Quantum fluctuations on top of a $\mathcal{PT}$-symmetric Bose-Einstein Condensate

We investigate the effects of quantum fluctuations in a parity-time($\mathcal{PT}$) symmetric two-species Bose-Einstein Condensate(BEC). It is found that the $\mathcal{PT}$-symmetry, though preserved by the macroscopic condensate, can be spontaneously broken by its Bogoliubov quasi-particles under quantum fluctuations. The associated $\mathcal{PT}$-breaking transitions in the Bogoliubov spectrum can be conveniently tuned by the interaction anisotropy in spin channels and the strength of $\mathcal{PT}$ potential. In the $\mathcal{PT}$-unbroken regime, the real Bogoliubov modes are generally gapped, in contrast to the gapless phonon mode in Hermitian case. Moreover, the presence of $\mathcal{PT}$ potential is found to enhance the mean-field collapse and thereby intrigue the droplet formation after incorporating the repulsive force from quantum fluctuations. These remarkable interplay effects of $\mathcal{PT}$-symmetry and interaction can be directly probed in cold atoms experiments, which shed light on related quantum phenomena in general $\mathcal{PT}$-symmetric systems.


I. INTRODUCTION
The parity-time(PT ) symmetry governs a fascinating class of non-Hermitian Hamiltonians whose energy spectra can be purely real and bounded below [1], analogous to the Hermitian ones. Nevertheless, very different from the Hermitian counterpart, their eigenstates are generally non-orthogonal and can even coalesce at exceptional points(EPs), where the PT -breaking transition occurs and the spectra after the transition become complex [2]. The single-particle PT -symmetric Hamiltonian and the associated breaking transitions have been successfully explored earlier in various photonic, electronic and acoustic systems (see reviews [3,4]), and recently also in the quantum walk interferometer [5], superconducting circuit [6], nitrogen-vacancy center [7], trapped ions [8,9] and ultracold gases [10][11][12].
In this work, we explore the effects of quantum fluctuations on top of a PT -symmetric two-species(↑, ↓) Bose-Einstein Condensate(BEC) with the non-Hermitian po- * xlcui@iphy.ac.cn tential where σ α (α = x, y, z) are the Pauli matrices. Obviously V PT commutes with the PT operator, with P flipping the spin(↑↔↓) and T changing i to −i. In the singleparticle level, the physics of V PT has been well studied in literature [3][4][5][6][7][8][9][10][11][12] and the PT -symmetry is preserved for γ < 1. Here we show that when turn on boson-boson interactions, quantum fluctuations can significantlly affect the elementary excitation of the system even far from the single-particle EP. Specifically, our main findings are listed as below: (I) The PT -symmetry, though preserved by the condensate, can be spontaneously broken by the Bogoliubov quasi-particles. The PT -breaking transition in the Bogoliubov spectrum can be conveniently tuned by the strength of V PT and the interaction anisotropy in spin channels.
(II) The quasi-particle in the PT -unbroken regime is generally gapped, on contrary to the gapless mode in the Hermitian case. Moreover, the mean-field instability of a non-Hermitian system does not necessarily lead to imaginary excitations therein.
(III) The presence of V PT can enhance the mean-field collapse of the BEC, and thereby extend the droplet formation to a broader interaction regime than the Hermitian counterpart.
The experimental relevance of our results and the implication to a general PT -symmetric system will also be discussed in this paper.
The rest of the paper is organized as follows. In section II we present the basic model of the system, including the single-particle physics and mean-field treatment. The followed section III is contributed to the mean-field ground state. In section IV, we build up a systematic theory for the Bogoliubov analysis of the non-Hermitian BEC. The resulted excitation spectrum and the droplet properties are presented, respectively, in section V and VI. Finally, we discuss the experimental relevance of our results in section VII and summarize the whole work in section VIII.

II. MODEL
We consider the following Hamiltonian for the interacting two-species bosons under the PT -symmetric potential (we take = 1 throughout the paper) Here α, β = {↑, ↓}, and {Ψ † α , Ψ α } are the field operators of spin-α bosons. In order to ensure the PT -symmetry of (2), we take the equal mass m ↑ = m ↓ ≡ m and equal intra-species coupling g ↑↑ = g ↓↓ ≡ g. In this case, the property of a homogeneous BEC is determined by three dimensionless parameters: the dissipation parameter γ, and two dimensionless combinations η ≡ g ↑↓ /g andΩ ≡ Ω/(gn) (n the total density of the BEC).

A. Single-particle physics
The non-interacting part of (2) can be diagonalized as where ν = {+, −} is the index of single-particle eigenstate with eigen-energy ν;k = k 2 /(2m) + νΩ 1 − γ 2 ; Ψ † ν;k,R (Ψ ν;k,L ) is the associated creation (annilation) operator of the right (left) eigenstate, which satisfies the commutation relation This relation is equivalent to the bi-orthogonality of right and left eigen-states, which is crucially important for building the theories of non-Hermitian BECs as presented later.
Since V PT decouples from the kinetic term, the right/left eigenstates can be decoupled as where |0 is the vacuum, |k is the plane-wave state with momentum k, and |ν R/L is the spin part of the eigenstate that is solely determined by V PT . Specifically, the right and left eigenstates are defined through the Schrödinger equations: with ν = νΩ 1 − γ 2 . In the regime γ < 1, |ν R/L can be expressed as with the parameter Here C ν,R , C ν,L are all normalization factors. For the Hermitian case (γ = 0 and u = 1), we can see that the right and left eigenvectors become identical, i.e., |+ R ∼ |+ L , |− R ∼ |− L , and different levels are orthogonal to each other R,L −|+ R,L = 0. In comparison, for the non-Hermitian case (γ = 0 and u is complex), these relations are no longer satisfied, i.e., |ν R = |ν L and R −|+ R = 0, L −|+ L = 0. However, given the definition of right/left eigenstates in Eq.6, the bi-orthogonality can be satisfied as long as + = − : Therefore, the normalization can be carried out between the right and left eigenvectors: which gives Note that Eqs.(9,10) guarantees the commutation relation (4). In this work, we choose a specific gauge such that the normalization factors are all real and identical: In this way, when the PT operator acts on these eigenvectors, we have This demonstrates that, in the γ < 1 regime, |ν R/L are both the eigenstates of PT -operator with eigenvalue νu * . If one chooses a different gauge other than (12), the eigenvalues in (13) will be changed. However, we have checked that the gauge choice will not affect the physical quantities studied in this work, given that (11) is satisfied.
B. Mean-field treatment of the PT -symmetric BEC In the mean-field framework, we can write down a general coherent ansatz for the right state of the BEC: Here N ν and θ ν are respectively the mean number and the phase of the condensate at level ν. In the regime γ < 1, since the single-particle state Ψ † ν;k,R/L |0 preserves the PT -symmetry, it is natural to require the condensate (14) equally preserve such symmetry. Given Eq. (13), this requirement leads to e 2iθν = νu * .
Following the same strategy, we can obtain the left state of the BEC, |Ψ 0 L , which shares the same form as (14) except replacing Ψ † ν;k=0,R by Ψ † ν;k=0,L . Given the commutation relation (4) and the coherent ansatz (14), we can obtain the following expectation values under the bi-orthogonal basis: This shows that in the mean-field framework under the bi-orthogonal basis, one can replace the zero-momentum operators Ψ † ν;k=0,R and Ψ ν;k=0,L with their mean values: In this way, we can go on to study the mean-field ground state and examine the effects of quantum fluctuations on top of it.
Here we would like to emphasize that the mean-field treatment is only valid under the bi-orthogonal basis, but not if only use one of the basis (right or left). For instance, we cannot obtain the expectation values as the form in Eq. (16) if only under the right basis ( R ... R ) or the left basis ( L ... L ), and as a result we cannot replace the operators by their according mean-field values as in (17).

C. Interaction channels
To facilitate later discussions, we rewrite the interaction part of (2) in the following form: with Here u − , u + and u + are the coupling constants for evenparity channels, and u represents the coupling for oddparity ones. u is non-zero and purely imaginary only for the non-Hermitian case with spin-independent interaction, i.e., when γ = 0 and η = 1. As shown later, the presence of these odd-parity channels will greatly affect the elementary excitation of the BEC.

III. MEAN-FIELD GROUND STATE
To determine the mean-field ground state, we examine the total mean-field energy E mf = L Ψ 0 |H|Ψ 0 R . It is found that under the phase constraint (15), E mf solely depends on the parameter x ≡ N − /N , where N = N − +N − is the total number. Explicitly, the energy per particle For simplicity, in this work we will focus on the η < 1 regime, where the minimum of mf (x) locates at x = 1, i.e., the bosons condense at the lower branch with energy Accordingly, we can obtain the chemical potential µ ≡ ∂E mf /∂N and further the compressibility χ ≡ ∂n/∂µ as The mean-field stability against density fluctuations would require χ > 0 and therefore This condition is more stringent than the Hermitian case (η > −1). In other words, a non-Hermitian BEC (with finite γ) can undergo mean-field collapse more easily than its Hermitian counterpart (γ = 0). This will be responsible for the γ-induced droplet formation as discussed later.

IV. BOGOLIUBOV ANALYSIS
Given the PT -symmetric BEC at k = 0 and ν = −, we now study its elementary excitations due to quantum fluctuations. Following the standard Bogoliubov approach, we assume Ψ † ν;k,R and Ψ ν;k,L (except for {ν = −, k = 0}) are all small fluctuation operators and only keep in the Hamiltonian all the bi-linear terms of these operators, which gives H = N mf + H BG with Here H BG naturally inherits PT -symmetry from the full Hamiltonian (2), since we have taken the condensate (14) as PT -symmetric. The first line in H BG is reduced from even-parity channels, and the second line from odd-parity ones. Obviously, the effect of odd-parity channels is to couple fluctuations in different branches (− ↔ +), and the coupling constant u is purely imaginary in the presence of both non-Hermiticity and interaction anisotropy.
To facilitate the diagonalization of the bilinear Hamiltonian (25), we rewrite it as where implies the summation be taken over half of k-space to avoid the double counting; the vectors are (27) and the matrix M is We aim to diagonalize H BG as the following form: where E ik are the four eigen-modes for Bogoliubov quasi-particles, and the two eigen-vectors arẽ The eigen-operators are required to satisfy the commutation relation To find out eigen-spectra E ik as well as the relation betweenF k ,G k and F k , G k , we start from the equation of motions(EoM) of these vectors. Based on the Heisenberg equation for non-Hermitian system (see derivation in Appendix A), we can write down the EoM of G k and G k : This implies that by diagonalizing the matrix Diag(1, −1, 1, −1)M(k), we can obtain the four Bogoliubov modes from its eigen-energies. Explicitly, by introducing a transformation matrix A in G k = AG k , we have Similarly, we can write down the EoM for F k andF k , and by introducing a transformation matrix B in Therefore, the Bogoliubov modes can also be obtained by diagonalizing the matrix M(k)Diag(−1, 1, −1, 1). The two diagonalization schemes, i.e., one is based on (33) and the other is based on (34), produce the same solution of E ik , which satisfy The four eigen-modes in (35) fall into two identical pairs, and we choose E 1k = E 2k and E 3k = E 4k . This is also a natural choice since in non-interacting limit, M(k) can exactly reduce to the diagonal matrix In fact, based on the commutation relations (4) and (31), we can find out the relation between the two trans-formation matrixes: and then one can prove straightforwardly that BM(k)A = Diag(E 1k , E 2k , E 3k , E 4k ). It follows that the first term in Eq.(26) is equal to the first term in Eq. (29). Therefore the constant terms in (26) and (29) are also identical. Now we can rewrite Eq. (29) as Further incorporating the regularization of bare couplings g and g 12 from the mean-field interaction energy (22), we can obtain the Lee-Huang-Yang(LHY) energy as: We have checked that the summation in above equation converges at large k and the ultraviolet divergence can be avoided.

V. EXCITATION SPECTRUM
In this section, we present the result of Bogoliubov excitation spectrum for the PT -symmetric BEC. Since E 1k = E 2k and E 3k = E 4k , we will only show the results of E 1k and E 3k .
To highlight the effect of non-Hermiticity to Bogoliubov excitations, we first go through the Hermitian case (γ = 0). In this case, all odd-parity terms in (25) are absent (u = 0) and the fluctuations in + and − branches are well decoupled. This leads to a gapless spectrum E 1k = (k 2 /2m) 2 + 2µ − k 2 /(2m) and a gapped one E 3k = (k 2 /2m + 2Ω) 2 + 2µ + (k 2 /2m + 2Ω), with µ ± = gn(1 ∓ η)/2. Clearly, in the mean-field collapse regime with η < −1, the lower spectrum E 1k becomes purely imaginary near k ∼ 0, signifying the dynamical instability. In addition, we note that under certain condition the two spectra become degenerate, i.e., E 1k0 = E 3k0 at: The according plot is given in Fig.2 (a). This feature will lead to interesting excitation property when turn on γ. In the presence of non-Hermiticity(γ = 0), the inter-branch fluctuations give two important impacts on the Bogoliubov modes, namely, the spontaneous PTsymmetry breaking and the gapped excitation, as detailed below.

A. Spontaneous PT -symmetry breaking
Although PT -symmetry is preserved by H, H BG and the condensate |Ψ 0 R,L , it can be spontaneously broken by the Bogoliubov quasi-particles, as manifested by the appearance of complex E ik . The PT -broken region in kspace sensitively depends on parametersΩ, η and γ. In Fig.1, we have divided (γ, η) plane into different regions (I-VI) according to different PT -breaking properties in the Bogoliubov spectra for a fixedΩ = 0.2. The complex spectra occur in regions III-VI.
Let us start from region III with a small γ and 2Ω < η < 1 (satisfying the condition in (40)). In this case, a finite γ will lead to the PT -breaking of excitation spectra near k 0 . As shown in Fig.2(b1,b2), for γ = 0.15, E 1k and E 3k are complex and conjugate to each other within a finite window |k| ≡ k ∈ (k 1 , k 2 ). Thus, as increasing k from zero, the PT -symmetry breaks at k 1 and then revives at k 2 . The critical boundaries k 1 , k 2 , which are determined by the solutions to b 2 k − 4c k = 0, sensitively depend on γ, see Fig.2(c). At small γ, we find that k 1,2 deviate from k 0 by a small shift δ ≡ |k − k 0 |, with Herek 0 = k 0 / √ 2mgn. As shown by the dashed lines in Fig.2(c), the dominant linear shifts based on above equation fit well to k 1,2 in small γ limit.
Continuously increasing γ, k 2 and k 1 respectively flow to ∞ and 0 at γ ∞ and γ 0 . This tells that the spectra at large k become complex if γ > γ ∞ , and the complex spectra extend to k = 0 if γ > γ 0 . Numerically, γ 0 is determined by satisfying b 2 k=0 = 4c k=0 . To find out γ ∞ accurately, we expand the function F k ≡ b 2 k − 4c k at large k → ∞ and only keep its leading order ∼ k 4 . Then γ ∞ is determined by the coefficient of this leading term crossing zero, which gives the equation: We can see that the above equation support a solution γ ∞ = 0 at η = 2Ω. This is also consistent with Eq. (40), which tells that the degenerate point k 0 goes to ∞ in the Hermitian case if η = 2Ω.
In Fig.1, γ 0 and γ ∞ are plotted as functions of η, and accordingly regions III-IV are separated. Specifically, the PT -breaking of Bogoliubov modes occur within a finite k-window in III (with γ < γ 0 , γ ∞ ), at large k in IV (γ ∞ < γ < γ 0 ), at small k in V (γ 0 < γ < γ ∞ ), and extend the whole k-space in VI(γ > γ 0 , γ ∞ ). The typical spectra in regions IV and V are given in Fig.2(d,e). Therefore, the PT -breaking transition takes place twice in III, once in IV and V, and no transition in VI. This shows that the PT -symmetry of Bogoliubov modes can be conveniently tuned by γ and η.

B. Gapped excitation
In the PT -unbroken region, such as II in Fig.1, the real Bogoliubov modes are gapped, instead of gapless as in Hermitian case. For γ 1, we find that the excitation gap scales linearly with γ: Such a gapped spectrum is in distinct contrast to the gapless mode in the Hermition BEC. It is closely related to the presence of imaginary odd-parity terms in (25), such as Ψ † +;k,R Ψ −;k,L which directly couple the condensed atoms at − -branch with higher + -branch crossing a finite energy gap. Such coupling takes no effect for a PT -symmetric BEC in the mean-field level but plays an important role in its quantum fluctuations. Because such imaginary coupling only exists for γ = 0 and η = 1, the quasi-particle is gapped in the same regime (see (43)). In Fig.3, we extract the energy gap as a function of γ for two typical η, which fit well to (43) in small γ regime.
Interestingly, the gapped excitation appears not only in the mean-field stable regime (region II(b)), but can also extend to the collapse regime (II(a)). This is in distinct contrast to the Hermitian case where the low-k spectrum is purely imaginary in the mean-field collapse side. It is to say, the mean-field instability in non-Hermitian system does not necessarily lead to imaginary excitations. In fact, for a given η < −1, the excitation spectra can turn from purely imaginary to purely real as increasing γ across a critical γ c , at which point the spectrum is gapless E 1,k=0 = 0. In Fig.1, we mark the γ < γ c region ('I') as shaded area, where the low-energy excitation spectra are purely imaginary. Numerically, γ c is determined by c k=0 = 0, and thus We can see that γ c = 0 when η = −1, reproducing the mean-field collapse point for the Hermitian case. When η slightly deviates from −1, we have which shows that γ c scales as the square root of the deviation, as displayed by the orange dash-dot line in Fig.1.

VI. γ-INDUCED DROPLET
The fact that the non-Hermiticity γ enhances the mean-field collapse (as inferred by Eq.(24)) renders the formation of a self-bound droplet after incorporating the LHY correction from quantum fluctuations. In general, Eq.(39) gives E LHY ≡ E LHY /V as: where f is a dimensionless functional. In Fig.4, we show the contour plot of f in (γ, η) plane given a fixedΩ = 0.2.
We can see that f , or equivalently E LHY , decreases continuously as γ increases and can even turn negative. Fortunately, in region II(a), which is the mean-field collapse regime with real and gapped spectra, the LHY force is always repulsive. A self-bound droplet state can then be supported in this region with zero pressure, i.e., ∂(E/N )/∂n = 0, with E = E mf + E LHY . This gives the equilibrium density of the droplet as At small particle number N , the quantum pressure becomes important and drives the droplet to gas transition. To estimate the critical number N c at the transition, we take the similar strategy as in Ref. [27] and write down the extended Gross-Pitaevskii(GP) equation as where Ψ(r) is the wave function of the BEC and the particle number is determined by N = d 3 r|Ψ(r)| 2 . By rescaling r, Ψ, t through we can reduce the GP equation to It is found that Eq.51 shares the same structure as the reduced GP equation in Hermitian case [27], which leads to the rescaled critical numberÑ c ≡ d 3r |Ψ| 2 = 18.65 at the vanishing of droplet solution (droplet-gas transition). Given the scaling relation in (49), we can obtain the critical N c = n eq ξ 3Ñ c in our system as We can see that both n eq and N c can be conveniently tuned by γ and η.
For the PT -symmetric potential (1), the σ x term can be implemented through the two-photon microwave and rf transition [31], and iσ z can be realized using the laserinduced state-selective dissipation up to a constant loss term iΩγ [10][11][12]. For realistic atomic system with such constant loss, the overall number of the system decays with time. However, the physics governed by the effective non-Hermitian Hamiltonian can still be probed under the post-selection scheme, as have been successfully explored in the non-interacting atomic gases [10,12]. The validity of the effective non-Hermitian Hamiltonian requires a short-time dynamics within timescale t 1/(Ωγ), where the impact of quantum jump can be neglected. As here we consider the weak coupling regime with na 3 1, which is a natural extension of and can be smoothly connected to the non-interacting regime, we do not expect the validity of the effective non-Hermitian Hamiltonian would alter too much. Moreover, it should be noted that the existing experiments on quantum droplet have exactly made use of the atom loss to observe the dropletgas transition [32][33][34][35][36]. We thus expect that the γ-induced droplet can be directly probed in realistic experiments.
The property of excitation spectrum can be explored by the Bragg spectroscopy as implemented previously in various cold atoms systems [37][38][39][40]. Since such spectroscopy detects the linear response of the system to external perturbations, we expect it can directly probe the excitation spectrum of non-Hermitian system as predicted in this work. Our results, which are directly relevant to atomic gases confined in a uniform trap [41][42][43][44][45], can also be utilized for the trapped system under local density approximation, as successfully implemented in previous experiments [37,38,40].

VIII. SUMMARY AND DISCUSSION
In summary, we have revealed the ground state and excitation properties of a PT -symmetric BEC, including the spontaneous PT -breaking and gapped spectrum for Bogoliubov quasi-particles, and the enhanced mean-field collapse and the facilitated droplet formation. These results show that the quantum fluctuations on top of a PT -symmetric BEC can lead to important and visible collective phenomena even far from the single-particle EPs, thus demonstrating the significant interplay of interaction and non-Hermiticity in bosonic system.
Finally, we point out that the intriguing excitation properties revealed in this work can be traced back to the fundamental character of non-Hermitian systems, i.e., the non-orthogonality of eigenstates. Such nonorthogonality character covers both the single-particle states and the elementary quasi-particles. This is why the PT -symmetry breaking can also occur in the latter. We thus expect the phenomena revealed here are not limited to the specific PT -potential considered in this work, but applicable to a broad class of non-Hermitian systems with PT -symmetry. Indeed, a recent study has pointed out the spontaneous PT -breaking of elementary excitations on top of a fermion superfluid [25]. These PTbreaking phenomena are generally associated with the collective many-body EP and may lead to giant fluctuation effect [22,23]. In future, it is worth to explore the impact of collective EPs in the quantum and thermal depletions, as well as the property of BEC in other parameter regime (γ, η,Ω) beyond the scope of this work.