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Practical realization of Quantum Computation Superconducting qubits Electrons on liquid Helium Cavity QED Lecture 19 QC implementation proposals Bulk spin Resonance (NMR) Optical Atoms Solid state Linear optics Cavity QED Trapped ions Optical lattices Electrons on He Semiconductors Nuclear spin Electron spin Orbital state qubits qubits qubits http://courses.washington.edu/bbbteach/576/ Superconductors Flux qubits Charge qubits Superconductivity Superconductivity is a phenomenon occurring in certain materials at extremely low temperatures, characterized by exactly zero electrical resistance and the exclusion of the interior magnetic field (the Meissner effect). A magnet levitating above a high-temperature superconductor, cooled with liquid nitrogen. Persistent electric current flows on the surface of the superconductor, acting to exclude the magnetic field of the magnet (the Meissner effect). This current effectively forms an electromagnet that repels the magnet. 1911 Walter Meissner “Meissner effect” Heike Kamerlingh Onnes Superconductivity in He 1933 Martinis (NIST) phase qubit Martinis (UCSB) two-qubit gate (87% fidelity) 1957 1962 Schnirman et al. – theoretical proposal for JJ qubits Devoret group (Saclay) first Cooper Pair Box qubit Nakamura, Tsai (NEC) Rabi oscillations in CPB Lukens, Han (SUNY SB) Flux qubit Supercurrent through a nonsuperconducting gap Bardeen, Cooper, Schrieffer Theory of Superconductivity Superconducting qubits – a timeline 1997 1998 1999 2000 2002 2006 Superconductivity The electrical resistivity of a metallic conductor decreases gradually as the temperature is lowered. However, in ordinary conductors such as copper and silver, impurities and other defects impose a lower limit. Even near absolute zero a real sample of copper shows a nonzero resistance. The resistance of a superconductor, on the other hand, drops abruptly to zero when the material is cooled below its "critical temperature", typically 20 kelvin or less. An electrical current flowing in a loop of superconducting wire can persist indefinitely with no power source. Like ferromagnetism and atomic spectral lines, superconductivity is a quantum mechanical phenomenon. It cannot be understood simply as the idealization of "perfect conductivity" in classical physics. Superconductivity Superconductors are also able to maintain a current with no applied voltage whatsoever. Experimental evidence points to a current lifetime of at least 100,000 years, and theoretical estimates for the lifetime of persistent current exceed the lifetime of the universe. In a normal conductor, an electrical current may be visualized as a fluid of electrons moving across a heavy ionic lattice. The electrons are constantly colliding with the ions in the lattice, and during each collision some of the energy carried by the current is absorbed by the lattice and converted into heat (which is essentially the vibrational kinetic energy of the lattice ions.) As a result, the energy carried by the current is constantly being dissipated. This is the phenomenon of electrical resistance. Superconductivity The situation is different in a superconductor. In a conventional superconductor, the electronic fluid cannot be resolved into individual electrons. Instead, it consists of bound pairs of electrons known as Cooper pairs. This pairing is caused by an attractive force between electrons from the exchange of phonons. Due to quantum mechanics, the energy spectrum of this Cooper pair fluid possesses an energy gap, meaning there is a minimum amount of energy ∆E that must be supplied in order to excite the fluid. Therefore, if ∆E is larger than the thermal energy of the lattice (given by kT, where k is Boltzmann's constant and T is the temperature), the fluid will not be scattered by the lattice. The Cooper pair fluid is thus a superfluid, meaning it can flow without energy dissipation. Superconductivity Superconductivity- the persistence of the resistantless electric currents. Certain metals lose their resistance when the temperature is lowered below a certain critical temperature ( which is different for different metals). Main point of the theory, known as Bardeen-Cooper-Schrieffer (BCS) theory is that in normal metals the electrons behave as fermions, while in superconducting state they form “Cooper pairs” and behave like bosons. - - - - - - Singe electrons- the wave function is antisymmetric under exchange Cooper pairs - the wave function is symmetric under exchange Superconductivity Normally electrons do not form pairs as they repel each other. However, inside the material the electrons interact with ions of the crystal lattice. Very simplify, the electron can interact with the positively charged background ions and create a local potential disturbance which can attract another electron. The binding energy of the two electrons is very small, 1meV, and the pairs dissociate at higher temperatures. At low temperatures, the electrons can exists in the bound states (from Cooper pairs). From BCS theory we learn that the lowest state of the system is the one in which Cooper pairs are formed. Superconductivity - - - - - - Singe electrons- only one electron can occupy a particular state Cooper pairs – the above restriction no longer applies as electron pairs are bosons and very large number of pairs can occupy the same state 1. Therefore, the electron pairs do not have to move from an occupied state to unoccupied one to carry current. 2. The normal state is an excited state which is separated from the ground state (in which electrons form Cooper pairs) by an energy gap. Therefore, electrons do not suffer scattering which a source of resistance as there is an energy gap between their energy and the energies of the states to which they can scatter. Flux quantization in superconductors We consider a superconductor in form of a hollow cylinder which is placed in an external magnetic field, which is parallel to the axis of the cylinder. ∫ dr The magnetic field is expelled from the superconductor (Meissner effect) and vanishes within it. Therefore, Cooper pairs move in the region of B=0, and we can apply the results which we previously developed. If the wave function of the Cooper pair in the absence of the field is ψ(0), then in the presence of the field we have r ψ '(r ) = ψ (0) (r )e (i 2e / h ) ∫ A (r ') dr ' r0 Flux quantization in superconductors r ψ '(r ) = ψ (0) (r )e S r0 ∫ dr (i 2e / h ) ∫ A (r ') dr ' r0 When we consider a closed path S around the cylinder which starts at point r0 we get ψ '(r ) = ψ (0) (r )e (i 2e / h ) ∫ A (r ')dr ' = ψ (0) (r )ei 2e Φ / h As the electron wave function should not be multivalued as we go around the cylinder we get the condition 2e Φ nπ h = 2nπ → Φ = , e h n = 0, 1, 2,... And the flux enclosed by the superconducting cylinder (or ring) is quantized! This effect has been experimentally verified which confirmed that the current in superconductors is carried by the pair of the electrons and not the individual electrons. How this effect can be used? The main attraction of the Aharonov-Bohm effect is the possibility to use it in switching devices, i.e. to use the change in magnetic filed to change the state of the device from 0 to 1. How much do we have to change the magnetic field to switch from the constructive to destructive electron interference? ∆Φ = πh e πh π ×1.05 ×10−34 J ⋅ s −6 ∆B = ≈ ≈ 5.1 × 10 T −19 −6 2 eA (1.6 ×10 C )( 20 × 10 m ) for 20µm x 20µm device This is a very small field! The Earth’s magnetic field is about 40µT. It is very difficult to practically use. Josephson junction superconductors J = J 0 sin δ insulator Josephson junction: a thin insulator sandwiched between two superconductors phase difference δ = θ 2 − θ1 Depends on the tunneling probability of the electron pairs There is a current flow across the junction in the absence of an applied voltage! Superconducting devices Extremely interesting devices may be designed with a superconducting loop with two arms being formed by Josephson junctions. The operation of such devices is based on the fact that the phase difference around the closed superconducting loop which encloses the magnetic flux Φ is an integral product of 2e Φ / h . eΦ = nπ . The current will vary with Φ and has maxima at h The control of the current through the superconducting loop is the basis for many important devices. Such loops may be used in production of low power digital logic devices, detectors, signal processing devices, and extremely sensitive magnetic field measurement instruments . SQUID magnetometer (Superconductind QUantum Interference Device) Superconducting quantum computing This promising implementation of quantum information involves nanofabricated superconducting electrodes coupled through Josephson junctions. Possible qubits are charge qubits, flux qubits, and hybrid qubits. Josephson Junction Charge (NEC) One-qubit device can control the number of Cooper pairs of electrons in the box, create superposition of states. Superconducting device, operates at low temperatures (30 mK). Nakamura et al., Nature, 398(786), 1999 Two-qubit device Pashkin et al., Nature, 421(823), 2003 JJ Flux (Delft) The qubit representation is a quantum of current (flux) moving either clockwise or counter-clockwise around the loop. Charge qubit In quantum computing, a charge qubit is a superconducting qubit whose basis states are charge states (ie. states which represent the presence or absence of excess Cooper pairs in the island). A charge qubit is formed by a tiny superconducting island (also known as a Cooper-pair box) coupled by a Josephson junction to a superconducting reservoir (see figure). The Circuit diagram of a cooper pair box circuit. state of the qubit is determined by the number of Cooper pairs which have tunneled across the junction. In contrast The island (dotted line) with the charge state of an atomic or molecular ion, the is formed by the charge states of such an "island" involve a macroscopic superconducting number of conduction electrons of the island. The quantum electrode between the superposition of charge states can be achieved by tuning gate capacitor and the the gate voltage U that controls the chemical potential of the island. The charge qubit is typically read-out by junction capacitance. electrostatically coupling the island to an extremely sensitive electrometer such as the radio-frequency single-electron transistor. Flux qubits In quantum computing, flux qubits (also known as persistent current qubits) are micrometer sized loops of superconducting metal interrupted by a number of Josephson junctions. The junction parameters are engineered during fabrication so that a persistent current will flow continuously when an external flux is applied. The computational basis states of the qubit are defined by the circulating currents which can flow either clockwise or counterclockwise. These currents screen the applied flux limiting it to multiples of the flux quanta and give the qubit its name. When the applied flux through the loop area is close to a half integer number of flux quanta the two energy levels corresponding to the two directions of circulating current are brought close together and the loop may be operated as a qubit. Flux qubits Computational operations are performed by pulsing the qubit with microwave frequency radiation which has an energy comparable to that of the gap between the energy of the two basis states. Properly selected frequencies can put the qubit into a quantum superposition of the two basis states, subsequent pulses can manipulate the probability weighting that qubit will be measured in either of the two basis states, thus performing a computational operation. http://qist.lanl.gov/qcomp_map.shtml “Scalable physical system with well-characterized qubits” The system is physical – it is a microfabricated device with wires, capacitors and such The system is in principle quite scalable. Multiple copies of a qubit can be easily fabricated using the same lithography, etc. But: the qubits can never be made perfectly identical (unlike atoms). Each qubit will have slightly different energy levels; qubits must be characterized individually. http://courses.washington.edu/bbbteach/576/ http://courses.washington.edu/ bbbteach/576/ “ability to initialize qubit state” Qubits are initialized by cooling to low temperatures (mK) in a dilution refrigerator. This is how: Energy splittings between qubit states are of the order of f = 1 - 10 GHz (which corresponds to T = hf/kB = 50 - 500 mK) If the system is cooled down to T0 = 10 mK, the ground state occupancy is, according to Boltzmann distribution: P|0> = exp(-hf/kBT0) = 0.82 – 0.98 Lower temperature dilution refrigerators mean better qubit initialization! “(relative) long coherence times” Coherence times from a fraction of a nanosecond (charge qubits) to tens of nanoseconds (flux) to microseconds (“quantronium”). Correspond to about 10 – 1000 operations before decoherence. Many sources of noise (it’s solid state!) “universal set of quantum gates” Single qubit gates: applying microwaves (1 – 10 GHz) for a prescribed period of time. Two-qubit gates: via capacitive or inductive coupling of qubits. Science 313 313, 1432 (2006) – entanglement of two phase qubits (Martinis’ group – UCSB) “qubit-specific measurement” Measurement depends on the type of qubit. Charge qubit readout: amplifier with bimodal response corresponding to the state of the qubit. Flux and phase qubits readout: built-in DC-SQUID that detects the change of flux. Superconducting qubits - pros and cons • Cleanest of all solid state qubits. • Fabrication fairly straightforward, uses standard microfab techniques • Gate times of the order of ns (doable!) • Scaling seems straightforward • Need dilution refrigerators (and not just for noise reduction) •No simple way to couple to flying qubits (RF photons not good) • Longer coherence needed, may be impossible Superconducting qubits – what can we expect in near term? • More research aimed at identifying and quantifying the major source(s) of decoherence. • Improved control of the electromagnetic environment – sources, wires, capacitors, amplifiers. • Entanglement demonstrations in other types of SC qubits. • Integration of the qubit manipulation electronics (on the same chip as the qubits themselves). http://qist.lanl.gov/qcomp_map.shtml Electrons on Liquid Helium http://wwwhttp://www-drecam.cea.fr/Images/astImg/375_1.gif Electrons on Helium Electrons are weakly attracted by the image charge (ε = 1.057 for LHe); the 1-D image potential along z is: -∑/z , where ∑ = (ε-1)e2/4(ε+1) They are prevented from penetrating helium surface by a high (~ 1eV) barrier. Bound states in this potential in 1-D look like hydrogen: En = −R/n2 (n = 1, 2, . . .), R = ∑2m/2ħ2 Rydberg energy is about 8K, and the effective Bohr radius is about 8 nm. Electrons on Helium - 2 Liquid helium film must be cooled down to mK temperatures in order to reduce the vapor pressure (which would otherwise wreak havoc with among the electrons) It is well known that below about 2.2 K He-4 turns superfluid. At few mK it is pure He II. These features are crucial for the QC proposal with electrons on LHe. The main source of noise (heating) for the electrons trapped on the surface is the ripplons. http://silvera.physics.harvard.edu/bubbles.htm The original proposal “Quantum Computing with Electrons Floating on Liquid Helium” P. M. Platzman, M. I. Dykman, Science 284 pp. 1967 – 1969 (1999). The qubit is formed by the two lowest energy states of the trapped electron. Given R = 8K = 170 GHz, the n = 1 and the n = 2 levels are split by about 125 GHz. Presence of electric fields from bias electrodes introduces Stark shift of the levels. Single qubit operations are performed by applying microwaves at the Stark-shifted frequency. Expected Rabi frequencies of the order of hundreds of MHz Patterned bottom electrodes Electrons on surface of LHe of thickness d (typically about 1 micron) will form a 2-D solid with lattice constant approximately equal to d. (This is because the Coulomb energy e2/d d is of the order 20 K >> kbT at 10 mK). In order to control the locations of the electrons, as well as to be able to individually address each qubits, the bottom electrode of the capacitor is patterned. This also provides confinement in the plane of the LHe film. Electrons can be physically raised and lowered by controlling the voltages on the patterned electrodes. Two-qubit gates Two-qubit gates via dipole-dipole interaction (similar to the liquid state NMR QC). For a dipole moment (er), the interaction energy between qubits separated by distance d is (er)2/d3. At 1 micron separation the interaction energy is estimated to be about 10 MHz. The frequency of the coupling is qubit state-dependent (because (er) is state-dependent). This forms the basis of the quantum logic gates like the CNOT gate. However, it is strongly distance-dependent. Thus, interactions are limited to nearest neighbors. The readout “In order to read out the wave function at some time tf , when the computation is completed, we apply a reverse field E+ to the capacitor...” Qubit readout relies on state-dependent electron tunneling when a reversed bias field is applied to the capacitor. Problems: reading out the whole system at once; need to detect single electrons reliably Conclusions.... • A “neat” and certainly very unique approach • Builds on ideas from the superconducting qubits, trapped ions, quantum dots • The experiment is harder than theory. Some theoretical predictions unrealistic. http://www.quantumoptics.ethz.ch http://www.quantumoptics.ethz.ch// http://courses.washington.edu/ bbbteach/576/ http://www2.nict.go.jp/ http://www.wmi.badw.de/SFB631/tps/dipoletrap_and_cavity.jpg http://qist.lanl.gov/qcomp_map.shtml Cavity Quantum ElectroDynamics • In cavity QED we want to achieve conditions where single photon interacts so strongly with an atom that it causes the atom to change its quantum state. • This requires concentrating the electric field of the photon to a very small volume and being able to hold on to that photon for an extended period of time. • Both requirements are achieved by confining photons into a small, highfinesse resonator. F = 2√R/(1 – R), where R is mirror reflectivity power in circulating power loss Microwave resonators • Microwave photons can be confined in a cavity made of good metal. Main source of photon loss (other than dirt) is electrical resistance. • Better yet, use superconductors! Cavity quality factors (~ the finesse) reach ~ few × 108 for microwave photons at several to several tens of GHz. • Microwave cavities can be used to couple to highly-excited atoms in Rydberg states. There are proposals to do quantum computation with Rydberg state atoms and cavities. S. Haroche, “Normal Superior School” The optical cavity • The optical cavity is usually a standard Fabry-Perot optical resonator that consists of two very good concave mirrors separated by a small distance. • The length of the cavity is stabilized to have a standing wave of light resonant or hear-resonant with the atomic transition of interest. • Making a good cavity is part black magic, part sweat and blood... G. Rempe - MPQ • These cavities need to be phenomenally good to get into a regime where single photons trapped inside interact strongly with the atoms. M. Chapman - GATech The technology: mirrors • To make g >> κ we need: • a small-volume cavity to increase g • a very high-finesse cavity to reduce κ M. Chapman - GATech • “clean” cavity to reduce other losses • Strong-coupling cavities use super-polished mirrors (surface roughness less order of 1 Å, flatness λ/100) to reduce losses due to scattering at the surface. • Mirrors have highly-reflective multi-layer dielectric coatings (reflectivity at central wavelength better than 0.999995, meaning finesse higher than 500000). • Mirrors have radius of curvature of 1 – 5 cm, and small diameter. Mirror spacing is 100 micron down to 30 micron. These features of the cavities make for stronger confinement of photons for higher g. Qubits: single atoms or ions (also, artificial atoms) • A cavity QED system is usually combined with and atom or ion trap D5/2 D3/2 • Two-level system formed by either the hyperfine splitting of the ground state (“hyperfine” qubit) or by the ground state and a metastable excited state (“optical” qubit) • The atom can interact with the laser field (“classical” field) and the cavity field (“quantum” field) • Qubit state preparation and detection techniques are well established and robust Qubit preparation and detection • Initialization of the qubits state is via optical pumping: applying a laser light that is decoupled from a single quantum state • Detection by selectively exciting one of the qubit states into a fast cycling transition and measuring photon rate. May also start by “shelving” one of the qubit states to a metastable excited state, then applying resonant laser light. The qubit state that ends up scattering laser light appears as “bright”, while the other state |1,1〉 |2,2〉 P3/2 appears as “dark”. • Both the preparation and the detection steps have been demonstrated to work with over 99% efficiency with trapped ions. 111Cd+ π P1/2 Cycling transition (cooling/detection) σ+ |0,0〉 14.5 GHz S1/2 |1,-1〉 |1,0〉 |1,1〉 Other qubits: photons • Cavity QED quantum computing makes use of photons to both mediate the atomic qubit entanglement and to transfer quantum information over long distances. • Photon detection: PBS (polarization beam splitter) and single photon counters Note on polarization. Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. Note on polarization Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. In electrodynamics, polarization is the property of electromagnetic waves, such as light, that describes the direction of their transverse electric field. The electric field vector may be arbitrarily divided into two perpendicular components labelled x and y (with z indicating the direction of travel). For a simple harmonic wave, where the amplitude of the electric vector varies in a sinusoidal manner, the two components have exactly the same frequency. However, these components have two other defining characteristics that can differ. First, the two components may not have the same amplitude. Second, the two components may not have the same phase, that is they may not reach their maxima and minima at the same time. Wikipedia Note on polarization The shape traced out in a fixed plane by the electric vector as such a plane wave passes over it, is a description of the polarization state. The following figures show some examples of the evolution of the electric field vector (blue) with time (the vertical axes), along with its x and y components (red/left and green/right), and the path traced by the tip of the vector in the plane (purple). Wikipedia Notes on polarization The polarization of a classical sinusoidal plane wave traveling in the z direction can be characterized by the Jones vector where the angle θ describes the relation between the amplitudes of the electric fields in the x and y directions. Polarization applet http://webphysics.davidson.edu/physlet_resources/dav_optics/Examples/polarization.html Combining atom trapping and cavity Optical lattice confining atoms inside a cavity (M. Chapman) ~100 µm Thin ion trap inside a cavity (Monroe/Chapman, Blatt) Cavity field used to trap atoms (G. Rempe) Other cavities: whispering gallery resonators • Quality factors of 108 and greater Whispering cavity resonator laser (http://physics.okstate.edu/shopova/research.html) J. Kimble (Caltech) • Simple (sort-of) technology – just make a nice, smooth glass sphere ~50 micron in diameter... • Evanescent field extends only a fraction of the wavelength (i.e. ~100 nm) outside the sphere – need to place atoms close to the surface. • “Artificial atoms” such as quantum dots can be used... Challenges of cavity QED QC • Cavity QED quantum computing attempts to combine two very hard experimental techniques: the high-finesse optical cavity and the single ion/atom trapping. This is not just doubly-very-hard, but may well be (very-hard)2 • Assuming “hard” > 1, we have “very hard” >> 1, and (“very hard”)2 >> “very hard” • However, the benefits of cavity QED, namely, the connection of static qubits to flying qubits, are very exciting and are well worth working hard for. Strengths 1. Ability to interconvert material and photonic qubits. 2. Source of deterministic single photons and entangled photons. 3. Cavity QED systems provide viable platforms for distributed quantum computing implementations for both neutral atom and trapped ions. 4. Well understood systems from a theoretical standpoint. The cavity QED system has been an important paradigm of quantum optics. Weakness 1. Ultimate performance of systems is dependent on advances in mirror coating and polishing technologies. Current mirror reflectivities, while adequate to achieve the strong coupling limit, are still ~!100 times lower than the theoretical limit imposed by Rayleigh scattering in the coating. Additionally, smaller mirror curvature would provide for large coherent coupling rates. 2. The role of the atomic motional degree of freedom in the cavity gate operation and subsequent evolution needs to be better understood both experimentally and theoretically. 3. Need to combine two already very hard to implement technologies. http://qist.lanl.gov/qcomp_map.shtml