- Letter
- Open Access

# Thermal noise in BEC-phononic gravitational wave detectors

- Carlos Sabín
^{1}Email author, - Jan Kohlrus
^{2}, - David Edward Bruschi
^{3, 4}and - Ivette Fuentes
^{2, 5}

**Received: **15 January 2016

**Accepted: **11 May 2016

**Published: **20 May 2016

## Abstract

Quasiparticles in a Bose-Einstein condensate are sensitive to space-time distortions. Gravitational waves can induce transformations on the state of phonons that can be observed through quantum state discrimination techniques. We show that this method is highly robust to thermal noise and depletion. We derive a bound on the strain sensitivity that shows that the detection of waves in the kHz regime is not significantly affected by temperature in a wide range of parameters that are well within current experimental reach.

## Keywords

- gravitational waves
- Bose-Einstein condensates

The detection of gravitational waves [1] remains an open problem and represents one of the most ambitious enterprises of science in the 21st century. After years of active efforts [2, 3], several large-scale experiments are still in operation around the globe, based both in laser interferometry - such as advanced LIGO, GEO 600 or VIRGO - and Weber bar detectors - such as AURIGA and Mario Schoenberg. However, no successful observation of a gravitational wave has been reported yet. Therefore, together with new upgrades of the existing setups, new major international projects are expected to start operations in the short and medium term, including space-based laser interferometers such as DECIGO and LISA. This enormous investment of resources is backed up by indirect evidences of the existence of gravitational waves as well as a number of experiments confirming the predictions of Einstein’s General Relativity, a theory from which the existence of spacetime ripples is a natural consequence [4]. However, since the Earth is very far from typical sources of gravitational waves, the intensity of the latter is so tiny when it reaches our detectors that gravitational wave detection is always a daunting task.^{1}

Recently, a new way of detecting spacetime distortions was proposed using a different physical principle [5]. The state of the quasiparticles of a Bose-Einstein condensate (BEC) is modified by the passing of the gravitational wave. If the frequency of the gravitational wave matches the sum of the frequencies of two BEC modes the transformation of the state is resonantly enhanced in a phenomenon resembling the Dynamical Casimir Effect [6, 7], characterized by a linear growing in time of the transformed state. Indeed, the scheme is equivalent to an artificial modulation of the length of the BEC trap, which can be implemented by a modulation of the atomic interaction strength [8, 9]. This gravitational quantum resonance is absent in laser interferometers since the frequency of a gravitational wave is very far from the optical regime and is also different of the vibrational resonances in Weber bars. In [5], we showed that the sensitivity of the setup is low enough to, in principle, enable the detection of gravitational waves in a certain experimental parameter regime.

^{−5}to \(10^{-2}~\mbox{Hz}\). Our setup operates in a completely different frequency range, ranging from Hz to KHz - the same frequency range as LIGO. Moreover, the sensitivity improves with the frequency while the sensitivity of LIGO is optimal in the range of 100 Hz and decreases at higher frequencies. In particular, this implies that models of spacetime waves generated by the merging of neutron star binary systems into massive neutron stars and black holes go beyond LIGO’s capabilities while being in principle within the reach of our proposal [12]. Detecting gravitational waves in the kHz regime would allow us to deepen our understanding of neutron stars by gathering information about their mass and radius. These parameters are necessary to describe the Neutron star state equation [13] and also allow cosmologists to compute distances that are key in the study of the cosmological constant and dark matter [14]. Other recent proposals of high-frequency gravitational-wave detectors can be found in [15, 16].

*p*is the total pressure,

*ρ*the total density and \(V_{a}\) is the so-called 4-velocity flow on the BEC, given by the gradient of Ψ. The pressure

*p*and the density

*ρ*differ from the their bulk counterparts only in a small linear perturbation. This description stems from the theory of linearized perturbations of fluids in a general relativistic background [17], and thus is valid as long as the BEC can be described as a quantum fluid, that is as long as it remains within the quantum hydrodynamic regime [18]. In the absence of background flows and considering a single spatial dimension, we obtain \(V_{a}=(c,0)\). In this case the effective metric reduces to

*k*is the mode’s momentum. This linear dispersion is valid as long as \(\hbar k\ll m_{0} c_{s}\), where \(m_{0}\) is the mass of the BEC’s atoms.

We consider that the BEC is contained in a 1-dimensional cavity trap. Therefore, we choose to impose close to hard-wall boundary conditions [22–24] that give rise to the spectrum, \(\omega_{n}=\frac{n \pi c_{s}}{L}\), where *L* is the cavity length and \(n\in\{1,2,\ldots\}\).

The phononic field \(\Pi(t,x)\) is then quantized by associating creation and annihilation operators \(a^{\dagger}_{k}\) and \(a_{k}\) to the mode solutions of the Klein-Gordon equation in the effective metric [5, 20], and can be expanded as \(\Pi(t,x)= \sum_{k} [ \phi_{k}(t,x) a_{k} + \phi^{*}_{k}(t,x) a^{\dagger}_{k}]\). The bosonic operators \(a_{k}\) and \(a^{\dagger}_{k}\) obey the canonical commutation relations.

By restricting our analysis to phononic Gaussian states, we are able to use the covariance matrix formalism to describe the dynamics of Π. Gaussian states of bosonic fields and their transformations take a very simple form in the covariance matrix formalism. This simplifies the application of quantum metrology techniques to relativistic quantum fields [25, 26]. Considering a collection of *N* bosonic modes, we define the quadrature operators \(X_{2n-1}=\frac{1}{\sqrt {2}}(a_{n}+a^{\dagger}_{n})\) and \(X_{2n}=\frac{1}{\sqrt{2} i}(a_{n}-a^{\dagger}_{n})\) where \(n=1,\ldots,N\), which correspond to the generalised position and momentum operators of the field, respectively. In the covariance matrix formalism Gaussian states are completely defined by the field’s first and second moments. Furthermore, quadratic linear unitary operators, such as Bogoliubov transformations, are represented by symplectic matrices S that satisfy \(\boldsymbol{S}^{T} \boldsymbol{\Omega} \boldsymbol{S}= \boldsymbol {\Omega}\). Here, the matrix **Ω** is the symplectic form defined by \(\boldsymbol{\Omega}=\bigoplus_{k=1}^{n}\boldsymbol {\Omega}_{k}\), \(\boldsymbol{\Omega}_{k}=-i\sigma_{y}\) and \(\sigma_{y}\) is the corresponding Pauli matrix. The first moments of the state are \(\langle X_{i}\rangle\) and the second moments are encoded in the covariance matrix σ defined by \(\sigma_{ij}=\langle X_{i} X_{j}+X_{j}X_{i}\rangle-2\langle X_{i}\rangle\langle X_{j}\rangle\). Without loss of generality, we restrict our analysis to initial Gaussian states with vanishing first moments, i.e. \(\langle X_{i}\rangle=0\).

For relatively high frequencies, such as \(\omega_{1}=2 \pi\times5\cdot 10^{3}~\mbox{Hz}\) - which corresponds to \(L=1~\mu\mbox{m}\) and \(c_{s}=10~\mbox{mm/s}\) - the above condition entails that we can consider temperatures up to 150 nK - which corresponds to \(e^{2 \beta}\simeq10\). This is a relatively high temperature for a BEC where *T* can be even lower than 1 nK [27, 28]. Note however that we should be cautious in extending our analysis beyond a few nK, since we are neglecting thermal depletion in the condensate bulk, that is temperature is much smaller than the chemical potential \(k_{B} T\ll\mu \). For typical values of the chemical potential \(\mu/k_{B}>100~\mbox{nK}\), which implies that it is reasonable to consider temperatures as large as 10 nK. As the temperature grows, the effect of the thermal cloud on the dynamics of the quantum field modes might become relevant via the Beliaev damping mechanism [29], which we are not considering here.

*ϵ*and Ω are the amplitude and frequency of the spacetime ripple, respectively. The change of the real spacetime metric \(g_{\mu\nu}\) induces a change of the effective metric \(\mathfrak{g}_{ab}\). This in turn generates a Bogoliubov transformation S on the quantum field. The flat - spacetime field operators \({a_{k}}\) are transformed into, \(\hat{a}_{k}=\sum_{j} (\alpha^{*}_{kj}a_{j}+\beta^{*}_{kj}a^{\dagger }_{j} ) \), where \(\hat{a}^{\dagger}_{k}\) and \(\hat{a}_{k}\) are creation and annihilation operators associated to the mode solutions in the perturbed spacetime and \(\alpha_{kj}(h_{+}(t))\) and \(\beta_{kj}(h_{+}(t))\) are Bogoliubov coefficients that depend on the wave’s spacetime parameters. They were computed in [5] in the case of a box-like BEC trap under the following assumptions.

We first consider that beam-pointing laser noise is negligible in the range of kHz frequencies [31–33] and thus the trap can be considered as rigid, and the intensity of the gravitational wave is so small that it remains rigid under its action. Second, the frequency of the gravitational wave matches the sum of two modes of interest \(\Omega=\omega_{m}+\omega_{n}\). Third, the BEC-wave interaction time *t* is long enough \(\omega_{1} t\gg1\). The latter two conditions hold for typical sources of gravitational waves. Under all the above conditions, the effect of the spacetime ripple is equivalent to a two-mode squeezing operation characterized by the coefficient \(\beta_{mn}\), which grows linearly in time in a phenomenon resembling the Dynamical Casimir Effect.

*ϵ*.

The main goal of this work is to analyze the impact of the temperature *T* in the bound on the optimal precision that can be achieved when estimating the wave amplitude *ϵ* through measurements on the state \(\boldsymbol{\sigma}_{\epsilon}\). The quantum Cramer-Rao theorem states that the error in the estimation of the parameter *ϵ* is bounded by \(\langle(\Delta\hat{\epsilon})^{2}\rangle\geq\frac {1}{MH_{\epsilon}}\), where \(H_{\epsilon}\) is the Quantum Fisher Information (QFI) and *M* the number of probes. The QFI can be computed using the Uhlmann fidelity \(\mathcal{F}\) between the state \(\boldsymbol {\sigma}_{\epsilon}\) and a state \(\boldsymbol{\sigma}_{\epsilon +d\epsilon}\) with an infinitesimal increment in the parameter. In particular one has \(H_{\epsilon}=\frac{8 (1-\sqrt{\mathcal {F}(\boldsymbol{\sigma}_{\epsilon},\boldsymbol{\sigma}_{\epsilon +d\epsilon})} )}{d\epsilon^{2}}\).

*linearly*to the zero order QFI with a term proportional to \(x_{n}\), \(x_{m}\).

Let us now review the validity of some underlying approximations. We have considered that thermal depletion is negligible in the atomic bulk and is not significantly modified by the action of the gravitational wave. Indeed, in the case of negligible thermal depletion, the wave function of the condensate would acquire a phase \(\Psi(t)=-\hbar k^{2}/(2 m_{0}) (t-\epsilon\cos{\Omega t}/\Omega)\), where in a 1D box-like potential \(k=\pi c/L\) and \(m_{0}\) is the mass of the BEC’s atoms [21]. Since the velocity flows are defined by \(V^{\mu}=c u^{\mu}/\|u\|\), where \(u^{\mu}=\hbar/m_{0} \partial^{\mu}\Psi\) [18], this means that \(V=(c,0)\) both in the absence of the gravitational wave and under its action - in agreement with our assumptions. Moreover, since the spacetime ripple only generates a phase shift, it does not change the number of condensed and depleted atoms. So if the condensate is initially prepared in a state where thermal depletion is negligible, the thermal depletion would remain negligible when the condensate undergoes interaction with the wave. Finally, it is interesting to analyze whether the phase shift of the condensate bulk could be used in order to detect the spacetime ripple or not. Indeed the QFI \(H_{\epsilon}\) associated to a quantum state \(\phi(t)=\phi_{0} e^{i\Psi(t)}\) is given by \(H_{\epsilon}=|\partial _{\epsilon}\Psi(t)|^{2}\). Therefore, in this case \(H_{\epsilon}=(\hbar k^{2}/(2 m_{0}\Omega)\cos{\Omega t})^{2}\). Thus, we first note that, unlike the quadratic growth in time of the QFI in our scheme, the QFI of the bulk would only oscillate in time. Furthermore, considering the mass of \({}^{87}Rb\) and the same values of *L* and Ω that we are considering in the rest of the work, the maximum value of this atomic QFI is \(H_{\epsilon}\simeq10^{-2}\), which would provide a bound for the strain sensitive of approximately \(10^{-8}~\mbox{Hz}^{-1/2}\) with the same parameters of Figure 2(a). Therefore, the atomic phase shift is extremely less sensitive than our mechanism and cannot be used for gravitational wave detection.

Summarizing, we have shown that the performance of our scheme for BEC-phononic gravitational wave detection is not significantly affected by the presence of initial thermal noise in the state of the phonons. The gravitational quantum resonance between the spacetime ripple and the quasiparticle modes in the BEC trap, is still present in a wide regime of temperatures, well within reach of cutting-edge cold-atoms technology. This represents a step further in the feasibility analysis of this novel scheme of gravitational wave astronomy, aiming to complement the open ambitious quest for gravitational waves.

After the submission of this work, a first event of gravitational wave detection was reported by the LIGO collaboration, which has been heralded as the opening of a new era in gravitational wave astronomy.

## Declarations

### Acknowledgements

We thank Kenta Hotokezaka and Lucia Hackermuller for useful comments and discussions. DE Bruschi was supported by the I-CORE Program of the Planning and Budgeting Committee and the Israel Science Foundation (grant No. 1937/12), as well as by the Israel Science Foundation personal grant No. 24/12. Financial support by Fundación General CSIC (Programa ComFuturo) is acknowledged by CS.

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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