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Information about Quantum information probes

Talk delivered at the Reletivistic Quantum Metrology, University of Nottingham 7-8 March 2014.

Prepared by the DScien team: www.dscien.com

Prepared by the DScien team: www.dscien.com

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Relativistic Quantum Metrology 7-8 March 2008, Nottingham

Quantum Wonderland

Quantum Wonderland

?

M. Steiner et al., Phys. Rev. Lett. (2013) single trapped ion in optical ﬁber cavity

H. Ott’s group, Kaiserslauten Rb atoms in 2D optical lattice

Quantum simulators

initialize

initialize

engineer H

engineer H

read out

read out

Condensed Matter systems Superﬂuid Superﬂuid Mott insulator I. Bloch’s group, 2002

Condensed Matter systems Single-site addressing S. Kuhr’s and I. Bloch’s group

Open Quantum Systems trapped ions quantum simulator An open-system quantum simulator with trapped ions, Julio T. Barreiro, Markus Müller, Philipp Schindler, Daniel Nigg, Thomas Monz, Michael Chwalla, Markus Hennrich, Christian F. Roos, Peter Zoller and Rainer Blatt, Nature 470 , 486-491 (2011)

Dirac Equation trapped ions quantum simulator Quantum simulation of the Dirac equation R. Gerritsma, G. Kirchmair, F. Zähringer, E. Solano, R. Blatt and C.F. Roos, Nature 463, 68 (2010)

2D Ising Model trapped ions quantum simulator 100 N 350 Engineered two-dimensional Ising interactions in a trapped-ion quantum simulator with hundreds of spins J.W. Britton, B.C. Sawyer, A.C. Keith, C.-C.J. Wang, J.K. Freericks, H. Uys, M.J. Biercuk, and J.J. Bollinger, Nature 484, 489 (2012)

Problem: Read out

Problem: Read out

Problem: Read out

Problem: Verification

Benchmarking

Benchmarking Problems with known solutions

Benchmarking Problems with known solutions Alternative measurement strategies

What if...

indirectly

indirectly

with minimal disturbance

m m Co ex pl te ys S

1 KEY IDEA Local Probe em ex m Co pl t ys S

2 KEY IDEA ENVIRONMENT em ex m Co pl t ys S

PROBE DECOHERENCE Depends on the state/properties of the complex systems m ex pl m Co te ys S

SHIFT in PERSPECTIVE

3 KEY IDEA New Tools

⇢(t) = t ⇢(0) dynamical map quantum channel

t,0 = t,s s,0 divisibility

t,0 = t,s s,0 Markovian dynamics Master equation in Lindblad form

Non-Markovian dynamics t,0 6= t,s s,0 Entanglement and Non-Markovianity of Quantum Evolutions Ángel Rivas, Susana F. Huelga, and Martin B. Plenio Phys. Rev. Lett. 105, 050403 (2010) On the degree of non-Markovianity of quantum evolution Dariusz Chruściński, Sabrina Maniscalco arXiv:1311.4213, in press in Phys. Rev. Lett.

Information ﬂow

Markovian dynamics

Non-Markovian dynamics re-coherence

Quantum information and distinguishability between quantum states Increase of information Increase of distinguishability Measure for the Degree of Non-Markovian Behavior of Quantum Processes in Open Systems H.-P. Breuer, E.-M. Laine, and J. Piilo, Phys. Rev. Lett. 103, 210401 (2009) Measure for the non-Markovianity of quantum processes, Elsi-Mari Laine, Jyrki Piilo, and Heinz-Peter Breuer Phys. Rev. A 81, 062115 (2010)

Quantum information and distinguishability between quantum states Decrease of information Decrease of distinguishability

Distinguishability between two states of the Q probe 1 D(⇢1 , ⇢2 ) = Tr|⇢1 2 ⇢2 |, Rate of change of distinguishability d (t, ⇢1,2 (0)) = D(⇢1 (t), ⇢2 (t)) dt

Markovian dynamics (t, ⇢1,2 (0)) 0 at all times Non-Markovian dynamics (t, ⇢1,2 (0)) > 0 for some time intervals

MAXIMUM Information Backﬂow N ( ) = max ⇢1,2 (0) Z dt (t, ⇢1,2 (0)) >0 Measure for the Degree of Non-Markovian Behavior of Quantum Processes in Open Systems H.-P. Breuer, E.-M. Laine, and J. Piilo, Phys. Rev. Lett. 103, 210401 (2009) Measure for the non-Markovianity of quantum processes, Elsi-Mari Laine, Jyrki Piilo, and Heinz-Peter Breuer Phys. Rev. A 81, 062115 (2010)

MAXIMUM Information Backﬂow NC = NQ = Z Z C (t)dt C >0 Q (t)dt Q >0 Non-Markovianity and reservoir memory: A quantum information theory perspective B. Bylicka, D. Chruściński, S. Maniscalco, arXiv:1301.2585

Q Information probes

Q PROBE strategy Quantifying information ﬂow between the Q probe and the complex system / quantum simulator

Ability of a quantum probe to indirectly extract information on a complex quantum system

1 Ultracold bosonic gas dimensionality

2 Ising model in a transverse ﬁeld

3 Trapped ion crystals

1 Ultracold bosonic gas dimensionality

2D 1D

Probing dimensionality phase ﬂuctuations density ﬂuctuations

Immersed probe atomic quantum dot Atomic Quantum Dots Coupled to a Reservoir of a Superﬂuid Bose-Einstein Condensate A. Recati, P. O. Fedichev, W. Zwerger, J. von Delft, and P. Zoller, Phys. Rev. Lett. 94, 040404 (2005) Probing BEC phase ﬂuctuations with atomic quantum dots M. Bruderer, and D. Jaksch, New J. Phys. 8, 87 (2006)

Immersed probe atomic quantum dot Atomic Quantum Dots Coupled to a Reservoir of a Superﬂuid Bose-Einstein Condensate A. Recati, P. O. Fedichev, W. Zwerger, J. von Delft, and P. Zoller, Phys. Rev. Lett. 94, 040404 (2005) Probing BEC phase ﬂuctuations with atomic quantum dots M. Bruderer, and D. Jaksch, New J. Phys. 8, 87 (2006)

4 Impurity atom VA x 2L 2D BEC p VB x Figure 1. A Bose–Einstein condensate (yellow region) co harmonic trap VB (x) interacts with cold impurity atoms each Quantifying, characterizing and controlling information ﬂow circle). The distance b in a double well Haikka, S. McEndoo,A (x) (grey in ultracoldS.atomic gases potential V G. De Chiara, M. Palma, and Maniscalco, P. Phys. Rev. A 84, 031602R (2011) the same trap is 2L and the distance between adjacent traps

4 Impurity atom VA x 2L 2D BEC VB x Figure 1. A Bose–Einstein condensate (yellow region) conﬁn harmonic trap VB (x) interacts with cold impurity atoms each of in a double well potential V A (x) (grey circle). The distance betw the same trap is 2L and the distance between adjacent traps is 2 QUANTUM PROBE HA = Z HB = describes the interactions between the impurities and the bath; here gAB = is the coupling constant of impurities–gas interaction, with aAB the scatteri impurities–gas collisions and m AB = m A m B /(m A + m B ) their reduced mass. B bath atoms are described in the second-quantized formalism. The ﬁeld operat impurities ⇧ ˆ ⌥(x) = ai, p ⇧i, p (x) ˆ p2 A d3 x ˆ † (x) + VA (x) ˆ (x) 2mA QUANTUM GAS Z i, p can be decomposed in terms of the real eigenstates ⇧i, p (x) of impurity atoms double well i of the potential VA (x) in the p th state, with energy h ⌅i, p and th ¯ annihilation operator ai, p . We assume that the wavefunctions of different dou ˆ negligible common support, i.e. ⇧i, p (x)⇧ j⌅=i,m (x) ⇤ 0 at any position x. We treat the gas of bosons following Bogoliubov’s approach (see, for in assuming a very shallow trapping potential VB (x), such that the bosonic gas c homogeneous. In the degenerate regime, the bosonic ﬁeld can be decomposed ⇧ ⌃ ⌃ ⇥ ˆ ˆ ⌃(x) = N0 ⌃0 (x) + ⌃(x) = N0 ⌃0 (x) + u k (x)ˆ k vk ( c p2 gB ˆ † 3 ˆ† B d x (x) + VB (x) + (x) ˆ (x) ˆ (x) 2mB 2 INTERACTION HAB = gAB Z k where ⌃0 (x) is the condensate wave function (or order parameter), N0 < N atoms in the condensate and ck , ck are the annihilation and creation operators o ˆ ˆ† ⇧ modes with momentum k. For a homogeneous condensate ⌃0 (x) = 1/ V , V b Its Bogoliubov modes ⌥ ⇤ ⌅ ik·x 1 ⇥k + n 0 gB e uk = +1 ⇧ , 2 Ek V d3 x ˆ (x) ˆ † (x) ˆ (x) ˆ (x) ⌥ ⇤ 1 ⇥k + n 0 gB vk = 2 Ek ⌅ ik·x e 1 ⇧ V

Qubit Probe |Li |Ri 4 Impurity atom VA x 2L 2D BEC p VB x Figure 1. A Bose–Einstein condensate (yellow region) co Pure DEPHASING harmonic trap VB (x) interacts with cold impurity atoms each in a double well potential V A (x) (grey circle). The distance b the same trap is 2L and the distance between adjacent traps

fo 4 a background gas particle. Furthermore, gk and ξk are te x coupling constants that depend on the spatial form Vof the p states |L and |R and on the shape of the Bogoliubov e modes. Their speciﬁc form is elaborated in Ref. [13]. is V When the background gas is at zero temperature the xreA condensate (yellow duced dynamics of the impurity atom harmonic trapBose–Einstein (x) (grey circle). Theatoms eachth is capturedwith cold impurity region) conﬁn V (x) interacts by the of in a double well potential V distance betw N the same trap following time-local master equation (ME):is 2L and the distance between adjacent traps is 2 F describes the interactions between the impurities and the bath; here g = is the coupling constant of impurities–gas interaction, with a the scatteri if (t) dρ(t) impurities–gas 1 collisions and m = m m /(m + m ) their reduced mass. B ⇢ij (t) = e z , ρ] + γ(t)[σz ρ(t)σatoms are describedzinσzsecond-quantized formalism. The ﬁeld operat ⇢ij (0) bath z − {σ the , ρ(t)}]. (2) M = Λ(t)[σ impurities dt 2 ⇧ aˆ ⇧ (x) p ˆ ⌥(x) = p Z t renormalizes the can be decomposed in termstherealqubit⇧ but atoms eigenstates (x) of impurity Quantity Λ(t) energy potentialofV the in the p state, with energy h¯ ⌅ andv double well i of the of (x) th annihilation operator a . We assume that the wavefunctions of different dou ˆ (t) qualitative (s) on the dissipativebosons ⇧following Bogoliubov’sany position(see, form ds eﬀect negligible common support,dynamics. Ini.e. (x)⇧ (x) ⇤ 0 at x. has no = We treat the gas of approach in assuming very potential (x), such that 0 work we are interested a in shallow trappingregime, theVbosonic ﬁeld canthe bosonic gasis stead in this the decay rate be decomposedc homogeneous. In the degenerate ⇧ ⌃ ⌃ ˆ ˆ ⌃(x) = N ⌃ (x) + ⌃(x) = N ⌃ (x) + u (x)ˆ a( c v 2 dk sin2 (k · L) where ⌃ (x) is thek t/¯ )wave function 2 τorder parameter), N < N sin(E condensate −k (or 2 /2 h 4gAB n0 atoms in the condensate and c , c aree annihilation and creation operators o ˆ ˆ the ,(x) = 1/⇧V , Vob γ(t) = (D) modes with momentum k. For a homogeneous condensate ⌃ h ¯ (2π)D n Its Bogoliubov modes + 2gB nD k ⌥ ⇤ ⌅ 1 ⇥ +n g e th u = +1 ⇧ , (3) 2 E V ⌥ ⇤ q ⌅ 1 ⇥ +n g e A Impurity atom 2L 2D BEC B Figure 1. B A AB AB AB A B A B i, p i, p i, p i, p th A i, p i, p i, p j⌅=i,m B 0 0 0 k 0 k k 0 0 † k k 0 k k vk = 0 B ik·x k k 2 0 B Ek ik·x 1 ⇧ V ⇥ k

Non-Markovianity: information ﬂow Ndeph recoherence: 3D information backﬂow 2D 1D aB /aRb FIG. 2. (Color online) Non-Markovianity measure Ndeph as information lost in background gas aB a function of the scattering length of the the environment when the background gas is three dimensional (red dashed line), quasi-two dimensional (blue dotted line) and quasi-one decoherence: that the d reversed. Conclu in an ultr mersed in how preci fects the p s the manip tion ﬂux. tally acce regimes, tion back for inform fundamen quantum for the re This w

Ndeph 3D 2D 1D aB /aRb FIG. 2. (Color online) Non-Markovianity measure Ndeph as a function of the scattering length of the background gas aB when the background gas is three dimensional (red dashed line), quasi-two dimensional (blue dotted line) and quasi-one dimensional (black solid line). The inset shows a longer range of the scattering length aB . In all ﬁgures the well separation Markovian to non-Markovian crossover that the d reversed. Conclu in an ultr mersed in how preci fects the s the manip p tion ﬂux. tally acce regimes, tion back for inform fundamen quantum for the re This w dation, th MICINN

Ndeph 3D 2D 1D aB /aRb FIG. 2. (Color online) Non-Markovianity measure Ndeph as a function of the scattering length of the background gas aB when the background gas is three dimensional (red dashed line), quasi-two dimensional (blue dotted line) and quasi-one dimensional (black solid line). The inset shows a longer range of the scattering length aB . In all ﬁgures the well separation 3D 2D 1D that the d reversed. Conclu in an ultr mersed in how preci fects the s the manip p tion ﬂux. tally acce regimes, tion back for inform fundamen quantum for the re This w dation, th MICINN

2 Ising model in a transverse ﬁeld em ex m Co pl t ys S

m ex pl te ys S m Co Spin chain

Hamiltonian of the spin chain H( ) = J X j z z j j+1 + x j

Hamiltonian of the spin chain H( ) = J X z z j j+1 + x j j Quantum phase transition /J ⌧ 1 /J = 1 /J 1 critical point (anti)ferromagnetic paramagnetic

Ising model trapped ions quantum simulator 16 spins quantum simulator H=J X i>j x x cij i j X y i i Emergence and Frustration of Magnetism with Variable-Range Interactions in a Quantum Simulator, R. Islam, C. Senko, W.C. Campbell, S. Korenblit, J. Smith, A. Lee, E.E. Edwards, J.C.C. Wang, J.K. Freericks, C. Monroe, Science, 340, 583 (2013)

paramagnetic phase | "y "y "y . . . i H=J X i>j /J = 5 x x cij i j X i y i

ferromagnetic phase | "x "x "x . . . i H=J X i>j cij x x i j X i /J = 0.01 | #x #x #x . . . i y i

16 spins quantum simulator collective spin-dependent ﬂuorescence measurements DESTRUCTIVE

16 spins quantum simulator collective spin-dependent ﬂuorescence measurements DESTRUCTIVE N=30 LIMIT TO CALCULATIONS OF DYNAMICS

can we measure the quantum phase transition indirectly, locally, and with minimal disturbance? ?

Q probe |eihe| |gihg| H( ) = J X z z j j+1 + x j j Hint ( ) = |eihe| X x j j H. T. Quan et al., Phys. Rev. Lett. 96, 140604 (2006)

@ Imperial http://youtu.be/RV1wykqg6rM Control of the conformations of ion Coulomb crystals in a Penning trap, S. Mavadia et al., Nature Communications 4, 2571 (2013)

PROBE Hint ( ) = |eihe| XSPINS x j j

Renormalised ﬁeld ⇤ = ( + )/J

Critical point ⇤ =1

1 2 3 qubit probe initialisation probe dynamics probe read out

1 qubit probe initialisation 1 (|ei + |gi) 2

2 probe dynamics ⇢t = t ⇢0 DEPHASING

1 2 change t 3 3 probe read out measure coherences N information ﬂow

Number of spins Contour plot of N P. Haikka, J. Goold, S. McEndoo, F. Plastina, and S. Maniscalco, Phys. Rev. A 85, 060101(R) (2012)

N information ﬂow dynamics of state distinguishability accessible information on the Q probe channel capacities DEPENDS ON THE SPIN CHAIN STATE

N =0 NO information backﬂow ONLY at critical point

3 Trapped ion crystals

N ions in a linear trap ⌫T transverse trap frequency ⌫C critical frequency

⌫ T > ⌫C ⌫T = ⌫C ⌫ T < ⌫C critical point phase transition

16 ions in a linear trap - Mainz experiment Observation of the Kibble–Zurek scaling law for defect formation in ion crystals S. Ulm et al Nature Communications 4, 2290 (2013)

Kibble–Zurek

collective ﬂuorescence measurements

Can we detect the structural phase transition by means of a local probe? ?

G. De Chiara, T. Calarco, S. Fishman, and G. Morigi, Phys. Rev. A 78, 043414 (2008)

G. De Chiara, T. Calarco, S. Fishman, and G. Morigi, Phys. Rev. A 78, 043414 (2008)

G. De Chiara, T. Calarco, S. Fishman, and G. Morigi, Phys. Rev. A 78, 043414 (2008)

Probe Open Quantum System

1 2 3 qubit probe initialisation probe dynamics Dephasing and dissipation probe read out Ramsey fringe interferometry

1 2 change t 3 N information ﬂow

100 ions 1000 ions critical point M. Borrelli, P. Haikka, G. De Chiara, S. Maniscalco, Phys. Rev. A 88, 010101(R) (2013)

long range interaction Ion crystal 1000 short range interaction Ising model 800 600 N N 6= 0 400 200 structural phase transition quantum phase transition

Where we are now....

Quantum simulators Complex systems

Quantum simulators Complex systems

Quantum probes information ﬂow between Q probe and complex system reveals properties of the latter one

properties of complex system (quantum simulator) are mapped into the decoherent dynamics of the Q probe

New tools Non-Markovianity measures Open Quantum System theoretical approaches

Outlook Relativistic quantum information probes ?

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