Heisenberg uncertainty of spatially gated electromagnetic fields

The electric and the magnetic field operators $E^(r)$ and $H^(r′)$ at two different points in space do not commute. This implies the existence of a Heisenberg uncertainty relation between them. In his 1927 Chicago lecture notes,¹ Heisenberg had calculated this uncertainty for fields averaged over a box of size l and obtained (ΔE)(ΔH) > ℏc/l⁴. This implies that for a sufficiently small box, electromagnetic field fluctuations are strong enough so that the quantum nature of the fields may not be ignored. Heisenberg did not have a particular application in mind but was rather interested in clarifying a fundamental issue: the corpuscular vs wave nature of photons. Thanks to recent advances in nano-optics,² this uncertainty may be tested experimentally in nanostructures. Here, we examine it for spatially gated² electric and magnetic fields. An important consequence of our study is that spectroscopy of nanostructures may not be fully described by classical fields since this uncertainty may not be neglected. Spatially gated fields thus have an intrinsic quantum nature that should have experimental signatures.

To derive the Heisenberg uncertainty relation for spatially gated electric and magnetic fields, we introduce two vector gate functions, η and γ, associated with the electric and magnetic fields that give rise to two gate function dependent gauge invariant Hermitian gated electric and magnetic field operators, defined by

In Sec. I and  Appendix A, we shall derive the commutation relation and the corresponding Heisenberg uncertainty relation that have the form

In Eq. (2), we have used a standard notation $(ΔQ^)=(ΔQ^)Ψ$ for the uncertainty of a Hermitian operator acting in Hilbert space (and thus defining an observable) in a quantum state |Ψ⟩,

Details of the derivation of Eq. (2), based on standard theory of light–matter interactions,^3,4 are presented in  Appendixes A and  B.

The connection between the uncertainty relation and commutator depends on the particle statistics and holds for any system of bosons. It applies in our case since the EM field is a system of free bosons (photons). By applying a proper quantization procedure, we can compute the commutator in Eq. (2), resulting in

In deriving Eq. (4), one should note the fact that the electromagnetic field is a system with first class constraints (using Dirac’s terminology⁵) that give rise to gauge invariance or, stated differently, that photons have transverse polarization.

The contribution to the measurement of the electric or the magnetic field at point r in space comes from some small region around r due to the spatial error bar of a measurement device. Here, we describe this by using Gaussian gates for both electric and magnetic fields. To this end, we introduce a family of Gaussian approximations for the Dirac δ-function,

and the corresponding Gaussian-gated electric and magnetic fields,

The Heisenberg uncertainty relation for (ΔE_j(r; l))(ΔH_s(r′; l)) is completely determined by the commutator of the corresponding gated operators,^3,6

and these immediately lead to the Heisenberg uncertainty relation

To derive Eq. (7), we applied Eq. (A15) and computed h(η, γ) for Gaussian gate functions making use of Eq. (4). The Gaussian integrations are easily performed. A similar calculation shows that in the case of not necessarily identical Gaussian gates, the Heisenberg uncertainty relation for (ΔE_j(r; l))(ΔH_s(r′; l′)) still has a form of Eq. (8), with the gate size l in the rhs replaced by

Equation (8) provides a clear physical interpretation for the 1/l parameter in the Heisenberg’s uncertainty formula. Equation (8) depends on a single parameter: the gate size l. The $(ℏc)/(2l)$ factor represents the 1/l Heisenberg’s parameter in energy units; the $2$ factor is specific to the Gaussian form of the gate. More precisely, the $(ℏc)/(2l)$ factor is the energy of a photon, whose wavelength is given by the gate size. The second factor $1(4πl)3$ is of the order of the inverse volume of the gate region. Their product thus represents the energy density of a photon, restricted to the gate region (according to the Heisenberg uncertainty principle for a photon, considered as a quantum particle). It is worth noting that (ΔE)(ΔH) has units of energy density. The dimensionless product of the last two factors in Eq. (8) characterizes the overlap of the gates so that when |r − r′|≫ l, the uncertainty vanishes. It is also interesting to note that at r = r′, the uncertainty vanishes as well, provided that the gate profiles are identical. One way to look at this dependence is that even if the original field is classical and contains many photons, the nano-gated field may contain only a few photons, and for a sufficiently short gate, it will, therefore, show quantum effects.

To get a sense of the fluctuation magnitudes, we compare (ΔE)(ΔH) for a Gaussian gate with l = 1 nm = 10⁻⁹ m with the EH product in a pulse with the intensity I = 10¹⁵ J s⁻¹ cm⁻² = 10¹⁹ J s⁻¹ m⁻². The maximal value of the dimensionless function [the product of the last two factors in the rhs of Eq. (8)] is $1/e≈0.6$ so that we have for l = 1 nm = 10⁻⁹ m and hc ≈ 2 × 10⁻²⁵ J m,

Making use of the Poynting vector,

we have

so that

Finally setting E = H for a plane wave and ΔE = ΔH for a coherent state of the electromagnetic field, we obtain

implying that for a nanometer gate, quantum effects are non-negligible even for a very strong pulse, which is usually thought of as classical.

We now compute the uncertainties of the gated and electric and magnetic fields prepared in a coherent sate. Since, in the coordinate representation, this state is simply a displaced ground state, the problem is reduced to computing the variance of the quantum fluctuations of the gated fields in vacuum, i.e., we have

The computation is carried out by introducing the photon creation/annihilation operators, see  Appendix B for details.

Applying the continuum limit to Eqs. (B12) and (B13), we obtain

Hereafter, we assume a Gaussian gate, i.e., η(r) = uδ(r; l) and γ(r) = $v$δ(r − r₀; l), with δ(r; l) given by Eq. (5), and u and $v$ are unit vectors. A simple inspection of Eq. (16) shows that the uncertainties do not depend on a particular choice of u, $v$⁠, and r₀ (reflecting the Poincare symmetry of the vacuum state), and, for our case, we have ΔH(γ) = ΔE(η) so that we need only compute ΔE(η). Choosing the z-axis along u and applying the spherical coordinates, we obtain upon substitution of the Fourier transform

of δ(r; l) into Eq. (16),

so that

In Sec. II, we identified the maximal value of the rhs of the Heisenberg uncertainty principle for the EM field for Gaussian gates of identical shape, with respect to the shift between the gates [see Eq. (10)], so that the uncertainty Heisenberg principle is, indeed, satisfied,

The commutator of $Ê$ and $Ĥ$ is a number that determines the lower bound of their uncertainty. This bound does not depend on the quantum state of the field and exists also in the vacuum state. We can then associate it with vacuum fluctuations. The ΔE and ΔH uncertainties do depend on the quantum state of light. We thus found that the vacuum state (and, hence, the coherent state) fluctuations are larger than the minimal uncertainty. This is similar to the results, reported in Ref. 7, that the minimal product of precision in intensity and degree of spatial localization is achieved not at coherent states but rather at the Epanechnikov distributions. Points of future interest include (i) finding a quantum state that satisfies the minimum uncertainty; (ii) investigating the connection of the presented uncertainty relations to the intensity vs spatial localization counterparts, explicitly considered in Ref. 7; and (iii) minimizing ΔE or ΔH to create squeezed fluctuations.^8,9 Another future goal is to identify an experimental signal that is sensitive to this uncertainty, e.g., chirality in a nanostructure.^10–12 

An alternative experimental realization of the present ideas is possible using nano-structures rather than spatial gating. Let us consider a setup whereby a nanosample interacts with a small part of a beam. The entire beam has many photons and is classical, but the relevant part of the beam that interacts with matter in this measurement is only a nanoslice that contains very few photons and should, therefore, be described by a quantum light. The measurement should thus show quantum light effects even though it employs a classical field! A simple analogy of this state of affairs exits in the time/frequency (rather than space) domain. When a short broadband pulse interacts with a system that has a narrow spectral line, only a few of its spectral components participate, and the experiment can be described by an effective narrow band (and long) pulse. The temporal resolution is eroded and not given by the pulse duration. The experiment may be then described by an effective pulse, which is different from the incoming pulse.¹³ This is the temporal analog of the spatial gating discussed here.

V.Y.C. gratefully acknowledges the support of the U.S. Department of Energy, Office of Sciences, Material Sciences and Engineering Division, Condensed Matter Theory Program. S.M. gratefully acknowledges the support of the National Science Foundation (Grant No. CHE-1953045).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

In this appendix, we derive the commutation relation [Eq. (4)] between the gated electric and magnetic field operators. We start with quantization of the electromagnetic field in vacuum, considered as a dynamical system. Its classical action is given by

To switch to the (classical) Hamilton formalism, we identify the conjugate momenta as variational derivatives,

and the second relation π₀(r) = 0 indicates that we are dealing with a dynamical system with constraints. The classical Hamiltonian is obtained in a standard way,

where we have made use of Eq. (A2). The Poisson bracket has a standard form

For the constraint E₀(r) to be preserved by the dynamics, we need to have $Ė0(r)=0$⁠, which combined with the Hamilton equation $Ė0(r)={H,Ė0(r)}$ and upon a direct computation of the rhs of the latter [making use of Eqs. (A3) and (A4)], yields div E(r) = 0, referred to as the secondary constraints. Computation of the time derivative of the secondary constraints yields 0, meaning that there are no higher-order constraints so that the complete set of them is given by

Obviously, the Poisson bracket between constraint is zero so that all constraints are type-1 in Dirac’s classification.

Gauge invariant quantization is performed using a canonical quantization of a dynamical system with type-2 (weak) constraints. A state in the extended Hilbert space is represented by a wavefunction Ψ[A(r), A₀(r)] with the electric field operators naturally defined as variational derivatives,

The constraints are applied in a weak sense, i.e., we introduce a physical subspace of the wavefunctions that satisfy the conditions $Ê0(r)Ψ=0$ and $divE^(r)Ψ=0$⁠, or explicitly

which means that a physical wavefunction is the one that does not depend on the scalar potential and the longitudinal component of the vector counterpart. A physical, or equivalently a gauge invariant, operator is the one that acts closely in the physical subspace.

To simplify the application of constraints, we switch to the momentum domain. To this end, we consider the space as a box of size L (with volume V) with periodic boundary conditions so that the quantization conditions for the photon wavevector are kL = 2πn, with $n∈Z3$⁠, allowing us to represent

We further associate with each allowed k an orthonormal basis set $(ek(α)|α=1,2,3)$ with $ek(3)=n=|k|−1k$ and represent

resulting in

Applying the constraints, we have $Ek(3)=0$ so that

and denoting with (u_j|j = 1, 2, 3) a lab frame, we compute

where the summation in Eq. (A12) runs over α, β = 1, 2. We further compute

Note that in the first expression in the second line in Eq. (A13), the summation runs over α, β = 1, 2, whereas to obtain the next equality in the chain, the summation has to be extended to its full range α, β = 1, 2, 3. This is, indeed, allowed since the summand with β = 3 turns out to 0, which also implies that the scalar product of the summand with α = 3 with the sum over β also turns out to zero so that the summations can be safely extended to their full ranges. Upon the substitution of Eq. (A13) into Eq. (A12), we obtain

Finally, we make use of Eq. (A14) to compute the Poisson bracket of the gated variables,

with h(η, γ) given by Eq. (4).

Upon canonical gauge invariant quantization of the electromagnetic field, Eq. (A15) provides the expression for the commutator of the gated operators, given by Eq. (2).

We start with representing the EM field operators in terms of the creation/annihilation operators,

together with a choice $e−k(α)=ek(α)$⁠, u_−k = u_k, and u_−k = u_k so that the conditions $Âα,−k=Âα,k†$ and $Êα,−k=Êα,−k†$ that reflect the real nature of the electromagnetic field are satisfied. Upon quantizing the expression for the Poisson bracket [Eq. (A10)] and postulating the commutation relations for the photon operators, we obtain

and by substituting Eq. (B1) into Eq. (B2), we obtain

We further have

and the Hamiltonian $H$ of the EM field in vacuum reads

with

Combining the condition that the unwanted terms [the last two terms in Eq. (B6)] disappear with Eq. (B3), we obtain a system of two equations,

whose solution

results in the explicit expressions,

Upon substitution of Eq. (B8) into Eqs. (B5) and (B6), we obtain the standard expression for the Hamiltonian $H$ of the EM field, recast in terms of the photon creation/annihilation operators,

Upon the substitution of Eqs. (A11) and (B9) into Eq. (1), we obtain

and further, making use of Eq. (B11) and applying abbreviated notation ⟨$•$⟩ = ⟨ Ω| $•$ |Ω⟩, we obtain

A similar computation for the uncertainty of the magnetic field yields

These result in Eq. (16) in the main text.