Colloquium: Multiparticle quantum superpositions and the quantum-to-classical transition

This work reports on an extended research endeavor focused on the theoretical and experimental realization of a macroscopic quantum superposition (MQS) made up of photons. This intriguing, fundamental quantum condition is at the core of a famous argument conceived by Schrödinger in 1935. The main experimental challenge to the actual realization of this object resides generally in unavoidable and uncontrolled interactions with the environment, i.e., “decoherence,” leading to the cancellation of any evidence of the quantum features associated with the macroscopic system. The present scheme is based on a nonlinear process, “quantum-injected optical parametric amplification,”which, by a linearized cloning process maps the quantum coherence of a single-particle state, i.e., a microqubit, onto a macroqubit consisting of a large number $M$ of photons in quantum superposition. Since the adopted scheme was found resilient to decoherence, a MQS demonstration was carried out experimentally at room temperature with $M\ge {10}^4$. The result led to an extended study of quantum cloning, quantum amplification, and quantum decoherence. The related theory is outlined and several experiments are reviewed, such as the test of the “no-signaling theorem” and the dynamical interaction of the photon MQS with a Bose-Einstein condensate. In addition, the consideration of the microqubit-macroqubit entanglement regime is extended to macroqubit-macroqubit conditions. The MQS interference patterns for large $M$ are revealed in the experiment and bipartite microqubit-macroqubit entanglement was also demonstrated for a limited number of generated particles: $M\precsim 12$. Finally, the perspectives opened by this new method for further studies on quantum foundations and quantum measurement are considered.

Figure 1

Three different configurations for the amplification of quantum states. (a) Schematic diagram of a noncollinear quantum-injected optical parametric amplifier (OPA). The injection is provided by an external spontaneous parametric downconversion source of polarization-entangled photon states. From 32. (b) Double injection of the optical parametric amplifier. From 11). (c) Collinear quantum-injected optical parametric amplifier. From 33.

Figure 2

Scheme of the optimal cloning process. The input and output qubits are represented on the Bloch sphere. The vectors associated with the output states are shrunk compared to the input state $|\varphi ⟩$.

Figure 3

Schematic diagram of the universal optimal cloning machine realized on the cloning ($C$) channel (mode ${\mathbf{k}}_1$) of a self-injected OPA and of the universal-NOT gate realized on the anticloning (AC) channel, ${\mathbf{k}}_2$. From 37.

Figure 4

Linear methods: (a) schematic diagram of the linear optics multiqubit symmetrization apparatus realized by a chain of interconnected Hong-Ou-Mandel interferometers; (b) symmetrization of the input photon and the ancilla polarization-entangled pairs. Nonlinear methods: (a) universal quantum cloning machine by optical parametric amplification, flipping by two wave plates and projection over the symmetric subspace; (b) collinear optical parametric amplification. From 140.

Figure 5

Configuration of the quantum-injected optical parametric amplifier. The SPDC quantum injector (crystal 1) is provided by a type-II generator of polarization-entangled photon pairs. From 31.

Figure 6

Scheme of the experimental setup. The main ultraviolet laser beam provides the excitation field beam at ${\lambda }_P=397.5\mathrm{nm}$. A type-II BBO crystal (crystal 1: C1) generates a pair of photons with $\lambda =795\mathrm{nm}$. The photon belonging to ${\mathbf{k}}_B$, together with the pump laser beam ${\mathbf{k}}_p^{\prime }$, is fed into a high-gain optical parametric amplifier consisting of a crystal 2 (C2), cut for collinear type-II phase matching. The residual UV pump beam and the amplified quantum states are spectrally separated by exploiting a dichroic mirror (DM). Measurement apparatus: the field is analyzed by two photomultipliers (${\mathrm{PM}}_1$ and ${\mathrm{PM}}_2$) and then discriminated through an O-filter device (OF), whose action is described in the text. From 42.

Figure 7

Average signal vs the phase of the input qubit. From 111.

Figure 8

The experimental apparatus adopted for the amplification of entangled pairs of photons (Quantum Optics Group, Dipartimento di Fisica, Sapienza Universita di Roma).

Figure 9

Coincidence counts $[L_B;D_A]$ vs the phase of the injected qubit for the diagonal (circles) and circular (squares) polarization bases. From 42.

Figure 10

Hybrid nonlocality and entanglement test on an optical microscopic-macroscopic state generated by a black box. The single-photon mode ${\mathbf{k}}_A$ is measured by a polarization detection apparatus, while the multiphoton mode ${\mathbf{k}}_B$ undergoes both polarization and homodyne measurements.

Figure 11

Bures distance for various classes of MQSs for $⟨n⟩=12.5$. The lower dash-dotted curve corresponds to $\mathcal{D}\mathbf{(}\mathcal{E}(|{\varphi }^{+}⟩),\mathcal{E}(|{\varphi }^{-}⟩)\mathbf{)}$, while the upper dotted curve is relative to the distinguishability $\mathcal{D}\mathbf{(}\mathcal{E}(|{\alpha }^{+}⟩),\mathcal{E}(|{\alpha }^{-}⟩)\mathbf{)}$. The solid curve corresponds to the MQS generated by phase-covariant cloning $\mathcal{D}\mathbf{(}\mathcal{E}(|{\Phi }_{PC}^{+}⟩),\mathcal{E}(|{\Phi }_{PC}^{-}⟩)\mathbf{)}$. From 40, 41. The dashed curve corresponds to the universal-cloning-based MQS $\mathcal{D}\mathbf{(}\mathcal{E}(|{\Phi }_U^{+}⟩),\mathcal{E}(|{\Phi }_U^{-}⟩)\mathbf{)}$. From 153.

Figure 12

Wigner function of a single-photon amplified state in a single-mode degenerate OPA for $g=3$. (a) ($R=0$.) Unperturbed case. (b) ($R=0.005$.) For small reflectivity, the Wigner function remains negative in the central region. (c) ($R=0.1$.) The Wigner function progressively evolves as a positive function in all the phase space. (d) ($R=0.5$.) Transition from a nonpositive to a completely positive Wigner function. From 154.

Figure 13

Setup for the generation and detection of a bipartite macroscopic field. The high laser pulse in mode ${\mathbf{k}}_P$ excites a type-II noncollinear source in the high-gain regime ($g=3.5$). The two spatial modes ${\mathbf{k}}_A$ and ${\mathbf{k}}_B$ are spectrally and spatially selected by interference filters (IFs) and single-mode fibers. After fiber compensation $(C)$, the two modes are analyzed in polarization and detected by four photomultipliers. From 163.

Figure 14

Theoretical (left plots) and experimental (right plots) density matrices ${\rho }_{\mathrm{SPDC}}^{\mathrm{HG}}$ for different gain values. The experimental density matrices have been reconstructed by measuring 16 two-qubit observables. From 24.

Figure 15

Implementation of macro-macro entanglement (a) via entanglement swapping on two QI-OPAs and (b) through two optical parametric amplifiers. From 34.

Figure 16

Layout of the QI-OPA and mirror-BEC experimental apparatus. The upper left inset shows the interference patterns detected at the output of the PBS shown in the upper right inset for two different measurement bases $\{+,-\}$ and $\{L,R\}$. Alternating slabs of condensate and vacuum are shown in the lower left inset. From 43.

Figure 17

Scheme for phase measurement. (a) Interferometric scheme adopted to estimate the phase ($\varphi$) introduced in the mode ${\mathbf{k}}_2$. (b) Interferometric scheme using a single photon and the optical parametric amplifier: the amplification of the single-photon state is performed before dominant losses. From 162.