Squeezed Displaced Schrödinger-cat state as a signature of the PT-symmetry phase transition
Yuetao Chen Shoukang Chang Shaoyan Gao























 

gaosy@xjtu.edu.cn
MOE Key Laboratory for Nonequilibrium Synthesis and
	Modulation of Condensed Matter, Shaanxi Province Key Laboratory of Quantum
	Information and Quantum Optoelectronic Devices, School of Physics, Xi'an
	Jiaotong University, 710049, People's Republic of China.



















Parity-time (𝒫𝒯) symmetric systems are gain-loss systems whose dynamics are governed by non-Hermitian Hamiltonians with degeneracies at exceptional-points (EPs) and has been studied in various photonic, electrical, mechanical systems, and so on. However, it is still an open question how to capture 𝒫𝒯 symmetry phase transition in electronic system where the transport properties of electron will be dramatically effected. Fortunately, the hybridization between photon and electron offers a novel way not only to control but also probe material properties. Here, we investigate a cavity coupled to a non-Hermitian Su-Schrieffer-Heeger (SSH) chain within mean-field ansatzs. We find that Squeezed Displaced Schrodinger cat (SDSc) will emerge with high fidelity in cavity ground state when 𝒫𝒯-symmetry is broken and the fidelity will experience a sharp drop from almost 1 to 0 as 𝒫𝒯 symmetry recovers. Additionally, in semiclassical limit, we find that there exists local extrema at two sides of x=0 in semiclassical photon Hamiltonian H_eff(x,p), a clear signature of the emergence of SDSc state in cavity ground state. Thus, the appearance of SDSc state can be used to capture 𝒫𝒯-symmetry phase transition which can not be modified by cavity mode. Besides, we exploit the cavity ground state to estimate the phase in the optical interferometer, and show that the quantum Fisher information and nonclassicalty will sharply decline at EPs. This reveals that 𝒫𝒯-symmetry breaking in electronic materials can also be captured by the quantum Fisher information and nonclassicalty in phase estimation.  

Squeezed Displaced Schrödinger-cat state as a signature of the 𝒫𝒯-symmetry phase transition
    Shaoyan Gao
    April 29, 2024
===========================================================================================
















§ INTRODUCTION

The 𝒫𝒯-symmetry phase transition, as a subset of non-Hermiticity which represents a fundamental departure from conventional quantum mechanics [1–10], manifesting a transition from a real energy spectrum to a complex one. Intrinsic 𝒫𝒯-symmetry breaking spontaneously emerges at EPs where nonreciprocal hopping strength is equal to reciprocal one in a sublattices, and eigenstates and eigenvalues coincide [11–19]. 
The presence of EPs gives rise to rich physical phenomena, including enhancing precision in quantum sensors [20], topological phase transitions [21–25], nonadiabatic transitions [26,27], unidirectional light propagation [28], and energy transport at macroscopic [29], among others. 
In recent years, significant attention has been drawn to realize 𝒫𝒯-symmetry phase transition, with demonstrations in optical system [30–32], electrical system [33], and mechanical setting [34]. However, apart from the interest of realizing 𝒫𝒯-symmetry phase transition systems in various fields, probing 𝒫𝒯-symmetry breaking is still an open question, particularly in electronic systems, where such breaking can exert a significant influence on electron transport. 

Fortunately, quantum Floquet presents a novel way for probing, manipulating, and fine-tuning material properties, leading for instance to the emergence of polaritons—hybrid light-matter excitations—exhibiting non-trivial topological characteristics [35–41], or to an anomalous Hall response in the presence of a circularly polarized field [42]. As the quantum version of classical Floquet engineering, quantum Floquet has been proposed to control materials
through quantum light without detrimental heating [43]. The fundamental idea involves embedding a material within an optical cavity and amplify the light-matter interaction [44, 45] due to the inverse square-root relationship between the coupling strength and the effective mode volume [45, 46]. This amplification can be achieved, for instance, through near-field enhancement effects [47]. Through this enhancement of the coupling, one can achieve the aim of classical Floquet engineering with only quantum vacuum fluctuations or a few photons in cavity. To meet the theoretical requirements, ultra-strongly coupled light-matter systems have been achieved through various implementation schemes, beginning with initial findings utilizing microwave and optical cavities [48, 49] where the properties of electronic materials can be well manipulated. Apart from controlling materials, the interaction between cavity photons and material electron also has an effect on cavity ground state, leading for example to the emergence of squeezed vacuum state [50] and displaced squeezed vacuum state [51] where cavity Hamiltonian can be renormalised by hopping parameter in hermitian electronic materials within mean-field ansatzs. Naturally, a basic question to be asked here is what is the fate of the cavity ground state coupling to electron in the presence of 𝒫𝒯-symmetry phase transition in non-Hermitian electronic materials and whether one can capture 𝒫𝒯-symmetry phase transition sensitively by a probe of cavity ground state. 

In this paper, we address this fundamental question by investigating the interplay of electron and quantum light by considering the non-Hermitian electronic SSH chain coupled to a single mode cavity through the full gauge
invariant Peierls phase. Electronic energy spectrum and cavity ground state can be obtained by diagonalizing electronic Hamiltonian and photon Hamiltonian within mean-field ansatzs where there is no correlation between electron and photon. From view of electronic degree, we shows that quantum fluctuations of the light field have no effect on 𝒫𝒯-symmetry phase transition but exhibit dramatic effects on the topological properties of the system which can be probed by photonic spectrum. However, from the perspective of photon degree, the results show that Squeezed Displaced Schrödinger-cat (SDSc) state with high fieldity emerges as 𝒫𝒯-symmetry broken and then disappears immediately when 𝒫𝒯 symmetry recovers. This reveals that the emergence of SDSc state can be regarded as a signature of 𝒫𝒯-symmetry phase transition in electronic materials and is possible due to 
the appearance of local extrema at two sides of x=0 in semiclassical photon Hamiltonian H_eff(x,p).  

With the cavity ground state in our scheme, we also estimate the linear phase in the optical interferometer. We find that the quantum Fisher information (QFI) about the phase and the nonclassicality of cavity ground will experience dramatic drop at EPs. Thus, QFI and the nonclassicality of cavity ground can be also regarded as signatures of 𝒫𝒯-symmetry phase transition. 
 
This paper is organized as follows. In <ref>, we introduce
we describe the Hamiltonian of the light-matter coupled system. In <ref>, we show the effect of quantum vacuum fluctuation in cavity on electronic spectrum in open boundary condition. In <ref>, we  discuss about the generation of SDSc state as 𝒫𝒯-symmetry breaking and investigate the influence of 𝒫𝒯-symmetry phase transition on the fidelity between exact SDSc state and cavity ground state. Semiclassical photon Hamiltonian H_eff(x,p) will be studied for explaining the physics behind the emergence of SDSc state. In <ref>, we exploit the cavity ground state to estimate the phase of the optical interferometer, and study the QFI and nonclassicality of cavity ground state at two sides of EPs. Finally, <ref> presents the discussion and summary of the main findings in this work.




§ HAMILTONIAN


Model. We consider a 1D tight-binding non-Hermitian SSH model
coupled to a single mode cavity which is pictorially shown in Fig. 1. 

The above model describes spin-free electron nonreciprocal nearest-neighborhopping in a one-dimensional chain of L units with two sublattices A and B. 
The corresponding Hamiltonian is

    H_non-Hermitian-SSH =(v+γ/2)∑_j=1^Lc_j,A^†c_j,B+(v-γ/2)∑_j=1^Lc_j,B^†c_j,A
    -w∑_j=1^L-1c_j+1,A^†c_j,B-w∑_j=1^L-1c_j,B^†c_j+1,A,

where v(w) are lattice intracell (intercell) hopping strength and γ
is the nonreciprocal intracell hopping strength, meanwhile c_j,ζ^†(c_j,ζ) are the fermionic creation (annihilation)
operators at site j and sublattice ζ =A,B. a_0 and b_0 are
the lattice constant and intracell distance between A and B sites
correspondingly as shown in Fig. 1. The non-Hermitian SSH model displays
topological phase transitions (v=±√(w^2+(γ/2)^2))
and 𝒫𝒯-symmetry breaking phase transitions (v=±γ/2) in
the absence of any light-matter interaction. It's worth noting that the two
phase transitions mentioned above are independent of the intracell distance b_0 without cavity. The Hamiltonian of the single mode cavity which is
coupled to the non-Hermitian SSH model reads

    H_phot=ω _c(a^†a+1/2),

where ω _c is the mode frequency and the single cavity mode
annihilation (creation) operators are represented by a(a^†)
satisfying the Bose-Einstein commutation relation [ a,a^†
] =1. We couple the 1D non-Hermitian SSH model to the single cavity
mode by applying a unitary transformation U to electronic Hamiltonian[47],
i.e.


    H=H_phot+U^†H_non-Hermitian-SSHU,

where the unitary transformation U is defined as


    U=exp (ieA∑_j,ζr_j,ζc_j,ζ^†c_j,ζ),

where r_j,ζ represents the position of the sublattice, with r_j,A=ja_0 and r_j,B=ja_0+b_0. The full Hamiltonian under this
transformation can be written as (setting a_0=1)

    H=(ν+γ/2)e^ig_coupling/√(L)b_0(a+a^†)∑_j=1^Lc_j,A^†c_j,B
        +(ν-γ/2)e^-ig_coupling/√(L)b_0(a+a^†)∑_j=1^Lc_j,B^†c_j,A
        -we^ig_coupling/√(L)(1-b_0)(a+a^†)∑_i=1^L-1c_j+1,A^†c_j,B
        -we^-ig_coupling/√(L)(1-b_0)(a+a^†)∑_i=1^L-1c_j,B^†c_j+1,A
        +ω_c(a^†a+1/2),

where A=g_coupling/√(L)(a+a^†) are related to the quantized
electromagnetic vector potential with the convention e=ħ =c=1. It's
worth noting that the full light-matter interaction Hamiltonian obtained by
applying unitary transformation is equivalent to the Hamiltonian  under Peierls substitution[] where the hoppings amplitudes in Eq. (
<ref>) are dressed as v→ vexp (ieAb_0),γ→γexp (ieAb_0) and w→ wexp[ -ieA(a_0-b_0)
]. This phenomenon will play a key role in modifying the electron energy spectrum under open boundary conditions. Furthermore, the presence of non-Hermitian SSH chain will exert a discernible impact on the ground state of the cavity depending on the hopping parameters in non-Hermitian SSH chain. Strikingly,
we will show that the presence of nonreciprocal hopping in non-hermitian SSH
model can lead to the emergency of SDSc states with high
fidelity which can be regarded as a signature of the
𝒫𝒯-symmetry breaking phase transition in non-Hermitian SSH chain and have great potential for applications in quantum information
processing and quantum metrology.



§ THE OPEN-BOUNDARY ENERGY SPECTRUM OF NON-HERMITIAN SSH MODEL IN
	THE PRESENCE OF CAVITY


The finite-length energy spectrum of the non-Hermitian SSH model exhibits a
𝒫𝒯-symmetry-breaking phase transition in the non-Hermitian SSH model due to
the nonreciprocal hopping, manifested through the emergence of complex eigenenergy spectra. The other investigation we carry out is therefore what is the fate of the 𝒫𝒯 symmetry with a finite coupling to the cavity mode serving as the primary focus of our inquiry. Meanwhile, there exists a distinctive signature of its non-trivial topology, the presence of exponentially localized zero modes near the boundaries. Hence, we also investigate the behavior of the zero modes in the presence of cavity. To this extent we study the model within a mean field ansatz (see appendix A) assuming that there is no entanglement between the cavity ground modes |ϕ⟩ and electronic ground state |φ⟩. The renormalised non-Hermitian SSH Hamiltonian dressed by
cavity obtained within this ansatz reads

    H_non-Hermitian-SSH^mf =⟨ϕ| H|ϕ⟩
       =(v^'+γ ^'/2
    			)∑_j=1^Lc_j,A^†c_j,B+(v^'-γ ^'
    			/2)∑_j=1^Lc_j,B^†c_j,A
       -w^'∑_j=1^L-1c_j+1,A^†c_j,B-w^'∑_j=1^L-1c_j,B^†c_j+1,A,

where v^'=vξ (g_coupling,b_0), γ ^'=γξ (g_coupling,b_0)
and w^'=wξ (g_coupling,1-b_0) are the dressed hoppings strength by
cavity with the factor


    ξ (g_coupling,l)=⟨ϕ| e^-ig_coupling/√(L)
    		l(a+a^†)|ϕ⟩ .

Meanwhile, a renormalised photon Hamiltonian H_phot is obtained within
the mean field ansatz (see Methods for the specific expression). By
self-consistently solving the corresponding mean-field Hamiltonian, we
obtain the photonic ground state |ϕ⟩ and the
electronic ground state |φ⟩, and find that
the renormalization factor ξ (g_coupling,l) is only real due to the lack of a
finite current in the ground state of electron-photon coupling Hamiltonian.
It's worth noting that the numerical methods employed in obtaining the
ground state achieve a high level of accuracy, limited only by the cutoff of
the maximum photon number in the Fock space N_max^phot, and we
choose N_max^phot=60 for accuracy and simplicity.

The finite-length energy spectrum of the non-Hermitian SSH model can be
obtainde by orthogonalizing H_non-Hermitian-SSH and is plotted in Fig.
2(a)-(i) as functions of lattice intracell hopping amplitude v for
different values of the light-matter coupling strength g_coupling and intracell
distance b_0. As shown in Fig. 2(a), consistent with expectations for g_coupling=0, we observe a TPT when v≈± 1.2 (red dashed line). During
this transition, a pair of exponentially small almost zero modes emerge
within the central region of the gap in the bulk in Fig. 2(c). Meanwhile,
there exists 𝒫𝒯-symmetry phase transition at EPs (v≈± 0.6) which leads to emergence of imaginary energy spectrum as depicted
in Fig. 2(b). At 𝒫𝒯 symmetric region, the real part of energy spectrum in
Fig. 2(c) can captures the properties of non-Hermitian SSH model. Upon
considering light-matter coupling, the energy spectrum changes for different
intracell distance b_0. Especially, the TPT point(red dashed line) is
shifted to smaller values of v for b_0=0.2 as shown in Fig. 2(d),
revealing that the TPT of non-Hermitian SSH model can be modified by the
quantum fluctuation of the cavity mode. In additional, In Fig. 2(g), it is
evident that the TPT point(red dashed line) is pushed to larger values of v
when b_0=0.8, and zero modes edge states can appear over a wider value
range of v compared to the situation when b_0=0.2. This indicates that coupling to the quantum fluctuation of the light field can affect the TPT in the open boundary condition and this is consistent with the situation in the periodical boundary condition (see appendix B). In addition, the photonic spectral function's behavior which encoding how the light emitted from the cavity also show the TPT in the presence of cavity (see appendix C). However, unlike TPT point, EPs remain stable for different value of b_0 in Fig. 2(e) and Fig. 2(h), indicating the robustness of 𝒫𝒯-symmetry phase transition to
quantum fluctuation of cavity mode. However, the ground state of cavity will experience a sharp change when the 𝒫𝒯 symmetry of the non-Hermitian SSH model is broken, as we are going to discuss below.




§ THE GENERATION OF SQUEEZED DISPLACED SCHRÖDINGER-CAT STATES


As is known to all, the photon groundstate of a pure cavity is vaccum state
where there only exists vacuum fluctuations without any photon. However,
different photon groundstate will emerge by considering the interaction
between an one-dimension electronic chain and cavity mode. Hence, we
investigate the groundstate of the electron-dressed cavity and show that the
 SDSc states can be obtained with
high fidelity when there exists a 𝒫𝒯-symmetry phase transition in
non-Hermitian SSH chain. To this extent we study the full Hamiltonian within a mean
field ansatz assuming that there is no correlation between the photon
ground state |ϕ⟩ and electronic ground state |φ⟩. Thus, the photonic mean Hamiltonian
reads


    H_phot =⟨φ| H|φ⟩ 
    			=(v+γ/2)e^ig_coupling/√(L)b_0(a+a^†)D_AB^jj+(v-γ/2)e^-ig/√(L)
    				b_0(a+a^†)(D_AB^jj)^†
    
    			-we^ig_coupling/√(L)(1-b_0)(a+a^†)D_AB^j+1,j-we^-i
    				g_coupling/√(L)(1-b_0)(a+a^†)(D_AB^j+1,j)^†
    			+ω _c(a^†a+1/2),


where D_AB^jj= ∑_j=1^L⟨φ|
c_j,A^†c_j,B|φ⟩ and D_AB^j+1,j= ∑_j=1^L-1⟨φ|
c_j+1,A^†c_j,B|φ⟩ are the
hopping matrix element in the electronic ground state |φ⟩. By calculating the eigenvectors of the photonic Hamiltonian
H_phot, we find the photon ground state |ϕ⟩
. To see the quantum features of the photon ground states, we now calculate
the Wigner function. The Wigner function, as a phase-space quasiprobability
distribution [3, 55], is defined in the (q,p) position and momentum space
as

    W(q,p)=1/2π∫_-∞^∞dx⟨ q-x/2
    	|ρ| q+x/2⟩ e^ipx,

where ρ is the density matrix of the photon state |ϕ⟩.

As shown in Fig. 3(a) and Fig. 3(b), we plot the Wigner function of the
photon ground state |ϕ⟩ and the exact SDSc
state |ϕ ^SDSc⟩ =S(r)D(λ )(
	|α⟩ +| -α⟩/√(2)
) in the 𝒫𝒯-symmetry phase of non-Hermitian SSH model (v<
	γ/2). S(r)=exp [(ra^† 2-r^∗a^2)/2] is the
squeeze operator, D(λ )=exp (λ a^†-λ ^∗a)
is the displacement operator and |α⟩
	+| -α⟩/√(2) is the Schrödinger-cat
states. By comparing fig. 3(a) with fig. 3(b), we find that the phase-space
quasiprobability distribution of |ϕ⟩ and |ϕ ^SDSc⟩ are almost the same. Similar to Schr
öinger's cat paradox, the presence of two peaks in the Wigner function
in fig. 3(a) and fig. 3(b) corresponds to both the “dead
cat” and the “alive
cat” states, and the observation of negative values in the
Wigner function signifies the nonclassical feature in SDSc state. We define
the fidelity F between the photon ground (GS) states |ϕ⟩ in our system and the exact Squeezed Displaced Schrö
dinger-cat states state |ϕ ^SDSc⟩ as

    F=Tr[√(√(ρ __SDSc)ρ _GS√(ρ __SDSc))],

where ρ __SDSc(ρ __GS) is the density matrix of the photon
state |ϕ⟩ (|ϕ ^SDSc⟩ )
. As the fidelity F approaches 1, it indicates that the state generated
by our system closely resembles the ideal state, show a high degree of
similarity. After the numerical calculation, we find that the fidelity F
between the photon ground state |ϕ ^GS⟩ and
the exact SDSc state |ϕ ^SDSc⟩ is 0.994
revealing that the cavity ground state is steered into a SDSc state with
high similarity.



However, the fidelity F approaches 0 when the 𝒫𝒯 symmetry of
non-Hermitian SSH model recovers (v≥γ/2). We
showing this by plotting the fidelity F as functions of coupling strength g_coupling and the intracell distance b_0 for different values of
intracell hopping strength v in Fig. 4(a)-(c). Fig. 4(a) shows a high
fidelity in the area of large value b_0, with a region of low fidelity
in the remaining area of examined parameter when the 𝒫𝒯 symmetry in non-Hermitian SSH chain is broken. Futhermore, from Fig. 4(a) we find that
the SDSc state |ϕ ^SDSc⟩ with high fidelity
can be obtained with weaker coupling strength g_coupling as the value of b_0 increased. When the 𝒫𝒯 symmetry recovers (v≥γ/2
) the fidelity F approaches 0 although there exists maximum value of
the fidelity F in Fig. 4(a) and 4(b). This implies a dramatic change in
the ground state of the cavity |ϕ ^GS⟩ in the
presence of 𝒫𝒯 symmetry of non-Hermitian SSH chain. We further show this by
plotting the fidelity F in Fig. 4(d) as a function of intracell hopping
strength v at fixed values of coupling strength g_coupling and the
intracell distance b_0 depicted. It can be vividly seen from Fig. 4(d)
that the fidelity F approaches 1 as the 𝒫𝒯 symmetry is broken (v<¦Ã
/2), undergoes an abrupt change at the EPs (v=¦Ã /2), and
ultimately decreases to 0 with the recover of the 𝒫𝒯 symmetry. This implies that the emergency of the SDSc state can be regarded as a signature of the 𝒫𝒯-symmetry phase transition in the non-Hermitian SSH model. It is worth to note that the ground state of the cavity is no longer the SDSc state but a cavity state with the vanishing of two peaks and negative values in Wigner function with the recover of the 𝒫𝒯 symmetry as depicted in the insets and it is an exact displaced squeezed vacuum state in the thermodynamic limit (see appendix D). 




The cavity ground state corresponds to the state with the lowest eigenvalue
of the electron-dressed cavity Hamiltonian. By examining the distribution of
the Hamiltonian in phase space, we can gain insight into the physical
mechanism underlying the alteration of the cavity ground state's properties.
The cavity Hamiltonian can be conveniently described as a harmonic
oscillator, allowing us to represent it using dimensionless position and
momentum operators in the semiclassical limit [] x≡ (a^†+a)/2
and p≡ (a^†-a)/2i. Thus, the semiclassical limit of Eq. (<ref>) can be rewritten as

    H_eff(x,p) =⟨φ| H|φ⟩
       =(v+γ/2)e^ig_coupling/√(L)
    				b_02xD_AB^jj+(v-γ/2)e^-ig_coupling/√(L)
    				b_02x(D_AB^jj)^†
       -we^ig_coupling/√(L)(1-b_0)2xD_AB^j+1,j-we^-ig_coupling/√(L)
    				(1-b_0)2x(D_AB^j+1,j)^†
       +ω _c(x^2+p^2+1/2),

where (x,p)∈ R^2 are now continuous (nonquantized) classical
variables. Therefore, the model described by a 1D tight-binding
non-Hermitian SSH model coupled to a single mode cavity exhibits a
semiclassical analogy with a single effective degree of freedom, and its
phase space is denoted as M=R^2. 

Mean-field properties of the quantum
model Eq. (<ref>) such as, e.g., the ground-state energy, the photon
population can be captured by studying the distribution of energy of
semiclassical Hamitonian. Especially, the emergence of SDSC state |ϕ ^SDSc⟩ can be understood through the
structure of the classical phase space. In Fig. 5(a), 5(b) and 5(c), we show
the energy contours of Hamitonian Eq. (<ref>), while the slice figure (p=0) of energy are shown in Fig. 5(d), 5(e) and 5(f), accordingly. In Fig.
5(a), we observe that H_eff(x,p) has global extrema at nonzero values of x and p. The classical phase space structure undergoes a complete
transformation when 𝒫𝒯 symmetry is broken (v<γ /2), revealing the emergence of two minima corresponding to the ground-state energy. These minima are positioned at symmetric values relative to x=0 can be specifically shown in Fig. 5(d), and it makes the ground state of the cavity
exhibits a distribution in phase space similar to that of a SDSc statewhich is actually the two peaks in the Wigner function in Fig. 3(b).
However, as shown in Fig. 5(b) and 5(e) when v=γ /2, H_eff(x,p)
allows for a single, global minimum at x=p=0, corresponding to the
ground-state energy. Although the energy of H_eff(x,p) in Fig. 5(c) and
5(f) has local extrema at nonzero values of x and p, it has no
connection with the ground-state energy and has no effect on cavity ground
state. Therefore, one can see that phase-space quasiprobability distribution
of cavity ground state in the presence of 𝒫𝒯 symmetry has no two-peaks
compared to that of SDSc state due to the single, global minimum of energy
at x=p=0 when v≥γ /2.




§ PHASE ESTIMATION WITH CAVITY GROUND STATE


The ground state, as the lowest energy eigenstate of the system, can be
prepared as stable a quantum resource for quantum metrology without
excitating a system to high energy level. In non-Hermitian systems, the 𝒫𝒯 symmetry of the system may influence the ground state, as previously
discussed, thereby impacting the quality of the ground state as a quantum
resource. To this extend, we utilize the generated cavity ground states to
estimate the phase in the optical interferometer and investigate the Quantum
Fisher Information (QFI). Furthermore, the nonclassicality measured using an
operational resource theory (ORT)[] will be discussed.

As depicted in Fig. 6, the optical interferometer comprises two linear phase
shifters, incorporating an unknown relative phase Φ _-=(Φ _1-Φ
_2) between the arms. The phase information will be encoded within the
initial quantum state (cavity ground state |ϕ
^GS⟩) at the input of the interferometer through unitary
evolution, i.e.,

    |Ψ⟩ =U_phase-shift|Ψ
    	_0⟩ ,


where U_phase-shift=exp [i(Φ _1a_1^†a_1+Φ
_2a_2^†a_2)], and a_1,2^†(a_1,2) represent
annihilation (creation) operators in the arms 1 and 2. |Ψ
_0⟩ =|ϕ ^GS⟩⊗|ϕ ^GS⟩ is the input state of the interferometer. Here, we
concentrate on the relative phase between the two arms.



Considering a pure input state with path-symmetric, the unitary evolution
can be simplified to exp [iΦ _-(a_1^†a_1-a_2^†a_2)/2]. Then, the QFI of the unitary evolution of the initial state |Ψ _0⟩ can be defined as[]


    F_QFI=4⟨Ψ _0|Δ ^2H|Ψ
    	_0⟩ ,


where H=i(∂ _Φ _-U_phase-shift^†)U_phase-shift
and Δ ^2H=(H-⟨ H⟩ )^2. With the given
operator U_phase-shift expression, the QFI can be calculated as

    F_QFI=2[Var_Ψ _0(a_i^†a_i)-Cov_Ψ
    		_0(a_1^†a_1,a_2^†a_2)],i=1 or 2,

where Var_Ψ _0(· ) and Cov_Ψ _0(· ) are the
variance and the covariance in the input state |Ψ
_0⟩, correspondingly. It can be clearly seen from Eq. (<ref>) that QFI only depends on the properties of the input state (|ϕ ^GS⟩⊗|ϕ
^GS⟩ ). Considering the initial state here is separable Cov_Ψ _0(a_1^†a_1,a_2^†a_2)=0, thus the QFI
can be reduced to 2Var_Ψ _0(a_i^†a_i). In addition,
a nonclassicality measure for quantum state rooted in the framework of ORT
has been introduced, demonstrating various essential properties as a
valuable resource for phase estimation. For pure states, nonclassicality
measured based on ORT reads


    N_n=⟨ a^†a⟩ -|⟨
    	a⟩| ^2+|⟨ a⟩
    	-⟨ a⟩ ^2| .


In order to study the impact of 𝒫𝒯 symmetry on phase estimation utilizing
the cavity ground state as a quantum resource, we plot the QFI F_QFI and
the nonclassicality N_n as functions of intracell hopping stength v in
Fig. 7(a) and 7(b), accordingly. From Fig. 7(a), a clear observation can be
made that the value of QFI F_QFI is large as the 𝒫𝒯 symmetry is broken (v<¦Ã /2) which indicates that the cavity ground state is an
appropriate quantum resource for phase estimation. However, there exists a
sudden transition in value of F_QFI at the exceptional point (v=¦Ã
/2), and it suddenly become small as the 𝒫𝒯 symmetry recovers. Thus, the cavity ground state with 𝒫𝒯 symmetry in non-Hermitian SSH model is not
suitable for estimatin phase. From Fig. 7(b), it is evident that the value
of the nonclassicality N_n experiences a significant shift at the EPs. Simultaneously, the cavity ground state with broken 𝒫𝒯-symmetry
exhibits higher nonclassicality, whereas the cavity ground state will lose
some nonclassicality in the presence of 𝒫𝒯 symmetry. Therefore, the ground
state of the cavity can be viewed as a good quantum resource for phase
estimation when 𝒫𝒯 symmetry in non-Herimtian SSH model is broken. 

§ CONCLUSION


In this work, we have investigated a non-Hermitian SSH chain coupled to a single spatially constant cavity mode. The open-boundary energy spectrum shows that topological phase transition in the non-Hermitian SSH chain can be modified by the fluctuation of cavity mode and captured in the photonic spectral function's
behavior. However, 𝒫𝒯-symmetry phase transition can not be modified in the presence of cavity and is difficult to detect. Fortunately, the emergency of SDSc state with high fidelity can capture 𝒫𝒯-symmetry phase transition in non-Hermitian SSH model coupled to the quantum light field of a cavity. When 𝒫𝒯 symmetry is broken the fidelity between exact SDSc state and cavity ground state approaches to almost 1, and then experiences a sharp drop to 0 as 𝒫𝒯 symmetry recovers. The emergency of SDSc state can naturally be understood by inspecting the effective Hamiltonian H_eff(x,p) in semiclassical limit: H_eff(x,p) has local extrema at two sides of x=0 when 𝒫𝒯 symmetry is broken. Therefore, our work therefore provides a convenient way to capture the 𝒫𝒯-symmetry phase transition in an electronic system. 
Besides, we exploit the ground state of cavity to estimate the phase in an optical interferometer, and find that QFI F_QFI and the nonclassicality N_n of the ground state will experience a sharp drop at the 𝒫𝒯-symmetry phase transition point. This indicates it is beneficial for quantum metrology and quantum information processing when considering the 𝒫𝒯-symmetry is broken in electronic system coupled to a cavity. Therefore, except for the flexible control and convenient probe of topological properties of the electronic systems provided by cavity photon, the ground state of cavity photon can be used to capture the 𝒫𝒯-symmetry phase transition in electronic system. 




9
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	52



§  MEAN-FIELD METHOD TO DECOUPLE ELECTRONS AND PHOTONS IN FINITE-LENGTH NON-HERMITIAN SSH CHAIN IN THE PRESENCE OF CAVITY

We present a detailed analysis of the solution to the light-matter Hamiltonian H Eq. (<ref>) within the mean field approximation, which neglects correlations between cavity modes |ϕ⟩ and electrons |φ⟩.

    |Ψ⟩=|ψ⟩|ϕ⟩.

Due to the mean field decoupling, we are tasked with solving an electronic mean field Hamiltonian as described by Eq. (<ref>), alongside a photonic mean field Hamiltonian characterized expressed as Eq. (<ref>). We determine the electronic spectrum of the non-Hermitian SSH model in the presence of coupling to cavity photons through the self-consistent solution of Eq. (<ref>) and Eq. (<ref>). Specifically, we fix g = 0 in H_non-Hermitian-SSH^mf and numerically compute D_AB^jj= ∑_j=1^L⟨φ|
c_j,A^†c_j,B|φ⟩ and D_AB^j+1,j= ∑_j=1^L-1⟨φ|
c_j+1,A^†c_j,B|φ⟩. Then we insert
D_AB^jj and D_AB^j+1,j into H_phot in Eq. (<ref>) with  g ≠ 0, and evaluate v^', γ ^' and w^'. Subsequently, we insert v^', γ ^' and w^' (computed for (g ≠ 0)) into H_non-Hermitian-SSH^mf, then insert them into H_non-Hermitian-SSH^mf to calculate D_AB^jj and D_AB^j+1,j again, and repeat the procedure until convergence. We find that the energy spectrum of the non-Hermitian SSH chain is modified in presence of cavity as shown in Fig. 2.



§ TPT IN NON-BLOCH ENERGY SPECTRUM


In the electron lattices, the topological phase transition in the periodic
boundary condition would be associated with a closing and reopening of the
gap in the energy spectrum. Thus, we proceed to explore the energy
spectrum under
the periodic boundary condition, focusing on the occurrence of topological
phase transitions. Before this, we note that the usual Bloch waves carry a
pure phase factors e^ik where k is Bloch wave vector in Bloch
Brillouin zone. However, the role of the phase factors e^ik is now
played by β =√((v-γ /2)/(v+γ /2))e^ik which has a
modulus |β|≠ 1 in general. This allow us to
write the Hamitonian in non-Bloch Brillouin zone(see Methods). With the
non-Bloch ferminoic field ψ _β^†=(c_β ,A^†,c_β ,B^†), the renormalized Hamitonian under the periodic
boundary condition can be written in compact form as


    H=∑_βψ _β^†h(β )/iβψ
    	_β+ω _c(a^†a+1/2),

where the non-Bloch Hamitonian h(β ) has this form

	
    h(β )=[ e^-ig_coupling/√(L)b_0(a+a^†)(v-γ
    			/2)+e^ig_coupling/√(L)(1-b_0)(a+a^†)β w] σ
    			^-
    +[ e^ig_coupling/√(L)b_0(a+a^†)(v+γ /2)+e^-i
    				g_coupling/√(L)(1-b_0)(a+a^†)β ^-1w] σ ^+,


where σ ^±=(σ _x± iσ _y)/2 are pseudo-spin
up(down) operator. In fact, the Hamiltonian of such a bipartite lattice can
be written in the form of pseudo-spin components h(β
)=∑_i=x,y,zd_β iσ _i by considering periodic
boundary conditions. Here, at the first glance, it is apparent that the
non-Bloch Hamiltonian h(β ) in Eq. (<ref>) lacks the presence of an
electronic mass term, i.e. d_β z=0, indicating the preservation of
chiral symmetry in the presence of cavity, i.e. g≠ 0. This guarantees
the existence of eigenstate pairs with opposite energies.

By applying the mean-field method described in the previous section(see
Methods), we perform calculations of the bulk energy spectrum while
considering the interaction between electron and photons. Through the
self-consistent solution of the electronic and photonic mean-field
Hamiltonian, we observe that the influence of photons on the electronic
hoppings which is characterized by purely real renormalization values ξ
(g,l), so the electronic bulk energy spectrum reads



    ε =±√(R_+R_-)
    	=±√((v^')^2-(γ
    		^')^2/4+(w^')^2-2w^'cosk·√(
    			(v^'+γ ^'/2)(v^'-γ ^'/2))),



where R_±=(v^'±γ ^'/2)+β ^∓ 1w^' are the pseudo-spin up(down) components of the non-Bloch Hamitonian h(β ) with v^', γ ^' and w^' defined
before Eq. (<ref>). It is worth noting that ε is real at
outside the EP point (| v^'| >|γ
^'/2| ).

Solving the photonic and electronic Hamiltonian self consistently we find
numerically the Non-Bloch energy spectrum that we plot for three values of
coupling strength g in Fig. 8 (panel (a) g=5, panel (b) g=14 and
panel (c) g=20). 

It is clear that there exists eigenstate pairs with
opposite energies due to the preservation of chiral symmetry. Fig. 8(a) was
exhibited to demonstrate a frequency gap between two bands when g=5, but
Fig. 8(b) incidentally shows that the gap is closed at the center of the
Brillouin zone as g=14, where they touch at Dirac points. When g is
increased to 20, the gap is reopened with topological non-triviality which
can be seen in the trajectories of R_± as shown in Fig. 8(d), (e) and
(f). Figures 8(d) show that R_± have no wrapping to the origin and
thus the electron system is topological trivial as g=5. When g is
increased to 14, the trajectories of R_± touch the origin and TPT
happens. Then an anticlockwise and a clockwise wrappings to the origin are
formed by R_+ and R_-, respectively, and thus the electron system is
topological non-trivial as g increases across the value of 14. This
gives a geometric picture to the TPT in Fig. 8(a), (b) and (c).







§ PHOTONIC SPECTRUM WITH TOPOLOGICAL PHASE TRANSITION


The excitaion of electron in the non-Hermitian SSH system coupled to a
cavity mode acts as hybrid light-matter polariton quasiparticles whose
energies are affected by the topological phase transition in the electronic
system. Meanwhile, the change in the properties of the polartion
quasiparticles can be read out from a photonic spectrum which also unveils
pivotal indications of the emission of the light from cavity. Thus, the
topological property of the electronic system can be captured from the
photonic spectrum. Particularly, we specifically investigate the photon
spectral function A(ω )=-1/πIm∫ dte^-iω
	t(-iθ (t))⟨ [a(t),a^†]⟩ influenced
by the electronic system, taking into account 1/L Gaussian fluctuations.
By using photon Green's function method[] A(ω ) can be expressed as


    A(ω )=-1/πχ ^''(ω )(ω
    		+ω _c)^2/(ω ^2-ω _c^2-2ω _cχ ^'(ω ))^2+(2ω _cχ ^''(ω ))^2,

where χ (ω )=K(ω )-⟨ J_d⟩ is the
current-current correlation function comprises both paramagnetic and
diamagnetic contributions, and χ ^'(ω )(χ ^^''(ω )) are the real(imaginary) part of χ (ω )(see
Methods),

	
    K(ω ) =∫_-∞^∞⟨ T_cJ_p(τ
    			)J_p(0)⟩ e^iωτdτ ,  
    
    			J_p =g/√(L)∑_βψ _β^†(iβ ^-1wσ ^+-iβ wσ ^-)ψ _β, 
    
    			J_d =-g^2/√(L)∑_βψ _β^†(β wσ ^-+β ^-1wσ ^+)ψ _β,


where J_p(τ ) andJ_p(0) are the current operators in the
Heisenberg picture.

As depicted in Fig. 9, we illustrate the photonic spectral function's
behavior in the presence of cavity. As shown in Fig. 9(a), in the
topological trivial phase (v>√((w)^2+(γ )^2/4)), there only
exists one peak in the photonic spectral function in the absence of cavity (g_coupling=0), corresponding the cavity mode with frequency ω _c
as depicted by the black dashed line. As the coupling strength g_coupling
increasing, the peak splits into two branches indicating the appearance of
the two hybrid light-matter polartion modes and the cavity mode with the
frequency higher than 2E_k=π (white dashed line) can hybrid with the
electron as shown in Fig. 9(b). As we fix the hopping parameters such that v=√(
	(w)^2+(γ )^2/4), the lowest branch of the photonic spectral
function disappear revealing the absence of the polartion modes with the
lowest frequency. This is directly linked to the closure of the bulk gap as
depicted in Fig. 8(b), serving as a clear indication of the topological
phase transition. When  v<√((w)^2+(γ )^2/4), that is
topological non-trivial phase, the lowest branch of the photonic spectral
function reappear indicating the reemergence of the lowest frequency
polartion branch depicted in Fig. 9(c). Thus, the topological property of non-hermitian SSH chain can be probed by investigating the photonic spectral function's behavior in the presence of cavity. 


§ GENERATION OF DISPLACED SQUEEZED STATE 


In addition to being able to generate squeezed displaced Schrö
dinger-cat states, other photonic states can emerge in the system where
non-Hermitian SSH chain coupled to a single mode cavity. Therefore, we focus on the emergency of the displaced squeezed vacuum states. We expand H_phot in Eq. (<ref>) to second order in the field that gave the only non-vanishing contribution in the thermodynamic limit to the ground state energy. It reads


    H_phot^2nd=ω _c(a^†a+1/2)+g_coupling/√(L)
    	(a+a^†)Π -g_coupling^2/2L(a+a^†)^2Λ


where Λ =(v+γ/2)D_AB^jjb_0^2+(v-γ/2
)(D_AB^jj)^†b_0^2-wD_AB^j+1,j(1-b_0)^2-w(D_AB^j+1,j)^†(1-b_0)^2 and Π =i(v+γ/2)D_AB^jjb_0+i(v-
	γ/2)(D_AB^jj)^†b_0-iwD_AB^j+1,j(1-b_0)-iw(D_AB^j+1,j)^†(1-b_0). It can be diagonalized using a combined squeezing and displacement transformation


    H^D    =e^S^d[Λ,Π]e^S^sq[Λ]H^A,A^2e^-S^d[Λ,Π]e^-S^sq[Λ]
    
    			S^d[Λ,Π]    =g_coupling/√(L)ω_0(𝒲[Λ]/ω_0)^-3/2(a^†-a)Π,  
    
    			S^sq[Λ]    =1/4ln(𝒲[Λ]/ω_0)(a^2-(a^†)^2).


where 𝒲[Λ]=ω _c√(1-2g_coupling^2/Lω _cΛ). 
Thus, the
cavity ground state |ϕ ^GS⟩ is given by the
ground state of Hamitonian H^D which is a displaced squeezed vacuum state |ϕ ^GS⟩ ^D that is connected to the bare
cavity vacuum | 0⟩ through a combined squeezing and displacement transformation,


    |ϕ ^GS⟩ ^D=e^S^d[Λ,Π]e^S^sq[Λ]| 0⟩ ,

Therefore, we come to the fact that the cavity ground state is a displaced squeezed vacuum state in the thermodynamic limit when 𝒫𝒯 symmetry is not broken.



















































































