# Polarons in supersolids:
path-integral treatment of an impurity in a one-dimensional dipolar supersolid

Laurent H. A. Simons  Theory of Quantum and Complex Systems, Physics
Department, Universiteit Antwerpen, B-2000 Antwerpen, Belgium  Michiel Wouters
Theory of Quantum and Complex Systems, Physics Department, Universiteit
Antwerpen, B-2000 Antwerpen, Belgium  Jacques Tempere  Theory of Quantum and
Complex Systems, Physics Department, Universiteit Antwerpen, B-2000 Antwerpen,
Belgium

###### Abstract

The supersolid phase of a dipolar Bose-Einstein condensate has an intriguing
excitation spectrum displaying a band structure. Here, the dressing of an
impurity in a one-dimensional dipolar supersolid with the excitations of the
supersolid is studied. The ground-state energy of the supersolid polaron is
calculated using a variational path integral approach, which obtained accurate
results for other polaron systems within the Bogoliubov and Fröhlich
approximations. A divergence is observed at the superfluid-supersolid phase
transition. The polaron radius is also computed, showing that for strong
impurity-atom interactions, the polaron can become localized to a single
droplet, behaving like a small solid-state polaron.

##  I Introduction

A supersolid is a state of matter that exhibits at the same time properties of
a superfluid, such as frictionless flow, and properties of a solid, such as
periodic density modulations unrelated to external potentials [1, 2, 3, 4, 5].
There are three main features associated with supersolidity: spontaneously
arising periodic density modulations, global phase coherence (due to the
superfluid fraction of the supersolid), and phase rigidity [5]. Solid helium
has been a candidate system for supersolidity. However, early experimental
claims of supersolidity in solid helium [6] were later disproved [7], and no
new evidence has emerged at the time of writing [1, 5, 8]. Recently,
supersolidity has been observed in ultracold gases such as spin-orbit-coupled
Bose-Einstein condensates (BECs) and dipolar BECs [8, 5, 9, 1, 10]. The
dipolar supersolid realization is of great interest as the supersolidity comes
from the intrinsic dipole-dipole interactions and is not due to, for example,
a superimposed optical lattice. Different systems and properties of dipolar
supersolids have been studied recently, such as vortices, excitations, and its
superfluid fraction [11, 12, 2, 13]. However, the effect of an impurity has
yet to be studied in the context of a dipolar supersolid. The problem of an
impurity in a Bose-Einstein condensate has been studied extensively for
superfluid contact and dipolar Bose gases [14, 15, 16, 17, 18, 19, 20, 21, 22,
23, 24, 25, 26, 27, 28, 29]. The polaron effect, where the impurity gets
dressed by the excitations of the system, is particularly interesting as in
the supersolid regime the excitation spectrum displays a band structure with
two Goldstone phonon modes compared to the single phonon mode of a superfluid
Bose gas [2].

This paper studies the dressing of a neutral impurity, which can have a finite
magnetic moment, with the excitations of a one-dimensional dipolar supersolid.
The polaronic energy is calculated within the Fröhlich and Bogoliubov
approximations suited for weak impurity-atom and atom-atom interactions. The
variational Feynman path-integral method is used, which has been shown to give
results comparable to Monte Carlo methods while being numerically less
intensive and time-consuming for other polaron problems [21, 25, 30]. The
effect of different system parameters on the polaronic energy is studied. A
divergence in the polaronic energy is observed when crossing into the
supersolid region, which can be another indication of supersolidity
experimentally. The radius of the polaron is calculated, and it is found that
for specific values, the polaron becomes localized to a single droplet of the
supersolid and behaves like a small solid-state polaron, hopping from one
droplet to another.

The paper is organized as follows. In Section II, the Hamiltonian describing
the supersolid polaron is derived within the Bogoliubov and Fröhlich
approximations. Section III discusses the description of the one-dimensional
dipolar supersolid itself. The wavefunction is calculated, and the Bogoliubov-
de Gennes equation is solved to obtain the excitations. The polaronic energy
and polaron radius are calculated in Section IV using the variational path-
integral method, and the results are discussed. The conclusion and outlook are
given in the last section, Section V.

##  II Polaron Hamiltonian in the supersolid regime

The Hamiltonian describing an impurity with mass $m_{I}$ in a one-dimensional
dipolar Bose gas is given by

$\displaystyle\hat{H}$
$\displaystyle=\frac{\hat{p}_{I}^{2}}{2m_{I}}+\sum_{q}\left(\frac{\hbar^{2}q^{2}}{2m_{B}}-\mu+E_{t}\right)\hat{a}^{\dagger}_{q}\hat{a}_{q}$
$\displaystyle+\frac{1}{2L}\sum_{k,k^{\prime},q}V(q)\hat{a}^{\dagger}_{k^{\prime}-q}\hat{a}^{\dagger}_{k+q}\hat{a}_{k}\hat{a}_{k^{\prime}}$
$\displaystyle+\frac{1}{L}\sum_{k,q}V_{IB}(q)\hat{\rho}_{I}(q)\hat{a}^{\dagger}_{k-q}\hat{a}_{k}.$
(1)

The first two terms represent the free impurity and a gas of non-interacting
bosons, respectively, where $\hat{a}^{\dagger}_{q}$ and $\hat{a}_{q}$ are the
creation and annihilation operators of a dipolar atom of mass $m_{B}$ with
momentum $q$, $\mu$ is the chemical potential, and $E_{t}$ is the contribution
of the transverse confinement to the energy. The third term represents the
interaction between the dipolar atoms characterized by an interaction
amplitude given by the effective one-dimensional dipolar interaction potential
in momentum space $V(q)$. The interaction between the impurity and the bosons
is described by the last term, where $\hat{\rho}_{I}(q)=\exp(iq\hat{z}_{I})$
is the impurity density and $V_{IB}(q)$ is the Fourier transform of the
impurity-atom interaction potential. The one-dimensional dipolar Bose gas is
formed by tightly confining the condensate in the $x$ and $y$ directions using
a harmonic trap with oscillator length $l_{\perp}$. The transverse degrees of
freedom can be integrated out by using the following ansatz [31, 32]

$\psi_{\perp}(x,y)=\frac{1}{\sqrt{\pi}\sigma l_{\perp}}\exp\left(-\frac{\nu
x^{2}+y^{2}/\nu}{2(\sigma l_{\perp})^{2}}\right).$ (2)

$\sigma$ and $\nu$ are variational parameters that can be obtained by
minimizing the energy of the condensate.
$E_{t}=\hbar^{2}(1/\nu+\nu)(1/\sigma^{2}+\sigma^{2})/(4m_{B}l_{\perp})$ is the
transverse trap energy corresponding to the above transverse wavefunction[31,
32]. There is no analytical form for the Fourier transform of the effective
one-dimensional dipolar interaction potential. Nonetheless, it has been shown
that it can be described well by the following analytical expression [31, 32]

$V(q)=g_{1D}^{(B)}\left(1+\epsilon^{(B)}_{dd}\left(\frac{3\left(1+Qe^{Q}\text{Ei}(-Q)\right)}{1+\nu}-1\right)\right),$
(3)

with $g_{1D}^{(B)}=2\hbar^{2}a^{(B)}_{s}/(m_{B}\sigma^{2}l_{\perp}^{2})$ the
1D s-wave interaction strength of the dipolar bosons and $a^{(B)}_{s}$ the
s-wave scattering length of the dipolar bosons. The strength of the dipole-
dipole interaction is characterized by
$\epsilon^{(B)}_{dd}=\mu_{0}\mu_{B}^{2}m_{B}/(12\pi\hbar^{2}a^{(B)}_{s})$
where $\mu_{0}$ is the vacuum permeability, and $\mu_{B}$ is the dipole moment
of the atoms, $Q=\sqrt{\nu}(q\sigma l_{\perp})^{2}/2$, and $\text{Ei}(Q)$ the
exponential integral.

The impurity-atom interaction potential in momentum space, $V_{IB}(q)$, has
the same form as $V(q)$, expression (3), but with $g_{1D}^{(B)}$ and
$\epsilon^{(B)}_{dd}$ replaced by their impurity-boson counterparts
$g_{1D}^{(IB)}$ and $\epsilon^{(IB)}_{dd}$, respectively. In particular,
$g_{1D}^{(IB)}=\hbar^{2}a^{(IB)}_{s}/(m_{r}\sigma^{2}l_{\perp}^{2})$ is the 1D
s-wave interaction strength of the dipolar boson-impurity interaction,
$a^{(IB)}_{s}$ the s-wave scattering length for the impurity-atom
interactions,
$\epsilon^{(IB)}_{dd}=\mu_{0}\mu_{I}\mu_{B}m_{r}/(6\pi\hbar^{2}a^{(IB)}_{s})$
where $\mu_{I}$ is the impurity dipole moment, and
$m_{r}=(m_{B}^{-1}+m_{I}^{-1})^{-1}$ the reduced mass.

In the supersolid regime where there is a density modulation, the ground state
wavefunction of the dipolar gas is no longer given by the constant $\sqrt{n}$
but can be described by [31]

$\psi_{0}(z)=\sqrt{n_{0}}\left(1+\sum_{l=1}^{\infty}\Delta_{l}\cos(lk_{s}z)\right),$
(4)

with $n_{0}=N_{0}/L$ the density of atoms in the zero-momentum mode, $k_{s}$
the momentum characterizing the density modulation, and the order parameters
$\Delta_{l}$ describing the contribution of mode $l$ to the wavefunction. As
translational symmetry is broken, a dipolar atom of the supersolid can be
described by $q$, which is restricted to the first Brillouin zone (BZ) given
by $[-k_{s}/2,k_{s}/2]$ and an integer $l$ [31]. The momentum of the atom is
then given by $q+lk_{s}$. The Hamiltonian becomes

$\displaystyle\hat{H}$
$\displaystyle=\frac{\hat{p}_{I}^{2}}{2m_{I}}+\sum_{q,l}\left(\frac{\hbar^{2}(q+lk_{s})^{2}}{2m_{B}}-\mu+E_{t}\right)\hat{a}^{\dagger}_{q,l}\hat{a}_{q,l}$
$\displaystyle+\frac{1}{2L}\sum_{k,k^{\prime},q,l,m,s}V(q+sk_{s})\hat{a}^{\dagger}_{k^{\prime}-q,m-s}\hat{a}^{\dagger}_{k+q,l+s}\hat{a}_{k,l}\hat{a}_{k^{\prime},m}$
$\displaystyle+\frac{1}{L}\sum_{k,q,l,l^{\prime}}V_{IB}(q+lk_{s})\hat{\rho}_{I}(q+lk_{s})\hat{a}^{\dagger}_{k-q,l^{\prime}-l}\hat{a}_{k,l^{\prime}}.$
(5)

Here, $\hat{a}^{\dagger}_{q,l}$ and $\hat{a}_{q,l}$ are the creation and
annihilation operators of a dipolar atom of mass $m_{B}$ with momentum $q$ in
mode $l$. Similarly to the case of a neutral [18] and charged [33] impurity in
a Bose gas, the Bogoliubov approximation and transformation can be applied.
However, because of the broken translational symmetry, other modes with
momentum $q=lk_{s}$ apart from the zero-momentum mode are also macroscopically
occupied [31]. The Bogoliubov approximation is now given by
$\hat{a}_{0,l}\rightarrow\Delta_{l}\sqrt{N_{0}}/2$ with $\Delta_{0}=2$ and
$\Delta_{-l}=\Delta_{l}$ [31]. A Bogoliubov transformation

$\hat{a}_{q,l}=\sum_{l^{\prime}}u^{l^{\prime}}_{l}(q)\hat{\alpha}_{q,l^{\prime}}+v^{l^{\prime}}_{l}(q)\hat{\alpha}^{\dagger}_{-q,l^{\prime}},$
(6)

can be found such that the supersolid term of the Hamiltonian diagonalizes
with $u^{l^{\prime}}_{l},v^{l^{\prime}}_{l}$ the Bogoliubov coefficients and
$\hat{\alpha}^{\dagger}_{q,l},\hat{\alpha}_{q,l}$ the Bogoliubov creation and
annihilation operators of a dipolar boson with momentum $q$ in mode $l$. The
Bogoliubov coefficients are normalized by the following relation, which is
derived by enforcing the bosonic commutation relation

$\sum_{l^{\prime}}[(u_{l}^{l^{\prime}}(q))^{2}-(v_{l}^{l^{\prime}}(q))^{2}]=1.$
(7)

Applying it to the Hamiltonian results in

$\displaystyle\hat{H}=$
$\displaystyle\frac{\hat{p}_{I}^{2}}{2m_{I}}+\sum_{q,l}\epsilon_{l}(q)\hat{\alpha}^{\dagger}_{q,l}\hat{\alpha}_{q,l}+U(\hat{z_{I}})$
$\displaystyle+\sum_{q,l}V_{l}(q,\hat{z_{I}})\left(\hat{\alpha}_{q,l}+\hat{\alpha}^{\dagger}_{-q,l}\right),$
(8)

where

$\displaystyle V_{l}(q,\hat{z}_{I})=$
$\displaystyle\frac{\sqrt{N_{0}}}{2L}\sum_{m,m^{\prime}}V_{IB}(q+mk_{s})e^{i(q+mk_{s})\hat{z}_{I}}$
$\displaystyle\times\Delta_{m^{\prime}}\left(u^{l}_{m-m^{\prime}}(q)+v^{l}_{m-m^{\prime}}(q))\right),$
(9)

and

$U(\hat{z}_{I})=\frac{n_{0}}{4}\sum_{l,l^{\prime}}V_{IB}(lk_{s})\Delta_{l^{\prime}-l}\Delta_{l^{\prime}}e^{ilk_{s}\hat{z}_{I}}.$
(10)

This is the Hamiltonian describing an impurity in a one-dimensional dipolar
supersolid within the Bogoliubov and Fröhlich approximation. Terms with two
creation and/or annihilation operators, beyond-Fröhlich terms characterizing
the absorption and emission of two Bogoliubov excitations, are neglected and
out of scope for this paper. Compared to the neutral and charged Bose polaron
problems, the term without any creation or annihilation operators is not
constant and acts as an external periodic potential on the impurity.
Therefore, it cannot be discarded and will be denoted by $U(z_{I})$ from now
on. The problem of an impurity in a dipolar supersolid has been reduced to an
impurity interacting with multiple excitation bands via a Fröhlich-like vertex
in a periodic background potential $U(z_{I})$. The order parameters will be
calculated in the next section, and the Bogoliubov-de Gennes eigenproblem will
be solved.

##  III  Supersolid wavefunction and excitations, without impurity

To calculate the wavefunction and order parameters, the total energy of the
supersolid needs to be minimized. The (extended) Gross-Pitaevskii energy can
be used as an energy functional, which is given by [31]

$\displaystyle E_{GP}[\psi]$ $\displaystyle=-\int
dz\psi^{*}(z)\frac{\hbar^{2}}{2m_{B}}\frac{d^{2}}{dz^{2}}\psi(z)+NE_{t}$
$\displaystyle+\frac{1}{2}\int dz\int
dz^{\prime}V(z-z^{\prime})|\psi(z)|^{2}|\psi(z^{\prime})|^{2}$
$\displaystyle+\frac{2}{5}\gamma\int dz|\psi(z)|^{5}.$ (11)

The first line is the kinetic energy and the transverse trap energy, while the
second and third lines are the dipolar interaction energy and Lee-Huang-Yang
(LHY) beyond-mean-field contribution, respectively. The parameter $\gamma$
quantifies the LHY correction and is given by [31, 34]

$\gamma=\frac{256}{15\pi}\frac{\hbar^{2}(a_{s}^{(B)})^{5/2}}{m_{B}(\sigma
l_{\perp})^{3}}\int_{0}^{1}du(1-\epsilon_{dd}^{(B)}+3\epsilon_{dd}^{(B)}u^{2})^{5/2}.$
(12)

The LHY contribution is taken into account as it prevents collapse and is
essential to describe the supersolid state. Using the ansatz (4) results in an
expression for the energy in terms of the order parameters $\Delta_{l}$ and
the variational parameters $\sigma$ and $\nu$. To obtain an analytical
expression for the LHY contribution to the energy, it is assumed that the
wavefunction is always positive or zero:

$1-\sum_{l=1}^{\infty}\Delta_{l}\geq 0,$ (13)

such that the absolute value can be discarded. This is valid close to the
superfluid-supersolid transition. The energy divided by the number of atoms
$N$ is given by

$\displaystyle\frac{E_{GP}[\Delta_{l},k_{s},\sigma,\nu]}{N}=\frac{\hbar^{2}k_{s}^{2}}{4m_{B}}\frac{\sum_{l=1}^{\infty}l^{2}\Delta_{l}^{2}}{1+\sum_{l=1}^{\infty}\Delta_{l}^{2}/2}+E_{t}$
$\displaystyle+\frac{n}{32}\frac{\sum_{l,m,p=-\infty}^{\infty}\Delta_{l}\Delta_{m}\Delta_{p}\Delta_{l+m+p}V[(l+m)k_{s}]}{(1+\sum_{l=1}^{\infty}\Delta_{l}^{2}/2)^{2}}$
$\displaystyle+\frac{\gamma
n^{3/2}}{80}\frac{\sum_{l,m,p,s=-\infty}^{\infty}\Delta_{l}\Delta_{m}\Delta_{p}\Delta_{s}\Delta_{l+m+p+s}}{(1+\sum_{l=1}^{\infty}\Delta_{l}^{2}/2)^{5/2}}.$
(14)

The above expression can be minimized numerically for $\Delta_{l}$, $k_{s}$,
$\sigma$, and $\nu$, which can be substitued into (4) to obtain the ground-
state wavefunction of the supersolid.

The excitation spectrum and Bogoliubov coefficients can be obtained by
expanding the supersolid part of the Hamiltonian up to second order in
creation and annihilation operators. The LHY contribution can be taken into
account by using the following Hamiltonian, which results in the correct speed
of sound and energy correction [31]

$\hat{H}_{LHY}=\frac{2}{5}\gamma\int
dz\left(\hat{\psi}^{\dagger}(z)\hat{\psi}^{\dagger}(z)\hat{\psi}(z)\hat{\psi}(z)\right)^{5/4},$
(15)

with

$\hat{\psi}(z)=\psi_{0}(z)+\frac{1}{\sqrt{L}}\sum_{q\neq
0,l}e^{i(q+lk_{s})z}\hat{a}_{q,l}.$ (16)

The resulting quadratic Hamiltonian can be written as

$\displaystyle\hat{H}_{B}=$
$\displaystyle\sum_{q,l,m}\left(h_{q,l,m}\hat{a}^{\dagger}_{q,l}\hat{a}_{q,m}+\frac{1}{2}g_{q,l,m}\hat{a}^{\dagger}_{q,l}\hat{a}^{\dagger}_{-q,m}\right.$
$\displaystyle\left.+\frac{1}{2}g_{q,l,m}\hat{a}_{-q,m}\hat{a}_{q,l}\right),$
(17)

with

$\displaystyle h_{q,l,m}=$
$\displaystyle\left(\frac{\hbar^{2}(q+lk_{s})^{2}}{2m}-\mu+E_{t}\right)\delta_{l,m}$
$\displaystyle+\frac{n_{0}}{4}\sum_{s}V(q+sk_{s})\Delta_{m-s}\Delta_{l-s}$
$\displaystyle+\frac{n_{0}}{4}\sum_{s}V((m-l)k_{s})\Delta_{s}\Delta_{s-(l-m)}$
$\displaystyle+\frac{5}{2}\gamma
n_{0}^{3/2}\sum_{s,s^{\prime}}\frac{\Delta_{s}\Delta_{s^{\prime}}\Delta_{l-m+s+s^{\prime}}}{8},$
(18)

and

$\displaystyle g_{q,l,m}=$
$\displaystyle\frac{n_{0}}{4}\sum_{s}V(q+sk_{s})\Delta_{m+s}\Delta_{l-s}$
$\displaystyle+\frac{3}{2}\gamma
n_{0}^{3/2}\sum_{s,s^{\prime}}\frac{\Delta_{s}\Delta_{s^{\prime}}\Delta_{l+m+s+s^{\prime}}}{8}.$
(19)

The Bogoliubov-de Gennes eigenproblem is then given by

$\begin{pmatrix}h_{q}&g_{q}\\\
-g_{q}^{T}&-h_{-q}^{T}\end{pmatrix}\begin{pmatrix}u\\\
v\end{pmatrix}=\hbar\omega_{q,l}\begin{pmatrix}u\\\ v\end{pmatrix}.$ (20)

Solving the above eigenproblem results in the excitation spectrum and
Bogoliubov coefficients for the dipolar supersolid. In Figure 1, the
excitation spectrum of a one-dimensional dipolar supersolid is shown for
realistic parameters $n=2386.2/l_{\perp}$ and $a^{(B)}_{s}=0.005l_{\perp}$
(which are taken from Ref. [31]) for different dipolar interaction strengths
$\epsilon_{dd}^{(B)}$. In this case, the critical value of
$\epsilon_{dd}^{(B)}$ is approximately $1.340$. For
$\epsilon_{dd}^{(B)}<\epsilon_{dd,c}^{(B)}$, the excitation spectrum is the
same as the analytical Bogoliubov dispersion, and a roton minimum is visible,
as expected. If $\epsilon_{dd}^{(B)}$ becomes larger than the critical value,
a band structure appears with two phononic modes: a crystal phonon mode and a
superfluid phonon mode. The larger $\epsilon_{dd}^{(B)}$ becomes, the smaller
the superfluid phonon mode becomes until the independent droplet regime is
reached, where only the crystal phonon mode remains. For the values shown, the
supersolid can be described by three order parameters $\Delta_{1}$,
$\Delta_{2}$, and $\Delta_{3}$, and the condition (13) is fulfilled.

Figure 1: The excitation spectrum of a one-dimensional dipolar supersolid with
parameters $n=2386.2$ and $a_{s}^{(B)}=0.005$ in units of
$\hbar=m_{B}=l_{\perp}=1$ for different values of $\epsilon^{(B)}_{dd}$. The
critical value is given by $\epsilon^{(B)}_{dd,c}=1.34$. The results are
equivalent to the ones from Refs. [31, 35].

##  IV Polaronic energy and polaron radius

For a system with action $S$, an upper bound on the free energy can be found
by introducing an exactly solvable variational trial action $S_{0}$ and
minimizing for the variational parameters. It is given by the Feynman-Jensen
inequality

$F\leq F_{0}+\frac{1}{\hbar\beta}\langle S-S_{0}\rangle_{0},$ (21)

where $F_{0}$ is the free energy of the trial action and $\beta=1/(k_{B}T)$
the inverse temperature. For the trial action, a general exactly solvable
quadratic trial action is used, which was also applied to other polaron
problems and gave results comparable to numerically exact Monte Carlo methods
(within the Fröhlich and Bogoliubov approximations) [36, 30],

$S_{0}=\int\limits_{0}^{\hbar\beta}\frac{m_{I}}{2}\dot{\mathbf{r}}^{2}(\tau)d\tau+\frac{m_{I}}{2\hbar\beta}\int\limits_{0}^{\hbar\beta}\int\limits_{0}^{\hbar\beta}x(\tau-\sigma)\mathbf{r}(\tau)\cdot\mathbf{r}(\sigma)d\tau
d\sigma.$ (22)

Here $x(\tau-\sigma)$ is a variational memory kernel [36, 30]. The kernel
statisfies two conditions: it is imaginary-time periodic
$x(\hbar\beta-\tau)=x(\tau)$ and $\int_{0}^{\hbar\beta}x(\tau)d\tau=0$ [30].
The free energy and expectation value of the trial action can be found in Ref.
[30].

An imaginary-time action corresponding to Hamiltonian (8) can be derived
similarly as for the Bose polaron problem [18], and the phonon degrees of
freedom can be integrated out as it is the path integral of a forced harmonic
oscillator. The resulting effective action is

$\displaystyle S=$
$\displaystyle\int_{0}^{\hbar\beta}\left(\frac{m_{I}}{2}\dot{z}_{I}(\tau)^{2}+U(z_{I}(\tau))\right)d\tau$
$\displaystyle-\frac{1}{\hbar}\sum_{q,l}\int_{0}^{\hbar\beta}d\tau\int_{0}^{\hbar\beta}d\sigma\frac{\cosh(\omega_{q,l}(\hbar\beta/2-|\tau-\sigma|))}{\cosh(\omega_{q,l}\hbar\beta/2)}$
$\displaystyle\times V^{*}_{l}(q,z_{I}(\tau))V_{l}(q,z_{I}(\sigma)).$ (23)

Using the Feynman-Jensen inequality, an upper bound is found for the free
energy depending on the kernel used in the trial action $x(\nu)$. An equation
for the kernel can be derived by taking the functional derivative of the free
energy with respect to the kernel, and putting it equal to zero. The free
energy can then be calculated iteratively numerically. The set of equations
are

$\displaystyle F$
$\displaystyle\leq\frac{\hbar}{2\pi}\int_{0}^{\infty}d\nu\left(\log\left(1+\frac{x(\nu)}{\nu^{2}}\right)-\frac{x(\nu)}{x(\nu)+\nu^{2}}\right)+\frac{n_{0}}{4}\sum_{l,l^{\prime}}V_{IB}(lk_{s})\Delta_{l-l^{\prime}}\Delta_{l^{\prime}}\exp\left(-\frac{\hbar
l^{2}k_{s}^{2}}{2m_{I}\pi}\int_{0}^{\infty}d\nu\frac{1}{x(\nu)+\nu^{2}}\right)$
$\displaystyle-\frac{n_{0}}{4\pi\hbar}\int_{0}^{k_{s}/2}dq\sum_{l,m,m^{\prime},s,s^{\prime}}V_{IB}(q+mk_{s})V_{IB}(q+sk_{s})\Delta_{m^{\prime}}\Delta_{s^{\prime}}(u^{l}_{m-m^{\prime}}(q)+v^{l}_{m-m^{\prime}}(q))(u^{l}_{s-s^{\prime}}(q)+v^{l}_{s-s^{\prime}}(q))$
$\displaystyle\int_{0}^{\infty}du\exp\left(-\omega_{q,l}u-\frac{\hbar}{2m_{I}\pi}\int_{0}^{\infty}\frac{d\nu}{x(\nu)+\nu^{2}}((q+mk_{s})^{2}+(q+sk_{s})^{2}-2(q+mk_{s})(q+sk_{s})\cos(\nu
u))\right),$ (24)

and

$\displaystyle x(\nu)$
$\displaystyle=-\frac{n_{0}k_{s}^{2}}{4m_{I}}\sum_{l,l^{\prime}}l^{2}V_{IB}(lk_{s})\Delta_{l-l^{\prime}}\Delta_{l^{\prime}}\exp\left(-\frac{\hbar
l^{2}k_{s}^{2}}{2m_{I}\pi}\int_{0}^{\infty}d\nu\frac{1}{x(\nu)+\nu^{2}}\right)+\frac{n_{0}}{4\pi\hbar
m_{I}}\int_{0}^{k_{s}/2}dq\sum_{l,m,m^{\prime},s,s^{\prime}}V_{IB}(q+mk_{s})$
$\displaystyle
V_{IB}(q+sk_{s})\Delta_{m^{\prime}}\Delta_{s^{\prime}}(u^{l}_{m-m^{\prime}}(q)+v^{l}_{m-m^{\prime}}(q))(u^{l}_{s-s^{\prime}}(q)+v^{l}_{s-s^{\prime}}(q))\int_{0}^{\infty}du((q+mk_{s})^{2}+(q+sk_{s})^{2}-2(q+mk_{s})$
$\displaystyle(q+sk_{s})\cos(\nu
u))\exp\left(-\omega_{q,l}u-\frac{\hbar}{2m_{I}\pi}\int_{0}^{\infty}\frac{d\nu}{x(\nu)+\nu^{2}}((q+mk_{s})^{2}+(q+sk_{s})^{2}-2(q+mk_{s})(q+sk_{s})\cos(\nu
u))\right).$ (25)

The iterative procedure is started using the Lee-Low-Pines kernel given by
$x(\nu)=0$. We consider the upper bound to be converged if the difference with
the last value is less than $0.1\%$. A Gauss-Legendre quadrature with $20$
points with $100$ subintervals was used as in Refs. [36, 33] for the polaronic
energy calculation. The polaronic energy $F-nV_{IB}(0)$ for
$n=2386.2/l_{\perp}$, $a^{(B)}_{s}=0.005l_{\perp}$,
$a^{(IB)}_{s}=0.007l_{\perp}$, and $m_{I}=m_{B}$ across the transition as a
function of $\epsilon^{(B)}_{dd}-\epsilon^{(B)}_{dd,c}$ is shown in Figure 2.
The blue line is for $\epsilon^{(IB)}_{dd}=0$ where the impurity does not have
a magnetic moment and is not dipolar. In the superfluid regime, the energy
decreases as the system gets closer to the phase transition. There is a
divergence visible at the transition
$\epsilon^{(B)}_{dd}=\epsilon^{(B)}_{dd,c}$ which was also seen in [26] using
the Lee-Low-Pines method in the superfluid regime for a two-dimensional
dipolar gas. The divergence from the superfluid regime is driven by the
softening of the roton mode [26], while the divergence coming from the
supersolid regime is driven by the lowering of the first excitation band. This
can be seen by studying the contribution of the different modes $l$ to the
ground-state energy. For the values used here, the contribution of the
periodic background potential to the ground-state energy is zero except for
the mean-field energy contribution $nV_{IB}(0)$. In the supersolid regime, the
energy increases as the first excitation band and others become larger. The
green dash-dotted line shows the polaronic energy when the impurity is dipolar
and has a finite magnetic moment resulting in $\epsilon^{(IB)}_{dd}=0.1$. The
additional dipole-dipole interaction for the impurity decreases the impurity-
atom interaction potential resulting in an increase of the polaronic energy
and a weaker polaron. In Figure 3, the dependence of the polaronic energy on
the s-wave scattering length $a^{(IB)}_{s}$ in the supersolid regime is shown
for $n=2386.2/l_{\perp}$, $a^{(B)}_{s}=0.005l_{\perp}$, $m_{I}=m_{B}$,
$\epsilon^{(B)}_{dd}=1.345$, and $\epsilon^{(IB)}_{dd}=0$ or $0.1$. Similar to
the charged Bose polaron problem [33], the energy is always negative and
decreases as $a^{(IB)}_{s}$ increases. Even when the impurity has a finite
magnetic moment, which increases the polaronic energy, the polaronic energy is
always negative within the Fröhlich approximation used here. In the supersolid
regime, it can be possible that for strong interaction strengths, the polaron
becomes localized to a single droplet (or single “lattice point”) of the
supersolid. To study the localization of the polaron, the polaron radius needs
to be calculated. The polaron radius can be calculated using [37, 33]

$r_{p}=\sqrt{\frac{\pi\hbar}{4m_{I}}\left(\int_{0}^{\infty}d\nu\frac{x(\nu)}{x(\nu)+\nu^{2}}\right)^{-1}}.$
(26)

The above equation was used to calculate the polaron radius for the charged
Bose polaron [33] and the anharmonic solid-state polaron [37]. Other methods
are also possible [18, 38]. In Figure 4, the polaron radius multiplied by the
density modulation momentum $k_{s}$ is plotted across the transition for
$n=2386.2/l_{\perp}$, $a^{(B)}_{s}=0.005l_{\perp}$,
$a^{(IB)}_{s}=0.007l_{\perp}$, and $m_{I}=m_{B}$ against
$\epsilon^{(B)}_{dd}-\epsilon^{(B)}_{dd,c}$. The blue line is for a non-
dipolar impurity $\epsilon^{(IB)}_{dd}=0$, while the green dash-dotted line is
for an impurity with a finite magnetic moment $\epsilon^{(IB)}_{dd}=0.1$. The
polaron radius behaves similar to the ground-state energy as a divergence is
observed at the transition due to softening of the roton mode and lowering of
the first excitation band. The finite magnetic moment of the impurity
increases the size of the polaron. For $k_{s}$ in the superfluid regime, the
value of $k_{s}$ for the supersolid close to the transition is used. For the
values used here, $r_{p}k_{s}$ is larger than one, indicating a polaron that
is not localized to a droplet. Figure 5 shows the polaron radius as a function
of $a^{(IB)}_{s}$ for $n=2386.2/l_{\perp}$, $a^{(B)}_{s}=0.005l_{\perp}$,
$m_{I}=m_{B}$, $\epsilon^{(B)}_{dd}=1.345$, and $\epsilon^{(IB)}_{dd}=0$. The
polaron radius decreases as $a^{(IB)}_{s}$ increases as expected, and at a
certain value of $a^{(IB)}_{s}$, the polaron radius falls below $1/k_{s}$, and
the polaron is localized to a single droplet. In this case, polaron motion
will be best described by hopping from one droplet to another, similar to a
small solid-state polaron.

Figure 2: The polaronic energy for $n=2386.2/l_{\perp}$,
$a^{(B)}_{s}=0.005l_{\perp}$, $a^{(IB)}_{s}=0.007l_{\perp}$, and $m_{I}=m_{B}$
as a function of $\epsilon^{(B)}_{dd}-\epsilon^{(B)}_{dd,c}$. The blue line is
the polaronic energy for the non-dipolar impurity, while the green dash-dotted
line is the polaronic energy of an impurity with a finite magnetic moment. The
grey dashed line indicates the transition from the superfluid to the
supersolid phase. Figure 3: The polaronic energy as a function of the s-wave
scattering length $a^{(IB)}_{s}$ in the supersolid regime for
$n=2386.2/l_{\perp}$, $a^{(B)}_{s}=0.005l_{\perp}$, $m_{I}=m_{B}$,
$\epsilon^{(B)}_{dd}=1.345$, and $\epsilon^{(IB)}_{dd}=0$ (blue line) or $0.1$
(green dash-dotted line). Figure 4: The polaron radius $r_{p}$ in units of
$1/k_{s}$ as a function of $\epsilon^{(B)}_{dd}-\epsilon^{(B)}_{dd,c}$ for
$n=2386.2/l_{\perp}$, $a^{(B)}_{s}=0.005l_{\perp}$,
$a^{(IB)}_{s}=0.007l_{\perp}$, and $m_{I}=m_{B}$. The blue line is the
polaronic energy for the non-dipolar impurity, while the green dash-dotted
line is the polaronic energy of an impurity with a finite magnetic moment. The
grey dashed line indicates the transition from the superfluid to the
supersolid phase. Figure 5: The polaron radius $r_{p}$ in units of $1/k_{s}$
in the supersolid regime as a function of $a^{(IB)}_{s}$ for
$n=2386.2/l_{\perp}$, $a^{(B)}_{s}=0.005l_{\perp}$, $m_{I}=m_{B}$,
$\epsilon^{(B)}_{dd}=1.345$, and $\epsilon^{(IB)}_{dd}=0$. The grey line
indicates the $r_{p}=1/k_{s}$ threshold where the impurity gets localized to a
single droplet of the supersolid.

##  V Conclusion and Outlook

The wavefunction of a one-dimensional dipolar supersolid and the corresponding
excitations have been reviewed and discussed. The ground-state energy of an
impurity dressed by the excitations of the supersolid has been calculated
using the Feynman path-integral method within the Bogoliubov and Fröhlich
approximations. A divergence was observed when the supersolid regime was
entered. An additional impurity-atom dipole interaction from a finite impurity
magnetic moment results in a larger ground-state energy. The polaron radius
has also been calculated, showing that for most values, the polaron radius is
larger than the size of a single droplet of the supersolid. However, for large
enough impurity-atom interaction strengths, the polaron can get localized to a
single droplet, behaving like a small polaron. An interesting next step would
be to experimentally study the system of an impurity in a dipolar supersolid
and compare the results to the path-integral method shown here. Additionally,
there are no numerically exact Monte Carlo results for comparison yet for the
supersolid polaron system.

###### Acknowledgements.

We gratefully acknowledge fruitful discussions with T. Bland and T.
Ichmoukhamedov. We acknowledge funding by the Research Foundation — Flanders,
projects nos. GOH1122N, G061820N, G060820N, and by the University Research
Fund (BOF) of the University of Antwerp.

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# Snapshot hyperspectral imaging with quantum correlated photons

Yingwen Zhang  National Research Council of Canada, 100 Sussex Drive, Ottawa
ON Canada, K1A0R6  Duncan England<EMAIL_ADDRESS>National
Research Council of Canada, 100 Sussex Drive, Ottawa ON Canada, K1A0R6
Benjamin Sussman  National Research Council of Canada, 100 Sussex Drive,
Ottawa ON Canada, K1A0R6  Department of Physics, University of Ottawa, Ottawa,
Ontario, K1N 6N5, Canada

###### Abstract

Hyperspectral imaging (HSI) has a wide range of applications from
environmental monitoring to biotechnology. Current snapshot HSI techniques all
require a trade-off between spatial and spectral resolution and are thus
unable to achieve high resolutions in both simultaneously. Additionally, the
techniques are resource inefficient with most of the photons lost through
spectral filtering. Here, we demonstrate a snapshot HSI technique utilizing
the strong spectro-temporal correlations inherent in entangled photons using a
modified quantum ghost spectroscopy system, where the target is directly
imaged with one photon and the spectral information gained through ghost
spectroscopy from the partner photon. As only a few rows of pixels near the
edge of the camera are used for the spectrometer, almost no spatial resolution
is sacrificed for spectral. Also since no spectral filtering is required, all
photons contribute to the HSI process making the technique much more resource
efficient.

##  I Introduction

Utilizing the properties of quantum entangled photons to enhance the
performance of imaging techniques has been an active area of research in
recent decades. Sub-shot noise imaging Brida et al. (2010); Samantaray et al.
(2017), quantum ghost imaging Pittman et al. (1995); Shapiro and Boyd (2012);
Padgett and Boyd (2017), imaging with undetected photons Lemos et al. (2014),
and infra-red microscopy Paterova et al. (2020) are just some examples of the
imaging techniques made possible by the unique properties of entangled
photons. With recent improvements in source and detector technologies, these
techniques are becoming increasingly practical, and integration timescales
have improved from hours to minutes, or even seconds Zhang et al. (2022). In
this work, we showcase another advantage of quantum correlated imaging that
may provide a viable alternative to classical methods. By applying the
concepts of ghost spectroscopy Scarcelli et al. (2003); Yabushita and
Kobayashi (2004); Janassek et al. (2018); Ryczkowski et al. (2021) to quantum
imaging, we show that snapshot hyperspectral imaging can be achieved by
measuring the position of one photon from an entangled pair, and inferring its
wavelength by making a spectral measurement on the other photon. With the use
of a time-tagging camera, we demonstrate that a $\sim$20 kilopixel image with
$\sim$2 nm spectral resolution over an 85 nm range can be achieved at the
single photon level within a reasonable data acquisition time of a few
minutes.

Hyperspectral imaging (HSI) is a class of imaging techniques used to obtain
the spectral information for each spatial location on an image and has a wide
range of applications ranging from environmental and agricultural monitoring,
biotechnology, medical diagnostics to food analysis Lu and Fei (2014); Bacon
et al. (2004); Gowen et al. (2007); Orth et al. (2015); Slonecker et al.
(2010); Delalieux et al. (2009); Zhu et al. (2020). Traditional HSI techniques
rely on scanning either the spatial or spectral degree of freedom Green et al.
(1998); Gat (2000), and can provide high resolution in both position and
spectrum, at the expense of long acquisition times due to the precision
scanning required. Alternatively, so-called snapshot hyperspectral imaging
(SHSI) techniquesHagen and Kudenov (2013) can acquire spectral data in a
single shot, by sacrificing spatial resolution. For example SHSI can be
achieved using spectral filter arrays either directly on the camera sensor
Lapray et al. (2014); Wang et al. (2019); Monakhova et al. (2020), or in
conjunction with a microlens array Shogenji et al. (2004); Mathews (2008);
Horstmeyer et al. (2009). In these schemes, because an array of spectral
pixels are required for each spatial pixel, a trade-off exists, and the
spatial resolution is reduced in proportion to the desired spectral
resolution. SHSI techniques without relying on spectral filter arrays have
also been demonstrated, such as using the spatial and spectral correlations in
speckle patterns Sahoo et al. (2017), but the trade-off between the spatial
and spectral information still exists. Finally, it is worth noting that, in
terms of the number of photons collected, all of these schemes are resource
inefficient. The vast majority of photons arriving at conventional HSI
apparatus will be lost; either blocked in the scanning apparatus, or removed
by spectral filtering. In the weak illumination regime, this inefficiency will
lead to significantly increased integration time.

In our SHSI demonstration, we show that the issues with resolution and
efficiency can be overcome by using entangled photon pairs. We will term this
technique quantum correlation hyperspectral imaging (QCHSI). In QCHSI, we
employ the temporal and spectral (anti)correlations in entangled photon pairs
created through the process of spontaneous parametric down-conversion (SPDC).
One photon, the signal, is sent to illuminate the target, which is imaged by a
time-tagging camera. Its entangled partner photon, the idler, which had no
interaction with the target, is sent onto a spectrometer built around the same
camera. The photon pairs are then identified through a second order timing
correlation measurement based on the photon detection times. Through this
process, the spectral information at each pixel will automatically emerge as a
result of the inherent spectral anti-correlation between the two photons. As
the spectral information can be collected on just a few rows of pixels on the
edge of the camera, almost no spatial resolution is sacrificed in exchange for
the spectral information. Furthermore, in this scheme, every photon within the
HSI apparatus can contribute to the image, and none are removed by scanning or
filtering. Hence the efficiency of the measurement is not fundamentally
limited by the scheme and is only limited by practical considerations such as
camera and grating efficiency.

##  II Experimental setup

Figure 1: (a) A 405.5 nm CW pump laser beam is focused onto a 1 mm thick Type
0 ppKTP crystal to generate temporal and spectrally (anti)correlated photon
pairs. A long-pass (LP) spectral filter is used to filter out the pump laser.
The generated photon pairs are first collimated by a lens before split into
two paths by a D-shaped mirror. The signal photon is sent to illuminate a
target that is imaged onto the time-tagging camera. The idler photon is sent
to a diffraction grating with the diffracted photons focused onto the camera
by a lens, placed at one focal length from the grating and camera. The two
photons are each imaged onto a different section of the camera with the aid of
a polarizing beam-splitter (PBS) and a half-wave plate (HWP) rotated at
$45^{\circ}$ placed in the path of the signal photon. (b) shows an raw image
of the two beams captured on the camera accumulated over 10 seconds with no
target present and the spectrum obtained from the cross-section of the
spectrum arm.

The experimental setup for QCHSI is shown in Fig. 1. A 405.5 nm continuous
wave (CW) laser beam is focused onto a 1 mm thick Type 0 periodically poled
potassium titanyl phosphate (ppKTP) crystal to generate, through the process
of SPDC, temporally correlated and spectrally anti-correlated photon pairs,
namely the signal and idler photons. The SPDC photons have a spectral
bandwidth of $\sim$100 nm centered around 811 nm. After filtering out the pump
with a long-pass spectral filter, the SPDC beam is collimated by a 10 cm focal
length lens. The signal and idler photons are then separated using a D-shaped
mirror (a 50:50 beam-splitter can also be used here to create a circular
shaped beam but at the cost of losing 50$\%$ of all coincidence events). The
signal photons are used to illuminate a target sample which is then imaged
onto the time-tagging camera (TPX3CAM Nomerotski (2019); ASI  where the
arrival time and position of each photon is tagged. The idler photon is sent
onto a spectrometer built around a diffraction grating, lens, and the camera.

The spectrometer is calibrated (see supplementary information) so that pixel
number can be mapped to wavelength, therefore we can tag both the arrival time
and wavelength of each idler photon. The two photon beams are recombined, but
slightly displaced, by a a half-wave plate rotated at $45\deg$ in one of the
paths and a polarizing beam-splitter placed just before the camera, such that
each beam will be imaged onto a different location of the camera. A $809\pm
41$ nm spectral filter (Semrock FF01-809/81-25) is placed over the camera to
block-off background light from outside the spectral range of interest. The
TPX3CAM system has a spatial resolution of $256\times 256$ pixels with a pixel
pitch of 55 $\mu$m and an effective temporal resolution of $\sim 7$ ns on each
pixel. We utilize this feature to time-tag every photon detected on each pixel
and use this information to identify temporally correlated photon pairs. After
identifying the coincident photons, the per-pixel spectral information of the
sample will automatically emerge as a result of the inherent spectral anti-
correlation between the signal and idler photons.

To recover the spectral response of the target, some corrections to the
measured coincidence spectrum need to be made. The first is to remove
accidental coincidence events due to random detection overlap between
uncorrelated photons. For accidental coincidences between pixel $n$ of the
idler beam and pixel $m$ of the signal beam, this is given by

$C_{n,m}^{\text{acc}}=N_{n}N_{m}\tau,$ (1)

where $N_{n(m)}$ is the number of photons detected per second in pixel $n(m)$
and $\tau$ is the coincidence gating time, which for this experiment is set at
20 ns. Thus, the total number of accidental coincidences in a pixel $m$ is
given by

$C_{m}^{\text{acc}}=\sum_{n}N_{n}N_{m}\tau.$ (2)

The second is that since the measured spectrum is of the idler photon and the
two photon spectrum is anti-correlated, the coincidence spectrum obtained is
in fact inverted from what is seen by the signal photon. This inversion is
corrected using the energy conservation relation

$\frac{1}{\lambda_{p}}=\frac{1}{\lambda_{s}}+\frac{1}{\lambda_{i}},$ (3)

where the $\lambda_{p}$, $\lambda_{s}$ and $\lambda_{i}$ are the wavelengths
of the pump, signal and idler photons. Also due to this wavelength inversion,
one should not send both photons to the target and split them afterwards in
the hopes of getting a better signal to noise ratio. The coincidence spectrum
obtained this way will be an overlapped spectrum of the target’s spectral
response and its inverse spectral response. The spectrum is calibrated by
matching the measured spectral shape to the fall-off edge of various spectral
filters.

Finally, we note that, in common with all spectrometers, the spectrum
determined after these corrections are convolutions of the target spectrum
with the instrument response (mainly from the camera and grating), to correct
for this will require a careful calibration of the detection system, which is
beyond the scope the this work. Details on this effect are outlined in the
supplementary information.

##  III Results

Figure 2: a - Orientation of the 3 band-pass filters ($786\pm 11$ nm, $810\pm
5$ nm, $850\pm 5$ nm) with respect to the imaging beam. b - Image of the twin
beams after temporal correlation analysis. c - Average signal photon spectrum
as calculated by energy conservation. d1-d3 - Correlated spectrum of selected
regions (highlighted by the yellow circle) in the imaging signal arm with the
corresponding signal photon spectrum shown in the insets. As reference, a
faint background of the full beam is shown in the images. e1-e3 Correlated
image of selected regions (highlighted by the yellow circle) in the
spectrometer arm with the insets showing the corresponding wavelength region
selected for the signal photon. A faint background of the full correlated
spectrum is shown in the images as reference. Data acquisition time is 100 s
and all displayed images and spectrum are background subtracted. As the
maximum intensity of the idler spectrum is much brighter than the signal beam,
the two are normalized separately in all the displayed images in order to make
features in the signal beam more visible. The displayed signal spectrum have
been smoothed through a cubic spline.

As a proof of principle demonstration of QCHSI, we placed three different
spectral band-pass filters - $786\pm 11$ nm (Semrock FF01-784/22-25), $810\pm
5$ nm (Semrock FF01-810/10-25) and $850\pm 5$ nm (Semrock FF01-850/10-25) - in
the path of the signal photon, their orientation with respect to the beam is
shown in Fig. 2(a). The image obtained after coincidence measurement is
performed between the full signal beam and the idler spectrum is shown in Fig.
2(b) with the corresponding signal photon spectrum shown in Fig. 2(c).

In Fig. 2(d), we show the spectrum at various regions of the target. This is
obtained when coincidence events between the highlighted region of interest
(ROI) on the signal beam and the full idler beam spectrum is considered. The
reverse is shown in Fig. 2(e), where coincidence events between a small ROI in
the idler spectrum and the full signal beam is considered. Here, only regions
on the target with signal photons spectrally correlated with the idler photons
in the ROI will be lit up.

In Fig. 3, a piece of paper with small punctured holes of $\sim$0.2 mm in
diameter is placed in front of the spectral filters to simulate a situation
where the signal photons are reflected by or transmitted through very small
targets (relative to the beam size). We see in Fig. 2(b) and (c) that due to
the high noise from the small amount of transmitted signal photons, only a few
bright spots can be seen on the target and features in the spectrum are also
mostly washed out by noise. However, as seen in Fig. 2(d), if coincidences
from just a few bright spots are considered, the spectrum can still be
recovered with good visibility. On the other hand, as seen in Fig. 2(e), if
only coincidences from a small ROI in the idler spectrum is considered, all
points in the target with correlated wavelength is lit up, some of which are
too faint to be visible in Fig. 2(b).

Figure 3: a - Orientation of the 3 band-pass filters ($786\pm 11$ nm, $810\pm
5$ nm, $850\pm 5$ nm) with respect to the imaging beam. Over the beam is a
piece of paper with small punctured holes of $\sim$0.2 mm in diameter. b -
Image of the twin beams after time correlation analysis. c - Average signal
photon spectrum as calculated by energy conservation. d1-d3 - Correlated
spectrum of selected regions (highlighted by the yellow circle) in the imaging
signal arm with the signal spectrum shown in the insets. As reference, a faint
background of the full beam is shown in the images. e1-e3 Correlated image of
selected regions (highlighted by the yellow circle) in the spectrometer arm
with the insets showing the corresponding wavelength region selected for the
signal photon. A faint background of the full correlated spectrum is shown in
the images as reference. Data acquisition time is 270 s and all displayed
images and spectrum are background subtracted. As the maximum intensity of the
idler spectrum is much brighter than the signal beam, the two are normalized
separately in all the displayed images in order to make features in the signal
beam more visible. The displayed signal spectrum have been smoothed through a
cubic spline.

##  IV Discussion

The data acquisition time of the current system is in the order of a few
minutes, this is a direct result of the limited capability of the camera
system which has a quantum efficiency of $\sim$5$\%$, a temporal resolution of
$\sim$7 ns, and a saturation limit of $\sim$107 photons per second. With
expected advances in camera technology, each parameter could be improved by a
factor of 10 in the near future; this would be enough to allow real-time
spectral imaging using QCHSI.

The spectral resolution of the QCHSI technique is determined by the bandwidth
of the pump laser and the resolution of the spectrometer. In this case, the
pump is a narrow linewidth continuous wave laser so the resolution limit is
dominated by the spectrometer setup. Based on the thickness of the spectral
band and the width of the spectral peak in the overlap region between the
$786\pm 11$ nm and $810\pm 5$ nm filters, we estimate the spectral resolution
of our system to be $\sim$2 nm. With improvements to the pair source and the
imaging optics, we expect the resolution of this setup could approach $0.7$ nm
as measured in a previous experiment Zhang et al. (2021).

In this initial demonstration, type-0 degenerate SPDC was used such that each
photon mode is centred at 810 nm, with a spectral bandwidth of $\sim$100 nm.
The bandwidth could be further increased by reducing the crystal thickness for
larger spectral coverage or, for increased spectral resolution, narrower
bandwidth photons could be generated by type-II SPDC. Furthermore, non-
degenerate pair sources could be used to interrogate samples in very different
spectral windows Aspden et al. (2015). In this demonstration, the same camera
is used for both the image and the spectrum for simplicity. But, more
generally, the spectral information could be obtained using a second camera,
or a separate device such as a 1D avalanche photo-diode array Johnsen et al.
(2014), or a time-of-flight fiber spectrometer Avenhaus et al. (2009).
Finally, we note that QCHSI could also be operated in ghost imaging mode,
where the target is placed in the spectrometer arm instead of the imaging arm,
however it can be difficult to accurately find the image plane of the “ghost”
target in the imaging arm using this approach.

Though we have demonstrated significant advantages of using QCHSI, there are
still inherent limitations in the technique which cannot be solved by improved
detectors. Principally, QCHSI is an active imaging technique where the target
must be illuminated with quantum correlated photons generated by the observer,
it cannot work passively using background light. Most conventional SHSI
techniques can work both with active and passive illumination. Secondly, QCHSI
it is not suitable for imaging fluorescent targets because the wavelength
change imparted by fluoresence in the signal arm will not be detected by
observing the idler arm.

Our demonstration can be potentially be performed using classical ghost
spectroscopy which has been demonstrated in recent years, using either the
spectral correlation present in specialized thermal sources Janassek et al.
(2018) or through computational methods using a programmable spectral filter
Ryczkowski et al. (2021). Due to the nature of classical ghost spectroscopy
needing to scan hundreds of spectral patterns, we expect the data acquisition
time will be similar to that presented in this work.

##  V Conclusion

In conclusion, we have demonstrated a proof of concept SHSI system utilizing
the strong temporal and spectral (anti)correlations inherent in entangled
photon pairs generated through the process of SPDC. With this system, almost
no spatial resolution will need to be sacrificed in exchange for the spectral
as in most conventional SHSI techniques, permitting for the maximum spatial
and spectral resolution allowed by the detection camera and imaging system to
be achieved simultaneously. Utilizing the background noise reduction provided
by the temporal correlation measurement and the higher photon efficiency of
the technique (no photon losses through scanning or spectral filtering) makes
this technique well suited for applications requiring weak illumination such
as covert illumination or imaging of light sensitive samples. Due to
limitations in detection technology, the performance of QCHSI is currently
still far from matching commercial SHSI cameras in terms of spatial resolution
and data acquisition time, but we believe this issue can be overcome in the
near future with the advent of improved single photon detection technologies
Tim ; Nomerotski et al. (2020); Morimoto et al. (2020).

## Supplementary Information

Figure 4: Background subtracted coincidence image (c) is obtained by
subtracting the background image (b), determined from eq. (2) of the main
text, from the raw coincidence image ((a).

Background subtraction in the coincidence image is demonstrated in Fig. 4
where the background image is determined pixel wise according to eq. (2) of
the main text.

Figure 5: The directly measured idler photon spectrum (b) is divided by the
raw coincidence idler photon spectrum (a) to obtain a corrected coincidence
spectrum of the idler photon (c). The pixel number is then converted to
wavelength and inverted according to energy conservation, given by eq. (3) of
the main text, to obtain the spectrum of the signal photon. The pixel to
wavelength conversion is calibrated according to the edges of multiple
spectral filters as seen in (d). The final spectrum (e) is obtained by
smoothing (d) with a cubic spline fit in order to remove the large
fluctuations seen on the edges of the spectrum, a result from division by
small numbers during the correction from (a) to (c).

The steps to obtain the final signal photon spectrum, as discussed at the end
of the “Experimental setup” section of the main text, is illustrated in Fig.
5.

To calculate the absorption spectrum of the target
$S_{\text{targ}}(\lambda_{s})$, we must account for the spectral response of
the optics, and camera, and for the spectral profile of the SPDC photons. The
measured coincidence spectrum $S_{\text{coinc}}(\lambda_{i},\lambda_{s})$ in
Fig. 5(a) is related to the spectral profile of each element as follows

$S_{\text{coinc}}(\lambda_{i},\lambda_{s})=S_{\text{SPDC}}(\lambda_{i},\lambda_{s})\left[\eta_{\text{cam}}(\lambda_{i})\eta_{\text{grat}}(\lambda_{i})\right]\left[\eta_{\text{cam}}(\lambda_{s})S_{\text{targ}}(\lambda_{s})\right],$
(4)

where $\eta_{\text{cam}}$ and $\eta_{\text{grat}}$ are the wavelength-
dependent quantum efficiency of the camera, and grating efficiency
respectively. We have ignored the spectral profile of simple optical elements
such as lenses and mirrors which are mostly flat within the spectral region of
this experiment. $\lambda_{s}$ and $\lambda_{i}$ are related through the
conservation of energy as given by eq.(3) of the main text, i.e.
$\frac{1}{\lambda_{p}}=\frac{1}{\lambda_{s}}+\frac{1}{\lambda_{i}}$. The
measured idler photon spectrum as seen in Fig. 5(b) is given by the product of
the SPDC spectrum with the camera and grating efficiencies:

$S(\lambda_{i})=S_{\text{SPDC}}(\lambda_{i},\lambda_{s})\left[\eta_{\text{cam}}(\lambda_{i})\eta_{\text{grat}}(\lambda_{i})\right].$
(5)

By dividing the coincidence spectrum by the idler spectrum as shown in Fig. 5,
this returns an output spectrum Fig. 5(c) of the form:

$S_{\text{out}}(\lambda_{i})=\frac{\eta_{\text{cam}}(\lambda_{s})S_{\text{targ}}(\lambda_{s})}{\eta_{\text{cam}}(\lambda_{i})\eta_{\text{grat}}(\lambda_{i})}.$
(6)

So we can see that this technique does not directly measure the spectrum of
the target $S_{\text{targ}}$ but rather, in common with all spectrometers,
measures the product of the target spectrum and the instrument response. With
careful intensity calibration of the camera system and measurements of the
grating efficiency, $S_{\text{targ}}$ could be determined, but this is beyond
the scope of this work.

Conversion from position to wavelength in a spectrometer is slightly
nonlinear, thus after inversion of the idler photon spectrum to obtain the
spectrum of signal photon, a third order polynomial fit, given by
$\lambda=ax^{3}+bx^{2}+cx+d$ (with $\lambda$ the wavelength and $x$ the pixel
number), is made in order to match the obtained spectrum to the edges of the
various spectral filters (with spectral response obtained from Semrock), this
is seen in Fig. 5(d). The obtained fitting parameters are $a=-4.83\times
10^{-6}$ nm, $b=1.78\times 10^{-3}$ nm, $c=0.199$ nm, $d=770.5$ nm.

## Acknowledgements

The authors are grateful to Antony Orth, Andrew Ridsdale, Denis Guay, and Doug
Moffatt for stimulating discussions and technical support. This work was
partly supported by Defence Research and Development Canada.

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