--- title: Quadratic Bosonic PT-Symmetric Systems url: https://www.emergentmind.com/topics/quadratic-bosonic-parity-time-symmetric-system-ptss type: topic --- # Quadratic Bosonic PT-Symmetric Systems Searching arXiv for the cited papers and closely related PT-symmetric bosonic classification work. Quadratic Bosonic Parity-Time-Symmetric System (PTSS) denotes a bosonic system whose dynamics is generated by a Hamiltonian quadratic in creation and annihilation operators and that is invariant under an appropriate parity-time operation. In the Hermitian Nambu/Bogoliubov setting, PT symmetry is induced by the coexistence of bosonic parity $P$ and time-reversal $T$ and constrains the excitation Hamiltonian through $U_{PT}\mathcal H^*(-k)U_{PT}^\dagger=\mathcal H(k)$; in open or effective non-Hermitian settings, PT symmetry is realized through balanced gain and loss or through quadrature-space drift matrices satisfying a PT condition. Across these settings, PTSSs are studied as stable bosonic BdG systems with excitation band gaps, as embedded PT-symmetric Bose–Einstein-condensate subsystems in closed Hermitian models, as quadrature-PT twin-beam amplifiers, and as photonic coupled dimers with flux-controlled PT phase transitions [1905.07989] [1503.01660] [2405.15612] [1809.10364]. ## 1. Operator formulations and bosonic dynamical structure For Hermitian quadratic bosonic systems with $N$ bosonic modes per unit cell, the standard formulation uses the Nambu spinor $$ \Psi(k)=(b_1(k),\ldots,b_N(k),b_1^\dagger(-k),\ldots,b_N^\dagger(-k))^T, $$ with quadratic Hamiltonian $$ H=\frac12\sum_k \Psi^\dagger(k)\mathcal H(k)\Psi(k), $$ where $\mathcal H(k)$ is a $2N\times 2N$ Hermitian matrix that is continuous in $k$ and semi-positive definite for stability. The canonical commutation relations are encoded by the symplectic metric $$ \Sigma=\tau=\operatorname{diag}(I_N,-I_N), $$ and the bosonic dynamical matrix is $D(k)=\Sigma\mathcal H(k)$. Excitation eigenmodes satisfy $D(k)\phi_n(k)=\omega_n(k)\phi_n(k)$, with paraunitary diagonalization implemented by a $k$-dependent matrix $T_B(k)$ obeying $T_B^\dagger(k)\Sigma T_B(k)=\Sigma$ and $T_B^\dagger(k)\mathcal H(k)T_B(k)=\Lambda(k)$, where $\Lambda(k)$ is positive diagonal. The excitation band gap is the topological obstruction, and physical normalization is imposed in the indefinite $\Sigma$-metric through $\phi_n^\dagger(k)\Sigma\phi_m(k)=\pm\delta_{nm}$ [1905.07989]. Open and effective non-Hermitian PTSSs are typically written in Heisenberg-Langevin or few-mode matrix form. In multimode bosonic systems one introduces $$ R=(a_1,a_1^\dagger,a_2,a_2^\dagger,\ldots,a_N,a_N^\dagger)^T, $$ and the dynamics is $$ \frac{dR}{dt}=MR+\xi(t), $$ with $M$ the drift matrix and $\xi(t)$ the Langevin forces. Linear beam-splitter couplings and parametric two-mode-squeezing couplings enter through canonical $2\times 2$ blocks, while damping and amplification are added through the drift matrix and diffusion rather than through a Hermitian Hamiltonian [2405.01667]. In continuous-variable twin-beam PTSSs, the space-evolution Hamiltonian density can take the quadratic form $$ H=\frac{i\hbar g}{2}(a_i^2-a_i^{\dagger 2})+\hbar\kappa(a_i a_s+a_i^\dagger a_s^\dagger), $$ which leads to block-diagonal quadrature drift matrices in the $\{q_i,p_s\}$ and $\{p_i,q_s\}$ subspaces [2405.15612]. A distinct operator realization appears in the Bargmann-Fock representation on $F^2(\mathbb C^2)$, where $a_j\to \partial/\partial z_j$ and $a_j^\dagger\to z_j$. Arindam Chakraborty studies the non-Hermitian quadratic boson operator $$ B(c_1,c_2,i\alpha,i\alpha)=c_1 z_1\partial_{z_1}+c_2 z_2\partial_{z_2}+i\alpha(z_1\partial_{z_2}+z_2\partial_{z_1}), $$ with $c_1,c_2,\alpha\in\mathbb R$, as a number-conserving quadratic model endowed with partial PT symmetry in Fock space [2206.01472]. ## 2. Symmetry operations: bosonic $P$, $T$, $C$, combined PT, and partial PT In the Hermitian Nambu framework, time-reversal $\mathcal T$, charge conjugation $\mathcal C$, and the composite symmetry $\mathcal P=\mathcal T\cdot\mathcal C$ are defined directly on bosonic operators and then re-expressed as constraints on $\mathcal H(k)$. With redefined operators $$ T=U_T^{-1}\mathcal K,\qquad C=U_C^{-1}\mathcal K,\qquad P=U_P^{-1}, $$ the symmetry constraints are $$ TH(k)T^{-1}=H(-k),\qquad CH(k)C^{-1}=H(-k),\qquad PH(k)P^{-1}=H(k). $$ A specifically bosonic feature is their action on the commutation metric: $$ T\tau T^{-1}=\tau,\qquad C\tau C^{-1}=-\tau,\qquad P\tau P^{-1}=-\tau. $$ Thus $T$ preserves $\tau$, while $C$ and $P$ flip the sign of $\tau$. The composite of $P$ and $T$ gives the bosonic PT constraint $$ U_{PT}\mathcal H^*(-k)U_{PT}^\dagger=\mathcal H(k). $$ This construction differs from the fermionic case because the composite $T\cdot C$ behaves as a parity symmetry $PH(k)P^{-1}=H(k)$ rather than as a chiral anticommuting symmetry [1905.07989]. In two-mode non-Hermitian dimers, parity and time reversal take their more familiar operational form. For modes $1$ and $2$, parity exchanges the modes, $$ P a_1P^{-1}=a_2,\qquad P a_2P^{-1}=a_1, $$ while time reversal complex conjugates coefficients and sends $i\mapsto -i$. In the basis $(\psi_1,\psi_2)$, PT acts as $$ (\psi_1,\psi_2)\mapsto (\psi_2^*,\psi_1^*). $$ The effective PT-symmetric dimer Hamiltonian $$ H_{PT}= \begin{pmatrix} \epsilon+i\gamma & -J\\ -J & \epsilon-i\gamma \end{pmatrix} $$ obeys $[H_{PT},PT]=0$, and in position space the corresponding condition is $V(x)=V^*(-x)$, which translates into an even real part and an odd imaginary part in the two-mode reduction [1503.01660]. The Fock-space formulation introduces partial PT symmetry. With weighted composition conjugations $W^{(j)}$ acting by $z_j\mapsto -z_j$ while leaving the other coordinate invariant, partial PT is defined by $$ W^{(j)}BW^{(j)}=B,\qquad j=1,2. $$ For the operator $B(c_1,c_2,i\alpha,i\alpha)$, partial PT symmetry holds, whereas global PT does not: $$ W_gBW_g\neq B. $$ At the same time, the operator is $W_g$-self-adjoint in the reproducing-kernel Hilbert-space sense, $$ W_g B^* W_g=B. $$ This provides a concrete distinction between PT invariance and pseudo-Hermiticity in the RKHS setting [2206.01472]. ## 3. Homotopy reduction and topological phases in Hermitian PTSSs A central structural result for stable Hermitian quadratic bosonic systems is that every such system is homotopic to a direct sum of two single-particle subsystems. Using paraunitary diagonalization $V^\dagger H V=\Lambda$ and the factorization $$ V=e^W(u_a\oplus u_b), $$ with $$ W=\begin{pmatrix}0&w\\ w^\dagger&0\end{pmatrix}, $$ one obtains $$ H(k)=e^{-W(k)}H_0(k)e^{-W(k)},\qquad H_0(k)=h_0(k)\oplus h_0'(k). $$ Exact reconstruction is given by $$ e^{2W}=H^{-1/2}[H^{1/2}\tau H\tau H^{1/2}]^{1/2}H^{-1/2},\qquad H_0=e^WHe^W, $$ and the homotopy $$ H_\epsilon(k)=e^{-\epsilon W(k)}H_0(k)e^{-\epsilon W(k)},\qquad \epsilon\in[0,1], $$ preserves the excitation spectra and symmetries. When only $T$ is present, $h_0$ and $h_0'$ are independent; when $C$ or $P$ is present, they are related by $$ u_c^{-1}h_0^*(-k)u_c=h_0'(k),\qquad u_p^{-1}h_0(k)u_p=h_0'(k), $$ so the topology is fully determined by $h_0(k)$ [1905.07989]. This homotopy reduces the bosonic classification to Altland-Zirnbauer classes carried by the effective subsystem $h_0(k)$. The relevant inherited single-particle classes are A, AI, and AII. In the paper’s setting, $d=1$ is trivial, $d=2$ supports $\mathbb Z$ phases derived from class A and $\mathbb Z_2$ phases of AII type, and $d=3$ supports AII-like $\mathbb Z_2$ topology. When $h_0$ and $h_0'$ are independent, the classification doubles to structures such as $\mathbb Z\oplus\mathbb Z$ or $\mathbb Z_2\oplus\mathbb Z_2$. The bosonic Berry connection is defined with the $\Sigma$-metric, $$ \mathcal A_n(k)=i\langle \phi_n(k)|\Sigma|\nabla_k\phi_n(k)\rangle, $$ with Berry curvature $\Omega_n(k)=\nabla_k\times \mathcal A_n(k)$ and Chern number $$ C_n=\frac{1}{2\pi}\int_{BZ} d^2k\,\Omega_n(k). $$ The paraunitary Berry connection obtained from the full $V(k)$ is continuously deformable to the subsystem Berry connection, so both yield identical invariants [1905.07989]. Concrete models make the PTSS content explicit. A two-dimensional bosonic BdG model in the paper’s class CI yields a Chern insulator after the $\Omega\to\infty$ homotopy trick, with $$ CN=\operatorname{sgn}[\operatorname{Re}(\xi^*m)] $$ for $0<|\operatorname{Re}(\xi^*m)|<2|\operatorname{Re}\xi|$, and $CN=0$ for $2|\operatorname{Re}(\xi^*m)|<|\operatorname{Re}(\xi^*m)|$; under open boundary conditions, two chiral edge modes appear inside the excitation gap. A second family, the paper’s class HI with $T^2=-I$ and both $C$ and $P$ present, realizes a PT-symmetric Hermitian bosonic system whose effective subsystem is the single-particle AII topological-insulator Hamiltonian. In $d=2,3$, the nontrivial phase occurs for $d-2<|m|d$ or $|m|0$, the pitchfork bifurcation shifts to smaller $\gamma$; and for $|g|>2v$ the pitchfork disappears and PT-broken states exist for all $\gamma$ [1503.01660]. The same paper shows that a PT-symmetric two-mode subsystem can be embedded into a fully closed Hermitian four-mode system. Two inner modes form subsystem $A$, while two outer modes act as coherent reservoirs. The full $4\times 4$ stationary matrix is Hermitian, with real tunnel couplings $v$ inside the subsystems and opposite-sign couplings $\pm\gamma$ between each inner well and its reservoir counterpart. Under the PT conditions $$ \psi_{j,1}=\psi_{j,2}^*,\quad j=A,B,\qquad \psi_{A,i}=\psi_{B,i}^*,\quad i=1,2, $$ together with inter-subsystem phase difference $\phi_{\rm rel}=0$, one has $\psi_B=\psi_A^*$ and the reservoir term acts as an effective imaginary on-site potential for subsystem $A$. The inner subsystem then obeys an effective non-Hermitian PT dimer Schrödinger equation with diagonal $\pm i\gamma$, even though the full four-mode Hamiltonian remains Hermitian [1503.01660]. The analytic PT-symmetric branches of the four-mode embedding are $$ \mu=-\frac g2\pm \sqrt{v^2+\gamma^2}, $$ which remain real for all $\gamma$. Current balance fixes the stationary phase relations. For the inner dimer, $$ J_{12}=2v\,\operatorname{Im}(\psi_1^*\psi_2), $$ and the exchange currents with the reservoirs are $$ J_{A1\leftrightarrow B1}=2\gamma\,\operatorname{Im}(\psi_{A,1}^*\psi_{B,1}),\qquad J_{A2\leftrightarrow B2}=2\gamma\,\operatorname{Im}(\psi_{A,2}^*\psi_{B,2}). $$ Stationarity requires these to balance. When $\phi_{\rm rel}\neq 0$, the reservoir contribution acquires a real component, PT symmetry is destroyed, and a cusp bifurcation replaces the pitchfork. The four-mode matrix model agrees qualitatively with one-dimensional Gross-Pitaevskii descriptions using both double-$\delta$ and smooth double-well potentials, and quantitative agreement is excellent when the wells are sufficiently isolated [1503.01660]. ## 5. Quadrature PT, exceptional points, and multimode non-Hermitian dynamics In continuous-variable PTSSs, PT symmetry can be formulated directly in quadrature space. For the type-II PSA-only twin-beam system, the quadratures $$ q_j=\frac{a_j^\dagger+a_j}{2},\qquad p_j=\frac{i(a_j^\dagger-a_j)}{2} $$ obey two independent first-order systems, $$ \frac{d}{dz}\begin{pmatrix} q_i\\ p_s\end{pmatrix} = \begin{pmatrix} g & \kappa\\ -\kappa & 0\end{pmatrix} \begin{pmatrix} q_i\\ p_s\end{pmatrix}, \qquad \frac{d}{dz}\begin{pmatrix} p_i\\ q_s\end{pmatrix} = \begin{pmatrix} -g & \kappa\\ -\kappa & 0\end{pmatrix} \begin{pmatrix} p_i\\ q_s\end{pmatrix}. $$ These are the active and passive blocks of a “dual opposing” quadrature PT symmetry. With $$ b=\frac{g}{2\kappa},\qquad \beta=\kappa\sqrt{1-b^2}, $$ the exceptional point is at $b=1$. For $b<1$, $\beta$ is real and the quadrature variances oscillate with PT-induced phase shifts $\pm\epsilon$, where $\epsilon=\tan^{-1}(2\beta/g)$; for $b>1$, periodicity ceases and one quadrature is steadily squeezed while its conjugate is anti-squeezed. The same framework yields dynamical and stationary classical-to-quantum transitions, exact cross-quadrature correlation coefficients, a covariance matrix $V_Q$, logarithmic negativity $$ E_N=\max[0,-\ln(4\eta)], $$ and anti-Hermiticity-enhanced sensing whose optimum lies in the unbroken-PT region away from the exceptional point; the passive PT quadratures outperform the active ones at the same parameters [2405.15612]. Multimode non-Hermitian bosonic PTSSs exhibit both conventional and “nonconventional” PT-symmetric dynamics. In the latter, PT-like behavior is confined to a subspace of the full Liouville/operator space. For the two-mode bidirectional dimer, the $4\times 4$ dynamical matrix has an $\mathrm{EP}_2$ threshold determined by $$ \zeta^2=\gamma_-^2 \quad\Longleftrightarrow\quad \frac{\kappa^2}{\epsilon^2}+\frac{\gamma_-^2}{\epsilon^2}=1, $$ with $\zeta=\sqrt{\epsilon^2-\kappa^2}$. Four-mode circular and tetrahedral configurations produce PT-like subspaces with equal imaginary parts and symmetric real-part splitting $\pm\beta$, and unidirectional block-triangular concatenation enables inherited exceptional degeneracies $\mathrm{EP}_4$, $\mathrm{EP}_5$, and $\mathrm{EP}_6$. The trade-off is physicality: bidirectional models with Markovian Gaussian noise preserve bosonic commutators for all times, but unidirectional models are intrinsically short-time and require $$ t\ll \min\{1/\gamma,1/\epsilon\} $$ for commutator deviations to remain small [2405.01667]. Photonic four-mode PTSSs supply a complementary exceptional-point geometry. In L. Jin’s coupled asymmetric dimers, the single-particle non-Hermitian Hamiltonian carries balanced gain and loss and a synthetic magnetic flux $\Phi$ through Peierls phases in the inter-dimer couplings. The eigenvalues are $$ E_{\pm,\pm}=\pm \sqrt{\Delta_1\pm \sqrt{\Delta_2}}, $$ with $$ \Delta_1=\kappa^2+J^2-\frac{\gamma^2}{2},\qquad \Delta_2=4J^2\kappa^2\cos^2\!\left(\frac{\Phi}{2}\right)-J^2\gamma^2+\frac{\gamma^4}{4}. $$ The system exhibits exact PT, partially broken PT, and completely broken PT phases; for intermediate $J$ it can reenter the exact PT phase as $\gamma$ increases. Two-state coalescences may have one or two defective eigenstates, phase-rigidity exponents are $\nu=1/2$ for isolated $\mathrm{EP}_2$, $\nu=1$ for coinciding $\mathrm{EP}_2$, and $\nu=3/4$ for the $\mathrm{EP}_4$ triple point, and these exponents are independent of $\Phi$. At the $\mathrm{EP}_4$, the perturbation-induced mode splitting scales as a fourth root, $\Delta\omega\propto \varepsilon^{1/4}$, which is the sensing signature of the four-state coalescence [1809.10364]. ## 6. Quantumness hierarchy, partial-PT variants, and scope of the concept A recent extension of quadratic bosonic PTSSs studies the hierarchy of quantum correlations in two-mode down-conversion models with matched coherent spectra but different damping and amplification patterns. The hierarchy includes global nonclassicality, entanglement, asymmetric quantum steering, and Bell nonlocality. Three variants are compared: the standard PTSS with balanced gain and loss, the passive PTSS with solely damping, and the active PTSS with solely amplification. They share identical eigenvectors and the same real parts of eigenfrequencies, but their noise statistics differ because damping contributes $l_j=2\gamma_j,\tilde l_j=0$, whereas amplification contributes $l_j=0,\tilde l_j=2|\gamma_j|$. The result is a systematic ordering in quantumness generation: the passive PTSS yields the most strongly nonclassical states, the standard PTSS is typically weaker than its passive and active counterparts, and Bell nonlocality is robust only in broad passive parameter regions. Under suitable conditions, $$ \epsilon^2-\kappa^2\gtrsim \gamma^2,\qquad \kappa/\epsilon\to 1,\qquad \gamma/\epsilon\ll 1, $$ the standard PTSS can still generate highly nonclassical states [2510.06171]. Partial PT symmetry defines another branch of the subject. In the Bargmann-Fock model, the right eigenfunctions of $B(c_1,c_2,i\alpha,i\alpha)$ in fixed-degree homogeneous polynomial subspaces have alternating real and purely imaginary coefficients when the eigenvalue is real, and then satisfy partial PT symmetry or antisymmetry under $W^{(j)}$. The spectral condition is controlled by the deformation parameter $\alpha$: for $|\alpha|<1$ the spectrum is real, at $\alpha=1$ all eigenvalues coalesce to zero as an exceptional point, and for $\alpha>1$ the spectrum becomes purely imaginary and partial PT symmetry of the eigenfunctions is lost. This setting is explicitly tied to bi-orthogonal systems in $\mathbb C^2$, a deformed $\mathfrak{su}(2)$ algebra, and RKHS-based pseudo-Hermiticity [2206.01472]. The literature therefore uses PTSS in more than one technically distinct sense. In the Hermitian quadratic-boson setting, PT symmetry is a symmetry constraint on a stable, gapped excitation problem with real spectrum, paraunitary diagonalization, and topological classification inherited from Altland-Zirnbauer classes. In non-Hermitian gain/loss settings, PT symmetry is a constraint on effective Hamiltonians or drift matrices and is associated with exceptional points, PT-unbroken and PT-broken phases, bifurcations, squeezing, entanglement, and sensing. A common misconception is that bosonic PT symmetry is exhausted by non-Hermitian gain/loss physics. The Hermitian classification program shows instead that PT symmetry can organize bosonic band topology without any gain/loss interpretation, while the same papers also emphasize that not every pseudo-Hermitian system with complex spectrum arises from a bosonic quadratic Hamiltonian [1905.07989]. The unifying feature across these variants is not a single universal normal form but a shared quadratic bosonic structure together with parity-time constraints adapted to the chosen representation. In Nambu space this leads to $\Sigma$-metric geometry, paraunitary transformations, and inherited topological invariants; in few-mode and continuous-variable systems it leads to gain-loss balance, phase constraints, and exceptional-point physics; in Fock-space partial PT models it leads to bi-orthogonal spectral theory and deformation-controlled symmetry breaking. Source: https://www.emergentmind.com/topics/quadratic-bosonic-parity-time-symmetric-system-ptss