Controlling and Measuring Heat Transport in Ion Traps April 22nd 2015 Alejandro Bermudez Martin Bruderer Martin B. Plenio Heat Transport on the Nanoscale Carbon Nanotubes Molecular Nanojunctions Ultracold Atomic Gases Quantum Dots & Tunnel Junctions Ion Crystals Overview Physics in Ion Traps Experimental Tools Heat Transport Quantum Dots Fourier’s Law Trapped Ion Set-up • Singly-charged ions (Be+, Ca+, Mg+, …) • Coulomb interaction between ions • Single ions & ion crystals • Control over motional & internal states (QIP) H.C. Nägerl, W. Bechter, J. Eschner, F. Schmidt-Kaler & R. Blatt, Appl. Phys. B 66, 603 (1998) J. Eschner, G. Morigi, F. Schmidt-Kaler & R. Blatt, J. Opt. Soc. Am. B 20, 1003 (2003) ted Ising spins with Examples - Experiments y spin having unit magnetic andset wePlanck’s have setconstant, Planck’s constant, , having pin unit magnetic moment,moment, and we have 3 For three define ; as J1,2the 5nearestJ2,3 as the nearest,.equalh,toequal one. to Forone. three spins, wespins, definewe J1 ; J1,2 5J1J2,3 neighbour interaction J1,3next-nearest-neighbour as the next-nearest-neighbour s eighbour interaction and J2 ;and J1,3 Jas 2 ;the (i) (i) interaction (Fig. sa the denote Pauliof matrices of the ith spin. n nteraction (Fig. 1b), and1b), sa and denote Pauli the matrices the ith spin. We initialize each spin parallel to atransverse strong transverse field and then n initialize We each spin parallel to a strong field and then Observing Frustration in Spin Systems on Heat Transport diabatically lower fieldthe relative the Ising couplings so that theso that Experiments adiabatically lower field relative to the Ising the e 2 the 3 to 3 couplings 1 2 Freericks , G.-D. Lin , L.-M. Duan & C. Monroe e d a a b b J 2 J2 n AF AF 4 MHz and n3 5 3.674 MHz. Off-resonance laser beams s 3 1 2 3 1 2 ye illuminate the ions, driving stimulated Raman transitions the forces in e spin states AFand also imparting Jspin-dependent J 1 1 J1 J1 quantum ction. As discussed in Methods, this allows simu10–12 n cthe Zigzag Ising Hamiltonian with Tilt a transverse field CM c n Zigzag X TiltX CM (i) (j) (i) e H~ Ji,j sx sx zBy sy ð1Þ - 3 ivj III i IV 3 II I t J1 IV III II I fis an effective J2 J1 uniform transverse magnetic field with each s J2 • Internal states = effective spins ? ? J (kHz) J (kHz) ng unit magnetic moment, and we have set Planck’s constant, 0 oe one. For• three spins, we J1 ;modes J1,2 5 J2,3 as the nearestInteractions viadefine collective 0 er interaction and J2 ; J1,3 as the next-nearest-neighbour ground states • Determine a (i) n (Fig. 1b), and sa denote the Pauli matrices of the ith spin. ol –3 each spin parallel to a strong transverse field and then ylize M. Ramm et al., New J. Phys. 16, 063062 (2014) K. Kim et al., 590 –3 –2 Nature 465, –1 (2010) 0 ~ µ ally to the e lower Islamfield et al.,relative Science 340, 583 Ising (2013) couplings so that the –3R. the A. Bermudez et al., PRL 111, 040601 (2013) –3 –2 New J. Phys.~ 14,–1093042 (2012) 0 ngure 1 | Frustrated A. Bermudez et al., Ising spins. a, Simplest case µ of spin frustration, with d hree antiferromagnetic spins on a triangle. b, Image of three trapped atomic 1 bspins. | Frustrated a, intensified Simplest case of spin frustration, with Yb1 Figure ions in 1the experiment,Ising taken with an charge-coupled- this Otto engine runs in- finite time and has thus Examples Proposals Verifying the Quantum Jarzynski Equality Single Ion Heat Engine (Otto Cycle) work on system !(0) F = he W/kT !(⌧ ) kT lnhe i= Z W/kT i dW p(W )e W/kT 1 over (color Control initial online). & final states •FIG. (a) Energy-frequency diagram of the • Control over trapping cycle ion. and The continuous line Resolve singlefor Fockthe statesradial mode of the cooling of single ions •Otto • Heating thedone ideal process, while •the dots motion showofthe of Determine work on system Generate axialresults mode •represents the Monte Carlo simulations. (b) The pictograms illustrate the G. Huber, F. Schmidt-Kaler, S. Deffner & E. Lutz, PRL 101, 070403 (2008) R. Dorner, S.R. Clark, L. Heaney, R. Fazio, J. Goold & V. Vedral, PRL 110,for 230601the (2013) radial state. four individual strokes of Energy-frequency the cycle 1 (color online). (a) diagram of the lor FIG. online). (a) Energy-frequency diagram of the O. Abah, …, F. Schmdt-Kaler, … & E. Lutz, PRL 109, 203006 (2012) (c) the Paul trap: theThe rf electrodes cycle mode for of theof radial mode ofcontinuous the ion. continuoushave line for Otto the Geometry radial thetapered ion. The line What are the basic tools? Motional and Internal States • Motional and internal states are decoupled • Manipulate internal states via laser coupling |f i |ni |2i |1i |0i 1 V (x) = m! 2 x2 2 motional states (harmonic trap) 1 p (|gi + |ei) 2 ⌦⇠I |ei |gi internal states (electron configuration) laser !L = !eg Optical Pumping - State Preparation • Resonant driving of transition |ei to |f i • Radiative decay into |ei and |gi |ni |2i |1i |0i 1 V (x) = m! 2 x2 2 motional states (harmonic trap) |f i !ef |ei |gi internal states (electron configuration) Resonance Fluorescence - Read Out • Resonant driving of transition |ei to |f i • See that level |ei is populated |ni |2i |1i |0i 1 V (x) = m! 2 x2 2 motional states (harmonic trap) |f i !ef |ei |gi internal states (electron configuration) Resonance Fluorescence - Read Out • Resonant driving of transition |ei to |f i • See that level |ei is populated |ni |2i |1i |0i 1 V (x) = m! 2 x2 2 motional states (harmonic trap) |f i !ef |ei |gi internal states (electron configuration) State-dependent Potential • AC-Stark shift from far-detuned driving • Position-dependent intensity ⌦(x) |ei { |ei |ei ⌦2 (x) Ve (x) = 2 !e > ! ⌦(x) |gi |gi |gi !g < ! Vg (x) = ⌦2 (x) 2 Couple Motional and Internal States • Drive transitions that change HO state • Requires narrow HO states |0, ei |1, ei carrier |2, ei blue sideband red sideband |0, gi |1, gi |2, gi ··· Spin-Mode Coupling • Internal states act as spins |ei ⌘ | "i |gi ⌘ | #i • Spin-spin interactions via common mode) |0, ei a† +a |2, ei |1, ei a† + + +a + |0, gi |1, gi = |eihg| = |gihe| |2, gi Hint ⇠ † (a + a ) x Sideband Cooling • Combine coherent driving & spontaneous decay • Control temperature down to hni ~ 0.05 • Doppler cooling hni ~ 3 - 10 |0, ei |0, gi |1, ei |1, gi |2, ei |2, gi hni ' ✓ ◆2 ! ⌧1 Temperature Measurement • Thermal population of motional state hni • Start in |gi and observe population in |0, ei |0, gi |1, ei |1, gi |ei after fixed time |2, ei |2, gi Pered hni = blue Pe 1 + hni Temperature Measurement • Thermal population of motional state hni • Start in |gi and observe population in |0, ei |0, gi |1, ei |1, gi |ei after fixed time |2, ei |2, gi Pered hni = blue Pe 1 + hni Collective Modes Hamiltonian for motional ion states H= X ⇣ p2 i ⌘ e2 X 1 1 i 2 + ri ! ri + 2m 2 2 |ri rj | i6=j Axial modes Radial modes harmonic oscillator modes ~ collective modes. Fourier’s Law Fourier’s Law J. Fourier, Théorie Analytique de la Chaleur (1822) Fourier’s Law J. Fourier, Théorie Analytique de la Chaleur (1822) Fourier’s Law • How does Fourier’s law emerge from microscopic laws? • When do temperature gradients occur? • Observe Fourier’s law on nanoscales? Z. Rieder, J. L. Lebowitz & E. Lieb, J. Math. Phys. 8, 1073 (1967) M. Michel, G. Mahler & J. Gemmer PRL 95, 180602 (2005) A. Asadian, D. Manzano, M. Tiersch & H. J. Briegel, Phys. Rev. E 87, 012109 (2013) Trapped-Ion Crystal Toolbox Functionalities of ions • Full control over internal states • Separation of time scales • Different atomic species • Bulk ions • Heat reservoir ions • Multi purpose ions Vibron Hopping Model Hamiltonian for motional ion states H= X ⇣ p2 i ⌘ e2 X 1 1 i 2 + ri ! ri + 2m 2 2 |ri rj | i6=j Tight-binding model for vibron hopping H= X i !i a†i ai + X Jij a†i aj + h.c. i6=j ! ⇠ 1MHz J ⇠ 10kHz • Small radial oscillations above ground state (vibrons) • Coupling from dipole-dipole interaction J ⇠ 1/d3 • Heat transport by vibron hopping D. Porras & J. I. Cirac, PRL 93, 263602 (2004) chain of weakly coupled harmonic oscillators Heat Reservoirs Tight-binding model for vibron hopping H= X !i a†i ai + i X Jij a†i aj + h.c. i6=j n ¯L n ¯R Continuous heating/cooling of edge ions L ↵ ⇢↵ = ↵ 2 (¯ n↵ + 1)(2a⇢↵ a† • Effective cooling rate {a† a, ⇢↵ }) + ↵ 2 n ¯ ↵ (a $ a† ) ↵ = {L, R} ↵ • Reservoirs at constant temperature n ¯↵ • Cooling much faster than hopping ↵ Jij Spin-Vibron Coupling Tight-binding model for vibron hopping H= X !i a†i ai i + X Jij a†i aj + h.c. i6=j n ¯L n ¯R Couple internal states to vibrons HiSV 1 = (A + 2 z i ) cos(⌫t 1 A cos(⌫t)a†i ai 2 1 z † = i ai ai 2 )a†i ai (1) HiSV = ⊳ Photon-assisted tunneling (2) HiSV ⊳ Probing & Disorder Ballistic Transport Vibron occupations Jij and project Assume ↵ th dynamics onto state ⇢th ⌦ ⇢ ⌦ ⇢ bulk L R hni iss = hI vib iss = ¯L Ln + L+ L R L + R ¯R Rn R (¯ nL n ¯R) • Ballistic transport of vibrons • Anomalous heat transport Spin-Induced Disorder Strong spin-vibron coupling HiSV 1 = 2 Spins in superposition 1 p |+ii = (| "ii + | #ii ) 2 z † i ai ai Random binary alloy (RBA) model (bosonic) H RBA = X "i a†i ai + i X Jij a†i aj + h.c. i6=j Binary diagonal disorder "i 2 {!i 1 2 , !i + 1 2 } Exploit “quantum parallelism” B. Velicky, S. Kirkpatrick & H. Ehrenreich, Phys. Rev. 175, 747 (1968) B. Paredes, F. Verstraete & J. I. Cirac, PRL 95, 140501 (2005) Fourier’s Law Ballistic Disorder Homogeneous Inhomogeneous Clear signature of Fourier’s law Dephasing from Electrodes Dynamic fluctuations from noisy electrodes Voltage Tight-binding model with dephasing noise H = X "i (t)a†i ai i + X Jij a†i aj + h.c. i6=j Markovian noise with correlation length ⇠c "i (t) = !i + !(t) Description as Lindblad operator L ⇢= X i,j e dij /⇠c (2ni ⇢nj {nj ni , ⇢}) Fourier’s Law Ballistic Dephasing Disorder Homogeneous Inhomogeneous Thermal Quantum Dot Single site & thermal leads | # · · · #i| i| " · · · "i !# ⌧ ! !" Photon-assisted tunneling HiSV H 1 = A(1 + 2 PAT = X ! =A † z ) cos(⌫t)a i i ai JiPAT a†i a + h.c i Full control of heat current through TQD Single-spin heat switch | i = | "i | i = | #i Nonequilibrium Schrödinger’s Cat Single site & thermal leads | # · · · #i| i| " · · · "i !# ⌧ ! !" Photon-assisted tunneling HiSV H 1 = A(1 + 2 PAT = X ! =A † z ) cos(⌫t)a i i ai JiPAT a†i a + h.c i Full control of heat current through TQD Single-spin heat switch | i = | "i | i = | #i 1 | i = p (| "i + | #i) 2 Verify Fluctuation Theorems Single site & thermal leads | # · · · #i| i| " · · · "i !# ⌧ ! !" ! Verify fluctuation theorems for bosons (switch on) p(N, t) lim = e! N ( t!1 p( N, t) D S) N = number of vibrons Use Ramsey probe to measure current M. Esposito, U. Harbola & S. Mukamel, Rev. Mod. Phys. 81, 1665 (2009) Non-Invasive Ramsey Probe |↑〉 Operator couples weakly to spin HiSV = |↓〉 π/2 pulse x ii 1 hOi iss t e = cos S(!) = π/2 pulse z i Spin evolution h free induction 1 Oi 2 Z 2 S(0)t 1 0 dt hhOi (t)Oi (0)iiss e ⊳ Oscillations with frequency ⊳ Damping by fluctuations i!t O δO² Measure occupations and thermal currents -1 MB and D. Jaksch, New J. Phys. 8, 87 (2006) G. B. Lesovik, F. Hassler & G. Blatter, PRL 96, 106801 (2006) Summary Implement thermal reservoirs Measure currents & local temperatures Control & manipulate heat flow Analytical Results Local vibron hopping and dephasing Vibron occupations Vibron current Long-range coupling 1/d³ probably not essential A. Asadian, D. Manzano, M. Tiersch & H. J. Briegel, Phys. Rev. E 87, 012109 (2013) Zig-zag Crossover Reduce radial trapping for zig-zag crossover linear zig-zag zag-zig • Anomalous transport for linear chain • Zig-zag results in temperature gradient and suppressed heat current • Nonlinearity & coupling of axial and radial modes K. Pyka et al., Nat. Commun. 4, 2291 (2013) A. Ruiz, D. Alonso, M. B. Plenio & A. del Campo, Phys. Rev. B 89, 214305 (2014) N. Freitas, E. Martinez & J. P. Paz, Preprint arXiv:1312.6644 (2013) Experimental Status • Heat transport in chain with up to 37 ions • Measure vibron occupation via cooling time • Coupling 10kHz stronger than cooling rate (need ↵ Jij ) M. Ramm, T. Pruttivarasin & H. Häffner, New J. Phys. 16, 063062 (2014) T. Pruttivarasin, M. Ramm, I. Talukdar, A. Kreuter & H. Häffner, New J. Phys. 13 075012, (2011)
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