Exceptional entanglement phenomena non-Hermiticity meeting non-classicality

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Exceptional entanglement phenomena: non-Hermiticity meeting non-classicality

Pei-Rong Han1,∗Fan Wu1,∗Xin-Jie Huang1,∗Huai-Zhi Wu1, Chang-Ling Zou2,3,8,

Wei Yi2,3,8, Mengzhen Zhang4, Hekang Li5, Kai Xu5,6,8, Dongning Zheng5,6,8,

Heng Fan5,6,8, Jianming Wen7,†Zhen-Biao Yang1,8,‡and Shi-Biao Zheng1,8§

1Fujian Key Laboratory of Quantum Information and Quantum Optics,

College of Physics and Information Engineering,

Fuzhou University, Fuzhou, Fujian, 350108, China

2CAS Key Laboratory of Quantum Information,

University of Science and

Technology of China, Hefei 230026, China

3CAS Center for Excellence in Quantum Information and Quantum Physics,

University of Science and Technology of China,

Hefei 230026, China

4Pritzker School of Molecular Engineering,

University of Chicago,

Chicago, IL 60637, USA

5Institute of Physics,

Chinese Academy of Sciences, Beijing 100190,

China

6CAS Center for Excellence in Topological Quantum Computation,

University of Chinese Academy of Sciences,

Beijing 100190, China

7Department of Physics,

Kennesaw State University, Marietta, Georgia

30060, USA

8Hefei National Laboratory, Hefei 230088, China

Non-Hermitian (NH) extension of quantum-mechanical Hamiltonians represents one of the most

signiﬁcant advancements in physics. During the past two decades, numerous captivating NH phe-

nomena have been revealed and demonstrated, but all of which can appear in both quantum and

classical systems. This leads to the fundamental question: What NH signature presents a radical

departure from classical physics? The solution of this problem is indispensable for exploring genuine

NH quantum mechanics, but remains experimentally untouched upon so far. Here we resolve this

basic issue by unveiling distinct exceptional entanglement phenomena, exempliﬁed by an entangle-

ment transition, occurring at the exceptional point (EP) of NH interacting quantum systems. We

illustrate and demonstrate such purely quantum-mechanical NH eﬀects with a naturally-dissipative

light-matter system, engineered in a circuit quantum electrodynamics architecture. Our results lay

the foundation for studies of genuinely quantum-mechanical NH physics, signiﬁed by EP-enabled

entanglement behaviors.

When a physical system undergoes dissipation, the Hermiticity of its Hamiltonian dynamics is broken down. As

any system inevitably interacts with its surrounding environment by exchanging particles or energy, non-Hermitian

(NH) eﬀects are ubiquitous in both classical and quantum physics. Such eﬀects were once thought to be detrimental,

and needed to be suppressed for observing physical phenomena of interest and for technological applications, until

the discovery that NH eﬀects could represent a complex extension of quantum mechanics [1–3]. Since then, increasing

eﬀorts have been devoted to the exploration of NH physics, leading to ﬁndings of many intriguing phenomena that are

uniquely associated with NH systems. Most of these phenomena are closely related to the exceptional points (EPs),

where both the eigenenergies and the eigenvectors of the NH Hamiltonian coalesce [4–6]. In addition to fundamental

interest, such NH eﬀects promise the realization of enhanced sensors [7–11].

Hitherto, there have been a plethora of experimental investigations on genuinely NH phenomena, ranging from

real-to-complex spectral transition to NH topology [12–20], as well as on relevant applications [21–25], most of which

were performed with classically interacting but non-entangled systems. The past few years have witnessed a number

of demonstrations of similar NH phenomena in diﬀerent quantum systems, ranging from photons [26–28] to atoms [29–

31] and ions [32, 33], and from nitrogen-vacancy centers [34–36] to superconducting circuits [37–39]. However, these

experiments have been conﬁned to realizations of NH semiclassical models, where the degree of freedom either of the

light or of the matter was treated classically in the eﬀective NH Hamiltonian describing the light-matter interaction,

and consequently, the observed NH eﬀects bear no relation to quantum entanglement. Indeed, all the genuinely NH

eﬀects demonstrated so far can occur in both quantum and classical systems. This naturally leads to an imperative

arXiv:2210.04494v4 [quant-ph] 31 Dec 2023

issue: What feature can simultaneously manifest non-Hermiticity and non-classicality? Recently, there has been

a signiﬁcant focus on nonequilibrium quantum phase transitions in the entanglement dynamics of NH many-body

systems [40–42], but the purely quantum-mechanical NH features associated with the Hamiltonian eigenstates have

remained unexplored.

We here perform an in-depth investigation on this fundamental problem, ﬁnding that dissipative interacting quan-

tum systems can display exceptional entanglement phenomena. In particular, we discover an EP-induced entanglement

transition, which represents a purely quantum-mechanical NH signature, with neither Hermitian nor classical analogs.

We illustrate our discovery with a dissipative qubit-photon system, whose NH quantum eﬀects are manifested by

the singular entanglement behaviors of the bipartite entangled eigenstates. We experimentally demonstrated these

singular behaviors in a circuit, where a superconducting qubit is controllably coupled to a decaying resonator. The

exceptional entanglement signatures, inherent in the qubit-photon static eigenstates, are mapped out by a density

matrix post-casting method, which enables us to extract the weak nonclassical signal from the strong noise back-

ground. In addition to the quantum character, the demonstrated NH phenomena originate from naturally occurring

dissipation, distinct from that induced by an artiﬁcially engineered reservoir [12–39]. Our results are universal for

composite quantum systems immersed in pervasive Markovian reservoirs, endowing NH quantum mechanics with

genuinely non-classical characters, which are absent in NH classical physics.

The theoretical model, used to illustrate the NH entanglement transition, is composed of a two-level system (qubit)

resonantly coupled to a quantized photonic mode, as sketched in Fig. 1a. The quantum state evolution trajectory

without photon-number jumps is governed by the NH Hamiltonian (setting ℏ= 1)

HNH = Ω(a†|g⟩⟨e|+a|e⟩⟨g|)−i

2κq|e⟩⟨e| − i

2κfa†a, (1)

where |e⟩(|g⟩) denotes the upper (lower) level of the qubit, a†(a) represents the creation (annihilation) operator for

the photonic mode, κq(κf) is the energy dissipation rate for the qubit (ﬁeld mode), and Ω is the qubit-ﬁeld coupling

strength. In the n-excitation subspace, the system has two right entangled eigenstates, given by

|Φn,±⟩=Nn,±(√nΩ|e, n −1⟩+En,±|g, n⟩),(2)

where Nn,±= (nΩ2+|En,±|2)−1/2and En,±=−iγ/4±∆En/2 are the corresponding eigenenergies, with γ=κf+κq,

∆En= 2pnΩ2−κ2/16, and κ=κf−κq. These two eigenstates are separated by an energy gap of ∆En. The inherent

quantum entanglement makes the system fundamentally distinct from previously demonstrated NH semiclassical

models [29–39], where the qubit is not entangled with the classical control ﬁeld in any way. When Ω > κ/(4√n), the

system has a real energy gap, and undergoes Rabi-like oscillations, during which the qubit periodically exchanges a

photon with the ﬁeld mode. With the decrease of Ω, the energy gap is continuously narrowed until reaching the EP,

where the two energy levels coalesce. After crossing the EP, the gap becomes imaginary, and the population evolution

exhibits an over-damping feature.

Unlike previous investigations, here each eigenenergy is possessed by the two entangled components, neither of

which has its own state. The nonclassical feature of each eigenstate is manifested by the light-matter entanglement,

which can be quantiﬁed by the concurrence [43]

E±=2√nΩ|En,±|

|En,±|2+nΩ2.(3)

When the rescaled coupling strength η= 4√nΩ/κ is much smaller than 1, the two eigenstates are respectively

dominated by |e, n −1⟩and |g, n⟩. With the increase of η, these two populations become increasingly balanced

until reaching the EP η= 1, where both eigenstates converge to the same maximally entangled state. During this

convergence, E±exhibit a linear scaling with η. In the Bloch representation, this corresponds to a rotation of the

eigenvector |Φn,+⟩(|Φn,−⟩) around the x(−x) axis from pointing at the north (south) polar, until merging at the y

axis, as shown in the left panel of Fig. 1b. After crossing the EP, E±become independent of Ω, which implies both

two eigenstates keep maximally entangled. However, |Φn,+⟩(|Φn,−⟩) is rotated around the z(−z) axis, progressively

approaching the x(−x) axis (right panel of Fig. 1b). This sudden switch of the rotation axis manifests an entanglement

transition at the EP, where the derivative of the concurrence with respect to ηpresents a discontinuity, jumping from

1 to 0. By measuring the time-evolving output states associated with the no-jump trajectory, we can extract the

information about both the energy gap and entanglement regarding the “static” eigenstates. It should be noted that

the resonator plays a radically diﬀerent role from the ancilla used in the previous experiments [34–37], which was

introduced as an artiﬁcially engineered environment to the test qubit, but whose dynamics was not included in the

eﬀective NH Hamiltonian dynamics. In distinct contrast, in the present system the degree of freedom of Rconstitutes

a part of the Hamiltonian, whose NH term is induced by the natural environment.

The experimental demonstration of the NH physics is performed in a circuit quantum electrodynamics architecture,

where the qubit-photon model is realized with a superconducting qubit Qand its resonator Rwith a ﬁxed frequency

ωr/2π= 6.66 GHz (see Supplemental Material, Sec. S2 [45]). The decaying rates of Qand Rare κq≃0.07 MHz

and κf= 5 MHz, respectively. The accessible maximum frequency of Q,ωmax = 2π×6.01 GHz, is lower than ωrby

an amount much larger than their on-resonance swapping coupling gr= 2π×41 MHz (Fig. 1c). To observe the EP

physics, an ac ﬂux is applied to Q, modulating its frequency as ωq=ω0+εcos(νt), where ω0is the mean |e⟩-|g⟩energy

diﬀerence, and εand νdenote the modulating amplitude and frequency. This modulation enables Qto interact with

Rat a preset sideband, with the photon swapping rate Ω tunable by ε(see Supplemental Material, Sec. S3 [45]).

Our experiment focuses on the single-excitation case (n= 1). Before the experiment, the system is initialized to

the ground state |g, 0⟩. The experiment starts by transforming Qfrom the ground state |g⟩to excited state |e⟩with

aπpulse, following which the parametric modulation is applied to Qto initiate the Q-Rinteraction (see Fig. S4 of

Supplementary Material for the pulse sequence). This interaction, together with the natural dissipations, realizes the

NH Hamiltonian of Eq. (1). After the modulating pulse, the Q-Rstate is measured with the assistance of an ancilla

qubit (Qa) and a bus resonator (Rb), which is coupled to both Qand Qa. The subsequent Q→Rb,Rb→Qa, and

R→Qquantum state transferring operations map the Q-Routput state to the Qa-Qsystem, whose state can be

read out by quantum state tomography (see Supplemental Material, Sec. S5 [45]).

A deﬁning feature of our system is the conservation of the excitation number under the NH Hamiltonian. This

Hamiltonian evolves the system within the subspace {|e, 0⟩,|g, 1⟩}. A quantum jump would disrupt this conversion,

moving the system out of this subspace. This property in turn enables us to post-select the output state governed the

NH Hamiltonian simply by discarding the joint Qa-Qoutcome |g, g⟩after the state mapping. With a correction of the

quantum state distortion caused by the decoherence occurring during the state mapping, we can infer the quantum

Rabi oscillatory signal, characterized by the evolution of the joint probability |e, 0⟩, denoted as Pe,0. Fig. 2a shows

thus-obtained Pe,0as a function of the rescaled coupling ηand the Q-Rinteraction time t. The results clearly show

that both the shapes and periods of vacuum Rabi oscillations are signiﬁcantly modulated by the NH term around the

EP η= 1, where incoherent dissipation is comparable to the coherent interaction. These experimental results are in

well agreement with numerical simulations (see Supplemental Material, Sec. S4 [45]).

The Rabi signal does not unambiguously reveal the system’s quantum behavior. To extract full information of

the two-qubit entangled state, it is necessary to individually measure all three Bloch vectors for each qubit, and

then correlate the results for the two qubits. The z-component can be directly measured by state readout, while

measurements of the x- and y-components require y-rotations and x-rotations before state readout, which breaks

down excitation-number conservation, and renders it impossible to distinguish individual jump events from no-jump

ones. We circumvent this problem by ﬁrst reconstructing the two-qubit density matrix using all the measurement

outcomes, and then discarding the matrix elements associated with |g, g⟩(see Supplemental Material, Sec. S5 [45]).

This eﬀectively post-casts the two-qubit state to the subspace {|e, g⟩,|g, e⟩}. We note that such a technique is in

sharp contrast with the conventional post-selection method, where some auxiliary degree of freedom (e.g., propagation

direction of a photon) enables reconstruction of the relevant conditional output state, but which is unavailable in the

NH qubit-photon system. With a proper correction for the state mapping error, we obtain the Q-Routput state

governed by the non-Hermitian Hamiltonian. In Fig. 2b, we present the resulting Q-Rconcurrence, as a function of

ηand t. To show the entanglement behaviors more clearly, we present the concurrence evolutions for η= 5 and 0.5

in Fig. 2c and d, respectively. The results demonstrate that the entanglement exhibits distinct evolution patterns in

the regimes above and below the EP.

To reveal the close relation between the exceptional entanglement behavior and the EP, we infer the eigenvalues and

eigenstates of the NH Hamiltonian from the output states, measured for diﬀerent evolution times (see Supplemental

Material, Sec. S7 [45]). The concurrences associated with the two eigenstates, obtained for diﬀerent values of η, are

shown in Fig. 3a and b (dots), respectively. As expected, each of these entanglements exhibits a linear scaling below

the EP, but is saturated at the EP, and no longer depends upon ηafter crossing the EP. The singular features around

the EP are manifested by their derivatives to η, which are shown in the insets. The discontinuity of these derivatives

indicates the occurrence of an entanglement transition at the EP. Such a nonclassical behavior, as a consequence of

the competition between the coherent coupling and incoherent dissipation, represents a unique character of strongly

correlated NH quantum systems, but has not been reported so far. These results demonstrate a longitudinal merging

of the two entangled eigenstates at the EP (left panel of Fig. 2b). Accompanying this is the onset of the transverse

splitting, which can be characterized by the relative phase diﬀerence between |Φ1,±⟩, deﬁned as φ=φ+−φ−, with

φ±representing the relative phases between |g, 1⟩and |e, 0⟩in the eigenstates |Φ1,±⟩. Such a phase diﬀerence, inferred

for diﬀerent values of η, is displayed in Fig. 3c. This quantum-mechanical NH signature is universally applicable

to dissipative interacting quantum systems, a feature absent in earlier superconducting-circuit-based single-qubit NH

experiments [38, 39].

The demonstrated exceptional entanglement transition is a purely quantum-mechanical NH eﬀect. On one hand,

quantum entanglement, which has no classical analogs, represents a most characteristic trait that distinguishes quan-

tum physics from classical mechanics [46]. On the other hand, this eﬀect, uniquely associated with the EP, is absent

in Hermitian qubit-photon systems [47–50]. We further note that such an NH eﬀect can occur in other dissipative

quantum-mechanically correlated systems, e.g., a system composed of two or more coupled qubits with unbalanced

decaying rates (see Supplemental Material, Sec. S8 [45]). This implies that such a quantum-mechanical NH signature

is universal for dissipative interacting quantum systems.

The real and imaginary parts of the extracted quantum Rabi splitting ∆Enare shown in Fig. 3d. As theoretically

predicted, above the EP the two eigenenergies have a real gap, which is continuously narrowed when the control

parameter ηis decreased until reaching the EP, where the two levels coalesce. After crossing the EP, the two levels

are re-split but with an imaginary gap, which increases when ηis decreased. This corresponds to the real-to-imaginary

transition of the vacuum Rabi splitting between the Q-Rentangled eigenstates, which is in distinct contrast with the

previous experiments on PT symmetry breaking [29–39], realized with the semiclassical models, where the exceptional

physics has no relation with quantum entanglement.

In conclusion, we have discovered an exceptional entanglement transition in a fundamental light-matter system

governed by an NH Hamiltonian, establishing a close connection between quantum correlations and non-Hermitian

eﬀects. This transition has neither Hermitian nor classical analogs, representing the unique feature of NH quantum

mechanics. The experimental demonstration is performed in a circuit, where a superconducting qubit is controllably

coupled to a resonator with a non-negligible dissipation induced by a natural Markovian reservoir. The NH quantum

signatures of the eigenstates are inferred from the no-jump output state, measured for diﬀerent evolution times.

Our results push NH Hamiltonian physics from the classical to genuinely non-classical regime, where the emergent

phenomena have neither Hermitian nor classical analogs. The post-projection method, developed for extracting the no-

jump output density matrix, would open the door to experimentally explore purely quantum-mechanical NH eﬀects

in a broad spectrum of interacting systems, where excitation number is initially deﬁnite and conserved under the

NH Hamiltonian. Such systems include resonator-qubits arrays [51], fully-connected architectures involving multiple

qubits coupled to a single resonator [52], and lattices composed of many qubits with nearest-neighbor coupling [53].

When the system initially has nexcitations, the no-jump trajectory can be post-selected by discarding the outcomes

with less than nexcitations.

Acknowledgments: We thank Liang Jiang at University of Chicago for valuable comments. Funding: This work

was supported by the National Natural Science Foundation of China (Grant No. 12274080, No. 11875108, No.

12204105, No. 11774058, No. 12174058, No. 11974331, No. 11934018, No. 92065114, and No. T2121001), Innovation

Program for Quantum Science and Technology (Grant No. 2021ZD0300200 and No. 2021ZD0301200), NSF (Grant

No. 2329027), the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB28000000),

the National Key R&D Program under (Grant No. 2017YFA0304100), the Key-Area Research and Development

Program of Guangdong Province, China (Gran No. 2020B0303030001), Beijing Natural Science Foundation (Grant

No. Z200009), and the Project from Fuzhou University (Grant No. 049050011050).

Author contributions: S.B.Z. predicted the exceptional entanglement transition and conceived the experiment.

P.R.H. and X.J.H., supervised by Z.B.Y. and S.B.Z., carried out the experiment. F.W., P.R.H., J.W., and S.B.Z.

analyzed the data. S.B.Z., J.W., Z.B.Y., and W.Y. cowrote the paper. All authors contributed to interpretation of

observed phenomena and helped to improve presentation of the paper.

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摘要：

Exceptionalentanglementphenomena:non-Hermiticitymeetingnon-classicalityPei-RongHan1,∗FanWu1,∗Xin-JieHuang1,∗Huai-ZhiWu1,Chang-LingZou2,3,8,WeiYi2,3,8,MengzhenZhang4,HekangLi5,KaiXu5,6,8,DongningZheng5,6,8,HengFan5,6,8,JianmingWen7,†Zhen-BiaoYang1,8,‡andShi-BiaoZheng1,8§1FujianKeyLaboratoryofQuantumI...

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