Jean-Pierre Zendri INFN-Padova

Planck Scale

Quantum Field Theory Compton wavelength:

General Relativity Schwarzschild radius:

𝜆𝑐=h𝑚𝑐

𝑟 𝑆=2𝐺𝑚𝑐2

𝑚𝑝=√ℏ𝑐𝐺

𝐿𝑎𝑟𝑔𝑒𝐻𝑎𝑑𝑟𝑜𝑛𝐶𝑜𝑙𝑙𝑖𝑑𝑒𝑟 𝐸≈ 1013𝑒𝑉𝐺𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙𝑤𝑎𝑣𝑒𝑑𝑒𝑡𝑒𝑐𝑡𝑜𝑟𝑠 ∆ 𝑥≈ 10−21𝑚

𝐴𝑡𝑜𝑚𝑖𝑐𝑐𝑙𝑜𝑐𝑘𝑠 ∆ 𝑡≈ 10−18 𝑠

Is the Planck scale accessible in earth based Experiments ?

Limits to distance (D) measurements

Quantum Mechanics

General Relativity

Quantum mechanics + General Relativity

∆ 𝑝 ℏ2 ∆𝑥0

𝛿𝐷 ∆𝑥0+𝑇

2𝑀∆ 𝑥0𝛿𝐷𝑀𝑖𝑛 √ ℏ𝑇2𝑀=√ 2ℏ𝐷

𝑐𝑀→𝑀=∞

0

To avoid a Black-Hole formation 𝐷≥𝑅 h h𝑆𝑐 𝑤𝑎𝑟𝑧𝑠 𝑖𝑙𝑑=2𝐺𝑀𝑐2

𝜹𝑫𝑴𝒊𝒏≥√ ℏ𝑮𝒄𝟑 =𝑳𝒑

Body A Mass:MSize:<DBA D T:Round trip time

𝐷=𝑐𝑇2

∆ 𝑥0

Ligth Pulse

Uncertainty Principle and Gravity

Newtonian Gravity + Special Relativity +Equivalence principle

𝑎≈𝐺(𝐸 /𝑐2)𝑅2 ∆ 𝑥 ≈

(𝐿𝑝❑)2 ∆𝑝ℏ

∆ 𝑥 ≥ 12𝜋𝜔𝑆𝑖𝑛(𝜀) ∆ 𝑝≥ h𝜔𝑆𝑖𝑛(𝜔)

∆ 𝑥 ∆𝑝≥ℏ

Heisembeg Telescope

∆ 𝑥 ∆𝑝≈ℏ+¿¿

E: photon energy Tint= Interaction time

+Photon unknown direction (e)

Uncertainty Principle and Gravity

Different approaches to Quantum GravityString Theory and loop quantum gravityAmati, D., Ciafaloni, M. & Veneziano, G. Superstring collisions at planckian energies. Phys. Lett. B 197, 81-88 (1987)Gross, D. J. & Mende, P. F. String theory beyond the Planck scale. Nucl. Phys. B 303, 407-454 (1988)

Thought experimentsLimits to the measurements of BH horizon area Maggiore, M. A generalized uncertainty principle in quantum gravity. Phys. Lett. B 304, 65-69 (1993).Scardigli, F. Generalized uncertainty principle in quantum gravity from micro-black holegedanken experiment. Phys. Lett. B 452, 39-44 (1999).Jizba, P., Kleinert, H. & Scardigli, F. Uncertainty relation on a world crystal and its applications to micro black holes. Phys. Rev. D 81, 084030 (2010).

∆ 𝑥 ∆𝑝≈ ¿

Generalized Uncertainty Principle (GUP)

• Including the clock wave function spread (quantum clock) R.J.Adler, et all., Phys. Lett. B477, 424 (2000) . W.A.Christiansen et all., Phys. Rev. Lett. 96, 051301 (2006).• Including clock rate in the Schwarzschild geometry and holographic principle

E.Goklu, G. Lammerzhal Gen. Rel. and Grav. 43, 2065 (2011). Y.J.Ng et all. Acad. Sci.755, 579 (1995).• String theory S.Abel and J.Santiago, J.Phys. G30, 83 (2004)

∆ 𝑥 ∆𝑝≈ ¿At the Planck scale 𝛽0 ≈ 1

The length uncertaincy should be larger than Lp

GUP and harmonic oscillator ground state

GUP and AURIGAHigher ground state energy for a quantum oscillator

Test on low-temperature oscillators set limits

GUP effects expected to scale with the mass m

Massive cold oscillators

(Sub millikelvin cooling of ton-scale oscillator)

• Strain sensitivity 2 10-

21<Shh<10-20 Hz-1/2

over 100 Hz band (FWHM ~ 26 Hz) • Burst Sensitivity hrss ~ 10-20 Hz-1/2

• Duty-cycle ~ 96 %• ~ 20 outliers/day at SNR>6

The AURIGA detector

3m long Al5056 2200 kg 4.5 K

Effective mass vs reduced mass

Readout measures the axial displacement of a bar face corresponding to the first longitudinal mode

Meff depends on the modal shape and interrogation point of the readout (e.g. Meff ∞ if the measurement is performed on a node of the vibration mode)

Really moving mass

1) Modal motion implies an oscillation of each half-bar center-of-mass, to which is associated a reduced mass M/2

xcm1 xcm2

2) The energy associated to the oscillation of the couple of c.m.’s, having a reduced mass Mred = M/2 is about 80% of that of the modal motion

• FSig Signal Force• FTh Langevin thermal

force

(t)

(

Equipartion theorem

Active Cooling: principle

∝𝑇

()

𝑇=4.5𝐾𝑒𝑙𝑣𝑖𝑛

Not significant for GUT

• FSig Signal Force• FTh Langevin thermal

force• FCD Feedback Force

(t)

(

Equipartion theorem

Active Cooling: principle

• FSig Signal Force• FTh Langevin thermal

force• FCD Feedback Force

(t)

(

Equipartion theorem

Active Cooling: principle

Cold damped distribution

¿

∝𝑇

/

Cooling down to the ground state

Active or passive feedback cooling of one (few) oscillator mode

Displacement sensitivity improvement

Prepare oscillator in its fundamental stateNOYES

KBKT

KLC

MB MT MLC

System of three coupled resonators: the bar and the transducer mechanical resonators and the LC electrical resonator

1) The bar resonator is coupled to a lighter resonator, with the same resonance frequency to amplify signals 2) A capacitive transducer, converts the differential motion between bar and the ligher resonator into an electrical current, which is finally detected by a low noise dc SQUID amplifier3) The transducer efficiency is further increased by placing the resonance frequency of the electrical LC circuit close to the mechanical resonance frequencies, at 930 Hz.

F Back-action force

- At increasing feedback gain, the 3 modes of the detector reduce their vibration amplitude.- The equivalent temperature of the vibration was reduced down to

Teff=0.17 mK

System of three coupled resonators: the bar and the transducer mechanical resonators

and the LC electrical resonator

AURIGA minimal energy

Modified commutators IGUP can be associated to a deformed canonical commutator

Planck scale modifications of the energy spectrum of quantum systems

Lack of observed deviations fromtheory at the electroweak scale

Lamb shift in hydrogen atoms

1S-2S level energy differencein hydrogen

Active Cooling: Fundamental limit𝑇𝑒𝑓𝑓 ∝ 𝑇𝛼

𝑇𝑒𝑓𝑓 𝑀𝑖𝑛=𝑇𝑛√1+ 2𝑄𝑇𝑇 𝑛𝑘𝑘𝑛

≤𝑇𝑛

𝑇 𝑛=1𝑘𝐵𝜔

√𝑆𝐹𝑛𝐹 𝑛𝑆𝑥𝑛𝑥𝑛

𝑘𝑛=√ 𝑆𝐹𝑛𝐹 𝑛𝑆𝑥𝑛𝑥𝑛

𝑇 𝑛≤ ℏ𝜔2𝑘𝐵Amplifier Noise temperature

Amplifier Noise Stiffness

Quantum   Mechanics→

≤ ℏ𝜔2𝑘𝐵

xn:Amplifier additive noise Fn: Amplifier back-action noise

𝑛𝑡=𝑘𝐵𝑇𝑒𝑓𝑓 𝑀𝑖𝑛ℏ𝜔 =

𝑘𝐵𝑇 𝑛ℏ𝜔 √1+ 2

𝑄𝑇𝑇 𝑛𝑘𝑘𝑛

Auriga possible upgrading

Temper. [K] Tn [quanta] nt b0

Auriga Now 4.2 400 25000 <3x1033

Auriga Cooled 0.1 27 * 1000 <1.2x1032

Auriga cooled+new SQUID

0.1 10 ** 600 <7x1031

* P.Falferi et al. APS 88 062505 (2006)** P.Falferi et al. APS 93 172506 (2008)

Auriga just cooling down

Auriga cooled down + New SQUID

Further improvements expected decreasing the LC thermal noise (but never nt<1/2)

μ0 < 4 x 10 -13

Modified commutators IIModifications of commutators are not unique

Experiments could distinguish between the various approaches

M. Maggiore Phys. Lett. B 319, 83-86 (1993)

Spacetime granularity (Quantum Foam)

Property of the spacetime geometry and not of physical objects(Soccer ball problem ?)

Apparatus independent (not based on a specific QG model)

General Relativity Quantum mechanics

Mass (energy) curves spacetime Vacuum energy

Energy of the virtual particles gives space time a "foamy" character at L ≈ Lp

(Wheeler, 1955)

AURIGA: re-interpretationAURIGA is not the “coolest” oscillator, but is the most motionless

Xrms= (kT/mω2)1/2 = (Eexp/mω2)1/2 ≈ 6 X 10-19 m

Macroscopic oscillators in their quantum ground state

(ω0 = 1 GHz T ≈ 50 mK)

Experimental proposals I/1

A sequence of 4 pulse is applied to the mechanical oscillator such that in the mechanical moves in the phase space around a loop: +Xm –Pm - Xm +Pm

Quantum mechanics: [Xm ,Pm]≠0

Experimental set-up

Experimental proposals Ib

• Classical phase rotation• Short cavity is chalangin (10-6 m !)

Critical Points

h𝜈 /𝑐

h𝜈 /𝑐

Experimental proposals IIa

1) The photon has to discard momentum into the crystal 2) This momentum will be returned to the photon upon exit. 3) The crystal moves for a distance scaling with the energy of the incoming photon

v=0

v=0

v ≠ 0

∆ 𝑥 ≈ 𝐿 h𝜈 (𝑛−1)𝑀𝑐2

𝑀 ,𝑛

h𝜈 /𝑛𝑐

Experimental proposals IIb

If the energy of the photon is so low that the crystal should move less than the Planck length, the photon cannot cross the crystal, leading to a decrease in the

transmissionCritical points

Thermal noiseOptical dissipations

Experimental proposals III

1. Space time is described with a wave function frequency limited (the Plank frequency).

2. If one end point of the particle position is limited by an aperture with size D the uncertaincy of the other points at distance L is limited by diffraction to be lpL/D

3. The possible orientation are minimized for DlpL/D thus D(LpL)1/2

There is an unavoidable transverse uncertaincy ∆ 𝑋 1 ∆ 𝑋 2≈ 𝐿𝑝×𝐿

In a Michelson interferometer the effect appears as noise that resembles a random Planckian walk of the beam splitter for durations up to the light-crossing time.

Measurable (according to Hogan) using two cross-correlated two nearly coolacated ng Michelson interferometers with arms length of about L=40 meters

Conclusions

HUMORHeisenberg

Uncertainty

Measured with

Opto-mechanical

Resonator

•The Auriga detector constrained the plank scale for a macorscopic body. Further improvements are possible but still far from the “traditional” plank scale.•Recent experiments on macroscopic mechanical oscillators showed that they behaves as quantum oscillator. •Put in prespective a new generation of experiment with macroscopic body should be abble to approach the plank scale.•It is not clear if these experiments would be abble to constrain the quantum gravity because the describtion of the macroscopic objects in the framework of quantumgravity models is still lacking.•A INFN group is in charge to examine the feasibility of the new proposed experiments and in case to propose new and more realistic set-up

Soccer ball problem

Possible (ad hoc) solutions1) Effects scale as the number of constituent (Atoms, quarks... ?)

Quesne, C. & Tkachuk, V. M. Phys. Rev. A 81, 012106 (2010).Hossenfelder, S. Phys. Rev. D 75, 105005 (2007) and Refs. therein

2) Decoherence ? Magueijo, J. Phys. Rev. D 73, 124020 (2006)

Doubly Special Relativity Amelino-Camelia, G. Int. J. Mod. Phys. D 11, 1643-1669 (2002).

Modified SR with two invariants: speed of light c, and minimal length Lp

Extremely huge for particles, but small for planets, stars and ... soccer balls

Problem of macroscopic (multiparticle) bodies

Minimal length Problems with standard special relativity (SR)

1) Physical momenta sum nonlinearly2) Correspondence principle