Building an NMR Quantum Computer

Transcription

1 Buildig a NMR Quatum Computer Spi, the Ster-Gerlach Experimet, ad the Bloch Sphere Kevi Youg Berkeley Ceter for Quatum Iformatio ad Computatio, Uiversity of Califoria, Berkeley, CA 9470 Scalable ad Secure Systems Research ad Developmet, Sadia Natioal Laboratories, Livermore, CA (Dated: April 4, 01) I. INTRODUCTION For the past few weeks you have bee learig about the mathematical ad algorithmic backgroud of quatum computatio. Over the course of the ext couple of lectures, we ll discuss the physics of makig measuremets ad performig qubit operatios i a actual system. I particular we cosider uclear magetic resoace (NMR). Before we get there, though, we ll discuss a very famous experimet by Ster ad Gerlach. II. SPINS AS QUANTIZED MAGNETIC MOMENTS The quatum two level system is, i may ways, the simplest quatum system that displays iterestig behavior (this is a very subjective statemet!). Before divig ito the physics of two-level systems, a little o the history of the caoical example: electro spi. I 191, Ster proposed a experimet to distiguish betwee Larmor s classical theory of the atom ad Sommerfeld s quatum theory. Each theory predicted that the atom should have a magetic momet (i.e., it should act like a small bar maget). However, Larmor predicted that this magetic momet could be orieted alog ay directio i space, while Sommerfeld (with help from Bohr) predicted that the orietatio could oly be i oe of two directios (i this case, aliged or ati-aliged with a magetic field). Ster s idea was to use the fact that magetic momets experiece a liear force whe placed i a magetic field gradiet. To see this, ot that the potetial eergy of a magetic dipole i a magetic field is give by: U = µ B Here, µ is the vector idicatig the magitude ad directio of the magetic momet. The directio of the momet is aalogous to the orietatio of a bar maget. This expressio for the potetial eergy ca also be used to derive a force that acts o the dipole (dipole is just aother ame for somethig that posesses a magetic momet). Recall that the force is defied as the egative gradiet of the potetial: ( F = U = µ B ) Let s suppose that the magetic field looks like B = B 0 zẑ; this field does t satisfy Maxwell s equatios, but it makes the aalysis easier. We get a force, F = ( µ B) = (µ z Bz) = µb cos(θ) ẑ The cos θ term comes from the dot product, ad θ is the agle betwee the magetic field ad the magetic momet of the atom. If, for example, the dipole is iitially aliged with the field, it will experiece a upward force, ad if it is atialiged, it will experiece a dowward force. Now cosider a beam of dipoles passig through this field gradiet. Larmor s classical theory predicts that the dipole momet could poit i ay directio, so the beam would spread out homogeously. The Bohr-Sommerfeld Electroic address: kcyoug@berkeley.edu

2 θ μ B Figure 1: A magetic momet i a magetic field. The magetic field is idicated by the vector B, the magetic momet of the atom is give by µ, ad the agle betwee the two vectors is θ. theory, though, predicts that the dipole momet ca take oly two values, aliged or ati-aliged with the field, so the beam would be split ito two beams. I 19 Gerlach performed this experimet usig silver atoms. (It turs out that electros are a bad choice for this experimet because they are also affected by the Loretz force, which is proportioal to their velocity. Ay spread i the iitial velocity causes a spread i the output that overwhelms the spi-gradiet force.) Gerlach saw his beam split ito two distict beams, thereby demostratig the spatial quatizatio of the magetic momet ad falsifyig the Loretz theory. I a iterestig twist, the Sommerfeld theory was also icorrect, eve though it predicted the correct result of this experimet. I 195/196, Uhlebeck ad Goudspit postulated that the electro carried its ow spi magetic momet idepedet of its orbital agular mometum. I ay case, the Ster-Gerlach experimet provides a toy model that we ca use to lear about quatum two level systems. Let s make the laguage a but more precise by labelig some thigs: Ove + - Figure : Schematic of Ster-Gerlach device. I this diagram we have a cartoo picture of the Ster-Gerlach device. A ove produces a beam of particles which eters a regio with a ihomogeous magetic field, that gradiet of which poits i the ˆ directio. The two beams that emerge we label ˆ+ ad ˆ. These symbols are what we use to label a quatum state, ad what is writte iside gives some iformatio about this particular state. However, we should stress that what we write iside the ket is simply a label that we give to help us remember how the state behaves. We could have just as easily called the two states Bob ad Alice. Here, though, ˆ+ deotes the state that is directed upwards whe the field gradiet is alog the vector ˆ, ad ˆ is the state directed dowwards. Ove Figure 3: Cascaded Ster-Gerlach devices. This all becomes much more iterestig if we cosider multiple, cascaded Ster-Gerlach devices. Let s add aother idetical device, which we ll deote as (ˆ), after the first, but we ll discard the ˆ state. Notice that if we measure the output of the first (ˆ) with a secod (ˆ) the we oly get oe beam out, the

3 3 Ove + - m m + m - Figure 4: Cascaded Ster-Gerlach devices. ˆ+ state agai. This should t be surprisig, as we have established with the first device that the dipole momet poits alog ˆ. But what would happe if we rotated the device? Now we get two beams! The probability that ˆ+ ˆm± is foud experimetally to be ad also P ( ˆ+ ˆm+ ) = 1 (1 + ˆ ˆm), P ( ˆ+ ˆm ) = 1 (1 ˆ ˆm). These probabilities ca also be cosidered as the relative itestates of the two outgoig beams, ˆm±, give a icomig beam ˆ+. Our task ow is to seek a quatum mechaical descriptio of this experimet. Oe way to do this is to search for the simplest descriptio we ca come up with, addig complexity oly as we eed to. Because the result of ay measuremet we ca do is either aliged or atialiged, the simplest model we ca try is that the Hilbert space is two-dimesioal. Sice we ca make a measuremet for which the states ˆ± are eigestates (measuremet with a device i the ˆ directio) with differet eigevalues (aliged or atialiged), they must be orthogoal states. Because we have two orthogoal states i a two-level space, these states ca form a basis. The states ˆm± the form a differet basis. Let s pick a special basis, ẑ± ad express all other states as liear combiatios of these vectors. To idicate that this basis is special, we relabel the vectors: ẑ+ 0 ẑ 1 We ca ow represet ay other state as a liear combiatio of the { 0, 1 } states: ˆ+ = α 0 + β 1, where α ad β are complex umbers. Now that we kow, roughly, how to express our states, we eed to figure out what the α ad β are. We ca start by cosiderig the followig double Ster-Gerlach experimet. Ove + - z 0 1 Figure 5: Cascaded Ster-Gerlach devices. So, give the iitial state ˆ+, the probability of measurig 0 is P 0 (ˆ+) = 0 ˆ+ = α β 0 1 = α = 1 (1 + ˆ ẑ) We have used that fact that 0 1 = 0 ad 0 0 = 1. To make this a bit icer, we are goig to rewrite ˆ ẑ = cos θ,

4 where θ is the agle betwee the two uit vectors ˆ ad ẑ. This agle is also equal to the spherical coordiate, θ, defied by the uit vector ˆ, which, i cartesia coordiates, becomes: ˆ = (si θ cos φ, si θ si φ, cos θ) Applyig the trig idetity, (1 + cos θ)/ = cos (θ/), we ca say that: α = cos(θ/) A similar aalysis shows that β = si(θ/). So this simple argumet has give us the magitudes of α ad β, but what about their phases? Because they are complex, they ca be writte as: 4 α = α e iψ β = β e iχ givig, ˆ+ = α e iψ 0 + β e iχ 1 = cos(θ/)e iψ 0 + si(θ/)e iχ 1 However, a quatum state is oly defied up to a overall phase, so we ca multiply this state by e iψ to get ˆ+ = cos(θ/) 0 + si(θ/)e i(χ ψ) 1 cos(θ/) 0 + si(θ/)e iφ 1, where φ = χ ψ is the relevat phase. So what is φ for a give ˆ+? Let s look at ˆx+ ad ŷ+ : ˆx+ = e iφx 1 ŷ+ = e iφy 1 If a double device is set up, with ˆx first, the ŷ, the probability of seeig ŷ+ give that the first beam emerged i ˆx+, is P (x y) = (1 + ˆx ŷ)/ = 1/. But this is also the overlap betwee the two states: P (ˆx ŷ) = ˆx + ŷ+ = ei(φy φx) 1 1 = cos(φ y φ x ) = 1 So, cos(φ y φ x ) = 0 φ y φ x = π/. This implies that we ca associate the phase agle, φ, with the secod spherical coordiate, φ, i: ˆ = (si θ cos φ, si θ si φ, cos θ) So, for ay vector ˆ, ˆ+ = cos(θ/) 0 + si(θ/)e iφ 1, where (θ, φ) are the polar coordiates of the vector ˆ. But what about ˆ? The states ˆ+ ad ˆ must be orthogoal. This gives a uique solutio, up to the stadard quatum mechaical phase, But this is the same as ˆ = si(θ/) 0 + cos(θ/)e iφ 1 ( ˆ)+ = cos((π θ)/) 0 + si((π θ)/)e iφ 1 = ˆ So, ˆ+ is orthogoal to ( ˆ)+! This represetatio we have bee usig (θ, φ as parameters for a qubit state) is kow as the Bloch Sphere represetatio. Every poit o the Bloch Sphere (a uit sphere i R 3 ) correspods to a uique state, with the orthogoal state beig represeted by the atipodal poit o the sphere.

5 5 z ψ φ y x Figure 6: Bloch sphere represetatio of a qubit state. We ll see a lot more of the Bloch Sphere represetatio over the ext few lectures, so you ll have time to get accustomed to it.