Quantum Hypercube States

We introduce quantum hypercube states, a class of continuous-variable quantum states that are generated as orthographic projections of hypercubes onto the quadrature phase space of a bosonic mode. In addition to their interesting geometry, hypercube states display phase-space features much smaller than Planck’s constant, and a large volume of Wigner negativity. We theoretically show that these features make hypercube states sensitive to displacements at extremely small scales in a way that is surprisingly robust to initial thermal occupation and to small separation of the superposed state components. In a high-temperature proof-of-principle optomechanics experiment we observe, and match to theory, the signature outer-edge vertex structure of hypercube states.

Figure 1

Petrie polygon orthographic projections of first four $n$ cubes. Red dots represent the projection of a single vertex onto a unique point in phase space, and orange dots the projection of two vertices onto the same point. $n=1$ is the projection of a line segment; $n=2$ of a square; $n=3$ of a cube; and $n=4$ of a tesseract.

Figure 2

Density plots of the Wigner functions of the first four hypercube states. All axes are in terms of the ground state width.

Figure 3

$n=2$ hypercube state for a mechanical resonator as produced by our scheme. Left-hand side: Theoretical density plots of the Wigner function in position-momentum ($X-P$) phase space. The color bar on the bottom shows the value of the Wigner function where blue and green indicate areas of negative quasiprobability. All axes are in terms of the ground state width. Right-hand side: Corresponding ${\ell }_1$ sensitivity, $d{\ell }_1/d\sigma$, vs position or momentum displacement plots (position, pink horizontal triangles) or (momentum, black vertical triangles). The blue dotted line indicates ${\ell }_1$ sensitivity of a coherent state at zero temperature. Top row: Cold temperature: small $\to$ large coupling ($\overline{n}=0$; $\mu =6\to \mu =8\times {10}^{-9}$). Middle row: small coupling: cold $\to$ hot temperature ($\mu =8\times {10}^{-9}$; $\overline{n}=0\to \overline{n}={10}^{15}$; the colors in the last panel are scaled by $10^8$ in accordance with the axis scaling). The ${\ell }_1$ sensitivity plot shows that even hypercube states with vanishing separation of coherent states are more sensitive than thermal states (red-shaded region) and at least as sensitive as squeezed (3 dB) thermal states (blue shaded region) with the same mean phonon number and more sensitive than a ground state coherent state for $\overline{n}\lesssim 0.5$. Bottom row: Large coupling: cold $\to$ warm temperature ($\mu =6$; $\overline{n}=0\to \overline{n}=6$). $n=2$ hypercube states are shown by blue triangles; for comparison, $n=3$ and $n=4$ hypercube states are shown, respectively, by purple squares and red circles. Note that in the right plot $\mu =12$ was used to ensure clear separation of coherent states across all the orders of the hypercube state.

Figure 4

A weak coherent state $|\alpha ⟩$ is inserted and split using a half wave plate (HWP) and a beam displacer (BD) into two arms of a folded interferometric setup. The optical field in the top arm interacts with a membrane mechanical resonator via radiation pressure, described by $e^{i\mu a_1^{†}{a}_1{X}_M}$. Here $a$ and $a^{†}$ are the annihilation and creation operators, $\mu$ is the optomechanical coupling strength, and $X_M$ is the mechanical position operator. The bottom arm interacts with the static frame of the membrane, obtaining a controllable phase shift $e^{i\varphi }$, which we set to $\varphi =\pi$. Post these interactions, the optical fields in both arms are interfered, split into the polarization components, and detected in an avalanche photon detector (APD). The detector signal is then split into two, three, or four paths depending on whether detection of a second-, third-, or fourth-order hypercube is being made; with each path past the first adding a time delay of $T/2n$. Coincidence counting between the paths is made at $C_n$, and upon detection of a coincidence event an oscilloscope is triggered to record a trace of the membrane’s position from the back (not shown).

Figure 5

Normalized probability densities of hypercube states; left to right columns correspond to second-, third- and fourth-order hypercube states. Top row: Ideal densities, from Eq. (1): Populations match the outer ring of vertices of the Petrie projections in Fig. 1; quantum features such as sub-Planck structure are absent due to high temperature. Middle row: Densities predicted from a model that accounts for experimental drift, timing uncertainties, and counting statistics; see Supplemental Material [18]. Note the predicted variations in peak height, width, and shape. Bottom row: Measured probability densities, the Bhattacharyya coefficient (a classical analogue of the fidelity) between the middle row of density plots and the experimentally measured density plots are $0.95\pm 0.02$, $0.90\pm 0.02$, and $0.82\pm 0.02$ for the second-, third-, and fourth-order hypercube states, respectively. The uncertainties represent 1 standard deviation obtained from the statistical uncertainties in determining the values of $X$ and $P$. The signature of hypercube states in the thermal regime—the ring of outer-edge-vertices—is clearly visible in all plots. All axes are in units of ${\lambda }_R/4=158.2\mathrm{nm}$, where ${\lambda }_R$ is the readout wavelength used to measure the mechanical resonators’ phase-space position.