# Correlation Effects in a Trapped Bose-Fermi Mixture: Exact Results

O E Alon1,2 and L S Cederbaum3 1 Department of Physics, University of Haifa,
3498838 Haifa, Israel  2 Haifa Research Center for Theoretical Physics and
Astrophysics, University of Haifa, 3498838 Haifa, Israel  3 Theoretical
Chemistry, Physical Chemistry Institute, Heidelberg University, D-69120
Heidelberg, Germany<EMAIL_ADDRESS>
###### Abstract

Many-body properties of a fermionic impurity embedded in a Bose-Einstein
condensate are analyzed analytically using a solvable model, the harmonic-
interaction model for Bose-Fermi mixtures. The one-particle and two-particle
densities, reduced density matrices, and correlation functions of the fermions
and bosons, both in position and momentum spaces, are prescribed in closed
form. The various coherence lengths are analyzed. We show that the first-order
coherence lengths in position and momentum spaces are equal whereas the
second-order quantities can differ substantially. Illustrative examples where
the sole interaction is between the impurity and the condensate are presented.
Implications are briefly discussed.

##  1 Introduction

Ultracold atomic Bose-Fermi mixtures have generated a vast and extensive
experimental and theoretical research for more than two decades, see, e.g.,
Refs. [1-32]. There are endless combinations, if one adds together
intraspecies and interspecies interactions, numbers of particles of each
species and their masses, and trapping potentials, to the inherent different
nature of the Bose and Fermi statistics. On the theory side, this wealth
offered by the ultracold realm makes solvable models rather scarce, especially
in traps. The utility of exactly-solvable models of quantum many-particle
systems is indispensable as tools to emulate such systems and to gain
instincts on realistic scenarios. These are additional to the mere
mathematical aesthetics which usually accompanies such models.

Harmonic-interaction models have amply been used to emulate many-particle
systems for many years [33-52]. Another usage of harmonic-interaction models
is to benchmark many-body theories and their numerical implementations
[53-57]. The harmonic-interaction models have been mainly used for single-
species bosons, single-species fermions, and Bose-Bose mixtures so far. In the
present work we wish to add another brick to the building of harmonic-
interaction models, and treat Bose-Fermi mixtures with it. Explicitly, we
investigate analytically first-order and second-order correlations of a
trapped fermionic impurity in a Bose-Einstein condensate both in position and
momentum spaces. We build on derivations and results using the harmonic-
interaction model for Bose-Fermi mixtures which will be reported in an
extended format elsewhere [58].

##  2 Theory

The Bose-Fermi Hamiltonian within the harmonic-interaction model reads

$\displaystyle\hat{H}(x_{1},\ldots,x_{N_{F}},y_{1},\ldots,y_{N_{B}})=$
$\displaystyle=\sum_{j=1}^{N_{F}}\left(-\frac{1}{2m_{F}}\frac{\partial^{2}}{\partial
x_{j}^{2}}+\frac{1}{2}m_{F}\omega^{2}x_{j}^{2}\right)+\sum_{j=1}^{N_{B}}\left(-\frac{1}{2m_{B}}\frac{\partial^{2}}{\partial
y_{j}^{2}}+\frac{1}{2}m_{B}\omega^{2}y_{j}^{2}\right)+$
$\displaystyle+\lambda_{F}\sum_{1\leq
j<k}^{N_{F}}\left(x_{j}-x_{k}\right)^{2}+\lambda_{B}\sum_{1\leq
j<k}^{N_{B}}\left(y_{j}-y_{k}\right)^{2}+\lambda_{BF}\sum_{j=1}^{N_{F}}\sum_{k=1}^{N_{B}}\left(x_{j}-y_{k}\right)^{2},\
$ (1)

where $N_{F},N_{B}$ are the numbers of fermions and bosons, $m_{F},m_{B}$
their masses, and $\lambda_{F},\lambda_{B}$ the intraspecies and
$\lambda_{BF}$ the interspecies interactions strengths, respectively. Herein,
the fermions are polarized.

Diagonalizing (2) leads to four distinct eigenfrequencies, also see the Bose-
Bose mixture [45, 50]

$\displaystyle\Omega_{F}=\sqrt{\omega^{2}+\frac{2}{m_{F}}\left(N_{F}\lambda_{F}+N_{B}\lambda_{BF}\right)},\qquad\Omega_{B}=\sqrt{\omega^{2}+\frac{2}{m_{B}}\left(N_{B}\lambda_{B}+N_{F}\lambda_{BF}\right)},$
$\displaystyle\Omega_{BF}=\sqrt{\omega^{2}+2\left(\frac{N_{B}}{m_{F}}+\frac{N_{F}}{m_{B}}\right)\lambda_{BF}},\qquad\omega.\
$ (2)

The ground-state energy of $N_{F}$ fermions and $N_{B}$ bosons is [58]

$\displaystyle
E=\frac{1}{2}\Bigg{[}\left(N_{F}^{2}-1\right)\sqrt{\omega^{2}+\frac{2}{m_{F}}(N_{F}\lambda_{F}+N_{B}\lambda_{BF})}+$
$\displaystyle+\left(N_{B}-1\right)\sqrt{\omega^{2}+\frac{2}{m_{B}}(N_{B}\lambda_{B}+N_{F}\lambda_{BF})}+\sqrt{\omega^{2}+2\left(\frac{N_{B}}{m_{F}}+\frac{N_{F}}{m_{B}}\right)\lambda_{BF}}+\omega\Bigg{]},$
(3)

and the wavefunction in Cartesian coordinates reads

$\displaystyle\Psi(x_{1},\ldots,x_{N_{F}},y_{1},\ldots,y_{N_{B}})=$
$\displaystyle={\mathcal{N}}\left(\frac{m_{F}\Omega_{F}}{\pi}\right)^{\frac{N_{F}^{2}-1}{4}}\left(\frac{m_{B}\Omega_{B}}{\pi}\right)^{\frac{N_{B}-1}{4}}\left(\frac{M_{BF}\Omega_{BF}}{\pi}\right)^{\frac{1}{4}}\left(\frac{M\omega}{\pi}\right)^{\frac{1}{4}}\times$
$\displaystyle\times\left[\prod_{1\leq j<k\leq
N_{F}}\left(x_{j}-x_{k}\right)\right]e^{-\frac{\alpha_{F}}{2}\sum_{j=1}^{N_{F}}x_{j}^{2}-\beta_{F}\sum_{1\leq
j<k}^{N_{F}}x_{j}x_{k}}e^{-\frac{\alpha_{B}}{2}\sum_{j=1}^{N_{B}}y_{j}^{2}-\beta_{B}\sum_{1\leq
j<k}^{N_{B}}y_{j}y_{k}}\times$ $\displaystyle\times
e^{+\gamma\sum_{j=1}^{N_{F}}\sum_{k=1}^{N_{B}}x_{j}y_{k}},\ $ (4)

where $\prod_{1\leq j<k\leq
N_{F}}\left(x_{j}-x_{k}\right)=V(x_{1},\ldots,x_{N_{F}})$ is the Vandermonde
determinant and ${\mathcal{N}}$ is a dimensionless normalization constant that
appears in the derivation due to the relative Jacoby coordinates of the
fermions [58]. The center-of-mass and relative center-of-mass degrees-of-
freedom have the masses $M=m_{F}N_{F}+m_{B}N_{B}$ and
$M_{BF}=\frac{m_{F}n_{B}}{M}$, respectively.

There are five coefficients in the wavefunction, also see [50],

$\displaystyle\alpha_{F}=m_{F}\Omega_{F}\left[1+\frac{1}{N_{F}}\left(\frac{m_{F}N_{F}}{M}\frac{\omega}{\Omega_{F}}+\frac{m_{B}N_{B}}{M}\frac{\Omega_{BF}}{\Omega_{F}}-1\right)\right],\qquad\beta_{F}=\alpha_{F}-m_{F}\Omega_{F},$
$\displaystyle\alpha_{B}=m_{B}\Omega_{B}\left[1+\frac{1}{N_{B}}\left(\frac{m_{B}N_{B}}{M}\frac{\omega}{\Omega_{B}}+\frac{m_{F}N_{F}}{M}\frac{\Omega_{BF}}{\Omega_{B}}-1\right)\right],\qquad\beta_{B}=\alpha_{B}-m_{B}\Omega_{B},$
$\displaystyle\gamma=\frac{m_{F}m_{B}}{M}(\Omega_{BF}-\omega),\ $ (5)

which are used to express and evaluate all the correlation properties in the
Bose-Fermi mixture. Explicitly, the reduced one-particle and two-particle
density matrices [59, 60] and Glauber’s normalized first-order and second-
order correlation functions [61-63] of the fermions and bosons, both in
position and momentum spaces, are put forward in closed form and investigated.

###  2.1 Position and momentum correlation functions of the bosons

Consider an impurity consisting of $N_{F}=2$ fermions embedded in a Bose-
Einstein condensate made of $N_{B}$ bosons. Two fermions in the impurity
already lead to rich results. Of course, this is the minimal number of
indistinguishable particles to exhibit the antisymmetry of the wavefunction
(2) under their exchange. The results for fermions obtained below are hence
explicit to two fermions, although some of them hold for a general number
$N_{F}$ of fermions. It is instructive to analyze the condensate first and
than the impurity, since the bosons’ correlation functions are relatively
simpler than the fermions’ ones.

The reduced one-particle density matrix and the density of the bosons in
position space are integrated from the all-particle density matrix and read
[50, 58]

$\displaystyle\rho^{(1)}_{B}(y,y^{\prime})=N_{B}\left(\frac{\alpha_{B}+C^{\prime}_{0,1}}{\pi}\right)^{\frac{1}{2}}e^{-\frac{\alpha_{B}+C^{\prime}_{0,1}}{4}\left(y+y^{\prime}\right)^{2}}e^{-\frac{\alpha_{B}}{4}\left(y-y^{\prime}\right)^{2}},$
$\displaystyle\rho^{(1)}_{B}(y)=N_{B}\left(\frac{\alpha_{B}+C^{\prime}_{0,1}}{\pi}\right)^{\frac{1}{2}}e^{-\left(\alpha_{B}+C^{\prime}_{0,1}\right)y^{2}}.\
$ (6)

We can now compute explicitly from (2.1) the first-order correlation function
in position space,

$\displaystyle
g_{B}^{(1)}(y,y^{\prime})=\frac{\rho^{(1)}_{B}(y,y^{\prime})}{\sqrt{\rho^{(1)}_{B}(y)\rho^{(1)}_{B}(y^{\prime})}}=e^{-\frac{-C^{\prime}_{0,1}}{4}\left(y-y^{\prime}\right)^{2}},\
$ (7)

where $C^{\prime}_{0,1}$ is negative and given in closed form as a function of
the coefficients (2) [50, 58], see below.

It is useful to define a dimensionless measure, the first-order coherence
length in position space, as the ratio of the off-diagonal decay length of the
first-order correlation function (7) and the diagonal decay length of the
reduced one-particle density matrix (2.1), see [58],

$\displaystyle
l_{B}^{(1)}=\sqrt{\frac{\alpha_{B}+C^{\prime}_{0,1}}{-C^{\prime}_{0,1}}}=\frac{1}{\sqrt{\left(\frac{1+\rho_{B}^{(1)}}{1-\rho_{B}^{(1)}}\right)^{2}-1}}=$
(8)
$\displaystyle=\frac{1}{\sqrt{\left[1+\frac{1}{N_{B}}\left(\frac{m_{B}N_{B}}{M}\frac{\omega}{\Omega_{B}}+\frac{m_{F}N_{F}}{M}\frac{\Omega_{BF}}{\Omega_{B}}-1\right)\right]\left[1+\frac{1}{N_{B}}\left(\frac{m_{B}N_{B}}{M}\frac{\Omega_{B}}{\omega}+\frac{m_{F}N_{F}}{M}\frac{\Omega_{B}}{\Omega_{BF}}-1\right)\right]-1}}.\
$

The second expression of $l_{B}^{(1)}$ in (8) is because it is possible to
express the first-order coherence length using the depleted fraction
$0\leq\rho_{B}^{(1)}<1$. The depleted fraction is given in closed-form using
the above coefficients,
$\rho_{B}^{(1)}=\frac{\sqrt{\alpha_{B}}-\sqrt{\alpha_{B}+C^{\prime}_{0,1}}}{\sqrt{\alpha_{B}}+\sqrt{\alpha_{B}+C^{\prime}_{0,1}}}$,
see [51]. Clearly the coherence length $l_{B}^{(1)}$ diverges for non-
interacting bosons when there is no depletion [$\rho_{B}^{(1)}=0$].
Illustrative examples and further discussion are given in Sec. 3.

Let us proceed and compute the reduced one-particle density matrix and the
density of the bosons in momentum space,

$\displaystyle\tilde{\rho}^{(1)}_{B}(q,q^{\prime})=\frac{1}{2\pi}\int
dydy^{\prime}e^{-i\left(qy-q^{\prime}y^{\prime}\right)}\rho^{(1)}_{B}(y,y^{\prime})=$
$\displaystyle=N_{B}\left(\frac{1}{\pi\alpha_{B}}\right)^{\frac{1}{2}}e^{-\frac{1}{4\alpha_{B}}\left(q+q^{\prime}\right)^{2}}e^{-\frac{1}{4\left(\alpha_{B}+C^{\prime}_{0,1}\right)}\left(q-q^{\prime}\right)^{2}},$
$\displaystyle\tilde{\rho}^{(1)}_{B}(q)=N_{B}\left(\frac{1}{\pi\alpha_{B}}\right)^{\frac{1}{2}}e^{-\frac{1}{\alpha_{B}}q^{2}}.\
$ (9)

Consequently, just like (7), the first-order correlation function in momentum
space is readily obtained

$\displaystyle\tilde{g}_{B}^{(1)}(q,q^{\prime})=\frac{\tilde{\rho}^{(1)}_{B}(q,q^{\prime})}{\sqrt{\tilde{\rho}^{(1)}_{B}(q)\tilde{\rho}^{(1)}_{B}(q^{\prime})}}=e^{-\frac{-C^{\prime}_{0,1}}{4\alpha_{B}\left(\alpha_{B}+C^{\prime}_{0,1}\right)}\left(q-q^{\prime}\right)^{2}}.\
$ (10)

Analogously to (8), we define a dimensionless measure, the first-order
coherence length in momentum space, as the ratio of the off-diagonal decay
length of the first-order correlation function (10) and the diagonal decay
length of the reduced one-particle density matrix (2.1), both in momentum
space,

$\displaystyle\tilde{l}_{B}^{(1)}=\sqrt{\frac{\alpha_{B}+C^{\prime}_{0,1}}{-C^{\prime}_{0,1}}}=l_{B}^{(1)}.$
(11)

We find that the first-order coherence lengths in position (8) and momentum
(11) spaces are equal.

To facilitate the Fourier transform of second-order quantities to momentum
space later on, the reduced two-particle density matrix in position space is
written in a quadratic form using normal coordinates as follows

$\displaystyle\rho_{B}^{(2)}(y_{1},y^{\prime}_{1},y_{2},y^{\prime}_{2})=N_{B}(N_{B}-1)\left[\frac{\left(\alpha_{B}-\beta_{B}\right)\left(\alpha_{B}+\beta_{B}+2C^{\prime}_{0,2}\right)}{\pi^{2}}\right]^{\frac{1}{2}}\times$
$\displaystyle\times
e^{-\frac{\alpha_{B}+\beta_{B}+2C^{\prime}_{0,2}}{8}\left(y_{1}+y_{2}+y^{\prime}_{1}+y^{\prime}_{2}\right)^{2}}e^{-\frac{\alpha_{B}-\beta_{B}}{8}\left(y_{1}-y_{2}+y^{\prime}_{1}-y^{\prime}_{2}\right)^{2}}\times$
$\displaystyle\times
e^{-\frac{\alpha_{B}+\beta_{B}}{8}\left(y_{1}+y_{2}-y^{\prime}_{1}-y^{\prime}_{2}\right)^{2}}e^{-\frac{\alpha_{B}-\beta_{B}}{8}\left(y_{1}-y_{2}-y^{\prime}_{1}+y^{\prime}_{2}\right)^{2}},$
$\displaystyle\rho_{B}^{(2)}(y_{1},y_{2})=N_{B}(N_{B}-1)\left[\frac{\left(\alpha_{B}-\beta_{B}\right)\left(\alpha_{B}+\beta_{B}+2C^{\prime}_{0,2}\right)}{\pi^{2}}\right]^{\frac{1}{2}}\times$
$\displaystyle\times
e^{-\frac{1}{2}\left(\alpha_{B}+\beta_{B}+2C^{\prime}_{0,2}\right)\left(y_{1}+y_{2}\right)^{2}}e^{-\frac{1}{2}\left(\alpha_{B}-\beta_{B}\right)\left(y_{1}-y_{2}\right)^{2}},\
$ (12)

from which its diagonal, the two-body density in position space, is given [51,
58].

Next, the second-order correlation function in position space reads

$\displaystyle
g_{B}^{(2)}(y_{1},y_{2})=\frac{\rho^{(2)}_{B}(y_{1},y_{2})}{\rho^{(1)}_{B}(y_{1})\rho^{(1)}_{B}(y_{2})}=\frac{N_{B}-1}{N_{B}}\left[\frac{\left(\alpha_{B}+C^{\prime}_{0,2}\right)^{2}}{\left(\alpha_{B}-\beta_{B}\right)\left(\alpha_{B}+\beta_{B}+2C^{\prime}_{0,2}\right)}\right]^{\frac{1}{2}}\times$
$\displaystyle\times
e^{-\frac{1}{2}\frac{-\left(\alpha_{B}-\beta_{B}\right)\left(\beta_{B}+C^{\prime}_{0,2}\right)}{\alpha_{B}+C^{\prime}_{0,2}}\left(y_{1}-y_{2}\right)^{2}}e^{+\frac{1}{2}\frac{-\left(\alpha_{B}+\beta_{B}+2C^{\prime}_{0,2}\right)\left(\beta_{B}+C^{\prime}_{0,2}\right)}{\alpha_{B}+C^{\prime}_{0,2}}\left(y_{1}+y_{2}\right)^{2}}.\
$ (13)

For $\beta_{B}+C^{\prime}_{0,2}<0$ the off diagonal decays and the diagonal
grows, and for $\beta_{B}+C^{\prime}_{0,2}>0$ vice versa, also see the
applications in Sec. 3 and A.

We can now proceed and define a dimensionless measure, the second-order
coherence length in position space, as the ratio of the off-diagonal decay or
growth length of the second-order correlation function (2.1) and the diagonal
decay length of the two-particle density (2.1),

$\displaystyle
l_{B}^{(2)}=\sqrt{\frac{\left(\alpha_{B}+\beta_{B}+2C^{\prime}_{0,2}\right)\left(\alpha_{B}+C^{\prime}_{0,2}\right)}{\left(\alpha_{B}-\beta_{B}\right)|\beta_{B}+C^{\prime}_{0,2}|}}=$
(14)
$\displaystyle=\sqrt{\frac{1+\frac{1}{N_{B}}\left(\frac{m_{B}N_{B}}{M}\frac{\Omega_{B}}{\omega}+\frac{m_{F}N_{F}}{M}\frac{\Omega_{B}}{\Omega_{BF}}-1\right)}{\left[1+\frac{2}{N_{B}}\left(\frac{m_{B}N_{B}}{M}\frac{\Omega_{B}}{\omega}+\frac{m_{F}N_{F}}{M}\frac{\Omega_{B}}{\Omega_{BF}}-1\right)\right]\left|\frac{1}{N_{B}}\left(\frac{m_{B}N_{B}}{M}\frac{\Omega_{B}}{\omega}+\frac{m_{F}N_{F}}{M}\frac{\Omega_{B}}{\Omega_{BF}}-1\right)\right|}}.\
$

It is instrumental to compare the first-order $l_{B}^{(1)}$ and second-order
$l_{B}^{(2)}$ coherence lengths, also see the illustrative examples in Sec. 3.

We now proceed to compute the the reduced two-particle density matrix and its
diagonal, the two-body density of the bosons, in momentum space which take on
the form

$\displaystyle\tilde{\rho}_{B}^{(2)}(q_{1},q^{\prime}_{1},q_{2},q^{\prime}_{2})=\frac{1}{\left(2\pi\right)^{2}}\int
dy_{1}dy_{2}dy^{\prime}_{1}dy^{\prime}_{2}e^{-i\left(q_{1}y_{1}+q_{2}y_{2}-q^{\prime}_{1}y^{\prime}_{1}-q^{\prime}_{2}y^{\prime}_{2}\right)}\rho_{B}^{(2)}(y_{1},y^{\prime}_{1},y_{2},y^{\prime}_{2})=$
(15)
$\displaystyle=N_{B}(N_{B}-1)\frac{1}{\pi\sqrt{\alpha_{B}^{2}-\beta^{2}_{B}}}e^{-\frac{1}{8\left(\alpha_{B}+\beta_{B}\right)}\left(q_{1}+q_{2}+q^{\prime}_{1}+q^{\prime}_{2}\right)^{2}}e^{-\frac{1}{8\left(\alpha_{B}-\beta_{B}\right)}\left(q_{1}-q_{2}+q^{\prime}_{1}-q^{\prime}_{2}\right)^{2}}\times$
$\displaystyle\times
e^{-\frac{1}{8\left(\alpha_{B}+\beta_{B}+2C^{\prime}_{0,2}\right)}\left(q_{1}+q_{2}-q^{\prime}_{1}-q^{\prime}_{2}\right)^{2}}e^{-\frac{1}{8\left(\alpha_{B}-\beta_{B}\right)}\left(q_{1}-q_{2}-q^{\prime}_{1}+q^{\prime}_{2}\right)^{2}},$
$\displaystyle\tilde{\rho}_{B}^{(2)}(q_{1},q_{2})=N_{B}(N_{B}-1)\frac{1}{\pi\sqrt{\alpha_{B}^{2}-\beta^{2}_{B}}}e^{-\frac{1}{2\left(\alpha_{B}+\beta_{B}\right)}\left(q_{1}+q_{2}\right)^{2}}e^{-\frac{1}{2\left(\alpha_{B}-\beta_{B}\right)}\left(q_{1}-q_{2}\right)^{2}}.\
$

Then, the second-order correlation function of the bosons in momentum space
can readily be evaluated as

$\displaystyle\tilde{g}_{B}^{(2)}(q_{1},q_{2})=\frac{\tilde{\rho}^{(2)}_{B}(q_{1},q_{2})}{\tilde{\rho}^{(1)}_{B}(q_{1})\tilde{\rho}^{(1)}_{B}(q_{2})}=$
$\displaystyle=\frac{N_{B}-1}{N_{B}}\sqrt{\frac{\alpha_{B}^{2}}{\alpha_{B}^{2}-\beta_{B}^{2}}}e^{-\frac{1}{2}\frac{\beta_{B}}{\alpha_{B}\left(\alpha_{B}-\beta_{B}\right)}\left(q_{1}-q_{2}\right)^{2}}e^{+\frac{1}{2}\frac{\beta_{B}}{\alpha_{B}\left(\alpha_{B}+\beta_{B}\right)}\left(q_{1}+q_{2}\right)^{2}}.\
$ (16)

For $\beta_{B}>0$ the off diagonal decays and the diagonal grows, and for
$\beta_{B}<0$ vice versa, also see the illustrative examples and A below.

Analogously to (14), we define a dimensionless measure, the second-order
coherence length in momentum space, as the ratio of the off-diagonal decay or
growth length of the second-order correlation function (2.1) and the diagonal
decay length of the two-particle density (15), both in momentum space,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\tilde{l}_{B}^{(2)}=\sqrt{\frac{\alpha_{B}\left(\alpha_{B}-\beta_{B}\right)}{|\beta_{B}|\left(\alpha_{B}+\beta_{B}\right)}}=$
(17)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!=\sqrt{\frac{1+\frac{1}{N_{B}}\left(\frac{m_{B}N_{B}}{M}\frac{\omega}{\Omega_{B}}+\frac{m_{F}N_{F}}{M}\frac{\Omega_{BF}}{\Omega_{B}}-1\right)}{\left[1+\frac{2}{N_{B}}\left(\frac{m_{B}N_{B}}{M}\frac{\omega}{\Omega_{B}}+\frac{m_{F}N_{F}}{M}\frac{\Omega_{BF}}{\Omega_{B}}-1\right)\right]\left|\frac{1}{N_{B}}\left(\frac{m_{B}N_{B}}{M}\frac{\omega}{\Omega_{B}}+\frac{m_{F}N_{F}}{M}\frac{\Omega_{BF}}{\Omega_{B}}-1\right)\right|}}\neq
l_{B}^{(2)}.\ $

It is found that the second-order coherence lengths in position (14) and
momentum (17) spaces are different, unlike the respective first-order
coherence lengths. The reason is that different parts of the many-particle
wavefunction contribute to the second-order quantities in position and
momentum spaces. Although $l_{B}^{(1)}\neq\tilde{l}_{B}^{(1)}$, their
respective expressions are ‘reciprocal’ to each other in the sense
$\frac{\Omega_{B}}{\omega},\frac{\Omega_{B}}{\Omega_{BF}}\Leftrightarrow\frac{\omega}{\Omega_{B}},\frac{\Omega_{BF}}{\Omega_{B}}$,
see above and compare (14) and (17). Additionally and on the other hand, there
seems to be no simple relation between the second-order coherence lengths and
the pair depleted fraction $0\leq\rho_{B}^{(2)}<1$. For reference, the pair
depleted fraction is given in closed-form using the above coefficients and
reads
$\rho_{B}^{(2)}=\frac{\sqrt{\alpha_{B}+\beta_{B}}-\sqrt{\alpha_{B}+\beta_{B}+2C^{\prime}_{0,2}}}{\sqrt{\alpha_{B}+\beta_{B}}+\sqrt{\alpha_{B}+\beta_{B}+2C^{\prime}_{0,2}}}$,
see [51]. Last but not least, it is deductive to compare the first-order
$\tilde{l}_{B}^{(1)}$ and second-order $\tilde{l}_{B}^{(2)}$ coherence lengths
in momentum space as well, also see the illustrative examples in Sec. 3.

###  2.2 Position and momentum correlation functions of the fermions

The reduced one-particle density matrix and the density of the $N_{F}=2$
fermions in position space read [58]

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\rho_{F}^{(1)}(x,x^{\prime})=2\left(\frac{\alpha_{F}+C_{1,0}}{\pi}\right)^{\frac{1}{2}}\times$
(18)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\times\left\\{1-\frac{1}{2}\frac{\alpha_{F}+C_{1,0}}{\alpha_{F}-\beta_{F}}+\frac{1}{4}\left[\frac{\left(\alpha_{F}+C_{1,0}\right)^{2}}{\alpha_{F}-\beta_{F}}\left(x+x^{\prime}\right)^{2}-\left(\alpha_{F}-\beta_{F}\right)\left(x-x^{\prime}\right)^{2}\right]\right\\}\times$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\times
e^{-\frac{\alpha_{F}+C_{1,0}}{4}\left(x+x^{\prime}\right)^{2}}e^{-\frac{\alpha_{F}}{4}\left(x-x^{\prime}\right)^{2}},$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\rho_{F}^{(1)}(x)=2\left(\frac{\alpha_{F}+C_{1,0}}{\pi}\right)^{\frac{1}{2}}\left[1-\frac{1}{2}\frac{\alpha_{F}+C_{1,0}}{\alpha_{F}-\beta_{F}}+\frac{\left(\alpha_{F}+C_{1,0}\right)^{2}}{\alpha_{F}-\beta_{F}}x^{2}\right]e^{-\left(\alpha_{F}+C_{1,0}\right)x^{2}}.\
$

Using (18), the first-order correlation function of the fermions in position
space is computed as

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!g_{F}^{(1)}(x,x^{\prime})=\frac{\rho^{(1)}_{F}(x,x^{\prime})}{\sqrt{\rho^{(1)}_{F}(x)\rho^{(1)}_{F}(x^{\prime})}}=$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!=\frac{1-\frac{1}{2}\frac{\alpha_{F}+C_{1,0}}{\alpha_{F}-\beta_{F}}+\frac{1}{4}\left[\frac{\left(\alpha_{F}+C_{1,0}\right)^{2}}{\alpha_{F}-\beta_{F}}\left(x+x^{\prime}\right)^{2}-\left(\alpha_{F}-\beta_{F}\right)\left(x-x^{\prime}\right)^{2}\right]}{\sqrt{\left[1-\frac{1}{2}\frac{\alpha_{F}+C_{1,0}}{\alpha_{F}-\beta_{F}}+\frac{\left(\alpha_{F}+C_{1,0}\right)^{2}}{\alpha_{F}-\beta_{F}}x^{2}\right]\left[1-\frac{1}{2}\frac{\alpha_{F}+C_{1,0}}{\alpha_{F}-\beta_{F}}+\frac{\left(\alpha_{F}+C_{1,0}\right)^{2}}{\alpha_{F}-\beta_{F}}{x^{\prime}}^{2}\right]}}e^{-\frac{-C_{1,0}}{4}\left(x-x^{\prime}\right)^{2}},\
$ (19)

where $C_{1,0}$ is negative and given in closed form as a function of the
coefficients (2) [50, 58], see for further discussion below.

Similarly to the bosons, it is instructive to define a dimensionless measure,
the first-order coherence length of the fermions in position space, as the
ratio of the off-diagonal decay length of the first-order correlation function
(2.2) and the diagonal decay length of the reduced one-particle density matrix
(18), giving

$\displaystyle l_{F}^{(1)}=\sqrt{\frac{\alpha_{F}+C_{1,0}}{-C_{1,0}}}=$ (20)
$\displaystyle=\frac{1}{\sqrt{\left[1+\frac{1}{N_{F}}\left(\frac{m_{F}N_{F}}{M}\frac{\omega}{\Omega_{F}}+\frac{m_{B}N_{B}}{M}\frac{\Omega_{BF}}{\Omega_{F}}-1\right)\right]\left[1+\frac{1}{N_{F}}\left(\frac{m_{F}N_{F}}{M}\frac{\Omega_{F}}{\omega}+\frac{m_{B}N_{B}}{M}\frac{\Omega_{F}}{\Omega_{BF}}-1\right)\right]-1}}.\
$

Formally, it is possible to re-express the first-order coherence length (20)
using the quantity
$\bar{\rho}_{F}^{(1)}=\frac{\sqrt{\alpha_{F}}-\sqrt{\alpha_{F}+C_{1,0}}}{\sqrt{\alpha_{F}}+\sqrt{\alpha_{F}+C_{1,0}}}$,
which satisfies $0\leq\bar{\rho}_{F}^{(1)}<1$ and which would have been the
depleted fraction if the impurity were bosonic, see [51]. The final result is
$l_{F}^{(1)}=\left[\left(\frac{1+\bar{\rho}_{F}^{(1)}}{1-\bar{\rho}_{F}^{(1)}}\right)^{2}-1\right]^{-\frac{1}{2}}$,
compare to (8). Obviously, the first-order coherence length $l_{F}^{(1)}$
diverges for non-interacting fermions.

The reduced one-particle density matrix and the density of the fermions in
momentum space are readily evaluated from the reduced one-particle density
matrix in position space,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\tilde{\rho}_{F}^{(1)}(k,k^{\prime})=\frac{1}{2\pi}\int
dxdx^{\prime}e^{-i\left(kx-k^{\prime}x^{\prime}\right)}\rho^{(1)}_{F}(x,x^{\prime})=$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!=2\left(\frac{1}{\pi\alpha_{F}}\right)^{\frac{1}{2}}\left\\{1-\frac{1}{2}\frac{\alpha_{F}-\beta_{F}}{\alpha_{F}}+\frac{1}{4}\left[\frac{\alpha_{F}-\beta_{F}}{\alpha_{F}^{2}}\left(k+k^{\prime}\right)^{2}-\frac{1}{\alpha_{F}-\beta_{F}}\left(k-k^{\prime}\right)^{2}\right]\right\\}\times$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\times
e^{-\frac{1}{4\alpha_{F}}\left(k+k^{\prime}\right)^{2}}e^{-\frac{1}{4\left(\alpha_{F}+C_{1,0}\right)}\left(k-k^{\prime}\right)^{2}},$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\tilde{\rho}_{F}^{(1)}(k)=2\left(\frac{1}{\pi\alpha_{F}}\right)^{\frac{1}{2}}\left[1-\frac{1}{2}\frac{\alpha_{F}-\beta_{F}}{\alpha_{F}}+\frac{\alpha_{F}-\beta_{F}}{\alpha_{F}^{2}}k^{2}\right]e^{-\frac{1}{\alpha_{F}}k^{2}}.\
$ (21)

Afterwards, the first-order correlation function in momentum space is given by

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\tilde{g}_{F}^{(1)}(k,k^{\prime})=\frac{\tilde{\rho}^{(1)}_{F}(k,k^{\prime})}{\sqrt{\tilde{\rho}^{(1)}_{F}(k)\tilde{\rho}^{(1)}_{F}(k^{\prime})}}=$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!=\frac{1-\frac{1}{2}\frac{\alpha_{F}-\beta_{F}}{\alpha_{F}}+\frac{1}{4}\left[\frac{\alpha_{F}-\beta_{F}}{\alpha_{F}^{2}}\left(k+k^{\prime}\right)^{2}-\frac{1}{\alpha_{F}-\beta_{F}}\left(k-k^{\prime}\right)^{2}\right]}{\sqrt{\left[1-\frac{1}{2}\frac{\alpha_{F}-\beta_{F}}{\alpha_{F}}+\frac{\alpha_{F}-\beta_{F}}{\alpha_{F}^{2}}k^{2}\right]\left[1-\frac{1}{2}\frac{\alpha_{F}-\beta_{F}}{\alpha_{F}}+\frac{\alpha_{F}-\beta_{F}}{\alpha_{F}^{2}}{k^{\prime}}^{2}\right]}}e^{-\frac{-C_{1,0}}{4\alpha_{F}\left(\alpha_{F}+C_{1,0}\right)}\left(k-k^{\prime}\right)^{2}}.\
$ (22)

Just like (11), we define a dimensionless measure, the first-order coherence
length of the fermions in momentum space, as the ratio of the off-diagonal
decay length of the first-order correlation function (2.2) and the diagonal
decay length of the reduced one-particle density matrix (2.2), both in
momentum space,

$\displaystyle\tilde{l}_{F}^{(1)}=\sqrt{\frac{\alpha_{F}+C_{1,0}}{-C_{1,0}}}=l_{F}^{(1)}.$
(23)

It is found that the first-order coherence lengths in position (20) and
momentum (23) spaces are equal.

To ease the Fourier transform of second-order quantities to momentum space
below, the reduced two-particle density matrix in position space, including
the term $\left(x_{1}-x_{2}\right)\left(x^{\prime}_{1}-x^{\prime}_{2}\right)$,
is expressed in a quadratic form using normal coordinates. The final result is
given by

$\displaystyle\rho_{F}^{(2)}(x_{1},x^{\prime}_{1},x_{2},x^{\prime}_{2})=2\left[\frac{\left(\alpha_{F}-\beta_{F}\right)^{3}\left(\alpha_{F}+\beta_{F}+2C_{2,0}\right)}{\pi^{2}}\right]^{\frac{1}{2}}\times$
$\displaystyle\times\left(x_{1}-x_{2}\right)\left(x^{\prime}_{1}-x^{\prime}_{2}\right)e^{-\frac{\alpha_{F}+\beta_{F}+2C_{2,0}}{8}\left(x_{1}+x_{2}+x^{\prime}_{1}+x^{\prime}_{2}\right)^{2}}e^{-\frac{\alpha_{F}-\beta_{F}}{8}\left(x_{1}-x_{2}+x^{\prime}_{1}-x^{\prime}_{2}\right)^{2}}\times$
$\displaystyle\times
e^{-\frac{\alpha_{F}+\beta_{F}}{8}\left(x_{1}+x_{2}-x^{\prime}_{1}-x^{\prime}_{2}\right)^{2}}e^{-\frac{\alpha_{F}-\beta_{F}}{8}\left(x_{1}-x_{2}-x^{\prime}_{1}+x^{\prime}_{2}\right)^{2}},$
$\displaystyle\rho_{F}^{(2)}(x_{1},x_{2})=2\left[\frac{\left(\alpha_{F}-\beta_{F}\right)^{3}\left(\alpha_{F}+\beta_{F}+2C_{2,0}\right)}{\pi^{2}}\right]^{\frac{1}{2}}\times$
$\displaystyle\times\left(x_{1}-x_{2}\right)^{2}e^{-\frac{1}{2}\left(\alpha_{F}+\beta_{F}+2C_{2,0}\right)\left(x_{1}+x_{2}\right)^{2}}e^{-\frac{1}{2}\left(\alpha_{F}-\beta_{F}\right)\left(x_{1}-x_{2}\right)^{2}},\
$ (24)

from which the two-body density in position space is prescribed.

The second-order correlation function in position space is now evaluated as

$\displaystyle
g_{F}^{(2)}(x_{1},x_{2})=\frac{\rho^{(2)}_{F}(x_{1},x_{2})}{\rho^{(1)}_{F}(x_{1})\rho^{(1)}_{F}(x_{2})}=\frac{1}{2}\left[\frac{\left(\alpha_{F}+C_{2,0}\right)^{2}}{\left(\alpha_{F}-\beta_{F}\right)^{3}\left(\alpha_{F}+\beta_{F}+2C_{2,0}\right)}\right]^{\frac{1}{2}}\times$
$\displaystyle\times\frac{\left(x_{1}-x_{2}\right)^{2}}{\left[\frac{1}{2\left(\alpha_{F}+C_{2,0}\right)}+\left(1+\frac{\beta_{F}+C_{2,0}}{\alpha_{F}+C_{2,0}}\right)^{2}x_{1}^{2}\right]\left[\frac{1}{2\left(\alpha_{F}+C_{2,0}\right)}+\left(1+\frac{\beta_{F}+C_{2,0}}{\alpha_{F}+C_{2,0}}\right)^{2}x_{2}^{2}\right]}\times$
$\displaystyle\times
e^{-\frac{1}{2}\frac{-\left(\alpha_{F}-\beta_{F}\right)\left(\beta_{F}+C_{2,0}\right)}{\alpha_{F}+C_{2,0}}\left(x_{1}-x_{2}\right)^{2}}e^{+\frac{1}{2}\frac{-\left(\alpha_{F}+\beta_{F}+2C_{2,0}\right)\left(\beta_{F}+C_{2,0}\right)}{\alpha_{F}+C_{2,0}}\left(x_{1}+x_{2}\right)^{2}}.\
$ (25)

For $\beta_{F}+C_{2,0}<0$ the off diagonal decays and the diagonal grows
whereas the opposite occurs for $\beta_{F}+C_{2,0}>0$, also see Sec. 3 and A
below.

We define and compute a dimensionless measure, the second-order coherence
length of the fermions in position space, as the ratio of the off-diagonal
decay or growth length of the second-order correlation function (2.2) and the
diagonal decay length of the two-particle density (2.2),

$\displaystyle
l_{F}^{(2)}=\sqrt{\frac{\left(\alpha_{F}+\beta_{F}+2C_{2,0}\right)\left(\alpha_{F}+C_{2,0}\right)}{\left(\alpha_{F}-\beta_{F}\right)|\beta_{F}+C_{2,0}|}}=$
(26)
$\displaystyle=\sqrt{\frac{1+\frac{1}{N_{F}}\left(\frac{m_{F}N_{F}}{M}\frac{\Omega_{F}}{\omega}+\frac{m_{B}N_{B}}{M}\frac{\Omega_{F}}{\Omega_{BF}}-1\right)}{\left[1+\frac{2}{N_{F}}\left(\frac{m_{F}N_{F}}{M}\frac{\Omega_{F}}{\omega}+\frac{m_{B}N_{B}}{M}\frac{\Omega_{F}}{\Omega_{BF}}-1\right)\right]\left|\frac{1}{N_{F}}\left(\frac{m_{F}N_{F}}{M}\frac{\Omega_{F}}{\omega}+\frac{m_{B}N_{B}}{M}\frac{\Omega_{F}}{\Omega_{BF}}-1\right)\right|}}.\
$

It would be instructive to contrast the first-order $l_{F}^{(1)}$ and second-
order $l_{F}^{(2)}$ coherence lengths, which is done in the illustrative
examples of Sec. 3.

To continue, the reduced two-particle density matrix and two-body density of
the fermions in momentum space are calculated,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\tilde{\rho}_{F}^{(2)}(k_{1},k^{\prime}_{1},k_{2},k^{\prime}_{2})=\frac{1}{\left(2\pi\right)^{2}}\int
dx_{1}dx_{2}dx^{\prime}_{1}dx^{\prime}_{2}e^{-i\left(k_{1}x_{1}+k_{2}x_{2}-k^{\prime}_{1}x^{\prime}_{1}-k^{\prime}_{2}x^{\prime}_{2}\right)}\rho_{F}^{(2)}(x_{1},x^{\prime}_{1},x_{2},x^{\prime}_{2})=$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=2\frac{1}{\pi\sqrt{\left(\alpha_{F}^{2}-\beta^{2}_{F}\right)\left(\alpha_{F}-\beta_{F}\right)^{2}}}\left(k_{1}-k_{2}\right)\left(k^{\prime}_{1}-k^{\prime}_{2}\right)e^{-\frac{1}{8\left(\alpha_{F}+\beta_{F}\right)}\left(k_{1}+k_{2}+k^{\prime}_{1}+k^{\prime}_{2}\right)^{2}}\times$
(27) $\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\times
e^{-\frac{1}{8\left(\alpha_{F}-\beta_{F}\right)}\left(k_{1}-k_{2}+k^{\prime}_{1}-k^{\prime}_{2}\right)^{2}}e^{-\frac{1}{8\left(\alpha_{F}+\beta_{F}+2C_{2,0}\right)}\left(k_{1}+k_{2}-k^{\prime}_{1}-k^{\prime}_{2}\right)^{2}}e^{-\frac{1}{8\left(\alpha_{F}-\beta_{F}\right)}\left(k_{1}-k_{2}-k^{\prime}_{1}+k^{\prime}_{2}\right)^{2}},$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\tilde{\rho}_{F}^{(2)}(k_{1},k_{2})=2\frac{1}{\pi\sqrt{\left(\alpha_{F}^{2}-\beta^{2}_{F}\right)\left(\alpha_{F}-\beta_{F}\right)^{2}}}\left(k_{1}-k_{2}\right)^{2}e^{-\frac{1}{2\left(\alpha_{F}+\beta_{F}\right)}\left(k_{1}+k_{2}\right)^{2}}e^{-\frac{1}{2\left(\alpha_{F}-\beta_{F}\right)}\left(k_{1}-k_{2}\right)^{2}}.\
$

Thereafter, the second-order correlation function in momentum space is
evaluated and reads

$\displaystyle\tilde{g}_{F}^{(2)}(k_{1},k_{2})=\frac{\tilde{\rho}^{(2)}_{F}(k_{1},k_{2})}{\tilde{\rho}^{(1)}_{F}(k_{1})\tilde{\rho}^{(1)}_{F}(k_{2})}=\frac{1}{2}\sqrt{\frac{\alpha_{F}^{2}}{\left(\alpha_{F}^{2}-\beta^{2}_{F}\right)\left(\alpha_{F}-\beta_{F}\right)^{2}}}\times$
$\displaystyle\times\frac{\left(k_{1}-k_{2}\right)^{2}}{\left[1-\frac{1}{2}\frac{\alpha_{F}-\beta_{F}}{\alpha_{F}}+\frac{\alpha_{F}-\beta_{F}}{\alpha_{F}^{2}}k_{1}^{2}\right]\left[1-\frac{1}{2}\frac{\alpha_{F}-\beta_{F}}{\alpha_{F}}+\frac{\alpha_{F}-\beta_{F}}{\alpha_{F}^{2}}k_{2}^{2}\right]}\times$
$\displaystyle\times
e^{-\frac{1}{2}\frac{\beta_{F}}{\alpha_{F}\left(\alpha_{F}-\beta_{F}\right)}\left(k_{1}-k_{2}\right)^{2}}e^{+\frac{1}{2}\frac{\beta_{F}}{\alpha_{F}\left(\alpha_{F}+\beta_{F}\right)}\left(k_{1}+k_{2}\right)^{2}}.\
$ (28)

For $\beta_{F}>0$ the off diagonal decays and the diagonal grows, and for
$\beta_{F}<0$ the other way around, also see the applications below.

Finally, a dimensionless measure, the second-order coherence length of the
fermions in momentum space, is defined as the ratio of the off-diagonal decay
or growth length of the second-order momentum correlation function (2.2) and
the diagonal decay length of the two-particle momentum density (2.2) and found
to be

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\tilde{l}_{F}^{(2)}=\sqrt{\frac{\alpha_{F}\left(\alpha_{F}-\beta_{F}\right)}{|\beta_{F}|\left(\alpha_{F}+\beta_{F}\right)}}=$
(29)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!=\sqrt{\frac{1+\frac{1}{N_{F}}\left(\frac{m_{F}N_{F}}{M}\frac{\omega}{\Omega_{F}}+\frac{m_{B}N_{B}}{M}\frac{\Omega_{BF}}{\Omega_{F}}-1\right)}{\left[1+\frac{2}{N_{F}}\left(\frac{m_{F}N_{F}}{M}\frac{\omega}{\Omega_{F}}+\frac{m_{B}N_{B}}{M}\frac{\Omega_{BF}}{\Omega_{F}}-1\right)\right]\left|\frac{1}{N_{F}}\left(\frac{m_{F}N_{F}}{M}\frac{\omega}{\Omega_{F}}+\frac{m_{B}N_{B}}{M}\frac{\Omega_{BF}}{\Omega_{F}}-1\right)\right|}}\neq
l_{F}^{(2)}.\ $

This implies that that the second-order coherence lengths in position (26) and
momentum (29) spaces are different, just as for the bosons. Although
$l_{F}^{(1)}\neq\tilde{l}_{F}^{(1)}$, their respective expressions are
‘reciprocal’ to each other in the manner
$\frac{\Omega_{F}}{\omega},\frac{\Omega_{F}}{\Omega_{BF}}\Leftrightarrow\frac{\omega}{\Omega_{F}},\frac{\Omega_{BF}}{\Omega_{F}}$,
see above and compare (26) and (29). Finally and unlike the above analysis for
the first-order coherence lengths, there is probably no simple relation
between the second-order coherence lengths and between what would have been
the pair depleted fraction
$0\leq\frac{\sqrt{\alpha_{F}+\beta_{F}}-\sqrt{\alpha_{F}+\beta_{F}+2C_{2,0}}}{\sqrt{\alpha_{F}+\beta_{F}}+\sqrt{\alpha_{F}+\beta_{F}+2C_{2,0}}}<1$,
see [51], if the impurity were made of bosons. It would hence be rewarding to
compare the first-order $\tilde{l}_{F}^{(1)}$ and second-order
$\tilde{l}_{F}^{(2)}$ coherence lengths in momentum space as well, also
examine the application section below.

##  3 Application

We present illustrative examples. The results are shown in Fig. 1. Consider an
impurity made of $N_{F}=2$ fermions in a condensate made of $N_{B}=10,000$
bosons. The mass of a fermion is $m_{F}=1.0$ and that of a boson $m_{B}=0.01$.
The frequency of the trap is $\omega=1.0$. We explore the case where there are
no intraspecies interactions, namely, $\lambda_{F}=0$ and $\lambda_{B}=0$. The
correlation effects are generated only by the interspecies interaction which
can assume the values $-\frac{M_{BF}\omega^{2}}{2}<\lambda_{BF}$. We
investigate the above-derived eight coherence lengths,
$l_{B}^{(1)},l_{B}^{(2)},l_{F}^{(1)},l_{F}^{(2)}$ in position space in Fig. 1a
and
$\tilde{l}_{B}^{(1)},\tilde{l}_{B}^{(2)},\tilde{l}_{F}^{(1)},\tilde{l}_{F}^{(2)}$
in momentum space in Fig. 1b. It is convenient to plot quantities as a
function of the relative center-of-mass frequency
$\Omega_{BF}=\sqrt{\omega^{2}+2\left(\frac{N_{B}}{m_{F}}+\frac{N_{F}}{m_{B}}\right)\lambda_{BF}}$,
see Eq. (2), which in the attractive sector is not bound from above and
satisfies $1<\Omega_{BF}$ and in the repulsive sector obeys
$0^{+}<\Omega_{BF}<1$. For completeness, the interspecies interaction strength
$\lambda_{BF}=\frac{1}{2}M_{BF}\left(\Omega^{2}_{BF}-\omega^{2}\right)$ is
also plotted in Fig. 1 (in the repulsive sector for the above parameters,
$|\lambda_{BF}|<\frac{M_{BF}\omega^{2}}{2}\approx 4.90\cdot 10^{-5}$ and is
not visible). Obviously, for $\Omega_{BF}=1$ the mixture is non-interacting
($\lambda_{BF}=0$), and correspondingly all coherence lengths diverge per
definition.

As the impurity-condensate interaction sets in, whether repulsive or
attractive, all coherence lengths become finite, see Fig. 1. A diverse
behavior is found. In position space, Fig. 1a, the impurity and condensate
coherence lengths decay to zero with increasing interspecies interaction,
repulsive or attractive, and the second-order coherence lengths are nearly the
same as the first-order ones. In momentum space, Fig. 1b, as shown above in
(11) and (23), the first-order coherence lengths are equal to their position-
space counterparts and thus decay to zero. On the other hand, both second-
order coherence lengths $\tilde{l}_{B}^{(2)}$ and $\tilde{l}_{F}^{(2)}$
saturate to (different) finite values for strong interspecies repulsion or
attraction. Analysis of the closed-form expressions for $\tilde{l}_{B}^{(2)}$
and $\tilde{l}_{F}^{(2)}$, Eqs. (17) and (29), straightforwardly show why the
limiting values of the second-order momentum-space coherence lengths are
finite, whereas the other six coherence lengths decay to zero in these limits.
We finish with a question and the beginning of its answer. Can one have
different behaviors of the coherence lengths than the above? The answer is
positive. Two other scenarios are analyzed in the A, demonstrating and
emphasizing further the richness of the exactly-solvable harmonic-interaction
many-body model for Bose-Fermi mixtures [58].

Figure 1: First-order and second-order dimensionless coherence lengths of the
impurity and the condensate. The number of fermions is $N_{F}=2$ and the
number of bosons $N_{B}=10,000$. The mass of a fermion is $m_{F}=1.0$ and that
of a boson $m_{B}=0.01$. Panel (a) depicts the coherence lengths in position
space and panel (b) in momentum space. The results are plotted against the
relative center-of-mass frequency
$\Omega_{BF}=\sqrt{\omega^{2}+2\left(\frac{N_{B}}{m_{F}}+\frac{N_{F}}{m_{B}}\right)\lambda_{BF}}$
[Eq. (2)] which satisfies $1<\Omega_{BF}$ for attractive and
$0^{+}<\Omega_{BF}<1$ for repulsive interspecies interactions. For reference,
in panel (a) the interspecies interaction strength
$\lambda_{BF}=\frac{1}{2}\frac{m_{F}m_{B}}{M}\left(\Omega^{2}_{BF}-\omega^{2}\right)$
is plotted. The frequency of the trap is $\omega=1.0$. There are no
intraspecies interactions, i.e., $\lambda_{F}=0$ and $\lambda_{B}=0$.
Correlations build up only due to impurity-condensate interaction. The first-
order coherence lengths in position and momentum spaces are equal whereas the
second-order quantities can differ considerably. See the text for more
details. The quantities shown are dimensionless.

##  4 Summary

A fermionic impurity embedded in a Bose-Einstein condensate is treated
analytically, at the many-body level of theory, within the exactly-solvable
harmonic-interaction model for trapped Bose-Fermi mixtures. In the present
work, we have focused on the construction and analysis of correlation
functions, both in position and momentum spaces, for $N_{F}=2$ fermions and a
general number $N_{B}$ of bosons. For the fermionic impurity as well as for
the condensate, we have prescribed the exact first-order and second-order
correlation functions as functions of the masses $m_{F},m_{B}$ and
intraspecies $\lambda_{F},\lambda_{B}$ and interspecies $\lambda_{BF}$
interaction strengths. Furthermore, we have studied properties of the various
coherence lengths, defined as the ratio between the off-diagonal decay or
growth length of a correlation function and the diagonal decay length of the
respective (reduced) density. It has been found that the first-order coherence
lengths in position and momentum spaces are equal, whereas the second-order
quantities can differ significantly. Illustrative examples were presented and
discussed. In particular, when the sole interaction is between the impurity
and the condensate, all coherence lengths decay to zero for strong attraction
or repulsion, except for the second-order coherence lengths in momentum space
which saturate at finite values. As an obvious yet challenging outlook, it
would be interesting to extend the Bose-Fermi harmonic-interaction model and
its applications to time-dependent scenarios.

This research was supported by the Israel Science Foundation (Grant No.
1516/19).

##  Appendix A Limiting cases of the bosons’ and fermions’ first-order and
second-order dimensionless coherence lengths in position and momentum spaces

Consider, first, the case of no interaction between the impurity and the
condensate, i.e., $\Omega_{BF}=\omega$. We analyze the bosons but the results
hold for the fermions as well. Then,

$\displaystyle
l_{B}^{(1)}=\tilde{l}_{B}^{(1)}=\frac{1}{\sqrt{\left[1+\frac{1}{N_{B}}\left(\frac{\omega}{\Omega_{B}}-1\right)\right]\left[1+\frac{1}{N_{B}}\left(\frac{\Omega_{B}}{\omega}-1\right)\right]-1}}\
$ (30)

and

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!l_{B}^{(2)}=\sqrt{\frac{1+\frac{1}{N_{B}}\left(\frac{\Omega_{B}}{\omega}-1\right)}{\left[1+\frac{2}{N_{B}}\left(\frac{\Omega_{B}}{\omega}-1\right)\right]\left|\frac{1}{N_{B}}\left(\frac{\Omega_{B}}{\omega}-1\right)\right|}},\quad\tilde{l}_{B}^{(2)}=\sqrt{\frac{1+\frac{1}{N_{B}}\left(\frac{\omega}{\Omega_{B}}-1\right)}{\left[1+\frac{2}{N_{B}}\left(\frac{\omega}{\Omega_{B}}-1\right)\right]\left|\frac{1}{N_{B}}\left(\frac{\omega}{\Omega_{B}}-1\right)\right|}}.$
(31)

The limits $\lim_{\Omega_{B}\to\infty}$ and $\lim_{\Omega_{B}\to 0^{+}}$ of
the coherence lengths (30) and (A) are all equal to zero expect for

$\displaystyle\lim_{\Omega_{B}\to
0^{+}}l_{B}^{(2)}=\lim_{\Omega_{B}\to\infty}\tilde{l}_{B}^{(2)}=\sqrt{\frac{1-\frac{1}{N_{B}}}{\left(1-\frac{2}{N_{B}}\right)\frac{1}{N_{B}}}}.\
$ (32)

The meaning of (32) is that the second-order position and momentum coherence
lengths are always finite (for a finite number of particles) at the limit of
strong attractive and repulsive interactions, respectively, expect for a
single pair of identical particles, $N_{B}=2$. Then, the coherence length
diverges because the diagonal of the two-particle density is nothing but the
pairs’ center-of-mass wavefunction (density) which does not depend on the
interaction.

Next, consider the regime of parameters in which the interspecies interaction
is ‘balanced’ by the intraspecies interactions, namely
$\lambda_{F}=-\frac{N_{B}}{N_{F}}\lambda_{BF}$ and
$\lambda_{B}=-\frac{N_{F}}{N_{B}}\lambda_{BF}$. Consequently,
$\Omega_{F}=\Omega_{B}=\omega$. Furthermore, the energy (2) depends explicitly
on the interspecies interaction only though the relative center-of-mass
degree-of-freedom,

$\displaystyle
E=\frac{1}{2}\left[\left(N_{F}^{2}+N_{B}-1\right)\omega+\sqrt{\omega^{2}+2\left(\frac{N_{B}}{m_{F}}+\frac{N_{F}}{m_{B}}\right)\lambda_{BF}}\right].$
(33)

Thus, we may call this regime of ‘balanced’ interactions the regime of
‘marginal interaction’, since only one out of $N_{F}+N_{B}$ degrees-of-freedom
is explicitly interaction dependent. Now,

$\displaystyle
l_{B}^{(1)}=\tilde{l}_{B}^{(1)}=\frac{1}{\sqrt{\left[1+\frac{1}{N_{B}}\frac{m_{F}N_{F}}{M}\left(\frac{\Omega_{BF}}{\omega}-1\right)\right]\left[1+\frac{1}{N_{B}}\frac{m_{F}N_{F}}{M}\left(\frac{\omega}{\Omega_{BF}}-1\right)\right]-1}}\
$ (34)

and

$\displaystyle
l_{B}^{(2)}=\sqrt{\frac{1+\frac{1}{N_{B}}\frac{m_{F}N_{F}}{M}\left(\frac{\omega}{\Omega_{BF}}-1\right)}{\left[1+\frac{2}{N_{B}}\frac{m_{F}N_{F}}{M}\left(\frac{\omega}{\Omega_{BF}}-1\right)\right]\left|\frac{1}{N_{B}}\frac{m_{F}N_{F}}{M}\left(\frac{\omega}{\Omega_{BF}}-1\right)\right|}},$
$\displaystyle\tilde{l}_{B}^{(2)}=\sqrt{\frac{1+\frac{1}{N_{B}}\frac{m_{F}N_{F}}{M}\left(\frac{\Omega_{BF}}{\omega}-1\right)}{\left[1+\frac{2}{N_{B}}\frac{m_{F}N_{F}}{M}\left(\frac{\Omega_{BF}}{\omega}-1\right)\right]\left|\frac{1}{N_{B}}\frac{m_{F}N_{F}}{M}\left(\frac{\Omega_{BF}}{\omega}-1\right)\right|}}.\
$ (35)

The limits $\lim_{\Omega_{BF}\to\infty}$ and $\lim_{\Omega_{BF}\to 0^{+}}$ of
the coherence lengths (34) and (A) are all equal to zero expect for

$\displaystyle\lim_{\Omega_{BF}\to\infty}l_{B}^{(2)}=\lim_{\Omega_{BF}\to
0^{+}}\tilde{l}_{B}^{(2)}=\sqrt{\frac{1-\frac{1}{N_{B}}\frac{m_{F}N_{F}}{M}}{\left(1-\frac{2}{N_{B}}\frac{m_{F}N_{F}}{M}\right)\frac{1}{N_{B}}\frac{m_{F}N_{F}}{M}}},\
$ (36)

which are finite for any number of particles $N_{B}$ including a pair, compare
to (32). The interspecies interaction dresses even a single pair of the other
species, making the pair’s center-of-mass degree-of-freedom interaction
dependent. Consequently, the respective second-order coherence lengths become
finite at the limit of strong attractive and repulsive interactions, see (36).

Finally, equivalent limiting relations as the above, Eqs. (30-32) and (34-36),
hold for the coherence lengths of the fermions,
$l_{F}^{(1)}=\tilde{l}_{F}^{(1)}$, $l_{F}^{(2)}$, and $\tilde{l}_{F}^{(2)}$,
with the obvious interchanges $N_{B}\Leftrightarrow N_{F}$,
$\Omega_{B}\Leftrightarrow\Omega_{F}$, and $m_{F}\Leftrightarrow m_{B}$.

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