Fourier decomposition of an electromagnetic radiation into a set of running or standing monochromatic waves is typical approach in classical electrodynamics. In its turn, quantum electrodynamics follows similar rules as its predecessor with the difference that the amplitudes in this decomposition are now treated as operators. The last are known as operators of creation and annihilation of a quantum of energy (photons). The quantized amplitudes, or photon number and phase when appropriate, form the usual playground in quantum optics, see e.g. [1].

This Letter proposes to look at the light from a non-conventional perspective. It is based on the observation that the radiation in the form of a wave packet demonstrates a new attribute which arises from the property of localization and therefore cannot be a characteristic for a single plane wave. This “collective” property of light is tied to the “center of gravity” of the coherent superposition of waves and is called position. The position immediately brings its conjugate partner — momentum (frequency) through commutation relation [X̂, P̂]=iħ. The two variables define the mechanical degree of freedom for a wave packet. In this mechanical sense the propagation of the wave packet can be treated as motion of a particle.

As photon creation and annihilation operators experience an interesting dynamics when the field propagates through a nonlinear system, in similar manner, the position and the momentum of wave packets will evolve non-trivially in a dispersive medium where group-velocity dispersion (GVD) or the dispersions of a higher order take non vanishing values. As the phase is correlated to the intensity through a nonlinearity, similarly, the position is correlated to the momentum (frequency) due to a frequency-dependent speed. Quantum state of the field changes during propagation in such dispersive medium. Thus, an initial coherent state will turn into a new state which is named here as a dispersed state. These dispersed states can find numerous applications. As an example, this Letter demonstrates how a certain combination of the momentum and position observables acquires the noise level below so-called Standard Quantum Limit (SQL) thus making ultra-precise optical measurements possible.

Positive- and negative-frequency parts of a linearly polarized electric field propagating in vacuum in z direction, obey commutation relations

where S is the cross-sectional area of the beam and c the speed of light in vacuum. We assume the quasi-monochromatic approximation and separate the electric field into slowly varying amplitude and rapidly oscillating exponential as

where k ₀ and ω ₀ are correspondingly central wave number and central frequency of the wave packet. Given the quasi-monochromatic approximation one can use Eq. (1) for deriving equal-space commutation relations for $\widehat{\varphi }$ and $\widehat{\varphi }$ †:

In line with our mechanistic approach we now introduce electromagnetic position and momentum operators. The position operator is defined as

i.e. as a direct analog of the classical definition of the center of gravity of a pulse with shape ϕ(z, t). In our quantum-mechanical description the c-numbers ϕ and ϕ* turn into field operators $\widehat{\varphi }$ and $\widehat{\varphi }$†. They are normalized such that the integral ∫ϕ̂†$\widehat{\varphi }$ dt gives the photon number operator N̂. Limit operation in Eq. (4) regularizes the definition for a vacuum state. Note that well-known contradictions with localizing photons discovered first in [2] and reviewed later with emphasis on quantum optics in [1], vanish in the quasi-monochromatic approximation (which is exploited through this Letter). The definition Eq. (4) is formulated as the integral over time rather than space, and therefore the position has dimensions of time. The use of time integration will prove convenient at the detection stage, see also [3]. An operator is canonically conjugate to position if it obeys the commutation relation:

where we omit ħ for brevity. It is easy to check that such an operator P̂ has the form

and as a consequence can be called (total) momentum. The pair of variables (X̂, P̂) represents a collective mode, namely the mechanical degree of freedom of the wave packet. Given this formulation we approach the main goal of the Letter and follow the evolution of the two conjugate variables in a dispersive medium.

The fully canonical quantization scheme of the electromagnetic field in Hopfield’s dielectric was developed by Huttner and Barnett in [4]. They derived an evolution equation for monochromatic annihilation and creation operators for forward- and backward-propagating polariton waves, ĉ±(z,ω) and ĉ†_±(z,ω), in the form which is most relevant for treating propagation problems:

with the polariton operators obeying commutation rules

and [ĉ±(z,ω), ĉ†_∓(z,ω′)]=0. Eq. (7) is similar to that which appears in classical electrodynamics for the forward- and backward-propagating classical fields amplitudes. The difference is in second term which has no classical equivalent and represents a Langevin fluctuation term corresponding to the absorption of the electromagnetic field. The noise operators obey characteristic commutation relations [f̂(z,ω), f̂†(z′, ω′)]=δ(z-z′)δ(ω-ω′). Eq. (7) is derived for a complex k(ω) and is fully consistent with the Kramers-Kronig relations. Effects of losses and causality are irrelevant for the present consideration, so we shall neglect the absorption. As a simple example of the dispersive medium, we choose fibers with the dispersion relation given by

where k(ω) is expanded around central wave number k ₀ of the wave packet, and the prime denotes differentiation with respect to ω evaluated at ω ₀.

From now on, we shall consider waves traveling only in the forward direction. According to Ref. [4], the annihilation operator for the forward-traveling electric field is reconstructed from polariton amplitudes as

The particular form of dispersion relation Eq. (9) implies a weak coupling between the medium such that each subsequent term is smaller than the previous one. Then, the polariton wave appears as consisting of mainly the electromagnetic field and of only a very small fraction of the medium. Keeping the leading term in Eq. (9) and applying the quasi-monochromatic approximation, k(ω) can be approximated as k ₀ and taken out of the integral Eq. (10). The evolution equation Eq. (7) can be then reformulated in terms of the field operator ϕ̂(z, τ):

with τ=t-k′z. Commutation relations for ϕ̂(z, τ) and ϕ̂†(z, τ) are derived from (8) giving same result as for vacuum (3) but with t changed now to τ. The field operators obey the equal-space commutation relations in contrast to familiar looking equal-time ones. However, these conditions are not imposed phenomenologically. They are derived from the fully canonical quantization scheme and are fully consistent with the usual equal-time commutation relations for the conjugate vector potential and electric field. Formulation of the equal-space commutation rules brings in significant simplifications in treating propagation problems [4, 3].

Before deriving evolution equations for the momentum Eq. (4) and position Eq. (6) from Eq. (7), we introduce a decomposition of the field operator into classical and quantum parts as ϕ̂=ϕ+v̂, and disregard all terms nonlinear in v̂. Then, definitions Eq. (4) and Eq. (6) are approximated as

in the reference frame where mean position and mean momentum are zero: P ₀≡i∫(∂ϕ*/∂τ)ϕdτ=0 and X ₀≡$n_0^{-1}$∫τ|ϕ|² dτ=0. Here, n ₀≡∫|ϕ|² dτ is the mean photon number. The classical variable ϕ(z, τ) appearing in the equations stands for the “mean shape” of the wave packet ϕ(τ) at a distance z.

The linearization approximation is introduced with the only purpose to demonstrate a homodyne measurement of the collective variables of the quantum wave packets. The approximation works fairly well for fields with a large number of photons and as long as the mean shapes remain well determined. With this restriction in mind, Eq. (11) is used for deriving the evolution equations for linearized versions of the position and momentum operators, Eqs. (12). The resultant equations are solved giving

with N̂(z)=N̂₀, and label 0 marks quantities evaluated at z=0. It is quite remarkable that commutation relation Eq. (5) remains intact during the propagation:

Therefore, the two variables preserve their status as a collective mode.

Let us consider the propagation of an initially transform-limited Gaussian wave packet in a coherent state with a mean shape ϕ(0, τ)=(n ₀/π ^1/2 τ_p )^1/2 exp(-τ ²/2${\tau }_p^2$ ) through the dispersive fiber. Here, n ₀=∫|ϕ(0, τ)|² dτ is taken as the mean number of photons and τ_p as the mean duration. The different frequencies constituting the wave packet propagate with different speeds and the initial shape continuously changes:

where Z_D ≡${\tau }_p^2$ /k″ is introduced as a characteristic dispersion length. As a result, the Gaussian wave packet experiences well known in classical fiber optics [5] temporal lengthening with duration T_p growing with distance as $T_p\left(z\right)={\tau }_p\sqrt{1+{(z⁄Z_D)}^2}$.

The field is initially (i.e. at z=0) in a coherent state. The last is defined as an eigenstate of the annihilation operator ϕ̂. Then, 〈v̂v̂^†〉=δ(τ-τ′) while the other variances vanish. Using linearized definitions Eqs. (12) we now calculate the initial momentum and position variances of the Gaussian wave packet:

The non-zero variances indicate that neither the momentum nor the position are sharply determined in the coherent state. Their product is $\frac{1}{4}$ (i.e. minimum) of Heisenberg’s uncertainty relation. The variances give the characteristic values of experimental errors in a quantum measurement and are often referred to as SQL. A measurement with the accuracy below the SQL is the principal goal in quantum optics. Usually, squeezed states [6] are employed for reaching this purpose. Here, we show that the dispersed states can perform this same task.

In the Heisenberg picture in use the changes in quantum state show up in evolution of the collective mode (P̂, X̂), Eq. (13). It is convenient to introduce quadratures q̂ and q _̂⊥ by rotating the (P̂, X̂) vector on angle ψ:

The quadratures are as “good” as P̂ and X̂. They obey the same commutation relation [q̂, q̂_⊥]=i as Eq. (14) and give the same product of variances in the coherent state 〈Δq̂²${〉}_{\text{coh}}^{\left(\text{in}\right)}$ 〈Δ${\widehat{q}}^2{〉}_{\mathrm{coh}}^{\left(\text{in}\right)}$ = 〈ΔX̂²${〉}_{\text{coh}}^{\left(\text{in}\right)}$ 〈ΔP̂ ²${〉}_{\text{coh}}^{\left(\text{in}\right)}$. Normalization t_c is arbitrary. We choose here $t_c\equiv {\tau }_p⁄\sqrt{n_0}$ for symmetrizing the SQLs of the quadratures so that ${〈\Delta {\hat{q}}^2〉}_{\mathrm{coh}}^{\left(\mathrm{in}\right)}={〈\Delta {\hat{q}}_{\perp }^2〉}_{\mathrm{coh}}^{\left(\mathrm{in}\right)}=\frac{1}{2}$. The goal will be reached if we show that there exists an angle ψ for which the variance 〈Δq̂²〉 is smaller than the SQL, i.e. $〈\Delta {\hat{q}}^2〉<\frac{1}{2}$. Calculations yield

where Eqs. (17), (13) and (16) are used, and $\psi ={\psi }_{min}=\frac{1}{2}\mathrm{arcctg}(-z⁄2Z_D)$ is chosen for minimizing the variance. The farther the pulse propagates into the medium the deeper the variance fall below the SQL (provided the linearization approximation is still valid). Thus, 10 dB decrease in the quantum noise of q̂ is achieved after a 10-fs pulse propagated 10 cm in a conventional fiber with k″=10 ps²/km (ZD=1 cm).

Eq. (18) demonstrates the suppression of quantum noise below SQL 〈Δq̂²${〉}_{\text{coh}}^{\left(\text{in}\right)}$ which is associated with the initial wave packet with mean shape ϕ(0, τ). One can notice that changes of the quantum state in the dispersive medium take place on the background of evolution of the classical shape governing by Eq. (15). It is therefore instructive to relate variance 〈Δq̂²〉min in the dispersed state to the variance of same quadrature q̂ provided the wave packet with output shape ϕ(z, τ) is prepared in a coherent state. First, we evaluate SQLs 〈ΔP̂²${〉}_{\text{coh}}^{\left(\text{out}\right)}$ and 〈ΔX̂²${〉}_{\text{coh}}^{\left(\text{out}\right)}$ for output shape ϕ(z, τ):

then formulate quadrature q̂ according to Eq. (17) with ψ=ψ _min, and finally compute its SQL as 〈Δq̂²${〉}_{\text{coh}}^{\left(\text{out}\right)}$≈1 for z≫Z _D. Comparison of 〈Δq̂²${〉}_{\text{coh}}^{\left(\text{out}\right)}$ and Eq. (18) shows the suppresion of noise relative to the SQL for the wave packet with mean shape ϕ(z, τ).

Let us estimate the efficiency of the suppression of quantum noise by use of the dispersed states when compared to the conventional methods based on squeezed states. Generation of the latter necessitates a nonlinear type of interaction. So, in general, the figure of merit is given by the ratio of nonlinear to dispersion length F=Z _NL/Z _D. The shorter and/or the weaker the pulse, the better the relative efficiency provided by the dispersed states. For example, F≈10¹⁰ for a one-photon 10-fs optical in a single-mode fiber. For the estimation, we use Z _D≈1 cm, and $Z_{\mathrm{NL}}=\frac{c{A}_{\mathrm{eff}}}{n_2{\omega }_0}\frac{1}{P_0}\approx 1.2\times {10}^{10}$ cm with ω ₀=0.6×10¹⁵ rad/s as central frequency, n ₂=3.2×10¹⁶ cm²/W as nonlinear index of refraction, A _eff=π(4µm)² as cross-sectional area, ħ=10^-34 W s ² as the Planck constant, and P ₀=ħω/τ_p as the power of the one-photon pulse.

 

Fig. 1. Sketch of the balanced homodyne detection for measuring position-momentum quadrature q̂ Eq. (20). A Gaussian pulse is splitted (BS1) into signal and local oscillator (LO). Both propagate through the same fiber (and thereby broaden similarly). Phase shift introduced by PS and shape transformation produced by “transformer” (e.g. by means of Fourier optics) shape the LO as L_q (τ)∝f_q (τ), Eq. (20). Delay lines serve for spatial separating the signal and the LO.

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Measurement of the momentum-position quadratures can be performed by balanced homodyning, see Fig. 1, which is a modification of the schemes reviewed by Haus in [7] in context of quantum solitons. The shape of the pulsed local oscillator L _q(τ) is dictated by the condition that, upon colliding the classical pulse with the quantized wave packet ϕ̂(z, τ) on beam splitter BS2 and detecting the outcome by two photodetectors D _± and integrating the difference of the photocurrents, the final result matches definition (17). That is, outcome of the setup Ô=∫[L*_q(τ) v̂(z, τ)+L _q(τ) v̂^†(z, τ)] dτ is expected to reproduce definition of the quadrature q̂:

where the projector f _q is evaluated by substituting Eqs. (12) in Eq. (17) and with ϕ(z, τ) as Gaussian shape Eq. (15) at the output of the fiber. Comparing expressions for Ô and q̂, one immediately relates L _q(τ) and f _q(τ) as L _q(τ) ∝ f _q(τ). Sometimes, spectral measurements may be preferable. Then, a collective frequency operator ω̂=P̂N̂^-1 as in [8] should be considered instead of the total momentum P̂.

Though the dispersed states demonstrate many similarities with well known squeezed states, the two should not be confused. A squeezed state is characterized by a reduction of noise below SQL for a quadrature-phase amplitude that is not possible for the linear dispersive medium. The two types of states, squeezed and dispersed, are complementary. First refers to the wave properties of light while the other exploits its particle content.

The dispersed states are characterized by: (i) no need in a nonlinearity for their generation; (ii) larger generation efficiency when pulses are shorter; (iii) a need of a collective measurement over the pulse as a whole rather than a single mode (plane wave). We note also that (a) a variety of media with dispersion relations different than Eq. (9) and arbitrary pulse shapes can be used for producing the dispersed states; (b) losses are easily incorporated into the formalism, see Eq. (7); (c) other non-classical states can be generated, e.g. one can entangle a pair of pulses dispersed in orthogonal directions by colliding them on a beam splitter, analogous to the proposal of Braunstein and Kimble [9] for entangling two amplitude-squeezed single-mode beams.

I gratefully acknowledge discussions with G. Alber, V. B. Braginsky, J. H. Eberly, M. Freyberger, G. Leuchs, W. P. Schleich, and I. Walmsley. I thank Prof. Eberly and RTC members for warm hospitality in Rochester. This work was supported by a grant to RTC by Corning Inc., and the programme “QUIBITS” of the European Commission.