DESY-23-223

Exact Tunneling Solutions in Multi-Field Potentials

J.R. Espinosaa, T. Konstandinb

a Instituto de Física Teórica, IFT-UAM/CSIC,
C/ Nicolás Cabrera 13-15, Campus de Cantoblanco, 28049, Madrid, Spain
b Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany

The tunneling potential formalism makes it easy to construct exact solutions
to the vacuum decay problem in potentials with multiple fields. While some
exact solutions for single-field decays were known, we present the first
nontrivial analytic examples with two and three scalar fields, and show how
the method can be generalized to include gravitational corrections. Our
results illuminate some analytic properties of the tunneling potential
functions and can have a number of uses, among others: to serve as simple
approximations to realistic potentials; to learn about parametric dependencies
of decay rates; to check conjectures on vacuum decay; as benchmarks for multi-
field numerical codes; or to study holographic interpretations of vacuum
decay.

##  1 Introduction

Vacuum decay via quantum tunneling is a ubiquitous phenomenon which is
relevant in many areas of particle physics and cosmology (as well as condensed
matter systems). A cursory look at the references to the pioneering work of
Coleman [1] (without gravity) and Coleman and De Luccia [2] (with gravity)
corroborates this. There is a vast range of physical systems where decay of
metastable states is of central importance, from cosmological phase
transitions to particle physics scenarios beyond the standard model featuring
multiple vacua; from the string landscape to the AdS/CFT correspondence, etc.
Exact solutions to the tunneling problem (beyond the thin-wall limit) are
useful for several reasons:

1) They can be good approximations to more complicated situations and allow a
parametric understanding of the real problem. One relevant example is the
(negative) quartic potential $V=-(\lambda/4)\varphi^{4}$ which is a simple
approximation to the potential of the Higgs field at field values much larger
than the electroweak scale (see e.g. [3]). This potential admits an exact
tunneling solution (the Fubini instanton, in the Euclidean formulation of the
decay problem) which allows to calculate the tunneling action for decay,
$S=8\pi^{2}/(3\lambda)$.

2) They can be used to learn about the analytic structure and parametric
dependencies of decay rates. As one particular example, such analytically
solvable potentials were used in [4] to study how the tunneling action scales
with the size and widths of barriers both for single-field potentials and
additive multi-field potentials. Another example can be found in [5], which
explores analytically the parametric dependence of the tunneling action for
the decay of the Standard Model vacuum including running of $\lambda$,
gravitational corrections and a non-minimal coupling of the Higgs field to
gravity.

3) They can be used to check (and discard) conjectures about some type of
vacuum decay. For example, the negative quartic potential discussed in 1)
above admits a family of tunneling instantons with degenerate action as a
result of the scale invariance of the potential. Thus, one might conjecture
that scale invariance is necessary to have such action degeneracy. However,
[6] found a simple analytic counterexample to this (see subsection 7.5 of that
paper).

4) They can serve as testbeds for numerical codes [7, 8, 9, 10, 11, 12] that
are often employed in practice, particularly for the study of multi-field
potentials, for which one cannot rely on the overshoot/undershoot method to
find solutions.

5) For the case with gravity, vacuum decay is associated to Coleman-De Luccia
(CdL) geometries [2]. Analytic CdL geometries beyond thin-wall can be useful
[13] to study the connection between bubble nucleation in eternal inflation
and a Euclidean conformal field theory via a holographic duality [14].
Similarly, such examples can also be useful to study in detail Maldacena’s
interpretation [15] of decays of a false vacuum into an AdS region in terms of
a dual field theory living on an end of the world brane expanding into the
false vacuum.

The aim of this paper is to provide a constructive method to generate exact
solutions starting from the single-field case. The main tool is the tunneling
potential method that provides an alternative description to tunneling [16]
and takes the equations to a particularly simple form in the case with gravity
[17].

##  2 Exact Tunneling Solutions for a Single Field

The tunneling potential method poses the determination of the tunneling action
for the decay of a false vacuum of $V(\varphi)$ at $\varphi_{+}$ in the
following form: find the (tunneling potential) function $V_{t}(\varphi)$,
which goes from $\varphi_{+}$ to some $\varphi_{0}$ on the basin of the true
vacuum at $\varphi_{-}$, and minimizes the action functional [16]

$S[V_{t}]=54\pi^{2}\int_{\varphi_{+}}^{\varphi_{0}}\frac{(V-V_{t})^{2}}{(-V^{\prime}_{t})^{3}}\
d\varphi\ ,$ (1)

where we assume $\varphi_{+}<\varphi_{0}<\varphi_{-}$ and define
$x^{\prime}\equiv dx/d\varphi$. The method reproduces the Euclidean bounce
result [1] and has a number of good properties discussed elsewhere. The Euler-
Lagrange equation, $\delta S/\delta V_{t}=0$, gives the“equation of motion”
(EoM) for $V_{t}$ as

$(4V_{t}^{\prime}-3V^{\prime})V_{t}^{\prime}+6(V-V_{t})V_{t}^{\prime\prime}=0\
.$ (2)

The boundary conditions for the tunneling solution are

$V_{t}(\varphi_{+})=V(\varphi_{+})\ ,\quad
V_{t}^{\prime}(\varphi_{+})=V^{\prime}(\varphi_{+})=0\ ,\quad
V_{t}(\varphi_{0})=V(\varphi_{0})\ ,\quad
V_{t}^{\prime}(\varphi_{0})=\frac{3}{4}V^{\prime}(\varphi_{0})\ .$ (3)

Without loss of generality we can set $\varphi_{+}=0$ and $V(\varphi_{+})=0$.

To get exact tunneling solutions for a single scalar field $\varphi$ we solve
the equation

$V(\varphi)=V_{t}(\varphi)+\frac{V_{t}^{\prime}(\varphi)^{2}}{3}\int_{\varphi_{0}}^{\varphi}\frac{d\bar{\varphi}}{V_{t}^{\prime}(\bar{\varphi})}\
,$ (4)

where $V_{t}(\varphi)$ is a tunneling function simple enough for the integral
to be solvable. The function $V_{t}$ is a monotonically decreasing function
that connects the false vacuum at $\varphi=0$ to a point $\varphi_{0}$ in the
basin of the true vacuum of $V$. It can be checked that (4) is indeed a
solution to the tunneling potential EoM (2). The integration constant is fixed
by the condition $V(0)=V_{t}(0)$.

If the integral in (4) can be performed, one can also obtain analytically the
field profile, in the inverse form $r(\varphi)$. This follows from the
relation $r^{2}=18(V-V_{t})/V_{t}^{\prime}{}^{2}$ [16] which, used in (4),
gives

$r^{2}(\varphi)=6\int_{\varphi_{0}}^{\varphi}\frac{d\bar{\varphi}}{V_{t}^{\prime}(\bar{\varphi})}\
.$ (5)

In some cases this relation can be inverted to get $\varphi(r)$ explicitly, as
we will show below, but having (5) analytically is equally useful. In terms of
$r^{2}(\varphi)$ we can rewrite the potential (4) as

$V(\varphi)=V_{t}(\varphi)+\frac{1}{18}V_{t}^{\prime}{}^{2}(\varphi)r^{2}(\varphi)\
.$ (6)

We show below some analytic examples for $V(\varphi)$ and the reader can
produce new ones easily. Previous exact solutions with a single field,
obtained via different methods, can be found in [18].

###  2.1 Example A: Polynomial $V_{t}$

The simple monotonic tunneling potential function [16]

$V_{t}=\varphi^{2}(2\varphi-3)\ ,$ (7)

leads to

$V=\varphi^{2}\left[2\varphi-3+(1-\varphi)^{2}\log\frac{(1-\varphi)^{2}\varphi_{0}^{2}}{(1-\varphi_{0})^{2}\varphi^{2}}\right]\
,$ (8)

with $0<\varphi_{0}<1$. The tunneling action is

$S=-\frac{\pi^{2}}{3}\left[\varphi_{0}+{\rm
Li}_{2}\left(\frac{\varphi_{0}}{\varphi_{0}-1}\right)\right]\ ,$ (9)

where ${\rm Li}_{2}(x)$ is the dilogarithm function, and the bounce profile is
obtained as

$\varphi_{B}(r)=\frac{\varphi_{0}}{\varphi_{0}+(1-\varphi_{0})e^{r^{2}}}=\frac{1-2e^{-R^{2}}}{1-2e^{-R^{2}}+e^{r^{2}-R^{2}}}\
,$ (10)

where $R$ is the radius of the bounce, defined by
$\varphi_{B}(R)=\varphi_{0}/2$. This example has a thin-wall limit for large
radius $R$ with $\varphi_{0}$ getting exponentially close to 1 (where $V_{t}$
has a minimum). The setup will reach the thin-wall limit by producing larger
and larger barriers once the scalar potential $V$ is reconstructed using (4).

Figure 1:  For the example of subsection 2.1 and for several values of
$\varphi_{0}$, potentials $V(\varphi)$ (left plot) and corresponding Euclidean
bounce profiles (right plot).

This is shown in Figure 1.

A small drawback of this example is that the small-field expansion of
$V(\varphi)$ contains a $\varphi^{2}\log\varphi$ term, so that the mass at the
false vacuum is not properly defined. Although this is not a problem to test
numerical codes, some examples below remedy this.

###  2.2 Example B: Trigonometric tunneling potential

The tunneling potential

$V_{t}=-\sin^{2}\varphi\ ,$ (11)

gives

$V=\left[-1+\frac{2}{3}\cos^{2}\varphi\log\left(\frac{\tan\varphi_{0}}{\tan\varphi}\right)\right]\sin^{2}\varphi\
,$ (12)

with $0<\varphi_{0}<\pi/2$. The tunneling action is

$S=\frac{\pi^{2}}{4}\left[\frac{\pi^{2}}{2}+6\log^{2}(\cot\varphi_{0})+3{\rm
Li}_{2}(-\cot^{2}\varphi_{0})\right]\ ,$ (13)

and the bounce profile reads

$\varphi_{B}(r)={\rm arcot}\left(e^{r^{2}/3}\cot\varphi_{0}\right)={\rm
arcot}\left(\frac{e^{r^{2}/3-R^{2}/6}}{\sqrt{e^{R^{2}/3}-2}}\right)\ ,$ (14)

where $R$ is the radius of the bounce, defined by
$\varphi_{B}(R)=\varphi_{0}/2$ and given in terms of $\varphi_{0}$ by
$R^{2}=3\log(1+{\rm sec}\varphi_{0})$. This example also has a thin-wall limit
for large radius $R$ (with $\varphi_{0}\to\pi/2$). As in the previous example,
the small field expansion of $V(\varphi)$ contains a $\varphi^{2}\log\varphi$
term.

###  2.3 Example C: Finite mass

For

$V_{t}(\varphi)=\frac{\varphi^{2}}{-1/2+\log\varphi}\ ,$ (15)

we obtain

$V(\varphi)=\frac{\varphi^{2}}{-1/2+\log\varphi}+\frac{8\varphi^{2}(1-\log\varphi)^{2}}{3(1-2\log\varphi)^{2}}\left(2\log^{2}\varphi-2\log^{2}\varphi_{0}+\log\frac{1-\log\varphi}{1-\log\varphi_{0}}\right)\
,$ (16)

with $0<\varphi_{0}<\sqrt{e}$. It can be checked that
$V(\varphi)=\varphi^{2}/3+{\cal O}(\varphi^{2}/\log\varphi)$ so that the mass
at the false vacuum is well defined.

In this case the tunneling action cannot be calculated analytically but it can
be simply calculated numerically. The field profile $\varphi(r)$ cannot be
obtained analytically either, but its inverse function is

$r^{2}(\varphi)=\frac{3}{4}\left(2\log^{2}\varphi-2\log^{2}\varphi_{0}+\log\frac{1-\log\varphi}{1-\log\varphi_{0}}\right)\
,$ (17)

as can be read off from (16) using (6). This example does not admit a thin-
wall limit though, since the tunneling potential does not feature a broken
minimum.

###  2.4 Example D: Derivative of the tunneling potential

Another class of exact solutions can be obtained by starting from the
derivative of the tunneling potential. For

$V_{t}^{\prime}(\varphi)=\frac{\varphi}{\log\varphi}\ ,$ (18)

we get

$V_{t}(\varphi)={\rm Ei}(2\log\varphi)\ ,$ (19)

where ${\rm Ei}(x)$ is the exponential integral function, which leads to

$V(\varphi)={\rm
Ei}(2\log\varphi)+\frac{1}{6}\varphi^{2}\left(1-\frac{\log^{2}\varphi_{0}}{\log^{2}\varphi}\right)\
,$ (20)

with $0<\varphi_{0}<1$. This potential has the small-field expansion
$V(\varphi)=\varphi^{2}/6+{\cal O}(\varphi^{2}/\log\varphi)$, so that the mass
at the false vacuum is well defined. In this case the tunneling action can be
obtained as

$S=\frac{3\pi^{2}}{16}\left[-8\ {\rm
Ei}(2\log\varphi_{0})\log^{4}\varphi_{0}+\varphi_{0}^{2}(3-6\log\varphi_{0}+2\log^{2}\varphi_{0}+4\log^{3}\varphi_{0})\right]\
,$ (21)

and the bounce profile can be obtained explicitly as

$\varphi_{B}(r)=e^{-\sqrt{r^{2}/3+\log^{2}\varphi_{0}}}\ .$ (22)

Again, this example does not admit a thin-wall limit, since the tunneling
potential does not feature a broken minimum.

###  2.5 Example E: Finite mass with thin-wall limit

Although the last two examples do not admit a thin-wall limit, this is easily
remedied by constructing a second local minimum in $V_{t}$. For this purpose,
take

$V_{t}^{\prime}(\varphi)=\frac{\varphi(\varphi-1)}{1-\log\varphi}\ ,$ (23)

which is finite in $[0,1]$ and has local minima at $\varphi=0$ and
$\varphi=1$. This function integrates to

$V_{t}=e^{2}\ {\rm Ei}(-2+2\log\varphi)-e^{3}\ {\rm Ei}(-3+3\log\varphi)\,,$
(24)

and we finally get

$V(\varphi)=e^{2}\ {\rm Ei}(-2+2\log\varphi)-e^{3}\ {\rm
Ei}(-3+3\log\varphi)+\frac{(1-\varphi)^{2}\varphi^{2}}{18(1-\log\varphi)^{2}}r^{2}(\varphi)\
,$ (25)

with the bounce profile given implicitly by

$r^{2}(\varphi)=3\left[2\log\frac{(1-\varphi)\varphi_{0}}{(1-\varphi_{0})\varphi}+\log^{2}\varphi-\log^{2}\varphi_{0}+2{\rm
Li}_{2}(1-\varphi)-2{\rm Li}_{2}(1-\varphi_{0})\right]\ .$ (26)

The thin-wall limit is obtained by choosing $\varphi_{0}$ close to the second
minimum, $\varphi_{0}\to 1$.

##  3 From Single-Field Solution to Two-Field Solutions

###  3.1 Paths via curvature and transverse coordinate

Consider a multi-field potential with a false vacuum at $\phi_{i}=0$, with
$i=1,...,N$. A tunneling $V_{t}$ solution satisfies two equations [19]. One is
the scalar single-field equation (2) where $\varphi$ is now the arc-length
along the decay trajectory, with

$d\varphi^{2}=\sum_{i=1}^{N}d\phi_{i}^{2},$ (27)

and $\varphi=0$ at the false vacuum. The second (transverse and vectorial)
equation reads

$2(V-V_{t}){\mbox{\boldmath{$\phi$}}}^{\prime\prime}={\mbox{\boldmath{$\nabla$}}}_{T}V\
,$ (28)

where ${\mbox{\boldmath{$\phi$}}}=\\{\phi_{1},\phi_{2},...,\phi_{N}\\}$,
primes denote as before derivatives with respect to $\varphi$, and

${\mbox{\boldmath{$\nabla$}}}_{T}V\equiv{\mbox{\boldmath{$\nabla$}}}V-V^{\prime}{\mbox{\boldmath{$\phi$}}}^{\prime}\
.$ (29)

Equation (28) shows that the curvature of the decay path is determined by the
gradient of the potential along the direction orthogonal to the path. In the
two-field case, this latter equation reduces to a scalar equation that can be
written as

$2(V-V_{t})\theta^{\prime}=-\partial V/\partial\phi_{T}\ ,$ (30)

where $\theta(\varphi)$ is the angle between
${\mbox{\boldmath{$\phi$}}}^{\prime}$ and $\phi_{1}$ at point $\varphi$ and
$\phi_{T}$ is a (properly normalized) field variable transverse to the $\phi$
trajectory. In other words, we have

${\mbox{\boldmath{$\phi$}}}^{\prime}=(\cos\theta,\sin\theta)\
,\quad{\mbox{\boldmath{$\phi$}}}^{\prime\prime}=(-\sin\theta,\cos\theta)\theta^{\prime}\
,$ (31)

and $\theta^{\prime}$ is nothing but the curvature of $\phi$.

Now we assume that we know $V$ and $V_{t}$ along the decay trajectory, just by
solving a single-field problem, as done in the previous section. Such single-
field decay can be associated to an infinite number of two-field potentials.
First, note that the path function $\theta(\varphi)$ is totally undetermined
by the single-field solution and we can choose it at will. Second, as (30)
holds only at the decay path, we have much freedom in integrating it to extend
$V$ along directions transverse to the decay path. A convenient family of
solutions is

$V_{2}(\varphi,\phi_{T})=V(\varphi)-2f(\phi_{T})[V(\varphi)-V_{t}(\varphi)]\theta^{\prime}(\varphi)+g(\phi_{T},\varphi)\
,$ (32)

where the functions $f(\phi_{T})$ and $g(\phi_{T},\varphi)$ satisfy
$f(\phi_{T})=\phi_{T}+{\cal O}(\phi_{T}^{2})$ and
$g(\phi_{T},\varphi)=\phi_{T}^{2}g_{0}(\varphi)+{\cal O}(\phi_{T}^{3})$. We
have used the notation $V_{2}$ to stress that this is a two-field potential.

The previous procedure shows the key elements required to get a two-field
example but often it is difficult to find a system of $(\varphi,\phi_{T})$
coordinates that covers the two-field plane (or a large enough part of it) in
a consistent way. Below we present two methods that can be used to avoid that
problem.

###  3.2 Paths via injective mappings

In the first method, we start from a path that can be parametrized
unambiguously by one field. That is, the path contains every value of e.g.
$\phi_{1}$ at most once. Consider a function $\Phi_{2}(\phi_{1})$ that defines
the vacuum decay path via

$(\phi_{1},\phi_{2})\in(\phi_{1},\Phi_{2}(\phi_{1}))\,.$ (33)

Given a tunneling potential $V_{t}(\varphi)$, one can construct the potential
along the path $V(\varphi)$ using (4). A useful ansatz for the two-field
potential in the full parameter space is then given by

$V_{2}(\phi_{1},\phi_{2})=V(\varphi(\phi_{1}))+W(\phi_{1})\left[\phi_{2}-\Phi_{2}(\phi_{1})\right]+{\cal
O}([\phi_{2}-\Phi_{2}(\phi_{1})]^{2})\,.$ (34)

This involves expressing the arc length along the path, $\varphi$, as a
function of $\phi_{1}$. By definition of the arc length one has

$\left(\frac{d\varphi}{d\phi_{1}}\right)^{2}=1+\left(\frac{d\Phi_{2}}{d\phi_{1}}\right)^{2}\
.$ (35)

The function $W(\phi_{1})$ is matched using the transverse EoM (28). One gets

$W(\phi_{1})=V^{\prime}(\varphi(\phi_{1}))\frac{\Phi_{2}^{\prime}(\phi_{1})}{\varphi^{\prime}(\phi_{1})}+2[V(\varphi(\phi_{1}))-V_{t}(\varphi(\phi_{1}))]\left[\frac{\Phi_{2}^{\prime\prime}(\phi_{1})}{\varphi^{\prime}(\phi_{1})^{2}}-\frac{\Phi_{2}^{\prime}(\phi_{1})\varphi^{\prime\prime}(\phi_{1})}{\varphi^{\prime}(\phi_{1})^{3}}\right]\
,$ (36)

which in this two-field case can be further simplified using

$\frac{\Phi_{2}^{\prime\prime}(\phi_{1})}{\varphi^{\prime}(\phi_{1})^{2}}-\frac{\Phi_{2}^{\prime}(\phi_{1})\varphi^{\prime\prime}(\phi_{1})}{\varphi^{\prime}(\phi_{1})^{3}}=\frac{\Phi_{2}^{\prime\prime}(\phi_{1})}{\varphi^{\prime}(\phi_{1})^{4}}\
.$ (37)

The quadratic and higher orders in (34) can be used to make the potential
bounded from below. To explain why this is important, consider what happens in
a thin-wall case. In such case, both endpoints of the decay trajectory are the
two vacua of the potential and one can calculate explicitly the matrix of
second derivatives of $V_{2}$ at these endpoints, either at $\varphi=0$ (false
vacuum) or at $\varphi=\varphi_{0}$ (true vacuum). Without ${\cal
O}([\phi_{2}-\Phi_{2}(\phi_{1})]^{2})$ terms in $V_{2}$, one would get the two
mass eigenvalues $m_{L}^{2}=V^{\prime\prime}>0$ (corresponding to the field
direction tangent to the decay path) and
$m_{T}^{2}=-(d\Phi_{2}/d\phi_{1})^{2}V^{\prime\prime}<0$ (corresponding to the
field direction transverse to the decay path), leading to a tachyonic
instability.

###  3.3 Paths via conformal mappings

The second method uses the power of conformal mappings to obtain orthogonal
curvilinear coordinates that cover the plane. As an example, from the complex
function $F(z)=z^{2}/2$, writing $z=x+iy$ and $F=u+iv$ we find the relations
$u=(x^{2}-y^{2})/2$ and $v=xy$. Lines of constant $u$ and constant $v$ provide
such a system of orthogonal coordinates covering the $(x,y)$ plane. This can
be used for our purpose by selecting e.g. one of the constant $u$ curves for
the decay trajectory, while constant $v$ lines orthogonal to the path give a
natural definition for the transverse coordinate. This orthogonal coordinate
is then quite useful to add in a simple way the quadratic corrections on top
of the decay path as discussed above. In the next section we provide a
concrete example realizing this construction.

##  4 Two-Field Examples

For the first example of a two-field solution we take example A of one-field
solutions described in Subsect. 2.1 and, following the method of Subsect. 3.1,
we choose $\theta^{\prime}(\varphi)=1$ so that the decay trajectory is the arc
of a circle in the $(\phi_{1},\phi_{2})$ plane. We define the transverse field
$\phi_{T}$ by

$\phi_{1}=(1+\phi_{T})\sin\varphi\ ,\quad\phi_{2}=1-(1+\phi_{T})\cos\varphi\
,$ (38)

so that the decay trajectory corresponds to $\phi_{T}=0$. We then have

$\varphi(\phi_{1},\phi_{2})=\arctan\left(\frac{1-\phi_{2}}{\phi_{T}+1},\frac{\phi_{1}}{\phi_{T}+1}\right)\
,\quad\phi_{T}(\phi_{1},\phi_{2})=\sqrt{\phi_{1}^{2}+(1-\phi_{2})^{2}}-1\ .$
(39)

Selecting next the functions $f$ and $g$ in (32) as

$f(\phi_{T})=\phi_{T}-20\phi_{T}^{2}\ ,\quad g(\phi_{T},\varphi)=\phi_{T}^{2}\
,$ (40)

we have all the ingredients needed for a two-field potential.

Figure 2: For the potential of eq. (32) with $\theta^{\prime}(\varphi)=1$, the
functions $f$ and $g$ chosen as in (40) and $\varphi_{0}=0.999$, the left plot
shows contour lines of $V_{2}$ (green) and the decay path between minima both
analytically (red) and as found by the FindBounce [12] package (blue-dashed).
The right plot shows a 3D version of the same.

In figure 2 we show the two-field potential as well as the decay trajectory,
for the choice $\varphi_{0}=0.999$ (corresponding to a bounce radius
$R=2.63$). In the contour plot we compare the analytical trajectory (red
curve) with the trajectory found using the public code FindBounce [12] (blue
dashed curve). The numerical action is computed to be $S=80.70$ while the
analytical result is $S=80.59$. As already mentioned, we can choose other
$\theta,f,g$ functions to generate other examples. The particular choice we
made helps to clarify the difficulties that can be encountered in trying to
find a global definition of $\phi_{T}$. In this example, the point
$(\phi_{1}=0,\phi_{2}=1)$ corresponds to $\phi_{T}=-1$ and any value of
$\varphi$, and this implies that the potential $V_{2}(\phi_{1},\phi_{2})$ is
undefined at that point. One could ignore the region of field space near that
point, or regularize the potential by writing e.g.

$V_{2}(\varphi,\phi_{T})=(1-\phi_{T}^{3})\left\\{V(\varphi)-2\phi_{T}[V(\varphi)-V_{t}(\varphi)]\right\\}+g_{0}\phi_{T}^{2}\
.$ (41)

Nevertheless, it is simpler to use the method introduced in Subsect. 3.2,
which we do next.

As a simple example, consider the decay path
$\Phi_{2}(\phi_{1})=\alpha\cosh(\phi_{1}/\alpha)-\alpha$. This leads to
$d\varphi=\cosh(\phi_{1}/\alpha)\,d\phi_{1}$ and hence
$\varphi=\alpha\sinh(\phi_{1}/\alpha)$ [we set the false minimum at $(0,0)$
and $\varphi=0$ at it]. Parametrized by $\varphi$, the decay path is

$(\phi_{1},\phi_{2})\in(\alpha\,\textrm{arcsinh}(\varphi/\alpha),\sqrt{\alpha^{2}+\varphi^{2}}-\alpha)$
(42)

Hence

${\mbox{\boldmath{$\phi$}}}^{\prime}=\frac{1}{\sqrt{\alpha^{2}+\phi^{2}}}\left(\alpha,\phi\right)\
,\quad\quad{\mbox{\boldmath{$\phi$}}}^{\prime\prime}=\frac{\alpha}{(\alpha^{2}+\phi^{2})^{3/2}}\left(-\phi,\alpha\right)$
(43)

For this path we get, from (36),

$W(\phi_{1})=V^{\prime}\tanh(\phi_{1}/\alpha)+\frac{2(V-V_{t})}{\alpha\cosh(\phi_{1}/\alpha)^{3}}\
,$ (44)

with $V$ and $V_{t}$ considered as functions of $\varphi$ and the function
$\varphi(\phi_{1})$ as given above.

Finally, to illustrate the use of conformal mappings as a tool to get
orthogonal curvilinear coordinates, we take, as in Subsect. 3.3,
$F(z)=z^{2}/2$, giving $u+iv=(x^{2}-y^{2})/2+ixy$. Identifying $x=\phi_{1}$
and $y=\phi_{2}$, the lines of constant $u$ and $v$ give a system of
orthogonal curvilinear coordinates in the $(\phi_{1},\phi_{2})$ plane. For our
example we take the decay path to lie in the line $u=-1/2$, with

$\Phi_{2}(\phi_{1})=\sqrt{1+\phi_{1}^{2}}\ .$ (45)

Figure 3 shows the path (blue line) and the system of coordinates (dashed
black lines). From $\Phi_{2}(\phi_{1})$ above one gets the arc length

$\varphi(\phi_{1})=-i\,{\rm E}[i{\rm Arcsinh}(\phi_{1}),2]\ ,$ (46)

where $E[\phi,m]$ is the incomplete elliptic function of the second kind. We
then write the two-field potential as

$V_{2}(\phi_{1},\phi_{2})=V(\varphi(\phi_{1}))+W(\phi_{1})[\phi_{2}-\Phi_{2}(\phi_{1})]+\frac{1}{2}M_{T}^{2}(\phi_{1},\phi_{2})(\phi_{2}^{2}-\phi_{1}^{2}-1)^{2}\
,$ (47)

where the second order term is proportional to the square of the transverse
distance from the path, with $M_{T}^{2}>0$ large enough to avoid instabilities
of the potential.

In Figure 3 we give such an example, for the same single-field potential used
above and taking $M_{T}^{2}=9(2-\phi_{1})$, which is positive in the region of
interest. The action obtained numerically with the FindBound package is
$S\simeq 80.66$ (while the exact one is $S=80.59$).

Figure 3: For the potential of (47) with the parameters chosen as explained in
the text, the left plot shows contour lines of $V_{2}$ (green) and the decay
path between minima both analytically (red) and as found by the FindBounce
[12] package (blue-dashed). The right plot shows a 3D version of the same.

##  5 From Single-Field Solution to Three-Field Solutions

Just as in the case with two fields, one can construct solutions with three
fields using a path that is injective in one variable. (In fact, the following
construction can be trivially extended to any number of fields.) Writing the
decay path as

$(\phi_{1},\phi_{2},\phi_{3})\in(\phi_{1},\Phi_{2}(\phi_{1}),\Phi_{3}(\phi_{1}))\
,$ (48)

with arc length field $\varphi$ satisfying

$\left(\frac{d\varphi}{d\phi_{1}}\right)^{2}=1+\left(\frac{d\Phi_{2}}{d\phi_{1}}\right)^{2}+\left(\frac{d\Phi_{3}}{d\phi_{1}}\right)^{2}\
,$ (49)

we take

$V_{3}(\phi_{1},\phi_{2},\phi_{3})=V+W_{2}(\phi)(\phi_{2}-\Phi_{2}(\phi_{1}))+W_{3}(\phi)(\phi_{3}-\Phi_{3}(\phi_{1}))+...$
(50)

where the dots stands for higher order terms that stabilize the trajectory.
The functions $W_{2,3}(\phi_{1})$ are obtained from the transverse EoM (28).
One gets

$\displaystyle W_{2}(\phi_{1})$ $\displaystyle=$ $\displaystyle
V^{\prime}(\varphi(\phi_{1}))\frac{\Phi_{2}^{\prime}(\phi_{1})}{\varphi^{\prime}(\phi_{1})}+2[V(\varphi(\phi_{1}))-V_{t}(\varphi(\phi_{1}))]\left[\frac{\Phi_{2}^{\prime\prime}(\phi_{1})}{\varphi^{\prime}(\phi_{1})^{2}}-\frac{\Phi_{2}^{\prime}(\phi_{1})\varphi^{\prime\prime}(\phi_{1})}{\varphi^{\prime}(\phi_{1})^{3}}\right],$
$\displaystyle W_{3}(\phi_{1})$ $\displaystyle=$ $\displaystyle
V^{\prime}(\varphi(\phi_{1}))\frac{\Phi_{3}^{\prime}(\phi_{1})}{\varphi^{\prime}(\phi_{1})}+2[V(\varphi(\phi_{1}))-V_{t}(\varphi(\phi_{1}))]\left[\frac{\Phi_{3}^{\prime\prime}(\phi_{1})}{\varphi^{\prime}(\phi_{1})^{2}}-\frac{\Phi_{3}^{\prime}(\phi_{1})\varphi^{\prime\prime}(\phi_{1})}{\varphi^{\prime}(\phi_{1})^{3}}\right],$
(51)

This construction is quite straightforward and can be further simplified by
choosing an appropriate path. This is demonstrated in the example of the next
section.

##  6 Three-Field Example

Figure 4:  Equipotential surfaces for the three-field potential of (54). The
red line shows the decay trajectory between the red points of false vacuum, at
$(0,0,0)$, and tunneling endpoint. The black dot is a potential saddle point.

To construct a 3D example we first select a decay trajectory. The helix is a
particularly interesting example

$(\phi_{1},\phi_{2},\phi_{3})\in\left(\alpha\varphi,\rho\sin(\sqrt{1-\alpha^{2}}\varphi/\rho),\rho-\rho\cos(\sqrt{1-\alpha^{2}}\varphi/\rho)\right)\
,$ (52)

where $\varphi$ is the arc length along the path. The winding of the helix
typically precludes a fully numerical search for the bounce, while the
construction is simple enough to track the solution analytically. We can use
$\phi_{1}$ to parametrize this path and write

$\Phi_{2}(\phi_{1})=\rho\sin\left(\frac{\sqrt{1-\alpha^{2}}}{\rho\alpha}\phi_{1}\right)\
,\quad\Phi_{3}(\phi_{1})=\rho-\rho\cos\left(\frac{\sqrt{1-\alpha^{2}}}{\rho\alpha}\phi_{1}\right)\
,$ (53)

as well as $\varphi(\phi_{1})=\phi_{1}/\alpha$. We then define the three-field
potential as

$\displaystyle V_{3}(\phi_{1},\phi_{2},\phi_{3})$ $\displaystyle=$
$\displaystyle
V(\varphi(\phi_{1}))+W_{2}(\phi_{1})(\phi_{2}-\Phi_{2}(\phi_{1}))+W_{3}(\phi_{1})(\phi_{3}-\Phi_{3}(\phi_{1}))$
(54)
$\displaystyle+25(\phi_{2}-\Phi_{2}(\phi_{1}))^{2}+25(\phi_{3}-\Phi_{3}(\phi_{1}))^{2}\
,$

where $V$ is a single-field potential like those discussed in Section 2 and
$W_{2,3}(\phi_{1})$ have been defined in the previous section.

For our numerical example we take $\alpha=1/2$, $\rho=1/8$ and use the
potential $V$ of example A in Subsection 2.1, with $\varphi_{0}=0.999$ (a
thin-wall case). Figure 4 shows equipotential surfaces of
$V_{3}(\phi_{1},\phi_{2},\phi_{3})$ and the decay trajectory (red line).

##  7 Including Gravity

Gravity can change qualitatively vacuum decay [2] and such effects can be
described in a simple way by the tunneling potential method, see [17]. The
tunneling action is now given by

$S[V_{t}]=\frac{6\pi^{2}}{\kappa^{2}}\int_{\varphi_{+}}^{\varphi_{0}}\frac{(D+V_{t}^{\prime})^{2}}{DV_{t}^{2}}\
d\varphi\ ,$ (55)

with $\kappa=1/m_{P}^{2}$, where $m_{P}$ is the reduced Planck mass, and

$D\equiv\sqrt{V_{t}^{\prime}{}^{2}+6\kappa(V-V_{t})V_{t}}\ .$ (56)

The Euler-Lagrange equation for $V_{t}$ is

$(4V_{t}^{\prime}-3V^{\prime})V_{t}^{\prime}+6(V-V_{t})\left[V_{t}^{\prime\prime}+\kappa(3V-2V_{t})\right]=0\
.$ (57)

The $V_{t}$ method is also useful to generate exact examples of vacuum decay
including gravitational corrections. We review here how this works, see [17,
20] for more details.

Starting from a simple enough $V_{t}(\varphi)$ we construct $V(\varphi)$ as

$V(\varphi)=V_{t}+\frac{V_{t}^{\prime}{}^{2}}{6\kappa(1/F-V_{t})}\ ,$ (58)

where the function $F$ is given by

$F(\varphi)=\frac{2\kappa}{E(\varphi)}\int_{\varphi_{0}}^{\varphi}\frac{E(\tilde{\varphi})}{V_{t}^{\prime}(\tilde{\varphi})}d\tilde{\varphi}\
,\quad E(\varphi)={\rm
exp}\left[2\kappa\int_{\varphi_{0}}^{\varphi}\frac{V_{t}(\tilde{\varphi})}{V_{t}^{\prime}(\tilde{\varphi})}d\tilde{\varphi}\right]\
.$ (59)

Alternatively [20], one can try to get $F$ directly by solving

$F^{\prime}V_{t}^{\prime}=2\kappa(1-FV_{t})\ .$ (60)

In those cases that admit an exact solution of the integrals for $E$ and $F$
above, one can always obtain the related metric function
$\rho(\varphi)$.111The field profile of the Euclidean bounce can be obtained
in some cases by integrating the relation $d\varphi/d\xi=-\sqrt{2(V-V_{t})}$,
where $\xi$ is the radial coordinate in the Euclidean metric
$ds^{2}=d\xi^{2}+\rho^{2}(\xi)d\Omega_{3}^{2}$, where $d\Omega_{3}^{2}$ is the
line element of the unit 3-sphere. The metric function $\rho(\xi)$ can also be
recovered from $V_{t}$ using the relation $\rho=\sqrt{2(V-V_{t})}/D$. In terms
of this function the potential reads

$V(\varphi)=V_{t}(\varphi)+\frac{1}{18}D^{2}(\varphi)\rho^{2}(\varphi)\ ,$
(61)

which reduces to (6) for $\kappa=0$.

Next, we provide an example of potential for which vacuum decay can be treated
analytically using the method just discussed. (Previous exact examples, found
using different approaches, can be found in [13, 21].) Let us take [17]

$V_{t}=V_{+}-\sin^{2}\varphi\ ,$ (62)

with $V_{+}\leq 0$ (Minkowski or AdS vacua). Setting $\kappa=1$, one gets

$V=V_{t}-\frac{s_{2\varphi}^{2}}{6V_{t}}\left\\{1+\frac{c_{\varphi}^{\alpha}s_{\varphi}^{-(1+\alpha)}}{2[A(\varphi)+C]V_{t}}\right\\}\
,$ (63)

where $s_{\varphi}\equiv\sin\varphi$, $c_{\varphi}\equiv\cos\varphi$,
$\alpha\equiv V_{+}-1$. The function $A(\varphi)$ is given in terms of the
Appell hypergeometric function of two variables, $F_{1}(a;b_{1},b_{2};c;x,y)$,
as

$A(\varphi)\equiv\frac{c_{\varphi}^{2+\alpha}}{\alpha^{2}(2+\alpha)}F_{1}\left(a;b_{1},b_{2};c;c_{\varphi}^{2};-c_{\varphi}^{2}/\alpha\right)\
,$ (64)

with $a=1+\alpha/2$, $b_{1}=(1+\alpha)/2$, $b_{2}=2$, and $c=2+\alpha/2$.
Finally, $C$ is an integration constant that can be expressed in terms of
$\phi_{0}$ solving $V(\phi_{0})=V_{t}(\phi_{0})$.

The Minkowski limit ($V_{+}=0$) of the previous result can be expressed in
terms of elementary functions and reads

$V=-s_{\varphi}^{2}+\frac{2}{3}c_{\varphi}\left\\{c_{\varphi}-\frac{1}{s_{\varphi}^{2}\left[{\rm
arctanh}(c_{\varphi})-{\rm
arctanh}(c_{\varphi_{0}})+1/c_{\varphi_{0}}\right]+c_{\varphi}}\right\\}\ .$
(65)

Additional examples for false dS vacua, obtained using the same technique, can
be found in [20].

##  8 Conclusions and Outlook

We have constructed various exact solutions to the tunneling problem with
several scalar fields. Our method to generate such examples relies on exact
solutions for a single scalar field uplifted to more field dimensions along an
almost arbitrary path in scalar multi-field space. Our examples and general
technique can be used for a number of applications as has been the case in the
past (see a partial list in the introduction) opening the scope to multifield
problems.

We leave possible applications of our techniques for future work, but comment
briefly on the use as benchmarks for the numerical codes that are frequently
used for phenomenological analyses of vacuum decay [7, 8, 9, 10, 11, 12].
Numerical codes often struggle with multi-field setups, in particular if a
good first guess for the path cannot be provided. We tried several public
codes on the three-field helix example of Section 6 and all of them failed
except CosmoTransitions [8], which was able to converge to the correct path
but only after providing a good starting path for the algorithm. The final
action was correct up to a few per-mille with about a hundred support points
for the path.

We also provided some exact non-trivial examples for tunneling including
gravity. Public codes to solve this problem numerically are not yet available
and the exact examples provided here should be helpful for their development.

## Acknowledgments

The work of J.R.E. has been funded by the following grants: IFT Centro de
Excelencia Severo Ochoa CEX2020-001007-S and by PID2022-142545NB-C22 funded by
MCIN/AEI/10.13039/ 501100011033 and by “ERDF A way of making Europe”. T.K.
acknowledges support by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy – EXC 2121 “Quantum
Universe” – 390833306.

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# Reconfigurable phase contrast microscopy with correlated photon pairs

Hazel Hodgson  National Research Council of Canada, 100 Sussex Drive, Ottawa,
Ontario K1A 0R6, Canada  Yingwen Zhang  Nexus for Quantum Technologies,
University of Ottawa, Ontario, K1N 6N5, Ottawa  National Research Council of
Canada, 100 Sussex Drive, Ottawa, Ontario K1A 0R6, Canada  Duncan England
National Research Council of Canada, 100 Sussex Drive, Ottawa, Ontario K1A
0R6, Canada<EMAIL_ADDRESS>Benjamin Sussman  National Research
Council of Canada, 100 Sussex Drive, Ottawa, Ontario K1A 0R6, Canada
Department of Physics, University of Ottawa, Ottawa, Ontario, K1N 6N5, Canada

###### Abstract

A phase-sensitive microscopy technique is proposed and demonstrated that
employs the momentum correlations inherent in spontaneous parametric down-
conversion. One photon from a correlated pair is focused onto a microscopic
target while the other is measured in the Fourier plane. This provides
knowledge of the position and angle of illumination for every photon striking
the target, allowing full post-production control of the illumination angle
used to form an image. The versatility of this approach is showcased with
asymmetric illumination and differential phase contrast imaging, without any
beam blocks or moving parts.

††preprint: AIP/123-QED

Optical microscopy was first demonstrated over 400 years ago Smith (2014), and
the field has been developing and evolving ever since. The majority of these
advancements have taken place in the realm of classical optics, where the
incident and scattered light are treated as electromagnetic waves. However,
recent years have seen the emergence of a series of techniques in which
quantum properties of light such as entanglement Defienne  _et al._ (2021), or
squeezing Casacio  _et al._ (2021), can be used to enhance imaging techniques.
When two photons from an entangled pair interact with a sample, image
resolution can be improved beyond the diffraction limit D’Angelo, Chekhova,
and Shih (2001); Israel, Rosen, and Silberberg (2014). Alternatively, one
photon from the pair interacts with the sample, while the other is measured on
a separate detector to gain extra information. In this configuration, the
resolution cannot be enhanced D’Angelo  _et al._ (2005), but new capabilities
are offered including ghost imaging Padgett and Boyd (2017), noise supression
Gregory  _et al._ (2020), sub-shot-noise imaging Brida, Genovese, and Berchera
(2010), interaction-free imaging Lemos  _et al._ (2014), hyperspectral imaging
Zhang, England, and Sussman (2022), and many more Genovese (2016).

In this work, we employ the photon momentum correlations inherent in
spontaneous parametric down-conversion (SPDC) to perform phase-sensitive
imaging on the micron scale, interrogating both a resolution target and a
biological sample. In our setup, photon pairs are produced by SPDC and one
photon from each pair is imaged onto the sample and then relay-imaged onto a
time-tagging camera to measure the position at which it was generated in the
crystal. The second photon from the pair is measured on the same camera in the
Fourier plane (far-field) to determine the angle at which it was generated.
Photon pairs are identified through temporal coincidences and, through
momentum anti-correlation, we can determine the angle at which each photon
arrived at the target as well as the position Zhang  _et al._ (2022). By
applying spatial filtering in the Fourier plane digitally in post-processing,
we can achieve full reconfigurability over the range of illumination angles
that are used to generate an image. In previous work Zhang  _et al._ (2022) we
used this information to reconstruct a 3-dimensional scene. Here we show that
the same principle can be used to make images that are sensitive to the phase
gradient of the sample, which modify the photon momentum and can therefore be
measured through correlations.

Our setup is analogous to classical computational imaging techniques in which
an active element is used to dynamically control the illumination angle of the
scene. Active elements used can include spatial light modulators Maurer  _et
al._ (2011), or LED arrays Liu  _et al._ (2014); Tian, Wang, and Waller
(2014); Tian and Waller (2015); Lee  _et al._ (2015); Jung  _et al._ (2017).
Active illumination has been shown to be advantageous in various schemes
including dark field imaging Goedhart  _et al._ (2007), Fourier Ptycography
for digital refocusing and depth sectioning Zheng, Kolner, and Yang (2011);
Horstmeyer  _et al._ (2016), motion deblurring Ma  _et al._ (2015), and many
more Maurer  _et al._ (2011). In our work, however, there are no active
elements or moving parts, instead the angle of each photon varies randomly due
to the stochastic nature of SPDC, and this angle is inferred through momentum
anti-correlation measured on the entangled partner photon.

Of particular relevance to this work is a scheme known as asymmetric imaging
in which a partial beam-block placed before the condenser lens leads to a
phase-like image. The partial block can be made by physically blocking half of
the beam Mehta and Sheppard (2009), or can be digitally generated using the
active element Tian, Wang, and Waller (2014). In either case, phase gradients
in the target lead to bright or dark highlights in the image, thereby
enhancing the phase contrast. A related technique known as differential phase
contrast (DPC) imaging subtracts two asymmetric images obtained using
complementary illumination patterns to remove the background and further
improve the phase contrast Tian and Waller (2015). In this work we use photon
correlation measurements to demonstrate both asymmetric and DPC imaging using
arbitrary illumination patterns. Because the angle of each illuminating photon
is known, this can be achieved entirely in post-processing after the image has
been acquired.

Figure 1: Experimental setup. Spatial-temporal correlated photon pairs with
orthogonal polarization at 810 nm are generated through SPDC by pumping a 1 mm
thick type II ppKTP crystal with a 405 nm CW laser. With the 405 nm light
filtered out using a longpass filter (not shown), the 810 nm photon pairs are
separated into two paths using a PBS. In one path, the image plane of the
crystal is first imaged onto the sample and then the camera. In the other
path, the Fourier plane of the crystal is directly imaged onto a different
part of the same camera. Post-selections in the Fourier plane are used to
generate digital apertures. The physical analogue of our post-processing
digital selection is shown in the highlighted blue box.

The mathematical formalism behind asymmetric imaging is discussed in detail
elsewhere Hamilton and Sheppard (1984); Yi, Chu, and Mertz (2006); Tian, Wang,
and Waller (2014); Tian and Waller (2015), so only a brief conceptual
introduction is given here. The classic setup for asymmetric imaging is shown
in the inset of figure 1 where an aperture $P_{c}$ is placed before the
condenser lens and an aperture $P_{o}$ is placed after the objective lens
Hamilton and Sheppard (1984). These apertures can be actual beam blocks, as
shown in the case of $P_{c}$, or can simply be defined by the physical size of
the optic, as in $P_{o}$. A sample is placed in the focal point between the
condenser and objective, and contains local phase gradients $m$ and $n$ in the
horizontal and vertical directions, the effect of which is to tilt the beam
slightly after it passes through the sample, by an amount that is proportional
to $m$ and $n$. When using a smooth aperture, the intensity difference between
the tilted and un-tilted beam is negligible and no change is visible in the
image. However, if an aperture has a sharp edge then a small tilt can have a
significant effect on the intensity of that point in the image. See for
example $P_{c}$ in figure 1 where the left half the beam has been blocked
before the condenser producing a sharp vertical edge. A positive gradient
would shift the beam in one direction, and a negative gradient would shift the
beam in the other direction resulting in, for example, the left side of a
phase object appearing light, and the right side dark. If the aperture were
flipped such that the right half of the beam is blocked, then the highlighting
would be inverted and the right side of the object would appear light, and the
left side dark. This effect is apparent in figures 2 and 3.

Asymmetric images formed from complimentary apertures can in turn be used to
create DPC images $I_{DPC}$, using the following Mehta and Sheppard (2009):

$I_{DPC}=2\frac{I_{R}-I_{L}}{I_{R}+I_{L}}$ (1)

Where $I_{L}$ and $I_{R}$ are images formed by asymmetric illumination from
the left and right side respectively. Here a right-left asymmetric example is
used for clarity, but any appropriate aperture could be used.

In this paper, we show that, by making appropriate post-selective measurements
on correlated photon pairs, we can apply a digital aperture $P_{d}$ which is
exactly equivalent to a physical aperture $P_{c}$. This equivalence is shown
in figure 1.  The contrast and resolution that can be achieved with with this
digital aperture is no better than could be achieved with a physical aperture,
but rather the advantage in this technique comes from the fact that $P_{d}$
can be dynamically reconfigured in post-processing after the image is taken.
To quantify the phase gradient, the numerical apertures and phase gradients
must all be appropriately calibrated and normalised. In this proof-of
principle demonstration, the stability and repeatability of the microscope was
insufficient to perform this calibration. Instead we show qualitatively that
phase gradients are highlighted in the output image.

Our experimental setup centres on three elements: a time-tagging single photon
sensitive camera, a phase target, and a correlated photon pair source. These
components are discussed sequentially below. The camera used is the TPX3CAM
Nomerotski (2019), which is an event-driven camera with $256\times 256$
pixels, each $55\times 55$ $\mu m$ in size. The camera is not single photon
sensitive so a fast image intensifier is appended. Because the intensifier
produces many photons for each incident photon, cluster identification and
centroiding is required to identify a single event. After these corrections
(see Ianzano  _et al._ (2020) for details), a 3D data set emerges containing
the ($x,y$) coordinate and arrival time of every photon arriving at the
intensifier. The total detection efficiency is 7% and is approximately uniform
across the camera. The arrival-time jitter is dominated by effects related to
the intensifier, and in this case is 16 ns FWHM Zhang  _et al._ (2020);
Vidyapin  _et al._ (2022).  The imaging optics produce a $40\times$
magnification of the image, with a $\sim$40 $\mu$m field of view (FoV). Due to
the need of using a single camera to image both photons, the signal beam was
demagnified such that it only illuminates a $\sim 80\times 80$ pixel area of
camera. In the ideal situation where two cameras are used, one for each
photon, the imaging optics could be changed such that the FoV could be
extended by a factor of $\sim 3$ for the same resolution. Or, equivalently,
the resolution could be increased by a factor of $\sim 3$ for the same FOV.

Two types of sample were used, both exhibiting features on the micron scale.
First two standardized phase targets, with etching depths of 150 nm and 350 nm
(Fig. 2(a) ). Second, an unstained cell removed from the inside of a human
cheek using a swab and transferred to a microscope slide (Fig. 2(b) ). The
data shown in all figures was obtained in a 250 s exposure time. Exposure
times as short as 1 s can produce visible images, but longer data sets produce
better quality images and allow more restrictive digital apertures in post-
processing.

The photon pairs are generated by SPDC in a 1 mm thick type II periodically
poled potassium titanyl phosphate (ppKTP) crystal, pumped by a $\sim$100 mW
405 nm continuous wave (CW) diode laser. The emitted photons have orthogonal
polarization and are correlated in position and anti-correlated in their
momentum. The photons are first separated with a polarizing beam splitter
(PBS) and in the path of the horizontally polarized ‘signal’ photons, the
plane of the crystal is first imaged onto the target, and then re-imaged onto
the camera. Meanwhile, in the path of the vertically polarized ‘idler’
photons, the fourier plane of the crystal is imaged onto a different part of
the same camera. See Fig. 1 for details.

Figure 2: Raw data to asymmetric and DPC image transformation. (a) Images of a
1951 USAF resolution phase target with 350 nm etching depth and (b) images of
an unstained cheek cell. (a1) and (b1) are the raw image data showing the
idler photon image taken in the crystal far field (angle) and the signal
photon image taken in the crystal near field (position). (a2) and (b2) are
coincidence images obtained through various asymmetric aperture selections
across the diagonal axes of the far field image. (a3) and (b3) are the
resultant DPC images.

The first step in the analysis is to identify coincident pairs of photons in
the image. A coincidence window of 20 ns is chosen due to the temporal jitter
of the apparatus. If two photons are incident on the camera, one in the near
field and one in the far field, and their arrival time difference is $<20$ ns,
then they are considered as a pair. All non-paired events are discarded. The
result is a ‘coincidence image’ made up entirely of pairs of photons. Knowing
that the photon pairs are momentum anti-correlated, the illumination angle of
each photon on the target can be deduced based on where its partner photon was
detected in the far-field. The analysis proceeds by making a post-selection in
the far-field such that only a subset of the photon pairs are used to form an
image. In this way, we can control the illumination angles which are used to
form the image, this is the direct analog of placing a physical aperture
before the condenser lens so we consider this to be a digital aperture.

Figure 2 shows how asymmetric and DPC images can be formed from this data, and
illustrates the advantages compared to full-field illumination. Figures 2(a1)
and (b1) show typical raw images with the far-field (angle) and near-field
(position) modes of the 1951 USAF resolution phase target and an unstained
cheek cell, respectively. In the near-field we see a low-contrast image of the
sample. In Fig. 2(a2) and (b2), 4 different hemispherical digital apertures
have been applied to the far-field and coincidence images corresponding to
each aperture are shown.  It is apparent that the digital apertures have
altered the appearance of the images, particularly the edge gradients are
highlighted. Furthermore, there are clear differences between the images
formed using apertures at different angles, revealing different sections of
the target.  This illustrates an interesting feature of this scheme: the
aperture can be manipulated in post-processing after the image is taken, so
one or more different apertures can be used to highlight particular features.
Figure 2(a3) and (b3) shows how the asymmetric images can be used to form a
DPC image according to Eq. (1). From the DPC image of the resolution target,
we see that the phase gradients parallel to the edge of the digital aperture
produces the highest contrast.

In Fig. 3, a raw image is compared to asymmetric images formed from the right
(RHS) and left (LHS) for the phase resolution targets with an etching depth of
Fig. 3(a) 350 nm and Fig. 3(b) 150 nm . To quantify these differences, we
integrate the photon counts across the same 3 lines in the image for each
selection. As can be seen, the left-hand side and right-hand side images shows
higher contrast level compared to the raw image. The left and right-hand side
images having shifted contrast patterns as opposite sides of the lines are
highlighted. As expected, this comparison not only indicates that the
asymmetric selections improves the contrast levels, but that the highlighted
areas shift as the angle of illumination changes. We use the following
expression to quantify the visibility of the lines:

$V=\frac{\bar{I}_{\text{max}}-\bar{I}_{\text{min}}}{\bar{I}_{\text{max}}+\bar{I}_{\text{min}}},$
(2)

where $\bar{I}_{\text{max}}$ is the average maximum intensity of the 3 lines
and $\bar{I}_{\text{min}}$ is the average mimimum intensity in between the 3
lines. Using this we get for the 350 nm target $V_{\text{LHS}}=16$%,
$V_{\text{RHS}}=16$%, $V_{\text{raw}}=7.4$% and for the 150 nm target
$V_{\text{LHS}}=7.8$%, $V_{\text{RHS}}=9.4$%, $V_{\text{raw}}=1.9$%.

Figure 3: Contrast comparison between direct and asymmetric imaging.
Asymmetric imaging taken with left-hemisphere selection (LHS), right-
hemisphere selection (RHS) and raw direct imaging for a resolution target with
etching depth of (a) 350 nm and (b) 150 nm. The region along which the cross-
section contrast is compared is highlighted in color with the corresponding
cross-section shown in (c) and (d).

In conclusion,  we have proposed and demonstrated a technique using photon
pairs produced by SPDC which is sensitive to local phase gradients in a
sample. Digital apertures in the Fourier plane of any shape can be applied in
post-processing taking advantage of the spatial-temporal quantum correlations.
The technique works for unstained or phase objects, and offers freedom to the
experimentalist in choosing the shape of the Fourier-filter in post-
processing.  Noise, in the form of electrical dark current or background
light, is uncorrelated. Conversely, the arrival time of photon pairs produced
by SPDC is highly correlated. Accordingly, by selecting only pairs of events
that arrive at the same time and discarding all single events, these
correlations can be used to suppress undesirable noise in the image England,
Balaji, and Sussman (2019). These noise-suppression features, enabled by the
quantum correlations in the SPDC source, mean that our technique can operate
with just picowatts of illuminating power and will be well-suited to non-
invasive imaging applications.

In the future, such a post-processing based image system could allow bright-
field, dark-field, phase contrast, and 3D images Zhang  _et al._ (2022) to all
be drawn from a single data set.  This approach is complimentary to existing
depth-sensitive microscopy techniques based on Hong-Ou-Mandel (HOM)
interference of photon pairs Ndagano  _et al._ (2022); Torre  _et al._ (2022).
These use two-photon interference to measure absolute path length differences
in reflective samples, whereas our technique employs position-momentum
correlations in transmissive samples to highlight local phase gradients.

The main limiting factor in our demonstration is the detection technology, in
particular the detection efficiency of our camera which only 7% Vidyapin  _et
al._ (2022). Since the required data acquisition time scales quadratically for
coincidence measurements, improvements in efficiency would greatly increase
the speed of this technique. For example, a camera with $50\%$ efficiency
would acquire similar images in just 5 seconds. Furthermore, the timing
resolution of this camera is $\sim 10$ ns and is a limiting factor to the
noise suppression gained through the temporal correlations. With the timing
resolution improved to 0.1 ns, the signal to noise ratio could be increased by
a factor of 10. Single photon event camera technologies satisfying these
specifications are already under development Morimoto  _et al._ (2020); Can ,
so integration times of less than 1 s could be possible in the near future.
Higher detector efficiency and resolution will also unlock new
functionalities. For example, digital apertures that result in high loss (such
as a high-pass filter for dark-field imaging) cannot be effectively
implemented with this setup, but could be possible with improved detectors.

###### Acknowledgements.

The authors are grateful to Andrei Nomerotski, Antony Orth, Adrian Pegoraro,
Andrew Ridsdale, Victor Vidyapin, Denis Guay, and Doug Moffatt for stimulating
discussions and technical support. This work was partly supported by Defence
Research and Development Canada and the National Research Council’s Quantum
Sensors Challenge Program.

## Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## Author Declarations

The authors have no conflicts to disclose.

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# Extended JC-Dicke model for a two-component atomic BEC inside a cavity

Yong Li  Department of Physics and Center of Theoretical and Computational
Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China  Peng
Zhang  ERATO, JST, Macroscopic Quantum Control Project, Hongo, Bunkyo-Ku,
Tokyo 113-8656, Japan  Z. D. Wang  Department of Physics and Center of
Theoretical and Computational Physics, The University of Hong Kong, Pokfulam
Road, Hong Kong, China

###### Abstract

We consider a trapped two-component atomic Bose-Einstein condensate (BEC),
where each atom with three energy-levels is coupled to an optical cavity field
and an external classical optical field as well as a microwave field to form
the so-called $\Delta$-type configuration. After adiabatically eliminating the
atomic excited state, an extended JC-Dicke model is derived under the
rotating-wave approximation. The scaled ground-state energy and the phase
diagram of this model Hamiltonian are investigated in the framework of mean-
field approach. A new phase transition is revealed when the amplitude of
microwave field changes its sign.

###### pacs:

73.43.Nq, 03.75.Kk, 42.50.Pq

##  I Introduction

The so-called Dicke model Dicke:1954  describes the interaction of a large
number of two-level systems (e.g., atoms) with a single optical mode. Since
the effective light-matter coupling strength is dependent on the number of
atoms $N$ ($\propto\sqrt{N}$), a sufficiently large $N$ would lead to a
classical phase transition Hepp:1973 ; Wang:1973  (at finite temperatures)
from the normal state, which corresponds to the atomic ground state associated
with the vacuum state of the optical mode, to the superradiant-phase state,
where the phenomenon of superradiance occurs with the finite scaled mean
numbers of both photons and excited-state atoms. Recently, exploration of
quantum phase transitions in the Dicke model at zero temperature has attracted
significant attentions Emary:2003 ; Lambert ; Hou:2004 ; Emary:2004 ; Lee:2004
; Li:2006 ; Vidal:2006 ; Alcalde:2007 ; Dimer:2007 ; Tolkunov:2007 ;
Chen:2007a ; Chen:2007b ; Chen:2008 ; Goto:2008 ; Tsyplyatyev:2008 ;
QHChen:2008 ; Alcalde:2009 ; Larson:2009 ; Huang:2009 . The drastic change
across the critical point due to a qualitative change of the ground state of
the Dicke model has been investigated, in the frameworks of the scaled ground
state energy, macroscopic atomic excited-state population, quantum
entanglement, Berry phase, quantum chaos and so on.

Apart from the standard Dicke model, several more generalized Dicke models
have also been proposed and studied Emary:2004 ; Lee:2004 ; Li:2006 ;
Tolkunov:2007 ; Chen:2007b ; Chen:2008 ; Goto:2008 ; Tsyplyatyev:2008 ;
Alcalde:2009 ; Larson:2009 . In some of them Chen:2007b ; Chen:2008 , the free
atoms in the standard Dicke models are replaced by the atomic Bose-Einstein
condensate (BEC). A BEC describes a collective quantum state of a large number
of atoms and may be used to generate a macroscopic quantum object with a
longer lifetime compared to the free atoms. It has attracted much interest to
combine the BEC with optical cavity in the strong coupling regime Ottl:2006 ;
Brennecke:2007 ; Brennecke:2008 ; Murch:2008 . On the other hand, comparing
with the case of free atoms in the standard Dicke model, there is an
additional atom-atom collision interaction in the BEC, which could lead to the
change of critical phenomenon Chen:2007b . More recently, under some
approximations, Chen et al. Chen:2008  proposed an extended Dicke model based
on a two-component atomic BEC in an optical cavity with the atoms coupled to
both the quantized optical cavity filed and an external classical driving
optical field, where an interesting phase diagram covering phenomena from
quantum optics to atomic BEC is addressed.

In this paper, we consider a trapped two-component atomic BEC where the two
condensated states and an ancillary excited state of each atom form a three-
level $\Delta$-type configuration via the couplings to the optical cavity and
an externally controlled classical optical field as well as an external
microwave field. Such a three-level configuration can be reduced to a two-
level configuration (i.e. the two condensated states) after adiabatically
eliminating the atomic excited state under the large sigle-photon detuning
condition. Under the so-called rotating-wave approximation (RWA), we derive an
effective, extended JC-type JC  Dicke model for such a two-component BEC. The
phase diagram for the derived extended JC-Dicke model is also investigated in
detail.

##  II Model and Hamiltonian

The setup of the JC-Dicke model under consideration is depicted in Fig. 1. An
optically-trapped Rb atomic BEC under the two-mode approximation with the
atomic states $5^{2}S_{1/2}$ $\left|F=1,m_{f}=-1\right\rangle$ (ground state
$\left|1\right\rangle$) and $\left|F=1,m_{f}=0\right\rangle$ (metastable state
$\left|2\right\rangle$) is placed in a single-mode quantized optical cavity.
These two states of the atomic BEC are coupled via both an external microwave
field and a two-photon process two-photon BEC  mediated by an ancillary
excited state $\left|3\right\rangle$ (from $5^{2}P_{3/2}$), where the single-
photon transition between $\left|3\right\rangle$ and the ground state
$\left|1\right\rangle$ (or the metastable state $\left|2\right\rangle$) is
coupled to the quantized optical cavity field (or the external classical
optical field). Here we assume that both the corresponding single-photon
detunings are large and the corresponding two-photon detuning is very small.
For such a case of two-photon Raman process Raman , the ancillary excited
state can be adiabatically eliminated and the effective Hamiltonian of the
two-component BEC reads ($\hbar=1$ hereafter)

$H_{\mathrm{eff}}=H_{at}+H_{ph}+H_{at-ph}+H_{at-cl}+H_{at-at}$ (1)

with

$\displaystyle H_{at}$
$\displaystyle=\nu_{1}c_{1}^{\dagger}c_{1}+(\nu_{2}+\omega_{2}-\omega_{1})c_{2}^{\dagger}c_{2},$
(2a) $\displaystyle H_{ph}$ $\displaystyle=\omega a^{\dagger}a,$ (2b)
$\displaystyle H_{at-op}$
$\displaystyle=\lambda_{\mathrm{eff}}e^{i\omega_{\mathrm{cl}}t}c_{2}^{\dagger}c_{1}a+h.c.,$
(2c) $\displaystyle H_{at-mw}$ $\displaystyle=\Omega
e^{-i\omega_{\mathrm{mw}}t}c_{2}^{\dagger}c_{1}+h.c.,$ (2d) $\displaystyle
H_{at-at}$
$\displaystyle=\frac{\eta_{1}}{2}c_{1}^{\dagger}c_{1}^{\dagger}c_{1}c_{1}+\frac{\eta_{2}}{2}c_{2}^{\dagger}c_{2}^{\dagger}c_{2}c_{2}+\eta_{12}c_{1}^{\dagger}c_{1}c_{2}^{\dagger}c_{2}$
(2e) denoting the energies of the free atoms in the BEC, the free cavity
field, the reduced effective interaction between the BEC with the optical
fields, the interaction of the BEC with the microwave field, and the atom-atom
collision interaction of the BEC, respectively. Here, $c_{1,2}$
($c_{1,2}^{\dagger}$) are the annihilation (creation) bosonic operators for
$\left|1\right\rangle$ and $\left|2\right\rangle$, respectively; $\omega_{i}$
($i=1,2,3$) is the corresponding internal level-energy for atomic state
$\left|i\right\rangle$; $\nu_{l}=\int
d^{3}\mathbf{r}\phi_{l}^{\ast}(\mathbf{r})[-\nabla^{2}/2m+V(\mathbf{r})]\phi_{l}(\mathbf{r})$
($l=1,2$) is the trapped frequency for the states $\left|1\right\rangle$ and
$\left|2\right\rangle$ with $V(\mathbf{r})$ being the trapped potential, $m$
the atomic mass, and $\phi_{l}(\mathbf{r})$ the corresponding condensate
wavefunction; $a$ ($a^{\dagger}$) is the annihilation (creation) operator of
the cavity mode with the frequency $\omega$;
$\lambda_{\mathrm{eff}}=g_{13}\Omega_{23}/\Delta$ is the reduced effective
coupling strength for two-photon Raman process, where $g_{13}$ ($\Omega_{23}$)
is the corresponding coupling strength between the BEC and the quantized
cavity field (classical optical field), $\Delta$ is the large single-photon
detuning: $\Delta\equiv\omega_{3}-$
$(\nu_{2}+\omega_{2})-\omega_{\mathrm{cl}}\gg\\{g_{13},\Omega_{23}\\}$ with
$\omega_{\mathrm{{cl}}}$ the frequency of the classical optical field;
$\Omega$ is the corresponding coupling strength between the BEC and the
microwave field (with the frequency $\omega_{\mathrm{{mw}}}$);
$\eta_{l}=(4\pi\rho_{l}/m)\int
d^{3}\mathbf{r}\left|\phi_{l}(\mathbf{r})\right|^{4}$ and
$\eta_{12}=(4\pi\rho_{12}/m)\int
d^{3}\mathbf{r}\left|\phi_{1}^{\ast}(\mathbf{r})\phi_{2}(\mathbf{r})\right|^{2}$
with $\rho_{l}$ and $\rho_{12}$ ($=\rho_{21}$) the intraspecies and the
interspecies $s$-wave scattering lengths, respectively. It is remarked that
the RWA has been used for all optical/microwave fields coupling to the atomic
BEC.

Figure 1: (Color online) (a) Schematic diagram of the experimental setup for a
trapped BEC of 87Rb atoms. (b) The internal atomic level configuration of BEC:
The $\left|1\right\rangle$$\leftrightarrow$$\left|2\right\rangle$ transition
of the BEC atoms is coupled to the external microwave field with the coupling
strength $\Omega$; The
$\left|1\right\rangle$$\leftrightarrow$$\left|3\right\rangle$ and
$\left|2\right\rangle$$\leftrightarrow$$\left|3\right\rangle$ transitions are
coupled largely-detuned to the optical cavity and the external classical
optical field with the corresponding coupling strengths $g_{13}$ and
$\Omega_{23}$, respectively. (c) When the single-photon detuning $\Delta$ is
large (the corresponding two-photon detuning is assumed to be small), a
reduced two-level configuration can be obtained after eliminating the
ancillary excited state $\left|3\right\rangle$. $\lambda_{\mathrm{eff}}$ is
the effective coupling strength for the two-photon Raman process.

By using the Schwinger relations

$\displaystyle J_{+}$ $\displaystyle=c_{2}^{\dagger}c_{1},\text{ \ \
}J_{-}=c_{1}^{\dagger}c_{2},$ (3a) $\displaystyle J_{z}$
$\displaystyle=\frac{c_{2}^{\dagger}c_{2}-c_{1}^{\dagger}c_{1}}{2},$ (3b)
which fulfill  $\left[J_{+},J_{-}\right]=2J_{z},\text{ \
}\left[J_{z},J_{\pm}\right]=\pm J_{\pm},$ (4)

the Hamiltonian (1) can be written as

$\displaystyle H_{\mathrm{eff}}$ $\displaystyle=\omega
a^{\dagger}a+\omega_{0}J_{z}+\frac{\eta}{N}J_{z}^{2}$ $\displaystyle+[(\Omega
e^{-i\omega_{\mathrm{mw}}t}+\frac{\lambda}{\sqrt{N}}e^{i\omega_{\mathrm{cl}}t}a)J_{+}+h.c.]+\text{const},$
(5)

where

$N=c_{2}^{\dagger}c_{2}+c_{1}^{\dagger}c_{1}$ (6)

is the number of the atoms,

$\displaystyle\omega_{0}$
$\displaystyle=\nu_{2}+\omega_{2}-\nu_{1}-\omega_{1}+\frac{N-1}{2}\left(\eta_{2}-\eta_{1}\right),$
(7a) $\displaystyle\eta$
$\displaystyle=(\frac{\eta_{1}+\eta_{2}}{2}-\eta_{12})N,\text{ }$ (7b)
$\displaystyle\lambda$ $\displaystyle=\lambda_{\mathrm{eff}}\sqrt{N},$ (7c)
and the constant term
$\text{const}=\frac{N}{2}[(\nu_{2}+\omega_{2}-\omega_{1}-\frac{\eta_{2}}{2}+\gamma_{2}N)+(\nu_{1}-\frac{\eta_{1}}{2}+\gamma_{1}N)]$

can be neglected in the following consideration.

For $\Omega\neq 0$, to eliminate the time-dependence in Hamiltonian (5), we
perform a unitary transformation
$U=\exp[-i\omega_{\mathrm{{mw}}}J_{z}t-i(\omega_{\mathrm{{mw}}}+\omega_{\mathrm{{cl}}})a^{\dagger}at]$
and obtain an effective Hamiltonian

$\displaystyle H$
$\displaystyle=\omega_{a}a^{\dagger}a+\omega_{b}J_{z}+\frac{\eta}{N}J_{z}^{2}$
$\displaystyle+(\frac{\lambda}{\sqrt{N}}aJ_{+}+\Omega J_{+}+h.c.),$ (8)

where
$\omega_{a}=\left(\omega-\omega_{\mathrm{mw}}-\omega_{\mathrm{cl}}\right)$ and
$\omega_{b}=\left(\omega_{0}-\omega_{\mathrm{mw}}\right)$ are the effective
frequencies in the rotating frame, in which $H$ is independent of time. To the
best of our knowledge, Hamiltonian (8) appears to be a new one in literatures,
and thus we call it as the extended JC-Dicke model.

##  III Mean-field ground state energy

We now look into the ground-state properties of Hamiltonian (8) and the
corresponding quantum phases as well as their transitions. We here consider
the case of positive $\omega_{a}$, where the stable ground state is
anticipated for Hamiltonian (8). By using the Holstein-Primakoff
transformation HP

$\displaystyle J_{+}$ $\displaystyle=b^{\dagger}\sqrt{N-b^{\dagger}b},\text{\
}J_{-}=\sqrt{N-b^{\dagger}b}b,$ $\displaystyle J_{z}$
$\displaystyle=b^{\dagger}b-\frac{N}{2}$ (9)

with $[b,b^{\dagger}]=1$, the effective Hamiltonian reads

$\displaystyle H$
$\displaystyle=\omega_{a}a^{\dagger}a+\omega_{b}(b^{\dagger}b-\frac{N}{2})+\frac{\eta}{N}(b^{\dagger}b-\frac{N}{2})^{2}$
$\displaystyle+[(\lambda
a+\Omega\sqrt{N})b^{\dagger}\sqrt{1-\frac{b^{\dagger}b}{N}}+h.c.].$ (10)

Similar to that used in the standard Dicke model Emary:2003 , we here
introduce the displacements for the two shifting boson operators as
$c^{\dagger}=a^{\dagger}-\sqrt{N}\alpha^{\ast}$ and
$d^{\dagger}=b^{\dagger}+\sqrt{N}\beta^{\ast}$ with the complex displacement
parameters $\alpha$ and $\beta$ describing the scaled collective behaviors of
both the atoms and the photons Emary:2003 ; Li:2006 ; Chen:2007b ; Chen:2008 .
In fact, the current method of introducing the displacements is equivalent to
the mean field approach. In this framework, it is clear that
$0\leq\left|\beta\right|\leq 1$.

After expanding the terms in the square in (10), the scaled Hamiltonian can be
written up to the order of $N^{-1}$ as

$H/N=H_{0}+N^{-1/2}H_{1}+N^{-1}H_{2},$ (11)

where

$\displaystyle H_{0}$
$\displaystyle=\omega_{a}\alpha^{\ast}\alpha+\omega_{b}(\beta^{\ast}\beta-\frac{1}{2})+\eta(\beta^{\ast}\beta-\frac{1}{2})^{2}$
$\displaystyle-[\left(\lambda\alpha+\Omega\right)\beta^{\ast}\sqrt{1-\beta^{\ast}\beta}+c.c.]$
(12)

denotes the scaled constant energy, and $H_{1,2}$ denote the linear and
bilinear terms, respectively. It is noted that $H_{0,1,2}$ are independent of
the number of atoms $N$.

The scaled ground-state energy is just given by the scaled constant energy in
the Hamiltonian

$E_{g}^{N}(\alpha,\beta)\equiv\frac{E_{g}(\alpha,\beta)}{N}=H_{0},$ (13)

where the displacements $\alpha$ and $\beta$ should be determined from the
equilibrium condition

$\displaystyle\partial[E_{g}(\alpha,\beta)/N]/\partial\alpha^{\ast}$
$\displaystyle=0,$ (14a)
$\displaystyle\partial[E_{g}(\alpha,\beta)/N]/\partial\beta^{\ast}$
$\displaystyle=0.$ (14b) After some derivations, we find that $\alpha$ is
given by
$\alpha=\frac{\lambda^{\ast}}{\omega_{a}}\beta\sqrt{1-\beta^{\ast}\beta}.$
(15)

and $\beta$ satisfies

$\displaystyle\Omega\sqrt{1-\beta^{\ast}\beta}=\beta[$
$\displaystyle\omega_{b}+w\left(2\beta^{\ast}\beta-1\right)$
$\displaystyle+(\frac{\Omega\beta^{\ast}}{2\sqrt{1-\beta^{\ast}\beta}}+c.c.)],$
(16)

where $w=\eta+\left|\lambda\right|^{2}/\omega_{a}$.

From the above equation (16), it is obvious that $\Omega/\beta$ should be real
since all of the parameters, except for the coupling strengths $\lambda$ and
$\Omega$, are real. Here, we assume the case of real $\Omega$ for simplicity
Note:Rabi frequency . That means, $\beta$ should also be real:
$-1\leq\beta\leq 1$ and satisfy

$0=\omega_{b}\beta\sqrt{1-\beta^{2}}+\Omega\left(2\beta^{2}-1\right)+w\beta\left(2\beta^{2}-1\right)\sqrt{1-\beta^{2}}.$
(17)

For a general real $\Omega$, the scaled ground state energy in Eq. (13) is
given by the displacement $\beta$ (by using Eq. (15) to eliminate the
displacement $\alpha$) as

$E_{g}^{N}(\beta)=\omega_{b}(\beta^{2}-\frac{1}{2})-2\Omega\beta\sqrt{1-\beta^{2}}+w(\beta^{2}-\frac{1}{2})^{2}.$
(18)

The displacement $\beta$ is the non-trivial real solution for equation (17).
In general, there are more than one real solutions for Eq. (17), only the one
that leads to the minimal scaled ground state energy should be chosen.

We would like to remark that the displacements determined from the equilibrium
equations could just make the linear term $H_{1}$ be $0$. At the same time,
the bilinear term $H_{2}$ makes no contribution to the scaled ground state
energy. Therefore, the exact forms of $H_{1,2}$ are not needed in our
analysis, and thus only the constant term $H_{0}$ determines fully the scaled
ground state energy of the current system.

Figure 2: (Color online) (a) The atomic displacement $\beta$ for the ground
state, (b) the square of atomic displacement $\beta^{2}$ for the ground state
(or the scaled magnetization plus $1/2$: $m+1/2$), and (c) the scaled ground
state energy $E_{g}^{N}$ versus $\omega_{b}$ for different coupling strength
$\Omega$. The energy and frequencies are in units of $w$ ($w>0$).

##  IV Numerical results and analysis

In this section, we focus on the numerical calculations of the scaled ground
state energy, e.g., $E_{g}^{N}(\beta)$ in Eq. (18), and the corresponding
displacement $\beta$ that makes $E_{g}^{N}(\beta)$ minimal. The minimum
$E_{g}^{N}(\beta)$ (as well as the corresponding $\beta$) is determined by the
three parameters: $\omega_{b}$, $\Omega$, and $w$.

Figure 2 plots $\beta$, $\beta^{2}$, and $E_{g}^{N}(\beta)$ (corresponding to
the ground state of the JC Dicke model) as a function of $\omega_{b}$ for
several values of $\Omega$. All the energies/frequencies in Fig. 2 are in
units of the positive $w$. We mentioned “positive $w$” here since $w$ may be
positive or negative, while the negative $w$ will lead to different results.
Seen from Fig. 2, a second-order normal-superradiant phase transition at
$\omega_{b}/w=1$ should be expected in the absence of microwave field
($\Omega=0$), as in the standard Dicke model Emary:2003 ; while in the
presence of microwave field ($\Omega\neq 0$), the normal phase and the
corresponding transition disappear.

Figure 3: (Color online) The atomic displacement $\beta$ for the ground state
versus the coupling strength $\Omega$ for different $\omega_{b}$. The
frequencies are in units of $w$ ($w>0$). Figure 4: (Color online) (a) The
atomic displacement $\beta$ for the ground state and (b) the scaled ground
state energy $E_{g}^{N}$ versus $\Omega$ and $w$. The frequencies/energies are
in units of $\omega_{b}$ (here we keep $\omega_{b}>0$).

The displacement $\beta$ for the ground state is plotted as a function of
$\Omega$ (in units of positive $w$) for several values of $\omega_{b}$ in Fig.
3. For a positive/negative $\Omega$, the corresponding $\beta$ is also
positive/negative. At the point of $\Omega\rightarrow 0$, the ground-state
$\beta$ may have a jump. In order to look the possible transition phenomenon
at the point of $\Omega\rightarrow 0$, we plot the ground-state $\beta$ and
$E_{g}^{N}(\beta)$ as the function of the parameters $\Omega$ and $w$ (in
units of positive $\omega_{b}$) in the 3-dimensional (3D) Fig. 4. In
experiments, the parameters $\Omega$ and $w$ are controllable, e.g., the
former one can be easily controlled by changing the strength of the microwave
field, the latter one can be controlled by adjusting the atom-atom interaction
interactions by the magnetic-field or/and optical Feshbach resonance
techniques m-Feshbach ; o-Feshbach ; Zhang:2009 . From Fig. 4(a), it is clear
that the displacement $\beta$ has a jump when $w>\omega_{b}$ at the point of
$\Omega\rightarrow 0$. Notably, the scaled ground-state energy is always
continuous at the jump point, but its first derivative with respect to
$\Omega$ does not, which implies a new kind of the first order phase
transition at the parameter point whenever $\Omega$ changes its sign. Making
the replacement: $\Omega\rightarrow-\Omega$, we can find the corresponding
replacements $\beta\rightarrow-\beta$ and $E_{g}^{N}(\beta)\rightarrow
E_{g}^{N}(-\beta)=E_{g}^{N}(\beta)$ according to Eqs. (17) and (18), just as
seen in Fig. 4.

Figure 5: (Color online) The atomic scaled magnetization $M$ versus $\Omega$
and $w$. The parameters are in units of $|\omega_{b}|$ for positive (a) and
negative (b) $\omega_{b}$. Figure 6: (Color online) The $\Omega$-$w$ phase
diagram for (a) the positive $\omega_{b}$ and (b) negative $\omega_{b}$.

We may define the scaled ’magnetization’ as $M\equiv\langle
J_{z}\rangle/N=\beta^{2}-1/2$ for the ground state. A positive magnetization
$M$ means the atomic inversion population: more atoms stay in the upper state
$\left|2\right\rangle$ than those in $\left|1\right\rangle$, while a negative
$M$ means the opposite case. Especially, $M=-1/2$ corresponds to the normal
phase: all atoms stay in the lower states. Figure 5 plots the magnetization
against the parameters $\Omega$ and $w$, where Figs. 5(a) and (b) correspond
respectively to the positive and negative $\omega_{b}$. From Fig. 5, we may
define phases $P_{1,2,3,4}$, as denoted in Fig. 6 and Table 1. The line
$L_{0}$ of normal phase and $L_{12}$ of the superradiant phase, appearing only
in the absence of microwave field, are the board lines of the phases $P_{1,2}$
in Fig. 6(a); so are $L_{0}^{\prime}$ and $L_{34}$ the board lines of
$P_{3,4}$ in Fig. 6(b). Table 1 shows the differences of relevant quantities
for all phases in Fig. 6 note .

We now address the points/line of quantum phase transition in the phase
diagram. Point $A$ in Fig. 6(a) denotes the well-known normal-superradiant
phase transition (along the transverse arrow) in the standard JC Dicke model,
while point $D$ represents a similar one (corresponding to point $A$) for the
case with the atomic near-inversion population when $\omega_{b}$ is negative.
The line $L_{12}$ corresponds to the phase transition line segment from
$P_{1}$ to $P_{2}$, which is of the first-order, as indicated above.
Intuitively, the phases $P_{1}$ and $P_{2}$ may be viewed as the para- and
dia- “magnetic” phases, because $\frac{\partial M}{\partial\Omega}>0$ and $<0$
for $P_{1}$ and $P_{2}$, respectively. Correspondingly, the line segment of
$L_{34}$ is the first-order phase transition line that separates $P_{3}$ and
$P_{4}$, which may be viewed respectively as the dia- and para- “magnetic”
phases.

Table 1: Relevant quantities for different phases $P_{1,2,3,4}$ in the ground state. quantity | P1 | P2 | P3 | P4
---|---|---|---|---
$\omega_{b}$ | $>0$ | $>0$ | $<0$ | $<0$
$\Omega$ | $>0$ | $<0$ | $>0$ | $<0$
$\beta$ | $(0,\frac{\sqrt{2}}{2})$ | $(-\frac{\sqrt{2}}{2},0)$ | $(\frac{\sqrt{2}}{2},1)$ | $(-1,-\frac{\sqrt{2}}{2})$
$M$ | $(-\frac{1}{2},0)$ | $(-\frac{1}{2},0)$ | $(0,\frac{1}{2})$ | $(0,\frac{1}{2})$
$\frac{\partial E_{g}^{N}}{\partial\Omega}$ | $<0$ | $>0$ | $<0$ | $>0$
$\frac{\partial M}{\partial w}$ | $>0$ | $>0$ | $<0$ | $<0$
$\frac{\partial M}{\partial\Omega}$ | $>0$ | $<0$ | $<0$ | $>0$

It is worth pointing that we have neglected the anti-resonant terms and the
corresponding $\hat{A}^{2}$ terms ($\hat{A}$ is the vector potential of the
optical field) in the original Hamiltonian. In the standard Dicke model, it
was pointed out that the anti-resonant terms would also bring an un-neglectful
influence near the critical point and lead to the modification of the result
of the quantum phase transition Duncan:1974 ; Emary:2003 . It was also pointed
out that the quantum phase transition in the standard Dicke model happens in
the effective ultra-strong matter-light coupling regime, where the
$\hat{A}^{2}$ term could also become very strong and may not be omitted.
Notably, if the effect of the $\hat{A}^{2}$ term were not neglected, the
quantum phase transition would be impossible to happen Rzazewski:1975  in the
standard Dicke model. In order to obtain the quantum phase transition in the
Dicke model, it was proposed in Ref. Dimer:2007  to get an effective Dicke
model which does not include the $\hat{A}^{2}$ term. In the current scheme,
similar to Ref. Dimer:2007 , we have obtained an effective extended JC-Dicke
model that does not have the anti-resonant terms and the $\hat{A}^{2}$ terms.
In our original Hamiltonian, it is assumed that the matter-light couplings are
much smaller than the corresponding atomic transition/optical carrier
frequencies, then it is safe to neglect the anti-resonant terms and the
$\hat{A}^{2}$ terms. After the unitary transformation, the time-independent
effective Hamiltonian may lead to the quantum phase transition when the
matter-light coupling is comparable to the effective carrier frequencies.

##  V Conclusion

In conclusion, we have derived an extended JC-Dicke model for a two-component
BEC coupled to the quantized optical cavity and the external classical optical
field as well as a microwave field. The scaled ground-state energy and the
phase diagram of this model Hamiltonian have been investigated in the
framework of mean-field approach. A new first-order phase transition has also
been revealed when the amplitude of micromave field changes its sign.

###### Acknowledgements.

We thank Ming-Yong Ye, Zi-Jian Yao, Gang Chen, and Zheng-Yuan Xue for helpful
discussions. This work was supported by the RGC of Hong Kong under Grant No.
HKU7051/06P, the URC fund of HKU, and the State Key Program for Basic Research
of China (No. 2006CB921800).

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