School of somethingFACULTY OF OTHERDepartment of Physics

Collapse and revival dynamics in spinsystems and preparation of states forquantum metrology

Tim SpillerYork Centre for Quantum Technologies

Collaborators:

Mark Everitt, Derek Harland, Shane Dooley, Francis McCrossan…

Outline of the talk

Motivation

Introduction to quantum metrology

Standard quantum limit and Heisenberg limit

Creating the physical resources for quantum metrology

Examples:

“Cat states” and collapse and revival in coupled spin systems

Summary and comments

Motivation

Introduction to quantum metrology

Standard quantum limit and Heisenberg limit

Creating the physical resources for quantum metrology

Examples:

“Cat states” and collapse and revival in coupled spin systems

Summary and comments

Motivation

Quantum information technologies rely on fundamentalfeatures of quantum physics (superposition, entanglement,measurement, no-cloning…) to offer advantages over theirconventional IT counterparts.

Quantum random number and communication technologiesare already in the shops!

Large scale quantum computers are still a long way off.

What is there in between? What (else) can be done withrelatively modest quantum resources, practically availabletoday or in the near future? One example: Metrology

Quantum information technologies rely on fundamentalfeatures of quantum physics (superposition, entanglement,measurement, no-cloning…) to offer advantages over theirconventional IT counterparts.

Quantum random number and communication technologiesare already in the shops!

Large scale quantum computers are still a long way off.

What is there in between? What (else) can be done withrelatively modest quantum resources, practically availabletoday or in the near future? One example: Metrology

Introduction to quantum metrology

Much of metrology can be reduced to the concept of measuringa phase acquired by a system.

This phase arises due to the physical quantity being measurede.g. the effect of an electric field on a charge; the effect of amagnetic field on a spin; the effect of gravity on a mass; theeffect of a medium on light; rotation of a system; the passage oftime itself…

Quantum metrology technologies enable enhanced accuracy ofphase measurement. A non-classical state of quantumresources can do better than the same resources usedclassically or conventionally.

Much of metrology can be reduced to the concept of measuringa phase acquired by a system.

This phase arises due to the physical quantity being measurede.g. the effect of an electric field on a charge; the effect of amagnetic field on a spin; the effect of gravity on a mass; theeffect of a medium on light; rotation of a system; the passage oftime itself…

Quantum metrology technologies enable enhanced accuracy ofphase measurement. A non-classical state of quantumresources can do better than the same resources usedclassically or conventionally.

Interferometer

In qubit language…

“Classical” version

Interferometer

Standard Quantum Limit (SQL)and Heisenberg Limit

• We should use the same (or same average when there is numberuncertainty) resources (N photons/qubits) for a fair comparison.

• “Classical” coherent state: Standard Quantum Limit

• Entangled N00N state: Heisenberg Limit

• Attaining the Heisenberg Limit gives a gain of in phase sensitivitycompared to the SQL.

• Beating the SQL is often referred to as phase supersensitivity – theregime of Quantum Metrology.

• To quantify the minimum possible with any quantum state into whichthe unknown phase has been imprinted, the Quantum Fisher Informationcan be used to give a lower bound.NOTE: This does not (necessarily) provide or suggest the measurementscheme needed to attain the bound, but it quantifies the potentialusefulness of states for metrology.

N/1

N/1

N

• We should use the same (or same average when there is numberuncertainty) resources (N photons/qubits) for a fair comparison.

• “Classical” coherent state: Standard Quantum Limit

• Entangled N00N state: Heisenberg Limit

• Attaining the Heisenberg Limit gives a gain of in phase sensitivitycompared to the SQL.

• Beating the SQL is often referred to as phase supersensitivity – theregime of Quantum Metrology.

• To quantify the minimum possible with any quantum state into whichthe unknown phase has been imprinted, the Quantum Fisher Informationcan be used to give a lower bound.NOTE: This does not (necessarily) provide or suggest the measurementscheme needed to attain the bound, but it quantifies the potentialusefulness of states for metrology.

F/1

Quantum Fisher informationProgress meetingMunich28-29 Jan 2011

1QF

is quantum Fisher information for an outcome state.QF

- Annals of Physics 247, 135 (1996)

Main background theorem: Quantum Cramer-Rao lower bound for metrology

Pure state:

Mixed state:

12 12

out inK KU '

12 12/out

K K

2' ' '

12 1212 124Q out

K K K K KF

12

2

,

2/Q K

K i ji j i j

F

12

and are the eigenvalues and eigenvectors of .Ki i

• We are interested in CV entangled coherent states (ECS) for metrology.

Recent Experiments (1)

A four-photon “N00N”state showsinterference fringeswith (fourfold) phasesupersensitivity andbeats the SQL.

Recent Experiments (2)

A ten-spin “N00N” stateshows (phase)supersensitivity to anexternal magnetic fieldof ~ 9.4 compared tothat for a single H spin.

Recent Experiments

Neither of these experiments is perfect…

The required entangling operation for photons is not in place (so abeam-splitter is used). The N00N state has to be post-selectedfrom data using specific firing patterns of the detectors.

The initial state of the spins is a mixture (although the C-NOT gateis in place). The N00N state has to be “post-selected” from themolecular ensemble (from MSSM) by frequency selection.

Neither provide prototype technology, but they are very impressivedemonstrations of the principle.

New methods are needed to generate robust non-classicalresources for quantum metrology.

Neither of these experiments is perfect…

The required entangling operation for photons is not in place (so abeam-splitter is used). The N00N state has to be post-selectedfrom data using specific firing patterns of the detectors.

The initial state of the spins is a mixture (although the C-NOT gateis in place). The N00N state has to be “post-selected” from themolecular ensemble (from MSSM) by frequency selection.

Neither provide prototype technology, but they are very impressivedemonstrations of the principle.

New methods are needed to generate robust non-classicalresources for quantum metrology.

Other applications of “supersensitivity”Improved timing or frequency metrology using atom/ion

entanglement ~

S. F. Huelga, et al., Phys. Rev. Lett. 79, 3865 (1997); atom/ionexperiments:

“Quantum lithography” using supersensitive spatial interferencefringes; A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C.P. Williams, and J. P. Dowling, Phys. Rev. Lett. 85, 2733(2000);

but needs selective N-photon absorber.

......2

1eeeggg

Improved timing or frequency metrology using atom/ionentanglement ~

S. F. Huelga, et al., Phys. Rev. Lett. 79, 3865 (1997); atom/ionexperiments:

“Quantum lithography” using supersensitive spatial interferencefringes; A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C.P. Williams, and J. P. Dowling, Phys. Rev. Lett. 85, 2733(2000);

but needs selective N-photon absorber.

Non-classical resources formetrology

Optical systems:[1] J Joo, W J Munro and T P Spiller, Quantum metrology with entangled coherent states, Phys.

Rev. Lett. 107, 083601 (2011); arXiv:11015044.

[2] J Joo, K Park, H Jeong, W J Munro, K Nemoto and T P Spiller, Quantum metrology for non-linear phase shifts with entangled coherent states, Phys. Rev. A 86, 043828 (2012);arXiv:12032099.

Spin systems:[3] S Dooley, F McCrossan, D Harland, M J Everitt, T P Spiller, Collapse and Revival and Cat States

with an N Spin System, Phys. Rev. A 87, 052323 (2013); arXiv:1302.2806.

[4] S Dooley, T P Spiller, Fractional revivals, multiple-Schr¨odinger-cat states, and quantum carpetsin the interaction of a qubit with N qubits, Phys. Rev. A 90, 012320 (2014); arXiv:1404.4296.

Superconducting devices:[5] M J Everitt, T P Spiller, G J Milburn, R D Wilson, A M Zagoskin, Cool for Cats, arXiv:1212.4795.

Rotating atoms:[6] L M Rico-Gutierrez, T P Spiller, J A Dunningham, Engineering entanglement for metrology with

rotating matter waves, New J. Phys. 15 (2013) 063010, arXiv:1304.7348.

Optical systems:[1] J Joo, W J Munro and T P Spiller, Quantum metrology with entangled coherent states, Phys.

Rev. Lett. 107, 083601 (2011); arXiv:11015044.

[2] J Joo, K Park, H Jeong, W J Munro, K Nemoto and T P Spiller, Quantum metrology for non-linear phase shifts with entangled coherent states, Phys. Rev. A 86, 043828 (2012);arXiv:12032099.

Spin systems:[3] S Dooley, F McCrossan, D Harland, M J Everitt, T P Spiller, Collapse and Revival and Cat States

with an N Spin System, Phys. Rev. A 87, 052323 (2013); arXiv:1302.2806.

[4] S Dooley, T P Spiller, Fractional revivals, multiple-Schr¨odinger-cat states, and quantum carpetsin the interaction of a qubit with N qubits, Phys. Rev. A 90, 012320 (2014); arXiv:1404.4296.

Superconducting devices:[5] M J Everitt, T P Spiller, G J Milburn, R D Wilson, A M Zagoskin, Cool for Cats, arXiv:1212.4795.

Rotating atoms:[6] L M Rico-Gutierrez, T P Spiller, J A Dunningham, Engineering entanglement for metrology with

rotating matter waves, New J. Phys. 15 (2013) 063010, arXiv:1304.7348.

Non-classical resources formetrology

Optical systems:[1] J Joo, W J Munro and T P Spiller, Quantum metrology with entangled coherent states, Phys.

Rev. Lett. 107, 083601 (2011); arXiv:11015044.

[2] J Joo, K Park, H Jeong, W J Munro, K Nemoto and T P Spiller, Quantum metrology for non-linear phase shifts with entangled coherent states, Phys. Rev. A 86, 043828 (2012);arXiv:12032099.

Spin systems:[3] S Dooley, F McCrossan, D Harland, M J Everitt, T P Spiller, Collapse and Revival and Cat States

with an N Spin System, Phys. Rev. A 87, 052323 (2013); arXiv:1302.2806.

[4] S Dooley, T P Spiller, Fractional revivals, multiple-Schr¨odinger-cat states, and quantum carpetsin the interaction of a qubit with N qubits, Phys. Rev. A 90, 012320 (2014); arXiv:1404.4296.

Superconducting devices:[5] M J Everitt, T P Spiller, G J Milburn, R D Wilson, A M Zagoskin, Cool for Cats, arXiv:1212.4795.

Rotating atoms:[6] L M Rico-Gutierrez, T P Spiller, J A Dunningham, Engineering entanglement for metrology with

rotating matter waves, New J. Phys. 15 (2013) 063010, arXiv:1304.7348.

Optical systems:[1] J Joo, W J Munro and T P Spiller, Quantum metrology with entangled coherent states, Phys.

Rev. Lett. 107, 083601 (2011); arXiv:11015044.

[2] J Joo, K Park, H Jeong, W J Munro, K Nemoto and T P Spiller, Quantum metrology for non-linear phase shifts with entangled coherent states, Phys. Rev. A 86, 043828 (2012);arXiv:12032099.

Spin systems:[3] S Dooley, F McCrossan, D Harland, M J Everitt, T P Spiller, Collapse and Revival and Cat States

with an N Spin System, Phys. Rev. A 87, 052323 (2013); arXiv:1302.2806.

[4] S Dooley, T P Spiller, Fractional revivals, multiple-Schr¨odinger-cat states, and quantum carpetsin the interaction of a qubit with N qubits, Phys. Rev. A 90, 012320 (2014); arXiv:1404.4296.

Superconducting devices:[5] M J Everitt, T P Spiller, G J Milburn, R D Wilson, A M Zagoskin, Cool for Cats, arXiv:1212.4795.

Rotating atoms:[6] L M Rico-Gutierrez, T P Spiller, J A Dunningham, Engineering entanglement for metrology with

rotating matter waves, New J. Phys. 15 (2013) 063010, arXiv:1304.7348.

Non-classical resources formetrology

Spin systems:[3] S Dooley, F McCrossan, D Harland, M J Everitt, T P Spiller, Collapse and Revival and Cat

States with an N Spin System, Phys. Rev. A 87, 052323 (2013); arXiv:1302.2806.

[4] S Dooley, T P Spiller, Fractional revivals, multiple-Schr¨odinger-cat states, and quantumcarpets in the interaction of a qubit with N qubits, Phys. Rev. A 90, 012320 (2014);arXiv:1404.4296.

These papers give examples of entangled spin resource generation through theunitary collapse and revival dynamics of coupled spin systems. The system couldbe a molecule (see earlier), or some other natural or fabricated set of two-statedevices coupled to a central device or bus.

Note: Other examples provide the generation of useful and interesting non-classicalstates through non-unitary decoherence dynamics, e.g. for superconducting circuits.

Spin systems:[3] S Dooley, F McCrossan, D Harland, M J Everitt, T P Spiller, Collapse and Revival and Cat

States with an N Spin System, Phys. Rev. A 87, 052323 (2013); arXiv:1302.2806.

[4] S Dooley, T P Spiller, Fractional revivals, multiple-Schr¨odinger-cat states, and quantumcarpets in the interaction of a qubit with N qubits, Phys. Rev. A 90, 012320 (2014);arXiv:1404.4296.

These papers give examples of entangled spin resource generation through theunitary collapse and revival dynamics of coupled spin systems. The system couldbe a molecule (see earlier), or some other natural or fabricated set of two-statedevices coupled to a central device or bus.

Note: Other examples provide the generation of useful and interesting non-classicalstates through non-unitary decoherence dynamics, e.g. for superconducting circuits.

Introduction – one qubit and a fieldA qubit coupled to an oscillatoryclassical field exhibitscontinuous Rabi oscillations

A qubit coupled to thecorrresponding quantum fieldexhibits collapse and revival ofthe Rabi oscillations

Resonance:

Coherent state of field:

A qubit coupled to an oscillatoryclassical field exhibitscontinuous Rabi oscillations

A qubit coupled to thecorrresponding quantum fieldexhibits collapse and revival ofthe Rabi oscillations

Resonance:

Coherent state of field:

Single qubit collapse and revival

There are three timescales for the collapse and revival:

The Rabi oscillation period:

The Gaussian envelope collapse time:

The revival time:

Scaling the coupling simply corresponds to a time scaling, butchanging the photon number lengthens the revival time.

“Rotating frame” dynamics:

Approximate:

There are three timescales for the collapse and revival:

The Rabi oscillation period:

The Gaussian envelope collapse time:

The revival time:

Scaling the coupling simply corresponds to a time scaling, butchanging the photon number lengthens the revival time.

“Rotating frame” dynamics:

Approximate:

Collapse and revival andsingle qubit “attractor” state

Half way to revival the qubit and field are in a product state,with the quantum information of the initial qubit state mappedinto a cat state of the field and the qubit in an “attractor” state

set by the initial phase of the field.

Collapse and revival incoupled spin systems

Can the field mode be replaced by a “big spin”?

Yes: take N qubits/spins in their maximum j=N/2 symmetricsubspace.

Can the “big spin” be prepared in a suitable initial state, whichis straightforward to prepare and generates interestingdynamics?

Yes: spin coherent state –>

Can the field mode be replaced by a “big spin”?

Yes: take N qubits/spins in their maximum j=N/2 symmetricsubspace.

Can the “big spin” be prepared in a suitable initial state, whichis straightforward to prepare and generates interestingdynamics?

Yes: spin coherent state –>

Collapse and revival incoupled spin systems

A regime of behaviour existswhere collapse and revival(demonstrated by the “specialqubit”) mimics that of the fieldcase. This can be shownformally with .

What about finite, practicalvalues of N? What does thestate of the “big spin” look likepart way to the revival time?Does it resemble a cat stateand is it useful?

Scaling:

A regime of behaviour existswhere collapse and revival(demonstrated by the “specialqubit”) mimics that of the fieldcase. This can be shownformally with .

What about finite, practicalvalues of N? What does thestate of the “big spin” look likepart way to the revival time?Does it resemble a cat stateand is it useful?

Scaling:

Collapse and revival incoupled spin systems

Approximate (for finite N) analyticsgenerate a “spin cat” state at halfthe revival time.

There is high fidelity of the actualstate of the big spin against the catstate for modest and practicalvalues of N.

Approximate (for finite N) analyticsgenerate a “spin cat” state at halfthe revival time.

There is high fidelity of the actualstate of the big spin against the catstate for modest and practicalvalues of N.

Magnetic field metrology witha spin cat state

As we quantify theprecision of magnetic fieldmeasurement by

Quantum Fisher Information

For relatively modest values ofN, the states of the big spinproduced at half the revivaltime can be used for near-Heisenberg-limited metrology.

As we quantify theprecision of magnetic fieldmeasurement by

Quantum Fisher Information

For relatively modest values ofN, the states of the big spinproduced at half the revivaltime can be used for near-Heisenberg-limited metrology.

Fractional revivals

For an initial spin coherent state of allN+1 spins on the “Bloch equator”

the revival time is

for even N and 2T for odd N. Spinsqueezing occurs at small t and

Superpositions of multiple cat statesoccur at these fractional revivals. Plotthe Q-function:

For an initial spin coherent state of allN+1 spins on the “Bloch equator”

the revival time is

for even N and 2T for odd N. Spinsqueezing occurs at small t and

Superpositions of multiple cat statesoccur at these fractional revivals. Plotthe Q-function:

Quantum carpetsAs the operatorcommutes with theHamiltonian (5), it (and allpowers) are conserved. Soif the Q-function is initiallypeaked at the Blochequator, it remains so. Insuch cases canbe plotted as a function oftime and to give aconcise picture of theevolution. A “quantumcarpet” results, clearlyshowing the fractionalrevivals.

As the operatorcommutes with theHamiltonian (5), it (and allpowers) are conserved. Soif the Q-function is initiallypeaked at the Blochequator, it remains so. Insuch cases canbe plotted as a function oftime and to give aconcise picture of theevolution. A “quantumcarpet” results, clearlyshowing the fractionalrevivals.

Fractional revivals even forsmall N

Although the analyticapproximations employed toconstruct the fractionalrevivals require N to be largeand the fidelity of the exactevolution against the analyticresults lowers for small N, thisis partially due to distortion ofthe spin coherent states.There is still clear evidence ofthe fractional revivals and catstates for rather small N,potentially accessible incurrent experiments.

Although the analyticapproximations employed toconstruct the fractionalrevivals require N to be largeand the fidelity of the exactevolution against the analyticresults lowers for small N, thisis partially due to distortion ofthe spin coherent states.There is still clear evidence ofthe fractional revivals and catstates for rather small N,potentially accessible incurrent experiments.

Quantum carpets even forsmall N

The small N carpets forthe exact evolution maynot be as exotic as theirlarger N counterparts;however, patternscorresponding to thefractional revivals arestill clearly visible.

The small N carpets forthe exact evolution maynot be as exotic as theirlarger N counterparts;however, patternscorresponding to thefractional revivals arestill clearly visible.

Summary and comments

- Quantum states enable metrology beyond the SQL.

- The phenomenon of collapse and revival occurs in multi-qubitspin systems.

- A parameter regime exists where the natural dynamics (halfway to revival) produces N-spin cat states that can effect near-Heisenberg-limited quantum metrology.

- Other new collapse and revival regimes exist – squeezed andmulti-cat states and carpets exist, also with metrology potential.

- More detailed decoherence studies for candidate physicalsystems are needed. Multi-cat states and carpets may bevisible for relatively modest values of N. Additional to thefundamental interest, these states may be useful for metrology.

- Quantum states enable metrology beyond the SQL.

- The phenomenon of collapse and revival occurs in multi-qubitspin systems.

- A parameter regime exists where the natural dynamics (halfway to revival) produces N-spin cat states that can effect near-Heisenberg-limited quantum metrology.

- Other new collapse and revival regimes exist – squeezed andmulti-cat states and carpets exist, also with metrology potential.

- More detailed decoherence studies for candidate physicalsystems are needed. Multi-cat states and carpets may bevisible for relatively modest values of N. Additional to thefundamental interest, these states may be useful for metrology.