Gravitational waves and coherent phonon states in elastic media: 1D-analysis

Longitudinal acoustic phonons in a one-dimensional elastic string are studied in the field of a weak gravitational wave. Working in the analogue gravity framework, a lagrangian describing longitudinal wave propagation along the string is derived in the background of a suitable acoustic metric, the latter encoding both the spacetime corrections due to the gravitational wave and the elastic properties of the string. After quantization, a phonon Hamiltonian is obtained, having the structure of the Luttinger liquid Hamiltonian. Such Hamiltonian is subsequently employed to analyze some effects induced by a gravitational wave, as time evolution of phonon vacuum state, gravitational vacuum squeezing and time evolution of two-mode choerent phonon states, with particular concern to ionic density fluctuations. These findings are in good agreement with similar results appeared in the literature, although in quite different material media. This suggests that the above cited effects do represent a quite general manifestation of the peculiar response to gravitational waves in macroscopic particle ensembles whose internal structure admits some kind of wave propagation.


Introduction
Recent experimental detection at LIGO [1,2] of gravitational wave (GW) signals from several binary black hole mergers (the first of them was BBH GW150914) has strongly renewed the interest of researchers working in the field of General Relativity and observational astrophysics, both from a theoretical and experimental point of view.
The starting of such new exciting GW astronomy era could hopefully allow to discriminate among Einstein's theory and other alternative theories of gravity. Such issue has been anticipated and discussed in detail in [3], where it has been pointed out that an accurate analysis of gravitational wave signals from highly sensitive detectors could give a definitive test for General Relativity.
Over the years, a lot of effort has been devoted to the analysis of the interaction between gravitational waves and matter fields. Several papers have dealt with the interaction between gravitational waves and superfluids [4][5][6][7]. Coupling between superconductors and gravitational waves has been studied in [8][9][10]. Also, influence of electromagnetic, gravitational and gravitoelectromagnetic waves on superconductors has been investigated in [11][12][13][14][15][16][17][18][19][20]. In recent years some Authors have considered the behaviour of Bose-Einstein condensates (BEC) in presence of weak gravitational waves, discussing possible phonon excitation inside the condensate [21,22] as an example of gravitationally induced particle creation [23]. Also, density fluctuations induced by gravitational waves in a BEC have been studied in [24].
However, to the best of our knowledge, little attention has been devoted to a general analysis of the interaction between gravitational waves and elastic media as e.g., crystal lattices. Such topic has been considered in the past [25] mainly through a classical approach, essentially based upon the geodesic deviation. In the present paper we will propose a quantum-mechanical approach, in which collettive lattice vibrations are described in terms of phonons interacting with a gravitational wave. Such semi-classical approach seems indeed the most adequate, as the expected gravitational effects are so small to be comparable to other quantum fluctuations.
Elastic media have usually a discrete structure. However, as far as long wavelength phonons are taken into account, they can be considered a continuous media, obeying a linear dispersion relation w = ( ) c k work in such long wavelength limit, also confining our analysis to a long, thin metallic string, modelled as a onedimensional, continuous elastic line. In such a simplified model we will neglect transverse vibrational modes, taking into account the longitudinal acoustic phonons only. We will show that several effects predicted in quite different macroscopic particle ensembles (as BECs [21,24]), such as phonon creation or parametric amplification induced by gravitational waves, arise quite generally in our 1D-elastic model. In that respect, wave propagation in material media does appear a relevant physical property allowing for an effective interaction between matter fields and gravitational waves.
The paper is organized as follows. In section 2 we focus on a one-dimensional continuous elastic line, described from a lagrangian point of view. In section 3 we generalize the theory to a weak gravitational wave background, introducing a suitable acoustic metric in the analogue gravity framework [21,26,27] and deriving the phonon Hamiltonian for the elastic line. In section 4 we quantize the Hamiltonian, analyzing the time evolution of the phonon vacuum state and particle creation effect in presence of a gravitational wave. In section 5 we study the interaction of a gravitational wave with a two-mode coherent phonon state. Finally, in section 6 we briefly consider ionic density fluctuations induced by a gravitational wave. Section 7 is devoted to some concluding remarks.
Throughout the paper, unless otherwise specified, use has been made of natural units: c=1, ÿ=1, G=1. Greek indices take values from 0 to 3; latin ones take values from 1 to 3. The metric signature is −2, with determinant g.

Elastic line in a GW background
According to the standard theory of ideal elastic media [28], all internal stresses are the result of strain. The most familiar class of elastic media is the one obeying Hooke's law: stress is proportional to strain. For a continuous medium, the resulting potential energy is translationally and rotationally invariant. In the case of a discrete medium (as a crystalline solid) these symmetries are broken at the lattice scales; they are-however-preserved in the long wavelength limit, in which we are mainly interested, as we will see. Before we proceed, let us point out some relevant assumptions: • we will assume a continuous, elastic medium; this assumptions is a good approximation as far as long wavelength phonons are taken into account; • we will consider a 1D-model, namely an elastic line, hence neglecting the transverse dimensions; this allows us to discard possible transverse vibrational modes, hence symplifying the further analysis; • in order to clearly disentangle the gravitational effects from other non-gravitational contributions, here we will not take into account dissipative effects, as higher order phonon-phonon or electron-phonon interactions. Such contributions will be considered in a future work.

Flat background analysis
For a crystalline solid, considered as an elastic medium, the action S 0 up to quadratic order can be written in terms of the symmetric combination of the spatial derivatives of the displacement field  u , where ρ is the density of the solid, μ and λ are the Lamé coefficients and is the strain tensor. Assume the solid to have, e.g., a cylindrical shape-an elastic string -with section A and length  L A . Let us introduce a reference frame having the x-axis along the elastic line. Limiting ourselves to longitudinal strain, we may discard the transverse displacement, hence retaining only the u x component. In such a case, we may integrate out the remaining spatial coordinates in (2.1), thus obtaining where Y=(2μ+λ) appears as the Young modulus 1 . Equation (2.3) can be regarded as the action for a onedimensional continuous elastic line. The corresponding 1+1 lagrangian density  0 is [29] 1 Recall the general relationship between the Lamé coefficients and the Young modulus: is the speed of propagation of the longitudinal sound waves along the elastic line 2 . Actually, from the Euler-Lagrange equations the following equation of motion is readily deduced just describing wave propagation along an elastic medium with velocity c L . Recall that, in the assumed long wavelength limit, the following linear dispersion relation holds

Weak gravitational wave background
The above results hold in a flat space-time. In a slightly curved background, h = + mn mn mn g h , the lagrangian (2.4) will acquire a contribution stemming from the interaction between the matter field and gravity. In this paper we are interested in the interaction with a weak gravitational plane wave.
Let us begin considering a gravitational (plane) wave propagating along an arbitrary direction f q f q q W =  ( ) cos sin , sin sin , cos in the reference frame {x, y, z} of a static observer u, with θ and f being the usual polar coordinates on the 2-sphere [30]. In the TT-gauge frame { } x y z , , of the wave, in whichẑ is assumed as the direction of propagation W  , the line element is Defining

Gravitational interaction in the analogue gravity formalism
In this section we will describe the gravitational interaction following the so-called analogue gravity approach [21,26,27,31]. Let us start observing that (2.4) can be written (hereafter we will suppress the subscript in the displacement field u x , simply writing it u) where the acoustic flat metric h , according to an observer u at rest with respect to the elastic line, is (see [31] and the almost exhaustive review [26] for details) The perturbed acoustic metricg in the u frame reads

Phonon Hamiltonian
In what follows we will suppose that the typical wavelength of the gravitational wave is much larger than the size (L) of the elastic line.
all along the line. In linearized gravity, it is customary to write    = + ; g 0 taking into account (3.4), the gauge-invariant interaction lagrangian density  g is where mn T 0 is the stress-energy tensor of the matter field in absence of gravitational perturbation, namely Introducing the amplitude polarizations A + and A × such that = So, the total lagrangian in presence of a gravitational wave reads The hamiltonian density  with respect to the static observer From (3.10) we get where  0 is the flat hamiltonian density represents the contribution due to the interaction with the gravitational wave. We notice that  g generally depends both on the polarization states of the gravitational wave through the quantity Ξ(Ω).

Quantization
We now move to quantize the theory [32]. The displacement field u becomes an operator 3  Making use of (4.1) the HamiltonianĤ 0 in absence of gravitational perturbation reads Integrating all over the elastic line and making the replacement  A similar a similar relation is obviously found forˆ † c p . Limiting ourselves to the lowest approximation order, we substituteĉ p and - † c p in the r.h.s. of (4.12) with their freely evolving counterparts (4.3), thus writing 3 Throughout the paper, a caret will be used to characterize an operator.
whose solution is a Bogolubov transformation . Looking at (4.15) and (4. 16) we see that, on lack of gravitational perturbation (i.e., and the free evolution of the canonical operatorĉ p (4.3) is recovered.
Armed with the above machinery, we will now move to explore the time evolution of two relevant phonon states, namely the vacuum state ñ |0 and-in the next section-a two-mode coherent state (TMCS).

Vacuum evolution
Let us consider the phonon vacuum state evolution under the influence of a gravitational wave. The phonon vacuum state ñ |0 is defined as Suppose we have initially no phonons in the elastic line at t=0. Using (4.14) we evaluate the expectation value of the phonon number after the interaction with the gravitational wave (  +¥ t ) We see that the gravitational interaction caused the excitation of a small number of phonon out of the vacuum state. Looking at (4.16), we recognize this is a second order ( ( O h 2 )) effect in the small gravitational wave strain h.

Gravitational squeezing of vacuum state
To gain further insight in the evolution of the phonon vacuum under the influence of a gravitational wave, let us consider the evolution operatorˆ( ) U t , which, in the present O(h) approximation, reads [37] Inspection of (4.19) reveals that the effect of the gravitational interaction is encoded in the action of twomode quadrature squeezing operators [38,39] * , followed by a phase rotation. The squeezing parameter ζ k (t) is indeed very small, due to the intrinsic weakness of the gravitational wave.

Two-Mode Coherent State evolution
Our main goal is the anaysis of the behaviour of a coherent phonon state interacting with a weak gravitational wave.
A coherent state ñ |z p is generated out from the vacuum under the action of the displacement operator , which holds true for any two operatorsÂ andB whose commutator is a c-number, we have We see that the gravitational interaction gives rise to a first-order (O(h)) change in the phonon number e i e dth t e N 0 , 5.13 proportional to the initial phonon number á ñ ( ) N 0 .

Response to a continuous sinusoidal wave
As an example, consider the response of the elastic line to a sinusoidal gravitational wave, having frequency ω g , produced by a steady source. In the TT reference frame we have .

5.15
Assume that the measurement has started at t=0 in the u frame. Then, from (5.13) The most interesting condition is ω g =2ω p . In this case, after an integration time t=T we get , the last term in square brackets becomes dominant with respect to the remaining terms Hence The above result is a nice example of parametric amplification of a two-mode coherent phonon state induced by a gravitational wave. We point out that we are facing a first-order effect in the gravitational strain h(t) As a rough numerical estimate, take a steel string with´- c 5 10 ms After an integration time T=1 month, we would get SNR;7. Obviously, this is an highly idealized scenario. We have discarded all the dissipative processes which unavoidably would cause a relevant attenuation in the SNR. Nevertheless, the result seems of some interest from a theoretical point of view, deserving-in our opinion-further careful investigation. In particular, we point out that the signal-to-noise ratio could be improved (even in a shorter integration time) provided the initial coherent state were prepared in a suitable phonon number-squeezed state. This latter issue is currently a field of intensive research (see [37,40] and refs. cited therein).

Density fluctuations: gravitational wave-induced phase shift
We complete the present analysis studying the behaviour of the ionic density á ñ n due the the interaction of a TMCS with a gravitational wave. We use again (5.3), where the operatorsĉ k andˆ † c k evolve in time according to (4.15) and (4.16). Hence á ñ = á ñ = á -

Concluding remarks
In this paper we have analyzed the influence of a weak gravitational wave on a one-dimensional elastic string (an elastic line). Working in the long wavelength limit we have considered the string as a continuous medium. Following a lagrangian approach, we have described the longitudinal wave propagation due to the elastic string properties. We have then generalized the theory to the case of a slightly curved background, in presence of a weak gravitational wave. In the analogue gravity framework, we have represented the matter-gravity interaction by means of a suitable acoustic metric, hence obtaining the corresponding wave Hamiltonian for the elastic line.
After quantization, we have employed the resulting phonon Hamiltonian (having the structure of the Luttinger liquid Hamiltonian) to explore time evolution of phonon vacuum state in presence of a gravitational wave. We have thus proven that gravitational waves can be effective in phonon excitation out from the vacuum state. Although derived in a quite different physical context, such result basically agrees with those obtained elsewhere in the literature [21]. Also, we have shown that the gravitational interaction gives rise to a weak squeezing in the phonon vacuum. We have further considered the behaviour of a two-mode coherent phonon state, proving the occurrence of parametric amplification effects when suitable resonance conditions are met. Finally, we have investigated ionic fluctuations in a coherent phonon standing wave, in which gravitational waves induce a time-dependent phase shift.
Our results suggest that phonon creation and parametric amplification as well as other related effects induced by gravitational waves do represent a quite general manifestation of the peculiar response of macroscopic particle ensembles whose internal structure admits some kind of wave propagation. Actually, the above discussed effects are likely to be described not only in elastic media but also in other physical systems allowing for wavelike propagation of small perturbations, as (super-)fluids and BECs, although in the latter case several experimental challenges have to be overcome as, e.g., the typical very short condensate lifetime.
Experimental observation of the above described effects in a thin, elastic string is presumably still out of reach with the present technology. Also when ultra-low temperature are reached (in order to reduce thermal noise), the main challenge still remains handling the quantum noise. In that respect, phonon squeezing in coherent states [39,40] could be a promising route, although recent claims about the experimental production of such squeezed phonon states [41] have not been confirmed yet.
Apart from present-day experimental observation, the proposed analysis seems however theoretically interesting, as it further contributes to shed light on the interaction between gravity and matter fields described