  This review provides a brief and quick introduction to the quantum Langevin
equation for an oscillator, while focusing on the steady-state thermodynamic
aspects. A derivation of the quantum Langevin equation is carefully outlined
based on the microscopic model of the heat bath as a collection of a large
number of independent quantum oscillators, the so-called independent-oscillator
model. This is followed by a discussion on the relevant `weak-coupling' limit.
In the steady state, we analyze the quantum counterpart of energy equipartition
theorem which has generated a considerable amount of interest in recent
literature. The free energy, entropy, specific heat, and third law of
thermodynamics are discussed for one-dimensional quantum Brownian motion in a
harmonic well. Following this, we explore some aspects of dissipative
diamagnetism in the context of quantum Brownian oscillators, emphasizing upon
the role of confining potentials and also upon the environment-induced
classical-quantum crossover. We discuss situations where the system-bath
coupling is via the momentum variables by focusing on a gauge-invariant model
of momentum-momentum coupling in the presence of a vector potential; for this
problem, we derive the quantum Langevin equation and discuss quantum
thermodynamic functions. Finally, the topic of fluctuation theorems is
discussed (albeit, briefly) in the context of classical and quantum cyclotron
motion of a particle coupled to a heat bath.
