Igor Pikovski Experim rimental S l Searc rch for or Quant ntum - - PowerPoint PPT Presentation

Experim rimental S l Searc rch for

• r Quant

ntum um G Gravity SISSA SA/ISA ISAS, S, Tri ries este, , Ita taly

Igor Pikovski

01.09.2014

High-energy scattering experiments Astrophysics and cosmology Novel systems, that allow for precision measurement in a „quantum gravtitational“ paramter regime? High-precision quantum metrology

• Measurable effects of classical gravity in

quantum mechanics

• Time dilation in quantum mechanics
• Universal decoherence due to gravitational

time dilation

Quantum mechanics

• n fixed background

space-time

• Pulsed quantum opto-mechanics
• Opto-mechanical scheme to experimentally

test possible quantum gravitational deformations of the center-of-mass canonical commutator

• I. Pikovski, M. Vanner, M. Aspelmeyer, M. S. Kim, Č. Brukner.

Pr Probing ng Pl Planc nck-Scale ale P Phys ysic ics wit ith Quant ntum Optics cs. . Nature Physics 8, 393 (2012); arxiv:1111.1979.

• I. Pikovski, M. Zych, F. Costa, Č. Brukner.

Univ iversal al Decp ecpherence ce due to Grav ravit itat atio ional al Tim ime Dilat ilatio ion. arXiv:1311.1095 (2013).

Quantum gravity phenomenology

Outline

Yes! Earth’s gravity affects matter waves.

|𝜔⟩ = 1 2 |𝜔𝑒𝑒𝑒𝑒⟩ + 𝑓−𝑗Δ𝜚 |𝜔𝑣𝑣⟩

Δ𝜚 = 𝑛𝑛𝑛𝑛/ℏ

𝐼𝑗𝑗𝑗 = 𝑛𝑛𝑛

Tested with:

• Neutron interferometry

e.g. R. Colella, A. W. Overhauser, S. A. Werner, PRL 34, 1472-1474 (1975)

• Atomic fountains

e.g. H. Müller, A. Peters, S. A. Chu, Nature 463, 926-929 (2010)

Aharonov-Bohm-type phase due to the Newtonian gravitational potential: Δ𝜚 = 𝑛𝑛𝑛𝑛/ℏ Our w r work rk: I Incorp rporate t tim ime d dila ilation in into desc scription o

• f QM syst

systems

Time dilation in QM

𝑣𝜈𝑣𝜈 = − 𝐼𝑠𝑠𝑠𝑗

2

𝑑2 , 𝑗ℏ 𝜖 𝜖𝜖 |𝜔⟩ = 𝐼𝑠𝑠𝑠𝑗|𝜔⟩ 𝑑𝑣0 = 𝐼

• low-energy limit:

Hamiltonian in the weak-field limit 𝑃(𝑑−2):

𝑗ℏ 𝜖 𝜖𝑛 |𝜔⟩ = 𝐼0 + 𝑛𝑑2 + 𝑣2 2𝑛 + 𝑛Φ 𝑛 + 𝑛Φ2 𝑛 2𝑑2 − 𝑣4 8𝑛3𝑑2 + Φ 𝑛 𝑑2 − 𝑣2 2𝑛2𝑑2 𝐼0 |𝜔⟩

𝐼𝑠𝑠𝑠𝑗 = 𝐼0 + 𝑛𝑑2

remaining static part internal dynamics Coupling between internal and external d.o.f.

QM on fixed (classical) background space- time with time dilation 𝐼𝑗𝑗𝑗 = 𝑛𝑛 𝑑2 𝐼0 Gravitational part of interaction with Φ 𝑛 = 𝑛𝑛:

= 𝐼 𝑣𝑗, 𝑛𝜈𝜈, 𝐼𝑠𝑠𝑠𝑗 = −𝑛00 𝐼𝑠𝑠𝑠𝑗

2

+ 𝑛𝑗𝑗𝑑2𝑣𝑗𝑣𝑗

Time dilation in QM

• composite systems:

Drop in interference visibility! Time dilation: |𝜔⟩ = 1 2 |𝜔𝑒𝑒𝑒𝑒⟩|𝐷𝑒𝑒𝑒𝑒⟩ + 𝑓−𝑗Δ𝜚 |𝜔𝑣𝑣⟩|𝐷𝑣𝑣⟩

𝐷𝑒𝑒𝑒𝑒 𝐷𝑣𝑣 < 1

Classically: General relativity entangles any clock to the path due to gravitational time dilation.

𝐼 ≈ 𝑛𝑛𝑛 + 𝐼0 + 𝑛𝑛 𝑑2 𝐼0

Experimental implications:

• Matter wave interferometry with additional internal clock-states |𝐷⟩

(e.g. |𝐷⟩ = |𝑛⟩ + |𝑓⟩) (M. Zych, F. Costa, I. Pikovski, Č. Brukner. Nature comm. 2, 505 (2011)) Quantum mechanically:

Time dilation decoherence

• Shapiro delay: Photons slowed down by gravity

(M. Zych, F. Costa, I. Pikovski, T.C. Ralph, Č. Brukner.

• Class. Quant. Grav. 29, 224010 (2012))

Arbitrary composite system in Earth‘s gravitational field.

𝐼0 = 𝑒𝑗ℏ𝜕𝑗

𝑂 𝑗=1

𝐼𝑗𝑗𝑗 = 𝑛𝑛𝑛 + ℏ𝑛𝑛 𝑑2 𝑒𝑗𝜕𝑗

𝑂 𝑗=1

𝜍𝑗 = 1 𝜌𝑒 𝑗 𝑒2𝛽𝑗 𝑓− 𝛽𝑗 2 𝑗

𝑗 ⁄ |𝛽𝑗⟩⟨𝛽𝑗|

Each constituent in equilibrium at temperature T: 𝑒

𝑗 = (𝑓ℏ𝜕𝑗 𝑙𝐶𝑈

⁄

− 1)−1

GR time dilation induces interaction with center-of-mass position x :

Time dilation decoherence

𝑛 Simple model: Particle has N internal harmonic oscillators:

𝜖𝑒𝑠𝑒 = 2 𝑂

• ℏ𝑑2

𝑙𝐶𝑈𝑛Δ𝑛

• Universal for all composite systems
• Gaussian decay of quantum coherence (for t ≪

𝑂𝜖𝑒𝑠𝑒)

• Decoherence mediated by time dilation, depends on internal

composition

• Relativistic, thermodynamic and quantum mechanical effect
• Regular quantum theory and general relativity

Δ𝑛 𝑊 𝑛 ≈ 1 + 𝑙𝐶𝑈𝑛Δ𝑛 𝑛 ℏ𝑑2

−𝑂 2 ⁄

≈ 𝑓

− 𝑗 𝜐𝑒𝑒𝑒

2

|𝜔𝑒𝑑⟩ = 1

2 |𝑛1⟩ + |𝑛2⟩ ,

𝜍 0 = |𝜔𝑒𝑑⟩⟨𝜔𝑒𝑑| ⊗ ∏ 𝜍𝑗

𝑂 𝑗=1

Spatial superposition, internal temperature T : Quantum coherence of center-of- mass reduces due to time-dilation: Evovles under 𝐼 = 𝑛𝑛𝑛 + (1 +

𝑕𝑦 𝑒2) ∑

ℏ𝑒𝑗𝜕𝑗

𝑂 𝑗=1

Time dilation decoherence

𝜍̇𝑒𝑑 𝑛 = − 𝑗 ℏ 𝐼𝑒𝑑 + 𝑛 + 𝑂𝑙𝐶𝑈 𝑑2 𝑛𝑛, 𝜍𝑒𝑑 𝑛 − 𝑂𝑛 𝑙𝐶𝑈𝑛 ℏ𝑑2

2

𝑛, 𝑛, 𝜍𝑒𝑑 𝑛

Off-diagonal elements supressed:

𝑛1 𝜍𝑒𝑑(𝑛) 𝑛2 ~ 𝜍𝑒𝑑 0 𝑓

− 𝑗 𝜐𝑒𝑒𝑒

2

• No dissipation
• Gaussian decay

𝜖𝑒𝑠𝑒 = 2 𝑂 ℏ𝑑2 𝑙𝐶𝑈𝑛Δ𝑛

Include full dynamics of center-of mass in Born approximation:

Unitary part. „A piece of iron weighs more when red-hot than when cool“ Decoherence into position basis

• No hidden assumptions,

relies only on time dilation

𝜍̇𝑒𝑑 𝑛 = − 𝑗 ℏ 𝐼𝑒𝑑 + 𝐼0 𝑑2 Γ(𝑛, 𝑣), 𝜍𝑒𝑑 𝑛 − Δ𝐼0 ℏ𝑑2

2

𝑒𝑒

𝑗

Γ(𝑛, 𝑣), Γ(𝑛, 𝑣), 𝜍𝑒𝑑 𝑛 − 𝑒

𝑠

Master equation with only gravitational interaction after build-up of superosition:

Γ 𝑛, 𝑣 = 𝑛𝑛 −

𝑞2 2𝑑2

where:

Γ, 𝜍

𝑠 = 𝑓−𝑗𝑠𝐼𝑒𝑑/ℏ Γ, 𝜍 𝑓𝑗𝑠𝐼𝑒𝑑/ℏ

and

• No „external“ environment
• Position pointer-basis

Time dilation decoherence

μm-scale object on Earth at room temperature, Δx=10−6m: 𝜖𝑒𝑠𝑒 = 2 𝑂 ℏ𝑑2 𝑙𝐶𝑈𝑛Δ𝑛

𝜖𝑒𝑠𝑒 = 10−3s

Green region: decoherence due to time dilation dominates over BB-emission. Other decoherence sources also present. Main competing mechanism black- body radiation:

Time dilation decoherence

𝜖𝑠𝑑 ~ 𝐽𝑛 𝜁 + 2 𝜁 − 1 ℏ𝑑 𝑙𝐶𝑈

6 10−3

𝑑𝑠3Δ𝑛2

• Despite small redshift, decoherence is substantial.
• Fundamental limit for spatial superpositions on Earth.
• Decoherence universally present on curved space-time.
• For strong gravitational fields / high accelerations: Very strong

decoherence

• Gravitational effect in quantum theory
• General relativistic time dilation leads to entanglement

between position and internal degrees-of-freedom

• Quantum Hamiltonian can be probed with matter waves with

clock-states or with photons via Shapiro delay

• Time dilation leads to decoherence of all composite particles,

timescale: 𝜖𝑒𝑠𝑒 =

2 𝑂 ℏ𝑒2 𝑙𝐶𝑈𝑕Δ𝑦

• No breakdown of quantum mechanics, as opposed to

collapse-theories

• Time dilation on Earth decoheres mesoscopic systems
• Could be verified in future experiments with molecules or

trapped nanospheres

Time dilation decoherence

• I. Pikovski, M. Zych, F. Costa, Č. Brukner.

Univ ivers rsal l Decphe herenc nce du due t to Gra Gravit itational T l Tim ime Dila Dilatio ion. . ArXiv:1311.1095 (2013).

(L. Garay, Int. J. Mod. Phys. A10, 145 (1995))

Δ𝑛 Δ𝑣 ≥ ℏ

2 (1+𝛾0 Δ𝑞2 𝑁𝑄𝑄

2 c2 )

response of the space-time, MPl ≈ 22𝜈𝑛 Planck-mass, 𝛾0 dimensionless parameter standard QM Usual quantum experiments

Current experimental bound from quantum systems : 𝛾0 < 1033

(S. Das & E. C. Vagenas, PRL 101, 221301 (2008))

Can quantum gravity have signatures in low-energy quantum mechanics? Novel effects seem inevitable at some scale. Modification of Heisenberg uncertainty relation, common to many approaches to QGR:

Probing Planck- scale physics



𝑌 , 𝑄

• 𝛾=i(1+𝛾0

𝑞0

2𝑄

2 𝑁𝑄𝑄

2 𝑒2 + ⋯ )



𝑌 , 𝑄

• 𝜈=i 1 + 2𝜈0

𝑞0𝑄 /𝑒 2+𝑑2 𝑁𝑄𝑄

2

+ ⋯ (M. Maggiore, Phys. Lett. B, 319 (1993))

(A. Kempf, G. Mangano and R. Mann, PRD, 52, 2 (1995))

• A. Kempf, M. Maggiore: Δ𝑛 Δ𝑣 ≥ ℏ

2 (1+𝛾Δ𝑣2+⋯) implies a modified

• commutator. E.g.:

Examples:



𝑌 , 𝑄

• 𝛿=i(1−𝛿0

𝑞0𝑄

• 𝑁𝑄𝑄𝑒 +𝛿02 𝑞0

2𝑄

2 𝑁𝑄𝑄

2 𝑒2 + ⋯ ) (A. F. Ali, S. Das and E. C. Vagenas,

• Phys. Lett. B, 678 (2009)

very small

ω/2𝜌 = 10 kHz, 𝑛 = 10−27kg → 𝑣0

2

𝑁𝑄𝑄

2 𝑑2 ~ 10−60

Probing Planck- scale physics

Ions in harmonic trap: ω/2𝜌 = 100 MHz, 𝑛 = 10−12kg → 𝑣0

2

𝑁𝑄𝑄

2 𝑑2 ~ 10−40

Optomechanics:

Opto-mechanical interaction: i. Free cavity: 𝐼 = ℏ𝜕0𝑒 𝑀, 𝜕0 = 4𝜌𝑒

𝑗 𝑀

ii. Radiation pressure changes 𝑀 → 𝑀 + 𝑛: 𝜕0 → 𝜕0 (1 − 𝑦

𝑀)

• iii. Quantize 𝑛 → 𝑛
• 𝐼

= ℏ𝜕𝑑𝑒 𝑑 + ℏ𝜕𝑀𝑒 𝑀 − ℏ𝑛0𝑒 𝑀𝑌 𝑑,

𝑛𝑑 𝜕0 → 𝜕(𝑛) 𝑀

Light in a cavity displaces a small mirror by radiation pressure: Massive mechanical

• scillator interacts with light.

𝑛0 =

𝜕𝑀 𝑀 ℏ 𝑑𝜕𝑑 coupling rate

Can add laser drive with detuning Δ = 𝜕𝑗𝑗 − 𝜕𝑀: Photons will scatter depending on the detuning: anti-Stokes (red sideband, Δ = −𝜕𝑑) → cooling, state swap Stokes (blue sideband, Δ = 𝜕𝑑) → heating, squeezing

Probing Planck- scale physics

O‘Connell et al., (2010) - UCSB Kleckner et al, (2006) - UCSB Teufel et al., (2011) - NIST Thompson et al., (2008) - Yale Schliesser et al., (2009) - EPFL

 Ability to create and control collective

quantum states of massive objects.

(Vanner, IP et al., PNAS 108, (2011)

 Optical cooling to the ground state.

(Chan et al., Nature 478 (2011))

 Entanglement between light and matter  Probing limits of quantum mechanics

(Marshall et al., PRL 91 (2003))

Gröblacher et al., (2009) – Vienna Probing Planck- scale physics

Phase space quadratures:

𝐸 𝛽 = 𝑓𝛽𝑏

+ − 𝛽∗𝑏

• 𝐸

𝛾 𝐸 𝛽 = 𝐸 𝛽 + 𝛾 𝑓𝑗 𝐽𝑑[𝛽∗𝛾] 𝐸 −𝛾 𝐸 −𝛽 𝐸 𝛾 𝐸 𝛽 = 𝑓2 𝑗 𝐽𝑑[𝛽∗𝛾]

Implemented to create a phase gate for ions

Leibfried et al., Nature 422, 27 (2003)

X

𝜒

𝜒 = 2 𝐽𝑛[𝛽∗𝛾] 2|𝛽| 2|𝛾| enclosed area P

• Results in an overall phase
• State independent
• No classical analogue
• Arises due to [𝑛, 𝑣] ≠ 0

= 𝑓𝑗 2Re 𝛽 𝑌

−𝑗 2Im[𝛽] 𝑄

• 𝑌

= 1 2 𝑏 + 𝑏 + , 𝑄 = 1 2𝑗 𝑏 − 𝑏 +

Displacements in phase space:

Can be used for quantum computing

Milburn, Schneider, James, Fortschr. Phys. 48, 801(2000), Sørensen, Mølmer, Phys. Rev. A 62, 022311 (2000)

Probing Planck- scale physics

𝜊 ̂ = 𝑓𝑗𝜇𝑗

𝑀𝑄 𝑓−𝑗𝜇𝑗 𝑀𝑌 𝑓−𝑗𝜇𝑗 𝑀𝑄 𝑓𝑗𝜇𝑗 𝑀𝑌

• = 𝑓−𝑗 𝜇2𝑗

𝑀

2

P X

𝜒 = 𝜇2𝑒 𝑀

2

• Resulting phase changes the

ancilla, but is state-independent

• System remains unaffected, and

is fully disentangled from the light

⇒ The system remains quantum

system and does not decohere Displacements of a quantum system around a loop in phase space via an ancillary system (photon):

enclosed area

Probing Planck- scale physics

 In quantum mechanics: [𝑌

, 𝑄 ]=i

 Alternative theories:

[𝑌

, 𝑄 ]=i F(𝑌 , 𝑄 )

𝜊 ̂ = 𝑓𝑗𝜇𝑗

𝑀𝑄 𝑓−𝑗𝜇𝑗 𝑀𝑌 𝑓−𝑗𝜇𝑗 𝑀𝑄 𝑓𝑗𝜇𝑗 𝑀𝑌 = 𝑓∑

(−𝑗𝜇𝑜 𝑀)𝑙+1 𝑙! ∞ 𝑙=1

[𝑌 , 𝑄 ]𝑙

[𝑌 , 𝑌 , … , 𝑄

• 𝜊

̂𝑅𝑁 = 𝑓− 𝑗𝜇2𝑗

𝑀

2

⇒

Probing Planck- scale physics

𝜊 ̂ = 𝑓− 𝑗𝜇2𝑗

𝑀

2+𝜗(𝑗

𝑀)

⇒

By measuring the ancilla one can

• btain a measure of the commutator.

𝑏 𝑀 = 𝛽 𝜊 ̂+𝑏 𝑀𝜊 ̂ 𝛽 ≅ 𝑏 𝑀 𝑅𝑁 𝑓−𝑗 Θ( 𝑌

, 𝑄 𝑑𝑛𝑒)

For initial coherent state (𝑂𝑣: # photons):

𝑆𝑓 𝑏𝑀 |𝛽⟩ 𝐽𝑛 𝑏𝑀 Θ 𝜊|𝛽⟩ ⟨𝑏𝑀⟩𝑟𝑑 𝜏𝑝𝑝𝑗

⇒

• Harmonic evolution: 𝑌

𝑑 𝑛 = 𝑌 𝑑cos(𝜕𝑑𝑛) − 𝑄

• 𝑑sin(𝜕𝑑𝑛)
• Pulsed interactions (duration

𝜖 ≪ 𝜕𝑑−1) (Vanner, IP et al., PNAS

108, (2011) :

𝐼 ≈ −ℏ𝑛0𝑒 𝑀𝑌 𝑑

𝜊 ̂ = 𝑓𝑗𝜇𝑗

𝑀𝑄 𝑓−𝑗𝜇𝑗 𝑀𝑌 𝑓−𝑗𝜇𝑗 𝑀𝑄 𝑓𝑗𝜇𝑗 𝑀𝑌

• Four roundtrips seperated

by 𝜕𝑑𝑛 = 𝜌/2:

Probing Planck- scale physics

Noise and losses analyzed (theoretically):

• Optical losses
• Cavity dynamics and temporal mode mismatching
• Coupling of the mechanics to external environment
• Finite harmonic evolution during interaction

Is it possible to see quantum gravitational modifications of the commutator?



𝑌 , 𝑄

• 𝛾=i(1+𝛾0

𝑞0

2𝑄

2 𝑁𝑄𝑄

2 𝑒2 )



𝑌 , 𝑄

• 𝜈=i 1 + 𝜈0

𝑞0𝑄 /𝑒 2+𝑑2 𝑁𝑄𝑄

2



𝑌 , 𝑄

• 𝛿=i(1−𝛿0

𝑞0𝑄

• 𝑁𝑄𝑄𝑒 +𝛿0

𝑞0

2𝑄

2 𝑁𝑄𝑄

2 𝑒2 )

Improvement by up to 33

• rders of magnitude,

Possibility to probe possible Planck-scale deformations. Deformations of center-of-mass

• bservable even for

𝛾0, 𝜈0, 𝛿0 ≲ 1

• Finesse

𝐺 ~105 − 106

• Mass

m ~ 0.1 − 1𝑒𝑛

• Frequency 𝜕𝑑~ 1 − 100𝑙𝐼𝑙
• Intentisity 𝑂𝑞~108 − 1012
• Bath temperature 𝑈 < 100𝑛𝑛
• Qualtiy factor

Q > 107

• Opt. cooling

𝑒𝑑 < 30

• Opt. loss

𝜃 > 0.9 Challenging, but realistic parameters. Requires different parameters for different theories:

Probing Planck- scale physics

Model: N identical particles, composition rule unmodified: The analysis relied on modifications of the center-of-mass mode of a quantum mechanical system. Maybe modification apply only to „elementary“ particles? Then (simple argument):

𝑛 𝑒𝑑 = 1 𝑂 𝑛 𝑗,

𝑂 𝑗=1

𝑣̂𝑒𝑑 = 𝑣̂𝑗

𝑂 𝑗=1

𝑛 𝑒𝑑, 𝑣̂𝑒𝑑 𝛾=i 1+ 1 𝑂 𝛾0 𝑁𝑄𝑄

2 𝑑2

𝑣̂𝑒𝑑

2

− 𝑣̂𝑗𝑣̂𝑗

𝑗≠𝑗

Compare to direct center-of-mass deformation: 𝑛 𝑒𝑑, 𝑣̂𝑒𝑑 𝛾=i 1+ 𝛾0 𝑁𝑄𝑄

2 𝑑2 𝑣̂𝑒𝑑 2

→

If only „elementary“ particles are modified, then composite systems have a modification with an an effective strength 𝛾0 →

𝛾0

𝑂 or

𝛾0 𝑂2

(depending on momentum correlations of the elemntary particles). Experiment would still put bounds, but is less sensitive

Probing Planck- scale physics

Present in many other proposals for lab tests

• f quantum gravity phenomenology, e.g.:
• C. Hogan, Phys. Rev. D 77, 104031 (2008);
• J. D. Bekenstein, Phys. Rev. D, 86, 124040 (2012);
• F. Marin et al., Nature Phys. 9, 71 (2013)

Currently no theory predicts the „correct“ approach. Good reasons for either assumption: Elementary particles:

• “Natural building blocks”
• No extensions into macroscopic

and classical domain (effect remains hard to detect) Wavefunction of composite system:

• “Natural” in quantum domain,

independent of composition

• Deformation-induced restriction

to localization applies universally

Probing Planck- scale physics

• Overlap between quantum mechanics and general relativity is

accessible in the lab, even at low energies

• Phenomenological predictions of quantum gravity can be

tested or constrained using quantum optics

• Assumptions:
• Quantization of space-time leads to deformations of the

canonical commutator

• Deformations apply to the center-of-mass
• Instead of Planck-scale position measurements, use a pulsed
• pto-mechanical scheme to amplify the effect and imprint

information of commutator deformations onto optical field

Probing Planck- scale physics

• I. Pikovski, M. Vanner, M. Aspelmeyer, M. S. Kim, Č. Brukner.

Pro robin bing P Pla lanck-Scal Scale P Physics cs with th Quantu ntum Opt ptic ics. . Nature Physics 8, 393 (2012); arxiv:1111.1979.

Opto-mechanics and tests of quantum gravity phenomenology Time dilation in quantum mechanics and implications for photons, matter waves and decoherence

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lsed d op

• pto

to-mech chan anics cs (with G. Cole, K. Hammerer, G. Milburn): PNAS 108, 16182-16187 (2011)

• Pro

robin bing P Pla lanck-Scale ph physic ics w wit ith qua uant ntum m opt ptic ics: Nature Physics 8, 393-397 (2012)

• Quantum

ntum inter erfer eromet etric vis isibili ibility a as a wit itness o

• f general re

l rela lativ ivis istic ic pro prope per t tim ime: Nature Communications 2, 505 (2011)

• Ge

General re l rela lativ ivis istic ic e effects in in qu quantum inte nterference of

• f photon

hotons (with T. C. Ralph):

• Class. Quantum Grav. 29, 224010 (2012)
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ivers rsal l decohe

• herenc

nce due ue to to gravitati tiona

• nal

tim ime dila dilatio ion: arxiv:1311.1095

Thank you for your attention!

I.P. Myungshik Kim Michael Vanner Časlav Brukner Časlav Brukner I.P. Markus Aspelmeyer Magdalena Zych Fabio Costa