Topological Phases of Quantum Matter September 08 2014 ESI Vienna, Austria Dissipatively Induced Quantum Phases of Atomic Fermions Sebastian Diehl UNIVERSITY OF INNSBRUCK Institute for Theoretical Physics, Innsbruck University, and Institute for Theoretical Physics, Technical University Dresden Collaborations: J. C. Budich, M. A. Baranov, P. Zoller (Innsbruck) C. Bardyn (Caltech), A. Imamoglu (ETH)
Motivation Many-body physics with cold atoms Vortices Fermion superfluid Bose-Einstein Condensate Mott Insulator (1999) (2003) (1995) (2002) Common theme: ➡ thermalization/equilibration (PennState, many-body Berkeley, Chicago, ...) system Temperature T, ➡ sweep and quench many-body dynamics particle number N (Munich, Vienna) ➡ metastable excited many-body states • closed system (isolated from environment) (Innsbruck, MIT, ...) • stationary states in ➡ ... thermodynamic equilibrium
Motivation Many-body physics with cold atoms Vortices Fermion superfluid Bose-Einstein Condensate Mott Insulator (1999) (2003) (1995) (2002) Common theme: Novel Situation: Cold atoms as open many-body systems drive (e.g. laser) many-body many-body system system Temperature T, particle number N dissipative environment • closed system (isolated from environment) • natural occurrences • use manipulation tools of • stationary states in of dissipation quantum optics thermodynamic equilibrium ➡ no immediate condensed ➡ drive/dissipation as dominant matter counterpart resource of many-body dynamics!
Outline Many-body physics with • basic idea + Ω − Ω tailored dissipation .. • pairing mechanism Dissipatively induced J α - = 0 • potential application: Cooling of + + i .. fermionic pairing atomic Fermi-Hubbard model - • targeted cooling into topological states Topology by • phys. realization with cold atoms dissipation 3 • characteristic many-body 2 1 properties in 1 and 2 dimensions 0 - 1 - 2 - 3 - 3 - 2 - 1 0 1 2 3 k x
Many-body physics with tailored dissipation + Ω − Ω SD et al., Nature Physics (2008) B. Kraus, SD, et al PRA (2008)
Many-Body Physics with Dissipation: Description • Many-Body master equations Lindblad operators J i ρ J † 2 { J † X ( i − 1 ∂ t ρ = − i [ H, ρ ] + κ i J i , ρ } ) system bath i coherent evolution dissipative evolution -- Liouvillian operator ➡ extend notion of Hamiltonian engineering to dissipative sector ➡ microscopically well controlled non-equilibrium many-body quantum systems ➡ here: focus on H = 0 • Important concept: Dark states J i | D i = 0 8 i ) L [ | D ih D | ] = 0 ρ = | D ih D | ➡ time evolution stops when
Many-Body Physics with Dissipation: Description • Many-Body master equations Lindblad operators J i ρ J † 2 { J † X ( i − 1 ∂ t ρ = − i [ H, ρ ] + κ i J i , ρ } ) system bath i coherent evolution dissipative evolution -- Liouvillian operator • Interesting situation: unique dark state solution B. Kraus, SD et al. PRA 08 Hilbert space • dark subspace one-dimensional • no other stationary solutions dark subspace t →∞ � ! | D ih D | ➡ directed motion in Hilbert space ρ ➡ dissipation increases purity
SD et al. Nat. Phys. (2008) Dark states: An analogy F. Verstraete et al. Nat. Phys. (2009) • optical pumping: three internal (electronic) levels (Aspect, Cohen-Tannoudji; Kasevich, Chu) dark state bright state • 1 atom on 2 sites: external (spatial) degrees of freedom ( a † 1 + a † ( a † 1 � a † 2 1 2 ) | vac � 2 ) | vac ⇥ anti-symmetric symmetric • N atoms on M sites | BEC i = 1 ⌘ N ⇣ X a † | vac i ` N ! ` ➡ combination of drive and dissipation enables purification (no conflict with second law of thermodynamics)
Sketch of implementation with cold bosonic atoms • Lindblad operators for BEC dark state: locally mapping any antisymmetric component into the symmetric one J i = ( a † i + a † i +1 )( a i − a i +1 ) by immersion of driven system into + Ω − Ω BEC reservoir (i) Drive: coherent coupling to auxiliary system with double wavelength Raman laser Rabi frequency b driving laser auxiliary system 2 1 system of interest a 1 a 2 λ laser = 2 λ lattice long times
Sketch of implementation with cold bosonic atoms • Lindblad operators for BEC dark state: locally mapping any antisymmetric component into the symmetric one J i = ( a † i + a † i +1 )( a i − a i +1 ) by immersion of driven system into + Ω − Ω BEC reservoir (ii) Dissipation: phonon emission into superfluid reservoir reservoir superfluid b driving laser reservoir auxiliary system 2 1 system of interest a 1 a 2 long times
Summary: Dissipative Many-Body State Preparation • Lindblad operators for BEC dark state: J i = ( a † i + a † J i | BEC i = 0 i +1 )( a i − a i +1 ) ➡ Long range phase coherence/ boson condensation builds up from quasilocal dissipative operations • Uniqueness of stationary solution can be shown (for fixed particle number) ➡ Ordered phase reached from arbitrary initial state � � � � = ) ρ ( t ) � ! | BEC ih BEC | for t ! 1 � � � � � � � � ��� � � � � � � � � ���� ������� First experimental �� � � � � � � � realizations � � � � � � � � � � � 0.4 Re � � 4 � � 0.3 0.2 0.1 0 � � � � � 0000 � � � � � � 0001 � � � � � � � � � � 1 � 1 1 ��� ��� ��� ��� ��� 1 1 1 � � � � � � � � � � � � 1 � 1110 � � 1 � � � � � � � � � � 1 � � � � 1 1 � � � � � � � � � � 1111 � 0 � � � � � 0 � � � � 0 � � � � � � 0 0 0 � � � � 0 1 � � � � 0 � � � � � Entanglement by dissipation � � � � � in atomic spin system Open-system simulator (Polzik group, Kopenhagen, PRL 2011) with trapped ions (Blatt group, Innsbruck, Nature 2011)
Dissipatively Induced Fermion Pairing ... J α +- = 0 + i ... - SD, W. Yi, A. Daley, P. Zoller, PRL (2010); W. Yi, SD, A. Daley, P. Zoller, New J. Phys. (2012);
Motivation: Fermi-Hubbard Model Quantum Simulation • Goal: finding ground state of Fermi-Hubbard model • Clean realization of fermion Hubbard model possible • Detection of Fermi surface in 40K (M. Köhl et al. PRL 05) • Fermionic Mott Insulators (R. Jördens et al. Nature 08; U. Schneider et al., Science 08) • Cooling problematic: small d-wave gap sets tough requirements T/E F BCS superconductors Unitary continuum Fermi gas SF transition Critical temperature Current lattice experiments for d-wave SF ➡ Still need to be 10-100x cooler • Roadmap via dissipative quantum state preparation approach : (1) Dissipatively prepare pure (zero entropy) state close to the expected ground state (2) Adiabatic passage to the Hubbard ground state
The State to Be Prepared High-Tc cuprate phase ... diagram +- d-wave SC + ... y - product state x d † = | BCS N ⇧ ⇥ ( d † ) N/ 2 | vac ⇧ [ c † i + e x , ↑ + c † i − e x , ↑ − ( c † i + e y , ↑ + c † i − e y , ↑ )] c † X i, ↓ i • Features shared with expected Hubbard ground state : (1) Quantum numbers ➡ no phase transition crossed in preparation process: gap protection (2) Energetically close? ➡ off-site pairing avoids excessive double occupancy ➡ Task: find “parent Liouvillian” for this state ➡ “cooling” into the d-wave
SD, W. Yi, A. Daley, P. Zoller, PRL 105 (2010) Pairing mechanism Antiferromagnet • Consider 1D cut only d-wave SC • Half filling: Neel state for antiferromagnetism full set: • Lindblad operators (1D): e.g. flip! ➡ dark state based on Fermi statistics flip! 0 • D-wave (analog) state: interpret the state as a symmetrically delocalized Neel order | BCS 1 i = ( d † ) N | vac i , d † = ( c † i +1 , ↑ + c † i − 1 , ↑ ) c † X i, ↓ i i, − = ( c † i +1 , ↑ + c † J + i = j + i, + + j + i − 1 , ↑ ) c i, ↓ • Lindblad operators (1D): e.g. phase locking ➡ Combine fermionic Pauli blocking with delocalization as for bosons
Dissipative Pairing: The d-wave jump operators • The full set of Lindblad operators is found from i , G † ] = 0 [ J α ∀ i, α | D ( N ) i ⇠ G † N | vac i • given by Pauli matrices ✓ ◆ c ↑ ,i c i = i = ( c † i +1 + c † c ↓ ,i i − 1 ) σ α c i J α en ,8. ' % 637,/012345. & /012345 ! • Discussion: These operators �� % �� • form exhaustive set: d-wave steady state unique, reached $ !"" # $! for arbitrary initial state !"" # $! ()*+",$- � . • bilinear: describe the redistribution of the superposition of a 1 �� (b) �� single particle 0.8 0 log(1 ! Fidelity) Fidelity 0.6 • � generalization to arbitrary symmetries possible ! 2 0.4 ! 4 0.2 0 25 50 0 0 20 40 60 80 Time (1/ ! ) ➡ Projective pair condensation mechanism, does not rely on attractive conservative forces
Recommend
More recommend
