Analog quantum simulation of gravitational waves in a Bose-Einstein condensate
© Bravo et al.; licensee Springer on behalf of EPJ 2014
Received: 30 May 2014
Accepted: 26 November 2014
Published: 4 January 2015
We show how to vary the physical properties of a Bose-Einstein condensate (BEC) in order to mimic an effective gravitational-wave spacetime. In particular, we focus in the simulation of the recently discovered creation of particles by a real spacetime distortion in box-type traps. We show that, by modulating the speed of sound in the BEC, the phonons experience the effects of a simulated spacetime ripple with experimentally amenable parameters. These results will inform the experimental programme of gravitational wave astronomy with cold atoms.
Quantum simulators  were originally conceived by Feynman as experimentally amenable quantum systems whose dynamics would mimic the behaviour of more inaccessible systems appearing in Nature. Together with the exploration of this visionary insight in countless quantum platforms at an increasingly accelerated rate, last years had witnessed the birth of alternate approaches to quantum simulation. For instance, a quantum simulator can be used to emulate a phenomenon predicted by a well established theory but very hard to test in the laboratory, such as Zitterbewegung . Going a step further, it can also be used to materialise an artificial dynamics that has never been observed in Nature while being theoretically conceivable [3, 4] or even to emulate the action of a mathematical transformation .
Einstein’s theory of general relativity  predicts the existence of gravitational waves , namely perturbations of the spacetime generated by accelerated mass distributions. Since sources are typically very far from Earth, the theory predicts that the amplitude of gravitational waves reaching our planet is extremely small and thus, finding experimental evidence of their existence is a difficult task. Indeed the quest for the detection of these spacetime distortions  has been one of the biggest enterprises of modern science and the focus of a great amount of work during the last decades, both in theory and experiment. Recently, some of the authors of this manuscript have proposed a novel method of gravitational wave detection based on the generation of particles in a Bose-Einstein condensate produced by the propagation of the gravitational ripple . Detection is carried out through a resonance effect which is possible because the range of frequencies of typical gravitational waves is similar to the one of the Bogoliubov modes of a BEC confined in a box-like potential. Then the gravitational wave is able to hit a particle creation resonance, in a phenomenon resembling the Dynamical Casimir Effect [10, 11]. Due to the frequencies involved, this effect is completely absent in optical cavities. The use of our technique would relax some of the most daunting demands of other programmes for gravitational wave detection such as the use of highly-massive mirrors and km-long interferometer arms. However, due to the extremely small amplitude of the gravitational waves when they reach the Earth, their detection is always challenging, since it requires an experimental setup extremely well isolated from possible sources of noise. It would be of great benefit for the experiment if the amplitude of the spacetime ripples were larger.
In this paper, we show how to realise a quantum simulation of the generation of particles by gravitational waves in a BEC. We exploit the fact that the Bogoliubov modes of a trapped BEC satisfy a Klein-Gordon equation on a curved background metric. The metric has two terms [9, 12–14], one corresponding to the real spacetime metric and a second term, corresponding to what we call the analogue gravity metric, which depends on BEC parameters such as velocity flows and energy density. While in  we analyse the effect of changes in the real spacetime metric, in this case we consider the manipulation of the analogue gravity [15–18] metric, assuming that the real spacetime is flat. Since in this case the experimentalist is able to manipulate artificially the parameters of the condensate, we are able to simulate spacetime distortions with a much larger amplitude, as if the laboratory were closer to the source of the gravitational ripples. We show that with realistic experimental parameters, a physically meaningful model of gravitational wave can be simulated with current technology.
The paper is organised as follows. First we review the effects of a real gravitational wave in the Bogoliubov modes of the BEC. Then we consider the case in which there are non-zero initial velocity flows in the BEC, showing that there is always a reference frame in which we can modulate the speed of sound in such a way that the phonons experience an effect analog to the one produced by the propagation of a real spacetime wave. Finally, for the sake of simplicity we assume that there are no velocity flows in the BEC and we show that the gravitational wave can be simulated with current technology.
Gravitational waves in a BEC
Ignoring the conformal factor - which can always be done in 1D or in the case in which is time-independent - we notice that the metric is the flat Minkowski metric with the speed of light being replaced by the speed of sound . By considering a rescaled time coordinate we recover the standard Minkowski metric . This means that the phonons live on a spacetime in which, due to the BEC ground state properties, time flows in a different fashion and excitations propagate accordingly.
In , it is shown that the propagation of the gravitational wave generates particles in a BEC confined in a box-like trap. This particle creation is characterised by the Bogoliubov coefficients , where , are the m and n mode solutions of Eq. (5) given by the metrics in Eq. (10) and Eq. (9) respectively. In particular, if we model the gravitational wave by a sinusoidal oscillation of frequency Ω that matches the sum of the frequencies of a certain pair of modes m and n, the number of particles grows linearly in time. We refer the reader to  for more details.
The aim of this section is to show how to get a line element similar to the one in Eq. (11) by manipulating the parameters of the BEC while the real spacetime is assumed to be flat.
The case with no background flows. Experimental implementation
In the last section, we have shown how the speed of sound can be modified to mimic a gravitational wave in a coordinate system in which there are no background flows. Now, we will consider for simplicity that this coordinate system is the lab frame and relate our results directly with experimental parameters.
This is much larger that the gravitational waves that we expect to see in the Earth , due to the fact that the Earth is far from typical sources of gravitational waves. Therefore it is interesting to think of the physical meaning of the gravitational waves that can be simulated with our techniques.
respectively, where G is Newton’s gravitational constant, m and M are the masses of the Earth and the Sun respectively, r the distance between the two bodies and R the distance between the detector and the centre of the mass of the system, which is assumed to be much larger than the wavelength λ of the gravitational wave . Thus, if we consider the real masses of the Sun and the Earth, a distance between them of and , we find Ω in the kHz range - which is very convenient to generate phonons in the BEC - and the desired value of . In  it is predicted that the changes in the covariance matrix of the Bogoliubov modes induced by gravitational waves of typical amplitudes are in principle detectable. Therefore, the changes generated by a simulated wave of much larger amplitude, should be observed in an experiment with current cold-atoms technology.
We have shown how to generate an artificial gravitational wave spacetime for the quantum excitations of a BEC. In the case in which there are no initial background flows, we show that the simulated ripple is obtained through a modulation of the speed of sound in the BEC. In the laboratory, this can be achieved with current technology by exploiting the dependence of the scattering length on the external magnetic field around a Feshbach resonance. With realistic experimental parameters, we find that simulated gravitational waves that can resonate with the Bogoliubov modes of the BEC. The amplitude of these artificial ripples is much larger than the typical amplitude expected for gravitational waves reaching the Earth, due to the fact that the Earth is very far from typical sources. Thus our simulated waves would mimic the waves generated by sources much closer to the BEC. This feature would enhance the effects of the ripple in the system, facilitating their detection. The experimental test of our predictions would be a proof-of-concept of the generation of particles by gravitational waves and would pave the way for the actual observation of real spacetime ripples in a BEC. More generally, our low-cost Earth-based tabletop experiment will inform the whole programme of gravitational wave astronomy.
TB acknowledges funding from CONACYT. IF and CS acknowledge funding from EPSRC (CAF Grant No. EP/G00496X/2 to IF).
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