Senitzky Coherent Excited States

• Senitzky coherent excited states are displaced Fock states that blend quantum particle-like and classical wave-like attributes, preserving the nodal structure of the original number state.
• They exhibit an oscillatory probability density along a classical trajectory while retaining nonclassical photon statistics and sub-Poissonian behavior.
• Their transformation properties under beamsplitter operations and experimental generation using cavity QED highlight their significance in quantum state engineering.

Senitzky coherent excited states form a class of quantum states defined as displaced number (Fock) states of the harmonic oscillator. In these states, an energy eigenstate is acted upon by the Weyl displacement operator so that the resulting probability density oscillates in phase space along a classical trajectory while retaining the nodal structure of a number state. This construction extends the standard (Glauber) coherent state—obtained when the displacement is applied to the ground state—to arbitrary energy eigenstates and provides a framework in which quantum (particle‐like) and classical (wave-like) features coexist in a single formalism.

1. Mathematical Structure and Definition

Senitzky coherent excited states, also known as generalized coherent states (GCS), are defined by the operation

|n, α⟩ = D₍α₎ |n, 0⟩,

where the Weyl displacement operator is given by

D₍α₎ = exp(α a† − α* a).

Here |n, 0⟩ is the number (Fock) state with n quanta, and the complex amplitude α = |α| e^iθ determines the phase-space displacement. In the position representation the wavefunction takes the form

ψₙ,α(x, t) = (ω/π)^1/4 (2^n/2 √(n!))^–1 exp[−ω (x − ⟨x⟩)²⁄2] Hₙ(√ω (x − ⟨x⟩)) × exp{ i [−(n + ½) ωt + x⟨p⟩ − ½ ⟨x⟩⟨p⟩] },

with the time-dependent expectation values

⟨x⟩ = √(2/ω) |α| cos(ωt − θ), ⟨p⟩ = −√(2ω) |α| sin(ωt − θ).

In the special cases α = 0 or n = 0 these states recover the number state and the usual coherent state, respectively.

2. Physical Interpretation in Quantum Optics

The framework of quantum optics maps single-mode field states to harmonic oscillator states. In this context, Senitzky coherent excited states exhibit the following features:

• The envelope of the probability density oscillates as in standard coherent states, with the displacement determined solely by α,
• The nodal structure inherent to the Fock state |n, 0⟩ is preserved, yielding a spatial profile with n nodes,
• The electric field expectation value is given by

⟨n, α| E(x) |n, α⟩ = √(2/ω) |α| cos(kx − ωt + θ + π/2),

indicating that the phase of α controls the macroscopic field pattern while the underlying quantum fluctuations are inherited from the number state.

3. Nonclassical Properties and Photon Number Statistics

Senitzky coherent excited states blend nonclassical features with classical oscillatory behavior. Their photon number properties are characterized by:

• A mean photon number

⟨N⟩ = n + |α|²,

which includes a contribution from the Fock state and from the displacement amplitude,

• A photon number variance

(ΔN)² = |α|²,

leading to sub-Poissonian statistics when n > 0,

• A fractional number fluctuation that vanishes in the large amplitude limit:

ΔN⁄⟨N⟩ = |α|⁄(n + |α|²).

In addition, the probability Pₖ(n, α) of measuring k photons is given by

Pₖ(n, α) = [k!/n!] e^–|α|²|α|^2(n–k) [Lₖ^n–k(|α|²)]²,

with Lₖ^n–k being the generalized Laguerre polynomial. These expressions highlight the oscillatory non-Poissonian statistics and clearly distinguish the quantum attributes of the state.

4. Transformation Properties and Beamsplitter Behavior

A notable aspect of Senitzky coherent excited states is their well-defined transformation under passive linear operations. When a state |n, α⟩ is subjected to a beamsplitter transformation, it splits into a superposition of displaced number states. In the ideal beamsplitter scenario with reflection coefficient R and transmission coefficient T the state transformation can be expressed as

|n, α⟩₁ |0, 0⟩₂ = Σₘ₌₀ⁿ √[C(n, m) R^m T^n–m] |m, Rα⟩₃ |n–m, Tα⟩₄.

Here the “number part” (indexed by n) undergoes a binomial-like split while the “coherent” displacement remains additive. This separation of the quantum and classical aspects under beamsplitter operations is crucial for applications in quantum state engineering and nonclassical light generation.

5. Experimental Generation and Applications

Senitzky coherent excited states can be generated dynamically by applying a classical drive to a quantum harmonic oscillator initially prepared in a Fock state. For example, in cavity quantum electrodynamics (QED) experiments, driving a mode initially in state |n, 0⟩ with a classical current produces the displaced state

|n, α⟩ = D₍α₎ |n, 0⟩.

This approach facilitates the experimental exploration of nonclassical photon statistics and has potential applications in quantum measurement, quantum communication, and quantum state engineering. Their hybrid classical-quantum character makes these states particularly interesting when considering protocols that require both high field intensity (controlled by α) and distinct quantum signatures (dictated by n).

6. Relation to Other Coherent and Photon-Added States

Senitzky coherent excited states generalize the notion of the standard coherent state (obtained for n = 0) and are closely related to photon-added coherent states as introduced by Agarwal and Tara. In this perspective, while conventional photon-added states involve the successive application of the creation operator on a coherent state to introduce nonclassical features, Senitzky states are constructed by displacing an arbitrary Fock state. Similar constructions extend to systems with continuous spectra where analogous strategies are used to define excited coherent states under the Gazeau–Klauder framework. Moreover, in certain contexts such as two-mode Landau levels, the use of two-variable coherent states enables further generalizations that interpolate between standard coherent states, excited coherent states, and number states. These relationships underscore the flexibility of the displacement operator method in bridging classical and quantum descriptions across diverse systems.

7. Overview of Key Properties

The following table summarizes select features of standard coherent states versus Senitzky coherent excited states:

Property	Standard Coherent State (n=0)	Senitzky Coherent Excited State (n>0)
Field Expectation	Gaussian profile oscillating along a classical trajectory	Oscillatory pattern with built-in nodal structure
Photon Number Statistics	Poissonian	Sub-Poissonian with oscillatory deviations
Transformation at Beamsplitter	Splits as a coherent state	Splits into a superposition of displaced number states

In summary, Senitzky coherent excited states provide an important bridge between purely classical and quantum descriptions of optical fields. Their mathematical construction as displaced number states underlies a host of nonclassical phenomena, and their versatile transformation properties make them a valuable tool in both theoretical investigations and experimental quantum optics.