Nonlinear Quantum Interferometry with Bose Condensed Atoms

Transcription

1 ACQAO Regional Workshop 0 onlinear Quantum Interferometry with Bose Condensed Atoms Chaohong Lee State Key Laboratory of Optoelectronic Materials and Technologies, School of Physics and Engineering, Sun Yat-Sen University, Guangzhou 5075, P. R. China lichaoh@mail.sysu.edu.cn; chleecn@gmail.com ResearcherID: Supported by: SFC, MOE, MOST

2 Where is SYSU? Shanghai Macau Hongkong Sun Yat-Sen First revolutionist in modern China; Establisher of R China

3 orth Entrance

4 Lecture Hall

5 Current research interests of my group Quantum interferometry (Bose-condensed atoms, ultracold trapped ions) Cavity-QED with Bose condensed atoms on-equilibrium many-body quantum dynamics (TEBD, quantum spin model, Bose-Hubbard model, etc) Theoretical studies of quantum technology with ultracold atoms (high-precision measurements, quantum simulation, atom clocks, etc)

6 Introduction Outline - Quantum metrology - Interferometry with Bose condensed atoms Matter-wave interferometry - Atomic matter-wave interference - onlinear excitations - Bose-Josephson junctions Many-body quantum interferometry - Quantum spin squeezing and many-particle entanglement - High-precision interferometry via spin squeezed state - High-precision interferometry via OO state Summary and open problems

7 . Introduction W. J. Ashworth, Metrology and the State: Science, Revenue, and Commerce, Science 306, 34 (004) Measurement standards defined by human body (old-time UK people) or rice length (old-time Chinese people) are not accurate. They may change case by case.

8 .. Quantum Metrology time mass length Quantum-Enhanced Measurements: Beating the Standard Quantum Limit V. Giovannetti, S. Lloyd, and L. Maccone, Science 306, 330 (004)

9 Mach-Zehnder interferometry

10 Ramsey interferometers ()prepare an initial state >; ()apply the first half-pi pulse to create an equal superposition of > and >; (3)accumulate a relative phase between > and > in the free evolution; (4)recombine > and > via the second half-pi pulse; (5)detect the final state. Atom Interferometry, edited by P. Berman (Academic Press, San Diego, 997)

11 Ramsey interferometry via independent particles (general phase measurement) Wineland et al., Spin squeezing and reduced quantum noise in spectroscopy. Phys. Rev. A 46, R6797-R6800 (99). Braunstein, Quantum limits on precision measurements of phase. Phys. Rev. Lett. 69, (99).

12 Frequency measurement via independent particles input readout independent particles standard quantum limit (shot noise limit)

13 Ramsey interferometry via cat state (OO state) (general phase measurement) Wineland et al., Spin squeezing and reduced quantum noise in spectroscopy. Phys. Rev. A 46, R6797-R6800 (99). Braunstein, Quantum limits on precision measurements of phase. Phys. Rev. Lett. 69, (99).

14 Frequency measurement via cat state (OO state) Cat-state preparation and read-out require distinguishing all atoms. How about indistinguishable systems? (BEC) Fringe pattern with period π/ Heisenberg limit cat-state atoms

15 .. Interferometry with Bose condensed atoms Michelson interferometers () The BEC is split at t=0 into two momentum components ±kl using a double pulse of a standing light wave. () A Bragg scattering pulse at t=t/ then reverses the momentum of the atoms and the wave packets propagate back. (3) At t=t the split wave packets overlap and a third recombining double pulse completes the interferometer. To apply a phase shift between the two paths, a magnetic field gradient was turned on for a short time while the atom clouds were separated. Wang et al., 005, An atom Michelson interferometer on a chip using a Bose-Einstein condensate, Phys. Rev. Lett. 94,

16 Double-well interferometers () Preparation: a single BEC coherently splits into two by increasing the potential barrier. () Phase shift: an interaction may be used to induce the phase shift between two BECs. (3) Interference: the split BECs in the two wells are recombined to observe the interference. (I) Optical potentials (optical trap + laser barrier) Shin et al., 004, Atom interferometry with Bose-Einstein condensates in a double-well potential, Phys. Rev. Lett. 9, (II) Magnetic potentials (atom chips) Schumm et al, 005, Matter-wave interferometry in a double well on an atom chip, ature Phys., 57.

17 (I) Optical potentials (optical trap + laser barrier) (II) Magnetic potentials (atom chip)

18 Ramsey interferometers with two-component systems JILA, LES, AU, Heidelberg Uni., SUT, etc.

19 Potential Applications () High-precision quantum frequency standards (atom clocks) Atomic transitions are very useful to measure time or frequency with very high accuracy that the definition of a second is based on them. Starting with a system of non-interacting atoms in the ground state 0>, an electromagnetic pulse is applied to create equal superposition of 0> and of an excited state > for each atom. A subsequent free evolution of the atoms for a time t introduces a phase factor between the two states, wt, where w is the frequency of the transition between 0> and >. At the end of the free evolution, a second electromagnetic pulse is applied and then the probability for the final state in 0> (Ramsey interferometry) is measured. > probe laser > coupling laser 0>

20 () High-precision measurements of physical constants Gravimeters (gravity), gryroscopes (rotation), and gradiometers ewton s constant G Tests of relativity Interferometers in orbit (GPS) Fine structure constant and ħ/m Cronin, Schmiedmayer, Pritchard, Rev. Mod. Phys. 8, 05 (009)

21 . Matter-wave interferometry.. Atomic matter-wave interference Coherent beam splitting

22 Interference of two freely expanding condensates

23 .. onlinear excitations onlinear excitations in D matter-wave interference

24 .3. Bose-Josephson junction (BJJ) Schematic diagrams for Bose-Josephson junctions: (a) an external Bose-Josephson junction linked by quantum tunneling, and (b) an internal Bose-Josephson junction via a twocomponent BEC linked by Raman fields.

25 Unified MF model for both external and internal BJJs

26 Rabi oscillation and macroscopic quantum self-trapping (MQST) Rabi oscillation onlinear systems, Ec 0 Linear systems, Ec=0

27 Experimental observation of MQST classical non - rigid z = n n pendulum n + n H = χm Ω cos( φ) Theory: Smerzi et al, PRL 79, 4950 (997) m Experiment: Oberthaler et al., PRL 95,0040 (005); PRL 05, 040 (00)

28 Shapiro resonance and chaos Poincare sections for a BJJ with a driving

29 Symmetry-breaking transition Theory: Lee et al., PRA 69, 0336 (004); Lee, PRL 0, (009); etc. Experiment: Oberthaler et al., PRL 05, 040 (00). Ω / ( χ)

30 Two characteristic time scales for slow dynsmics across the crtical point, () reaction time (how fast the system follows its ground state), τ = h / Δ r () transition time (how fast the system is driven), τ t = Δ g g ( t) / Universal dynamics near critical point ( t) dδ g dt ( t) The excitation gap over the ground state hω( hω + ECL) for Δ g ( t) = ( ECL) ( hω) for where, L = /, E = χ C hω / E hω / E ( g + g g ) < 0 C C L L τ = r τ t τ t τ r τ τ r r < τ, t > τ, t adiabatic evolution non - adiabatic evolution critical point (t=0)

31 Kibble-Zurek scalings near critical point Ω / ( χ) Lee, PRL 0, (009)

32 Ground states for symmetric systems, ) ( ) ( 8 ) ( / z C J J B n n n n E a a a a H χ δ + = Ω = + + v v h Regime State form Coherent matrix Fluctuations 0 / > Ω >> χ χ / 0 Ω χ 0 / < Ω >> χ χ ( ) ( ) )! / ( 0 / / a a + + ( )! 0 / a a ( ) ( ) ( )! 0 / a a i a j a ~ 0 i Δ i ~ Δ Δ i ~ / z J x J H χ + h = Ω 3. Many-body quantum interferometry

33 Resonant tunneling and interaction blockade in asymmetric systems Theory C. Lee, L.-B. Fu, and Yu. S. Kivshar, EPL 8, (008); Carr et al., Experiment P. Cheinet, I. Bloch, et al., Phys. Rev. Lett. 0, (008)

34 3.. Quantum spin squeezing and many-particle entanglement Quantum spin squeezing Jian Ma, Xiaoguang Wang, C. P. Sun, and Franco ori, arxiv:0.978

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36 Preparing spin squeezing by nonlinear interactions Kitagawa and Ueda

37 Spin squeezing and entanglement A symmetric state is entangled if and only if it violates the inequality,

38 3.. High-precision interferometry via spin squeezed states J ( J / φ) Δ Δ z ( J ) z ( φ) = z = = cosφ, Δ J max ξ, ( J ) z =, R z = / φ ξ R ξ ξ R R =, spin coherent state <, spin Dependent squeezed on ξ, R Δ state ( φ) achieves from standard quantum limit, Heisenber g limit, to super - Heisenberg limit. phase sensitivity

39 (Kitagawa, Ueda) strong nonlinearity via controlling spatial overlap strong nonlinearity via using Feshbach resonance

40 pair-correlated states from spin dynamics

41 H 3.3. High-precision interferometry via OO states δ Ω + + C / h = ( n n) ( a a + a a) + ( n n) = δj z ΩJ x + χj z Fock basis : spin basis : OO OO = = E 8 ( n =,n = 0 + n = 0,n = ) The OO state is a ground state for system of δ = J =,J z = + J =,J z 0, χ < = + 0 and Ω/ χ << Adiabatic preparation of OO state via dynamical bifurcation J z ( χ) Ω/ C. Lee, PRL 97, 5040 (006)

42 Beam splitting and recombination via dynamical bifurcation Ω=40 Ω=0 For a system of δ = 0 and χ < GS = CS SU() Ω OO = 0, if Ω = 40 Ω = 0, ( P + P ) Here, P = J = /, M = / and P =. J = /, M = + / are the ground and first - excited states for the system of Ω = 0 and 0 < δ < respectively. They can be used as two paths of a MZ interferometer. / χ,

43 Phase accumulation via the term of δjz Switch on the term δj z for a period of time T, ( -iδt (/) + iδt (/) OO e P + e P ) with ϕ = δt, which is the phase accumulated in a single - atom system. Extract the relative phase from the population information via a dynamical bifurcation from Ω/χ<< to Ω/χ >> Due to the indistinguishability, we can not use the proposals of Wineland et al. and Caves et al. At the side of Ω/ χ <<, the ground [first excited] states will be ( P + P / ) [( P P ) Therefore, the state after the dynamical bifurcation becomes cos ( ϕ / ) GS i sin( ϕ / ) whose populations are P and P GS FS = = / cos sin ] even for a very small Ω. FS, ( ϕ / ) = ( + cos( ϕ ))/ ( ϕ / ) = ( cos( ϕ ))/.

44 Schematic diagram for MZ interferometry via OO states of indistinguishable systems / n, Hamiltonia z x z J J J H χ δ + Ω = h, P and, P with P sin i P cos FS sin i GS cos P e P e P P CS, i i SU() + = = ϕ ϕ ϕ ϕ ϕ ϕ State Evolution

45 Keynotes negative coupling two paths nonlineari beam splitting/ path entangled tunnelling Raman ty ( χ < transitio two degenerate recombinat 0) (double ion Feshbach - well n (two - component d ground system), states dynamical state (OO state) resonance or for the dynamical condensate ) bifurcatio system of n bifurcatio n χ < 0 Advantages large total number of particle (in order of 0 3, 0 for systems of photons and trapped ions) reduced influence of environment (adiabatic evolution and closed sub- Hilbert space) measurement precision of Heisenberg limit (path entangled states) experimental possibility (double-well or two-component systems) Challenge adiabatic evolution requests long coherent time C. Lee, PRL 97, 5040 (006)

46 Summary 4. Summary and open problems In interferometers of Bose condensed atoms, the atom-atom interaction brings the nonlinearity to the system. Tuning the effective nonlinearity, symmetry-breaking transitions appear and the dynamics near the critical point obey the universal Kibble-Zurek mechanism. The spin squeezed states and OO state can be prepared by controlling the nonlinearity and these states can used for highprecision interferometry beyond the standard quantum limit. Open Problems - noises (quantum fluctuations and technical noises) - imperfect effects (atom loss and environment) - coupling between internal and external degrees of freedom - finite-temperature effects More details in, C. Lee, et al., arxiv:0.4734v3 (a review article)

47 Thanks for your attention! International senior scientist (000- talent program, our university) and postdoctoral positions available (my group)!