Stark Effect

Definition

The Stark effect is the shifting and splitting of atomic energy levels — and therefore spectral lines — in an external electric field. Place an atom in a field and its levels no longer sit at their field-free energies: some climb, some drop, and degenerate levels fan out into a pattern that grows with the field.

The headline result is that the size of the shift depends on the field in one of two ways:

Linear   (hydrogen):    ΔE ∝ E        (first-order)
Quadratic (most atoms):  ΔE = −½ α E²  (second-order)

Here E is the applied electric field and α is the atom's polarizability — how readily the field stretches its electron cloud into a dipole. The Stark effect is the electric counterpart of the Zeeman effect, which splits the same levels using a magnetic field instead.

How it works

An atom in a field E gains an interaction energy H' = −d·E, where d = −e·r is the electric dipole operator. Perturbation theory then asks: what does this extra energy do to each level?

First order — the linear Stark effect. The first-order shift is the expectation value ⟨ψ|H'|ψ⟩. Because H' is odd under spatial inversion (it flips sign when you flip r → −r), this expectation value is zero for any state of definite parity. A non-zero first-order shift therefore demands states of opposite parity that share the same energy — a parity degeneracy the field can mix. Hydrogen has exactly this: the 2s and 2p levels are degenerate, so the field blends them and the level splits linearly. The result is a permanent dipole that lines up with the field.

Second order — the quadratic Stark effect. Most atoms have no such degeneracy. There the first-order term vanishes and the leading effect is second order — the field induces a dipole, and that induced dipole interacts back with the field:

ΔE = −½ α E²

The minus sign means the polarizable atom is always pulled toward stronger field (it lowers its energy by polarizing) — the same reason a neutral object is attracted to a charged comb. The polarizability α is a sum over all other states, dominated by the nearest ones:

α = 2 Σ_(k≠0)  |⟨k|d|0⟩|² / (E_k − E_0)

Closely spaced levels (small denominators) make α large — which is exactly what happens in highly excited atoms.

A worked example — hydrogen n = 2

Take hydrogen's n = 2 manifold (the 2s and 2p states) and switch on a field. Working in atomic units, the four states split into a symmetric fan. The linear shift for the n = 2 stretched state is:

ΔE = 3 · n · E   (atomic units, for the extreme state)
   = 3 · 2 · E = 6E

Convert to lab numbers. One atomic unit of field is 5.14 × 10¹¹ V/m, and one atomic unit of energy (a Hartree) is 27.2 eV. A modest lab field of 10⁶ V/m is therefore E ≈ 1.9 × 10⁻⁶ atomic units, giving a shift of about 6 × 1.9 × 10⁻⁶ ≈ 1.2 × 10⁻⁵ Hartree ≈ 3 × 10⁻⁴ eV. That is small but absolutely measurable — it is exactly the order of splitting Stark saw smeared across hydrogen's Balmer lines in 1913.

Now contrast the ground state, which has no degeneracy and shifts quadratically. Hydrogen's ground-state polarizability is α = 4.5 atomic units (about 0.67 cubic ångströms). At the same 10⁶ V/m field:

ΔE = −½ α E² = −½ · 4.5 · (1.9 × 10⁻⁶)²
   ≈ −8 × 10⁻¹² Hartree ≈ −2 × 10⁻¹⁰ eV

Six orders of magnitude smaller than the linear n = 2 shift, at the same field. That gulf — linear versus quadratic — is the whole story of the Stark effect in one comparison.

Variants and regimes

Regime	Field dependence	When it applies	Set by
Linear (first-order)	ΔE ∝ E	Hydrogen & degenerate parity states	Permanent / mixed dipole
Quadratic (second-order)	ΔE = −½ α E²	Most atoms, non-degenerate levels	Polarizability α
Rydberg quadratic	ΔE = −½ α E², α ∝ n⁷	Highly excited atoms (large n)	Enormous α
Strong-field mixing	Non-perturbative	Field comparable to internal field	Full diagonalization
Field ionization	Tunneling / barrier suppression	E > threshold ≈ 1/(16 n⁴) a.u.	Tilted potential
AC (dynamic) Stark	∝ intensity / detuning	Oscillating (laser) field	Dynamic polarizability

The same atom moves through several of these regimes as you turn up the field. A ground-state atom starts quadratic; a Rydberg atom is quadratic but with a colossal α; push harder and perturbation theory fails, levels cross and mix non-perturbatively; push harder still and the atom field-ionizes.

Why Rydberg atoms steal the show

The single most dramatic fact about the Stark effect is how it scales with excitation. Almost every length and energy scale of a Rydberg atom follows a power of the principal quantum number n: the orbit radius grows as n², the level spacing shrinks as 1/n³, and — combining those — the polarizability grows as:

α ∝ n⁷

Going from the ground state to n = 50 multiplies α by roughly 50⁷ ≈ 8 × 10¹¹ — close to a trillion-fold increase in sensitivity. A field of a few V/cm, completely invisible to a ground-state atom, produces large, cleanly resolved Stark maps in a Rydberg atom. This is why:

• Rydberg atoms are used as field sensors — they can measure electric fields and even microwave fields with extraordinary precision, because the shift is huge and the resonance is narrow.
• Neutral-atom quantum computers use Rydberg states — the very polarizability that makes them field-sensitive also makes two nearby Rydberg atoms interact strongly (the Rydberg blockade), the basis for two-qubit gates.
• Field ionization detects them state-selectively — because the ionization threshold ≈ 1/(16 n⁴) drops fast with n, ramping a field ionizes high-n states first, reading out which level the atom was in.

Applications

• Spectroscopy and plasma diagnostics. Stark broadening of hydrogen lines is a standard, calibration-free thermometer for plasma electron density — the local microfields from ions and electrons broaden the lines by an amount that depends on density.
• Atomic clocks. The DC and AC (blackbody) Stark shifts are among the largest systematic frequency errors in optical lattice clocks; they must be measured and subtracted to reach the 18th decimal place.
• Optical traps and tweezers. The AC Stark shift from a focused laser creates a potential well that holds neutral atoms — the foundation of optical lattices and single-atom tweezer arrays.
• Quantum control. Stark-shifting a transition with a controllable field tunes atoms in and out of resonance, used for addressing individual qubits and for Stark-switching spectroscopy.
• Quantum-confined Stark effect. In semiconductor quantum wells the same physics shifts excitonic absorption with applied voltage, the operating principle of high-speed electro-optic modulators in fiber telecom.
• Field metrology. Rydberg-atom electrometers turn the n⁷ polarizability into self-calibrated, SI-traceable measurements of RF electric fields.

Common pitfalls and misconceptions

• "The Stark effect is always linear." Only hydrogen (and accidentally degenerate parity states) shift linearly. For every other atom the leading effect is quadratic — the linear term is forbidden by parity. Reaching for ΔE ∝ E on a sodium or helium line is a classic error.
• "The shift goes up with the field." The quadratic shift is negative: ΔE = −½ α E². A polarizable atom lowers its energy in a field, which is why it is attracted to high-field regions — the whole basis of optical trapping.
• Confusing polarizability with permanent dipole. Atoms have no permanent electric dipole in a non-degenerate state (parity forbids it); the field induces one. The induced dipole p = αE is what produces the quadratic shift. The linear hydrogen case effectively behaves as if it had a permanent dipole only because the field mixes degenerate states.
• Treating it as a magnetic effect. The Stark effect is electric (couples to charge displacement); the Zeeman effect is magnetic (couples to magnetic moment). They split the same levels but through entirely different couplings, and only the Stark effect can be quadratic.
• Forgetting field ionization. Perturbation theory eventually breaks. Above E ≈ 1/(16 n⁴) atomic units the bound state is no longer bound — it tunnels out. Quoting a clean Stark shift in a field that already ionizes the atom is meaningless.
• Ignoring the n⁷ scaling. A "tiny" field can be a huge perturbation for a Rydberg atom. The same field that shifts the ground state by 10⁻¹⁰ eV can completely scramble an n = 50 manifold.

Derivation and scaling analysis

Start from the perturbation H' = eEz (field along z). Standard time-independent perturbation theory gives the energy correction as a series in the field:

E_n = E_n⁽⁰⁾ + ⟨n|H'|n⟩ + Σ_(k≠n) |⟨k|H'|n⟩|² / (E_n⁽⁰⁾ − E_k⁽⁰⁾) + ...
            └── 1st order ──┘   └────────── 2nd order ──────────┘
                 (∝ E)                        (∝ E²)

First-order term. ⟨n|z|n⟩ vanishes for any parity eigenstate, so the linear term is zero unless degenerate opposite-parity states force you into degenerate perturbation theory. For hydrogen's n = 2, diagonalizing H' in the {2s, 2p₀} subspace yields eigenvalues ±3eEa₀ — the linear splitting, symmetric about the unshifted level.

Second-order term. When the first order vanishes, the surviving correction is the quadratic sum above. Pulling out the field gives ΔE = −½ α E² with α defined by the matrix elements and energy denominators. Two scaling facts fall straight out:

• Why hydrogen is the exception. Only the Coulomb 1/r potential produces the "accidental" ℓ-degeneracy that places 2s and 2p at the same energy. Any deviation from pure 1/r — which every multi-electron atom has — lifts that degeneracy and kills the linear term.
• Why α ∝ n⁷. The dipole matrix element scales as the orbit size, ⟨d⟩ ∝ n²; squared that is n⁴. The relevant energy denominator is the level spacing, ∝ 1/n³. Multiply: α ∝ n⁴ · n³ = n⁷. This single estimate explains the entire Rydberg sensitivity story.

Field ionization scaling. The total potential along the field is −1/r − Ez (atomic units). The downhill side develops a saddle point; setting the barrier height equal to the binding energy −1/(2n²) gives a threshold field E_ion ≈ 1/(16 n⁴). The fourth-power suppression is why a Rydberg atom that needs only a few V/cm to show a Stark map needs only a slightly larger field to ionize entirely — the perturbative and non-perturbative regimes sit close together at high n.