We propose a new approach to axion detection based on precision frequency measurements as opposed to the traditional power detection techniques[arXiv:1806.07141]. The approach utilises a high Quality Factor cavity supporting two mutually orthogonal modes. We demonstrate how axion modified Maxwell equations lead to either a beam splitter or parametric interaction terms in the axion up- or downconversion cases respectively. The term couples two modes of different frequencies with the axion frequency (mass) being either the difference (upconversion) or sum (downconversion) of the frequencies of the modes. The derivation introduces a unitless two-mode geometric coefficient characterising the coupling between two particular modes. The Hamiltonian term in the rotating wave approximation is proportional to axion complex amplitude and the axion-photon coupling constant.
The double mode cavity could be used for the traditional power detection when one or both modes are strongly pumped. Although such a method would be inefficient compared to the common DC magnet technique. On the other hand a possibility to employ a highly sensitive cross correlation technique may lead to comparable results. Instead of measuring tiny amounts of power deposited in the cavity modes, we propose to measure frequency shifts associated with the axion coupling terms. Based on the equation of motion of axion coupling modes, we calculate frequencies shifts of the modes that can be observed with one of a few frequency control techniques. We predict both real and imaginary parts of the resonance frequency to be sensitive to axions depending on the type of coupling. We also calculate transfer functions from axion induced couplings to phase noise of two orthogonally polarised modes in both open loop and closed loop regimes. Based on these transfer functions, we are able to estimate axion sensitivity of the dual mode axion detection approach. Even a room temperature microwave oscillator realisation of the proposed techniques will result in new limits on the coupling between axions and photons. The cryogenic realisation, if implemented, could exclude axion-photon couplings below the predicted DFSZ coupling. The power of the new approach relies on the fact that unlike in the power detection method where the sensitivity is limited by the thermal (or quantum) noise in the readout, the frequency sensing is limited by resonator linewidths and their internal fluctuations. For modern cryogenic microwave resonators, Quality factors exceed $10^9$ giving fractional frequency stability better than $10^{-16}$ have been demonstrated. Such levels of frequency stability provide better signal-to-noise ratio that the best power detection techniques. The main advantages of the proposed frequency control
method are: 1. magnet-free. Unlike traditional haloscopes, the proposed method does not require a strong DC magnetic fields; 2. SQUID-free. All sensitivities calculated in this work are based on usage of traditional low noise semiconductor amplifiers. Though superconducting technology might be used in the future, its presence is not crucial contrary to traditional methods; 3. cavity volume independence. Although cavity volume influences many parameters of the experiment such as resonance frequencies and quality factors, the sensitivity is not directly proportional to this parameter unlike in traditional haloscopes. This removes the major obstacle for higher mass ($f_a >10$GHz) axion searches. Moreover, optical cavities might be used to probe otherwise unaccessible regions of THz and infrared spectrum as well as millimiter-wave and microwave frequencies; 4. Liquid-Helium temperature operation (> 4K) where only a limited number of components such as cavity and amplifiers have to be at low temperature. This factor removes the need of dilution refrigeration that is a key component in traditional haloscopes making the whole experiment available to a broader audience. Although dilution refrigeration might give some incremental improvement in the axion search, all ultra-stable microwave and optical clocks and oscillators do not require temperatures below 4K. 5. access to higher and lower frequency ranges. The fact that actual axion mass is either the sum or difference of working frequencies opens a possibility to search for axions in less accessible frequency rages. For instance, working around 20GHz, one is able to probes axion masses in the vicinity of 40GHz, where experiments are significantly more difficult; 6. limited power levels. Although, the sensitivity does explicitly depend on power levels, on the current calculation only limited power levels are used unlike in some other proposals; 7. axion phase sensitive. Comparing to DC magnet haloscopes, the dual frequency method is able to provide additional information about axions, particularly its phase relative to pump signals, although that might lead to more complicated detection schemes; 8. KSVZ/DSFZ achievable. It is estimated that the cryogenic dual mode experiment is able to achieve the limit of the widely accepted axion dark matter models. On the other hand, even a tabletop search may lead to competitive limits on dark matter; 9. broadband search for low mass axions is possible. It is demonstrated that a wideband search that does not require tuning is possible.