quantum information probes

What do we gain?

Quantum probes versus direct measurements

Sabrina Maniscalco

Institute of Photonics and Quantum SciencesHeriot-Watt University

Edinburgh

S.Maniscalco@hw.ac.uk

Relativistic Quantum Metrology7-8 March 2008, Nottingham

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

engineer H

read out

Condensed Matter systemsSuperfluid

Mott insulator

Superfluid

I. Bloch’s group, 2002

Condensed Matter systems

S. Kuhr’s and I. Bloch’s group

Single-site addressing

Open Quantum Systems

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)

trapped ions quantum simulator

Dirac Equationtrapped 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

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)

trapped ions quantum simulator

100 N 350

Problem: Read out

Problem: Verification

Benchmarking

Problems with known solutions

Benchmarking

Problems with known solutions

Alternative measurement

strategies

What if...

indirectly

with minimal disturbance

Comple

x Syst

Comple

x Syst

KEY IDEA1

Local Probe

Comple

x Syst

KEY IDEA2

ENVIRONMENT

Comple

x Syst

PROBE DECOHERENCE

Depends on the state/properties of the complex systems

SHIFT inPERSPECTIVE

KEY IDEA3

New Tools

⇢(t) = �t⇢(0)

dynamical mapquantum channel

divisibility

�t,0 = �t,s�s,0

Markovian dynamicsMaster equation in Lindblad form

�t,0 = �t,s�s,0

Non-Markovian dynamics

Entanglement and Non-Markovianity of Quantum EvolutionsÁngel Rivas, Susana F. Huelga, and Martin B. Plenio

Phys. Rev. Lett. 105, 050403 (2010)

On the degree of non-Markovianity of quantum evolutionDariusz Chruściński, Sabrina Maniscalco

arXiv:1311.4213, in press in Phys. Rev. Lett.

�t,0 6= �t,s�s,0

Information flow

Markovian dynamics

re-coherence

Quantum information and distinguishability between quantum states

Increase of informationIncrease of distinguishability

Measure for the Degree of Non-Markovian Behavior of Quantum Processes in Open SystemsH.-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 BreuerPhys. Rev. A 81, 062115 (2010)

Quantum information and distinguishability between quantum states

Decrease of informationDecrease of distinguishability

D(⇢1, ⇢2) =1

2Tr|⇢1 � ⇢2|,

Distinguishability between two states of the Q probe

Rate of change of distinguishability

�(t, ⇢1,2(0)) =d

dtD(⇢1(t), ⇢2(t))

�(t, ⇢1,2(0)) 0 at all times

�(t, ⇢1,2(0)) > 0 for some time intervals

Markovian dynamics

N (�) = max

⇢1,2(0)

�>0dt�(t, ⇢1,2(0))

MAXIMUM Information

Backflow

Measure for the Degree of Non-Markovian Behavior of Quantum Processes in Open SystemsH.-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 BreuerPhys. Rev. A 81, 062115 (2010)

MAXIMUM Information Backflow

�Q>0�Q(t)dt

�C>0�C(t)dt

Non-Markovianity and reservoir memory: A quantum information theory perspectiveB. 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

Ultracold bosonic gas dimensionality

Ising model in a transverse field

Trapped ion crystals

Ultracold bosonic gas dimensionality

Probing dimensionality

density fluctuations

phase fluctuations

Immersed probeatomic quantum dot

Atomic Quantum Dots Coupled to a Reservoir of a Superfluid Bose-Einstein CondensateA. 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 probeatomic quantum dot

Atomic Quantum Dots Coupled to a Reservoir of a Superfluid Bose-Einstein CondensateA. 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)

picture of BEC (2D)

Impurity atom

Figure 1. A Bose–Einstein condensate (yellow region) confined in a shallowharmonic trap VB(x) interacts with cold impurity atoms each of which is trappedin a double well potential VA(x) (grey circle). The distance between two wells inthe same trap is 2L and the distance between adjacent traps is 2D.

describes the interactions between the impurities and the bath; here gAB = 2⇤ h2aAB/mAB

is the coupling constant of impurities–gas interaction, with aAB the scattering length of theimpurities–gas collisions and mAB = mAmB/(mA + mB) their reduced mass. Both impurity andbath atoms are described in the second-quantized formalism. The field operator of the atomicimpurities

⌥(x) =⇧

ai,p⇧i,p(x) (5)

can be decomposed in terms of the real eigenstates ⇧i,p(x) of impurity atoms localized on thedouble well i of the potential VA(x) in the pth state, with energy h⌅i,p and the correspondingannihilation operator ai,p. We assume that the wavefunctions of different double wells have anegligible 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 instance, [17]) andassuming a very shallow trapping potential VB(x), such that the bosonic gas can be consideredhomogeneous. In the degenerate regime, the bosonic field can be decomposed as

⌃(x) =⌃

N0 ⌃0(x) + �⌃(x) =⌃

N0 ⌃0(x) +⇧

�uk(x)ck � v⇥

k(x)c†k

⇥, (6)

where ⌃0(x) is the condensate wave function (or order parameter), N0 < N is the number ofatoms in the condensate and ck, c†

k are the annihilation and creation operators of the Bogoliubovmodes with momentum k. For a homogeneous condensate ⌃0(x) = 1/

⇧V , V being the volume.

Its Bogoliubov modes

⇤⇥k + n0gB

⌅eik·x⇧

V, (7)

⇤⇥k + n0gB

Ek� 1

⌅eik·x⇧

New Journal of Physics 11 (2009) 103055 (http://www.njp.org/)

Quantifying, characterizing and controlling information flow in ultracold atomic gases P. Haikka, S. McEndoo, G. De Chiara, M. Palma, and S. Maniscalco,

Phys. Rev. A 84, 031602R (2011)

Impurity atom

⌥(x) =⇧

ai,p⇧i,p(x) (5)

⌃(x) =⌃

N0 ⌃0(x) + �⌃(x) =⌃

N0 ⌃0(x) +⇧

�uk(x)ck � v⇥

k(x)c†k

⇥, (6)

⇤⇥k + n0gB

⌅eik·x⇧

V, (7)

⇤⇥k + n0gB

Ek� 1

⌅eik·x⇧

Zd3x †(x)

p2A2mA

+ VA(x)

� (x)

Zd3x �†(x)

p2B2mB

+ VB(x) +gB2�†(x)�(x)

��(x)

QUANTUM PROBE

QUANTUM GAS

INTERACTION

HAB = gAB

Zd3x (x)�†(x)�(x) (x)

picture of BEC (2D)

Impurity atom

⌥(x) =⇧

ai,p⇧i,p(x) (5)

⌃(x) =⌃

N0 ⌃0(x) + �⌃(x) =⌃

N0 ⌃0(x) +⇧

�uk(x)ck � v⇥

k(x)c†k

⇥, (6)

⇤⇥k + n0gB

⌅eik·x⇧

V, (7)

⇤⇥k + n0gB

Ek� 1

⌅eik·x⇧

|Li |RiQubit Probe

Pure DEPHASING

picture of BEC (2D)

Impurity atom

⌥(x) =⇧

ai,p⇧i,p(x) (5)

⌃(x) =⌃

N0 ⌃0(x) + �⌃(x) =⌃

N0 ⌃0(x) +⇧

�uk(x)ck � v⇥

k(x)c†k

⇥, (6)

⇤⇥k + n0gB

⌅eik·x⇧

V, (7)

⇤⇥k + n0gB

Ek� 1

⌅eik·x⇧

where !z = |R!"R|#|L!"L| and Ek =!

"k["k + 2g(D)B nD]

is the energy of k-th Bogoliubov mode ck of the con-

densate with boson-boson coupling frequency g(D)B and

condensate density nD. D denotes the e!ective dimen-sion of the environment. The energy of a free mode is"k = h2k2/(2mB) where k = |k| and mB is the mass ofa background gas particle. Furthermore, gk and #k arecoupling constants that depend on the spatial form of thestates |L! and |R! and on the shape of the Bogoliubovmodes. Their specific form is elaborated in Ref. [13].When the background gas is at zero temperature the re-duced dynamics of the impurity atom is captured by thefollowing time-local master equation (ME):

dt= "(t)[!z , $]+%(t)[!z$(t)!z#

2{!z!z , $(t)}]. (2)

Quantity "(t) renormalizes the energy of the qubit buthas no qualitative e!ect on the dissipative dynamics. In-stead in this work we are interested in the decay rate

%(t) =4g2ABn0

dk sin2(k · L)(2&)D

sin(Ekt/h)

"k + 2g(D)B nD

e!k2!2/2,

where gAB is the impurity-boson coupling frequency, ' isa trap parameter, and L is half the distance between thetwo wells of the double well potential.We have derived ME (2) using the time-convolutionlessprojection operator technique to second order in thecoupling constant gAB [6]. Remarkably, in this casethe second order ME describes the reduced dynamicsexactly [14]. Solving the ME reveals that the impu-rity atom dephases without exchanging energy with thebackground gas. More precisely, $ii(t) = $ii(0) and$ij(t) = e!!(t)$ij(0) when i $= j, where $ij = "i|$|j!,i, j = R,L. The decoherence function #(t) =

# t0 ds %(s)

coincides with that derived in Ref. [13], however here wewish to stress the connection between the decay rate andthe non-Markovian features. The authors of Ref. [13] dis-covered situations when the decoherence function #(t) isnon-monotonic and conjectured that this is due to non-Markovian e!ects in the reduced dynamics. Already theform of the ME (2) supports this intuition; the theory ofnon-Markovian quantum jumps has shown that there is aprofound connection between non-Markovian e!ects andnegative regions of the decay rates of Lindblad-structuredMEs as the one of Eq. (2) [15]. In the following we con-firm that this is indeed true for this model and, moreover,expose the physical mechanisms at the root of this non-Markovian phenomena.Non-Markovianity measure- Breuer, Laine and Pi-

ilo (BLP) have proposed a rigorous definition for non-Markovianity of a quantum channel $ based on thedynamics of the so-called information flux !(t) =dD[$1(t), $2(t)]/dt [8]. This is the temporal change in

the distinguishability D[$1(t), $2(t)] =12 ||$1(t)# $2(t)||1

of two evolving quantum states $1,2(t) = $(t)$1,2(0) asmeasured by the trace distance. Negative informationflux describes information leaking from the system to itsenvironment and it is associated to Markovian dynamics.Instead, if it is possible to find a pair of states $1,2(0)for which the information flux is positive for some in-terval of time, that is, the system regains some of thepreviously lost information, then process $ is consid-ered non-Markovian. The amount of non-Markovianityis defined to be the maximal amount of information thatthe system may recover from its environment, formallyNBLP = max"1,2

#>0 ds!(s).For the model studied in this Letter we find that !(t) > 0if and only if %(t) < 0, that is, the process is non-Markovian precisely when the decay rate can take tem-porarily negative values. Within experimentally relevantvalues of the physical parameters we have discovered atmost a single time-interval t % [a, b] when the decay rateis negative and information flows back to the system afteran initial period of information loss. Therefore, insteadof using the original measure faithfully and quantifyingnon-Markovianity as the maximal amount of informationthat the system may recover, we introduce a normalizedquantity that reveals the maximal fraction of the previ-ously lost information that the system can recover:

N = max"1,2

D[$1(b), $2(b)]#D[$1(a), $2(a)]

D[$1(0), $2(0)]#D[$1(a), $2(a)]. (4)

Unlike NBLP , the modified quantifier N is bounded be-tween zero (system only leaks information) and one (sys-tem regains all previously lost information) and is there-fore more meaningful as a number. We have confirmednumerically that in the relevant case of dephasing noisethe above quantity is maximized for the same pair ofinitial states that maximize NBLP . These are the stateswhose Bloch vectors lie on the opposite sides of the equa-tor of the Bloch sphere [16]. Using these states in thegeneral expression of Eq. (4) we find the analytic expres-sion of the non-Markovianity measure for a dephasingqubit to be

Ndeph =e!!(b) # e!!(a)

e!!(0) # e!!(a), #(t) =

0ds %(s). (5)

We are now ready to study how changes in the back-ground scattering length and in the dimensionality of theBEC a!ect the dynamics of information flow.Three-dimensional BEC- As a first step we consider

a 3D background BEC with equal confinement of thebackground gas in all directions. We consider a 87Rb-condensate of density n3 = n0 = 1020m!3 and 23Na im-purity atoms trapped in an optical lattice with latticewavelength ( = 600nm and trap parameter ' = 45nm.The impurity-boson coupling is gAB = 2&h2aAB/mAB,where mAB = mAmB/(mA +mB) and mA and mB are

⇢ij(t) = e��(t)⇢ij(0)

�(t) =

0ds �(s)

picture of BEC (2D)

aB/aRb

FIG. 2. (Color online) Non-Markovianity measure Ndeph asa function of the scattering length of the background gas aB

when the background gas is three dimensional (red dashedline), quasi-two dimensional (blue dotted line) and quasi-onedimensional (black solid line). The inset shows a longer rangeof the scattering length aB . In all figures the well separationis L = 75nm.

spectrum is sub-Ohmic when s < 1, Ohmic when s = 1or super-Ohmic when s > 1. Introducing an ad hoc ex-ponential cut-o! so that J(!) = !s exp{!!2/!2

C}, where!C is the cut-o! frequency, it is straightforward to showthat the dynamics is non-Markovian when s > scrit = 2.Therefore, in a general setting, a qubit dephasing underthe e!ect of either a sub-Ohmic or an Ohmic environ-ment can only leak information to its environment. Ifthe environment has a super-Ohmic spectrum the issueis less straightforward: only if the spectrum is su"cientlysuper-Ohmic with s > scrit, information can flow backto the system from the environment.The qubit in an ultracold bosonic environment consid-ered in this work is a special case of the model abovewith J(!) =

k|gk|2"(h!!Ek). The reservoir spectrum

J(!) is very complex due to the complicated form of thecoupling constant gk. However, it can be shown that inthe case of a free background gas in one, two or threedimensions the spectrum is sub-Ohmic, Ohmic or super-Ohmic, respectively [13]. The spectrum changes criti-cally when one considers the boson-boson coupling quan-tified by the scattering length aB. In this case increasingthe scattering length e!ectively increases the value of s.Hence when we increase aB in the 1D case, the spectrumchanges from sub-Ohmic to Ohmic to super-Ohmic andonce a critical threshold of super-Ohmicity is reached theenvironment can feed information back to the system. Inthe 2D non-interacting case the spectrum is Ohmic anda weaker interaction is required to reach the crossoverpoint scrit, leading to acrit,2DB < acrit,1DB . Finally, in the3D case the spectrum is already super-Ohmic in the non-interacting case, although not super-Ohmic enough togive rise to non-Markovian dynamics. Already a smallincrease in the scattering length modifies the spectrum so

that the direction of information flow can be temporarilyreversed.

Conclusion-We have studied quantum information fluxin an ultracold hybrid system of an impurity atom im-mersed in a BEC environment. We have shown explicitlyhow precise control of the ultracold background gas af-fects the spectrum felt by the qubit and therefore enablesthe manipulation of the qubit dynamics and the informa-tion flux. In particular, we have discovered experimen-tally accessible means to reach non-Markovian dynamicalregimes, where the background gas may feed informa-tion back to the qubit instead of acting only as a sinkfor information. Such quantum reservoir engineering isfundamental for understanding decoherence processes inquantum information processing and, more specifically,for the realization of quantum simulators.

This work was supported by the Emil Aaltonen foun-dation, the Finnish Cultural foundation and the SpanishMICINN (Juan de la Cierva, FIS2008-01236 and QOIT-Consolider Ingenio 2010), Generalitat de CatalunyaGrant No. 2005SGR-00343. We acknowledge MarkusCirone, Francesco Plastina and John Goold for usefuldiscussions.

! pmehai@utu.fi; www.openq.fi[1] J. Billy et al. Nature 453, 891 (2008); G. Roati et al.,

Nature 453, 895 (2008).[2] M. Greiner et al. Nature 415, 39 (2002).[3] B. Paredes et al. Nature 429, 277 (2004).[4] B. P. Anderson and M. A. Kasevich, Science 282, 1686

(1998); E.W. Hagley et al., Science 283, 1706 (1999).[5] See, for example, A. Recati, P. O. Fedichev, W. Zwerger,

J. Von Delft and P. Zoller, Phys. Rev. Lett. 94, 040404(2005); A. Griessner, A. J. Daley, S. R. Clark, D. Jakschand P. Zoller, Phys. Rev. Lett. 97 220403 (2006); M.Bruderer and D. Jaksch, New J. Phys. 8 87 (2006); C.Zipkes, S. Palzer, C. Sias and M. Kohl, Nature 464, 388(2010); S. Schmid, A. Harter and J. H. Denschlag, Phys.Rev. Lett. 105, 133202 (2010); S. Will, T. Best, S. Braun,U. Schneider and I. Bloch, Phys. Rev. Lett. 106, 115305(2011).

[6] H.-P. Breuer and F. Petruccione The Theory of OpenQuantum Systems (Oxford University Press, Oxford,2001).

[7] C. Chin, R. Grimm, P. Julienne and E. Tiesinga, Rev.Mod. Phys. 82, 1225 (2010).

[8] H-P. Breuer, E.-M. Laine, and J. Piilo, Phys. Rev. Lett.103, 210401 (2009); E.-M. Laine, J. Piilo and H-P.Breuer, Phys. Rev. A 81, 062115 (2010).

[9] A. Rivas, S. F. Huelga and M. Plenio, Phys. Rev. Lett.105, 050403 (2010).

[10] D. Chruscinski, A. Kossakowski, A. Rivas,arXiv:1102.4318v2 [quant-ph].

[11] X.-M. Lu, X. Wang and C. P. Sun, Phys. Rev. A 82,042103 (2010).

[12] M. A. Nielsen and I. L. Chuang Quantum Computationand Quantum Information (Cambridge University Press,

Non-Markovianity: information flow

decoherence: information lost in the environment

recoherence: information backflow

picture of BEC (2D)

aB/aRb

Markovian to non-Markovian crossover

picture of BEC (2D)

aB/aRb

3D 2D 1D

Comple

x Syst

Ising model in a transverse field

Comple

x Syst

Spin chain

Hamiltonian of the spin chain

H(�) = �JX

j+1 + ��x

Hamiltonian of the spin chain

Quantum phase transition

critical point(anti)ferromagnetic paramagnetic

�/J ⌧ 1 �/J � 1�/J = 1

H(�) = �JX

j+1 + ��x

Ising model trapped ions quantum simulator

16 spins quantum simulator

H = JX

� �X

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)

�/J = 5paramagnetic phase| "y"y"y . . . i

H = JX

� �X

ferromagnetic phase

. . . i | #x

. . . i

H = JX

� �X

�/J = 0.01

collective spin-dependent fluorescence measurements

DESTRUCTIVE

collective spin-dependent fluorescence measurements

DESTRUCTIVE

N=30LIMIT TO CALCULATIONS

OF DYNAMICS

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

disturbance?

H(�) = �JX

j+1 + ��x

(�) = �|eihe|X

Q probe|eihe|

|gihg|

H. T. Quan et al., Phys. Rev. Lett. 96, 140604 (2006)

http://youtu.be/RV1wykqg6rM

@ Imperial

Control of the conformations of ion Coulomb crystals in a Penning trap, S. Mavadia et al., Nature Communications 4, 2571 (2013)

(�) = �|eihe|X

PROBE SPINS

Renormalised field�⇤ = (�+ �)/J

Critical point

�⇤ = 1

1 qubit probe initialisation

2 probe dynamics

3 probe read out

2(|ei+ |gi)

2 probe dynamics

⇢t = �t⇢0

DEPHASING

3 probe read outmeasure coherences

N information flow

3 change t

Contour plot ofN

P. Haikka, J. Goold, S. McEndoo, F. Plastina, and S. Maniscalco, Phys. Rev. A 85, 060101(R) (2012)

N information flow

state distinguishabilityaccessible information on the Q probe channel capacities

dynamics of

DEPENDS ON THE SPIN CHAIN STATE

ONLY at critical point

N = 0NO information backflow

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

Observation of the Kibble–Zurek scaling law for defect formation in ion crystalsS. Ulm et al

Nature Communications 4, 2290 (2013)

16 ions in a linear trap - Mainz experiment

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)

Open Quantum SystemProbe

2 probe dynamics

3 probe read outRamsey fringe interferometry

Dephasing and dissipation

3 change t

N information flow

critical point

100 ions

1000 ions

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

Ising modelIon crystal

N 6= 0

structural phase transition quantum phase transition

long range interaction short range interaction

Where we are now....

Quantum simulators

Complex systems

Quantum simulators

Complex systems

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

Quantum probes

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

New toolsNon-Markovianity measures

Open Quantum System theoretical approaches

Outlook

quantum information probes Relativistic?

www.dscien.com

Funding:

quantum information probes

quantum probes

quantum wonderland

quantum simulators

open quantum systems

complex quantum system

relativistic quantum

trappedion quantum simulator

dynamical map quantum

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