Bipolar stepper motor driving by Micro-controller Xmega

You could do this with linear amplifiers such as high power op-amps, but that's not what would commonly be done today - it's very challenging (/expensive/inefficient) to build high power linear amplifiers.

Instead, it is more common to use full-on/full-off switching devices such as MOSFETs (or at very high voltages, IGBTs) in motor drivers. These are switched at a rate many, many times the commutation frequency, with a varying duty cycle for each pulse. The duty cycle varation - called pulse width modulation - effectively approximates the desired sine wave. Since the actual switching frequency is faster than the system can mechanically respond, it ends up averaging it out*

That said, a stepper would be an odd choice for a spindle motor. More common would be to use a brushless AC servo motor. Much as with a stepper this would be driven by several distinct phases of PWM-approximated sine waves. A main difference is that a servo motor typically has fewer poles - in effect fewer "steps" (electrical cycles) per mechanical revolution - and so needs a lower commutation frequency for a given rotational speed.

*though some brushless motor controllers intentionally modulate the applied voltage at audio frequency and make the motor itself emit audible beeps while you work through their options menu.

One possible solution to this is to use two L298 DUAL FULL-BRIDGE DRIVER ICs wired in parallel. For instructions on the correct way to do this, refer to "APPLICATIONS OF MONOLITHIC BRIDGE DRIVERS" page 2, the section titled "PARALLELING OUTPUTS". The L298 will allow supply voltages up to 46V and the parallel combination is recommended for stepper motors up to 3.5A

To control the L298, consider using the L297 STEPPER MOTOR CONTROLLER which will handle the chopper-drive control better than you can do with the Arduino.

There are reference schematics in the datasheets and application note referenced above.

You can use your Arduino to handle the step and direction (/CLOCK and CW/CCW) signals to the L297. Be aware that to run your motors at the highest speeds, you will need to ramp-up and ramp-down the step rates in order to accelerate and decelerate the motor. You will also need to limit any "jitter" in the step (clock) timing as much as possible or the motor may slip (lose steps) or not turn at all. This "jitter" is often caused by latency due to other interrupts. This may be improved by keeping interrupt routines short and (even better) giving your stepper motor timer interrupt the highest priority (not sure if that is possible as I have no experience with Arduinos).