Encoding complex-balanced thermalization in quantum circuits

TL;DR

The paper tackles the challenge of realizing complex-balanced thermalization (CBT) in quantum devices within a Markovian framework. It introduces a quantum-circuit platform where a system interacts with engineered reservoir qubits via non-unitary two-qubit gates and partial-trace operations, yielding a quantum master equation description with a short-time limit $\bar{t}\ll 1$. A key contribution is the introduction of dual spectral functions $\gamma^{(n)}_{\omega}$ and $\bar{\gamma}^{(n)}_{\omega}$ arising from non-orthogonal reservoir eigenstates, accompanied by a modified KMS relation $\eta^{(n)}_{\omega} = e^{-{\bar{\beta}} \omega}$, enabling non-uniform heating and CBT. The authors demonstrate two applications— temporally-correlated dichromatic emission and LEP-protected quantum synchronization at finite temperature—highlighting enhanced temporal correlations and robust synchronization beyond conventional reservoirs.

Abstract

We propose a protocol for effectively implementing complex-balanced thermalization via Markovian processes on a quantum-circuit platform that couples the system with engineered reservoir qubits. The non-orthogonality of qubit eigenstates facilitates non-uniform heating through a modified Kubo-Martin-Schwinger relation, while simultaneously supports amplification-dissipation dynamics by violating microscopic time-reversibility. This offers a new approach to realizing out-of-equilibrium states at given temperatures. We show two applications of this platform: temporally-correlated dichromatic emission and Liouvillian exception point protected quantum synchronization at finite temperatures, both of which are challenging to achieve with conventional thermal reservoirs.

Figures (4)

• Figure 1: A protocol for quantum circuits. (a) An overview: A quantum system interacts with a qubit set over $N$ time periods. (b) A period: The system collides in turn with $N_q$ reservoir qubits, labeled $Q_1$, $\cdots$, $Q_{N_\text{q}}$. (c) Collision step $n = (m-1)N_q + l - 1$: The non-unitary two-qubit gate $U^{(n)}$ couples the system to the qubit q$_l$. After performing the "trace out" operation, only the resulting system state participates in subsequent collisions.
• Figure 2: Under time reversal, microscopic subprocesses at (a) positive time $t>0$ and (b) negative time $t<0$.
• Figure 3: (a) Dichromatic photon alternative emission setup. In the setup, a three-level system is coupled to two photon modes $p_x$ ($x=1$, $2$) through Jaynes-Cummings terms $g_\text{int} (\mathinner{|{2}\rangle \! \langle{1}|} p_1 + \mathinner{|{1}\rangle \! \langle{0}|} p_2 + \textrm{h.c.} )$. These photon modes carry energies $\omega_{21}$ and $\omega_{10}$, respectively, without detuning. Photon emission is modeled using additional Lindblad operators $L_x = \sqrt{\kappa} p_x$ in Eq. \ref{['eq:GKSL']}. The three-level system interacts with $N_\text{q}=3$ qubits through the quantum-circuit platform [Fig. \ref{['fig:fig1']}]. (b) Time-evolving photon numbers $\braket{n_{1/2}}$. (c) Second-order time correlation functions $G_{x_1 x_2}^{(2)}$ for $t=+\infty$. Inset: data for small $\tau$ near the LEP ($\cosh \theta_\text{q}^\text{LEP}=2$). We used spin-$1$ operator $A^{(n)}=\mathcal{S}^x$, and parameters $\theta^{(n)}=\pi/3$, $\beta=1$, $g=1$, $g_\text{int}=0.4$, $\kappa=0.1$ and $\bar{t}=0.05$. For (b, c), $\theta_\text{q}=\pi/6$.
• Figure 4: (a) A LEP-protected quantum synchronization setup. In the setup, two spins $\mathbf{s}_1$ and $\mathbf{s}_2$ (momenta $|\mathbf{s}_1|=|\mathbf{s}_2|=1/2$) are coupled with an Ising-type interaction and modulated by external magnetic fields along both the $x$ and $z$ axes. The Hamiltonian is given by $H_\text{s} = J s^z_1 s^z_2 + h_x (s^x_1 + s^x_2) + h_z (s^z_1 + s^z_2)$, with $J$, $h_x$ and $h_z$ being the strength of interaction and fields, provides a spectrum of four energy levels: $\ket{0}$ (ground state), $\ket{1}$ (first excited state), $\ket{2}$ and $\ket{3}$. They interact with $N_\text{q}=6$ reservoir qubits through the quantum-circuit platform [Fig. \ref{['fig:fig1']}]. (b) Phase diagram when $\theta_\text{q}=0.55$. The QS region is shaded according to the long-term oscillation amplitude of $\braket{s^x_1}$. (c) Time-evolving $\braket{s^x_1}$, $\braket{s^x_2}$ and Pearson coefficient $C_{12}$ obtained from the collision map with $\theta_\text{q}=0.55$ and $\varphi_0=\pi/3$. (d) Time-evolution trajectory of spin $\braket{\mathbf{s}_1}=(\braket{s^x_1},\braket{s^y_1},\braket{s^z_1})$ on the Bloch sphere for different $\theta_\text{q}$ with $\varphi_0=\pi/3$ fixed. At the $4$th LEP $\theta_\text{q} = \text{arctanh}\left(\sin\varphi_\text{c}\right)\approx 1.317$, the system can no longer sustain balanced amplification-dissipation driving. We choose $A^{(n)}=s^x_1$, $J = 0.2$, $h_z = 2 h_x = 1$, $\beta=1$, $g=2$, and $\bar{t}=0.05$.