--- title: Shifted Optimized Boson Basis url: https://www.emergentmind.com/topics/shifted-optimized-boson-basis type: topic --- # Shifted Optimized Boson Basis Searching arXiv for the cited papers and closely related terminology. arXiv search query: "shifted optimized boson basis spin-boson Dicke model extended bosonic coherent basis" Shifted Optimized Boson Basis denotes a class of bosonic truncation schemes in which the local bosonic Hilbert space is first displaced so that the basis is centered on the physically relevant mean oscillator excursion, and is then, when required, further compressed by an optimized local basis. In the finite-size Dicke model, the closely related extended bosonic coherent basis uses displaced Fock states to encode the macroscopic field displacement of the superradiant regime and thereby accelerate convergence in energy. In the spin-boson model, the shifted-OBB protocol combines a static displacement with an optimized boson basis inside a TEBD workflow, reducing the local dimension needed for accurate real-time dynamics [1312.1954], [2508.18622]. ## 1. Definition and conceptual basis The elementary construction begins from the Glauber displacement operator $$ D(\alpha)=\exp[\alpha a^\dagger-\alpha^* a], $$ which is unitary and implements $$ D^\dagger(\alpha)aD(\alpha)=a+\alpha,\qquad D^\dagger(\alpha)a^\dagger D(\alpha)=a^\dagger+\alpha^*. $$ From the ordinary Fock basis $\lvert n\rangle$, one defines displaced Fock states $$ \lvert n;\alpha\rangle \equiv D(\alpha)\lvert n\rangle. $$ Because the transformation is unitary, these states form an orthonormal basis for the single-mode boson Hilbert space: $$ \langle n;\alpha\vert m;\alpha\rangle=\delta_{nm}. $$ In the Dicke-model setting, this basis is also called an extended bosonic coherent-state basis. Its physical motivation is that the field acquires a macroscopic displacement in the superradiant regime, so a suitable $\alpha$ incorporates that displacement directly into the basis [1312.1954]. In the spin-boson chain formulation, the same principle is applied sitewise. For each chain site $k$, one defines the quadrature $$ \hat x_k\equiv(\hat b_k^\dagger+\hat b_k)/\sqrt{2}, $$ chooses a real shift $\alpha_k$ equal to the equilibrium expectation $\langle \hat x_k\rangle$ in the bath ground state with a fully polarized spin, and introduces $$ D_k(\alpha_k)\equiv \exp[\alpha_k(\hat b_k^\dagger-\hat b_k)], $$ so that $$ D_k^\dagger(\alpha_k)\hat b_k D_k(\alpha_k)=\hat b_k+\alpha_k\equiv \hat b_k'. $$ The 2025 formulation then applies an optimized boson basis step by diagonalizing a local density matrix and retaining a compressed local basis adapted to the shifted problem. This suggests that the phrase “shifted optimized boson basis” refers not merely to a displaced Fock basis, but to the combination of displacement and adaptive local truncation [2508.18622]. ## 2. Displaced representations in the finite-\(N\) Dicke model The Dicke Hamiltonian is $$ H_D=\omega a^\dagger a+\omega_0 J_z+\frac{2\gamma}{\sqrt{N}}(a+a^\dagger)J_x, $$ with $j=N_{\text{atoms}}/2$. Conjugating by $D(\alpha)$ gives the displaced Hamiltonian $$ H(\alpha)\equiv D^\dagger(\alpha)H_DD(\alpha), $$ and, for real $\alpha$, $$ H(\alpha)=\omega(a^\dagger a+\alpha(a+a^\dagger)+\alpha^2)+\omega_0J_z+\frac{2\gamma}{\sqrt N}\bigl((a+a^\dagger)+2\alpha\bigr)J_x. $$ After collecting terms, $$ H(\alpha)=\omega a^\dagger a+\frac{2\gamma}{\sqrt N}(a+a^\dagger)J_x+\left[\omega\alpha^2+\frac{4\gamma\alpha}{\sqrt N}J_x\right]+\omega_0J_z. $$ The $\omega\alpha^2$ term can be dropped or absorbed, while the term proportional to $J_x$ acts like an additional static field on the atoms [1312.1954]. Two strategies for choosing the shift are specified. One uses the exact shift in the integrable limit $\omega_0\to0$, namely $G\cdot m$, where $G=2\gamma/(\omega\sqrt N)$ and $m$ is an eigenvalue of $J_z$, so that a different shift is associated with each atomic-spin sector. The other treats $\alpha$ as a real variational parameter and, at each truncation $N$, minimizes the approximate ground-state energy $E_0^{(N)}(\alpha)$ with respect to $\alpha$. The second strategy is the direct antecedent of the “optimized” terminology in later shifted-basis work, although the 2025 spin-boson protocol implements optimization through local density-matrix compression rather than a single global variational displacement [1312.1954]. In the combined basis $\{\lvert n;\alpha\rangle\otimes \lvert j,m\rangle\}$, the nonzero bosonic matrix elements are $$ \langle n;\alpha\vert a^\dagger a\vert m;\alpha\rangle = m\,\delta_{n,m}, $$ and $$ \langle n;\alpha\vert a+a^\dagger\vert m;\alpha\rangle =\sqrt{m+1}\,\delta_{n,m+1}+\sqrt m\,\delta_{n,m-1}. $$ The atomic matrix elements are the usual $$ \langle j,m\vert J_z\vert j,m'\rangle=m\,\delta_{m,m'}, $$ and $$ \langle j,m\vert J_x\vert j,m'\rangle =\frac12\left[\sqrt{j(j+1)-m(m+1)}\,\delta_{m',m+1}+h.c.\right]. $$ This yields a sparse Hamiltonian matrix truncated to $n,n'=0,\dots,N_{\max}$, which is then diagonalized numerically [1312.1954]. ## 3. Convergence properties and efficiency gains For a bosonic truncation $n=0,\dots,N_{\max}$, one diagonalizes the Hamiltonian block of size $(N_{\max}+1)\times(2j+1)$ and defines the ground energy $E_0^{(N_{\max})}(\alpha)$. Convergence is monitored through $$ \Delta E(\alpha;N_{\max})\equiv \left|E_0^{(N_{\max}+1)}(\alpha)-E_0^{(N_{\max})}(\alpha)\right|\to0, $$ and a result is deemed converged to $\varepsilon$ when $\Delta E<\varepsilon$ [1312.1954]. The reported convergence pattern is strongly phase dependent. In the normal phase, $\gamma<\gamma_c=\sqrt{\omega\omega_0}/2$, the Fock basis and the shifted basis perform similarly. In the superradiant regime, $\gamma>\gamma_c$, the displaced basis typically achieves $\Delta E<10^{-6}$ with $N_{\max}\lesssim10\ldots20$, whereas the Fock basis may require $n_{\max}\gtrsim100$ for the same precision. If $\alpha$ is optimized variationally at each $N_{\max}$, convergence in $N_{\max}$ is accelerated further and is described as roughly exponential, $$ \Delta E(N_{\max})\sim C e^{-\kappa N_{\max}}, $$ with $\kappa\approx0.7$–$1.0$ in typical parameter regimes [1312.1954]. The practical efficiency claims are equally specific. In the strong-coupling or superradiant region $\gamma\gtrsim\gamma_c$, the required boson-truncation dimension in the shifted basis grows only slowly with $j$, and is reported even to decrease, whereas the Fock truncation grows roughly linearly in $N_{\text{atoms}}$ and quadratically in $\gamma$. Memory and CPU time then drop by orders of magnitude once $\gamma>\gamma_c$, making $j=20\ldots40$ fully feasible in the displaced basis. The best efficiency is reported for $\gamma/\gamma_c\gtrsim1$ and moderate $\omega_0/\omega\lesssim1$, although even out of resonance the shifted basis outperforms Fock so long as $\gamma$ is not vanishingly small [1312.1954]. A common misconception is that the advantage comes from altering the algebra of the boson mode. In fact, the algebra is unchanged; the gain comes from recentering the basis on the dominant coherent displacement so that the residual fluctuations are represented with fewer states. ## 4. Shifted-OBB construction for the spin-boson model The 2025 spin-boson formulation applies the same recentering principle locally along a bosonic chain and then compresses each shifted local Hilbert space by the optimized boson basis procedure of Guo et al. For site $k\ge1$, one starts from the shifted local Fock basis $\{\lvert n_k'\rangle\}$ of $\hat b_k'$ with occupation $n_k'=0,1,\dots,d_k-1$, constructs the MPS $A$-tensors in that uncompressed basis, and then performs the OBB step via local density-matrix diagonalization. The local tensor factorization is written as $$ (A^k[n_k])_{\alpha\beta} =\sum_{\ell=0}^{d_{\mathrm{opt}}-1}(\tilde A^k[\ell])_{\alpha\beta}V^k_{\ell,n_k}, $$ with $$ \lvert \ell\rangle \equiv \sum_{n_k=0}^{d_k-1}V^k_{\ell,n_k}\lvert n_k'\rangle, $$ where $d_{\mathrm{opt}}\ll d_k$ is chosen so that the discarded weight in the local density matrix is below some tolerance, typically $\sim10^{-8}$ [2508.18622]. The reason this compression is effective is explicit in the formulation. In a polarized bath, the low-frequency modes acquire a large coherent displacement $\langle \hat x_k\rangle\sim O(\alpha_k)$. In the unshifted Fock basis, one needs $d\gg \alpha_k^2$ to capture these large photon numbers. After shifting out the mean displacement, $\hat b_k'$ has zero mean and a much narrower variance, so a small $d_{\mathrm{opt}}$ already faithfully represents the distribution. In the published calculations, one usually takes $d_{\mathrm{opt}}=d$ with $d=3$–$10$ and bond dimension $D_c\simeq60$–$130$ [2508.18622]. The empirical benchmark given in the same work makes the reduction concrete. Monitoring the first local minimum of $\langle \sigma_z(t)\rangle$, denoted $\langle \sigma_z\rangle_{\min}$, the unshifted basis requires $d\approx50$ to match a benchmark curve, whereas in the shifted basis $d\approx10$ already suffices. By fitting $\langle \sigma_z\rangle_{\min}$ versus $d$ to a logarithm, one extracts an effective boson number $N_{\mathrm{eff}}$, and the shifted protocol with $d=10$ yields $N_{\mathrm{eff}}\approx53$ [2508.18622]. ## 5. TEBD implementation and observables The shifted-OBB method is embedded into TEBD by combining precomputed local shifts with standard two-site Trotter gates. The shifts $\alpha_k=\langle \hat x_k\rangle$ are first obtained from a static DMRG or static TEBD solution of the shifted problem with spin fully up. One then builds the initial MPS in the shifted basis with OBB matrices $V^k$. An optional extra, described as an “infinite” shift of strength $\varepsilon$, is applied by replacing $\alpha_k\to(1+\varepsilon)\alpha_k$ in the initial state so as to inflate the local boson number by a factor $(1+\varepsilon)^2$; in practice $\varepsilon=0.1$ is used [2508.18622]. Time evolution proceeds by decomposing the unshifted Hamiltonian as $H=\sum_k h_{k,k+1}$ and, on each even or odd bond, forming $$ g_{k,k+1}=\exp[-i\,h_{k,k+1}\Delta t] $$ in the unshifted basis, then transforming it to the shifted basis through $$ g'=(U_k\otimes U_{k+1})\,g\,(U_k\otimes U_{k+1})^\dagger. $$ The transformed gate is applied to the MPS tensors on sites $k,k+1$, followed by SVD, truncation of the bond dimension to $D_c$, and recompression of the local legs by OBB truncation to $d_{\mathrm{opt}}$. Repeating this on even and odd bonds completes one Trotter-Suzuki layer [2508.18622]. Accuracy is assessed by several observables. Dynamical fidelity is measured by comparing $\langle \sigma_z(t)\rangle$ against benchmark data or variational predictions. In the Ohmic case, long-time bath mode occupations $\langle n_p(t)\rangle$ as a function of frequency $\omega_p$ are inspected, and the peak position is required to approach the renormalized tunnel splitting $$ \Delta_r\sim \Delta(\Delta/\omega_c)^{\alpha/(1-\alpha)} $$ from the Silbey-Harris formula. The effective boson number $N_{\mathrm{eff}}$ is extracted by fitting $\langle \sigma_z\rangle_{\min}$ versus $d$ to a logarithmic form in the sub-Ohmic case or a power-law form in the super-Ohmic case. At fixed accuracy, shifted-OBB typically reduces $d$ by a factor $\sim5$–$10$, does not require increasing the bond dimension $D_c$, and leaves the most expensive gates as two-site objects of size $(d_{\mathrm{opt}}D_c)^2$ [2508.18622]. ## 6. Orthogonality, misconceptions, and model-dependent limitations Orthogonality depends on whether one uses a single global shift or sector-dependent shifts. With a single global $\alpha$, the displaced Fock states remain orthonormal and no overlap matrix or orthonormalization is needed. If one instead uses different $\alpha_m$ for each atomic-spin sector $m$, then overlaps $\langle n;\alpha_m\vert n';\alpha_{m'}\rangle\neq0$ arise for $m\neq m'$, and one must orthonormalize by a small generalized eigenvalue routine. In practice, a single optimized $\alpha$ is reported to suffice for the Dicke problem [1312.1954]. The spin-boson implementation is explicitly model dependent. In the sub-Ohmic regime $01$, the shifted initial bath reveals a new aperiodic pseudo-coherent phase at strong coupling, specified as $\alpha\gtrsim4$ for $s=3$. Because no simple benchmark exists there, a heuristic measure based on the oscillatory behavior of $d\langle \sigma_z\rangle/dt$ is introduced. The extra shift parameter $\varepsilon=0.1$ is empirical: larger $\varepsilon$ speeds up convergence but introduces errors in the initial state that must be checked against a smaller-$\varepsilon$ run [2508.18622]. These caveats delimit a common misunderstanding. Shifted optimized boson bases are not universal black-box truncations; their success depends on whether the dominant bosonic physics is a large mean displacement with comparatively compressible residual fluctuations. ## 7. Position within bosonic truncation strategies Across the two settings represented here, the shifted optimized boson basis can be understood as a displacement-centered alternative to raw Fock-space truncation. In the Dicke model, the crucial step is to encode the superradiant field displacement directly in the basis and, optionally, optimize the displacement variationally. In the spin-boson model, the crucial step is to determine sitewise equilibrium shifts and then compress the shifted basis dynamically by local density-matrix optimization inside TEBD [1312.1954], [2508.18622]. The unifying principle is that truncation should be performed around the physically occupied region of bosonic phase space rather than around the vacuum of the bare operators. In the Dicke setting, this yields rapid convergence of the ground-state energy and substantial savings in Hilbert-space dimension in the superradiant regime. In the spin-boson setting, it yields accurate polarization dynamics at significantly reduced computational cost and makes it possible to resolve long-time bath features and preparation-dependent dynamical phases within small local dimensions [2508.18622]. Within that broader perspective, “shifted optimized boson basis” names a family of methods rather than a single fixed algorithm: a static displacement may already be sufficient in some equilibrium problems, while nonequilibrium tensor-network simulations benefit from combining the shift with a dynamical optimized local basis. Source: https://www.emergentmind.com/topics/shifted-optimized-boson-basis