Tutorials

From Lindblad dynamics to the quantum Mpemba effect — conceptual and computational building blocks

These tutorials build the conceptual scaffolding needed to understand, simulate, and eventually test the quantum Mpemba effect in a trapped-ion system. They are written for readers with a graduate quantum mechanics background who are not specialists in open quantum systems. Familiarity with density matrices and the harmonic oscillator is assumed.

T1 Open Quantum Systems: the Lindblad Master Equation

Why open systems?

A trapped ion is never truly isolated. It couples to motional modes, laser fields, and background electromagnetic fluctuations. The correct framework is an open quantum system: a small system of interest (the spin, or spin + motion) coupled to a large environment (the bath).

The Lindblad form

When the bath is Markovian (memoryless) and the system-bath coupling is weak, the reduced density matrix of the system evolves as:

d/dt ρ = -i[H, ρ] + Σ_k γ_k ( L_k ρ L_k† - ½ L_k†L_k ρ - ½ ρ L_k†L_k ) ↑ coherent ↑ jump ↑ anti-commutator (probability conserving)

The L_k are jump operators encoding the system-bath interaction. γ_k are the corresponding rates. The first term is coherent (Hamiltonian) evolution. The second is dissipation.

Example: single spin with decay

H = (ω_0/2) σ_z L = √γ σ_- (spontaneous decay to ground state) In matrix form (basis {|↑⟩, |↓⟩}): d/dt ρ_↑↑ = -γ ρ_↑↑ d/dt ρ_↓↓ = +γ ρ_↑↑ d/dt ρ_↑↓ = -(γ/2 + iω_0) ρ_↑↓

The steady state is ρ_ss = |↓⟩⟨↓|. The off-diagonal (coherence) decays at rate γ/2. The population decays at rate γ.

T2 Liouvillian Spectra and Relaxation Modes

The Lindblad equation as a linear map

Writing ρ as a column vector (vectorisation), the Lindblad equation becomes:

d/dt |ρ⟩⟩ = ℒ |ρ⟩⟩ where ℒ is the Liouvillian superoperator (an n²\timesn² matrix for an n-dimensional Hilbert space).

Eigendecomposition

ℒ R_k = λ_k R_k (right eigenoperators) General solution: ρ(t) = ρ_ss + Σ_{k≥1} c_k e^{λ_k t} R_k where λ_0 = 0 (steady state), Re(λ_k) < 0 for all k ≥ 1.

Relaxation gap and slowest mode

The eigenvalue λ₁ closest to zero (smallest |Re(λ₁)|) determines the long-time relaxation. Define the relaxation gap:

Δ = |Re(λ_1)| τ_relax = 1/Δ

The Mpemba condition in spectral language

c_k = ⟨⟨L_k | ρ(0) - ρ_ss ⟩⟩ (overlap with left eigenoperators) Mpemba condition: c_1 \approx 0 When c_1 = 0, the slow mode R_1 is absent from the initial deviation. Relaxation is dominated by faster modes R_2, R_3, ... The system appears to equilibrate faster despite starting farther away.

Geometric Interpretation

The set of initial states with c₁ = 0 forms a hyperplane in state space — the Mpemba manifold. States on this manifold reach the neighbourhood of ρ_ss faster than states off it, even when the latter start closer. Finding the Mpemba manifold for the spin-motion system is a core numerical task (see Numerics).

T3 The Reduced Steady State (RSS)

What is the RSS?

In a coupled spin-motion system, the full steady state ρ_ss^{total} lives in the joint Hilbert space. When we measure only the spin (via fluorescence or state tomography), we access the reduced steady state:

ρ_ss^{spin} = Tr_motion[ ρ_ss^{total} ]

The RSS is generally not the maximally mixed state, and it is not a thermal state in the simple sense. Its structure depends on ω_0, ω_m, g, T, and bath parameters. This is a non-trivial feature of the problem.

Why the RSS is central to the Mpemba experiment

The quantum Mpemba scenario we are testing requires:

Initialise the spin in state ρ_A farther from ρ_ss^{spin} (in trace distance).
Initialise the spin in state ρ_B closer to ρ_ss^{spin}.
Evolve both under the same dynamics.
Show that ρ_A(t) passes below ρ_B(t) in distance to ρ_ss at some crossing time t*.

Step 1 requires knowing ρ_ss^{spin} accurately — which requires either a long simulation or careful analysis of the Liouvillian fixed point.

A diagnostic check: initialise in the RSS itself

A first simulation check: if the spin is initialised in ρ_ss^{spin} but the motional mode is at a thermal state with temperature T ≠ 0, the spin will evolve away from ρ_ss^{spin} and then return at long times. This is the signature that ρ_ss^{spin} is not a local fixed point of the spin-only dynamics — it is a global fixed point of the full coupled dynamics. This check is implemented in the Numerics toolbox.

T4 Lamb-Dicke Approximation and its Limits

The Lamb-Dicke parameter

η = k \cdot x_zpf = k \cdot \sqrt(ℏ / 2mω_m) where k is the laser wavevector and x_zpf is the zero-point motion amplitude. Lamb-Dicke regime: η \sqrt(⟨n⟩ + 1) ≪ 1

What changes outside Lamb-Dicke?

In the Lamb-Dicke (LD) regime, the spin-motion coupling is linear and the effective decay rates are frequency-independent. Outside LD, higher sideband orders contribute, the effective spectral density becomes non-trivial, and the mapping to a simple spin-boson model acquires corrections.

Numerical Check

The simulation pipeline (see Numerics) explicitly checks how much the predicted Mpemba crossing changes when the LD approximation is relaxed. This is a parameter scan over η at fixed g and T. If the crossing is robust beyond LD, it strengthens the experimental case.

Typical values in the Freiburg platform

η \approx 0.05 - 0.15 (depending on mode and laser geometry) ⟨n⟩ at T \approx 300 µK motional temperature: ⟨n⟩ \approx 5-20 LD criterion: η \sqrt(⟨n⟩ + 1) \approx 0.05 \cdot \sqrt21 \approx 0.23 [borderline] \to LD corrections should be checked, not assumed negligible.

T5 Trace Distance: Computation and Measurement

Definition

D(ρ, σ) = ½ Tr|ρ - σ| = ½ Σ_i |s_i| where s_i are the singular values of (ρ - σ). For qubit states (2\times2 matrices), this simplifies to: D(ρ, σ) = ½ \sqrt[ (ρ_↑↑ - σ_↑↑)² + |ρ_↑↓ - σ_↑↓|² \cdot 4 ] = ½ |Bloch vector difference|

Experimental access

For a qubit spin, trace distance to a known target state ρ_ss requires measuring all three Bloch vector components: ⟨σ_x⟩, ⟨σ_y⟩, ⟨σ_z⟩. This is done via standard quantum state tomography: three sets of measurements with pulses before readout.

Statistical considerations

Trace distance is a non-linear function of the density matrix. Statistical error propagation from shot noise is non-trivial. The pre-registration will specify the bootstrapping procedure used to obtain confidence intervals on D(ρ(t), ρ_ss) at each time point.