Atomic force microscopy has established itself as one of the most powerful tools in nanotechnology. With meticulous setups amassing techniques such as ultra high vacuum, cryogenic temperatures, and CO-terminated tips, it is able to create a wonderful vista of surfaces, not missing the atoms for the topographical features [1-6]. There is, however, room for improvement in cutting-edge AFM experiments, as the standard quantum limit in sensitivity, represented by a minimum between detection noise and backaction noise, has not been reached [7,8]. Beyond this limit, techniques exist that can even break this quantum barrier by redirecting noise from one quadrature to another [9-11]. Yet, there is even opportunity in revitalising the accessibility of standard AFM, as performing experiments at cryogenic temperatures and under ultra-high vacuum [12,13] requires years of expertise.

For inspiration, we turn to quantum optomechanics and its sister field of quantum electromechanics, as they both report outstanding signal-to-noise ratios [14]. In the former, a reflective mechanical resonator constitutes half of a Fabry–Pérot cavity, converting photons to phonons and vice versa. Thus, the mechanical position can be read through the optical cavity. Upon this basic interaction, many emerging kinds of behaviour were found: sideband cooling down to quantum levels [15,16], parametric amplification [17] before signal detection, state squeezing [18-20], and Bogoliubov modes [21,22] for drastically reducing noise and directional amplifiers [23,24]. The group of proposed applications is even larger and hosts ideas such as quantum circulators [23,24], Ising model simulators [25], and improved gravity wave detection experiments [8]. All these techniques can be migrated to AFM, with the main hurdle being the integration of an optical Fabry–Pérot cavity with an elastic microcantilever. We chose to use purely mechanical coupling, an alternative mirroring our source of inspiration. It relies on non-linear elastic coupling between different vibrational eigenmodes of a mechanical resonator. As the stress field of one mode stiffens the vibrational motion of another, an energy exchange is established between them. This phenomenon is referred to as intermodal coupling [26]. It allows to replace the optical cavity from optomechanics with a mechanical eigenmode.

So far, intermodal coupling was proven in doubly clamped beams, square membranes and circular membranes [18,26-31]. For atomic force microscopy imaging, a slight angle between the sensing mechanical resonator and the sample of interest is required, ensuring that the only contact occurs between the sample surface and the tip of the mechanical resonator. This promotes cantilevers as the chosen geometry for this task, as building a clamped beam or a square membrane at the edge of a chip is considerably more challenging. In the following, we will explore intermodal coupling in a microcantilever as an opportunity to bring intermodal coupling techniques, derived from optomechanics, to AFM. It is easily accessible, with no hardware modifications and only requiring multifrequency excitation applied to the cantilever by either a piezoshaker or a modulated laser, found in many AFM setups.

The field of multifrequency AFM has improved both imaging contrast and the amount of extracted information from AFM experiments by exploiting the nonlinearity of the tip–surface interaction [32-36]. The methods applied excel in both their creativity and engineering prowess. A first example is on-resonance excitation of the first mode of a cantilever with measurements being performed at its harmonics [37]. Another method involved clever designs such as T-shaped cantilevers [38] and inner-paddled cantilevers [39] aiming at reducing the noise impact on force reconstruction. Bimodal AFM is another addition to the field, where two eigenmodes are excited and read simultaneously [40]. Last, intermodulation products, created by two signals close to the fundamental cantilever mode, form a sea of evenly spaced tones to be measured [35,41,42]. All of these rely on the nonlinear tip–surface force to create these multitonal responses, from which the force is reconstructed.

In this paper, we are building towards a hybrid multifrequency approach different from the ones described above. The on-resonance measurement would follow frequency-modulated AFM or bimodal AFM while being assisted by a new off-resonance excitation, which would activate intermodal coupling between two or more eigenmodes. With this geometric nonlinearity, we can circumvent the use of tip–sample forces and apply techniques from optomechanics. Sideband cooling will reduce thermal noise of the fundamental mode. Parametric amplification relies on coherent bimodal drive to amplify the signal of the fundamental mode. Both increase the signal-to-noise ratio of the measurement, creating opportunity for either improved sensitivity or increased speed. Furthermore, sideband cooling has a secondary use in ultrahigh-vacuum AFM as a tool for controlling the Q-factor of the fundamental mode.

Intermodal coupling requires a strong drive tone, referred to as a pump, at either the frequency difference between or the sum of two cantilever eigenmodes of interest. Using the difference, also known as a red sideband or anti-Stokes pump, leads to sideband cooling and mode splitting. Applying the sum, referred to as blue sideband pump, will cause either mode squeezing or parametric amplification [22], provided that the amplitude is optimally chosen. We will focus on the red sideband, as sideband cooling is useful for reducing thermal noise in standard AFM and mode splitting is a good way to measure the coupling rates. Here, the phonons from the first mode will have their frequency upconverted to the same as the second mode’s phonons, thus allowing them to interact. This pump effectively amplifies the single phonon–phonon coupling rate of the mode combination and linearly increases the overall coupling strength , where Xpump is the pump amplitude, thus giving us the following Hamiltonian for one eigenmode i coupled to another eigenmode j

(1)

where ωi and ωj are the frequencies of the i-th mode, henceforth known as the sense mode, and j-th mode, taking the role of the cavity mode in cavity optomechanics, respectively. Xi and Xj are their respective amplitudes, is the amplitude of the pump in meters, is the directional single phonon–phonon parametric coupling rate in Hz/meters. The last term describes a small signal Vsense, proportional to the voltage applied to the piezoshaker, with the frequency swept close to ωi, used to amplify the spectral response of the sense mode above the thermal excitation level.

The above Hamiltonian is a modified version of the one used in [27]. In contrast to this previous work, we do not exclude the possibility of asymmetrical coupling. This refers to an energy transfer either easier or harder from the first mode to the second compared to a transfer from second to first. Two directional coupling terms were introduced to account for this possibility, later to be investigated in detail. Equation 1 only shows the energy of two modes and their interaction, amplified by the red sideband pump, which is set at the frequency difference of the two modes in question. A main advantage of working with continuous mechanical systems, such as microcantilevers, is the plethora of eigenmodes available [43]. For every combination of two eigenmodes, a pump frequency can be applied to activate that intermodal coupling. Thus, the Hamiltonian can be expanded to include more eigenmode combinations including their individual energies as well as the interaction terms (the latter is only relevant if a pump is applied). We will focus only on a finite number of eigenmodes due to our equipment limitations. The full Hamiltonian is given by

(2)

If this coupling is a direct analogue to optomechanics, the coupling matrix should be symmetric, that is, . Expanding the experiment to multiple eigenmodes will elucidate if this symmetry is respected or not in these purely mechanical interactions and provide a spectroscopy map of intermodal coupling.

The coupling presented so far, using a red sideband signal, has two ways for manifesting itself, namely sideband cooling, where the mode of interest has its quality factor reduced alongside its effective temperature, and mode splitting, where two hybridized eigenmodes replace the original. The latter is useful in estimating the coupling strength, but the former is more applicable to AFM. It can not only control the quality factor of cantilevers, but it can also reduce the thermal noise of the measurement. These two kinds of behaviour have a regime associated to each, both directly related to the overall coupling strength . The i-th mode, as the sense mode, is in the weak regime if is smaller than Γj/2, the linewidth of the cavity mode. In this case its susceptibility (spectral response) can be written as

(3)

where δ is the frequency offset from the eigenfrequency ω1, and Γ1 and Γ2 are the linewidths of the modes. The equation can be further simplified to a Lorentzian with an increasing effective linewidth as per equation , enabling us to extract the coupling strength. If Gij > Γj/2, the sense mode is in the strong regime. Here the susceptibility equation is

(4)

In this case, the distance between peaks can be approximated as .

The effective temperature of the mode is calculated by normalizing the integral of the measured amplitude squared to the case when the pump is off when the system is at room temperature as follows:

(5)

where X is the spectral response amplitude with respect to the frequency offset from eigenfrequency δ and pump amplitude Vpump, Tambient is the temperature of the room where the experiment was performed, and δstart and δend are the start and end frequencies, respectively, of the lock-in measurement.

An AFM microcantilever (Bruker RFESP-75) is glued to a piezoshaker and placed in a vacuum chamber (between 1.2 × 10−6 and 5 × 10−7 mbar) under a laser Doppler vibrometer (LDV) (Polytech MSA 500) to measure the cantilever’s resonance frequencies and mode shapes (Figure 1). A lock-in amplifier (Intermodulation Products MLA-3 [36]) is used to control the piezoshaker and measure multiple frequencies from the vibrometer. For each possible mode combination, we activated the anti-Stokes pump and used a smaller sweeping signal to amplify the sense mode.

Figure 1: (a) Schematic drawing of the experimental setup. The cantilever is glued to the macrosized piezo driver. The LDV can either send data to the MSA to determine the eigenmode shapes or to the lock-in amplifier for higher bandwidth measurements. The latter also synthesizes the signal applied to the piezo driver. (b) Schematic of the signals used. Three signals are in effect at all times: the red sideband pump ωp, an off-set red sideband pump ωh ensuring even heating across the sample and a small one, compared to the previous, sweeping over the sense mode. (c) Comparison between a two-signal measurement (left) and a three-signal measurement (right) ensuring thermal stabilisation. The second flexural mode is coupled with the fifth flexural mode. The sum of heating signal and pump is constant. The stabilisation signal was 3 kHz higher than the red sideband pump, which was set at 3176.9 kHz. Due to their large frequency distance from the observed mode compared to its linewidth, ωp and ωh were not included in the graph.