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The macroscopic quantum superpositions. The concept of macroscopic quantum superposition (MQS) and entanglement dates back to 1935 when E. Schrödinger formulated his famous thought experiment: the quantum formalism was applied to macroscopic object – a cat and microscopic object – a bottle of poison, both closed in a box, describing them in a joint quantum superposition, the Schrödinger cat state [1]. The idea was overwhelming: the cat was simultaneously dead and alive, as long as no one knew if the bottle was broken by a random mechanism in the box. It pointed out the phenomenon of quantum entanglement and posed a question to what extent the macroscopic world obeys the quantum mechanical laws, opening new chapters in the field of quantum mechanics: quantum optics and information. Over the years the efforts were taken to turn this abstract idea into practice by creating MQS in superconductors, nanoscale magnets, laser-cooled trapped ions, photons in a microwave cavity and C60 molecules [2]. The first realizations of Schrödinger cat states for truly macroscopic objects were performed with superconducting devices [2] and macroscopic-size diamonds in room temperature (2011) [3,4]. There, however, the superposition was not macroscopically populated. The study of MQS is one of the principal directions of development in quantum physics, due to their striking properties, impossible to explain within the classical physics. MQS allow to explore the quantum-to-classical transition [5,6], interact efficiently with matter and photons [7], form a non-Gaussian class of quantum states, necessary for fault tolerant quantum computing [8]. They help understanding the principle of quantum measurement and give hope for a loophole-free Bell inequality test [9]. They are promising alternative for quantum technologies.

MQS illustrate quantum chaotic systems which are very fragile for disturbances. They decohere exponentially fast with their size [8]. Thus, their realizations in condensed matter systems require extraordinary physical conditions which isolate them from the environment.

Outline of the system and method of its generation. In 2007, robust macroscopically populated quantum superpositions of light were generated in the room conditions by optimal quantum cloning based on optical parametric amplification [10,11,12]. Parametric down conversion (PDC) joined with a strong laser pumping and quantum state seeding of a χ(2) nonlinear crystal are necessary for their creation. Up to date, seeding was performed with vacuum [13] (Fig. 1a), single photon [11] (Fig. 1b) and coherent state (July 2011; Fig. 1c) [14]. Depending on the seeded state, the outcome has slightly different properties. For vacuum, it is a bright squeezed vacuum (BSV) and is Gaussian. This is a symmetrical with respect to its two subsystems macroscopic-macroscopic high-dimensional singlet, a macroscopic analog of two-photon Bell states [15]. In case of a single photon, the outcome depends on its polarization and is non-Gaussian. If it is a part of a biphoton (a pair of polarization entangled photons), there are two possible mutually orthogonal states emerging from the crystal. From the quantum informational point of view, these two states form a macroscopic quantum bit (qubit), and together with the other (not amplified) part of the biphoton, form an unsymmetrical microscopic-macroscopic singlet. The common feature of these states is macroscopic entanglement between the polarization and photon number. Coherent state seeding leads to outcome being squeezed coherent state, which is Gaussian. All these states have macroscopic photon population 105-1013. They form hybrid systems, merging properties of both discrete and continuous variables.

Figure 1
Fig. 1. Macroscopic quantum states of light.

Theoretical description and physical properties. The quantum no-cloning theorem [16], states that an unknown quantum state cannot be copied perfectly and deterministically at the same time. It is however possible to make imperfect copies deterministically, characterized by a cloning fidelity F lower than one, by the optimal quantum cloning [17]. For light, the optimal cloning was realized for polarization qubits using PDC process. For wide range of experimental parameters, PDC process is well described by the single mode approximation and a classical (not depleted) pump [17]. The gain Γ, the product of the nonlinearity and pump power denoted by g and integrated over time, fixes the mean photon number in its output which is proportional to sinh2 Γ.

For BSV the cloner comprises two PDC crystals with perpendicular optical axes and Hamiltonian in the linear polarization basis H, V [15]

Hamiltonian Formula 1

The unitary operation generated by this Hamiltonian Û = eiĤt describes the universal quantum cloner with the qubit cloning fidelity F = 2/3 [18]. It corresponds to two-mode squeezing and creates entanglement: the photon number in a given polarization in mode a is random, but equals the number of photons in the orthogonal polarization in mode b. It is shown by Taylor expansion of the exponent in Û. The higher g is, the more terms in the expansion are significant, i.e. the higher pump power is, the more populated state is produced.

For the macroscopic qubit the cloner consists of one PDC crystal and for its symmetry, it is best to consider its Hamiltonian in equatorial plane of Poincarè sphere [11]

Hamiltonian Formula 2

This is the phase-covariant cloner with qubit cloning fidelity F = 3/4. The equatorial plane consists of polarization states φ = H + eV given by the angle φ ∈ ⟨0, 2π) and the form of Hamiltonian is invariant here. It describes a single mode squeezing and creates a fine structure in the photon number distributions for the macroscopic qubit: occupations of polarizations φ and φ have different parity, revealing chessboard pattern and important asymmetry in its shape, see Fig. 2.

Figure 2
Fig. 2. Photon number distribution for the macroscropic qubit.

The quantum properties in MQS are often inaccessible due to inefficient detection what significantly limits their study and applications. They would be well distinguishable in measurement if the detectors were single photon resolving and sources where ideal [19]. As a solution, a specific state filtering which modifies certain properties of MQS was applied [11,12]: the Orthogonality Filter (OF). It increases distinguishability, but destroys their quantum character.

Experimental methods. BSV is produced by two 2 mm beta barium borate (BBO) crystals cut for a PDC collinear type I phase matching, located in the arms of a Mach-Zehnder interferometer [13,20], see Fig. 3. The pump (shown blue), a pulsed laser beam with the wavelength 355 nm, repetition rate 1 kHz, pulse duration 18 ps and energy per pulse 0.2 mJ, enters the interferometer through a polarizing beam splitter (PBS), and each beam pumps one of the crystals. Their optical axes are orthogonal, which leads to the production of horizontally polarized photons from one crystal and vertically polarized photons from the other one. The PDC is non-degenerate and both crystals emit beams at two wavelengths: the signal at 635 nm and the idler at 805 nm (shown by red and orange lines). After the crystals, the pump radiation is cut off. Next, the beams are superimposed at a PBS. The variable phase φ, introduced by a mirror on a piezoelectric element, and a dichroic plate (DP) allow producing all four Bell states, not only the singlet [15]. In this setup, a gain of only about was achieved but by improving the configuration, values of g = 8 corresponding to 106 photons per mode are expected. The output Bell states are multi-mode with the mode number ca. 106, determined by the angular filtering: if a solid angle of about 10-4 sr is selected, 102 longitudinal modes and 104 transverse modes are collected. The states are registered using charge integrating detectors with the quantum efficiency ca. 90%. The total detection efficiency of the setup is 70%.

Figure 3
Fig. 3. Experimental scheme for generation of the bright squeezed vacuum (BSV) [15].

The macroscopic qubit and micro-macro singlet state are produced with single photon seeding [11], Fig. 4. The pulsed pump with wavelength 397.5 nm, repetition rate 250 kHz and pulse power ca. 800 mW, is split in two orthogonally polarized beams by PBS. One of these beams pumps a 1.5 mm type II BBO crystal which produces a noncollinear pair of orthogonally polarized photons at 795 nm wavelength. One of these photons, e.g. A, is used as a trigger: preparation of the photon B in equatorial polarization is conditionally determined by detecting the photon A after polarization analysis with half (λ/2), quarter wave plates (λ/4) and phase shifters (PS). It is registered with quantum efficiency ca. 5% and detection rate 5 kHz. Photon B is then coupled to a single-mode optical fiber and is sent to the other crystal together with the second pump beam. The second crystal is 1.5 mm and cut for collinear type II phase matching. It performs optical amplification. The gain g = 4.5 and 104 of photons per pulse were obtained. The output state has also multi-mode character, but the mode number is much lower, ca. 10, due to the character of PDC type II process. The output is registered with photon multipliers (PM) and OF filter in front with efficiency 2%. If the photon A was not detected, a micro-macro singlet would be produced. A limitation in this method is small efficiency of single photon seeding, which requires improvement [21].

Figure 4
Fig. 4. Experimental scheme for generation of macroscopic qubit and micro-macro singlet [11].

Major developments. Macroscopic qubit was analyzed in terms of Wigner function displaying its negative values and a non-Gaussian nature [22]. Anomalous lack of decoherence for it was reported [23,24]. Entanglement test with the OF for the micro-macro states was performed [11], but later refuted [25]. New criteria for the test were formulated [25,26], but not tested. BSV is formally equivalent to a singlet state of large spin. Entanglement witness in the form of spin inequality for BSV was discussed [27,28]. Proposal for violation of Bell-type inequalities for both MQS was considered [27,30,31,32]. Experimental realization of entanglement measures e.g. Schmidt number, logarithmic negativity as well as constructing a feasible Bell inequality for entangled MQS remains an open issue. It was shown that a Bell test with MQS and inefficient detection may reveal entanglement in separable states [33]. Such detection is incapable of grasping any quantum character of MQS without prior filtering [34].

It was recognized that other kinds of filtering than OF, described by POVM which preserve quantum properties, or efficient detection techniques are necessary for further development of the field. Recently, we proposed a new POVM filter which implementation relies on the interference of MQS entering a beam splitter, each polarization mode via a different input port [35]. It consists of the linear optical elements: polarizers, beam splitters and the feed-forward loop. Attempts to involve other kinds of more efficient detection: homodyne detection [36] and human eyes were taken [37].

Also strategies genuine to accurate measurements for continuous variables, which do not require filtering, were adopted. Using Stokes variables higher-order correlations in polarization [38,39] and entanglement test for BSV were shown [39]. Estimation of the Schmidt number was based on measurement of conditional photon number distribution [40].

All MQS of light have applications in enhanced optical phase estimation with noisy detection which was shown by quantum Fisher information [14,41] and facilitate efficient nonlinear optics [17].

Some measures of the MQS size [40] and of non-Gaussian character of an arbitrary quantum state based on quantum relative entropy [42] and on the Hilbert-Schmidt distance [43] were proposed. A new measure quantifying the size and coherence in MQS within phase space was suggested [44]. Nonclassicality measures: quantum discord [45], entanglement potential [46], Wigner function negativity [47,48] and interference fringe peak [49,50], P-function [51], nonclassical depth [52,53], normally ordered characteristic function [55] and Klyshko criterion [55], which expresses the positivity of the phase-averaged P-function were proposed. The latter has the advantage of being the most sensitive and experimentally feasible. Generally, such measures with operational meaning are currently being prospected [56,57].

The interaction with Bose-Einstein condensate was investigated [54]. Another class of decoherence stable MQS, superpositions of GHZ states, was found [55]. The assessment that large MQS decohere faster was recently challenged [56].

Originality of the project. The MQS of light are the recent development of quantum optics. These states are macroscopically populated and possess a complex structure. The intuition based on experience with the two-photon (microscopic) singlets, has been proven to be often misleading in case of macroscopic entanglement. The originality of this project is based on developing novel theoretical and numerical tools for efficient investigation of those states. Usual analytical and computational methods are impractical: either they take into account too many details and become intractable or too less, and the results are false. The situation is even more difficult if inefficient photodetection comes into play. In this case, the quantum properties of MQS are inaccessible. Proper measurement strategy, involving quantum state engineering adapted to various available measurement techniques plays key role in their study and answering the questions related to the foundations of quantum mechanics. The novel solution to these problems is to apply POVM filtering to MQS. Filtering may serve for different purposes: efficient detection, engineering of the sources, preselection strategy for Bell test, entanglement distillation. Another novel aspect of this project is the quest for entanglement and nonclassicality measures, especially experimentally feasible. Efficient interaction between the light and matter remains an open issue. One of the ideas of the project is to take advantage of the macroscopic population of MQS and investigate their interaction with polaritonic condensate and biomolecules. In this sense, the results obtained in this project will be entirely new and will shape our understanding of macroscopic superpositions and entanglement as well as transition between the quantum and the classical.

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