STIRAP

Stimulated Raman adiabatic passage (STIRAP) transfers population between two quantum states by adiabatically rotating a dark superposition while keeping a lossy intermediate state ideally unoccupied. A pump field couples the initial state to the intermediate state, and a Stokes field couples the target state to the same intermediate state.

The defining feature is the counterintuitive pulse order: the Stokes field acts first, before there is population in the target state. During the pulse overlap, the instantaneous dark state changes continuously from the initial state to the target state:

$$|D(t)\rangle = \cos\theta(t)|1\rangle - \sin\theta(t)|2\rangle,$$

with

$$\tan\theta(t) = \frac{ \Omega_P(t) }{ \Omega_S(t) }.$$

If nonadiabatic coupling, two-photon detuning, decoherence, and unwanted levels are sufficiently small, an initial $|1\rangle$ follows this dark path to $-|2\rangle$.

STIRAP is robust against moderate pulse-area variation after the adiabaticity margin is large. It is not immune to relative phase noise, two-photon detuning, inadequate overlap, intermediate-state loss, lower-state dephasing, differential light shifts, or multilevel interference. “No intermediate population” is an ideal limiting statement, not a license to omit the loss model.

Required background. Electromagnetically Induced Transparency introduces the Λ dark state, Adiabatic Approximation explains following an instantaneous eigenstate, and Raman Spectroscopy fixes the two-photon frequency difference.

Three-State Model

State labels

Use:

• $|1\rangle$ for the initially populated terminal state;
• $|2\rangle$ for the desired terminal state;
• $|3\rangle$ for the radiative or otherwise lossy intermediate state.

The labels match the Λ convention on the EIT page: $|1\rangle$ and $|2\rangle$ are the two terminal states, while $|3\rangle$ is the shared intermediate.

The pump field couples

$$|1\rangle \longleftrightarrow |3\rangle$$

with Rabi frequency $\Omega_P(t)$. The Stokes field couples

$$|2\rangle \longleftrightarrow |3\rangle$$

with Rabi frequency $\Omega_S(t)$.

The name “Stokes” comes from the Raman frequency ordering. Its field can be present before target-state population exists; the protocol does not require initial population in either $|2\rangle$ or $|3\rangle$.

Detunings

Let

$$\Delta_P = \omega_{31}-\omega_P, \qquad \Delta_S = \omega_{32}-\omega_S$$

be atom-minus-field detunings. Define

$$\Delta \equiv \Delta_P,$$

and the two-photon detuning

$$\delta = \Delta_P-\Delta_S.$$

Two-photon resonance means

$$\omega_P-\omega_S = \omega_{21}.$$

Frequencies, detunings, and Rabi couplings use angular units. Decay rates are inverse times; a population lifetime is $1/\Gamma$, not $2\pi/\Gamma$.

Rotating-frame Hamiltonian

For constant optical phases, choose fixed state phases so the pulse envelopes are real and nonnegative. A time-dependent rephasing contributes an additional diagonal term and cannot be silently discarded. In the ordered basis

$$\left( |1\rangle, |2\rangle, |3\rangle \right),$$

the rotating-wave Hamiltonian, suppressing time arguments inside the matrix, is

$$\frac{H(t)}{\hbar} = \begin{pmatrix} 0 & 0 & \Omega_P/2 \\ 0 & \delta & \Omega_S/2 \\ \Omega_P/2 & \Omega_S/2 & \Delta \end{pmatrix}.$$

This minimal model assumes:

1. the selected states and polarizations isolate one effective Λ system;
2. both rotating-wave approximations are controlled;
3. field envelopes vary slowly relative to the optical carriers;
4. spatial motion and ensemble inhomogeneity are either negligible or modeled separately;
5. dissipation is specified rather than inferred from a non-Hermitian Hamiltonian alone.

Instantaneous Dark State

Mixing angle

At exact two-photon resonance,

$$\delta=0,$$

define the generalized coupling

$$\Omega(t) = \sqrt{ \Omega_P^2(t) + \Omega_S^2(t) }.$$

The mixing angle is

$$\begin{aligned} \sin\theta(t) &= \frac{\Omega_P(t)}{\Omega(t)}, \\ \cos\theta(t) &= \frac{\Omega_S(t)}{\Omega(t)}. \end{aligned}$$

The dark and bright terminal-state superpositions are

$$\begin{aligned} |D(t)\rangle &= \cos\theta(t)|1\rangle - \sin\theta(t)|2\rangle, \\ |B(t)\rangle &= \sin\theta(t)|1\rangle + \cos\theta(t)|2\rangle. \end{aligned}$$

Direct multiplication gives

$$H(t)|D(t)\rangle=0.$$

The intermediate-state amplitudes cancel:

$$\frac{\hbar}{2} \left[ \Omega_P\cos\theta - \Omega_S\sin\theta \right] = 0.$$

Bright eigenvalues

In the basis

$$\left( |D\rangle, |B\rangle, |3\rangle \right),$$

the resonant two-photon Hamiltonian is

$$\frac{H}{\hbar} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & \Omega/2 \\ 0 & \Omega/2 & \Delta \end{pmatrix}.$$

The bright eigenvalues are

$$\lambda_\pm = \frac{ \Delta \pm \sqrt{ \Delta^2+\Omega^2 } }{ 2 }.$$

At one-photon resonance,

$$\Delta=0,$$

they reduce to

$$\lambda_\pm = \pm\frac{\Omega}{2}.$$

One convenient eigenvector convention is

$$|\pm\rangle = \frac{ |B\rangle \pm |3\rangle }{ \sqrt2 }$$

at $\Delta=0$.

The instantaneous dark state has zero dynamical eigenvalue in this gauge. Its time dependence still produces geometric and nonadiabatic terms.

Counterintuitive Pulse Order

Endpoint mapping

For transfer from $|1\rangle$ to $|2\rangle$, require the asymptotic ratios

$$\frac{ \Omega_P }{ \Omega_S } \longrightarrow 0$$

at the beginning of the interaction, and

$$\frac{ \Omega_S }{ \Omega_P } \longrightarrow 0$$

at the end.

Then

$$|D(t_i)\rangle \simeq |1\rangle,$$

while

$$|D(t_f)\rangle \simeq -|2\rangle.$$

The Stokes pulse must therefore precede the pump pulse. During their overlap, $\theta$ rotates from $0$ to $\pi/2$.

Top: Gaussian Stokes and pump pulses with $\tau=1.2\sigma$ and $\Omega_0\sigma=20$. Middle: direct Schrödinger evolution from $|1\rangle$ at $t=-5\sigma$ (solid) compared with the ideal dark-state target population (dashed). Bottom: the finite-speed passage occupies $|3\rangle$, with a small residual excitation after the pulses; the vertical scale is percent. The target population at $5\sigma$ is about $99.13\%$. This closed, resonant example has $\Delta=\delta=\Gamma_3=0$. Loss requires the corresponding open-system calculation, rather than simply reading the final population from this plot.

Why the intuitive order fails

If the pump arrives first, the initial state is directly coupled to $|3\rangle$ before a dark superposition aligned with $|1\rangle$ exists. The system can undergo ordinary Rabi excitation, spontaneous loss, or detuning-sensitive Raman dynamics.

An intuitive sequence can still transfer population under specially chosen pulse areas or detunings. That transfer is not the defining dark-state adiabatic path of STIRAP.

Overlap is essential

“Stokes first” does not mean two nonoverlapping pulses. If both couplings are negligible while the mixing angle must rotate, the bright–dark gap closes and adiabatic following fails. Useful sequences have:

• a Stokes-leading entrance;
• substantial simultaneous coupling during rotation;
• a pump-dominated exit.

If the pulses overlap almost completely with a nearly constant ratio, the angle barely changes and little endpoint transfer occurs. If their separation is too large, the gap collapses between them.

Endpoint degeneracy

Both fields eventually vanish, so all bare terminal states become degenerate in the rotating interaction Hamiltonian. The elementary nondegenerate adiabatic theorem cannot be applied blindly at $t\to\pm\infty$.

The practical statement is about a finite interval: one pulse dominates while the state is aligned with a decoupled endpoint, the mixing angle rotates only while a sufficient gap exists, and the state is frozen into the target before the remaining field vanishes.

Adiabaticity

Geometric coupling

The instantaneous basis transformation contributes

$$H_{\mathrm{geom}} = - i\hbar U^\dagger\dot U.$$

At $\Delta=\delta=0$,

$$\frac{H_{\mathrm{ad}}}{\hbar} = \begin{pmatrix} 0 & -i\dot\theta/\sqrt2 & -i\dot\theta/\sqrt2 \\ i\dot\theta/\sqrt2 & \Omega/2 & 0 \\ i\dot\theta/\sqrt2 & 0 & -\Omega/2 \end{pmatrix}$$

in the basis $(|D\rangle,|+\rangle,|-\rangle)$, up to basis-phase conventions. The physically invariant magnitudes are

$$\left| \langle\pm|\dot D\rangle \right| = \frac{ |\dot\theta| }{ \sqrt2 }.$$

Nonadiabatic transfer out of the dark state is suppressed when these couplings are small relative to the bright gaps.

Mixing-angle derivative

Differentiating

$$\tan\theta = \frac{\Omega_P}{\Omega_S}$$

gives

$$\dot\theta = \frac{ \Omega_S\dot\Omega_P - \Omega_P\dot\Omega_S }{ \Omega_P^2+\Omega_S^2 }.$$

For resonant STIRAP, an exact local dimensionless ratio is

$$\epsilon_{\mathrm{ad}}(t) = \frac{ \sqrt2|\dot\theta(t)| }{ \Omega(t) }.$$

A useful requirement is

$$\max_{t\in I} \epsilon_{\mathrm{ad}}(t) \ll 1.$$

Here $I$ is the finite interval over which the mixing angle changes appreciably. The factor $\sqrt2$ follows from normalized resonant bright eigenvectors and gaps $\Omega/2$; rephasing those eigenvectors does not change the ratio. Small local ratios are a diagnostic, not by themselves a complete bound on accumulated error. The scale comparison

$$|\dot\theta| \ll \Omega$$

is universal for the resonant three-state model.

Global pulse-area criterion

A common global diagnostic is

$$\mathcal S = \int \Omega(t) \,dt \gg 1.$$

Large generalized pulse area is helpful but not sufficient. A sequence can have large area and still fail because:

• the pulse ratio has the wrong endpoint order;
• the angle rotates where $\Omega$ is small;
• two-photon detuning couples dark and bright states;
• phase noise changes the dark vector;
• an unwanted level closes the relevant gap.

Local and global checks should be reported together.

Gaussian Pulse Pair

Delayed envelopes

Consider equal-peak Gaussian pulses:

$$\Omega_S(t) = \Omega_0 \exp \left[ - \frac{ \left( t+\tau/2 \right)^2 }{ 2\sigma^2 } \right],$$

and

$$\Omega_P(t) = \Omega_0 \exp \left[ - \frac{ \left( t-\tau/2 \right)^2 }{ 2\sigma^2 } \right].$$

For $\tau>0$, the Stokes pulse is centered at $-\tau/2$ and the pump at $+\tau/2$.

Their ratio is

$$\frac{\Omega_P}{\Omega_S} = \exp \left( \frac{ t\tau }{ \sigma^2 } \right).$$

Therefore

$$\theta(t) = \arctan \left[ \exp \left( \frac{ t\tau }{ \sigma^2 } \right) \right].$$

The angle derivative is

$$\dot\theta(t) = \frac{ \tau }{ 2\sigma^2 } \operatorname{sech} \left( \frac{ t\tau }{ \sigma^2 } \right).$$

Its maximum occurs at $t=0$:

$$\dot\theta_{\max} = \frac{\tau}{2\sigma^2}.$$

The generalized coupling there is

$$\Omega(0) = \sqrt2\, \Omega_0 \exp \left( - \frac{ \tau^2 }{ 8\sigma^2 } \right).$$

The center adiabatic ratio is consequently

$$\epsilon_{\mathrm{ad}}(0) = \frac{ \tau }{ 2\Omega_0\sigma^2 } \exp \left( \frac{ \tau^2 }{ 8\sigma^2 } \right).$$

This expression displays the pulse-separation tradeoff. Larger $\tau$ improves endpoint ordering but reduces overlap and the central gap. This is a center diagnostic, not the maximum over an infinite Gaussian pulse: the gap vanishes faster than the angle derivative in the remote tails. Choose a finite interval with negligible omitted angle rotation and check the actual dynamics and endpoint preparation.

Ideal populations

If the state follows $|D(t)\rangle$ exactly,

$$\begin{aligned} P_1(t) &= \cos^2\theta(t), \\ P_2(t) &= \sin^2\theta(t), \\ P_3(t) &= 0. \end{aligned}$$

For the Gaussian pair,

$$P_2(t) = \frac{ 1 }{ 1+ \exp \left( - \frac{ 2t\tau }{ \sigma^2 } \right) }.$$

The logistic form is the ideal adiabatic path, not a prediction that remains exact at finite passage speed, even without detuning or decay. Indeed, $H|D\rangle=0$ cannot generate a nonzero $|\dot D\rangle$ by itself. A finite-speed realization needs small bright-state admixtures, unless additional counterdiabatic couplings are applied.

Detuning Sensitivity

One-photon detuning

At $\delta=0$, the algebraic dark state remains exact for any common one-photon detuning $\Delta$. One-photon detuning is therefore less destructive than two-photon detuning in the ideal model.

It still changes the bright gaps:

$$\lambda_\pm = \frac{ \Delta \pm \sqrt{ \Delta^2+\Omega^2 } }{ 2 }.$$

For

$$|\Delta|\gg\Omega,$$

the nearest bright eigenvalue has magnitude

$$|\lambda_{\mathrm{near}}| \simeq \frac{ \Omega^2 }{ 4|\Delta| }.$$

Large one-photon detuning can therefore make adiabatic following harder by shrinking the nearest gap, even though it suppresses direct intermediate excitation in a far-detuned Raman picture.

Two-photon detuning

The term

$$\hbar\delta |2\rangle\langle2|$$

is not diagonal in the dark–bright basis. Since

$$|2\rangle = - \sin\theta|D\rangle + \cos\theta|B\rangle,$$

it contributes

$$\frac{H_\delta}{\hbar} = \delta\sin^2\theta |D\rangle\langle D| + \delta\cos^2\theta |B\rangle\langle B| - \delta\sin\theta\cos\theta \left( |D\rangle\langle B| + |B\rangle\langle D| \right).$$

The dark–bright coupling is largest near $\theta=\pi/4$:

$$\left| \frac{ \delta\sin2\theta }{ 2 } \right| \le \frac{|\delta|}{2}.$$

Two-photon detuning therefore directly spoils the decoupling. The allowed detuning depends on pulse duration and gap; there is no universal bandwidth independent of pulse shape.

Differential light shifts

Additional off-resonant levels produce terminal-state energy shifts $\Delta_{\mathrm{LS},j}=\delta E_j/\hbar$. For the Hamiltonian above the effective two-photon detuning becomes

$$\delta_{\mathrm{eff}}(t) = \delta + \Delta_{\mathrm{LS},2}(t) - \Delta_{\mathrm{LS},1}(t).$$

Because the pump and Stokes intensities change differently in time, $\delta_{\mathrm{eff}}$ can sweep during the passage. Chirp compensation, polarization choice, auxiliary fields, or an expanded multilevel model may be required. Include only extra couplings absent from the displayed three-state Hamiltonian here; its own pump and Stokes shifts emerge from that Hamiltonian and must not be counted twice.

Optical Phase and Coherence

Complex couplings

Write

$$\Omega_P = |\Omega_P|e^{i\phi_P}, \qquad \Omega_S = |\Omega_S|e^{i\phi_S}.$$

With $H_{31}/\hbar=\Omega_P/2$, $H_{32}/\hbar=\Omega_S/2$, and their Hermitian conjugates in the opposite entries, the dark state is

$$|D\rangle = \cos\theta|1\rangle - e^{i\phi} \sin\theta|2\rangle,$$

where

$$\phi = \phi_P-\phi_S.$$

For constant relative phase, ideal complete transfer produces

$$|1\rangle \longrightarrow - e^{i\phi} |2\rangle.$$

For a pure target population this phase is global. It becomes observable when STIRAP prepares a superposition, forms one arm of an interferometer, or is embedded in a gate sequence.

Phase noise

A time-dependent relative phase contributes geometric coupling and behaves like a fluctuating two-photon detuning. The dark-state derivative contains

$$\frac{d}{dt}|D\rangle \supset - i\dot\phi\, e^{i\phi} \sin\theta |2\rangle.$$

Its orthogonal derivative has norm

$$\|(1-|D\rangle\langle D|)|\dot D\rangle\| =\sqrt{\dot\theta^2+\sin^2\theta\cos^2\theta\,\dot\phi^2}.$$

Adiabatic following also accumulates the geometric phase $\gamma_D=-\int\sin^2\theta\,\dot\phi\,dt$. This overall phase matters when comparing with a reference arm. Phase locking and optical-path stability therefore affect both transfer and interferometric phase.

For overlapping traveling-wave beams, the relative phase includes

$$\left( \mathbf k_P-\mathbf k_S \right) \mathbin{\cdot} \mathbf r.$$

Motion through that phase pattern creates two-photon Doppler shifts and ensemble dephasing.

Dissipation and Transfer Error

Intermediate-state decay

Let the intermediate population decay at rate $\Gamma_3$. A minimal master equation is

$$\dot\rho = - \frac{i}{\hbar} [H,\rho] + \sum_j \mathcal D[L_j]\rho,$$

with

$$\mathcal D[L]\rho = L\rho L^\dagger - \frac12 \left\{ L^\dagger L, \rho \right\}.$$

For the normalized state conditioned on no spontaneous jump, write its intermediate population as $P_{3,c}(t)$. The exact no-jump probability for these decay channels is

$$P_{\mathrm{no\ jump}} = \exp \left[ - \Gamma_3 \int P_{3,c}(t) \,dt \right].$$

This is not generally the final population remaining in the Λ system: jumps may return to either terminal state. If every jump enters an external sink, it is the survival probability. Alternatively, the lost population is exactly $\Gamma_3\int\rho_{33}(t)dt$ for unconditional evolution with that sink. Replacing the conditional occupation by the closed-system occupation gives only a weak-loss estimate.

Nonadiabatic corrections commonly give

$$P_3 = O \left[ \left( \frac{\dot\theta}{\Omega} \right)^2 \right]$$

on resonance, with pulse-shape-dependent coefficients.

Even a small peak intermediate population can cause appreciable loss if $\Gamma_3$ is large or the passage is long.

Terminal-state decoherence

The dark state is a coherence between $|1\rangle$ and $|2\rangle$. Dephasing of that coherence acts directly even when $P_3=0$. If the terminal coherence decays at rate $\gamma_{12}$, making the protocol arbitrarily slow is not beneficial.

The useful operating window is schematically

$$\frac1{\Delta_{\mathrm{gap}}} \ll T \ll \frac1{\gamma_{12}},$$

where $T$ is the passage time and $\Delta_{\mathrm{gap}}$ is the relevant minimum bright–dark gap during the active rotation interval, excluding aligned pulse tails. The first inequality is adiabatic; the second limits accumulated dephasing.

Branching and repumping

Spontaneous emission from $|3\rangle$ may return population to $|1\rangle$, to $|2\rangle$, or to states outside the modeled Λ system. A high final $|2\rangle$ population after many attempts does not prove coherent STIRAP if optical pumping can produce the same result.

Time-resolved intermediate fluorescence, reversal tests, phase-sensitive measurements, and master-equation fits help separate coherent passage from dissipative accumulation.

Far-Detuned Raman Limit

Eliminating the intermediate amplitude

For

$$|\Delta| \gg |\Omega_P|, |\Omega_S|, |\delta|,$$

and with envelope bandwidth and $\Gamma_3$ also small relative to $|\Delta|$, the intermediate amplitude follows the terminal amplitudes after fast transients have been excluded or appropriately prepared:

$$c_3 \simeq - \frac{ \Omega_Pc_1+\Omega_Sc_2 }{ 2\Delta }.$$

Define the pump and Stokes light-shift scales and the effective Raman coupling by

$$\begin{aligned} S_P &= \frac{\Omega_P^2}{4\Delta}, \\ S_S &= \frac{\Omega_S^2}{4\Delta}, \\ \Omega_{\mathrm{eff}} &= - \frac{\Omega_P\Omega_S}{2\Delta}. \end{aligned}$$

Substitution gives the compact effective terminal-state Hamiltonian

$$\frac{H_{\mathrm{eff}}}{\hbar} = \begin{pmatrix} - S_P & \Omega_{\mathrm{eff}}/2 \\ \Omega_{\mathrm{eff}}/2 & \delta - S_S \end{pmatrix}.$$

The same elimination generates differential AC Stark shifts. Dropping those diagonal terms while retaining the Raman coupling gives an inconsistent effective Hamiltonian. For an ordinary Raman Rabi pulse, the diagonal detuning is $\delta+S_P-S_S$ and resonance sets it to zero. For dark-state STIRAP, however, $\delta=0$ already leaves the exact null vector $(\Omega_S,-\Omega_P)$ even when $S_P\ne S_S$. The correlated diagonal and off-diagonal terms form a rank-one matrix. Canceling its differential diagonal term by an added chirp generally destroys that null vector; the Raman π-pulse condition is not the STIRAP dark condition.

Relation to STIRAP

Far-detuned Raman transfer and STIRAP overlap but are not synonyms.

• STIRAP emphasizes adiabatic following of a dark eigenstate and can operate near one-photon resonance.
• Adiabatic elimination emphasizes a virtual intermediate state at large detuning and produces an effective two-state model.
• A far-detuned pulse sequence can be both adiabatically eliminated and STIRAP-like if it follows the appropriate instantaneous eigenstate.
• A resonant $\pi$ pulse of the effective Raman coupling is not STIRAP merely because two fields are used.

Robustness and Its Limits

What amplitude robustness means

Once

$$\epsilon_{\mathrm{ad}}\ll1,$$

moderate common rescaling of both pulse amplitudes leaves the endpoint dark state unchanged and often leaves transfer near unity. This is the familiar plateau of STIRAP efficiency versus pulse area.

The plateau has boundaries. Reducing intensity eventually violates adiabaticity; increasing it can activate off-resonant levels, ionization, AC Stark shifts, power broadening, heating, or technical nonlinearities.

Timing robustness

Transfer can tolerate a range of pulse delays when:

• Stokes still leads;
• overlap remains substantial;
• the angle rotates while the gap is large;
• endpoint ratios remain well separated.

Timing is not irrelevant. A scan over delay is one of the most useful experimental diagnostics because STIRAP should show a broad counterintuitive-order transfer region distinct from the intuitive order.

Detuning robustness

STIRAP is commonly more tolerant of one-photon detuning than two-photon detuning. The statement is qualitative. Large one-photon detuning shrinks the nearest gap, and multilevel systems can contain accidental resonances. The transfer map should be measured or simulated across both detuning axes.

Shortcuts to adiabaticity

Counterdiabatic or superadiabatic protocols add controls designed to cancel nonadiabatic transitions. In the three-state problem, the exact counterdiabatic term can require direct coupling between $|1\rangle$ and $|2\rangle$, a transition that may be forbidden or technically difficult.

Shortcuts can reduce duration, but they introduce calibration, bandwidth, phase, and model-dependence costs. They belong to the broader Optimal Control landscape rather than replacing the baseline STIRAP criteria.

Multilevel Systems

Unwanted intermediate states

Atoms and molecules rarely contain one isolated intermediate state. If several states $|3,\alpha\rangle$ participate, each has pump and Stokes couplings:

$$\Omega_{P,\alpha} \propto \langle3,\alpha| \mathbf d \mathbin{\cdot} \boldsymbol\epsilon_P |1\rangle,$$

and

$$\Omega_{S,\alpha} \propto \langle3,\alpha| \mathbf d \mathbin{\cdot} \boldsymbol\epsilon_S |2\rangle.$$

A common dark vector requires a nonzero pair $(c_1,c_2)$ satisfying $\Omega_{P,\alpha}c_1+\Omega_{S,\alpha}c_2=0$ for every intermediate state. With exactly two terminal states, the coupling matrix must have rank one (or zero for a completely uncoupled pair); rank two leaves no dark vector. Thus all nonzero coupling rows must be proportional, including their complex phases. Additional terminal states can support a larger dark subspace. Extra intermediate levels can instead remove darkness, shift resonances, interfere in Raman amplitudes, or open leakage channels.

The nearest-level rule is insufficient when a farther state has a much larger transition moment.

Degenerate terminal manifolds

Zeeman, hyperfine, rotational, or vibrational degeneracy can turn one target into a manifold. Polarization and magnetic fields select which coherent superposition is transferred. Unresolved target states can make a population measurement appear efficient while reducing state fidelity.

A controlled calculation should specify:

1. quantization axis and field geometry;
2. polarization components;
3. all relevant Clebsch–Gordan or molecular line-strength factors;
4. differential shifts and decoherence;
5. preparation and detection efficiencies for each sublevel.

Continuum and lossy targets

STIRAP ideas extend to ionization, dissociation, waveguides, cavities, and continuum-assisted transfer, but a normalizable dark eigenstate and adiabatic gap may not survive unchanged. Effective non-Hermitian eigenvectors require biorthogonal analysis and a complete accounting of jump channels.

The three-state unitary formulas should be treated as a model, not a universal theorem about every delayed-pulse transfer.

Molecular Applications

Rovibrational state transfer

Molecules provide dense rovibronic structure and terminal states separated by large changes in vibrational or rotational character. STIRAP can connect an initially accessible state to a deeply bound target through an electronically excited intermediate while suppressing spontaneous emission.

The practical design problem includes:

• Franck–Condon factors for both legs;
• rotational and parity selection rules;
• intermediate-state lifetime;
• nearby rovibronic levels;
• laser phase coherence across the Raman difference;
• differential AC Stark shifts;
• spatially varying pulse ratio in a molecular sample.

Large one-leg transition strength is not enough. Both pump and Stokes matrix elements must support an adequate gap during overlap.

Ultracold molecule preparation

STIRAP is widely used to convert weakly bound atom-pair molecules into a selected rovibrational ground state. High transfer efficiency is crucial because spontaneous scattering heats or removes particles, while imperfect state purity complicates dipolar-collision and many-body experiments.

A round-trip transfer measurement can reduce some number-calibration uncertainties. It yields the product $\eta_f\eta_r$ when only the correct forward target can return, survival and detection are accounted for, and no alternative return path contributes. Symmetry is needed only to infer one-pass efficiency as the square root of that product.

Superposition preparation

If the pulse ratio approaches a finite value instead of infinity at the end, and both pulses then vanish together with that ratio and relative phase held fixed, fractional STIRAP can prepare (up to an overall phase)

$$|\psi_f\rangle = \cos\theta_f|1\rangle - e^{i\phi_f} \sin\theta_f|2\rangle.$$

This turns pulse-ratio and optical-phase control into amplitude and phase control. Unlike complete transfer, the relative phase is directly observable.

For practical molecular constraints, see Cold Molecules and Molecular Control Frontiers.

Exercises

1. Test a common dark state

Two intermediate states have pump/Stokes coupling rows $(1,2)$ and $(3,6)$. Find a normalized terminal dark vector. Does it survive if the second row becomes $(3,-6)$?

Solution

For proportional rows, $(c_1,c_2)=(2,-1)/\sqrt5$ annihilates both. The altered rows have determinant $-12$, so the only common null vector is zero. Making both intermediate states far detuned may suppress their occupation, but it does not restore an exact three-state-style dark vector.

2. Explain the Gaussian tail problem

For the equal-width Gaussian pulses above, compare the large positive-time decay of $\dot\theta$ and $\Omega$. Explain why the infinite-time maximum of $\epsilon_{\rm ad}$ is not a useful requirement.

Solution

The pulse ratio is $e^{t\tau/\sigma^2}$, hence $\dot\theta\sim(\tau/\sigma^2)e^{-t\tau/\sigma^2}$ while $\Omega\sim\Omega_0e^{-(t-\tau/2)^2/(2\sigma^2)}$. The Gaussian gap therefore vanishes faster, and the local ratio diverges. Yet the remaining rotation is $\pi/2-\theta\sim e^{-t\tau/\sigma^2}$ and can already be negligible. A finite-interval calculation must check the residual endpoint error and actual transfer rather than impose a nonclosing gap in the remote tails.

3. Estimate local adiabaticity

Take

$$\sigma = 1.0\ \mathrm{\mu s}, \qquad \tau = 1.2\ \mathrm{\mu s},$$

and

$$\frac{\Omega_0}{2\pi} = 10.0\ \mathrm{MHz}.$$

At the pulse midpoint:

1. calculate $\dot\theta_{\max}$;

2. calculate $\Omega(0)$;

3. calculate the local adiabaticity ratio

   $$\epsilon_{\mathrm{ad}}(0) = \frac{ \sqrt2\dot\theta_{\max} }{ \Omega(0) }.$$

Solution

The maximum angle speed is

$$\begin{aligned} \dot\theta_{\max} &= \frac{\tau}{2\sigma^2} \\ &= \frac{ 1.2\times10^{-6} }{ 2(1.0\times10^{-6})^2 } \\ &= 6.0\times10^5\ \mathrm{s^{-1}}. \end{aligned}$$

The midpoint coupling is

$$\Omega(0) = \sqrt2\, \Omega_0 \exp \left( - \frac{\tau^2}{8\sigma^2} \right).$$

Since $\tau^2/(8\sigma^2)=0.18$,

$$\begin{aligned} \Omega(0) &= \sqrt2 \left[ 2\pi(10.0\times10^6) \right] e^{-0.18} \\ &\simeq 7.42\times10^7\ \mathrm{s^{-1}}. \end{aligned}$$

Hence

$$\begin{aligned} \epsilon_{\mathrm{ad}}(0) &= \frac{ \sqrt2(6.0\times10^5) }{ 7.42\times10^7 } \\ &\simeq 0.0114. \end{aligned}$$

The midpoint is strongly adiabatic by this local criterion. A complete assessment still checks the pulse wings, detuning, and decoherence.

4. See how one-photon detuning shrinks the gap

At one instant, suppose

$$\frac{\Omega}{2\pi} = 10\ \mathrm{MHz}.$$

Compare the nearest bright–dark gap for:

1. $\Delta=0$;
2. $\Delta/(2\pi)=100\ \mathrm{MHz}$.

Use the large-detuning approximation for the second case.

Solution

At resonance,

$$\lambda_\pm = \pm\frac{\Omega}{2},$$

so the nearest gap is

$$\frac{ \Delta_{\mathrm{gap,res}} }{ 2\pi } = 5.0\ \mathrm{MHz}.$$

For large positive detuning,

$$|\lambda_{\mathrm{near}}| \simeq \frac{\Omega^2}{4|\Delta|}.$$

In frequency units,

$$\begin{aligned} \frac{ \Delta_{\mathrm{gap,far}} }{ 2\pi } &\simeq \frac{ (10\ \mathrm{MHz})^2 }{ 4(100\ \mathrm{MHz}) } \\ &= 0.250\ \mathrm{MHz}. \end{aligned}$$

The nearest gap is smaller by a factor of $20$. The ideal dark vector still exists at two-photon resonance, but the same pulse envelopes are much less adiabatic relative to this gap.

5. Track the transferred optical phase

Let

$$\phi = \phi_P-\phi_S = \frac{\pi}{3}$$

be constant during a complete STIRAP sequence.

1. What target state follows from an initial $|1\rangle$?
2. Is this phase observable in a final population measurement?
3. When would it become observable?

Solution

The complex dark state is

$$|D\rangle = \cos\theta|1\rangle - e^{i\phi} \sin\theta|2\rangle.$$

At the end of complete transfer,

$$|\psi_f\rangle = - e^{i\pi/3}|2\rangle = e^{i4\pi/3}|2\rangle.$$

For a state occupying only $|2\rangle$, this overall phase does not change the population. It becomes observable if the target interferes with a reference amplitude, if fractional STIRAP prepares a superposition, if the passage is reversed in a phase-sensitive sequence, or if STIRAP is part of a quantum gate.

Fluctuation of $\phi$ during the sequence is more serious than a fixed phase: it changes the instantaneous dark vector and can cause transfer error.

6. Estimate spontaneous loss from residual occupation

Suppose the normalized no-jump intermediate population averages

$$\overline{P_3} = 1.0\times10^{-3}$$

during a $2.0\ \mathrm{\mu s}$ overlap. Let

$$\frac{\Gamma_3}{2\pi} = 6.0\ \mathrm{MHz}.$$

Estimate the no-jump probability and the probability of at least one spontaneous emission. When is the latter also irreversible loss?

Solution

Approximate

$$\int P_3(t)\,dt \simeq \overline{P_3}T.$$

The integrated decay hazard is

$$\begin{aligned} \Lambda &= \Gamma_3\overline{P_3}T \\ &= \left[ 2\pi(6.0\times10^6) \right] \left( 1.0\times10^{-3} \right) \times \left( 2.0\times10^{-6} \right) \\ &\simeq 0.0754. \end{aligned}$$

Therefore

$$P_{\mathrm{surv}} \simeq e^{-\Lambda} \simeq 0.927,$$

and

$$P_{\mathrm{at\ least\ one\ jump}} \simeq 1-P_{\mathrm{surv}} \simeq 0.073.$$

An average intermediate population of only $10^{-3}$ still produces about $7\%$ probability of emission because the radiative rate is fast on the pulse timescale. This is irreversible population loss only if the jump channels leave the retained system; returning jumps can instead reduce coherence or repopulate the target.

7. Derive the far-detuned Raman Hamiltonian

For

$$|\Delta| \gg |\Omega_P|, |\Omega_S|, |\delta|,$$

start from the amplitude equations generated by the three-state Hamiltonian.

1. Set $\dot c_3\simeq0$ and solve for $c_3$.
2. Derive the effective $2\times2$ Hamiltonian.
3. Identify the effective Raman Rabi frequency and differential light shift.

Solution

The intermediate amplitude equation is

$$i\dot c_3 = \Delta c_3 + \frac{\Omega_P}{2}c_1 + \frac{\Omega_S}{2}c_2.$$

Adiabatic elimination gives

$$c_3 \simeq - \frac{ \Omega_Pc_1+\Omega_Sc_2 }{ 2\Delta }.$$

The terminal equations are

$$i\dot c_1 = \frac{\Omega_P}{2}c_3,$$

and

$$i\dot c_2 = \delta c_2 + \frac{\Omega_S}{2}c_3.$$

Introduce

$$\begin{aligned} S_P &= \frac{\Omega_P^2}{4\Delta}, \\ S_S &= \frac{\Omega_S^2}{4\Delta}, \\ \Omega_{\mathrm{eff}} &= - \frac{\Omega_P\Omega_S}{2\Delta}. \end{aligned}$$

Substitution then yields

$$\frac{H_{\mathrm{eff}}}{\hbar} = \begin{pmatrix} - S_P & \Omega_{\mathrm{eff}}/2 \\ \Omega_{\mathrm{eff}}/2 & \delta-S_S \end{pmatrix}.$$

The differential shift of the two terminal diagonal energies is

$$\Delta_{\mathrm{diff}} = \frac{ \Omega_P^2-\Omega_S^2 }{ 4\Delta }$$

with the fixed sign convention used here. The full diagonal difference is $\delta+\Delta_{\mathrm{diff}}$. Setting that to zero is the ordinary Raman Rabi-resonance condition. In contrast, at $\delta=0$ the entire effective matrix annihilates $(\Omega_S,-\Omega_P)$ even for unequal pulse strengths: this is the dark-state condition for STIRAP.

8. Interpret a round trip

A calibrated round trip returns $81\%$ of the prepared molecules. Suppose only correctly transferred molecules can return and no particle loss occurs between passes. What is determined without assuming identical passes? What changes if independent spectroscopy gives $\eta_f=0.95$?

Solution

The measurement gives $\eta_f\eta_r=0.81$, not either factor separately. Because each lies between zero and one, both are at least $0.81$. Identical passes would give $0.90$ each; the independent forward result instead implies $\eta_r=0.81/0.95\simeq0.853$. A population round trip alone does not determine the transferred phase or rule out incoherent population transfer. An interferometric comparison supplies that distinct information.

References

• K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Reviews of Modern Physics 70, 1003–1025 (1998).
• X. Chen and J. G. Muga, “Engineering of fast population transfer in three-level systems,” Physical Review A 86, 033405 (2012).
• U. Gaubatz, P. Rudecki, S. Schiemann, and K. Bergmann, “Population transfer between molecular vibrational levels by stimulated Raman scattering with partially overlapping laser fields,” Journal of Chemical Physics 92, 5363–5376 (1990).
• D. Grischkowsky, M. M. T. Loy, and P. F. Liao, “Adiabatic following model for two-photon transitions: Nonlinear mixing and pulse propagation,” Physical Review A 12, 2514 (1975).
• K. S. Kumar, A. Vepsäläinen, S. Danilin, and G. S. Paraoanu, “Stimulated Raman adiabatic passage in a three-level superconducting circuit,” Nature Communications 7, 10628 (2016).
• T. A. Laine and S. Stenholm, “Adiabatic processes in three-level systems,” Physical Review A 53, 2501–2512 (1996).
• J. Martin, B. W. Shore, and K. Bergmann, “Coherent population transfer in multilevel systems with magnetic sublevels. III. Experimental results,” Physical Review A 54, 1556–1569 (1996).
• K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Julienne, D. S. Jin, and J. Ye, “A high phase-space-density gas of polar molecules,” Science 322, 231–235 (2008).
• N. V. Vitanov, A. A. Rangelov, B. W. Shore, and K. Bergmann, “Stimulated Raman adiabatic passage in physics, chemistry, and beyond,” Reviews of Modern Physics 89, 015006 (2017).