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12 Solid State Quantum Computers

12.1 Solid state NMR/EPR

12.1.1 Scaling behavior of NMR quantuminformation processors

1000

0.001

1

0 10 20# qubits

Signal

Figure 12.1: Loss of signal amplitude due to prepa-ration of pseudo-pure states as a func-tion of the quantum register size.

Liquid state NMR was the first experimental tech-nique that allowed the implementation of quantumalgorithms and is still the basis for the most ad-vanced quantum information processors. Neverthe-less, there are serious obstacles to advancing thissystem much farther. One difficulty is associatedwith the preparation of pseudo-pure states [169]:The procedure averages all populations but one. Aslong as the spin system can be described by the high-temperature approximation, the population of an in-dividual spin state is inversely proportional to thenumber of states. It therefore decreases as 2�N withthe number of spins N. The detectable signal sizetherefore limits the possible number of spins to beused in such a quantum information processor.

The reduction of sensitivity associated with thepreparation of pseudo-pure states can be avoided byusing algorithms that do not require pure states towork with. For this purpose, variations of algorithmshave been developed that can be applied directly tomixed states [240, 172, 241, 173]. For the purpose ofdatabase search, such modified algorithms can even

be exponentially faster [172, 173] than the originalalgorithm developed by Grover [107].

Another approach to beating the exponential de-crease of the signal size due to the pseudo-pure statepreparation would be to work with sufficiently highspin polarization that one can create good approx-imations of pure states. Virtually complete polar-ization of the electron spins by thermal relaxationcan be achieved at a temperature of 100 mK in amagnetic field of 2 T, where h̄w

kBT = 27 � 1. Highenough nuclear spin polarization, in contrast, cannotbe achieved in thermal equilibrium within the cur-rently accessible experimental conditions.

Highly spin polarized hydrogen nuclei can be ob-tained by several non-equilibrium techniques, e.g.,by separating the ortho and para components inmolecular hydrogen gas [242]. When the symme-try between the two nuclei in the molecule is bro-ken, e.g., through a chemical reaction, it is possi-ble to achieve truly entangled nuclear spin states[243, 244]. Other approaches to pure state prepa-ration include optical pumping [245, 246] or polar-ization exchange with electron spins at very low tem-perature [247, 248]. All these techniques require thatthe system be kept at low temperature to avoid com-peting processes that reduce the polarization. Thisalso implies that the material that contains the spinsbe a solid rather than a liquid.

Another aspect of liquid state NMR that may makeit difficult to scale up to larger numbers of qubits, isthe addressing of the individual qubits. Current im-plementations use the natural chemical shift rangeof the nuclear spins to distinguish them by their res-onance frequency. Since the chemical shift range islimited, this procedure cannot be extended to arbi-trarily large numbers of spins. The larger the numberof qubits, the smaller is therefore the separation oftheir resonances and therefore the slower the switch-ing speed. It appears therefore necessary to design

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an addressing scheme that does not rely on chemicalshift differences.

12.1.2 Spins in solids

Some solid state implementations of spin-qubits maybe considered direct extensions of liquid state NMR:Kampermann and Veeman used a quadrupolar sys-tem [249], much like a similar system in a liquidcrystal [250].

Figure 12.2: System for coupling 4 nuclear spins(red) via an electron spin (blue).

A potentially more powerful scheme was demon-strated by Mehring et al. [251]. Their system usedan electron spin coupled to different nuclear spins byhyperfine interaction. They also introduced the ideaof using electron spins as “bus-qubits” to allow nu-clear spins to efficiently exchange information. Thiswas introduced as the “S-bus concept”. As for allother spin-based quantum computers demonstratedso far, there is no straightforward extension of thisscheme to large (> 100) numbers of qubits.

A scheme that is intermediate between liquid stateNMR and the single-spin solid state NMR approachis the “crystal-lattice quantum computer" [252, 253,254], where arrays of identical nuclear spins are usedas a single qubit. Compared to liquids, these solidsoffer the possibility of increasing the spin polariza-tion, not only by lowering the temperature, but alsoby polarization transfer from electronic spins, e.g.,by dynamic nuclear polarization. Addressability ofindividual qubits could be obtained by a strong fieldgradient produced by a micrometer-sized ferromag-net. Furthermore, solids are required for some de-tection schemes that offer higher sensitivity than theusual inductive detection [255].

12.1.3 31P in silicon

This should be possible, however, if the system pro-posed by Kane can be implemented [256]. He pro-posed to use 31P impurities in Si, the only I = 1/2shallow (group V) donor in Si. The 31P:Si systemwas exhaustively studied 40 years ago in the firstelectron-nuclear double-resonance experiments. Atsufficiently low 31P concentrations at temperature T= 1.5 K, the electron spin relaxation time is thou-sands of seconds and the 31P nuclear spin relaxationtime exceeds 10 hours. This system would thereforeallow for a large number of gate operations within adecoherence time.

Figure 12.3: Proposed scheme for a quantum com-puter that uses 31P atoms in a 28Si ma-trix [256].

Figure 12.3 shows the principle of this scheme: the31P atoms are to be placed in a matrix of 28Si (whichhas no nuclear spin). Operation of these qubitswould be identical to that of a liquid state NMR sys-tem, i.e., by radio frequency pulses. However, sinceall qubits see the same chemical environment, theirresonance frequencies are identical. As a way of ad-dressing them, it may be possible to use small elec-trodes, which are labeled “A-gates" and “J-gates",respectively, in Fig. 12.3.

The Hamiltonian of the 31P system can be written as

H = WSSz �WIIz +AIzSz.

Here, the first two terms represent the Zeeman inter-action of the electron and nuclear spin, respectively.

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The third term is the hyperfine coupling between thetwo spins.

|0i|1i

Qubit

Figure 12.4: States of the electron-nuclear spinsystem.

The four eigenstates of this system are | #"i, | ##i, | "#i, and | ""i, where the first position refers tothe electron and the second to the nuclear spin state.Their energies are

�WS �WI

2� A

4�WS +WI

2+

A4

WS +WI

2� A

4WS �WI

2+

A4.

The hyperfine coupling constant A between electronsand nuclei depends on the electron density at the siteof the nucleus. If the voltage applied to the gate elec-trodes changes the electrostatic potential near thedonor sites, it shifts the electrons closer or fartherfrom the gates and thereby changes the electron den-sity at the site of the nucleus and therefore its hyper-fine coupling. Kane’s proposal uses this possibilityto tune the resonance frequency of the qubit. Forthis purpose, it uses a two-dimensional subspace ofthe 4-dimensional Hilbert space of the 31P electron-nuclear spin system: the electron spin is fixed in the| "i-state, while the nuclear spin represents the qubit.The two qubit states are thus

|0i = | #"i , |1i = | ##i

and their energies (�WS �WI)/2�A/4 and (�WS +WI)/2 + A/4. For this two-dimensional subspace,we can write an effective Hamiltonian

H2 = (A2

�WI)Iz.

Adjusting the hyperfine constant A therefore adjuststhe transition frequency of the qubit.

Figure 12.5: Dependence of the hyperfine couplingconstant on the gate voltage, accordingto [256].

The electrodes labelled “A-gates" could therefore beused for addressing the individual qubits by shiftingtheir energies in and out of resonance. Kane showedthat gate voltages of ⇡ 0�1 V should be able to tunethe nuclear spin Larmor frequency between 50 and100 MHz. Similarly J-gates would move electrondensity between the donor sites (see fig. 12.7), thusinducing an indirect coupling between qubits and al-lowing the addressing of pairs of qubits.

12.1.4 Qubit operations

The Kane proposal has a number of very attractivefeatures. In particular, the long relaxation timesof both spins in the system indicate the possibil-ity of performing many gate operations before de-coherence leads to the loss of quantum informa-tion. The reason for these long decoherence timescan be traced to the fact that 28Si, which formsthe main component of natural abundance Si, hasno nuclear spin. Accordingly, it does not perturbthe electron spin by hyperfine interaction. The ef-fect of the 4.6% natural abundance 29Si can be re-duced by isotopic enrichment. Since the nuclei in-volved are relatively light, spin-orbit interaction isweak, which also contributes to the long decoher-

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ence times. While the manufacturing poses signif-icant challenges, the enormous investments of thesemiconductor industry in technological develop-ments of Si-based circuits has led to a highly ad-vanced technology base for this system.

One of the main challenges of this approach is theplacement of the individual donor atoms, whichshould occur with atomic precision. Significantprogress has been made in this direction by pattern-ing the surface with a scanning tunneling microscopeand subsequent overgrowth [257].

To meet the di Vincenzo criteria, it is necessary toinitialize the qubits. This cannot be done by simplycooling the system to the ground state: The Boltz-mann factor for electron spins at a temperature of 0.1K, in a magnetic field of 2 T, is close to 1. For thenuclear spins under the same condition is ⇡ 5 ·10�3

- clearly too low for initialization of the qubit intothe ground state.

Figure 12.6: Initialization of the 31P qubit. Blue isthe electron spin, red the nuclear spin.

While thermal polarization is not sufficient to ini-tialize the nuclear spin qubit, it can be used to ini-tialize the electron spin. From the state, where bothqubit states are equally populated, initialization intothe ground state can be achieved by applying a mi-crowave pulse to the transition between the | "#i $| #"i states. From this state, thermal relaxation willprimarily populate the | ""i state, thus enhancing thepopulation of the |0i state of the qubit. The remain-ing population, which decays into the |1i state of thequbit, can be excited again, until the vast majorityof the atoms have accumulated in the logical groundstate.

Barrier

Si31P qubit

J Gate

Figure 12.7: Principle of operation of J-gates.

While each A-gate operates on the transition fre-quency of an individual qubit, the J-gates are de-signed to affect primarily the interaction betweentwo qubits. For this purpose, it draws electron den-sity of both neighboring qubits into the region be-tween them. The resulting overlap between the twoelectron densities results in a spin-dependent ex-change interaction. Through the hyperfine interac-tions of both qubits, this also mediates an effec-tive exchange interaction between the nuclear spinqubits. The strength of this interaction is of the or-der of 75 kHz.

|??> |B?+?B>

|BB>

|B?-?B>

electron spin pair

Coupling J

Ener

gy

|0> =|1> =

Figure 12.8: Conversion of |0i state to electronic sin-glet state.

Another requirement is the detection of the individ-ual spins. Proposals for this purpose exist. They allinvolve some conversion of the nuclear spin qubitstates to electronic states. This can start, e.g., witha sweep of the coupling constant between the qubitelectron and a readout electron, controlled by a J-gate. Depending on the state of the nuclear spin,the electron spin pair ends up either in the singletor in one of the triplet states. The singlet state canthen be converted into a charge state, by transferringthe electron to the readout donor. According to thePauli principle, this is only possible for the singlet

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state, but not for the triplet state. The charge statecan then be detected via a single electron transistor[256, 258]. In a related experiment, the signature ofa single 31P nuclear spin was measured in a Si-FET[259]. In a slightly different system, the coherentevolution of an ensemble of 31P nuclear spins wasmeasured [260]. It remains to combine the coherentevolution with the single spin detection.

12.1.5 Si/Ge heterostructures

GeSi0.23Ge0.77 barrierSi0.15Ge0.85Si0.4Ge0.6Si0.23Ge0.77 barriern-Si0.4Ge0.6 ground planeSi-Ge buffer layerSi substrate

Figure 12.9: A proposal for a 31P qubit on the basisof a SiGe heterostructure.

The concept of using donor atoms in silicon can alsobe modified by using Si/Ge heterostructures [261],rather than bulk Si. An attractive feature of such het-erostructures is that the g-factor of the electron spindepends on the material.

g D=2.00

g T=0.82

gav=2.00

z

Cond

uctio

n ba

nd e

nerg

y / m

eV

gate o!<111> substrate

biased gate

gav ~ 1.5

Figure 12.10: Proposed Si/Ge heterostructure withdifferent g-factors, depending on gatepotential.

Using electrodes, the electrons can be pushed intothe Si or Ge material, thereby changing their res-

onance frequency and providing addressability forsingle-qubit gates. For this purpose, the defect mustbe positioned in Si0.4Ge0.6, where the electronic g-factor is close to 2. In the neighboring layer con-sisting of Si0.15Ge0.85, the g-factor is of the orderof 0.82. If a suitable field is applied, which lowersthe potential of the Si0.15Ge0.85 layer, the electronicwavefunction is partially pushed over into that layerand the effective g-factor is lowered correspond-ingly, to an estimated value of 1.5. The overlap ofthe wave function with the Si0.15Ge0.85 also changesthe Bohr radius and thereby affects the overlap of theelectron with neighboring electrons. The gate there-fore also has the function of the J-gate, making theimplementation of separate J-gates unnecessary.

4 qubits2 qubits

Figure 12.11: Electrostatically confined electronqubits in a Si/SiGe quantum well.

Friesen [262] goes into another direction: he pro-posed to define the qubits by electrostatic confine-ment of electrons in Si / SiGe quantum wells.

J=0

J!0gate

Figure 12.12: Change of overlap between two elec-tron wave packets.

Since the Bohr radius of the electron can be changedby applying potentials to the electrodes, it becomespossible to create non-vanishing overlaps betweenneighboring wave functions.

Using donors in silicon may not require patterningon the nm scale. Using random doping and tailored

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optical excitation, it may be possible to control atleast small groups of qubits [263]. In this proposal,the qubits would be the spin states of deep donors,like Si:Bi. An additional control impurity, e.g. Er,would be excited by a suitable laser. Depending onits electronic state, its wavefunction would overlapwith the qubits and thereby mediate a coupling be-tween them. This proposal obviously requires a sig-nificant amount of fine-tuning for every qubit andcontrol impurity. However, to some degree such finetuning will be required for every nano-fabricated de-vice, since the parameters of every artificial structurevary to some degree.

12.1.6 N@C60

Figure 12.13: Array of N@C60 molecules forming aquantum register.

Among the most attractive qubit materials are the en-dohedral fullerenes N@C60 and P@C60 [264]. Theendohedral atom is trapped inside the highly sym-metric fullerene molecule, which can be considereda nanometer-sized trap for a neutral atom. The nitro-gen atom has an electron spin of S=3/2, while the nu-cleus has spin I=1 (for 14N or I=1/2 for 15N and 31P).In the context of quantum computing, the main inter-est in them arises from the possibility to use them asroom-temperature, nanometer-sized traps for neutralatoms [265]. In particular, nitrogen and phospho-rous atoms are attractive candidates, which are hardto trap with other methods. Their p-shell is half full,which results in a total electron spin S = 3/2. Theelectron spin is coupled to the nuclear spin by hy-perfine interaction. The relevant Hamiltonian of thespin system can be written as

HS = gµB~B0 ·~S � gn~B0 ·~I +A~S ·~I.

Here, ~S is the electron spin,~I the nuclear spin, g, µBand gn are the electron g-factor, Bohr’s magneton andthe nuclear gyromagnetic ratio, ~B0 is the magneticfield and A the hyperfine coupling constant. For theatoms trapped in a C60 molecule, the correspondingvalues are

nucleus spin / h̄ A/MHz14N 1 15.8815N 1/2 22.2631P 1/2 138.4

Using the electronic as well as the nuclear spin de-grees of freedom allows one, in principle, to encodeup to three qubits in each molecule.

Fig. 12.13 shows a possible use of these moleculesas qubits: each C60 molecule acts as a trap for anitrogen or phosphorus atom, whose spins encodethe quantum information. The major properties thatmake this system so attractive for quantum informa-tion processing is that (i) the spins have very longlifetimes, with the longitudinal relaxation time T1 ex-ceeding 1s at low temperature [266] and (ii) they areeasier to manipulate. It would be possible, e.g., todeposit them on the surface of a suitable material,such as silicon, and manipulate them by a scanningtunneling microscope [267].

Wires(not to scale)

Figure 12.14: Scheme for resonant addressing ofN@C60 molecular spins: the wirescary copropagating currents, whichgenerate a magnetic field gradient su-perimposed over a static external mag-netic field.

Gate operations can be performed by resonant mi-crowave pulses applied to the electron spins andradio-frequency pulses applied to the nuclear spintransitions [268]. Addressing of the individual

206


molecules can be achieved, e.g., by applying a mag-netic field gradient that shifts the resonances of theindividual molecules [269]. By depositing copperwires on the Si surface and running currents of theoder of 1 A through two parallel wires generates amagnetic field that combines with the homogeneousbackground magnetic field to a magnetic field gradi-ent that between the two wires, as shown schemati-cally in Fig 12.14. For a distance between the wiresof the order of 1 µm, the resulting gradient wouldbe of the order of 4 ⇥ 105 T/m. For two N@C60molecules separated by 1.14 nm (the diameter of themolecules), this results in a frequency splitting of12.7 MHz. This should allow precise qubit address-ing in frequency space. If larger distances are cho-sen between the molecules, the frequency differenceis correspondingly larger.

One major difficulty of the system is that the mag-netic dipole couplings between the molecules arestatic, i.e. they cannot be switched as required bythe algorithm. This problem can be solved by usingthe electron and nuclear spin for encoding a singlelogical qubit. Figure 12.15 shows the relevant en-ergy level scheme for the the 15N@C60 or 31P@C60electron-nuclear spin system. The four nuclear spintransitions and the electron spin transitions are splitby the hyperfine coupling of 22 or 138 MHz.

Figure 12.15: Energy levels of the 15N@C60 or31P@C60 electron-nuclear spin sys-tem. I refers to the nuclear spin, S tothe electron spin.

Using both degrees of freedom allows one to storethe information in the nuclear spin degree of free-

dom. Since the nuclear spin couples only weaklyto other degrees of freedom, the quantum informa-tion stored in it has a long lifetime. It is also effec-tively isolated from the other molecules, since themagnetic dipole-dipole couplings between nuclearspins is ⇡ 109 times smaller than that between elec-tron spins. When the algorithm requires an activecoupling between two qubits, it can be generated byswitching both qubits into the electron spin degreesof freedom, thereby switching the coupling on. Thetwo-qubit operation is then performed on the elec-tron spins and the qubits are switched back to thenuclear spins when the gate operation is complete[269, 265, 270].

While several elements of this scheme have beentested, the readout of the qubits remains a signifi-cant challenge. Experimental evidence [271] showsthat it is possible to electrically contact individualmagnetic N@C60 molecules and measuring spin ex-citations in their electron tunneling spectra. The tun-neling spectra allow the identification of the chargeand spin states of the molecule. If such measure-ments can be combined with the other elements, aquantum computer based on endohedral fullerenesappears possible.

12.1.7 Rare Earth Ions

The electronic properties of rare earth ions, i.e. theelements from Lanthanum (Z = 57) to to Lutetium(Z = 71) distinguish them from almost all other el-ements. The states that are responsible for thesespecial properties are the partly filled 4 f orbitals.The relevant transitions that fall into the visible ornear-IR range of the spectrum are all forbidden byparity and often also by spin selection rules. Thisresults in long lifetimes and narrow natural linewidths Vleck [272]. Furthermore, the states are onlyweakly affected by crystal field effects, which resultsalso in relatively small inhomogeneous broadening.These properties have fascinated physicists workingin atomic spectroscopy as well as physicists and en-gineers interested in optical data storage [273] or op-tical data processing [274]. Rare earth ions werealso found to be useful qubits for quantum infor-mation processing, either stored in electromagnetic

207


traps also [275] or as dopant ions in dielectric crys-tals [276].

An additional use for rare earth ions in solid mate-rials came with the search for quantum memories[277]. These devices must store the complete quan-tum state of a photon in a suitable material for timesof µs to s [278, 279]. For this purpose, it is necessaryto convert ‘flying qubits’ into stationary qubits andvice versa. This is achieved when the photons inter-act with an optical transition. These processes cannot only proceed directly, they can also be assistedby different experimental techniques, such as elec-tromagnetically induced transparency (EIT) [280].Compared to conventional optical storage, quantummemories require storage of the complete quantumstate of a photon. This is nontrivial, since it is notpossible to convert the quantum state into classi-cal information; this is usually specified in terms ofthe “no-cloning theorem” Wootters and Zurek [44].Quantum storage thus requires that not only the pop-ulations of the relevant states are conserved, but alsothe relative phases between them. This requirementis extremely difficult to meet in almost all solid-statematerials, with crystals containing doped rare-earthions as the major exception [281, 282, 277]: due totheir electronic structure, the optical dephasing timesare unusually long.

The materials used for this purpose are mostly basedon Pr3+ or Eu3+ substituting for La3+ or Y3+, suchas Pr:La2(WO4)3, Pr:YAlO3, or Pr:Y2SiO5. Rele-vant criteria include the accessibility of suitable tran-sition frequencies by available lasers, the line widthof these transitions, the lifetimes of the electronicand nuclear spin states, the transition strengths andabsorption depths for a given amount of doping.High levels of doping can generate stress in the crys-tal and therefore broadening of the resonance lines,in particular if the ionic radii of the host and guestion differ significantly.

Free ion Crystal field Hyperfine

Ener

gy /

cm-1 1D2

3H5

3H40

2200

16375

....

....

0+51

+146+155

+274 0.9 MHz

1.6 MHz

7.1 MHz

14.1 MHz

Figure 12.16: Energy levels of the Pr3+ ion substi-tuting for Y3+ in YAlO3.

Fig. 12.16 shows, as a typical example, the simpli-fied energy level scheme of the Pr3+ ion substitutingfor Y3+ in an YAlO3 crystal. Among the many pos-sible optical transitions, that between the 3H4 elec-tronic ground state and the 1D2 electronically ex-cited state is easily accessible by high-resolution ringdye lasers, with a transition energy of 16375 cm�1,which corresponds to a wavelength of 610.7 nm. Thehighly degenerate states of the free ion split in thepresence of a crystal field. At the same time, thecrystal field also quenches the orbital angular mo-mentum of the electrons. On a much smaller energyscale, the states split further due to the interaction ofthe nuclear spin with external magnetic fields and thenuclear quadrupole moment with the electric fieldgradient tensor of the crystal. Both interactions areenhanced by the second-order hyperfine interaction.

12.1.8 Photonic quantum memories

Laser Input Opt. refocusing

Echo Time

Figure 12.17: Photon echo as a short-time opticalmemory.

As illustrated in Fig. 12.17, the simplest optical stor-

208


age scheme can be realized as a photon echo: Ab-sorption creates a superposition state of two elec-tronic states |gi and |ei, which contains the informa-tion about the quantum state of the absorbed photon.The inhomogeneous dephasing in the material canbe reversed by an echo pulse, resulting in the emis-sion of a photon at a later time, in a direction whichis determined by the directions of the incident pulseand the refocusing pulse. This simple photon echoexperiment has several limitations: its efficiency isquite small - only a few percent of the incident lightis typically recovered, the rest is lost to absorption,and the storage times are relatively short. Differ-ent solutions have been proposed for these problems[283, 284, 285, 277, 286], and a number of theseimprovements have been tested experimentally (see,e.g. [287, 288]). The ultimate goal is to store thestate of a single photon with fidelity close to unityfor a time of the order of seconds.

Rare earth ions can be used not only for informa-tion storage: once the information has been inputinto the system, it can also be processed by theusual quantum gate operations. Optical pulses aswell as radio frequency fields can be used to gen-erate the quantum logical gate operations. In theform of trapped atomic ions, rare earth ions wererelatively quickly adopted for quantum computingapplications (see, e.g., [289]). Rare earth ions insolid state materials offer in principle the same po-tential [290]. Compared to many other solid-statesystems, they can be operated at relatively “warm”temperatures close to 4.2 K. The optical as well asthe magnetic dipole degrees of freedom offer manypossibilities for generating gate operations. A num-ber of demonstration experiments has verified thispotential. Important milestones include the demon-stration of optical coherence lifetimes of 4.4 ms inEr3+:Y2SiO5[291]. Significantly longer coherencelifetimes can be achieved, if the phase information istransferred from the optical transition to a hyperfinetransition, where coherence lifetimes of several sec-onds can be achieved with suitable techniques [278].All these experiments involve the use of coherentmagneto-optics, discussed in the following subsec-tion.

The transfer of the information from the electronic

Laser Input Opt. refocusing

Echo

Time / µs0 2 4 6 8 10 12 14 16La

ser

puls

esEc

ho s

igna

l

Pr:LaWO, B0=0, T 1.7K

Figure 12.18: Experimentally measured photonechoes in Pr3+:La2(WO4)3 as afunction of the delay between thepulses. The fitted curve correspondsto a dephasing rate of T2 = 9.34 µs.

degrees of freedom into the nuclear spin state in-creases the lifetime by several orders of magni-tude. Even more important, however, is that itnow becomes easier to use established experimen-tal techniques for further extending the lifetime. Themain limitation for the decay of quantum informa-tion stored in the nuclear spin degrees of freedomof rare earth ions are the magnetic fluctuations ofthe environment. Their influence can be suppressedby different techniques, including the application ofsuitable magnetic fields, which suppress the effectof magnetic field fluctuations on the transition fre-quency to first order [292], or by sequences of radio-frequency pulses that refocus the dephasing inducedby the environment [293].

Figure 12.18 summarizes some results on the storageof optical states. The left hand panel shows a seriesof photon echoes, measured with a p/2 � t � p � t

sequence. The decay of the echoes with increasingpulse separation can be fitted as an exponential decayµ e�4t/T2 with a dephasing time T2 = 9.34 µs. Thiscorresponds to the lifetime of quantum states in theelectronic degrees of freedom for this material.

The transfer from the electronic to the nuclear spindegrees of freedom by itself is not sufficient for a sig-nificant increase of the decoherence time. The blackcurve in the right-hand side panel of Fig. 12.18, la-

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0.2

0.4

0.6

0.8

1

Time

Sign

al a

mpl

itude

1 µs 1 ms 1 s

FID

Hahn echoCPMG

Figure 12.19: Increase of the lifetime of nuclear spincoherence by several orders of magni-tude, using either Hahn echoes or theCarr-Purcell-Meiboom-Gill sequence.

beled ’FID’ shows the decay after the transfer. Thisdecay is dominated by magnetic interactions and canbe refocused either by a simple refocusing pulse(’Hahn-echo’ in Fig. 12.18) or, more effectively, bya series of pulses (points labeled ’CPMG’ in Fig.12.19. Clearly, these refocusing techniques extendthe lifetime of the coherence in the material by morethan 5 orders of magnitude.

12.1.9 Molecular Magnets

Mn

O

Figure 12.20: Mn12O12 cluster, forming the centralpart of a “molecular magnet” qubit.

Molecules containing clusters of transition metalions have also been proposed as possible qubit sys-tems [294]. The ions in these “molecular magnets”are strongly coupled by exchange interaction andhave large total spins. Examples include clusterslike Mn12 [295, 296] shown in Figure 12.20 and Fe8[297] with total spin S = 10. The spin interacts withits anisotropic environment, resulting in a large zero-

field splitting energy

HZF = �DS2z ,

which stabilizes the ground states with mS = ±S.In most cases, the environment does not have axialsymmetry and the Hamiltonian therefore contains anadditional anisotropy term,

HZF2 = �DS2z �E

12�S2

+ +S2��.

If a magnetic field is applied along the z-axis, thetotal Hamiltonian becomes

Hmm = �DS2z �E

12�S2

+ +S2�� h̄wLSz. (12.1)

0.0

Zeeman energy0.1 0.2 0.3 0.4 0.5

Ener

gy /

0

-50

-100

Figure 12.21: Energy levels of an S = 10 sys-tem, corresponding to the Hamiltonian(12.1), with E/D = 0.05.

As shown in Fig. 12.21, this completely lifts the de-generacy of the system. The 2S+1 energy levels of-fer a wide range of possible schemes for storing thequantum information in this system. However, sincethe energies are spread over a range of > 100GHz h̄,it is an enormous challenge to implement coherentcontrol for the complete system. As an alternative, itwas proposed to use only the two lowest energy lev-els, corresponding to mS = ±S. This does not elimi-nate the challenge of implementing coherent control,however, since these states are not directly coupledby a magnetic dipole transition.

A major challenge of these systems for quantum in-formation applications is the coupling between the

210


molecules and their environment, which leads to rel-atively fast decoherence, compared to the more rigidsolid-state systems discussed above. For some sys-tems, it is possible to deposit them as monolayers ona surface (often gold layers, see, e.g. [298]), withoutsignificantly changing their magnetic properties.

12.1.10 The NV�-center in diamond

N

V

1 µm

Figure 12.22: Structure of the N/V center in dia-mond. The right-hand side repre-sents an image of a diamond surface,recorded by a scanning confocal mi-croscope. Each bright spot representsa single N/V center.

Diamond has a number of well characterized defects,of which the most prominent one is the nitrogen-vacancy center [299, 300]. It consists of a nitrogen ata carbon site and an adjacent vacancy, i.e. a missingcarbon. The electrons of the defect combine to anS = 1 total spin. The attractive properties of this cen-ter include the long coherence times at room temper-ature, and the special optical properties: the photo-stability is very high, allowing experiments on singlecenters for months.

For the purpose of this section, we will not discussbulk experiments on N/V centers, but only experi-ments with single centers. Each of the bright spots inthe right-hand part of Fig. 12.22 represents a singleN/V center. While it is not possible to determine thisfrom the image alone, which was taken by scanningconfocal microscopy with a resolution of ⇡ 300 nm,it is possible to estimate it from the observed countrate. A much cleaner signature, however, if obtainedby measuring the correlation function of the arrival

times of the photons on the detector. If we measurethe delays t between the arrival times of individualphotons, we find that the probability to detect a sec-ond photon immediately after the first drops to zerofor short times [301]. This is easy to understand byconsidering that after the emission of a photon, thecenter is in the ground state and cannot emit anotherphoton until it has absorbed one.

Is this a single center ?

Antibunching

100 200 3000Delay / ns

Nor

mal

ized

# p

hoto

ns

0

1

Figure 12.23: Photon correlation function for a sin-gle N/V center.

Fig. 12.23 shows an example: the emission ratedrops almost to zero for short delays, and it takes⇡ 15 ns for the emission probability to rise again toits average value. This rise time decreases with in-creasing laser intensity.

Initialization as well as readout rely on absorption-emission cycles between the 3A2 electronic groundstate and the 3E electronically excited state, whosezero phonon line has a wavelength of l0 = 637nm.The phonon sidebands can be excited by green laserlight (e.g. l = 532 nm). Between these two elec-tronic triplet states are two singlet states, whichcan be populated by intersystem crossing processes.These processes are spin dependent. Pumping thesystem for ⇡ 0.5 µs with 1 mW of green laser lightleaves it with high probability in the mS = 0 spinstate. When the system is in the mS = 0 state, thescattering rate for unpolarized green light is abouttwice that of the mS = ±1 states, which allows arelatively straightforward detection of the individualspin states.

In the absence of a magnetic field, the mS = ±1 spinstates are degenerate, but separated from the mS = 0state by the zero field splitting of D = 2.87 GHz. A

211


magnetic field lifts the degeneracy of the mS = ±1states. In addition, the electron spin is coupled tothe nitrogen nuclear spin (usually 14N, I = 1) andto a those carbon sites that are occupied by a 13Cisotope (I = 1/2) with a hyperfine coupling constantthat starts at 130 MHz for the carbon sites adjacentto the vacancy and decreases with the distance [302].The most important terms in the ground-state Hamil-tonian of the NV defect are therefore

H = DS2z + µ0g~B~S +AN~S~IN +Â

kAk

C~S~Ik

C,

where the sum runs over all sites occupied by 13Cisotopes.

Frequency / MHz0 2 4 6 8

14N13C

Figure 12.24: Spectrum from a single N/V centershowing resolved hyperfine couplingsto the 14N and one 13C nuclear spin.

Figure 12.24 shows a typical spectrum: the mS =0 ! mS = �1 transition of the electron spin is splitby the hyperfine interaction with the 14N nuclearspin (AN = 2.17 MHz) and one 13C nuclear spin(AC = 0.58 MHz). Many additional nuclear spinscouple to the electron spin with hyperfine couplingconstants <= 0.3 MHz, which do not lead to re-solved splittings, but to a broadening of the reso-nance line.

The decay of electron spin coherence by the hyper-fine interaction with the 13C nuclear spins can berefocused by the usual spin-echo experiments. Asshown in Fig. 12.25, a single refocusing pulse, cor-responding to the Hahn echo, can generate echoesfor delays of up to 10 µs. For longer times, the re-focusing does not work, because fluctuations in theenvironment make the refocusing inefficient. Likein the case of molecular diffusion, it becomes thennecessary to apply multiple refocusing pulses withshorter delays between them [97, 98]. As shown by

Time

τ

µ-w

ave

Lase

r! !" !"" !"""

Decay time / µs

Sign

al

Hahn echo2 pulses4

81632

64128256

Figure 12.25: Refocusing of electron spin coherenceby spin-echo experiments. The curvesin the lower panel show the decay ofthe echo amplitude as a function ofthe total measurement time for differ-ent experiments with increasing num-ber of refocusing pulses.

the other curves in Fig. 12.25, sequences of refocus-ing pulses can extend the coherence time up to about1 ms.

The experimental data of Fig. 12.25 show that thedecay is not exponential. The curves drawn throughthe experimental points were obtained by fitting a“stretched exponential” e�(t/T2)b to the experimentaldata. For small number of refocusing pulses, the ex-ponent is close to 1, but it becomes smaller for largernumber of pulses, indicating a complex dynamicsin the environment. One contribution to this comesfrom the Larmor precession of the 13C nuclear spins,which is synchronized by the microwave pulses ap-plied to the electron spins. At the start of the experi-ment, the laser pulse initializes the electron spin intothe mS = 0 state. In this state, the secular part SzIzof the hyperfine interaction vanishes and the nuclearspin interacts only with the external magnetic field.

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The p/2 microwave pulse then puts the system intoa superposition of the mS = 0 and mS = 1 state. If theelectron is in the mS = 1 state, the nuclear spins in-teract not only with the external magnetic field, butalso experience an effective field from the electronspin, which is oriented along the symmetry axis ofthe NV-center. If this axis does not coincide withthe direction of the external magnetic field, the twostates have different quantization axes for the nu-clear spin and the microwave pulse creates not onlya superposition of electron spins, but also a super-position of the nuclear spins, which evolves betweenthe p/2 and p pulse. The refocusing pulse cannotcompletely refocus such a time-dependent environ-ment and the echo amplitude decreases. However,this evolution of the environment is partly coherent,since the Larmor frequency is the same for all 13Cspins. The environment therefore refocuses after aLarmor period and if the electron spin refocusingpulse is applied at this particular time t = 2p/WC(or a multiple thereof), the echo amplitude recovers[303, 166].

Defects with similar properties have also been iden-tified in SiC [304], although they have not beenequally well characterized as the NV defect in dia-mond.

12.1.11 Single-spin readout

A difficult problem in all spin-based quantum com-puter concepts is the readout of the result. Whilesome of the concepts try to simplify this task bycoding the qubits in ensembles of spins, it wouldbe preferable to be able to read out individual spin.Several successful single-spin measurements havebeen reported that were based on optical readout[162, 161, 163, 164], or scanning tunneling mi-croscopy [305, 306]. A number of different ap-proaches have been proposed [307, 308, 309].

The optical readout of spin is based on the opti-cal readout of electronic states, but the details arestrongly system-dependent. Early optical readoutexperiments concentrated on excited triplet states.Since the lifetime of the individual triplet states dif-fers, a resonance microwave field that exchanged

populations between them can “short-circuit" the de-cay of long-lived states. If a laser drives a transi-tion from the ground state to an excited singlet state,some of the molecules undergo inter-system cross-ing to the lower lying triplet state. Since its lifetimeis rather long, molecules get trapped in this state,thus reducing the ground state population. The ob-served fluorescence is a measure of the ground statepopulation. Resonant irradiation of triplet transitionschanges the fraction of spins in the electronic groundstate and is therefore observed as an increase in thefluorescence. Optical detection of fluorescence has,e.g., made it possible to perform and observe quan-tum gates on individual electronic and nuclear spinsin diamond, using optical excitation of a nitrogen-vacancy (N/V)-center [310, 165, 166, 311].

Another experimental approach to single-spin detec-tion uses a scanning tunneling microscope (STM)[305, 312, 306]. While the details of the experimentmust be considered unknown, it appears that the tun-neling current contains an oscillating component atthe Larmor frequency if the tip is placed over a para-magnetic molecule. The oscillating signal compo-nent is separated from the dc component and fed intoa microwave spectrum analyzer.

10 nm

Figure 12.26: Spatial distribution of STM-EPR sig-nal on graphite surface. The elevatedregions correspond to four adsorbedBDPA molecules. The right-hand partof the figure shows the STM-detectedEPR spectrum of TEMPO clusters.The three resonance lines are due tothe hyperfine interaction with the 14Nnuclear spin [306].

By setting the detection frequency to the EPR fre-quency, it is possible to map the spin density on thesurface. The example shown in Figure 12.26 repre-sents the signal from four organic radical molecules

213


(BDPA) that were deposited on a graphite surface[306]. The right-hand part shows the STM-detectedEPR signal from TEMPO molecules, another stableradical. In this case, the electron spin couples to thenuclear spin of the 14N nuclear spin. The hyperfineinteraction splits the EPR resonance into three reso-nance lines, corresponding to the three nuclear spinstates. A related technique is the mechanical detec-tion of magnetic resonance [313], which has beenshown to be capable of single-spin detection in suit-able systems [314].

Both techniques – optical and scanning probe mi-croscopy – allow for the detection of individual elec-tronic spins. While this is not a readout of the spinstate, it can be used as such if the spin being de-tected is not the qubit to be read out, but coupled tothe computational qubit: the coupling shifts the EPRfrequency, allowing one to detect the spin state ofthe computational qubit through the EPR frequencyof the readout qubit.

A difficulty of the optical readout is that the spa-tial resolution is limited by the optical wavelength.Near-field optical techniques reach better spatial res-olution, but their collection efficiency is too low forefficient readout of qubit states. STM-based systemsrequire mechanical motion, resulting in a slow read-out process. For an all solid state system, electronicreadout would provide the possibility to eliminateexternal optical and mechanical (STM) accessories.A possible approach is to use single electron tran-sistors (SET’s), in combination with spin-dependenttunneling processes [315, 316], but their viability forsingle-spin readout has still to be verified.

12.2 Superconducting systems

12.2.1 Basics

Qubits can in principle be implemented as harmonicoscillators. If we consider an LC oscillator, it can bedescribed classically by the differential equation

∂

2Q∂ t2 +

QLC

= 0.

Quantum mechanically, this corresponds to theHamiltonian

H =F2

2L+

Q2

2C= h̄w0(n+

12).

Qubits made from ordinary electrical circuits woulddecohere quickly owing to resistive power loss[317]. In superconductors at low temperature, how-ever, electrons bind into Cooper pairs that condenseinto a state with zero-resistance current and a well-defined phase. In superconducting circuits, the po-tential for the quantum variables of that Cooper-paircondensate may be changed by controlling macro-scopically defined inductances (L), capacitances (C),and so on, allowing the construction of qubits. Like-wise, the potential may be dynamically altered byelectrical signals to give complete quantum con-trol. These devices therefore resemble classicalhigh-speed integrated circuits and can be readily fab-ricated using existing technologies.

Typical parameters of these systems are dimensionson the order of ⇡ 10 µm, inductances of 10�10H, ca-pacitance C ⇡ 10�12F and resulting resonance fre-quencies w0 ⇡ 2p 16GHz. These parameters de-pend on the details of the manufacturing process.It is therefore important to control these parame-ters precisely, but also to analyze the actual de-vices and calibrate the required control fields. For(near-)dissipation-free operation, the devices mustbe cooled to <50 mK in a dilution refrigerator. Thethermal energy kBT must be small compared to thephoton energy h̄w0, which again must be small com-pared to the gap energy D of the superconductor:

kBT ⌧ h̄w0 ⌧ D.

12.2.2 Charge qubits

Superconducting materials owe their specific proper-ties to a liquid formed by Cooper pairs, i.e., pairs ofelectrons held together by a coupling to lattice vibra-tions. The pairs have zero total spin and are thereforeBosons that can occupy a single quantum state sub-ject to a simple effective Hamiltonian. As shown inFigure 12.27, typical qubit systems consist of a small“box" of superconducting material that is in contact

214


with a reservoir of Cooper pairs through a Joseph-son junction (i.e., a thin layer of insulating material)[318]. In addition, a control electrode can change theelectrostatic potential of the box.

Box

ReservoirTunnel junction

Control electrode Gate voltage

n n+1

Figure 12.27: Components of a superconductingqubit (left) and its lowest energy lev-els as a function of the gate voltage(right).

The Coulomb energy required to bring a single elec-tron with charge �e onto a neutral qubit island is

EC =e2

2(Cg +CJ),

where Cg and CJ are the capacitances to the controlelectrode and the reservoir. In addition to the mutualrepulsion of the electrons, the Coulomb energy de-pends on the potential applied through the controlelectrode. Since this energy contribution also de-pends on the net charge on the box, it is convenientto write the electrostatic part of the Hamiltonian as

H0 = 4EC(n�ng)2, (12.2)

where n is the number of excess Cooper pairs inthe box1 and ng = CgVg/2e parametrizes the controlvoltage. The control electrode therefore changes thenumber of excess Cooper pairs that makes the islandeffectively neutral.

The so-called charge qubits are defined by the num-ber n of excess Cooper pairs on the island. Eachn value yields one of the dashed parabolas in Fig-ure 12.27, showing the quadratic dependence on thecontrol voltage for each of the Cooper pair numbereigenstates |ni. These states are coupled by Cooperpair tunneling to the reservoir, represented by the

1It is assumed that the box contains no unpaired conductionelectrons. To suppress states with broken Cooper pairs, pa-rameters can be chosen such that the superconducting en-ergy gap D is the largest energy scale in the problem.

Josephson coupling energy EJ . Choosing states |niand |n+1i as the qubit states (and neglecting allother states), we can write an effective Hamiltonianfor the qubit as

H = 4EC

h̄(1�2ng)Sz � EJ

h̄Sx, (12.3)

where we have shifted the origin of the energy axis tothe mean of the two states. The pseudo-spin definedby the qubit therefore interacts with an adjustablemagnetic field along its z-axis that is defined by thecontrol electrode’s potential, plus an effective fieldalong the x-axis, which is determined by the Joseph-son splitting.

An obvious difficulty for this type of qubit is thatthe Hamiltonian is not diagonal in the chosen ba-sis: the transverse field, which is determined by thetunnel splitting, cannot be switched off. The con-trol voltage, which affects the longitudinal field, canbe used to apply gates, but the qubits are never ina completely quiet state where the information doesnot evolve. A way to circumvent this problem wassuggested by Makhlin et al. [319]: if the junctionto the reservoir is replaced by a loop with two junc-tions, the magnetic flux through this loop can adjustthe effective tunnel splitting.

12.2.3 Flux qubits

Rather than encoding the information in the chargedegrees of freedom of small superconducting is-lands, it is also possible to associate the qubit stateswith two states of distinct magnetic flux through asuperconducting ring [320]. Compared to the chargequbits, flux qubits should offer longer decoherencetimes, since they are not subject to electrostatic cou-plings to stray charges.

Figure 12.28 shows the basic element of a flux qubit,a superconducting ring with a Josephson junction.The energy of the system is

Hfl = �EJ cos✓

2p

FF0

◆+

(F�Fx)2

2L+

Q2

2CJ,

where EJ is the Josephson energy, F0 = h/2e is theflux quantum, Fx is an external flux bias, L the self-inductance of the loop, Q the charge, and CJ the

215


ΦΦ’ΦEj

Flux Φ

Figure 12.28: A simple flux qubit (left) consists ofa loop that includes a Josephson junc-tion. The second version allows con-trol of the Josephson energy by theflux F0. The total energy forms a dou-ble well potential as a function of theflux.

capacitance of the junction. The first term repre-sents the Josephson coupling energy of the junction,which is a periodic function of the flux F through theloop. The second term is the magnetic field energyof the flux, and the third the Coulomb energy of thecharge over the junction.

For suitable parameters, the total energy forms adouble well potential, as shown on the right-handside of Figure 12.28. The two minima correspondto the two basis states of the flux qubit, which arecoupled by the junction energy EJ . The longitudi-nal component of the effective magnetic field is nowdetermined by the external flux, while the transversecomponent depends on the junction energy. In closeanalogy to the charge qubit, it is again possible totune the junction energy by inserting a small loopand adjusting the flux through this control loop, asshown in the center of Figure 12.28.

12.2.4 Gate operations

As discussed above, the Hamiltonians that describethe charge as well as the flux qubits can be broughtinto the form of effective spin-1/2 systems, which areacted upon by effective magnetic fields. Dependingon the details of the implementation, the componentsof this effective field can be changed over a certainrange by suitable control parameters. Two differentapproaches have been used to implement gate oper-ations: the control parameters can be switched be-tween different values and left there at constant val-ues for the suitable duration, or they can be mod-

ulated to resonantly excite a transition between thebasis states.

Tuning the z-‘ eld’

Gate voltage

Ener

gy

Figure 12.29: Gate operation for a charge qubit.

If dc (unmodulated) pulses are used, the whole pro-cess of switching the control field on, letting the sys-tem evolve, and switching back, should be fast onthe timescale of the unperturbed evolution of the sys-tem. With dc pulses, a coherent superposition of thetwo states can be created by initialization of the sys-tem into the ground state and then suddenly pulsingthe control field to equalize the energy of the twostates [321]. Leaving them in the degenerate statesfor a quarter of the tunneling cycle time creates anequal superposition of the two states. This super-position remains if the control field is switched offsufficiently rapidly.

Occ

upat

ion

prob

abili

ty o

f |1>

Rabi pulse duration / ns

Figure 12.30: Resonant excitation of a super-conducting qubit showing Rabioscillations.

These very demanding requirements can be relaxedif resonant irradiation is used [322, 323]. The result-ing evolution is then exactly that of a spin-1/2 under

216


resonant irradiation.

Like in any other implementation, two-qubit gatesrequire a coupling between qubits. This can be im-plemented directly between qubits either through theCoulomb interaction between charges, which yieldsa coupling operator S j

zSkz , in the basis of eq. (12.3),

or through inductive coupling between flux states,which can be written in the form S j

ySky. For flux

qubits, gate operations can be implemented by suit-ably time-dependent bias currents [324]: Such two-qubit gates were demonstrated by Yamamoto et al.[325], Berkley et al. [326], and by Plantenberg et al.[327].

For larger systems, it may be advantageous not to usepairwise couplings, but rather to couple each qubitto a common degree of freedom, such as an LC os-cillator. The resulting system has a common “bus”qubit, in analogy to the trapped ions, where the mo-tion is used as a common bus qubit. Such a pro-cedure may simplify the coupling network and alsolower the amount of decoherence introduced into thesystem by the gate electrodes.

Apart from the systems discussed here, supercon-ducting qubits have also been implemented thatare intermediate between the charge and flux qubit.Choosing such an intermediate state allows one tooptimize, in particular, the decoherence by choosingthe basis states such that the effects of external noisesources are minimized.

While early implementations of superconductingqubits had short coherence times and low fidelityquantum gate operations, multi-qubit systems arenow possible that implement full quantum algo-rithms with useful fidelity. Fig. 12.31 shows, as anexample, an implementation of Grover’s algorithmusing a pair of superconducting qubits [328].

12.2.5 Readout

In most cases, readout should convert the popula-tions of the computational basis states into a mea-surable signal. For charge qubits, readout can beperformed by an SET, which is very sensitive tosmall changes in the electric field. For flux qubits,

Figure 12.31: Grover algorithm implemented with 2superconducting qubits.[328]

SQUIDs (superconducting quantum interference de-vices) represent the most sensitive detection device.An early experiment [321] used a probe electrodethat was coupled to the box by a tunnel junction,which provides an escape route for excess electronsin the box: if an excess Cooper pair is in the box, atunnel current is registered through the probe gate.This electrode was also used to initialize the sys-tem into the ground state. In this experiment, theelectrode was permanently coupled to the qubit box.The escape path for the electrons therefore presenteda significant contribution to the decay of the coher-ence in the system. Since the coupling is an efficientsource of decoherence for the system, it will have tobe switched off for an actual quantum informationprocessing device.

In the system displayed in Figure 12.32, Rabi oscil-lations have been initiated with an intense electricalfield pulse. While the readout is done on a single sys-tem, it represents an average over a large number ofpulse cycles. The measured quantity was thereforethe probe current, not the number of electrons. It isproportional to the probability of finding the qubitin the upper state, from where electrons can tunnelout into the probe electrode. The oscillation periodis given by the tunnel splitting, which can be tunedwith the flux f through the loop that includes the two

217


Figure 12.32: Signal from superconducting qubitundergoing Rabi oscillations as afunction of control charge [321].

tunnel junctions between the reservoir and the box.It agrees with the splitting that was measured by mi-crowave spectroscopy. At larger offsets, the cycleRabi frequency increases, but the oscillation ampli-tude decreases. To reduce noise, the experiment wasperformed at a temperature of 30 mK in a dilution re-frigerator. Coherent dynamics of a single flux qubithave also been observed by [329].

While these readout schemes are destructive (i.e.they change the state of the qubit), it is also pos-sible to read out the state nondestructively, using adispersive coupling [330, 331]. In these schemes,the coupling between the qubit and the readout sys-tem is such that the readout circuit changes the fre-quency (or phase) of the qubit, but not the popula-tions. These readout schemes can therefore be con-sidered as quantum non-demolition measurements[142, 143].

12.3 Semiconductor qubits

Semiconductor systems have been proposed veryearly for implementing quantum information pro-cessing [332]. One of their main attractions is thatthe technological requirements for building devicesthat are structured in the nanometer range have been

extremely well developed by the semiconductor in-dustry. Many of those technologies can be applieddirectly to QIP devices.

12.3.1 Materials

Semiconductor materials provide the richest set oftools for constructing qubits. Some of the proposedsolid state spin based implementations use semicon-ductor materials in some form and were discussedin Section 12.1. Here we concentrate on other sug-gested systems that do not use impurity spins for thequbit implementation.

The extensive use and associated technology basefor semiconductor materials in conventional elec-tronics is also one of the attractive features for quan-tum computing implementations: no other mate-rial base has a similar range of tools available, notonly for generating structures with dimensions inthe nanometer range, but also for adjusting materialproperties like conductivity, potential, bandgap etc.

Apart from the impurity spins discussed in Section12.1, semiconductor materials offer a range of ad-ditional possibilities for defining qubits. This in-cludes excitons, electron spins, nuclear spin, elec-tric charges, and more. Most of these systems, how-ever, have only been suggested as implementationsand only a few, if any of them, are likely to be im-plemented for more than one qubit.

While the group IV materials Si and Ge were mostlyused in implementations on the basis of impurityspins, III/V materials like GaAs are preferred formost of the other approaches. Using III/V materi-als is particularly important for implementations thatuse optical excitation or readout, which requires di-rect bandgap materials. In addition, the high elec-tron mobilities that can be reached in high-purity 2Delectron systems, promise slow decoherence.

12.3.2 Quantum Dots

One possible basis for semiconductor qubits arequantum dots, i.e., structures that confine electronsin three dimensions in such a way that the energies

218


become discrete. Typical sizes of these structuresrange from 5 to 50 nm.

Figure 12.33: Two coupled quantum dots as qubits;left: schematic representation; right:transmission electron micrograph;height of dots is 1–2 nm, dot sep-aration 4 nm, dot radius 8–12 nm[333, 334].

Quantum dots form spontaneously when some semi-conductor materials are deposited on a substrate witha different lattice constant, e.g., during the growth ofInAs on a GaAs substrate. The difference in latticeconstant implies that the material grown on top issignificantly strained. The elastic energy associatedwith this strain can be minimized if the layer growsnot as a film, but assembles into islands; this processis called Stranski–Krastanow growth.

Volmer-Weber Frank-van der Merwe

Stranski-Krastanov # Mono-layers

<1

1..2

>2

Figure 12.34: Growth modes of semiconductors dur-ing MBE.

Stopping the growth process at the right momentleaves an assembly of mesa-like structures behind,whose dimensions can be adjusted to match therange where quantum confinement effects are sig-nificant. If additional layers of GaAs and InAs aregrown over the quantum dots, the dots in the secondlayer tend to align with the existing dots. One hastherefore a good chance to obtain coupled dots, as inthe example shown in Figure 12.33.

Figure 12.35: Electrostatically defined quantum dotpair. [335]

Quantum dots can also be defined electrostatically.By varying the electrostatic potential, it is then pos-sible to define quantum states that correspond to dif-ferent numbers of electrons in each quantum dot anduse these states as computational basis states [335].

12.3.3 Excitons in quantum dots

The confinement of the electrons in the quantum dotsmakes the energy levels discrete, thus offering thepossibility of using them for encoding quantum in-formation. One possibility is to use excitonic states[336, 333], i.e., electron–hole pairs, which are cre-ated by the absorption of light. The energy Eex ofexcitons is determined by Eex = Eg � Eb, where Egis the bandgap and Eb the binding energy of theelectron–hole pair.

Dot 1 Dot 2

|00>

Dot 1 Dot 2

|01>

Dot 1 Dot 2

|10>

Dot 1 Dot 2

|11>

Physicalstate

Logicalstate

Figure 12.36: Possible encoding of two qubits by asingle electron–hole pair in two quan-tum dots. State |0i is identified withthe particle being in dot 1, state |1iwith the particle in dot 2.

Using an exciton in a pair of coupled qubits, quan-tum information may be encoded into the electronand hole being in one or the other quantum dot: iden-tifying the logical |0i with the left quantum dot, the

219


four states shown in Figure 12.36 correspond to |00i,|10i, |11i, and |01i, respectively. At a separationof 4–8 nm, the electron wave-functions of the twoquantum dots overlap, allowing electrons and holesto tunnel between them. The eigenstates are there-fore the symmetric and antisymmetric linear combi-nations that are observed in the photoluminescencespectrum.

The excitons are usually generated by a short laserpulse. For single quantum dots, this process can bemade coherent, as indicated by the observation ofRabi oscillations [337, 338]. Using the presence orabsence of an exciton in a single quantum dot as thequbit, Bianucci et al. demonstrated a single-qubitDeutsch-Jozsa algorithm [132]. If two excitons arepresent in the same quantum dot, their interaction al-lows one to implement two qubits. Gates can againbe performed by optical excitation, with differentfrequencies for the different transitions [339].

Readout of excitonic states is relatively straightfor-ward in principle: the electron–hole pairs recombineafter a time of the order of 1 ns [340], emitting a pho-ton that can be detected. The wavelength of the pho-ton indicates the state occupied by the particles be-fore their decay. Depending on the coding scheme,the eigenstates of the system, which determine thephoton wavelength, may not be the qubit states, but amodification of the algorithm could still make use ofthe information gained from the photoluminescencedata. Unfortunately, the recombination destroys thequantum information stored in the exciton and theprobability that the photon emitted by the electron–hole pair is subsequently detected is too low to al-low for reliable readout in a single event. Insteadof detecting an emitted photon, it is also possible toconvert the photo-excited electrons into free carriers,which can then be detected electrically [338].

12.3.4 Electron spin qubits

Using the spin degree of freedom rather than thecharge has two essential advantages. The Hilbertspace consist only of the two spin states, thus min-imizing any “leakage” of quantum information intoother states. Second, while the lifetime of an exci-ton is limited by radiative recombination to ⇡ 1 ns

[340], observed spin lifetimes have increased frommicroseconds [341] to milliseconds [342].

Compared to nuclear spins, electron spins offerstronger couplings to magnetic fields and thereforefaster gate operation, and they may be controlled byelectric fields also [343]. The advantages of elec-tron spins (fast gates) and nuclear spin (slow deco-herence) may also be combined by storing the infor-mation in nuclear spin and switching it into electronspins for processing [344].

Specific spin states of electrons in quantum dots canbe created either by optical excitation with circularlypolarized light or by spin injection [345, 346, 347,348] from magnetic materials. Manipulation of thespin states can be achieved either optically, by mi-crowave pulses, or electrically. In the case of op-tical excitation, one uses Raman pulses that coupleone of the qubit states to virtual states in the vicinityof trion2 states [349, 350]. If the Raman laser fieldis kept well off-resonance, it creates only little ex-cited state population and the associated decay rateremains small. Electrical excitation is possible if thequantum dot structures are defined by electrostaticpotentials. Modulation of the potentials then mod-ulates the tunnel splittings, which may be exploitedfor logical gate operations [351]. Coupling to themagnetic moment of the spin, it is also possible todrive the system by resonant microwave fields [352],in close analogy to NMR experiments.

The disadvantage of the III/V materials for spin-based qubits is that the natural abundance materialsall have nuclear spins with which the electron spininteracts via the hyperfine interaction

Hh f = Sz Âk

AkIkz ,

where we have assumed that the electron spin isquantized in a strong magnetic field k z. The sumruns over all nuclear spins Ik and the hyperfine cou-pling constant Ak is proportional to the electron den-sity at the location of the corresponding nucleus.While the interaction of the individual nuclear spinwith the electron is relatively weak, the number of

2A trion consists of an electron plus an exciton, i.e. two elec-trons and one hole in a bound state.

220


interacting nuclei is very large. As a result, the com-bined interaction of the nuclear spins within the en-velope of the electron wave function generates an ef-fective magnetic field BN ⇡ hÂk AkIk

z i. This “nuclearfield” adds to the Larmor precession of the electronspin with frequencies in the GHz range. Since theorientation of the nuclear spins is not constant intime, this effective field fluctuates and leads to a lossof coherence. This is a much smaller problem inSi, where the most abundant species (28Si) does nothave a nuclear spin and therefore does not interactwith the electron spin.

In contrast to silicon-based systems, where isotopi-cally enriched 28Si material is free of nuclear spins,GaAs has three nuclear isotopes with spin I=3/2.Electron spins therefore always are subject to hyper-fine interaction with the nuclei over which the elec-tron wavefunction extends. This interaction there-fore yields a significant contribution to the dephas-ing of electron spins in GaAs [353, 354].

12.3.5 Readout

Readout of single electronic spins presents a signifi-cant challenge.

3) Optical2) mini-SQUID1) Spin-SET

Figure 12.37: Some readout schemes forsemiconductor-based qubits.

The approaches that are currently investigated in-clude optical readout, similar as in the case of ex-citons, or electrical detection. In the case of opti-cal readout, the process can be amplified by drivinga transition where, after excitation, the system fallsback into the same initial state, thus allowing one toscatter many photons [355] and thus increasing thedetection probability. However, in quantum dots, thenumber of photons that can be scattered in this man-ner is much smaller than for free atoms. Using dis-persive optical detection, such as Kerr rotation mea-

surements [356], it is again possible to minimize thedisturbance of the electron spin.

Readout of electron spin qubits can be performedby converting the spin state to a charge state andusing single electron techniques for readout or op-tically, e.g. by Kerr rotation measurements [356].The conversion into a charge state can be performedby spin-dependent tunneling, which is then detected[357, 358]. Like in superconducting systems, read-out may be easier in intermediate systems that donot rely on individual spins, but on ensembles withpseudo-spin, such as “quantum hall droplets” [359].

Asymmetric QW

|1i|0i

Figure 12.38: Readout scheme for a spin-qubit in anasymmetric quantum well.[360]

Friesen proposed a readout scheme for electron spinqubits in SiGe heterostructures [360]. In the asym-metric quantum well, an excitation of the electronfrom the ground state of the well to the first ex-cited state shifts the charge laterally by an amountDy. This change of the electric charge distributioncan be detected, e.g., by a single electron transistor.The excitation can be driven resonantly, and sincethe frequency depends on the spin state, it can bemade spin-selective.

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