Decoherence effects in non-classicality tests of gravity

The experimental observation of a clear quantum signature of gravity is believed to be out of the grasp of current technology. However several recent promising proposals to test the possible existence of non-classical features of gravity seem to be accessible by the state-of-art table-top experiments. Among them, some aim at measuring the gravitationally induced entanglement between two masses which would be a distinct non-classical signature of gravity. We explicitly study, in two of these proposals, the effects of decoherence on the system's dynamics by monitoring the corresponding degree of entanglement. We identify the required experimental conditions necessary to perform successfully the experiments. In parallel, we account also for the possible effects of the Continuous Spontaneous Localization (CSL) model, which is the most known among the models of spontaneous wavefunction collapse. We find that any value of the parameters of the CSL model would completely hinder the generation of gravitationally induced entanglement.


I. INTRODUCTION
Testing the quantumness of gravity represents an outstanding challenge that has been approached from different perspectives [1]. Clearly the direct detection of the graviton -the quantum mediator of the gravitational interaction -would undeniably confirm the quantumness of gravity. However, the current state-of-art technology is not yet sufficiently advanced to allow for such a detection [2,3], and alternative paths need to be explored. Several low-energy table-top experiments were proposed over the years, which aim at testing effects resulting from either of a quantum [4][5][6][7][8] or a classical theory of gravity [9]. Among such proposals, we focus on two of them, one by Bose et al. [10] and Marletto-Vedral [11] (BM) and one by Krisnanda et al. [12], which seem to be within reach of the current technology. They rely on the generation of a gravitationally induced entanglement between two masses, which would work as a witness of the nonclassical nature of the gravitational mediator [10,11]. This new approach to witnessing non-classical features of gravity has been subject of intense study and debate [13][14][15][16][17][18][19][20][21][22][23][24], and is at the core of several recent experimental proposals [25][26][27][28][29]. Since the gravitational interaction is weak, one needs to employ large masses to achieve a measurable amount of entanglement in an experimentally reasonable time. However, larger masses are strongly affected by the environmental noises. Such effects, in particular decoherence, suppress quantum superpositions and thus hinder the entanglement generated by gravity. Therefore, it is crucial to consider explicitly decoherence effects in the dynamics of the system. Moreover, since these setups are very sensitive to any source of decoherence, one should account also for non-standard decoherence sources as those described by models of spontaneous wavefunction collapse. These models represent possible solutions to the quantum measurement problem. They modify the standard evolution due to quantum mechanics by introducing stochastic and non-linear terms in the Schrödinger equation [30,31]. While these models are still under active testing [32][33][34][35], it is instructive to account for their possible effects in these setups.
In this work, by building on the first analysis performed in [36,37] for the setup of Bose et al., we study the decoherence effects on the entanglement allegedly induced by non-classical gravity and identify the experimental conditions required to perform successfully the BM [10,11] and Krisnanda et al. [12] proposals. Moreover, we also quantify the effective decoherence due to the action of the Continuous Spontaneous Localization (CSL) model [38,39], which is the most studied among the models of spontaneous wavefunction collapse.

II. WITNESSING NON-CLASSICAL FEATURES OF GRAVITY
The proposals in Refs. [10,11] and that in Ref. [12] to test non-classical features of gravity are based on the generation of gravitationally induced entanglement. Indeed, a classical mediator cannot induce entanglement between two systems that are not directly interacting [40]. Therefore, the observation of gravitationally induced entanglement between two massive systems implies that the gravitational mediator displays non-classical features. We briefly review the two proposals.  [10] and Marletto-Vedral [11]. Two diamond particles with an embedded spin are initially prepared in the state (|↑ + |↓ )/ √ 2 and subsequently sent through two adjacent Stern-Gerlach interferometers. The particles are initially at a distance d and the size of the superpositions in the interferometer is ∆x. (b) Setup proposed by Krisnanda et al. [12]. Two osmium spheres of radius R and mass m are initially trapped in two harmonic potentials with frequency ω0 at a distance d. The spheres are initially cooled down close to the ground state of the harmonic potentials and the traps are subsequently removed, thus letting the masses freely interact. Here,xA andxB denote the displacement of the centres of mass of the two spheres from their initial equilibrium position.
perimental schemes, as for example that exploiting magnetic levitation [44], can be also considered. Two diamond particles of mass m and radius R embedded with a single spin 1/2 are initially prepared in the superposition state (|↑ + |↓ )/ √ 2 and sent simultaneously through two Stern-Gerlach interferometers as depicted in Fig. 1(a). After passing through an inhomogenous magnetic field, the spin superposition induces a position superposition for each of the particles, thus entangling the spin and the position. The composite state of the system thus becomes is the single particle state, and |L and |R are the localized states of the particles in the left and right branch of the interferometer, respectively. This assumption about the states of the particles has been shown to be a valid approximation for this setup [24]. Now gravity comes in play, mediating the interaction between the two masses. If gravity is to obey the quantum superposition principle, it induces a different phase for each of the parts of the superposition, which depends on the distance from the other branches of the superposition [10]. This interaction leads to an entangled state of the two masses [10,11].
An estimate of the gravity-induced phases can be obtained by assuming that the dominant effect can be computed via Newtonian interaction [10,11,13,17]; this approximation is valid also in the linearised quantum grav-ity model [1,45]. In particular, with reference to the setup represented in Fig. 1(a), we assume that the gravitational interaction is sufficiently weak to not modify the relative distances between the branches but at the same time strong enough to modify the corresponding relative phases. Moreover, we assume that all other interactions among the two particles are negligible and we account for the action of gravity only in the parallel branches of the interferometers. In such a way, the problem is significantly simplified. The corresponding driving Hamiltonian, in the {|i 1 ⊗ |j 2 } with i, j = L, R basis representation, reads where we defined with d denoting the distance between the center of mass of the two particles and ∆x the superposition distance. By assuming that the initial state is that in Eq. (1), the corresponding density matrix at time t reads: with The entanglement of the density matrix ρ is verified by applying the Peres-Horodecki criterion [46,47]. In particular, we calculate the eigenvaluesλ i of the partially transposed density matrix, which we explicitly derive in Appendix A. They read: where we defined the characteristic time According to the Peres-Horodecki criterion, the system is in an entangled state if at least one among the eigenvaluesλ i is negative. The condition is both necessary and sufficient for 2×2 dimensional systems as in our case. We notice thatλ − 2 is always negative ∀t = kπτ G with k ∈ Z. We quantify the corresponding entanglement by exploiting the logarithmic negativity E = log 2 ||ρ|| 1 , where ||·|| 1 is the trace norm [40]. This gives The entanglement of the system has a periodic dynamics and reaches its first maximum value at time t MAX = π 2 τ G . By inserting the values of the parameters chosen by Bose et al. [10], which are reported in Tab. I, we have t MAX 25 s. We remark that the value of d has been chosen by Bose et al. such that the Casimir-Polder interactions between the two spheres are 10 times smaller than the gravitational interaction. In this way, it is easier to discriminate the contribution of gravity to entanglement from that of these short-range forces. We note that a slightly different setup with mitigated Casimir-Polder interactions has been recently proposed in [36]. This setup proposes to insert a conducting plate between the two spheres and provides an improvement of one/two orders of magnitude on the mass and superposition size with respect to [10]. The corresponding analysis of decoherence acting on the experimental apparatus and that due to external acceleration noises were performed in [48] and [49] respectively, while the decoherence effects on the system were not quantified. In an attempt to simplify the treatment of the decoherence effects, here we will not consider this modified version of the setup, although our analysis can be applied also to this scenario.

B. Krisnanda et al. proposal
The second setup we consider was recently proposed by Krisnanda et al. in [12]. The setup is schematically depicted in Fig. 1(b) and consists of two osmium spheres of mass m and radius R that are initially trapped in harmonic potentials with frequency ω 0 separated by a distance d. The masses are assumed to be cooled down close to the ground state of their harmonic potentials, which is achievable with current technology [50,51]. The particles are subsequently released from the traps and are let free to interact gravitationally and entangle. Again, the Newtonian potential can be exploited to effectively describe such an interaction. The quantification of the degree of entanglement can be provided through well established continuous variables techniques applied to bipartite Gaussian states [52]. In particular, in such a case, the separability of the state can be simply verified through its 4 × 4 covariance matrix in position and momentum.
In the proposal of Krisnanda et al., one assumes that the two masses are initially prepared in Gaussian thermal states of the harmonic potentials. Moreover, one can approximate the Newtonian potential to second order in the quantum fluctuations (x A −x B ) of the relative distance d [12]: where the last term entangles the two masses [12,53]. Under this approximation, which is valid for |x A −x B | d according to the parameters reported in Tab. I and for a duration of the experiment of around 1 s [12], the dynamics is Gaussian. Krisnanda et al. derived the analytic expression for the logarithmic negativity of the system in the limit 2Gm ω 2 0 d 3 and √ 2Gmωt ω 0 d 3/2 , which reads [12]: where σ(t) = 4G 2 m 2 ω 2 0 t 6 /9d 6 . Using the parameters reported in Tab. I, after 1 s, the logarithmic negativity is 10 −2 , a value which can be detected with current technology [54].

III. DECOHERENCE EFFECTS
To measure the entanglement created by the quantum gravitational interaction one must ensure that environmental decoherence is sufficiently weak to not spoil such an effect. The main sources of decoherence are the scattering of residual gas and the scattering, emission and absorption of thermal photons. Here, we compute explicitly these effects on the dynamics of the system in the two setups above described, and their influence on the gravitationally-induced entanglement.

A. BM proposal
The effect of decoherence on the BM setup can be conveniently described in terms of a master equation, which in the position representation reads [55,56]: whereĤ is the free Hamiltonian, ρ(x, x , t) = x|ρ(t)|x and we consider the common ansatz [56] Such a master equation leads to an exponential suppression in time of the off-diagonal terms of the density matrix in the position representation. According to Eq. (12), decoherence is characterized by the localization strength Γ 0 and localization distance a, whose explicit forms are reported in Tab. II. In the proposals under consideration, there are only four possible position configurations, thus the problem is strongly simplified to a discrete description. Moreover, we assume that the decoherence acts independently on the two masses. By starting from the common initial state defined in Eq. (1), the density matrix ρ of the system at time t reads: where Γ = Γ air (∆x)+Γ ph,sc (∆x)+Γ ph,abs (∆x)+Γ ph,em (∆x) is the sum of the effects due to the collisions of the residual air molecules and the scattering and absorption and emission of thermal photons. The entanglement is then quantified as in the free case via the Peres-Horodecki criterion [46,47]. While the entire derivation is reported in Appendix A, here we show the obtained eigenvaluesλ i of the partially transposed density matrix: where τ G is defined in Eq. (7). We remind that entanglement can be achieved only for negative values of one of the above eigenvalues. Since cosh Γt ≥ 1 and sinh Γt ≥ 0, negative values can be achieved only byλ − 2 . This happens when  (12) which quantify the decoherence effects induced by collisions with air molecules (Air) and scattering (Sc), absorption (Ab) and emission (Em) of thermal photons on a sphere of radius R, dielectric constant and bulk temperature Ti [55,56]. Here, mair denotes the mass of the molecules of the residual air, T and P are respectively the temperature and the pressure at which the experiment is performed, and ζ(n) is the Riemann zeta function. To quantify the effects, one has = 5.7 + i × 10 −4 [57] for diamond used in the BM proposal, while = 0.6 + i × 2.5 [58] for osmium used in the Krisnanda proposal. Moreover, for the sake of simplicity, we take Ti = T and mair ∼ 6.6 × 10 −27 kg corresponding to a atom of helium. or, equivalently, when where τ C = 1/Γ is the coherence time of the system [55]. The corresponding logarithmic negativity reads: In the upper panel of Fig. 2, we reportλ − 2 for different values of time and of Γ. In the bottom panel, we report the logarithmic negativity of the system for different values of Γ. In Fig. 3, we have plotted the environmental conditions of temperature and pressure required to reach different amounts of entanglement with the parameters of Tab. I. Given these parameters, we observe in Fig. 2 that for Γ greater than 1/τ G 0.06 s −1 the system is separable at all times. This corresponds to pressures and temperatures highlighted by the blue line named "Min" in Fig. 3. Moreover, we notice that one needs temperatures lower than 4 K and residual air pressures lower than  17) at timet. The respective shaded areas correspond to values of pressure and temperature that prevent observing the targeted value of E within timet. The black line identified with Min denotes the minimum experimental conditions required to generate entanglement. We note that E = 10 −2 is compatible with recent experimental observations [54]. The residual gas is assumed to be helium. In the setup of Krisnanda et al., the effect of environmental decoherence on the system can be described by the Heisenberg-Langevin equations for the positionx j and momentump j operators of the two masses. In one dimension, these equations read [59]: whereĤ is the free Hamiltonian of the system, γ ε quantifies the dissipation of the environment and it is related to Γ 0 through [60] with Λ = Γ 0 /4a 2 , m denoting the mass of the particle and T the temperature of the environment.ξ j (t) is an environmental noise operator, which can be described in terms of its mean ξ j (t) ε = 0, where · ε denotes the average over the environmental degrees of freedom, and the two-time correlation function. In particular, the latter can be strongly simplified in the Markovian regime which can be achieved in the limit of high temperatures of the environment: k B T ω. In such a case, it reads [59,60]: where δ(·) is the Dirac delta. Using the parameters considered by Krisnanda et al., which are reported in Tab. I, we see that the high-temperature limit holds for T 10 −6 K. We now apply Eq. (18) to the setup under consideration. By denoting with ω 0 the frequency of the harmonic potentials where the two particles are initially trapped, the Langevin equations for the adimensional quadratureŝ where we defined γ ε = γ air + γ ph,sc + γ ph,abs + γ ph,em and ξ j = ξair j +ξ ph,sc j +ξ ph,abs j +ξ ph,em j / √ mω 0 , and introduced the parameters η = 2Gm ω 2 0 d 3 , and k = which characterize the strength of the gravitational attraction [12]. We notice that, since the separation between the two spheres is much greater than the localization distances of the various processes, the noises acting on the two spheres can be safely considered as independent. Therefore, the two-time correlation function becomes where Λ = Λ air + Λ ph,sc + Λ ph,abs + Λ ph,em . The procedure that we use to calculate the logarithmic negativity of the system in this setup is reported thoroughly in Appendinx B and is here briefly outlined. First of all, we use the Langevin equations (21) to compute the covariance matrix σ for the quadraturesX j andP j . Using this matrix we can easily find the covariance matrix σ of the partially transposed state of the systemρ. The entanglement of the system can be calculated by using the so-called symplectic eigenvaluesν i ofσ. For bipartite Gaussian states, the minimumν min of these eigenvalues is smaller than 1/2 if and only if the system is in an entangled state. Finally, the logarithmic negativity of the system can be expressed as [52] E K-dec = −log 2 (2ν min ). (24) The explicit expression forν min is cumbersome and we report it in Appendix B. Here, we report its behaviour for small times: Comparing the term 2 Λ/mω 0 with the value of γ ε given in Eq. (19), we notice that γ ε 2 Λ/mω 0 when T ω/k B ≈ 10 −6 K. If we work in this limit, we can safely neglect γ ε in Eq. (25) and we have thatν min < 1/2, thus indicating that the system is in an entangled state, when 2 Λ/mω 0 < ηω 0 . Notably, this latter condition can be rewritten as which coincides with the condition in Eq. (15) in the limit of ∆x d once one considers the short-wavelength approximation in Eq. (12), i.e. Γ Λ∆x 2 which is valid for ∆x a. Thus, both Eq. (15) and Eq. (26) can be synthesised as Eq. (16): to generate gravitational induced entanglement, the characteristic time of entanglement creation must be shorter than the decoherence time characterizing the interaction of the system with its surrounding environment.
For later times, the behaviour of the entanglement can be seen in the upper panel of Fig. 4 where we reportν min for different values of time and of Λ, with γ ε = 0. A numerical analysis shows that the results do not change appreciably for values of γ ε up to 10 −5 s −1 , which correspond to pressures around 10 −2 Pa and temperatures around 500 K. In the bottom panel of Fig. 4 we show the time evolution of the logarithmic negativity E K-dec of the system for different values of Λ. Finally, in Fig. 5 we show the environmental conditions required to obtain a specific amount of entanglement at fixed times with the parameters of Tab. I. We notice that one needs temperatures lower than 9 K and residual air pressures around 10 −16 Pa to have entanglement. With these pressures and temperatures, considerable amounts of entanglement are reached in about 10 s.

IV. CSL MODEL
In addition to environmental decoherence, we study also the possible effects of the Continuous Spontaneous Localization (CSL) model on the two setups. This is the most studied among collapse models [30,31]. The CSL model proposes a non-linear and stochastic modification of the Schrödinger equation. This modification induces the collapse of the wavefunction for macroscopic systems while it leaves the dynamics of microscopic systems almost unaffected [30,31]. The collapse of the wavefunction causes the suppression of quantum superpositions and thus of any non-classical correlation , such as entanglement. CSL is quantified in terms of two parameters λ and r C , which describe respectively the frequency of the collapse and its spatial resolution. Theoretical proposals include [38,39,61,62]: r C = 10 −7 m, and λ = (10 −17 ÷ 10 −9 ) s −1 .
Larger values of λ are experimentally excluded. For further details we refer to [34] and references therein. We now apply CSL to the two setups we are considering. Min denotes the minimum experimental conditions required to have any amount of entanglement in the system. The residual gas is assumed to be helium.

A. BM proposal
Since, in the BM proposal, the distance d between the two masses is greater than r C [cf. Tab. I], one can safely neglect possible correlations of the noise and describe the CSL effect by simply using Eq. (11). According to CSL, a = r C , while Γ 0 depends on the geometry of the system. In particular, for a homogeneous sphere of mass m and radius R, Γ 0 reads [63]: Consequently, by exploiting the relation in Eq. (15), one finds that gravitationally induced entanglement can be generated only for λ ≤ 10 −24 s −1 at r C = 10 −7 m. This value is 7 orders of magnitude smaller than the theoretical estimates on the lower bound on λ. Therefore we conclude that any value of the CSL parameters proposed in the literature would prevent the creation of entanglement in the BM setup.

B. Krisnanda et al. proposal
For the setup of Krisnanda et al., we quantify the effect of the CSL model using the Langevin equations (18), where we substitute the second equation with [63,64]: where ξ CSL (t) is a real-valued white noise with zero mean and correlation function reading [65] ξ CSL with · denoting the average over the realizations of the noise, and Λ CSL = Γ CSL 0 /4r 2 C . Since the distance between the two spheres is much larger than r C , the CSL noise acts independently on the two masses. Accordingly to CSL, there is no dissipation. Therefore, Eq. (26), where Λ is substituted by Λ CSL , holds true for small times and defines the regime where gravitationally induced entanglement can be generated. By using the values of the parameters in Tab. I, the latter condition is satisfied for λ ≤ 10 −23 s −1 at r C = 10 −7 m, which is 6 orders of magnitude smaller than the lower value in Eq. (27). Therefore, the same conclusion as for the BM proposal holds: the presence of the CSL noise would prevent the creation of entanglement in the setup.

V. CONCLUSIONS
In this paper, we have studied decoherence effects on the gravitationally induced entanglement in the setups of BM [10,11] and Krisnanda et al. [12]. We have found an analytic expression for the entanglement when decoherence is explicitly considered in the dynamics of the system. Moreover, we have determined the maximum strength of decoherence, quantified by the expression in Eq. (16), which allows for gravitationally-induced entanglement generation in these setups. Such an expression also provides the ground to set the requirements of temperature and pressure necessary to perform the experiment.
The numerical analysis has shown that temperatures and pressures as low as 1 K and 10 −16 Pa are sufficient  [66]. However, the time-scale involved requires a free-falling particle to fall for 10 cm and 6.2 m respectively for the BM and the Krisnanda proposals. Clearly, maintaining the above conditions of temperature and pressure can be technically challenging. As pointed out in [10], milder environmental conditions such as P = 10 −15 Pa would lead to a decoherence time of the order of a few seconds, which is comparable to the time of the experiment. However, such conditions would invalidate the BM proposal, while the Krisnanda proposal would work only for lower temperatures ≤ 10 mK. Moreover the two particles should remain aligned during the experiments, one should also prepare the system without any horizontal and vertical relative velocity, which, over time, would change the relative distance thus potentially disrupting the experiments. Table III compares the free-fall times and corresponding heights necessary for the two proposals at different environmental conditions. While entanglement could be difficult to generate, one could rely on other non-classical correlations such as discord [6], but this was beyond the aim of the current work.
In addition to environmental decoherence, we have also considered the effect of the CSL model on the two setups. We have shown that any proposed value of the CSL parameters would prevent the creation of entanglement in the two setups. We note that this was previously noted by Bose et al. [10] for the BM proposal. Conversely, if gravitationally induced entanglement were observed in such setups, it would rule out the CSL model by 6 ∼ 7 orders of magnitude. Such an achievement would improve the current bounds [34] by more than 13 orders of magnitude, showcasing the necessity of a strong improvement of the state-of-art technology to successfully perform these experiments. On the contrary, one expects no effects from a stochastic gravitational wave background [67].
Finally, given the generality of the approaches used to consider explicitly the decoherence effects in the dynamics of the system, the results that we have obtained can be easily generalized to similar setups [25,26] or to different decohering mechanisms, such as gravitational decoherence [68].
Eq. (2), reads: Let us consider the partial transposition of ρ with respect to the second system: The eigenvalues ofρ can be easily evaluated and are those reported in Eq. (6). b. Case with decoherence. When the system is subject to decoherence, the density matrix of the system at time t can be calculated using Eq. (11): (A4) Again, if we consider the partial transposition of ρ with respect to the second system we get: whose correspondent eigenvalues are given by Eq. (14). In order to calculate the logarithmic negativity of the system, we first find its covariance matrix. This can be done in the following way. First, if we definê in terms of which we can rewrite Eqs. (21) in a more compact way: where we introduced The solution of Eq. (B2) reads: where W ± = e ±Kt . This expression can be used to calculate the covariance matrix of the system. Its elements are defined as: where · denotes the average over the initial state and over the environmental degrees of freedom if the case with decoherence is considered. Using Eq. (B4), we can calculate σ(t) as: where W ± denotes the transposed of W ± and D = diag(0, µω 0 , 0, µω 0 ), with µ = 2 Λ/mω 2 0 . Now, if we divide σ(t) in the following 2 × 2 submatrices: we can calculate the symplectic eigenvalues of the covariance matrix of the partially transposed state in the following way [52]: where we defined ∆(σ) = Detα + Detβ − 2Detγ.
The smallest of these symplectic eigenvalues is related to the logarithmic negativity of the system via E = − log 2 (2ν min ).