Introduction

After more than 70 years of focusing on the theoretical and mathematical aspects of the theory of quantum gravity (QG), in recent decades we have encountered some serious proposals on the project of the QG phenomenology1. Indeed, the first phenomenological proposition for QG from the experimental point of view has been set up very recently2. This has given rise to the possibility of bringing other areas like astrophysics, particle physics, cosmology, together in a same footing along with some feasibilities for experiment. Thereby, guided by the empirical facts into the context of different approaches of QG, the novel concepts are growing rapidly day by day3. Today, we are dealing with deliberately genuine schemes of QG phenomenology in different contexts like string theory4,5, loop quantum gravity (LQG)6,7, doubly special relativity8,9, etc., which are not only successful from the theoretical aspects but also they made a significant impact on relevant experiments10.

Several approaches of QG including those mentioned above support the fact that in the QG regime the space-time has a discrete structure and the Planck length is the smallest measurable and invariant length with respect to all inertial observers in the nature, which plays the role of a regulator for quantum field theories. In ordinary Schrödinger representation of quantum mechanics, the spectra of the position and momentum operators being continuous, such natural cut-off can not be implemented by using the standard quantum mechanical theory. However, one can present an effective model of QG to introduce the Planck length through the modification of the Heisenberg’ uncertainty principle. More specifically, given that gravity is not an ordinary force, but a property of space-time, we can deal with an effective framework of QG by considering a fundamental and minimal characteristic length for the geometry of space-time that is probed by a moving quantum particle. Within such context the standard Heisenberg uncertainty principle and relativistic dispersion relation are replaced by the generalized uncertainty principle (GUP)11,12,13,14,15,16,17,18 and the modified dispersion relation (MDR)19,20, respectively, where the geometry of momentum space is not trivial anymore and sometimes the Lorentz symmetry may also be broken.

Polymer quantum mechanics is one of the most recent phenomenological approaches to the problem of QG, which is suggested in the symmetric sector of LQG21. The original version of polymer quantum mechanics is a quantum mechanics on a lattice such that the lattice length plays the role of a minimal length22. In standard quantum mechanical theory, the position and momentum operators are unbounded operators, however, in order to insert a natural cut-off in the quantum gravity regime, one requires to deal with the bounded observables. Polymer quantum mechanics is an excellent framework to achieve this23,24,25,26. It has also been demonstrated that polymer quantization and GUP have the same physical consiquences27. Within this scheme, one implements the Weyl algebra21,22 by preserving the standard canonical form of the commutation relations such that the momentum operator becomes ill-defined. However, it can be regularized later by introducing an exponential shift operator so that the momentum operator becomes well-defined21. As a consequence, one obtains a modified version of the momentum operator and, hence, the Hamiltonian, which supports the existence of a minimal length. Meanwhile, an alternative scheme for this also exists. One can implement a noncanonical representation of the Heisenberg algebra such that the Hamiltonian operator remains in its standard functional form but the commutation relation is modified as28,29

$${[x,p]}_{{\mu }_{0}}=i\hslash \sqrt{1-{\mu }_{0}^{2}{(\frac{p}{{M}_{{\rm{P}}}c})}^{2}}\mathrm{.}$$
(1)

Here Mp~1019 GeV is the Planck mass and μ0 labels a numerical coefficient which determines the precise value of the polymer length μ0l P . It is expected that μ0 should be of the order of unity \({\mu }_{0}={\mathscr{O}}\mathrm{(1)}\), but its precise value can only be determined by experiments. In this paper, we are interested to constrain this quantity through an opto-mechanical (OM) experimental setup. It should be noted that there exist several different phenomenological aspects to test the short distance effects of QG14,18,30,31, however, a more recent approach based on an OM experimental scheme serves a more interesting way to the problem32. The biggest advantage of the method discussed in the latter scenario is that one does not require an expansive high energy scattering experiment to probe the Planck length accuracy in position measurement, which is not possible anyway since the highest energy possible to reach by the resources available to us is about 15 orders of magnitude away from the Planck energy33. The system32, rather, based on a simple table-top experiment which can be utilized cleverly to serve the purpose. Being inspired by the optical layout designed in32, we setup a similar scheme to test the polymer deformation, which emerges from the symmetric sector of LQG, while in32 the authors explored the existence of minimal length in QG instead. One of the interesting results in our article is that we notice that the polymer deformation parameter μ0 can be measured with a remarkable sensitivity under this scheme, which we discuss in the following section. Moreover, it should be emphasized that following the GUP model considered in32, we apply the polymer modified commutation relation (1) to the center of mass rather than to each single particle individually. Otherwise, the underlying OM scheme would not be able to put a stringent bound on μ0.

Opto-mechanical scheme for polymer-modified commutation relation

To start with, let us first represent the position and momentum operators x, p corresponding to our system (1) as dimensionless observables \({X}_{m}=x\sqrt{m{\omega }_{m}/\hslash },\,{P}_{m}=p/\sqrt{\hslash m{\omega }_{m}}\), which are familiar as quadratures in quantum optics. Consequently, the polymer modified commutation relation (1) obtains the following form

$${[{X}_{m},{P}_{m}]}_{{\mu }_{0}}=i\hslash \sqrt{1-{\mu }^{2}{P}_{m}^{2}},$$
(2)

with \(\mu ={\mu }_{0}\sqrt{\hslash m{\omega }_{m}}/{M}_{{{\rm{P}}}^{{c}}}\). The OM scheme is designed in a way that the mechanical oscillator of mass m and angular frequency ω m (2) interacts with an optical pulse in a very efficient way inside an optical cavity. In order to understand this interaction in reality, one requires a unitary displacement operator34 \({U}_{m}={e}^{i{n}_{L}\lambda {X}_{m}}\), which displaces the quadrature X m of a mechanical oscillator in phase space induced by the optical field of interaction length λ. Here n L represents the photon number operator. A sequence of four such radiation pressure interactions causes the mechanical state to be displaced around a loop in phase space, which effectively forms an optical cavity (resonator) yielding the total interaction operator

$$\xi ={e}^{i{n}_{L}\lambda {P}_{m}}\,{e}^{-i{n}_{L}\lambda {X}_{m}}\,{e}^{-i{n}_{L}\lambda {P}_{m}}\,{e}^{i{n}_{L}\lambda {X}_{m}}\,\mathrm{.}$$
(3)

After a complete sequence, apparently it seems that none of the two systems would be affected, since the operations are likely to neutralize the effects of each other classically. However, interestingly, it was argued in35 that both the classical and quantum mechanical cases contribute to the effect apart from some differences in values. So, in principle one can consider any of the effects for the purpose. However, since our work is based on the quantum mechanical scenario, it is more natural that we consider the quantum effects only. Consequently, the two subsequent displacements in phase space governed by (3) cause an additional phase to the state under consideration, especially when the commutation relation between X m and P m is modified as (2). This additional phase shift in the oscillator will create a change in the optical field correspondingly, which is what we need to measure. Thus, a finite shift in the optical field will confirm the existence of the polymer modified commutation relation. In order to obtain the desired change in the optical field, let us analyze the mean of the optical field operator

$$\langle {a}_{L}\rangle =\langle \alpha |{\xi }^{\dagger }a\xi |\alpha \rangle ={\langle {a}_{L}\rangle }_{{\rm{QM}}}{e}^{-i{\rm{\Theta }}},$$
(4)

where |α〉 is the coherent state of the input optical field with mean photon number N p , a L being the annihilation operator and Θ describes the additional optical phase emerging due to the polymer modified commutation relation. Meanwhile, the average of the standard quantum mechanical field operator can be computed as36

$${\langle {a}_{L}\rangle }_{qm}={e}^{-i{\lambda }^{2}-{N}_{p}(1-{e}^{-2i{\lambda }^{2}})}\mathrm{.}$$
(5)

Now, the only thing that remains is to compute 〈α|ξ†aξ|α〉 as described in (4). For this, we first simplify the interaction operator (3). Using the standard relation \({e}^{aA}B{e}^{-aA}={\sum }_{k=0}^{\infty }\frac{{i}^{k}{a}^{k}}{k!}{c}_{k}\), with ic k  = [A, ck − 1] and c0 = B, one can re-express the total interaction operator (3) as follows

$$\xi ={e}^{-i{n}_{L}\lambda \sum _{k}\frac{{c}_{k}{(\lambda {n}_{L})}^{k}}{k!}}\mathrm{.}$$
(6)

This relation clearly depends on the commutation relation related to the mechanical oscillator, but not with that of the optical field. It is easy to demonstrate that within the context of the standard quantum mechanics one achieves \(\xi ={e}^{-i{\lambda }^{2}{n}_{L}^{2}}\)1. This means that in the absence of QG, the optical field is affected only by a self-Kerr-nonlinearity term as \({n}_{L}^{2}\) operation and the mechanical state stays unchanged. However, it is clear that c k with k > 1 are nonzero in polymer setup, while all of them are vanishing in the standard quantum mechanics. Considering A = X m and B = c0 = P m , it is easy to show that all the coefficients c k with k > 1 are nonzero and can be represented by the recursive relation as

$${c}_{k}=-\mu \,{c}_{k-2},\quad k=\mathrm{2,3,....,}$$
(7)

with

$${c}_{0}={P}_{m},\quad {c}_{1}=\sqrt{1-{\mu }^{2}{P}_{m}^{2}}\mathrm{.}$$
(8)

This implies that all the coefficients c k are determined only by two independent entities c0 and c1. Therefore, we can simply write the displacement operator (6) in terms of these two independent coefficients as

$$\xi ={e}^{-i\lambda {n}_{L}(-\mu \frac{{\lambda }^{2}{n}_{L}^{2}}{\mathrm{2!}}){c}_{0}-i{\lambda }^{2}{n}_{L}^{2}(1-\mu \frac{{\lambda }^{2}{n}_{L}^{2}}{\mathrm{3!}}){c}_{1}},$$
(9)

which can be rewritten in a simpler form as

$$\xi ={e}^{-i{\lambda }^{2}{n}_{L}^{2}{c}_{1}}{e}^{i\mu (\frac{{\lambda }^{3}{n}_{L}^{3}}{2}{P}_{m}+\frac{{\lambda }^{4}{n}_{L}^{4}}{6}{c}_{1})}\mathrm{.}$$
(10)

Note that, we have not considered the contribution of c k for k ≥ 4, since these terms have negligible effects for the order of \({\mathscr{O}}({\mu }^{2})\) and higher. As we discussed earlier, in the absence of the polymer deformation parameter, i.e. for μ = 0, the above equation reduces to the relation corresponding to the standard quantum mechanics. However, for the case considered here, we notice a deviation from that, which is given by the second exponential. Consequently, from (4) we compute the deviation in the average value of the field operator with c1 ≈ 1 and follow the similar steps as in32,36 to obtain the deviation of the optical phase as

$${\rm{\Theta }}(\mu )\simeq \frac{2}{3}\mu {N}_{p}^{3}{\lambda }^{4}\,{e}^{-in{\lambda }^{2}},$$
(11)

Schematic for experimental setup

Let us now discuss a realistic experimental scenario that can measure the additional phase (11) emerging form the polymer modified commutation relation. The OM scheme that we shall explore here have been utilized for different purposes32,37,38,39,40,41,42. In our case, the device couples the polymer modified mechanical oscillator with an optical pulse via radiation pressure inside a high-finesse optical cavity as depicted in Fig. 1. More specifically, the input light is created from a tunable diode laser and, subsequently, amplified by an erbium doped fibre amplifier. Then, it is divided into two parts, one part is kept as a reference beam for future use. While the other is passed through a polarized beam splitter (PBS) followed by an electro-optic modulator (EOM) to allow for an interaction with a mechanical oscillator having position X m inside a high-finesse optical cavity with cavity field a L . The light is then retro-reflected and enters into the delay line with a vertical polarization. After interacting with the PBS again, its polarization turns into the horizontal direction so that it can interact with the mechanical oscillator again. The same process is repeated four times in total so that the canonical commutator is mapped onto the optical field as described in (3). Finally, the EOM is operated in such a way that it does not rotate the polarization and the light is taken out of the system. The output beam, thus, contains all the information of the OM interaction and, therefore, it must possess an additional phase Θ as described before in (11). This will be evident when it is tested with respect to the reference beam by any standard interferometer like Mach-Zehnder (MZ) interferometer.

Figure 1
figure 1

Experimental setup inspired by32 to probe the induced effects of polymer modified commutation relation in a macroscopic mechanical resonator.

Full size image

The EOM that we use here has to be very efficient. We design it in a way that it uses a particular type of electro-optic effect (Pockels effect) to modulate the phase of the incident beam of light. We use a refractive modulator (whose refractive index changes due to the application of electric signal that causes the modulation) that produces an output beam of light whose phase is modulated with the electric signal applied to the Beta Barium borate (BBO) electro-optic crystal. The modulation can be controlled by the source of electric signal according to our requirement.

A new tight constraint on the polymer deformation parameter μ 0

Let us now realize how the above scheme can be implemented in reality with the available resources in a standard optics lab. The above OM interaction can be described by the inter-cavity Hamiltonian H = ħω m n m  − ħg0n L X m , where n m is the mechanical number operator and \({g}_{0}={\omega }_{c}\sqrt{\hslash }/(L\sqrt{m{\omega }_{m}})\) denotes OM coupling rate with the mean cavity frequency ω c and mean cavity length L43. By assuming the optical pulses to be short enough, one can approximate the intra-cavity dynamics via the unitary operation \(U={e}^{i\lambda {n}_{L}{X}_{m}}\)44. Here \(\lambda \simeq {g}_{0}/\kappa =4 {\mathcal F} \sqrt{\hslash }/({\lambda }_{L}\sqrt{m{\omega }_{m}})\) refers to the effective interaction strength in which κ and λ L are the amplitude decay rate and the wavelength of optical pluses, respectively, with the cavity finesse \( {\mathcal F} \)1. In this framework, the OM phase (11) is modified as

$${\rm{\Theta }}=\frac{512{\mu }_{0}{\hslash }^{\mathrm{5/2}}{ {\mathcal F} }^{4}{N}_{p}^{3}}{3{M}_{{\rm{P}}}c{\lambda }_{L}^{4}{(m{\omega }_{m})}^{\mathrm{3/2}}}\mathrm{.}$$
(12)

In realistic scenarios, we face many different sources of background noise which may limit the ability to detect our measurable quantity in the laboratory. Thus, it is essential to study the signal to noise ratio (SNR) for our system, SNR = Θ/δΦ, where δΦ is the uncertainty in measuring the optical phase shift. For a proper experimental setup, it is required that SNR > 1. The inaccuracy of the measurement δΦ depends on the quantum noise of the outgoing optical pulse σout. In an ideal experiment with coherent state of light of mean photon number N p , the phase uncertainty turns out to be \(\delta {\rm{\Phi }}={\sigma }_{{\rm{out}}}/\sqrt{{N}_{p}{N}_{r}}\), where N r is the number of independent runs of the experiment. Therefore, the precision of the measurements are not limited and can be enhanced further by adjusting the strength of the optical field N p as well as the number of experimental runs N r , from which one can directly compute the resolution of δμ0. For every measurement, there exists a realistic parameter regime of μ0. For instance, let us to set the values ω m  = 2π × 105 Hz, m = 10−11 kg and \( {\mathcal F} ={10}^{5}\) which are the mechanical oscillator frequency, oscillator mass and optical cavity of finesse, respectively, with a wavelength of λ L  = 1064 nm, all of which are within the the range of current experiments45,46,47,48,49. Now by performing a single run of the experiment, i.e. N r  = 1, as well as by fixing the pulse sequence of mean photon number N p  = 108, within a relevant basic inaccuracy δΦ we reach at \(\delta {\mu }_{0} \sim {10}^{4}\) for the resolution of the dimensionless polymer modified parameter. Interestingly, by decreasing the mechanical oscillator frequency and oscillator mass to ω m  = 2π × 103 Hz, m = 10−13 kg, respectively, while other involving experimental parameters remain unchanged, one finds \(\delta {\mu }_{0} \sim 10\), which is very tighter than the formerly obtained results. Also by increasing the photon number as well as the number of measurement runs to N p  = 1010 and N r  = 102,104, 106, one acquires very impressive resolution of the order of magnitudes \(\delta {\mu }_{0} \sim {10}^{2},\,\mathrm{10,}\,1\), respectively. However, it should be noted that the above experimental parameters are strongly affected by disorders such as noise sources, mechanical damping, etc., which are subject to more technical discussion and, therefore, we refer the readers to the supplementary information provided in32. Some detailed experimental analysis has also been performed in the spirit of the underlying scheme in refs50,51,52, which provide a deeper understanding of the subject. Although at the first glance it seems to be challenging to achieve some values attributable to the above experimental parameters, however, as we showed with the resources available to us it is not that difficult. In fact, with some fine adjustments in our experimental setup and/or parameters, it may also be possible to reach an impressive resolution where the polymer length deformation may turn out to be of the order of Planck length. However, that requires a lot more sophistication.

Conclusions

We have explored a detailed and elegant procedure to understand the effects of the modified commutation relation thorough polymer quantization in laboratory. Our method utilizes an OM setup designed originally in1 that helps us to transfer the information of a polymer modified mechanical oscillator to the high intensity optical pulse in terms of a sequence of OM interaction inside an optical resonator. Consequently, we end up with an optical phase shift that is easily measurable with a very high accuracy through a MZ interferometric system. This makes the whole procedure much easier to collect the information of polymer deformations thorough an elegant optical system already available to us. Moreover, we obtained a new bound on the deformation parameter μ0, which may lead us towards an advanced understanding of the polymer quantization as well as the problem of quantum gravity. In what follows, we point out our contribution in comparison to the seminal work32 conducted on the framework of GUP models. While the ref.32 is in the framework of GUP, our study is based on polymer quantization scheme which is a relatively new and dynamic field towards the quantum gravity phenomenology. Although, our modified commutation relation (1), which addresses a maximum value for the momentum as p < M p /μ0 have also been studied in the context of GUP, however, they are not conceptually equivalent in QG ground. In fact, the origin of the fundamental length scale arising from the frameworks of polymer quantization and GUP are different, while GUP setups are suggested phenomenologically, our setup emerges naturally in the non-relativistic limit of the symmetric sector of LQG. Thus, one may argue that the polymer setup is more appealing at least in this respect. Furthermore, our contribution explicitly shows that the polymer length, like the fundamental length scale emerging from the GUP schemes, can be detected with a remarkable resolution as expected from the theory via some fine adjustments on the experimental parameters through the OM scheme proposed in32.