Displaced Oscillator Model

• Displaced Oscillator Model is a quantum system with a shifted harmonic potential that alters energy eigenstates, spectral properties, and state dynamics.
• It finds applications in quantum optics, condensed matter, and decoherence studies, enabling controlled tunneling and state engineering in double-well and photonic lattice setups.
• The model’s analytical tractability through closed-form solutions and explicit secular equations makes it a valuable benchmark for testing numerical, perturbative, and semiclassical methods.

The displaced oscillator model encompasses a range of quantum systems where the canonical harmonic oscillator potential or its energy eigenstates are subject to displacement in position or phase space. Such displacements yield modified algebraic, spectral, and dynamical properties with high relevance in quantum mechanics, quantum optics, and condensed matter. The mathematical and physical consequences of these displacements depend on the context: the static double-well potential with tunable minima, the phase-space displacement of Fock states in bosonic modes, or the driven oscillator under intrinsic decoherence. Displaced oscillator models play a critical role as exactly or quasi-exactly solvable benchmarks and as prototypes for quantum state engineering, decoherence studies, and photonic simulations.

1. Double-Well Displaced Oscillator: Potential and Spectral Structure

The archetypal displaced oscillator potential is defined as

$V(x; d) = \min[(x-d)^2, (x+d)^2]$

where $d\geq 0$ is the displacement parameter. For $d=0$, the system reduces to a single harmonic well $V(x)=x^2$. For $d>0$, the potential displays two symmetric minima at $x=\pm d$ separated by a barrier of height $V(0)=d^2$, interpolating between a single-well and a fully decoupled double-well as $d\to\infty$ (Znojil, 2016).

The stationary Schrödinger equation for this model reads:

$-\psi''(x) + V(x; d)\psi(x) = E \psi(x)$

The potential is piecewise analytic with a cusp at $x=0$, requiring solutions on each half-line matched by the continuity of $\psi$ and $\psi'$ at the origin.

Key spectral and analytic features include:

• For special displacements $d$, the model admits finite sets of closed-form quasi-exactly solvable (QES) states, constructed as exponentials multiplied by degree-$N$ polynomials. These solutions exist when the recurrence relations on the polynomial coefficients truncate, yielding energies $E=2N+1$ and $d$ determined by algebraic QES equations.
• For generic $d$, non-QES states are analytically accessible as "non-polynomially exactly solvable" (NES): each half-line solution is built from confluent hypergeometric functions, with the global spectrum fixed by matching conditions at $x=0$.

Physical insights include:

• As $d$ increases, low-lying energy levels split into nearly degenerate pairs due to tunneling, with the gap decreasing exponentially $\Delta E \sim \exp(-d^2)$.
• The model realizes an analytically tractable cusp catastrophe, providing a quantum mechanical analog of the bifurcation in Ginzburg-Landau potentials.
• For all $d$, wavefunctions and spectra are available in closed form or via explicit transcendental equations, cementing this model as a powerful benchmark for analytic, perturbative, and numerical methods (Znojil, 2016).

2. Phase Space Displacement: Displaced Fock States and Hamiltonian Picture

In the algebraic (operator) context, the displaced oscillator Hamiltonian is

$H = \omega a^\dagger a + \lambda(a + a^\dagger)$

where $\omega$ is the oscillator frequency, $\lambda$ is the displacement strength, and $a$, $a^\dagger$ are bosonic ladder operators. The displacement operator $D(\alpha) = \exp(\alpha a^\dagger - \alpha^* a)$ with $\alpha=-\lambda/\omega$ diagonalizes $H$ up to an additive constant. The displaced Fock states are defined as $|n,\alpha\rangle = D(\alpha) |n\rangle$ and serve as the eigenmodes of the displaced Hamiltonian (Urzúa et al., 2021, Keil et al., 2011).

Displacement in phase space is realized in quantum optics and waveguide lattices. In the Glauber-Fock photonic lattice, the tight-binding Hamiltonian with couplings $C_{n+1}=C_1\sqrt{n+1}$ produces dynamics identical to the application of $D(iC_1z)$ to an initial Fock state $|k\rangle$, mapping the evolution coordinate $z$ to displacement magnitude along the imaginary axis in phase space. The correspondence between real-space waveguide index and oscillator excitation number facilitates direct emulation of displaced Fock state dynamics in classical photonic systems (Keil et al., 2011).

This framework underpins the study of nonclassical correlations, quantum walks, and phase-space engineering. Displaced Fock states and their associated dynamics are foundational in quantum state manipulation, optical coherent control, and benchmarking of quantum information protocols.

3. Quasi-Exact and Non-Exact Solvability in the Displaced Oscillator

For the double-well displaced oscillator, QES states arise at isolated values of $d$ where the ansatz

$\psi_N(x) = e^{-x^2/2 + dx} P_N(x)$

yields a finite polynomial $P_N$. The polynomial coefficients satisfy explicit recurrence relations, terminating when $a_{N+1}=0$. This yields a finite algebraic condition for $d$ (the QES polynomial equation). Parity conditions at $x=0$ (even: $\psi(0)=1,\,\psi'(0)=0$; odd: $\psi(0)=0,\,\psi'(0)=1$) further constrain allowable solutions. Explicit examples include:

• $N=0$ ($E=1$): trivial Hermite solution at $d=0$,
• $N=1$ ($E=3$): even closed-form state at $d=\pm 1$,
• $N=2$ ($E=5$): odd QES states at $d=\pm 1/\sqrt{2}$, even QES at $d=\pm \sqrt{5/2}$.

For generic (non-QES) $d$, solutions on each half-line require confluent hypergeometric functions:

• On $x>0$: $\psi^+(x;E) = e^{-z/2} U(a,b;z)$ with $z=(x+d)^2$,
• On $x<0$: $\psi^-(x;E) = \pm e^{-z/2} U(a,b;z)$ with $z=(-x+d)^2$, where $U(a,b;z)$ is the Tricomi function, $a=(1-E)/4$, $b=1/2$, and matching at $x=0$ yields secular equations for the spectrum. This distinction between algebraic solvability and transcendental exact solvability captures the richness of the displaced oscillator's spectral theory (Znojil, 2016).

4. Dynamical Evolution and Decoherence in the Displaced Oscillator

The driven oscillator subjected to Milburn’s intrinsic decoherence evolves according to

$$\frac{d\rho}{dt} = \gamma\left(e^{-iH/\gamma} \rho\, e^{iH/\gamma} - \rho\right)$$

where $\gamma$ is the intrinsic decoherence rate. The exact solution is a Poissonian sum over unitary "kicks", with each pure-state component evolving as $\exp(-i k H/\gamma)$. Averaging over these kicks replaces pure unitary evolution by phase-damped dynamics without energy exchange with an external bath (Urzúa et al., 2021).

Key features:

• Expectation values of quadrature operators and the number operator in coherent and squeezed-coherent initial states display oscillations at frequency $\omega$ modulated by Gaussian-Poisson damping envelopes.
• The displacement parameter $\lambda$ rigidly shifts the oscillator equilibrium and appears as a constant additive offset in observables.
• In the large $\gamma$ limit, decoherence vanishes and pure oscillatory dynamics are restored; for finite $\gamma$, suppression of revivals and decohering of superpositions can be explicitly quantified.

Analytic formulas for $\langle a + a^\dagger\rangle_t$ and $\langle N\rangle_t$ capture the combined effect of displacement, squeezing, and intrinsic decoherence, providing an exact reference for the interplay between coherent quantum dynamics and intrinsic dephasing processes (Urzúa et al., 2021).

5. Photonic Lattice Realizations and Quantum Correlations

The semi-infinite Glauber-Fock photonic lattice, characterized by coupling constants $C_n = C_1\sqrt{n}$, realizes the displacement operator $\hat D(iC_1 z)$ in a spatially extended system. Light propagation in this lattice enacts a continuous displacement in Fock space, with intensity patterns directly mapping to quantum number distributions of the displaced oscillator model (Keil et al., 2011).

The breakdown of shift invariance, a direct consequence of $C_n$ scaling as $\sqrt{n}$, leads to spatially sensitive quantum correlation patterns, distinguishing this structure from uniform (translationally invariant) arrays. Two-photon coinicidence probabilities $\Gamma_{qr}$, both for product and entangled inputs, depend non-trivially on the initial waveguide locations, enabling studies of position-dependent quantum walks and nonclassical correlation dynamics specific to the displaced-oscillator Hamiltonian.

Classical–quantum correspondences emerge:

• Photonic site index $n \leftrightarrow$ oscillator number state $|n\rangle$,
• Propagation distance $z \leftrightarrow$ phase-space displacement parameter $\alpha$,
• Optical intensity $\leftrightarrow$ quantum number distribution of displaced Fock state,
• Measured correlations encode quantum features such as bosonic bunching, fermionic exclusion, and path entanglement in the displaced-basis.

This mapping not only facilitates the simulation of displaced oscillator dynamics in photonic platforms but also provides direct classical analogues of fundamental quantum optical processes.

6. Benchmark Status, Physical Insights, and Applications

The displaced oscillator model serves as a canonical benchmark for analytic and numerical methods:

• All eigenfunctions (QES or NES) are known in closed form, enabling arbitrary-precision evaluation and direct testing of semiclassical or matrix-diagonalization algorithms.
• The tunable parameter $d$ or $\lambda$ provides continuous control over the double-well structure, tunneling splitting, and phase-space dynamics, supporting controlled studies of quantum critical phenomena and decoherence.
• Analytical accessibility extends to applications in quantum optics (implementation of displaced states and operators), photonic lattice engineering, and quantum information protocols.
• The model provides a standard for the calibration and validation of perturbation theories, instanton estimates (for tunneling splitting), and comparison of dephasing versus dissipative effects.

Due to the availability of explicit spectra, wavefunctions, and dynamical quantities, the displaced oscillator model occupies a unique position as a tractable yet nontrivial testbed for fundamental and applied quantum theory (Znojil, 2016, Keil et al., 2011, Urzúa et al., 2021).