Multiphoton entanglement and interferometry

Multiphoton interference reveals strictly nonclassical phenomena. Its applications range from fundamental tests of quantum mechanics to photonic quantum information processing, where a significant fraction of key experiments achieved so far comes from multiphoton state manipulation. The progress, both theoretical and experimental, of this rapidly advancing research is reviewed. The emphasis is given to the creation of photonic entanglement of various forms, tests of the completeness of quantum mechanics (in particular, violations of local realism), quantum information protocols for quantum communication (e.g., quantum teleportation, entanglement purification, and quantum repeater), and quantum computation with linear optics. The scope of the review is limited to “few-photon” phenomena involving measurements of discrete observables.

Figure 1

A two-photon interferometer with variable phase shifts ${\varphi }_1$ and ${\varphi }_2$. Before being combined at the $50:50$ beam splitter (BS) and then subject to single-photon detections, the two paths of each photon acquire a relative phase shift. For experimental realization of such a two-photon interferometer, see Fig. 11. Adapted from 204.

Figure 2

A three-photon interferometer with variable phase shifts ${\varphi }_1$, ${\varphi }_2$, and ${\varphi }_3$. Adapted from 175.

Figure 3

The function of a BS. (a) The BS coherently transforms two input spatial modes (a), (b) into two output spatial modes (c), (d). (b) A Mach-Zehnder interferometer consists of two 50–50 BSs, mirrors and phase shift (as a whole it forms a universal tunable beam splitter).

Figure 4

The operation of a polarizing beam splitter (PBS). (a) In the usual configuration, the PBS transmits horizontal, and reflects vertical, polarization. (b) Two photons, each entering via a different input: If the two photons incident onto the PBS have identical polarization, then they will always go out along different directions, so there will be one photon in each of the two output modes. (c) If the two incident photons have opposite polarizations, they will always go out along the same direction, so there will be two photons in one of the two outputs and none in the other. In essence, a PBS can thus be used as a polarization parity checker. From 356; 360.

Figure 5

(a) Individually incoming photons can be transmitted or reflected. However, since the two photons are identical, we cannot distinguish between the two cases when both are transmitted or both reflected. The BS introduces a phase difference of $\pi$ between the two amplitudes describing such possibilities. This leads to a destructive interference. Thus, no coincidence detection can be found. (b) Data from 202: Pairs of photons impinging on a BS are produced by spontaneous parametric down-conversion (see Sec. 4a) and have the same polarization and frequency distributions. The measured number of coincidence counts as a function of relative path displacement (temporal distinguishability) shows a “Hong-Ou-Mandel” dip, for equal optical paths. From 202.

Figure 6

(a) Shih-Alley type polarization entanglement. One combines two orthogonally polarized photons at a BS and registers the coincidence events behind it. (b) In a similar way, by superposing two ($H+V$) polarized photons at a PBS, one can observe an entangled state $|H{⟩}_c|H{⟩}_d+|V{⟩}_c|V{⟩}_d$ in the postselected coincidence counts. From 352; 432.

Figure 7

(a) A Bell-state analyzer using a BS. (b) A modified Bell-state analyzer. The angle between the half-wave plate axis and the horizontal direction is 22.5°. It corresponds to a 45° rotation of the polarization. (c) A GHZ-state analyzer. Adapted from 356.

Figure 8

Type II parametric down-conversion. Inside a specially cut (birefringent) nonlinear crystal (e.g., BBO), a pump photon can convert spontaneously into a pair of photons of lower frequencies. The polarization of signals is orthogonal with respect to the one of idlers. One can attempt to pick only photons of frequency which is one-half of the frequency of the pumping field. In such a case, if the crystal is suitably oriented, $H$ polarized photons are emitted into one cone (in the figure, the lower one), while $V$ polarized photons are emitted into the other cone, and the two cones intersect. Due to the phase-matching pairs of photons pop up only along an intersection of a plane containing the pump beam with the cones, emissions along the directions at which the two cones intersect have undefined polarizations. However, as one of the photons is from the upper cone and the other one is from the lower one, and due to the indistinguishability of the two processes, we have no way to establish which is which (except from a direct polarization measurement), photons in beams A and B are polarization entangled. This holds under condition that one erases all possible distinguishing features of upper and lower cone photons (which arise because one of them is an ordinary ray and the other one an extraordinary one) which can easily be done via compensation methods described in the text.

Figure 9

A Franson-type experiment allowing interference of time-entangled photon pairs (under the condition of a strict coincidence). The main feature of the scheme is a pair of two identical unbalanced Mach-Zehnder interferometers. Adapted from 151.

Figure 10

Schematic of the pulsed time-entangled twin-photon source. The interferometers are represented by curvy lines. This is to stress that practical realizations of this type of interferometers are usually built out of optical fibers and couplers. From 67.

Figure 11

The Rarity-Tapster experiment with momentum entanglement from a type I SPDC. Using double apertures A two correlated pairs of modes are selected from the emission spectrum of a type I down-conversion source. On each of the two BS beams of the same wavelength are superimposed. The detectors $D_{a3}$, $D_{b3}$, $D_{a4}$, and $D_{b4}$ register two-photon coincidences between the $a$ and $b$ outputs. From 394.

Figure 12

(a) Experimental setup for the creation and analysis of hyperentangled photons. (b) Spatial filtration (spa): hologram (holo) and single-mode fiber (smf). (c) Energy time transformation (e–t): thick quartz decoherer (dec) and liquid crystal (LC). (d) Polarization filtration (poln): quarter-wave plate (qwp), half-wave plate (hwp), and polarizer (pol). From 23.

Figure 13

Super-resolving phase measurement with $N00N$ states. (a) A schematic of the experimental setup of 336: a Mach-Zehnder interferometer consisting of two $50:50$ beam splitters. (b) Single-photon count rate in mode $e$ as a function of phase plate (PP) angle with single-photon input $|10{⟩}_{ab}$. (c) Two-photon count rate in modes $e$ and $f$ for input state $|11{⟩}_{ab}$. (d) Four-photon count rate of three photons in mode $e$ and one photon in mode $f$ for the input state $|11{⟩}_{ab}$. The visibility of the fringes is 0.91(6), greater than the threshold to beat the standard quantum limit. Adapted from 336.

Figure 14

Four-particle interference effects for two pairs of particles originating from two independent sources. Adapted from 540.

Figure 15

A three-photon polarization-entanglement source. The photon sources $A$ and $B$ are pumped by short pulses. The PBS1 transmits 45° polarization and reflects $-45°$ polarization, $F$ is a narrow filter, and ${\mathrm{D}}_{T1}$ and ${\mathrm{D}}_{T2}$ are two single-photon detectors. A single-photon trigger event in one of the detectors signals that coincident detections in channels 1, 2, and 3 would result in GHZ correlations. The setup can also be used for observation of four-photon GHZ interference. From 519.

Figure 16

Entanglement among three single photons. (a). Experimental setup for observing three-photon GHZ entanglement. The UV-pulse incident on the BBO crystal produces four photons, two in each mode. Conditioned on the registration of one photon at the trigger detector T, the three photons registered at D1, D2, and D3 exhibit the desired GHZ correlations. (b). Experimental data. The photon at D1 polarized at 45° and the photon at detector D2 polarized at $-45°$. The two curves show the fourfold coincidences for a polarizer in front of detector D3 oriented at $-45°$ and 45°, respectively, as a function of the spatial delay in path a. (c). Experimental data for the case of the polarizer in front of detector D1 set to 0°. Adapted from 56.

Figure 17

One of the two observation stations. A random number generator is driving the electrooptic modulator. Silicon avalanche photodiodes are used as detectors. A “time tag” is stored for each detected photon together with the corresponding random number 0 or 1 and the code for the detector “$+$” or “$-$” corresponding to the two outputs of the polarizer. From 488.

Figure 18

(a) Predictions of quantum mechanics and (b) of local realism, and (c) observed results for the final ${\widehat{x}}_1{\widehat{x}}_2{\widehat{x}}_3$ measurement in the first GHZ-type experiment. $H^{\prime }$ and $V^{\prime }$ denote here diagonal and antidiagonal linear polarizations. From 355.

Figure 19

The experimental violation of Leggett-type nonlocal realism in 181. The dashed line shows the bound of inequality (68) for the particular class of nonlocal realistic theories. The solid line is the quantum prediction reduced to fit the experimentally achieved visibility. The experimental data were taken for various difference angles of local measurement settings (parametrized as on the Poincaré sphere). From 181.

Figure 20

Experimental setup for quantum dense coding. From 318.

Figure 21

Experimental quantum teleportation. (a) Principle of quantum teleportation. Alice has particle 1, whose qubit state she wants to teleport to Bob. Alice and Bob share an ancillary entangled pair of qubits carried by particles 2 and 3. Alice performs a Bell-state measurement jointly on particles 1 and 2. She sends a two-bit classical information informing Bob about the result of her measurement. Based on this, he performs one of four unitary transformations (U) on his ancillary particle. The transformations are such that particle 3 ends up in the state identical to the original state of particle 1. (b) Setup of the Innsbruck teleportation experiment. A pulse of ultraviolet light passing through a nonlinear crystal creates an ancillary pair of entangled photons 2 and 3 in a polarization state $|{\psi }^{-}{⟩}_{12}$. The pulse is reflected, and during its second passage through the crystal another pair of photons can be created. One of them plays the role of particle 1. The other one serves as a trigger indicating that photon 1 is under way. Alice looks for coincidences after the BS at which photons 1 and 2 are superposed. Upon receiving a classical information indicating that Alice obtained a coincidence count in detectors f1 and f2, which implies a collapse into $|{\psi }^{-}{⟩}_{12}$, Bob knows that his photon 3 is in the original polarization state of photon 1. He can check whether this is indeed so by a polarization analysis with a PBS and the detectors d1 and d2 behind it. From 58.

Figure 22

The measured threefold coincidence rates at d2f1f2 ($+45°$) and d1f1f2 ($-45°$) for two cases of the state to be teleported being polarized under $+45°$ (a) and (b) or $-45°$ (c) and (d), respectively. The coincidence rates are plotted as a function of the delay (in $\mu \mathrm{m}$) between the arrival of photon 1 and 2 at Alice’s BS [see Fig. 21b]. The threefold coincidence rates are plotted after subtracting the spurious threefold background contribution. The data, together with similar results for other polarizations, constitute a positive result of a test for teleportation of an arbitrary state. From 58.

Figure 23

Experimental setup for the “two qubit at one subsystem scheme” of quantum teleportation. From 48.

Figure 24

The principle of open-destination teleportation. Adapted from 528.

Figure 25

Two-qubit quantum teleportation. Adapted from 524.

Figure 26

Principle of entanglement swapping. From 356.

Figure 27

Experimental setup of entanglement swapping. The generation scheme for photon pairs 1–2 and 3–4, and the Bell-state measurement identifying the $|{\psi }^{-}{⟩}_{23}$ state is identical as in the Innsbruck teleportation experiment. The Bell-state measurement on photons 2 and 3 projects the two remaining photons 1 and 4 onto an entangled state. To analyze their entanglement, one looks at coincidences between detectors $\mathrm{D}{1}^{+}$ and D4, and between detectors $\mathrm{D}{1}^{-}$ and D4, for different polarization angles $\Theta$. Note that, since the detection of coincidences between detectors $\mathrm{D}{1}^{+}$ and D4 and $\mathrm{D}{1}^{-}$ and D4 are conditioned on the detection of the $|{\psi }^{-}{⟩}_{23}$ state (which happens whenever one has a coincidence behind the beam splitter), one in fact is looking for fourfold coincidences to signify the event of entanglement swapping. Narrow bandwidth filters (F) are positioned in front of each detector to make photons from different emissions indistinguishable. From 356.

Figure 28

Scheme of entanglement purification by 33; 360. (a). Two poorly entangled (source and target) pairs are initially shared by Alice and Bob. They both perform a (local) CNOT operation on the two particles at their hands, measure the particles belonging to the target pair in the $0/1$ basis, and compare the measured results via classical communication. If the results are the same, then the remaining pair will have a higher degree of entanglement than the original two pairs. In this case, they keep the source pair. In the case of obtaining opposite results, they discard it. By repeating the same procedure, always starting from the pairs produced in the former purification step, it is possible to distill pairs of arbitrarily high entanglement quality [for more details, see 33]. (b). An alternative and more feasible scheme which does not require a CNOT operation but only polarizing beam splitters. The PBS transmits horizontal ($|H⟩$), and reflects vertical ($|V⟩$), polarization. By selecting only those events for which there is one, and only one, photon in each output mode of the PBS, one can project two photons input from different spatial modes into the subspace spanned by $|H⟩|H⟩$ and $|V⟩|V⟩$ (for more details see text). Adapted from 360.

Figure 29

Experimental setup for entanglement purification. UV laser pulses pass through a BBO crystal twice to produce two polarization-entangled photon pairs, i.e., pair 1 in a1–b1 and pair 2 in a2–b2. Four compensators (Comp.) are used to offset the birefringent effect caused by the BBO crystal during parametric down-conversion. The photons in modes b1 and b2 pass through a half-wave plate ($\lambda /2$) to simulate a noise that reduces the entanglement quality. Next, the two pairs are sent via local PBSs. This results in entanglement purification. Adjusting the positions of the delay mirrors $\Delta 1$ and $\Delta 2$ tunes the optical paths in such a way that the photons at local measurement stations arrive at their PBS simultaneously. Detections of exactly one photon in each of the four outputs (a3, a4, b3, and b4) behind a 45° polarizer (Pol.) lead to a successful purification act. From 359.

Figure 30

Experimental results. (a), (b) Measured fractions, in the $H/V$ and in the $+/-$ bases, for the original mixed state. (c), (d) Measured fractions for the purified state in modes a3 and b3, in the same pair of bases. A comparison of the fractions in (a) and (b), with the results shown in (c) and (d) indicates entanglement purification. From 359.

Figure 31

Quantum repeater scheme. (a) Creation of a sequence of entangled pairs. (b) Nested purification protocol which combines the methods of entanglement swapping and purification, assisted with repeated creation of auxiliary pairs. (c) Purification of entangled pairs stored in distant locations. Adapted from 71.

Figure 32

Schematic of a setup for generating entanglement between the two atomic ensembles L and R in the DLCZ scheme. The inset shows the relevant level structure of the atoms in the ensemble with the ground state, $|g⟩$, the metastable state for storing a qubit, $|s⟩$ and the excited state, $|e⟩$. The transition $|g⟩\to |e⟩$ is coupled by a classical laser light, with the Rabi frequency $\Omega$. The forward-scattered Stokes light comes from the transition $|e⟩\to |s⟩$. An off-resonant coupling with a large detuning $\Delta$ is assumed. Adapted from 123.

Figure 33

Quantum memory for photonic polarization qubits. Two identical ensembles are identically driven by a classical field, which is equally right-circularly and left-circularly polarized. Classical and quantized light fields are fed into the first ${\mathrm{PBS}}_{\mathrm{R}/\mathrm{L}}$ (“rotated” PBS which reflects left-circular photons and transmits right-circular photons) and leave at two different outputs of the second ${\mathrm{PBS}}_{\mathrm{R}/\mathrm{L}}$. Two half-wave plates (HWP), enabling the transformation $|\mathrm{R}⟩\leftrightarrow |\mathrm{L}⟩$, are placed along the $|\mathrm{L}⟩$ output of the first ${\mathrm{PBS}}_{\mathrm{R}/\mathrm{L}}$. As the atomic cell $r$ (cell $l$) works as a quantum memory for single photons with right-circular (left-circular) polarization, via the adiabatic transfer method, the whole setup is therefore a quantum memory of any single-photon polarization states. The inset shows the relevant level structure of the atoms, with the ground state $|b⟩$, the storage metastable state $|c⟩$, and the excited state $|e⟩$. From 95.

Figure 34

A quantum CNOT gate for photonic qubits. (a) A nondestructive CNOT gate constituting of a PBS, half-wave plates (HWP) and using an ancilla entangled photon pair $|{\psi }^{-}{⟩}_{34}$ (375). (b) Experimental results of two-photon CNOT gate. From 532.

Figure 35

Nondeterministic photonic qubit fusion gates. The type I fusion gate combines two input single photons on a PBS and one of the outputs is measured in the $H\pm V$ basis. The type II fusion gate combines two photons on a 45° rotated PBS (that is, both inputs and outputs are rotated using a HWP), and both outputs are detected. Adapted from 74.

Figure 36

Construction of two-dimensional cluster states using photon fusion (74; 124). (a) Certain measurements on a cluster qubit will leave the remaining qubits in a new cluster state with a different layout. (b) The effect of type I and type II fusion operations on successful connection of two linear cluster states. (c) One method of efficient construction of two-dimensional cluster states [see also 124].

Figure 37

Few-qubit cluster states and the quantum circuits they implement. For each three-qubit and four-qubit cluster, its quantum state ($|{\Phi }_{\mathrm{lin}3}⟩$, $|{\Phi }_{\mathrm{lin}4}⟩$, $|{\Phi }_{⊏4}⟩$, $|{\Phi }_{⊐4}⟩$, or $|{\Phi }_{\square 4}⟩$) and the computation carried out in the one-way quantum computer model is shown. Adapted from 483.