Adiabatic quantization of Andreev levels

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We identify the time T between Andreev reflections as a classical adiabatic invariant in a ballistic chaotic cavity (Lyapunov exponent λ), coupled to a superconductor by an N-mode point contact. Quantization of the adiabatically invariant torus in phase space gives a discrete set of periods Tₙ, which in turn generate a ladder of excited states εnm=(m+1/2)πℏ/Tₙ. The largest quantized period is the Ehrenfest time T₀=λ⁻¹ N. Projection of the invariant torus onto the coordinate plane shows that the wave functions inside the cavity are squeezed to a transverse dimension W/√N, much below the width W of the point contact.