# Reduced cross sections of electron and neutrino charged current quasielastic
scattering on nuclei

A. V. Butkevich  Institute for Nuclear Research, Russian Academy of Sciences,
Moscow 117312, Russia

###### Abstract

The semi-exclusive averaged reduced cross sections for (anti)neutrino charged
current quasi-elastic scattering on carbon, oxygen, and argon are analyzed
within the relativistic distorted wave impulse approximation. We found that
these cross sections as functions of missing nucleon energy are similar to
those of electron scattering and are in agreement with electron scattering
data for three nuclei. The difference between the electron and neutrino cross
sections can be attributed to Coulomb distortion on the electron wave
function. The averaged reduced cross sections depend slowly upon incoming
lepton energy. The approach presented in this paper provide novel constraints
on nuclear models of quasi-elastic neutrino-nucleus scattering and can be
easily applied to test spectral functions and final state interactions,
employed in neutrino event generators.

###### pacs:

25.30.-c, 25.30.Bf, 25.30.Pt, 13.15.+g

##  I Introduction

For current NOvA1 ; T2K  and future DUNE ; HK2T ; SBN  accelerator-based
neutrino experiments the primary physics goals are measuring the lepton CP
violation phase, determing neutrino mass ordering and testing the three flavor
paradigm. In these experiments to evaluate the oscillation parameters, the
probabilities of neutrino oscillations as functions of neutrino energy are
measured. The neutrino beams are not monoenergetic and have broad
distributions that range from tens of MeVs to a few GeVs. The accuracy to
which neutrino oscillation parameters can be extracted depends on the ability
of experiments to determine the individual energy of detected neutrino.

Measurements at neutrino energy 1 GeV are critical for the T2K T2K  and HK
HK2T  programs, which are carbon and water (oxygen) detectors as well as for
the SBN (argon) SBN  program. Measurements from 1 to 2 GeV are important for
the NOvA (carbon, chlorine) NOvA1  experiment, and measurements spanning from
1 to 10 GeV critical for the DUNE (argon) DUNE  program. At the GeV- scale
neutrino energies the neutrino can interact with a nucleus through a wide
range of reaction channels. These include the charged-current (CC)
quasielastic (QE) scattering, two-body meson exchange current (MEC) channels,
resonance production and deep inelastic scattering.

The incident neutrino energy is reconstructed using kinematic or calorimetric
methods. At energy about 1 GeV, where the CCQE scattering is dominant, the
incoming neutrino energy can be derived from lepton kinematics alone. The
calorimetric method relies not only on the visible energy measured in the
detector, but also on the models of the neutrino-nucleus interactions that are
implemented in neutrino event generators. In addition the neutrino-nucleus
scattering model is critical for obtain background estimates, and for correct
extrapolations of the near detector constraints to the far detector in
analyses aimed at determing the neutrino oscillation parameters.

Unfortunately, due to wide range of neutrino energy beams and poor statistics
available from current experiments, it is very difficult to measure
differential neutrino-nucleus cross sections for specific energies and to test
beam energy reconstruction techniques. On the theoretical side, many studies
have been presented aiming at improving our knowledge on lepton-nucleus
interaction BAV1 ; BAV2 ; Martini1 ; Martini2 ; Nieves1 ; Nieves2 ; BAV4 ;
Martini3 ; Simo ; Megias1 ; Megias2 ; Megias3 ; Rocco ; BAV5 ; BAV6 ; Gon1 ;
Gon2 ; Gon3 ; BAV7 ; BAV8 ; BAV9 ; Kim . However, it is extremely challenging
to provide reliable and consistent predictions for the diversity of processes
that can take place in the energy range covered by the neutrino beams. Various
contributions to the cross sections can significantly overlap with each other
making it difficult to identify, diagnose and remedy shortcoming of nuclear
models.

While electron and neutrino interactions are different at the primary vertex,
many underlying physics process in the nucleus are the same, and electron
scattering data collected with precisely controlled kinematics (initial and
final energies and scattering angles) and large statistic allows validation
and improvement of the description of nuclear effects. There are a large body
of electron-scattering data on carbon and calcium and only a few data sets
available for scattering on argon.

All of the above reaction mechanisms are very similar for electrons and for
neutrinos. From the nuclear point of view the influence of nuclear medium
effects such as the nuclear ground state and interaction of the outgoing
nucleon with the residual nucleus can be expected to be largely the same for
electron as for neutrino-induced processes. We can exploit this similarity and
use electron scattering data with known beam energies to test the neutrino
energy reconstruction methods CLAS  and interaction models. The vector part of
the electroweak interaction can be inferred directly from the electron
scattering data. Because electron and neutrino scattering are strongly linked
in theory, any model of neutrino interactions (vector+axial) should also be
able to reproduce electron (vector) interactions. A model unable to reproduce
electron measurements cannot be expected to provide accurate prediction for
neutrino cross sections.

It is therefore unsurprising that recent years have seen a plethora of
analyses of electron-scattering data to test the vector current part of the
lepton-nucleus interaction against existing inclusive electron scattering
cross sections for different target nuclei at several incident beam energies
and scattering electron angles. The relativistic distorted wave impulse
approximation (RDWIA), initially designed for description of exclusive
$(e,e^{\prime}p)$ data Pick ; Udias ; JKelly  and then adopted for neutrino
reactions was successfully tested against of inclusive $(e,e^{\prime})$ data
BAV7 ; Gon3 . The SuSAv2 model exploits the similarities between both
interaction types to guide the description of weak scattering process Megias2
; Megias3 . The utility of validating neutrino events generators against
inclusive electron scattering data that they had not been tuned to was
demonstrated in Refs. Ankow ; e4v1 ; NEUT ; Dytman .

Such inclusive reactions involve total hadronic cross sections and typically
are relatively insensitive to the details of the final nuclear states. Rather
simple models may yield cross sections that are not very different from those
found in the most sophisticated models. Typically, the inclusive predictions
using different models are rather similar and agree to about 10-20%, but they
cannot make predictions on both leptons and hadrons in final states. The semi-
exclusive $(l,l^{\prime}p)$ lepton scattering process involves not the total
cross sections, but the specific asymptotic states and allows to test more in
detail the nuclear model. Microscopic and unfactorized models like the RDWIA
can be used to model both lepton-boson and boson-nucleus vertexes in the same
detail and compare the results to semi-exclusive observables. The comparison
of the results of the RDWIA approach and cascade models employed in the
neutrino event generators provides constraints on cascade models from proton-
nucleus scattering Udias .

The reduced cross section, obtained from the measured differential semi-
exclusive electron scattering cross section dividing on the kinematic factor
and the off-shell electron-proton cross section, can be identified with the
distorted spectral function. Final state interactions between the ejected
nucleon and the residual nucleus make the reduced cross sections depend upon
the initial and ejectile nucleon’s momenta and angle between them (depends
upon momentum transfer). Thus, irrespective of the type of interaction
(electromagnetic or weak) the distorted spectral function is determined mainly
by the intrinsic properties of the target and the ejected nucleon interaction
with residual nucleus.

The purpose of the present work is calculation of the CCQE neutrino scattering
reduced cross sections averaged over phase space as functions of the missing
nucleon momentum and incoming neutrino energy, and comparison of them with
ones obtained from measurements of $(e,e^{\prime}p)$ scattering on carbon,
oxygen and argon targets. The direct comparison of the spectral functions used
in the factorized approach in neutrino event generators to the measured
reduced cross sections of the electron-nucleus scattering can provide an
additional test of the nuclear models employed in these generators.

The outline of this paper is the following. In Sec.II we introduce the
formalism needed to describe the semi-exclusive lepton-nucleus CCQE scattering
process. The RDWIA model is briefly introduced in Sec.III. Results of the
calculations are presented in Sec.IV. Our conclusions are summarized in Sec.V.

##  II Formalism of quasi-elastic scattering

We consider the formalism used to describe electron and neutrino quasi-elastic
exclusive

$l(k_{i})+A(p_{A})\rightarrow l^{\prime}(k_{f})+N(p_{x})+B(p_{B}),$ (1)

scattering off nuclei in the one-photon (W-boson) exchange approximation. Here
$l$ labels the incident lepton [electron or muon (anti)neutrino], and
$l^{\prime}$ represents the scattered lepton (electron or muon),
$k_{i}=(\varepsilon_{i},\mbox{\boldmath$k$}_{i})$ and
$k_{f}=(\varepsilon_{f},\mbox{\boldmath$k$}_{f})$ are the initial and final
lepton momenta, $p_{A}=(\varepsilon_{A},\mbox{\boldmath$p$}_{A})$, and
$p_{B}=(\varepsilon_{B},\mbox{\boldmath$p$}_{B})$ are the initial and final
target momenta, $p_{x}=(\varepsilon_{x},\mbox{\boldmath$p$}_{x})$ is the
ejectile nucleon momentum, $q=(\omega,\mbox{\boldmath$q$})$ is the momentum
transfer carried by the virtual photon (W-boson), and
$Q^{2}=-q^{2}=\mbox{\boldmath$q$}^{2}-\omega^{2}$ is the photon (W-boson)
virtuality.

###  II.1 CCQE lepton-nucleus cross sections

In the laboratory frame the differential cross section for exclusive electron
($\sigma^{el}$) and (anti)neutrino ($\sigma^{cc}$) CC scattering can be
written as

$\displaystyle\frac{d^{6}\sigma^{el}}{d\varepsilon_{f}d\Omega_{f}d\varepsilon_{x}d\Omega_{x}}$
$\displaystyle=\frac{|\mbox{\boldmath$p$}_{x}|\varepsilon_{x}}{(2\pi)^{3}}\frac{\varepsilon_{f}}{\varepsilon_{i}}\frac{\alpha^{2}}{Q^{4}}L_{\mu\nu}^{(el)}\mathcal{W}^{\mu\nu(el)}$
(2a)
$\displaystyle\frac{d^{6}\sigma^{cc}}{d\varepsilon_{f}d\Omega_{f}d\varepsilon_{x}d\Omega_{x}}$
$\displaystyle=\frac{|\mbox{\boldmath$p$}_{x}|\varepsilon_{x}}{(2\pi)^{5}}\frac{|\mbox{\boldmath$k$}_{f}|}{\varepsilon_{i}}\frac{G^{2}\cos^{2}\theta_{c}}{2}L_{\mu\nu}^{(cc)}\mathcal{W}^{\mu\nu(cc)},$
(2b)

where $\Omega_{f}$ is the solid angle for the lepton momentum, $\Omega_{x}$ is
the solid angle for the ejectile nucleon momentum, $\alpha\simeq 1/137$ is the
fine-structure constant, $G\simeq$ 1.16639 $\times 10^{-11}$ MeV-2 is the
Fermi constant, $\theta_{C}$ is the Cabbibo angle ($\cos\theta_{C}\approx$
0.9749), $L^{\mu\nu}$ is the lepton tensor, and $\mathcal{W}^{(el)}_{\mu\nu}$
and $\mathcal{W}^{(cc)}_{\mu\nu}$ are correspondingly the electromagnetic and
weak CC nuclear tensors.

For exclusive reactions in which only a single discrete state or a narrow
resonance of the target is excited, it is possible to integrate over the peak
in missing energy and obtain a fivefold differential cross section of the form

$\displaystyle\frac{d^{5}\sigma^{el}}{d\varepsilon_{f}d\Omega_{f}d\Omega_{x}}$
$\displaystyle=R\frac{|\mbox{\boldmath$p$}_{x}|\tilde{\varepsilon}_{x}}{(2\pi)^{3}}\frac{\varepsilon_{f}}{\varepsilon_{i}}\frac{\alpha^{2}}{Q^{4}}L_{\mu\nu}^{(el)}W^{\mu\nu(el)}$
(3a)
$\displaystyle\frac{d^{5}\sigma^{cc}}{d\varepsilon_{f}d\Omega_{f}d\Omega_{x}}$
$\displaystyle=R\frac{|\mbox{\boldmath$p$}_{x}|\tilde{\varepsilon}_{x}}{(2\pi)^{5}}\frac{|\mbox{\boldmath$k$}_{f}|}{\varepsilon_{i}}\frac{G^{2}\cos^{2}\theta_{c}}{2}L_{\mu\nu}^{(cc)}W^{\mu\nu(cc)},$
(3b)

where $R$ is a recoil factor

$R=\int d\varepsilon_{x}\delta(\varepsilon_{x}+\varepsilon_{B}-\omega-
m_{A})={\bigg{|}1-\frac{\tilde{\varepsilon}_{x}}{\varepsilon_{B}}\frac{\mbox{\boldmath$p$}_{x}\cdot\mbox{\boldmath$p$}_{B}}{\mbox{\boldmath$p$}_{x}\cdot\mbox{\boldmath$p$}_{x}}\bigg{|}}^{-1},$
(4)

$\tilde{\varepsilon}_{x}$ is solution to equation
$\varepsilon_{x}+\varepsilon_{B}-m_{A}-\omega=0,$ where
$\varepsilon_{B}=\sqrt{m^{2}_{B}+\mbox{\boldmath$p$}^{2}_{B}}$,
$~{}\mbox{\boldmath$p$}_{B}=\mbox{\boldmath$q$}-\mbox{\boldmath$p$}_{x}$ and
$m_{A}$ and $m_{B}$ are masses of the target and recoil nucleus, respectively.
Note, that missing momentum is
$\mbox{\boldmath$p$}_{m}=\mbox{\boldmath$p$}_{x}-\mbox{\boldmath$q$}$ and
missing energy $\mbox{\boldmath$\varepsilon$}_{m}$ is defined by
$\mbox{\boldmath$\varepsilon$}_{m}=m+m_{B}-m_{A}$.

All information about the nuclear structure and effects of final-state
interaction (FSI) between the ejectile nucleon and residual nucleus is
contained in the electromagnetic and weak CC hadronic tensors,
$W^{(el)}_{\mu\nu}$ and $W^{(cc)}_{\mu\nu}$, which are given by the bilinear
products of the transition matrix elements of the nuclear electromagnetic or
CC operator $J^{(el)(cc)}_{\mu}$ between the initial nucleus state $|A\rangle$
and the final state $|B_{f}\rangle$ as

$\displaystyle W^{(el)(cc)}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\sum_{f}\langle B_{f},p_{x}|J^{(el)(cc)}_{\mu}|A\rangle\langle
A|J^{(el)(cc)\dagger}_{\nu}|B_{f},p_{x}\rangle,$ (5)

where the sum is taken over undetected states.

In the exclusive reaction (1) the outgoing lepton and proton are detected and
the exclusive lepton scattering cross sections (3a) and (3b) in terms of
response functions can be written as

$\displaystyle\frac{d^{5}\sigma^{el}}{d\varepsilon_{f}d\Omega_{f}d\Omega_{x}}$
$\displaystyle=\frac{|\mbox{\boldmath$p$}_{x}|\tilde{\varepsilon}_{x}}{(2\pi)^{3}}\sigma_{M}R\big{(}V_{L}R^{(el)}_{L}+V_{T}R^{(el)}_{T}+V_{LT}R^{(el)}_{LT}\cos\phi+V_{TT}R^{(el)}_{TT}\cos
2\phi\big{)},$ (6a)
$\displaystyle\frac{d^{5}\sigma^{cc}}{d\varepsilon_{f}d\Omega_{f}d\Omega_{x}}$
$\displaystyle=\frac{|\mbox{\boldmath$p$}_{x}|\tilde{\varepsilon}_{x}}{(2\pi)^{5}}G^{2}\cos^{2}\theta_{c}\varepsilon_{f}|\mbox{\boldmath$k$}_{f}|R\big{\\{}v_{0}R_{0}+v_{T}R_{T}+v_{TT}R_{TT}\cos
2\phi+v_{zz}R_{zz}$ $\displaystyle+(v_{xz}R_{xz}-v_{0x}R_{0x})\cos\phi-
v_{0z}R_{0z}+h\big{[}v_{yz}(R^{\prime}_{yz}\sin\phi+R_{yz}\cos\phi)$
$\displaystyle-
v_{0y}(R^{\prime}_{0y}\sin\phi+R_{0y}\cos\phi)-v_{xy}R_{xy}\big{]}\big{\\}},$
(6b)

where

$\sigma_{M}=\frac{\alpha^{2}\cos^{2}\theta/2}{4\varepsilon^{2}_{i}\sin^{4}\theta/2}$
(7)

is the Mott cross section and $h$ is +1 for positive lepton helicity and -1
for negative lepton helicity. The coupling coefficient $V_{k}$ and $v_{k}$,
the expression of which are given in Ref. BAV1  are kinematic factors
depending on the lepton’s kinematics. The response functions $R_{i}$ are given
in terms of components of the exclusive hadronic tensors BAV1  and depend on
the variables $(Q^{2},\omega)$ or $(|\mbox{\boldmath$q$}|,\omega)$.

It is also useful define a reduced cross section

$\sigma_{red}=\frac{d^{5}\sigma^{(el)(cc)}}{d\varepsilon_{f}d\Omega_{f}d\Omega_{x}}/K^{(el)(cc)}\sigma_{lN},$
(8)

where $K^{el}=R{p_{x}\varepsilon_{x}}/{(2\pi)^{3}}$ and
$K^{cc}=R{p_{x}\varepsilon_{x}}/{(2\pi)^{5}}$ are phase-space factors for
electron and neutrino scattering and $\sigma_{lN}$ is the corresponding
elementary cross section for the lepton scattering from the moving free
nucleon normalized to unit flux. The reduced cross section is an interesting
quantity that can be regarded as the nucleon momentum distribution modified by
FSI, i.e. as the distorted spectral function. Final-state interactions make
the reduced cross sections
$\sigma_{red}({\mbox{\boldmath$p$}}_{m},{\mbox{\boldmath$p$}}_{x})$ depend
upon ejectile momentum ${\mbox{\boldmath$p$}}_{x}$, angle between the initial
and final nucleon momentum and upon incident lepton energy. These cross
sections for (anti)neutrino scattering off nuclei are similar to the electron
scattering apart from small differences at low beam energy due to effects of
Coulomb distortion of the incoming electron wave function as shown in Refs.
BAV1 ; BAV2 ; BAV4

The factorization approximation to the knockout cross section stipulates that

$\frac{d^{5}\sigma^{(el)(cc)}}{d\varepsilon_{f}d\Omega_{f}d\Omega_{x}}=K^{(el)(cc)}\times\sigma_{lN}\times\sigma_{red}(\mbox{\boldmath$\varepsilon$}_{m},{\mbox{\boldmath$p$}}_{m},{\mbox{\boldmath$p$}}_{x})$
(9)

This factorization implies that the initial nuclear sate and FSI effects are
decoupled from leptonic vertex with preserved the correlations between the
final lepton and nucleon.

The reduced cross section as a function of missing momentum $p_{m}$, averaged
over phase volume in $(\omega,\Omega_{f},\phi)$ coordinates, where
$\Omega_{x}=(\cos\theta_{pq},\phi)$, can be written as

$\langle\sigma^{red}(p_{m})\rangle=\frac{1}{V}\int d\phi\int
d\mbox{\boldmath$\varepsilon$}_{f}\int
d\Omega_{f}\frac{p_{m}}{p_{x}|\mbox{\boldmath$q$}|}R_{c}\sigma^{red}(\mbox{\boldmath$\varepsilon$}_{f},\Omega_{f},p_{m},\phi),$
(10)

where
$p_{m}=|\mbox{\boldmath$p$}_{m}|,~{}p_{x}=|\mbox{\boldmath$p$}_{x}|,~{}\mbox{\boldmath$p$}_{m}=\mbox{\boldmath$p$}_{x}-\mbox{\boldmath$q$}$,
and

$\displaystyle\cos\theta_{pq}$
$\displaystyle=\frac{\mbox{\boldmath$p$}^{2}_{x}+\mbox{\boldmath$q$}^{2}-\mbox{\boldmath$p$}^{2}_{m}}{2p_{x}|\mbox{\boldmath$q$}|},$
(11a) $\displaystyle R_{c}$
$\displaystyle=1+\frac{\varepsilon_{x}}{2p^{2}_{x}\varepsilon_{B}}(\mbox{\boldmath$p$}^{2}_{x}+\mbox{\boldmath$q$}^{2}-\mbox{\boldmath$p$}^{2}_{m}),$
(11b) $\displaystyle V$ $\displaystyle=\int d\phi\int
d\mbox{\boldmath$\varepsilon$}_{f}\int
d\Omega_{f}\frac{p_{m}}{p_{x}|\mbox{\boldmath$q$}|}R_{c}.$ (11c)

Precise electron reduced cross sections data can be used to validate the
neutrino reduced cross sections (spectral functions) that are implemented in
neutrino generators.

###  II.2  Nuclear current

Obviously, the determination of the response tensor $W^{\mu\nu}$ requires the
knowledge of the nuclear current matrix elements in Eq.(5). We describe the
lepton-nucleon scattering in the impulse approximation, assuming that the
incoming lepton interacts with only one nucleon, which is subsequently
emitted. The nuclear current is written as the sum of single-nucleon currents.
Then, the nuclear matrix element in Eq.(5) takes the form

$\displaystyle\langle p,B|J^{\mu}|A\rangle$ $\displaystyle=$
$\displaystyle\int d^{3}r~{}\exp(i\mbox{\boldmath$t$}\cdot\mbox{{\bf
r}})\overline{\Psi}^{(-)}(\mbox{\boldmath$p$},\mbox{{\bf
r}})\Gamma^{\mu}\Phi(\mbox{{\bf r}}),$ (12)

where $\Gamma^{\mu}$ is the vertex function,
$\mbox{\boldmath$t$}=\varepsilon_{B}\mbox{\boldmath$q$}/W$ is the recoil-
corrected momentum transfer,
$W=\sqrt{(m_{A}+\omega)^{2}-\mbox{\boldmath$q$}^{2}}$ is the invariant mass,
$\Phi$ and $\Psi^{(-)}$ are relativistic bound-state and outgoing wave
functions. For electron scattering, most calculations use the CC2
electromagnetic vertex function for a free nucleon deFor

$\Gamma^{\mu}=F^{(el)}_{V}(Q^{2})\gamma^{\mu}+{i}\sigma^{\mu\nu}\frac{q_{\nu}}{2m}F^{(el)}_{M}(Q^{2}),$
(13)

where $\sigma^{\mu\nu}=i[\gamma^{\mu}\gamma^{\nu}]/2$, $F^{(el)}_{V}$ and
$F^{(el)}_{M}$ are the Dirac and Pauli nucleon form factors. Because the bound
nucleons are off shell, the vertex $\Gamma^{\mu}$ in Eq.(13) should be taken
for an off-shell nucleon. We employ the de Forest prescription for the off-
shell vertex deFor

$\tilde{\Gamma}^{\mu}=F^{(el)}_{V}(Q^{2})\gamma^{\mu}+{i}\sigma^{\mu\nu}\frac{\tilde{q}_{\nu}}{2m}F^{(el)}_{M}(Q^{2}),$
(14)

where $\tilde{q}=(\varepsilon_{x}-\tilde{E},\mbox{\boldmath$q$})$ and the
nucleon energy
$\tilde{E}=\sqrt{m^{2}+(\mbox{\boldmath$p$}_{x}-\mbox{\boldmath$q$})^{2}}$ is
placed on shell. We use the approximation of MMD  on the nucleon form factors.
The Coulomb gauge is assumed for the single-nucleon current. Although the
experimental analysis usually employ the de Forest CC1 prescription for
$\sigma_{lN}$, consistency requires that calculation of $\sigma_{red}$ to
employ the $\sigma_{lN}$ that corresponds to the current operator used in the
RDWIA calculations.

The single-nucleon charged current has $V{-}A$ structure
$J^{\mu(cc)}=J^{\mu}_{V}+J^{\mu}_{A}$. For a free nucleon vertex function
$\Gamma^{\mu(cc)}=\Gamma^{\mu}_{V}+\Gamma^{\mu}_{A}$ we use CC2 vector current
vertex function

$\Gamma^{\mu}_{V}=F_{V}(Q^{2})\gamma^{\mu}+{i}\sigma^{\mu\nu}\frac{q_{\nu}}{2m}F_{M}(Q^{2})$
(15)

and the axial current vertex function

$\Gamma^{\mu}_{A}=F_{A}(Q^{2})\gamma^{\mu}\gamma_{5}+F_{P}(Q^{2})q^{\mu}\gamma_{5}.$
(16)

Weak vector form factors $F_{V}$ and $F_{M}$ are related to corresponding
electromagnetic ones for proton $F^{(el)}_{i,p}$ and neutron $F^{(el)}_{i,n}$
by the hypothesis of conserved vector current (CVC)

$F_{i}=F^{(el)}_{i,p}-F^{(el)}_{i,n}.$ (17)

The axial $F_{A}$ and psevdoscalar $F_{P}$ form factors in the dipole
approximation are parameterized as

$F_{A}(Q^{2})=\frac{F_{A}(0)}{(1+Q^{2}/M_{A}^{2})^{2}},\quad
F_{P}(Q^{2})=\frac{2mF_{A}(Q^{2})}{m_{\pi}^{2}+Q^{2}},$ (18)

where $F_{A}(0)=1.2724$, $m_{\pi}$ is the pion mass, and $M_{A}$ is the axial
mass. We use de Forest prescription for off-shell extrapolation of
$\Gamma^{\mu(cc)}$. Similar to electromagnetic current, the Coulomb gauge is
applied for the vector current $J_{V}$.

##  III Model

The semi-exclusive differential and reduced cross sections for neutrino
scattering were studied in Refs. BAV1 ; BAV2 ; BAV4 ; BAV8 ; BAV9 , using the
relativistic shell model approach and taking into account the FSI effects. A
formalism for the $A(e,e^{\prime}N)B$ reaction that describes the channel
coupling in the FSI of $N+B$ system was developed in Ref. JKelly .

In this work the independent particle shell model (IPSM) is assumed for the
nuclear structure. The model space for 12C$(l,l^{\prime}N)$ consists of
$1s_{1/2}$ and $1p_{3/2}$ nucleon-hole states in the 11B and 11C nuclei. The
model space for 16O$(l,l^{\prime}N)$ consists of $1s_{1/2}$, $1p_{3/2}$, and
$1p_{1/2}$ nucleon-hole states in the 15N and 15O nuclei. The model space for
40Ar$(l,l^{\prime}N)$ consists of $1s_{1/2}$, $1p_{3/2}$, $1p_{1/2}$,
$1d_{5/2}$, $2s_{1/2}$, and $1d_{3/2}$ nucleon-hole states in the 39Cl, and
$1s_{1/2}$, $1p_{3/2}$, $1p_{1/2}$, $1d_{5/2}$, $2s_{1/2}$, $1d_{3/2}$, and
$1f_{7/2}$ nucleon-hole states in the 39Ar nuclei. All states in these nuclei
are regarded as discrete states even though their spreading widths are
actually appreciable.

In the independent particle shell model the relativistic bound-state function
$\Phi$ in Eq.(12) is obtained as the self-consistent solutions of a Dirac
equation, derived within a relativistic mean-field approach, from a Lagrangian
containing $\sigma$, $\omega$, and $\rho$ mesons Serot . The nucleon bound-
state functions were calculated by the TIMORA code Horow  with the
normalization factors $S(\alpha)$ relative to full occupancy of the IPSM
orbitals. According the RDWIA analysis of the JLab 12C$(e,e^{\prime}p)$ data
Dutta ; Kelly1  $S(1p_{3/2})=84\%$, $S(1s_{1/2})=100\%$ and average factor
about $\approx 89\%$. We use also the following values of normalization
factors of 16O: $S(1p_{3/2})=66\%$, $S(1p_{1/2})=70\%$, and
$S(1s_{1/2})=100\%$, that were obtained in the RDWIA analysis of the JLab data
Fissum . From the RDWIA analysis BAV4  of NIKHEF data Kramer1 ; Kramer2 ;
Kramer3  follows that the occupancy of the

Figure 1:  Proton momentum distributions for the different single particle
states in 12C nucleus. Also shown is the total proton momentum distribution
(solid line).  Figure 2:  Same as Fig.1 but in 16O nucleus.  Figure 3:  Total
proton and neutron momentum distributions in 40Ar nucleus.

orbitals of 40Ca and 40Ar are approximately 87% on average. Proton and neutron
binding energies and the occupancy’s of the orbitals in 40Ar are given in
Table II of Ref. BAV4 . In this work we assume that the missing strength can
be attributed to the short-range nucleon-nucleon $(NN)$ correlations, leading
to the appearance of the high-momentum and high-energy nucleon distribution in
the target.

Figures 1 and 2 show the proton momentum distributions for occupied orbitals
in 12C and 16O, calculated within the mean-field approach. The neutron
momentum distributions in these nuclei are almost identical to proton ones.
The total proton and neutron momentum distributions in 40Ar are presented in
Fig.3. These distributions are normalized to the total number of
protons/neutrons on the IPSM shells.

For an outgoing nucleon, the simplest chose is to use plane-wave function
$\Psi$ in Eq.(12) that is, no interactions are between the ejected nucleon $N$
and the residual nucleus $B$, i.e. to use the so-called plane-wave impulse
approximation (PWIA). For a more realistic description, final state
interaction effects should be taken into account. In the RDWIA the distorted-
wave function of the knocked out nucleon $\Psi$ is evaluated as a solution of
a Dirac equations containing a phenomenological relativistic optical potential
Fissum . This potential consists of a real part, which describes the
rescattering of the ejected nucleon and an imaginary part for the absorption
of it into unobserved channels. We use the LEA program LEA  for numerical
calculation of the distorted-wave function with the EDAD1 parameterization
Cooper  of the relativistic optical potential for carbon, oxygen and calcium.

##  IV Results and analysis

The reduced cross sections of 12C$(e,e^{\prime}p)$ reaction in the range of
missing energy, that corresponds to knockout of $1s$ and $1p$-shells protons
were measured at Tokyo Tokyo , Saclay SaclayC1 ; SaclayC2 , NIKHEF NIKHEFC ,
SLAC SLAC , and JLab Dutta . The knockout of $1p$-shell protons in
16O$(e,e^{\prime}p)$ was studied at Saclay SaclayC1 ; SaclayO , NIKHEF
NIKHEFO1 ; NIKHEFO2 , Mainz Mainz , and JLab Fissum . In these experiments,
cross sections data for the lowest-lying fragments of each shell were measured
as functions of $p_{m}$, and normalization factors (relating how much the
measured cross section data were less than predicted in IPSM) were extracted.
The E12-14-012 experiment JLabE , performed in JLab has measured the
$(e,e^{\prime}p)$ reduced cross sections using 40Ar JLabAr  and 48Ti JLabTi
targets. The reduced cross sections measured in the missing momentum and
missing energy ranges $15\leq p_{m}\leq 300$ MeV/c and $12\leq E_{m}\leq 80$
MeV.

The distorted spectral function depends upon initial momentum
$\mbox{\boldmath$p$}_{m}$, ejectile momentum $\mbox{\boldmath$p$}_{x}$ and
angle between the initial and final nucleon momenta. Thus it depends upon
kinematical conditions and is different for parallel and perpendicular
kinematics.

Figure 4:  Comparison of the RDWIA calculations for electron, neutrino and
antineutrino reduced cross sections for the removal of nucleons from 1$s$ and
1$p$ shells of 12C as functions of the missing momentum. JLab data Dutta  for
beam energy $E_{beam}$=2.455 GeV, proton kinetic energy $T_{p}$=350 MeV, and
$Q^{2}$=0.64 (GeV/$c)^{2}$. The RDWIA calculations are shown for electron
scattering (dashed line) and neutrino (solid line) and antineutrino (dashed-
dotted line) scattering. This figure taken from Ref. BAV2 .

Furthermore, $\sigma_{red}$ depends upon initial electron energy due to
Coulomb distortion. The RDWIA approach with LEA code was successfully tested
against measured 12C$(e,e^{\prime}p)$ Kelly1 , 16O$(e,e^{\prime}p)$ Fissum ,
and 40Ca$(e,e^{\prime}p)$ BAV4  differential and reduced cross sections, and
the normalization factors $S(\alpha)$ for the IPSM orbitals were derived.

In Refs. BAV1 ; BAV2 ; BAV4  electron and CCQE (anti)neutrino scattering on
oxygen, carbon, calcium, and argon targets were studied. It was found that the
reduced cross sections for (anti)neutrino scattering are similar to those of
electron scattering, and the latter are in good agreement with electron data.
The difference between the electron and (anti)neutrino reduced cross sections
calculated for Saclay kinematics is less than 10%.

Figure 5:  The RDWIA calculation of neutrino (solid line) and antineutrino
(dashed-dotted line) averaged reduced cross sections compared with measured
exclusive cross section data for the removal of nucleons from 1$p$ and 1$s+1p$
shells of 12C as functions of the missing momentum. The data are from Saclay
SaclayC1  for $1p$ and the beam energy $E_{beam}=500$ MeV, SLAC SLAC  for
$1s+1p$ shells and $E_{beam}=2015$ MeV, and from JLab Dutta  for $1s+1p$
shells and $E_{beam}$=2455 MeV

This can be attributed to Coulomb distortion upon electron wave function which
is usually described as the effective momentum approximation (EMA) EMA . In
the EMA, the electron Coulomb wave function is replaced by a plane wave
function with effective momentum whose value is larger than the value of
electron momentum at infinity, because of Coulomb attraction. The flux is also
increased in the interaction zone by focusing of electron wave. This effect is
proportional to charge of the target and weakens as the beam energy increases.
The small difference between neutrino and antineutrino reduced cross sections
is due to the difference in the FSI of the proton and neutron with the
residual nucleus.

In this section we present the results of the RDWIA calculations of the
averaged reduced cross sections Eq.(10) for (anti)neutrino scattering off
carbon, oxygen, and argon as functions of the missing momentum $p_{m}$ and
compare them with the measured $(e,e^{\prime}p)$ reduced cross sections. In
Ref. BAV2  electron, neutrino, and antineutrino cross sections for the removal
protons from the $1s$, $1p$, and $1s+1p$ shells of 120C as functions of
missing momentum $p_{m}$ were calculated and compared with JLab data Dutta .
For illustration, Fig. 4 shows measured the removal cross sections as compared
with the LEA code calculations BAV2 .

Figure 6:  The RDWIA neutrino averaged reduced cross section for removal of
nucleons from the $1s+1p$ shells of 12C as a function of neutrino energy and
missing momentum $p_{m}$.

It should be note that negative value of $p_{m}$ corresponds to $\phi=\pi$ and
positive to $\phi=0$, where $\phi$ is the angle between the scattering
$(\mbox{\boldmath$k$}_{i},\mbox{\boldmath$k$}_{f})$ and reaction
$(\mbox{\boldmath$p$}_{x},\mbox{\boldmath$p$}_{B})$ planes. The data for beam
energy $E_{beam}=2.445$ GeV and $Q^{2}=0.64$ (GeV/c)2 were measured in the
quasi-perpendicular kinematics with constant $(\omega,\mbox{\boldmath$q$})$.
The electron and neutrino scattering off the nuclei are closely interrelated
and one can treat both processes within the same formalism. There is an
overall good agreement between the cross sections calculated in the RDWIA and
data.

The averaged reduced cross sections for removal of nucleons from the $1p$, and
$1s+1p$ shells of 12C$(\nu_{\mu},\mu p)$ and 12C$(\bar{\nu}_{\mu},\mu n)$
reactions are shown in Fig. 5 as functions of positive $p_{m}$ values together
with Saclay SaclayC2 , SLAC Kelly1 , and JLab Dutta  data.

Figure 7:  Comparison of the RDWIA calculations for neutrino (dashed line) and
antineutrino (dashed-dotted line) averaged reduced cross sections for the
removal of nucleons from 1$p$ shell of 16O with Saclay SaclayC1  and NIKHEF
NIKHEFO1  data as functions of $p_{m}$. Also shown are the RDWIA calculations
of the reduced cross section for electron scattering (solid line) from Ref.
BAV2 .

The data for beam energies $E_{beam}=500,2015$, and 2445 MeV were measured.
There is an overall agreement between the calculated averaged cross sections
and reduced cross sections of the $(e,e^{\prime}p)$ reaction measured in
different kinematics. The RDWIA averaged reduced cross section for removal
nucleon from $1s+1p$ shells in 12C$(\nu_{\mu},\mu p)$ is shown in Fig. 6 as a
function of incoming neutrino energy and missing momentum $p_{m}$. In the
range of the maximum at $60\leq p_{m}\leq 90$ MeV/c the cross section
increases slowly with neutrino energy, and slight changes at $p_{m}\geq 120$
MeV/c and $p_{m}\leq 40$ MeV/c.

The averaged reduced cross sections for the removal of nucleons from the $1p$
shell in 16O$(\nu_{\mu},\mu p)$ and 16O$(\bar{\nu}_{\mu},\mu n)$ reactions are
shown in Fig. 7 as functions of $p_{m}$ together with Saclay SaclayC1  and
NIKHEF NIKHEFO1  data. There is an overall agreement between calculated cross
sections and data, but the values of the calculated cross sections at maximum
is systematically higher (about 15-20%) than measured ones for NIKHEF
kinematics. Unfortunately, there are no data for removal protons from the $1s$
and $1s+1p$ shells of 16O. Therefore the models of lepton-nucleus interaction
that don’t take into account the shell structure of nucleus can not be tested
against the available reduced cross sections measured in 16O$(e,e^{\prime}p)$
reaction.

Figure 8:  Same as Fig. 6 but for the 16O.

The RDWIA averaged reduced cross section for removal nucleon from $1s+1p$
shells in 16O$(\nu_{\mu},\mu p)$ is shown in Fig. 8 as a function of incoming
neutrino energy and missing momentum $p_{m}$. As can be seen from figures 6
and 8 the dependence of these cross sections upon neutrino energy and $p_{m}$
are almost similar.

Figure 9:  Comparison of the RDWIA calculations for electron (solid line),
neutrino (dashed line), and antineutrino (dashed-dotted line) reduced cross
sections for the removal of nucleons from 1$d_{3/2}$, 2$s_{1/2}$, and
1$d_{5/2}$ shells of 40Ca with NIKHEF data Kramer3 . The cross sections are
presented as functions of missing momentum $p_{m}$. The figure taken from Ref.
BAV4 .

The structure of calcium and argon nuclei is similar, although unlike
${}^{40}_{18}$Ar, ${}^{40}_{20}$Ca is a symmetric and closed-shell nucleus.

Figure 10:  Missing momentum distribution in argon obtained by integrating
over the missing energy range of 0 - 30 MeV (left panel) and 30 - 54 MeV
(right panel), presented with the geometrical factor of $4\pi p_{m}^{2}$. The
gray band shows the measured spectral function including the full error.

In Ref. BAV4  the $(e,e^{\prime}p)$ reduced cross sections for removal of the
proton from $1d_{3/2}$ shell, for transition to the 1/2+ exited state of the
39K nucleus at excitation energy $E_{x}=2.522$ MeV, and for the transitions to
the 5/2+ excited states at $E_{x}$=5.258 MeV and $E_{x}$=6.328 MeV, obtained
by knocking out protons from the 2$s_{1/2}$ and 1$d_{5/2}$ orbitals,
correspondingly were calculated. The calculated reduced cross sections are
shown in Fig. 9 with the NIKHEF data Kramer1 ; Kramer2  and provide a good
description of the shape and magnitude of the measured distribution. Neutrino
and antineutrino calculated reduced cross sections of
40Ca$(\nu,\mu^{-}p){}^{39}$Ca and 40Ca$(\bar{\nu},\mu^{+}n){}^{39}$K reactions
also shown are in Fig. 9. There is an overall good agreement between
calculated cross sections, but the values of the electron cross sections at
the maximum is systematically higher than those for (anti)neutrino. This can
be attributed to Coulomb distortion upon the incident electron wave function.

The JLab experiment JLabE  has measured the $(e,e^{\prime}p)$ cross sections
using argon and titanium targets JLabAr ; JLabTi . The reduced cross sections
were obtained in the missing momentum $15\leq p_{m}\leq 300$ MeV/c and missing
energy range $12\leq p_{m}\leq 80$ MeV.

Figure 11:  Missing momentum distribution in titanium obtained by integrating
over the missing energy range of 0 - 30 MeV, presented with the geometrical
factor of $4\pi p_{m}^{2}$. The gray band shows the measured spectral function
including the full error.

The procedure to obtained information on neutron distribution in argon is
based on the observation that neutron spectrum of ${}^{40}_{18}$Ar is mirrored
by proton spectrum of the nucleus of titanium, having charge $Z=22$. Therefor
one can expect that the proton spectral function obtained from
Ti$(e,e^{\prime}p)$ data provides information on the neutron spectral function
of argon.

The ${}^{40}_{18}$Ar and ${}^{48}_{22}$Ti data were analyzed to obtain the
spectral functions, describing the energy and momentum distributions of
protons in the argon and titanium ground states. The effect of FSI, which are
known to be significant in $(e,e^{\prime}p)$ reactions, was taken into account
within the distorted-wave impulse approximation approach. Figure 10 shows the
missing momentum distributions of protons in argon obtained by integrating the
data over the missing energy ranges $0-30$ MeV and $30-50$ MeV. The proton
missing momentum distribution in titanium was obtained by integrating the data
over the missing energy range $0-30$ MeV is shown in Fig. 11. Also shown in
Figs. 10 and 11 are the results obtained without FSI effects in the
relativistic plane wave impulse approximation (RPWIA), with normalization
factors $S_{\alpha}$ from Ref. BAV4 . There is an overall agreement between
the RPWIA calculations and data within sizable uncertainties of the measured
proton momentum distributions. A more accurate determination of the distorted
spectral functions for different shells of 40Ar and 48Ti will improve the
testing of models using for description of neutrino interaction with these
nucleus.

Figure 12:  Same as Fig. 6 but for the 40Ar.

The averaged reduced cross section of 40Ar$(\nu_{\mu},\mu p)$ reaction
calculated in the RDWIA approach is shown in Fig. 12 as a function of neutrino
energy and missing momentum $p_{m}$. It has to be pointed out that unlike 12C
and 16O, in 40Ar the maximum of these cross sections is shifted to the range
of lower missing momentum $p_{m}\approx 15$ MeV/c. The cross section increases
very slowly with neutrino energy.

Neutrino event generators employ the factorization approach to make
predictions about the lepton and also the outgoing nucleon kinematics from
inclusive models. These models are aimed to describe inclusive cross section
that is only as function of the final lepton kinematics. This factorization
uses the spectral functions, which are generated from different nucleon
distributions in the initial nuclear state (local Fermi gas, shell model,
etc). While the behavior of the cross section against the lepton kinematics
may be described correctly, there is no guarantee that the correlations
between the final lepton and nucleon for a given event are preserved. The
comparison of the employed spectral function with the measured reduced cross
sections allows the estimation of the accuracy of the nuclear effects
calculations. On the other hand, the effective spectral functions can be
obtained within the microscopic and unfactorized models, like the RDWIA, that
successfully describes exclusive $(e,e^{\prime}p)$ cross sections and employed
in the neutrino event generators.

##  V Conclusions

In this article, we studied within the RDWIA approach the semi-exclusive
reduced cross sections of CCQE (anti)neutrino scattering on carbon, oxygen,
and argon. We calculated averaged over phase space reduced cross sections for
removal of nucleons from the $\mathrm{1p}$, and $\mathrm{1s+1p}$ shells of
12C$(\nu_{\mu},\mu p)$, 12C$(\bar{\nu}_{\mu},\mu n)$ and 16O$(\nu_{\mu},\mu
p)$, 16O$(\bar{\nu}_{\mu},\mu n)$ reactions as functions of missing momentum
and incoming neutrino energy, and compared them with the reduced cross
sections obtained from measurements of $(e,e^{\prime}p)$ scattering on 12C and
16O. We also calculated in the relativistic plane wave impulse approximation
the averaged reduced cross sections for single nucleon knocked out in
40Ar$(\nu_{\mu},\mu p)$ and 49Ti$(\bar{\nu}_{\mu},\mu n)$ reactions as
functions of $p_{m}$ and $\varepsilon_{\nu}$ and compared them with proton
momentum distributions in argon and titanium obtained from $(e,e^{\prime}p)$
scattering in the JLab experiments

We found that the shape and magnitude of the averaged reduced cross sections
for (anti)neutrino scattering as a function of missing momentum are similar to
measured reduced cross sections of electron scattering. The averaged removal
cross sections calculated for argon and titanium within RPWIA approach seem
mostly consistent with the data within sizable uncertainties of measured
proton momentum distributions. The difference less than 10% between the
electron and (anti)neutrino cross sections can be attributed to Coulomb
distortion upon incoming electron wave function. The small difference between
neutrino and antineutrino reduced cross sections is due to difference in the
FSI of proton and neutron with the residual nucleus. The averaged reduced
cross sections for removal nucleon from $\mathrm{1s+1p}$ shells in carbon and
oxygen have maximum in the range of missing momentum $60\leq p_{m}\leq 90$
MeV/c and in 40Ar the maximum is shifted in the range of $p_{m}\approx 15$
MeV/c. The cross sections increase very slowly with neutrino energy.

Some neutrino event generators employ the factorization approach to make
prediction about the lepton and outgoing nucleon kinematics, using different
nucleon distributions in the ground nuclear state. In this way the direct
comparison of the implemented spectral functions with the precise electron
reduced cross sections data allows to estimate the accuracy of the nuclear
effects calculations, such as the nuclear ground state and FSI.

## Acknowledgments

The author greatly acknowledge A. Habig for fruitful discussions and a
critical reading of the manuscript. A would like to thank S. Luchuk for his
constructive comments and suggestions.

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# Quantum state tomography of an itinerant squeezed microwave field

F. Mallet  JILA, National Institute of Standards and Technology and the
University of Colorado, Boulder, CO 80309, USA  M. A. Castellanos-Beltran
JILA, National Institute of Standards and Technology and the University of
Colorado, Boulder, CO 80309, USA  Department of Physics, University of
Colorado, Boulder, CO 80309, USA  H. S. Ku  JILA, National Institute of
Standards and Technology and the University of Colorado, Boulder, CO 80309,
USA  Department of Physics, University of Colorado, Boulder, CO 80309, USA  S.
Glancy  National Institute of Standards and Technology, Boulder, Colorado
80305, USA  E. Knill  National Institute of Standards and Technology, Boulder,
Colorado 80305, USA  K. D. Irwin  National Institute of Standards and
Technology, Boulder, Colorado 80305, USA  G. C. Hilton  National Institute of
Standards and Technology, Boulder, Colorado 80305, USA  L. R. Vale  National
Institute of Standards and Technology, Boulder, Colorado 80305, USA  K. W.
Lehnert<EMAIL_ADDRESS>JILA, National Institute of
Standards and Technology and the University of Colorado, Boulder, CO 80309,
USA  Department of Physics, University of Colorado, Boulder, CO 80309, USA

###### Abstract

We perform state tomography of an itinerant squeezed state of the microwave
field prepared by a Josephson parametric amplifier (JPA). We use a second JPA
as a pre-amplifier to improve the quantum efficiency of the field quadrature
measurement (QM) from 2 % to $36\pm 4~{}\%$. Without correcting for the
detection inefficiency we observe a minimum quadrature variance which is
$68^{+9}_{-7}~{}\%$ of the variance of the vacuum. We reconstruct the state’s
density matrix by a maximum likelihood method and infer that the squeezed
state has a minimum variance less than 40 % of the vacuum, with uncertainty
mostly caused by calibration systematics.

Josephson parametric amplifier, squeezed state, quantum state tomography

###### pacs:

42.50.Dv, 42.50.Lc, 03.67.Bg

Fundamental quantum optics experiments at microwave frequencies have been
recently performed with superconducting qubits or Rydberg atoms inside high-
quality microwave cavities. Examples include the reconstruction of the Wigner
functions of Fock states from one Houck et al. (2007) to a few photons and
coherent superpositions of few photons Deléglise et al. (2008); Hofheinz et
al. (2008, 2009). States such as these, which are manifestly nonclassical
light states, are crucial for quantum information processing, because they can
be used to generate entanglement. However, in the cited experiments, these
states are confined in cavities. Therefore distributing entanglement to
separate parties, as required in quantum communication protocols, remains
challenging for microwave implementations. In contrast to the discrete Fock
state approach, continuous variables quantum information (CVQI) strategy uses
another type of nonclassical states, the squeezed states, which are readily
created in itinerant modes. These states exhibit reduced noise, below the
vacuum fluctuations, in one of their quadrature components and amplified noise
in the other one. They are also easily generated at optical frequencies in the
itinerant output modes of parametric amplifiers made of optically nonlinear
crystals. At optical frequencies, CVQI has progressed rapidly from the initial
creation of squeezed states Slusher et al. (1985) and tomographic
reconstruction Smithey et al. (1993); Schiller et al. (1996); Breitenbach et
al. (1997) of those states to teleportation Furusawa et al. (1998); Yonezawa
et al. (2007) and quantum error correction Aoki et al. (2009); Lassen et al.
(2010).

At microwave frequencies, the field is less advanced. The generation of
microwave squeezed states using the nonlinear electrical response of
superconducting Josephson junctions has been reported Yurke et al. (1988),
with inferred squeezing down to $10~{}\%$ of vacuum variance Castellanos-
Beltran et al. (2008). Such states can be powerful tools for quantum
information processing and communication because microwaves and
superconducting qubits can mimic useful light–atom interactions, as
demonstrated in Wallraff et al. (2004). Furthermore, these devices are made of
compact and integrable electrical circuits, with much promise for building
complex quantum information processors. The lack of an efficient quadrature
measurement (QM) for itinerant modes has slowed the advancement of CVQI.
However, as demonstrated recently in Teufel et al. (2009), it is possible with
a JPA to realize an efficient single QM.

In this Letter, we report the tomography of an itinerant squeezed microwave
field. We demonstrate that our JPA based measurement scheme has a quantum
efficiency $20$ times greater than a QM employing state-of-art semiconductor
amplifiers. We infer the quantum state prepared by maximum likelihood
tomography, correcting for inefficiency in our QM. We discuss the achieved
degree of squeezing, from the perspective of generating entanglement on chip.

Homodyne tomography is a standard experimental tool to infer the quantum state
of a single mode of light. It was proposed in Vogel and Risken (1989) and
pioneered on a squeezed optical field in Smithey et al. (1993). Its principle
is depicted in the Fig. 1. A homodyne detection apparatus measures the value
of the quadrature $X_{\theta}$, where $\theta$ is set by adjusting the phase
of the local oscillator. The probability density function
$\textrm{pr}(X_{\theta})$ for measuring a particular value of $X_{\theta}$ is
the marginal density function of the Wigner function, i.e.
$\textrm{pr}(X_{\theta})=\int
dX_{\theta+\pi/2}W(X_{\theta},X_{\theta+\pi/2})$, as shown in Fig. 1 (b). Thus
by performing measurements of $X_{\theta}$ on many identical copies of the
state and varying $\theta$, the “hidden” quantum object can be seen from
different angles and its state inferred. Losses and other Gaussian noise
sources in the homodyne detector can be modeled with the insertion of a
fictitious beam splitter of transmissivity $\eta$, as shown in Fig. 1 (a). In
such a case, the measured $\textrm{pr}(X_{\theta})$ are no longer the
projections of the desired Wigner function $W$, but of a smoother distribution
which is the convolution of $W$ with a Gaussian Wigner function Leonhardt and
Paul (1993). However methods like maximum likelihood quantum state tomography
can be used to deconvolve the effect of inefficiency Lvovsky and Raymer
(2009).

Figure 1:  Principle of the experiment. (a-left): The squeezer (SQ, in red)
prepares a squeezed state whose quadrature distributions are measured for
different phases $\theta$ with an efficiency $\eta$. (a-right): Simulated
measurement results for 20,000 realizations of creating the squeezed state and
measuring it at a single $\theta$. The top graph shows the measured quadrature
value versus realization number. The bottom plot is a histogram (blue circles)
and Gaussian probability distribution $\textrm{pr}(X_{\theta})$ (red curve) of
this random process. (b): Graphical Interpretation: the probability
distribution $\textrm{pr}(X_{\theta})$ is simply the projection of the Wigner
function.

At optical frequencies, $\eta\geq 90~{}\%$ is routinely obtained using a pair
of balanced photodiodes Lvovsky and Raymer (2009). Such detectors are not
available for microwaves and until recently the best setup was a chain of
phase-insensitive amplifiers followed by a mixer, or two such chains in
parallel Menzel et al. (2010); Mariantoni et al. (2010); Bozyigit et al.
(2010). In such a case, noise $A_{n}$ greater than $1/2$ (the vacuum variance)
must be added to the QM Caves (1982). This noise can be modeled as an
effective efficiency by the relation $\eta=1/(1+2A_{n})$ Leonhardt and Paul
(1994), so the QM efficiency using phase-insensitive amplifiers is limited to
$50~{}\%$. State of the art microwave amplifiers, high-electron-mobility
transistors (HEMTs), have $A_{n}\approx 10-20$. In practice, the unavoidable
losses present in a microwave experiment typically result in $\eta\approx 2$%.
However, as demonstrated in Teufel et al. (2009), inserting a JPA used as a
single quadrature preamplifier before the HEMT increases the experimentally
achieved $\eta$ by a factor of approximately 20.

To perform a high-quality reconstruction of the Wigner function of a squeezed
microwave state we operate two JPAs in series, as shown in Fig. 2 (b). The
first JPA, referred to as the squeezer (SQ), prepares the squeezed state. The
second JPA, referred to as the pre-amplifier (AMP), amplifies the quadrature
of the squeezed state determined by the phase difference $\theta$ between the
AMP and the SQ pump tones. We vary $\theta$ by applying to the two cavities
pump tones slightly detuned from one another. The SQ stage is pumped at $7.45$
GHz, while the AMP stage is pumped at $100$ kHz higher frequency; therefore,
sweeping $\theta$ through $2\pi$ every $10$ µs.

Our implementation of an SQ or an AMP at microwaves, as shown in Fig. 2 (a),
requires three elements: (i) a JPA used in reflection, (ii) a directional
coupler and (iii) a circulator. As described in Castellanos-Beltran et al.
(2008), the JPAs are nonlinear resonant cavities built from coplanar
waveguides whose central conductor has been replaced by a series of many
Josephson junctions. The Josephson junctions’ nonlinearity causes the cavity’s
phase velocity to be intensity dependent. Therefore when the cavity is pumped
it becomes a phase sensitive amplifier for input modes whose frequencies lie
within the bandwidth of the JPA centered on the pump frequency. Such microwave
modes incident on the JPA are reflected and exit the cavity with one
quadrature amplified and the other squeezed, depending on their phase relative
to the pump’s phase. A directional coupler is used to add the pump tone to the
incident signal and remove the pump tone from the reflected signal. Finally
the incident and reflected modes are separated into different cables using a
circulator.

Figure 2:  (a): To implement a SQ or AMP at microwaves, three microwaves
components are required: (i) a JPA, (ii) a directional coupler and (iii) a
circulator. Taking port (1) of the directional coupler as reference, (2) is
the weakly coupled port, (3) the isolated port and (4) the direct port. Port
(2) is used to pump the JPA. Port (3) is used to apply a cancelation tone
(adjusted with a room temperature attenuator and phase shifter) that nulls the
pump and displaces the output of the JPA back to the origin of the phase
space. (b): Schematic of the experiment. In this figure, all the microwave
components and cables are considered lossless; their imperfections are
absorbed into the experimentally determined total transmissivities $\xi$,
$\alpha$ and $\beta$.

Following Fig. 2 (b), in the limit of large HEMT power gain $G_{H}$, our
quantum efficiency can be cast as

$\eta=\frac{\alpha}{2+2A_{A}-\alpha+[2A_{H}-(1-\beta)]/G_{A}\beta},$ (1)

where $A_{A}$ ($A_{H}$) is the AMP (HEMT) added noise, $\alpha$ ($\beta$) is
the fraction of power transmitted by the microwave circuitry between the SQ
and the AMP (the AMP and the HEMT), and $G_{A}$ is the power gain of the AMP
stage. A detailed description of how we calibrate each of these parameters is
in the supplementary information. Briefly, we inject different amounts of
thermal noise into the amplifier chain while operating each JPA either as an
amplifier (ON) or as a noiseless element with unit gain (OFF). We then infer
the added noise and loss of the elements by observing the variation in the
noise at the output of the measurement chain. The thermal noise is varied by
connecting the input of the SQ through a switch to either a “hot load” (50
$\Omega$ microwave termination at $4.1$ K) or a “cold load” (at $20$ mK).
Although the tomography is only performed with the “cold load”, both are
required for calibration. We obtain $A_{A}=0.25\pm 0.06$, $A_{H}=17.3\pm 0.1$,
$\alpha=68\pm 2~{}\%$ and $\beta=74\pm 5~{}\%$. However, as the switch is
operated at the 4.1 K stage and is slightly lossy, the state presented at the
input of the SQ with the “cold load” is not pure quantum vacuum, but a low
occupancy thermal state with average photon number $\overline{n}\simeq 0.15\pm
0.15$. One quadrature of the resulting squeezed state is then amplified at the
AMP stage with sufficient gain $G_{A}=180$ such that the noise in the
amplified quadrature exceeds $A_{H}$ for any $\theta$. From Eq. (1), we
obtained an overall quantum efficiency of $36\pm 4~{}\%$, which can be
compared to $\eta\approx 2~{}\%$ without the AMP stage.

In this experiment our uncertainty in $\eta$ and $\overline{n}$ create a
systematic source of error. We thus perform our data analysis under three
assumptions (1) high efficiency ($\eta=0.40$) and high mean photon number
($\overline{n}=0.30$), (2) best estimate for both efficiency ($\eta=0.36$) and
mean photon number ($\overline{n}=0.15$), and (3) low efficiency ($\eta=0.33$)
and low mean photon number ($\overline{n}=0$). These three cases give us
“pessimistic”, “best-guess”, and “optimistic” analyses, in terms of the purity
of the squeezed state estimated by the tomography. Using a lower estimate for
$\eta$ and $\overline{n}$ as inputs to the tomography algorithm causes it to
return a more pure, more squeezed, and therefore a more “optimistic” estimate
of the squeezed state. Associated with each of these three cases, we also have
statistical uncertainty, so the given error bounds cover an interval that
includes both uncertainties around the “best-guess” estimate. They are
reported in the form $X_{-L}^{+U}$, where $X$ is the statistical mean using
the “best-guess” calibration and $L$ and $U$ are respectively the lower and
upper bounds of the one standard deviation uncertainty in the “pessimistic”
and “optimistic” cases.

We must also calibrate the QM to convert the measured voltage noise into units
of noise quanta. In optical homodyne tomography, this is usually done by
inserting the vacuum and observing the quadrature noise. Analogously, we
insert the weak thermal state with mean photons $\overline{n}$ (by simply
turning the SQ stage OFF) and measure voltages proportional to quadrature
values at many $\theta$, as shown (in blue) in Fig. 3. As expected this
voltage noise is $\theta$ independent, with a variance $\Delta
V_{\mathrm{SQ,OFF}}^{2}=3.2\times 10^{-5}~{}\textrm{mV}^{2}$. Under the
convention that vacuum has variance $1/2$ in unitless quadrature space (or in
units of “quanta”), we calibrate this voltage variance to $\Delta
X_{\mathrm{SQ,OFF}}^{2}=(1-\eta)/2+\eta(1/2+\overline{n})=0.55_{-0.05}^{+0.07}$
quanta. Therefore the desired conversion factor $\Delta
X_{\mathrm{SQ,OFF}}^{2}/\Delta
V_{\mathrm{SQ,OFF}}^{2}=1.71_{-0.17}^{+0.20}\times
10^{4}~{}\mathrm{quanta/mV}^{-2}$ is used to rescale the variances in Fig. 3
(c).

Figure 3:  (a): Density plot of number of occurrences in a $1~{}$µV bin size
of the amplified quadrature voltage $V_{\theta}$ versus $\theta/2\pi$, with
the SQ pump OFF (top) and ON (bottom). (b): In particular, histograms of
$V_{\theta}$ at the maximum of squeezing: data ($\circ$) and Gaussian fit
(continuous lines) for the SQ pump OFF (blue) and ON (red). (c): Noise
variance $\Delta X^{2}_{\theta}$ in quanta units on a log scale versus
$\theta/2\pi$ for the SQ pump ON (red) and OFF (blue). The (black) line
indicates our estimate of the vacuum noise level under the “best-guess”
calibration.

In Fig. 3 (a), we show QM data of the squeezed state. With SQ ON (red) we
observe the characteristic phase dependent noise for a squeezed state. At the
phase for which the variance is minimum, we show the histogram of quadrature
measurements in Fig. 3 (b). The SQ OFF histogram is clearly wider than the SQ
ON histogram, demonstrating our ability to observe squeezing directly at the
output of our measurement chain. In Fig. 3 (c) we plot the variance of the QM
with SQ ON and OFF as a function of $\theta$, expressed in units of quanta,
clearly showing squeezing below the vacuum level. Without correcting for
$\eta$, we observe a minimum quadrature variance which is $\Delta
X^{2}_{\mathrm{SQ,MIN}}=68_{-7}^{+9}~{}\%$ of the vacuum variance.

To infer the quantum state created by the squeezer, correcting for loss during
the QM, we used maximum likelihood quantum state tomography Hradil et al.
(2004). For each of the three calibration cases, we performed 35
reconstructions using independent subsets each containing 10,000 QMs of the
total measured data. We estimated statistical uncertainty from the spread of
properties (such as fidelity or minimum variance) of the set of 35
reconstructions. The statistical uncertainty was significantly lower than the
systematic uncertainty. In Fig. 4 we show the Wigner function of the “best-
guess” reconstructed state $\rho$. The pure squeezed vacuum state
$|\psi\rangle$ that has the highest fidelity with $\rho$ has minimum
quadrature variance $6.0_{-1.1}^{+1.4}~{}\%$ of the vacuum variance, and that
maximum fidelity is $F=\langle\psi|\rho|\psi\rangle=0.81_{-0.17}^{+0.16}$. As
explained in the supplementary information, the minimum variance of $\rho$ is
biased by an amount comparable to our systematic uncertainty, so we infer the
minimum variance $\Delta x^{2}_{\mathrm{SQ,MIN}}$ directly from the observed
minimum variance as $\Delta x^{2}_{\mathrm{SQ,MIN}}=(1/\eta)(\Delta
X^{2}_{\mathrm{SQ,MIN}}-(1-\eta)/2)$. We find $\Delta
x^{2}_{\mathrm{SQ,MIN}}=12^{+30}_{-12}~{}\%$ of the vacuum variance. For
comparison, the most highly squeezed optical state ever made has a variance of
only $7~{}\%$ of the vacuum variance Mehmet et al. (2010).

Figure 4: Mean of 35 reconstructions of the Wigner function of the state
exiting the SQ, inferred by maximum likelihood under the “best-guess”
assumption, in quanta units. The faint pattern of ripples extending from the
origin is caused by truncation at 30 photons of the density matrix used to
represent the state. The white circle at the origin shows the full-width at
half-maximum of the vacuum state.

Producing squeezed states of itinerant modes allows the generation of
distributable entanglement by sending two copies of a squeezed vacuum state
through the two input ports of a balanced beam splitter. The coherent
information Schumacher and Nielsen (1996) is one useful way to characterize
the entanglement between the two output modes. The asymptotic number of
maximally entangled qubit pairs (e-bits) that can be distilled per copy of the
noisy entangled state, by using local operations and one-way classical
communication, is at least as large as the coherent information Devetak and
Winter (2005). Given two copies of $\rho$, one could make two entangled modes
with $2.5^{+1.0}_{-0.4}$ e-bits of coherent information.

In conclusion, we have reconstructed the Wigner function of an itinerant
squeezed microwave field generated at the output of a Josephson Parametric
Amplifier. Using a second JPA as a preamplifier has increased the quantum
efficiency of the microwave homodyne detection from approximately $2~{}\%$ to
$36~{}\%$. The level of squeezing is primarily limited by noise added to the
squeezed state by the JPA. Improving the performance of the JPAs (as both
squeezers and phase-sensitive amplifiers) will require more detailed
investigation of the source of this noise. We used maximum likelihood quantum
state tomography to deconvolve the QM inefficiency in order to precisely
characterize the state generated. This is an important step toward generating
easily distributable microwave entanglement on chip.

Notes: A different method was recently used to obtain a similar state
reconstruction Eichler et al. (2010).

###### Acknowledgements.

The authors acknowledge support from the DARPA/MTO QuEST program.

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##  I Supplementary Materials for “Quantum state tomography of an itinerant
squeezed microwave field”

##  II Data acquisition and calibration

Determining the amplifier added noise and loss requires several calibration
steps that permit us to isolate the effect of a specific loss or added-noise
contribution to the overall efficiency of the homodyne measurement. The
crucial aspect that makes this calibration possible is that the JPA cavities
have widely-tunable resonance frequencies, adjusted by imposing a magnetic
flux Castellanos-Beltran et al. (2008, 2009). Far from resonance the JPA
cavities behave as open circuits. They are simply mirrors that reflect the
microwave field without otherwise transforming it; therefore, either the SQ or
AMP or both stages can effectively be bypassed.

We begin with both JPA stages bypassed, so that they have
$G_{\rm{S}}=G_{\rm{A}}=1$. If the switch were lossless, when it is connected
to the cold load, the noise power exiting the HEMT amplifier would be
$S=G_{H}(A_{H}+S_{f})$, where
$S_{f}=(1/2)+n_{f}=(1/2)+[\exp(\hbar\omega/k_{B}T_{f})-1]^{-1}$ and $T_{f}$ is
the refrigerator’s temperature. Notice that the result doesn’t depend on the
transmissivities $\alpha$, $\beta$, or $\xi$ because these are at the same
temperature as the cold load, consequently each loss component emits as much
power as it absorbs. However, with the switch connected to the hot load, the
expression for the total power at the output becomes
$S=G_{H}(A_{H}+(\xi\alpha\beta)S_{h}+(1-\xi\alpha\beta)S_{f})$, with
$S_{h}=(1/2)+n_{h}=(1/2)+[\exp(\hbar\omega/k_{B}T_{h})-1]^{-1}$ and
$T_{h}=4.1$ K. In both cases, we expect and observe that $S$ depends linearly
on $S_{f}$ with an offset. By fitting these linear dependencies we can extract
$G_{H}$, $A_{H}$, and the product $\xi\alpha\beta$.

We cannot assume that the switch is lossless. Because its loss sits at 4.1 K,
it will always emit noise power $S_{h}(1-\lambda)+S_{in}\lambda$, where
$S_{in}$ is the incident noise and $\lambda$ is the switch transmissivity. So,
even when $n_{f}\ll 1/2$, the state presented at the SQ stage will have
average thermal occupancy $\bar{n}=(1-\lambda)\xi n_{h}$. We write the noise
power at the output as function of $S_{f}$, for switches in both positions as

$S_{1c}=G_{H}A_{H}+S_{h}G_{H}(1-\lambda)\xi\alpha\beta+S_{f}[G_{H}\lambda\xi\alpha\beta+G_{H}(1-\xi\alpha\beta)]=b_{1c}+m_{1c}S_{f}$
(2)

$S_{1h}=G_{H}A_{H}+S_{h}G_{H}(\xi\alpha\beta)+S_{f}[G_{H}(1-\xi\alpha\beta)]=b_{1h}+m_{1h}S_{f},$
(3)

where the subscript $1c$ ($1h$) corresponds to the switch connected to the
cold (hot) load. Fitting our noise data to the right hand side of Eq. 2 and 3,
we can obtain the four parameters $b_{1h},b_{1c},m_{1h}$ and $m_{1c}$. However
as these parameters are not independent,
$S_{h}=(b_{1h}-b_{1c})/(m_{1c}-m_{1h})$, we cannot extract the switch loss
independently. We can nevertheless bound this unknown loss by taking a worst
case estimate as the manufacturers minimum specified transmission (at room
temperature) $\lambda=0.83$ and assuming it is less lossy at 4.1 K. We
moreover confirmed that at room temperature the frequency dependent loss of
the switch is within the manufacturer’s specification. Then by using
$1<\lambda<0.83$, we can bound the desired parameters using Eq. 2 and 3, with
the expressions $(\xi\alpha\beta)^{-1}=1+m_{1h}S_{h}\lambda/(b_{1h}-b_{1c})$,
and $G_{H}=m_{1h}/(1-\xi\alpha\beta)$, and
$A_{H}=(b{1c}/G_{H})-(1-\lambda)S_{h}(\xi\alpha\beta)$.

We then perform the same analysis, finding the linear dependence of the output
noise on $S_{f}$ and on the switch position, with the AMP ON and SQ OFF. From
these fits and knowledge of $A_{H}$ and $G_{H}$ we find $\xi\alpha$, $A_{A}$,
and$G_{A}\beta$. Finally, we operate the experiment with AMP OFF and SQ ON. A
third time we fit the linear dependence of $S$ on $S_{f}$ with the switch in
both positions, determining $\xi$, $\alpha$ and $\beta$ separately (Fig. 5b).
We evaluate the expressions for $\alpha$, $\beta$, $A_{A}$, $A_{H}$, $G_{H}$
and $\bar{n}$ at the bounds on $\lambda$, finding the range of values in the
main text. We also find $\xi=-9.9\pm 1$ dB, of which 6 dB arises from an
attenuator that has been placed at the input of the SQ stage.

Figure 5: The noise density $S$ in arbitrary units at the output of the
measurement versus refrigerator temperature $T_{f}$. a.) Data acquired with
the AMP and SQ OFF and the switch connected to the cold load (circles) and hot
load (squares). The lines are linear fits to $S$ versus $S_{f}$ for the case
of the switch connected to the cold load (solid) and hot load (dashed). b.)
Data acquired with the AMP ON and SQ OFF (blue) and AMP OFF and SQ ON (red),
with the switch connected to the cold load (circles) with a linear fit (solid)
and hot load (squares) with a linear fit (dashed). The arbitrary y-scale is
consistent between the six plots. The linear fits do not appear as lines
because we plot $S$ versus $T_{f}$ rather than $S_{f}$.

To acquire these calibration data sets, we regulate the refrigerator’s
temperature at 10 values between base temperature ($T<50$ mK) and 800 mK,
which requires about 7 hours to complete. For each temperature point we
measure the noise at the output under all six conditions, 2 switch positions,
and 3 amplifier configurations (AMP OFF SQ OFF, AMP ON SQ OFF, and AMP OFF SQ
ON). We inject a tone detuned from the AMP pump by 20 kHz. By dividing the
noise power at the output of the chain by the power in this tone, we become
insensitive to any variation in $G_{H}$ over the time needed to acquire the
data. At the end of the calibration, we immediately operate the experiment
with SQ ON and AMP ON, to acquire the data in the paper. In addition, we use
the tone to ensure that we do not saturate the amplifier chain.

Data is acquired by digitizing the output (IF port) of the mixer at rate of
$10^{7}$ samples per second. We filter the IF port with a 5 MHz anti-aliasing
low-pass filter. The digitized data is digitally filtered with a 3rd-order
Butterworth high-pass filter with a 500 kHz corner (3 dB) frequency. The noise
density $S$ is the average noise density in the frequency range between 500
kHz and 5 MHz.

##  III Maximum likelihood analysis of the squeezed state

Table 1:  Inferred properties of the squeezed state, upon our three analysis assumptions. | Pessimistic | Best guess | Optimistic
---|---|---|---
Fidelity | $0.66\pm 0.02$ | $0.807\pm 0.016$ | $0.960\pm 0.005$
Min. var. of comparison pure state $\left|\psi\right\rangle$111Ratio of the
variance of the squeezed quadrature of the pure squeezed vacuum state with
highest fidelity to the variance of the vacuum.:
| $0.065\pm 0.009$ | $0.060\pm 0.003$ | $0.0493\pm 0.0006$
$\rho$’s purity | $0.62\pm 0.02$ | $0.74\pm 0.02$ | $0.96\pm 0.01$
$\rho$’s sq. var.222Ratio of variance of most likely state $\rho$’s squeezed or anti-squeezed quadrature to the variance of the vacuum. | $0.918\pm 0.002$ | $0.484\pm 0.013$ | $0.304\pm 0.008$
$\rho$’s anti-sq. var.222Ratio of variance of most likely state $\rho$’s squeezed or anti-squeezed quadrature to the variance of the vacuum. | $25.54\pm 0.07$ | $20.17\pm 0.06$ | $19.18\pm 0.05$
Coherent info.333Coherent information (in e-bits) that could be produced with two copies of the squeezed state and a beam splitter. | $2.19\pm 0.08$ | $2.46\pm 0.09$ | $3.42\pm 0.05$
Linear sq. var.444Direct linear inference of the squeezed state’s minimum variance, relative to vacuum variance. | $0.40\pm 0.02$ | $0.12\pm 0.02$ | $-0.18\pm 0.02$

Table 1 shows the statistical errors in our estimates of inferred parameters
characterizing the squeezed state, for the three analysis cases, based upon
our systematic calibration uncertainties. The first line presents the fidelity
$F=\langle\psi|\rho|\psi\rangle$, where $\rho$ is the maximum likelihood
reconstructed density matrix of the field exiting the SQ and $|\psi\rangle$ is
the pure vacuum squeezed state that maximizes the fidelity. The second line
gives the ratio of the minimum variance of $|\psi\rangle$ to the variance of
vacuum. The third line gives the purity $\mathrm{Tr}({\rho^{2}})$ of $\rho.$
The fourth and fifth lines give the ratios of the squeezed and anti-squeezed
variances of the reconstructed state to the variance of vacuum. The sixth line
presents the coherent information that could be obtained by combining on a
beam splitter two copies $\rho$. The last line gives our estimate the the
experimental states’ minimum variance based on direct linear inference.

We have stated three variances that characterize the state created in this
experiment: the linear estimate of the experimental state’s minimum variance
($12~{}\%$), the most likely state $\rho$’s minimum variance ($48~{}\%$), and
the minimum variance of the pure squeezed vacuum state
$\left|\psi\right\rangle$ that maximizes the fidelity with $\rho$
($6.0~{}\%$). Here we give more discussion of these variances.

The quadrature measurements we observe are the linear combination of the
quantum state created by the squeezer and vacuum fluctuations:

$X_{\theta}=\sqrt{\eta}x_{\theta}+\sqrt{(1-\eta)}y_{\theta},$

where $x_{\theta}$ is the quadrature of the squeezed state, and $y_{\theta}$
is the quadrature of the vacuum state. Solving for $x_{\theta}$ gives

$x_{\theta}=\frac{1}{\sqrt{\eta}}\left(X_{\theta}-\sqrt{\left(1-\eta\right)}y_{\theta}\right).$

Therefore the inferred variance of the squeezed state’s quadrature $\Delta
x_{\theta}^{2}$ is

$\Delta x_{\theta}^{2}=\frac{1}{\eta}\left[\Delta
X_{\theta}^{2}-\left(1-\eta\right)\Delta y_{\theta}^{2})\right].$

The vacuum variance $\Delta y_{\theta}^{2}=1/2$, and we can easily calculate
an unbiased estimate of $\Delta X_{\theta}^{2}$ for every phase $\theta$. This
gives us an unbiased estimate of $\Delta x_{\theta}^{2}$ that does not depend
on the details (for example, Gaussianity) of the quantum state. We calculate
$\Delta x_{\theta}^{2}$ using 20,000 quadrature measurements at each of 100
evenly spaced $\theta$ and calculate the minimum value $\Delta
x_{\mathrm{SQ,MIN}}^{2}$, in Table 1. The statistical uncertainties show one
standard deviation in the estimate of $\Delta x_{\mathrm{SQ,MIN}}^{2}$. For
the “optimistic case” we calculate a negative variance, which is clearly
unphysical. This is a sign of inconsistency in the “optimistic” calibration
parameters. Because the “optimistic” estimate for the squeezed state is
computed using the lower bounds on $\eta$ and $\overline{n}$, this negative
variance is evidence that the detector’s true $\eta$ and / or effective gain
($\Delta X_{\mathrm{SQ,OFF}}^{2}/\Delta V_{\mathrm{SQ,OFF}}^{2}$) must be
larger than the lower bounds set by calibration.

The minimum variance of $\rho$ is significantly higher than this linear
estimate. This is caused by bias in the maximum likelihood method. Quantum
state estimation by maximum likelihood is biased toward more mixed states, and
the amount of bias increases with increasing purity of the state from which
the measurements are drawn Glancy et al. (2009). Based on numerical
experiments, the bias in our estimates of the fidelity should be well below
the uncertainty level set by systematic effects. However, the bias in our
estimates of the minimum variance of the inferred state could be larger. To
attempt to quantify this effect, we simulated measuring and performing maximum
likelihood tomography on a Gaussian state. This Gaussian state is chosen to
have minimum and maximum variances equal to those calculated by the linear
method described above for the “best-guess” case. By computer we simulate
10,000 quadrature measurements (the same number we used for ML analysis of the
true experiment) from this Gaussian state and perform maximum likelihood
tomography on those measurements. The inferred state has minimum variance
$40~{}\%$. Therefore it is possible that the experimental state has smaller
minimum variance than the most likely state inferred from only 10,000
measurements. Because we have some independent evidence for non-Gaussian
effects in the experiment, we cannot quantify this size of this bias using
this Gaussian simulation. Other numerical simulations have confirmed that this
bias decreases as the number of measurements analyzed increases and that this
bias is not caused by truncation of the Hilbert space at 30 photons.

The apparent discrepancy between the $6~{}\%$ for the variance of
$|\psi\rangle$ and the $48~{}\%$ for the variance of $\rho$ also deserves some
comments. It is important to note that one would not expect the minimum
variance of a mixed state to equal the minimum variance of its highest
fidelity pure state. The fidelity between a mixed Gaussian state (centered at
the origin of phase space) whose minimum and maximum variances are $v_{x}$ and
$v_{p}$ and a pure squeezed vacuum state with minimum variance $v_{s}$ is
given by

$F_{\mathrm{Gauss}}=\frac{2}{\sqrt{\frac{(1+4v_{s}v_{p})(v_{s}+v_{x})}{v_{s}}}}.$

The highest fidelity pure state has minimum variance
$v_{s}=\frac{1}{2}\sqrt{\frac{v_{x}}{v_{p}}}$, and the fidelity between these
two states is

$F_{\mathrm{Gauss,max}}=\frac{2}{1+2\sqrt{v_{x}v_{p}}}$

Consider the state $\sigma$ to be a Gaussian state with minimum variance of
$48~{}\%$ and maximum variance $2017~{}\%$. ($\sigma$ has variances equal to
those of our state $\rho$, but unlike $\rho$, $\sigma$ is guaranteed to be
Gaussian.) Then let $|\psi\rangle$ be the pure squeezed vacuum state that has
maximum fidelity with $\sigma$.
$F_{\mathrm{Gauss,max}}=\langle\psi|\sigma|\psi\rangle=0.49$, and the minimum
variance of $|\psi\rangle$ is $7.7~{}\%$. The difference between the minimum
variances of $\rho$ and $\left|\psi\right\rangle$ is to be expected. However,
the maximum fidelity of $\rho$ is significantly larger than we would expect if
it was perfectly Gaussian. This non-Gaussianity could be caused by bias in the
maximum likelihood inference and / or genuine non-Gaussian effects in the
experiment.

Tomographic reconstruction of a quantum state requires that the experimental
device always creates the same (potentially mixed) quantum state, that the
measurements are well described by inefficient quadrature measurements, and
that the calibration of those measurements is consistent. In this experiment
we have observed some evidence that at least one of these assumptions is
violated. The likelihood of the maximum likelihood state is significantly
lower than one should expect from simulated measurements on that state. That
is, if the tomographic assumptions above were true, we expect to find a
significantly higher value for the maximum likelihood. We believe this effect
could be caused by an interaction between the state preparation and
measurement stages of the experiment, such as a phase dependent efficiency of
the measurement JPA, and/or non linear processes in the measurement.