Observation of Three-Photon Spontaneous Parametric Down-Conversion in a Superconducting Parametric Cavity

Spontaneous parametric down-conversion (SPDC) has been a key enabling technology in exploring quantum phenomena and their applications for decades. For instance, traditional SPDC, which splits a high-energy pump photon into two lower-energy photons, is a common way to produce entangled photon pairs. Since the early realizations of SPDC, researchers have thought to generalize it to higher order, e.g., to produce entangled photon triplets. However, directly generating photon triplets through a single SPDC process has remained elusive. Here, using a flux-pumped superconducting parametric cavity, we demonstrate direct three-photon SPDC, with photon triplets generated in a single cavity mode or split between multiple modes. With strong pumping, the states can be quite bright, with flux densities exceeding 60 photons per second per hertz. The observed states are strongly non-Gaussian, which has important implications for potential applications. In the single-mode case, we observe a triangular star-shaped distribution of quadrature voltages, indicative of the long-predicted “star state.” The observed state shows strong third-order correlations, as expected for a state generated by a cubic Hamiltonian. By pumping at the sum frequency of multiple modes, we observe strong three-body correlations between multiple modes, strikingly, in the absence of second-order correlations. We further analyze the third-order correlations under mode transformations by the symplectic symmetry group, showing that the observed transformation properties serve to “fingerprint” the specific cubic Hamiltonian that generates them. The observed non-Gaussian, third-order correlations represent an important step forward in quantum optics and may have a strong impact on quantum communication with microwave fields as well as continuous-variable quantum computation.

Popular Summary

For over 30 years, spontaneous parametric down-conversion (SPDC) has been a workhorse for quantum optics. By splitting one “pump photon” into two daughter photons, SPDC has had a crucial role in fundamental tests of quantum theory as well as many applications in quantum information processing. From the early days, researchers have explored splitting the pump photon into three photons (as a possible resource in quantum computation, for example), but it has proven extremely difficult to realize experimentally—until now. Here, we report on an implementation of three-photon SPDC in the microwave domain.

To split one microwave photon into three daughter photons, we use a flux-pumped, superconducting parametric resonator. Our triplet source is bright, producing a propagating photon flux comparable to ordinary two-photon SPDC. We clearly see strong three-photon correlations in the output photons, even in the absence of normal two-photon correlations. The symmetry properties of these correlations allow us to “fingerprint” how the photons were created, clearly demonstrating little contamination from typical SPDC processes.

These results form the basis of an exciting new paradigm of three-photon quantum optics. One can only hope that this new paradigm will be as successful as two-photon quantum optics.

Figure 1

Meandered SQUID design with improved pump coupling. The narrow, meandered path is shared between the SQUID and the ground plane of the pump line. The kinetic inductance of the meander then directly couples the pump current in the ground plane to the SQUID phase. The center conductor of the pump line is narrowed to reduce its magnetic coupling to the SQUID, which contributes with the opposite sign of the kinetic coupling. Following microwave simulations, a mirrored structure is patterned in the top ground plane, such that the ground current divides evenly between the two ground planes. Compared to previous designs, we find a reduction in the pump strength required for TPDC by 3 orders of magnitude. The reduction greatly reduces the spurious effects of strong pumping and is critical to realizing three-photon down-conversion.

Figure 2

Measured histograms of trisqueezed states in a single mode. (a) A histogram of a bright trisqueezed state with $F\approx 66$. The triangular symmetry of the histogram is apparent, clearly indicating its non-Gaussian character. The inset shows the projection of the 2D histogram onto the $x$ quadrature, which displays a significant amount of asymmetry (skewness), despite the system noise. (b) A histogram of a trisqueezed state with $F\approx 1$ after subtracting the system noise histogram. The triangular symmetry is still apparent despite the low photon flux. (c) A polar plot of the skewness ${\gamma }_{\phi }$ of the generalized quadrature $x_{\phi }$ (see the text) as a function of the phase angle $\phi$, showing data and theory. The small rotation of ${\gamma }_{\phi }$ is caused by a relative phase between the microwave pump and the local oscillator of the heterodyne receiver. The theory curve is normalized to the maximum experimental value and rotated appropriately but otherwise has no adjustable parameters. The plot is produced from the same raw data as (b) but without subtracting the system noise. This statistic essentially measures the asymmetry of the quadrature distribution with respect to the symmetry axis perpendicular to $x_{\phi }$. We see that the triangular symmetry of the underlying state is very visible even though it is completely obscured in a histogram of the raw data, which suggests that ${\gamma }_{\phi }$ is a good way to analyze the non-Gaussian character of our trisqueezed states. This approach is particularly effective because the skewness of the Gaussian system noise is zero (within our measurement error).

Figure 3

Covariance and coskewness of three-photon SPDC to (top) two modes and (bottom) three modes measured at room temperature. In the two-mode case, we have a degeneracy between two of the three generated photons. This degeneracy leads to one mode participating twice in the nonzero coskewness term. For our choice of pump phase, the only significant coskewness terms are then $⟨I_1^2{I}_2⟩$ and $⟨Q_1^2{I}_2⟩$. In the three-mode case, the only nonzero coskewness term must contain all three modes, e.g., $⟨I_1{I}_2{I}_3⟩$. To show that the observed coskewness is coherently generated by the pump, we sweep the pump phase from $-2\pi$ to $2\pi$ and observe the effect. We see a clear oscillation in the coskewness. The oscillations are fit well by a sinusoid. Meanwhile, all covariance terms are essentially zero throughout the sweep, indicating that the generated states do not contain second-order correlations.

Figure 4

Measured coskewness in (top) the two-mode case and (bottom) the three-mode case. In the two-mode case, we consider the three quadratures ${\widehat{I}}_1$, ${\widehat{Q}}_1$, and ${\widehat{I}}_2$ as a representative sample. The pump frequency is chosen such that mode 1 is the degenerate mode. Accordingly, we expect only $⟨I_1^2{I}_2⟩$, $⟨Q_1^2{I}_2⟩$, and $⟨I_1{Q}_1{I}_2⟩$ to be nonzero, which is what is observed. Their relative values are determined by the pump phase [Eq. (b5)]. In the three-mode case, we consider the $\widehat{I}$ quadratures of all three modes. As expected, the only nonzero term includes all three modes, i.e., $⟨I_1{I}_2{I}_3⟩$.

Figure 5

Third-order correlation analysis of multimode trisqueezed states. Rows 1 and 2 show results for, respectively, the two-mode and three-mode trisqueezed states, generated by ${\widehat{H}}_{2\mathrm{M}}$ and ${\widehat{H}}_{3\mathrm{M}}$. The spherical plots show ${\gamma }_{\varphi \theta }$, which is the skewness of the generalized multimode quadrature ${\widehat{A}}_{\varphi \theta }$ [see Eq. (6)], which mixes the mode quadratures through symplectic symmetry operations. The 6D phase space of the three modes is projected into a 3D space for the purposes of visualization. Generally, ${\gamma }_{\varphi \theta }$ combines the contributions of the skewness and coskewness of the three modes involved in each case. (a) Experimental data for the two-mode case, with a maximum skewness of 0.057. (b) The theoretical prediction for the two-mode cubic Hamiltonian ${\widehat{H}}_{2\mathrm{M}}$. The clear agreement shows that the observed state is generated by that specific Hamiltonian. (c) A plane cut of the spherical plots through the plane at $\varphi =0$. The theory curve is normalized to the maximum experimental value, but otherwise the theory has no adjustable parameters. There are no lobes apparent on the right half of the plot, because ${\gamma }_{\varphi \theta }$ is negative for $\varphi \in [\pi /2,3\pi /2]$, causing these lobes to overlap those of $\varphi \in [-\pi /2,\pi /2]$. (d) Experimental data for the three-mode case, with a maximum skewness of 0.024. (e) The theoretical prediction for the three-mode cubic Hamiltonian ${\widehat{H}}_{3\mathrm{M}}$. (f) A plane cut of the spherical plots through the plane at $\varphi =0.61$ (35°). Again, we see a clear agreement between the observed state and the target Hamiltonian. In particular, the antinodes (lobes) of ${\gamma }_{\varphi \theta }$ appear only at angles where all three modes are mixed, as expected for genuine three-mode interference.

Figure 6

A cartoon of the correlation feed-forward protocol for the three-mode trisqueezed state. This scheme tests our hypothesis on the conditional structure of two- and three-mode correlations. In the three-mode protocol shown here, we first measure the local phase of the reference mode (mode 1 in this example), followed by applying a rotation by ${\varphi }_{\mathrm{ref}}/2$ on the other two modes. The result is an observable correlation between mode 2 and mode 3 with the characteristic structure of two-photon SPDC, i.e., two-mode squeezing.

Figure 7

The resultant single-mode histogram of mode 1 after applying the correlation feed-forward protocol to the two-mode trisqueezed state. We take mode 2 as the reference mode, correcting the phase of mode 1. Then, we compute the histogram of the data and subtract from it the histogram of the system noise (with the pump turned off) after also applying the protocol. We clearly see that the subtracted histogram is stretched along the $x$ axis, indicating squeezing of the on state. The small rotation of the figure can be explained by a small misalignment of the pump and digitizer phases (see Fig. 4).

Figure 8

Lumped-element description of a dc SQUID.

Figure 9

Lumped-element description of a transmission line resonator terminated by a SQUID (adapted from Ref. [50]).