Quantum Thermodynamics Toy Models

This notebook guide specifies a reproducible finite-dimensional calculation for quantum thermodynamics toy models. The baseline notebook should build the two-point measurement work distribution for a driven two-level system, compute exact branch probabilities, sample synthetic experimental records, and verify a simple quantum Jarzynski equality and Crooks relation numerically.

As of this review, no executable notebook under notebooks/density-open-systems/quantum-thermodynamics-toy-models/ is promoted as a reproduced artifact. This page is the admission contract for that notebook: it states the model, work convention, fluctuation-relation checks, Monte Carlo diagnostics, and accepted outputs required before thermodynamic numerical claims should be cited.

Purpose

The notebook should demonstrate how to:

• construct thermal populations for a finite two-level Hamiltonian;
• implement the Two-Point Measurement Scheme;
• compute a discrete Work Distribution from branch probabilities;
• drive the system with a simple unitary transition matrix;
• verify normalization, positivity, and detailed branch bookkeeping;
• compute $\langle e^{-\beta W}\rangle$ and compare it with $e^{-\beta\Delta F}$;
• verify a Crooks branch-ratio check for the matching reverse protocol;
• compare exact sums with finite-sample Monte Carlo estimates;
• record conventions clearly enough that signs of work and free energy are not ambiguous.

The first version should be a closed-system TPM notebook. Lindblad heat exchange, quantum-jump thermodynamics, feedback, and continuously monitored work definitions should be added only after the closed finite-dimensional baseline is reproducible.

Directory Plan

Use a dedicated directory:

notebooks/density-open-systems/quantum-thermodynamics-toy-models/
  quantum-thermodynamics-toy-models.ipynb
  README.md

The opening notebook cell or README.md should state:

• Python and package versions;
• basis ordering;
• energy convention and units;
• inverse temperature $\beta$;
• forward and reverse Hamiltonians;
• transition-unitary convention;
• sign convention for work;
• number of Monte Carlo samples and random seed;
• numerical tolerances for fluctuation-relation residuals;
• date and commit identifier when the notebook is promoted.

Baseline Two-Level Hamiltonians

Use two Hamiltonians with the same energy basis:

$$H_0 = \epsilon_0 \lvert e\rangle\langle e\rvert, \qquad H_\tau = \epsilon_\tau \lvert e\rangle\langle e\rvert .$$

The ground energy is set to zero in both Hamiltonians. This convention is simple but not mandatory; if another zero of energy is used, the notebook should show that the work differences are unchanged by a common energy shift applied consistently.

The partition functions are

$$Z_0 = 1+e^{-\beta\epsilon_0}, \qquad Z_\tau = 1+e^{-\beta\epsilon_\tau}.$$

The initial Gibbs probabilities for the forward process are

$$p_g^0 = \frac{1}{Z_0}, \qquad p_e^0 = \frac{e^{-\beta\epsilon_0}}{Z_0}.$$

The equilibrium free-energy difference is encoded by

$$e^{-\beta\Delta F} = \frac{Z_\tau}{Z_0}, \qquad \Delta F = - \frac{1}{\beta} \ln \frac{Z_\tau}{Z_0}$$

for $\beta\ne0$. The notebook should use the partition-function ratio directly when testing the Jarzynski equality, especially if it includes high-temperature limits.

Drive Model

Use a unitary drive that produces transition probability $r$ between the two energy states:

$$P_{g\to e} = P_{e\to g} = r, \qquad P_{g\to g} = P_{e\to e} = 1-r, \qquad 0\le r\le1.$$

A concrete implementation is the rotation

$$U(\theta) = e^{-i\theta\sigma_x/2}, \qquad r=\sin^2\left(\frac{\theta}{2}\right).$$

The transition matrix is doubly stochastic:

$$\sum_m P_{m|n}=1, \qquad \sum_n P_{m|n}=1.$$

The second condition is the unitarity ingredient needed in the closed-system Jarzynski proof. The notebook should check both sums numerically rather than assuming that a matrix of numbers called probabilities is physically valid.

Forward TPM Work Distribution

Use the work sign convention from Energy, Heat, and Work:

$$W = E_m^\tau-E_n^0,$$

where $n$ is the first measured energy outcome and $m$ is the final measured energy outcome. The four forward branches are:

Branch	Probability	Work
$g\to g$	$p_g^0(1-r)$	$0$
$g\to e$	$p_g^0r$	$\epsilon_\tau$
$e\to g$	$p_e^0r$	$-\epsilon_0$
$e\to e$	$p_e^0(1-r)$	$\epsilon_\tau-\epsilon_0$

Thus

$$\begin{aligned} P_F(W) = {}& p_g^0(1-r)\delta(W) + p_g^0r\,\delta(W-\epsilon_\tau) \\ &+ p_e^0r\,\delta(W+\epsilon_0) + p_e^0(1-r) \delta\!\left( W-\epsilon_\tau+\epsilon_0 \right). \end{aligned}$$

The exact mean work is

$$\begin{aligned} \langle W\rangle_F = {}& p_g^0r\,\epsilon_\tau - p_e^0r\,\epsilon_0 \\ &+ p_e^0(1-r) \left( \epsilon_\tau-\epsilon_0 \right). \end{aligned}$$

The notebook should compute the distribution from arrays of branch data, not from hard-coded symbolic formulas only. The symbolic formulas are validation targets.

Jarzynski Check

For the exact branch table,

$$\left\langle e^{-\beta W} \right\rangle_F = \sum_{m,n} p_F(m,n) e^{-\beta(E_m^\tau-E_n^0)} .$$

The notebook should compare this number with

$$e^{-\beta\Delta F} = \frac{Z_\tau}{Z_0}.$$

Report the residual

$$R_J = \left| \left\langle e^{-\beta W}\right\rangle_F - \frac{Z_\tau}{Z_0} \right|.$$

For the exact finite table, $R_J$ should be near floating-point roundoff. If the residual is large, common causes are:

• the transition matrix is not doubly stochastic;
• the sign of work is reversed;
• $Z_0$ and $Z_\tau$ are swapped;
• thermal probabilities were assigned to the wrong Hamiltonian;
• the notebook sampled records but compared one noisy Monte Carlo estimate as though it were exact.

Reverse Protocol and Crooks Check

The reverse protocol starts in the thermal state of $H_\tau$ and drives back to $H_0$ with the reverse unitary. For this toy model, use the same transition probability $r$ for matched reverse branches:

$$P_R(n|m) = P_F(m|n).$$

The reverse branch probability is

$$p_R(n,m) = \frac{e^{-\beta E_m^\tau}}{Z_\tau} P_R(n|m),$$

and the reverse work is

$$W_R = E_n^0-E_m^\tau = -W_F .$$

For each branch with nonzero matched transition probability,

$$\frac{p_F(m,n)}{p_R(n,m)} = e^{\beta(W_{mn}-\Delta F)} .$$

The distribution-level Crooks relation is

$$\frac{P_F(W)}{P_R(-W)} = e^{\beta(W-\Delta F)}$$

when the work value identifies matching probability weight. If different branches share the same work value, the notebook should aggregate probabilities at equal work before forming the distribution ratio.

Report a branch-ratio residual such as

$$R_C = \max_{m,n} \left| \ln \frac{p_F(m,n)}{p_R(n,m)} - \beta(W_{mn}-\Delta F) \right|$$

over branches whose forward and reverse probabilities are both nonzero.

Monte Carlo Records

After the exact branch table works, sample synthetic TPM records:

1. draw the initial energy $n$ from the Gibbs probabilities for $H_0$;
2. draw the final energy $m$ from $P_{m|n}$;
3. store $n$, $m$, $W_{mn}$, and the branch label;
4. estimate the empirical distribution of $W$;
5. estimate $\langle W\rangle$ and $\langle e^{-\beta W}\rangle$ with standard errors.

For $N$ independent records, the sample mean

$$\widehat A_N = \frac{1}{N} \sum_{j=1}^N e^{-\beta W_j}$$

has standard error estimated by

$$\operatorname{SE}(\widehat A_N) = \frac{s_A}{\sqrt N},$$

where $s_A$ is the sample standard deviation of the values $e^{-\beta W_j}$.

The notebook should compare the Monte Carlo estimate with the exact sum. It should not declare the fluctuation relation failed merely because rare work values are undersampled. Exponential averages are often dominated by rare branches, especially at large $\beta$.

Suggested Parameter Sweep

Use dimensionless parameters such as

$$\beta\epsilon_0\in\{0.5,1,2,5\}, \qquad \frac{\epsilon_\tau}{\epsilon_0}\in\{0.5,1,1.5,2\}, \qquad r\in\{0,0.1,0.5,1\}.$$

For each point, save:

• $Z_0$, $Z_\tau$, and $Z_\tau/Z_0$;
• the branch-probability table;
• exact $\langle W\rangle$;
• exact $\langle e^{-\beta W}\rangle$;
• $R_J$ and $R_C$;
• Monte Carlo estimates and standard errors for selected sample sizes.

The sweep should include both gentle and strongly nonequilibrium cases. For $r=0$, the state follows the same energy label. For $r=1$, the population branches swap. Intermediate $r$ gives a simple model of nonadiabatic transitions.

Validation Tests

The promoted notebook should pass these checks.

Probability normalization

Verify

$$\sum_{m,n}p_F(m,n)=1, \qquad p_F(m,n)\ge0$$

within numerical tolerance. Do the same for the reverse process.

Transition-matrix unitarity

Check both column and row sums:

$$\sum_m P_{m|n}=1, \qquad \sum_n P_{m|n}=1.$$

Column sums alone make conditional probabilities valid. Row sums are what make the transition matrix compatible with a closed finite-dimensional unitary in this simple nondegenerate setting.

Work-distribution aggregation

If two branches produce the same work value, aggregate their probabilities before plotting a histogram or comparing $P_F(W)$ and $P_R(-W)$. Floating-point work values should be grouped by a declared tolerance.

Jarzynski residual

The exact table should satisfy

$$R_J \lesssim 10^{-12}$$

in ordinary double precision for moderate parameter values. A larger residual can be acceptable only if the notebook explains the numerical source, such as intentionally low precision.

Sampling convergence

For Monte Carlo estimates, show convergence with sample size. The exact result should fall within a few estimated standard errors once $N$ is large enough for the chosen parameters.

Sign-convention reversal

As a deliberate negative test, compute the Jarzynski residual with $W$ replaced by $-W$. The result should generally fail unless the protocol is specially symmetric. This test catches accidental sign conventions.

Interpreting the Toy Model

This notebook is intentionally small. Its value is not that a driven two-level system exhausts quantum thermodynamics, but that every branch, probability, work value, and fluctuation-relation residual can be inspected directly.

Several limitations should be stated clearly:

• the baseline model is closed during the drive, so there is work but no heat exchange with a bath during the protocol;
• the initial Gibbs state is an assumption, not the result of simulated thermalization;
• projective energy measurements are part of the TPM definition and can disturb coherent initial states;
• the Jarzynski equality checks the whole distribution, not just the mean work;
• Monte Carlo failure at small sample size is not a physical violation;
• adding a Lindblad bath requires a trajectory-level heat and entropy convention, not just the same closed-system formula.

For open trajectory formulations, use Quantum-Jump Trajectories and Fluctuation Theorems as the conceptual entry points. The present notebook should first make the closed TPM bookkeeping airtight.

Accepted Outputs

A promoted version should save:

• a branch table for each parameter point;
• an exact work-distribution table with aggregated work values;
• a plot of $P_F(W)$ for selected parameters;
• exact Jarzynski and Crooks residual tables;
• Monte Carlo convergence plots for $\langle e^{-\beta W}\rangle$;
• a warning or diagnostic for undersampled rare branches;
• a summary file recording all conventions and tolerances.

Accepted outputs should be regenerated from a clean run. Scratch plots and exploratory parameter scans may remain in the notebook, but they should not be cited as reproduced results.

Common Mistakes

• Replacing the final nonequilibrium state by a final thermal state in the forward process.
• Treating work as an ordinary Hermitian observable independent of the measurement protocol.
• Forgetting that $W=E_m^\tau-E_n^0$ is a signed energy difference.
• Swapping $Z_0$ and $Z_\tau$ in $\Delta F$.
• Verifying the Jarzynski equality with exact sums, then claiming that a small Monte Carlo sample should match equally well.
• Using a conditional probability matrix that is column-stochastic but not compatible with a unitary transition matrix.
• Comparing Crooks ratios before aggregating branches with the same work value.
• Adding an open-system dissipator and expecting the closed-system TPM relation to apply without additional heat and bath entropy terms.

Exercises

Normalization

Show that the four forward branch probabilities sum to one.

Solution

The branch probabilities are

$$p_g^0(1-r), \quad p_g^0r, \quad p_e^0r, \quad p_e^0(1-r).$$

Their sum is

$$p_g^0[(1-r)+r] + p_e^0[r+(1-r)] = p_g^0+p_e^0 = 1.$$

Jarzynski Equality

Using the branch table, verify

$$\left\langle e^{-\beta W}\right\rangle_F = \frac{Z_\tau}{Z_0}.$$

Solution

The exponential average is

$$\begin{aligned} \left\langle e^{-\beta W}\right\rangle_F = {}& p_g^0(1-r) + p_g^0r e^{-\beta\epsilon_\tau} \\ &+ p_e^0r e^{\beta\epsilon_0} + p_e^0(1-r) e^{-\beta(\epsilon_\tau-\epsilon_0)} . \end{aligned}$$

Substitute

$$p_g^0=\frac{1}{Z_0}, \qquad p_e^0=\frac{e^{-\beta\epsilon_0}}{Z_0}.$$

Then

$$\begin{aligned} \left\langle e^{-\beta W}\right\rangle_F = {}& \frac{1-r}{Z_0} + \frac{r e^{-\beta\epsilon_\tau}}{Z_0} + \frac{r}{Z_0} + \frac{(1-r)e^{-\beta\epsilon_\tau}}{Z_0} \\ = {}& \frac{1+e^{-\beta\epsilon_\tau}}{Z_0} = \frac{Z_\tau}{Z_0}. \end{aligned}$$

Crooks Branch Ratio

Derive

$$\frac{p_F(m,n)}{p_R(n,m)} = e^{\beta(W_{mn}-\Delta F)}$$

for a matched branch.

Solution

For a matched branch,

$$p_F(m,n) = \frac{e^{-\beta E_n^0}}{Z_0} P_F(m|n),$$

and

$$p_R(n,m) = \frac{e^{-\beta E_m^\tau}}{Z_\tau} P_R(n|m).$$

Microscopic reversibility in this toy model gives

$$P_R(n|m)=P_F(m|n).$$

Therefore

$$\frac{p_F(m,n)}{p_R(n,m)} = \frac{Z_\tau}{Z_0} e^{\beta(E_m^\tau-E_n^0)}.$$

Since

$$\frac{Z_\tau}{Z_0} = e^{-\beta\Delta F}, \qquad W_{mn}=E_m^\tau-E_n^0,$$

the ratio is

$$e^{\beta(W_{mn}-\Delta F)}.$$

Rare Branches

Why can $\langle e^{-\beta W}\rangle$ converge slowly in Monte Carlo sampling even when the exact branch table is tiny?

Solution

The exponential average weights negative-work branches strongly when $\beta$ is large. Those branches may have small probability, so a finite sample can miss them or sample them poorly. The exact sum includes their contribution automatically, but Monte Carlo estimates need enough records to resolve the rare weighted events. This is a sampling problem, not a violation of the fluctuation relation.

Cross-Links

• Energy, Heat, and Work
• Two-Point Measurement Scheme
• Work Distributions
• Jarzynski Equality and Crooks Relation
• Fluctuation Theorems
• Entropy Production
• Quantum-Jump Trajectories
• Solving Lindblad Equations
• Formula Sheet
• Approximation Checklist

References

• C. Jarzynski, “Nonequilibrium equality for free energy differences,” Physical Review Letters 78, 2690-2693 (1997).
• G. E. Crooks, “Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences,” Physical Review E 60, 2721-2726 (1999).
• H. Tasaki, “Jarzynski relations for quantum systems and some applications,” arXiv:cond-mat/0009244 (2000).
• P. Talkner, E. Lutz, and P. Hänggi, “Fluctuation theorems: Work is not an observable,” Physical Review E 75, 050102(R) (2007).
• M. Campisi, P. Hänggi, and P. Talkner, “Colloquium: Quantum fluctuation relations: Foundations and applications,” Reviews of Modern Physics 83, 771-791 (2011).
• U. Seifert, “Stochastic thermodynamics, fluctuation theorems and molecular machines,” Reports on Progress in Physics 75, 126001 (2012).