Nonreciprocity in cavity magnonics at millikelvin temperature

Numerous technologies working at microwave frequency, including test and measurement circuits, simultaneous transmit-and-receive architecture, and wireless transmission, require non-reciprocal components such as isolators and circulators. Their function in the classical regime is either to protect delicate components from high power reflections or to route the outgoing and incoming signals to the appropriate transmitter and receiver.¹ Meanwhile, isolators and circulators are also employed in cryogenic quantum mechanical experiments to shield sensitive signals from backscattering noise.^2–5 

Traditionally, non-reciprocal components are designed using ferrite materials, which lose their Lorentz reciprocity under the application of an external magnetic field.⁶ One of the widely adopted configurations is the stripline Y-junction type.⁷ The design is renowned for its power handling ability (tens to hundreds of Watts) and incredibly low loss (⁠ $<1$ dB). The isolation and bandwidth are, however, limited to a point.⁸ Consequently, devices are cascaded in some applications to reach the desired performance. Because of this, there have been significant efforts over the past few decades on alternative non-reciprocal technology to engineer cost-efficient and small-form-factor devices with improved performance.

On the one hand, there are magnet-free approaches that can be integrated into a chip-scale structure while providing considerable port isolation (⁠ $>20$ dB). Transistors and temporal modulation are the two categories.^9–13 Even though the former is readily MMIC (Monolithic Microwave Integrated Circuit) compatible, the transistor’s added noise and nonlinear distortion have prevented them from being widely deployed.¹⁴ Reciprocity can also be violated through spatiotemporally modulated waveguides.^12,15,16 In practice, the material being modulated is typically associated with considerable insertion loss at a higher frequency regime. The rapidly expanding optomechanical and electromechanical systems also have shown great potential with lately discovered isolation and circulation effect.^17–20 

On the other hand, coupling magnons (collective excitation of the spin ensemble) with the microwave cavity can produce non-reciprocity. Both circulators and isolators have been realized recently.^21–24 In a coherently coupled system, the magnon mode can strongly couple with a selected chiral cavity mode to produce an asymmetrical transmission profile that spans nearly $500$ MHz.²² A circulator utilizing the similar scheme achieved over 50 dB of port isolation.²¹ Alternatively, the cooperation of coherent and dissipative coupling produces non-reciprocity that offers complete isolation (⁠ $>80$ dB).²³ The eigenmodes in such a system couple differently with the microwave traveling in the opposite direction. In addition, the repulsive behavior of the linewidth is exploited to fully compensate for the hybridized mode’s damping. A sharp unidirectional rejection band develops as a result. The working principle is later found applicable to a circulator.²⁴ 

Considering the current ferrite non-reciprocal devices occupy substantial space inside the dilution refrigerator and, hence, limiting the number of qubits to be incorporated, the demand for compact and effective circulators and isolators still remain.²⁵ Cavity magnonic’s tunability,^26,27 combined with recent efforts to push toward on-chip integration, has proven to be beneficial for device design.²⁸ The nearly perfect isolation found in Ref. 23 is also attractive for sensitive signal detection. While the coherent coupling has been proven to exist throughout a wide temperature range,^29–31 it is yet to be validated whether the same is true for dissipative coupling. Hereof, this work investigates the bidirectional transmission coefficient of a cavity magnonic system at millikelvin temperature to explore its non-reciprocal properties.

The real part of $ω ~ ±$ portrays the energy-level dispersion, whereas the imaginary part controls linewidth behavior. By carefully examining Eq. (2), it can be seen that $Re( ω ~ ±)$ are the same regardless of the chosen $Θ 1 ( 2 )$ value. Thus, the system exhibits an identical dispersion for both forward and backward transmission measurements. In contrast, the imaginary part is different due to the $( J − i Γ e i Θ ) 2$ term, implying that the energy dissipation of hybridized modes depends on the measurement direction.

A single yttrium iron garnet (YIG) sphere placed close to the intersection of a planar cross-shaped microwave cavity makes up the device under test (DUT). The sample holder, depicted in Fig. 1(b) as a transparent box, fixes the location of the YIG. A permanent magnet is fastened to the end of an actuator to control the magnetic field $H → z$⁠. In this way, the ferrite is assumed to be saturated, and $ω m$ can be tuned by varying the actuator’s position. The entire setup is housed inside a dilution refrigerator and the sample is anchored to the mixing plate of the refrigerator, as shown in Fig. 1(c).

Inside the dilution fridge [Fig. 1(c)], microwave signals (⁠ $V 1 , 2 in , out$⁠) are from a vector signal generator (not shown). Attenuators are positioned between different cooling stages to regulate the heat flow and simultaneously adjust signal power to a predetermined level. Cryogenic circulators and isolators are installed within the mixing chamber (7 mK) to redirect transmission and reflection signals to their appropriate ports. The high electron mobility transistors (HEMTs) serve as sensitive amplifiers in the system.

The YIG sphere, with a diameter of $1$ mm, was provided by Ferrisphere, Inc.³³ The Kittel mode is the focus of this work since it not only has the highest number of spins coupled with the system’s photons but also has a straightforward dispersion, i.e., $ω m=γ( H → A+ H → z)$⁠, where $γ=2π×26.3 μ 0 GHz / T$ is the gyromagnetic ratio, $H → z$ is the static bias magnetic field perpendicular to the cavity board, and $H → A$ is the magnetocrystalline anisotropy field. During the experiment, the magnon mode’s frequency is tuned from $4.57$ to $8.89 GHz$⁠, and the intrinsic damping rate is found to be $α m/2π=1 MHz$⁠.

A ROGERS RO4003C $( ε r=3.55)$ laminate measuring 0.81 mm thick is used to construct the microwave cavity. The 50  $Ω$ transmission line that connects the two ports allows for bi-directional transmission measurements. Its midpoint joins two open-ended 6.6 mm long stubs functioning as half-wavelength resonators. A direction-dependent magnetic chirality is produced nearby the cross junction as a result of the interaction between the fields of the transmission line and the stub resonators. The cavity resonant frequency and intrinsic damping rate at millikelvin are $ω c/2π=6.354 GHz$ and $α c/2π=11 MHz$⁠, respectively (see supplementary material for Appendix A).

The coherent interaction occurs because the YIG sphere is located inside the magnetic field of the stub resonator. An extrinsic damping rate of $γ c/2π=3 GHz$ results from the electrical connection between the stub and transmission line. Given the magnon mode’s proximity to the transmission line, there also exists a dissipation channel of a similar kind at rate $γ m/2π=0.16 MHz$⁠. It is also important to mention that both coupling strengths are maintained constant throughout the experiment due to the fixed YIG position. Magnon mode frequency is the only tuned variable.

Two-way transmission profiles of the DUT are measured first, and then their difference is calculated. The measured forward and backward transmission coefficients $| S 21 , 12 | 2$ are shown in Figs. 2(a) and 2(c) as a function of frequency detuning: $(ω− ω c)/2π$ and field detuning: $( ω m− ω c)/2π$⁠. At the degeneracy point, a normal mode splitting of approximately 110 MHz is present in both directions, orders of magnitude wider than the intrinsic damping rates, indicating that the magnon mode is strongly coupled to the cavity mode.

Several obscure anti-crossings are found near $( ω m− ω c)/2π=−0.5$ and $1.0 GHz$⁠, which can be related to the weak coupling between the cavity mode and other magnetostatic modes induced by the inhomogeneity of magnetic fields. They are not considered in the scope of this work.

Next, the coupling strengths are evaluated by fitting Eq. (3) to each measured transmission spectra. The average of coherent coupling strength is $J/2π=63 MHz$⁠, whereas $Γ/2π=22 MHz$ for the dissipative. $Θ$ takes $π$ for $| S 21 | 2$ and $0$ for $| S 12 | 2$⁠. A good agreement between the fitting results and the raw measurement data is seen in Figs. 2(b) and 2(d).

One notices that the mode splitting, being 110 MHz, is less than that of a pure coherent case $2J/2π=126 MHz$⁠.²⁷ It is, in fact, altered by the dissipative interaction, which causes energy-level attraction instead.³⁴ Nevertheless, the transmission mapping still exhibits anti-crossing behavior since $J>Γ$⁠.

Figures 2(e) and 2(f) display the absolute value of isolation (⁠ $ISO=| S 21 | 2−| S 12 | 2$⁠). Two bright peaks at the field detuning $( ω m− ω c)/2π=−0.26,0.21 GHz$ (white arrows) suggest a non-reciprocal transmission, further confirming that two subsystems are not interacting in a purely coherent manner.

The forward and backward transmission shown in Figs. 3(a), 3(c) and 3(b), 3(d) correspond to the two isolation peaks identified previously. According to Figs. 3(e) and 3(f), the respective isolation is $−26.9$ and $25.9 dB$⁠, emerging at frequency detuning conditions $0.012$ and $−0.016 GHz$⁠.

Due to inadequate calibration on the feed and return signal path [ $V 1 , 2 in / out$ in Fig. 1(c)], there is an imperceptible offset between forward and backward measurements, with an average of $1.49 dB$ across the sweep frequency span (see supplementary material for Appendix A). Since the isolation is substantially greater than the offset, the DUT is nonreciprocal.

For practical applications, the performance characteristics of a non-reciprocal device, such as isolation, insertion loss (IL), and bandwidth (BW), are examined across the range of field detuning.

As depicted by the two solid lines in Fig. 4(a), the calculated isolation is maximum at $( ω m− ω c)/2π=±0.23$ GHz, implying mirror symmetric non-reciprocity with regard to the cavity mode frequency. The attenuation of the forward traveling wave causes the left peak, whereas the attenuation of the backward traveling wave results in the right peak. Away from the detuning conditions mentioned above, the isolation drastically decreases. The DUT is anticipated to exhibit even larger isolation if the tuning resolution of the magnetic field can be improved.

The 3 dB bandwidth is shown in Fig. 4(b). The width is obtained by counting up $3$ dB from the transmission minimum. When comparing Figs. 4(b) to 4(a), a trade-off between the isolation and bandwidth becomes obvious. Maximum isolation is achieved with minimum bandwidth at $( ω m− ω c)/2π=±0.23 GHz$ (marked by dashed lines). This is consistent with previously discovered zero damping condition (ZDC) at room temperature.³² From the oscillator’s point of view, a zero bandwidth oscillator absorbs all the energy from the transmission line. The two isolation peaks found in the measurement correspond to bandwidths of 2.5 and 2.7 MHz, respectively. Once the field detuning surpasses $±1.3$ GHz, the isolation becomes less than 3 dB, and the bandwidth is indeterminate.

Figure 4(c) illustrates device insertion loss, defined as $IL=| S 12 | 2$ for negative field detuning and $IL=| S 21 | 2$ for positive field detuning. Interestingly, there is no apparent trade-off between ISO and IL. Therefore, the system can be biased nearby but not at the ZDC for an optimized performance. When the field detuning is large, system is in the dispersive regime. Consequently, the insertion loss is an approximate to the cavity’s resonant dip. The reason for reduced insertion loss at the lower detuning range is explained below.

The real part of hybridized frequencies is highlighted in Fig. 4(d), which is extracted by reading off transmission zeros. There is a high degree of agreement with Fig. 2. Imaginary parts are presented in Figs. 4(e) and 4(f). The linewidth is extracted by fitting the reciprocal of power amplitude to a superposition of two Lorentzian curves (see supplementary material for Appendix B). Notably, the imaginary part of the system depends on the direction of the traveling wave. This is due to the complex coupling strength $J±iΓ$⁠, whose directional dependence determines the sign of the $iΓ$ term. Near the zero field detuning, there is a negative damping regime where attenuation on the traveling wave is weakened. Consequently, two effects will be observed in the resonance dip: reduced insertion loss and linewidth broadening.

As the final step of the experiment, the traveling wave’s power is varied from $10$ to $−40 dBm$⁠. The system parameters are found to be irrelevant of probing microwave power. It is expected here since the number of photons is much smaller than the number of spins such that the system sits inside the linear response regime. Furthermore, the number of spins and photons involved far exceeds the scale of quantum-level phenomena.

In summary, the performance of an isolator based on cavity magnonic interaction was investigated at millikelvin temperature. The interference between coherent and dissipative coupling causes nonreciprocity. Experimental results confirm that earlier discovery²³ applies to all temperature ranges. The cryogenic performance of such a cavity magnonic isolator can be tuned and improved in the same way as the room temperature experiments have demonstrated.³⁵ 

This work has two additional implications. One is that at low temperatures, dissipative coupling exists. Creating novel hybrid systems for quantum information science may benefit from such validation. Second, normal mode splitting is impacted by dissipative coupling. The ferrite material may interact with the traveling wave when cavity magnonic platforms adopt a more compact profile. Thus, evaluating the forward and backward transmission is necessary for better calibrated coupling strength.

See the supplementary material for detailed procedures for fitting parameter extractions and a comparison between cavity magnonic isolators and conventional ferrite-loaded Y-junction isolators.

This work has been funded by NSERC Discovery Grants and NSERC Discovery Accelerator Supplements (C.-M.H.). We would like to thank Dr. Yongsheng Gui for discussions. S.B. acknowledges funding by the Natural Sciences and Engineering Research Council of Canada (NSERC) through its Discovery Grant, funding and advisory support provided by Alberta Innovates through the Accelerating Innovations into CarE (AICE)—Concepts Program, and support from Alberta Innovates and NSERC through Advance Grant. This project is funded (in part) by the Government of Canada.

The authors have no conflicts to disclose.

Mun Kim: Conceptualization (equal); Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal). Armin Tabesh: Data curation (equal); Methodology (equal); Software (equal); Writing – review & editing (equal). Tyler Zegray: Data curation (equal); Software (equal); Writing – review & editing (equal). Shabir Barzanjeh: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Project administration (equal); Resources (equal); Supervision (equal); Validation (equal). Can-Ming Hu: Conceptualization (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.