ARTICLESPUBLISHED ONLINE: 18 MARCH 2012 | DOI: 10.1038/NPHYS2262

Probing Planck-scale physics with quantum opticsIgor Pikovski1,2*, Michael R. Vanner1,2, Markus Aspelmeyer1,2, M. S. Kim3* and Caslav Brukner2,4

One of the main challenges in physics today is to merge quantum theory and the theory of general relativity into a unifiedframework. Researchers are developing various approaches towards such a theory of quantum gravity, but a major hindranceis the lack of experimental evidence of quantum gravitational effects. Yet, the quantization of spacetime itself can haveexperimental implications: the existence of a minimal length scale is widely expected to result in a modification of theHeisenberg uncertainty relation. Here we introduce a scheme to experimentally test this conjecture by probing directly thecanonical commutation relation of the centre-of-mass mode of a mechanical oscillator with a mass close to the Planck mass.Our protocol uses quantum optical control and readout of the mechanical system to probe possible deviations from the quantumcommutation relation even at the Planck scale. We show that the scheme is within reach of current technology. It thus opens afeasible route for table-top experiments to explore possible quantum gravitational phenomena.

It is at present an open question whether our underlying conceptsof space–time are fully compatible with those of quantummechanics. The ongoing search for a quantum theory of gravity

is therefore one of the main challenges in modern physics. Amajor difficulty in the development of such theories is the lackof experimentally accessible phenomena that could shed light onthe possible route for quantum gravity. Such phenomena areexpected to become relevant near the Planck scale, that is, atenergies on the order of the Planck energy EP = 1.2× 1019 GeVor at length scales near the Planck length LP = 1.6× 10−35 m,where space–time itself is assumed to be quantized. However,such a minimal length scale is not a feature of quantum theory.The Heisenberg uncertainty relation, one of the cornerstones ofquantummechanics1, states that the position x and themomentump of an object cannot be simultaneously known to arbitraryprecision. Specifically, the indeterminacies of a joint measurementof these canonical observables are always bound by 1x1p≥ h/2.Yet, the uncertainty principle still allows for an arbitrarily precisemeasurement of only one of the two observables, say position,at the cost of our knowledge about the other (momentum). Instark contrast, in many proposals for quantum gravity the Plancklength constitutes a fundamental bound below which positioncannot be defined. It has therefore been suggested that theuncertainty relation should be modified to take into account suchquantum gravitational effects2. In fact, the concept of a generalizeduncertainty principle is found in many approaches to quantumgravity, for example in string theory3,4, in the theory of doublyspecial relativity5,6, within the principle of relative locality7 andin studies of black holes8–10. A generalized uncertainty relationalso follows from a deformation of the underlying canonicalcommutator [x,p] ≡ xp− px (refs 11–15), as they are related via1x1p≥ (1/2)|〈[x,p]〉|.

Preparing and probing quantum states at the Planck scaleis beyond today’s experimental possibilities. Current approachesto test quantum gravitational effects mainly focus on high-energy scattering experiments, which operate still 15 orders ofmagnitude away from the Planck energy EP, or on astronomicalobservations16,17, which have not found any evidence of quantum

1Vienna Center for Quantum Science and Technology (VCQ), Boltzmanngasse 5, A-1090 Vienna, Austria, 2Faculty of Physics, University of Vienna,Boltzmanngasse 5, A-1090 Vienna, Austria, 3Quantum Optics and Laser Science (QOLS) group, Blackett Laboratory, Imperial College London, SW7 2BW,UK, 4Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria.*e-mail: igor.pikovski@univie.ac.at; m.kim@imperial.ac.uk.

gravitational effects as of yet18,19. Another route would be toperform high-sensitivity measurements of the uncertainty relation,as any deviations from standard quantum mechanics are, at leastin principle, experimentally testable13–15. However, with the bestposition measurements being of order1x/Lp∼1017 (refs 20,21), atpresent sensitivities are still insufficient and quantum gravitationalcorrections remain unexplored.

Here we propose a scheme that circumvents these limitations.Our scheme allows one to test quantum gravitational modificationsof the canonical commutator in a novel parameter regime, therebyreaching a hitherto unprecedented sensitivity in measuring Planck-scale deformations. The main idea is to use a quantum opticalancillary system that provides a directmeasurement of the canonicalcommutator of the centre of mass of a massive object. In this wayPlanck-scale accuracy of position measurements is not required.Specifically, the commutator of a very massive quantum oscillatoris probed by a sequence of interactions with a strong optical fieldin an optomechanical setting, which uses radiation pressure insidean optical cavity22,23. The sequence of optomechanical interactionsis used to map the commutator of the mechanical resonatoronto the optical pulse. The optical field experiences a measurablechange that depends on the commutator of the mechanicalsystem and that is nonlinearly enhanced by the optical intensity.Observing possible commutator deformations thus reduces to ameasurement of the mean of the optical field, which can beperformed with very high accuracy by optical interferometrictechniques. We show that, already with state-of-the art technology,tests of Planck-scale deformations of the commutator are withinexperimental reach.

Modified commutation relationsA commonmodification of theHeisenberg uncertainty relation thatappears in a vast range of approaches to quantum gravity2–4,24,25is 1x1p ≥ h(1 + β0(1p/(MPc))2)/2. Here, β0 is a numericalparameter that quantifies the modification strength, c is the speedof light andMP'22 µg is the Planckmass. Theminimalmeasurablelength scale appears as a natural consequence with 1xmin= LP

√β0

(Fig. 1). Such a modification alters the allowed state-space and can

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ARTICLES NATURE PHYSICS DOI: 10.1038/NPHYS2262

Forbidden byδ δquantum gravity

proposalshx p

2

Δp

Δxmin = LPΔx1017LP

MPc

ΔxΔp = h 2

ΔxΔp = Δp 2 01 + h 2 M2

Pc2

Δxmin

Δx

β

0β

0β

Figure 1 | The quantum uncertainty relation and a quantum gravitationalmodification. The minimum Heisenberg uncertainty (red curve) is plottedtogether with a modified uncertainty relation (dashed blue curve) withmodification strength β0. MP and LP are the Planck mass and Planck length,respectively. The shaded region represents states that are allowed inregular quantum mechanics but are forbidden in theories of quantumgravity that modify the uncertainty relation. The inset shows the two curvesfar from the Planck scale at typical experimental position uncertainties1x�1xmin. An experimental precision of δxδp is required to distinguishthe two curves, which is beyond experimental possibilities at present.However, this can be overcome by our scheme, which allows one to probethe underlying commutation relation in massive mechanical oscillators andits quantum gravitational modifications.

be seen as amanifestation of a deformed canonical commutator, forexample of the form12

[x,p]β0 = ih

(1+β0

(p

MPc

)2)

(1)

So far, no effect of a modified canonical commutator has beenobserved in experiments. At present the best availablemeasurementprecision (Table 1) allows one to put an upper bound on themagnitude of the deformation of β0 < 1033 (ref. 13). For theoriesthat modify the commutator this rules out the existence of anintermediate fundamental length scale on the order of x∼ 10−19 m.Note that the Planck-scale modifications correspond to β0∼ 1 andare therefore untested. Furthermore, the above modification ofthe commutator is not unique and experiments can, in principle,distinguish between the various theories. In particular, a generalizedversion of the commutator deformation is11

[x,p]µ0 = ih

√1+2µ0

(p/c)2+m2

M 2P

(2)

Here, m is the rest mass of the particle and µ0 is again a freenumerical parameter. For small masses m� p/c ∼<MP, and forµ0 = β0, the above modified commutator reduces to equation (1).However, an important difference is that the commutation relationin equation (2) depends directly on the rest mass of the particle.In the limit p/c�m ∼<MP, the commutator reduces to [x,p]µ0 ≈

ih(1+µ0m2/M 2P ), which can be seen as a mass-dependent rescaling

of h. It is worth noting that a modified, mass-dependent Planckconstant h = h(m) also appears in other theories, some ofwhich predict that the value of Planck’s constant can decreasewith increasing mass (h→ 0 for m� MP), in contrast to theprediction above. Such a reduction would also account for atransition to classicality in massive systems or at energies close tothe Planck energy6,10.

Table 1 | Current experimental bounds on quantumgravitational commutator deformations.

System/experiment β0,max γ0,max References

Position measurement 1034 1017 20,21Hydrogen Lamb shift 1036 1010 13,15Electron tunnelling 1033 1011 13,15

The parameters β0 and γ0 quantify the deformation strengths of the modification given inequation (1) and of the modification given in equation (3), respectively. For electron tunnellingan electric current measurement precision of δI∼ 1 fA was taken.

Among the various proposals for different commutatordeformations, we choose as a last example the recently proposedcommutator14 which also accounts for a maximum momentumthat is present in several approaches to quantum gravity5,6

[x,p]γ0 = ih

(1−γ0

pMPc+γ 2

0

(p

MPc

)2)

(3)

Here, γ0 is again a free numerical parameter that characterizesthe strength of the modification. Experimental bounds on γ0are more stringent than in the case of equation (1) and wereconsidered in ref. 15. The best bound at present can be obtainedfrom Lamb shift measurements in hydrogen, which yield γ0 ∼< 1010(Table 1).

The strength of the modifications in all the discussed examplesdepends on the mass of the system. For a harmonic oscillatorin its ground state the minimum momentum uncertainty isgiven by p0 =

√hmωm, where m is the mass of the oscillator

and ωm is its angular frequency. The deformations are thereforeenhanced in massive quantum systems. We note that theories ofdeformed commutators have an intrinsic ambiguity as to whichdegrees of freedom it should apply to for composite systems(see Supplementary Information). For the centre of mass mode,the mass dependence of the deformations suggests that usingmassive quantum systems allows easier experimental access to thepossible deformations of the commutator, provided that precisequantum control can be attained. Optomechanical systems, wherethe oscillator mass can be around the Planck mass and evenlarger, therefore offer a natural test-bed for probing commutatordeformations of its centre of mass mode.

Scheme to measure the deformationsIn the following we will outline a quantum optical schemethat allows one to measure deformations of the canonicalcommutator of a mechanical oscillator with unprecedentedprecision. For simplicitywe use dimensionless quadrature operatorsXm and Pm. They are related to the position and momentumoperators via x = x0Xm and p = p0Pm, where x0 =

√h/(mωm)

and p0=√hmωm.

The scheme relies on displacements of the massive mechanicaloscillator in phase space, where the displacement operator isgiven by26 D(z/

√2)= ei(Re[z]Xm−Im[z]Pm). The action of this operator

displaces the mean position and momentum of any state by Im[z]and Re[z], respectively. In quantum mechanics, two subsequentdisplacements provide an additional phase to the state, whichcan be used to engineer quantum gates27–29. Here we considerdisplacements of the mechanical resonator that are induced by anancillary quantum system, the optical field, with an interactionstrength λ. A sequence of four optomechanical interactions ischosen such that the mechanical state is displaced around a loop inphase space, described by the four-displacement operator

ξ = eiλnLPme−iλnLXme−iλnLPmeiλnLXm (4)

394 NATURE PHYSICS | VOL 8 | MAY 2012 | www.nature.com/naturephysics

© 2012 Macmillan Publishers Limited. All rights reserved.

NATURE PHYSICS DOI: 10.1038/NPHYS2262 ARTICLESIn classical physics, after the whole sequence, neither of the twosystems would be affected because the four operations canceleach other. However, for non-commuting Xm and Pm there isa change in the optical field depending on the commutator[Xm,Pm] = iC1. We can rewrite equation (4) using the well-knownrelation30 eaXmPme−aXm =

∑∞

k=0(ikak/k!)Ck , where iCk = [Xm,Ck−1]

and C0 = Pm. This yields ξ = exp(−iλnL∑

k(λnL)kCk/k!), which

depends explicitly on the commutation relation for the oscillator,but not on the commutator of the optical field. For the quantummechanical commutator, that is C1 = 1, we obtain ξ = e−iλ

2n2L . Inthis case, the optical field experiences a self-Kerr nonlinearity, thatis an n2L operation, and the mechanical state remains unaffected.However, any deformations of the commutator would show in ξ ,resulting in an observable effect in the optical field.

As an example we consider the modification given byequation (1). To first order in β ≡ β0hωmm/(MPc)2 � 1 oneobtains C1 = 1+βP2

m, C2 ≈ β2Pm, C3 ≈ 2β and Ck ≈ 0 for k ≥ 4.Equation (4) thus becomes ξβ = e−iλ

2n2L e−iβ(λ2n2LP

2m+λ

3n3LPm+(1/3)λ4n4L)

(this approximation has the physical meaning that one can neglectcontributions which are higher order in β for the observablesconsidered below). As one can see immediately, a deformedcommutator affects the optical field differently owing to non-vanishing nested commutators Ck , k> 1. In addition to a Kerr-typenonlinearity the optical field experiences highly non-Gaussian n3Land n4L operations. The additional effect scales with β and thereforeallows a direct measure of the deformations of the canonicalcommutator of the mechanical system via the optical field. To seethat explicitly, let us denote the optical field by aL, with the realand imaginary parts representing its measurable amplitude andphase quadratures, respectively. Also, for simplicity, we restrictthe discussion to coherent states |α〉 with real amplitudes of theoptical input field and we neglect possible deformations in thecommutator of the optical field during read-out31,32 as those areexpected to be negligible compared with the deformations of themassive mechanical oscillator (see refs 33,34 for schemes that canprobe the non-commutativity of the optical field). For a largeaverage photon numberNp�1, and for a mechanical thermal statewith mean phonon occupation n� λNp, the mean of the opticalfield becomes (for |2|� 1):

〈aL〉' 〈aL〉qme−i2 (5)

where 〈aL〉qm = αe−iλ2−Np(1−e−i2λ

2) is the quantum mechanical value

for the unmodified dynamics. The β-induced contribution causesan additional displacement in phase space by

2(β)'43βN 3

p λ4e−i6λ

2

(6)

The resulting optical state is represented in Fig. 2. We notethat the magnitude of the effect is enhanced by the opti-cal intensity and the interaction strength. For the µ- and theγ -deformation of the commutator, referring to equations (2)and (3), respectively, the effect on the optical field is simi-lar, but shows a different scaling with the system parameters(see Table 2 and Supplementary Information for the derivation).Probing deviations from the quantum mechanical commuta-tor of the massive oscillator thus boils down to a precisionmeasurement of the mean of the optical field, which can beachieved with very high accuracy via interferometric means, suchas homodyne detection.

Experimental implementationWe now discuss a realistic experimental scenario that can attainsufficient sensitivity to resolve the deformation-induced changein the optical field even for small values of β0, µ0 and γ0,that is in a regime that can be relevant for quantum gravity.

Im [aL]

Re [aL]

⟨aL⟩ qm

⏐ ⟩ out

ΦΘ

⏐ ⟩α

αξ σ

Figure 2 | Changes to the optical field following the pulsedoptomechanical interactions. The effect of the four-displacementoperation ξ onto an optical state for the experimentally relevant case λ� 1.In this case, an initial optical coherent state |α〉 is rotated in phase-spacethrough an angle8. A part2 of the rotation is due to a possible quantumgravitational deformation of the canonical commutator of the mechanicalresonator (see equation (6)). Measuring the mean of the optical field 〈aL〉

and extracting the2-contribution allows one to probe deformations of thecanonical commutator. Optical interferometric schemes can provide ameasurement of the overall mean rotation with a fundamental imprecisionδ〈8〉= σout/

√NpNr, (Nr: number of measurement runs, Np: number of

photons, σout: quadrature width of the optical state, which remains veryclose to the coherent state value 1/2). To resolve the2-contribution, themeasurement imprecision must fulfill δ〈8〉<2, which we show can beachieved in quantum optomechanical systems even for deformations onthe Planck scale.

The optomechanical scheme proposed here can achieve such aregime: it combines the ability to coherently control large masseswith strong optical fields. From a more general perspective,optomechanical systems provide a promising avenue for preparingand investigating quantum states of massive objects rangingfrom a few picograms up to several kilograms22,23. Significantexperimental progress has been recently made towards this goal,including laser cooling of nano- and micromechanical devices intotheir quantum ground state35,36, operation in the strong-couplingregime37–39 and coherent interactions39,40. Owing to their highmass they have also been proposed for tests of so-called collapsemodels41,42, which predict a breakdown of the quantummechanicalsuperposition principle for macroscopic objects. For our purposehere, which is the high-precision measurement of the canonicalcommutator of a massive oscillator, we focus on the pulsed regimeof quantum optomechanics43.

We consider the set-up depicted in Fig. 3, where a mechanicaloscillator is coupled to the optical input pulse via radiationpressure inside a high-finesse optical cavity. This is describedby the intra-cavity Hamiltonian44 H = hωmnm− hg0nLXm, wherenm is the mechanical number operator and g0 = ωc(x0/L) is theoptomechanical coupling rate with the mean cavity frequencyωc and mean cavity length L. For sufficiently short opticalpulses the mechanical harmonic evolution can be neglected andthe intracavity dynamics can be approximated by the unitaryoperation U = eiλnLXm (ref. 43). Here, the effective interactionstrength is (see Supplementary Information) λ' g0/κ = 4Fx0/λL,where κ is the optical amplitude decay rate, F is the cavityfinesse and λL is the optical wavelength. To realize the desireddisplacement operation in phase-space it is also required to achievea direct optomechanical coupling, with the same optical pulse,to the mechanical momentum (equation (4)). Such a momentum

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ARTICLES NATURE PHYSICS DOI: 10.1038/NPHYS2262

Table 2 | Experimental parameters to measure quantumgravitational deformations of the canonical commutator.

[Xm,Pm] Equation (2) Equation (3) Equation (1)

|2| µ032hF2mNp

M2Pλ

2Lωm

γ096h2F3N2

p

MPcλ3Lmωm

β01024h3F4N3

p

3M2Pc

2λ

4L mωm

F 105 2× 105 4× 105

m 10−11 kg 10−9 kg 10−7 kgωm/2π 105 Hz 105 Hz 105 HzλL 1,064 nm 1,064 nm 532 nmNp 108 5× 1010 1014

Nr 1 105 106

δ〈8〉 10−4 10−8 10−10

The parameters are chosen such that a precision of δµ0∼ 1, δγ0∼ 1 and δβ0∼ 1 can be achieved,which amounts to measuring Planck-scale deformations.

coupling could be achieved for example via the Doppler effectby using mirrors with a strongly wavelength-dependent opticalreflectivity45. A more straightforward route is to use the harmonicevolution of the mechanical resonator between pulse round-trips (for example, ref. 43), which effectively allows Xm and Pmto be interchanged after a quarter of the oscillator period. Inthis case, the contribution from the commutator deformationhas a different pre-factor, but remains of the same form (seeSupplementary Information), and part of the phase-space rotationin the optical field is of classical nature. This has no effect onthe ability to distinguish and observe the rotation due to thedeformed commutator. After the four-pulse interaction has takenplace the optical field can be analysed in an interferometricmeasurement, which yields the phase information of the light withvery high precision.

As in previous approaches to measure possible modificationsof the canonical commutator13,15, the relevant question is whichultimate resolution δβ0, δµ0, δγ0 the experiments can provide. Inthe case of a null result, these numbers would set an experimentalbound for β0, µ0, γ0 and hence provide an important empiricalfeedback for theories of quantum gravity. We restrict our analysisto the experimentally relevant case λ< 1, for which the effect ofa deformed commutator resembles a pure phase-space rotationof the optical output state by angle 8 (Fig. 2). The inaccuracyδ8 of the measurement outcome depends on the quantumnoise σout of the outgoing pulse along the relevant generalizedquadrature and can be further reduced by quantum estimationprotocols46. For our purposes we only require to measure the meanoptical field, equation (5). The precision of this measurement isnot fundamentally limited and is enhanced by the strength ofthe field and the number of experimental runs Nr via δ〈8〉 =σout/√NpNr, from which one directly obtains the fundamental

resolutions δβ0, δµ0, δγ0. For each of the discussed deformationsit is possible to find a realistic parameter regime (Table 2) withmarkedly improved performance compared with existing bounds.In particular, we assume a mechanical oscillator of frequencyωm/2π = 105 Hz and mass m = 10−11 kg, and an optical cavityof finesse F = 105 at a wavelength of λL = 1,064 nm, which isin the range of current experiments37,47–50. To test a µ-modifiedcommutator (equation (2)), a pulse sequence of mean photon-number Np = 108 is sufficient to obtain a resolution δµ0 ∼ 1already in a single measurement run (Nr = 1). For the caseof a γ -modified commutator (equation (3)), the same sequencewould result in δγ0 ∼ 109. By increasing the photon-number toNp = 5× 1010, the finesse to F = 2× 105 and the number ofmeasurement runs to Nr = 105 (this would require stabilizing theexperiment on a timescale of the order of seconds) one obtainsδγ0 ∼ 1. Note that this would improve the existing bounds for γ0

Reference

EOM

Delay line

/4

[X m, Pm] = ?

In HPBS

First: HHV

VOutnth: H

VLast: V

Interferometricmeasurement

aL

Xmλ

Figure 3 | Proposed experimental set-up to probe deformations of thecanonical commutator of a macroscopic mechanical resonator. Anincident pulse ‘In’ is transmitted through a polarizing beam splitter (PBS)and an electro-optic modulator (EOM) and then interacts with amechanical resonator with position Xm via a cavity field aL. The optical fieldis retro-reflected from the optomechanical system and then enters a delayline, during which time the mechanical resonator evolves for one quarter ofa mechanical period. The optical pulse, now vertically polarized, is rotatedby the EOM to be horizontally polarized and interacts again with themechanical resonator. This is repeated for a total of four interactions, suchthat the canonical commutator of the resonator is mapped onto the opticalfield. Finally, the EOM does not rotate the polarization and the pulse exits inthe mode labelled ‘Out’, where it is then measured interferometrically withrespect to a reference field such that the commutator deformations can bedetermined with very high accuracy.

(ref. 15) by ten orders of magnitude. To obtain similar bounds fora β-modification is more challenging. The pulse sequence with theprevious parameters yields δβ0∼ 1012, which already constitutes animprovement by about 20 orders of magnitude compared with thecurrent bound for β0 (ref. 13). This can provide experimental accessto a possible intermediate length-scale or a meaningful feedbackto theories of quantum gravity in the case of a null result. Byfurther pushing the parameters toNp= 1014,Nr= 106, F = 4×105,m=10−7 kg and λL=532 nm it is even possible to reach δβ0∼1, thatis, a regime where Planck-scale deformations are relevant and 33orders of magnitude beyond current experiments. To achieve suchexperimental parameters is challenging, but is well within the reachof current technology.

The above considerations refer to the ideal case in whichexperimental noise sources can be neglected. Effects such asmechanical damping and distortions of the effective interactionstrength impose additional requirements on the experimentalparameters, which are discussed in detail in the SupplementaryInformation. In summary, being able to neglect the effects ofpulse shape distortion and optical loss requires that the mechanicalmode is optically cooled close to a thermal occupation of aboutn< 30. Similarly, decoherence effects are negligible when the wholemechanical system is in a bath of temperature T < 100mK forresonators with a quality factor of Q> 106, which can be achievedwith dilution refrigeration. In general, the scheme is very robustagainst many noise sources, as it relies on the measurement of themean of the optical field and the noise sources can be isolated byindependent measurements. We also note that contributions froma modified commutator scale in a different way with the systemparameters as compared with deleterious effects. It is thereforepossible to distinguish these by varying the relevant parameters,such as optical intensity and the oscillator mass. The proposedscheme thus offers a feasible route to probe the possible effectsof quantum gravity in a table-top quantum optics experimentand hence to provide important empirical feedback for theoriesof quantum gravity.

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© 2012 Macmillan Publishers Limited. All rights reserved.

NATURE PHYSICS DOI: 10.1038/NPHYS2262 ARTICLESReceived 4 November 2011; accepted 13 February 2012;published online 18 March 2012; corrected online26 March 2012 and 10 April 2012

References1. Heisenberg, W. Über den anschaulichen Inhalt der quantentheoretischen

Kinematik und Mechanik. Z. Phys. 43, 172–198 (1927).2. Garay, L. G. Quantum gravity and minimum length. Int. J. Mod. Phys. A10,

145–165 (1995).3. Amati, D., Ciafaloni, M. & Veneziano, G. Superstring collisions at planckian

energies. Phys. Lett. B 197, 81–88 (1987).4. Gross, D. J. & Mende, P. F. String theory beyond the Planck scale.Nucl. Phys. B

303, 407–454 (1988).5. Amelino-Camelia, G. Doubly special relativity: First results and key open

problems. Int. J. Mod. Phys. D 11, 1643–1669 (2002).6. Magueijo, J. & Smolin, L. Generalized Lorentz invariance with an invariant

energy scale. Phys. Rev. D 67, 044017 (2003).7. Amelino-Camelia, G., Freidel, L., Kowalski-Glikman, J. & Smolin, L. Principle

of relative locality. Phys. Rev. D 84, 084010 (2011).8. Maggiore, M. A generalized uncertainty principle in quantum gravity.

Phys. Lett. B 304, 65–69 (1993).9. Scardigli, F. Generalized uncertainty principle in quantum gravity from

micro-black hole gedanken experiment. Phys. Lett. B 452, 39–44 (1999).10. Jizba, P., Kleinert, H. & Scardigli, F. Uncertainty relation on a world crystal and

its applications to micro black holes. Phys. Rev. D 81, 084030 (2010).11. Maggiore, M. The algebraic structure of the generalized uncertainty principle.

Phys. Lett. B 319, 83–86 (1993).12. Kempf, A., Mangano, G. & Mann, R. B. Hilbert space representation of the

minimal length uncertainty relation. Phys. Rev. D 52, 1108–1118 (1995).13. Das, S. & Vagenas, E. C. Universality of quantum gravity corrections. Phys.

Rev. Lett. 101, 221301 (2008).14. Ali, A. F., Das, S. & Vagenas, E. C. Discreteness of space from the generalized

uncertainty principle. Phys. Lett. B 678, 497–499 (2009).15. Ali, A. F., Das, S. & Vagenas, E. C. A proposal for testing quantum gravity in

the lab. Phys. Rev. D 84, 044013 (2011).16. Amelino-Camelia, G., Ellis, J., Mavromatos, N. E., Nanopoulos, D. V. &

Sarkar, S. Tests of quantum gravity from observations of gamma-ray bursts.Nature 393, 763–765 (1998).

17. Jacob, U. & Piran, T. Neutrinos from gamma-ray bursts as a tool to explorequantum-gravity-induced Lorentz violation. Nature Phys. 7, 87–90 (2007).

18. Abdo, A. A. et al. A limit on the variation of the speed of light arising fromquantum gravity effects. Nature 462, 331–334 (2009).

19. Tamburini, F., Cuofano, C., Della Valle, M. & Gilmozzi, R. No quantumgravity signature from the farthest quasars. Astron. Astrophys. 533, A71 (2011).

20. Grote, H. & the LIGO Scientific Collaboration. The status of GEO 600. Class.Quantum Grav.25, 114043 (2008).

21. Abbott, B. P. et al. LIGO: The Laser Interferometer Gravitational-waveObservatory. Rep. Prog. Phys. 72, 076901 (2009).

22. Kippenberg, T. J. & Vahala, K. J. Cavity optomechanics: Back-action at themesoscale. Science 321, 1172–1176 (2008).

23. Aspelmeyer, M., Groeblacher, S., Hammerer, K. & Kiesel, N. Quantumoptomechanics—throwing a glance. J. Opt. Soc. Am. B 27, A189–A197 (2010).

24. Milburn, G. J. Lorentz invariant intrinsic decoherence. New J. Phys.8, 96 (2006).

25. Kempf, A. Information-theoretic natural ultraviolet cutoff for spacetime.Phys. Rev. Lett. 103, 231301 (2009).

26. Barnett, S. M. & Radmore, P. M. Methods in Theoretical Quantum Optics(Oxford Univ. Press, 2002).

27. Milburn, G. J., Schneider, S. & James, D. F. V. Ion trap quantum computingwith warm ions. Fortschr. Phys. 48, 801–810 (2000).

28. Sørensen, A. & Mølmer, K. Entanglement and quantum computation withions in thermal motion. Phys. Rev. A 62, 022311 (2000).

29. Leibfried, D. et al. Experimental demonstration of a robust, high-fidelitygeometric two ion-qubit phase gate. Nature 422, 412–415 (2003).

30. Wilcox, R. M. Exponential operators and parameter differentiation in quantumphysics. J. Math. Phys. 8, 962–982 (1967).

31. Nozari, K. Some aspects of Planck scale quantum optics. Phys. Lett. B. 629,41–52 (2005).

32. Ghosh, S. & Roy, P. ‘‘Stringy’’ coherent states inspired by generalizeduncertainty principle. Preprint at http://arxiv.org/hep-ph/11105136 (2011).

33. Parigi, V., Zavatta, A., Kim, M. S. & Bellini, M. Probing quantum commutationrules by addition and subtraction of single photons to/from a light field. Science317, 1890–1893 (2007).

34. Kim, M. S., Jeong, H., Zavatta, A., Parigi, V. & Bellini, M. Scheme forproving the bosonic commutation relation using single-photon interference.Phys. Rev. Lett. 101, 260401 (2008).

35. Teufel, J. D. et al. Sideband cooling of micromechanical motion to the quantumground state. Nature 475, 359–363 (2010).

36. Chan, J. et al. Laser cooling of a nanomechanical oscillator into its quantumground state. Nature 478, 89–92 (2011).

37. Gröblacher, S., Hammerer, K., Vanner, M. R. & Aspelmeyer, M. Observationof strong coupling between a micromechanical resonator and an optical cavityfield. Nature 460, 724–727 (2009).

38. Teufel, J. D. et al. Circuit cavity electromechanics in the strong-couplingregime. Nature 471, 204–208 (2011).

39. O’Connell, A. D. et al. Quantum ground state and single-phonon control of amechanical resonator. Nature 464, 697–703 (2010).

40. Verhagen, E., Deléglise, S., Weis, S., Schliesser, A. & Kippenberg, T. J.Quantum-coherent coupling of a mechanical oscillator to an optical cavitymode. Nature 482, 63–67 (2012).

41. Marshall, W., Simon, C., Penrose, R. & Bouwmeester, D. Towards quantumsuperpositions of a mirror. Phys. Rev. Lett. 91, 130401 (2003).

42. Romero-Isart, O. et al. Large quantum superpositions and interference ofmassive nanometer-sized objects. Phys. Rev. Lett. 107, 020405 (2011).

43. Vanner, M. R. et al. Pulsed quantum optomechanics. Proc. Natl Acad. Sci. USA108, 16182–16187 (2011).

44. Law, C. K. Interaction between a moving mirror and radiation pressure: AHamiltonian formulation. Phys. Rev. A 51, 2537–2541 (1995).

45. Karrai, K., Favero, I. & Metzger, C. Doppler optomechanics of a photoniccrystal. Phys. Rev. Lett. 100, 240801 (2008).

46. Boixo, S. et al. Quantum-limited metrology and Bose–Einstein condensates.Phys. Rev. A 80, 032103 (2009).

47. Corbitt, T. et al. Optical dilution and feedback cooling of a gram-scale oscillatorto 6.9mK. Phys. Rev. Lett. 99, 160801 (2007).

48. Thompson, J. D. et al. Strong dispersive coupling of a high-finesse cavity to amicromechanical membrane. Nature 452, 72–76 (2008).

49. Verlot, P., Tavernarakis, A., Briant, T., Cohadon, P-F. & Heidmann, A.Scheme to probe optomechanical correlations between two optical beamsdown to the quantum level. Phys. Rev. Lett. 102, 103601 (2008).

50. Kleckner, D. et al. Optomechanical trampoline resonators. Opt. Express 19,19708–19716 (2011).

AcknowledgementsThis work was supported by the Royal Society, by the Engineering and Physical SciencesResearch Council, by the European Comission (Quantum InterfacES, SENsors, andCommunication based on Entanglement (Q-ESSENCE)), by the European ResearchCouncil Quantum optomechanics: quantum foundations and quantum information onthe micro- and nanoscale (QOM), by the Austrian Science Fund (FWF) (ComplexQuantum Systems (CoQuS), START program, Special Research Programs (SFB)Foundations and Applications of Quantum Science (FoQuS)), by the FoundationalQuestions Institute and by the John Templeton Foundation. M.R.V. is a recipient of aDOC fellowship of the Austrian Academy of Sciences. I.P. and M.R.V are members ofthe FWF Doctoral Programme CoQuS and they are grateful for the kind hospitalityprovided by Imperial College London. The authors thank S. Das, S. Gielen, H. Grosse, A.Kempf, W. T. Kim and J. Lee for discussions.

Author contributionsI.P. and M.S.K. conceived the research, which was further developed by Č.B. and allco-authors. M.R.V. conceived the experimental scheme. M.A. analysed the feasibility ofthe scheme with input from all co-authors. All authors performed the research under thesupervision of Č.B. and all authors wrote the manuscript.

Additional informationThe authors declare no competing financial interests. Supplementary informationaccompanies this paper on www.nature.com/naturephysics. Reprints and permissionsinformation is available online at www.nature.com/reprints. Correspondence andrequests for materials should be addressed to I.P. or M.S.K.

NATURE PHYSICS | VOL 8 | MAY 2012 | www.nature.com/naturephysics 397© 2012 Macmillan Publishers Limited. All rights reserved.

In the version of this Article originally published online, ref. 5 should have read: Amelino-Camelia, G. Doubly special relativity: First results and key open problems. Int. J. Mod. Phys. D 11, 1643–1669 (2002). This has been corrected in all versions of the Article.

Probing Planck-scale physics with quantum opticsIgor Pikovski, Michael R. Vanner, Markus Aspelmeyer, M. S. Kim and Časlav Brukner

Nature Physics http://dx.doi.org/10.1038/nphys2262 (2012); published online 18 March 2012; corrected online 26 March 2012.

CORRIGENDUM

In the version of this Article originally published online, the experimental parameters for testing equations (1) and (3) given in Table 2 and subsequently used in the text on the same page were incorrect. These errors have been corrected in all versions of the Article.

Probing Planck-scale physics with quantum opticsIgor Pikovski, Michael R. Vanner, Markus Aspelmeyer, M. S. Kim and Časlav Brukner

Nature Physics http://dx.doi.org/10.1038/nphys2262 (2012); published online 18 March 2012; corrected online 10 April 2012.

CORRIGENDUM

© 2012 Macmillan Publishers Limited. All rights reserved.