Single-photon source

Our single-photon source is based on a Sagnac interferometer, commonly used to create polarization-entangled photon pairs, but we generate photon pairs in a separable polarization state. Our Sagnac loop is built using a dual-wavelength polarizing beamsplitter (dPBS) and two mirrors. A type-II collinear periodically-poled Potassium Titanyl Phosphate (PPKTP) crystal of length 20 mm is placed inside the loop and pumped by a 23.7 mW diode laser centred at 395 nm. This results in photon pairs at a degenerate wavelength of 790 nm. The pump beam polarization is set to horizontal in order to generate the down-converted photons in a separable polarization state . The dichroic mirror (DM) transmits the pump beam and reflects the down-converted photons, and the half wave plate (HWP) and quarter waveplate (QWP) are used to adjust the polarization of the pump beam. Long (LP) and narrow band (BP) pass filters block the pump beam and select the desired down-converted wavelength. Polarizers are aligned to transmit only down-converted photons with the desired polarization. After this, the down-converted photon pairs are coupled into single-mode fibres (SMF), and one photon from the pair is used as a herald while the other single photon is sent to the rest of the experiment using a fibre collimator (FC).

Theoretical treatment of the Sagnac interferometer

Here we derive the probability of a photon incident on an imperfect Sagnac interferometer to exit the ‘dark port’ if two phases internal to the Sagnac interferometer do not commute.

We start with a single-photon incident on a 50:50 beamsplitter. Ideally, given a 50:50 beamsplitter and a reflection phase of π/2, the state of a photon after reflecting is:

where CW and CCW refer to the clockwise and counter-clockwise modes in Fig. 1a, respectively. Next, applying two phases (as in Fig. 1a), represented by operators A and B, we have

To be completely general we will assume that the ‘i’ does not commute with A and B.

The operators A and B can be represented as

where is the identity operator. In complex quantum mechanics and , where φ A and φ B are real numbers. In this case, α and β are complex numbers so A and B commute. However, in quaternionic quantum mechanics the phase φ A is generalized to vector , where , , and are real numbers. Then iφ A is replaced with i +j +k , where {i, j, k} is a basis over the imaginary part of the quaternionic space. With these definitions α and β in equation (7) become unit quaternions

Now, the operators A and B of equation (7) no longer commute in general. In fact, α and β could be even more general hyper-complex numbers, consisting of more than three imaginary components.

Next, by applying the form of the operators defined in equation (7), we can write the state in equation (6) as

In complex quantum mechanics, αβ=βα and the two complex numbers describe a global phase, so they have no effect on experimental outcomes. However, if α and β do not commute, the output state is

Thus the probability for an incident photon to exit the Sagnac interferometer via the dark port (the amplitude of the second term) is

This quantifies the degree of commutativity between α, β, and i. If α, β, and i all mutually commute it is zero. Moreover, if i commutes with α and β it simply becomes the commutator of α and β, as shown in equation (1) of the main text.

We will next treat the imperfect alignment of our interferometer. We start by writing the state from equation (9) as a density matrix

where the denotes the conjugate of a quaternion or complex number. Let our Sagnac interferometer have a visibility of v=(P B −P D )/(P B +P D ), where P D and P B are the intensities of the dark and bright ports, respectively. We can model this by simply scaling the coherences by v, as

This reduced coherence can be derived by coupling the CW and CCW modes to additional modes, and then tracing out those additional modes. This is a very general method to model imperfections since it does not require any assumptions on the types of imperfections: the CW and CCW modes could couple to additional spatial modes, temporal modes, etc.

Again, we can compute the probability to find the photon in the dark port by applying the beamsplitter transformation. Doing so yields

Then the probability of the photon to exit the bright port is simply P B =1−P D . Notice that P D defined here differs slightly from the equation (2) of the main text, in that |iαβ−βαi| replaces |[α, β]|. However, a non-zero value of this new quantity would also signify a deviation of QQM from CQM, and is, thus, also interesting to study.

Theoretical treatment of the Mach-Zehnder interferometer

After the Sagnac interferometer, the bright and dark ports are interfered in our Mach–Zehnder interferometer (Fig. 1b). Interfering two optical fields, with intensities of P B and P D , on a 50:50 beamsplitter results in a signal with a visibility of.

The same result holds if P B and P D are instead the probabilities of finding a photon in either path. Thus, the visibility of the Mach–Zehnder interferometer with both phases inserted in the Sagnac interferometer, can be computed from P D (equation (14)). After simplifying, we arrive at

where

This visibility V BOTH is a function of both the degree of commutativity |iαβ−βαi| and the visibility of the Sagnac interferometer v. To compare to our experimental procedure imagine that we turn off the liquid–crystal phase (which we represent by α) and leave the negative-index metamaterial inserted. Then α drops out and the degree of commutativity becomes the commutator of i and β, so equation (16) becomes

where

This visibility that depends on the commutation of the negative-index metamaterial with the reflection phase inside the Sagnac interferometer, and on the visibility v of the Sagnac interferometer.

By combining equations (16 and 18) we arrive at a result which does not depends only on two measurable visibilities of the Mach-Zehnder interferometer, and not on the visibility v of the Sagnac interferometer:

Thus we can experimentally determine the ratio Γ BOTH /Γ NIM from two visibilities of the Mach–Zehnder interferometer with and without the liquid–crystal phase turned on. The left-hand side of equation (20) simplifies to the Γ defined after equation (3) in the main text if i commutes with both α and β. Notice also that if |iαβ−βαi|=|[i, β]|≠0 this ratio will be one. Thus this parameter is insensitive to a very specific type of non-commutativity between α, β, and i where- in |iαβ−βαi|=|iβ−βi|. Physically this would be the case, for example, if α commutes with β and i, but β and i do not commute. The reason for defining the quantity Γ BOTH /Γ NIM will become clear in the next section.

Converting the visibility change into a phase change

In this section we will derive a figure of merit which provides additional physical intuition into our results. Namely, a difference in the net phase between the NIM phase being applied before the LC phase, and vice versa. In our experiment we measure the visibility of an interference signal which is proportional to the commutator of the two phases. This signal arises from interference between the dark and bright output ports of the Sagnac interferometer. As we show above, if two phases inside the Sagnac do no commute, light will leak into the dark port. Then interfering the bright and dark modes leads to an interference signal which has a visibility given by equation (15).

Imagine that leakage into the dark port arises from of a phase shift θ between the clockwise and the counter-clockwise modes of the Sagnac. Physically, this means that there is a different phase shift if the photon sees the metamaterial before or after the liquid-crystal. It is straightforward to show, within CQM, that if the two modes of a Sagnac interferometer experience a phase shift θ the probabilities of the photon exiting either port become

where v is the visibility of the Sagnac interferometer. Now substituting equation (21) into equation (15) we arrive at the visibility of the Mach-Zehnder interferometer as a function of the phase inside the Sagnac interferometer

Experimentally, we measure two visibilities of the Mach–Zehnder interferometer, which we now attribute to a phase change in the Sagnac interferometer. In the present picture, V(θ) and V(0) are the visibilities of the Mach-Zehnder interferometer with and without a phase difference between the clockwise and counter-clockwise modes. Thus, we will equate V(0) to the visibility when only one phase is inside the Sagnac interferometer V NIM ≡V(0), and V(θ) to the visibility when both phases are in the Sagnac interferometer V BOTH ≡V(θ). Then we will substitute equation (22) into (20), simplifying and solving for θ. Doing this yields

We can then understand this θ as an effective phase shift between the clockwise and counter-clockwise modes, arising from the non-commutativity of the phases. So we see that measuring these two visibilities allows us to use equation (23) to convert our result into this phase. Doing this, and using Gaussian error propagation on equation (23) results in θ<0.03°.

Fitting to extract visibility

To extract the visibility from the normalized data we fit a sinusoid to the data, and calculate the visibility from the fit parameters. The explicit form of our fitting equation is

where ɛ, κ, f, and p are all free parameters. The visibility of this curve in equation (24) in terms of the fit parameters is

We compute the error on each visibility using Gaussian error propagation, starting with the fitting uncertainties.

Negative-index metamaterial

We use a fishnet metamaterial to achieve an optical medium with a negative refractive-index. Our fishnet negative index metamaterial (NIM) consists of seven physical layers of silver (Ag, 40 nm) and magnesium fluoride (MgF 2 , 50 nm), with a 15 nm capping layer of MgF 2 . The metamaterial is suspended to avoid any positive phase contribution from the substrate. Figure 1c shows the resulting negative phase shift of our NIM as a function of the wavelength of the light, and the inset shows an SEM image of the surface of our NIM. Supplementary Notes 1–3 contains complete details of the design, fabrication, and characterization of our NIM.

In our experiment, the NIM is mounted on an automated translation stage so that it can be reliably and repeatably removed and inserted. It has a clear aperture of approximately 20 μm, thus we focus the beam sufficiently to pass through it. To find the optimal position of the NIM, we scan the translation stage, while monitoring the transmission of both the clockwise and counter-clockwise modes of the Sagnac interferometer. We align the sample, relative to the focus of the lenses, such that the transmission of both modes is maximized at the same position. Another point of concern is the significant back reflection (≈50%) of the NIM for our wavelength range. Since this back reflection can couple to our detectors, we slightly tilt the NIM, by 0.44°, to reduce this background signal. We tilt the NIM along a carefully chosen axis so as to keep the polarization parallel to the thinner lines of the fishnet nanostructures, it has be shown that in this configuration such metamaterials still work optimally54.

Liquid crystal retarder

We use a commercial nematic liquid crystal cell whose molecules orient to an applied electrical field. We characterize the LC by placing it between two polarizing beamsplitter cubes with its optical axis at an angle of 45°. We then measure the light intensity transmitted through the second PBS as we vary the voltage applied to the LC. Since the transmitted intensity is proportional to (1+cos ζ), where ζ is the relative phase imparted by the LC, this measurement allows us to determine the relative phase (modulo 2π) effected by the LC as a function of the applied voltage. The measured relative phase of the LC is shown in Fig. 1d.

Data availability

The data that support the findings of this study and the computer code to analyse it are available from the corresponding authors on request.