278 35 Cooper Pairs; Superconductors and Superfluids This means that below this temperature there is a component of the liquid which flows without energy dissipation. He 4 -atoms (2 protons, 2 neutrons, 2 electrons) as composite particles are bosons. Thus, even though there is a strong interaction between the bo- son particles, they can condense without pairing, and the transition to superfluidity consists in the formation of such a condensate. The case of He 3 is fundamentally different. This atom consists of two protons, only one neutron, and again two electrons; thus as a composite particle it is a fermion. Originally one did not expect superfluidity at all in He 3 . All the greater was the surprise when in 1972 Osheroff et al., [26], found that also a He 3 -liquid becomes a superfluid, but admittedly at much lower temperatures, i.e. below 2.6 mK. In fact at these low temperatures (not yet ultralow, see Part IV) also in this liquid Cooper pairs of He 3 - atoms are formed, where, however, this time the interaction favors triplet pairing, i.e. now the position factor of a Cooper pair is antisymmetric, e.g., with l = 1, and the spin function is symmetric, i.e., a triplet function.As a consequence, in the superfluid state of He 3 both the expectation value of ˆ L and that of ˆ S can be different from zero at the same time, which makes the theory very complex. In fact in 2003 a physics Nobel prize was awarded to Anthony Legget, 2 who was involved in the theory of He 3 -superfluidity. iii) Finally, some words on high-temperature superconductors.Thisisacer- tain class of non-metallic superconductors with many CuO 2 -planes, and values of T s around 120 K, i.e., six times higher than usual. In these sys- tems the electrons also seem to form pairs; the “elementary charge” of the carriers of superconductivity is again 2e, but the underlying mech- anism is not yet clear, although the systems have been under scrutiny since 1986/87. Perhaps one is dealing with Bose-Einstein condensation (see Part IV) of electron-pair aggregates 3 . The “phononic mechanism” of conventional metallic superconductivity (see above) is apparently re- placed here by electronic degrees of freedom themselves. Moreover it is also not accidental that the planar radius of the respective Cooper pairs is much smaller than usual. The electronic degrees of freedom also include antiferromagnetic spin cor- relations, which seem to play an essential role. Furthermore, the position- space factor seems to correspond to d x 2 −y 2 and/or d xy pairing, i.e., to attractive and repulsive interactions, respectively, at four alternating pla- nar axes distinguished by 90 ◦ . 2 Nobel prize winners in 2003: Alexej A. Abrikosov, Vitalij L. Ginzburg, Anthony J. Legget, see below. 3 There may be a relation to the crossover from Bardeen-Cooper-Schrieffer behav- ior of (weakly coupled) ultracold molecules of fermions to Bose-Einstein conden- sation of (strongly coupled) preformed fermion pairs, when a so-called Feshbach resonance is crossed. See PartIV. 36 On the Interpretation of Quantum Mechanics (Reality?, Locality?, Retardation?) 36.1 Einstein-Podolski-Rosen Experiments In a famous paper published in 1935 [29] Einstein, Podolski and Rosen dis- cussed the implications of quantum mechanics, and correctly pointed out several consequences that do not agree with common sense. For the following two decades in Princeton, Einstein and his group tried to obtain an “ac- ceptable” theory by augmenting quantum mechanics with so-called “hidden variables”. Only a few years after Einstein’s death, however, John Bell at CERN showed that this is not possible (see below). The phenomena related to the “unacceptable consequences” involve the notion of entangled states. In the paper of Einstein, Podolski and Rosen entangled states are constructed from eigenstates of the position and mo- mentum operators ˆx and ˆp x . However, it is more obvious to form entangled states from spin operators. For example, in the singlet state |S =0,M =0 of two interacting electrons, see (34.5), the spin states of the single particles are entangled, which means that the result cannot be written as a product of single-particle functions.Forthetriplet states, in (34.4), two of them, |S =1,M = ±1,arenot entangled, while the third state, |S =1,M =0,is as entangled as the singlet state. 1 Thus, if one has a two-fermion s-wave decay of a system (the “source”), with an original singlet state of that system, where two identical fermi- ons leave the source in diametrically opposed directions, and if one finds by measurements on these particles (each performed with a single-particle measuring apparatus at large distances from the source) a well-defined z- component ˆ S z of the first particle (e.g., the positive value + 2 ), then one always finds simultaneously(!), with a similar apparatus applied to the sec- ond diametrically-opposed particle, the negative value − 2 ,andvice versa,if quantum mechanics is correct (which it is, according to all experience; see Fig. 36.1). However, since the same singlet wave function |S =0,M =0diagonalizes not only the z-component of the total spin, 1 For three particles, there are two different classes of entangled states, viz |ψ 1 ∝ |↑↑↑+|↓↓↓ and |ψ 2 ∝↑↑↓ + ↓↑↑ + ↑↓↑,andforfourormoreparticles,there are even more classes, which have not yet been exploited. 280 36 On the Interpretation of Quantum Mechanics S z := S z (1) + S z (2) , but also (as can easily be shown, → exercises) the x-component S x := S x (1) + S x (2) , with the same eigenvalue ·M (= 0), identical statements can also be made for a measurement of ˆ S x instead of ˆ S z (this is true for a singlet state although the operators do not commute). These consequences of quantum mechanics seemed “unacceptable” to Ein- stein, since they violate three important postulates, which according to Ein- stein an “acceptable” theory should always satisfy: a) the postulate of reality, i.e., that even before the measurement the state of the system is already determined (i.e., it appears unacceptable that by figuratively switching a button on the measuring equipment from ˆ S z to ˆ S x the observed spins apparently switch from alignment (or anti- alignment) along the z-direction to alignment (or anti-alignment) along the x-direction; b) the locality of the measuring process (i.e., a measurement at position 1 should not determine another property at a different location 2), and c) retardation of the propagation of information (i.e., information from a po- sition 1 should reach a position 2 only after a finite time (as in electrody- namics, see Part II, with a minimum delay of Δt := |r 1 −r 2 | c ). Fig. 36.1. Singlet decay processes (schematically). A quantum system has decayed from a state without any angular momentum. We assume that the decay products are two particles, which travel diametrically away from each other in opposite di- rections, and which are always correlated despite their separation. In particular |ψ always has even symmetry w.r.t. exchange of the spatial positions of the two par- ticles, whereas the spin function is a singlet; i.e., if a measuring apparatus on the r.h.s. prepares a spin state with a value 2 in a certain direction, then a simultanous measurement on the opposite side, performed with a similar apparatus, prepares the value − 2 . (This is stressed by the small line segments below the r.h.s. and above the l.h.s. of the diagram, where we purposely do not present arrows in the z or x direction, since any direction is possible. Purposely also no origin has been drawn to stress that this correlation is always present.) 36.2 The Aharonov-Bohm Effect; Berry Phases 281 In contrast, in quantum mechanics the individual state of the system is determined (i.e., prepared) just by the measurement 2 . This concerns the problem of reality. Above all, however, quantum mechanics is non-local, i.e., |ψ already con- tains all the information for all places, so that no additional information must be transferred from place 1 to 2 or vice versa 3 . This simultaneously concerns problems 2 and 3. Of course, Einstein’s objections must be taken very seriously, the more so since they really pinpoint the basis of the formalism. The same applies to the efforts, although inappropriate after all (see below), to modify Schr¨odinger’s theory by adding hidden variables to obtain an acceptable theory (i.e. “real- istic”, “local”and “retarded”). Ultimately, nine years after Einstein’s death, it was shown by Bell, [30], a theorist at CERN, that the situation is more complex than Einstein assumed. Bell proved that Einstein’s “acceptability” postulates imply certain inequalities for the outcome of certain experiments (the so-called “Bell inequalities” for certain four-site correlations), which sig- nificantly differ from the prediction of quantum mechanics. As a consequence, predictions of either quantum mechanics or (alternatively) the quasi-classical “acceptability” postulates of Einstein, Podolski and Rosen, can now be veri- fied in well-defined experiments. Up to the present time many such Bell ex- periments have been performed, and quantum mechanics has always “won” the competition. 36.2 The Aharonov-Bohm Effect; Berry Phases In the following, both the non-locality of quantum mechanics and wave- particle duality appear explicitly. We shall consider a magnetized straight wire along the z-axis. Since we assume that the magnetization in the wire points in the z-direction, the magnetic induction outside the wire vanishes everywhere, B outside ≡ 0 . Thus, an electron moving outside should not take any notice of the wire, as it “passes by” according to the classical equation of motions m · dv dt = e · v ×B . 2 For example, a singlet two-spin function contains a coherent superposition of both cases (m s )(1) = +/2and(m s )(1) = −/2; they only become mutually exclusive by the (classical) measurement equipment. 3 It is perhaps not accidental that difficulties are encountered (e.g., one has to perform renormalizations) if one attempts to unify quantum mechanics with special relativity, which is a local theory (in contrast to general relativity,which is, however, not even renormalizable). 282 36 On the Interpretation of Quantum Mechanics However, quantum mechanically, electrons are represented by a wave func- tion ψ(r,t), i.e., due to this fact they also “see” the vector potential A(r) of the magnetic field, and this is different from zero outside the wire, giving rise to a nontrivial magnetic flux Φ 0 (see below), a gauge invariant quantity, which causes a visible interference effect. In detail: For a so-called “symmet- rical gauge” we have A ≡ e ϕ A ϕ (r ⊥ ) where the azimuthal component A ϕ (r ⊥ )= Φ 0 2πr ⊥ , and r ⊥ := x 2 + y 2 , where Φ 0 = BπR 2 is the magnetic flux through the cross-section of the wire. As already mentioned, the magnetic field vanishes identically outside of the wire, i.e., B z = d(r ⊥ ·A ϕ ) r ⊥ dr ⊥ =0. However, it is not B, but A, which enters the equation of motion for the quantum-mechanical probability amplitude i dψ dt = Hψ = ∇ i − eA 2 ψ/(2m) . Hence the probability amplitude is influenced by A, and finally one ob- serves interferences in the counting rate of electrons, which pass either side of the wire and enter a counter behind the wire. The results are gauge invariant, taking into account the set of three equa- tions (24.16). The decisive fact is that the closed-loop integral W A · dr = Φ 0 (=0), which determines the interference is gauge-invariant. The wave function ef- fectively takes the wire fully into account, including its interior, although the integration path W is completely outside. Aharonov-Bohm interferences have indeed been observed, e.g., by B¨orsch and coworkers, [27], in 1961, thus verifying that quantum mechanics is non- local, in contrast to classical mechanics. In this connection we should mention the general notion of a so-called Berry phase; this is the (position dependent) phase difference 4 that results in an experiment if the wave function not only depends on r but also on parameters α (e.g., on an inhomogeneous magnetic field), such that these parameters change adiabatically slowly along a closed loop. This formulation is consciously rather lax, since the concept of a Berry phase is based on general topological relations (e.g., parallel transport in manifolds), see [28]. 4 We remind ourselves that wavefunctions ψ(r), which differ only by a global complex factor, describe the same state. 36.3 Quantum Computing 283 36.3 Quantum Computing The subject of quantum computing is based on the fact that quantum- mechanical wave functions are superposable and can interfere with each other: ψ = ψ 1 + ψ 2 , |ψ| 2 = |ψ 1 | 2 + |ψ 2 | 2 +2·Re(ψ ∗ 1 · ψ 2 ) . 5 This is a field of research which has seen great progress in the last few years. Firstly we should recall that classical computing is based on binary digits; e.g., the decimal number “9” is given by 1 × 2 3 +1×2 0 , or the bit sequence 1001; as elementary “bits” one only has zero and unity; hence N-digit binary numbers are the vertices of a 2 N -dimensional hypercube. In contrast, quan- tum computing is based on the ability to superimpose quantum mechanical wave functions. One considers N-factor product states of the form |Ψ = N ν=1 c (ν) 0 ψ (ν) 0 + c (ν) 1 ψ (ν) 1 ; i.e., as in Heisenberg spin systems with S = 1 2 one uses the Hilbert space H N “2-level” , a (tensor) product of N “2-level systems”, where the orthonormalized basis states ψ (ν) 0 and ψ (ν) 1 are called “quantum bits” or qubits.Ofcourse|Ψ is, as usual, only defined up to a complex factor. The way that large amounts of computation velocity can be gained is illustrated by the following example. Assume that (i) the unitary operator ˆ U, i.e., a complex rotation in Hilbert- space, is applied to the initial state |Ψ 0 to generate the intermediate state |Ψ 1 = ˆ U|Ψ 0 , before (ii) another unitary operator ˆ V is applied to this intermediate state |Ψ 1 to generate the end state |Ψ 2 = ˆ V |Ψ 1 . Now, the unitary product operator ˆ W := ˆ V ˆ U 5 Several comprehensive reviews on quantum computing and quantum cryptogra- phy can be found in the November issue 2005 of the German “Physik Journal”. 284 36 On the Interpretation of Quantum Mechanics directly transforms the initial state |Ψ 0 into the final state |Ψ 2 . In general a classical computation of the product matrix ˆ W is very complex; it costs many additions, and (above all!) multiplication processes, since W i,k = j V i,j U j,k . In contrast, sequential execution by some kind of “experiment”, |Ψ 0 →|Ψ 1 →|Ψ 2 , may under certain circumstances be a relatively easy and fast computation for a skilled experimenter. One could well imagine that in this way in special cases (in particular for N →∞) even an exponentially difficult task (i.e., growing exponentially with the number of digits, N, or with the size of the system, L) can be transformed to a much less difficult problem, which does not grow exponentially but only polynomially with increasing N and L. In fact there are computation which are classically very difficult, e.g., the decomposition of a very large number into prime factors. One can easily see that 15 = 3 × 5and91=7×13; but even for a medium-sized number, e.g. 437, factorization is not easy, and for very large numbers the task, although systematically solvable, can take days, weeks, or months, even on modern computers. In contrast, the same problem treated by a quantum computer with a special algorithm (the Shor algorithm), tailored for such computers and this problem, would be solved in a much shorter time. This is by no means without importance in daily life: it touches on the basis of the encoding principles used by present-day personal computers for secure messages on the internet, i.e., so-called PGP encoding (PGP ˆ=“pretty good privacy”). According to this encoding, every user has two “keys”, one of which, the “public key” of the receiver, is used by the sender for encoding the message. This “public key” corresponds to the afore-mentioned “large number” Z. But for fast decoding of the message the receiver also needs a private key, which corresponds to the decomposition of Z into prime factors, and this key remains known only to the receiver 6 . A “spy”, knowing the computer algorithms involved and the “public key” of the receiver, e.g., his (or her) “large number”, can thus in principle calculate the corresponding “private key”, although this may take weeks or months and would not matter for a lot of short-term transactions by the receiver until the “spy” has finished his computation. But if the “spy” could use a “quantum computer”, this would be a different matter. 6 According to this so-called PGP-concept (PGP ˆ= “Pretty Good Privacy”) the private key is only “effectively private”, see below, similar to a very large integer being factorized by a finite-time computation. 36.4 2d Quantum Dots 285 Fortunately, even now (i.e., in our age of classical computers), quantum mechanics offers a totally different way of encoding, called quantum crypto- graphy (see below) which, in contrast to the classical encoding schemes is ab- solutely secure. However, because of the “coherency” requirement 7 of quan- tum mechanics the method suffers from the problem of range, so that at present it can only be applied over distances smaller than typically ∼ 10 to 100 km 8 . After these preliminary remarks on quantum cryptography, we return to quantum computing. There are other relevant examples where quantum computing would be much more effective than classical computing, e.g., in the task of sorting an extremely large set of data, where on a quantum computer the Grover algo- rithm would be much faster than comparable algorithms on classical com- puters. One may ask oneself why under these circumstances have quantum com- puters not yet been realized, even though intense work has been going on in this field for many years. The difficulties are hidden in the notion of “a skilled experimentalist”, used above, because it is necessary to ensure that, (i), dur- ing the preparation of the initial state |ψ 0 ; then, (ii), during the execution of all operations on this state; and finally, (iii), during all measurements of the results the coherence and superposability of all signals should be essentially undisturbed. This implies inter alia that errors (due to the necessary auto- matic correction) should only happen with an extremely small probability, e.g., < 10 −4 , and that all experiments performed in the course of a quantum computation should be extremely well-controlled. Additionally, the system should be “scalable” to N 1. Many different suggestions have indeed been made for producing a quan- tum computer, one of which will be outlined in the next section. 36.4 2d Quantum Dots Two-dimensional quantum dots can be regarded as artificial atoms with di- ameters in the region of ∼ 100 ˚ A(= 10 nm) in a two-dimensional electron (or hole) gas [“2DEG” (or “2DHG”)]. A two-dimensional electron gas is formed, e.g., at the planar interface between a GaAs-semiconductor region (for z<0) and an Al 1−x Ga x As-region (for z>0). At the interface there is a deflexion of the energy bands, and as a consequence an attractive potential trench V (x, y, z) forms, which is, however, attractive only w.r.t. the z-coordinate, 7 The coherency demands do not allow, for quantum mechanical purposes, the usual amplification of the signals, which is necessary for the transmission of electromagnetic signals over hundreds or thousands of kilometers. 8 A quantum cryptographical encoding/decoding software has been commercially available since the winter of 2003/2004. 286 36 On the Interpretation of Quantum Mechanics whereas parallel to the interface (in the x-andy-directions) the potential is constant. Although the electrons are thus in a bound state w.r.t. the z coordinate, they can move freely parallel to the interface, however, with an effective mass m ∗ that can be much smaller than the electron mass in vacuo, m e (e.g., m ∗ =0.067m e ). Similarly to the case of a transistor, where one can fine tune the current between the source and drain very sensitively via the gate voltage, it is ex- perimentally possible to generate a local depression region of the potential energy V (r). Often the local region of this depression is roughly circular and can be described by a parabolic confining potential, as for a 1d-harmonic os- cillator, but modified to 2d (the range of the confined region is x 2 + y 2 ≤ R 2 , with R ≈ 50 ˚ A(= 5 nm), and the magnitude of the depression is typically −3.5meV). As a consequence, in the resulting two-dimensional potential well one has a finite number of bound states of the “2DEG” (so-called confined electrons). For simplicity the confinement potential is (as already mentioned) usually described by a parabolic potential (see below). In this way one obtains a kind of artificial 2d-atom, or so-called 2d-quantum dot 9 , with a characteristic radius R in the region of 5 nm, which, on the one hand, is microscopically small, but conversely (in spite of the suggestive name “quantum dot”) very large compared to a normal atom, which has a diameter in the range of 0.1nm. The electrons of the 2d-quantum dot are described by the following single- particle Hamiltonian (where the spin is neglected, which makes sense 10 ): H = p 2 r ⊥ 2m ∗ + (p ϕ − e · A ϕ ) 2 2m ∗ + m ∗ ω 2 0 2 r 2 ⊥ , (36.1) where r ⊥ := x 2 + y 2 and ϕ =arctan y x are the usual planar polar coordinates. The azimuthal quantity A ϕ is the only component of our vector potential belonging to the constant magnetic induction B (= curlA 11 ),whichpointsinthez-direction. If more electrons are confined to the dot one must write down a sum of such terms, and additionally the Coulomb repulsion of the electrons and the Pauli principle must be taken into account. 9 There are also 3d-quantum dots. 10 Theeffectivemassm ∗ is much smaller than the free-electron mass m e ,which enters into the Bohr magneton μ B = μ 0 e 2m e . Thus in the considered “artificial atoms” the (orbital) second term on the r.h.s. of (36.1) dominates the spin term −g ∗ ·μ B ˆ S ·(B/μ 0 ). Here g ∗ = −0.44 is the effective Land´e factor of the system, and the other quantities have their usual meaning (see above). 11 We remind ourselves that A is not unique. 36.5 Interaction-free Quantum Measurement; “Which Path?” Experiments 287 An intermediate calculation results in an even simpler expression for the magnetic induction: with the ansatz A ϕ ≡ B 0 2 · r ⊥ and the formula B z = d(r ⊥ · A ϕ ) r ⊥ ·dr ⊥ (B r ⊥ and B ϕ vanish) , one obtains B ≡ e z ·B 0 . Again a product ansatz is helpful: ψ(r ⊥ ,ϕ) ! = R(r ⊥ ) · e imϕ . With this ansatz one obtains an explicit solution of the Schr¨odinger equation corresponding to (36.1). Due to the 2π-periodicity w.r.t. ϕ the variable m must of course be an integer. However, for B z = 0 one obtains only chiral cylindrical symmetry (i.e., m and −m are not equivalent). This implies that with B = 0 there is of course a preferred chirality, i.e., clockwise or counter- clockwise. The equations around (36.1) are known as Fock-Darwin theory. This yields one of the few explicit solutions of nontrivial quantum mechanical problems, and in 2001 there was even a set of written examination questions on this theory for teacher students (file 6 of U.K.’s exercises in winter 2003). Finally, from such artificial atoms (i) one can create artificial molecules; (ii) the spin state of these molecules can be calculated in a kind of Heitler- London theory, see above; and (iii) these states can be fine-tuned in a very controlled way as is necessary for quantum computing. In fact, one of the most promising suggestions of quantum computation is based on these systems. (In this context we should mention the theoretical work of Loss and DiVicenzo, [33], plus the experiments of Vandersypen and Kouwenhoven, [34].) 36.5 Interaction-free Quantum Measurement; “Which Path?” Experiments In principle the Aharonov-Bohm experiment can already be considered as a form of interaction-free quantum measurement, since the electron “mea- sures” the presence of the magnetized wire without coming into direct contact with it. But there is an indirect interaction via the vector potential, which is influenced by the wire. Interaction-free quantum measurements in another sense (see below) have been performed more recently by Anton Zeilinger’s group at Innsbruck 12 in 12 Now at the university of Vienna. . is (as already mentioned) usually described by a parabolic potential (see below). In this way one obtains a kind of artificial 2d-atom, or so-called 2d-quantum dot 9 , with a characteristic radius. We assume that the decay products are two particles, which travel diametrically away from each other in opposite di- rections, and which are always correlated despite their separation. In particular. , may under certain circumstances be a relatively easy and fast computation for a skilled experimenter. One could well imagine that in this way in special cases (in particular for N →∞) even an