QQuantum detection of wormholes
Carlos Sab´ın Instituto de F´ısica Fundamental, CSIC, Serrano 113-bis 28006 Madrid, Spain ∗ We show how to use quantum metrology to detect a wormhole. A coherent state of the electromagnetic fieldexperiences a phase shift with a slight dependence on the throat radius of a possible distant wormhole. We showthat this tiny correction is, in principle, detectable by homodyne measurements after long propagation lengthsfor a wide range of throat radii and distances to the wormhole, even if the detection takes place very far awayfrom the throat, where the spacetime is very close to a flat geometry. We use realistic parameters from state-of-the-art long-baseline laser interferometry, both Earth-based and space-borne. The scheme is, in principle, robustto optical losses and initial mixedness. ∗ Correspondence to: csl@i ff .csic.es a r X i v : . [ qu a n t - ph ] A p r l = q r b l = q r b ✓ ( b ) l = 0 ; r = b FIG. 1. Sketch of the idea. A coherent single-mode state of a laser propagates along a large distance l − l , e. g. between two satellites. Ifthe spacetime is totally flat the distance is just r − r . However, if the spacetime contains a distant wormhole at l = b ,the traveled distance will be slightly di ff erent even if the spacetime is almost completely flat in that region, namely r , r >> b . This smallcorrection of the distance generates a slight phase shift, making the phase θ of the coherent state dependent on b . In principle, b could beestimated by means of homodyne detection. Note that the length scale of the figure is not realistic. We have not observed any wormhole in our Universe, although observational-based bounds on their abundance have beenestablished [1]. The motivation of the search of these objects is twofold. On one hand, the theoretical implications of theexistence of topological spacetime shortcuts would entail a challenge to our understanding of deep physical principles such ascausality [2–5]. On the other hand, typical phenomena attributed to black holes can be mimicked by wormholes. Therefore,if wormholes exist the identity of the objects in the center of the galaxies might be questioned [6] as well as the origin of thealready observed gravitational waves [7, 8]. For these reasons, there is a renewed interest in the characterization of wormholes[9–11] and in their detection by classical means such as gravitational lensing [12, 13], among others [14].Quantum metrology aims at providing enhancements to the measurements realised by classical means, by exploiting quantumproperties such as squeezing and entanglement. This approach has already proven useful in a wide range of physical problems,from timekeeping [15] to gravitational wave astronomy [16]. While the successful observations of gravitational waves byadvanced LIGO still did not leverage the benefits of squeezing, the technology is ready to be implemented in further upgrades,which is expected to significantly increase the detectors sensitivity [17]. However, advanced LIGO has already been able todetect the e ff ect of tiny spacetime oscillations on the phase of a coherent state of a laser containing a large number of photons.In the near future, several big scientific projects will be launched to space, including quantum metrology experiments and along-baseline laser interferometer for gravitational wave detection – LISA [18]. Inspired by these impressive technologicaldevelopments, it is natural to ask whether it is possible to use similar technology for the detection of another deviations from aflat-spacetime geometry.In this work, we propose to use a quantum measurement scheme for the detection and characterization of wormholes (seeFig.1. We show that a single-mode state of the electromagnetic field propagating in space will undergo a phase shift whichwould carry a slight dependence on the radius of the throat of a distant wormhole, in the case that the spacetime contains one ofthem. While this is of course a tiny correction with respect to the flat-spacetime case, we show that after large propagation lengthsa good sensitivity is, in principle, within experimental reach by using parameters of modern long-baseline laser interferometers,even if propagation takes place very far away from the wormhole – in a quasiflat spacetime geometry. We consider realisticinitial states and measurement protocols,finding that coherent states with large number of photons and homodyne detectionrespectively, are convenient choices. Indeed, for certain values of the phase shift, the homodyne measurement scheme achievesthe ultimate quantum bound in the case of a coherent state. We show that the sensitivity is proportional to the ratio betweenthe radius of the wormhole throat and the distance to the wormhole – which is assumed to be very large – providing a wideparameter window where the scheme is applicable. Furthermore, our results are, in principle, highly robust to the presence ofoptical losses and initial mixedness. Our aim is, of course, not to propose a detailed experimental setup, but to show a new ideathat is, in principle, possible and might inspire further investigations, which would ascertain its actual feasibility. METHODS
We start by considering a single-mode state of the electromagnetic field. By now, we will assume that the initial state is pureand we will discuss later the e ff ects of temperature. Therefore, the initial state is characterized by: | φ (cid:105) = D ( α ) S ( r ) | (cid:105) , (1)where S ( r ) = exp[( r / a − a † )] is a squeezing operator with real squeezing parameter r and D ( α ) = exp (cid:16) α a † − α ∗ a (cid:17) is adisplacement operator with parameter α . Here, a and a † are the standard annihilation and creation operators of the single mode.Now we consider that the initial state in Eq. (1) undergoes unitary evolution characterized by the operator U ( θ ) = exp (cid:16) − i θ a † a (cid:17) ,which depends on the phase θ . Thus the resulting state is | φ θ (cid:105) = U ( θ ) D ( α ) S ( r ) | (cid:105) , (2)which is now θ -dependent. The phase shift θ can be estimated by realising measurements on the state | φ θ (cid:105) . The ultimate quantumbound on the sensitivity ∆ θ of the state with respect to the parameter θ is given by the quantum Cramer-Rao bound [19]: ∆ θ ≥ √ H ( θ ) , (3)where H ( θ ) is the quantum Fisher information (QFI) of the state. Therefore, maximizing the QFI amounts to find the optimalbound for a given state. Indeed, it is well-known that it always exists an optimal measurement strategy which saturates theinequality in Eq. (3). In other words, the classical Fisher information (FI) of the optimal measurement matches the QFI.However, the optimal measurement might not be experimentally feasible in general.For the single-mode pure state | φ θ (cid:105) the QFI is given by the variance of the generator of the phase, in this case the numberoperator [20, 21]: H ( θ ) = (cid:104) ∆ a † a (cid:105) = (4) = (cid:104) | α | (cosh r − sinh r ) + r cosh r (cid:105) . The average number of photons of the state is (cid:104) n (cid:105) = (cid:104) a † a (cid:105) = | α | + sinh r . (5)It is well-known [20, 21] that for a fixed number of average photons the choice which maximizes the QFI is the squeezed vacuum α =
0, which attains a (cid:104) n (cid:105) -scaling – Heisenberg limit – in contrast with the (cid:104) n (cid:105) -scaling of the coherent state r = r = H ( θ ) = | α | = (cid:104) n (cid:105) . (6)Notice that the choice of the coherent state as a probe does not prevent us from using the benefits of squeezing or entanglementat a later stage of the measurement protocol. This is the same approach as in GEO 600 –which will soon be applied as wellin advanced LIGO–, where a coherent state with a large number of photons is then mixed with a squeezed vacuum in a beamsplitter. Indeed, this strategy has been shown to be very close to the optimal one – which involves an entangled NOON state oflarge N– in the experimentally relevant regime of optical losses and large number of photons [22]. Furthermore, in the case ofcoherent states we know that the experimentally standard homodyne measurements are optimal. Indeed, the FI F ( θ ) associatedto the homodyne measurement of the quadrature | p (cid:105) is [23]: F ( θ ) = | α | cos ( θ ) . (7)Therefore, homodyne measurements saturate the Cramer-Rao bound and provide the optimal sensitivity for particular values ofthe phase θ = m π, m = , , , ... . In the following, we will restrict ourselves to coherent states, homodyne measurements and θ = m π .If the coherent state propagates between two points separated by a distance L in flat spacetime, then the phase shift is simply θ f = ω t = ω L / c . Therefore, we can choose L in order to meet the condition ω L / c = m π , from which we can finally write: m = L λ , (8)where λ is the wavelength of the mode. Then, finally the phase shift in flat spacetime is: θ f = π L λ ; L λ = , , . . . . (9)Let us now consider that instead of a flat spacetime, the field is propagating along the radial direction of a spacetime containinga traversable massless wormhole – e.g. the paradigmatic Ellis wormhole [3, 24]. Thus, the spacetime geometry is given by ds = − c dt + − b r dr , (10)where we are not considering the angular part of the metric. There is a singular point of r at which r = b , which determines theradius of the wormhole’s throat and defines two di ff erent Universes or two asymptotically flat regions within the same Universe-as r goes from ∞ to b and then back from b to ∞ . RESULTS
In the wormhole spacetime, a free-falling observer possesses a spatial coordinate such that l = ± (cid:113) r − b . In the coordinates l , t light will describe the trajectory | l − l | = c t . Therefore, in the laboratory coordinates we will have that the propagationlength is di ff erent: L (cid:48) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:113) r − b − (cid:113) r − b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (11)The wavelength λ of the mode will be modified as well. Let us consider that the propagation takes place between two points r , r in the same branch of the Universe and very far from the wormhole throat r , r >> b . We assume that the separation L is small as compared to either point L << r , r r > r
1, thus L = r − r . Finally, thewavelength λ is negligible with respect to the separation L , L >> λ . Under these approximations and taking into account Eq. (9)the phase shift in the wormhole spacetime is θ = θ f − b r Lr , (12)where we see that the second term in the parenthesis represents a small correction to the flat spacetime phase shift θ f .Notice that under all the above approximations Eq. (10) reduces to a quasiflat spacetime given by the metric: g µν = η µν + (cid:99) g µν (13)where η µν is the Minkowski metric and we only have a very small correction (cid:99) g µν , where the only non-zero element is (cid:99) g rr = b / r <<
1. In other words, we are considering that the propagation of the coherent state takes place very far away from thewormhole, so that the spacetime is almost completely flat and the e ff ect of the wormhole would be just a tiny perturbation. Inthis sense, the scenario resembles gravitational wave detection, where a di ff erent small perturbation of Minkowski spacetime isdetected.In order to find out whether this small correction of the phase shift is in principle detectable, we use that, by construction, H ( θ ) –or equivalently, F ( θ ) of the homodyne measurement scheme, which is equal to the QFI for the values of the phase that weare discussing, as mentioned above– can be related to H ( b ) through H ( b ) = (cid:12)(cid:12)(cid:12)(cid:12) ∂θ∂ b (cid:12)(cid:12)(cid:12)(cid:12) H ( θ ), since both H ( x ) and F ( x ) are obtainedthrough partial derivatives with respect to x [19] Then, using Eqs. (3), (6) and (12) we find the following expression for thesensitivity relative to the value of b : ∆ b b = λ π L r b r L √(cid:104) n (cid:105) . (14)If instead of using the QFI, we use the FI in Eq. (7) we obtain: ∆ b b = λ π L r b r L √(cid:104) n (cid:105) cos θ , (15)where θ is given by Eq. (12) and thus cos θ is very close to 1.We can consider a laser of λ = nm and a few W of power, which amounts to an average number of photons of (cid:104) n (cid:105) (cid:39) per second, as in LIGO [22]. Using the average number of photons per second in Eqs. (14) and (15) gives rise to figures of meritwith units of Hz − / , as is standard in laser interferometry. For the length L , we can consider the arm length of long-baselineinteferometers, ranging from a few km –LIGO– to a few million km –LISA. The order of magnitude of b is unknown andthe only limitation in the current approach is the sensible approximation b << r , which implies that we stay in a quasiflatspacetime su ffi ciently far from the wormhole throat.In Fig. 2 we see that in the regime of lengths of the order of LISA’s interferometer arms, the QFI and the FI are practicallyindistinguishable and we are able to achieve a sensitivity which is significantly smaller than b for distances r = b and L = b . Note that for these values of L , then b is of the order of m , and r is of the order of 10 µ pc. The detection bygravitational lensing e ff ects requires a much larger value of the throat radius 0 . − pc [1]. Considering this range of values,our Fig. (2) would correspond to r = − pc. Note that the estimated distance between the Earth and the rotational centerof the Milky Way is around 10 pc [25]. Indeed, the sensitivity would be much better if we consider smaller values of r / b ,always respecting r >> b , L and taking into account that it does not seem realistic to assume that r is close to the detectionpoint. If we consider that the maximum error that we want to tolerate corresponds to ∆ b / b = . − / , then with the valuesof Fig. 2 we can tolerate distances up to r / b = with r = L which amounts to a minimum throat radius of b (cid:39)
10 cm.These values for b are significantly smaller than the ones detectable by gravitational lensing. FIG. 2. Figure of merit of the sensitivity of our scheme vs. average number of photons per second for λ = nm, r / b = , r / L = and two di ff erent values of L in the range of millions of km. In both cases we plot both the optimal bound provided by the QFI and the FI ofhomodyne detection, which turn out to be practically indistinguishable. In Fig. (3) we consider much smaller lengths L (cid:39) m, as the interferometer arms in LIGO. This amounts to a restrictionof the maximum allowed values of the parameter r / b , if we keep the same values for r – thus r / L is larger than in Fig. (2).We consider 10 in the figure and we could go up to 10 given the threshold for the error established above. This increases theminimum values of the throat radius that we could consider for a fixed large distance, as compared to the scenario in Fig. (2). FIG. 3. Figure of merit of the sensitivity of our scheme vs. average number of photons per second for λ = nm, r / b = , r / L = and two di ff erent values of L in the range of km. DISCUSSION
Let us discuss the e ff ect of some possible deviations from the theoretical scenario described so far. Since we are within thestandard scheme of phase estimation with coherent states and homodyne detectors, we can take advantage of some results in theliterature. For instance, the presence of optical losses would a ff ect the sensitivity by multiplying the FI by a factor η [26], where η is a number between 0 and 1 accounting for the optical e ffi ciency. In modern laser interferometers the value of η = .
62 [22]is achieved, which implies that optical losses are 38%. We can also consider that the initial state is not the pure state resultingfrom the action of the displacement operator onto the vacuum, but onto a mixed state characterized by a thermal distribution withaverage number of excitations n T . Although for the optical frequencies that we are considering the actual number of thermalphotons would be negligible even at room temperature on Earth, it can be useful to consider a relatively high n T in order to seethe impact of any possible initial mixedness. The FI in this case would be reduced by a factor (1 + n T ) − [23]. Putting alltogether, the expected sensitivity in the presence of optical losses and initial mixedness would be given by: ∆ b b = η + n T λ π L r b r L √(cid:104) n (cid:105) cos θ . (16)In Fig. (4) we see the robustness of our scheme in the presence of realistic optical losses and relatively high initial mixedness.Further analysis of error sources would strongly depend on the particular details of the experimental setup and lies beyond thescope of this work. We can anticipate however, that an eventual experimental test would be of course extremely challenging,e.g. an extremely careful calibration of any gravity gradient noise would be required, as in LIGO. However, notice that LIGOhas been able to detect a phase sift of approximately 10-10 rad. Using Eqs. (9) and (12) and the values of L and λ consideredin this work, we find that in our case this means π b r (cid:39) − m − for L = π b r (cid:39) − m − for L = km. While, inprinciple, there is no restriction on the value of b ,we can focus on a case of interest, such as a ”black-hole mimicker” [7, 8].Then b would be slightly larger than the Schwarzschild radius of the black hole, which in the case of the first LIGO detectionwould be around 200 km. Putting all the numbers together, we find that detection would be possible at a distance ranging from r (cid:39) µ Pc (for L = r (cid:39) . L = km). Along these lines, it might be useful to recall that a quantum FIG. 4. Figure of merit of the sensitivity of our scheme vs. average number of photons per second in the presence of optical losses and initialmixedness for λ = nm, r / b = , r / L = and L = m. We see that the combined e ff ect of optical loss and mixedness has amoderate impact on the sensitivity. simulator of a wormhole spacetime for the electromagnetic field in the GHz regime has been recently proposed [27], whichcould be an important low-cost Earth-based source of information for the actual –presumably space-borne– experiments.In summary, we have shown that it is, in principle, possible to detect a distant wormhole in our spacetime by measuring thecorrection to the phase shift of an electromagnetic field propagating over large distances. We have considered a standard scenariowith a single-mode coherent state with a large average number of photons and homodyne detection, as well as state-of-the-artnumbers in modern laser interferometry. The relevant parameters turn out to be the ratio between the radius of the wormholethroat b and the distance to the wormhole r , together with the ratio between the propagation distance L and r . While theseratios cannot be large –since r >> b , L – we take advantage of the large average number of photons per second and the largevalue of L /λ . We show that detection is in principle possible for r / b up to 10 − , which allows us to consider a wide rangeof throat radii and distances. The scheme is in principle highly robust to optical losses and initial mixedness. While of coursewe are aware that an actual experimental test based on these ideas would be extremely challenging, we hope that the results ofthis work are promising enough to open a new avenue of research on the detection of one of the most fascinating objects thatmight –or might not– exist in Nature. ACKNOWLEDGEMENTS.
Financial support by Fundaci´on General CSIC (Programa ComFuturo) is acknowledged by C.S as well as additional supportfrom Spanish MINECO / FEDER FIS2015-70856-P and CAM PRICYT Project QUITEMAD + S2013 / ICE-2801.
ADDITIONAL INFORMATIONCompeting financial interests
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