrument value pairs that involve the lowest instrument value.
This condition is stronger than $\operatorname{LATE}$ condition but still
considerably weaker than Imbens-Angrist monotonicity—monotonicity is allowed
to be violated for all instrument value pairs that do not include the lowest
instrument value. For MTE analysis to identify $\operatorname{LATUT}$, on the
other hand, monotonicity must hold for all instrument value pairs that involve
the _highest_ instrument value.

###  2.2 Estimating ATE by Extrapolating the MTE curve

When $f\left(u\right)\equiv\operatorname{E}\left[Y\mid Z=u\right]$ is
estimated parametrically, one might seek to extrapolate beyond the support of
$Z$ to estimate the average treatment effect,
$\operatorname{ATE}\equiv\operatorname{E}\left[\beta\right]$. In particular,
let $\hat{f}:\left[0,1\right]\rightarrow\mathbb{R}$ be an extrapolation of $f$
that covers the full interval $\left[0,1\right]$. The corresponding MTE curve
is defined by
$\hat{\operatorname{LIV}}\left(u\right)\equiv\hat{f}^{\prime}\left(u\right)$.
One can then estimate $\operatorname{ATE}$ by

$\tilde{\operatorname{ATE}}\equiv\int_{0}^{1}\hat{\operatorname{LIV}}\left(u\right)du$

How do monotonicity violations influence such analysis? By the fundamental
theorem of calculus,
$\tilde{\operatorname{ATE}}=\hat{f}\left(1\right)-\hat{f}\left(0\right)$. If
the extrapolation is well specified, $\hat{f}\left(1\right)$ can be thought of
as the average outcome for agents in the hypothetical case of receiving
$Z=1$.666In the context of the judge IV design, it would correspond to the
average outcomes for defendants randomly assigned a hypothetical supremely
stringent judge that always incarcerates. In that case,
$\hat{f}\left(1\right)=\operatorname{E}\left[Y\left(1\right)\right]$.
Similarly, $\hat{f}\left(0\right)$ can be thought of as the average outcome
for agents had they been assigned $Z=0$ which gives
$\hat{f}\left(0\right)=\operatorname{E}\left[Y\left(0\right)\right]$. We thus
get that
$\tilde{\operatorname{ATE}}=\operatorname{E}\left[Y\left(1\right)-Y\left(0\right)\right]=\operatorname{ATE}$
if the extrapolation is well specified—$\hat{f}$ is able to identify the
average outcome for agents in the hypothetical cases of receiving $Z=1$ and
$Z=0$. Formally

######  Proposition 1.

$\tilde{\operatorname{ATE}}=\operatorname{ATE}$ if
$\hat{f}\left(1\right)=\operatorname{E}\left[Y\left(1\right)\right]$ and
$\hat{f}\left(0\right)=\operatorname{E}\left[Y\left(0\right)\right]$.

The MTE-based estimator of $\operatorname{ATE}$ is equivalent to the estimator
proposed by Arnold et al.,  (2021), who extrapolate towards a supremely
lenient judge to estimate the ATE of pre-trial release on pre-trial misconduct
in a judge IV setting. As pointed out by Arnold et al.,  (2021), this approach
does not require any monotonicity assumptions. Thus, monotonicity violations
do not affect the validity of this approach.

###  2.3 Using MTE to Analyze Heterogeneous Effects

The literature also uses the MTE framework to assess heterogeneous treatment
effects based on the treatment propensity $U$ by directly inspecting
$\operatorname{LIV}\left(u\right)$—the “MTE curve” (_e.g._ , Doyle Jr,  2007;
Maestas et al.,  2013; French & Song,  2014). But
$\operatorname{LIV}\left(u\right)$ is difficult to interpret when Imbens-
Angrist monotonicity is violated. To see this, it is instructive to consider
$\operatorname{LIV}\left(u\right)$ as the limit of a standard Wald estimand:

$\operatorname{LIV}\left(u\right)=\lim_{v\rightarrow
u}\tilde{\operatorname{LATE}}_{u,v}$

where

$\tilde{\operatorname{LATE}}_{u,v}\equiv\frac{\operatorname{E}\left[Y\mid
Z=u\right]-\operatorname{E}\left[Y\mid Z=v\right]}{u-v}$

Under Imbens-Angrist monotonicity, $\tilde{\operatorname{LATE}}_{u,v}$
identifies

$\operatorname{LATE}_{u,v}\equiv\operatorname{E}\left[\beta\mid
D\left(u\right)>D\left(v\right)\right]$

the local average treatment effect for cases where receiving treatment at
$Z=u$ but not at $Z=v$. As $v$ approaches $u$, however, Imbens-Angrist
monotonicity between $v$ and $u$ might be unlikely.777In the context of
judges, Imbens-Angrist monotonicity is less likely for judge pairs with
similar stringencies than for judge pairs with more different stringencies
(Sigstad, , 2024).  Individual points of the MTE curve are then hard to
interpret. But looking at more aggregate properties of the MTE curve could
still be meaningful. For instance, the average of
$\operatorname{LIV}\left(u\right)$ across a range
$u\in\left[\underline{u},\bar{u}\right]$ identifies LATE for agents receiving
treatment at $Z=\bar{u}$ but not at $Z=\underline{u}$ when monotonicity holds
between these instrument values:

######  Proposition 2.

$\operatorname{E}\left[\operatorname{LIV}\left(U\right)\mid\underline{u}\leq
U\leq\bar{u}\right]=\operatorname{LATE}_{\underline{u},\bar{u}}$

for all $g\in\mathcal{G}$ if and only if $s\left(\bar{u}\right)\geq
s\left(\underline{u}\right)$ for all $s\in\mathcal{S}$.

As $\bar{u}$ and $\underline{u}$ become more distant, monotonicity between
these two values typically becomes more plausible.888This is at least true for
the random-judge design (Sigstad, , 2024).

###  2.4 Identifying Meaningful Treatment Effects without MTE

While MTE analysis gives correct results under weaker assumptions than Imbens-
Angrist monotonicity, the “MTE curve” $\operatorname{LIV}\left(u\right)$ is
not a meaningful object when monotonicity is violated. A more honest approach
is to estimate aggregate treatment effects directly, without first estimating
an MTE curve. The following results show how LATE, LATT, and LATUT can be
directly identified without first estimating
$\operatorname{LIV}\left(u\right)$.

######  Theorem 2.

(Identifying meaningful treatment effects without MTE analysis).

i)

$\operatorname{LATE}=\frac{\operatorname{E}\left[Y\mid
Z=\bar{z}\right]-\operatorname{E}\left[Y\mid
Z=\underline{z}\right]}{\bar{z}-\underline{z}}$ if $s\left(\bar{z}\right)\geq
s\left(\underline{z}\right)$ for all $s\in\mathcal{S}$.

ii)

$\operatorname{LATT}=\frac{\operatorname{E}\left[Y\right]-\operatorname{E}\left[Y\mid
Z=\underline{z}\right]}{\operatorname{E}\left[Z\right]-\underline{z}}$ if
$s\left(z\right)\geq s\left(\underline{z}\right)$ for all $s\in\mathcal{S}$
and $z$.

iii)

$\operatorname{LATUT}=\frac{\operatorname{E}\left[Y\mid
Z=\bar{z}\right]-\operatorname{E}\left[Y\right]}{\bar{z}-\operatorname{E}\left[Z\right]}$
if $s\left(\bar{z}\right)\geq s\left(z\right)$ for all $s\in\mathcal{S}$ and
$z$.

iv)

$\operatorname{LATE}_{z_{1},z_{2}}=\frac{\operatorname{E}\left[Y\mid
Z=z_{1}\right]-\operatorname{E}\left[Y\mid Z=z_{2}\right]}{z_{1}-z_{2}}$ if
$s\left(z_{1}\right)\geq s\left(z_{2}\right)$ for all $s\in\mathcal{S}$.

In particular, $\operatorname{LATE}$ is identified by the standard Wald
estimand of the effect of receiving the highest instrument value compared to
receiving the lowest instrument value.999A similar estimand is discussed by
Frölich,  (2007) (Theorem 8). Furthermore, $\operatorname{LATT}$ and
$\operatorname{LATUT}$ are identified by the difference between the mean
outcome and the expected outcomes for agents receiving the lowest and highest
instrument values, respectively. The only parameters that need to be estimated
are thus $\bar{z}$, $\underline{z}$, $\operatorname{E}\left[Y\mid
Z=\underline{z}\right]$ and $\operatorname{E}\left[Y\mid Z=\bar{z}\right]$.
There are two advantages of this approach. First, it is not needed to estimate
a full MTE curve. Estimating an MTE curve is difficult in practice due to data
limitations and typically requires parametric assumptions. When the aim is to
estimate only $\operatorname{E}\left[Y\mid Z=\underline{z}\right]$ and
$\operatorname{E}\left[Y\mid Z=\bar{z}\right]$ instead of the full MTE curve,
one can do this non-parametrically.101010For instance, one can directly
estimate $\operatorname{E}\left[Y\mid Z=\bar{z}\right]$ using the sample
analog, or one can estimate it using a local linear regression (with, _e.g._ ,
a triangular kernel) on a sample of the highest instrument values. Note that
such LATE estimates (obtained either through MTE analysis or the Wald
approach) essentially ignore agents receiving medium instrument values. These
estimates will thus typically be less precise than 2SLS estimates which
exploit all instrument values. Also, note that in finite samples, the sample
analog of $\bar{z}-\underline{z}$ will be upward biased. For instance, even if
all instrument values have the same treatment propensity
($\bar{z}=\underline{z}$), the sample analog of $\bar{z}-\underline{z}$ will
still be positive. To avoid this bias, one could use a split-sample approach:
Estimate which instrument values are associated with the highest and lowest
treatment propensities in one sample and estimate the treatment propensities
associated with these instrument values in another sample. I leave a thorough
discussion of inference to future research. Second, the results above are
valid also for discrete instruments when MTE analysis is not
applicable.111111For instance, applying MTE-analysis in the judge IV setting
formally requires a continuum of judges.

##  3 Conclusions

Marginal treatment effects can be used to estimate more meaningful treatment
parameters than two-stage least squares but require Imbens-Angrist
monotonicity. In this note, I have presented conditions under which MTE-based
estimates still identify the parameters of interest when Imbens-Angrist
monotonicity is violated. I also showed how the same parameters can be
identified without relying on the MTE framework. I leave questions of
estimation for future work.

## References

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##  Appendix A Proofs

###### Proof.

(Theorem 1.) _Part i)._

$\displaystyle\tilde{\operatorname{LATE}}$ $\displaystyle=$
$\displaystyle\frac{1}{\bar{z}-\underline{z}}\int_{\underline{z}}^{\bar{z}}\frac{d\operatorname{E}\left[Y\mid
Z=u\right]}{du}du$ $\displaystyle=$
$\displaystyle\frac{1}{\bar{z}-\underline{z}}\left(\operatorname{E}\left[Y\mid
Z=\bar{z}\right]-\operatorname{E}\left[Y\mid Z=\underline{z}\right]\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\bar{z}-\underline{z}}\left(\operatorname{E}\left[DY\left(1\right)+\left(1-D\right)Y\left(0\right)\mid
Z=\bar{z}\right]-\operatorname{E}\left[DY\left(1\right)-\left(1-D\right)Y\left(0\right)\mid
Z=\underline{z}\right]\right)$ $\displaystyle=$
$\displaystyle\frac{\Pr\left[D\left(\bar{z}\right)>D\left(\underline{z}\right)\right]}{\bar{z}-\underline{z}}\operatorname{E}\left[Y\left(1\right)-Y\left(0\right)\mid
D\left(\bar{z}\right)>D\left(\underline{z}\right)\right]$ $\displaystyle-$
$\displaystyle\frac{\Pr\left[D\left(\bar{z}\right)<D\left(\underline{z}\right)\right]}{\bar{z}-\underline{z}}\operatorname{E}\left[Y\left(1\right)-Y\left(0\right)\mid
D\left(\bar{z}\right)<D\left(\underline{z}\right)\right]$

The first equality invokes the fundamental theorem of calculus and the fourth
equality uses Assumption 1. This expression equals $\operatorname{LATE}$ for
all $g\in\mathcal{G}$ if and only if
$\Pr\left[D\left(\bar{z}\right)<D\left(\underline{z}\right)\right]=0$.

_Part ii)._ _Let $f\left(u\right)$ denote the density of $Z$ at $u$. Then_

$\displaystyle\tilde{\operatorname{LATT}}$ $\displaystyle=$
$\displaystyle\frac{1}{\operatorname{E}\left[Z\right]-\underline{z}}\int_{\underline{z}}^{\bar{z}}\Pr\left[Z>u\right]\frac{d\operatorname{E}\left[Y\mid
Z=u\right]}{du}du$ $\displaystyle=$
$\displaystyle\frac{1}{\operatorname{E}\left[Z\right]-\underline{z}}\int_{\underline{z}}^{\bar{z}}\left(\operatorname{E}\left[Y\mid
Z=u\right]-\operatorname{E}\left[Y\mid
Z=\underline{z}\right]\right)f\left(u\right)du$ $\displaystyle=$
$\displaystyle\frac{1}{\operatorname{E}\left[Z\right]-\underline{z}}\left(\operatorname{E}\left[Y\right]-\operatorname{E}\left[Y\mid
Z=\underline{z}\right]\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\operatorname{E}\left[Z\right]-\underline{z}}\operatorname{E}\left[\operatorname{E}\left[Y\mid
S\right]-\operatorname{E}\left[Y\mid Z=\underline{z},S\right]\right]$
$\displaystyle=$
$\displaystyle\frac{1}{\operatorname{E}\left[Z\right]-\underline{z}}\operatorname{E}\left[\operatorname{E}\left[DY\left(1\right)+\left(1-D\right)Y\left(0\right)\mid
S\right]-\operatorname{E}\left[Y\mid Z=\underline{z},S\right]\right]$
$\displaystyle=$
$\displaystyle\frac{1}{\operatorname{E}\left[Z\right]-\underline{z}}\operatorname{E}\left[\operatorname{E}\left[D\mid
S\right]\operatorname{E}\left[Y\left(1\right)\mid
S\right]+\left(1-\operatorname{E}\left[D\mid
S\right]\right)\operatorname{E}\left[Y\left(0\right)\mid
S\right]-\operatorname{E}\left[Y\mid Z=\underline{z},S\right]\right]$
$\displaystyle=$
$\displaystyle\frac{1}{\operatorname{E}\left[Z\right]-\underline{z}}\operatorname{E}\left[\operatorname{E}\left[D\mid
S\right]\operatorname{E}\left[Y\left(1\right)-Y\left(0\right)\mid
S\right]+\operatorname{E}\left[Y\left(0\right)\mid
S\right]-\operatorname{E}\left[Y\mid Z=\underline{z},S\right]\right]$

The second equality uses that both integrals represent the area between the
curve $\operatorname{E}\left[Y\mid Z=u\right]$ and
$\operatorname{E}\left[Y\mid Z=\underline{z}\right]$ (weighted by the density
$f$). The fourth equality uses the law of iterated expectations, and the sixth
equality invokes Assumption 1. For this to equal

$\displaystyle\operatorname{LATT}$ $\displaystyle=$
$\displaystyle\operatorname{E}\left[Y\left(1\right)-Y\left(0\right)\mid
D\left(\underline{z}\right)<D\left(\bar{z}\right),D=1\right]$ $\displaystyle=$
$\displaystyle\frac{1}{\operatorname{E}\left[D\mid
D\left(\underline{z}\right)<D\left(\bar{z}\right)\right]}\operatorname{E}\left[D\left(Y\left(1\right)-Y\left(0\right)\right)\mid
D\left(\underline{z}\right)<D\left(\bar{z}\right)\right]$

for all $g\in\mathcal{G}$, we need that for each $s\in\mathcal{S}$, either
$s\left(\underline{z}\right)=0$ or $\operatorname{E}\left[D\mid
S=s\right]=1$.121212If $\operatorname{E}\left[D\mid S=s\right]<1$ and
$s\left(\underline{z}\right)=1$ for a response type $s\in\mathcal{S}$,
$\tilde{\operatorname{LATT}}$ will—unlike $\operatorname{LATT}$—place a
negative weight on $\operatorname{E}\left[Y\left(1\right)-Y\left(0\right)\mid
S=s\right]$. It is straightforward to check that the expressions for
$\tilde{\operatorname{LATT}}$ and $\operatorname{LATT}$ coincide when either
$s\left(\underline{z}\right)=0$ or $\operatorname{E}\left[D\mid S=s\right]=1$
for all $s\in\mathcal{S}$. In other words, we need $D\left(z\right)\geq
D\left(\underline{z}\right)$ for all $z$. The proof of part iii) is analogous
to part ii). ∎

###### Proof.

(Theorem 2.) This follows from the proof of Theorem 1. ∎

###### Proof.

(Proposition 2.)

$\displaystyle\operatorname{E}\left[\operatorname{LIV}\left(U\right)\mid\underline{u}\leq
U\leq\bar{u}\right]$ $\displaystyle=$
$\displaystyle\frac{1}{\bar{u}-\underline{u}}\int_{\underline{u}}^{\bar{u}}\operatorname{LIV}\left(u\right)du$
$\displaystyle=$ $\displaystyle\frac{\operatorname{E}\left[Y\mid
Z=\bar{u}\right]-\operatorname{E}\left[Y\mid
Z=\underline{u}\right]}{\bar{u}-\underline{u}}$

The latter Wald estimand identifies
$\operatorname{LATE}_{\underline{u},\bar{u}}$ for all $g\in\mathcal{G}$ if and
only if there are no “defiers”,
$\Pr\left[D\left(\underline{u}\right)>D\left(\bar{u}\right)\right]=0$. ∎