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Teleportation of Nonclassical Wave Packets of Light

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Science  15 Apr 2011:
Vol. 332, Issue 6027, pp. 330-333
DOI: 10.1126/science.1201034

Abstract

We report on the experimental quantum teleportation of strongly nonclassical wave packets of light. To perform this full quantum operation while preserving and retrieving the fragile nonclassicality of the input state, we have developed a broadband, zero-dispersion teleportation apparatus that works in conjunction with time-resolved state preparation equipment. Our approach brings within experimental reach a whole new set of hybrid protocols involving discrete- and continuous-variable techniques in quantum information processing for optical sciences.

In the early development of quantum information processing (QIP), a communication protocol called quantum teleportation was discovered (1) that involves the transportation of an unknown arbitrary quantum state |ψ〉 by means of entanglement and classical information. Experimental realizations of quantum teleportation (2, 3) and more advanced related operations (4) in the continuous-variable regime have been achieved by linear optics methods, although only for Gaussian states so far. However, at least third-order nonlinear operations are necessary for building a universal quantum computer (5)—something that Gaussian operations and Gaussian states alone cannot achieve. Photon subtraction techniques based on discrete-variable technology can provide useful nonlinearities and are used to generate Schrödinger’s-cat states and other optical non-Gaussian states (6). Schrödinger’s-cat states are of particular interest in this context, as they have been shown to be a useful resource for fault-tolerant QIP (7). It is therefore necessary to extend the continuous-variable technology to the technology used in the world of non-Gaussian states.

We have combined these two sets of technologies, and here we demonstrate such Gaussian operations on nonclassical non-Gaussian states by achieving experimental quantum teleportation of Schrödinger’s-cat states of light. Using the photon subtraction protocol, we generate quantum states closely approximating Schrödinger’s-cat states in a manner similar to (811). To accommodate the required time-resolving photon detection techniques and handle the wave-packet nature of these optical Schrödinger’s-cat states, we have developed a hybrid teleporter built with continuous-wave light yet able to directly operate in the time domain. For this purpose we constructed a time-gated source of Einstein-Podolsky-Rosen (EPR) correlations as well as a classical channel with zero phase dispersion (12). We were able to bring all the experimental parameters up to the quantum regime, and we performed successful quantum teleportation in the sense that both our input and output states are strongly nonclassical.

A superposition of the quasi-classical coherent state |α is one of the consensus definitions of a Schrödinger’s-cat state |cat, typically written |cat|α±|α. Such optical Schrödinger’s-cat states are known to be approximated by multiple photon subtractions from a squeezed vacuum state (6). In these protocols, a squeezed vacuum state S^(s)|0 is weakly tapped via a subtraction channel, where |0 is the vacuum state and S^(s) is the squeezing operator with squeezing parameter s. When a photon detection event occurs in the subtraction channel, S^(s)|0 is projected by the quantum action of the photon detector onto a non-Gaussian state, which can be tuned to approximate a Schrödinger’s-cat state (810). The approximation is not perfect and can be quantified by means of the fidelity figure Fcat=|cat|a^S^(s)|0|2 (13).

To represent the superposition nature of these states, we use the Wigner formalism where for any quantum state |ϕ one associates a quasi-probability distribution W(x, p), where x and p are the phase-space position and momentum parameters. W(x, p) is called the Wigner function and holds information exactly equivalent to |ϕ (14). Although the position x^ and momentum p^ quadrature operators and the vector state |ϕ are abstract objects, W(x, p) is always a definite real-valued function that can be numerically reconstructed if one performs a complete phase-resolved sequence of homodyne measurement x^cosθ+p^sinθ, a process called quantum tomography (15, 16). W(x, p) is not a true probability distribution, however, as there exist quantum states whose Wigner functions are not positive. |ϕ is defined to be a strongly nonclassical state when its Wigner function W(x, p) fails to be a positive distribution. Negativity in W(x, p) turns out to be an especially useful description of the nonclassicality of a Schrödinger’s-cat state |cat; |α and |α induce two “classical” Gaussians in phase space, the superposition of which creates an oscillating interference pattern inducing negativity in W(x, p). In contrast, a statistical mixture of |α and |α would never show such negativity.

In a quantum teleportation process, the input Win and output Wout Wigner functions are related by the convolution (denoted ○)Wout=WinGexp(r)(1)where r is the EPR correlation parameter, Gσ is a normalized Gaussian of standard deviation σ, and ħ (Planck’s constant divided by 2π) has been set to 1 (17). When finite quantum entanglement r is used, Wout will be a thermalized copy of Win. Only with infinite r will Gσ become a delta function so that Win = Wout. The quality of quantum teleportation is usually evaluated according to the teleportation fidelity Ftele=ϕin|ρ^out|ϕin, which can be written as Ftele = 1/[1 + exp(–2r)] for Gaussian states (18). More important for our case, negative features of Win (if any) can only be teleported and retrieved in Wout when Ftele ≥ 2/3 (19), a threshold also known as the no-cloning limit (20). However, the practical lower bound on Ftele will be higher because of decoherence and experimental imperfection of Win (21). We have thus defined the success criterion of Schrödinger’s-cat–state teleportation as the successful transfer of its nonclassical features, or alternatively, successful teleportation of the Wigner function Win negativity.

Our experimental quantum teleporter and Schrödinger’s-cat–state source (Fig. 1) upgrade the experiments described in (3) and (10), respectively. We use three optical parametric oscillators to generate the necessary squeezed vacua. One is used for the Schrödinger’s-cat–state preparation; the other two are combined together on a half beam splitter whose two exit ports are the resulting pair of EPR-correlated beams. The teleportation is conducted in three steps. Alice first receives both the input state and her EPR beam and performs two joint quadrature measurements, obtaining results x0 and p0. Bob then receives Alice’s measurements β=(x0+ip0)/2 through the classical channels and applies the displacement operator D^(β) on his EPR beam. A final stage consists of a third homodyne detector for tomography at the teleporter output. We emphasize that Alice and Bob do not assume any prior knowledge of the input state and adhere to unity-gain teleportation, so that the teleporter does not have any restriction regarding the specific family of quantum states it can faithfully teleport.

Fig. 1

Experimental setup. OPO, optical parametric oscillator; APD, avalanche photodiode; HD, homodyne detector; LO, local oscillator; EOM, electro-optical modulator; ADC, analog-to-digital converter; FC, filtering cavity. See (12) for details.

To benchmark our teleporter, we first evaluate the fidelity Ftele of teleportation of the vacuum state |0, the coherent state of amplitude zero. At quantum optical frequencies where the mean thermal photon number is virtually 0, this is simply done by blocking the input port of the teleporter. The teleported vacuum photocurrent is expected to have uniform Gaussian statistics with a variance σ2 = ½ + [exp(–2r)] (ħ = 1) from which we can deduce teleportation fidelity (Fig. 2). The blue traces are the shot-noise level, the noise spectrum of the input vacuum |0. The red traces are the classical limit of teleportation obtained by turning off the entanglement between Alice and Bob (r = 0). We measure 4.8 dB of added noise above the shot noise, in agreement with the expected teleportation fidelity of 0.5. When Alice and Bob share entanglement, the added noise drops to that shown by the green traces: 1.4 dB above the shot noise around 1 MHz, corresponding to a fidelity of 0.83. This is in agreement with the experimental figure of −6.9 dB that we observe in direct measurement of the EPR correlations shared between Alice and Bob.

Fig. 2

Broadband teleportation of the vacuum state Embedded Image. (A and B) Experimentally measured power spectra of the photocurrents calculated by Fourier transform are shown for the position (A) and momentum (B) quadratures. Blue, shot-noise input; green, quantum teleportation output; red, teleportation output without entanglement. (C to E) Reconstructed Wigner functions of the input state Embedded Image (C), quantum teleported vacuum (D), and classically teleported vacuum (E).

In contrast to quantum teleportation experiments conducted to date for narrow sidebands of light (2, 3), our setup operates over a wide frequency bandwidth, as required by the nature of our input state. Because its generation relies on the detection of a single photon and the induced projection, a Schrödinger’s-cat state made via photon subtraction is a short wave packet of light. A phenomenological way to picture these wave packets is to consider them as the closed boxes containing the macroscopic superposition states as in Schrödinger’s original idea. This requires Alice and Bob to teleport every frequency component of these “box-like” wave packets for faithful teleportation to occur. In this way, Alice and Bob do not need to actually teleport the Schrödinger’s-cat states directly, but merely the potential boxes containing them. Consequently, Alice and Bob do not need to know when a detection event occurs; rather, they are only concerned with continuous and faithful “box” wave-packet teleportation, whichever state lies in the box. In fact, Alice and Bob actually teleport most of the time a squeezed vacuum state S^(s)|0.

In essence, our teleporter is a time-resolved apparatus that deconstructs the input wave packets into a stream of infinitely small time bins and reconstructs them at the output, within the extent of what we refer to as the teleportation bandwidth. This bandwidth is clearly visible in both of the green experimental traces where the added noise slowly increases with frequency (Fig. 2). This is a direct consequence of the finite bandwidth of squeezing used for entanglement. However, across the frequencies relevant to our input state, teleportation fidelity is always greater than the no-cloning limit of 2/3, a necessary regime for negativity teleportation. A very careful implementation of the classical channel has been required (12) to achieve experimental realization of this fidelity.

To verify the success of Schrödinger’s-cat–state teleportation, we perform experimental quantum tomography of the input and output states independently (Fig. 3). Both input and output marginal distributions exhibit the characteristic eye shape of photon-subtracted squeezed states, with a clear lack of detection events around the origin for any phase. Although necessary, this feature alone is not sufficient to confirm the presence of negativity in Win or Wout. The reconstructed input Wigner function Win shows the two positive Gaussians of |α and |α together with a central negative dip [Win(0, 0) = −0.171 ± 0.003] caused by the interferences of the |α and |α superposition. The output Wigner function Wout retains the characteristic non-Gaussian shape as well as the negative dip [Wout(0, 0) = −0.022 ± 0.003] to a lesser degree. The degradation of the central negative dip and the full evolution of Win toward Wout can be fully understood using Eq. 1 with a model of Win, as was done in (21). Given the measured input state negativity of Win(0, 0) = −0.171 and −6.9 dB of squeezing, the results of (21) predict an output negativity value of −0.02, in good agreement with measured output negativity. Although this figure does not take into account the input-state squeezing, a more detailed model shows that a squeezing parameter s = 0.28 affects output negativity in the third decimal place only (12). The experimental input and output states have an average photon number n^ equal to 1.22 ± 0.01 and 1.33 ± 0.01, respectively (12). The increase in the output-state size is due to teleportation-induced thermalization. We calculate that the fidelity Fcat is as high as 0.750 ± 0.005 for the input Wigner function Win, with the nearest Schrödinger’s-cat state having an amplitude |αin|2 = 0.98 (12). However, after the teleportation Wout, fidelity is reduced to 0.46 ± 0.01, with the nearest Schrödinger’s-cat state having an amplitude |αout|2 = 0.66. If Wout fidelity is calculated with |αin|2 = 0.98, then Fcat = 0.45 ± 0.01.

Fig. 3

Teleportation of Schrödinger’s-cat states. (A to C) Experimentally measured input state’s Wigner function Win (A), marginal distribution (B), and photon number distribution (C). (D to F) Experimentally measured output state’s Wigner function Wout (D), marginal distribution (E), and photon number distribution (F).

We have demonstrated an experimental quantum teleporter able to teleport full wave packets of light up to a bandwidth of 10 MHz while at the same time preserving the quantum characteristic of strongly nonclassical superposition states, manifested in the negativity of the Wigner function. Although Fcat and W(0, 0) drop in the teleportation process, there is no theoretical limitation other than available squeezing, and stronger EPR correlations would achieve better fidelity and negativity transmission. The various more complex states generated as an application of photon subtraction so far (22, 23) can be, in principle, readily sent through our broadband quantum teleporter. This opens the door to universal QIP and further hybridization schemes between discrete- and continuous-variable techniques (24).

Supporting Online Material

www.sciencemag.org/cgi/content/full/332/6027/330/DC1

Materials and Methods

References and Notes

  1. See supporting material on Science Online.
  2. Supported by the Strategic Information and Communications R&D Promotion (SCOPE) program of the Ministry of Internal Affairs and Communications of Japan, Special Coordination Funds for Promoting Science and Technology, Grants-in-Aid for Scientific Research, Global Center of Excellence, Advanced Photon Science Alliance, and Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST) commissioned by the Ministry of Education, Culture, Sports, Science and Technology of Japan, Academy of Sciences of the Czech Republic, Japanese Society for the Promotion of Science, and the Australian Research Council, Center of Excellence (grant CE11E0096).
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