Anharmonicity is the departure of an oscillating system from the harmonic approximation, in which potential energy is quadratic in displacement and the restoring force is linear. It appears in mechanical oscillators, molecular vibrations, crystal lattices, and quantum circuits. Its consequences include amplitude-dependent oscillation frequencies, unequally spaced quantum energy levels, and coupling between vibrations. Anharmonicity describes the underlying dynamics; it does not by itself imply damping or chaotic motion. (ecommons.cornell.edu)
Mathematical description
Near a smooth, stable equilibrium, the potential can be expanded as a Taylor series in displacement (x):
[ V(x)=V_0+\frac12kx^2+\frac{1}{3!}a_3x^3 +\frac{1}{4!}a_4x^4+\cdots . ]
The linear term vanishes at equilibrium, and (k>0) gives the local stiffness. Retaining only the quadratic term yields harmonic motion. Cubic and higher terms represent anharmonic corrections. Because the restoring force is (F=-dV/dx), a cubic potential term produces a quadratic force term, while a quartic potential term produces a cubic force term. If the potential is symmetric under (x\mapsto -x), all odd-order terms vanish, but even-order anharmonicity can remain. (ocw.mit.edu)
The approximation is local: higher-order terms become important when the system explores larger displacements. A truncated expansion need not describe the potential globally. For example, a cubic correction alone makes a polynomial potential unbounded on one side, even when it accurately approximates a stable physical potential near its minimum. Weak corrections can be treated with perturbation theory, whose usefulness depends on the states and displacement range considered. (ocw.mit.edu)
Classical oscillations
In classical mechanics, an undamped harmonic oscillator has a frequency independent of amplitude. An anharmonic oscillator generally does not. A standard example is the Duffing oscillator, whose unforced equation can be written
[ m\ddot{x}+kx+\beta x^3=0, \qquad V(x)=\frac12kx^2+\frac14\beta x^4, ]
where (m) is the mass. For weak nonlinearity and displacement amplitude (A), its leading frequency correction is
[ \omega(A)\simeq\omega_0 \left(1+\frac{3\beta A^2}{8k}\right), \qquad \omega_0=\sqrt{k/m}. ]
Positive (\beta) produces hardening: frequency increases with amplitude. Negative (\beta) produces local softening; such a truncated potential is not globally confining. (ecommons.cornell.edu)
Nonlinear oscillations can contain higher harmonics rather than a purely sinusoidal waveform. With periodic forcing and damping, amplitude-dependent frequency shifts can also create multiple steady responses and hysteresis. These driven-system effects should be distinguished from the conservative anharmonicity of the potential itself. (galileoandeinstein.phys.virginia.edu)
Quantum energy levels
In quantum mechanics, the harmonic oscillator has equally spaced levels,
[ E_n=\hbar\omega_0(n+\tfrac12), ]
where (\hbar) is the reduced Planck constant. Anharmonic terms shift different levels by different amounts. For a weak quartic perturbation (V_1=\lambda x^4), first-order perturbation theory gives
[ \Delta E_n^{(1)} =3\lambda \left(\frac{\hbar}{2m\omega_0}\right)^2 (2n^2+2n+1). ]
Consequently, positive quartic anharmonicity increases successive level spacings at this order. A cubic perturbation has zero first-order diagonal correction because harmonic eigenstates have definite parity, although it changes the states and contributes to energies at higher orders. (ocw.mit.edu)
Thus, anharmonicity does not universally mean decreasing level spacing. The sign and magnitude of the shifts depend on the potential. Quantum corrections also affect low-lying states, not only highly excited motion. (ocw.mit.edu)
Molecular vibrations and spectroscopy
A chemical bond is approximately harmonic near its equilibrium length, but its potential becomes asymmetric and approaches a dissociation limit at large separation. The Morse potential captures these features:
[ V(r)=D_e[1-e^{-a(r-r_e)}]^2, ]
where (r_e) is the equilibrium separation, (D_e) the well depth, and (a) controls its width. Unlike the harmonic oscillator, the Morse model supports only finitely many bound vibrational levels. (ocw.mit.edu)
In molecular spectroscopy, diatomic vibrational term values are commonly expressed as
[ G(v)=\frac{E_v}{hc} =\omega_e(v+\tfrac12) -\omega_ex_e(v+\tfrac12)^2+\cdots . ]
Here (v) is the vibrational quantum number; (\omega_e) and (\omega_ex_e) are conventionally reported in inverse centimetres, not angular-frequency units. Positive (x_e) gives decreasing successive spacings at this order. NIST tabulates these constants for many diatomic molecules. (srd.nist.gov)
Anharmonic corrections modify vibrational transition probabilities and can make overtone transitions nonzero. Measured intensities depend on both the vibrational states and the displacement dependence of the molecular dipole moment; a nonquadratic potential is therefore only part of the spectroscopic description. (ocw.mit.edu)
Crystal lattices and thermal properties
In condensed-matter physics, harmonic lattice vibrations resolve into independent normal modes, whose quantum excitations are phonons. Higher-order interatomic force constants couple these modes. The resulting phonon interactions shift frequencies, broaden spectral lines, and contribute to finite lifetimes and resistance to heat transport, making anharmonicity central to lattice thermal conductivity. (arxiv.org)
Anharmonicity also underlies thermal expansion. An asymmetric bond potential can shift the average atomic separation as temperature rises. This simple picture explains ordinary positive expansion, although collective lattice behavior requires a mode-based treatment. (ocw.mit.edu)
The quasiharmonic approximation retains harmonic phonons at each fixed volume while allowing their frequencies to depend on volume. It can describe thermal expansion through volume-dependent vibrational free energy, but omits explicit phonon–phonon interactions at fixed volume. Strongly anharmonic systems may require corrections beyond this approximation. (arxiv.org)
Quantum circuits
Anharmonicity enables selective control in superconducting qubits. Defining the frequency anharmonicity as
[ \alpha=\omega_{12}-\omega_{01}, ]
unequal transition frequencies allow a drive to address the lowest transition without equally resonantly exciting the next. In a transmon, the cosine potential of a Josephson junction supplies this nonlinearity. In the large (E_J/E_C) regime, the leading result is (\alpha\simeq-E_C/\hbar), where (E_C) is the charging energy. Its finite anharmonicity distinguishes the device from a harmonic resonator and sets a frequency scale relevant to unwanted excitation of higher levels. (arxiv.org)