GREENBERGER- HORNE- ZEILINGER (GHZ) STATES IN QUANTUM DOT MOLECULE

Transcription

1 GREENERGER- ORNE- ZEILINGER (GZ) STATES IN QUANTUM DOT MOLECULE A. SARMA and P. AWRYLAK QUANTUM TEORY GROUP INSTITUTE FOR MICROSTRUCTURAL SCIENCES NATIONAL RESEARC COUNCIL OF CANADA OTTAWA, KAOR,CANADA

2 Outlne : Introducton (what are GZ states and why are they mportant? ) Realzng GZ states n : Three Spn System (esenberg amltonan) Quantum Dot Molecule (etended-ubbard amltonan) Summary & Concluson

3 Introducton : Prncple of superposton and entanglement are the basc ngredents n developng a quantum computer. Entanglement plays an mportant role n applcatons lke quantum teleportaton, etc. and s also of fundamental nterest to test quantum mechancs aganst ssues lke non-localty, etc. Necessary to generate hghly entangled states lke two partcle ell states and the three (or mult-) partcle Greenberger- orne- Zelnger * (GZ) states ; GZ ± [ ± ] S S S y z * D.M.Greenberger, M.A.orne and A.Zelnger n ell s theorem, Quantum Theory and Conceptons of the Unverse. (989)

4 Generatng GZ States n a three spn system

5 S / -/4 J 4 I) Isotropc esenberg amltonan J S. S+ (S 4 S ) y y z z J ( σ σ + σ σ + σ σ where ; J > 0 ferromagnetc nteracton y z S (,, ) σ + σ σ + + ) S S S y z h 0 y h 0 σ ( ), σ 0 0 z h 0 ( ) and σ ( ) 0 II) Intal bass : III) Dagonalze : S / /4 J 4 / J

6 Egenstates correspondng to 4- fold degenerate ground and ected states,,,, - [ + + ] [ + + ],, 4,, - Two dstnct classes of mamally entangled states* π 4π [ + e + e ] 5,, π 4π [ + e + e ],, π 4 π [ + e + e ] π 4π [ + e + e ] 7,, - 8,, - ± ( n ) ± - K e π Sn n ± + π ( n ) S n n ± L e

7 Proposed* setup for generatng entangled states n three spn system Apply n- plane radal magnetc feld,, of same strength and summng to zero. z y z S y S N S S N S where ; r ê g e μ b π π y π π y [ σ - sn σ - cos σ - sn σ + cos σ ] ; r. S π π π π (- sn ê - cos ê ) ; (- sn ê + cos ê ) + + y r ; b geμ h y *. Roethlsberger et al, Phys. Rev. Lett. 00, 0050 (008)

8 Role of n- plane radal magnetc feld z y z S y acts as a perturbaton. S [ ] b + e + e N π 4π S S N b S And smlarly b 7 7 b So the transton from to s a thrd order process n b..e., 7 7 b

9 Constructng the full amltonan usng the egenstates of S/ The full amltonan s b whch s of type b b b J b 0 b J S/ We apply degenerate parttonng perturbaton theory on the upper block up to rd order to obtan effectve amltonan n the subspace S/. eff 0 8b J 0 8b J Dagonalzng the above effectve amltonan yelds one of the two GZ states, GZ ± >, as the ground state wth egenvalue -8b /J. The energy splttng between the two GZ states s b / J

10 Generatng GZ States n Quantum Dot Molecule

11 GS of TQD Molecule wth three and four electrons * QDM + + ε nσ tj cσ c jσ U nσ n- σ Vj ρρ j where n σ c σ cσ and ρ σ j σ j σ n σ I) N elec. At half- fllng ubbard amltonan reduces to so. esenberg amltonan wth J AF ~ 4t /(U-V) where t < V < U t, V GS s n subspace of total spn S/ but we need t to be n mamally spn polarzed subspace of total spn S/. I) N elec. 4 asng the dot GS n S GS n S0 * Y.-P.Shm and P.awrylak, Phys. Rev., 78, 57 (008)

12 TQDM wth four electrons and GS n subspace of total spn S

13 SQDM wth nne electrons Can the GS of SQDM be tuned n a mamally spn polarzed subspace of total spn S /?

14 4 t, V 5 S Quantum Dot (SQD) molecule wth nne electrons. t, V System descrbed wth n an etended ubbard amltonan as gven below wth U» (V, V', t, t') and t > t', V > V'. Central dots (,,) are on resonance wth each other at energy zero. And the on- ste energes of aulary dots (4,5,) s vared by ε such that ε 0. amltonan matr s block dagonal n subspace of S tz wth dmensons 0 (S tz /, -/) and 90 (S tz /, -/). SQD σ ε n σ + j σ t j c + σ c jσ + U n σ n -σ + j V j ρ ρ j

15 Domnant Charge Confguratons US [ ] DS [ ] We evaluate the contrbuton of domnant charge confguratons n formng the ground state (GS) by defnng an overlap probablty, P, whch s gven as ; P 0 < GS US >

16 Phase dagram of overlap probablty, P, as a functon of t'/t and V'/V. ubbard parameters (n unts of eff. Rydberg) U.0 ; V0. For gven ponts n phase dagram we evaluate J

17 Setup for generatng GZ states n Quantum Dot Molecule I) X Y Z SQD + r r ê MAG ; r e MAG g μ. S π π ( ê -cos ê ) -sn π π π ( ê cos ê ); ( sn ê cos ) π + + -sn y 4 êy r r r π π ( ê - cos êy ); - 5 sn ê y II) Dagonalze and obtanng GS we calculate the energy dfference, ΔE, between GS and FS.e., two GZ states, entanglement measure, tangle τ, and the overlap probablty : P 0 < GS GZ + > ± where GZ [ US ± DS ]

18 Energy splttng, ΔE

19 Mamum overlap probablty,p

20 Summary & Concluson : We analyzed the proposal on generatng entangled states n three spn system wth an eternal magnetc feld. We understood the role of the radal n- plane feld n generatng GZ state as GS of three spn system. We proposed a desgn of s quantum dot molecule wth nne electrons along wth an nplane radal magnetc feld as a generator of GZ state n quantum dot molecule system. We evaluated the overlap probablty of the ground state of the system wth GZ state and conclude that wth proper choce of ubbard parameters the overlap probablty can be mamzed thus generatng an appromate GZ state as ground state of SQD molecule.