Nuclear Coherent Population Transfer (NCPT)

• NCPT is a technique that uses dual coherent x-ray pulses in a three-level Λ-system to transfer population between nuclear states while minimizing losses from the decaying intermediate state.
• It employs a counterintuitive pulse sequence (Stokes before pump) to create an adiabatic dark state, facilitating nearly lossless transfer and reducing sensitivity to branching decay channels.
• Advanced protocols, including shortcut methods and physics-informed neural networks, are being developed to optimize pulse design and improve NCPT performance for applications like isomer depletion and nuclear clocks.

Searching arXiv for recent and foundational papers on Nuclear Coherent Population Transfer. Nuclear coherent population transfer (NCPT) denotes coherent transfer of population between two lower-lying nuclear states in a three-level nucleus using two x-ray laser pulses in a nuclear $\Lambda$-scheme, in direct analogy with stimulated Raman adiabatic passage (STIRAP) in atomic physics. The pump field couples $|1\rangle \leftrightarrow |3\rangle$, the Stokes field couples $|2\rangle \leftrightarrow |3\rangle$, and the intended pathway is not direct $|1\rangle \to |2\rangle$, but coherent transfer through the common upper state $|3\rangle$, while ideally avoiding significant occupation of $|3\rangle$ itself (Liao et al., 2010). In the nuclear context, the concept is motivated especially by isomer depletion, controlled pumping or release of energy stored in long-lived nuclear states, and, in later work, by nuclear clocks and nuclear batteries (Liao et al., 2010, Liu et al., 15 Aug 2025).

1. Definition and basic mechanism

The defining physical picture of NCPT is a nuclear three-level $\Lambda$-scheme with initial population in $|1\rangle$, target state $|2\rangle$, and excited triggering level $|3\rangle$. In the STIRAP realization, the Stokes field is applied first and the pump field second, i.e. the counterintuitive sequence. This prepares the system in an adiabatic dark state involving the two lower states and little or no occupation of the decaying upper state $|1\rangle \leftrightarrow |3\rangle$0; as the pulse amplitudes evolve, the dark state rotates from mostly $|1\rangle \leftrightarrow |3\rangle$1 to mostly $|1\rangle \leftrightarrow |3\rangle$2, enabling complete transfer (Liao et al., 2010).

A standard explicit form of the dark state used in the nuclear STIRAP literature is

$|1\rangle \leftrightarrow |3\rangle$3

which contains no $|1\rangle \leftrightarrow |3\rangle$4 component (Liao et al., 2013). This is the central reason NCPT is attractive for nuclei with lossy intermediate states: spontaneous decay, internal conversion, or other open decay channels from $|1\rangle \leftrightarrow |3\rangle$5 are suppressed by avoiding real population of that level.

The early NCPT formulation distinguishes two regimes. In regime (i), the upper-state lifetime is longer than the pulse duration; then NCPT can occur either by STIRAP or by sequential isolated $|1\rangle \leftrightarrow |3\rangle$6-pulses. In regime (ii), the upper-state lifetime is shorter than the pulse duration; then sequential pumping is ineffective because $|1\rangle \leftrightarrow |3\rangle$7 decays too rapidly, and STIRAP is the only viable route to NCPT (Liao et al., 2010). This distinction is especially important because many nuclear level schemes are open: $|1\rangle \leftrightarrow |3\rangle$8 can decay not only back into $|1\rangle \leftrightarrow |3\rangle$9 and $|2\rangle \leftrightarrow |3\rangle$0, but also to other states.

Within that framework, NCPT differs from incoherent nuclear pumping. In incoherent excitation, one excites a higher nuclear level and relies on spontaneous decay and branching ratios to populate a desired target state. NCPT instead uses two coherent fields to move population from $|2\rangle \leftrightarrow |3\rangle$1 to $|2\rangle \leftrightarrow |3\rangle$2 coherently, ideally nearly losslessly, and with far less sensitivity to the branching structure of the upper state when adiabatic passage is maintained (Liao et al., 2013).

2. Relativistic x-ray driving and theoretical framework

The central obstacle for NCPT is that coherent $|2\rangle \leftrightarrow |3\rangle$3-ray lasers do not exist, while many relevant nuclear transitions lie in the keV–MeV range. The standard solution is to work in the rest frame of a relativistically accelerated nucleus, so that an x-ray photon of laboratory energy $|2\rangle \leftrightarrow |3\rangle$4 is Doppler upshifted into resonance with the nuclear transition (Liao et al., 2010). In the nuclear rest frame, the detunings are written as

$|2\rangle \leftrightarrow |3\rangle$5

with $|2\rangle \leftrightarrow |3\rangle$6, $|2\rangle \leftrightarrow |3\rangle$7, $|2\rangle \leftrightarrow |3\rangle$8 for the pump beam, and $|2\rangle \leftrightarrow |3\rangle$9 for the Stokes beam (Liao et al., 2010).

The pump resonance condition is

$|1\rangle \to |2\rangle$0

Once $|1\rangle \to |2\rangle$1 is fixed this way, the Stokes resonance can be met either by changing the Stokes photon energy or by changing its angle (Liao et al., 2010). This leads to two standard geometries. In the copropagating-beam geometry, $|1\rangle \to |2\rangle$2, both pulses see the same Doppler factor, and the two nuclear transitions generally require different laboratory photon energies; this is therefore a two-color XFEL scheme. In the crossed-beam geometry, $|1\rangle \to |2\rangle$3, a single-color x-ray beam is split into two branches, and the different Doppler factor for the Stokes beam allows the same laboratory photon energy to match a different nuclear transition in the nuclear rest frame (Liao et al., 2010, Liao et al., 2013).

The nuclear dynamics are treated in the nuclear rest frame by a density-matrix master equation,

$|1\rangle \to |2\rangle$4

with initial condition

$|1\rangle \to |2\rangle$5

and rotating-wave Hamiltonian

$|1\rangle \to |2\rangle$6

in the notation of the 2010 treatment (Liao et al., 2010). A closely related formulation writes the spontaneous-emission term as

$|1\rangle \to |2\rangle$7

with $|1\rangle \to |2\rangle$8 and $|1\rangle \to |2\rangle$9 the decay branching ratios from $|3\rangle$0 (Liao et al., 2013).

A specifically nuclear feature is the effective resonant intensity. If the nuclear linewidth $|3\rangle$1 is much smaller than the laser bandwidth $|3\rangle$2, only a small fraction of laser photons are resonant, and the usable intensity is reduced. One expression given is

$|3\rangle$3

for the narrow-line case (Liao et al., 2010). This is one reason the literature repeatedly finds high-$|3\rangle$4, higher-energy nuclear transitions more favorable in practice than low-energy narrow lines.

The STIRAP adiabaticity condition appears in two closely related forms. One statement is

$|3\rangle$5

with $|3\rangle$6 (Liao et al., 2010). Another is

$|3\rangle$7

noted as an empirical threshold (Liao et al., 2013). Positive delay corresponds to the counterintuitive STIRAP sequence; negative delay corresponds instead to a $|3\rangle$8-pulse-like transfer regime.

3. Regimes, candidate nuclei, and principal applications

The first detailed NCPT study examined several specific nuclei. For $|3\rangle$9Re, with $|3\rangle$0 keV and $|3\rangle$1 keV, NCPT was analyzed as a non-isomer example in regime (i), where both $|3\rangle$2-pulse transfer and STIRAP are possible. For $|3\rangle$3Tc, with an isomeric initial state at $|3\rangle$4 keV, $|3\rangle$5 keV, $|3\rangle$6 keV, and isomer half-life $|3\rangle$7 d, the system served as the principal isomer-depletion example (Liao et al., 2010). For $|3\rangle$8Gd and $|3\rangle$9Er, with $\Lambda$0 keV and $\Lambda$1 keV respectively, the studies identified favorable regime-(ii) cases requiring strong acceleration but benefiting from broader transitions (Liao et al., 2010).

The contrast between low-energy and high-energy cases is structural rather than merely kinematic. Low-energy transitions require only modest $\Lambda$2, but often have widths of order $\Lambda$3eV or less, far narrower than the x-ray bandwidth, making the effective coupling tiny. MeV transitions need larger $\Lambda$4 ($\Lambda$5-100) but usually have widths $\Lambda$6 eV, which match the laser better and make NCPT much more realistic (Liao et al., 2010). This is the basis for the recurrent conclusion that the most promising cases lie in the high-$\Lambda$7 STIRAP regime.

A major application domain is isomer depletion. In the NCPT picture, a long-lived metastable state serves as the initially populated state, and population is transferred via a triggering level to a level that decays rapidly onward, thereby releasing stored nuclear energy (Liao et al., 2010). The original work explicitly connected this to controlled pumping or release of energy stored in long-lived nuclear states; later work broadened the application space to nuclear batteries and next-generation nuclear clocks (Liu et al., 15 Aug 2025).

A particularly important later specialization is $\Lambda$8Th. The 2022 analysis treated coherent population of the $\Lambda$9 $|1\rangle$0Th isomer via the second excited state at $|1\rangle$1, using the three lowest positive-parity states of $|1\rangle$2Th as a $|1\rangle$3-type system: ground state $|1\rangle$4, isomeric target state $|1\rangle$5, and the $|1\rangle$6 second excited state $|1\rangle$7 (Kirschbaum et al., 2022). In that case, the upper transitions are nearly degenerate in energy but differ strongly in nuclear structure strength, because $|1\rangle$8 is an in-band transition while $|1\rangle$9 is cross-band (Kirschbaum et al., 2022). The coherent driving therefore depends critically on the in-band and cross-band reduced transition probabilities $|2\rangle$0 and $|2\rangle$1.

4. Numerical performance and experimental feasibility

The foundational NCPT parameter study concluded that complete or near-complete NCPT is possible with coherent future XFEL sources and accelerated nuclei, and identified $|2\rangle$2 as the most promising intensity window for complete transfer in the favorable high-$|2\rangle$3 STIRAP regime (Liao et al., 2010). For $|2\rangle$4Gd, coherent transfer starts at about $|2\rangle$5 for XFELO and $|2\rangle$6 for SXFEL, while more than $|2\rangle$7 transfer is achieved up to $|2\rangle$8 for XFELO and $|2\rangle$9 for SXFEL (Liao et al., 2010). In regime (ii), $|3\rangle$0-pulse transfer does not work because $|3\rangle$1 decays too quickly; only STIRAP succeeds.

For $|3\rangle$2Tc isomer depletion, the reported requirements are substantially higher. Starting from the $|3\rangle$3 keV isomer with 91-day half-life, complete depletion in the crossed-beam SXFEL case occurs at $|3\rangle$4, while crossed-beam XFELO reaches only $|3\rangle$5 at $|3\rangle$6 because the longer pulse duration leads to greater spontaneous-decay losses from $|3\rangle$7; in the copropagating XFELO geometry, $|3\rangle$8 NCPT is reached at the same intensity (Liao et al., 2010).

Temporal coherence is treated as essential. Existing XFELs such as LCLS and the European XFEL were noted to have coherence times of order $|3\rangle$9 fs, far shorter than their $|1\rangle \leftrightarrow |3\rangle$00 fs pulse durations, and therefore not suitable for high-fidelity STIRAP-based NCPT. The most discussed future sources are SXFEL, with pulse duration $|1\rangle \leftrightarrow |3\rangle$01 ps and bandwidth $|1\rangle \leftrightarrow |3\rangle$02 meV, and XFELO, with pulse duration $|1\rangle \leftrightarrow |3\rangle$03 ps, meV-scale bandwidth, and coherence time comparable to pulse duration (Liao et al., 2010). In the crossed-beam single-color geometry, common central-frequency jumps are less damaging because equal detuning shifts $|1\rangle \leftrightarrow |3\rangle$04 preserve two-photon resonance better; variations up to

$|1\rangle \leftrightarrow |3\rangle$05

lead to less than a $|1\rangle \leftrightarrow |3\rangle$06 decrease in NCPT (Liao et al., 2010).

Accelerator tolerances are comparably stringent. FAIR was quoted as providing ion beams up to $|1\rangle \leftrightarrow |3\rangle$07 GeV/u, corresponding to $|1\rangle \leftrightarrow |3\rangle$08, with precision $|1\rangle \leftrightarrow |3\rangle$09, while the LHC can accelerate heavy ions to much larger $|1\rangle \leftrightarrow |3\rangle$10 with energy spread around $|1\rangle \leftrightarrow |3\rangle$11 (Liao et al., 2010). For NCPT, however, the resonance requirement translates into $|1\rangle \leftrightarrow |3\rangle$12 in the strong-acceleration regime and $|1\rangle \leftrightarrow |3\rangle$13 in the moderate-acceleration regime (Liao et al., 2010). The 2013 three-beam study derived an additional first-order two-photon resonance condition for the one-color crossed setup,

$|1\rangle \leftrightarrow |3\rangle$14

showing that off-design ions and off-angle photons can still satisfy two-photon resonance jointly (Liao et al., 2013).

Beam overlap is a decisive practical constraint. With a $|1\rangle \leftrightarrow |3\rangle$15 x-ray focus and LHC beam parameters, the 2010 feasibility estimate gave up to $|1\rangle \leftrightarrow |3\rangle$16 nuclei meeting the laser focus per bunch and pulse in the copropagating geometry, but only about $|1\rangle \leftrightarrow |3\rangle$17 in the crossed geometry for $|1\rangle \leftrightarrow |3\rangle$18 (Liao et al., 2010). The related 2013 estimate gave up to $|1\rangle \leftrightarrow |3\rangle$19 nuclei per bunch and pulse in the copropagating two-color setup, versus only about $|1\rangle \leftrightarrow |3\rangle$20 in the crossed one-color setup (Liao et al., 2013). These overlap limitations are one reason the literature repeatedly judges the two-color copropagating geometry experimentally more realistic than the crossed-beam alternative.

For $|1\rangle \leftrightarrow |3\rangle$21Th, the 2022 comparative study examined three setups: Gamma Factory ultraviolet pulses with highly relativistic ions, coherent XFEL pulses with moderately accelerated ions, and a fixed target plus coherent resonant x-rays (Kirschbaum et al., 2022). It identified the Gamma Factory as the most promising scenario. In that setup, with $|1\rangle \leftrightarrow |3\rangle$22, $|1\rangle \leftrightarrow |3\rangle$23 ions per bunch, and ultraviolet laser pulses Doppler-boosted to the $|1\rangle \leftrightarrow |3\rangle$24 nuclear transitions, STIRAP reaches $|1\rangle \leftrightarrow |3\rangle$25 transfer at nominal intensity over a broad delay range and remains essentially $|1\rangle \leftrightarrow |3\rangle$26 even at $|1\rangle \leftrightarrow |3\rangle$27 for $|1\rangle \leftrightarrow |3\rangle$28 (Kirschbaum et al., 2022). By contrast, successive $|1\rangle \leftrightarrow |3\rangle$29-pulses give only about $|1\rangle \leftrightarrow |3\rangle$30 transfer at the nominal beam spread $|1\rangle \leftrightarrow |3\rangle$31, improving to about $|1\rangle \leftrightarrow |3\rangle$32 for $|1\rangle \leftrightarrow |3\rangle$33, to $|1\rangle \leftrightarrow |3\rangle$34 for $|1\rangle \leftrightarrow |3\rangle$35, or to $|1\rangle \leftrightarrow |3\rangle$36 at the nominal spread if the intensity is scaled by a factor $|1\rangle \leftrightarrow |3\rangle$37 (Kirschbaum et al., 2022). The closing assessment is that STIRAP is the more robust mechanism under realistic detuning.

5. Shortcut protocols and machine-learned pulse design

Later work recast NCPT as a control problem for realistic open nuclear systems. A prominent development is the mixed-state inverse engineering (MIE) scheme for an open three-level nuclear system with spontaneous emission (Wang et al., 2024). In that approach, an additional direct coupling between $|1\rangle \leftrightarrow |3\rangle$38 and $|1\rangle \leftrightarrow |3\rangle$39 converts the $|1\rangle \leftrightarrow |3\rangle$40 system into a cyclic three-level system, and the control Hamiltonian contains an auxiliary field $|1\rangle \leftrightarrow |3\rangle$41 satisfying

$|1\rangle \leftrightarrow |3\rangle$42

The paper states that the amplitude of the additional field is related to the ratio of the pump and Stokes field amplitudes, and that complete transfer can be achieved even when the intensities of the pump and Stokes fields are exceedingly low, provided an appropriate additional field is selected (Wang et al., 2024).

The same study also considered a case without additional coupling, where the resonant three-level problem is reduced to an effective two-level model and the pump and Stokes pulses are modified by counterdiabatic driving. In that construction, the modified controls are

$|1\rangle \leftrightarrow |3\rangle$43

yielding almost $|1\rangle \leftrightarrow |3\rangle$44 transfer in explicit examples such as $|1\rangle \leftrightarrow |3\rangle$45Tc and $|1\rangle \leftrightarrow |3\rangle$46Gd (Wang et al., 2024). The MIE scheme was reported to maintain infidelity below $|1\rangle \leftrightarrow |3\rangle$47 in the shown examples and to exhibit excellent robustness with respect to laser peak intensity and pulse delay.

A further control-theoretic advance is the use of physics-informed neural networks (PINNs) for open-system NCPT (Liu et al., 15 Aug 2025). In that work, the system is again a three-level $|1\rangle \leftrightarrow |3\rangle$48-type nuclear system with spontaneous emission from $|1\rangle \leftrightarrow |3\rangle$49, and the network jointly approximates the state trajectory and control fields. The architecture is a multilayer perceptron with five fully connected hidden layers, each with 128 neurons, using $|1\rangle \leftrightarrow |3\rangle$50 activation; training uses the Adam optimizer with learning rate $|1\rangle \leftrightarrow |3\rangle$51 for 30,000 epochs (Liu et al., 15 Aug 2025). The control objective is the final target population

$|1\rangle \leftrightarrow |3\rangle$52

and the total pulse area is defined as

$|1\rangle \leftrightarrow |3\rangle$53

The reported performance gains are strong in both short- and long-lifetime regimes. For $|1\rangle \leftrightarrow |3\rangle$54, the table gives

$|1\rangle \leftrightarrow |3\rangle$55

versus

$|1\rangle \leftrightarrow |3\rangle$56

and

$|1\rangle \leftrightarrow |3\rangle$57

For $|1\rangle \leftrightarrow |3\rangle$58, the table gives

$|1\rangle \leftrightarrow |3\rangle$59

versus

$|1\rangle \leftrightarrow |3\rangle$60

and

$|1\rangle \leftrightarrow |3\rangle$61

The stated conclusion is that PINNs can achieve higher transfer efficiency with smaller pulse areas and shorter durations across different lifetime regimes (Liu et al., 15 Aug 2025).

These shortcut and learning-based methods do not alter the basic NCPT Hamiltonian picture. Their role is to improve pulse construction under spontaneous emission and finite state lifetimes. A plausible implication is that, for realistic NCPT, the decisive issue has shifted from showing that adiabatic passage is possible in principle to finding control protocols that suppress intermediate-state loss while relaxing intensity, duration, and bandwidth constraints.

6. Related phenomena, boundary cases, and conceptual delimitations

Several neighboring literatures are relevant to NCPT but are not identical to it. One example is optical coherent population trapping of donor-bound electron spins in GaAs, where coherent population trapping of an electronic $|1\rangle \leftrightarrow |3\rangle$62-system creates an autonomous feedback loop on the surrounding nuclear-spin bath through dynamic nuclear polarization (Onur et al., 2015). That work demonstrates CPT-assisted nuclear-state stabilization, redistribution, bistability, and potentially narrowing of the statistical Overhauser-field distribution, but it does not demonstrate coherent transfer between discrete nuclear states by a nuclear $|1\rangle \leftrightarrow |3\rangle$63-scheme (Onur et al., 2015). It is therefore best regarded as CPT-assisted nuclear-state preparation rather than NCPT in the strict sense.

A second related domain is coherent population trapping of a single $|1\rangle \leftrightarrow |3\rangle$64 nuclear spin coupled to an NV center in diamond at room temperature (Jamonneau et al., 2015). There the lower states are nuclear-spin states, but the $|1\rangle \leftrightarrow |3\rangle$65-system is engineered indirectly through hyperfine-mixed electron-spin ESR transitions. The experiment demonstrates preparation of a dark superposition,

$|1\rangle \leftrightarrow |3\rangle$66

with CPT dip contrast of approximately $|1\rangle \leftrightarrow |3\rangle$67 and steady-state dark-state population $|1\rangle \leftrightarrow |3\rangle$68 (Jamonneau et al., 2015). Yet it does not demonstrate deterministic adiabatic transfer of population from one bare nuclear state to the other. It is therefore a dark-state trapping protocol rather than full STIRAP-like NCPT.

A third boundary case is the 2026 proposal for coherent excitation exchange between the electron shell and nucleus in $|1\rangle \leftrightarrow |3\rangle$69Th ions (Tkalya, 1 Jul 2026). That work treats the electron shell and nucleus as two coupled qubits and analyzes underdamped oscillatory exchange between $|1\rangle \leftrightarrow |3\rangle$70 and $|1\rangle \leftrightarrow |3\rangle$71, with coherent coupling $|1\rangle \leftrightarrow |3\rangle$72 inferred to be of order $|1\rangle \leftrightarrow |3\rangle$73 in $|1\rangle \leftrightarrow |3\rangle$74 and of order $|1\rangle \leftrightarrow |3\rangle$75 in $|1\rangle \leftrightarrow |3\rangle$76 (Tkalya, 1 Jul 2026). In the resonant strong-coupling regime,

$|1\rangle \leftrightarrow |3\rangle$77

so the transfer is reversible and oscillatory rather than monotonic and adiabatic (Tkalya, 1 Jul 2026). This is highly relevant to coherent nuclear-state manipulation, but it is not a standard STIRAP/Raman NCPT proposal.

These distinctions matter because “coherent population transfer” can be used loosely in several communities. In the strict nuclear-quantum-optical sense established by the x-ray-driven $|1\rangle \leftrightarrow |3\rangle$78-scheme literature, NCPT refers to coherent transfer between nuclear levels through two phase-coherent fields, typically by STIRAP or related open-system control protocols, with the explicit goal of moving population $|1\rangle \leftrightarrow |3\rangle$79 while minimizing occupancy of the lossy intermediate state $|1\rangle \leftrightarrow |3\rangle$80 (Liao et al., 2010, Wang et al., 2024). Related CPT, DNP, and electron-bridge schemes illuminate dark-state engineering, feedback, and coherent shell–nucleus exchange, but they remain conceptually distinct from NCPT in that narrow sense.