Splitting and recombining an electron wave packet has been used to test relativity at a record sensitivity. The result heralds an era of precision measurements of relativity using quantum-information methods. See Letter p.592
Lorentz invariance is a fundamental symmetry of space-time that lies at the heart of Albert Einstein's special theory of relativity. Writing in this issue (page 592), Pruttivarasin et al.1 show how they have tested Lorentz invariance for electrons at an unprecedented sensitivity by splitting and recombining a superposition of electron wave functions — an electron wave packet — that is bound to calcium ions. This ingenious experiment opens the door to a new generation of precision tests of relativity using quantum-information techniques.
Lorentz invariance states that the laws of physics that govern a physical system are unchanged for different system orientations or velocities. Equivalently, the laws of physics exhibit rotation symmetry (spatial isotropy) and boost symmetry. In practice, it is easier to rotate an apparatus than to boost it, and therefore many tests of relativity are designed to explore the spatial isotropy of the behaviour of a physical system. Pruttivarasin and colleagues' experiment can be viewed as a quantum analogue of two famous tests of spatial isotropy: the Michelson–Morley experiment for electrodynamics2 and the Hughes–Drever experiment for matter3,4.
The Michelson–Morley experiment uses a device called an interferometer that first splits a light ray into two beams travelling along orthogonal paths (arms), and then reflects and recombines the beams to yield an interference pattern. Monitoring the interference pattern as the orientation of the apparatus changes can be understood as testing the constancy of the speed of light and hence the rotation symmetry of the laws of electrodynamics. If the speed of light is different in different directions, then the travel times of the two beams vary as the apparatus is rotated, changing the interference pattern.
By contrast, the Hughes–Drever experiment studies the spatial isotropy of the propagation of matter by examining the connection between its energy and its momentum. In essence, the experiment involves placing one or more atoms in a magnetic field, thereby splitting some atomic energy levels and imbuing the system with a definite orientation. Monitoring the frequency of transitions between certain energy levels for a changing orientation of the whole apparatus provides a check of isotropy. If the energy levels depend on the momentum directions of the atomic constituents, the transition frequency will change as the apparatus is rotated.
In their study, Pruttivarasin et al. take advantage of the quantum nature of the electron, realizing an electron analogue of the Michelson–Morley and Hughes–Drever experiments through quantum-information techniques that allow a suitable electron wave packet to be created and monitored. The authors confined a pair of calcium ions (40Ca+), about 16 micrometres apart, in an electromagnetic trap and applied a vertical magnetic field of 3.93 gauss to introduce a definite orientation of the system, which changes as Earth rotates. They then applied laser pulses to electrons bound to the calcium ions, creating an electron wave packet that combines two quantum states of different electron orientations relative to the magnetic field and that oscillates between two configurations. The researchers used further laser pulses to monitor the energy difference between the two states for 23 hours. The experimental procedure can be viewed as repeatedly splitting and then recombining the two quantum states 95 milliseconds later. Violations of spatial isotropy would be seen as variations in this energy difference as Earth rotates.
The experiment confirms spatial isotropy for electrons at the impressive level of one part in 1018. This represents a milestone sensitivity, because it is smaller than the dimensionless ratio of about 10−17 between the strengths of the electroweak and gravitational forces that could naturally be expected to govern violations of Lorentz invariance arising in unified theories of quantum physics and gravity5. The authors' experiment is thus the first to delve into this realm of sensitivity for electrons.
A crucial subtlety in interpreting tests of Lorentz invariance and hence of special relativity is that a physical reference system is required to define the lengths of rods, the ticking rates of clocks and the idea of orthogonality in space and time. For example, in a classic Michelson–Morley experiment, the lengths and orthogonality of the interferometer arms are established in terms of properties of matter. The interpretation of the experiment as a test of the isotropy of the speed of light therefore relies on the assumption that these properties are independent of the system's orientation. Analogously, the interpretation of a Hughes–Drever experiment as a test of the rotation symmetry with matter assumes isotropic transition frequencies of light. Indeed, any measurement in physics is really a comparison between two systems, with only the difference between them being physically meaningful.
It follows that Pruttivarasin and co-workers' experiment can be viewed equally as an isotropy test for light assuming conventional electrons or as an isotropy test for electrons assuming conventional electrodynamics. In the first scenario, the experiment is interpreted as searching for possible spatial anisotropies in the Coulomb force that binds the electron wave packet to the calcium ions, and hence in the laws of electrodynamics, and the results represent a fivefold improvement in sensitivity over current limits6,7. In the second picture, the energy of the electron wave packet depends on the direction of its momentum, and the new constraints sharpen existing bounds8 100-fold.
Possible experimental improvements include choosing different ions to yield a longer-lived electron wave packet, binding the wave packet to ions of greater charge, preparing the wave packet more directly, and taking data over a longer period. Another 100-fold improvement in sensitivity may lie within reach.
Other tests of Lorentz inariance are also feasible using these methods. Lorentz violations accessible in principle include those characterized by nine coefficients9, each corresponding to a different physical effect. Six coefficients govern violations of rotation symmetry, whereas three control boost violations. Pruttivarasin et al. obtained constraints involving four of the six types of rotation-symmetry violation (see Table 1 of the paper1). The remaining two could be studied in a similar experiment mounted on a turntable with its axis of rotation differing from that of Earth. The three coefficients controlling boost violations could also be measured with tenfold improved sensitivity using data acquired over many months, by taking advantage of the changing direction of Earth's velocity as it revolves around the Sun. Stay tuned for future cutting-edge tests of relativity using these quantum-information techniques.