Quantum technology continues to turn formerly unmeasurable effects into technologically important physics. For example, minuscule shifts of atomic energy levels due to room-temperature blackbody radiation have become leading influences in atomic clocks at or beyond the 10−14 level of accuracy2. They have thus become important to precision timekeeping3, and for applications such as improving time standards, relativistic geodesy and searches for variations of fundamental constants. Thermal radiation from a heated source should also result in a repulsive radiation pressure on atoms through absorption of photons4,5,6,7. However, the scattering rate for room-temperature blackbody radiation is small, leading to only mm s−1 velocity changes in hundreds of thousands of years for the caesium D line, for example. Here, we show that spatially inhomogeneous blackbody radiation produces a much higher acceleration at the μm s−2 level pointing towards the source, even near room temperature. It is well described by the intensity gradient of blackbody radiation that gives rise to a spatially dependent a.c. Stark shift1, similar to the dipole forces induced by lasers in optical tweezers8, atom trapping9, or coherent manipulation of atoms10 or of molecular clusters11. We expect it to be the dominant force on polarizable objects over a large temperature range1 and thus important in atom interferometry, nanomechanics or optomechanics12. Controlling this force will enable higher precision in atom interferometers, including tests of fundamental physics such as of the equivalence principle13,14,15, planned searches for dark matter and dark energy16, gravity gradiometry17,18, inertial navigation and perhaps even Casimir force measurements and gravitational wave detection19,20.

As shown in Fig. 1, we perform atom interferometry with caesium atoms21 in an optical cavity to measure the force induced by blackbody radiation. Our setup is similar to the one we used previously22,23. Caesium atoms act as matter waves in our experiment. They are laser-cooled to a temperature of about 300 nK and launched upwards into free fall, reaching 3.7 mm into the cylinder at their apex. During free fall, we manipulate them with counterpropagating laser beams, which “kick” the atoms with an impulse ℏk eff from two photons. The intensity and the duration of the laser pulses determine whether we transfer the atom with a 50% probability (a “π/2-pulse”) or nearly 100% (a “π-pulse”), respectively. We apply a π/2–π–π/2 pulse sequence, spaced by intervals of T = 65 ms, that splits, redirects and recombines the free-falling atomic wavefunction, forming a Mach–Zehnder atom interferometer. The matter waves propagate along the two interferometer arms while accumulating an acceleration phase difference Δϕ = k eff a tot T 2, where a tot is the total average acceleration experienced by the atom in the laboratory frame. For spatially varying accelerations, the phase shift is calculated by integrating the potential energy and taking the difference between the two paths. Since in our case, the separation between paths is negligible on the spatial scale of the potential variations, this amounts to integrating an acceleration profile a(z) over the atom’s trajectory in free fall z(t):

$$\bar{a}=\frac{1}{2T}\underset{0}{\overset{2T}{\int }}a(z(t)){\rm{d}}t$$

Fig. 1: Setup. a, Space–time diagram of each atom’s trajectories in our Mach–Zehnder interferometer. b, The intensity gradient of blackbody radiation surrounding a heated, hollow cylinder causes a force on atoms. The cylinder is made from non-magnetic metal (tungsten) and measures 25.4 mm in height and diameter. The cavity light passes the cylinder through a 10 mm bore at its centre to perform interferometry. c, Theoretical calculation of the acceleration of caesium atoms due to blackbody radiation, a BBR , as a function of the distance z along the cylindrical axis. The vertical axis is taken from the centre of the source mass. The grey shaded area marks the region inside the hollow core of the cylinder. Discontinuities in the predicted acceleration stem from simplifying edge effects at the entrance to the hollow cylinder. Full size image

The probability of the atom exiting the interferometer in one of the outputs is given by P = cos2(Δϕ/2).

At the start of each experimental run, we heat the cylinder to a temperature of about 460 K with an infrared laser, which is subsequently switched off. We then measure the acceleration of the atoms during the cool-down period of up to 6 h, while we monitor the temperature with an infrared sensor. When the source mass has cooled to near room temperature, we reheat it to start another run. Because the cylinder has a 5 mm slit on the side, we can change its position between a location close to the interferometer and a remote one23 without interrupting the cavity mode. This allows us to separate forces induced by the source mass from other forces, in particular the million-fold larger one from Earth’s gravity. The near position exposes the atoms to blackbody radiation arising from the source, while the far position serves as a reference. We then investigate the temperature dependence of the acceleration difference.

Figure 2 shows this measured acceleration a cyl as a function of the source mass temperature T s with a comparison to theory. The red dotted line in Fig. 2 shows the predicted acceleration a cyl = a BBR + a grav from both the gravitational pull a grav and the blackbody interaction a BBR = C(T s 4 − T 0 4) of atoms with the source mass. Here, a grav = 66 nm s−2 is calculated, T 0 = 296 K is the measured ambient temperature and C = −4.3 × 10−5 ± 0.6 (μm s−2 K−4) is calculated from the albedo and geometry of the source (see the Methods section). The model leaves no free parameters.

Fig. 2: Experimental data. Measured acceleration as a function of the source mass temperature T s . A quartic dependence on T s is observed for the acceleration experienced by caesium atoms towards the source mass. a, Data from 63 thermal cycles, about 2–5 h each, are binned in temperature with N bin = 65 measurements per bin. The black dots represent the weighted mean of each bin. Vertical error bars show the 1-sigma statistical uncertainty on the weighted mean. Systematic effects have been considered in detail and show no significant contributions to the error bars; see the Methods section. Horizontal bars show the temperature spread of the N bin measurements in the bin. The red dotted line presents a calculation of the average acceleration imparted to the atoms during interferometry. The error for this theoretical prediction is dominated by the approximately 10% uncertainty of the source mass emissivity. The gravitational pull of the cylinder gives the room-temperature offset of the acceleration, indicated by the black dotted line at a grav = −66 nm s−2. b, Residuals from the bulk acceleration data (cyan) to the zero-parameter theory model. c, A histogram of the bulk residuals is well described by a normal distribution. A Gaussian fit to the histogram (black dot–dashed curve) has mean compatible with zero within the standard error of 29 nm s−2. Full size image

It is important to rule out artifacts that could partially mimic a blackbody-induced acceleration. For example, spatially constant energy-level shifts induced by the blackbody radiation (rather than an a.c. Stark shift gradient, which produces a force) can be ruled out because they would be common to both interferometer arms, and thus cancel out (see the supplement). The pressure applied by hot background atoms from outgassing of the heated source mass removes a substantial fraction of the cold atoms from the detection region at its highest temperatures, so it is conceivably a component of the measured force on the remaining atoms. This, however, can be ruled out by multiple observations. First, this pressure should push the atoms away from the source, while the observed acceleration is towards the source. Second, it should depend exponentially on the source mass temperature; such an exponential component is not evident in the data. Finally, any scattering of hot background atoms with atoms that take part in the interferometer would be incoherent, and would reduce the visibility of our interference fringes. Figure 3, however, shows that the visibility is constant over our temperature range, ruling out scattering. This observation also confirms that absorption or stimulated emission of incoherent blackbody photons is negligible (see Fig. 4).

Fig. 3: Visibility. Visibility as a function of temperature, averaged in bins of 2 K for clarity. Scattering or absorption of photons would lead to a dephasing of the atomic ensemble, resulting in a reduction of visibility. No obvious loss of visibility is a strong indication that the contribution of scattering and absorption events is negligible. The inset shows interference fringes taken at T s = 437 K and T s = 326 K. Each fringe consists of 80 experimental runs with a cycle time of 1.2 s. The fitted fringe phase gives an acceleration measurement, contributing a data point to the bulk data seen in Fig. 2b,c. Full size image

Fig. 4: Blackbody radiation. The spectra of blackbody radiation for various temperatures compared with transition frequencies of ground-state caesium indicated by vertical lines. The dash–dotted line on the left refers to the hyperfine splitting of the ground state used in the current definition of the second. The dashed lines on the right are strong optical absorption lines of caesium, starting from the D 1 transition 62S 1/2 → 62P 1/2 at 894 nm. The coloured band indicates the visible spectrum as a guide for the eye. Full size image

We now explain the measured acceleration in terms of a force due to the gradient in the ground-state energy-level shift (a.c. Stark effect) induced by blackbody radiation, h × 15 Hz at our highest temperatures, where h is the Planck constant. For the relevant temperature range, nearly all thermal radiation has a frequency well below the caesium D line. The shift of the atomic ground-state energy can be approximated by using the atom’s d.c. polarizability24 α Cs ≈ h × 0.099 Hz (V cm−1)−2 as2 ΔE(r) = −α Cs u(r)/(2ε 0 ), where u(r) is the electromagnetic energy density for the thermal field measured at a distance r from the source, and ε 0 is the vacuum permittivity. For isotropic blackbody radiation at the temperature T s of the source, we have u = 4σT 4/c (where c is the speed of light), and

$${\rm{\Delta }}{E}_{0}=-2\frac{{\alpha }_{{\rm{Cs}}}\,\sigma {T}_{s}^{4}}{c{\varepsilon }_{0}}$$ (1)

where σ is the Stefan–Boltzmann constant. If the heated body is a sphere of radius R, then the sphere’s blackbody radiation will dilute with distance, with energy density u(r) proportional to R 2/(4r 2). Taking the gradient gives the acceleration from the blackbody radiation force1 in spherical geometry:

$$a=-{\alpha }_{{\rm{Cs}}}\frac{\sigma \left({T}_{s}^{4}-{T}_{0}^{4}\right)}{c{\varepsilon }_{0}{m}_{{\rm{Cs}}}}\frac{{R}^{2}}{{r}^{3}}$$ (2)

The force points radially inwards (except for atoms in an excited state, whose polarizability may be negative). To model the force in our experiment, we use analytical ray tracing to take the geometry of our setup into account (see the Methods section).

The force exerted on a polarizable object due to the intensity gradient of the blackbody radiation can be derived from the same fluctuation electrodynamic formalism as the temperature-dependent Casimir–Polder force25,26. Conventionally, thermal Casimir–Polder forces are considered in planar geometry, where the intensity gradient due to propagating radiation modes becomes zero, and only the contribution of evanescent fields remains. Such forces dominate at a length scale of the thermal wavelength λ T = hc/(k B T), and scale in different asymptotic regimes as the first or second power of the surface temperature26,27. In our experiment, the Casimir–Polder force is negligible due to the millimetre-scale distance between the atoms and the surface. However, the intensity of blackbody radiation of a finite-sized source body is spatially dependent, and the propagating-mode contribution must be taken into account1. This gives rise to a long-range force having the characteristic T 4 scaling of blackbody radiation, which we observe here for the first time.

Just as blackbody radiation affects atomic clocks2,3, the acceleration due to the blackbody field gradient observed here influences any high-precision acceleration measurements with polarizable matter, including atomic and molecular interferometers, experiments with nanospheres and potentially measurements of the Casimir effect and gravitational wave detectors. For example, inside a thin cylindrical vacuum chamber, the thermal radiation field nearly follows the local temperature T(z) of the walls, inducing an acceleration of atoms of

$$a\left(z\right)=\frac{1}{{m}_{{\rm{At}}}}\frac{\partial }{\partial z}\frac{2{\alpha }_{{\rm{At}}}\,\sigma {T}^{4}\left(z\right)}{c{\varepsilon }_{0}}$$ (3)