On the Heisenberg limit for detecting vacuum birefringence

Quantum electrodynamics predicts the vacuum to behave as a non-linear medium, including effects such as birefringence. However, for experimentally available field strengths, this vacuum polarizability is extremely small and thus very hard to measure. In analogy to the Heisenberg limit in quantum metrology, we study the minimum requirements for such a detection in a given strong field (the pump field). Using a laser pulse as the probe field, we find that its energy must exceed a certain threshold depending on the interaction time. However, a detection at that threshold, i.e., the Heisenberg limit, requires highly non-linear measurement schemes - while for ordinary linear-optics schemes, the required energy (Poisson or shot noise limit) is much larger. Finally, we discuss several currently considered experimental scenarios from this point of view.


I. INTRODUCTION
Classical electrodynamics is governed by the Maxwell equations, which are linear in the absence of sources. Thus electromagnetic waves in vacuum obey the superposition principle and do not interact [1]. Quantum electrodynamics, on the other hand, predicts deviations from this behavior: Even the quantum vacuum should be polarizable and thus behave as a non-linear medium due to the coupling to the fermionic modes, see, e.g., [2][3][4][5][6][7][8][9][10][11].
However, since this polarizability is extremely weak for available fields, this fundamental prediction of quantum electrodynamics has not been experimentally verified yet for electromagnetic waves in vacuum (i.e., real photons). Note that an analogous effect has been observed for the interaction of photons with the Coulomb field of atomic nuclei (referred to as Delbrück scattering) [12][13][14][15].

II. EULER-HEISENBERG LAGRANGIAN
Let us start with a brief recapitulation of the basic principles. For slowly varying electromagnetic fields E and B well below the Schwinger critical field determined by the electron mass m and the elementary charge q i.e., the range we are interested in, the propagation can be described by the lowest-order Euler-Heisenberg La-grangian density [3,7,8,11,48] with the vacuum permittivity ε 0 and the pre-factor where α QED ≈ 1/137 is the fine-structure constant. From now on, we shall employ natural units with in order to simplify the expressions.

A. Pump and probe field
We consider the standard situation where we have a strong (but sub-critical) pump field E 0 and B 0 acting on the vacuum plus a weaker probe pulse E 1 and B 1 in order to detect the induced polarizability. The pump field is supposed to be a solution of the equations of motion stemming from (2) and we study the propagation of the probe field in this background. Then, inserting the split (2) and linearizing the equations of motion in E 1 and B 1 , we obtain the effective Lagrangian for the probe field The polarizability of the vacuum is encoded in the change of the dielectric permittivity tensor δǫ and the magnetic permeability tensor δµ as well as the symmetry breaking tensor δΨ. These quantities depend on the strength of the pump field E 0 and B 0 (see the Appendix) and are suppressed as O(α QED [E 2 0 + B 2 0 ]/E 2 crit ). Since they are very small, we only keep their first order.
Note that we consider the modifications in the propagation of the probe pulse induced by the pump field. Thus, we do not consider other non-linear QED effects such as photon splitting or four-wave mixing, see, e.g. [49][50][51][52][53][54][55][56][57][58][59][60][61][62], which would correspond to linear or cubic powers of E 1 and B 1 in (5). In the following, we shall focus on the probe field and consider the tensors δǫ, δµ and δΨ as externally given. Thus, we shall drop the sub-scripts for the probe field E 1 and B 1 from now on.

B. Interaction Hamiltonian
In terms of the usual vector potential A, the canonically conjugate momentum (which equals the dielectric displacement field D) reads and thus the Hamiltonian density is given by After splitting off the undisturbed vacuum contribution H 0 = [Π 2 + B 2 ]/2, the remaining part describes the interaction between the probe field and the polarizability δǫ, δµ and δΨ induced by the pump field

III. HEISENBERG LIMIT
As announced above, let us now study the question of which requirements the probe pulse has to fulfill in order to detect the vacuum polarizability δǫ, δµ and δΨ. According to the laws of quantum mechanics, this effect is only detectable if the quantum state |ψ of the probe field after its interaction with the pump fieldÛ int |ψ deviates sufficiently from the quantum state |ψ without this interaction. As one possible signature, the no-signal fidelity given by (see also [63]) should sufficiently deviate from unity (T is the time ordering operator). Since the interaction Hamiltonian is linear in the small tensors δǫ, δµ and δΨ, let us apply first-order perturbation theory We see that the lowest-order contribution corresponds to a phase shift [63] ϕ = − dt ψ|Ĥ int (t)|ψ , which could be measured by interferometric means, for example (see below).

A. Classical fields
Let us estimate the maximum possible phase shift (12) for a given probe pulse. First, we treat the probe pulse as a classical field, which should be a good approximation for laser pulses. Inserting Eqs. (10) and (8) into (12), we obtain space-time integrals over the terms Π · δǫ · Π/2, B · δµ · B/2 and Π · δΨ · B. Since the tensor δǫ is real and symmetric, we may diagonalize it and obtain the bound where the norm ||δǫ|| is the maximum of the absolute values of the eigenvalues of δǫ. In complete analogy, we can bound the term B · δµ · B by the same norm ||δµ|| multiplied by B 2 . Thus, we find at each space-time point, where E = (Π 2 + B 2 )/2 is the (lowest-order) energy density of the probe pulse. The remaining term Π·δΨ·B is a bit more complicated because the tensor δΨ is not symmetric in general. Thus, we employ the singular value decomposition with the non-negative singular values σ I and the two (left and right) orthonormal basis sets u I and v I . Then, using (u I · Π)(v I · B) ≤ |Π||B| ≤ (Π 2 + B 2 )/2, we arrive at where T denotes the interaction time and E the total energy of the probe pulse. The spatial integral can be bounded from above by the maximum over all positions r since all the involved quantities, such as the energy density E, are non-negative (for classical fields, quantum fields will be discussed in the next section). Turning the above argument around, we get a minimum energy E of the probe pulse required for detecting the vacuum polarizability δǫ, δµ and δΨ in a given interaction time T since the phase shift ϕ should not be too small in order to achieve a measurable effect. Since the energy E scales linearly with the number N of probe photons, we refer to (16) as the Heisenberg limit.

B. Quantum fields
In the previous section, we treated the probe pulse as a classical field in order to derive the bound (16). In the following, let us study whether an analogous bound can be established for quantum fields. As a crucial difference, expectation values such as Π 2 or Π · δε ·Π are divergent and thus require renormalization. As usual, we achieve this by subtracting the vacuum expectation value Of course, this requires appropriate regularization. Here, we use the normal mode decomposition. To this end, we introduce a complete set of orthonormal and transversal ∇·f I = 0 basis functions f I (r) and expand the field operates into this basis set Inserting this normal mode decomposition, we find After diagonalizing this real and symmetric matrix M IJ via the orthogonal matrix O IJ , we may introduce a new set of basis functions via F I = J O IJ f J and expand the field operator in this new setΠ(t, r) = IP I (t)F I (r) which gives the simplified expression where λ I are the eigenvalues of the matrix M IJ . Since thê P 2 I are positive operators, we may even derive a (formal) bound on the operator level where ||M || = max I |λ I | is the norm of the matrix M IJ in analogy to the previous section. It can be estimated by the maximum "expectation value" d 3 r f · δε · f for normalized functions f and thus agrees with max r ||δε||.
The above operator-valued bound (22) seems to be the proper quantum generalization of the Heisenberg limit (16)  I ) in terms of the eigenmodes I with the eigenfrequencies Ω I . However, this bound is of limited use since the expectation value diverges due to the infinite zero-point energy (as mentioned above). After subtracting this zero-point energy (17), we cannot deduce the inequality p 2 I ren ≤ p 2 I ren + Ω 2 I q 2 I ren anymore since the renormalized expectation values p 2 I ren and q 2 I ren can become negative. For example, in a squeezed state |0 → |r , we may increase the momentum variance p 2 I → exp{+r} p 2 I while decreasing the position variance q 2 I → exp{−r} q 2 I , such that it is below its ground-state value q 2 I < 0|q 2 I |0 which means that q 2 I ren becomes negative. Thus, if we naively replace the classical energy E in the Heisenberg limit (16) by the renormalized expectation value E ren for quantum fields, it would be possible to violate this bound by squeezing many modes just a little bit r ≪ 1 such that their p 2 I ren ∼ r increase while the growth of the energy E ren ∼ r 2 is suppressed. One could suspect that this enhancement would be compensated by the other terms such asB · δµ ·B which contains q 2 I ren but this is not the case since different modes contribute differently to these terms. Thus, one could squeeze those modes where the first termΠ · δε ·Π dominates in one way p 2 I → exp{+r} p 2 I and the other modes where the second termB · δµ ·B dominates in the opposite way p 2 I → exp{−r} p 2 I . For simplicity, we have omitted the third term ∝ δΨ since it has yet another mode structure, see also Sec. V A.
In summary, the divergent zero-point energy invalidates a bound like (16) for quantum electrodynamics. To obtain a generalized bound, one would have to limit the number of involved modes I as well as their eigenfrequencies Ω I , which is difficult [84].

IV. COMPARISON TO POISSON LIMIT
Coming back to the Heisenberg limit (16), one might object that a global phase ϕ cannot be measured. While this is correct in principle, this objection could be circumvented by considering a scenario involving a quantum superposition of two paths of the probe pulse, one interacting with the pump field and the other one not. This state corresponds to a NOON state [64][65][66] where either all N probe photons take the one path |N, 0 or all N probe photons take the other path |0, N . Note that this is a highly non-classical state, in analogy to the Greenberger-Horn-Zeilinger (GHZ) state [67,68]. After interaction with the pump field (in one path only), this state evolves into (|N, 0 + e iϕ |0, N )/ √ 2 which becomes orthogonal to the initial state (23) for ϕ = π. Note, however, that both, preparing the initial state (23) as well as measuring the final state (|N, 0 + e iϕ |0, N )/ √ 2 would require effectively N -photon interactions, i.e., a highly non-linear optics scheme.
In a typical linear optics set-up the (coherent) state of a laser is described by the factorizing state where each photon ℓ individually either takes the one path |1, 0 or the other path |0, 1 . In this case, one would obtain a Poisson distribution of the photon numbers in the output channel and thus the accuracy scales with 1/ √ N instead of 1/N , which is the well-known classical Poisson (shot-noise) limit.
Let us illustrate this distinction in terms of the scaling of the phase with photon number N . According to the Heisenberg limit (16), we find where ∆ϕ 1 is the phase shift experienced by a single photon with frequency ω Since ∆ϕ N must be of order unity to obtain a measurable detection probability, we get the well-known Heisenberg scaling ∆ϕ 1 ∼ 1/N . In contrast, the Poisson distribution of the photon numbers in the classical (i.e., coherent) state (24) results in a relative accuracy of 1/ √ N which yields the well-known Poisson limit ∆ϕ 1 ∼ 1/ √ N , see [69].

A. Static magnetic pump field
There are several running or planned experiments where the pump field is an approximately constant magnetic field of a few Tesla, see, e.g., [36,70]. For a purely magnetic field, the symmetry-breaking term δΨ vanishes (see the Appendix). The maximum eigenvalues of the remaining terms δε and δµ are given by 10ξB 2 0 and 12ξB 2 0 , respectively, which are then of order 10 −22 . Thus, the accuracy requirements are roughly comparable to those for the detection of gravitational waves at LIGO [71].
As in LIGO, the signal can be amplified by having the probe photons bounce back and forth many times (in a cavity, for example), which facilitates a large integration time T . Assuming an optimized cavity finesse of order 10 6 and length scales of order meter, we get an integration time of order ωT = O(10 12 ) periods for optical or near-optical photons. Again in analogy to LIGO, the remaining orders of magnitude should be compensated by a sufficiently large number of probe photons. Using the Heisenberg limit (16), we would get N = O(10 10 ) which is a comparably low number. However, as explained above, this detection scheme would require an effective N -photon interaction involving this number of photons, which is currently out of reach.
With laser fields and linear optics schemes, we can only reach the Poisson limit, which gives N = O(10 20 ) corresponding to a probe pulse energy E in the Joule regime. This shows that such an experiment is not impossible with present-day-technology but still quite challenging.
Note that the actual limit is even a bit larger because these experiments typically do not measure the polarizabilities δε and δµ directly, but only the induced rotation of polarization -which measured their difference in the different directions. Otherwise, the rotation of polarization is very similar to an interferometric set-up, where the two arms correspond to the two polarizations.

B. Optical pump and XFEL probe
Another popular scheme envisions a strongly focused ultra-strong laser pulse (again in the optical or nearoptical regime) where intensities of order 10 22 W/cm 2 or more should be reachable with present-day or nearfuture technology see, e.g., [72][73][74]. This corresponds to electric fields above 10 14 V/m which generate polarizabilities δǫ, δµ and δΨ of order 10 −11 . This illustrates a major advantage in comparison to the static set-up in the previous section V A, as the pump field is much stronger in a laser focus. As a drawback, the interaction time T is limited to the pump pulse length of a few (say ten) optical cycles.
However, for a probe pulse generated by an x-ray free electron laser (XFEL) with photon energies in the 10 keV range, this corresponds to O(10 5 ) XFEL cycles, see, e.g., [24]. The Heisenberg limit (16) then gives N = O(10 6 ) photons, i.e., an energy of O(10 10 eV) or O(10 −9 J). Again, as an N -photon interaction with these numbers seems unrealistic, the Poisson limit yields N = O(10 12 ) photons, corresponding to an energy of O(10 16 eV) or O(10 −3 J). As before, this shows that the detection is challenging but not completely out of reach.
In analogy to the previous section V A, the envisioned scheme is based on the rotation of the polarization which offers experimental advantages in comparison to an interferometric set-up with x-rays, but decreases the signal a bit. Note that, with N = O(10 11 ) photons in an initially polarized probe beam (see also [75][76][77][78][79]), the signal may consist of a single photon with flipped polarization after several runs. This necessitates a careful study of potentially competing effects in order to distinguish the signal from the background.

C. Optical pump and optical probe
In contrast to the scenario described above, one could also consider an optical (or near-optical) probe pulse, see also [28]. Using the same parameters for the pump pulse, the Heisenberg limit (16) would yield the same energy E as in the previous section V B. However, the probe pulse would now contain N = O(10 10 ) photons because the interaction time corresponds to a few optical cycles only. The Poisson limit then yields N = O(10 20 ) corresponding to an energy of O(10 20 eV) or O(10 J). The fact that this is of the same order as the pump pulse itself shows the challenges of this detection scheme.
On the other hand, for this all-optical scheme, it is not necessary to have the optical laser close to an XFEL. Thus, it might be possible to reach even higher intensities in the 10 23 W/cm 2 regime, which reduces the requirements on the probe pulse to N = O(10 18 ) photons, i.e., an energy of O(10 18 eV) or O(10 −1 J). As one possibility, one could use a dual-beam facility (see, e.g., [80]) or spit off a small part of the pump pulse before focusing and use it as probe pulse. This could help ensuring the necessary temporal overlap between pump and probe pulse, which can pose a challenge for the scheme described in the previous section V B. Still, performing interference experiments with such O(10 −1 J) pulses containing N = O(10 18 ) photons is highly non-trivial. Again in analogy to LIGO, it might be advantageous to operate the interferometer not exactly at the dark spot, but close to it -corresponding to a small phase mismatch ϕ 0 between the two arms. For example, using a phase mismatch of ϕ 0 = 10 −3 (corresponding to a precision of placing the mirrors in the nanometer regime), the output at the darker port would contain O(10 12 ) photons. A single-photon phase shift of ∆ϕ 1 = O(10 −9 ) would then generate a signal of O(10 6 ) photons difference on top of the background of O(10 12 ) photons. Measuring such a large photon number with a relative accuracy of O(10 −6 ) is challenging and probably requires an advanced detector with many mega-pixels.
On the other hand, this set-up could also offer an advantage as the signal would now contain many O(10 6 ) coherent photons, which could help distinguishing it from the background, especially from incoherent noise.

D. Angular dependence
Especially for the latter all-optical scenario, it is probably unrealistic to assume a head-on collision between pump and probe pulse. Thus, let us estimate the angular dependence of the phase shift. For simplicity, we model pump and probe pulse as plane waves (as a first step). Then, their relative directions can be described in terms of the three Euler angles ψ, θ, and φ. Without loss of generality, the pump field is supposed to propagate in z direction with E 0 and B 0 pointing in x and y directions, respectively. After starting with the same orientation, the probe field directions are obtained by three rotations: First, a rotation around the z axis with the angle φ, second, a rotation around the new x axis with the angle θ, followed by a third rotation around the new z axis with the angle ψ.

Then, the interaction Hamiltonian reads
For the co-propagating case θ = 0, it vanishes identically (as is well known). In this case, the angles ψ and φ just rotate the polarization and one can transform into a comoving Lorentz frame where all fields E 0 and B 0 as well as E 1 and B 1 become arbitrarily small. The maximum is obtained in the counter-propagating case θ = π where such a Lorentz boost diminishing all fields is not possible. In this limiting case, the angles ψ and φ again just rotate the polarization and the maximum signal is obtained by ψ − φ = −π/4. Apart from these well-known limiting cases, we see that this condition ψ−φ = −π/4 does also give the maximum signal for arbitrary given θ, which may be unavoidable due to experimental constraints. For small deviations ∆θ = θ − π from the optimal counter-propagating case θ = π, we find

VI. CONCLUSIONS
We study the general requirements for detecting the weak vacuum polarizability (5) predicted by quantum electrodynamics. We find that the lowest-order effect is a phase shift (12) which could, at least in principle, be detected by interferometric means. Approximating the probe pulse by a classical field, we obtain an upper bound (16) for the phase shift depending on the interaction time T and the total energy E of the probe pulse.
Since this phase shift must be of order unity for a measurable detection probability, this inequality does also give the Heisenberg limit (note that ET equals the number of photons N times the number of periods ωT ). However, such a measurement would require a highly nonlinear optics scheme, which is out of reach for realistic parameters. For a linear-optics scheme, we recover the well-known Poisson (shot-noise) limit.
Going beyond the classical field approximation, the failure of proving a bound as (16) for quantum fields hints at the interesting (theoretical) possibility to reach an even higher accuracy by exploiting the zero-point fluctuations. One option could be to squeeze many field modes a little bit such that their quadratures are modified (in order to increase the sensitivity) while the total energy expectation value does not change significantly. For optical frequencies, such a squeezing could be achieved in nonlinear crystals in analogy to parametric down conversion, while for XFEL frequencies, a corresponding undulator set-up could serve the same goal, see also [81]. However, apart from preparing this squeezed state initially, reading out the final state poses grand experimental challenges.
Altogether, we obtain three regimes: the linear-optics regime corresponding to the Poisson (shot-noise) limit, the Heisenberg limit (16) for (locally) classical fields, and a regime beyond that limit for quantum fields.
As a demonstration, we apply these limits to three experimental scenarios, which offer different advantages (e.g., control of polarization for XFEL fields) and disadvantages. For all cases, we find that the detection of the vacuum polarizability is quite challenging but not completely out of reach. Note that, apart from the verification aspect, the vacuum polarizability could also provide a clean way to measure the peak intensity of the laser, which is a highly non-trivial task.
This effective Lagrangian provides information about the vacuum polarization tensors (δǫ, δµ and δψ) which can be extracted after comparing with (5) Therefore these tensors are given in terms of the electric and magnetic components of the pump pulse which is a symmetric tensor, δµ is obtained from δǫ by interchanging E and B and finally the symmetry-breaking tensor As it has been already discussed in Section III, we need to compute the eigenvalues of δǫ and δµ and the singular values of δψ since the latter is in general an asymmetric tensor. It is straightforward to obtain the eigenvalues λ 1,2,3 of the symmetric tensors. For a general pump field, we obtain for δǫ In the limit of constant crossed fields they simplify to Then the spectral representation of δǫ is where v I are the eigenvectors of δǫ (after a siutable rotation of the coordinate system) v 1 = (1, 0, 0) , v 2 = (0, 1, 0) , v 3 = (0, 0, 1) , and for δµ (with its eigenvalues are denoted by Λ I ) the eigenvalues are simply obtained from the ones for δǫ by the following replacements In the limit of plane-wave background (or constant crossed fields), the two sets of eigenvalues for δǫ and δµ become equivalent -so to obtain the upper bound, it suffices to consider the maximum value of one of them. Bounding the terms in the Hamiltonian with δǫ and δµ is simple since they have real eigenvalues, therefore one can consider the maximum value of their eigenvalues as discussed in Section III A. The most nontrivial term is the one with δψ since it has no symmetry in general then a direct eigenvalue computation may lead to imaginary values which correspond to non-orthogonal set of eigenvectors. Therefore one can rely on singular value decomposition which are defined based on the following theorem [82]: If A is a real m × n matrix then there exist two orthogonal matrices U = [u 1 , · · · , u m ] ∈ R m×m and V = [v 1 , · · · , v n ] ∈ R n×n such that where p = min{m, n} and In other words the singular values σ 1 , · · · , σ p of a m × n matrix A are the positive square roots, σ I = √ λ I > 0, of the nonzero eigenvalues of the associated Gram matrix K = A T A. The corresponding eigenvectors of K are known as the singular vectors of A (note that for m = n or rectangular matrices there is no eigenvalues in its general definition and that is why one finds the singular values).
This theorem lead to the following singular decomposition for a non-symmetric matrix A ∈ R n×n and from here in which σ I are the singular values and the left and right singular vectors u I and v I for I = 1, 2, · · · , n respectively. Applying this theorem to δψ we get For a constant crossed field they give After having the eigenvalues of δǫ and δµ as well as the singular values of δψ one can easily compute the phase given in (16).

Appendix B: The rotation matrix
Let us consider two different frames for the pump and probe pulses in which the former is fixed to be denoted by xyz and the latter XY Z. We need three Euler angles to rotate XY Z [83]. The sequence of the rotations are the following: XY Z rotates by an angle φ about the Z-axis to obtain ξηζ with corresponding rotation matrix D. For the second rotation, ξηζ is rotated about the ξ-axes by an angle θ to obtain new axes called ξ ′ η ′ ζ ′ with rotation matrix C. Finally in the last step the later is rotated by an angle ψ about ζ ′ to obtain xyz with rotation matrix B. The three successive rotations lead to a transformation matrix A cos ψ cos φ − cos θ sin φ sin ψ cos ψ sin φ + cos θ cos φ sin ψ sin ψ sin θ − sin ψ cos φ − cos θ sin φ cos ψ − sin ψ sin φ + cos θ cos φ cos ψ sin θ cos ψ sin θ sin φ − sin θ cos φ cos θ Therefore we have the following equation where x = (x, y, z) and X = (X, Y, Z). To obtain a general form for the interaction Hamiltonian we need the probe electric and magnetic fields (E ω , B ω ). For a general probe field we have the following transformation if we consider a constant crossed background for the probe pulse in which the propagation direction lies again in Z and E ω and B ω in X and Y directions accordingly then the interaction Hamiltonian defined in (7) with the polarization tensors obtained in Appendix A one arrives at which has a maximum at (θ = π, ψ − φ = −π/4).