## Objective

The goal is to design a detector that will use gravitational effects to provide key cosmological information necessary for understanding the origin of our Universe and open up the possibility of exploring physics at regimes that have not been accessible with any current technology so far. This will be achieved through a paradigm change in the detection mechanism for Gravitational Waves (GWs). The new detection scheme is underpinned by quantum field theory in curved space-time, which is a theoretical framework that has not been used before in GW detection. We will not base our detection scheme on interferometry like most advanced detectors, instead the detector will exploit quantum resonances at nanoKelvin (nK) temperatures in a micrometer-size quantum system. Previous theoretical computations have shown that distortions of space-time produce observable effects on the phononic excitations of a BEC, making the system suitable for the detection of GWs at frequencies of order 10 - 1000 Hz. The physical implementation of the detector will require Earth-based BEC setups that are of very low cost in comparison to interferometry-based detectors.

## Methodology

The methodology involves the synergy of theoretical methods at the overlap of quantum optics, quantum metrology, general relativity, quantum field theory (QFT) in curve spacetime, and cosmology.

## QFT

In QFT, Bogoliubov transformations relate the states of the field as described by different observers and different space-time regions that are asymptotically flat. The Bogoliubov transformations depend on the parameters of the spacetime, such as the expansion rate of the universe; the amplitude of a gravitational wave; the Schwarzschild radius of a black hole; and, in the case of moving localized fields, the proper acceleration or proper time. We recently developed techniques to apply quantum information and quantum metrology techniques to relativistic quantum fields by representing general Bogoliubov transformations in the symplectic group, which naturally lends to the covariance matrix formalism applied to bosonic quantum fields. The covariance matrix formalism is frequently used in quantum optics.

Changing the boundary conditions of a QFT, in general, creates correlated particles from the vacuum. This is the dynamical Casimir effect. We have developed a technique to characterize these effects which is based on exactly solving the time evolution of the field with boundaries that move arbitrarily but with relatively low accelerations. This allows our techniques to be applied to a wide variety of scenarios where boundaries can oscillate or undergo more complicated trajectories. In particular, we have applied our techniques to calculate field transformations induced by GWs.

Changing the boundary conditions of a QFT, in general, creates correlated particles from the vacuum. This is the dynamical Casimir effect. We have developed a technique to characterize these effects which is based on exactly solving the time evolution of the field with boundaries that move arbitrarily but with relatively low accelerations. This allows our techniques to be applied to a wide variety of scenarios where boundaries can oscillate or undergo more complicated trajectories. In particular, we have applied our techniques to calculate field transformations induced by GWs.

## Phonons of Bose-Einstein Condensates

Under certain conditions, a Bose-Einstein condensate (BEC) can be described by a mean classical mean field background together with small collective quantum excitations, which can behave as phonons. In a relativistic framework, the phonons propagate like the quanta of a massless quantum field satisfying a Klein-Gordon equation on an effective curved background metric. This effective metric has two terms, one term induced by the real space-time metric, and a second term induced by the parameters of the BEC, such as velocity flows and energy density. In the field of analogue gravity, the real space-time metric is considered to be flat while the experimentalists gauge the parameters of the BEC in order to mimic desired spacetime dynamics. In this way, it is possible to simulate sonic black hole horizons and expanding universes in a BEC. However, our work has shown that the real spacetime metric on the phononic field can also have very interesting effects: changes in the real spacetime metric produce changes in the state of phononic excitations that can be, in principle, detected.

## Quantum Metrology

In quantum metrology, quantum properties are exploited to achieve greater statistical precision than purely classical strategies when determining a parameter of the system that is not an observable. Examples of such parameters include temperature, time, acceleration, and coupling strengths. As with any measurement strategy, we can divide the process into the following phases: input of the initial state, evolution through a channel, and measurement of the output state. The parameter to be estimated typically controls the evolution of the input state and is encoded in the output state. One then looks for the most optimal measurement strategy achievable such that the error in the estimation process is minimised.

In classical physics, the mean squared error of a measurement can, in general, scale at most as 1/(NM) on average where N is the number of input probes used in each estimation process, and M is the total number of estimation processes used in the experiment. This scaling is just a consequence of the central limit theorem. However, it has been shown that the ``quantumness'' of physical states and resources, such as entanglement, can be exploited in order to achieve an enhancement of 1/N in this scaling, known as Heisenberg scaling. This improved scaling can become extremely advantageous when the available number of input probes is high but the number of repetitions is constrained, and provides a signature of genuine quantum effects.

Quantum metrology techniques have already proven to give fruitful results when applied to general quantum field theoretical setups, and here we will be applying quantum metrology to probe space-time effects produced by astrophysical sources. So far we have looked at initial two-mode squeezed states of the phononic field, which provides Heisenberg-scaling in our detector.

In classical physics, the mean squared error of a measurement can, in general, scale at most as 1/(NM) on average where N is the number of input probes used in each estimation process, and M is the total number of estimation processes used in the experiment. This scaling is just a consequence of the central limit theorem. However, it has been shown that the ``quantumness'' of physical states and resources, such as entanglement, can be exploited in order to achieve an enhancement of 1/N in this scaling, known as Heisenberg scaling. This improved scaling can become extremely advantageous when the available number of input probes is high but the number of repetitions is constrained, and provides a signature of genuine quantum effects.

Quantum metrology techniques have already proven to give fruitful results when applied to general quantum field theoretical setups, and here we will be applying quantum metrology to probe space-time effects produced by astrophysical sources. So far we have looked at initial two-mode squeezed states of the phononic field, which provides Heisenberg-scaling in our detector.

## Basic detection scheme

We have shown that, in theory, phononic excitations of a BEC produced by spacetime distortions can be used to detect GWs. The GW resonates with a specifically chosen set of frequencies of the phononic modes of the BEC creating observable changes in the state of the field mode. Two modes of the phononic field of the BEC can be prepared that are in resonance with the GW in a two-mode squeezed state, thus enhancing the effects by exploiting quantum properties. The amplitude of a GW that is imprinted in the phononic field can then be estimated by measuring the phononic field using standard measurement techniques, such as absorption imaging, or more novel techniques, such as using impurities.