Photon Number Wave Packets in Quantum Optics

• Photon number wave packets are quantized radiation states with defined spectral-temporal envelopes that link photon number to mode structure in quantum systems.
• They enable engineered atom–photon interactions through spectral encoding and pulse shaping, achieving near-unity excitation probabilities in reversible state transfers.
• Advanced measurement and interference techniques reveal nontrivial dynamics in cavity QED and driven systems, driving innovations in quantum optical control.

Photon number wave packets are quantized radiation states in which photon number is tied to a nontrivial mode structure rather than to a single monochromatic oscillator. In continuous-mode quantum optics, they appear as single-photon and $N$-photon states with specified spectral-temporal envelopes and, more generally, as states with arbitrary spectral distribution functions over one or multiple spatial and polarization modes. In cavity QED, the same expression is also used for localized peaks of finite width propagating through the discrete photon-number distribution $P_n(t)$. Across these usages, the central issue is the joint organization of photon number, mode structure, and dynamics in interference, emission, absorption, propagation, control, and measurement (Amaral et al., 2017, Baragiola et al., 2012, Nimmesgern et al., 2023).

1. Formal definitions and state-space descriptions

A standard continuous-mode single-photon wave packet is written as

$$|1_{\xi}\rangle = a^{\dagger}_{\xi} |0\rangle, \qquad a^{\dagger}_{\xi} = \int d\omega\, \xi(\omega)\, a^{\dagger}(\omega),$$

where $\xi(\omega)$ is the spectral amplitude. In the Hong–Ou–Mandel setting, an analogous representation is

$$|1\rangle_j = \int_{-\infty}^{\infty} d\omega\, X_j(\omega)\, \hat{a}_j^\dagger(\omega) |0\rangle,$$

with $j=1,2$ labeling the two paths and $X_j(\omega)$ the spectral mode functions. This description makes explicit that a photon number state is simultaneously a statement about excitation number and about the occupied spectral-temporal mode (Naeij, 20 Jun 2025, Amaral et al., 2017).

The same framework extends to definite higher photon number. For a continuous-mode $N$-photon Fock state with all photons in the same temporal mode,

$$|N_\xi\rangle = \frac{1}{\sqrt{N!}} [B^\dagger(\xi)]^N |0\rangle, \qquad B^\dagger(\xi)=\int ds\, \xi(s)\, b^\dagger(s).$$

More generally, an arbitrary $N$-photon state can be written as

$P_n(t)$0

where $P_n(t)$1 is a symmetric spectral distribution function that need not factorize. This occupation-number formulation is the basis for master-equation treatments of arbitrary photon-number inputs, arbitrary spectral distribution functions, and multiple spatial or polarization modes (Baragiola et al., 2012).

Concrete envelopes show the time–frequency reciprocity directly. For a rectangular spectral envelope,

$P_n(t)$2

the temporal envelope is

$P_n(t)$3

This example is representative of the general principle that spectral support fixes temporal structure through Fourier transformation (Naeij, 20 Jun 2025).

A distinct but related usage arises in strongly driven cavity systems. There the basic object is the photon-number distribution

$P_n(t)$4

or equivalently with $P_n(t)$5 and $P_n(t)$6 in the two-level notation. A photon number wave packet in this sense is a localized peak of finite width moving through the discrete $P_n(t)$7-space of cavity occupation numbers rather than through physical space or frequency (Nimmesgern et al., 2023, Nimmesgern et al., 3 Sep 2025).

2. Interaction with atoms and arbitrary quantum systems

A general theoretical framework for photon-number wave packets interacting with arbitrary quantum systems is given by the input–output and QSDE formalism for continuous-mode Fock states. Within the white-noise, Markov, and rotating-wave approximations, the interaction Hamiltonian is

$P_n(t)$8

and the dynamics of an arbitrary system driven by $P_n(t)$9-photon input are captured by a finite set of coupled master equations for generalized density operators $$|1_{\xi}\rangle = a^{\dagger}_{\xi} |0\rangle, \qquad a^{\dagger}_{\xi} = \int d\omega\, \xi(\omega)\, a^{\dagger}(\omega),$$0. The same framework yields output quantities such as photon flux and quadratures, and its multimode version treats multiple spatial and polarization channels. Because absorption or emission changes the field state, the resulting system dynamics are intrinsically non-Markovian in the Fock-state setting (Baragiola et al., 2012).

For a single two-level atom, Stobińska, Alber, and Leuchs analyzed the multimode QED problem and emphasized the time-reversal relation between spontaneous emission and perfect absorption. The maximum excitation probability for an incident one-photon packet can be written as

$$|1_{\xi}\rangle = a^{\dagger}_{\xi} |0\rangle, \qquad a^{\dagger}_{\xi} = \int d\omega\, \xi(\omega)\, a^{\dagger}(\omega),$$1

and for an exponentially rising input, which is the time reverse of spontaneous decay, $$|1_{\xi}\rangle = a^{\dagger}_{\xi} |0\rangle, \qquad a^{\dagger}_{\xi} = \int d\omega\, \xi(\omega)\, a^{\dagger}(\omega),$$2 approaches unity; with a long enough pulse of duration $$|1_{\xi}\rangle = a^{\dagger}_{\xi} |0\rangle, \qquad a^{\dagger}_{\xi} = \int d\omega\, \xi(\omega)\, a^{\dagger}(\omega),$$3, it approaches $$|1_{\xi}\rangle = a^{\dagger}_{\xi} |0\rangle, \qquad a^{\dagger}_{\xi} = \int d\omega\, \xi(\omega)\, a^{\dagger}(\omega),$$4. This establishes wave-packet engineering as a prerequisite for reversible atom–photon state transfer in free space (Stobińska et al., 2010).

Spectral encoding modifies this interaction substantially. When a spectral phase mask

$$|1_{\xi}\rangle = a^{\dagger}_{\xi} |0\rangle, \qquad a^{\dagger}_{\xi} = \int d\omega\, \xi(\omega)\, a^{\dagger}(\omega),$$5

is applied, the encoded spectral amplitude becomes $$|1_{\xi}\rangle = a^{\dagger}_{\xi} |0\rangle, \qquad a^{\dagger}_{\xi} = \int d\omega\, \xi(\omega)\, a^{\dagger}(\omega),$$6, and the temporal envelope is correspondingly restructured by interference among the encoded frequency components. In binary spectral encoding, the spectrum is divided into frequency bins, each assigned phase $$|1_{\xi}\rangle = a^{\dagger}_{\xi} |0\rangle, \qquad a^{\dagger}_{\xi} = \int d\omega\, \xi(\omega)\, a^{\dagger}(\omega),$$7 or $$|1_{\xi}\rangle = a^{\dagger}_{\xi} |0\rangle, \qquad a^{\dagger}_{\xi} = \int d\omega\, \xi(\omega)\, a^{\dagger}(\omega),$$8, producing a temporal envelope that is a sum of modulated sinc functions. The encoded packet spreads in time, its peak intensity decreases, and its ability to excite a two-level atom is reduced. The excitation probability,

$$|1_{\xi}\rangle = a^{\dagger}_{\xi} |0\rangle, \qquad a^{\dagger}_{\xi} = \int d\omega\, \xi(\omega)\, a^{\dagger}(\omega),$$9

drops as code length increases, indicating a direct trade-off between spectral encoding for multiplexing or concealment and efficient atom–photon coupling (Naeij, 20 Jun 2025).

Time-symmetric emission and absorption protocols pursue the opposite objective. In free space, STIRAP-based control of spontaneous emission and absorption makes it possible to generate and retrieve time-reversal-symmetric single-photon wave packets, including Gaussian temporal envelopes, for quantum state transfer between matter and light without cavity mode selection. This suggests that photon number wave packets are not only carriers of quantum information but also matching conditions for reversible interfaces (Trautmann et al., 2015).

3. Interference, spectral distinguishability, and complementarity

The two-photon interference of frequency-displaced photonic wave packets in a Hong–Ou–Mandel interferometer shows that photon number wave packets are governed not only by occupation number but also by accessible distinguishability in the spectral domain. The fidelity between two single-photon states is

$\xi(\omega)$0

and in the experimental implementation the distinguishability parameter is obtained from measured spectra as

$\xi(\omega)$1

Here $\xi(\omega)$2 corresponds to complete spectral indistinguishability and $\xi(\omega)$3 to complete distinguishability (Amaral et al., 2017).

When weak-coherent states are used instead of ideal single photons, the usual HOM visibility must be corrected because the maximum achievable visibility is $\xi(\omega)$4. With

$\xi(\omega)$5

the experiment tests the complementarity relation

$\xi(\omega)$6

The reported behavior is that narrowing the amplitude-modulator gate broadens the spectra, increases spectral overlap, decreases $\xi(\omega)$7, and restores interference, so that $\xi(\omega)$8 rises (Amaral et al., 2017).

A persistent misconception is that distinguishability is irrelevant if detectors do not directly resolve the distinguishing degree of freedom. The interference measurements show the opposite: even when detectors are not directly sensitive to frequency, the in-principle accessibility of which-way information in the spectral domain suppresses HOM interference. Photon number wave packets therefore encode interference-relevant structure across all accessible degrees of freedom, not only in those explicitly read out (Amaral et al., 2017).

Related interference control occurs in tripod-type EIT memories. By storing a single-photon-level pulse in a superposition of two collective excitations and retrieving the components sequentially, the output state can be prepared as

$\xi(\omega)$9

Frequency detuning between the two control fields produces temporal beating and frequency-domain interference, making time-bin splitting and coherent recombination part of the wave-packet interference repertoire (Yang et al., 2015).

4. Generation, shaping, conversion, and scattering control

Several distinct mechanisms exist for tailoring photon number wave packets. In a cavity-assisted source based on a classically pumped three-level $$|1\rangle_j = \int_{-\infty}^{\infty} d\omega\, X_j(\omega)\, \hat{a}_j^\dagger(\omega) |0\rangle,$$0-type emitter, the shape of the external pump pulse $$|1\rangle_j = \int_{-\infty}^{\infty} d\omega\, X_j(\omega)\, \hat{a}_j^\dagger(\omega) |0\rangle,$$1 and the emitter–cavity interaction $$|1\rangle_j = \int_{-\infty}^{\infty} d\omega\, X_j(\omega)\, \hat{a}_j^\dagger(\omega) |0\rangle,$$2 jointly determine the spatio-temporal mode of the outgoing one-photon field. The output mode is described by a Wigner function

$$|1\rangle_j = \int_{-\infty}^{\infty} d\omega\, X_j(\omega)\, \hat{a}_j^\dagger(\omega) |0\rangle,$$3

so the generated field is a mixture of vacuum and a one-photon Fock state in the desired nonmonochromatic mode. For given $$|1\rangle_j = \int_{-\infty}^{\infty} d\omega\, X_j(\omega)\, \hat{a}_j^\dagger(\omega) |0\rangle,$$4, the pump can be designed to produce a desired wave-packet shape, while the overall efficiency is bounded by

$$|1\rangle_j = \int_{-\infty}^{\infty} d\omega\, X_j(\omega)\, \hat{a}_j^\dagger(\omega) |0\rangle,$$5

In the vSTIRAP regime high efficiency is possible, whereas weak driving more closely reproduces the pump envelope but with low efficiency (Khanbekyan et al., 2017).

Tripod-EIT storage provides a complementary route based on reversible light–matter mapping. Two control beams allow a signal pulse to be stored as a superposition of two spin waves and later retrieved into two temporally distinct modes, with the splitting ratio determined by the control-field amplitudes. By varying timing, amplitude, phase, and detuning, the retrieved single-photon-level pulse can be multi-split or shaped into customized pulse trains. This makes time-bin qubit preparation and waveform modulation part of the same control problem (Yang et al., 2015).

Biphoton wave packets can also be shaped collectively in an atomic medium. In a four-level double-$$|1\rangle_j = \int_{-\infty}^{\infty} d\omega\, X_j(\omega)\, \hat{a}_j^\dagger(\omega) |0\rangle,$$6 cold-atom system, an electromagnetically induced grating spatially modulates both linear and nonlinear susceptibilities. The anti-Stokes field is diffracted, most strongly into the zeroth order, and the joint biphoton spectrum can be broadened or narrowed by adjusting the grating period, control-field Rabi frequency, optical depth, and medium length. The notable feature is that this spectral shaping occurs without a cavity (Wen et al., 2010).

Frequency conversion and spectral compression add another layer of control. A slow-light sum-frequency-generation scheme predicts that a $$|1\rangle_j = \int_{-\infty}^{\infty} d\omega\, X_j(\omega)\, \hat{a}_j^\dagger(\omega) |0\rangle,$$7-ps pulse can be converted into a pulse with duration in the ns regime, with a spectral compression factor of the order of $$|1\rangle_j = \int_{-\infty}^{\infty} d\omega\, X_j(\omega)\, \hat{a}_j^\dagger(\omega) |0\rangle,$$8 and a useful intrinsic efficiency up to $$|1\rangle_j = \int_{-\infty}^{\infty} d\omega\, X_j(\omega)\, \hat{a}_j^\dagger(\omega) |0\rangle,$$9. Independent of the input pulse shape, the converted pulse approaches a near-exponential rising shape, which is suitable for temporal-mode matching into an optical cavity. This suggests a practical bridge between broadband photonic sources and narrowband cavity or memory interfaces (Raymer, 11 Apr 2025).

Scattering itself can be used as a shaping mechanism. In a two-dimensional photonic waveguide coupled nonlocally to a giant atom, arbitrary target scattering single-photon wave packets can be generated by adjusting the coupling strengths at different lattice sites. The essential control variable is the momentum-space coupling function $j=1,2$0, which encodes interference among the multiple connection points. Compared with a pointlike emitter, the giant-atom geometry introduces nonlocal interference as a wave-packet design resource (Cheng et al., 2024).

5. Photon-number-space wave packets in driven cavities

In the driven Jaynes–Cummings model, photon number wave packets appear in a mathematically different sense: as localized distributions moving through discrete photon-number space. If a two-level system coupled to a single-mode cavity is strongly driven, the mean photon number does not grow continuously. Instead, the cavity exhibits oscillations in $j=1,2$1, corresponding to peaks of finite width running up and down in the distribution $j=1,2$2. For resonant driving, a single packet appears; for finite detuning, two or more packet structures can coexist and oscillate at different frequencies and amplitudes (Nimmesgern et al., 2023).

The resonant large-drive regime yields an analytic upper scale for the packet motion,

$j=1,2$3

so the wave packet oscillates between $j=1,2$4 and $j=1,2$5 rather than accumulating photons indefinitely. At finite detuning, the laser-dressed-state and cavity-dressed-state pictures explain the coexistence of multiple packets, their different oscillation frequencies, and their turning points in $j=1,2$6-space. This replaces the intuition of monotonic cavity filling by a tight-binding or WKB-like propagation picture in photon-number space (Nimmesgern et al., 2023).

Dissipation reshapes these structures in qualitatively different ways. Cavity loss damps the oscillations and can produce a stationary bimodal photon-number distribution near a critical detuning. Radiative decay and pure dephasing of the two-level system instead broaden the packets and generate trails in phase space. A notable result is that pure dephasing can push photon occupations toward higher photon numbers than in the non-dissipative case (Nimmesgern et al., 2023).

Rapid temporal control of the driving strength turns these dynamics into a wave-packet synthesis protocol. By suddenly changing $j=1,2$7, the dressed-state basis changes, and existing packets can spawn additional packets that then evolve independently. The resulting dynamics are classified into four subclasses, A through D, determined primarily by the detuning $j=1,2$8 and drive $j=1,2$9. Class D is the LDS-decoupled regime in which the packets are well separated and robust over many cycles. Stepwise or alternating changes in the drive can therefore generate a specified number of photon number wave packets on demand, and the theory proposes that their existence can be inferred experimentally from the oscillation spectrum of the mean photon number alone (Nimmesgern et al., 3 Sep 2025).

6. Measurement, projection, and extensions of the concept

Photon number wave packets are operationally defined not only by preparation but also by the measurement they are projected onto. Propp and van Enk constructed POVMs that project onto arbitrary single-photon temporal or spectral wave packets by using a time-dependent two-level detector with tunable decay rate $X_j(\omega)$0 and detuning $X_j(\omega)$1. In the ideal case the POVM element has the form

$X_j(\omega)$2

while for a more realistic detector including transmission, amplification, inefficiency, and dark counts it becomes

$X_j(\omega)$3

As long as the transmission function is nonzero over the spectral support of interest, any smooth square-integrable single-photon wave packet can in principle be targeted by appropriate choice of $X_j(\omega)$4 and $X_j(\omega)$5 (Propp et al., 2020).

This measurement theory makes explicit the detector-side trade-offs. Long wave packets require the detector to remain active throughout the packet duration, so there is a trade-off between time resolution or count rate and efficiency. At the same time, the construction identifies minimum-uncertainty Gaussian target modes and discusses single-shot Heisenberg-limited time-frequency measurements with

$X_j(\omega)$6

The central implication is that time-frequency selectivity is not merely an analysis tool; it is part of the physical definition of which photon number wave packet has been measured (Propp et al., 2020).

Related QED work on matter-wave packets clarifies when photon-number observables do and do not retain wave-packet sensitivity. For free-electron radiation, the emitted photon number increment separates into a first-order term $X_j(\omega)$7 and a second-order term $X_j(\omega)$8. The first-order contribution, which depends on the electron wave-packet size and structure, is present only for coherent photon states; it vanishes for Fock states. Spontaneous emission in the vacuum state and stimulated emission into Fock states are wavepacket-independent, whereas coherent-state stimulation exhibits an exponential suppression factor $X_j(\omega)$9 when the electron wave packet becomes large compared with the radiation wavelength (Pan et al., 2018).

At second order in QED, inclusive photon measurements retain coherent dependence on matter-wave structure in three cases: stimulated radiation by an electron, coherent radiation from a beam of particles, and reradiation of a photon in the Compton process. In the small-recoil limit, a single-electron wave packet can even be assigned a susceptibility tensor of the same form as for an electron plasma. A further extension uses laser-synthesized free-electron wave packets to control entangled photon-pair emission in electron–atom collisions, so that the relative phases and delays of the electron pathways imprint themselves onto the angular and polarization correlations of emitted photon pairs (Kazinski et al., 2022, Goetz et al., 2020).

Taken together, these developments show that photon number wave packets are not a single narrowly defined object but a family of related constructs linking Fock-state mode structure, wave-packet-resolved measurement, cavity photon-number dynamics, and engineered light–matter interfaces. The common theme is that photon number acquires operational meaning only together with the mode, basis, or number-space structure in which it is localized and manipulated.