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
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
Contents Motivation 1 Quantum Interferometry 2 A Molecule in the Field of a Gravitational Wave 3 Gravitational Waves and Charged Point Masses Gravitational Waves and the HD + Molecule Gravitational Wave Detection 4 Conclusion 5 D. Lorek (ZARM) Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 3 / 17
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 of 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
Quantum Interferometers Atom Interferometric “Lever Arm” energy difference between paths: E IF assumption: effect causes a shift, that scales with optical frequency E s : ∆ E = ∆ E 2 − ∆ E 1 = h · E s sensitivity ∆ E / E > ǫ ref ∼ 10 − 15 ——– D. Lorek (ZARM) Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 5 / 17
Quantum Interferometers Atom Interferometric “Lever Arm” energy difference between paths: E IF assumption: effect causes a shift, that scales with optical frequency E s : ∆ E = ∆ E 2 − ∆ E 1 = h · E s sensitivity ∆ E / E > ǫ ref ∼ 10 − 15 ——– laser spectroscopy: h · E s / E s > ǫ ref atom interferometry: h · E s / E IF > ǫ ref minimal detectable h : E IF h min = ǫ ref E s D. Lorek (ZARM) Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 5 / 17
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
Molecular Quantum Interferometry Gravitational Wave Detection prepare molecules in a coherent superposition of two mutually orthogonal orientations 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
Gravitational Waves and Charged Point Masses Gravitational Waves linearized gravity: g µν = η µν + h µν η µν = diag( − 1 , 1 , 1 , 1) , | h µν | ≪ 1 TT gauge : ✷ h ij = 0 , h µ 0 = 0 , δ kl h ik , l = 0 , δ ij h ij = 0 D. Lorek (ZARM) Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 7 / 17
Gravitational Waves and Charged Point Masses Quantum Physics Klein–Gordon equation minimally coupled to gravity and to the Maxwell field: g µν D µ D ν ψ − m 2 c 2 � 2 ψ = 0 covariant derivative: D µ T ν = ∂ µ T ν + { ν µσ } T σ − ie � c A µ T ν 2 g νρ ( ∂ µ g σρ + ∂ σ g µρ − ∂ ρ g µσ ) µσ } := 1 Christoffel symbol: { ν Maxwell potential: A µ D. Lorek (ZARM) Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 8 / 17
Gravitational Waves and Charged Point Masses Quantum Physics Klein–Gordon equation minimally coupled to gravity and to the Maxwell field: g µν D µ D ν ψ − m 2 c 2 � 2 ψ = 0 insert: g µν = η µν + h µν � i � [ c 2 S 0 + S 1 + c − 2 S 2 + . . . ] � Ansatz [1]: ψ = exp compare equal powers of c 2 : [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
Gravitational Waves and Charged Point Masses odinger Equation: i � ∂ t ˜ φ = H ˜ Schr¨ φ Hamiltonian [1]: δ ij − h ij � δ ij − h ij � H = − � 2 ∂ i ∂ j − eA 0 + ie � A i � � ∂ j 2 m m c [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
Gravitational Waves and Charged Point Masses odinger Equation: i � ∂ t ˜ φ = H ˜ Schr¨ φ Hamiltonian: δ ij − h ij � δ ij − h ij � H = − � 2 ∂ i ∂ j − eA 0 + ie � A i � � ∂ j 2 m m c for non–relativistic systems A i ≪ A 0 → Interaction Hamiltonian: H I = � 2 2 m h ij ∂ i ∂ j D. Lorek (ZARM) Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 8 / 17
Gravitational Waves and Charged Point Masses Electric Potential A µ inhomogenous Maxwell equations coupled to gravity: 4 π j µ = D ν ( g µρ g νσ F ρσ ) Field-Strength Tensor: F ρσ = ∂ ρ A σ − ∂ σ A ρ insert: g µν = η µν + h µν point charge: j 0 = q δ 3 ( r ) and j i = 0 D. Lorek (ZARM) Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 9 / 17
Gravitational Waves and Charged Point Masses Electric Potential A µ inhomogenous Maxwell equations coupled to gravity: 4 π j µ = D ν ( g µρ g νσ F ρσ ) insert: g µν = η µν + h µν point charge: j 0 = q δ 3 ( r ) and j i = 0 µν · exp( i [ � periodic gravitational waves: h µν = h 0 k � x − ω t ]) Influence of the GW is adiabatic (low frequency) and quasi–constant (long wavelength) → potentials are static Ansatz: A 0 = q / r + qA (1) A i = qA (1) 0 , i D. Lorek (ZARM) Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 9 / 17
Gravitational Waves and Charged Point Masses Potential A 0 of a Point Charge q in the Field of a GW x i h 0 ij x j � x − ω t ) � 2 r 2 e i ( � A 0 = q k � 1 − r D. Lorek (ZARM) Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 9 / 17
Gravitational Waves and the HD + Molecule HD + Molecular Hamiltonian molecular Hamiltonian contains: H = T e + V en 1 + V en 2 + V nn + T n 1 + T n 2 + δ H T : kinetic energy V : potential energy e , n 1, n 2: electron, first nucleus, second nucleus . . . D. Lorek (ZARM) Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 10 / 17
Gravitational Waves and the HD + Molecule Perturbation to Molecular Hamiltonian electronic kinetic energy: Radial dependence of ( h · R ∞ ) δ ˜ T e ( R ) cos(2 φ )(1 − cos (2 θ )) perturbations electronic Coulomb energy: ( h · R ∞ ) δ ˜ V en ( R ) cos(2 φ )(1 − cos (2 θ )) nuclear Coulomb energy: − ( h · R ∞ ) (2 R ) − 1 cos(2 φ )(1 − cos (2 θ )) D. Lorek (ZARM) Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 11 / 17
Gravitational Waves and the HD + Molecule Perturbation to Molecular Hamiltonian electronic kinetic energy: Radial dependence of ( h · R ∞ ) δ ˜ T e ( R ) cos(2 φ )(1 − cos (2 θ )) perturbations electronic Coulomb energy: ( h · R ∞ ) δ ˜ V en ( R ) cos(2 φ )(1 − cos (2 θ )) nuclear Coulomb energy: − ( h · R ∞ ) (2 R ) − 1 cos(2 φ )(1 − cos (2 θ )) Perturbation Energy: ∼ 0 . 1 · h · R ∞ D. Lorek (ZARM) Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 11 / 17
Gravitational Waves and the HD + Molecule Total Perturbation Operator Perturbation Matrix GW can not drive rotational, vibrational or electronic transitions GW only couples states with | ∆ m | = 2 as expected from quadrupole nature of GW D. Lorek (ZARM) Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 12 / 17
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
Gravitational Waves and the HD + Molecule Eigenvalues Differential Energy Shift: 60 µ Hz for h = 10 − 19 D. Lorek (ZARM) Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 13 / 17
Gravitational Waves and the HD + Molecule State Spectra state spectrum for l = 1 state spectrum for some l = 5 eigenstates eigenstates D. Lorek (ZARM) Rovibrational QIs and Gravitational Waves Firenze, 24th Feb. 2009 14 / 17
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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