Rovibrational Quantum Interferometers and Gravitational Waves D. - - PowerPoint PPT Presentation

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The Galileo Galilei Institute for Theoretical Physics Rovibrational Quantum Interferometers and Gravitational Waves D. Lorek, A. Wicht, C. L ammerzahl, H. Dittus D. Lorek (ZARM) Rovibrational QIs and Gravitational Waves Firenze, 24th Feb.

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The Galileo Galilei Institute for Theoretical Physics

Rovibrational Quantum Interferometers and Gravitational Waves

  • D. Lorek, A. Wicht, C. L¨

ammerzahl, H. Dittus

  • D. Lorek (ZARM)

Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 1 / 17

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SLIDE 2

Motivation

Molecular Interferometry

basic features of Atom Interferometry coherent manipulation of internal molecular quantum states molecules can distinguish between different directions molecular states are sensibel to non–isotropic effects → Gravitational Wave Detectors

  • D. Lorek (ZARM)

Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 2 / 17

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Contents

1

Motivation

2

Quantum Interferometry

3

A Molecule in the Field of a Gravitational Wave Gravitational Waves and Charged Point Masses Gravitational Waves and the HD+ Molecule

4

Gravitational Wave Detection

5

Conclusion

  • D. Lorek (ZARM)

Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 3 / 17

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SLIDE 4

Quantum Interferometers

Atom and Molecular Interferometry

applications: fine structure constant, gravitational acceleration, gravity gradients, inertial sensors, test of GR, . . . frequency measurement ↔ second phase sensitive frequency measurement → sensitivity increases ∼ T (not ∼ √ T) truly differential phase (frequency) measurement - cancellation

  • f common mode phase evolution (common frequency)

Stanford atom interferometer: relative shift of 10−19

  • D. Lorek (ZARM)

Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 4 / 17

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Quantum Interferometers

Atom Interferometric “Lever Arm”

energy difference between paths: EIF assumption: effect causes a shift, that scales with optical frequency Es: ∆E = ∆E2 − ∆E1 = h · Es sensitivity ∆E/E > ǫref ∼ 10−15 ——–

  • D. Lorek (ZARM)

Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 5 / 17

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Quantum Interferometers

Atom Interferometric “Lever Arm”

energy difference between paths: EIF assumption: effect causes a shift, that scales with optical frequency Es: ∆E = ∆E2 − ∆E1 = h · Es sensitivity ∆E/E > ǫref ∼ 10−15 ——– laser spectroscopy: h · Es/Es > ǫref atom interferometry: h · Es/EIF > ǫref

minimal detectable h: hmin = ǫref

EIF Es

  • D. Lorek (ZARM)

Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 5 / 17

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SLIDE 7

Molecular Quantum Interferometry

Rovibrational Quantum Interferometers

coherent manipulation of different individual rotational–vibrational molecular quantum states molecules are not spherically symmetric → molecules can distinguish between different directions in space w. r. t. their internuclear axis → molecular spectra depend on the orientation of the molecule, if a non–isotropic situation is considered

  • D. Lorek (ZARM)

Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 6 / 17

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Molecular Quantum Interferometry

Gravitational Wave Detection

prepare molecules in a coherent superposition of two mutually orthogonal

  • rientations in the x–y–plane ↔ two

paths of a quantum interferometer linearly polarized GW (z–direction) the GW modifies the internuclear distance periodically free quantum evolution: non–isotropic perturbation removes the orientational degeneracy → states will acquire a quantum phase difference

  • D. Lorek (ZARM)

Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 6 / 17

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Gravitational Waves and Charged Point Masses

Gravitational Waves

linearized gravity: gµν = ηµν + hµν

ηµν = diag(−1, 1, 1, 1) , |hµν| ≪ 1

TT gauge : ✷hij = 0 , hµ0 = 0 , δklhik,l = 0 , δijhij = 0

  • D. Lorek (ZARM)

Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 7 / 17

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Gravitational Waves and Charged Point Masses

Quantum Physics

Klein–Gordon equation minimally coupled to gravity and to the Maxwell field:

g µνDµDνψ − m2c2

2 ψ = 0

covariant derivative: DµT ν = ∂µT ν + { ν

µσ } T σ − ie c AµT ν

Christoffel symbol: { ν

µσ } := 1 2gνρ (∂µgσρ + ∂σgµρ − ∂ρgµσ)

Maxwell potential: Aµ

  • D. Lorek (ZARM)

Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 8 / 17

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Gravitational Waves and Charged Point Masses

Quantum Physics

Klein–Gordon equation minimally coupled to gravity and to the Maxwell field: g µνDµDνψ − m2c2

2 ψ = 0

insert: gµν = ηµν + hµν Ansatz [1]: ψ = exp i

[c2S0 + S1 + c−2S2 + . . .]

  • compare equal powers of c2:

[1] C. Kiefer, T. P. Singh, Phys. Rev. D 44, 1067–1076 (1991)

  • D. Lorek (ZARM)

Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 8 / 17

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SLIDE 12

Gravitational Waves and Charged Point Masses

Schr¨

  • dinger Equation: i∂t ˜

φ = H˜ φ Hamiltonian [1]: H = − 2

2m

  • δij − hij

∂i∂j − eA0 + ie

m Ai c

  • δij − hij

∂j

[1] S. Boughn, T. Rothman, Class. Quantum Grav. 23, 5839–5852 (2006)

  • D. Lorek (ZARM)

Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 8 / 17

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SLIDE 13

Gravitational Waves and Charged Point Masses

Schr¨

  • dinger Equation: i∂t ˜

φ = H˜ φ Hamiltonian: H = − 2

2m

  • δij − hij

∂i∂j − eA0 + ie

m Ai c

  • δij − hij

∂j for non–relativistic systems Ai ≪ A0 →

Interaction Hamiltonian: HI = 2

2mhij∂i∂j

  • D. Lorek (ZARM)

Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 8 / 17

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Gravitational Waves and Charged Point Masses

Electric Potential Aµ

inhomogenous Maxwell equations coupled to gravity:

4πjµ = Dν (g µρg νσFρσ)

Field-Strength Tensor: Fρσ = ∂ρAσ − ∂σAρ

insert: gµν = ηµν + hµν point charge: j0 = qδ3(r) and ji = 0

  • D. Lorek (ZARM)

Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 9 / 17

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Gravitational Waves and Charged Point Masses

Electric Potential Aµ

inhomogenous Maxwell equations coupled to gravity:

4πjµ = Dν (g µρg νσFρσ)

insert: gµν = ηµν + hµν point charge: j0 = qδ3(r) and ji = 0 periodic gravitational waves: hµν = h0

µν · exp(i[

k x − ωt]) Influence of the GW is adiabatic (low frequency) and quasi–constant (long wavelength) → potentials are static Ansatz: A0 = q/r + qA(1)

0 ,

Ai = qA(1)

i

  • D. Lorek (ZARM)

Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 9 / 17

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Gravitational Waves and Charged Point Masses

Potential A0 of a Point Charge q in the Field of a GW A0 = q

r

  • 1 −

xih0

ijxj

2r 2 ei( k x−ωt)

  • D. Lorek (ZARM)

Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 9 / 17

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Gravitational Waves and the HD+ Molecule

HD+ Molecular Hamiltonian

molecular Hamiltonian contains:

H = Te + Ven1 + Ven2 + Vnn + Tn1 + Tn2 + δH

T: kinetic energy V : potential energy e, n1, n2: electron, first nucleus, second nucleus

. . .

  • D. Lorek (ZARM)

Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 10 / 17

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Gravitational Waves and the HD+ Molecule

Perturbation to Molecular Hamiltonian electronic kinetic energy:

(h · R∞) δ ˜ Te(R) cos(2φ)(1 − cos (2θ))

electronic Coulomb energy:

(h·R∞) δ ˜ Ven(R) cos(2φ)(1−cos (2θ))

nuclear Coulomb energy:

−(h·R∞) (2R)−1 cos(2φ)(1−cos (2θ)) Radial dependence of perturbations

  • D. Lorek (ZARM)

Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 11 / 17

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SLIDE 19

Gravitational Waves and the HD+ Molecule

Perturbation to Molecular Hamiltonian electronic kinetic energy:

(h · R∞) δ ˜ Te(R) cos(2φ)(1 − cos (2θ))

electronic Coulomb energy:

(h·R∞) δ ˜ Ven(R) cos(2φ)(1−cos (2θ))

nuclear Coulomb energy:

−(h·R∞) (2R)−1 cos(2φ)(1−cos (2θ)) Radial dependence of perturbations

Perturbation Energy: ∼ 0.1 · h · R∞

  • D. Lorek (ZARM)

Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 11 / 17

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Gravitational Waves and the HD+ Molecule

Total Perturbation Operator

GW can not drive rotational, vibrational or electronic transitions GW only couples states with

|∆m| = 2 as expected from

quadrupole nature of GW Perturbation Matrix

  • D. Lorek (ZARM)

Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 12 / 17

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Gravitational Waves and the HD+ Molecule

Eigenvalues

eigenvalues for the total perturbation operator for the vibrational ground state v = 0

  • D. Lorek (ZARM)

Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 13 / 17

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Gravitational Waves and the HD+ Molecule

Eigenvalues Differential Energy Shift: 60 µHz for h = 10−19

  • D. Lorek (ZARM)

Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 13 / 17

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Gravitational Waves and the HD+ Molecule

State Spectra

state spectrum for l = 1 eigenstates state spectrum for some l = 5 eigenstates

  • D. Lorek (ZARM)

Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 14 / 17

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Gravitational Wave Detection

Spherical Part of the Probability Distribution | R|ψ|2

  • D. Lorek (ZARM)

Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 15 / 17

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Comparison with “Classical” Detectors

Advantages

most accurate measurement methods available physics of the probe and the interaction between the probe and the environment is well understood exactly identical, well defined probes exist no storage time limit spectral-, polarization- and directional-sensitivity can be chosen and modified within milliseconds

Challenges

ultra-cold molecules (≪mK) to be prepared in a specific state multi-chromatic narrow linewidth laser-fields required control/suppress environmental perturb. (em. fields, vibrations)

  • D. Lorek (ZARM)

Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 16 / 17

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Conclusion

Quantum Interferometry Basic ideas of AI → Rovibrational states of molecules Molecular states are sensibel to non–isotropic effects Quantum sensor for fundamental physics Gravitational Waves and HD+ Prospects

  • D. Lorek (ZARM)

Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 17 / 17

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Conclusion

Quantum Interferometry Gravitational Waves and HD+ Hamiltonian of a charged particle in a GW field Electric potential of a point charge in a GW field → Perturbation operator of HD+ and energy shifts Adequate construction Prospects

  • D. Lorek (ZARM)

Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 17 / 17

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Conclusion

Quantum Interferometry Gravitational Waves and HD+ Prospects h = 10−19 → 60 µHz (current AI: ≈ 100 µHz) Further tests of fundamental physics AI

  • D. Lorek (ZARM)

Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 17 / 17

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Conclusion

Quantum Interferometry Gravitational Waves and HD+ Prospects

  • A. Wicht, C. L¨

ammerzahl, D. L., H. Dittus, Phys. Rev. A 78, 013610

Thanks for your attention!

  • D. Lorek (ZARM)

Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 17 / 17