Kuprov: Getting started guide for cavity QED and spin-boson simulations (Spinach 2.12, PR 195)

2026-08-06T04:45:34Z

Getting started guide for cavity QED and spin-boson simulations (Spinach 2.12, PR 195)

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This tutorial builds a spin-boson simulation from nothing: a spin coupled to an electromagnetic cavity mode, then the same cavity made lossy and given a temperature, and finally a pulse sequence run through the device context. The reference material is the [[Spin system specification#Bosonic modes|bosonic modes]] section of the spin system specification and the [[device.m]] page; the shipped examples live in the examples/quantum_tech folder of the ''Spinach'' distribution.

==Declaring a cavity mode==

''Spinach'' would refuse to run in script mode, so the calculation starts with a function declaration:

function cavity_tutorial()

Bosonic modes are declared in [[sys]].isotopes alongside spins. The label carries the number of Fock states to keep: 'C5' is a cavity mode truncated at five population levels. Here it is accompanied by an electron in a 0.33 Tesla magnet:

% Magnet field
[[sys]].magnet=0.33;

% An electron and a five-level cavity mode
[[sys]].isotopes={'E','C5'};

The number of levels is a convergence parameter. Five is generous for a calculation that never puts more than one photon into the cavity, and far too few for a coherent state of amplitude three; the safe procedure is to repeat the calculation with more levels and watch the answer stop moving.

The mode frequency goes into [[inter]].modes.frqs, in Hz, with an empty cell for every particle that is not a mode. Making the cavity resonant with the electron Larmor frequency is the standard starting point:

% Cavity resonant with the electron Larmor frequency
[[inter]].modes.frqs={[] -sys.magnet*spin('E')/(2*pi)};

The coupling between the spin and the mode is an exchange coupling, declared in the upper triangle of a cell array with one row and column per particle:

% Jaynes-Cummings coupling of 2.828 MHz
[[inter]].modes.exchange=cell(2,2);
[[inter]].modes.exchange{1,2}=2.828e6;

==Formalism and basis set==

A two-particle system of this size is comfortable in complete Hilbert space:

% Basis set
[[bas]].formalism='zeeman-hilb';
[[bas]].approximation='none';

% Spinach housekeeping
spin_system=[[create.m|create]](sys,inter);
spin_system=[[basis.m|basis]](spin_system,bas);

At this point ''Spinach'' prints three console tables that are worth reading: mode parameters, mode couplings by channel, and, when they are present, the modulation blocks. If a number is not where it was intended to be, it will be visible there.

==Choosing a frame==

Spin-boson systems accept three [[assume.m|assumption sets]]. The 'cavity' set places spins and modes in a common rotating frame and applies the rotating wave approximation, which is what cavity QED usually means:

% Rotating frame Hamiltonian
spin_system=[[assume.m|assume]](spin_system,'cavity');
H_JC=[[hamiltonian.m|hamiltonian]](spin_system);

In this frame the mode energy becomes a detuning from the carrier, and the exchange coupling keeps only its flip-flop part. The 'labframe' set retains everything, including the counter-rotating terms, and is the reference against which the approximation should be checked. The 'spin-phonon' set keeps the spins in their usual rotating frames and the modes in the laboratory frame; it is the one to use when longitudinal or modulation couplings are present, because those average to zero in the cavity frame and are refused there.

==The avoided crossing==

The physics of the Jaynes-Cummings model is visible in the one-excitation manifold: the spin excitation and the photon are degenerate at zero detuning, and the coupling splits them. Sweeping the spin detuning and diagonalising that manifold produces the familiar hyperbolae:

% Electron detuning operator
Ez=[[operator.m|operator]](spin_system,{'Lz'},{1});

% Locate the one-excitation manifold
spin_exc=[[state.m|state]](spin_system,{'ZL2','BL1'},{1,2});
cav_exc=[[state.m|state]](spin_system,{'ZL1','BL2'},{1,2});
one_quant=speye(size(H_JC,1));
one_quant=one_quant(:,[find(diag(spin_exc)>0.5)...
find(diag(cav_exc)>0.5)]);

% Detuning range and eigenvalue array
delta=2*pi*linspace(-15e6,15e6,100);
eig_array=zeros(2,100);

% Loop over detunings
for n=1:numel(delta)
H=delta(n)*Ez+H_JC; H=(H+H')/2;
eig_array(:,n)=eig(full(one_quant'*H*one_quant));
end

% Plot the one-photon case
kfigure(); plot(1e-6*delta/(2*pi),1e-6*eig_array/(2*pi));
axis tight; kxlabel('detuning, MHz');
kylabel('energy levels, MHz'); kgrid;

Bosonic states use their own labels: 'BL#' is the projector onto the #-th Fock level, counted from 1, so 'BL1' is the vacuum and 'BL2' is the one-photon state. Spin energy levels use the parallel 'ZL#' notation. The minimum splitting at zero detuning is twice the declared coupling.

==A lossy cavity at finite temperature==

Real resonators leak. Damping is declared with the mode, as a lifetime, a linewidth, or a quality factor, and a temperature must be supplied because the thermal occupation of the mode bath is part of the physics:

% Microwave resonator with five Fock levels
[[sys]].magnet=0; [[sys]].isotopes={'C5'};
[[inter]].modes.frqs={6.02e9};
[[inter]].modes.lifetimes={10e-9};
[[inter]].modes.t2_times={9.9e-9};
[[inter]].temperature=0.050;

% Dissipation needs Liouville space
[[bas]].formalism='zeeman-liouv';
[[bas]].approximation='none';

Dissipators cannot be represented in Hilbert space, so a Liouville space formalism is mandatory here; ''Spinach'' warns and omits the mode dissipation if this is forgotten. The evolution generator is assembled in the usual way:

% Dissipative evolution generator
H=[[hamiltonian.m|hamiltonian]](assume(spin_system,'labframe'));
R=[[relaxation.m|relaxation]](spin_system);
G=-1i*H+R;

A Fock state and a coherent state make instructive initial conditions: the Fock state cascades down the ladder, and the coherent state decays with its Poisson structure largely intact. Both settle into the thermal state of the mode.

% Fock state and coherent state
rho_fock=[[state.m|state]](spin_system,'BL5',1);
rho_coh=[[coherent.m|coherent]](spin_system,1,1.5);

[[coherent.m]] reports the norm that the Fock space truncation removed from the Poisson tail. If that number is not small, the mode needs more levels - this is the truncation convergence check in its most convenient form.

==Running a pulse sequence: the device context==

Spin-boson systems are devices rather than molecules in magnets: the spin subsystem has an orientation, the cavity does not. They are therefore run through [[device.m]] rather than [[liquid.m]] or [[powder.m]]. The example below is a spin coupled longitudinally to a phonon mode, which produces both spin coherence modulation and a spin-conditioned displacement of the oscillator:

% An electron and a seven-level phonon mode
[[sys]].magnet=0; [[sys]].isotopes={'E','V7'};
[[inter]].modes.frqs={[] 20e6};

% Longitudinal coupling: row 1 carries the spin
% projection, column 2 carries the quadrature
[[inter]].modes.longitudinal=cell(2,2);
[[inter]].modes.longitudinal{1,2}=4e6*sqrt(2);

% Sequence parameters
parameters.sweep=5e8; parameters.npoints=501;
parameters.rho0=state(spin_system,{'Lx','BL1'},{1,2});

% Trajectory through the device context
traj=[[device.m|device]](spin_system,@traject,parameters,'spin-phonon');

The context builds the Hamiltonian at the requested orientation of the spin subsystem, adds relaxation and kinetics, applies transmitter offsets to the spin channels and parameters.mode_offset detunings to the modes, and hands the generators to the sequence. Drives on a mode are not a context matter: the sequence builds them from the creation and annihilation operators of the mode it drives.

==Exercises==

# Repeat the avoided crossing calculation with 'labframe' instead of 'cavity' and compare the splitting. How large does the coupling have to become before the counter-rotating terms matter?
# Increase the Fock truncation of the resonator from five levels to ten and watch the coherent state decay again. At which amplitude does the truncation start to distort the answer?
# Replace the electron in the first calculation with a proton, adjust the cavity frequency accordingly, and confirm that the vacuum Rabi oscillation still transfers the excitation completely.
# Declare a transmon instead of a cavity mode, obtain its frequency and anharmonicity from Josephson and charging energies with [[ejec2duffing.m]], and look at the anharmonic ladder.
# Add a dispersive coupling between a spin and a cavity and watch the cavity frequency shift with the spin state - the mechanism behind circuit QED readout.

==See also==
[[Spin system specification]], [[Relaxation theory parameters]], [[Basis set specification]], [[device.m]], [[coherent.m]], [[rlx_modes.m]], [[assume.m]], [[create.m]], [[Kernel contexts]]

''Version 2.12, authors: [[Ilya Kuprov]]''

Cavity QED and spin-boson simulations - Revision history

Kuprov: Getting started guide for cavity QED and spin-boson simulations (Spinach 2.12, PR 195)