Homodyne Measurement of a Non-Hermitian Qubit Undergoing Fluorescence

TL;DR

The paper investigates a PT-symmetric non-Hermitian qubit realized via post-selection on a three-level quantum system and probed with continuous homodyne measurement. It analyzes the interplay between non-Hermitian decay and measurement backaction by comparing ensemble-averaged trajectory dynamics with Liouvillian evolution, deriving a no-jump stochastic master equation and employing a path-integral formalism to identify optimal measurement paths. Near the exceptional point ($EP$), deviations between trajectory-averaged and Liouvillian dynamics emerge and depend on the drive axis and measurement quadrature, highlighting the role of measurement backaction in shaping transients. The results offer insights into manipulating open quantum systems and performing controlled dynamics near EPs using measurement-based strategies.

Abstract

Implementation of a two-level non-Hermitian qubit via post-selection of a three-level system has been demonstrated. The post-selection procedure, which discards quantum jump to the ground-state manifold while retaining excitations in the first and second excited-state manifolds, effectively generates a non-Hermitian qubit exhibiting PT symmetry. In this work, we perform continuous homodyne measurement of this non-Hermitian qubit and analyze the interplay between decay introduced by post-selection and measurement backaction. We compare the ensemble-averaged dynamics obtained from measurement trajectories with the the Liouvillian average. We formulate the no-jump stochastic differential equation describing the post-selected non-Hermitian qubit and show that its ensemble-averaged dynamics agree with those of the jump-updated post-selected evolution at drive strengths far from the Liouvillian exceptional point (EP). The degree of deviation near the EP depends sensitively on the nature of the drive. This discrepancy is attributed to the interplay between measurement backaction and the non-Hermitian decay introduced by post-selection. Furthermore, we determine the optimal path of the non-Hermitian qubit by extremizing the action within the path-integral formulation of the quantum trajectory framework Our results provide insights into how measurement backaction and non-Hermitian dynamics together shape the transient behavior of open quantum systems and enable controlled manipulation of qubits near exceptional points.

Figures (9)

• Figure 1: Measurement schemes of the three-level system illustrating different unraveling models of its trajectories. The system emits photons that are directed to the detector via the environment or a cavity. The detector has three components. One part ($D_{ef}$) detects photons released from the $\lvert f \rangle \to \lvert e \rangle$ transition with frequency $\omega_{ef}$, and another ($D_{eg}$) detects photons from the $\lvert e \rangle \to \lvert g \rangle$ transition with frequency $\omega_{eg}$. The detected signals from these two components are sent to $D_{gef}$, where the final state update happens. The schematic shows two types of measurement. The top figure shows pure photon-counting detection: at some time $t_1$, when a jump from $\lvert f \rangle$ to $\lvert e \rangle$ occurs, $D_{ef}$ clicks; at some other time $t_2$, when a jump from $\lvert e \rangle$ to $\lvert g \rangle$ occurs, $D_{eg}$ clicks. In the former case, the detector $D_{gef}$ updates the state according to the Kraus operator $K_{1e}$, and in the latter according to $K_{1g}$. When no jump occurs, the state is always updated according to $K_0$. The bottom figure shows a hybrid measurement scheme: the jump from $\lvert f \rangle$ to $\lvert e \rangle$ is continuously and weakly monitored through a homodyne measurement setup, while the jump from $\lvert e \rangle$ to $\lvert g \rangle$ is monitored by photon counting measurement. The weak and continuous homodyne measurement is realized by inserting a beam splitter in front of the detector, and a local oscillator of frequency $\omega_{ef}$ and phase $\theta$ (with respect to the $\lvert f \rangle \to \lvert e \rangle$ photon) is sent to the beam splitter. We assume that, because of the frequency mismatch, the $\lvert e \rangle \to \lvert g \rangle$ photon is sent to the photon-counting channel. As long as there is no jump from $\lvert e \rangle$ to $\lvert g \rangle$, the detector $D_{gef}$ updates the state as per $K_H$, and upon detection it updates via $K_J$.
• Figure 2: (a)–(c) State-update trajectories obtained from Eq. \ref{['eq:state_uodate']}. The trajectories are shown in terms of the state populations $P_{f}$, $P_{e}$, and $P_{g}$, corresponding to the second excited, first excited, and ground states, respectively. These populations are the diagonal elements of the density matrix. (b) The ensemble-averaged values of the populations are shown as dotted lines. This ensemble average is compared with the average dynamics of the full three-level system, shown as dashed lines. The two averages match excellently. Parameters used: $\gamma_{g} = 1~\text{MHz}$, $\gamma_{e} = 0.2~\text{MHz}$, and $\omega = 3~\text{MHz}$, with the number of trajectories $= 10^{3}$.
• Figure 3: Eigenvalue spectrum of the Liouvillian, with the real part shown in (a) and the imaginary part shown in (b). Parameters used: $\gamma_{g} = 1~\text{MHz}$, $\gamma_{e} = 0.2~\text{MHz}$.
• Figure 4: The evolution of the normalized $\rho_{ff}$ element of the density matrix, i.e., the population of the $\lvert f \rangle$ state ($P_f$). The solid line corresponds to the result obtained from averaging the trajectories where both jump and no-jump state updates are considered, while the dotted line corresponds to the Liouvillian dynamics. (a)–(b) and (e)-(f) show the case where the drive strength is close to the Liouvillian EP, and (c)–(d) correspond to the case away from the EP. (a), (c) and (e) correspond to driving about the $x$-axis, while (b), (d) and (f) correspond to driving about the $y$-axis. All plots are generated at $\theta = 0$. A similar behavior is observed when $\theta = \pi/2$, except that in this case the $x$-axis drive resembles (b) and (f), while the $y$-axis drive resembles (a) and (e). Parameters used: for (a) and (b), $\omega = 0.3~\text{MHz}$, for (c) and (d), $\omega = 2~\text{MHz}$, and for (d) and (e) $\omega=0.7$. The common parameters are $\gamma_{e} = 0.2~\text{MHz}$ and $\gamma_{g} = 1~\text{MHz}$. The number of trajectories used is $10^4$ ad the time step is $\text{dt}=0.01\mu s$
• Figure 5: Trajectory evolution of the Bloch vector components $x$, $y$, and $z$ of the non-Hermitian qubit at $\theta = 0$. (a)–(c) The qubit is driven about the $x$-axis. The mean of the trajectories is shown as a solid black curve. The qubit, initialized in the excited state, undergoes unitary oscillations about the $x$-axis. This evolution drives the $y$ and $z$ components, leading to oscillations. Due to the measurement backaction associated with the $z$ evolution, the $x$ component trajectories also fluctuate, although their mean remains zero. (d)–(f) The qubit is driven about the $y$-axis. Here the $x$ and $z$ components oscillate unitarily, while the $y$ component remains zero. This is because the qubit starts in the excited state ($z = 1$, $x = 0$, $y = 0$), and the backaction on the $y$ evolution is independent of $z$, leaving $y$ unchanged. For $\theta = \pi/2$ measurements, the $y$-axis drive resembles the dynamics in (a)–(c), while the $x$-axis drive resembles those in (d)–(f); i.e., the roles are reversed. The common parameters used are $\gamma_{g} = 1~\text{MHz}$ and $\gamma_{e} = 0.2~\text{MHz}$. The drive frequency is set to $\omega = 2~\text{MHz}$.