For researchers across the sciences, spectroscopy is the main tool for uncovering the energetic structure of their object of study. These data in turn provide a range of information about the object. The idea is that some quantity or other can be characterized as a function of the frequency of radiation that the object absorbs or emits. Typically, the objects of interest, be they nuclei of complex molecules for study by nuclear magnetic resonance or electrons of unknown substances for optical spectroscopy, are probed by weak electromagnetic radiation over a range of frequencies. The object's response — its absorption of the signal or creation of new radiation — usually leads to a sequence of peaks in frequency (ν) called a line spectrum. The lines reveal energetic information through Planck's formula ΔE = hν, in which the separation between two of the system's energies (ΔE) is related to ν by Planck's constant (h). Work by Berns et al.1, described on page 51 of this issue, shows how this principle can be extended: data gained from information provided by the amplitude of the probe radiation can be used to map large parts of an energy spectrum without changing the radiation frequency.

Conventional frequency spectroscopy is a hugely successful technique, but it has its blind spots. First, the frequency and the energy scale of interest are linked by a constant of nature that cannot be changed. In consequence, some energies cannot be probed because radiation of the appropriate frequency, for example frequencies in the terahertz range, are difficult to generate and detect. Second, it may be difficult to deliver the probe radiation to a sample — for instance a cryogenic sample — kept in a protected environment.

The technique of amplitude spectroscopy, as introduced by Berns et al., offers a solution to the problem of reaching energies that correspond to these inaccessible frequencies. A central principle in the authors' approach is quantum interference at an 'avoided' energy-level crossing. An avoided crossing is a quantum phenomenon that has an analogy in mechanics: if two pendulums of identical frequency of oscillation are coupled, they will show new patterns of motion — in phase and in anti-phase — that will have differing frequencies depending on the strength of the coupling. In quantum physics, frequencies correspond to energies, and by the same token, energy levels that are brought close to each other do not cross but keep a minimum distance apart — they show an avoided crossing (Fig. 1a). The quantum physics involved at an avoided crossing is described by the Landau–Zener–Stückelberg (LZS) mechanism2,3,4. To explain this further, however, some comparisons are needed.

Figure 1: Use of avoided energy-level crossing for amplitude spectroscopy.
figure 1

a, The energies of two quantum states 0 and 1 (dashed lines) approach each other as an applied external field f is changed. Quantum coupling makes the energy levels avoid each other by an energy difference ΔE that is never smaller than Emin, reached at f0. The ground state (red line) smoothly crosses over to the excited state (blue line) and vice versa. b, Conventional frequency spectroscopy. A signal modulation of small amplitude A induces vertical transitions between the two states (filled circles). c, A modulation f0 large amplitude encompassing f0 at low frequency leaves the system in the ground state as it changes between 0 and 1 across the avoided crossing. d, The same modulation at high frequency leaves the system in state 0, thus crossing over from the ground to the excited state. e, The same modulation at intermediate frequency superimposes c and d and splits the state into two branches. This is the mode of operation in amplitude spectroscopy1 that makes it possible to track the energy spectrum.

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Suppose two quantum energy levels 0 and 1, whose energy is controlled by an external field f, should cross at some value f0 (Fig. 1a). However, quantum coupling between the levels keeps the energies above a minimum distance apart (Emin). Thus, the energy levels avoid crossing each other. Instead, at around f0 the ground state is neither 0 nor 1; it is a quantum superposition of both, and the same holds true for the excited state. These superpositions smoothly connect 0 and 1 at around f0. In frequency spectroscopy, ν is matched to E/h to map out this energy structure, keeping the signal amplitude (A) as small as possible (Fig. 1b).

The LZS formulation describes large signal amplitudes that sweep across the avoided crossing (Fig. 1c–e). The main parameter is the rate of change of energy V = hνA, a quantity of dimension E2. If V is small, the evolution is adiabatic (that is, no heat enters or leaves), and the system remains in the ground state as it changes between 0 and 1 across the avoided crossing (Fig. 1c). If V is large, the state has no time to change, and remains where it started, in 0 or 1, thus crossing over from the ground to the excited state (Fig. 1d). At intermediate values of V, comparable to E2min, the state splits into a quantum superposition between the two energy branches, creating a wealth of interference patterns (Fig. 1e). The situation in which V = hνA is comparable to E2min shows the principle of the frequency conversion: hν is multiplied by the large amplitude A to match E2min, allowing A to be traded for frequency.

To track the data for analysis, the energy spectrum is resolved by interferometry. Using detailed knowledge of the interference phenomena generated by a test device, Berns et al.1 reconstruct the full energy spectrum of the device. When an avoided crossing is reached, crossing over to an adjacent state is possible, leading to the 'spectroscopy diamonds' seen in Figure 1a of their paper (page 52)1. The diamonds, bounded by pairs of avoided crossings that can be located on the f axis from the size of the diamond, contain a periodic interference pattern from which the energy levels away from the avoided crossing can be extracted. Finally, use of short spectroscopy pulses allows the precise identification of Emin, providing access to previously inaccessible parts of the spectrum.

The experimental test chosen by Berns et al. was carried out on a well-controlled macroscopic quantum system known as a superconducting flux qubit5. This consists of a small loop of superconducting material interrupted by three Josephson tunnel junctions. Its physics is analogous to that of a particle in a double-well potential whose coordinate is the magnetic flux through the loop, which also serves as the control parameter f. This artificial, engineered device leads to the clean, precise data presented by Berns and colleagues. An advantage of this system for demonstrating amplitude spectroscopy is that because it is macroscopic, its magnetic moment is orders of magnitude larger than that of an atom, making it possible to drive the required large amplitude.

In principle, the amplitude-spectroscopy scheme described by Berns et al.1 will be widely applicable. But the system must satisfy two requirements: its spectrum must connect the energy levels by avoided crossings; and it must be possible to make long sweeps with the probe radiation without damaging the sample. These demands mean that certain atomic gases and molecular magnets are the most likely candidates for use in such an approach. Amplitude spectroscopy will not replace frequency spectroscopy, but it will complement that technique to complete the picture that researchers can extract from their samples.