Deterministic Quantum Teleportation with Feed-Forward in a Solid State System

[From left to right] Andreas Wallraff, Christopher Eichler, Yves Salathe, Markus Oppliger, Philipp Kurpiers, Lars Steffen

Authors: Lars Steffen, Yves Salathe, Markus Oppliger, Philipp Kurpiers, Matthias Baur^*, Christian Lang, Christopher Eichler, Gabriel Puebla-Hellmann, Arkady Fedorov^*, Andreas Wallraff

Affiliation: Department of Physics, ETH Zurich, Switzerland

^*Present address: ARC Centre for Engineered Quantum Systems, University of Queensland, Brisbane, Australia

Link to QUDEV-Lab >>

Transferring the state of an information carrier between two parties is an essential primitive in both classical and quantum communication and information processing. Quantum teleportation [1] describes the concept of transferring an unknown quantum state from a sender to a physically separated receiver without transmitting the physical carrier of information itself. To do so, teleportation makes use of the non-local correlations provided by an entangled pair shared between the sender and the receiver and the exchange of classical information.

In our recent publication [2] we report the first teleportation of information in a solid state system. We use a chip-based superconducting circuit architecture [3, 4] (Fig. 1) with three superconducting transmon qubits [5] and three superconducting coplanar waveguide resonators [6] which serve as a quantum bus [7, 8] and allow to perform qubit state readout [9, 10]. We have realized the full deterministic quantum teleportation protocol using quantum-limited parametric amplifiers [11] for single-shot readout, a crossed quantum bus technology and flexible real-time digital electronics.

Figure 1: In our chip-based superconducting circuit three qubits are coupled to three resonators to realize deterministic quantum teleportation.

The success of the teleportation protocol in every instance with unit fidelity is counterintuitive from a classical point of view [12]. The receiver’s quantum bit (qubit) does not interact with any other qubit after it has been entangled with one of two qubits in the sender’s possession. The input state (|ψ_in>) is prepared at a later time at the sender. The classical information sent by the sender is not sufficient to recreate |ψ_in> perfectly at the receiver. Indeed, assuming no entanglement between sender and receiver one can replicate the sender’s state at best with a process fidelity of 1/2. To always recover the original state |ψ_in> the sender performs a measurement in the basis of the Bell states, which projects the two qubits in the sender’s possession randomly onto one of the four Bell states. As a consequence the receiver’s qubit is projected instantaneously into a state related to |ψ_in> without ever having interacted with the sender’s qubit. The receiver’s qubit only differs from the input state by a single-qubit rotation which depends on the four possible measurement results. In the final step, the sender communicates the Bell measurement result as two bits of classical information via a classical channel and therefore the receiver can always obtain the original input state |ψ_in>. This final step is frequently referred to as feed-forward.

Figure 2: a) Standard protocol of quantum teleportation. The protocol starts with the preparation of a Bell state between Q2 and Q3 (blue box) followed by the preparation of an arbitrary state |ψ_in> (green box) and a Bell state measurement of Q1 and Q2 (red box). The classical information extracted by the measurement of Q1 and Q2 is transferred to the receiver to perform local gates conditioned on the measurement outcomes. At the end of the protocol Q3 is in a state |ψ_out> which ideally is identical to |ψ_in> (also colored in green). Here, H is the Hadamard gate, X and Z are Pauli matrices σ_x and σ_z, respectively. The cnot-gate is represented by a vertical line between the control qubit (•) and the target qubit (⊕). b) The protocol implemented in the experiment presented here uses controlled-phase gates indicated by vertical lines between the relevant qubits (•), and single qubit rotations R^θ_±y of angle θ about the ±y-axis. To finalize the teleportation we either post-select on any single one of the four measurement outcomes (00, 01, 10 and 11) acquired in a single shot, or we deterministically use all four outcomes, which we then may use to implement feed-forward. The feed-forward operators R^π_x and R^π_y are applied to Q3 conditioned on the four measurement outcomes according to the table presented in the framed box.

Our implementation of the protocol (Fig. 2b) uses single qubit rotations and controlled-phase gates [13, 14] and is equivalent to the original protocol shown in Fig. 2a. The teleportation process succeeds with order unit probability for any input state, as we reliably prepare entangled states as a resource and are able to distinguish all four maximally entangled Bell states in a single measurement.

Figure 3: The solid bars show the experimentally obtained process matrix describing the state transfer from the sender to the receiver. The ideal process matrix, which is the identity operation, is shown in wire frames.

To identify simultaneously the four outcomes of the Bell-state measurement in a deterministic way we use two-qubit single-shot read-out [15, 16] for which we achieve a success probability of (81.8±0.5) %. In the teleportation protocol we analyze the Bell-state measurement in real time by using fast electronics based on a field programmable gate array (FPGA) and realize the feed-forward step in about 500 ns. We have achieved an average process fidelity of (62.2±0.3) % for the full quantum teleportation algorithm, which is clearly above the classical threshold (Fig. 3). We have demonstrated teleportation at a rate of 10,000 quantum bits per second between two macroscopic systems separated by 6 mm.

The low transmission loss of superconducting waveguides is likely to enable the range of this and other schemes to be extended to significantly larger distances, enabling tests of non-locality and the realization of elements for quantum communication at microwave frequencies. The demonstrated feed-forward may also find application in error correction schemes.

References:
[1] Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters. "Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels". Physical Review Letters, 70, 1895 (1993). Abstract.
[2] L. Steffen, Y. Salathe, M. Oppliger, P. Kurpiers, M. Baur, C. Lang, C. Eichler, G. Puebla-Hellmann, A. Fedorov, and A. Wallraff. "Deterministic quantum teleportation with feed-forward in a solid state system". Nature, 500, 319 (2013). Abstract.
[3] A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf. "Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics". Nature, 431, 162 (2004). Abstract.
[4] F. Helmer, M. Mariantoni, A. G. Fowler, J. von Delft, E. Solano, and F. Marquardt. "Cavity grid for scalable quantum computation with superconducting circuits". Europhysics Letters, 85, 50007 (2009). Abstract.
[5] Jens Koch, Terri M. Yu, Jay Gambetta, A. A. Houck, D. I. Schuster, J. Majer, Alexandre Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf. "Charge-insensitive qubit design derived from the Cooper pair box". Physical Review A, 76, 042319 (2007). Abstract.
[6] M. Göppl, A. Fragner, M. Baur, R. Bianchetti, S. Filipp, J. M. Fink, P. J. Leek, G. Puebla, L. Steffen, and A. Wallraff. "Coplanar waveguide resonators for circuit quantum electrodynamics". Journal of Applied Physics, 104, 113904 (2008). Abstract.
[7] J. Majer, J. M. Chow, J. M. Gambetta, J. Koch, B. R. Johnson, J. A. Schreier, L. Frunzio, D. I. Schuster, A. A. Houck, A. Wallraff, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf. "Coupling superconducting qubits via a cavity bus". Nature, 449, 443 (2007). Abstract.
[8] Mika A. Sillanpää, Jae I. Park, Raymond W. Simmonds. "Coherent quantum state storage and transfer between two phase qubits via a resonant cavity". Nature, 449, 438 (2007). Abstract.
[9] R. Bianchetti, S. Filipp, M. Baur, J. M. Fink, M. Göppl, P. J. Leek, L. Steffen, A. Blais, and A. Wallraff. "Dynamics of dispersive single-qubit readout in circuit quantum electrodynamics". Physical Review A, 80, 043840 (2009). Abstract.
[10] S. Filipp, P. Maurer, P. J. Leek, M. Baur, R. Bianchetti, J. M. Fink, M. Göppl, L. Steffen, J. M. Gambetta, A. Blais, and A. Wallraff. "Two-qubit state tomography using a joint dispersive readout". Physical Review Letters, 102, 200402 (2009). Abstract,
[11] Bernard Yurke and Eyal Buks. "Performance of cavity-parametric amplifiers, employing Kerr nonlinearites, in the presence of two-photon loss". Journal of Lightwave Technology, 24, 5054 (2006). Abstract.
[12] S. Massar and S. Popescu. "Optimal extraction of information from finite quantum ensembles". Physical Review Letters, 74, 1259 (1995). Abstract.
[13] Frederick W. Strauch, Philip R. Johnson, Alex J. Dragt, C. J. Lobb, J. R. Anderson, and F. C. Wellstood. "Quantum logic gates for coupled superconducting phase qubits". Physical Review Letter, 91, 167005 (2003). Abstract.
[14] L. DiCarlo, J. M. Chow, J. M. Gambetta, Lev S. Bishop, B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf. "Demonstration of two-qubit algorithms with a superconducting quantum processor". Nature, 460, 240 (2009). Abstract.
[15] François Mallet, Florian R. Ong, Agustin Palacios-Laloy, François Nguyen, Patrice Bertet, Denis Vion & Daniel Esteve. "Single-shot qubit readout in circuit quantum electrodynamics". Nature Physics, 5, 79 (2009). Abstract.
[16] R. Vijay, D. H. Slichter, and I. Siddiqi. "Observation of quantum jumps in a superconducting artificial atom". Physical Review Letters, 106, 110502 (2011). Abstract.

Hybrid Quantum Teleportation

[From Left to Right] Shuntaro Takeda, Maria Fuwa, and Akira Furusawa

Authors: Shuntaro Takeda, Maria Fuwa, and Akira Furusawa

Affiliation: Department of Applied Physics, School of Engineering, University of Tokyo, Japan

Link to Furusawa Laboratory >>

The principles of quantum mechanics allow us to realize ultra-high-capacity optical communication and ultra-high-speed quantum computation beyond the limits of current technology. One of the most fundamental steps towards this goal is to transfer quantum bits (qubits) carried by photons through “quantum teleportation” [1]. Quantum teleportation is the act of transferring qubits from a sender to a spatially distant receiver by utilizing shared entanglement and classical communications.

After its original proposal in 1993 [1], a research group in Austria succeeded in its realization in 1997 [2]. However, this scheme involved several deficiencies. One is its low transfer efficiency, estimated to be far below 1%. This is due to the probabilistic nature of entanglement generation and the joint measurement of two photons. This scheme also required post-selection of the successful events by measuring the output qubits after teleportation [3]. The transferred qubits are destroyed in this process, and thus cannot be used for further information processing. Various other related experiments have been reported thus far, but most withhold the same disadvantages. This problem has been a major limitation in the development of optical quantum information processing.

In our recent publication [4], we demonstrated “deterministic” quantum teleportation of photonic qubits for the first time. That is, photonic qubits are always teleported in each attempt, in contrary to the former probabilistic scheme. In addition, it does not require post-selection of the successful events. The success of the experiment lies in a hybrid technique of photonic qubits and continuous-variable quantum teleportation [5,6,7]; this required the combination of two conceptually different and previously incompatible approaches.

Figure 1: Concept of our hybrid technique for quantum teleportation. Single-photon-based qubits are combined with continuous-variable quantum teleportation to transfer optical waves.

Continuous-variable quantum teleportation, first demonstrated in 1998 [7], has long been used to teleport the amplitude and phase signals of optical waves, rather than photonic qubits. The advantage of continuous-variable teleportation is its deterministic success due to the on-demand availability of entangled waves and the complete joint measurement of two waves. However, its application to photonic qubits had long been hindered by experimental incompatibilities: typical pulsed-laser-based qubits have a broad frequency bandwidth that is incompatible with the original continuous-wave-based continuous-variable teleporter, which works only on narrow frequency sidebands. We overcame this incompatibility by developing an innovative technology: a broadband continuous-variable teleporter [8] and a narrowband qubit compatible with that teleporter [9].

Figure 2: Configuration of the teleportation experiment. Laser sources and non-linear optical processes supply the qubit and the required entanglement. More than 500 mirrors and beam splitters constitute the teleportation circuit.

This hybrid technique enabled the realization of completely deterministic and unconditional quantum teleportation of photonic qubits. The transfer accuracy (fidelity) ranged from 79 to 82 percent in four different qubits. Another strength of our hybrid scheme lies in the fact that the qubits were teleported much more efficiently than the previous scheme, even with low degrees of entanglement. This is a decisive breakthrough in the field of optical teleportation 16 years after the first experimental realizations.

References:
[1] Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters, “Teleporting an Unknown Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels”, Physical Review Letters, 70, 1895 (1993). Abstract.
[2] Dik Bouwmeester, Jian-Wei Pan, Klaus Mattle, Manfred Eibl, Harald Weinfurter, Anton Zeilinger, “Experimental quantum teleportation”, Nature, 390, 575 (1997). Abstract.
[3] Samuel L. Braunstein and H. J. Kimble, “A posteriori teleportation”, Nature 394, 840 (1998). Abstract.
[4] Shuntaro Takeda, Takahiro Mizuta, Maria Fuwa, Peter van Loock, Akira Furusawa, “Deterministic quantum teleportation of photonic quantum bits by a hybrid technique”, Nature 500, 315 (2013). Abstract.
[5] Lev Vaidman, “Teleportation of quantum states”, Physical Review A 49, 1473 (1994). Abstract.
[6] Samuel L. Braunstein and H. J. Kimble, "Teleportation of Continuous Quantum Variables", Physical Review Letters, 80, 869 (1998). Abstract.
[7] A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional Quantum Teleportation”, Science 282, 706 (1998). Abstract.
[8] Noriyuki Lee, Hugo Benichi, Yuishi Takeno, Shuntaro Takeda, James Webb, Elanor Huntington, Akira Furusawa, “Teleportation of Non-Classical Wave-Packets of light”, Science 332, 330 (2011). Abstract.
[9] Shuntaro Takeda, Takahiro Mizuta, Maria Fuwa, Jun-ichi Yoshikawa, Hidehiro Yonezawa, Akira Furusawa, “Generation and eight-port homodyne characterization of time-bin qubits for continuous-variable quantum information processing”, Physical Review A 87, 043803 (2013). Abstract.

Atom Interferometry in a 10 Meter Atomic Fountain

[From left to right] Mark Kasevich; the 10 m fountain team in 2013: Alex Sugarbaker, Tim Kovachy, Jason Hogan, Susannah Dickerson, Sheng-wey Chiow; and Dave Johnson

Authors: Alex Sugarbaker, Susannah M. Dickerson, Jason M. Hogan, David M. S. Johnson, and Mark A. Kasevich

Affiliation: Department of Physics, Stanford University, USA

Link to Kasevich Group Website >>

The equivalence principle states that all objects fall with the same acceleration under the influence of gravity. It is the conceptual foundation of Einstein’s general relativity, but is it true exactly? If not, there are profound implications for our understanding of gravity and the nature of the universe. It is therefore important to continue to test the equivalence principle as precisely as we can.

Galileo reportedly tested it by dropping spheres from the Leaning Tower of Pisa. Apollo astronauts tested it by dropping a hammer and a feather on the moon. More recent measurements have shown that the accelerations of two falling objects differ by no more than one part in 10¹³ [1, 2]. We aim to test the equivalence principle to one part in 10¹⁵ by dropping atoms of two different isotopes of rubidium in a 10 meter tower.

We will precisely measure acceleration differences between the two isotopes using atom interferometry. According to quantum mechanics, atoms are waves. Just as in optical interferometry, it is possible to split and recombine them to form an interference pattern [3, 4, 5]. In our interferometer, we send each atom along two different paths through space – each is in two places at once. When the atom waves are brought back together, the interference pattern depends on the phase difference between the two paths taken.

This phase difference in turn depends sensitively on the forces that act differently on the two parts of the atom while they are separated. This sensitivity to forces is what makes atom interferometry so useful. Compact atom interferometers have been made that can precisely measure rotation and acceleration, which can aid in navigation, mineral exploration, and geophysics. Atom interferometers have also measured the gravitational and fine-structure constants [6, 7]. They could also be used to search for gravitational waves [8].

The sensitivity of an atom interferometer increases with longer interferometer durations. Therefore, as recently described in Physical Review Letters [9, 10, 11], we have built an atom interferometer in which ⁸⁷Rb atoms are separated for 2.3 seconds before being recombined and interfered (Fig. 1). Three times longer than previous records [12], this multiple-second duration is well into the range of macroscopic, human-perceivable timescales. Furthermore, the two halves of each atom are separated by 1.4 centimeters before recombination – that's enough for you to swing your hand between them!

Fig. 1 Photograph of the 10 meter atomic fountain in a pit in the basement of the physics building at Stanford University.

How do we make a long-duration atom interferometer? We prepare a cloud of atoms at the bottom of a 10 meter vacuum tower and then launch them to the top. The interferometry is performed while the atoms rise up and fall back down to the bottom of the tower. The atoms are in free-fall, isolated from the noisy environment.

The cloud of atoms used must be very cold – a few billionths of a degree above absolute zero. At room temperature, the atoms in a gas move at speeds of hundreds of meters per second. Room-temperature rubidium atoms would collide with the walls of our vacuum chamber long before they fell back to the bottom. We therefore cool a few million rubidium atoms to a few nanokelvin before launching them into the tower. (The cooling process builds upon the same techniques used to generate Bose-Einstein condensates [13].)

Even at a few nanokelvin, individual atoms follow slightly different trajectories through the interferometer (like the droplets in a fountain of water), experiencing different position- and velocity-dependent forces. This yields a spatially-dependent phase, which in turn yields a spatial variation in the output atom density distribution that we can observe directly with a CCD camera (Fig. 2). This might at first appear undesirable, but it actually reveals rich details about the forces that generate the spatial interference pattern. Similar spatial fringe patterns have been used to great benefit in optical interferometers for centuries, but it is only recently that the effect has been leveraged in atom interferometry.

Fig. 2 Atomic interference patterns observed at the output of the interferometer. The images are sorted by phase, which can be measured for each experimental shot.

The long drift time of our interferometer enables it to have an acceleration sensitivity of 7 X 10^-12 g for each experimental shot, a hundredfold improvement over previous limits [14]. This is roughly the same as the gravitational attraction you would feel towards a person 10 meters away from you. We have used the sensitive interferometer and the spatial fringe patterns mentioned above to make precise measurements of Earth's rotation [9, 10].

The high sensitivity of the interferometer also holds great promise for our design goal – testing the equivalence principle (as mentioned above). By averaging more measurements or implementing advanced interferometry techniques, we can achieve the desired 10^-15 g sensitivity. Adding a simultaneous ⁸⁵Rb interferometer and comparing the results for the two isotopes will then enable us to make a new precision test of the equivalence principle. This will probe the fundamental assumptions of our current theory of gravity.

References:
[1] S. Schlamminger, K.Y. Choi, T.A. Wagner, J.H. Gundlach, and E.G. Adelberger, “Test of the Equivalence Principle Using a Rotating Torsion Balance”, Physical Review Letters, 100, 041101 (2008). Abstract.
[2] James G. Williams, Slava G. Turyshev, Dale H. Boggs, “Progress in Lunar Laser Ranging Tests of Relativistic Gravity”, Physical Review Letters, 93, 261101 (2004). Abstract.
[3] Mark Kasevich and Steven Chu, “Atomic interferometry using stimulated Raman transitions”, Physical Review Letters, 67, 181 (1991). Abstract.
[4] Alexander D. Cronin, Jörg Schmiedmayer, David E. Pritchard, “Optics and interferometry with atoms and molecules”, Reviews of Modern Physics, 81, 1051 (2009). Abstract.
[5] We focus on light-pulse atom interferometry, where pulses of laser light are used to split, recombine, and interfere the atoms.
[6] G. Lamporesi, A. Bertoldi, L. Cacciapuoti, M. Prevedelli, and G.M. Tino, “Determination of the Newtonian Gravitational Constant Using Atom Interferometry”, Physical Review Letters, 100, 050801 (2008). Abstract.
[7] Rym Bouchendira, Pierre Cladé, Saïda Guellati-Khélifa, François Nez, and François Biraben, “New Determination of the Fine Structure Constant and Test of the Quantum Electrodynamics”, Physical Review Letters, 106, 080801 (2011). Abstract.
[8] Jason Hogan, “A new method for detecting gravitational waves”, SPIE Newsroom, 6 May (2013). Article.
[9] Susannah M. Dickerson, Jason M. Hogan, Alex Sugarbaker, David M. S. Johnson, Mark A. Kasevich, “Multiaxis Inertial Sensing with Long-Time Point Source Atom Interferometry”, Physical Review Letters, 111, 083001 (2013).  Abstract.
[10] Alex Sugarbaker, Susannah M. Dickerson, Jason M. Hogan, David M. S. Johnson, Mark A. Kasevich, “Enhanced Atom Interferometer Readout through the Application of Phase Shear”, Physical Review Letters, 111, 113002 (2013). Abstract.
[11] P. Bouyer, “Viewpoint: A New Starting Point for Atom Interferometry”, Physics, 6, 92 (2013). Article.
[12] H. Müntinga, H. Ahlers, M. Krutzik, A. Wenzlawski, S. Arnold, D. Becker, K. Bongs, H. Dittus, H. Duncker, N. Gaaloul, C. Gherasim, E. Giese, C. Grzeschik, T. W. Hänsch, O. Hellmig, W. Herr, S. Herrmann, E. Kajari, S. Kleinert, C. Lämmerzahl, W. Lewoczko-Adamczyk, J. Malcolm, N. Meyer, R. Nolte, A. Peters, M. Popp, J. Reichel, A. Roura, J. Rudolph, M. Schiemangk, M. Schneider, S. T. Seidel, K. Sengstock, V. Tamma, T. Valenzuela, A. Vogel, R. Walser, T. Wendrich, P. Windpassinger, W. Zeller, T. van Zoest, W. Ertmer, W. P. Schleich, E. M. Rasel, “Interferometry with Bose-Einstein Condensates in Microgravity”, Physical Review Letters, 110, 093602 (2013). Abstract.
[13] “Bose-Einstein condensate”. Past 2Physics Article.
[14] Holger Müller, Sheng-wey Chiow, Sven Herrmann, Steven Chu, Keng-Yeow Chung, “Atom-Interferometry Tests of the Isotropy of Post-Newtonian Gravity”, Physical Review Letters, 100, 031101 (2008). Abstract.

Coherently Manipulating Nanomechanical Oscillators

[From Left to Right] Hajime Okamoto, Imran Mahboob, Hiroshi Yamaguchi

Authors: Hajime Okamoto, Imran Mahboob, and Hiroshi Yamaguchi

Affiliation: NTT Basic Research Laboratories, Atsugi-shi, Kanagawa, Japan

Semiconductor nanomechanical oscillators enable the pursuit of new physical phenomenon that can only be observed through the tiny mechanical displacement as well as enabling the development of nanoscience and nanotechnology. These systems with sharp mechanical resonances can be manipulated by electrical or optical means permitting a wide range of applications such as metrology and information processing [1].

Coupling two such nanomechanical oscillators has recently emerged as a subject of interest. This is because the sympathetic oscillation dynamics in the coupled system expand the potential applications of nanomechanical objects such as highly precise sensors, high-Q band-pass filters, signal amplifier, and even logic gates [2-5]. However, an obstacle to the further development of this architecture arises from the usually weak elastic (or electric) coupling between the nanomechanical components. This limits the ability to coherently transfer the vibration energy between the oscillators within the oscillator’s ring-down time. In other words, the coherent manipulation of the vibrational states, e.g., Rabi cycle and Ramsey fringe as in two-level quantum systems [6], is highly challenging in this coupled phonon system.

In our recent publication [7], we demonstrated coherent coupling of nanomechanical oscillators. This was realized in two geometrically interconnected GaAs doubly-clamped beams [Fig. 1(a)], where the frequency of beam R is higher than that of beam L as shown in Fig. 1(b). Because of the frequency mismatch and the weak mechanical coupling, coherent energy exchange between the two beams is not possible by the geometric interconnection alone.

Figure 1. (a) Schematic drawing of the sample and the piezoelectric effect. The piezoelectric effect enables harmonic driving, pumping, and detection of the mechanical motion via gate voltage. (b) Schematic drawing of the mechanical resonance for beam L and beam R. The frequency of beam R is higher than that of beam L (293.93 kHz) by 440 Hz. The coherent energy exchange between the two beams is achieved by applying the pumping voltage to beam L with the frequency difference between the two beams. (c) Schematic of the pumping protocol in a mass and spring model.

Coherent energy exchange can be achieved by dynamically coupling the mechanical oscillations of the two beams. This is realized by periodically modulating the spring constant of one beam at the frequency difference between the two beams [Fig. 1(c)]. This periodic modulation, namely pumping, can be induced by applying gate voltage via the piezoelectric effect in this sample [Fig. 1(a)]. This pumping enables strong vibrational coupling, leading to the cyclic (Rabi) oscillations between the two vibrational states (the beam-L state and the beam-R state) on the Bloch sphere (Fig. 2). The Rabi cycle period, i.e., the coupling strength, is fully adjustable by changing the pump amplitude via the gate voltage. As a result, the vibration energy can be quickly transferred from one beam to the other enabling the vibration of the original beam to be switched off on a time-scale orders of magnitude shorter than its ring-down time. This quick energy transfer to the adjacent oscillator opens up the prospect of high-speed repetitive operations for sensors and logics using nanomechanical systems [8,9].

Figure 2. Schematic drawing showing the evolution of the coupled mechanical oscillators on the Bloch sphere when undergoing a Rabi cycle.

In terms of phonon populations, the present system with sub-megahertz frequency is still in the classical regime with large mode occupation. However, the above technique could also be extended to gigahertz frequency mechanical oscillators in which the average phonon number in the mechanical modes falls below one at cryogenic temperatures [10]. This in turn leads to the exciting possibility of phononic quantum bits and entanglement between distinct macroscopic mechanical elements [11].

References:
[1] A. N. Cleland, “Foundations of Nanomechanics” (Springer-Verlag Berlin Heidelberg New York, 2003). [2] Matthew Spletzer, Arvind Raman, Alexander Q. Wu, Xianfan Xu, Ron Reifenberger, “Ultrasensitive mass sensing using mode localization in coupled microcantilevers”, Applied Physics Letters, 88, 254102 (2006). Abstract.
[3] F. D. Bannon, J. R. Clark, and C. T.-C. Nguyen, “High-Q HF microelectromechanical filters”, IEEE J. Solid-State Circuits 35, 512-526 (2000). Abstract.
[4] R. B. Karabalin, Ron Lifshitz, M. C. Cross, M. H. Matheny, S. C. Masmanidis, and M. L. Roukes, “Signal amplification by sensitive control of bifurcation topology”, Physical Review Letters, 106, 094102 (2011). Abstract.
[5] Sotiris C. Masmanidis, Rassul B. Karabalin, Iwijn De Vlaminck, Gustaaf Borghs, Mark R. Freeman, Michael L. Roukes, “Multifunctional nanomechanical systems via tunably coupled piezoelectric actuation”, Science, 317, 780-783 (2007). Abstract.
[6] J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, A. C. Gossard, “Coherent manipulation of coupled electron spins in semiconductor quantum dots”, Science, 309, 2180-2184 (2005). Abstract.
[7] Hajime Okamoto, Adrien Gourgout, Chia-Yuan Chang, Koji Onomitsu, Imran Mahboob, Edward Yi Chang, Hiroshi Yamaguchi, “Coherent phonon manipulation in coupled mechanical resonators”, Nature Physics, 9, 480-484 (2013). Abstract.
[8] Hiroshi Yamaguchi, Hajime Okamoto, and Imran Mahboob, “Coherent control of micro/nanomechanical oscillation using parametric mode mixing”, Applied Physics Express, 5, 014001 (2012). Abstract.
[9] I. Mahboob, E. Flurin, K. Nishiguchi, A. Fujiwara, H. Yamaguchi, “Interconnect-free parallel logic circuits in a single mechanical resonator”, Nature Communications, 2, 198 (2011). Abstract.
[10] A. D. O’Connell, M. Hofheinz, M. Ansmann, Radoslaw C. Bialczak, M. Lenander, Erik Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, John M. Martinis, A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator”, Nature, 464, 697-703 (2010). Abstract.
[11] Simon Rips and Michael J. Hartmann, “Quantum information processing with nanomechanical qubits”, Physical Review Letters, 110, 120503 (2013). Abstract.

Studying Light Pulses By Counting Photons

Elizabeth A. Goldschmidt (left) 
and Alan Migdall (right) 
[Photo courtesy: JQI/NIST/University of Maryland]

The photodetectors in Alan Migdall’s lab often see no light at all, and that’s a good thing since he and his colleagues at the Joint Quantum Institute (JQI) perform physics experiments that require very little light, the better to study subtle quantum effects. The bursts of light they observe typically consist of only one or two photons--- the particle form of light---or (statistically speaking) even less than one photon. Their latest achievement is to develop a new way of counting photons to understand the sources and modes of light in modern physics experiments.

Migdall’s lab, located at the National Institute for Standards and Technology (NIST), is just outside Washington, DC in the US. The new light-measuring protocol is summarized in a recent issue of the journal Physical Review A [1]. The work reported there was performed in collaboration with NIST’s Italian counterpart, the Instituto Nazionale de Ricerca Metrologica (INRIM).

Light Modes

Generating light suitable for quantum mechanical applications such as quantum computing and quantum cryptography requires exquisite control over properties such as the frequency, polarization, timing, and direction of the light emitted. For instance, probing atoms with light involves matching the frequency of the light to the atoms’ natural resonance frequency often to within one part in a billion. Moreover, communicating with light means encoding information in the arrival time or frequency or spatial position of the light, so high-speed communication means using very closely- spaced arrival times for light pulses, and pinpoint knowledge of the light frequencies and positions of the arriving light in order to fit in as much information as possible.

When a light pulse contains a mixture of light photons with different frequencies (or polarizations, arrival times, emission angles, etc.) it is said to have multiple modes. In some cases, a single light source will naturally produce such multi-mode light, whereas in others multiple modes are a signature of the presence of additional, and generally unwanted, light sources in the system. Discriminating the different modes in a light field, especially a weak light field that has very few photons, can be extremely difficult as it requires very sensitive detection that can discriminate between modes that are very close together in frequency, space, time, etc.

For instance, to study a pair of entangled photons (created by shooting light into a special crystal where one photon is converted into a pair of secondary, related photons) detection efficiency is all important; and folded into that detection efficiency is a requirement that the arrival of each of the daughter photons be matched to the arrival of the other daughter photon. In addition to this temporal alignment, the spatial alignment of detectors, (each oriented at a specific angle respect to the beamline) must be exquisite. To correct for any type of less-than-perfect alignment, it is necessary to know how many different light modes are arriving at the detector.

Photon Number

The laws of quantum mechanics ensure that light always exhibits natural intensity fluctuations. Even from an ultra-stable laser, the number of photons arriving at a detector will vary randomly in time. By recording the number of photons in each pulse of light over a long time, however, the form of the fluctuations of a particular light field will become clear. In particular, we can learn the probability of generating 0, 1, 2, 3, etc. photons in each pulse.

The handy innovation in Migdall’s lab was to develop a method to use this set of probabilities to determine the modes in a very weak light field. This method is very useful because most light detectors that can see light at the level of a single photon cannot tell the exact frequency or position of the light, which makes determining the number of modes difficult for such fields.

The JQI-INRIM experiment used a detector “tree” that counts photon number. It did this by taking the incoming light pulse, using partial mirrors to divide the pulse into four, and then allowing these to enter four detectors set up to record individual photons. If the original pulse contained zero photons then none of the detectors will fire. If the pulse contained one photon, then one of the detectors will fire, and so on.

Elizabeth A. Goldschmidt, a JQI researcher and University of Maryland graduate student, is the first author on the research paper. “By looking at just the intensity fluctuations of a light field we have shown that we can learn about the underlying processes generating the light,” she said. “This is a novel use of higher-order photon-number statistics, which are becoming more and more accessible with modern photodetection.”

Goldschmidt believes that this method of counting photons and statistically analyzing the results as a way of understanding the modes present in light pulses will help in keeping tight control over light sources that emit single photons (where, for instance, you want to ensure that unwanted photons are not being produced). And those that emit pairs of entangled photons---where the quantum relation between the two photons is exactly right, such as in “heralding” experiments, where the detection of a photon in one detector serves as an announcement for the existence of a second, related, photon in a specially staged detector nearby.

Alan Migdall compares the photon counting approach to wine tasting. “Just as some experts can taste different flavors in a wine---a result of grapes coming from different parts of the Loire Valley---so we can tell apart various modes of light coming from a source.”

References:
[1] Elizabeth A. Goldschmidt, Fabrizio Piacentini, Ivano Ruo Berchera, Sergey V. Polyakov, Silke Peters, Stefan Kück, Giorgio Brida, Ivo P. Degiovanni, Alan Migdall, Marco Genovese, "Mode reconstruction of a light field by multi-photon statistics", Physical Review A, 88, (2013). Abstract.