Concepción clásica del derecho como ideología, dispositivo o instrumento de

All physical actions corresponds to positive maps, because they transform density operators into density operators, being always positive.

A linear map  ∈ M (H, H0_{) is a positive map (P M ) if for all ˆρ ≥ 0 ∈ A(H), (ˆρ) ≥ 0.}

For instance, the transposition is a P M .

A more general extension reads as follows. Consider a linear map  ∈ M (H, H0) and another Hilbert spaceH00, we define the extension of  as the linear map  ⊗ I ∈

M (H ⊗ H00,H0⊗H00): ( ⊗ I) X k Ak⊗ Bk ! =X k (Ak) ⊗ Bk (1.69) where Ak ∈ A(H) and Bk ∈ A(H 00

). In addition, we say that, a linear map is a completely positive map (CP M ) (all CP M corresponds to a physical action), if all the extensions are P M . For entangled states this is crucial, because we must ensure that changes made on the system will preserve the positive character of each subsystem.

Not all P M are CP M , for example the partial transpose. Such maps are particu- larly useful for detecting entanglement as we will exemplify below. Let us consider the Bell state ˆρ = |Φ+i hΦ+_{| (an entangled state)}

ˆ ρ = 1

2(|00i h00| + |00i h11| + |11i h00| + |11i h11|) (1.70)

the transposition only of the system A give us: ˆ

ρTA ₌ 1

2(|00i h00| + |10i h01| + |01i h10| + |11i h11|) (1.71)

where, the state ˆρTA _{has negative eigenvalues. The Peres-Horodecki partial trans-}

position criterion is based on the above idea. It reads as following : a density matrix ˆ

ρ ∈ A(C2⊗ CN_{) with N ≤ 3 is separable. If and only if ˆρ}T1,2 _{≥ 0 (T}

1,2 stands for

the partial transpose for the first or second subsystem).

In this thesis, we will consider the negativity as the main entanglement measures, as we will eventually also compute the same entanglement for an open system, and negativity is a measure also valid for that case. This quantity is based on the Peres- Horodecki criterion, as ˆρT1,2 _{will have negative eigenvalues for entangled states, we}

compute the degree of negativity as follows [Horodecki2009,Vidal2002]: N (t) =X

i

(|λi| − λi), (1.72)

where the λiare the eigenvalues of the partially transposed bipartite system at fixed

Chapter 2

Entanglement Stabilization in a

Non-Linear Qubit-Oscillator

System

In this chapter we will study a quartic (undriven) non-linear quantum mechanical oscillator coupled to a spin qubit. To elaborate upon our results, we will present a summary on the conditioned displacement Hamiltonian, i.e., when the position of the oscillator is monitored/conditioned by each spin qubit component.

Firstly, in order to produce an experimentally operational regime of the qubit-oscillator coupling, we will introduce several feasible architectures on hybrid qubit-oscillator (linear) systems under this type of interaction. Thereafter, we will briefly present some typical features on quartic non-linear classical oscillators otherwise known as the Duffing equation. We will also highlight both important features of its quantum counterpart and possible feasible implementations. Subsequently, we will investigate the main results of this study, which is the entanglement dynamics between a qubit coupled to a quartic non-linear mechanical oscillator. Lastly, we explore the entan- glement witnessing of the non-linear system, and how the Bell’s inequality might be violated under certain restrictions, resulting in a full benchmark of the hybrid quantum entanglement achieved.

The next section regarding two-level systems coupled to mechanics in solid-state physics is based on the academic paper “Hybrid mechanical systems” [Treutlein2015] where the reader can find a more detailed analysis.

2.1

Hybrid Linear Quantum-Oscillator Systems

Over the last few decades, the interaction between micro and nano-mechanical res- onators in the quantum regime and quantum two-level systems have shown remark- able milestones. For example, a macroscopic mechanical oscillator can be driven into a superposition of spatially separated states, which is a result interesting in itself. Furthermore, qubit-oscillator systems have been proven to be of paramount importance in other applications, such as fast, ultrasensitive force and displace- ment detectors [Blencowe2000], electrometers [Cleland1998, Erbe2001], and radio frequency signal processors [Wang2000]. In addition, the bare motion of the me- chanical oscillator can be used as a sensitive probe to extract information regarding static and dynamical properties of the microscopic quantum system. As we will study in Chapter 4, quantum mechanical oscillators can be used as suitable candi- dates for light-matter transducers.

In general, qubit-oscillator systems are quite topical in the quantum information field, and have been studied in different operational regimes under several types of interactions. For the purpose of this study we will just focus on a particular type of interaction, namely conditioned displacement Hamiltonian. Here, the interaction is given by ˆσz ⊗ ˆx, where ˆσz is the Pauli-z operator, and ˆx is the deviation of the

oscillator from its equilibrium position. Hence, the relevant Hamiltonian in the Schr¨odinger picture reads as:

ˆ H = ˆHq+ 1 2mpˆ 2₊1 2mω 2 mxˆ2− ~λˆσzxˆ (2.1)

in the above equation (~ is the Planck constant), ˆHqis the qubit Hamiltonian. The

single-mode mechanical oscillator is characterized by its position ˆx, momentum ˆp, mass m, and angular frequency ωm; λ is the qubit-mechanics coupling strength.

Using the expressions in 1.19, we can easily rewrite the Hamiltonian as: ˆ

H = ˆHq+ ~ωmˆb†ˆb − ~λˆσzx.ˆ (2.2)

Eq.2.2can be implemented under several schemes (see Fig.2.1for a list of solid-state qubits coupled to mechanical resonators). In particular, the linear qubit-oscillator interaction in solid-state systems can be justified due to the strong dependence on the local electrostatic and magnetic field with the qubit. Hence, the gap energy (Eeg) between the excited (e) and the ground (g) state of the qubit can be expanded

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in Taylor series as : Eeg(ˆx) = Eeg0 + ∂xEegx + 1/2 × ∂ˆ x2Eegxˆ2 + . . ., being ˆx =

xzpf× (ˆb†+ ˆb), as the zero-point fluctuation xzpf ≈ 10−13m is much smaller than

other relevant dimensions in the system, the linear approximation is enough to validate the Hamiltonian in Eq. 2.2.

Figure 2.1: Four architectures to couple a hybrid qubit-oscillator system via conditioned displacement Hamiltonian. a) A micromechanical resonator capaci- tively coupled to a quantized charge, b) a qubit encoded in circulating currents in a superconducting loop interacts with an arm of the loop, c) an electronic spin coupled to a quantized motion of a magnetized resonator tip, and d) a deformation potential coupled to quantum dots (a qubit embedded in the material resonator bar). As demonstrated later, the last two schemes have been found to be appro- priate candidates to include non-linearities. This figure was taken from the

paper review in Ref. [Treutlein2015].

Now, let us briefly explore the physical schemes shown in Fig. 2.1 to give an order of magnitude of the qubit-oscillator coupling parameter λ.

Starting with the micromechanical resonator capacitively coupled to a quantized charge shown in Fig.2.1-a, for instance, it could be a Cooper-pair box or an electron on a quantum dot. In particular, Cooper boxes have attracted much attention since they can mimic a controllable two-level system [Makhlin2001,Echternach2001], be- ing a perfect qubit candidate. Essentially, Cooper boxes are composed of a small su- perconducting island weakly coupled to a superconducting reservoir [Bouchiat1998,

Makhlin2001], characterized by Coulomb charging energy, and the strength of the Cooper-pair tunneling between the island and the reservoir.

When the Cooper box interacts with a cantilever (mechanical oscillator) it causes displacement in the cantilever, whose sign depends on which of the two charge states the Cooper box is in. As it is possible to prepare the Cooper box in a superposition of charge states, it leads to an entangled state between the Cooper box and the oscillator, being in a superposition of spatially separated states. The macroscopicity of this superposition depends on how strong the coupling strength

is. The Hamiltonian corresponds to: ˆ H = 4ECδnˆσz− 1 2EJσˆx+ ~ωmˆb †_{ˆb + λ} cbσˆz(ˆb + ˆb†), (2.3)

where δn = ng− (n + 1/2) with ng = −(CgcVgc+ CgmVgm)/2e the dimensionless, total

gate charge. The control gate voltage Vc

g and cantilever gate electrode voltage Vgm

ranges are restricted such that −1/2 ≤ δn ≤ 1/2 for some chosen n, so that only Cooper charge states |ni ≡ |−i ≡ 1₀
and |n + 1i ≡ |+i ≡ 0

1
play a role. Thus it

is natural to use spin notation where ˆσx and ˆσz are the usual Pauli matrices. The

coupling constant between the box and cantilever electrode is λcb = −4ECnmg ∆xzpf

d ,

where nm_g = −C_gmV_gm/2e, ∆xzpf is the zero-point fluctuation displacement uncer-

tainty of the cantilever, and d  ∆xzpf is the cantilever electrode-island gap. Only

the in-plane fundamental flexural mode of the cantilever, with frequency ωm and

operators ˆb and ˆb†, is taken into account. All other modes have a much weaker coupling to the box and will be neglected. We assume that the Josephson junction capacitance CJ  Cgcand Cgm, so that the charging energy of the box EC ≈ e2/2CJ.

Some of the values used in Ref. [Armour2002] are EC = 150µeV, nmg ≈ −63Vgm,

∆xzpf ≈ 1.4 × 10−13m, d = 0.1µm. λcb/ωm≈ 0.14 − 0.41, where ωm = 50 MHz.

In addition, a qubit system can alternatively be encoded in clockwise and anti- clockwise circulating currents in a superconducting loop as shown in Fig. 2.1-b. The coupling for this scheme is ~λc ≈ B0IqlxZPF. Some typical values are B0 ≤

10mT for the perpendicular magnetic field to the bending motion, Iq ≈ 100 nA

for the circulating current, and l ≈ 5µm for the length of the resonator. Hence, λc/2π ≈ 0.1 − 1MHz.

Qubit-oscillator schemes can also be prepared in systems where an electronic spin qubit is coupled to a quantized motion of a magnetized resonator tip as shown in Fig. 2.1-c. The spin qubit can be associated with a nitrogen-vacancy (NV) impu- rity in diamond and the nano-mechanical resonator to a cantilever of fundamental frequency bending mode ωr and dimensions (l, w, t) being its length, width, and

thickness, respectively. NV centers in diamond are essentially a lattice defect con- sisting of a nitrogen atom adjacent to a missing carbon, resulting in a defect that might occur naturally or with ion implantation. The long spin coherence time (≈ 20ms), scalability, and precision control (initialized - control - readout) make a perfect candidate for several quantum applications. As a result of the high sen- sitivity of the spin qubit to the magnetic tip motion, the coherent spin-motion

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exchange excitations can lead both to ground state cooling, as well as macroscopic quantum superposition states of the oscillator [Rabl2009], and also to quantum spin transducer [Rabl2010]

As the magnetic tip produces a field ≈ Gmz = Gˆ mz0(ˆb + ˆb†) (ˆb is the boson annihi-

lation operator for the oscillator), the Hamiltonian reads as: ˆ

H = ˆHN V + ~ωrˆb†ˆb + ~λmσˆz(ˆb + ˆb†), (2.4)

where ˆHN V =P_i=±1−~∆i|ii hi| + ~Ωi/2(|0i hi| + |ii h0|) is the spin qubit Hamilto-

nian, being ∆±and Ω±the detunings and the Rabi frequencies of the two microwave

transitions [Rabl2009].

The coupling strength λm = gsµBGmz0, where gs ≈ 2, µB is the Bohr magneton,

Gm the magnetic field gradient and z0 is the zero-point fluctuation. As an example,

we will refer to Ref. [Rabl2009], where the authors consider a Si-cantilever of dimensions (l, w, t) = (3, 0.05, 0.05)µm with a fundamental frequency of ωr ≈ 7

MHz and a z0 ≈ 5 × 10−13m. A magnetic tip of size ≈ 100 nm and a magnetic

gradient of Gm ≈ 7.8 × 106 T/m at a distance h ≈ 25 nm away from the tip and

results in a coupling strength λm/2π ≈ 115 kHz.

In Ref. [Kolkowitz2012] the NV centers are implanted ∼ 5 nm below the surface of a bulk diamond sample. The spin sublevels |ms= 0i and |ms= ±1i of the

electronic ground state exhibit a zero-field splitting of ∼ 2.87GHz. In the presence of a static magnetic field, the degeneracy between |+1i and |−1i is lifted, allowing us to selectively address the |0i → |1i transition with microwave radiation.

Finally, in Fig. 2.1-d the qubit (for instance, quantum dots or defect centers) can be coupled by changing the local lattice configuration of the host material. This deformation is illustrated in where flexural vibrations of the resonator induce a local stress σ ∼ z0xzpf/l2, where l is the resonator length and z0 is the the distance of

the defect from the middle of the beam. The associated level shift is then:

~λl ≈ (De− Dg)z0xzpf/l2 (2.5)

where De and Dg are deformation for the ground and excited electronic states. For