Ultra-High Cooperativity Interactions between Magnons and Resonant Photons in a YIG sphere
Resonant photon modes of a 5mm diameter YIG sphere loaded in a cylindrical cavity in the 10-30GHz frequency range are characterised as a function of applied DC magnetic field at millikelvin temperatures. The photon modes are confined mainly to the sphere, and exhibited large mode filling factors in comparison to previous experiments, allowing ultrastrong coupling with the magnon spin wave resonances. The largest observed coupling between photons and magnons is GHz for a 15.5 GHz mode, corresponding to a cooperativity of . Complex modifications beyond a simple multi-oscillator model, of the photon mode frequencies were observed between 0 and 0.1 Tesla. Between 0.4 to 1 Tesla, degenerate resonant photon modes were observed to interact with magnon spin wave resonances with different couplings strengths, indicating time reversal symmetry breaking due to the gyrotropic permeability of YIG. Bare dielectric resonator mode frequencies were determined by detuning magnon modes to significantly higher frequencies with strong magnetic fields. By comparing measured mode frequencies at 7 Tesla with Finite Element modelling, a bare dielectric permittivity of of the YIG crystal has been determined at about .
Hybrid photon-magnon systems in ferromagnetic spheres have recently emerged as a promising approach towards coherent information processing Chumak et al. (2015); Tabuchi et al. (2015, 2014); Zhang et al. (2014); Goryachev et al. (2014a); Bai et al. (2015); Huebl et al. (2013); Zhang et al. (2015a). Due to the large exchange interaction between spins in ferromagnets, they will lock together to form a macrospin that can be utilised for coherent information processing protocols Imamoğlu (2009); Zhang et al. (2015a). The quantised excitation of the collective spin is referred to as a magnon. Yttrium Iron Garnet (YIG) based magnon systems are attractive due to very high spin density, resulting in significant cooperativity as well as relatively narrow linewidths Goryachev et al. (2014a); Zhang et al. (2015b); Osada et al. (2015); Zhang et al. (2014). Furthermore, due to the possibility of coupling magnon modes to photons at optical frequencies Zhang et al. (2015b); Osada et al. (2015); Shen and Bloembergen (1966); Demokritov et al. (2001), magnon systems may be considered as a candidate for coherent conversion of microwave and optical photons Zhang et al. (2015b); Osada et al. (2015). In addition, magnons interact with elastic waves Kittel (1958); Zhang et al. (2015c) opening a window for combining mechanical and magnetic systems. These systems therefore possesses great potential as an information transducer that mediates inter-conversion between information carriers of different physical nature thus establishing a novel approach to hybrid quantum systems Tian et al. (2004); Imamoğlu (2009); Verdú et al. (2009); Xiang et al. (2013).
Among all magnon systems the central role is devoted to YIG, a material that possesses exceptional magnetic and microwave properties and has been used in microwave systems such as tuneable oscillators and filters for many decades Cherepanov et al. (1993); Gurevich (1963). Although, only recently Soykal and Flatté proposed and modelled the photon-magnon interaction based on YIG nano-spheres with application to quantum systems Soykal and Flatté (2010a, b). As predicted by the authors, extremely large coupling rates, , could be achieved in YIG spheres, which is favourable for coherent information exchange and has been demonstrated experimentally later Goryachev et al. (2014a); Kostylev et al. (2015); Zhang et al. (2014). For these experiments, the interaction is observed between photon and magnon resonances created correspondingly by photon cavity boundary conditions and spin precession under external DC magnetic field. A commonly used method is to place a relatively small YIG sphere in a local maxima of the magnetic field inside a much larger microwave cavity. This is done to achieve quasi uniform distribution of the cavity field over the sphere volume to avoid spurious magnon modes. Cavities can take oval Tabuchi et al. (2014); Zhang et al. (2014) or spherical shapes Zare Rameshti et al. (2015); Soykal and Flatté (2010a, b), and even re-entrant cavities with multiple posts have been used in an attempt to focus the microwave energy over the sphere Goryachev et al. (2014a); Kostylev et al. (2015). In this work we investigate a completely different regime in which the magnon and photon wavelengths are comparable, leading to considerably larger coupling strengths, but additional couplings to higher order modes. In general, for this case the strength of the photon-magnon interaction will be determined by an overlap integral of the two respective mode shape functions. Given the magnon mode shape is limited to the sphere’s volume, this integral will be maximised when the photon mode is confined to the same volume. To achieve the latter, we utilise an exceptionally large YIG sphere with diameter mm, matching magnon and microwave photon mode volumes, unlike previous microwave cavity experiments.
In order to investigate this regime we use common microwave spectroscopy techniques Farr et al. (2013); Goryachev et al. (2014b); Bourhill et al. (2013, 2015a); Goryachev et al. (2014a); Kostylev et al. (2015) to directly observe the mode splitting caused by the magnon interaction to determine the coupling values. Similar systems have been extensively utilised not only in the field of spintronics to investigate the interaction between microwave photons and paramagnetic spin ensembles Farr et al. (2013); Bourhill et al. (2015a); Goryachev et al. (2014b); Bourhill et al. (2013); Farr et al. (2014), but also to realise optical comb generation Del/’Haye et al. (2007), ultra-low threshold lasing Kippenberg et al. (2006), cavity-assisted cooling, control and measurement of optomechanical systems Aspelmeyer et al. (2014); Bourhill et al. (2015b), and extremely stable cryogenic sapphire oscillator clock technology Ivanov and Tobar (2009a, b). To date, exciting internal, highly-confined photonic modes in a YIG sphere has only recently been demonstrated in the optical regime using Whispering Gallery Modes (WGM) Zhang et al. (2015b); Osada et al. (2015) but has never before been achieved in the microwave domain. This is due to the typical sub-millimetre diameter of the spheres. As such, interactions with magnons must be observed via Brillouin scattering Serga et al. (2012), which has yielded high quality factors and also demonstrated a pronounced nonreciprocity and asymmetry in the sideband signals generated by the magnon-induced scattering.
Extremely large mode splittings () cause simultaneous coupling to a higher density of modes, with an overlap of avoided level crossings. Therefore, the model proposed by Soykal and Flatté Soykal and Flatté (2010a) becomes no longer applicable, as it assumes the interaction occurs between a single photonic and magnon mode. More recently, a paper by Rameshti et al. Zare Rameshti et al. (2015) simulated a similar scenario of the presented experiment, in which the ferromagnetic sphere is itself the microwave cavity. Our observed results may appear to be in good agreement with this work’s predictions, however, what is apparent is that in this specialised case, one must consider more than just the magnetostatic, uniform Kittel magnon mode, a limitation of Zare Rameshti et al. (2015). Indeed, due to the nonuniformity of both the microwave mode magnetic field energy density across the sphere, which is unique to this experiment, and the nonuniformity of the sphere parameters arising due to cryogenic cooling, the assumption that only the uniform Kittel magnon resonance participates is no longer valid. Despite this, in this paper we use a two mode model to obtain estimations of coupling strengths, and demonstrate how this results in inconsistent susceptibility values.
Ii Physical Realization
The mm YIG sphere was manufactured by Ferrisphere, Inc. with a quoted room temperature saturation magnetisation of T. It is placed on a small sapphire disk, with a concavity etched out using a diamond tipped ball grinder, to keep the sphere from rolling out of position, and reduce dielectric losses that would arise if the YIG were in direct contact with the conductive copper housing. Sapphire was chosen over teflon as an intermediary between the YIG and copper to improve the thermal conductivity to the sphere.
Together, the sapphire and YIG are housed inside a copper cavity with dimensions specified in figure 1. A loop probe constructed from flexible subminiature version A cable launchers is used to input microwaves and a second is used to make measurements, allowing the determination of parameters. The entire cavity is cooled to about 20 mK by means of a dilution refrigerator (DR) with a cooling power of about 500 W at 100 mK. The cavity is attached to a copper rod bolted to the mixing chamber stage of the DR that places it at the field center of a 7 T superconducting magnet, whose applied field is oriented in the direction of the cavity. The magnet is attached to the 4 K stage of the DR, with the copper cavity mounted within a radiation shield of approximately 100 mK that sits within the bore of the magnet.
Iii Experimental Observations
The transmission spectrum of the YIG was recorded for DC magnetic fields swept from 0 – 7 T using a vector network analyser (VNA), with partial results shown in figure 2. A host of magnon resonances/higher order magnon-polaritons can be observed originating from (0 T, 0 GHz) with an approximate gradient of 28 GHz/T. The more-or-less horizontal lines approaching the magnon resonances from either side correspond to resonant photon modes of the sphere. Importantly, we can observe that in the dispersive regime, far removed from any microwave resonant mode, there still exist multiple magnon modes. We observe that the anticrossing gaps are populated by unperturbed modes, which are remnant “tails” of both “higher” and “lower” mode interactions, as predicted by Rameshti et al. in the ultrastrong coupling regime Zare Rameshti et al. (2015).
For the remainder of this paper, we will focus on the six lowest frequency photon modes, whose resonant frequencies may only be accurately determined at large magnetic fields, when the entire spin ensemble has been detuned, as shown by figure 3.
The modes have been categorised into three distinct classes: mode “” is the lowest frequency and lowest factor mode, the two highest factor modes; “” and “”, and the three remaining highest frequency modes; “1”, “2” and “3”. Their asymptotic frequencies as T are summarised in table 1.
The behaviour of these modes as the magnon resonances are tuned via the applied magnetic field is shown in figure 4. It has been shown previously Goryachev et al. (2014a) that a standard model of two interacting harmonic oscillators can accurately determine the coupling values from such avoided crossings. However, we observe strong distortion around 0 T, and also an asymmetry of the mode splittings about the central magnon resonances due to the ultrastrong coupling of the photon modes to the magnon modes, as was observed previously in Ruby Farr et al. (2014). Therefore we fit only the curves to the right of the magnon resonance. These fits are shown as the dashed lines in figure 4. From these fits we can approximate the values of for each mode, as summarised in table 1. The linewidths, and frequencies, of the photon modes are determined from the transmission spectra taken at high field values (figure 3 (b)), whilst the magnon linewidth, can be determined by analysing the transmission spectra in the dispersive regime. We take a frequency sweep at T from 5.75–9 GHz in order to view the magnon resonance peaks far away from any interaction with the dielectric microwave modes, as shown in figure 5. There is a level of variation amongst the magnon linewidths as calculated by fitting the peaks with Fano resonance fits, as shown in figure 6. This variation and the presence of multiple peaks demonstrates the presence of higher order magnon modes. Taking the average and standard deviation of these linewidths gives a final estimate of magnon linewidth as MHz. Cooperativity is calculated as .
The cooperativity values in table 1 demonstrate that all modes are strongly coupled to the magnons, and all with the exception of modes 2 and 3 are in the ultrastrong coupling regime (i.e. Zare Rameshti et al. (2015)). The largest cooperativity value obtained is that of mode , which is, to the authors’ knowledge, the largest value ever reported to date in any previously studied spin system.
A transmission spectrum taken at T is shown in figure 7, demonstrating the mode splitting of mode 1, symmetric about the magnon resonance. Overlaid in red is the bare photon resonance at 7 T, i.e. the microwave mode unperturbed by the magnon modes. From this red curve, is determined to be 5.355 MHz, as shown in table 1. When one takes the average of MHz and , one obtains the line width of the resulting hybrid state when the magnon resonance is tuned coincident in frequency with the photon mode, as depicted by the dashed blue curve in figure 7, i.e., 4.4 MHz. This excellent agreement indicates that at this particular field, mode 1 exists as a hybrid magnon-polariton.
Around T, we observe a severe distortion of the cavity mode frequency dependence on magnetic field as demonstrated by Fig. 8. Around 16 GHz, we see there exist five modes, corresponding to modes , and 1–3, on the “left” side of the magnon resonances. These modes have been given a primed nomenclature to indicate their existence at fields lower than that required to tune the magnons to their frequencies. This phenomenon has been previously observed in single crystal YAG Farr et al. (2015) highly doped with rare-earth Erbium ions, and is explained by the influence on the ferromagnetic phase of the impurity ions on degenerate modes. The effect can be explained by the influence of the ensemble of strongly coupled spins on the centre-propagating waves of the near degenerate mode doublet. For large spin-photon interactions, tails of Avoided Level Crossings (ALCs) from the positive half plane () should still exist on the negative half plane () and vice a versa. Although, instead of gradual change of direction, the system demonstrates an abrupt transition to a “no coupling” state. It is worth mentioning that such effect has not been observed in photonic systems interacting with paramagnetic spin ensembles Farr et al. (2013); Goryachev et al. (2014b, c). In the present case, the effect is much more pronounced, with fractional frequency deviations and the magnetic field range of the effect both orders of magnitude larger than observed previously Farr et al. (2015), a result of the magnetic spin density.
COMSOL 3.5’s electromagnetic package was used to model the system. A 3D model was used so as to analyse the degeneracies in the axis of the dielectric modes. The internal copper wall of the cavity is modelled as a perfect electrical conductor, which for the purposes of the desired eigenfrequency study, is an appropriate simplification.
The results of the FEM using a value of and mm, where is the radius of curvature of the sapphire support’s concavity, are summarised in figure 10 and in table 2. The measured frequency of the doublet modes has been taken as the average of the two constituent’s frequencies at T.
From the FEM and the analytical mode shapes of spherical dielectric resonances described by le Floch et al. (2006), we can identify mode as an mode with no degeneracy. Therefore it is present as a singular resonance. The other five modes appear as modes. There should only exist a fold degeneracy for resonant spherical photon modes, which can be broken by internal impurities or by asymmetric boundary conditions set by a cylindrical enclosure, microwave loop probes, and the sapphire substrate, collectively termed “backscatterers”. This degeneracy arises from a Legendre polynomial in the mode’s and field analytical expressions of the form , where . The integers and represent the number of maxima of the mode’s energy density in the direction over 180, and the number in the direction over 180, respectively. This would imply that for we should observe three distinct modes corresponding to a single and two modes, rather than five modes. However, the FEM demonstrates that the use of the sapphire support base introduces a further degeneracy to the modes depending on the amount of field that permeates the sapphire. The modelling predicts four modes, existing as two sets of two, which are separated by approximately 500 MHz. This is in fair agreement with the separation of modes , with modes 1–3. Therefore it is apparent that modes and are a doublet pair with .
Given that modes 2 and 3 approach relatively similar frequencies at high magnetic fields, it is reasonable to assume that these modes correspond to the second doublet pair, which FEM predicts will have a larger proportion of microwave field inside the sapphire support. This means that mode 1 must be the single mode.
From figure 4, we can see that both the doublet pairs demonstrate a gyrotropic response when interacting with the magnon resonances, i.e., one mode interacts more than its doublet pair. This is a common occurrence in spin ensemble systems and has been observed in paramagnetic systems such as Fe in sapphire Benmessai et al. (2009); Goryachev et al. (2014b); Bourhill et al. (2013). This asymmetric interaction strength for doublet pairs has also been observed in ferromagnets by Krupka et al. Krupka (1991); Krupka et al. (1999) and predicted by Rameshti et al. Zare Rameshti et al. (2015), with the latter stating that , where a different notation to that used here is employed, in which . The notations are equivalent as an doublet in Zare Rameshti et al. (2015) corresponds to a doublet pair here.
The gyrotropic response is a result of the anisotropy of a ferromagnet’s permeability tensor; the same reason why these materials are used in circulators. The permeability tensor, containing off diagonal terms appears as
where is the permeability of free space, and is the magnetic susceptibility of the ferromagnet, which is related to the magnetic permeability tensor by .
When any resonant photonic mode exists as a doublet, it is because the degeneracy has been broken by some backscatterer, and the two resulting modes exist as counter propagating travelling waves Benmessai et al. (2009); Goryachev et al. (2014b). The overall effect is that one travelling wave will see an effective permeability of , whilst the other will see , which can be rewritten as , and we can state that , where is the “unperturbed” magnetic susceptibility that a standing wave would observe.
The effective susceptibility that a mode experiences will determine the interaction strength of that mode with a magnon resonance according to Goryachev et al. (2014a):
where is the total magnetic filling factor of the mode; i.e., the proportion of magnetic field within the ferromagnetic material compared to the entire system. This parameter is used in an attempt to quantify the overlap of the magnon and photon modes and is calculated as
It should be noted that typically it is only the magnetic field energy density perpendicular to the external magnetic field that is considered to interact with the spin system Goryachev et al. (2014a); Soykal and Flatté (2010a). However, the interaction of mode 1 is far larger than its perpendicular filling factor of 0.075 would suggest. So, in an attempt to account for the interaction with nonuniform magnon modes, the total magnetic filling factor has been used. These have been calculated from the FEM and the resulting values of are displayed in table 3.
Given our assumption that mode 1 represents the (1,0) dielectric mode, which will exist as a standing wave given no possible degeneracy, the calculated value for this mode should represent the unperturbed magnetic susceptibility of the YIG. Taking the average of the values for the doublet modes () and () yields a value of ; in reasonable agreement with the value obtained from mode 1.
The FEM predicts that modes , 2 and 3 will each contain a significant proportion of magnetic field energy within the sapphire support, so one would expect these modes to observe a lower effective magnetic susceptibility, which would appear true for the latter two modes (their average susceptibility yields an unperturbed susceptibility of ). However, mode demonstrates a much larger coupling strength than what should be afforded a mode with its filling factor, hence a value approximately three times larger than the unperturbed value obtained from modes , and 1. This suggests that our approximation of using the total magnetic filling factor to quantify the overlap of the magnon and photon modes is not entirely accurate. To accurately explain the origins of the differing interaction strengths of each mode, knowledge of higher order, nonuniform magnon mode shapes are required, in order to replace the filling factor approximation with an overlap value. Unlike Zhang et al.’s Zhang et al. (2014) ultrastrong coupling results with a mm YIG sphere, in which higher order magnon modes mostly couple weakly with the microwave cavity, here we excite internal, nonuniform electromagnetic resonances, so it is more likely than not that these modes will couple more strongly to nonuniform magnon modes if their mode shapes match up well spatially. The derived values of susceptibility in table 3 agree within an order of magnitude to previously measured results Krupka (1991) but have been underestimated due to the use of filling factor as opposed to a mode overlap integral.
Finally, we can use the predicted mode frequencies of the FEM to determine the permittivity of the YIG sample, by varying until the frequencies match the asymptotic values measured at high magnetic fields. At these magnetic field values, the matrix in equation (1) becomes the identity matrix Krupka (1991). By measuring the depth of the sapphire concavity and its width at the surface, the radius of curvature was determined to be mm. With this information, an iterative simulation was conducted mapping mode frequencies versus relative permittivity of YIG. It was found that mode 1 is relatively insensitive to the radius of curvature of the sapphire support. This is due to the absence of electric field density outside the YIG for this particular mode. Given that contains a significant amount of uncertainty, this mode is used to match with . A plot of versus permittivity is shown in figure 9. From this result, we can state that . This value agrees well with previous measurements taken using the so-called “Courtney” technique with YIG samples Krupka et al. (1999).
In conclusion, we observe ultrastrong coupling between internal dielectric microwave resonances and magnons inside a mm YIG sphere. The large diameter of the sphere results in not only an increased number of spins, but also the accessibility of the internal electromagnetic resonances due to their existence below K-band frequencies. The use of internal microwave modes instead of an external cavity resonance results in far larger magnetic filling factors than ever before achieved in such an experiment, hence the coupling values and cooperitivity values observed are, to the authour’s knowledge, the largest ever reported, with a maximum GHz, or mode linewidths, and . This implies an extremely high level of coherence in this system. Most importantly however, the numerous resonant magnon peaks in the dispersive regime and the discrepancies in calculated susceptibilities suggest that higher order magnon modes participate in this system. This implies that the previously theoretically analysed models of such systems are incomplete.
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