Understanding quantum work in a quantum many-body system

Based on previous studies in a single-particle system in both the integrable [Jarzynski, Quan, and Rahav, Phys. Rev. X 5, 031038 (2015)] and the chaotic systems [Zhu, Gong, Wu, and Quan, Phys. Rev. E 93, 062108 (2016)], we study the the correspondence principle between quantum and classical work distributions in a quantum many-body system. Even though the interaction and the indistinguishability of identical particles increase the complexity of the system, we find that for a quantum many-body system the quantum work distribution still converges to its classical counterpart in the semiclassical limit. Our results imply that there exists a correspondence principle between quantum and classical work distributions in an interacting quantum many-body system, especially in the large particle number limit, and further justify the definition of quantum work via two-point energy measurements in quantum many-body systems.

Figure 1

Energy spectrum of the 1D two-site BH Hamiltonian (38) as a function of the work parameter $J$ for $N=100$. Inset: The details of the energy spectrum in the red rectangle.

Figure 2

Quantum [Eq. (25)] and classical [Eq. (36)] transition probabilities for the 1D two-site BH model with different number of particles (a) $N=100$, (b) $N=200$. The solid blue curve represents the quantum case $P^Q\left(n^B\right|m^A)$, while the dashed red curve represents the classical case $P^C\left(n^B\right|m^A)$. For the quantum case, the initial state is the ground state of $H(t=0)$ with $|m^A\rangle =|{\mathbf{n}}^A\rangle =|N/2,N/2\rangle$. For the classical case, the initial state is a collection of microscopic states ${\mathrm{\Psi }}^A=\{{\psi }_1^A,{\psi }_2^A\}=\{(N/2,{\varphi }_1^A),(N/2,{\varphi }_2^A)\}$, with ${\varphi }_1^A$ and ${\varphi }_2^A$ the independent random numbers evenly sampled in the range $[0,2\pi )$. Here, due to $J(t=\tau )=0$, the finial energy eigenstates are given by the Fock states: $|n^B\rangle =|n_1^B,N-n_1^B\rangle$ with $n_1^B=0,1,...,N$. Inset: Time dependence of the work parameter $J\left(t\right)$.

Figure 3

Cumulative quantum and classical transition probabilities for the 1D two-site BH model with different number of particles: (a) $N=10$, (b) $N=50$, (c) $N=100$, (d) $N=200$. The jagged blue solid curve shows the quantum case, ${\sum }_{n^B}{P}^Q\left(n^B\right|m^A)$, while the smooth dashed red curve shows the classical case, ${\sum }_{n^B}{P}^C\left(n^B\right|m^A)$. For the quantum case, the initial state is chosen to be the ground state of $H(t=0)$ with $|m^A\rangle =|{\mathbf{n}}^A\rangle =|N/2,N/2\rangle$. The corresponding classical initial state is a collection of microscopic states ${\mathrm{\Psi }}^A=\{{\psi }_1^A,{\psi }_2^A\}=\left\{(N/2,{\varphi }_1^A)(N/2,{\varphi }_2^A)\right\}$, where ${\varphi }_1^A$ and ${\varphi }_2^A$ are the uniformly distributed random numbers in the range $[0,2\pi )$. Due to the fact that $J(t=\tau )=0$, the final energy eigenstates are Fock states: $|n^B\rangle =|n_1^B,N-n_1^B\rangle$ with $n_1^B=0,...,N$.

Figure 4

RMSE $R\left(N\right)$ (blue pentagrams) as a function of the number of particles $N$. The other parameters are $U=5/N,\tau =10,\hslash =1$.

Figure 5

Energy spectrum of the 1D three-site BH model (44) with the work parameter $J$ for $N=20$. Inset: The details of the red rectangle.

Figure 6

Quantum (solid blue curve) and classical (dashed red curve) transition probabilities of the 1D three-site BH model: (a) Transition probabilities between different energy eigenstates, (b) Cumulative transition probabilities. For the quantum case, the number of bosons is $N=20$ and the initial state is one of the three degenerate eigenstates of the 19th energy level of $H(t=0)$ with $|m^A\rangle =|{\mathbf{n}}^A\rangle =|5,5,10\rangle$. The classical counterpart of $|m^A\rangle$ is a collection of microscopic states ${\mathrm{\Psi }}^A=\{{\psi }_1^A,{\psi }_2^A,{\psi }_3^A\}=\{(5,{\varphi }_1^A),(5,{\varphi }_2^A),(10,{\varphi }_3^A)\}$, with ${\varphi }_j\text{"s}(j=1,2,3)$ the uniformly distributed random numbers in the range $[0,2\pi )$. Due to the fact that the final value of $J$ is zero, the energy eigenstates of $H(t=\tau )$ are the Fock states: $|n^B\rangle =|n_1^B,n_2^B,N-n_1^B-n_2^B\rangle$ with $n_1^B=0,n_2^B=0,...,N;n_1^B=1,n_2^B=0,...,N-1;...;n_1^B=N,n_2^B=0$.