Quantum Teleportation of Multiple Properties of A Single Quantum Particle

Jian-Wei Pan (left) and Chao-Yang Lu

Authors: Chao-Yang Lu, Jian-Wei Pan

Affiliation:
CAS Centre for Excellence and Synergetic Innovation Centre in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, China.

The science fiction dream of teleportation [1] is to transport an object by disintegrating in one place and reappearing intact in another distant location. If only classical information is of interest, or if the object could be fully characterized by classical information—which can in principle be precisely measured—the object can be perfectly reconstructed (copied) remotely from the measurement results. However, for microscopic quantum systems such as single electrons, atoms or molecules, their properties are described by quantum wave functions that can be in superposition states. Perfect measurement or cloning of the unknown quantum states is forbidden by the law of quantum mechanics.

In 1993, Bennett et al. [2] proposed a quantum teleportation scheme to get around this roadblock. Provided with a classical communication channel and shared entangled states as a quantum channel, quantum teleportation allows the transfer of arbitrary unknown quantum states from a sender to a spatially distant receiver, without actual transmission of the object itself. Quantum teleportation has attracted a lot of attention not only from the quantum physics community as a key element in long-distance quantum communication, distributed quantum networks and quantum computation, but also the general audience, probably because of its connection to the scientific fiction dream in Star Trek. An interesting question is frequently asked: “would it be possible in the future to teleport a large object, say a human?” Before attempting to seriously answer that question, let us take steps back, look at where we actually are and think about a much, much easier and fundamental question: have we teleported multiple, or all degrees of freedom (DOFs) that fully describe a single particle, thus truly teleporting it intact? The answer is NO.

Past 2Physics article by Chao-Yang Lu and/or Jian-Wei Pan :

January 04, 2015: "Achieving 200 km of Measurement-device-independent Quantum Key Distribution with High Secure Key Rate" by Yan-Lin Tang, Hua-Lei Yin, Si-Jing Chen, Yang Liu, Wei-Jun Zhang, Xiao Jiang, Lu Zhang, Jian Wang, Li-Xing You, Jian-Yu Guan, Dong-Xu Yang, Zhen Wang, Hao Liang, Zhen Zhang, Nan Zhou, Xiongfeng Ma, Teng-Yun Chen, Qiang Zhang, Jian-Wei Pan

June 30, 2013: "Quantum Computer Runs The Most Practically Useful Quantum Algorithm" by Chao-Yang Lu and Jian-Wei Pan.

Although extensive efforts have been undertaken in the experimental demonstrations of teleportation in various physical systems, including photons [3], atoms [4], ions [5], electrons [6], and superconducting circuits [7], all the previous experiments shared one fundamental limitation: the teleportation only transferred one degree of freedom (DOF). This is insufficient for complete teleportation of an object, which could naturally possess many DOFs. Even in the simplest case, for example, a single photon, the elementary quanta of electromagnetic radiation, has intrinsic properties including its frequency, momentum, polarization and orbital angular momentum. A hydrogen atom—the simplest atom—has principle quantum number, spin and orbital angular momentum of its electron and nuclear, and various couplings between these DOFs which can result in hybrid entangled quantum states.

Complete teleportation of an object would require all the information in various DOFs are transferred at a distance. Quantum teleportation is a linear operation applied to the quantum states, thus teleporting multiple DOFs should be possible in theory. Experimentally, however, it poses significant challenges in coherently controlling multiple quantum bits (qubits) and DOFs. Hyper-entangled states—simultaneous entanglement among multiple DOFs—are required as the nonlocal quantum channel for teleportation. Moreover, the teleportation also necessitates unambiguous discrimination of hyper-entangled Bell-like states from a total number of 4^N (N is the number of the DOFs). Bell-state measurements would normally require coherent interactions between independent qubits, which can become more difficult with multiple DOFs, as it is necessary to measure one DOF without disturbing another one. With linear operations only, previous theoretical work has suggested that it was impossible to discriminate the hyper-entangled states unambiguously.

We have taken a first step toward simultaneously teleporting multiple properties of a single quantum particle [8]. In the experiment, we teleport the composite quantum states of a single photon encoded in both the polarization—spin angular momentum (SAM) — and the orbital angular momentum (OAM). We prepare hyper-entangled states in both DOFs as the quantum channel for teleportation. By exploiting quantum non-demolition measurement, we overcome the conventional wisdom to unambiguously discriminate one hyper-entangled state out of the 16 possibilities. We verify the teleportation for both spin-orbit product states and entangled state of a single photon, and achieve an overall fidelity of 0.63 that well exceeds the classical limit.

Figure 1: Scheme for quantum teleportation of the spin-orbit composite states of a single photon. Alice wishes to teleport to Bob the quantum state of a single photon 1 encoded in both its SAM and OAM. To do so, Alice and Bob need to share a hyper-entangled photon pair 2-3. Alice then carries out an h-BSM assisted by teleportation-based QND measurement with an ancillary entangled photon pair.

Figure 1 illustrates our linear optical scheme for teleporting the spin-orbit composite state. The h-BSM is implemented in a step-by-step manner. First, the two photons, 1 and 2, are sent through a polarizing beam splitter (PBS). Secondly, the two single photons out of the PBS are superposed on a beam splitter (BS, see Fig.1a). Only the asymmetric Bell state will lead to a coincidence detection where there is one and only one photon in each output, whereas for the three other symmetric Bell state, the two input photons will coalesce to a single output mode. In total, these two steps would allow an unambiguous discrimination of the two hyper-entangled Bell states. To connect these two interferometers, we exploit quantum non-demolition (QND) measurement—seeing a single photon without destroying it and keeping its quantum information intact. Interestingly, quantum teleportation itself can be used for probabilistic QND detection. As shown in Fig.1 left inset, another pair of photons entangled in OAM is used as ancillary. The QND is a standard teleportation itself.

Figure 2: Experimental setup for teleporting multiple properties of a single photon. Passing a femtosecond pulsed laser through three type-I β-barium borate (BBO) crystals generates three photon pairs, engineered in different forms. The h-BSM for the photons 1 and 2 are performed in three steps: (1) SAM BSM; (2) QND measurement; (3) OAM BSM.

Figure 2 shows the experimental setup for the realization of quantum teleportation of the spin-orbit composite state of a single photon. We prepare five different initial states to be teleported (see Fig. 3 left inset), which can be grouped into three categories: product states of the two DoFs in the computational basis, products states of the two DoFs in the superposition basis, and a spin-orbit hybrid entangled state. To evaluate the performance of the teleportation, we measure the teleported state fidelity

defined as the overlap of the ideal teleported state (|φ >) and the measured density matrix. The teleportation fidelities for |φ >_A, |φ >_B, |φ >_C, |φ >_D and |φ >_E yield 0.68±0.04, 0.66±0.04, 0.62±0.04, 0.63±0.04, and 0.57±0.02, respectively. Despite these experimental noise, the measured fidelities of the five teleported states are all well beyond 0.40—the classical limit, defined as the optimal state estimation fidelity on a single copy of a two-qubit system. These results prove the successful realization of quantum teleportation of the spin-orbit composite state of a single photon. Furthermore, for the entangled state |φ >_E, we emphasize that the teleportation fidelity exceeds the threshold of 0.5 for proving the presence of entanglement, which demonstrates that the hybrid entanglement of different DoFs inside a quantum particle can preserve after the teleportation.

Figure 3: Experimental results for quantum teleportation of spin-orbit entanglement of a single photon. The fidelities are above the classical limit and entanglement limit.

Our methods can in principle be generalized to more DOFs, for instance, involving the photon’s momentum, time and frequency. The efficiency of teleportation can be enhanced by using more ancillary entangled photons, quantum encoding, embedded teleportation tricks, and high-efficiency single-photon detectors. The multi-DOF teleportation protocol is by no means limited to this system, but can also be applied to other quantum systems such as trapped electrons, atoms, and ions, which can be expected to be tested in the near future. Besides the fundamental interest, the developed methods in this work on the manipulation of quantum states of multiple DOFs will open up new possibility in quantum technologies.

References:
[1] Anton Zeilinger, "Quantum teleportation". Scientific American, 13, 34–43 (2003). Link.
[2] Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, William K. Wootters, "Teleporting an unknown quantum state via dual classic and Einstein-Podolsky-Rosen channels". Physical Review Letters, 70, 1895–1899 (1993). Abstract.
[3] Dik Bouwmeester, Jian-Wei Pan, Klaus Mattle, Manfred Eibl, Harald Weinfurter, Anton Zeilinger, "Experimental quantum teleportation". Nature, 390, 575–579 (1997). Abstract.
[4] Xiao-Hui Bao, Xiao-Fan Xu, Che-Ming Li, Zhen-Sheng Yuan, Chao-Yang Lu, Jian-Wei Pan, "Quantum teleportation between remote atomic-ensemble quantum memories", Proceedings of the National Academy of Sciences of the USA, 109, 20347–20351 (2012). Abstract.
[5] M. D. Barrett, J. Chiaverini, T. Schaetz, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, D. Leibfried, R. Ozeri, D. J. Wineland, "Deterministic quantum teleportation of atomic qubits". Nature, 429, 737–739 (2004). Abstract.
[6] W. Pfaff, B. J. Hensen, H. Bernien, S. B. van Dam, M. S. Blok, T. H. Taminiau, M. J. Tiggelman, R. N. Schouten, M. Markham, D. J. Twitchen, R. Hanson, "Unconditional quantum teleportation between distant solid-state quantum bits". Science, 345, 532–535 (2014). Abstract.
[7] L. Steffen, Y. Salathe, M. Oppliger, P. Kurpiers, M. Baur, C. Lang, C. Eichler, G. Puebla-Hellmann, A. Fedorov, A. Wallraff, "Deterministic quantum teleportation with feed-forward in a solid state system". Nature. 500, 319–322 (2013). Abstract.
[8] Xi-Lin Wang, Xin-Dong Cai, Zu-En Su, Ming-Cheng Chen, Dian Wu, Li Li, Nai-Le Liu, Chao-Yang Lu, Jian-Wei Pan, "Quantum teleportation of multiple degrees of freedom of a single photon". Nature, 518, 516-519 (2015). Abstract.

Nonlocality and Conflicting Interest Games

[From Left to Right] Anna Pappa, Niraj Kumar, Thomas Lawson, Miklos Santha, Shengyu Zhang, Eleni Diamanti, Iordanis Kerenidis

Authors: Anna Pappa^1,2, Niraj Kumar^1,3, Thomas Lawson¹, Miklos Santha^2,4, Shengyu Zhang⁵, Eleni Diamanti¹, Iordanis Kerenidis^2,4

Affiliation:
¹LTCI, CNRS–Télécom ParisTech, Paris, France, 
²LIAFA, CNRS–Université Paris 7, France, 
³Indian Institute of Technology, Kanpur, India, 
⁴CQT, National University of Singapore, Singapore, 
⁵Department of Computer Science and Engineering and ITCSC, The Chinese University of Hong Kong, Shatin, Hong Kong.

Nonlocality is a fundamental property of quantum mechanics that has puzzled researchers since the early formulations of quantum theory. Consider two parties, Alice and Bob, with inputs x_A and x_B respectively, who are positioned far from each other, and are asked to produce one output each (y_A for Alice and y_B for Bob). Even if the two players have pre-agreed on some local hidden variables, there exist quantum correlations that cannot be reproduced by any such set of variables [1,2]. These correlations allow the two parties to perform several computational tasks more efficiently, e.g. they can win specific games with probabilities strictly higher than allowed by any classical theory.

Till now, all known examples of quantum games considered players that have common interests, meaning that they either jointly win or lose the game. A famous such example is the CHSH game [3; CHSH stands for first letters of last names of the authors of this paper], where the players win if their outputs are different when both input bits are equal to 1, and if they are the same otherwise. It can be shown that classical resources provide a winning probability of 0.75, while the sharing of a maximally entangled pair can boost the winning probability to approximately 0.85. Another important type of games is conflicting interest games. A typical example is the “Battle of the Sexes”, where Alice and Bob want to meet, but Alice wants to go to the ballet, while Bob prefers theater. In case both go to the ballet, Alice is very pleased and Bob is fine with it; if they go to the theater, Bob is very pleased and Alice is fine with it, while if they go to different places, they are both very unhappy.

In our recent work [4], we examine whether the nonlocal feature of quantum mechanics can offer an advantage similar to the one observed in the CHSH game, but for games with conflicting interests. In order to observe a quantum advantage, we will study games with incomplete information (or Bayesian games), where each party receives some input unknown to the other party [5]. We present a Bayesian game with conflicting interests, and we show that there exist quantum strategies with average payoff for the two players strictly higher than that allowed by any classical strategy. The payoffs of the players for different inputs can be viewed as a table: the rows correspond to the outputs/actions of Alice (y_A), while the columns to the outputs/actions of Bob (y_B).

The players are interested in maximizing their average payoff over the probability distribution of their inputs, and they may use some advice from a third party (source) in order to achieve their goal. This advice can be in the form of classical bits or quantum states. In general, the classical bits received by the two players can be correlated between them (for example they can be either 00 or 11), and the quantum states may be entangled. By examining all possible strategies with classical advice, it is not difficult to verify that in our game, the sum of the average payoffs of the two players cannot be more than 1.125.

On the other hand, if we consider the case where quantum advice is given to the two players in the form of a maximally entangled state (Bell pair), the players can use projective measurements on their part of the state [6], in order to boost the sum of their average payoffs to 1.28, which is higher than any strategy with classical advice can achieve. It is very interesting to note here that the strategy that attains this payoff is also a quantum equilibrium, meaning that no player can gain a higher payoff by choosing a different strategy unilaterally.

Finally, we have demonstrated our game in practice, using the commercial entangled photon source quED by QuTools and taking a large number of independent runs of the game, in order to estimate each player’s average payoff. We found that the joint payoff is 1.246, which is well above the classical bound of 1.125, and slightly below the maximum value allowed by quantum strategies (1.28), because of experimental noise.

In conclusion, we demonstrated that the nonlocal feature of quantum mechanics can offer an advantage in a scenario where the two parties have conflicting interests. We examined a Bayesian game that attains higher payoffs for both players when using quantum advice compared to any classical strategy, and we experimentally verified the quantum advantage, by playing the game using a commercial photon source.

References:
[1] John Bell, "On the Einstein Podolsky Rosen paradox". Physics, 1, 195-200 (1964). Full Article.
[2] Nicolas Brunner, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, Stephanie Wehner, “Bell nonlocality”, Review of Modern Physics, 86, 419 (2014). Abstract.
[3] John F. Clauser, Michael A. Horne, Abner Shimony, Richard A. Holt, “Proposed Experiment to Test Local Hidden-Variable Theories”. Physical Review Letters, 23, 880–884 (1969). Abstract.
[4] Anna Pappa, Niraj Kumar, Thomas Lawson, Miklos Santha, Shengyu Zhang, Eleni Diamanti, Iordanis Kerenidis, "Nonlocality and Conflicting Interest Games", Physical Review Letters, 114, 020401 (2015). Abstract.
[5] J. C. Harsanyi, Management Science, 14 (3), 159-183 (Part I), 14 (5): 320-334 (Part II), 14 (7): 486-502 (Part III) (1967/1968).
[6] Richard Cleve, Peter Høyer, Ben Toner, John Watrous, "Consequences and limits of nonlocal strategies", Proceedings of the 19th Annual IEEE Conference on Computational Complexity, pages 236–249 (2004). Full Article.

Building A Better Quantum Interface

From left to right: (top row) Bernardo Casabone, Konstantin Friebe, Birgit Brandstätter, (bottom row) Klemens Schüppert, Rainer Blatt, Tracy E. Northup.

Author: Konstantin Friebe

Affiliation: Institut für Experimentalphysik, Universität Innsbruck, Austria.

One of the largest challenges in quantum information science stems from the fact that to date no truly scalable quantum computer has been built, i.e., current devices only contain a few quantum bits (qubits). However, in order to access regimes where the power of quantum computation really comes into play, one would need scalable architectures with many qubits.

One approach for solving this issue is called distributed quantum computing. It consists of many small-scale quantum computers that are linked via photonic channels in close analogy to the internet or the “cloud” [1]. In this way, many small quantum computers can be linked together to make one large computer.

In this approach, the challenge consists of faithfully transferring quantum information between remote small-scale quantum computers via photonic channels. Quantum interfaces for this purpose can be built using cavity quantum electrodynamics systems. In such a system, the stationary qubits for computation are kept inside a cavity, also known as an optical resonator, i.e., between two mirrors. These mirrors enhance the interaction between the stationary qubits and photons (“flying qubits”), so that faithful transfer of quantum information from matter to light becomes possible. In fact, the mirrors make it possible to generate a single information-carrying photon from a stationary qubit and to be able to send this photon to another quantum computer with high efficiency [2].

Figure 1: Schematic of the setup. Two calcium ions (green spheres) are trapped inside an optical resonator (mirrors). By addressing the ions with laser beams at wavelengths 729 nm (global 729, addressing 729) and 393 nm, it is possible to prepare them in an entangled state with controllable phase, and a single photon can be generated in the cavity (red standing wave profile). The polarisation of the photon carries one qubit of information. After the photon has left the resonator, it is analysed using waveplates (λ/2, λ/4) and a polarising beam splitter (PBS), which splits up the two orthogonal polarisations H (horizontal) and V (vertical). The photon is then detected at one of two avalanche photodiodes (APD1 and APD2). (This Figure is from reference [3]).

In our recent experiment at the University of Innsbruck, Austria, we trapped two calcium ions inside an optical resonator [3]. In this case, the ions constitute the small-scale quantum computer. Ions are well-suited for this task, as an extensive toolbox for their preparation, manipulation and readout exists. The two ions were laser-cooled and prepared in an entangled state by manipulating their electronic and motional states with a laser field. Entanglement means that the two ions have “lost their individuality” and have to be described as a collective system with collective qualities. In this case, it is the electronic states of the two ions that are entangled with one another. This entangled state can be characterized by a phase, i.e., a number between zero and 2π. By controlling the phase of the entangled state of the two ions, it was possible to either enhance the probability to generate a photon in the cavity (phase 0) or to suppress the generation of a photon (phase π). The first case is called superradiance, while the suppression is called subradiance.

Figure 2: Probability of detecting a photon as a function of the time after the photon generation is started. The blue circles show the photon detection probability for the superradiant case (entangled state with phase 0), while the brown diamonds represent the subradiant state (entangled state with phase π). For comparison, the case of the two individual ions is shown (open triangles). For the superradiant case, the photon is produced faster than for a single ion, while in the subradiant case, photon generation is suppressed. Lines are simulations. (This Figure is from reference [3]).

We next encoded one qubit of information in the state of two entangled ions, that is, we used two “physical qubits” as a single “logical qubit”. The information stored in this qubit was then mapped onto the polarisation state of a single photon. By analysing the polarisation of the photon after it had left the resonator, we were able to show that the transfer of information was more faithful if the two ions were in the state with phase 0 than for a single ion. The efficiency of the process was higher, too.

Figure 3: Process fidelity (upper panel), the measure for the faithfulness of the transfer of quantum information, and efficiency (lower panel) as a function of the time after the photon generation is started. Blue filled circles are data from the superradiant entangled state, while open black circles are data from a single ion. Both process fidelity and efficiency are higher for the case of two entangled ions in the superradiant state. Lines are simulations. (This Figure is adapted from reference [3]).

In conclusion, by encoding quantum information in entangled states of more than one qubit, a better, i.e., more faithful quantum interface can be built, which is important for distributed quantum computing. Additionally, by choosing the phase π instead, qubits can be completely decoupled from the interface. This is interesting for future quantum computers consisting of long arrays of ions, where one might need a selective interface for just certain qubits, while the others remain undisturbed.

References:
[1] H. J. Kimble, "The quantum internet", Nature, 453, 1023 (2008). Abstract. 
[2] T.E. Northup, R. Blatt, "Quantum information transfer using photons", Nature Photonics, 8, 356 (2014). Abstract. 
[3] B. Casabone, K. Friebe, B. Brandstätter, K. Schüppert, R. Blatt, T. E. Northup, "Enhanced quantum interface with collective ion-cavity coupling", Physical Review Letters, 114, 023602 (2015). Abstract.
[4] C. Russo, H. G. Barros, A. Stute, F. Dubin, E. S. Phillips, T. Monz, T. E. Northup, C. Becher, T. Salzburger, H. Ritsch, P. O. Schmidt, R. Blatt, "Raman spectroscopy of a single ion coupled to a high-finesse cavity", Applied Physics B, 95, 205 (2009). Abstract.

Achieving 200 km of Measurement-device-independent Quantum Key Distribution with High Secure Key Rate

[Left to Right] Hualei Yin, Tengyun Chen, Yanlin Tang.

Authors: Yan-Lin Tang^1,2, Hua-Lei Yin^1,2, Si-Jing Chen³, Yang Liu^1,2, Wei-Jun Zhang³, Xiao Jiang^1,2, Lu Zhang³, Jian Wang^1,2, Li-Xing You³, Jian-Yu Guan^1,2, Dong-Xu Yang^1,2, Zhen Wang³, Hao Liang^1,2, Zhen Zhang^2,4, Nan Zhou^1,2, Xiongfeng Ma^2,4, Teng-Yun Chen^1,2, Qiang Zhang^1,2, Jian-Wei Pan^1,2

Affiliation:
¹National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, Shanghai Branch, University of Science and Technology of China, Hefei, China,
²CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, Shanghai Branch, University of Science and Technology of China, Hefei, China,
³State Key Laboratory of Functional Materials for Informatics, Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai, China,
⁴Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China.

 Information security is a long-standing problem in history. Nowadays, with the developing requirement of information transmission, the security becomes a much more sensitive problem. By leveraging the laws of quantum Mechanics, Quantum Key Distribution (QKD) [1] can provide a solution for information-theoretical security. As the most practical application of quantum information technology, it is under rapid development in both theoretical and experimental aspects. Besides the standard BB84 protocol, various protocols are proposed subsequently to adapt to different situations. Meanwhile, the QKD systems are successfully transformed from controlled laboratory environments to real-life implementations, and quite a few commercial QKD systems are available in the market up till now.

Past 2Physics article by Jian-Wei Pan :
June 30, 2013: "Quantum Computer Runs The Most Practically Useful Quantum Algorithm" by Chao-Yang Lu and Jian-Wei Pan.

Despite these tremendous developments, real-world QKD systems still suffer from various attacks [2-4] which explore the loopholes rooted in the deviations of practical implementations from the theoretical models in security proofs. Most of these attacks are targeting at the measurement devices. Among them, the first successful attack is the time-shift attack [3] which explores the loophole of time-dependent efficiency mismatch of two detectors. The most powerful kind of attacks is the detector-blinding attack [4], which fools the detector to work in the unwanted linear mode and forces them to act according to Eve’s will. Although certain countermeasures are provided to close some specific side channels, there might still be some side channels which are hard to estimate and will cause potential threats. So we are looking for an effective solution to close these loopholes once and for all.

Fortunately, Measurement-Device-Independent Quantum Key Distribution (MDIQKD) was invented by H-.K-. Lo in 2012 [5] to remove all side channels from the most vulnerable measurement unit. This protocol is inspired by the time-reversed EPR protocol [6], and it does not rely on any measurement assumption and can thus close all the measurement loopholes once and for all. Since its invention, it has attracted worldwide attention, and has been successfully demonstrated based on various MDIQKD systems, including polarization encoding system [7,8] and time-bin phase-encoding system [9,10]. In view of the performance, these previous MDIQKD demonstrations have limitations as well, such as short distance and a poor key rate (the best is 0.1 bps@50km [10]). This is so because the critical element of MDIQKD protocol is the Bell-state measurement, which requires both perfect interference of two independent laser sources and efficient two-fold coincidence detection. It imposes severe technical challenges on the laser modulation, high-efficiency detection and system stabilization. Therefore, based on the previous results and the intrinsic requirements of MDIQKD, people might still wonder that this ingenious protocol is a fancy but impractical idea.

In this recent work published in Physical Review Letters [11] by our group, we have extended the MDIQKD secure distance to state-of-the-art 200 km, comparable with the limit of regular decoy-state BB84 protocol. The secure key rate is almost three orders of magnitude higher than the previous results of MDIQKD demonstrations. These results are achieved with a fully-automatic highly-stable 75 MHz system and high-efficiency superconducting single photon detectors (SNSPDs), as shown in Fig. 1. We also employ an optimized decoy-state scheme and new post-processing method with a much lower failure probability than previous ones.

Fig.1: (a) Schematic layout of our MDIQKD setup. Alice's (Bob's) signal laser pulses (1550 nm) are modulated into three decoy-state intensities by AM1. An AMZI, AM2~4 and one PM are to encode qubits. Charlie's setup consists of a polarization stabilization system and a BSM system. The polarization stabilization system in each link includes an electric polarization controller (EPC), a polarization beam splitter (PBS) and an InGaAs/InP single-photon avalanche photodiode (SPAPD). The BSM system includes an interference BS and two SNSPDs. (b) Time calibration system. Two synchronization lasers (SynL, 1570 nm) are adopted, with the 500 kHz shared time reference generated from a crystal oscillator circuit (COC) and with the time delayed by a programmable delay chip (PDC). Alice (Bob) receives the SynL pulses with a photoelectric detector (PD) and then regenerates a system clock of 75 MHz. WDM: wavelength division multiplexer, ConSys: control system. (c) Phase stabilization system. Circ: circulator, PC: polarization controller, PS: phase shifter.

This is the first time we increase the repetition rate to 75 MHz, compared with 1MHz of our previous demonstration [10]. The repetition rate improvement owes to the laser source with good waveform, the high-speed electrical control system, and the superconducting single photon detector with a small time jitter of a few 10 ps [12]. In terms of high-speed laser modulation, we remark that the speed improvement for MDIQKD is not as easy as that for regular BB84 protocol, since the indistinguishability of two independent laser sources has some subtle requirements for laser modulation. Firstly, we should adopt direct laser modulation to ensure the phase is intrinsically randomized to avoid the unambiguous-state-discrimination attack [13]. The problem is that in a high-speed situation, the current mutation will induce severe overshoot, ringing and chirp inside the laser pulse. Especially, the chirp adds an extra phase at the tail of our laser pulse. Thus, we cut off the tail part by an amplitude modulator (AM), to optimize the laser interference and ensure the waveform indistinguishability. Secondly, regarding the vacuum state modulation (based on the vacuum+weak decoy state scheme), we should take the influence of the direct laser modulation into consideration, which is not a severe problem for regular BB84 protocol. We find that when we randomly modulate some laser pulses into vacuum state by not sending triggering signal to the laser (namely direct laser modulation), the interference visibility will decrease to a very bad level. This is because of the aperiodic triggering signals to the laser which introduce large temperature fluctuation and wavelength fluctuation. The wavelength fluctuation thus causes imperfect interference. To avoid this effect, instead of direct vacuum modulation, we adopt an alternative method of external vacuum modulation by AM. We utilize three AMs, within which only one is for decoy state encoding, and the other two are mainly used for qubit encoding and are also beneficial to decrease the vacuum intensity. Thus a high extinction ratio of the vacuum state of more than 10000 : 1 is achieved.

This is also the first time superconducting nanowire single-photon detectors (SNSPD), one of the best single photon detectors at near-infrared (NIR) wavelengths, is applied in an MDIQKD system. Since the BSM, the essence of MDIQKD, requires two-fold coincidence detection, the key rate is proportional to the square of detection efficiency. In our experiment, operated below 2.2 K with a Gifford-McMahon cryocooler, two SNSPDs with detection efficiencies of 40% and 46% largely improve the key rate. Besides, the low dark count rate of 10 Hz helps to achieve an enough signal-to-noise ratio even at 200 km distance. Besides the high detection efficiency and low dark count, there is another important property, small timing jitter of a few 10 ps, which is beneficial for QKD performance, especially the system timing jitter and repetition rate. We can expect an improvement of 1 GHz up to 10 GHz MDIQKD system adopting the SNSPD in the near future.

Another important element for achieving 200 km distance is the system stabilization. Since the 200 km situation will make the system stability difficult because of the severe fiber fluctuation, and make it even harder with weak feedback signals due to large fiber attenuation. Besides, since the detection rate is slower in 200 km, we need more time to accumulate enough data required by strict fluctuation analysis. In short, we need our system to work in a worse environment for a longer time. Faced with these problems, we build a fully-automatic feedback system without manual efforts to precisely calibrate and stabilize all the parameters, such as the time, spectrum, polarization and the phase reference. Although the whole feedback system is a challenge in engineering, it is critical to enable continuous running and will be a necessary component in practical MDIQKD system.

Fig.2: Bird's-eye view of the field-environment MDIQKD. Alice is placed in Animation Industry Park in Hefei (AIP), Bob in an office building (OB), and Charlie in the University of Science and Technology of China (USTC). Alice (Bob) is on the west (east) side of Charlie. AIP-USTC link is 25 km (7.9 dB), and OB-USTC link is 5 km (1.3 dB).

To further show the practical value of MDIQKD in an unstable environment, we have moved the system into installed fiber network and implemented a field test as shown in Fig. 2 [14]. Previously, an MDIQKD field test was attempted over an 18.6 km deployed fiber, however, a secure key was not actually generated since random modulated decoy state was not performed. In comparison, our field test strictly adopts the decoy-state scheme to guarantee the source security. With optimized decoy-state parameters and Chernoff bound in strict fluctuation analysis with tight failure probability of 2×10⁻⁹, we have achieved secure key rates of 67 bps (@50km in the laboratory for 130.0 hours) and 17 bps (@30km in the field test for 18.2 hours), shown in Fig.3, which are at least two orders of magnitude higher than previous results.

Fig. 3: Secure key rates of experiments in the laboratory and in the field test, as well as the simulation results. The four dots correspond to the experimental results with the fiber transmitting loss of 9.9 dB (50 km), 19.9 dB (100 km), 29.8 dB (150 km) and 39.6 dB (200 km). The solid curve shows the result calculated by simulating the vacuum+weak decoy state scheme with the experimental parameters. The dashed curve represents the optimal result with infinite number of decoy states. The square marks the field test result, which is 17 bps. Also shown are results from the previous demonstration for comparison.

These technological advances in our work constitute a critical ingredient for quantum repeater [15], the core resource for long distance quantum communication. Besides, the MDIQKD protocol has an intrinsic property which is desirable for constructing quantum network [16] with the star-type structure. We can place the expensive detection system in the server node to perform the BSM operation, and all the users can share this system. Furthermore, the techniques of stable BSM we developed have many other applications, such as quantum teleportation [17] and quantum fingerprinting [18].

References:
[1] Charles H. Bennett and Gilles Brassard, "Quantum cryptography: Public key distribution and coin tossing”, in Proceedings of IEEE International Conference on Computers, Systems, and Signal Processing (Bangalore, India, 1984), pp. 175–179. Full Article.
[2] Chi-Hang Fred Fung, Bing Qi, Kiyoshi Tamaki, Hoi-Kwong Lo, “Phase-remapping attack in practical quantum-key-distribution systems”. Physical Review A, 75, 032314 (2007). Abstract.
[3] B. Qi, C.-H. F. Fung, H.-K. Lo, and X. Ma, “Time-shift attack in practical quantum cryptosystems”, Quantum Information & Computation, 7, 073 (2007).
[4] Lars Lydersen, Carlos Wiechers, Christoffer Wittmann, Dominique Elser, Johannes Skaar, Vadim Makarov, “Hacking commercial quantum cryptography systems by tailored bright illumination”, Nature Photonics, 4, 686 (2010). Abstract.
[5] Hoi-Kwong Lo, Marcos Curty, Bing Qi, “Measurement-Device-Independent Quantum Key Distribution”, Physical Review Letters, 108, 130503 (2012). Abstract.
[6] Eli Biham, Bruno Huttner, Tal Mor, “Quantum cryptographic network based on quantum memories”. Physical Review A, 54, 2651 (1996). Abstract.
[7] Zhiyuan Tang, Zhongfa Liao, Feihu Xu, Bing Qi, Li Qian, Hoi-Kwong Lo, “Experimental Demonstration of Polarization Encoding Measurement-Device-Independent Quantum Key Distribution”, Physical Review Letters, 112, 190503 (2014). Abstract.
[8] T. Ferreira da Silva, D. Vitoreti, G. B. Xavier, G. C. do Amaral, G. P. Temporão, J. P. von der Weid, “Proof-of-principle demonstration of measurement-device-independent quantum key distribution using polarization qubits”, Physical Review A, 88, 052303 (2013). Abstract.
[9] A. Rubenok, J. A. Slater, P. Chan, I. Lucio-Martinez, W. Tittel, “Real-World Two-Photon Interference and Proof-of-Principle Quantum Key Distribution Immune to Detector Attacks”, Physical Review Letters, 111, 130501 (2013). Abstract.
[10] Yang Liu, Teng-Yun Chen, Liu-Jun Wang, Hao Liang, Guo-Liang Shentu, Jian Wang, Ke Cui, Hua-Lei Yin, Nai-Le Liu, Li Li, Xiongfeng Ma, Jason S. Pelc, M. M. Fejer, Cheng-Zhi Peng, Qiang Zhang, Jian-Wei Pan, “Experimental Measurement-Device-Independent Quantum Key Distribution”. Physical Review Letters, 111, 130502 (2013). Abstract.
[11] Yan-Lin Tang, Hua-Lei Yin, Si-Jing Chen, Yang Liu, Wei-Jun Zhang, Xiao Jiang, Lu Zhang, Jian Wang, Li-Xing You, Jian-Yu Guan, Dong-Xu Yang, Zhen Wang, Hao Liang, Zhen Zhang, Nan Zhou, Xiongfeng Ma, Teng-Yun Chen, Qiang Zhang, Jian-Wei Pan, “Measurement-device-independent quantum key distribution over 200 km”. Physical Review Letters, 113, 190501 (2014). Abstract.
[12] Xiaoyan Yang, Hao Li, Weijun Zhang, Lixing You, Lu Zhang, Xiaoyu Liu, Zhen Wang, Wei Peng, Xiaoming Xie, Mianheng Jiang, “Superconducting nanowire single photon detector with on-chip bandpass filter”, Optics Express, 22, 16267 (2014). Abstract.
[13] Yan-Lin Tang, Hua-Lei Yin, Xiongfeng Ma, Chi-Hang Fred Fung, Yang Liu, Hai-Lin Yong, Teng-Yun Chen, Cheng-Zhi Peng, Zeng-Bing Chen, Jian-Wei Pan, “Source attack of decoy-state quantum key distribution using phase information”, Physical Review A, 88, 022308 (2013). Abstract.
[14] Yan-Lin Tang, Hua-Lei Yin, Si-Jing Chen, Yang Liu, Wei-Jun Zhang, Xiao Jiang, Lu Zhang, Jian Wang, Li-Xing You, Jian-Yu Guan, Dong-Xu Yang, Zhen Wang, Hao Liang, Zhen Zhang, Nan Zhou, Xiongfeng Ma, Teng-Yun Chen, Qiang Zhang, Jian-Wei Pan, “Field Test of Measurement-Device-Independent Quantum Key Distribution”, arXiv:1408.2330 [quant-ph] (2014).
[15] H.-J. Briegel, W. Dür, J. I. Cirac, P. Zoller, “Quantum Repeaters: The Role of Imperfect Local Operations in Quantum Communication”, Physical Review Letters, 81, 5932 (1998). Abstract.
[16] Jane Qiu, “Quantum communications leap out of the lab”, Nature, 508, 441 (2014). Article.
[17] Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, William K. Wootters, “Teleporting an unknown quantum state via dual classic and Einstein-Podolsky-Rosen channels”, Physical Review Letters, 70, 1895 (1993). Abstract.
[18] Harry Buhrman, Richard Cleve, John Watrous, Ronald de Wolf, “Quantum Fingerprinting”, Physical Review Letters, 87, 167902 (2001). Abstract.

Observation of Majorana Fermions in Ferromagnetic Atomic Chains on a Superconductor

From Left to Right: (top row) Stevan Nadj-Perge, Ilya K. Drozdov, Jian Li, Hua Chen ; (bottom row) Sangjun Jeon, Allan H. MacDonald, B. Andrei Bernevig, Ali Yazdani.

Authors: Stevan Nadj-Perge¹, Ilya K. Drozdov¹, Jian Li¹, Hua Chen², Sangjun Jeon¹, Jungpil Seo¹, Allan H. MacDonald², B. Andrei Bernevig¹, Ali Yazdani¹.

Affiliation
¹Joseph Henry Laboratories and Dept of Physics, Princeton University, USA.
²Department of Physics, University of Texas at Austin, USA.

Link to Yazdani Lab >>
Link to Allan H. MacDonald's Group >>

In 1937 Italian scientist Ettore Majorana, one of the most promising theoretical physicists at that time, proposed a hypothetical fermionic excitation, now called Majorana fermion, which has a property that it is its own anti-particle [1]. Ever since significant efforts were invested in finding an elementary particle described by Majorana. While present for many years in particle physics community, it was only in 2001 that Alexei Kitaev suggested an intriguing possibility that a type of a quasi-particle, a condensed matter analog of the Majorana fermion could exist [2]. Such quasi-particle would emerge as a zero energy excitation localized at the boundary of a one dimensional topological superconductor. Following his seminal work various systems were proposed as a potential platform for realization of Majorana quasi-particles. Apart from fundamental scientific interest, the motivation for investigating Majorana bound states partly relies on their potential for use for quantum computing [3,4].

Past 2Physics article by Ali Yazdani :
August 25, 2013: "Visualizing Nodal Heavy Fermion Superconductivity"
by Brian Zhou, Shashank Misra, and Ali Yazdani.

Previous to our experiments, the most promising experimental route to realize these elusive bound states was based on semiconductor-superconductor interfaces in which Majorana fermions would appear as conductance peaks at zero energy [5,6]. Indeed experiments in 2012 reported zero energy conductance peaks suggesting presence of Majorana modes in these type of systems [7,8], however, alternative explanations related to disorder and Kondo phenomena proposed latter could not be fully ruled out. It is worth noting that in these interfaces spatial information about localized excitations is very hard to obtain.

Fig. 1: (A) Schematic of the proposal for realization and detection of Majorana states: A ferromagnetic atomic chain is placed on the surface of strongly spin-orbit–coupled superconductor and studied using STM. (B) Band structure of a linear suspended Fe chain before introducing spin-orbit coupling or superconductivity. The majority spin-up (red) and minority spin-down (blue) d-bands labeled by azimuthal angular momentum m are split by the exchange interaction J (degeneracy of each band is noted by the number of arrows). a, interatomic distance. (C) Regimes for trivial and topological superconducting phases are identified for the band structure shown in (B) as a function of exchange interaction in presence of SO coupling. The value J for Fe chains based on density functional calculations is noted. μ is the chemical potential.

Building upon previous proposal to realize Majorana modes in an array of magnetic nanoparticles [9], we proposed to use a chain of magnetic atoms coupled to a superconductor [10]. The key advantage of this platform is that the experimentally properties of this system can easily be studied using standard scanning tunneling microscopy (STM) technique. While ours and other follow-up initial proposals [11-16] consider specific orientation of the magnetic moments, the approach works also for ferromagnetic atomic chains as long as it is coupled to a superconductor with strong spin-orbit coupling (Fig. 1A) [17]. In this case, the large exchange interaction results in a band structure of the chains such that majority spin band is fully occupied while the Fermi level is in the minority spin bands. For example the electronic structure of a linear Iron (Fe) ferromagnetic atomic chain is shown in Fig. 1B. Considering only d-orbitals which are spin-polarized it is easy to show that many of the bandstructure degeneracies are lifted and that for large range of parameters, chemical potential and the exchange energy, such chains are in topologically non-trivial regime characterized by the odd number of crossing at the Fermi level (Fig 1C). When placed on the superconducting substrate with strong-spin orbit coupling the resulting superconductivity on the chain will necessarily be topological in nature resulting in zero energy Majorana bound states located at the chain ends.

Fig. 2: (A) Topograph of the Pb(110) surface after growth of Fe, showing Fe islands and chains indicated by white arrows and atomically clean terraces of Pb (regions with the same color) with size exceeding 1000 Å. (Lower-right inset) Anisotropic atomic structure of the Pb(110) surface (Upper-left insets) images of several atomic Fe chains and the islands from which they grow (scale bars, 50 Å). (B) Topography of the chain colorized by the conductance at H= ±1 T from low (dark blue) to high conductance (dark red). (C) Difference between conductance on and off the chain showing hysteresis behavior. (D and E) Atomic structure of the zigzag chain, as calculated using density functional theory. The Fe chain structure that has the lowest energy in the calculations matches the structural features in the STM measurements, a_B is the Bohr radius.

We have developed a way to grow iron atomic chains on the surface of lead (Pb) which, due to heavy atomic mass, is expected to have strong spin-orbit coupling. For this purpose we used Pb(110) crystallographic surface orientation which has characteristic anisotropy (Fig. 2(A) lower left inset). When a sub-monolayer of Fe is evaporated and slight annealing, the anisotropy of the substrate could trigger growth of one-dimensional atomic chains. We investigated the resulting structures by using STM at cryogenic temperatures (temperature was 1.4K in the experiment). On relatively large atomically ordered regions of the Pb(110) surface we observed self-assembled islands as well as single atom wide chains of Fe. Depending on growth conditions, we find Fe chains as long as 500 Å with ordered regions approaching 200 Å. In order to confirm ferromagnetic order on the chain we have performed spin-polarized measurements using bulk antiferromagnetic Chromium STM tips. Tunneling conductance (dI/dV) at a low bias voltage as a function of the out-of-plane magnetic field shows contrast for opposite fields (Fig. 2B) and a hysteresis behavior (Fig. 2C, note that no hysteresis is observed on the Pb substrate). The observed hysteresis loop corresponds to the tunneling of electrons between two magnets with the field switching only one of them at around 0.25 T.

We also observed the variation of the spin-polarized STM signal along the chain which is likely due to its electronic and structural properties. Indeed, both topographic features and periodicity of signal variation could be very well explained by our theory collaborators who performed density functional theory modeling. Their calculations suggested that our chains have zig-zag structure which explains both topographic information obtained using STM and matches well with our spin-polarized measurements (Fig. 2D and 2E).

Fig. 3: (A) STM spectra measured on the atomic chain at locations corresponding to those indicated in (B) and (C). For clarity, the spectra are offset by 100 nS. The red spectrum shows the zero-bias peak at one end of the chain. The gray trace measured on the Pb substrate can be fitted using thermally broadened BCS superconducting density of states (dashed gray line, fit parameters Δs = 1.36 meV, T = 1.45 K). (B and C) Zoom-in topography of the upper (B) and lower end (C) of the chain and corresponding locations for spectra marked (1 to 7). Scale bars, 25 Å. (D and E) Spectra measured at marked locations, as in (B) and (C). (F) Spatial and energy-resolved conductance maps of another atomic chain close to its end, which shows similar features in point spectra as in (A). The conductance map at zero bias (middle panel) shows increased conductance close to the end of the chain. Scale bar, 10 Å.

After establishing basic properties of our chains we investigated low-energy excitations using spatial spectroscopic mapping, see Fig 3. While on the surface of bare Pb(110), there is clear structure of the superconducting gap on the Fe atomic chain, the presence of the in-gap states is predominant (Fig. 3A). Most notably a peak close to zero bias voltage is observed near the chain end together with asymmetric less-developed gap-like structure in the middle of the chain (Fig. 3D and Fig. 3E). Both spatially resolved spectra and the spectroscopic maps at low bias voltage show signatures expected from Majorana bound states (Fig. 3F). The ability to correlate the location of the zero bias conductance peak with the end of the atomic chains is one of the main experimental results of our work. This is one of the basic requirements for interpreting that this feature is associated with the predicted Majorana bound state of a topological superconductor. In addition to robust observation of the zero bias peaks in many chains, we have performed several control experiments to eliminate other potential effects which may give similar looking signatures. For example, when superconductivity is suppressed by applying small magnetic field, the spectrum on the chain becomes featureless in contrast to what would be expected for Kondo effect. Also for very short chains zero biased peaks were not observed, ruling out trivial effects related to the chain ends. Furthermore, in order to increase experimental resolution we took measurements with superconducting tip which also confirm the over picture consistent with Majorana bound states in this system.

The observed spectroscopic signatures are consistent with the existence of Majorana bound states in our system. An obvious extension of our experiments is to create two dimensional islands and search for propagating Majorana modes or, for example, investigate other systems with both even and odd number of band crossing at Fermi level in order to further test the concept behind our study. Ultimately the future experiments will focus on manipulation of Majorana bound states in this system [18].

References: 
[1] Ettore Majorana, "Teoria simmetrica dell’elettrone e del positrone", Il Nuovo Cimento, 171 (1937). Abstract.
[2] A. Yu. Kitaev, "Unpaired Majorana fermions in quantum wires". Physics Uspekhi, 44, 131 (2001). Full Article.
[3] A. Yu. Kitaev, "Fault-tolerant quantum computation by anyons". Annals of Physics. 303, 2 (2003). Abstract.
[4] Jason Alicea, Yuval Oreg, Gil Refael, Felix von Oppen, Matthew P. A. Fisher, "Non-Abelian statistics and topological quantum information processing in 1D wire networks". Nature Physics, 7, 412 (2011). Abstract.
[5] Yuval Oreg, Gil Refael, and Felix von Oppen, "Helical Liquids and Majorana Bound States in Quantum Wires", Physical Review Letters, 105, 177002 (2010). Abstract.
[6] Roman M. Lutchyn, Jay D. Sau, S. Das Sarma, "Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures". Physical Review Letters, 105, 077001 (2010). Abstract.
[7] V. Mourik, K. Zuo, S.M. Frolov, S.R. Plissard, E.P.A.M. Bakkers, L.P. Kouwenhoven, "Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices", Science 336, 1003 (2012). Abstract. 2Physics Article.
[8] Anindya Das, Yuval Ronen, Yonatan Most, Yuval Oreg, Moty Heiblum, Hadas Shtrikman, "Zero-bias peaks and splitting in an Al-InAs nanowire topological superconductor as a signature of Majorana fermions". Nature Physics, 8, 887 (2012). Abstract. 2Physics Article.
[9] T. P. Choy, J. M. Edge, A. R. Akhmerov, C. W. J. Beenakker, "Majorana fermions emerging from magnetic nanoparticles on a superconductor without spin-orbit coupling". Physical Review B, 84, 195442 (2011). Abstract.
[10] S. Nadj-Perge, I. K. Drozdov, B. A. Bernevig, Ali Yazdani, "Proposal for realizing Majorana fermions in chains of magnetic atoms on a superconductor". Physical Review B, 88, 020407 (2013). Abstract.
[11] Falko Pientka, Leonid I. Glazman, Felix von Oppen, "Topological superconducting phase in helical Shiba chains". Physical Review B, 88, 155420 (2013). Abstract.
[12] Jelena Klinovaja, Peter Stano, Ali Yazdani, Daniel Loss, "Topological Superconductivity and Majorana Fermions in RKKY Systems". Physical Review Letters, 111, 186805 (2013). Abstract.
[13] Bernd Braunecker, Pascal Simon, "Interplay between Classical Magnetic Moments and Superconductivity in Quantum One-Dimensional Conductors: Toward a Self-Sustained Topological Majorana Phase". Physical Review Letters, 111, 147202 (2013). Abstract.
[14] M. M. Vazifeh, M. Franz, "Self-Organized Topological State with Majorana Fermions". Physical Review Letters, 111, 206802 (2013). Abstract.
[15] Sho Nakosai, Yukio Tanaka, Naoto Nagaosa, "Two-dimensional superconducting states with magnetic moments on a conventional superconductor". Physical Review B, 88, 180503 (2013). Abstract.
[16] Younghyun Kim, Meng Cheng, Bela Bauer, Roman M. Lutchyn, S. Das Sarma, "Helical order in one-dimensional magnetic atom chains and possible emergence of Majorana bound states". Physical Review B, 90, 060401 (2014). Abstract.
[17] Stevan Nadj-Perge, Ilya K. Drozdov, Jian Li, Hua Chen, Sangjun Jeon, Jungpil Seo, Allan H. MacDonald, B. Andrei Bernevig, Ali Yazdani, "Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor". Science 346, 602-607 (2014). Abstract.
[18] Jian Li, Titus Neupert, B. Andrei Bernevig, Ali Yazdani, "Majorana zero modes on a necklace". arXiv:1404.4058 [cond-mat] (2014).

Optomechanical Coupling between a Multilayer Graphene Mechanical Resonator and a Superconducting Microwave Cavity

Left to Right: (top row) V. Singh, S. J. Bosman, B. H. Schneider, (bottom row) Y. M. Blanter, A. Castellanos-Gomez, G. A. Steele.

Authors: 
V. Singh, S. J. Bosman, B. H. Schneider, Y. M. Blanter, A. Castellanos-Gomez, 
G. A. Steele

Affiliation:
Kavli Institute of NanoScience, Delft University of Technology, The Netherlands.

Introduction:

Mechanical resonators made from two dimensional exfoliated crystals offer very low mass, low stress, and high quality factor due to their crystalline structure [1]. These properties make them very attractive for application in mass sensing, force sensing, and exploring the quantum regime of motion by providing large quantum zero-point fluctuations over a small bandwidth. The most studied exfoliated crystal so far is graphene, where a considerable progress has been made in exploring its properties for mass sensing, study of nonlinear mechanics, and voltage tunable oscillators [2-9]. These properties also make graphene attractive for exploring the quantum regime of motion.

Past 2Physics articles by Andres Castellanos-Gomez and Gary A. Steele:

July 20, 2014: "Few-layer Black Phosphorus Phototransistors for Fast and Broadband Photodetection" by Michele Buscema, Dirk J. Groenendijk, Sofya I. Blanter, Gary A. Steele, Herre S.J. van der Zant, Andres Castellanos-Gomez.

A possible route towards exploring the quantum regime of graphene motion is cavity optomechanics [10]. It has shown exquisite position sensitivity, enabled the preparation and detection of mechanical systems in the quantum ground state with conventional top-down superconducting mechanical resonators [11-18]. Therefore, a natural candidate for implementing cavity optomechanics with graphene resonator is to couple it to a high Q superconducting microwave cavity. However, coupling graphene resonators with superconducting cavities in such a way that both retain their excellent properties (such as their high quality factors) is technologically challenging. Using a deterministic dry transfer technique [19], we combine a multilayer graphene resonator to a high quality factor microwave cavity [20]. Although multilayer graphene has a higher mass than a mono-layer, it could be advantageous for coupling to a superconducting cavity because of its lower electrical resistance.

Results:

Device

To fabricate the superconducting cavities in coplanar waveguide geometry, we use an alloy of molybdenum and rhenium with superconducting transition temperature of 8.1 K. Using the dry transfer technique, we place a few layer thick graphene mechanical resonator near the coupler forming coupling capacitor for the cavity. Figure 1(a) shows a false color scanning electron microscope image of a device with a 10 nm thick multilayer graphene resonator coupled to a superconducting cavity. Figure 1(b) shows an equivalent schematic diagram with graphene resonator acting as a capacitor (C_g ) between the superconducting cavity (formed by L_sc and C_sc ) and the external microwave source. By cooling these cavities to very low temperatures (14 mK), we measured internal quality factor as high as 107,000.

FIG. 1: Coupling of a multilayer graphene mechanical resonator to a superconducting cavity. (a) A tilted angle scanning electron micrograph (false color) near the coupler showing 4 μm diameter multilayer (10 nm thick) graphene resonator (cyan) suspended 150 nm above the gate. (b) Schematic lumped element representation of the device with the equivalent lumped parameters as C_sc ≈ 415 fF and L_sc ≈ 1.75 nH.

Mechanical motion readout sensitivity

To the first order, the superconducting microwave cavity can be thought simply as motional transducer for the graphene resonator. To readout the motion of the graphene resonator, we inject a microwave near the cavity frequency given by

                                       
The motion of graphene resonator modulates the cavity frequency and hence its displacement gets imprinted on the phase of the reflected microwave signal from the cavity. By measuring the phase of the reflected signal (technically known as the homodyne detection), one can directly read the mechanical motion of the resonator [11]. The large quality factor of our cavity and its ability to sustain superconductivity with large number of the microwave photons enable us to measure the thermo-mechanical motion of the graphene resonator down to temperatures of 96 mK and a displacement sensitivity as low as 17 fm/√Hz.

Optomechanical coupling

In addition to detecting the motion of the graphene drum, we can also exert a force on the mechanical drum by using the radiation pressure of microwave photons trapped in the superconducting cavity. This force comes from the fact that light carries momentum: shining light from a flashlight at a piece of paper would in principle apply a force to it, pushing it away from the light source. The radiation pressure force that light exerts, however, is usually far too small to detect. Due to the tiny mass of the graphene sheet and the ability to detect small displacement, we could see the graphene sheet shaking in response to a "beat" set by the microwave light sent into the cavity.

By sending two microwave signals, a probe signal ω_p (near the cavity resonance frequency ω_c ) and another signal at ω_d (detuned by mechanical frequency ω_m , such that ω_d = ω_c +ω_m ), one can apply a a radiation pressure force on the mechanical resonator. This radiation pressure force beats at the mechanical resonance frequency, leading to coherent driven motion of the mechanical resonator, as shown schematically by process 1 in Figure 2(a). In presence of the significant optomechanical coupling, this coherent drive of the mechanical resonator down-converts the detuned drive photons exactly at the probe frequency (pink arrow) shown by process 2 in Figure 2(a). These two signals at probe frequency interfere with each other leading to a transmission window, appearing as a sharp peak in the cavity response, shown in Figure 2(b). This phenomena is known as "optomechanically induced transparency" (OMIT) and is a signature of the optomechanical coupling between the graphene mechanical resonator and the superconducting cavity [21-23]. As this effect rely on the coherent driven motion of the graphene mechanical resonator, the width of the transparency window is set by the mechanical resonator's linewidth as shown in the inset of Figure 2(b). Using the radiation pressure force driving, we measure the quality factor of the graphene resonator as high as 220,000.

FIG. 2: Optomechanically induced transparency (OMIT). (a) Schematic illustrate OMIT features in terms of the interference of the probe field (black arrow) with the microwave photons that are cyclically down- and then up- converted by the optomechanical interaction (pink arrow). (b) Measurement of the cavity reflection |S₁₁| in presence of sideband detuned drive tone. A detuned drive at ω_c+ω_m results in a window of optomechanically induced reflection (OMIR) in the cavity response. Inset: Zoom of the OMIR window. (c) Measurement of the cavity reflection |S₁₁| with a stronger detuned drive. At the center of the cavity response, the reflection coefficient exceeds 1, corresponding to mechanical microwave amplification of 17 dB by the graphene resonator.

By increasing the drive signal amplitude further, one can increase the strength of the optomechanical coupling. Using this, we make an amplifier in which microwave signals are amplified by the mechanical motion of the graphene resonator [16]. With a stronger detuned drive, we observed a microwave gain of 17 dB (equivalent to a photon gain of 50) as shown in Figure 2(c), before the nonlinear effects from the mechanical resonators come into play. Similarly, a different "beat" of the microwave photons (having ω_d = ω_c - ω_m ) allows one to store microwave photons into the mechanical motion of the resonator [24]. To this end we show a storage time up to 10 millisecond, which is equivalent to delay from a few hundreds of kilometer long coaxial cable.

The phenomena of OMIT also allow one to directly extract a quantity called "cooperativity" C without any fi t parameters. The quantity C is an important fi gure of merit in characterizing the optomechanical systems. For example, in sideband resolved limit (when mechanical frequency exceeds the cavity linewidth), the criteria for quantum-coherent regime can be simply written as C + 1 > n_th , where n_th is the average number of thermal phonon in the mechanical resonator. In our experiment, we have been able to achieve C = 8 close to the expected number of thermal phonon in the mechanical resonator at 14 mK, bringing this system close to the quantum coherent regime.

Summary and outlook:

In our work, we demonstrated the potential of exfoliated graphene crystal applied to form an optomechanical device, which so far have been realized using top-down technology. This opens up a new dimension to explore exfoliated two-dimensional crystals in optomechanical systems, and harnessing their unique properties such as extremely low mass and high quality factors. For future devices, two-dimensional superconducting exfoliated flakes could be of great interest for such applications. Superconducting cavity in our work is a very good detector for mechanical displacement with a bandwidth three orders of magnitude larger than the mechanical line-width. This would provide a new tool to study nonlinear restoring forces, nonlinear damping, and mode coupling in mechanical resonators from twodimensional crystals. The characterization of our device shows that in future by making little larger area mechanical resonators, devices operating in quantum regime can be easily realized, which can possibly be used as a memory element in a quantum computer. As many of the 2D crystals can be grown by chemical processes in large areas, they also hold the promise of scalability.

References:
[1] Andres Castellanos-Gomez, Vibhor Singh, Herre S.J. van der Zant, Gary A. Steele, "Mechanics of freely-suspended ultrathin layered materials". arXiv:1409.1173 [cond-mat] (2014).
[2] J. Scott Bunch, Arend M. van der Zande, Scott S. Verbridge, Ian W. Frank, David M. Tanenbaum, Jeevak M. Parpia, Harold G. Craighead, Paul L. McEuen, "Electromechanical resonators from graphene sheets". Science, 315, 490-493 (2007). Abstract.
[3] Changyao Chen, Sami Rosenblatt, Kirill I. Bolotin, William Kalb, Philip Kim, Ioannis Kymissis, Horst L. Stormer, Tony F. Heinz, James Hone, "Performance of monolayer graphene nanomechanical resonators with electrical readout". Nature Nanotechnology, 4, 861-867 (2009). Abstract.
[4] Vibhor Singh, Shamashis Sengupta, Hari S Solanki, Rohan Dhall, Adrien Allain, Sajal Dhara, Prita Pant, Mandar M Deshmukh, "Probing thermal expansion of graphene and modal dispersion at low-temperature using graphene nanoelectromechanical systems resonators". Nanotechnology, 21, 165204 (2010). Abstract.
[5] Robert A. Barton, B. Ilic, Arend M. van der Zande, William S. Whitney, Paul L. McEuen, Jeevak M. Parpia, Harold G. Craighead, "High, size-dependent quality factor in an array of graphene mechanical resonators". Nano Letters, 11, 1232{1236 (2011). Abstract.
[6] A. Eichler, J. Moser, J. Chaste, M. Zdrojek, I. Wilson-Rae, A. Bachtold, "Nonlinear damping in mechanical resonators made from carbon nanotubes and graphene". Nature Nanotechnology, 6, 339-342 (2011). Abstract.
[7] Xuefeng Song, Mika Oksanen, Mika A. Sillanpää, H. G. Craighead, J. M. Parpia, Pertti J. Hakonen, "Stamp transferred suspended graphene mechanical resonators for radio frequency electrical readout". Nano Letters, 12, 198-202 (2012). Abstract.
[8] Robert A. Barton, Isaac R. Storch, Vivekananda P. Adiga, Reyu Sakakibara, Benjamin R. Cipriany, B. Ilic, Si Ping Wang, Peijie Ong, Paul L. McEuen, Jeevak M. Parpia, Harold G. Craighead, "Photothermal self-oscillation and laser cooling of graphene optomechanical systems". Nano Letters, 12, 4681-4686 (2012). Abstract.
[9] Changyao Chen, Sunwoo Lee, Vikram V. Deshpande, Gwan-Hyoung Lee, Michael Lekas, Kenneth Shepard, James Hone, "Graphene mechanical oscillators with tunable frequency". Nature Nanotechnology 8, 923{927 (2013). Abstract.
[10] Markus Aspelmeyer, Tobias J. Kippenberg, Florian Marquardt, "Cavity optomechanics". arXiv:1303.0733 [cond-mat.mes-hall] (2013).
[11] C. A. Regal, J. D. Teufel, K. W. Lehnert, "Measuring nanomechanical motion with a mi- crowave cavity interferometer". Nature Physics, 4, 555-560 (2008). Abstract.
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