An implantable LVAD is provided that includes a stationary one-way outflow valve; a cardiac sensor, configured to sense one or more features of cardiac cycles; a left atrial pressure sensor, configured to sense left atrial pressure; and a pump shaped so as to define a pump chamber. The pump includes a tubular linear motor, which includes a magnetic piston, which includes a reciprocating one-way valve configured to allow downstream blood flow and inhibit upstream blood flow; and a stator, which is configured to magnetically drive the magnetic piston with reciprocating motion, so as to pump blood downstream during downstream motion of the magnetic piston while the reciprocating one-way valve is closed. Control circuitry is configured to activate the tubular linear motor to provide pulsatile flow synchronized with cardiac cycles having a volume per cardiac cycle set responsively to the sensed left atrial pressure. Other configurations are also described.