Resonant Regeneration in the Sub-Quantum Regime -- A demonstration of fractional quantum interference
John G. Hartnett, Joerg Jaeckel, Rhys G. Povey, Michael E. Tobar
aa r X i v : . [ qu a n t - ph ] J a n IPPP/10/92; DCPT/10/184
Resonant Regeneration in the Sub-Quantum Regime – A demonstration of fractional quantum interference
John G. Hartnett , Joerg Jaeckel , Rhys G. Povey and Michael E. Tobar School of Physics, University of Western Australia,Crawley W.A., Australia Institute for Particle Physics Phenomenology, Durham University,Durham DH1 3LE, United Kingdom
Abstract
Light shining through wall experiments (in the optical as well as in the microwaveregime) are a powerful tool to search for light particles coupled very weakly to pho-tons such as axions or extra hidden sector photons. Resonant regeneration, wherea resonant cavity is employed to enhance the regeneration rate of photons, is one ofthe most promising techniques to improve the sensitivity of the next generation ofexperiments. However, doubts have been voiced if such methods work at very lowregeneration rates where on average the cavity contains less than one photon. Inthis note we report on a demonstration experiment using a microwave cavity drivenwith extremely low power, to show that resonant amplification works also in thisregime. In accordance with standard quantum mechanics this is a demonstrationthat interference also works at the level of less than one quantum. As an addi-tional benefit this experiment shows that thermal photons inside the cavity causeno adverse effects. γX X
Figure 1: Schematic of a “light shining through wall” experiment. An incoming photon γ is converted into a new particle X which interacts only very weakly with the opaquewall. It passes through the wall and is subsequently reconverted into an ordinary photonwhich can be detected. One of the most intriguing possibility for physics beyond the Standard Model is theexistence of new light particles coupled only very weakly to the known particles. Indeed,such weakly interacting slim particles (WISPs) seem to be a feature of many extensions ofthe Standard Model based on field and string theory [1–12]. Prominent examples of theseare axions and extra U(1) gauge bosons. Several astrophysical puzzles could be explainedby the existence of such particles [13–26] giving more than just theoretical motivation forfuture experimental searches.Already the last few years have seen dramatic improvements in laboratory searches.Most of this progress [27–39] has been obtained using the so-called light-shining-through-walls (LSW) [1, 40, 41] technique shown in Fig. 1. The idea is as follows. If an incomingphoton is somehow converted into a WISP the latter can transverse an opaque wall withoutbeing stopped. On the other side of the wall the WISP could then reconvert into a photon .The “light” doesn’t have to be optical but could, for example, be electromagnetic wavesat microwave frequencies [37, 38, 43, 44].An important next step forward, promising many orders of magnitude improvementin sensitivity, will be the introduction of high quality resonators, i.e. cavities, both inthe production and in the regeneration regions [43–46]. In the production region this hasalready been pioneered by the ALPS experiment [36]. A high quality factor enhancesthe production and regeneration probability by the factor of the number of passes, N pass ,the light makes through the cavity. On the production side of the experiment this en-hancement simply arises due to the fact that the light has a chance to be converted ineach pass. Or, from a different point of view there is simply more light inside the cavity In principle it does not have to be exactly one WISP traversing the wall. The photon could alsoconvert into a number of virtual WISPs which after the wall recombine into a photon [42]. For examplethis could happen for minicharged particles. It should be noted that BFRT also used mirrors to enhance the production probability [28], but thesetup used was an optical delay line and not a cavity. N . However, only a fraction of this power ∼ /N pass leaves thecavity. In total the detectable power output of the cavity is therefore enhanced by N pass .The enhancement in this “resonant regeneration” is based on the positive interferenceof the photon wave inside the cavity. Now, one can ask whether this interference still worksif the regenerated power is so low that on average there is less than one photon insidethe cavity? [47] This is the main question addressed in this note. We begin with a shorttheoretical argument and then describe an experiment to demonstrate this sub-quantuminterference. Beyond its purpose to explicitly demonstrate that resonant regenerationworks even in the regime where the cavity contains less than one photon this experimentalsetup can also be viewed as a nice and simple demonstration of “less than one photoninterfering with itself”. In a way this experiment is an alternative version of a double slitexperiment [48] demonstrating interference on the single quantum level.Finally, it should be noted that as we are only concerned with the regeneration sidethe following arguments and demonstration not only apply to LSW experiments but theyare, of course, equally applicable to axion dark matter searches [49, 50] which rely on anexternal WISP source and consist of just the regeneration part of an LSW experiment. The first question one might have, of course, is if this is really relevant for current or nearfuture LSW experiments. To see that this is the case let us calculate the power output ofa cavity filled with 1 photon. Let us first consider the ideal case where the only way thecavity loses energy is by photons leaving the cavity towards the detector. In this case we2ave P out = P loss = ω E stored Q = ~ ω Q (2.1)= 4 . × − W (cid:18) f GHz (cid:19) (cid:18) Q (cid:19) = 1 . × − W (cid:16) µ m λ (cid:17) (cid:18) F (cid:19) (cid:16) m ℓ (cid:17) , where Q is the quality factor of the cavity and ω is the angular frequency of the light. Inthe second and third line we have calculated the power for typical values for a microwaveresonator of frequency f and for an optical system with wavelength λ and a Fabry-Perotcavity of finesse F and length ℓ . In both cases the power at which the sub-quantumregime is reached is well within achievable detector sensitivities.Therefore the question what happens when the power levels drop into the region whenthere is less than one quantum inside the cavity is indeed relevant for future experiments.Let us now give a brief argument why we think that resonant amplification also worksin this regime. To be specific let us consider an ideal laser generating a plane wave offixed frequency. This then turns into a plane wave of WISPs which arrive at our detectorcavity where a plane wave of regenerated photons appears. The plane wave of regeneratedphotons is a momentum (and energy) eigenstate and as such we have no information onwhere the photon actually is. Accordingly interference can happen between different“parts” of one photon (and this is all we need for the resonant amplification to occur).This is in complete analogy to what happens in a double slit experiment, where oneobserves interference even when at any given time only “half” of a photon goes througheach slit. Interference in a cavity is indeed very similar to the famous double-slit experiment [48].To see this let us consider a Fabry-Perot cavity as an example (in other resonators theresults would be analogous). Let us shine a plane wave of light at an angle θ into ourcavity as depicted in Fig. 2. The total transmitted amplitude will then be the sum of allthe transmission amplitudes T k where k denotes the number of passes through the cavity, T trans = T + T + . . . = T ∞ X k =0 R k exp(i kδ ) = T − R exp(i δ ) , (2.2) To be precise the Q here is the loaded quality factor of the cavity. In the idealized case consideredin this section all “losses” are due to the photons leaving the cavity towards the detector. Therefore thecoupling to the detector determines the loaded Q . T = (1 − R ) is the transmission amplitude in one pass, R is the amplitude forreflection back to the exit mirror and exp(i δ ) accounts for the phase accumulated in onepass through the cavity. The transmission probability is given by | T trans | .As we can see the transmission amplitude is exactly the sum over all possible paths/passesjust as in a double slit experiment. The phase difference between the passes is given by δ = 2 πλ nℓ cos( θ ) , (2.3)where λ is the incident (vacuum) wavelength, l is the spacing between the mirrors and n is the refractive index. For real R (pure reflection) close to 1 the transmission is stronglypeaked for δ = 0. This can be achieved by varying θ and we obtain a spatial interferencepattern just as in the original double slit experiment. Alternatively we can, however, alsoobserve the same interference pattern by varying the wavelength, i.e. the frequency.The higher the reflectivity R , i.e. the closer it is to 1 the stronger the interference pat-tern is peaked. Now, since N pass ≈ / (1 − R ), R closer to 1 means that more paths/passeseffectively contribute to the geometric sum, Eq. (2.2). In other words the (inverse) widthof the interference peak is a measure of the number of paths/passes that interfere. Thetransmission probability is given by (for simplicity we use cos( θ ) = 1 and n = 1 from nowon), | T trans | ( ω ) = 11 + R (1 − R ) (1 − cos( δ )) (2.4) ≈
11 + 4 (cid:16) ω − ω res ω res (cid:17) (cid:16) ωℓ/c − R (cid:17) = 11 + 4 Q (cid:16) ω − ω res ω res (cid:17) , for δ, (1 − R) ≪ , and the width ∆ ω of this resonance curve is∆ ωω = 1 − Rωℓ/c = 1 N pass ωℓ/c = 1 Q for (1 − R ) ≪ . (2.5)In the last equalities we have used that the quality factor is also directly related to thewidth of the resonance curve. So far we have completely ignored the issue of thermal noise. The thermal noise spectrumof a cavity is given by, dP noise , out dω = 12 π ~ ω exp (cid:16) ~ ωk B T (cid:17) − Q (cid:16) ω − ω res ω res (cid:17) . (2.6)4 T R R nℓ θ Figure 2:
Light path inside a Fabry-Perot cavity.
Integrating over frequencies we obtain the total noise power coming out of the cavity, P noise , out = ~ ω Q
14 1exp (cid:16) ~ ω res k B T (cid:17) − , for Q ≫ . (2.7)Note that the last factor is basically the Bose-Einstein occupation number of countingthe thermal photons inside the cavity whereas the first factor is the same as in Eq. (2.1).Therefore, if the number of photons inside the cavity is large, the thermal noise becomesrelevant, even before we reach the regime where there is less than 1 “signal” photon insidethe cavity.The occupation number strongly depends on the frequency and the temperature. Inthe optical regime ~ ω ∼ k B T ∼ /
40 eV the ther-mal noise is highly suppressed due to the exponential factor. However, in the microwaveregime we have ~ ω ∼ (1 − µ eV ≪ / P noise = dP noise , out dω δω, (2.8)where δω = Max( δω resolution , δω signal ) , (2.9)and δω resolution is the frequency resolution of the measurement and δω signal is the bandwidthof the signal.Note that the frequency resolution of the measurement is directly related to the mea-surement time t measure , δω resolution & πt measure . (2.10)In order to measure the transmission probability, Eq. (2.4), at powers in the(sub-)quantum regime, Eq. (2.1), we need to make sure that the resolution bandwidth issufficiently small. In the limit k B T ≫ ~ ω we need, δf resolution = δω resolution π ≤ ~ ω k B T Q (cid:18) SN (cid:19) − = 10 Hz (cid:18) f GHz (cid:19) (cid:18) KT (cid:19) (cid:18) Q (cid:19) (cid:18) SN (cid:19) − , (2.11)where S/N denotes the desired signal to noise ratio.
In the previous section we have seen that the small width of the resonance curve arisesbecause a large number of passes interferes which each other. Indeed the width of theresonance curve is a direct measure of the number of passes effectively contributing to theinterference.As discussed in the introduction the (positive) interference of a large number of passesis what is needed for resonant regeneration. Therefore, if we find a narrow resonancecurve even at very low input power (such that on average there is less than one photoninside the cavity) we establish this necessary prerequisite for resonant regeneration.6igure 3: Experimental setup to measure the resonance curve of a cavity at very lowinput power and correspondingly low power stored inside the cavity. The power outputof the generator is fed through a series of attenuators to achieve the desired low incidentpower into the cavity. The output signal is amplified using a very low noise amplifier andmixed (multiplied) with a fixed frequency signal from a second generator to down convertit to baseband. The resulting low frequency signal is then filtered, further amplified andanalyzed using a FFT spectrum analyzer.
From the above discussion we see that in order to demonstrate the necessary interferencefor resonant regeneration we need to measure the resonance curve at very low input powerssuch that on average there is less than one photon inside the cavity.To do this we used an experimental setup (as shown in Fig. 3) in the microwavefrequency range f ∼
10 GHz. At first glance this seems more difficult as the power belowwhich the experiment would be in the sub-quantum regime is smaller for lower frequencies(see Sect. 2.1). However, there are two advantages which more than offset this. Firstly,in the microwave regime even very low power of the order of 10 − W is well within thedetectable range. Secondly, and perhaps even more importantly, microwave generatorscan be tuned in frequency such that we can easily sweep through a band of frequenciesaround the resonance and measure the resonance curve.Therefore, we used the setup shown in Fig. 3 which allowed us to measure the resonancecurve of a fixed cavity at different levels of input power, corresponding to different numbersof photons inside the cavity. 7ur setup (shown in Fig. 3) consisted of a microwave generator which we used to gen-erate a high resolution signal with variable frequencies between 9 .
588 GHz and 9 .
593 GHz.Note, that although the frequency of the generator is variable, once a specific frequencyis chosen, the frequency width of the generated signal is extremely narrow and can beneglected in the analysis.The generated output goes through a chain of attenuators which reduce the signalpower by more than 12 orders of magnitude ( −
124 dB). This allowed us to achieve thevery low power levels necessary to probe the quantum regime. This signal was then fedinto a copper cavity (with resonance frequency f res = 9 .
590 GHz) for which we intendedto observe interference by measuring the resonance curve.To avoid any drift in the cavity resonance frequency the cavity was placed in a vacuumchamber and kept at a stable temperature T = 305 . .
584 GHz signal from a secondgenerator and filtered through a low pass filter to obtain a signal in the MHz range whichwas analyzed on a FFT spectrum analyzer.To achieve a sufficient reduction of the thermal noise we need sufficiently good fre-quency resolution on our spectrum analyzer. For the lowest input power we chose δf resolution = 1 Hz Before analyzing our results we note that to account for the finite coupling of the cavityon the input and the output ports we have to slightly modify Eqs. (2.4) and (2.1). Forthe transmitted power we have, P trans = P inc β β (1 + β + β )
11 + 4 Q L (cid:16) ω − ω res ω res (cid:17) , (3.1)where β and β are the couplings on the input and output ports, respectively, and Q L isthe loaded Q-factor. The energy stored inside the cavity is given by, E stored = P inc Q L ω β (1 + β + β )
11 + 4 Q L (cid:16) ω − ω res ω res (cid:17) . (3.2)8 ‰ Γ - - - - - ~ Γ~ Γ~ Γ - - - - - f @ GHz D P ou t ± . @ d B m D Figure 4: Measured resonance line shapes for various input powers . From top to bottomthe input power was −
55 dBm, −
125 dBm, −
135 dBm, and −
145 dBm. In the given setupthis corresponds to an average of ∼ × , ∼ ∼ .
3, and ∼ .
03 photons inside thecavity when the cavity is on resonance. The upper plot gives a check of the classical limitwhen the input power is high and the number of quanta is large. Comparing the lowerthree curves to this, we can see that the resonance curve has the same shape even whenthe number of photons inside the cavity is low, thereby demonstrating that interferenceis present also in this situation. We note that the output power is somewhat lower than expected from Eq. (3.1). This is probablydue to unaccounted for line losses. in [dBm] γ in cavity Q L -55 ∼ × ∼ ∼ . ∼ .
03 8200Table 1: Measured values for the loaded Q L at different input power. The input couplingcoefficient of the cavity was β = 0 . ± .
05 and for the output was β = 0 . ± .
05. Wenote that the errors in the determination of Q L are relatively large (of order 10% − Q L ∼ ∼ −
55 dBm=10 − . W. Using Eq. (3.2) we can see that this power corresponds to an averge of approx-imately 3 × photons in the cavity, certainly in the classical regime. The three lowercurves correspond to lower input powers in the quantum regime: −
125 dBm, −
135 dBm,and −
145 dBm corresponding to ∼ ∼ .
3, and ∼ .
03 photons inside the cavity. Al-ready on inspection the curves at very low input power are quite similar to the “classical”curve suggesting that interference works as expected. To check this we have fitted theoutput power to a Lorentzian curve. The results of our curve fits are given in Tab. 1.As we can see the measured Q L are in reasonable agreement within the uncertainty of∆ Q L ∼ Q L = 6100 is also in reasonable agreement with the other measurements. Forthis measurement we chose a frequency resolution of δf resolution = 625Hz. Comparing withEq. (2.11) we see that using this resolution the noise power in each frequency intervalactually corresponds to less than one thermal photon inside the cavity per frequency bin.In this sense the thermal noise spectrum itself can be viewed as a test of the interferencewith less than one photon inside the cavity. In this note we have investigated the question of whether resonant regeneration in light-shining-through-wall experiments still work in the regime where there is on average lessthan one photon inside the regeneration cavity. We have argued that the desired enhance-10 .587 ´ ´ ´ ´ ´ ´ ´ ´ - - - - - - - f @ GHz D PS D @ d B m (cid:144) H z D Figure 5: Measurement of the thermal noise spectrum. The frequency resolution was δf resolution = 625Hz. The red fitted curve is a Lorentzian fit with Q L = 6100.ment is directly related to the (positive) interference of the photons in the cavity. Thewidth of the resonance of a cavity or its quality factor, Q , are a direct measure of thisinterference. As a demonstration we have measured the Q -factor of a microwave cavityat different power levels from the classical regime, with many photons inside the cavity,to the “quantum” regime, with few or even less than one photon inside the cavity. Themeasured Q values are in reasonable agreement, demonstrating that the desired degree ofinterference is present even at very low power levels. This is in agreement with standardquantum mechanical arguments. Indeed, measuring the transmission curve of a cavityis very similar to a classic double (or better yet, multiple) slit experiment. Interferencetakes place between photons doing one pass, two passes and so on inside the cavity. Andthe transmission curve is the interference pattern in frequency space. In this sense oursetup can also be viewed as an alternative version of the double slit experiment.We have also discussed the role of thermal photons and noise. For frequencies in themicrowave range at room temperature the number of thermal photons inside the cavityis larger than 1 since k B T ≫ ~ ω . Our measurements show that this does not causeany adverse effects to interference, in agreement with the fact that photons only weaklyinteract with each other. 11 cknowledgements J. J. would like to thank Andrei Afanasev, Holger Gies and Axel Lindner for interestingdiscussions. Moreover, J. J. would like to than the Frequency Standards and MetrologyResearch Group at the University of Western Australia for their hospitality. This work issupported by the Australian Research Council grant DP1092690.
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