Leveraging the strong atom-light interaction, a Rydberg atomic receiver can measure radio waves with extreme sensitivity. Existing research primarily focuses on improving the signal detection capability of atomic receivers, while traditional signal processing schemes at the transmitter side have remained unchanged. As a result, these schemes fail to maximize the throughput of atomic receivers, given that the coupling between atomic dipole moment and radio-wave magnitude results in a nonlinear transmission model in contrast to the traditional linear one. To address this issue, our work proposes to design customized precoding techniques for atomic multiple-input-multiple-output (MIMO) systems to achieve the channel capacity. A strong-reference approximation is initially proposed to linearize the nonlinear transition model of atomic receivers. This facilitates the derivation of atomic-MIMO channel capacity as $\min(N_r/2, N_t)\log({\rm SNR})$ at high signal-to-noise ratios (SNRs) for $N_r$ receive atomic antennas and $N_t$ classic transmit antennas. Then, a new digital precoding technique, termed In-phase-and-Quadrature (IQ) aware precoding is presented, which features independent processing of I/Q data streams using four real-valued matrices. The design is shown to be capacity-achieving for the atomic MIMO system. In addition, for the case of large-scale MIMO system, we extend the preceding fully-digital precoding design to the popular hybrid precoding architecture, which cascades a classical analog precoder with a low-dimensional version of the proposed IQ-aware digital precoder. By alternatively optimizing the digital and analog parts, the hybrid design is able to approach the performance of the optimal IQ-aware fully digital precoding. Simulation results validate the superiority of proposed IQ-aware precoding methods over existing techniques in the context of atomic MIMO communication.