Abstract
When we want to consider physical realizations of quantum information models and protocols, we have to identify specific (experimental) settings, which allow to implement the fundamental building blocks of quantum information processing. Thus, first we have to identify these building blocks and the requirements they imply for an experimental realization.
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Notes
- 1.
Or rather, of some single physical entity. It is indeed possible to encode the two-level system in more than one particle, forming a “logical qubit” [53].
- 2.
“Mesoscopic” systems [40] are usually understood as composite quantum systems which mediate between the microscopic and the macroscopic worlds, being sufficiently small for quantum effects to emerge. As a matter of fact, one of the origins of this research area is the continuing miniaturization of integrated circuitry used in traditional, “classical” computers, what implies that the quantum granularity of matter – manifest, e.g., through conductance fluctuations [17] across mesoscopic conductors, and usually assumed to be smoothed on the macroscopic scale, due to decoherence effects – has to be taken into account. Note that the line of thought is somewhat exactly opposite in our present context: Here, the question is not how much we have to reduce the size of a device to witness quantum effects, but rather whether there exists a fundamental size limit beyond which quantum effects cannot be observed anymore [2].
- 3.
The coupling to uncontrolled degrees of freedom implies decoherence, noise, and decay, the characteristic features of “open system dynamics.” For example, strong resonant driving of an ionic two-level system may induce residual, non-resonant coupling to a third atomic level. If the latter remains unobserved in the specific experimental setting, what is equivalent of tracing over this part of the ionic Hilbert space, decoherence is induced on the two-dimensional subspace. Such a mechanism is believed to limit the maximally achieved gate fidelity, e.g., in the ion trap experiments in Innsbruck [44]. Possible strategies to reduce such detrimental effects are error correction and decoherence-free subspaces [50, 30].
- 4.
That is why proposals of “silicon quantum computing” [28] earned a lot of attention.
- 5.
Exercise: Why/when is this a good assumption?
- 6.
Exercise: Convince yourselves that the norm \(\vert c_0\vert^2+\vert c_1\vert^2=1\) is conserved for all times by (10) and (11).
- 7.
Verify this, as an exercise
- 8.
Note that, for the terms given by (20) and (21), \(\sigma_{ij}^{(k)}\) also incorporates the “trivial” time dependence \(\exp\Big(\mp\, {{\rm i}}\, (\omega_0 - \omega)\,t\Big)\).
- 9.
This uniquely defines the initial state – why?
- 10.
Exercise: Derive this result with the above ansatz!
- 11.
“AC” for “alternating current,” since induced by an oscillating electromagnetic field.
- 12.
Exercise: Have a look at [12], and there at the sections on avoided level crossings. Interpret the AC Stark shift in terms of the quantities which characterize an avoided level crossing. In [12], the avoided crossing results from solving the stationary eigenvalue equation for the two-level Hamiltonian \(H=H_0+W\), with \(H_0=E_0\vert 0\rangle\langle 0\vert +E_1\vert 1\rangle\langle 1\vert\) and \(W=W_{00}\vert 0\rangle\langle 0\vert + W_{11}\vert 1\rangle\langle 1\vert +W_{10}\vert 1\rangle\langle 0\vert +W_{01}\vert 0\rangle\langle 1\vert \), when the eigenvalues \(E_{\pm}\) of H are plotted as a function of the detuning \(\delta =(E_1-E_0)/2\). For the identification of that treatment with our present problem, identify δ with our present definition of the detuning Δ, as the frequency mis-match between the driving field frequency and the driven atomic transition.
- 13.
Exercise: Derive this result.
- 14.
Mind the analogy of (39) with \(\mathrm{d}\boldsymbol{L}/\mathrm{d}t=\boldsymbol{\varOmega}\times\boldsymbol{L}\), \(\boldsymbol{L}\) the angular momentum, \(\boldsymbol{\varOmega}\) the precession frequency in the classical dynamics of rigid bodies (e.g., motion of a top) [34].
- 15.
- 16.
Exercise: Express the Bloch vector \(\boldsymbol{S}\), (36), in terms of the Pauli matrices.
- 17.
Here is a short reminder: Given the Schrödinger equation
$${{\rm i}}\hbar\vert\dot{\psi}\rangle =H\vert\psi\rangle\, ,\, \mathrm{with}\, H=H_0+V\, ,$$((52))the “trivial” time evolution (which we suppose to be known) generated by \(H_{0}\) (which we assume to be autonomous, i.e., time-independent) can be transformed away by defining
$$\vert\tilde{\psi}(t)\rangle =T^{-1}\vert\psi (t)\rangle , \mathrm{where} T=\mathrm e^{-{{\rm i}} H_0 t/\hbar} ;$$((53))$$\tilde{V}(t)=T^{-1}VT , \,\tilde{H_0}=H_0 .$$((54))The time evolution of \(\vert\tilde{\psi}\rangle\) is now given by
$${{\rm i}}\,\hbar\vert\dot{\tilde{\psi}}\rangle =\tilde{V}\vert\tilde{\psi}\rangle$$((55))and describes the time evolution induced by the perturbation V, relative to the unperturbed, trivial dynamics induced by \(H_{0}\).
- 18.
Which the keen reader should prove, as an exercise.
- 19.
See also our earlier discussion following (15) and (16). Once again, \(g\sqrt{n+1}\) fixes the typical system timescale, as will become evident hereafter.
- 20.
Note that this is a special variant of the secular approximation, which is ubiquitous, e.g., also in the derivation of master equations or in nonlinear resonance analysis in classical mechanics.
- 21.
Convince yourselves, as an exercise.
- 22.
The “Floquet picture” [19, 49, 41, 58], where the field is once again treated classically, allows a completely analogous treatment as the “dressed state picture” [14] expressed in (43) and (47), with (essentially) the same tensor structure \(\mathcal{H}_{\mathrm{atom}}\otimes\mathcal{H}_{\mathrm{field}}\) of the underlying Hilbert space. In particular, it defines the appropriate framework when dealing with more complicated atomic/molecular specimens. The essential approximation with respect to the dressed state picture is the neglect of the change of the field state upon emission/absorption of a photon by the atom. The Floquet picture naturally incorporates co- and anti-rotating terms (the latter are neglected in the Jaynes–Cummings picture). It therefore defines an excellent framework to assess the limitations of RWA, as well as of the two-level approximation. A very nice discussion of the contribution of the anti-rotating terms can be found in Sect. III.B of [49].
- 23.
Exercise: Why, in the final state detection with detectors \(\mathrm{D}_{\mathrm{e,g}}\), is it important to measure the population of the excited state before that of the ground state?
- 24.
The Q factor is proportional to \(\tau_{\mathrm{cav}}\) [43].
- 25.
Circular Rydberg states [27, 15, 21, 20] belong to the most “classical” eigenstates of single electron atoms: The electronic density is localized along an eccentricity zero classical Kepler trajectory and looks much like a swimming ring centered around the nucleus, in the plane defined by the quantization axis.
- 26.
Note that, in principle, the simple verification of these conditional probabilities does not prove entanglement generation. The same results would be expected for the state \(\rho = ({|u\rangle}{\langle u|}\otimes{|d\rangle}{\langle d|}+{|d\rangle}{\langle d|}\otimes{|u\rangle}{\langle u|})/2\) – which is separable. This is the reason why in a “Bell experiment” one has to perform the measurement in different bases (see chapter “Quantum Probability and Quantum Information Theory”, Sect. 6.4 and references therein). In the present experiment, the coherence between \({|u_1\rangle}\otimes{|d_2\rangle}\) and \({|d_1\rangle}\otimes{|u_2\rangle}\) was therefore explicitly verified in a second, complementary measurement.
- 27.
This Hamiltonian implicitly assumes the “Lamb–Dicke approximation” [10, 55], which requires that the ions be spatially localized to dimensions smaller than the addressing laser wavelength, i.e., that the position uncertainty \(\varDelta\!x\) of the center-of-mass wavefunction of the atom in the trap potential is much smaller than \(\lambda_{L}\), the laser wavelength.
- 28.
Note that this evolution already implements a CPHASE (conditional phase) gate.
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Acknowledgments
Partial support by VolkswagenStiftung is gratefully acknowledged. F. de Melo also acknowledges financial support by Alexander von Humboldt Foundation.
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de Melo, F., Buchleitner, A. (2010). Physical Realizations of Quantum Information. In: Benatti, F., Fannes, M., Floreanini, R., Petritis, D. (eds) Quantum Information, Computation and Cryptography. Lecture Notes in Physics, vol 808. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11914-9_8
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