L–Single-Spin–Single-Mode Dynamics

An Operator's Handbook – Common views

author: U. Warring affiliation: Institute of Physics, University of Freiburg version: 0.2.3 last_updated: 2025-12-19 license: CC BY 4.0 Disclaimer – This handbook is a conceptual and pedagogical synthesis intended to clarify minimal models, regimes, and design principles for single-spin–single-mode quantum dynamics; it does not claim completeness, universality, or direct applicability to all experimental platforms. Any emulation of bath-like or power-law behaviour discussed here is finite, state-dependent, and explicitly bounded by the assumptions and cutoffs stated in the text.

Foreword

This handbook presents the Common View of single-spin–single-mode quantum dynamics: the shared understanding of how a two-level system coupled to a harmonic oscillator is modelled, controlled, and measured across trapped ions, cavity QED, circuit QED, and optomechanics.

The Common View encodes decades of insight—canonical Hamiltonians, well-defined coupling regimes, standard control techniques, and proven protocols. It remains indispensable as a reference frame between theory and experiment.

At the same time, the Common View primarily describes how phenomena appear under idealised conditions. Irreversibility, decoherence, and scaling behaviour often enter as secondary effects, grouped under labels such as "imperfections" or "technical limitations."

It is at this boundary that a complementary perspective becomes useful.

The Ordinans Perspective

In parallel, we are developing the Ordinans perspective (from Latin ordinare, "to arrange"). Rather than replacing existing models, it re-organises them according to a strict separation:

• A minimal, invariant system: the spin–mode composite $\mathcal{H}{\text{spin}} \otimes \mathcal{H}{\text{mode}}$, treated as fully deterministic and reversible.

• Explicitly engineered environments: clocks, baths, measurement channels, and feedback controllers, each coupled through defined interfaces.

The shift is subtle but consequential: from describing what happens to designing what is coupled. Decoherence, non-Markovian dynamics, and measurement-induced diffusion become properties of engineered surroundings, not intrinsic system behaviour.

How to Read This Handbook

This text is a map of the coastline: reliable, stable, intentionally conservative.

The Ordinans perspective is developed alongside, not within. Cross-references point to parallel documents that reinterpret the same physics under stricter system–environment separation. These are extensions, not corrections.

The long-term goal is to embed the Common View within a design framework where complex behaviour is constructed—and therefore falsifiable.

This foreword marks that transition in perspective.

How to Cite This Handbook

Stable citation format:

Warring, U. (2025). Single-Spin–Single-Mode Quantum Dynamics: An Operator's Handbook (Version 0.2).

Version-specific: Always cite the version number. Sections may evolve in subsequent versions; backward compatibility is maintained for numbered sections.

For specific results: When citing particular equations or claims, reference the section number (e.g., "Section 4.9, Eq. for IPR").

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PART I — INVARIANT CORE

1. Scope, Philosophy, and Minimality

1.1 Scope

Single-spin–single-mode quantum dynamics refers to the physics of a single two-level quantum system (a spin-½ or qubit) interacting with a single quantised harmonic oscillator mode. This minimal composite system—essentially the quantum Rabi model—is a cornerstone of quantum optics and quantum information, underlying phenomena from cavity QED to trapped-ion quantum logic.

1.2 Thesis

Even this ostensibly simple system generates complex quantum dynamics. By tuning system parameters, preparing different initial states, or allowing environment coupling, one can explore diverse regimes—from exactly solvable models to chaotic-like behaviour—and gain insight into larger many-body or open quantum systems.

This handbook develops a unified framework for single-spin–single-mode dynamics, organised into six parts:

Part	Title	Content
I	Invariant Core	Hamiltonians, symmetries, regime boundaries
II	Dynamics from Simplicity	Manifold-resolved dynamics, initial-state complexity
III	From Closed to Open	Single mode as environment, memory effects, dissipation
IV	Analytic Tools	Solution recipes, approximation validity
V	The Trapped-Ion Platform	Hamiltonian engineering, experimental knobs
VI	Horizons	Open questions, what this handbook enables

1.3 The Minimality Theorem

What Makes This Minimal?

The spin–mode duo is minimal in the information-theoretic sense: it is the smallest Hilbert space that admits (i) non-commuting observables on both subsystems, (ii) entanglement, and (iii) a tunable symmetry-breaking term (the coupling).

Adding spatial degrees of freedom or more modes does not introduce new classes of phenomena—only more parameters. This handbook shows that collapse, revival, effective non-Markovianity, and time-dependent energy renormalisation are already present here; they are not emergent from thermodynamic limits.

1.4 The Exportability Principle

The single-spin–single-mode system serves as a reference implementation. Any phenomenon demonstrated here can be exported to larger systems with confidence that the essential physics is understood. Conversely, claims about multi-mode systems should be benchmarked against this minimal case: if the phenomenon requires more than one mode, that is significant; if it does not, the single-mode setting offers cleaner analysis.

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2. Canonical Hamiltonians and Symmetries

2.1 The Quantum Rabi Hamiltonian

The generic Hamiltonian for a spin-½ (two-level atom or qubit) interacting with a single mode is:

$H = \frac{\hbar \omega_0}{2}\sigma_z + \hbar \omega \, a^\dagger a + \hbar g \, (\sigma^+ + \sigma^-)( a + a^\dagger)$

where:

• $\omega_0$ is the spin's transition frequency

• $\omega$ is the oscillator frequency

• $a^\dagger$, $a$ are the mode's creation/annihilation operators

• $g$ is the coupling strength

• $\sigma_z$, $\sigma^\pm$ are Pauli operators on the two-level system

This quantum Rabi Hamiltonian is invariant under a $\mathbb{Z}_2$ parity transformation (flipping the sign of both spin-$\sigma_x$ and oscillator quadrature)—a symmetry that yields a conserved parity operator:

$\Pi = \sigma_z (-1)^{a^\dagger a}$

2.2 The Jaynes–Cummings Model

In the rotating-wave approximation (valid when $g \ll \omega, \omega_0$ and $\omega \approx \omega_0$), one drops the non-energy-conserving terms $\sigma^+ a^\dagger$ and $\sigma^- a$, recovering:

$H_{\text{JC}} = \frac{\hbar \omega_0}{2}\sigma_z + \hbar \omega \, a^\dagger a + \hbar g \, (\sigma^+ a + \sigma^- a^\dagger)$

The Jaynes–Cummings model conserves the total excitation number:

$N = a^\dagger a + \frac{1}{2}(1 + \sigma_z)$

This exact invariant simplifies the solution: the Hilbert space decomposes into independent two-dimensional subspaces ${|e, n\rangle, |g, n+1\rangle}$ for each excitation number.

Exactness Disclaimer

The JC model is analytically diagonalisable but physically approximate. Its exactness is a mathematical property of the truncated Hamiltonian, not a claim about nature. The counter-rotating terms are always present; we are operating in a regime where their effect is emulable by a Lamb shift, not a dynamical process.

This is the first example of the handbook's core principle: exactness is a function of the question asked, not the Hamiltonian written down.

2.3 The Operator's Decision Tree (The Navigator)

START: Define System Parameters
           │
           ▼
    ┌──────────────────┐
    │  Calculate g/ω   │
    └──────────────────┘
           │
     ┌─────┴─────┬──────────────┐
     ▼           ▼              ▼
  g/ω ≪ 1    0.1 < g/ω < 1   g/ω > 1
     │           │              │
     ▼           ▼              ▼
┌─────────┐ ┌─────────┐    ┌─────────┐
│ Weak/   │ │   USC   │    │   DSC   │
│Resonant │ │         │    │         │
└────┬────┘ └────┬────┘    └────┬────┘
     │           │              │
     ▼           ▼              ▼
┌─────────┐ ┌─────────┐    ┌─────────────┐
│Is Δ≈0?  │ │  QRM    │    │ Adiabatic/  │
└──┬───┬──┘ │Z₂ Parity│    │ Polaron     │
   │   │    │(Braak)  │    │ Frames      │
   ▼   ▼    └────┬────┘    └─────────────┘
 Yes   No        │
  │    │         │
  ▼    ▼         │
┌───┐ ┌────┐     │
│JC │ │Disp│     │
│U(1)│ │σ_z │    │
└─┬─┘ └────┘     │
  │              │
  └──────┬───────┘
         ▼
┌─────────────────────┐
│SELECT INITIAL STATE │
│Complexity Generator │
└─────────┬───────────┘
    ┌─────┴─────┬─────────┬──────────┐
    ▼           ▼         ▼          ▼
 |n⟩ Fock   |α⟩ Coh.  |ξ⟩ Sqz.  Thermal
    │           │         │          │
    ▼           ▼         ▼          ▼
 Single     Collapse   Anomalous  Power-law
 Freq Ω_n  & Revival   Revival    Emulation

2.4 Regimes of Interaction

Regime	Condition	Symmetry	Hamiltonian	Approximation
Dispersive	$g \ll |\Delta|$, $g \ll \omega, \omega_0$	Eff. $U(1)$	$H_{\text{disp}}$	exact within RWA (no population transfer)
Resonant JC	$g \ll \omega$, $\Delta \approx 0$	$U(1)$	$H_{\text{JC}}$	RWA valid
Ultrastrong	$0.1 < g/\omega < 1$	$\mathbb{Z}_2$	$H_{\text{Rabi}}$	Braak exact
Deep Strong	$g/\omega > 1$	$\mathbb{Z}_2$	$H_{\text{Rabi}}$	Polaron frame

where $\Delta = \omega - \omega_0$ is the spin-mode detuning.

Dispersive Regime ($g \ll |\Delta|$): The spin and oscillator exchange virtual excitations, leading to energy shifts (Lamb/AC Stark) but not real energy exchange. The effective Hamiltonian is:

$H_{\text{disp}} \approx \hbar \omega \, a^\dagger a + \frac{\hbar}{2}\left(\omega_0 + \frac{g^2}{\Delta}(2 a^\dagger a + 1)\right)\sigma_z$

Validity Condition (Complete)

The dispersive approximation requires both $g \ll |\Delta|$ and $g \ll \omega, \omega_0$. The first ensures the perturbation expansion converges; the second ensures the RWA remains valid as a precursor step.

Resonant Strong Coupling ($g$ moderate, $\omega \approx \omega_0$): The dynamics feature Rabi oscillations. If the oscillator has $n$ quanta, the spin flips at the Rabi frequency $\Omega_n = 2g\sqrt{n+1}$. The energy spectrum splits into doublets separated by the vacuum Rabi splitting $2g$.

Ultrastrong Coupling ($0.1 < g/\omega < 1$): The counter-rotating terms $\sigma^+ a^\dagger$ and $\sigma^- a$ cannot be neglected. The ground state is no longer the spin-down vacuum $|g, 0\rangle$ but a squeezed entangled state containing virtual excitations. The Bloch–Siegert shift becomes appreciable.

Common Misconception Alert

"Ultrastrong coupling means non-perturbative." False. The Rabi model is analytically solvable at any $g/\omega$ (Braak 2011). The challenge is not solvability but interpretation: as $g \to \omega$, the notion of separate spin and mode excitations becomes a bad basis choice, not a physical mystery.

Deep Strong Coupling ($g/\omega > 1$): This extreme regime pushes the interaction beyond perturbative expansions. Even basic notions of separate "spin" and "oscillator" excitations become blurred.

2.5 Key Point: Symmetry as Signpost

The single-spin–single-mode system is governed by a simple Hamiltonian with symmetries that depend on coupling regime:

• JC limit: Approximate $U(1)$ symmetry (conservation of $N$) makes the problem exactly solvable via independent 2×2 blocks.

• Full Rabi: Only $\mathbb{Z}_2$ parity remains, yet the model is still integrable (Braak 2011).

Integrability Clarification

The Rabi model is integrable in Braak's sense: the spectrum can be determined by transcendental equations involving confluent Heun functions. This is not Liouville integrability (which requires $N$ independent constants of motion for $N$ degrees of freedom). The discrete parity symmetry suffices for exact solvability but does not simplify dynamics to independent one-dimensional motions.

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PART II — DYNAMICS FROM SIMPLICITY

3. Manifold-Resolved Dynamics

Within the Jaynes–Cummings approximation, the Hilbert space decomposes into independent manifolds:

$\mathcal{H} = \mathcal{M}_0 \oplus \mathcal{M}_1 \oplus \mathcal{M}_2 \oplus \cdots$

where $\mathcal{M}n = \text{span}{|e, n\rangle, |g, n+1\rangle}$ for $n \geq 0$, and $\mathcal{M}{-1} = \text{span}{|g, 0\rangle}$ is the ground state (decoupled).

Within each manifold $\mathcal{M}_n$, the Hamiltonian acts as a 2×2 matrix:

$H_n = \hbar \begin{pmatrix} (n+\tfrac{1}{2})\omega + \tfrac{\Delta}{2} & g\sqrt{n+1} \ g\sqrt{n+1} & (n+\tfrac{1}{2})\omega - \tfrac{\Delta}{2} \end{pmatrix}$

The eigenstates (dressed states) are:

$|n, \pm\rangle = \cos\theta_n |e, n\rangle \pm \sin\theta_n |g, n+1\rangle$

with $\tan(2\theta_n) = 2g\sqrt{n+1}/\Delta$, and eigenfrequencies:

$E_{n,\pm} = \hbar\left[(n+\tfrac{1}{2})\omega \pm \tfrac{1}{2}\sqrt{\Delta^2 + 4g^2(n+1)}\right]$

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4. Motional Initial States as Complexity Generators

The oscillator's initial state acts as a generator of complexity by controlling the distribution of interaction frequencies and quantum phases in the spin–mode evolution.

4.1 The Distribution Function

Define the Fock-state distribution:

$p_n = |\langle n | \psi_{\text{mode}}(0) \rangle|^2$

This distribution ${p_n}$ plays the role of a spectral weight in excitation space. The spin coherence evolves as:

$\langle \sigma_+(t) \rangle \approx \sum_{n=0}^{\infty} p_n , e^{-i \Omega_n t} \cdot (\text{amplitude factors})$

where $\Omega_n = 2g\sqrt{n+1}$. The statistics of $p_n$ determine the interference pattern.

4.2 Fock States: The Single-Frequency Benchmark

If the mode starts in a Fock state $|n\rangle$, the spin undergoes simple Rabi oscillations at a single frequency $\Omega_n = 2g\sqrt{n+1}$:

$P_e(t) = \cos^2\left(g\sqrt{n+1} , t\right)$

This is the reference case: predictable, oscillatory, and non-complex.

4.3 Coherent States: Collapse and Revival

A coherent state $|\alpha\rangle$ has Poisson-distributed Fock components:

$p_n = e^{-|\alpha|^2} \frac{|\alpha|^{2n}}{n!}$

The spin excitation probability becomes:

$P_e(t) = \frac{1}{2}\left[1 + \sum_{n=0}^\infty p_n \cos(2g\sqrt{n+1} , t)\right]$

Collapse: The spread of frequencies ${\Omega_n}$ causes dephasing on timescale:

$t_{\text{collapse}} \sim (g\sqrt{\langle n \rangle})^{-1}$

Revival: The discrete quantisation allows rephasing at:

$t_{\text{revival}} \sim \frac{2\pi\sqrt{\langle n \rangle}}{g}$

The discovery of revivals was a seminal demonstration of field quantisation: revivals are a direct consequence of discrete photon number states, absent in classical or continuum fields.

4.4 Squeezed States: Variance Control

A squeezed state has reduced number uncertainty in certain quadratures. This modifies collapse/revival patterns:

• Narrower $p_n$: Slower collapse (fewer frequencies interfere)

• Even/odd parity structure: Frequency-comb dynamics with substructure in revivals

4.5 Displaced-Squeezed States: Tunable Asymmetry

Combining displacement and squeezing allows independent control of:

• Mean excitation $\langle n \rangle$ (sets revival timescale)

• Variance $(\Delta n)^2$ (sets collapse rate)

• Parity structure (sets revival substructure)

4.6 Engineered Classical Mixtures: The Power-Law Emulator

Key Handbook Statement

In a single mode, power-law-like behaviour is realised in excitation space, not frequency space.

Motivation: Many-body physics studies power-law spectral densities $J(\omega) \sim \omega^s$. In a single mode, we cannot replicate a continuum, but we can engineer $p_n$ to emulate the dynamical signatures of non-Markovian baths.

Recipe: Prepare a mixed motional state:

$\rho_{\text{mode}} = \sum_n p_n |n\rangle\langle n|, \quad p_n \propto (n + n_0)^{-\beta}$

using stochastically applied sideband pulses. The spin coherence then evolves as:

$\langle \sigma_+(t) \rangle \approx \sum_n p_n , e^{-i 2g\sqrt{n+1} , t}$

For large $n$ and $\beta > 1$, the sum yields an envelope $\sim t^{-(\beta-1)}$ up to a cutoff time $t_c \sim n_{\max}/g$.

Guardian Warning: Emulation Boundary

This is emulation. The underlying Hamiltonian is still JC; there is no bath-induced backflow, only frequency mixing. The power-law envelope is a signature, not a physical spectral density. The cutoff $n_{\max}$ is not a bath bandwidth but a preparation limit.

4.7 Thermal States: The "Bath" Emulator

A thermal state has:

$p_n = \frac{\bar{n}^n}{(1+\bar{n})^{n+1}}$

This causes irreversible-looking damping of Rabi oscillations. The uncertain photon number acts like static disorder, washing out coherent oscillations without revival (since a thermal state has no fixed phase relationship to rephase).

This mimics decoherence even in a closed system, purely due to uncertainty in the initial mode state.

4.8 State Class Summary

Initial State	Distribution p_n	Dynamics	Complexity Source
Fock $|n\rangle$	$\delta_{n,n_0}$	Single-frequency Rabi	–
Coherent $|\alpha\rangle$	Poisson	Collapse & revival	–
Squeezed $|\xi\rangle$	Sub-Poisson	Modified revival pattern	–
Thermal	Bose–Einstein	Irreversible-like decay	No phase correlation
Power-law mixture	$(n+n_0)^{-\beta}$	Algebraic envelope	Engineered weight

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4.9 An Ordering Parameter in Excitation Space

4.9.1 Definition

For a motional state expanded in the Fock basis with probabilities $p_n = |\langle n | \psi \rangle|^2$, the inverse participation ratio (IPR) is:

$\text{IPR} = \sum_{n=0}^{n_{\max}} p_n^2$

The effective excitation number provides intuitive interpretation:

$N_{\text{eff}} = \frac{1}{\text{IPR}}$

This quantifies how many Fock states (equivalently, how many JC manifolds) actively participate in the dynamics.

4.9.2 Interpretation Rule

Scope Boundary

In this handbook, the IPR is not a measure of chaos, ergodicity, or thermalisation. It is an ordering parameter in excitation space that quantifies how many JC manifolds actively participate in the dynamics.

• Large IPR (small $N_{\text{eff}}$) → localisation in excitation space; few manifolds dominate; regular dynamics

• Small IPR (large $N_{\text{eff}}$) → delocalisation; many manifolds interfere; complex envelopes

This maps directly onto dynamical signatures:

IPR Regime	N_eff	Dynamical Character
$\text{IPR} \approx 1$	$\approx 1$	Single-frequency Rabi oscillation
$\text{IPR} \sim \bar{n}^{-1/2}$	$\sim \sqrt{\bar{n}}$	Collapse with clean revivals
$\text{IPR} \ll 1$	$\gg 1$	Extended collapse, complex envelope

4.9.3 IPR Across State Classes

Initial State	Typical IPR Scaling	N_eff	Dynamical Implication
Fock $|n\rangle$	$\text{IPR} = 1$	1	Single frequency, fully ordered
Coherent $|\alpha\rangle$	$\sim (2\pi\bar{n})^{-1/2}$	$\sim \sqrt{2\pi\bar{n}}$	Collapse + revival
Squeezed $|\xi\rangle$	Smaller than coherent at same $\bar{n}$	Larger	Enhanced interference
Thermal	$\sim (1 + 2\bar{n})^{-1}$	$\sim 1 + 2\bar{n}$	Strong dephasing-like decay
Power-law mixture	Depends on $\beta$; may diverge	Tuneable	Algebraic envelopes

4.9.4 Connection to Prior Work

Similar participation-based measures have been successfully used to characterise localisation and information spreading in trapped-ion spin–boson and spin–phonon models, where they quantify how strongly dynamics explore the available mode space (Clos, Porras, Warring & Schaetz 2016).

The conceptual move here is consistent but deliberately simpler:

• There: Participation across mode space in multi-ion chains

• Here: Participation across excitation space of a single mode

This reinforces the minimality thesis: the essential physics of localisation/delocalisation is already present in the single-mode setting.

4.9.5 Relation to Non-Markovianity Measures

The IPR quantifies static delocalisation in excitation space. Non-Markovianity measures quantify dynamical memory effects in the reduced spin dynamics.

In particular, information-backflow measures based on trace-distance revivals (Wittemer, Clos, Breuer, Warring & Schaetz 2018) provide a complementary, operational characterisation of memory that is sensitive to time-dependent correlations rather than initial-state structure.

Interface Boundary

In this handbook, non-Markovianity measures are used only as diagnostic overlays: they do not define regimes, but help interpret when excitation-space delocalisation translates into observable memory effects.

A low IPR (high $N_{\text{eff}}$) implies many frequencies interfere, which correlates with slower decay of distinguishability and longer memory times—but does not cause non-Markovianity in any formal sense.

4.9.6 Operational Use

The IPR can be:

1. Computed from any prepared distribution $p_n$

2. Measured via motional state tomography

3. Engineered by choosing state preparation protocols

This makes it a practical design parameter: before running an experiment, compute the IPR of your target state to predict whether dynamics will be regular (high IPR) or complex (low IPR).

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PART III — FROM CLOSED TO OPEN

5. The Single Mode as an Environment

5.1 Perspective Shift

The same spin–mode Hamiltonian admits two complementary perspectives:

1. Closed system: Track the full state $|\psi_{SM}(t)\rangle$; evolution is unitary, information is conserved.

2. Open system: Trace out the mode degrees of freedom; the spin's reduced state $\rho_S(t) = \text{Tr}_M[|\psi\rangle\langle\psi|]$ follows non-unitary dynamics.

The second perspective reveals phenomena invisible in the first: effective decoherence, relaxation, and—crucially—energy renormalisation.

5.2 The Master Equation in Minimal-Dissipation Form

For a thermal initial state of the mode with mean occupation $\bar{n}$, the exact time-convolutionless master equation in minimal-dissipation form reads:

$\dot{\rho}S = -i\left[\frac{\tilde{\omega}(t)}{2}\sigma_z, \rho_S\right] + \gamma_z(t)\mathcal{D}[\sigma_z]\rho_S + \gamma+(t)\mathcal{D}[\sigma_+]\rho_S + \gamma_-(t)\mathcal{D}[\sigma_-]\rho_S$

where $\mathcal{D}[O]\rho = O\rho O^\dagger - \frac{1}{2}{O^\dagger O\, \rho}$ is the Lindblad superoperator, and all coefficients $\tilde{\omega}, \gamma_{\pm,z}$ are time-dependent.

The emergent Hamiltonian is:

$K_S(t) = \frac{\tilde{\omega}(t)}{2}\sigma_z$

with renormalised frequency $\tilde{\omega}(t) = \omega_0 + \delta\tilde{\omega}(t)$.

5.3 Time-Dependent Energy Renormalisation

This section incorporates results from Colla et al. (2025), Nature Communications 16:2502

According to the minimal-dissipation Ansatz, the splitting between coherent (Hamiltonian) and incoherent (dissipator) evolution is uniquely determined by minimising the dissipator's action.

For the mode initially in vacuum ($\bar{n} = 0$), the energy shift has an explicit analytic form:

$\delta\tilde{\omega}(t) = -\frac{2g^2}{\Delta} \cdot \frac{1}{1 + \left(1 + \frac{4g^2}{\Delta^2}\right)\cot^2\left(\frac{\pi t}{T}\right)}$

where $T(\Delta) = 2\pi/\sqrt{\Delta^2 + 4g^2}$ is the JC oscillation period.

Key features:

• The shift is periodic in coupling duration with period $T$

• Maximum shift: $\delta\tilde{\omega}(T/2) = -2g^2/\Delta$

• Near resonance ($\Delta \to 0$), the effect is resonantly enhanced

Time-averaged shift:

$\langle \delta\tilde{\omega}(t) \rangle_T = \frac{-2g^2 \, \text{sgn}(\Delta)}{|\Delta| + 2\pi/T}$

This equals the dressed-state energy shift—the conventional Lamb shift emerges as a time average of fundamentally time-dependent dynamics.

5.4 Experimental Observation (Freiburg 2025)

Recent trapped-ion experiments observed these time-dependent shifts directly:

Parameter	Value
Mode frequency $\omega_m$	$2\pi \times 1.30$ MHz
Coupling $g/\omega$	$\approx 0.05$ (near USC threshold)
Detuning $\Delta/g$	$\approx 0.8$
Maximum shift $\delta\tilde{\omega}/\omega$	$\approx 15\%$
Mean shift	$\approx 4\%$

The observed modulation matches the minimal-dissipation prediction (Eq. above), providing direct evidence for time-dependent energy renormalisation in the JC model.

5.5 The Generalised Lamb Shift

Conceptual Reframing

The Lamb shift and AC Stark shift—traditionally viewed as static energy corrections—are time-averaged manifestations of fundamentally dynamic phenomena. They arise from correlation buildup between system and environment, appearing only constant when the environment is traced out and time-averaged.

In the dispersive limit ($|\Delta| \gg g$):

$\langle \delta\tilde{\omega} \rangle_T \to -\frac{g^2}{\Delta}$

recovering the standard Lamb shift formula. But this is a limiting case of a more general time-dependent structure.

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6. Mode Damping and Memory

6.1 Closed vs Open: The Fundamental Distinction

Property	Closed (single mode)	Open (continuum bath)
Spectrum	Discrete	Continuous
Collapse	Temporary (dephasing)	Permanent (decoherence)
Information	Preserved (entangled)	Lost (to environment)
Entropy	Constant	Increases
Revivals	Yes	No

In a closed system, the "collapse" of spin coherence is temporary—the full state remains pure, but the spin's reduced state loses coherence due to entanglement with the mode. Given sufficient time, coherence returns (revival).

In an open system coupled to many modes, collapse is permanent—information leaks into uncontrollable degrees of freedom.

6.2 Adding True Dissipation

When the mode leaks photons (rate $\kappa$) or the spin has spontaneous emission (rate $\gamma$):

$\dot{\rho} = -\frac{i}{\hbar}[H, \rho] + \kappa \mathcal{D}[a]\rho + \gamma \mathcal{D}[\sigma^-]\rho$

The competition between coherent dynamics and dissipation yields:

• Strong damping ($\kappa, \gamma \gg g$): Quantum coherence constantly "monitored," no entanglement buildup

• Moderate damping: Damped revivals—each revival smaller than the last

• Weak damping ($\kappa, \gamma \ll g$): Near-ideal JC dynamics with slow decay envelope

Overclaim Protection

The Lindblad master equation is Markovian by construction. It cannot capture non-Markovian backflow. However, a single damped mode with $\kappa \sim g$ can emulate memory effects on timescales $t \lesssim \kappa^{-1}$. Do not confuse this with true system-bath memory.

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7. Emulating Power-Law Baths: What Is and Is Not Possible

7.1 The Spin–Boson Benchmark

The full spin–boson model couples a two-level system to a continuum of oscillators characterised by spectral density $J(\omega) \sim \omega^s$:

• $s < 1$: Sub-Ohmic (strong low-frequency coupling)

• $s = 1$: Ohmic

• $s > 1$: Super-Ohmic

7.2 Single-Mode Emulation

A single mode cannot reproduce a continuum. However, the dynamical signatures can be emulated:

True Bath Property	Single-Mode Emulation
Spectral density $J(\omega)$	Distribution $p_n$ in excitation space
Power-law decay envelope	Power-law $p_n$ yields $t^{-(\beta-1)}$
Bath bandwidth	Preparation cutoff $n_{\max}$
Non-Markovian backflow	Mode recurrences (different mechanism)

7.3 The Emulation Boundary

What cannot be emulated:

• True information loss (single mode always recurs)

• Bath-induced localisation (requires continuum)

• Arbitrary correlation functions (limited by $H_{\text{JC}}$ structure)

Guardian Warning

Do not claim that a single-mode experiment demonstrates "non-Markovian dynamics" without qualifying: the recurrences arise from mode finiteness, not from structured bath memory. The physics is different even if signatures overlap.

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PART IV — ANALYTIC TOOLS

8. Solution Recipes

8.1 Decision Framework

CHOOSE YOUR TOOL

Question: What regime?
├── Dispersive (g ≪ |Δ|, g ≪ ω)
│   └── USE: Perturbation theory / Schrieffer-Wolff
│       ├── Output: H_eff with Lamb shifts
│       └── Validity: Check g²/Δ² ≪ 1
│
├── Resonant JC (g ≪ ω, Δ ≈ 0)  
│   └── USE: Exact JC diagonalisation
│       ├── Output: Dressed states, Rabi formula
│       └── Validity: Check g/ω ≪ 1
│
├── Ultrastrong (0.1 < g/ω < 1)
│   └── USE: Braak exact / Numerical
│       ├── Output: Spectrum via transcendental eq.
│       └── Validity: Check counter-rotating effects
│
└── Deep Strong (g/ω > 1)
    └── USE: Polaron frame / Numerical
        ├── Output: Transformed Hamiltonian
        └── Validity: Strong-coupling expansion

8.2 The Rotating-Wave Approximation

Procedure: Drop $\sigma^+ a^\dagger$ and $\sigma^- a$ terms.

When valid: $g \ll \omega, \omega_0$ and timescales $\gg 1/\omega$.

What it yields: The JC Hamiltonian decomposes into 2×2 blocks with exact eigenstates:

$|n, \pm\rangle = \cos\theta_n |e,n\rangle \pm \sin\theta_n |g, n+1\rangle$

What it misses: Bloch–Siegert shifts, virtual-photon physics, ground-state entanglement.

8.3 The Braak Solution

For the full Rabi Hamiltonian, Braak (2011) showed integrability via:

1. Parity decomposition: $H = H_+ \oplus H_-$

2. Bargmann representation: Map to differential equations

3. Transcendental eigenvalue equation: Zeros of confluent Heun functions

When to use: USC regime where RWA fails but numerical truncation is uncertain.

Practical note: The solution is implicit (eigenvalues via root-finding), not closed-form. For dynamics, numerical methods are often more practical.

8.4 Perturbative Expansions

Dispersive regime ($g/\Delta$ expansion):

$H_{\text{eff}} = H_0 + \frac{g^2}{\Delta}\sigma_z a^\dagger a + \frac{g^2}{2\Delta}\sigma_z + O(g^4/\Delta^3)$

Bloch–Siegert regime ($g/(\omega_0 + \omega)$ expansion):

$\delta_{\text{BS}} \approx \frac{g^2}{2(\omega_0 + \omega)}$

Strong-coupling regime (treat $\omega$ as perturbation on $g$-dominated interaction).

8.5 Numerical Methods

Truncated diagonalisation: Truncate oscillator space to $n \leq N_{\max}$, diagonalise numerically.

• Rule of thumb: $N_{\max} > \langle n \rangle + 5\sqrt{\langle n \rangle}$

• Check: Verify eigenvalues converge as $N_{\max}$ increases

Time-domain simulation: Integrate Schrödinger/master equation directly.

• For pure states: Fourth-order Runge-Kutta on state vector

• For mixed states: Lindblad equation or quantum trajectories

8.6 Open-System Tools

Markov limit: Standard Lindblad equation with constant rates.

Non-Markov (single mode): Exact TCL master equation (see Section 5.2).

Structured bath: Nakajima–Zwanzig projection, path integrals (beyond this handbook's scope).

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PART V — THE TRAPPED-ION PLATFORM

9. Hamiltonian Engineering with a Single Ion

9.1 The Physical Mapping

Theoretical Object	Ion Trap Realisation
Spin-½	Two hyperfine/Zeeman states (Laser dressed )
$|{\downarrow}\rangle$, $|{\uparrow}\rangle$	e.g., $|F=3, m_F=3\rangle, |F=2, m_F=2\rangle$ (Laser dressed)
Mode $a, a^\dagger$	Quantised motion (axial COM mode)
$\omega$	Trap frequency ($\sim 2\pi \times 1$ MHz)
$\omega_0$	Zeeman-like splitting (tunable via an effective $B$-field – Laser detuning)
$g$	$\eta \Omega$ (Lamb-Dicke × Laser Rabi rate)

9.2 Sideband Transitions as JC/AJC Engineering

Laser-ion interaction in the Lamb-Dicke regime:

Carrier ($\delta = 0$):

$H_{\text{car}} = \frac{\hbar \Omega}{2}(\sigma^+ + \sigma^-)$ Flips spin without changing motion.

Red sideband ($\delta = -\omega$):

$H_{\text{RSB}} = \hbar \eta \Omega (\sigma^+ a + \sigma^- a^\dagger)$ JC interaction: spin flip ↔ phonon annihilation.

Blue sideband ($\delta = +\omega$):

$H_{\text{BSB}} = \hbar \eta \Omega (\sigma^+ a^\dagger + \sigma^- a)$ Anti-JC interaction: spin flip ↔ phonon creation.

Bichromatic drive (both sidebands): $H_{\text{bi}} \approx \hbar g_{\text{eff}} (\sigma^+ + \sigma^-)(a + a^\dagger)$ Approximates the full Rabi Hamiltonian.

Operational Boundary

The Lamb-Dicke regime ($\eta \ll 1$) is not a fundamental limit—it is a convenience that linearises the coupling. Stronger coupling (large $\eta$) introduces higher-order terms $\sim \eta^2 (a + a^\dagger)^2 \sigma_z$, which can be accounted for but invalidate the simple mapping.

This handbook assumes $\eta \leq 0.2$ unless otherwise stated.

9.3 Lamb-Dicke Validity

The Lamb-Dicke parameter:

$\eta = k \sqrt{\frac{\hbar}{2 m \omega}} = k , x_0$

where $k$ is the laser wavevector component along the mode, and $x_0$ is the ground-state wavepacket size.

Lamb-Dicke regime: $\eta \sqrt{\langle n \rangle + 1} \ll 1$

This ensures the motional wavefunction samples only the linear part of the laser's spatial phase.

9.4 — Targeted Hamiltonian Engineering: The 1+1D Dirac Example

(Worked example; benchmark for operator-level control)

The possibility of emulating relativistic quantum dynamics with trapped ions was articulated early in the development of quantum simulation, motivated by the observation that bichromatic sideband driving naturally generates linear couplings between spin and motional quadratures. In a seminal proposal, Lamata et al. showed that a single trapped ion, driven on the red and blue sidebands with controlled detuning, realises an effective Hamiltonian formally equivalent to the 1+1D Dirac equation, with tunable effective mass and speed of light parameters [Lamata2007].

This work established three ideas that are central to the present chapter:

1. The Dirac Hamiltonian emerges within the same Hilbert space as the Jaynes–Cummings and quantum Rabi models; no additional degrees of freedom are required.

2. Relativistic features such as Zitterbewegung arise as interference phenomena, not as fundamentally new dynamics.

3. The mapping is parameter-exact within stated approximations, making it suitable as a benchmark for Hamiltonian engineering rather than a qualitative analogy.

Subsequent theoretical work generalised this approach to related relativistic models (e.g. the Dirac oscillator) and clarified the exact equivalence between Jaynes–Cummings–type interactions and relativistic operator structures under unitary transformations [Bermudez2007, Gerritsma2010].

In this handbook, the Dirac mapping is treated not as an excursion into relativistic quantum mechanics, but as a worked example of targeted Hamiltonian synthesis using the single-spin–single-mode toolbox developed in Part V. The emphasis is therefore on operational control, approximation bounds, and diagnostic observables, rather than on relativistic interpretation per se.

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9.4.1 Objective

Demonstrate that the standard single-ion, single-mode toolbox (red/blue sidebands) suffices to engineer an effective Hamiltonian formally identical to the 1+1D Dirac equation, thereby benchmarking precise Hamiltonian control without expanding the system's Hilbert space.

This example illustrates:

• Systematic use of the bichromatic drive (Section 9.2)

• Translation between theoretical and experimental parameter spaces

• Protocol validation through characteristic observable signatures

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9.4.2 The Recipe (RSB + BSB Synthesis)

Apply a bichromatic laser field tuned symmetrically around the carrier, addressing the first red and blue motional sidebands with equal Rabi frequencies $\Omega$ and effective detuning $\delta$.

In the interaction picture (with respect to the bare qubit and motional Hamiltonians), the driven interaction reads:

$H_{\mathrm{int}}(t) = \hbar \eta \Omega \left[\sigma^+ \left(a , e^{-i\delta t} + a^\dagger e^{+i\delta t}\right) + \sigma^- \left(a^\dagger e^{+i\delta t} + a , e^{-i\delta t}\right)\right]$

Under the Lamb–Dicke condition and rotating-wave approximation (see Section 9.4.5), this reduces to a time-independent effective Hamiltonian of Dirac form:

$H_{\mathrm{Dirac}} = \hbar c_{\mathrm{eff}} \, \hat{p} \, \sigma_x + \hbar m_{\mathrm{eff}} c_{\mathrm{eff}}^2 \, \sigma_z$

This Hamiltonian acts on the same two-level–plus–one-mode Hilbert space as the Jaynes–Cummings model; only the basis and interpretation differ.

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9.4.3 The Dictionary (Relativistic ↔ Experimental)

Relativistic Symbol	Meaning (Dirac Language)	Experimental Control
$c_{\mathrm{eff}}$	Effective speed of light	$c_{\mathrm{eff}} = 2 \eta \Omega x_0$
$m_{\mathrm{eff}} c_{\mathrm{eff}}^2$	Rest energy	$\hbar \delta$ (via symmetric detuning)
$\hat{p}$	Momentum operator	$\hat{p} = \dfrac{i\hbar}{2x_0}(a^\dagger - a)$
$\hat{x}$	Position operator	$\hat{x} = x_0 (a + a^\dagger)$
$\sigma_{x,z}$	Spinor components	Qubit Pauli operators

Here $x_0 = \sqrt{\hbar/(2 m_{\mathrm{ion}} \omega)}$ is the motional ground-state extent.

Tunable parameters:

• "Speed of light" $c_{\mathrm{eff}}$: Set via laser intensity ($\Omega$) and Lamb–Dicke factor ($\eta$)

• "Mass" $m_{\mathrm{eff}}$: Set via laser detuning ($\delta$); sign-reversible

• Trap frequency $\omega$: Sets length scale $x_0$ and momentum scale

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9.4.4 Minimality Checkpoint

Guardian Protection

In scope: 1+1D Dirac Hamiltonian realised with two internal states and one motional mode. This is a change of interpretation, not an enlargement of the system.

Out of scope:

• 3+1D Dirac (requires four spinor components and multiple modes)

• Lorentz invariance tests or particle physics claims

• Curved-space or field-theoretic extensions

If four spinor components or multiple spatial modes are required, the Single-Mode Harbour has been left.

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9.4.5 Approximation Bounds (Navigator Checklist)

The mapping $H_{\mathrm{int}} \to H_{\mathrm{Dirac}}$ is operationally exact when these conditions hold:

1. Lamb–Dicke regime: $\eta\sqrt{\langle n\rangle + 1} \lesssim 0.2$ (Section 9.3) Ensures linear coupling between motion and spin

2. Rotating-wave approximation: $|\delta| \gtrsim 5 \eta \Omega$ Suppresses counter-rotating terms (Section 2.2)

3. Excitation cutoff: $n_{\max} \ll (\delta / 2\eta\Omega)^2$ Ensures prepared motional state samples linear regime

Violation protocol: If any bound fails, revert to full quantum Rabi treatment (Section 8.3) or tighten experimental parameters before claiming Dirac mapping.

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9.4.6 Observable Signature: Zitterbewegung as Interference

Zitterbewegung ("trembling motion") appears here as interference between the two eigen-manifolds of $H_{\mathrm{Dirac}}$, not as a relativistic mystery.

Predicted Dynamics

Prepare the spinor-motional state:

$|\psi(0)\rangle = \tfrac{1}{\sqrt{2}}\left(|\uparrow\rangle + |\downarrow\rangle\right) \otimes |\alpha\rangle$

with $|\alpha\rangle$ a coherent motional state ($\bar{n} = |\alpha|^2$).

The position expectation value evolves approximately as:

$\langle \hat{x}(t) \rangle \simeq x_{\mathrm{ZB}} \sin!\left(\frac{2 m_{\mathrm{eff}} c_{\mathrm{eff}}^2}{\hbar} t\right) e^{-(\Omega_{\mathrm{col}} t)^2}$

with oscillation amplitude:

$x_{\mathrm{ZB}} = 2 \eta x_0 \sqrt{\bar{n} + 1}$

and collapse rate:

$\Omega_{\mathrm{col}} \sim \eta \Omega / \sqrt{\bar{n}}$

Connection to Part II

The envelope and collapse follow directly from the same $p_n$ distribution governing JC collapse and revival (Section 4.3). "Positive/negative energy branches" in Dirac language correspond to the $\sigma_z = \pm 1$ components in the spin–mode product basis.

Key Insight: Zitterbewegung is not new physics—it is the JC interference pattern viewed through a different coordinate representation. The frequency $2 m_{\mathrm{eff}} c_{\mathrm{eff}}^2 / \hbar = 2\delta$ is simply twice the commanded detuning.

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9.4.7 Operational Procedure (Benchmark Protocol)

Five-step validation sequence:

1. Initialise: Doppler cool + resolved sideband cool to $\bar{n} \lesssim 0.1$; apply displacement drive to prepare $|\alpha\rangle$ with target $\bar{n}$.

2. Spinor preparation: Apply carrier $\pi/2$ pulse to create $\tfrac{1}{\sqrt{2}}(|\uparrow\rangle + |\downarrow\rangle)$.

3. Evolve: Apply bichromatic RSB+BSB drive (equal amplitudes, symmetric detuning $\pm\delta$) for time $t$.

4. Readout:

   • Measure $\langle\sigma_z(t)\rangle$ via fluorescence detection

   • Reconstruct $\langle \hat{x}(t)\rangle$ via sideband spectroscopy or direct motional tomography

5. Validate:

   • Fit oscillation frequency → extract $\delta$ (compare to commanded value)

   • Fit amplitude → extract $c_{\mathrm{eff}}$ and $\bar{n}$

   • Fit envelope → characterise decoherence rates

This single sequence simultaneously benchmarks:

• Hamiltonian calibration (frequency/amplitude accuracy)

• Motional state preparation (coherent displacement fidelity)

• Coherence times (via envelope decay)

Typical Parameter Set

Sanity-Check Values (for experiment planning):

Parameter	Typical Value	Resulting Quantity
$\omega$	$2\pi \times 1.5$ MHz	$x_0 \approx 10$ nm (for $^{40}$Ca$^+$)
$\eta$	0.05–0.15	(geometry-dependent)
$\Omega$	$2\pi \times 50$ kHz	$c_{\mathrm{eff}} \approx 1$–$3$ µm/ms
$\delta$	$2\pi \times 100$ kHz	$2\delta = 2\pi \times 200$ kHz (ZB freq)
$\bar{n}$	5–20	$x_{\mathrm{ZB}} \approx 20$–$60$ nm
Evolution time	0–50 µs	~10 ZB cycles before collapse

Expected signal: Oscillating $\langle\hat{x}(t)\rangle$ with ~30 nm amplitude, 5 µs period, collapsing over ~20 µs (for $\bar{n} \approx 10$).

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9.4.8 Error Budget and Diagnostic Protocol

Operator Triage (First-Response Guide)

Use this checklist to prioritise debugging before consulting the full error budget:

• ZB frequency wrong or drifting? → Check detuning drift (Row 1) and AC Stark systematics (Row 2) first.

• Contrast lost early (clean decay)? → Check intensity noise (Row 3), bichromatic phase noise (Row 5), and qubit dephasing (Row 11).

• Waveform distorted (non-sinusoidal)? → Check RSB/BSB imbalance (Row 4) and Lamb–Dicke breakdown (Row 8).

• Slow parameter creep across the day? → Check $\omega(t)$ drift (Row 7) and your reference chain for $\delta(t)$ (Row 1).

Protocol: Identify symptom class → check listed rows → apply mitigation → rerun a short, high-SNR sequence (small $\bar{n}$, short $t$) → iterate.

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Table 9.1 — Dirac Protocol Error Budget (Zitterbewegung Benchmark)

#	Error Source	Hamiltonian Artifact	Signature in Zitterbewegung
1	$\delta$ drift (laser/reference instability)	Time-dependent mass term: $m_{\mathrm{eff}} c_{\mathrm{eff}}^2 \propto \delta(t)$	Frequency jitter/shift of ZB oscillation; phase wander between repetitions. Mitigation: lock $\delta$ to a stable reference; interleave Ramsey calibration runs.
2	Differential AC Stark shift (carrier/sideband imbalance; spectator levels)	Extra $\sigma_z$ term ("fake mass"): $+\tfrac{\hbar\delta_{\mathrm{Stark}}}{2}\sigma_z$	Systematic frequency offset (apparent mass) mismatched from commanded $\delta$. Mitigation: apply Stark-compensation tones; calibrate via carrier-only spectroscopy (Section 9.2).
3	Intensity / Rabi-rate noise $\Omega(t)$	Fluctuating $c_{\mathrm{eff}} \propto \eta\Omega(t)$; weak amplitude-to-phase conversion	Contrast loss; run-to-run envelope variability; fitted $c_{\mathrm{eff}}$ broadens. Mitigation: AOM intensity servo; normalise via simultaneous Rabi monitor.
4	Red/blue amplitude imbalance ($\Omega_{\mathrm{RSB}} \neq \Omega_{\mathrm{BSB}}$)	Rotated coupling axis; residual JC/AJC asymmetry; $\sigma_y$ admixture	Distorted waveform (non-sinusoidal); offset in $\langle \hat{x}(t)\rangle$; reduced dictionary fidelity. Mitigation: calibrate sideband Rabi rates separately; active balancing via DDS amplitude trim (Section 11).
5	Bichromatic phase noise / jumps	Drifting coupling axis: $\sigma_x \to \cos\phi,\sigma_x + \sin\phi,\sigma_y$	Apparent dephasing without heating; "rotating" quadrature readout; poor repeatability. Mitigation: phase-coherent DDS/AWG; fixed phase convention; periodic phase re-zero.
6	Motional heating (electric-field noise)	Stochastic drive + diffusion; effective Lindblad heating term $\kappa \mathcal{D}[a^\dagger]$	Faster envelope decay; reconstructed $\bar{n}$ grows; sideband asymmetry. Mitigation: filter/ground electrodes; shorten evolution; sympathetic cooling where available (Section 6.2).
7	Motional frequency drift $\omega(t)$(trap potential drift)	Changes $x_0 \propto \omega^{-1/2}$; perturbs interaction picture and scaling of $\hat{x}, \hat{p}$	Slow systematic drift of inferred $c_{\mathrm{eff}}$ and $x_{\mathrm{ZB}}$; possible beating. Mitigation: stabilise trap RF/DC; interleave $\omega$ measurement (Section 11).
8	Lamb–Dicke breakdown (too large $\eta\sqrt{\langle n\rangle + 1}$)	Higher-order couplings beyond $\hat{x}, \hat{p}$; dictionary becomes nonlinear	Non-Gaussian distortion; harmonics; parameter-dependent waveform. Mitigation: reduce $\bar{n}$; increase $\omega$; enforce $\eta\sqrt{\langle n\rangle + 1} \le 0.2$ (Section 9.3).
9	RWA violation ($\delta\not\gg \eta\Omega$)	Counter-rotating terms; Bloch–Siegert-like shifts	Extra fast components; frequency pulling; apparent mass deviates from $\delta$. Mitigation: increase $|\delta|$ or reduce $\Omega$; scaling check vs $\Omega$ (Section 2.2).
10	Off-resonant carrier coupling (imperfect spectral selectivity)	Unwanted $\sigma_x$ rotation during evolution; spurious spinor mixing	Baseline offsets; reduced ZB visibility; spurious oscillations at carrier frequency. Mitigation: tighten spectral shaping; verify with carrier-only control runs.
11	Qubit dephasing (magnetic noise, LO phase noise)	Dephasing channel on $\sigma_z$ (or effective quantisation axis)	Pure contrast loss; minimal frequency shift; envelope decays even at low heating. Mitigation: magnetic shielding; clock states; dynamical decoupling if compatible (Section 6.2).
12	SPAM errors (state prep, readout)	No Hamiltonian change; biases inferred $\langle \sigma_z(t)\rangle$ and reconstructed $\langle \hat{x}(t)\rangle$	Apparent reduced amplitude/offset; error floor on contrast. Mitigation: SPAM calibration matrix; repeat reference points at $t = 0$ (Section 10.2).
13	$\langle \hat{x}(t)\rangle$reconstruction bias (sideband spectroscopy model)	Measurement-model artefact; incorrect mapping from spectra to quadrature	Phase/amplitude bias in $\langle \hat{x}(t)\rangle$ while $\langle\sigma_z(t)\rangle$ appears consistent. Mitigation: cross-check with independent tomography; simulate reconstruction pipeline (Section 10.2).

Notation lock: $\delta$ (commanded detuning), $\Omega$ (Rabi rate), $\eta$ (Lamb–Dicke factor), $x_0$ (ground-state size), $\sigma_{x,z}$ (Pauli operators), $\omega$ (motional frequency).

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9.4.9 What This Example Demonstrates (and What It Does Not)

Demonstrates

• Precise Hamiltonian synthesis using the Part V toolbox (RSB, BSB, bichromatic drives)

• Equivalence of operator languages: JC/Rabi dynamics ↔ Dirac dynamics via basis transformation

• Parameter tunability: "Speed of light" and "mass" are experimental knobs, not fundamental constants

• Validation protocol: Zitterbewegung as a calibration-quality benchmark

• Error attribution: Table 9.1 provides systematic debugging framework

Does Not Demonstrate

• Fundamental relativistic effects: No Lorentz invariance, causality tests, or spacetime structure probed

• Particle physics: No pair creation, no second quantisation, no field-theoretic phenomena

• Multi-dimensional Dirac physics: 3+1D requires expanding beyond single-spin–single-mode (see Minimality Checkpoint)

Conceptual Boundary: This example shows that the trapped-ion platform can emulate the mathematical structure of the 1+1D Dirac equation with parameter-level control. It does not test whether nature's actual electrons obey this equation (they do, but for different reasons). The value lies in Hamiltonian engineering precision, not in simulating high-energy physics per se.

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Cross-Reference to Appendix B

For readers interested in the formal equivalence, Appendix B provides the explicit unitary transformation mapping the Dirac oscillator to the Jaynes–Cummings Hamiltonian. This confirms that:

$H_{\mathrm{Dirac\ oscillator}} \xrightarrow{U = e^{-i\pi\sigma_y/4}} H_{\mathrm{JC}}$

Operational takeaway: Every JC experiment is already a Dirac-oscillator experiment. The Hamiltonian is invariant; only the question asked of it changes.

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Note on Experimental Status and Prior Demonstrations

The Dirac-emulation proposal of Lamata et al. was followed by a series of experimental demonstrations using single trapped ions, most prominently by the group of Gerritsma et al. These experiments directly observed key signatures predicted by the effective Dirac dynamics, including Zitterbewegung and Klein-paradox–like behaviour, using a single ^{40}Ca^+ ion driven by bichromatic sideband fields [Gerritsma2010Nature, Gerritsma2011PRL].

Further experiments explored variations of relativistic Hamiltonians, including tunable effective masses, sign changes of the mass term, and extensions to Dirac-oscillator–type dynamics [Casas2016, Lamata2011]. In all cases, the physical implementation relied on the same core ingredients documented in this chapter: resolved sideband control, coherent bichromatic driving, and precise calibration of detunings and Rabi frequencies.

From an operational perspective, these experiments demonstrated that:

• The Dirac mapping is experimentally accessible with standard trapped-ion technology.

• Observables such as Zitterbewegung are robust diagnostics of Hamiltonian synthesis quality, sensitive to detuning, phase stability, and motional coherence.

• No fundamentally new hardware is required beyond that already used for Jaynes–Cummings and Rabi-model experiments.

The present handbook section does not aim to reproduce or extend these results. Instead, it abstracts their core lesson: the 1+1D Dirac Hamiltonian serves as a stringent, interpretable benchmark for single-ion Hamiltonian engineering, complementary to cat-state generation, squeezing, and collapse-and-revival protocols treated elsewhere in Part V.

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10. Engineering Initial States

10.1 Ground-State Cooling

Resolved sideband cooling: Drive red sideband repeatedly. $|{\downarrow}, n\rangle \xrightarrow{\text{RSB}} |{\uparrow}, n-1\rangle \xrightarrow{\text{decay}} |{\downarrow}, n-1\rangle$

Achieves $\bar{n} \lesssim 0.1$ routinely.

10.2 State Preparation Protocols

Target State	Preparation Method	Typical Fidelity
Fock $|n\rangle$	Sequential BSB $\pi$-pulses from $|0\rangle$	$>95\%$
Coherent $|\alpha\rangle$	Classical drive (displacement)	$>99\%$
Squeezed $|\xi\rangle$	Parametric modulation / pulse sequence	$>90\%$
Cat state	Conditional displacement + measurement	$>85\%$
Power-law mixture	Stochastic sideband pulses	<Did anyone do this?>

10.3 Cat State Generation

Protocol (Monroe et al. 1996):

1. Prepare $|{\downarrow}, 0\rangle$

2. Apply carrier $\pi/2$ to create $(|{\downarrow}\rangle + |{\uparrow}\rangle)/\sqrt{2}$

3. Apply state-dependent displacement: $|{\downarrow}\rangle \to |{\downarrow}, \alpha\rangle, |{\uparrow}\rangle \to |{\uparrow}, -\alpha\rangle$

4. Result: $(|{\downarrow}, \alpha\rangle + |{\uparrow}, -\alpha\rangle)/\sqrt{2}$

Normalisation Note

The cat state $(|\alpha\rangle + |-\alpha\rangle)/\sqrt{2}$ is properly normalised only for $|\alpha| \gg 1$ (where $\langle \alpha | -\alpha \rangle = e^{-2|\alpha|^2} \approx 0$). For small $\alpha$, include the overlap correction.

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11. Experimental Knob Map

Theoretical Parameter	Ion Trap Realisation	Typical Range	Calibration Method
$g$	$\eta \Omega$	$2\pi \times (1-300)$ kHz	Rabi flopping on sideband
$\omega$	Trap frequency	$2\pi \times (0.5-5)$ MHz	Secular frequency measurement
$\omega_0$	Zeeman shift	Tunable via effective $B$-field	Sideband spectroscopy
$\Delta = \omega - \omega_0$	Laser detuning	$2\pi \times (0-1)$ MHz	Ramsey spectroscopy
$\kappa$ (damping)	Sympathetic cooling	$10-10^4\, \text{s}^{-1}$	Sideband asymmetry
$\gamma_\phi$ (dephasing)	Engineered noise	Tuneable	Ramsey decay
$\bar{n}$	Cooling quality	$0.01-10$	Sideband ratio
$p_n$ distribution	Squeeze + displace + mix	Engineered	Motional tomography

11.1 Accessing Different Regimes

Regime	How to Access
Dispersive	Large detuning $\Delta \gg g$ (detune laser far from sideband)
Resonant JC	$\Delta \approx 0$, moderate $\eta \Omega$
Near-USC	Small $\Delta$, large $\eta \Omega$ (typical: $g/\omega \sim 0.05-0.1$)
Engineered dissipation	Add sympathetic cooling or electrode noise

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PART VI — HORIZONS

12. What Is Already Known

The single-spin–single-mode system is among the most thoroughly characterised quantum systems:

• JC spectrum and dynamics: Fully solved analytically (Jaynes & Cummings 1963)

• Collapse and revival: Observed in cavity QED and ion traps (1980s-present)

• Braak integrability: Full Rabi model solved (2011)

• USC phenomena: Observed in superconducting circuits (2010s)

• Time-dependent Lamb shift: Observed in trapped ions (Colla et al. 2025)

• State engineering: Fock, coherent, squeezed, cat states all demonstrated

13. What This Handbook Enables

13.1 For Graduate Students

• Conceptual bridge: Connect textbook Hamiltonians to laboratory reality

• Decision protocols: Know which approximation applies when

• Operational fluency: Translate theoretical parameters to experimental knobs

13.2 For Experimentalists

• Design guidance: Engineer specific dynamical signatures via state preparation

• Regime navigation: Identify which physics dominates in your parameter range

• Benchmark protocols: Test system calibration against exact predictions

13.3 For Theorists

• Minimal testbed: Verify new ideas in the simplest non-trivial setting

• Emulation boundaries: Know what can and cannot be simulated with one mode

• Open-system foundations: Use the minimal-dissipation framework as reference

14. Open Questions and Outlook

14.1 Within the Single-Mode Framework

1. Non-thermal initial states: How does $K_S(t)$ depend on squeezed, Fock, or cat initial states of the mode?

2. Beyond JC: Full characterisation of time-dependent renormalisation in the Rabi model.

3. Quantum thermodynamics: Connecting $K_S(t)$ to work and heat definitions at strong coupling.

14.2 Extensions

1. Multi-mode environments: How do signatures change with 2, 3, ..., $N$ modes?

2. Structured baths: Engineering effective spectral densities with multiple modes.

3. Autonomous quantum machines: Using emergent driving for heat engines.

14.3 The Handbook's Continuing Role

This handbook is a living document. As the field develops, new sections will address:

• Additional state classes and their dynamics

• Refined experimental protocols

• Connections to quantum error correction and fault tolerance

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References

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Appendix A: Notation Summary

Symbol	Meaning
$\omega_0$	Bare spin transition frequency
$\omega$	Oscillator/mode frequency
$g$	Spin-mode coupling strength
$\Delta = \omega - \omega_0$	Detuning
$\eta$	Lamb-Dicke parameter
$\Omega$	Laser Rabi frequency
$\bar{n}$	Mean thermal occupation
$p_n$	Fock-state probability distribution
$\Omega_n = 2g\sqrt{n+1}$	$n$-photon Rabi frequency
$T = 2\pi/\sqrt{\Delta^2 + 4g^2}$	JC oscillation period
$K_S(t)$	Emergent (renormalised) system Hamiltonian
$\tilde{\omega}(t)$	Renormalised spin frequency
$\text{IPR} = \sum_n p_n^2$	Inverse participation ratio
$N_{\text{eff}} = 1/\text{IPR}$	Effective excitation number

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Appendix B:

Version History

Version	Date	Changes
0.1	2025–12-17	Initial draft (review format)
0.2	2025–12-17	Modular restructure; Guardian protections; Navigator; minimal-dissipation Ansatz; power-law emulator; trapped-ion axis elevated
0.2.1	2025–12-17	Added Section 4.9 (IPR ordering parameter); citation apparatus; references for Clos et al. 2016, Wittemer et al. 2018
0.2.2	2025–12-18	Added Section 9.4 (1+1D Dirac worked example); bichromatic drive synthesis; Zitterbewegung as JC interference; Table 9.1 error budget with operator triage; Appendix C cross-reference
0.2.3	2025-12-19	Added clarifying Foreword