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Breaking time-reversal symmetry (TRS) in the absence of a net bias can give rise to directed steady-state non-equilibrium transport phenomena such as ratchet effects. Here we present, theoretically and experimentally, the concept of a Lissajous rocking ratchet as an instrument based on breaking TRS. Our system is a semiconductor quantum dot (QD) with periodically modulated dot-lead tunnel barriers. Broken TRS gives rise to single electron tunneling current. Its direction is fully controlled by exploring frequency and phase relations between the two barrier modulations. The concept of Lissajous ratchets can be realized in a large variety of different systems, including nano-electrical, nano-electromechanical or superconducting circuits. It promises applications based on a detailed on-chip comparison of radio-frequency signals.PACS numbers: 73.63. Kv, 85.35.Gv Ratchets cause directed particle motion due to a combination of broken symmetry and non-equilibrium forces, where the latter may be deterministic or fluctuating. The most famous example is Feynman's flashing ratchet which uses a pulsating spatially asymmetric potential to actively turn fluctuations into work [1][2][3][4]. Another species is the rocking ratchet driven by forces periodic in time but with broken spatio-temporal symmetry. A simple example of a rocking ratchet is a pump which transports electrons one-by-one through a QD driven by two external periodic forces with a relative phase breaking the symmetry [5,6]. This is in contrast to the somewhat simpler turnstile where the spatial symmetry is broken by a finite dc voltage [7]. In the non-adiabatic limit electron pumps are investigated for their suitability as current standard [8,9].In this article we restrict ourselves to the adiabatic regime and study a generic implementation of a rocking ratchet by applying two time-periodic forces, phaselocked at various commensurate frequencies. In our implementation we measure the dc current I through a QD embedded in the two-dimensional electron system (2DES) 90 nm beneath the surface of an etched 1 µm wide channel of GaAs/AlGaAs heterostructure. The 2DES is cooled to ∼100 mK where its carrier density is n e 2.83 × 10 15 m −2 and its mobility is µ e 320 m 2 V −1 s −1 . We control the QD by applying voltages to two metal gates [see lower inset in Fig. 1(a)].For a first orientation we present in Fig. 1(a) a stability diagram of our QD measured at finite dc voltage V = (µ R − µ L )/e = 100 µV applied between its two leads (at chemical potentials µ L,R ). Plotted is the current I as a function of gate voltages V L and V R applied to the left (L) versus right (R) gate [yellow in the rhs inset of Fig. 1(a)]. The axes V L,R are offset relative to the actually applied voltages, such that V L = V R = 0 at our working point, which is marked in Fig. 1(a) by a black cross. We define the chemical potential µ n of the QD as the energy needed to add the next electron to it, where n is an index number and we choose n = 0 for the dot level closest to the working point. For...