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

The parity-time($\mathcal{PT}$) symmetry governs a fascinating class of non-Hermitian Hamiltonians whose energy spectra can be purely real and bounded belowBender , 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 $\mathcal{PT}$-breaking transition occurs and the spectra after the transition become complexbook . The single-particle $\mathcal{PT}$-symmetric Hamiltonian and the associated breaking transitions have been successfully explored earlier in various photonic, electronic and acoustic systems (see reviewsreview1 ; review2 ), and recently also in the quantum walk interferometerQW_PT , superconducting circuitSC_PT , nitrogen-vacancy centerNV_PT , trapped ionsion_PT1 ; ion_PT2 and ultracold gasesLuo ; Gadway ; Jo .

Given the intriguing single-particle property of $\mathcal{PT}$-symmetry, its interplay with interaction has become a rapidly developing research frontier in recent yearsWunner ; Konotop ; Mott1 ; Ueda ; Das ; Pan ; Yu ; Zhou ; Littlewood ; Littlewood2 ; Mott2 ; Yi ; Zhang ; Scheurer . Previous studies have focused on the interacting $\mathcal{PT}$-symmetric bosons and fermions based on mean-field analysesWunner ; Konotop ; Das ; Zhou ; Zhang ; Pan ; Yu , and the phase transitions and critical phenomena near EPsMott1 ; Ueda ; Littlewood ; Littlewood2 ; Mott2 . To date, little has been said about beyond-mean-field effects with $\mathcal{PT}$-symmetry, and whether such effects can generate equally significant quantum phenomena far from EPs. Answering these questions will help to capture the very intrinsic physics in the interplay of interaction and $\mathcal{PT}$-symmetry, which will in turn shed light on related quantum phenomena in a much broader context.

In this work, we explore the effects of quantum fluctuations on top of a $\mathcal{PT}$-symmetric two-species($\uparrow,\downarrow$) Bose-Einstein Condensate(BEC) with the non-Hermitian potential

where $\sigma_{\alpha}$($\alpha=x,y,z$) are the Pauli matrices. Obviously $V_{\rm PT}$ commutes with the $\mathcal{PT}$ operator, with $\mathcal{P}$ flipping the spin($\uparrow\leftrightarrow\downarrow$) and $\mathcal{T}$ changing $i$ to $-i$. In the single-particle level, the physics of $V_{\rm PT}$ has been well studied in literaturereview1 ; review2 ; QW_PT ; SC_PT ; NV_PT ; ion_PT1 ; ion_PT2 ; Luo ; Gadway ; Jo and the $\mathcal{PT}$-symmetry is preserved for $\gamma<1$. 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 $\mathcal{PT}$-symmetry, though preserved by the condensate, can be spontaneously broken by the Bogoliubov quasi-particles. The $\mathcal{PT}$-breaking transition in the Bogoliubov spectrum can be conveniently tuned by the strength of $V_{\rm PT}$ and the interaction anisotropy in spin channels.

(II) The quasi-particle in the $\mathcal{PT}$-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 $V_{\rm PT}$ 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 $\mathcal{PT}$-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.

We consider the following Hamiltonian for the interacting two-species bosons under the $\mathcal{PT}$-symmetric potential (we take $\hbar=1$ throughout the paper)

Here $\alpha,\beta=\{\uparrow,\downarrow\}$, and $\{\Psi^{{\dagger}}_{\alpha},\Psi_{\alpha}\}$ are the field operators of spin-$\alpha$ bosons. In order to ensure the $\mathcal{PT}$-symmetry of (2), we take the equal mass $m_{\uparrow}=m_{\downarrow}\equiv m$ and equal intra-species coupling $g_{\uparrow\uparrow}=g_{\downarrow\downarrow}\equiv g$. In this case, the property of a homogeneous BEC is determined by three dimensionless parameters: the dissipation parameter $\gamma$, and two dimensionless combinations $\eta\equiv g_{\uparrow\downarrow}/g$ and $\tilde{\Omega}\equiv\Omega/(gn)$ ($n$ the total density of the BEC).

To determine the mean-field ground state, we examine the total mean-field energy $E_{\rm mf}=_{L}\langle\Psi_{0}|H|\Psi_{0}\rangle_{R}$. It is found that under the phase constraint (15), $E_{\rm mf}$ solely depends on the parameter $x\equiv N_{-}/N$, where $N=N_{-}+N_{-}$ is the total number. Explicitly, the energy per particle $\epsilon_{\rm mf}\equiv E_{\rm mf}/N$ reads

For simplicity, in this work we will focus on the $\eta<1$ regime, where the minimum of $\epsilon_{\rm mf}(x)$ locates at $x=1$, i.e., the bosons condense at the lower branch with energy

Accordingly, we can obtain the chemical potential $\mu\equiv\partial E_{\rm mf}/\partial N$ and further the compressibility $\chi\equiv\partial n/\partial\mu$ as

The mean-field stability against density fluctuations would require $\chi>0$ and therefore

This condition is more stringent than the Hermitian case ($\eta>-1$). In other words, a non-Hermitian BEC (with finite $\gamma$) can undergo mean-field collapse more easily than its Hermitian counterpart ($\gamma=0$). This will be responsible for the $\gamma$-induced droplet formation as discussed later.

Given the $\mathcal{PT}$-symmetric BEC at ${\mathbf{k}}=0$ and $\nu=-$, we now study its elementary excitations due to quantum fluctuations. Following the standard Bogoliubov approach, we assume $\Psi^{{\dagger}}_{\nu;{\mathbf{k}},R}$ and $\Psi_{\nu;{\mathbf{k}},L}$ (except for $\{\nu=-,\ {\mathbf{k}}=0\}$) are all small fluctuation operators and only keep in the Hamiltonian all the bi-linear terms of these operators, which gives $H=N\epsilon_{\rm mf}+H_{\rm BG}$ with

Here $H_{\rm BG}$ naturally inherits $\mathcal{PT}$-symmetry from the full Hamiltonian (2), since we have taken the condensate (14) as $\mathcal{PT}$-symmetric. The first line in $H_{\rm BG}$ 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 ($-\leftrightarrow+$), and the coupling constant $u^{\prime\prime}$ 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 $\sum^{\prime}$ implies the summation be taken over half of ${\mathbf{k}}$-space to avoid the double counting; the vectors are

and the matrix ${\cal M}$ is

We aim to diagonalize $H_{\rm BG}$ as the following form:

where $E_{i{\mathbf{k}}}$ are the four eigen-modes for Bogoliubov quasi-particles, and the two eigen-vectors are

The eigen-operators are required to satisfy the commutation relation

To find out eigen-spectra $E_{i{\mathbf{k}}}$ as well as the relation between $\tilde{F}_{\mathbf{k}},\tilde{G}_{\mathbf{k}}$ and $F_{\mathbf{k}},G_{\mathbf{k}}$, 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 $G_{\mathbf{k}}$ and $\tilde{G}_{\mathbf{k}}$:

This implies that by diagonalizing the matrix ${\rm Diag}(1,-1,1,-1){\cal M}({\mathbf{k}})$, we can obtain the four Bogoliubov modes from its eigen-energies. Explicitly, by introducing a transformation matrix ${\cal A}$ in $G_{\mathbf{k}}={\cal A}\tilde{G}_{\mathbf{k}}$, we have

Similarly, we can write down the EoM for $F_{\mathbf{k}}$ and $\tilde{F}_{\mathbf{k}}$, and by introducing a transformation matrix ${\cal B}$ in $F^{T}_{\mathbf{k}}=\tilde{F}^{T}_{\mathbf{k}}{\cal B}$, we have

Therefore, the Bogoliubov modes can also be obtained by diagonalizing the matrix ${\cal M}({\mathbf{k}}){\rm Diag}(-1,1,-1,1)$.

The two diagonalization schemes, i.e., one is based on (75) and the other is based on (85), produce the same solution of $E_{i{\mathbf{k}}}$, which satisfy

with

The four eigen-modes in (86) fall into two identical pairs, and we choose $E_{1{\mathbf{k}}}=E_{2{\mathbf{k}}}$ and $E_{3{\mathbf{k}}}=E_{4{\mathbf{k}}}$. This is also a natural choice since in non-interacting limit, ${\cal M}({\mathbf{k}})$ can exactly reduce to the diagonal matrix ${\rm Diag}(E_{1{\mathbf{k}}},E_{2{\mathbf{k}}},E_{3{\mathbf{k}}},E_{4{\mathbf{k}}})$ with $E_{1{\mathbf{k}}}=E_{2{\mathbf{k}}}$ and $E_{3{\mathbf{k}}}=E_{4{\mathbf{k}}}$.

In fact, based on the commutation relations (4) and (55), we can find out the relation between the two transformation matrixes:

and then one can prove straightforwardly that ${\cal B}{\cal M}({\mathbf{k}}){\cal A}={\rm Diag}(E_{1{\mathbf{k}}},E_{2{\mathbf{k}}},E_{3{\mathbf{k}}},E_{4{\mathbf{k}}})$. It follows that the first term in Eq.(26) is equal to the first term in Eq.(45). Therefore the constant terms in (26) and (45) are also identical. Now we can rewrite Eq.(45) 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 ${\mathbf{k}}$ and the ultraviolet divergence can be avoided.

In this section, we present the result of Bogoliubov excitation spectrum for the $\mathcal{PT}$-symmetric BEC. Since $E_{1{\mathbf{k}}}=E_{2{\mathbf{k}}}$ and $E_{3{\mathbf{k}}}=E_{4{\mathbf{k}}}$, we will only show the results of $E_{1{\mathbf{k}}}$ and $E_{3{\mathbf{k}}}$.

To highlight the effect of non-Hermiticity to Bogoliubov excitations, we first go through the Hermitian case ($\gamma=0$). In this case, all odd-parity terms in (25) are absent ($u^{\prime\prime}=0$) and the fluctuations in $+$ and $-$ branches are well decoupled. This leads to a gapless spectrum $E_{1{\mathbf{k}}}=\sqrt{({\mathbf{k}}^{2}/2m)^{2}+2\mu_{-}{\mathbf{k}}^{2}/(2m)}$ and a gapped one $E_{3{\mathbf{k}}}=\sqrt{({\mathbf{k}}^{2}/2m+2\Omega)^{2}+2\mu_{+}({\mathbf{k}}^{2}/2m+2\Omega)}$, with $\mu_{\pm}=gn(1\mp\eta)/2$. Clearly, in the mean-field collapse regime with $\eta<-1$, the lower spectrum $E_{1{\mathbf{k}}}$ becomes purely imaginary near ${\mathbf{k}}\sim 0$, signifying the dynamical instability. In addition, we note that under certain condition the two spectra become degenerate, i.e., $E_{1{\mathbf{k}}_{0}}=E_{3{\mathbf{k}}_{0}}$ at:

The according plot is given in Fig.2 (a). This feature will lead to interesting excitation property when turn on $\gamma$.

In the presence of non-Hermiticity($\gamma\neq 0$), the inter-branch fluctuations give two important impacts on the Bogoliubov modes, namely, the spontaneous $\mathcal{PT}$-symmetry breaking and the gapped excitation, as detailed below.

The fact that the non-Hermiticity $\gamma$ 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.(98) gives ${\cal E}_{\rm LHY}\equiv E_{\rm LHY}/V$ as: