Fluctuations of work in realistic equilibrium states of quantum systems with conserved quantities

1 The out-of-equilibrium dynamics of quantum systems is one of the most fascinating 2 problems in physics, with outstanding open questions on issues such as relaxation to 3 equilibrium. An area of particular interest concerns few-body systems, where quan4 tum and thermal fluctuations are expected to be especially relevant. In this contribu5 tion, we present numerical results demonstrating the impact of conserved quantities (or 6 ‘charges’) in the outcomes of out-of-equilibrium measurements starting from realistic 7 equilibrium states on a few-body system implementing the Dicke model. 8


Introduction
modynamics [4,11,12], that demand that the equilibrium state of such a system be described 48 by a density matrix of the form of the generalised Gibbs ensemble (GGE), namely Here, β is the usual inverse temperature, while {β k }(k = 1, . . . , N cons ) are called generalised 50 inverse temperatures.

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The fact that the equilibrium state is of the GGE form has implications for the expectation 52 values of measurements done on the system in equilibrium, as has been extensively analysed More recently, the present authors have generalised these QFRs to the case that the equilibrium 63 state is of the GGE form and for an arbitrary number of charges for the initial and final states, 64 thus notably expanding the rage of non-equilibrium problems that can be tackled [22]. In particular, our formalism is explicitly able to deal with processes where the number of charges Here, E is the energy of the initial state, and M k the expectation value of operatorM k in the 84 initial state. 85 We then submit the system to an out-of-equilibrium process by changing its Hamiltonian 86 from the initial valueĤ to some new final HamiltonianĤ . In general, we expect the set of 87 operators that commute withĤ to be different from that of charges ofĤ, and we label the  work, W, done on the system after a single run of these protocols as: These are stochastic quantities, as they depend on the result of projective measurements at where 〈·〉 stands for an average over many runs of the protocol. Eqs. (8) and (9) with the partition functions in the GGE, Z GGE , defined in (1), and ∆F = F − F the differ-123 ence in generalised (dimensionless) free energy functions, F = − ln Z GGE and F = − ln Z GGE .

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Analogously to above, if we multiply both sides of Eq. (10) by P BW (−W) and integrate over 125 W, we obtain the following equality: This is the generalised quantum Jarzynski equality [22]. and (9), and generalised QFRS, Eqs. (10) and (11). We found that when the initial state whereb † andb are the operators creating and annihilating excitations in the bosonic field,

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Our numerical studies testing the standard and generalised QFRS in Ref.
[22] were obtained 156 assuming that the system is initially equilibrated, and hence perfectly described by either a 157 Gibbs, with inverse temperature β, or a GGE density matrix, with two generalised tempera-

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To tackle this question we design the following protocol:  We calculate statistics of work for the forward process -i.e., the PDFs P FW (w) and P FW (W)from steps 1-2, and for the backward process from steps 2-3. We compare these statistics of   In this contribution, we have tested the validity of our generalised QFRs [22] to a more 207 stringent test by considering a more realistic situation, in which the system is not allowed an 208 infinite time to relax to its equilibrium state in contact to baths. Our robust numerical cal-209 culations support that, when the Hamiltonian describing the system has conserved charges, 210 2 To be sure that we start from an equilibrium state, we must let the system relax in the final Hamiltonian, α = 0 and g = 6ε 0 , before starting the backward part of the protocol. However, this relaxation time is irrelevant for our numerical simulation. All our results are based on the two-projective measurement scheme. Hence, if the actual state of the system at a certain time t is |Ψ(t)〉 = n C n (t) |Φ n 〉, where |Φ n 〉 are the eigenfunction of the Hamiltonian, only the square moduli of the coefficents, |C n | 2 , are relevant. Therefore, the dephasing introduced by the relaxation procedure does not play any role in the results.