1 Introduction

Since the first direct detection of gravitational waves (GW) in 2015 [1, 2], GW astronomy, which was hitherto inaccessible, has opened up as a new source of data for Astronomical as well as cosmological inference [3,4,5,6,7,8].

The GWs not only provide us an observational probe into some strong-gravity scenarios like black-hole and neutron star mergers but also may uncover signatures of new physics. For example, the response of ordinary matter to GW signals should differ in the framework of standard General Relativity (GR) vis-à-vis modified theory of gravity (MTG), especially since in the later framework additional GW polarization modes show up [9,10,11,12,13,14,15,16]. In a generic MTG framework where the matter couples to the gravity via the metric, linearisation of the theory yields GWs which may have up to four extra polarisation modes in addition to the two tensor modes predicted by standard GR [9,10,11,12]. Although the currently operational GW detectors are unable to extract the polarisation content of a GW signal with sufficient precision [17, 18] certain early findings [19,20,21,22] have indicated that a full-grown network of advanced LIGO [17,18,19,20,21,22], advanced Virgo [23], KAGRA [24, 25] and others together should be able to distinguish among the various polarization modes [26,27,28]. If the resonant detectors [29,30,31,32,33] reach the desired signal-to-noise ratio for successful detection they will strengthen this network further.

Therefore a theoretical study of matter-GW interaction in the modified-gravity framework seems relevant since it may uncover key features that, if observed in actual GW data, can serve as observational evidence in favor of MTG over standard GR. Specifically, among the additional polarization modes, the longitudinal scalar mode causes displacement of these test particles along the plane of propagation of the GW. Evidently, the scalar longitudinal mode can serve as a feature that distinguishes GW in MTG from that in standard GR. However, this is known in a classical setting whereas in any realistic scenario the effective spatial shift of a test particle under the influence of GW is \(\mathcal{{O}}\left[ 10^{-22} \, \textrm{m}\right] \) which is far smaller than the nuclear length scale. Naturally a quantum mechanical treatment is needed to check if it corroborates the classical result [34,35,36,37,38]. Therefore, in the present paper, we propose to study the response of a quantum mechanical test particle to GW in the MTG framework in a four-dimensional spacetime. We specifically consider the scalar longitudinal polarization mode present in the MTG scenario, since it is supposed to affect the response of the test particle beyond the transverse plane. Consequently the two-dimensional picture of matter-GW interaction that we usually come across, has to be extended to a three-dimensional one. In this paper we elaborate the procedure of this extension in detail, specifically for a test particle interacting with GW, but keeping the methodology fairly general such that it can be applied for more interesting systems like a harmonic oscillator influenced by GW and so on.

The organization of the paper is as follows: In the next section, we explain how the quantum mechanical description of a test particle interacting with GWs in MTG formalism can be obtained. In Sect. 3 we review the various polarization modes of GW in MTG framework. This helps us choose a suitable basis for the \(3 \times 3\) matrices describing the GWs in Sect. 4 where we solve for the quantum dynamics of the test particle. In Sect. 5 we test the limits of our solution for consistency-check, discuss the implication of our result and also the scope of application of the framework we have developed in this paper.

2 Methodology

Irrespective of the chosen framework being the standard Einstein–Hilbert gravity or modified gravity, in any metric theory of gravity the physically observable response of a test particle to incoming GW is determined by the geodesic deviation equation, which serves as the equation of motion of the test particle. The chosen framework only affects the components of curvature tensor and the metric [34, 39, 40]. The geodesic deviation equation incorporates those changes through the curvature tensor and determines the response of the particle accordingly. This geodesic deviation equation, in a proper-detector frame, i.e., in an earth-bound gravitational wave detector frame for a test particle of mass m in the long wavelength and low-velocity limit,Footnote 1 takes the form [34]

$$\begin{aligned} m\ddot{x}^{j}=-mR^{j}_{0,k0}x^{k} \end{aligned}$$
(1)

Here dot denotes the derivative with respect to the coordinate time of the proper detector frame, \(x^{j}\) is the proper distance of the particle from the origin and

$$\begin{aligned} R^{j}_{0,k 0} = -\frac{1}{2}\ddot{h}_{jk} \end{aligned}$$
(2)

are the components of the curvature tensor in terms of metric perturbation \(h_{\mu \nu }\) which are defined through the decomposition of the metric \(g_{\mu \nu }\) as

$$\begin{aligned} g_{\mu \nu }=\eta _{\mu \nu }+h_{\mu \nu };~~~|h_{\mu \nu }|<<1 \end{aligned}$$
(3)

in a linearised theory of gravity. \(h_{\mu \nu }\) behaves as a second rank tensor under background Lorentz transformation with the flat Minkowski background metric \(\eta _{\mu \nu }\). The metric perturbation \(h_{\mu \nu }\) satisfies a wave equation and therefore is referred to as the gravitational wave.

For a quantum mechanical description of this system, we need to quantize an appropriate classical Hamiltonian, which we can obtain from a suitable Lagrangian that, when extrimized, would lead to the classical equation of motion Eq. (1). Up to a total time-derivative term that Lagrangian is

$$\begin{aligned} \mathcal {L}=\frac{1}{2}m\dot{x}^{2}-m\Gamma ^{j}_{0k} x_{j}x^{k} \end{aligned}$$
(4)

where, the affine connection \(\Gamma ^{j}_{0k} \) is given by \(\Gamma ^{j}_{0k} =\dot{h}_{jk}/2\). Using the canonical momentum \(p_{j}=m\dot{x}_{j} - m\Gamma ^{j}_{0k}x^{k}\), we can write the classical Hamiltonian of the system as

$$\begin{aligned} \mathcal {H}=\frac{1}{2m}{\left( p_{j} + m\Gamma ^{j}_{0k}x^{k}\right) }^2 \end{aligned}$$
(5)

which, due to the linearised nature of the theory, simplifies to

$$\begin{aligned} \mathcal {H}=\frac{p_{j}^{2}}{2m}+ \Gamma ^{j}_{0k} x_{j} p_{k} \end{aligned}$$
(6)

To quantize this system we replace the canonical variables in (6) by the corresponding position and momentum operators and in terms of the raising and lowering operators \(a_{j}\) and \(a_{j}^{\dagger }\) defined by

$$\begin{aligned} \hat{x}_{j}=\left( \frac{\hbar }{2m\omega }\right) ^\frac{1}{2}\left( a_{j}+a_{j}^ {\dagger }\right) ; \end{aligned}$$
$$\begin{aligned} \hat{p}_{j}=-i\left( \frac{m\omega \hbar }{2}\right) ^{\frac{1}{2}}\left( a_{j}-a^{\dagger }\right) \end{aligned}$$
(7)

the quantum mechanical Hamiltonian takes the form

$$\begin{aligned} \hat{\mathcal {H}}=\frac{\hbar \omega }{4}\left( 2a_{j}a^{\dagger }_{j}+1-a^{\dagger 2}_{j}-a^{2}_{j}\right) -\frac{i\hbar }{4}\Gamma ^{j}_{0k}\left( a_{k}a_{j}-a^{\dagger }_{k}a^{\dagger }_{j}\right) \end{aligned}$$
(8)

Here the frequency \(\omega \) is determined by the initial uncertainty in either the position or the momentum of the particle [37]. The Hamiltonian operator (8) drives the time evolution of the system. This can be depicted in terms of the Heisenberg equation of motion for \(a_{j}(t)\)

$$\begin{aligned} \frac{da_{j}(t)}{dt}=\frac{i}{\hbar }\left[ \mathcal {H},a_{j}\right] \end{aligned}$$
(9)

and that of \(a^{\dagger }_{j}(t)\), given by the complex conjugate of Eq. (9).

Using (8) and employing the algebra satisfied by the raising and lowering operators

$$\begin{aligned} \left[ a_{j}(t), a^{\dagger }_{k}(t)\right]= & {} \delta _{jk} \nonumber \\ \left[ a_{j}(t),a_{k}(t)\right]= & {} 0= \left[ a^{\dagger }_{j}(t),a^{\dagger }_{k}(t)\right] \end{aligned}$$
(10)

the time-evolution Eq. (9) becomes

$$\begin{aligned} \frac{da_{j}(t)}{dt}= & {} -\frac{i\omega }{2}\left( a_{j}-a^{\dagger }_{j}\right) +\frac{\dot{h}_{jk}}{2}a_{k} \end{aligned}$$
(11)

Introducing the time-dependent Bogoliubov transformation

$$\begin{aligned} a_{j}(t)= & {} u_{jk}(t)a_{k}(0)+v_{jk}(t)a^{\dagger }_{k}(0) \nonumber \\ a^{\dagger }_{j}(t)= & {} a^{\dagger }_{k}(0)\bar{u}_{kj}(t)+a_{k}(0)\bar{v}_{kj}(t) \end{aligned}$$
(12)

which relates the operators \(a_{j}(t)\)  and  \(a^{\dagger }_{j}(t)\) with their initial values at \(t=0\), the time evolution of the system can be cast in terms of the Bogoliubov coefficients \(u_{jk}\left( t \right) \) and \(v_{jk}\left( t \right) \) which are \(3\times 3\) matrices. Owing to Eq. (10), the Bogoliubov coefficients must satisfy

$$\begin{aligned} uv^{\top }=u^{\top } v;~~~~ uu^{\dagger }-vv^{\dagger }=\mathcal {I} \end{aligned}$$
(13)

written in matrix form, where T and \(\dagger \) denote transpose and transpose of complex conjugate respectively. \(\mathcal {I}\) is the \(3 \times 3\) identity matrix.

Corresponding to the initial values in Eq. (12) the appropriate boundary conditions for the Bogoliubov coefficients are

$$\begin{aligned} u_{jk}(0)=\delta _{jk}, v_{jk}(0)=0 \end{aligned}$$
(14)

Using (12), the time-evolution in Eq. (11) is written in terms of a new pair of variables (\(\zeta , \xi \)) defined by

$$\begin{aligned} \zeta _{jk}=u_{jk}-v^{\dagger }_{jk};~~~\xi _{jk}=u_{jk}+v^{\dagger }_{jk} \end{aligned}$$
(15)

as

$$\begin{aligned} \frac{d\zeta _{jk}}{dt}= & {} -\frac{1}{2}\dot{h}_{jp}\zeta _{pk} \end{aligned}$$
(16)
$$\begin{aligned} \frac{d\xi _{jk}}{dt}= & {} -i\omega \zeta _{jk}+\frac{1}{2}\dot{h}_{jp}\xi _{pk} \end{aligned}$$
(17)

Solving this set of coupled differential equations we shall obtain the quantum dynamics of the test mass that will describe its response, not only to the standard plus and cross polarizations of the incoming GW but also to the extra polarization modes that appear in a modified gravity scenario. So before proceeding further, we need to elaborate briefly on the polarization modes of GW that appear in the MTG framework.

3 Polariztion modes in MTG

In four-dimensional spacetime, GWs in a modified theory of gravity can have a maximum of four extra polarization modes [9,10,11,12] in addition to the two tensor modes (known as the ‘cross’ and ‘plus’ polarizations) of GWs that appear in the standard GR framework [34]. Two of these additional polarizations are vector modes which are usually ignored since most of the gravity theories in cosmologically viable scenarios, do not have them [9]. The remaining two are scalar modes. One of them, known as the breathing mode, displaces a circular arrangement of test particles within the transverse plane, much like the two tensor modes of GW do in the standard GR framework. But the other scalar mode, known as the longitudinal one, causes displacement of these test particles along the plane of propagation of the GW. Evidently, the scalar longitudinal mode can serve as a feature that distinguishes GW in MTG from that in standard GR.

This result is known in a classical setting. However due to the extremely small length-scale at which matter interacts with GW, it is only natural to demand that a quantum mechanical treatment shuld corroborate the classical result. Therefore in the remainder of this paper, we shall consider only the scalar longitudinal polarization along with the standard ‘cross’ and ‘plus’ polarizations and check if results same as the classical ones can be reproduced in a quantum mechanical setting.

For our purpose the GW \(h_{jk}\) can be most conveniently represented by

$$\begin{aligned} h_{jk}(t)=\sum ^{3}_{I=1} h_{I}\left( t \right) e^{I}_{jk} \end{aligned}$$
(18)

where the polarization information is contained in the three independent \(3\times 3\) matrices [9]

$$\begin{aligned} e^{\times }_{jk}= & {} e^{1}_{jk} = \begin{pmatrix} 0 &{}\quad 1 &{}\quad 0 \\ 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 \end{pmatrix}, \nonumber \\ e^{+}_{jk}= & {} e^{2}_{jk}= \begin{pmatrix} 1 &{}\quad ~0 &{}\quad 0 \\ 0 &{}\quad -1 &{}\quad 0 \\ 0 &{}\quad ~0 &{}\quad 0 \end{pmatrix}, \nonumber \\ e^{\textrm{s}}_{jk}= & {} e^{3}_{jk}=\sqrt{2} \begin{pmatrix} 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1 \end{pmatrix} \end{aligned}$$
(19)

and \( h_{\times }\left( t \right) , \, h_{+}\left( t \right) , \, h_{\textrm{s}}\left( t \right) \) are the time-dependentFootnote 2 amplitudes of the ‘cross’, ‘plus’ and scalar longitudinal polarization modes respectively for \(I = 1,2,3\). While choosing the polarization matrices (19) we have taken z-axis as the direction in which GW propagates, thus \(x-y\) plane is referred to as the transverse one. In the next section we shall use this chosen form of GW (18) to solve for the quantum dynamics of the test particle.

4 Quantum dynamics of the test particle

To obtain the dynamics we need to solve the coupled differential Eqs. (16, 17) for the matrices \(\left( \zeta _{jk}, \xi _{jk} \right) \). It will be mathematically convenient to expand them in terms of a nine-component basis \(\left\{ e^{M}{}_{jk}\right\} , \, \left( M = 1,2,3,..., 9\right) \) for the space of all \(3 \times 3\) matrices, as following:

$$\begin{aligned} \zeta _{jk}(t)= & {} \sum ^{9}_{M=1}A_{M}e^{M}_{jk} \end{aligned}$$
(20)
$$\begin{aligned} \xi _{jk}= & {} \sum ^{9}_{M=1}B_{M}e^{M}_{jk} \end{aligned}$$
(21)

The basis \(\left\{ e^{}{}_{jk}\right\} \) has, along with \(e^{1}_{jk}\), \(e^{2}_{jk}\) and \(e^{3}_{jk}\) given in Eq. (19), six other independent matrices

$$\begin{aligned}&e^{4}_{jk}= \begin{pmatrix} 0 &{}\quad 0 &{}\quad 1 \\ 0 &{}\quad 0 &{}\quad 0 \\ 1 &{}\quad 0 &{}\quad 0 \end{pmatrix}, \,\, e^{5}_{jk}= \begin{pmatrix} 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1 \\ 0 &{}\quad 1 &{}\quad 0 \end{pmatrix}, \\&e^{6}_{jk}= \begin{pmatrix} 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 \end{pmatrix} , \,\, e^{7}_{jk}= \begin{pmatrix} ~0 &{}\quad 1 &{}\quad 0 \\ -1 &{}\quad 0 &{}\quad 0 \\ ~0 &{}\quad 0 &{}\quad 0 \end{pmatrix}, \end{aligned}$$
$$\begin{aligned} e^{8}_{jk}= \begin{pmatrix} ~0 &{}\quad 0 &{}\quad 1 \\ ~0 &{}\quad 0 &{}\quad 0 \\ -1 &{}\quad 0 &{}\quad 0 \end{pmatrix}, e^{9}_{jk}= \begin{pmatrix} 0 &{}\quad ~0 &{}\quad 0 \\ 0 &{}\quad ~0 &{}\quad 1 \\ 0 &{} \quad -1 &{}\quad 0 \end{pmatrix} \end{aligned}$$
(22)

Note that these six \(3 \times 3\) metrics are not polarisation modes of the gravitational wave and therefore do not need to be symmetric. They are merely members of the set \(\left\{ e^{M}{}_{jk}\right\} \) of basis matrices in the space of all \(3 \times 3\) metrics. The expansion (20, 21) helps reduce the equations of motion (16, 17), which are in the matrix form, to a set of coupled first-order differential equations of the coefficients \(A_{M}\) and \(B_{M}\):

$$\begin{aligned} \dot{A_{1}}= & {} -\frac{1}{2}\left( A_{6}\dot{h}_{\times }+A_{7}\dot{h}_{+}\right) \nonumber \\ \dot{A_{2}}= & {} \frac{1}{2}\left( A_{7}\dot{h}_{\times }-A_{6}\dot{h}_{+}\right) \nonumber \\ \dot{A_{3}}= & {} -\frac{A_{3}\dot{h}_\textrm{s}}{\sqrt{2}}\nonumber \\ \dot{A_{4}}= & {} -\frac{\dot{h}_{\times }}{4}\left( A_{5}+A_{9}\right) -\frac{\dot{h}_{+}}{4}\left( A_{4}+A_{8}\right) -\frac{\dot{h}_\textrm{s}}{2\sqrt{2}}\left( A_{4}-A_{8}\right) \nonumber \\ \dot{A_{5}}= & {} -\frac{\dot{h}_{\times }}{4}\left( A_{4}+A_{8}\right) +\frac{\dot{h}_{+}}{4}\left( A_{5}+A_{9}\right) -\frac{\dot{h}_\textrm{s}}{2\sqrt{2}}\left( A_{5}-A_{9}\right) \nonumber \\ \dot{A_{6}}= & {} -\frac{1}{2}\left( A_{1}\dot{h}_{\times }+A_{2}\dot{h}_{+}\right) \nonumber \\ \dot{A_{7}}= & {} \frac{1}{2}\left( A_{2}\dot{h}_{\times }-A_{1}\dot{h}_{+}\right) \nonumber \\ \dot{A_{8}}= & {} -\frac{\dot{h}_{\times }}{4}\left( A_{5}+A_{9}\right) -\frac{\dot{h}_{+}}{4}\left( A_{4}+A_{8}\right) +\frac{\dot{h}_\textrm{s}}{2\sqrt{2}}\left( A_{4}-A_{8}\right) \nonumber \\ \dot{A_{9}}= & {} -\frac{\dot{h}_{\times }}{4}\left( A_{4}+A_{8}\right) +\frac{\dot{h}_{+}}{4}\left( A_{5}+A_{9}\right) \nonumber \\ {}{} & {} +\frac{\dot{h}_\textrm{s}}{2\sqrt{2}}\left( A_{5}-A_{9}\right) \end{aligned}$$
(23)

and

$$\begin{aligned} \dot{B_{1}}= & {} -i\omega A_{1}-\frac{1}{2}\left( B_{6}\dot{h}_{\times }+B_{7}\dot{h}_{+}\right) \nonumber \\ \dot{B_{2}}= & {} -i\omega A_{2}+\frac{1}{2}\left( B_{7}\dot{h}_{\times }-B_{6}\dot{h}_{+}\right) \nonumber \\ \dot{B_{3}}= & {} -i\omega A_{3}-\frac{B_{3}\dot{h}_\textrm{s}}{\sqrt{2}}\nonumber \\ \dot{B_{4}}= & {} -i\omega A_{4} -\frac{\dot{h}_{\times }}{4}\left( B_{5}+B_{9}\right) \nonumber \\ {}{} & {} -\frac{\dot{h}_{+}}{4}\left( B_{4}+B_{8}\right) -\frac{\dot{h}_\textrm{s}}{2\sqrt{2}}\left( B_{4}-B_{8}\right) \nonumber \\ \dot{B_{5}}= & {} -i\omega A_{5}-\frac{\dot{h}_{\times }}{4}\left( B_{4}+B_{8}\right) +\frac{\dot{h}_{+}}{4}\left( B_{5} +B_{9}\right) \nonumber \\ {}{} & {} -\frac{\dot{h}_\textrm{s}}{2\sqrt{2}}\left( B_{5}-B_{9}\right) \nonumber \\ \dot{B_{6}}= & {} -i\omega A_{6} -\frac{1}{2}\left( B_{1}\dot{h}_{\times }+B_{2}\dot{h}_{+}\right) \nonumber \\ \dot{B_{7}}= & {} -i\omega A_{7}+\frac{1}{2}\left( B_{2}\dot{h}_{\times }-B_{1}\dot{h}_{+}\right) \nonumber \\ \dot{B_{8}}= & {} -i\omega A_{8}-\frac{\dot{h}_{\times }}{4}\left( B_{5}+B_{9}\right) \nonumber \\ {}{} & {} -\frac{\dot{h}_{+}}{4}\left( B_{4}+B_{8}\right) +\frac{\dot{h}_\textrm{s}}{2\sqrt{2}}\left( B_{4}-B_{8}\right) \nonumber \\ \dot{B_{9}}= & {} -i\omega A_{9}-\frac{\dot{h}_{\times }}{4}\left( B_{4}+B_{8}\right) \nonumber \\ {}{} & {} +\frac{\dot{h}_{+}}{4}\left( B_{5}+B_{9}\right) +\frac{\dot{h}_\textrm{s}}{2\sqrt{2}}\left( B_{5}-B_{9}\right) \end{aligned}$$
(24)

Noting from (3) that \(|h_{I}\left( t \right) | \ll 1\), these equations can be solved iteratively about their \(h_{I}\left( t \right) =0\) solutions. For a suitable boundary condition on \(h_{I}\) we assume that the GW hits the system at t = 0 so that

$$\begin{aligned} h_{I}\left( t \right) =0~~(I=1,2,3),~~~ for~~~ t\le 0 \end{aligned}$$
(25)

The first-order solutions obtained by this iterative method are:

$$\begin{aligned} A_{1}=-\frac{h_{\times }}{2},~~~ A_{2}=-\frac{h_{+}}{2},~~~ A_{3}=-\frac{h_\textrm{s}}{2}+\frac{1}{\sqrt{2}},~~~ A_{6}=1 \nonumber \\ A_{4}=0; ~~~~~ A_{5}=0;~~~~~ A_{7}=0;~~~~~ A_{8}=0;~~~~~ A_{9}=0 \end{aligned}$$
(26)

And,

$$\begin{aligned} B_{1}= & {} \frac{i\omega h_{\times }t}{2}+\frac{i\omega }{2}\int t \dot{h}_{\times }dt -\frac{h_{\times }}{2}\nonumber \\ B_{2}= & {} \frac{i\omega h_{+}t}{2}+\frac{i\omega }{2}\int t \dot{h}_{+}dt -\frac{h_{+}}{2}\nonumber \\ B_{3}= & {} \frac{i\omega h_\textrm{s}t}{2}-\frac{i\omega t}{\sqrt{2}}+\frac{i\omega }{2}\int t \dot{h}_{\times }dt -\frac{h_\textrm{s}}{2}+\frac{1}{\sqrt{2}}\nonumber \\ B_{4}= & {} 0;~~~ B_{5}=0;~~~ B_{6}=\left( 1-i\omega t\right) \nonumber \\ B_{7}= & {} 0;~~~ B_{8}=0;~~~ B_{9}=0 \end{aligned}$$
(27)

Using (20, 21) we can reexpress the solution (26, 27) in terms of the matrix pair (\(\zeta _{jk},\xi _{jk}\)) which can be further substituted in (12) via (15) to obtain the time-evolution of the raising and lowering operators \(a_{j}(t)\) and \(a^{\dagger }_{j}(t)\) in terms of their initial values \(a_{j} (0)\) and \(a^{\dagger }_{j}(0)\).

From the definition of the raising and lowering operators (7) at initial time \(t = 0\) we can relate \(a_{j}(0)\) and \(a^{\dagger }_{j}(0)\) with the initial position and momentum expectation values

$$\begin{aligned} \langle \hat{x}_{i}\left( 0\right) \rangle= & {} \left( X_{0}, Y_{0}, Z_{0} \right) \nonumber \\ \langle \hat{p}_{i}\left( 0\right) \rangle= & {} \left( P_{x0}, P_{y0}, P_{z0}\right) \end{aligned}$$
(28)

of the quantum mechanical particle that is used as the test body. The initial uncertainty in either the position or the momentum of this test body determines \(\omega \).

Using the definition of the raising and lowering operators (7) at any arbitrary time, now we can express the time evolution of the position and momentum expectation values \(<\hat{x}_{i}(t)>\) and \(<\hat{p}_{i}(t)>\) in terms of their initial values (28)

$$\begin{aligned}<\hat{x}_{1}(t)>&{=}&X_{0}\left[ 1{-}\frac{h_{+}}{2}\right] {+} \frac{P_{x0}}{m}\left[ t {-} \frac{t h_{+}}{2} {+} \frac{1}{2} \int t \dot{h}_{+} dt\right] \nonumber \\ {}{} & {} -\frac{h_{\times }}{2}Y_{0}-\frac{P_{y0}}{2m}\left[ th_{\times }+\int t \dot{h}_{\times } dt\right] \nonumber \\<\hat{x}_{2}(t)>&{=}&Y_{0}\left( 1{+}\frac{h_{+}}{2}\right) {+} \frac{P_{y0}}{m}\left[ t {+} \frac{th_{+}}{2}{+}\frac{1}{2}\int t \dot{h}_{+} dt\right] \nonumber \\ {}{} & {} -\frac{h_{\times }}{2}X_{0} - \frac{P_{x0}}{2m}\left[ th_{\times }-\int t \dot{h}_{\times } dt\right] \nonumber \\ <\hat{x}_{3}(t)>= & {} \left[ 1-\frac{h_\textrm{s}}{\sqrt{2}}\right] Z_{0} \nonumber \\ {}{} & {} + \frac{P_{z0}}{m}\left[ t - \frac{th_\textrm{s}}{\sqrt{2}} - \frac{1}{\sqrt{2}}\int t \dot{h}_\textrm{s}dt\right] \end{aligned}$$
(29)

and,

$$\begin{aligned}<\hat{p}_{1}(t)>= & {} \left( 1-\frac{h_{+}}{2}\right) P_{x0}-\frac{h_{\times }}{2}P_{y0}\nonumber \\<\hat{p}_{2}(t)>= & {} \left( 1+\frac{h_{+}}{2}\right) P_{y0}-\frac{h_{\times }}{2}P_{x0}\nonumber \\ <\hat{p}_{3}(t)>= & {} \left( 1- \frac{h_\textrm{s}}{\sqrt{2}}\right) P_{z0} \end{aligned}$$
(30)

This is the formal solution that depicts the response of a quantum mechanical test particle to incoming GW having the ‘plus” and “cross” polarization of standard GR and the extra “longitudinal scalar ” mode of polarization owing to the MTG framework. Let’s discuss whether it has consistent limits and how it differs from a similar test scenario in the framework of standard GR in the next section.

5 Results and discussion

We begin with the limiting scenario of no incoming GW, so the quantum particle essentially becomes free. This can be implemented by substituting \(h_{I}\left( t\right) = 0, \) for \( I = \times , +, \textrm{s}\) for all time t in the solutions (29, 30), which yield

$$\begin{aligned}<\hat{x}_{1}(t)>= & {} X_{0}+ \frac{P_{x0}}{m}t \nonumber \\<\hat{x}_{2}(t)>= & {} Y_{0}+\frac{P_{y0}}{m}t \nonumber \\ <\hat{x}_{3}(t)>= & {} Z_{0}+\frac{P_{z0}t}{m} \end{aligned}$$
(31)

and,

$$\begin{aligned}<\hat{p}_{1}(t)>= & {} P_{x0}\nonumber \\<\hat{p}_{2}(t)>= & {} P_{y0}\nonumber \\ <\hat{p}_{3}(t)>= & {} P_{z0}. \end{aligned}$$
(32)

Indeed, these are the general solutions for a free quantum particle with the chosen initial values of position and momenta (28).

With this consistency check in place we now compare our solution to that of particle-GW interaction in a standard GR framework where the only polarization modes present are the two tensorial ones \(\left( h_{\times }, \, h_{+} \right) \). The corresponding solution can be easily obtained from our result (29, 30) by switching off the scalar mode, i.e. by substituting \(h_{s}\left( t \right) = 0\) for all t. It is evident that the \(x-y\) sector of the position and momentum expectation values remain unaltered by this while the z-component of both position \(<\hat{x}_{3}(t)>\) and momentum \(<\hat{p}_{3}(t)>\) reduce to the free-particle like behaviour. This is the contrast between the two frameworks of matter-GW interaction that we have anticipated in the introduction and demonstrated in this paper using a quantum mechanical approach.

The particular purpose that our quantum mechanical treatment serves are twofold:

  • Since the effective spatial shift of a test particle under the influence of GW occurs at a sub-nuclear length-scale reproducing the classical result quantum mechanically reinforces our trust in it.

  • More importantly our treatement provides the necessary quantum mechanical platform to calculate the actual observables of a matter-GW interaction scenario in the MTG formalism.

To eleborate the second point, note that although the position and momentum expectation values of the test particle analytically show the effect of GW on it, owing to the extremely small coupling of matter with GW, the position or momentum of the test particle are not suitable observables in an actual GW detector. Rather one looks for the GW-induced transition of the test body to an excited state [41,42,43]. Our quantum mechanical approach is particularly suitable for calculating the transition probabilities.

Such transitions will be enhanced in a resonance scenario, which can be achieved if, instead of a free particle, the test body is a harmonic oscillator [44] with an intrinsic frequency close to that of the GW. In fact, GW interacting with a quantum mechanical harmonic oscillator is an apt theoretical description of resonant detectors of GWs [41,42,43, 45]. The framework elaborated in this paper can be easily extended for the same and this shall be taken up in a future communication.