#  Quantum signatures of black hole mass superpositions

Joshua Foo<EMAIL_ADDRESS>Centre for Quantum Computation &
Communication Technology, School of Mathematics & Physics, The University of
Queensland, St. Lucia, Queensland, 4072, Australia  Cemile Senem Arabaci
Department of Physics and Astronomy, University of Waterloo, Waterloo,
Ontario, Canada, N2L 3G1  Magdalena Zych  Centre for Engineered Quantum
Systems, School of Mathematics and Physics, The University of Queensland, St.
Lucia, Queensland, 4072, Australia  Robert B. Mann  Perimeter Institute, 31
Caroline St., Waterloo, Ontario, N2L 2Y5, Canada  Department of and Astronomy,
University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1

###### Abstract

We present a new operational framework for studying “superpositions of
spacetimes”, which are of fundamental interest in the development of a theory
of quantum gravity. Our approach capitalizes on nonlocal correlations in
curved spacetime quantum field theory, allowing us to formulate a metric for
spacetime superpositions as well as characterizing the coupling of particle
detectors to a quantum field. We apply our approach to analyze the dynamics of
a detector (using the Unruh-deWitt model) in a spacetime generated by a BTZ
black hole in a superposition of masses. We find that the detector exhibits
signatures of quantum-gravitational effects corroborating and extending
Bekenstein’s seminal conjecture concerning the quantized mass spectrum of
black holes in quantum gravity. Crucially, this result follows directly from
the approach, without any additional assumptions about the black hole mass
properties.

Introduction—Black holes continue to captivate physicists from a diverse array
of backgrounds, from cosmology and astroparticle physics, to quantum field
theory and general relativity. Because of the extreme gravitational
environments they generate, they are considered primary candidates for
studying regimes in which quantum gravity effects are present [1, 2, 3].
Indeed, the discoveries of Hawking radiation [4] and black hole evaporation
[5] gave rise to the well-known information paradox [6, 7, 8, 9] and an entire
field seeking its resolution, which aptly illustrates the existing conflicts
between quantum theory and general relativity.

Bekenstein [10, 11, 12] was among the first to recognise that a complete
theory of quantum gravity must account for the treatment of black holes as
quantum objects. His key insight was in demonstrating that the black hole
horizon area, and hence its mass, is an adiabatic invariant, with an
associated discrete quantization [13] and evenly spaced energy levels [14].
These pioneering studies have given rise to the burgeoning research field of
quantum black holes [15, 16, 17, 18, 19, 20].

The existence of mass-quantized black holes implies that they may also exist
in superpositions of mass eigenstates. A mass-superposed black hole is an
example of a quantum superposition of spacetimes; since the different masses
individually define a unique classical solution to Einstein’s field equations,
the resulting amplitudes of the mass superposition correspond to associated
“spacetime states”. Understanding the effects that arise in such spacetime
superpositions is an important stepping stone towards developing complete
description of a quantized spacetime. “Top-down” approaches – those seeking to
formally quantize general relativity – date back to the Wheeler-deWitt (WdW)
equation [21, 22], a wave functional of the metric that offered a conceptual
scheme for dealing with superpositions of spacetime geometries [23, 24].
Building upon this work, canonical quantization techniques were applied to the
metric variables [25, 26] leading to the development of loop quantization [27,
28] which in turn, has yielded solutions including black holes in a
superposition of masses [29, 30, 31, 32, 33, 34, 35].

Recently, there has been significant interest in the study of spacetime
superpositions from an operational viewpoint, grounded in the definition of
spacetime events using physical devices (clocks, rods, and detectors).
Examples include investigations into the resulting quantum causal structures
[36, 37] quantum reference frames and the equivalence principle [38, 39, 40,
41], and fundamental decoherence mechanisms [42], along with applications to
analog gravity [43], and tabletop experiments aimed at testing the quantum
nature of gravity [44, 45, 46]. A recent investigation developed an approach
where a particle detector (modelled as the Unruh-deWitt detector [47]) evolves
in a de Sitter spacetime in superpositions of spatial translations and
curvature, to study the quantum effects arising in such a background [48].
However, the conformal equivalence of de Sitter and Rindler spacetime (a
uniformly accelerated reference frame in Minkowski spacetime), meant that even
in such scenario some of the effects were equivalent to those experienced by a
detector in a superposition of semiclassical trajectories in Minkowski
spacetime [49, 50].

We are thus motivated by a problem of fundamental interest, namely the
detection of genuinely quantum-gravitational effects that do not arise for
relativistic quantum matter in a classical spacetime [51]. In this Letter we
propose an operational method for constructing and quantitatively analyzing
effects produced by general spacetime superpositions, and apply it to the
analytically tractable (2+1)-dimensional Banados-Teitelboim-Zanelli (BTZ)
black hole. We show that the black hole mass superposition elicits novel
effects with no direct analogue in a classical spacetime. We obtain a
“conditional” metric for the spacetime superposition, which builds upon
approaches in which correlations in quantum field theory act as a proxy for
spacetime distance [52, 53]. These (two-point) field correlations include
those evaluated between the different spacetime amplitudes in superposition
[54, 55]. In this way we are able to calculate the response of a particle
detector in the non-classical spacetime considered. As the detector interacts
with the Hawking radiation of the black hole, its response experiences
resonances at squared rational values of ratio between the superposed mass
values. This effect provides a novel, independent signature that supports and
extends Bekenstein’s conjecture regarding the discrete mass eigenspectrum of
quantum black holes. Throughout we utilize natural units,
$\hslash=k_{B}=c=8G=1$.

The BTZ black hole—The BTZ black hole is a (2+1)-dimensional solution to
Einstein’s field equations with a negative cosmological constant
$\Lambda=-{1/l^{2}}$ where $l$ is the anti-de Sitter (AdS) length scale (the
BTZ spacetime is asymptotically AdS) [56, 57, 58]. While our method of
coupling matter to a superposition of spacetimes applies to any metric and
spacetime dimension, the advantage of studying the BTZ spacetime first is that
its field correlation functions have analytic closed forms [59, 60, 61, 62,
63, 64, 65, 66], whereas in its (3+1)-dimensional counterparts (e.g.
Schwarzschild or Schwarzschild-AdS spacetimes), computationally expensive mode
sums are usually needed.

The BTZ spacetime is obtained as a quotient of AdS-Rindler spacetime [60, 67,
61] under the identification $\Gamma:\phi\to\phi+2\pi\sqrt{M}$ (see
Supplemental Materials for further details). The line element is

$\displaystyle\mathrm{d}s^{2}$
$\displaystyle=-f(r)\mathrm{d}t^{2}+f(r)^{-1}\mathrm{d}r^{2}+r^{2}\mathrm{d}\phi^{2},$
(1)

where $f(r)=(r^{2}/l^{2}-M)$ and $\sqrt{M}l<r<\infty$, $-\infty<t<\infty$,
$\phi\in[0,2\pi)$. The spacetime has a local Tolman temperature
$T_{L}=r_{H}/(2\pi l^{2}\sqrt{f(R)})$, where $R$ denotes the radial coordinate
and $r_{H}=\sqrt{M}l$ is the radial coordinate of the event horizon.

To construct a quantum field theory on this background, consider an
automorphic field $\hat{\phi}(\textsf{x})$, which for the BTZ spacetime is
constructed from an ordinary (massless scalar) field $\hat{\psi}$ in
(2+1)-dimensional AdS spacetime (AdS3) via the identification $\Gamma$,
yielding [68]

$\displaystyle\hat{\phi}(\textsf{x}):=\frac{1}{\sqrt{\mathcal{N}}}\sum_{n}\eta^{n}\hat{\psi}(\Gamma^{n}\textsf{x})$
(2)

where $\textsf{x}=(t,r,\phi)$, $\mathcal{N}=\sum_{n}\eta^{2n}$ and $\eta=\pm
1$ (corresponding to untwisted and twisted fields respectively). The Wightman
(two-point correlation) function between $\textsf{x},\textsf{x}^{\prime}$ is

$\displaystyle W_{\text{BTZ}}^{(D)}(\textsf{x},\textsf{x}^{\prime})$
$\displaystyle=\frac{1}{\mathcal{N}}\sum_{n,m}\eta^{n}\eta^{m}\langle
0|\hat{\psi}(\Gamma_{D}^{n}\textsf{x})\hat{\psi}(\Gamma_{D}^{m}\textsf{x}^{\prime})|0\rangle$
(3)

where $\Gamma_{D}^{n}:(t,r,\phi)\to(t,r,\phi+2\pi n\sqrt{M_{D}})$ in a BTZ
spacetime with black hole mass $M_{D}$. We have assumed that the field state
on the right-hand side of Eq. (3) is the AdS vacuum state; the thermal
properties of the black hole arise from the topological identifications (see
[59] for details of constructing the BTZ Wightman function as an image sum of
the vacuum AdS Wightman function). Although the normalisation factor
$\mathcal{N}$ is formally divergent, Eq. (3), is finite.

We consider the field quantized on a background arising from superposing BTZ
spacetimes with different black hole masses. The black hole–quantum field
system can be described in the tensor product Hilbert space
$\mathcal{H}=\mathcal{H}_{\text{BH}}\otimes\mathcal{H}_{F}$, where we consider
the black hole to be (without loss of generality) in a symmetric
superposition111Conceptually, a black hole mass-superposition may be generated
by sending a photon prepared in a wavepacket distribution of frequencies into
the black hole. This would in general create a black hole in a superposition
of energy (i.e. mass-energy) eigenstates. of two mass states $|M_{A}\rangle$,
$|M_{B}\rangle$ while the field is in the AdS vacuum $|0\rangle$. When
coupling a particle detector to the black hole–quantum field system, we will
require correlation functions between the fields on the different amplitudes
of the superposition; an analogous procedure (see Supplementary Material) as
in Eq. (3) yields

$\displaystyle W_{\text{BTZ}}^{(AB)}(\textsf{x},\textsf{x}^{\prime})$
$\displaystyle=\frac{1}{\mathcal{N}}\sum_{n,m}\eta^{n}\eta^{m}\langle
0|\hat{\psi}(\Gamma_{A}^{n}\textsf{x})\hat{\psi}(\Gamma_{B}^{m}\textsf{x}^{\prime})|0\rangle,$
(4)

noting the two different isometries, $\Gamma_{A}$, $\Gamma_{B}$, corresponding
to the superposed masses, $M_{A}$, $M_{B}$, that act on the coordinates of the
field operators.

Particle detectors—To couple matter to the quantum black hole—field system, we
utilize the Unruh-deWitt model, which considers a pointlike two-level system
linearly coupled to the field [47].

In our case the interaction Hamiltonian reads,

$\displaystyle\hat{H}_{\text{int.}}$
$\displaystyle=\lambda\eta(\tau)\hat{\sigma}(\tau)\sum_{D=A,B}\hat{\phi}(\textsf{x}_{D})\otimes|M_{D}\rangle\langle
M_{D}|$ (5)

where $|\lambda|\ll 1$ is the coupling constant, $\eta(\tau)$ is a time-
dependent switching function, and $\hat{\sigma}(\tau)=(|e\rangle\langle
g|e^{i\Omega\tau}+\text{H.c})$ is the SU(2) ladder operator between the
detector’s energy eigenstates $|g\rangle$ and $|e\rangle$ with energy gap
$\Omega$, $|M_{D}\rangle\langle M_{D}|$ is a projector on the black hole mass,
and $\hat{\phi}(x_{D})$ is the field operator for the spacetime associated
with black hole mass $M_{D}$. This interaction means that for each mass
$M_{D}$, the field is identified accordingly, i.e.
$(t,r,\phi)\to(t,r,\phi+2\pi n\sqrt{M_{D}})$. Finally, $\tau$ is a proper time
related to the coordinate time $t$ of the BTZ spacetime with mass $M_{D}$ by
$\tau=\gamma_{D}t$ where $\gamma_{D}=\sqrt{f(M_{D},R)}$ is a redshift factor
evaluated at the radial coordinate of the detector. We take the detector to be
prepared in its ground state $|g\rangle$ at a fixed radial coordinate $R$,
which for the superposed black hole means it is in a superposition of proper
distances from the horizon. The evolution of the system relative to the
coordinate time $t$ is described as

$\displaystyle|\psi(t_{f})\rangle$
$\displaystyle=e^{-i\hat{H}_{0,S}t_{f}}\hat{U}(t_{i},t_{f})e^{i\hat{H}_{0,S}t_{i}}|\psi(t_{i})\rangle$
(6)

where $\hat{H}_{0,S}$ is the free Hamiltonian of the whole system, including
the evolution of the mass eigenstates of the black hole between the initial
and final times $t_{i}$ and $t_{f}$. This evolution introduces a relative
phase between the constituent states of the superposition. Particular
attention must be paid to the definition of $t_{i}$ and $t_{f}$. Here, we
assume that the black hole–field–detector system is in a tensor product with
an external ‘clock state’ whose time coordinate defines the evolution of the
whole system. For example, for sufficiently large radial coordinate $r$, the
fractional difference in proper times of the clock between two spacetimes with
mass $M_{A}$ and $M_{B}$ respectively is $(M_{B}-M_{A})l^{2}/2r$. Given the
finite precision of any physical clock, such a factorizable clock state can
thus be associated with the proper time of a clock at sufficiently large $r$.
One should expect a theory of quantum gravity describing spacetime
superpositions to admit these kinds of factorizable clock states, as well as
clock states that are entangled with the spacetime [69].

Returning to Eq. (6), we expand the time-evolution operator to leading order
in the coupling strength,

$\displaystyle\hat{U}(t_{i},t_{f})$
$\displaystyle=I+\hat{U}^{(1)}+\hat{U}^{(2)}+\mathcal{O}(\lambda^{3})$ (7)

where the first- and second-order terms are

$\displaystyle\hat{U}^{(1)}$
$\displaystyle=-i\lambda\int_{t_{i}}^{t_{f}}\mathrm{d}\tau\>\hat{H}_{\text{int}}(\tau),\vphantom{\iint_{\mathcal{T}}}$
(8) $\displaystyle\hat{U}^{(2)}$
$\displaystyle=-\lambda^{2}\int_{t_{i}}^{t_{f}}\mathrm{d}\tau\int_{t_{i}}^{\tau}\mathrm{d}\tau^{\prime}\hat{H}_{\text{int.}}(\tau)\hat{H}_{\text{int.}}(\tau^{\prime}).$
(9)

We evolve the initial state according to Eq. (6) and then look at the final
state of the detector conditioned on measuring/projecting the black hole in a
mass-superposition basis. This scenario describes a Mach-Zehnder type
interference where the interferometric paths are associated with different
mass states of the black hole, and yields a detector transition probability
conditioned upon the measurement. If one considers, for example, a measurement
in the symmetric-antisymmetric superposition basis, $|\pm\rangle$, then one
finds that the ground and excited state probabilities of the detector are (see
Supplemental Materials)

$\displaystyle P_{G}^{(\pm)}$ $\displaystyle=\frac{1}{2}\Big{(}1\pm\cos(\Delta
E\Delta
t)\Big{)}\Big{[}1-\frac{\lambda^{2}}{2}\Big{(}P_{A}+P_{B}\Big{)}\Big{]},$ (10)
$\displaystyle P_{E}^{(\pm)}$
$\displaystyle=\frac{\lambda^{2}}{4}\Big{(}P_{A}+P_{B}\pm 2\cos(\Delta E\Delta
t)L_{AB}\Big{)},$ (11)

where $P_{D}$ $(D=A,B)$ is the transition probability of a static detector
outside a BTZ black hole with classical mass $M_{D}$, while $L_{AB}$ is a
cross-correlation term quantifying the correlations between the fields on the
spacetime superposition. Explicitly,

$\displaystyle P_{D}$
$\displaystyle=\int_{t_{i}}^{t_{f}}\mathrm{d}\tau\int_{t_{i}}^{t_{f}}\mathrm{d}\tau^{\prime}\chi(\tau)\bar{\chi}(\tau^{\prime})W_{\text{BTZ}}^{(D)}(\textsf{x},\textsf{x}^{\prime}),$
(12) $\displaystyle L_{AB}$
$\displaystyle=\int_{t_{i}}^{t_{f}}\mathrm{d}\tau\int_{t_{i}}^{t_{f}}\mathrm{d}\tau^{\prime}\chi(\tau)\bar{\chi}(\tau^{\prime})W_{\text{BTZ}}^{(AB)}(\textsf{x},\textsf{x}^{\prime}),$
(13)

where $\chi(\tau)=\eta(\tau)e^{-i\Omega\tau}$. We have also defined $\Delta
E=|\sqrt{M_{A}}-\sqrt{M_{B}}|$ as the energy difference between the black hole
mass eigenstates and $\Delta t=t_{f}-t_{i}$ as the time window over which the
interaction is switched on. Note that the probabilities in Eq. (10) and (11),
obtained for conditional measurements in the complete basis $|\pm\rangle$, add
to unity.

Calculations utilizing the Unruh-deWitt model typically take $\Delta
t\to\infty$; one integrates over the entire history of the detector. In this
case, the presence of an oscillating phase, dependent on the mass and time
difference, means that this limit is not well-defined222In the Supplemental
Materials, we show how one may utilize the rotating-frame transformation [70],
commonly used in quantum optics and atomic physics, to absorb any relative
phases into the definitions of the states.. In our analysis, we consider for
the sake of argument arbitrarily chosen values of $t_{f}$ such that
$\sigma\ll\Delta t$ where $\sigma$ is a characteristic timescale for which the
detector interacts with the field. Specifically, we consider transition
probabilities for Gaussian detector switching functions,
$\eta(\tau)=\exp(-\tau^{2}/2\sigma^{2})$, giving

$\displaystyle\frac{P_{D}}{\sigma}$
$\displaystyle=\frac{\sqrt{\pi}H_{0}(0)}{8}-\frac{i}{8\sqrt{\pi}}\text{PV}\int_{-t_{f}/2l}^{t_{f}/2l}\frac{\mathrm{d}z\>X_{0}(2lz)H_{0}(2lz)}{\sinh(z)}$
$\displaystyle+\frac{1}{4\sqrt{2\pi}\sum_{n}\eta^{2n}}\sum_{n\neq
m}\text{Re}\int_{0}^{t_{f}/l}\frac{\mathrm{d}z\>X_{0}(lz)H_{0}(lz)}{\sqrt{\beta_{nm}-\cosh(z)}}$
(14)

where $X_{0}(z)$, $H_{0}(z)$ are functions of the detector and spacetime
parameters derived in the Supplemental Materials. Likewise, the cross-term is
given by

$\displaystyle\frac{L_{AB}}{\sigma}$
$\displaystyle=\frac{Y_{0}}{\sum_{n}\eta^{2n}}\sum_{n,m}\text{Re}\int_{0}^{t_{f}/l}\frac{\mathrm{d}z\>Z_{0}(lz)Q_{0}(lz)}{\sqrt{\alpha_{nm}-\cosh(z)}},$
(15)

with $Y_{0}$, $Z_{0}(z)$ and $Q_{0}(z)$ derived in the Supplemental Materials.
The constants $\alpha_{nm}$, $\beta_{nm}$ are given by

$\displaystyle\beta_{nm}$
$\displaystyle=\frac{1}{\gamma_{D}^{2}}\left[\frac{R^{2}\cosh(2\pi(n-m)\sqrt{M_{D}})}{M_{D}l^{2}}-1\right],$
(16) $\displaystyle\alpha_{nm}$
$\displaystyle=\frac{1}{\gamma_{A}\gamma_{B}}\left[\frac{R^{2}\cosh(2\pi(m\sqrt{M_{A}}-n\sqrt{M_{B}}))}{\sqrt{M_{A}M_{B}}l^{2}}-1\right].$
(17)

In the Supplemental Materials, we also derive expressions for a compactly
supported detector switching.

Results—We can now analyze the response of the detector outside the mass-
superposed black hole. In Fig. 1, we have plotted the response of the detector
as a function of the mass ratio of the black hole superposition,
$\sqrt{M_{B}/M_{A}}$.

Figure 1: Transition probability of the detector as a function of
$\sqrt{M_{B}/M_{A}}$. The measurement basis corresponding to the relevant plot
is indicated by the legend. In (a), the dashed lines correspond to
$\sqrt{M_{B}/M_{A}}=1/n$ where $n=\\{1,\ldots 6\\}$. In (b), the dashed lines
correspond to $\sqrt{M_{B}/M_{A}}=(n-1)/n$ where $n=\\{3,\ldots 8\\}$.
Moreover, the oscillating cross term in (b) is $\pi/2$ out-of-phase with that
for the black hole measured in the (anti)symmetric basis. In all plots we have
also used $l/\sigma=5$, $R/\sigma=25$, $t_{f}=5\sigma$ and $M_{A}l^{2}=2$.

Fig. 1(a) shows the conditional transition probabilities for measurements in
the symmetric and antisymmetric superposition basis, while Fig. 1(b) considers
measurements in the $|i\pm\rangle=(|M_{A}\rangle\pm i|M_{B}\rangle)/\sqrt{2}$
basis. The most interesting features are the resonant peaks that occur at
rational values of the square root ratio of the superposed masses. In Fig.
1(a), we have denoted some of these ratios with dashed lines. We ascribe this
behaviour to a constructive interference between the field modes associated
with topologically closed AdS spacetimes, yielding resonances in the detector
response at integer values of $\sqrt{M_{B}/M_{A}}$. While we have shown the
transition probability for $M_{B}<M_{A}$, these resonances also occur for
$M_{B}>M_{A}$. The effect also occurs for other values of the square root mass
ratio – in Fig. 1(b), we have highlighted the ratios
$\sqrt{M_{B}/M_{A}}=(n-1)/n$ for $n\in\\{3,\ldots 8\\}$. To understand this,
note that in Eq. (17), the argument of $\cosh(x)$ vanishes when
$m\sqrt{M_{A}}=n\sqrt{M_{B}}$. These ‘coincidences’ occur for terms in the
image sum when the ratio $\sqrt{M_{A}/M_{B}}$ is a rational number, which
includes ratios allowed by Bekenstein’s quantum black hole conjecture.
Specifically, the allowed mass values of the BTZ black hole, assuming the
Bohr-Sommerfeld quantization scheme for the horizon radius, are given by [20]:

$\displaystyle r_{H}$ $\displaystyle=\sqrt{M}l=n,\qquad n=1,2,\ldots$ (18)

While our construction does not require that the superposed masses are
quantized in integer values, the form of Eq. (17) explains the origin of the
signatory resonances. The detector responds uniquely to black hole mass
superpositions with mass ratios corresponding to the masses predicted by
Bekenstein’s conjecture. This result provides independent support for
Bekenstein’s conjecture, demonstrating how the detector’s excitation
probability can reveal a genuinely quantum-gravitational property of a quantum
black hole.

Note also the low-frequency oscillation in the transition probability as
$\sqrt{M_{B}/M_{A}}$ is varied. This behaviour arises from the evolution of
the black hole superposition, which depends on the energy difference between
the superposed states. As $\sqrt{M_{B}/M_{A}}\to 1$, the transition
probability approaches that of a single black hole for a measurement in
$|+\rangle$, whereas it vanishes for $|-\rangle$. For measurements in
$|i\pm\rangle$, the transition probability approaches half of that of a
detector situated outside the black hole with a classical mass value
$M_{A}=M_{B}$. In Fig. 1(a), the amplitude of the oscillation decreases in the
limit of $\sqrt{M_{B}/M_{A}}\ll 1$ or $\gg 1$ due to the decay of the
correlation term $L_{AB}$.

Metric for spacetime superpositions—Our approach has allowed us to study the
quantum effects of a spacetime (black hole) superposition through the
calculation of a physical observable – the transition probability of a
detector coupling to the field. This observable is constructed from the two-
point correlations of the field, which remarkably can be done also on a non-
classical spacetime, such as that of a BTZ black hole in superposition of
masses. Furthermore, it is possible to write down the associated spacetime
metric using these correlators. We first review a related argument relating
metric and field correlators by Saravani et. al. and Kempf [52, 53]. Recall
that in general relativity, spacetime is described by the pair $(M,\sigma)$
where $M$ is a differentiable manifold and
$\sigma(\textsf{x},\textsf{x}^{\prime})$ is the Synge world function (or the
geodesic distance) for two points $(\textsf{x},\textsf{x}^{\prime})$ [71, 72,
73, 74]. The latter contains all the information about a spacetime, since it
allows one to reconstruct the metric [71, 72, 73, 74]:

$\displaystyle g_{\mu\nu}(x)$ $\displaystyle=-\lim_{x\to
x^{\prime}}\frac{\partial}{\partial x^{\mu}}\frac{\partial}{\partial
x^{\prime\nu}}\sigma(\textsf{x},\textsf{x}^{\prime}).$ (19)

Since the quantum correlations of a field decay with the magnitude of the
distance between spacetime points, these studies replaced the notion of proper
distance with that of an appropriate measure of such correlations. In [53],
the Feynman propagator is used, but other objects like the Wightman function
are also suitable. Using the Wightman function, the spacetime metric can be
expressed as

$\displaystyle g_{\mu\nu}$ $\displaystyle={\Upsilon(d)}\lim_{x\to
x^{\prime}}\frac{\partial}{\partial x^{\mu}}\frac{\partial}{\partial
x^{\prime\nu}}W(\textsf{x},\textsf{x}^{\prime})^{\frac{2}{d-2}}$ (20)

where $\Upsilon(d)=-(1/2)(\Gamma(d/2-1)/(4\pi^{d/2}))^{\frac{2}{d-2}}$ and
$d>2$ is the spacetime dimension333A unique expression exists for $d=2$. In
this context, building a spacetime in superposition occurs at the level of the
field operator. By taking
$\hat{\phi}(x)\to\hat{\phi}(\textbf{x})=\sum_{D}f_{D}\hat{\phi}(x_{D})$ (where
$\textbf{x}=\\{x_{D}\\}$, $\sum_{D}|f_{D}|^{2}=1$ and the relative phases
between $f_{D}$ are determined by the state in which the black hole is
measured), the Wightman function becomes a sum over all two-point correlators
between the fields $\hat{\phi}(x_{D})$, $\hat{\phi}(x_{D^{\prime}}^{\prime})$,
defined with respect to the coordinates of the spacetime states in
superposition. Equation (20) is then modified as follows, yielding a
conditional metric describing a superposition of spacetimes:

$\displaystyle g_{\mu\nu}$ $\displaystyle=\Lambda(d)\lim_{x\to
x^{\prime}}\frac{\partial}{\partial x^{\mu}}\frac{\partial}{\partial
x^{\prime\nu}}\sum_{D,D^{\prime}}f_{D}f_{D^{\prime}}^{\star}W(x_{D},x^{\prime}_{D^{\prime}})^{\frac{2}{d-2}}.$
(21)

Equation (21) involves correlations between the field operators parametrized
with coordinates covering black hole spacetimes associated with different
masses; it represents a ‘conditional metric’ effectively seen by a detector in
the quantum superposition of spacetimes.

Conclusion—In this Letter, we have derived genuine quantum-gravitational
phenomena via a simple particle detector model in a spacetime with quantum
degrees of freedom. In particular, we have developed an operational framework
for analyzing ‘superpositions of spacetimes’ via matter (represented by an
Unruh-deWitt detector) coupled to a quantum field. We applied this approach to
study the effects produced by a BTZ black hole in a superposition of masses.
The response of a detector in this spacetime features ‘cross-correlations’
between the different spacetime amplitudes.

The response of the detector is sensitive to the mass ratio of the superposed
black hole, in particular exhibiting signatory peaks at rational values of
$\sqrt{M_{B}/M_{A}}$. This description of a quantum-gravitational effect is
the first of its kind, and constitutes a new method for investigating effects
implied by Bekenstein’s conjecture about the quantization of a black hole’s
mass [14]. We also presented a description of a conditional spacetime metric
arising from such scenarios which is constructed using field correlations as a
measure of distance [52, 53]. Finally, while we have studied the analytically
tractable (2+1)-dimensional black hole, there is no in-principle obstruction
from applying our approach to more general cases in (3+1)-dimensions, which
are expected to provide additional insight into this important problem.

The scheme presented here opens a path to further understanding important
quantum gravity ideas like quantum black holes, spacetime superpositions, and
quantum-gravitational causal structures. Such notions are considered crucial
for a complete description of quantum gravity; however they have generally
only been considered using top-down perspectives in the literature. Our
approach allows for the calculation of observables, such as the response of a
first-quantized system to quantum fields, without the need for a complete
framework of quantizing gravity. By connecting the notion of quantum field
correlations with the superposed spacetime metric, we have also opened up the
possibility of studying the dynamics of spacetime superpositions and the
effects they may induce on systems such as low-energy particles, which
continues to be a topic of growing interest.

Acknowledgements—This work was supported in part by the Natural Sciences and
Engineering Research Council of Canada. M.Z. acknowledges support from
Australian Research Council (ARC) Future Fellowship, grant FT210100675, and
ARC Centre of Excellence EQUS CE170100009. The University of Queensland (UQ)
acknowledges the Traditional Owners and their custodianship of the lands on
which UQ operates.

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RBI-ThPhys-2021-021

# EXTENDED SCALAR SECTORS AT CURRENT AND FUTURE COLLIDERS

T. ROBENS  Ruder Boskovic Institute, Bijenicka cesta 54, 10000 Zagreb, Croatia

###### Abstract

We here give a brief overview on the current status of selected models that
extend the scalar sector of the Standard Model with additional matter states.
We comment on current bounds as well as possible discovery prospects at
current and future colliders.

##  1 Introduction

After the discovery of a particle that complies with the properties of the
Higgs boson predicted by the Standard Model, particle physics has entered an
exciting era. One important question is whether the scalar sector realized by
Nature indeed corresponds to the one predicted by the SM, or whether the
resonance at 125 GeV is a manifestation of a more extended scalar sector, and
additional scalar states could be observed at current or future collider
facilities.

##  2 Theoretical and experimental constraints

In general, the introduction of non SM-like terms in the potential can lead to
massive mass-eigenstates and, successively, induce additional collider
signatures. These predictions need to be confronted with constraints from both
theoretical and experimental sources. We here list them for brevity and refer
the reader to the corresponding references for more detail.

###  2.1 Theory constraints

Theoretical constraints include checks that the potential is bounded from
below, perturbative unitarity, typically via requirements on the maximal
eigenvalue for the $2\,\rightarrow\,2$ scalar scattering matrix using partial
wave expension, and perturbativity of the couplings. Tools which can be used
for these checks include, among others, 2HDMC [1, 2] and ScannerS [3, 4].

###  2.2 Experimental constraints

From the experimental side, one of the most important features that need to be
met is the availability of a scalar particle that complies with the SM Higgs
boson, including all current measurements (mass and signal strength).
Furthermore, one needs to ensure agreement with electroweak precision
measurements, null-searches at past and current colliders, and, for particles
providing a dark matter candidate, agreement with astrophysical measurements
such as relic density and direct detection bounds. Tools which enable
comparison with collider results are e.g. HiggsBounds [5, 6, 7, 8, 9, 10, 11]
and HiggsSignals [12, 13, 14, 10, 15], while dark matter predictions can be
obtained using MicrOMEGAs [16, 17]. These are confronted with experimental
findings, i.e. latest results from the GFitter collaboration [18, 19, 20] as
well as astrophysical observables from the Planck collaboration [21] and
XENON1T [22].

##  3 Real singlet extension: novel results

We here discuss the $\mathbb{Z}_{2}$ symmetric real singlet extension of the
SM, where we follow previous work on this model [23, 24, 25, 26, 27, 28]. We
discuss the case where the $\mathbb{Z}_{2}$ symmetry is softly broken by a
vacuum expectation value (vev) of the singlet field, inducing mixing between
the gauge-eigenstates which introduces a mixing angle $\alpha$. The model has
in total 5 free parameters, out of which 2 are fixed by the measurement of the
$125\,{\rm GeV}$ resonance mass and electroweak precision observables. This
leaves

$\sin\alpha,\,m_{2},\,\tan\beta\,\equiv\,\frac{v}{v_{s}}$ (1)

as free parameters of the model, where $v\,(v_{s})$ corresponds to the vev of
the SM-like doublet (singlet) field. We here concentrate on the case where
$m_{2}\,\geq\,125\,{\rm GeV}$; in this case, $\sin\alpha\,\rightarrow\,0$
corresponds to the SM decoupling. In figure 1, we show novel results for this
model using currently available constraints111Collider constraints are
implemented via the current versions of HiggsBounds/ HiggsSignals., including
a comparison of the currently maximal available rate of
$H\,\rightarrow\,h_{125}h_{125}$ with the current combination limits from
ATLAS [29]. These plots supersede the results shown in [27, 30, 28]. The most
constraining direct search bound depends on the mass considered; in general,
searches for diboson final states [31, 32, 33, 34] are most important,
although some regions are also constrained from the Run 1 Higgs combination
[35]. Especially [33, 34] currently correspond to the best probes of the
models parameter space222We include searches currently available via
HiggsBounds..

Figure 1: Novel results for the singlet extension. Left: comparison of current
constraints for a fixed value of $\tan\beta\,=\,0.1$. Right: maximal
$H\,\rightarrow\,h\,h$ allowed, with electroweak constraints at the
electroweak scale (blue) or including RGE running to a higher scale (red), in
comparison with results from the ATLAS combination.

##  4 Inert Doublet Model: sensitivity study

The Inert Doublet Model is a two Higgs doublet model (2HDM) that obeys an
exact $\mathbb{Z}_{2}$ symmetry, leading to a dark matter candidate from the
second scalar doublet [36, 37, 38]. The results discussed here build on
previous work [39, 26, 27, 40, 41, 42, 43]. The model features four additional
scalar states $H,\,A,\,H^{\pm}$, and has in total 7 free parameters prior to
electroweak symmetry breaking

$v,\,m_{h},\,\underbrace{m_{H},\,m_{A},\,m_{H^{\pm}}}_{\text{second
doublet}},\,\lambda_{2},\,\lambda_{345}\,\equiv\,\lambda_{3}+\lambda_{4}+\lambda_{5},$
(2)

where the $\lambda_{i}$s are standard couplings appearing in the 2HDM
potential. Two parameters ($m_{h}$ and $v$) are fixed by current measurements.

The discovery potential for ILC/ CLIC has been discussed in [41, 42, 44],
including detailed signal and background simulation, beam-strahlung, etc. We
here focus on the results of [43], where a sensitivity comparison for selected
benchmark points [40, 41, 43] using a simple counting criteria was presented:
a benchmark point is considered reachable if at least 1000 signal events are
produced using nominal luminosity of the respective collider (c.f. also [45]).
The summary of sensitivities in terms of mass scales is given in table 1, and
a graphical display in terms of production cross sections for pair-production
of the novel scalars at various collider options and center-of-mass energies
in figure 2, taken from [43]. All cross-sections have been calculated using
Madgraph5 [46] with a UFO input file from [47]333 Note the official version
available at [48] exhibits a wrong CKM structure, leading to false results for
processes involving electroweak gauge bosons radiated off quark lines. In our
implementation in [43], we corrected for this. Our implementation corresponds
to the expressions available from [49].. Results for CLIC were taken from [41,
42].

collider | all others | $AA$ |  $AA$ +VBF
---|---|---|---
HL-LHC | 1 TeV | 200-600 GeV | 500-600 GeV
HE-LHC | 2 TeV | 400-1400 GeV | 800-1400 GeV
FCC-hh | 2 TeV | 600-2000 GeV | 1600-2000 GeV
CLIC, 3 TeV | 2 TeV | - | 300-600 GeV
$\mu\mu$, 10 TeV | 2 TeV | - | 400-1400 GeV
$\mu\mu$, 30 TeV | 2 TeV | - | 1800-2000 GeV
Table 1: Sensitivity of different collider options, using the sensitivity
criterium of 1000 generated events in the specific channel. $x-y$ denotes
minimal/ maximal mass scales that are reachable.

Figure 2: Pair-production cross-section predictions at $pp$ colliders as a
function of the sum of produced particle masses. Left: For all considered
production channels at 13 TeV LHC. Right: for selected channels at 13 TeV, 27
TeV, and 100 TeV. Horizontal dashed lines denote the limit of the cross
section at which 1000 events are produced with the respective target
luminosity.

We also include predictions in the VBF-type mode, corresponding to

$p\,p\,\rightarrow\,X+j\,j,\;\mu^{+}\,\mu^{-}\,\rightarrow\,X\,+\,\nu_{\mu}\,\bar{\nu}_{\mu}$
(3)

for proton-proton collisions and processes at a muon collider, where $X$
signifies the corresponding final state. The cross sections are displayed in
figure 3, taken from [43].

Figure 3: As figure 2, but now considering the VBF-type production mode. Left:
for pp colliders, where two additional jets are produced and right: at
$\mu\mu$ colliders.

We see that especially for $AA$ production the VBF mode serves to
significantly increase the discovery reach of the respective machine. Using
the simple counting criterium above, we can furthermore state that a 27 TeV
proton-proton machine has a similar reach as a 10 TeV muon collider, while 100
TeV FCC-hh would correspond to a 30 TeV muon-muon machine.

##  5 2 real singlet extension: triple-Higgs final states

We now discuss the model introduced in [50], where the scalar sector of the SM
is extended by two real scalars with a discrete
$\mathbb{Z}_{2}\,\otimes\,\mathbb{Z}_{2}^{\prime}$ symmetry, that is
explicitely broken by the vevs of the two scalar fields, leading to mixing
between all scalar states. The model is characterized by 9 parameters after
electroweak symmetry breaking

$m_{1},\,m_{2},\,m_{3},\,v,\,v_{X},\,v_{S},\,\theta_{hS},\,\theta_{hX},\,\theta_{SX},$
(4)

where $m_{i},\,v,\,\theta$ denote masses444We use the convention
$m_{1}\,\leq\,m_{2}\,\leq\,m_{3}$., vevs, and mixing angles. As before, one
mass $m$ and $v$ are fixed.

In [50], various benchmark planes where proposed within this model, allowing
for processes which by that time had not been investigated by the LHC
experiments:

$p\,p\,\rightarrow\,h_{3}\,\rightarrow\,h_{1}\,h_{2},\;p\,p\,\rightarrow\,h_{a}\,\rightarrow\,h_{b}\,h_{b}$
(5)

where in the latter case $h_{a},\,h_{b}\,\neq\,h_{125}$ such that none of the
scalars is identified with the 125 GeV resonance. We here focus on BP3, which
features the first production mode, in the scenario with
$h_{1}\,\equiv\,h_{125}$. Depending on $m_{2}$, this allows for a
$h_{125}\,h_{125}\,h_{125}$ final state. For the subsequent decay
$h_{125}\,\rightarrow\,b\,\bar{b}$, this point was investigated in [51] at a
14 TeV LHC, including detailed signal and background simulation, where we made
use of a customized loop_sm model implemented in MadGraph5_aMC@NLO (v2.7.3)
[52, 53], and subsequently interfaced to HERWIG (v7.2.1) [54, 55, 56, 57, 58,
59, 60]. Results are shown in table 2. We see that using the analysis strategy
of [51], several benchmark points are already accessible with a relatively low
integrated luminosity.

$(m_{2},m_{3})$ | $\sigma(pp\rightarrow h_{1}h_{1}h_{1})$ | $\sigma(pp\rightarrow 3b\bar{b})$ | $\text{sig}|_{300\rm{fb}^{-1}}$ | $\text{sig}|_{3000\rm{fb}^{-1}}$
---|---|---|---|---
${[{\rm GeV}]}$ | ${[{\rm fb}]}$ | ${[{\rm fb}]}$ |  |
$(255,504)$ | $32.40$ | $6.40$ | $2.92$ |  $9.23$
$(263,455)$ | $50.36$ | $9.95$ |  $4.78$ |  $15.10$
$(287,502)$ | $39.61$ | $7.82$ |  $4.01$ |  $12.68$
$(290,454)$ | $49.00$ | $9.68$ |  $5.02$ |  $15.86$
$(320,503)$ | $35.88$ | $7.09$ |  $3.76$ |  $11.88$
$(264,504)$ | $37.67$ | $7.44$ |  $3.56$ |  $11.27$
$(280,455)$ | $51.00$ | $10.07$ |  $5.18$ |  $16.39$
$(300,475)$ | $43.92$ | $8.68$ |  $4.64$ |  $14.68$
$(310,500)$ | $37.90$ | $7.49$ |  $4.09$ |  $12.94$
$(280,500)$ | $40.26$ | $7.95$ |  $4.00$ |  $12.65$
Table 2: 6 b final state leading-order production cross sections at 14 TeV, as
well as significances for different integrated luminosities.

##  6 Conclusion

A detailed understanding of the scalar sector realized by Nature is one of the
most important tasks at current and future collider facilities. We have
presented results for several models that extend the scalar sector of the SM
by additional scalar states, and reported on the current status as well as
future collider prospects.

## Acknowledgments

The author thanks all her collaborators who made the achievements of the
results discussed here possible. This research was supported in parts by the
National Science Centre, Poland, the HARMONIA project under contract
UMO-2015/18/M/ST2/00518 and OPUS project under contract
UMO-2017/25/B/ST2/00496 (2018-2021), by the European Union through the
Programme Horizon 2020 via the COST actions CA15108 - FUNDAMENTALCONNECTIONS
and CA16201 - PARTICLEFACE, and by the UK’s Royal Society.

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# Quantum Illumination with a Parametrically Amplified Idler

Jonathan N. Blakely  U. S. Army Combat Capabilities Development Command
Aviation & Missile Center, Redstone Arsenal, Alabama 35898, USA

###### Abstract

Quantum illumination uses a quantum state of the electromagnetic field to
detect the presence of a target against a bright background more sensitively
than any classical state. Most often, the quantum state is a two-mode squeezed
vacuum consisting of signal and idler modes with a non-zero phase-sensitive
cross correlation, which serves as the signature for target detection, and a
zero phase-insensitive cross correlation, which means the modes produce no
fringes in second order interference. Here it is shown that applying phase-
sensitive amplification to the idler modes of a two-mode squeezed vacuum
results in a non-zero phase-insensitive cross correlation enabling reception
by a simple beam splitter and photodetectors. It is shown that quantum
illumination with a parametrically amplified idler has a lower probability of
error than an asymptotically optimal classical-state scheme in discriminating
between a present target and an absent target with equal prior probabilities.

###### keywords:

quantum illumination , second order interference , quantum radar , quantum
hypothesis testing , phase-insensitive cross correlation

††journal: Physics Letters A

##  1 Introduction

Using a quantum state of the electromagnetic field as a probe, the presence of
a target can be detected with lower probability of error than with any
classical state of equal energy [1, 2]. This approach, known as quantum
illumination, exploits entanglement between photons transmitted and photons
retained, even though this entanglement is destroyed by lossy reflection from
the target and the influence of background noise. Quantum illumination has
been the subject of extensive theoretical study [3, 4, 5, 6, 7, 8, 9, 10, 11,
12, 13, 14, 15, 16] and has been demonstrated experimentally [17]. However,
quantum illumination has not yet formed the basis of a practical quantum
sensing technology as several barriers exist to convenient implementations
[18]. Nonetheless, quantum illumination is fundamentally important as the
first example of a bosonic system that provides a quantum advantage in an
entanglement-breaking channel subject to an energy constraint. Thus it is
important to consider variations on the scheme to better understand quantum
sensing in noisy, lossy environments, and to overcome the barriers to
practical application [19, 20, 21, 22, 23, 24, 25, 26].

In that spirit, this article examines the consequences of applying parametric
amplification to the retained field modes (i.e. the idler). The parametric
amplification considered here is phase sensitive in that it increases one
quadrature component of the field while decreasing the other. This operation
is shown to change the nature of the correlations between the signal and idler
modes such that they become measureable in second order interference. Thus the
quantum illumination receiver can be a simple combination of a beam splitter
and two photodetectors.

Figure 1: (Color online) Schematic diagram of quantum illumination with a
parametrically amplified idler. (a) The source is a spontaneous parametric
down converter that produces signal and idler modes.The signal mode directly
interrogates a region of interest. The idler is input to a phase-sensitive
amplifier and delayed. (b) The amplified and delayed idler is combined on a
beam splitter with radiation received from the region of interest.
Photodetectors at each output port count photons and the difference signal is
formed for comparison with the decision threshold.

This variation on the quantum illumination theme is depicted schematically in
Fig. 1. A spontaneous parametric down conversion source, here depicted as a
strong pump laser driving a nonlinear medium (see Fig. 1(a)), generates
entangled signal and idler beams. The signal beam propagates to the target
region. The idler passes through a second pumped nonlinear medium where it
undergoes parametric amplification before entering a delay line for storage.
Due to the inclusion of the phase-sensitive amplfication stage, the receiver
architecture (see Fig. 1(b)) simply consists of a beam splitter and
photodetectors configured much like a Brown-Twiss interferometer [27]. The
optical implementation shown here is inspired by a recent experiment
demonstrating phase-sensitive amplification of one of two correlated beams
produced by four-wave mixing in a rubidium vapor cell [28]. Implementation at
microwave frequencies is also conceivable. Superconducting parametric
converters and amplifiers for microwave signal generation are now widely used
[29, 30], and microwave Hanbury Brown-Twiss experiments have been conducted
[31, 32].

In Sec. 2, the quantum state of the entangled fields generated by spontaneous
parametric down conversion and the correlations between them, which serve as
the signature of target presence, are described. In Sec.3, the effects of
parametric amplification of the idler beam on these correlations and their
detection are explained. The non-separability of the state is verified. In
Sec. 4, this state is used as the input to the binary-hypothesis test commonly
used to model quantum illumination. In Sec. 5, the probability of error in
target detection with equal prior probabilities is derived and compared to a
classical benchmark to establish the regime of quantum advantage. In Sec. 6,
the significance of these results is discussed.

##  2 Correlations in the Two-Mode Squeezed Vacuum State

The transmitted and retained fields in quantum illumination are entangled
pulses generated by the process of spontaneous parametric down conversion
[18]. In this nonlinear process, photons from a strong pump pulse are
converted into pairs of photons, one each in a signal mode and an idler mode,
eventually producing spatially separated, correlated signal and idler pulses.
Spontaneous parametric down conversion can occur at visible wavelengths, for
example in certain crystals or atomic vapors, or at microwave wavelengths, in
cryogenically cooled, superconducting circuits. Conversion is only efficient
over a limited bandwidth. Typically, it is assumed that down conversion
efficiency is roughly flat across a frequency band of width $W$, and that all
other processes can be neglected within this band. Each pulse is assumed to be
of duration $T$ and contain $M=TW$ temporal modes [18]. Thus, $M$ is both the
time-bandwidth product and the number of modes in the signal and idler pulses.
Alternatively, the bandwidth can be narrowed to $1/T$ to allow only a single
temporal mode. In this case, the time-bandwidth product is made equal to the
same $M$ by transmitting $M$ single-temporal-mode pulses, each of duration
$T$, in a total time of $MT$. This approach will be necessary in order to
implement the amplification and reception scheme described below.

Each single mode in either approach is characterized by the same mean photon
number $N_{S}$. Also, each signal mode is entangled with exactly one idler
mode, and independent of all others, in the two-mode quantum state known as a
two-mode squeezed vacuum. This Gaussian state can be represented by the Wigner
distribution covariance matrix [33]

$\displaystyle\mbox{\boldmath$V$}=\frac{1}{2}\begin{pmatrix}\nu&0&c&0\\\
0&\nu&0&-c\\\ c&0&\nu&0\\\ 0&-c&0&\nu\end{pmatrix}$ (1)

where $\nu=2N_{S}+1$ and $c=2\sqrt{N_{S}(N_{S}+1)}$.

The off-diagonal elements of a two-mode covariance matrix determine the
correlations between the modes. Two types of correlations occur which may be
distinguished by their response to a phase shift $\theta$ applied to both
modes [18]. The phase-insensitive cross correlation between two modes
$\hat{a}_{1}$ and $\hat{a}_{2}$, given by
$\left\langle\hat{a}_{1}^{\dagger}\hat{a}_{2}\right\rangle$, is invariant
under a phase shift. In terms of the elements of the covariance matrix, this
correlation can be expressed as

$\displaystyle\left\langle\hat{a}_{1}^{\dagger}\hat{a}_{2}\right\rangle=\frac{1}{2}\left\\{(\mbox{\boldmath$V$})_{13}+(\mbox{\boldmath$V$})_{24}+i\left[(\mbox{\boldmath$V$})_{14}-(\mbox{\boldmath$V$})_{23}\right]\right\\}$
(2)

The phase-sensitive cross correlation is
$\left\langle\hat{a}_{1}\hat{a}_{2}\right\rangle$. This correlation changes in
phase by $2\theta$ when the modes are shifted by $\theta$. In terms of the
elements of the covariance matrix.

$\displaystyle\left\langle\hat{a}_{1}\hat{a}_{2}\right\rangle=\frac{1}{2}\left\\{(\mbox{\boldmath$V$})_{13}-(\mbox{\boldmath$V$})_{24}+i\left[(\mbox{\boldmath$V$})_{14}+(\mbox{\boldmath$V$})_{23}\right]\right\\}$
(3)

From Eq. (1), the two-mode squeezed vacuum has phase-sensitive cross
correlation $\left\langle\hat{a}_{S}\hat{a}_{I}\right\rangle=c/2$, but zero
phase-insensitive cross correlation.

As will be seen later, the phase-sensitive correlation provides the signature
of target presence in quantum illumination. Unfortunately, this correlation is
the more difficult of the two to detect as it does not give rise to fringes in
second order interference. A receiver design based on interfering the idler
field with radiation received from the target vicinity must include some
additional element to produce fringes, such as a phase conjugation [4, 26]. An
alternative approach is to modify the two-mode squeezed vacuum state to obtain
a non-zero phase-insensitive cross correlation which may then take over as the
signature of target presence given an appropriate receiver design. To that
end, the effect of phase-sensitive, parametric amplification of the idler
field is now considered.

##  3 The Parametrically-Amplified Idler

Parametric or phase-sensitive amplification is a process that boosts one field
quadrature while attenuating the other [34]. In terms of input quadrature
operators $\hat{q}_{\text{in}}$ and $\hat{p}_{\text{in}}$, parametric
amplification (with appropriately chosen phase) affects a transformation
giving output operators of the form

$\displaystyle\hat{q}_{\text{out}}=G\hat{q}_{\text{in}},$ (4)
$\displaystyle\hat{p}_{\text{out}}=G^{-1}\hat{p}_{\text{in}}$ (5)

where $G$ is the gain. It is assumed that the amplifier bandwidth is matched
to the bandwidth $W$ of the idler beam. Applying this transformation to the
single idler mode of Eq. (1) produces an output characterized by the
covariance matrix

$\displaystyle\mbox{\boldmath$V$}=\frac{1}{2}\begin{pmatrix}\nu&0&Gc&0\\\
0&\nu&0&-G^{-1}c\\\ Gc&0&G^{2}\nu&0\\\ 0&-G^{-1}c&0&G^{-2}\nu\end{pmatrix}$
(6)

The important consequence of parametric amplification of the idler is that it
breaks the equality in magnitude of the off-diagonal terms so that Eq. (2) now
gives a phase-insensitive cross correlation
$\left\langle\hat{a}_{S}^{\dagger}\hat{a}_{I}\right\rangle=\left(G-G^{-1}\right)c/4$,
non-zero for any gain greater than unity. Moreover, increasing amplification
boosts the correlation from zero without requiring an increase in the mean
transmitted signal energy, $MN_{S}$. As shown below, the non-zero phase-
insensitive cross correlation provides a measurable signature for target
detection using second order interference.

It is important to note that other possible operations that might be performed
on the idler modes, such as phase-insensitive amplification or projective
measurement, can destroy the entanglement between the idler and the signal
modes. The signal-idler state then becomes a classical state and the
illumination is no longer ‘quantum’, strictly speaking. In contrast, phase-
sensitive amplification of the idler does not destroy the entanglement. The
covariance matrix of Eq. 6 represents a non-separable state. This fact can be
verified by the positive partial transpose test for non-separability. This
test requires a symplectic eigenvalue less than 1/2 for the covariance matrix
that results from a mirror reflection of the momentum operator of one mode
[35]. A straightforward calculation shows that for Eq. 6, the relevant
symplectic eigenvalue has the value
$2^{-1}\left(\sqrt{N_{S}}+\sqrt{N_{S}+1}\right)^{-2}$ which is less than $1/2$
for any non-zero value of $N_{S}$, and independent of the amount of
amplification of the idler. Thus a two-mode squeezed vacuum with a
parametrically amplified idler constitutes a true quantum state for
illuminating a target as described in the section that follows.

##  4 Quantum Illumination

Target detection using quantum illumination is a binary hypothesis test where
a quantum state interrogates a vicinity of interest to decide whether a target
is absent (hypothesis $H_{0}$) or present (hypothesis $H_{1}$). The quantum
state contains idler modes which are retained and signal modes which are
transmitted. The radiation received from the vicinity of interest consists of
either background thermal modes (under $H_{0}$) or a combination of background
and weakly reflected signal modes (under $H_{1}$). In the latter case, noise
and loss destroy the entanglement between signal and idler, but a remnant of
the correlation remains that can be a stronger signature of target presence
than would be achievable using any classical state [2].

Under hypothesis $H_{0}$, each mode $\hat{a}_{R}$ of the received field is
simply a thermal state $\hat{a}_{B}$ with mean photon number $N_{B}$. Then the
two-mode covariance matrix for a received mode and an idler mode is

$\displaystyle\mbox{\boldmath$V$}^{0}_{\text{R,I}}=\frac{1}{2}\begin{pmatrix}\omega&0&0&0\\\
0&\omega&0&0\\\ 0&0&G^{2}\nu&0\\\ 0&0&0&G^{-2}\nu\end{pmatrix}$ (7)

where $\omega=2N_{B}+1$. Under hypothesis $H_{1}$, the target weakly reflects
the signal modes with reflectance $\kappa<<1$, so a received mode is a mix of
signal $\hat{a}_{S}$ and background $\hat{a}_{B}$ such that
$\hat{a}_{R}=\sqrt{\kappa}\hat{a}_{S}+\sqrt{1-\kappa}\hat{a}_{B}$, where the
background mean photon number is set to $N_{B}/(1-\kappa)$ to avoid a passive
target signature. In this case, the covariance matrix for a pair of received
and idler modes is

$\displaystyle\mbox{\boldmath$V$}^{1}_{\text{R,I}}=\frac{1}{2}\begin{pmatrix}\gamma&0&\sqrt{\kappa}Gc&0\\\
0&\gamma&0&-\sqrt{\kappa}G^{-1}c\\\ \sqrt{\kappa}Gc&0&G^{2}\nu&0\\\
0&-\sqrt{\kappa}G^{-1}c&0&G^{-2}\nu\end{pmatrix},$ (8)

where $\gamma=2\kappa N_{S}+\omega$. The off diagonal elements of this matrix
represent the remnant of the correlations of the transmitted quantum state.
For realistic values of $N_{S}$, $N_{B}$, and $\kappa$, these correlations are
too small to constitute quantum entanglement [18]. But if they are measurable,
they may serve as the signature of target presence. Importantly, the
inequality of magnitudes of the off-diagonal elements implies a non-zero
phase-insensitive cross correlation that may be detected simply as second
order interference.

A standard apparatus for measuring second order interference consists of a
beam splitter with a photon-counting detector at each output port (see Fig.
1(a)). The difference in the photon counts is a measure of second order
interference. The action of a 50-50 beam splitter combining two modes with
covariance matrix $V$ is represented by the symplectic matrix

$\displaystyle\mbox{\boldmath$B$}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&0&1&0\\\
0&1&0&1\\\ 1&0&-1&0\\\ 0&1&0&-1\end{pmatrix}$ (9)

where the covariance matrix of the modes at the output ports, represented by
the operators $\hat{a}_{+}$ and $\hat{a}_{-}$, is

$\displaystyle\mbox{\boldmath$V$}_{\pm}=\mbox{\boldmath$B$}\mbox{\boldmath$V$}\mbox{\boldmath$B$}^{T}.$
(10)

Thus, combining the received field with the idler field on a beam splitter
gives the covariance matrix

$\displaystyle\mbox{\boldmath$V$}^{0}_{\pm}=\frac{1}{2}\begin{pmatrix}\frac{\omega+G^{2}\nu}{2}&0&\frac{\omega-G^{2}\nu}{2}&0\\\
0&\frac{\omega+G^{-2}\nu}{2}&0&\frac{\omega-G^{-2}\nu}{2}\\\
\frac{\omega-G^{2}\nu}{2}&0&\frac{\omega+G^{2}\nu}{2}&0\\\
0&\frac{\omega-G^{-2}\nu}{2}&0&\frac{\omega+G^{-2}\nu}{2}\end{pmatrix}$ (11)

under hypothesis $H_{0}$, and

$\displaystyle\mbox{\boldmath$V$}^{1}_{\pm}=\frac{1}{2}\begin{pmatrix}\frac{\gamma+G^{2}\nu+2\sqrt{\kappa}Gc}{2}&0&\frac{\gamma-G^{2}\nu}{2}&0\\\
0&\frac{\gamma+\frac{\nu}{G^{2}}-2\sqrt{\kappa}\frac{c}{G}}{2}&0&\frac{\gamma-G^{-2}\nu}{2}\\\
\frac{\gamma-G^{2}\nu}{2}&0&\frac{\gamma+G^{2}\nu-2\sqrt{\kappa}Gc}{2}&0\\\
0&\frac{\gamma-G^{-2}\nu}{2}&0&\frac{\gamma+\frac{\nu}{G^{2}}+2\sqrt{\kappa}\frac{c}{G}}{2}\end{pmatrix}$
(12)

under $H_{1}$. Photodetectors at the output ports measure $\hat{N}_{+}$ and
$\hat{N}_{-}$, the photon number of the output modes $\hat{a}_{+}$ and
$\hat{a}_{-}$, respectively. The difference between the counts is formed
electronically.

The receiver counts photons from all modes on each output of the beam
splitter. The photon count for any single mode is a non-Gaussian random
variable, but all the counts are identically distributed. For large enough
time-bandwidth product (i.e. number of modes) $M$, the central limit theorem
implies that the total photon count difference is Gaussian with mean and
variance given by

$\displaystyle\langle\hat{N}_{i}\rangle$
$\displaystyle=M\left(\langle\hat{N}_{+,i}\rangle-\langle\hat{N}_{-,i}\rangle\right)$
(13)

and

$\displaystyle\langle\Delta\hat{N}_{i}^{2}\rangle=M\left(\left\langle\left(\hat{N}_{+,i}-\hat{N}_{-,i}\right)^{2}\right\rangle-\left\langle\hat{N}_{+,i}-\hat{N}_{-,i}\right\rangle^{2}\right),$
(14)

respectively, with $i=0,1$ referring to the corresponding hypothesis.

From the Gaussian distributions for the two hypotheses, and setting the
decision threshold to

$\displaystyle\frac{M\left(\langle\hat{N}_{0}\rangle\sqrt{\langle\Delta\hat{N}_{1}^{2}\rangle}+\langle\hat{N}_{1}\rangle\sqrt{\langle\Delta\hat{N}_{0}^{2}\rangle}\right)}{\sqrt{\langle\Delta\hat{N}_{1}^{2}\rangle}+\sqrt{\langle\Delta\hat{N}_{0}^{2}\rangle}},$
(15)

the total error probability for equal probability discrimination is

$\displaystyle
P_{\text{SI}}=\frac{1}{2}\text{erfc}\left(\sqrt{M\cdot\text{SNR}_{QI}}\right)$
(16)

where

$\displaystyle\text{SNR}_{\text{QI}}=\frac{\left(G-G^{-1}\right)^{2}}{\left(G^{2}+G^{-2}\right)}\times\frac{\kappa
c^{2}}{\left(\sqrt{\gamma\nu+\kappa c^{2}}+\sqrt{\nu\omega}\right)^{2}}$ (17)

is the single-mode-pair signal-to-noise ratio [36]. Notably, the gain-
dependent leading factor is zero for unity gain. This is a direct reflection
of the fact that the phase-insensitive cross correlation is zero without
amplification. The leading factor is approximately unity for any gain $G>>1$
as seen in Fig. 2. Thus amplification of the idler beyond about 15 dB produces
only a negligible improvement in the signal-to-noise ratio.

The significance of Eq. (17) becomes apparent when compared to target
detection schemes based on classical states. In the next section, the
performance of quantum illumination with a parametrically amplified idler is
shown to exceed that of a classical-state detection scheme that asymptotically
matches the best possible classical-state scheme.

Figure 2: (Color online) The gain dependent prefactor in the signal-to-noise
ratio vanishes in absence of amplification but quickly rises to unity when
amplification is applied.

##  5 Advantage Over Classical-State Target Detection

In this section, quantum illumination with a parametrically amplified idler is
tested against an important classical-state benchmark. The benchmark system is
a specific detection scheme with an error exponent that approaches the optimal
value for coherent states in the strong background ($N_{B}>>1$) regime. This
system uses a multi-mode coherent state as the signal and coherently
integrated, mode-by-mode homodyne detection as the receiver. For brevity, this
detection scheme will be referred to as coherent-state homodyne detection.

Assuming the signal pulse contains $M$ modes each with mean photon number
$N_{S}$ (to match the energy in the quantum illumination signal), the total
probability of error for detection with equal prior probabilities is of the
form of Eq. (16) but with the single-mode signal-to-noise ratio [3, 4]

$\displaystyle\text{SNR}_{\text{CSH}}=\frac{\gamma-\omega}{4\omega}=\frac{\kappa
N_{S}}{4N_{B}+2}.$ (18)

When $N_{B}>>1$, $\text{SNR}_{\text{CSH}}\approx\kappa N_{S}/(4N_{B})$ which
approximates the quantum Chernoff bound error exponent for optimum reception
using a coherent state [2]. Importantly, this bound has been shown also to be
optimal for all classical state signals. Thus, coherent-state homodyne
detection is significant as a benchmark insofar as it is a structured receiver
that approaches optimal performance in this regime [3, 4, 16].

When comparing the single-mode signal-to-noise ratio in Eq. (18) to Eq. (17)
assuming a strong background (i.e. $N_{B}>>1$), three regimes may be
distinguished. First, consider the case where $N_{S}<<1$. This is the low-
signal, strong-background regime originally identified as the domain where
quantum illumination with a Gaussian state outperforms target detection using
any classical state [2]. Under these conditions, Eq. 17 gives
$\text{SNR}_{\text{QI}}\approx\kappa N_{S}/(2N_{B})$, a 3 dB (i.e. a factor of
2) improvement over coherent-state homodyne detection. This result is
comparable to the performance of phase conjugation and optical parametric
amplifier receivers [4].

Next, let $1<<N_{S}<<N_{B}/\kappa$. Then Eq. 17 gives
$\text{SNR}_{\text{QI}}\approx\kappa N_{S}/(4N_{B})$. Quantum illumination
only matches optimal classical-state detection in this regime. Since typically
it is assumed $\kappa<<1$, this regime can be quite broad. Given the relative
difficulty of implementing quantum illumination, it is unlikely that it would
have any practical advantage over a presumably much simpler classical-state
scheme for applications limited to this regime. However, for applications
requiring operation in both this regime and the previous one, it could be said
that quantum illumination would provide performance equal to or better than
any classical state scheme.

Finally, at even greater signal brightness levels where $N_{S}>>N_{B}/\kappa$,
the signal-to-noise ratio saturates with $\text{SNR}_{\text{QI}}\approx 1/2$.
In this regime, quantum illumination with or without an amplified idler is no
longer competitive with classical-state target detection. It is difficult to
imagine any rationale for using quantum illumination in this regime.

To illustrate these cases, the ratio of the signal-to-noise ratio for quantum
illumination with a parametrically amplified idler to that of coherent-state
homodyne detection is plotted in Fig. 3 with a gain of $G=15$ dB, background
photon number $N_{B}=10^{2}$, and reflectance $\kappa=10^{-3}$. All three
regimes are apparent as well as the transitions from a 3 dB advantage to 0 dB
around $N_{S}\approx 1$, and from 0 dB to a quantum disadvantage around
$N_{S}\approx N_{B}/\kappa=10^{5}$.

Figure 3: (Color online) Ratio of single-mode signal-to-noise ratios for
quantum illumination with a parametrically amplified idler and coherent-state
homodyne detection with $G=15$ dB, $N_{B}=10^{2}$, and $\kappa=10^{-3}$.

##  6 Discussion

The introduction of parametric amplification to the idler beam has been shown
to enable reception using a simple and familiar configuration of a beam
splitter and photodetectors. Phase-sensitive amplification modifies the
correlation structure of the original two-mode squeezed vacuum to produce a
non-zero phase-insensitive cross correlation detectable in second order
interference. The simple receiver of Fig. 1(b) enabled by parametrically
amplifying the idler may be viewed as coming at a cost of increasing the
complexity of the transmitter. This architecture trades a nonlinear process at
the receiver, for example phase conjugation or sum frequency generation [4,
9], for a nonlinear process in the transmitter, phase-sensitive amplification.
However, there may be cause to prefer the latter since it is reasonable to
expect that the parameters of the idler entering the parametric amplifier can
be more carefully controlled than those of received light entering a phase
conjugator or other nonlinear element.

## References

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