Proposal for a New Test of the Time Independence Of The Fine Structure Constant,

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1 Proposal or a New Test o the Time Independence O The Fine Structure Constant, α, Using Orthogonally Polarised Whispering Gallery Modes in a Single Sapphire Resonator Michael Edmund Tobar and John Gideon Hartnett Frequency Standards and Metrology Research Group, School o Physics, University o Western Australia, Crawley A new experiment to test or the time independence o the ine structure constant, α, is proposed. The experiment utilizes orthogonally polarized Transverse Electric and Transverse Magnetic Whispering Gallery Modes in a single sapphire resonator tuned to similar requencies. When conigured as a dual mode sapphire clock, we show that the anisotropy o sapphire makes it is possible to undertake a sensitive measurement rom the beat requency between the two modes. At inrared requencies this is possible due to the dierent eect o the lowest phonon requency on the two orthogonally polarized modes. At microwave requencies we show that the phonon eect is too small. We show that the Electron Spin Resonance o paramagnetic impurities (such as Cr 3+ ) in the lattice eects only one polarization with an α 6 dependence. This enables an enhancement o the sensitivity to temporal changes in α at microwave requencies. I. INTRODUCTION Two major tasks in undamental physics are the quantization o gravity and the uniication o all interactions. All approaches to achieve these tasks lead to deviations rom present day physics. For example, violations o the Einstein s Equivalence Principle are possible. Some violations may maniest as spatial and/or time variations o the undamental physical constants such as the ine structure constant α, and recently attempts to measure these eects have gained considerable attention. One o the most precise tools or testing these theories is the requency or time standard (clock). The measurement requires the comparison o two clocks with dierent power law dependence on α [1-3]. Alternative approaches examine astrophysical and geophysical data [4, 5]. A novel method using monolithic Fabry-Perot resonators to test or drit in α was recently proposed[6]. The index o reraction o ionic or molecular crystal depends on α; thus a drit in α will cause a change in the dispersion characteristic o the crystal. In particular [6] analyzed the dependence o optical phonons and electronic transitions o ionic impurities, and remarked that the technique was applicable down to microwave requencies. In this paper we show that the proposed technique needs to be modiied at lower requencies incorporating inrared to microwave. We show that the irst phonon resonance at Hz is in inherently low loss at 4 K and may oer the best requency or this type o measurement. At microwave requencies we show that the introduction o Cr 3+ ions to the lattice signiicantly enhance the dispersion and sensitivity due to an Electron Spin Resonance (ESR) at GHz. Temporal drit in α may be detected by exciting two modes in the same resonator that sample dierent dispersion. For an isotropic material this may only be achieved at dierent requencies. Since sapphire is anisotropic, the dual mode technique [7, 8] may be implemented to realize this experiment. Transverse Electric (TE) and Transverse Magnetic (TM) modes may be excited at nearly the same requency, and will sample dierent values o dispersion. This eect may be ampliied with the presence o paramagnetic impurities. Paramagnetic impurities result in an anisotropic magnetic susceptibility o the lattice. Thus, i both TE and TM modes

2 are excited simultaneously, a drit in α will be measurable rom the beat requency o the two modes. The proposed experiment is to create a 4 K cooled dual mode requency standard based on a sapphire resonator. The beat requency will be stabilized and measured. I a positive measurement is made that suggests a drit in α it will be necessary to veriy this by constructing two or more systems to distinguish the result rom a systematic drit. II. DIELECTRIC RESONATOR DEPEDENCE ON FINE STRUCTURE CONSTANT: MICROWAVE FREQUENCIES TO THE INFRARED To calculate the dependence o the ine structure constant on the reractive index at optical requencies, in re [6] the reractive index was related to the electric susceptibility through the local ield model. To model a macroscopic resonator we choose to use the one phonon model, and we relate the permittivity, ε r, to the electronic susceptibility χ, with the ollowing relation ε r =1+χ. This model is well known to describe the relative permittivity accurately in sapphire, in the radio requency to inrared regions, and at temperatures below 50 K[9-11]. At requencies above the inrared, the susceptibility is summed over more than one lattice vibrational state, but in this case we only consider the region where the lowest phonon requency inluences the results, which avoids the necessity o a summation o states. In this case the complex permittivity is given by; χ ƒε r( )= (1) j 1 i Q i i Where i is the phonon requency, Qi = i is the dimensionless quality actor o the phonon i transition and χ is the electronic susceptibility, which was shown to be independent o α [6]. Thus, ollowing the same approach we ind that the real part o the permittivity o the material may be expressed as; χ 1 2 α Ki εr(, α)= α Ki 2 α KQ i i Here the phonon requency is o the orm, i = α 2 Ki, where K i is a constant also shown to be independent o α [6]. Also, we assume that the quality actor o the phonon transition is irst order independent o α and requency independent. The requency independence will be maintained as long as, < c, where c is the cut o requency o the validity o the model, which extends to the inrared[9]. Phonon resonances with > c have an exponential requency dependence on loss, which still satisies Kramers Kronig relation, in this case (2) takes a more complex orm. This lead Braixmaier et al. [6] to concluded that sampling an ionic crystal near a phonon resonance added too much loss to make use o the high dispersion. On the contrary, in this paper we show that the undamental phonon resonances in sapphire remains low loss near the high sensitivity region close to the phonon resonance requency. By implicitly dierentiating (2) it is straightorward to show the ollowing relation; 2 2 (2)

3 dε r dε = 2 r (3) dα α d Thus we can relate a ractional change in permittivity to a ractional change in ine structure constant by; ε r Ε α = 2 (4) εr α where Ε is the normalised dispersion coeicient o permittivity given by; Ε = d εr (5) d εr Next we must relate the permittivity change to a requency change. For a dielectric resonator o resonant requency, r, with 100% o its energy conined in the dielectric the simple relationship holds. r = 1 εr (6) r 2 εr Thus by combining (6) and (4) the relationship between the dielectric resonant requency and the ine structure constant is given by; r = Ε α (7) r α Thus, it is clear that i two requencies are measured with dierent values o Ε, then the beat requency will be proportional to changes in the ine structure constant. III EXPLOITING THE ANISOTROPIC PERMITTIVITY OF SAPPHIRE Sapphire is a uniaxial anisotropic material, in general both the real and imaginary parts o the permittivity are anisotropic. The loss tangent, Tanδ, o the material is an important parameter or a dielectric resonator, as the quality actor o the resonance, Q r, is equal to the inverse o Tanδ i 100% o the energy is stored in the dielectric. The loss tangent is calculated rom the ratio o the imaginary and real permittivity, and is given by; χ Q Tanδ i i ( )= (8) χ χ i i 1 Q i This is related to the absorption coeicient by; Α( ) = 2π εr Tanδ (9) c The loss tangent or absorption o a low loss material is particularly hard to measure. For example at microwave requencies the Whispering Gallery (WG) mode method was developed and remains the only method capable o measuring loss tangents o order 10-6 or less[12, 13]. This is particularly true or low temperature measurement near 4 K where the loss tangent is smaller than The WG mode method is such a sensitive method that it enabled the irst determination o anisotropy in the loss tangent o uniaxial anisotropic crystals[14]. Near 4 K, paramagnetic impurities also eect the result over a broad requency range even i the transition requency is as ar away as a ew THz, which is the case or titanium impurities[15]. This is because the transitions are thermally excited at cryogenic temperatures. Thus, a measurement o the phonon-induced losses is quite diicult unless a very pure sample has been

4 obtained. We have measured many samples in the microwave region, and the losses due to paramagnetic impurities usually limit the Q r o the resonance around 4 K to a temperature independent value close to The exception is the most pure grade o HEMEX rom Crystal systems. In these samples a Q r o at 12 GHz has been measured, which still exhibit a power law o T 1, which means the Q r limit is not due to paramagnetic impurities [16]. We believed that this limit is due to the dielectric lattice. This result was used in combination with the low temperature permittivity versus requency measurements made by Lowenstein et. al. [17], to calculate the properties o the phonon resonances at 4K. Fitting this data to (1) and (8), the calculated parameters parallel and perpendicular to the c-axis are given in table 1, and the real permittivity, loss tangent and absorption coeicient are plotted as a unction o requency in igures 1 to 3. Table I Fit parameters to the dielectric properties o the irst phonon resonances in sapphire Fit Parameter (~4K) Parallel to c-axis Perpendicular to c-axis ε = 1+χ i Q i ε ε Figure 1. Real permittivity versus requency calculated rom (1) with the parameters presented in table I. Values are plotted rom microwave to inrared or the permittivity parallel, ε, and perpendicular, ε, to the c-axis. Permittivity increases markedly around the irst phonon resonance.

5 Tanδ Figure 2 Loss tangent versus requency calculated rom (8) with the parameters presented in table I. Values are plotted rom microwave to inrared. A() [cm-1] Figure 3 Absorption coeicient [cm -1 ] versus requency calculated rom (9) with the parameters presented in table I. Values are plotted rom microwave to inrared. There is a lack o data on the properties o the irst phonon resonance at cryogenic temperatures. However, the value o permittivity compares well with excepted values, and the phonon requency compares well with other published values at room temperature[18]. It is likely that most loss measurements have not been sensitive enough to measure true absorption coeicients or loss tangent at 4 K (except in the case o the WG method), and we intend to veriy the calculated losses shown in ig 2 and 3. However we should note that it is well known that the sapphire loss tangent has 1/ dependence as shown. The idea o this experiment is to exploit the anisotropy o sapphire and construct a Dual Mode interrogation system or oscillator operating on a TE and TM mode at nearly the same requency, ie r [8]. The beat requency can then be monitored to test or the ine structure constant. For this case the sensitivity o the beat requency to a drit in ine structure constant is calculated rom (7) to be; Ε α = di Εdi Ε Ε, where = ( ) ( ) (10) r α

6 Figure 4 shows the sensitivity unction, Ε di, as a unction o requency. Figure 4 Calculated Ε di, as a unction o requency calculated rom (1) and (5) with the parameters presented in table I Because the modes are excited in the same crystal, some orm o common mode rejection o noise sources could be expected as discussed in [6]. Also, the dierence requency will be much lower than the resonance requency. This down conversion will lower the noise requirements o the readout o the beat requency. The key point to this analysis is that close to the phonon requency a sensitivity unction o order unity can be obtained with low loss. One would like to excite modes in the sensitive requency range o to Hz (300 to 30 µm wavelengths). The requency o a pure WG mode to irst approximation is given by; cm r = (11) 2πr ε Where c is the speed o light, m is the azimuthal mode number, r is the sapphire radius and ε is either the permittivity parallel (or a TM mode) or permittivity perpendicular (or a TE mode). For example, orthogonal WG modes could be excited in a 5 mm diameter sapphire cylinder with m varying rom 160 to 2600, and a Free Spectral Range o 4 to 6 GHz. Single mode sapphire clocks with low ractional requency instabilities o order to have been already demonstrated at microwave and optical requencies[19, 20]. This method should be possible to adapt to the suggested requencies and has the potential or high sensitivity tests o drits in the ine structure constant o order /yr as discussed in [6]. IV. PARAMAGNETICALLY DOPED SAPPHIRE At microwave requencies paramagnetic impurities supply transitions between electron spin states. This changes the susceptibility o the sapphire lattice, which is typically an anisotropic eect. The magnitude o the susceptibility and the requency o the Electron Spin Resonance (ESR) depend on the ine structure constant. In the ollowing we derive this dependence and calculate the inluence on the requency o a dielectric resonance in the sapphire crystal due to the inluence o the Cr 3+ ESR transition in sapphire. A. Determination o the Dispersion Added by Cr 3+ ions

7 Paramagnetic Cr 3+ ions in sapphire only change the magnetic susceptibility perpendicular to the c-axis. The orm o the complex susceptibility added by the ESR transition may be written as; χ ƒ χ χ ( ) χ ( ) ( )= + = j 1 j 2 2 (12) Q Q0 Here χ is the magnetic susceptibility, 0 is the requency o the ESR (11.45 GHz or Cr 3+ in sapphire[21]),q0 = 0 = π0τ 0, where 0 is the bandwidth o the ESR and τ 0 is the 0 relaxation time o the ESR. The loss added to the lattice can be characterized by the magnetic loss tangent, tanδ m and is given by; χ χ Q tanδm( ) = + χ 0 0 (13) 1 2 χ To evaluate the magnetic susceptibility we use data taken at the University o Western Australia by Mann and analyzed by Krupka[22]. The experiment consisted o analyzing the WG modes in a 5cm diameter sapphire cylinder with a ew parts per million impurity ions o Cr 3+, enclosed in a 8cm diameter copper cavity. The measured real part o the susceptibility is shown in igure 5. Mann and Krupka itted the data with χ and τ0 as ree parameters and calculated χ = ±2% and τ 0 = ±6% seconds. We point out here that itting to the real part o the permittivity can not give an accurate determination o the ESR relaxation rate or bandwidth as a it to the real part is very insensitive to this parameter. Thus we believe the quoted errors are likely to be wrong, especially so or τ 0. To obtain a more accurate it the magnetic loss tangent must be calculated. To calculate this eect, WG mode Q-actors, Q r, must be measured and the magnetic illing actor perpendicular to the c-axis, p m, calculated. In general the Q r o a TM dielectric resonance is given by; 1 R Qr = p s m tanδ m + (14) G Here R S is the surace resistance o the Copper cavity and G is the geometric actor o the mode. Thus to calculate tanδ m one needs to measure Transverse Magnetic modes with p m close to unity and a high G-actor. The highest G-actor TM modes are the undamental WGH m,0,0 mode amily. Only these modes are selected rom the measured modes o Mann and Krupka[22] shown in igure 5, as all other modes are either limited by the copper cavity or have a low sensitivity to the perpendicular magnetic ield. The Q-actors and illing actors o the measured WGH m,0,0 modes are shown in table II. Table II WGH m,0,0 mode amily data Mode Frequency p m G-actor Q-actor WGH 12,0, WGH 13,0, WGH 14,0, WGH 15,0,

8 I the data was only limited by the magnetic loss tangent one would expect an increasing Q-actor the urther rom the GHz resonance. This is not the case due to the cavity losses eecting the measurement due to smaller coninement or lower m numbers. Figure 6 clearly shows this eect, ignoring the R S, the eective tanδ m is plotted as a unction o requency. The two points closest to the GHz resonance are identiied to be limited by the magnetic loss tangent, while the two low requency points are limited by the copper surace resistance. Hence by undergoing a simultaneous it to the real susceptibility and the loss tangent, a more accurate determination o the ESR parameters can be obtained. We calculate χ = ±6% and 0 = , which gives a τ 0 = seconds. Note that the susceptibility is very close to Mann and Krupka's calculation, however the spin-spin relaxation rate was calculated to be a actor o 5 longer. Susceptibility Frequency [GHz] Figure 5 Magnetic susceptibility versus requency. Dots show calculations due to requency shits in WG modes (data taken rom Mann and Kruka[22]), while the bold line shows the best it rom (12) and (13). Eective Loss Tangent Figure 6 The magnetic loss tangent versus requency in Hz. Dots show calculated eective loss tangent due to Q-actors o the WGH m,0,0 mode amily, which is calculated assuming Rs =0 in (14). The bold line shows the best it rom (12) and (13). Note the two lower requency calculations diverge orm the predicted due to the smaller G-actors, which make the measurements limited by the surace resistance o the copper cavity. We have calculated rom (14) that a surace resistance o the order o 50 mω explains this eect.

9 In the case o magnetic spin-spin transitions, we show later that unlike the electronic case calculated previously[6], the susceptibility depends on α. Thus we take a more general approach to calculating the sensitivity. First we must relate the real magnetic susceptibility change to a requency change. For a dielectric resonator o resonant requency, r, with 100% o its energy conined in the dielectric, the simple relationship holds. dr 1 = r (15) dχ 21+ χ Using implicit dierentiation we can show the ollowing relationship holds; r Μ d d = α Μ α r χ where = (16) r α r dχ dα Here Μ is the sensitivity conversion o the mode requency with respect to ine structure constant. By combining (15) and (16) it ollows that; 1 dχ α Μ = (17) 2 dα 1+ χ To calculate the sensitivity conversion (17) must be evaluated over all parameters that vary with α. In the ollowing analysis we show that the susceptibility and ESR requency are highly dependent on α, and we do not consider any dependence on the spin-spin relaxation time as this will have no irst order eect on the requency o the ESR transition or the dielectric resonance. B. Cr 3+ ESR Frequency Dependence on α The Cr 3+ ion has three unpaired electrons, with the ree-ion ground state o 4 F 3/2. When the ion is present in a tetragonal lattice like sapphire, the non-cubic symmetry o the ield causes a zero applied ield splitting o the spin states (or Electron Spin Resonance) due to the crystal ield. Orton has used crystal ield theory [23] to calculate the energy associated with the transition between spin states using second order perturbation theory. The zero ield splitting was calculated to be; 2 8δλ E = E± 3/ 2 E± 1/ 2 2 (18) Here λ is the spin-orbit coupling, is the eective isotropic ield splitting between the ground ( 4 A 2 ) and irst excited state ( 4 T 2 ) o the 4 F orbital states and δ is the splitting o the irst excited state due to the anisotropic crystal ield. To calculate crystal ield eects, the potential experienced by an electron wave unction due to the nearest neighbor electrons is calculated[23]. From this calculation one can show the ollowing. b 2 4 e r e = Χ = Χ a r and δ 5 δ 4 5 (19) πεo a 4πεo a Here a is the atomic spacing perpendicular to the c-axis, b is the atomic spacing parallel to the c-axis, r 4 is the mean radius to the power o our o the electron wave unction and Χ and Χ δ are dimensionless constants o order unity. Now we use similar reasoning to Braixmaier et. al. [6], that the inter-atomic spacing and the wave unction size is proportional to the Bohr radius, a o, o the electron, and by substituting (19) into (18) we obtain; 2 a E o4πε λ o 2 (20) e The spin-orbit coupling and Bohr radius may be written in terms o the ine structure constant by;

10 2 4 4 h λ = mc e Z α and ao = (21) α me c By substituting (21) into (20) one may show that the requency o the transition is given by: c h o Z 8 6 α Λ = (22) Λ me c Here Λ is the Compton wavelength. Thus the requency o the paramagnetic resonance is proportional to the sixth power on α. Experimentally the resonance has been conirmed to be at GHz, and thus a microwave sapphire clock operating close to this requency is suited or this type o experiment. C. Cr 3+ ESR Susceptibility Dependence on α The susceptibility o the transition is proportional to the susceptibility given by [21]; 2 1 eh χ N µ B N ~ µ B a 3 ~ (23) o 2me c Here N is the number density o ions, k is Boltzmann's constant and µ B is the Bohr magneton. Thus, it is straightorward to show that; χ 4 α (24) Unlike the case o a phonon resonance, the susceptibility due to the spin-spin interaction is dependent on the ine structure constant. D. Sensitivity o Cr3+ Doped Sapphire to α Braxmaier et. al. [6] considered electronic transitions in doped ionic crystals to increase the electronic dispersion in the crystal. They assumed that the normalized sensitivity (in our case Μ ) was unity, in the ollowing we do some similar analysis or doped sapphire at microwave requencies. By combining (22) and (24) with (17) the normalized sensitivity o a dielectric mode in doped sapphire is given by; χ 3 2χ Μ ( ) = (25) 1 + χ For the doped sapphire investigated in this paper, the sensitivity and inverse magnetic loss tangent as a unction o requency is plotted in ig 7 and 8. The sensitivity is unity at the resonance requency with a loss tangent o order The sensitivity could be urther enhance with the deliberate addition o more Cr3+ ions (ruby). However, the enhancement in sensitivity compromises the loss tangent o the material. This means a better lock to the resonance is needed or the experiment to work. Another option is to ind a transition in sapphire with a longer relaxation time.

11 Q sapphire ~ 1 Tanδ Figure 7 Calculated magnetic Q-actor limit versus requency or a TM mode in the doped sapphire presented in this paper. ( ) Μ Sensitivity Figure 8 Calculated sensitivity, Μ, as a unction o requency using (25) and (12) or the doped sapphire presented in this paper V. DISCUSSION The simple requirement or this technique requires a low-loss crystal resonator with large dispersion. The two requirements are in act contradictory due to the Kramers-Kronig relations between the real and imaginary part o a resonance system. We have shown that the irst phonon resonance at Hz is in inherently low loss at 4 K and may oer the best requency or this type o measurement. At microwave requencies zero ield splitting o the ESR due to Cr 3+ ions in the sapphire lattice signiicantly enhances the dispersion and hence sensitivity. However we show that due to Kramers Kronig relation Cr 3+ adds magnetic losses due to a short spin-spin relaxation time o s. Improvement o this technique requires the addition o paramagnetic impurities with longer spin-spin relaxation times. Braxmaier et. al. [6] showed a similar technique at optical requencies is capable o measuring drits in the Fine Structure Constant o the order /year. We have proposed two new experiments that can achieve a similar result at lower requencies.

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