aa r X i v : . [ phy s i c s . op ti c s ] J un Planckian signatures in optical harmonic generationand supercontinuum
Claudio Conti , Department of Physics, University Sapienza, Piazzale Aldo Moro 2, 00185, Rome,IT Institute for Complex Systems, National Research Council (ISC-CNR), Via deiTaurini 19, 00185, Rome, ITE-mail: [email protected]
Abstract.
Many theories about quantum gravity, as string theory, loop quantumgravity, and doubly special relativity, predict the existence of a minimal lengthscale and outline the need to generalize the uncertainty principle. This generalizeduncertainty principle relies on modified commutation relations that - if applied tothe second quantization - imply an excess energy of the electromagnetic quanta withrespect to ¯ hω . Here we show that this “dark energy of the photon” is amplified duringnonlinear optical process. Therefore, if one accepts the minimal length scenario, onemust expect to observe specific optical frequencies in optical harmonic generationby intense laser fields. Other processes as four-wave mixing and supercontinuumgeneration may also contain similar spectral features of quantum-gravity. Nonlinearoptics may hence be helpful to falsify some of the most investigated approaches to theunification of quantum mechanics and general relativity. Keywords : Optical harmonic generation, supercontinuum, quantum gravity, Planckianphysics, generalized uncertainty principle. lanckian signatures in optical harmonic generation and supercontinuum
1. Introduction
The search for observable signatures of the new physics at the Planck scale is a route wemust follow to test the many ideas for the unification of quantum mechanics and generalrelativity. [1, 2, 3, 4] As reviewed in [5], there are many proposed tests of quantum-gravityphenomenology, including quantum-optics [6, 7, 8].Despite these investigations, it is hard to believe to the possibility of probingPlanckian physics in the laboratory. The use of intense laser fields and the correspondingimpressive technological developments of recent years may support our imagination.This possibility was discussed in particular by Magueijo. [9] At the moment, most ofthe emphasis is in laser-driven particle acceleration, laser induced particle-antiparticlegeneration, and related quantum-field processes. This activity gains momentum by therealization of novel extreme light infrastructures. [10]Here we consider a different perspective, and try to show that a particular effectpredicted by many different quantum-gravity theories may be falsified by experimentsin nonlinear optics.The considered effect is related to the existence of a minimal length scale, which has beenpredicted by string-theory, loop quantum gravity, doubly-special relativity, polymerquantization, black hole physics and related investigations.[11]The minimal length scenario has important consequences. In order to include thisscenario in our fundamental models, we need to change the Heisenberg uncertaintyprinciple. In simple terms, the existence of a minimal length is due to granularity ofthe spacetime quantified by the Planck scale ℓ P (or other unknown length scale). Wecannot localize particles at a length scale smaller than ℓ P . This fact - as explored inthe vast related literature - is at odds with the standard uncertainty principle, whichdoes not predict any minimal value for the position uncertainty ∆ X (we neglect hererelativistic effects [12]).The most studied generalized uncertainty principle (GUP) reads as∆ X ∆ P X ≥ ¯ h (cid:16) β X ∆ P X (cid:17) (1)with ˆ P X the momentum, and β X a unknown parameter. β X fixes the minimal positionuncertainty ∆ X min = ¯ h √ β X . Since the original proposals, [13, 14, 15, 16, 17, 18] thepossibility of a generalized uncertainty principle has attracted a lot of attention, rangingfrom theoretical works to proposed experimental tests. [19, 20, 21, 20, 22, 23, 24, 25, 26,27, 28, 29]A key-strategy to account for Eq. (1) by “minimal changes” to the standardquantum mechanics is to consider modified commutators. Standard quantum mechanicswith the generalized commutator[ ˆ X, ˆ P X ] = ı ¯ h (1 + β X ˆ P X ) (2)readily implies Eq. (1). [17] These generalized quantum mechanics have been largelystudied in recent years, looking for observable phenomena in the laboratory, or at lanckian signatures in optical harmonic generation and supercontinuum
2. Modified Quantization of the Electromagnetic Field
We start from the standard quantization procedure for the electromagnetic (EM)field. [37, 38] In a cavity, the energy of the classical electrical field with angular frequency ω k is written as E = 12 X k P k + ω k Q k , (3)where P k and Q k are the field quadratures, and k is the mode index. In the following,we consider a single mode and omit the index k : E = 12 (cid:16) P + ω Q (cid:17) . (4)As reported in quantum optics textbooks (see, e.g., [38]), following the originalDirac strategy [37], one recognizes in Eq.(4) a formal equivalence with the harmonicoscillator, and quantizes the field by converting the classical quantities P and Q in operators with the commutation relation [ ˆ Q, ˆ P ] = i ¯ h . Correspondingly, the EMquantized energy is E n = ¯ hω (cid:18) n + 12 (cid:19) . (5)For each photon with frequency ω , there is a quantum of energy ¯ hω .We hence follows the same argument, and identify in Eq. (4) a quantum harmonicoscillator. However, with the prescription of the mentioned quantum gravity theories, lanckian signatures in optical harmonic generation and supercontinuum P and ˆ Q , namely[ ˆ Q, ˆ P ] = ı ¯ h (cid:16) β ˆ P (cid:17) . (6)In the GUP literature, the quantum harmonic oscillator has been revised and largelystudied. Pedram [39] has shown that one can write ˆ Q and ˆ P in terms of the standardˆ q and ˆ p , with [ˆ q, ˆ p ] = ı ¯ h , asˆ Q = ˆ q ˆ P = tan( √ β ˆ p ) √ β , (7)and determine analytically the energy of the modified harmonic oscillator (see appendix).
3. The “dark energy” of the photon
In the modified quantum mechanics with the generalized commutator in Eq. (6), theeigenstates and eigenvalues of the energy are also modified. We follow refs. [39, 32] forthe harmonic oscillator, the energy eigenvalues are [17] E n ( ω ) = ¯ hω (cid:18) n + 12 (cid:19) s β ¯ h ω β ¯ hω + β ¯ h ω n . (8)For β = 0, Eq. (8) gives the well known E n = ( n + 1 / hω . At the lowest order in β ,we have E n ( ω ) ≃ ¯ hω (cid:18) n + 12 (cid:19) + β ¯ h ω (cid:16) n + 2 n + 1 (cid:17) . (9)Eq. (9) shows the modified dispersion expected to be valid at the Planck scale, followingthe recipe of quantum-gravity theories, as string-theory. [40] Eq. (9) also shows that onecan retain the standard expression E n = n ¯ hω n , if the angular frequency is assumed tobe dependent on the number of photons, with ω n ≡ E n ( ω ) n ¯ h . (10)For n >> / ω n ≃ ω nβ ¯ hω ! , (11)which signals a blue-shift of the photon energy when the number of photons grows.This result may be related to similar phenomena within doubly-special relativity (see,for example, [41] and references therein).From Eq. (8), we have E n ( ω ) > n ¯ hω . This implies that a single photon at frequency ω has an excess energy with respect to the standard quantum mechanics. Apparently,this excess energy does not correspond to any detectable electromagnetic color, we hencerefer to it as the “dark-energy of the photon”. For the single photon, we have δ E ( ω ) = E ( ω ) −
32 ¯ hω ≃ β ¯ h ω , (12) lanckian signatures in optical harmonic generation and supercontinuum n photons with n >> δ n E ( ω ) = E n ( ω ) − (cid:18) n + 12 (cid:19) ¯ hω ≃ βn ¯ h ω . (13)In general, this dark energy is very small and - at equilibrium - is seemingly impossibleto observe. We may argue if δE n has some observable signature.One can consider many different effects as, for example, cosmological dark-energy,black-body radiation, quantum noise, spontaneous and stimulated emission processes. δE n is very small, and lab-top experiments aimed to test this excess energy are difficultto imagine and realize. Looking at Eq.(13), it is natural to consider high-field effects, asnonlinear electromagnetic processes, because δE n scales with the square of the numberof photons n .In the next section, we consider the second-harmonic generation (SHG) of opticalradiation. This process was also previously considered for lab-tests of the modifieddispersion relation predicted by doubly special relativity.[5] Following the standard model of cosmology, about 70% of the total energy of the universeis unknown and has very low density. The density of the dark energy is estimated to be10 − kg m − . [42]We want to compare the density of the dark energy with the density of δE . Thelargest contribution to the photons in the universe comes from the cosmic microwavebackground (CMB), with ω CMB ≃ rad/s and a total number of photons n CMB =10 . By using Eq. (12), we have that the “dark energy of the CMB” is δE ′ CMB ≡ n CMB δE ( ω CMB ) ≃ n CMB β (¯ hω CMB ) . (14)In (14), we assume that the CMB photons belong to different modes. For β = 1 / E P ,with E P the Planck energy, taking the volume of the universe as V u = 10 m , we havedensity ρ ′ CMB = δE ′ CMB / ( c V u ) = 10 − kg m − , with c the vacuum light velocity. ρ ′ CMB is much smaller than the density of the dark energy.Recent models of dark matter accounted for the possibility of large scale Bose-Einstein condensates (see, for example, [43] and reference therein). If we hence take allthe CMB photons condensed in the same mode (Eq. (13)) δE ′′ CMB ≡ δE n CMB ( ω CMB ) ≃ n CMB β (¯ hω CMB ) , (15)which gives the too large ρ ′′ CMB = 10 kg m − .These calculations are done with the arbitrarily chosen β = 1 / E P ; on the contrary,one can use the β parameter to fit the density of dark-energy. For example, in the caseof the condensate of dark matter, when choosing β = 10 − J − , the dark energy of theCMB photons δE ′′ CMB has the same density of the dark energy in the universe. lanckian signatures in optical harmonic generation and supercontinuum
4. Excess energy in second harmonic generation
In optically nonlinear media, intense electromagnetic fields may combine and generatenovel frequencies. [44] The simplest process is the second-harmonic generation: 2photons with frequency ω generate a single photon at frequency 2 ω . In the standardquantum mechanics, such a process readily conserves energy, that is, the energy of thegenerated photon 2¯ hω is the sum of the energies ¯ hω of the 2 original photons.This simple scenario is however more complicated in the modified quantummechanics here considered. As detailed in the following, excess energy is to be expected. We start considering the case in which two photons at the fundamental frequency (FF) ω have different polarization, and hence belong to distinguishable modes. The twophotons combine to generate a second-harmonic photon (SH). In the literature aboutnonlinear optics, this case is conventionally named “type II” SHG. The energy of thetwo photons is hence 2 E ( ω ), and it turns out to be different from the SH photon energy E (2 ω ). Specifically, we have E (2 ω ) > E ( ω ) . (16)Therefore, we have the paradox that optical second harmonic generation does notconserve energy .If we still assume that the GUP is valid, the solution for this paradox may be thatthe excess energy δE SHG − II = E (2 ω ) − E ( ω ) (17)is lost in other degrees of freedom as, e.g., thermal or acoustic energy of the crystaladopted for the SHG.This possibility is very difficult to explore in experiments. For example, one canimagine to make a SHG experiment at very low temperature and look for excess heat.However, even at very low temperature, it may be impossible to distinguish this heatfrom the one due to the linear absorption of photons, which is always present.Here we consider the possibility that the excess energy δE SHG − II is still available aselectromagnetic degrees of freedoms. This corresponds to emission of frequency-shiftedexcess photons during SHG.We consider the case in which 2 n photons at frequency ω are converted to n photonsat 2 ω . The excess energy for n second-harmonic photons reads δE SHG − IIn = E n (2 ω ) − E n ( ω ) . (18)For n >>
1, at lowest order in β , we have δE SHG − IIn ≃ β (¯ hω ) n (19)Eq. (19) shows that the excess energy scales quadratically with the number of photons,which is a very convenient situation as β can be very small. lanckian signatures in optical harmonic generation and supercontinuum The theories for quantum gravity do not predict the value for β . However, this valueis expected to be related to the Planck scales. In particular, as the β here consideredhas the dimension of the inverse of energy, we consider the Planck energy E P ∼ = 10 Jand assume, as a representative case, β ≃ / E P . SHG and related experiments may beadopted to set limits to β .Our hypothesis is that, if the GUP scenario is correct, the excess energy is convertedin emitted radiation. We can estimate the wavelength of the emitted radiation byassuming that the excess energy δE SHG − IIn is found in N photons in modes at angularfrequency Ω. We can estimate Ω by the equation E N (Ω) = δE SHG − IIn ( ω ) . (20)Considering Eq.(19), and neglecting the β correction for the energy of the photons atΩ, we have N ¯ h Ω = β (¯ hω ) n . (21)Eq. (21) predicts the energy of the generated modes and shows that, even if β is verysmall, a sufficiently large number of pumping photons may lead to an observable emissiondue to the Planck scale effect.Assuming that a observable amount of photons at Ω is generated (we take N = 100),figure 1 shows the emitted frequency when varying the photon number n and taking ω = 2 πc /λ with λ = 1 µ m and β = 1 / E P .It is remarkable that, with the small value chosen for β , one obtains observablefrequencies in the optical domain at a moderate pump photon number n . For example, n = 10 photons are contained in a laser pulse with energy of the order of 100 µ J, whichis routinely employed in ultra-fast nonlinear optics.The experimental signature is given by the shift of the generated frequencies Ω / π with the pulse energy (or photon number n ). The slope of the shift gives a estimatefor the unknown β as in Eq. (21). Notably the emitted frequency may fall in the UVregion, or lower wavelengths, when increasing n (for the considered value of β ). In this case, we assume that FF photons belonging to the same mode are converted tothe second harmonic. In nonlinear optics, this is commonly indicated as “type 0,” or“type I,” SHG. In this case, we have 2 n photons with angular frequency ω in a modewith energy E n ( ω ).By using Eq. (8), we have δE SHG − In = E n (2 ω ) − E n ( ω ) . (22)For n >>
1, at the lowest order in β , we have δE SHG − In ≃ β (¯ hω ) n, (23) lanckian signatures in optical harmonic generation and supercontinuum Photon number E m i tt e d f r e q u e n c y ( T H z ) Photon number ( 10 ) E m i tt e d w a v e l e n g t h ( m ) Figure 1.
Estimated emission frequency Ω / π (left panel) and wavelength (rightpanel) for N = 100 output photons versus the number of photons in the pump pulsefor type-II second-harmonic generation. which scales linearly with the number of photons n . Hence, when n ≃ , δE SHG − In isseveral order of magnitudes smaller than δE SHG − IIn considered above.
5. Third order nonlinearity and Planck-scale Kerr effect
At this stage, we identify a further paradox: we see that a nonlinear-optical processat the second order, which commonly involves 3 photons, resembles a third orderprocess and involves 4 photons. Indeed, we are claiming that we generate photonsat frequency Ω, during second harmonic generation with couples of photons at ω andgenerated photons at 2 ω . The paradox is solved by the fact that the generalizedcommutators actually induce a third-order nonlinearity in the field evolution and, hence,new frequencies can be generated in addition to ω and 2 ω .We introduce the wave-vector as k n = ω n /c and, defining the refractive index n R ,we set k n = ω n /c = ωn R /c , which gives the refractive index of vacuum as n R = 1 + 12 β ¯ hωn. (24)Eq. (24) resembles the well known third-order optical Kerr effect, [45], i.e., a energydependent refractive index. Eq. (24) also reveals the vacuum polarizability arising fromthe existence of a minimal length. Vacuum Kerr effect due to high-energy effects (asparticle generation) was considered by many authors in the past (see, e.g., [46]).In a different perspective, we can consider the Hamiltonian in Eq. (7) in terms ofthe conventional ˆ q and ˆ p :ˆ H = 12 (cid:16) ˆ P + ω ˆ Q (cid:17) = 12 " tan( √ β ˆ p ) √ β + ω ˆ q . (25)At the lowest order in β , we have H = 12 (cid:16) ˆ p + ω ˆ q (cid:17) + β p . (26) lanckian signatures in optical harmonic generation and supercontinuum a and ˆ a † ˆ q = q ¯ h ω (cid:16) ˆ a † + ˆ a (cid:17) ˆ p = ı q ¯ hω (cid:16) ˆ a † − ˆ a (cid:17) , (27)and we have an Hamiltonian with third order nonlinearity :ˆ H = ¯ hω (cid:18) ˆ a † ˆ a + 12 (cid:19) + βH . (28)Indeed, after normal ordering, H = (¯ hω ) h ˆ a − a † a + 6(ˆ a † ) (ˆ a ) − a † ) a + (ˆ a † ) i . (29)Eq.(29) shows that modified uncertainty relations introduce higher order four-wavemixing terms [47]; therefore, we may expect the generation of novel photons.
6. Link with supercontinuum generation
The previous arguments are obviously not limited to the process of SHG, but they readilyapply to more complex nonlinear process as, e.g., supercontinuum, high-harmonicgeneration, etc. To show the way these arguments may be extended, let us consider afour-wave-mixing process - the onset of supercontinuum - in which 3 angular frequencies ω , ω and ω combine to generate a fourth wave at angular frequency ω . In the standardformulation ( β = 0), energy conservation implies ω = ω + ω + ω . However, in ourframework, the excess energy reads δE W M ( ω ) = E ( ω ) − E ( ω ) − E ( ω ) − E ( ω ) , (30)which can be used to generalize the arguments above to other nonlinear processes.
7. Conclusions
We have assumed the validity of the arguments concerning the existence of a minimallength scale coming from string theories, loop quantum gravity, doubly special relativity,and other theories attempting to unify general relativity and quantum mechanics. Asdiscussed by several authors, these arguments imply a generalization of the uncertaintyprinciple, and hence of the standard commutation relations. The way the commutationrelation has to be changed (Eq. (6)) follows the most investigated formulation.We hence postulated that these generalized commutators must be also valid for theelectromagnetic field quadratures and - in a more general perspective - to any otherclassical field quantization.This approach leads to a perturbation to the energy of the quanta of theelectromagnetic field, according to Eq.(8). This perturbation is extremely small,corresponds to an excess “dark energy” with respect to the commonly accepted value¯ hω , and links to the discussion about dark energy and related cosmological phenomena.We tried to identify other phenomena that can be tested in the laboratory, andwe considered nonlinear optics. By simple arguments, it turn outs that - if Eq.(8) is lanckian signatures in optical harmonic generation and supercontinuum n of the pumping photons. This circumstance gives a direct smoking gun of the “Planckianemission” and may be searched in the experiments. In general, the very large numberof photons attainable with modern laser technology may allow to unveil the very smalleffects weighted by the unknown constant β .Highly nonlinear optical processes, as supercontinuum generation, driven by moderndevices, like micro-structured optical fibers, may furnish novel roads for looking at exotic- but fundamental - phenomena, as the effects of quantum-gravity in the electromagneticfield propagation. There are open questions in recent experiments concerning ultra-wideband generation with ultra-short pulses. Looking for out-of-shelf explanations may bean interesting adventure.
8. Acknowledgments
This publication was made possible through the support of a grant from the JohnTempleton Foundation (grant number 58277). The opinions expressed in thispublication are those of the author and do not necessarily reflect the views of the JohnTempleton Foundation.
Appendix : The modified harmonic oscillator
Various authors in the literature about the generalized uncertainty principle investigatedthe modifications to the quantum harmonic oscillator since the early papers. [17, 30, 19]Pedram [39] reported on a elegant formulation of the problem by expressing thegeneralized quadratures ˆ Q and ˆ P in terms of the standard ones ˆ q and ˆ p by Eq. (7).If one considers the Hamiltonianˆ H = 12 (cid:16) ˆ P + ω ˆ Q (cid:17) (A.1)with the modified commutation relation h ˆ P , ˆ Q i = ı ¯ h (cid:16) β ˆ P (cid:17) , (A.2)it turns out that the eigenproblem is exactly solvable by lettingˆ Q = ˆ q ˆ P = tan( √ β ˆ p ) √ β , (A.3)with [ ˆ p, ˆ q ] = ı ¯ h . In the momentum representation, the stationary Schr¨odinger equationreads − ¯ h ω φ d p + 12 tan (cid:16) √ βp (cid:17) β φ = Eφ. (A.4) lanckian signatures in optical harmonic generation and supercontinuum
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