Superconducting Quantum Devices: Quantum BitsSuperconducting Quantum Devices: Quantum Bits, Quantum Optics, and More 

Minicourse, NTU, TaiwanJuly 2010

Lin TianUniversity of California,  Merced

ltian@ucmercedhtt //f lt d d /lti /http://faculty.ucmerced.edu/ltian/

Before I start --

Progress in quantum optical system for quantum computingand quantum communication

• Ion trap – teleportation, quantum Fourier transformationp p q• Optical lattice – Feshbach resonance, many body Hamiltonians• Atom and photon in cavity -- quantum repeater, entanglement

Hamiltonian tool bo : Applications:Hamiltonian tool box:• controlled Hamiltonian and transitions• controlled decoherence

l li b i l i

Applications:• quantum state engineering• precision measurement• quantum information• laser cooling by optical pumping • quantum information

Progress in nanoscale systems for quantum computingand quantum communication

• Josephson junction – macroscopic quantum effect, controlled logic gatesp j p q g g• Quantum dot – single bit control, coupling with cavity mode• Nanomechanical modes -- approaching the quantum limit• Many other systems: graphene, nanotube, photonic crystal,Many other systems: graphene, nanotube, photonic crystal,

exotic systems such as electrons on liquid helium ...

Artificial/Macroscopic atoms and oscillators can now be achievedArtificial/Macroscopic atoms and oscillators can now be achievedBetter quantum engineering, control, and probing wanted!

quantum information/technology/fundamentalgy

solid-statequantum solid-statestructures

quantumoptics

• flexibility&scalability• coupling and gates• rich physics

• well developed techniques• advanced experiments • identical system - simplification

1. Applying AMO approach to nanoscale system2 Impact on quantum technology - metrology, photon source

• rich physics• identical system - simplification

2. Impact on quantum technology metrology, photon source3. Impact on quantum information processing4. Impact on fundamental physics - many body systems

Smaller But Better• high Q - low decoherence• nano-scale fabricated systems• flexible design

• Josephson junction resonator mode• nanomechanical mode

• spins in solids• solid-state qubits

• flexible design

nanomechanical mode• transmission line mode …

q• impurity/defects …

Quantum Q tQuantum oscillator

Quantum bit

a new generation of quantum optical systems !!a new generation of quantum optical systems !!

Modern ComputersModern Computers

Computation and computers have a long historyp p g yAncient computer

first appears in Babylon12th and 13th century AD

Mechanical computing

y

p g

Leibnitz (1646 1716)•Leibnitz (1646-1716)

Modern ComputersModern Computers

ENIAC: vacuum tube ``calculator’’ -1945, WarElectronic Numerical Integrator and Calculator

500 000 ld d j i t500,000 soldered joints, 18,000 vacuum tubes, 6 000 switches6,000 switches and 500 terminals

10 decimal digits- simulate nuclear bombs1949 bold prediction:1949 bold prediction:Future 1000 vacuum tubes, 1.5 tons

Comp ter histor m se m in Mo ntain Vie CA• Computer history museum in Mountain View, CA

The new World of Quantum Mechanics

• Spooky quantum effect – quantum entanglement

The new World of Quantum Mechanics

Spooky quantum effect quantum entanglement

AliceAliceBob• quantum correlation

• in different basis(EPR l ti )• (EPR correlation)

• test of quantum mechanics

Different from classical correlation

Quantum ComputingQuantum Computing

Basic ideas – 5 ingredients in making a quantum computer as c deas 5 g ed e ts a g a qua tu co pute• Logic element: qubit – quantum two level systems

• Preparation of initial states – quantum memory element• Logic gates: unitary evolution following Schrodinger eq• Logic gates: unitary evolution following Schrodinger eq.

• Measurement on selected qubits to extract results• Quantum error well controlled/corrected• some people argue there are more …

Superconducting QubitsSuperconducting Qubits

Charging Energy Josephson EnergyJosephsonjunction

Charge Qubits EJ/Ec<1Quantum Hamiltonian

Makhlin, Schoen, Shnirman, RMP 2002Various qubits have been designed, demonstrated with coherence time > sJosephson junction resonator has been tested Q~103-4.

Quantum OscillatorsQuantum OscillatorsQuantum resonator modes in nanoscale• motional states in ion traps• Josephson junction resonators• superconducting transmission line• nanomechanical modes

Smaller & more coherent (macroscopic)Smaller & more coherent (macroscopic) systems in their quantum limit!

Quantum applications – Quantum information,

Transmission line

Metrology, foundations of quantum physics … Nanomechanical systems

NEMS

Nanomechanical ResonatorNanomechanical Resonator

1. Sometime ago• Vibration of strings• Dynamics – Euler-Bernoulli Eq.

2. Now, the decrease of size provides:, phigh frequency -- GHzhigh Q – 103 –5 & =0/Q

E: Young ModulusE: Young ModulusI: moment of inertiaa : linear density

3. quantum mechanics – (?) => cooling a doubly clamped beam,flexural modes

u(z,t)L

ll h b f hWe will start the basics of the quantum circuits where qubits and resonator modes will be discussed

We will then process through a number of Interesting effects in quantum optics in such devicessuch devices

We will touch on more advanced topics on quantum emulation in such systemsquantum emulation in such systems

Outline of LecturesOutline of Lectures

1. Quantum circuit and Hamiltonian2 S d ti t bit ( bit )2. Superconducting quantum bits (qubits)3. Decoherence (spin‐boson model and circuit model) 4. Recent progress (may come back)5. Circuit quantum electrodynamics (QED) and Transmission line resonator6. Two‐level system fluctuators and Superconducting Josephson junction resonator7. Nanomechanical systems, resonator, and laser cooling8. Coherent frequency conversion

• LC oscillator (see notes) Classical eq. of motion, Kirchhoff’s Law q ,Quantization, Lagrangian approach, canonical variablesGeneral theory, free energy

• Josephson junction devicesClassical property, dc JJ effect, ac JJ effectQuantum regime, criterion, single junction quantizationEarlier work on JJASmaller circuits, qubits and resonators

di i i l (l )• dissipative elements (later)Circuit approach, see Part 3Transmission line, see Part 4

Josephson junction - ClassicalJosephson junction - Classicalp jp j1. insulator/tunneling junction/break junction sandwiched betweenSuperconductorsp

si

junction

is

Capacitor: important in quantum regimePhase variable relation to

2. Josephson junction devices have been studied as sensors of magnetic fields,weak forces, amplifier for gravitational wave detection

Most famous JJ devices: SQUID – quantum interference device

Metrology applicationMetrology application

Josephson junction - ClassicalJosephson junction - Classical3. Two important relation for Josephson junctions – for quantum devices as well

p jp j

Flux quantum

4. Josephson effects:

DC ff t t t t h lt 0 b t t t/ tDC effect: at constant phase, voltage=0, but constant/non-zero current

AC effect: = t, ac current, but constant voltage

Phase variable relation to

In BCS, it has meaning of order parameter of superconductingThe phase here is phase difference between two sides

Josephson junction - ClassicalJosephson junction - Classicalp jp j5. SQUID – quantum interference device

Flux inside the loop

See notes

6. Sometimes, resistance can be important

Josephson junction - QuantumJosephson junction - Quantump j Qp j Q

Charging EnergyJosephsonjunction Josephson Energy

1. now, the capacitor becomes smaller – improved fabrication technologyAround 1990

2. Capacitance energyQuantum Hamiltonian

3. Quantum circuit approach to derive Hamiltonian (see notes)

4. Quantum particle in a periodicalpotential

Josephson junction - QuantumJosephson junction - Quantum

5. How does it become quantum mechanical?

p j Qp j Q

When Ec is not negligible

Quantum fluctuation of charges becomes significantg g

6. Single junction with voltage bias – see notes

Free energy is derived from Lagrangian approach

A better picture- Island- Gated by voltage

ground

Gated by voltage- charge on island- free energy vs energy

g

Josephson junction - QuantumJosephson junction - Quantump j Qp j Q

7. Canonical variable ???

Charge quantizedFlux periodicity

Josephson junction ArraysJosephson junction Arraysp j yp j yJosephson junction arrays have been studied for many-body physics• classical JJA – two-dimensional XY, observe BKT transition

t JJA M tt i l t t iti f t• quantum JJA – Mott insulator transition for vortex• quantum phase model • dissipative quantum phase transition

2D array of JJ2D array of JJEJ >>Ec superconductingEJ <<Ec Mott insulator

Josephson junction ArraysJosephson junction Arraysp j yp j y

Josephson junction ArraysJosephson junction Arraysp j yp j y

n – vortex density

R0=0, when vortex localized

Localizing at commensurate densityLocalizing at commensurate density

From Many to FewFrom Many to Fewyy

Recent progress in superconducting devices brings moreRecent progress in superconducting devices brings more ……• fabrication of small junctions with strong quantum effects• low sub-gap resistance, low temperature, better device• superconducting qubitssuperconducting qubits • high-Q cavity mode• strong coupling between qubits and cavity• using cavity to measure qubitsg y q

• single Cooper pair box (charge qubits)Circuit, how to obtain quantum two-level system, q yQuantum logic gatesQuantronium – a spin offUniversal degeneracy pointg y pQubit operationsNakamura experiment

• superconducting flux qubitsCircuit with three junctions, HamiltonianSpectrumG l d liGate control and coupling

Single Cooper Pair BoxSingle Cooper Pair Boxg pg p

I will follow their notation too

Single Cooper Pair BoxSingle Cooper Pair Boxg pg p

A better pictureI l d- Island

- Gated by voltage- charge on island

fground

- free energy vs energy

New notation

Where is the “qubit”?

Single Cooper Pair BoxSingle Cooper Pair Boxg pg p

1. Charging regime

The first term in the Hamiltonian dominant

Energy vs. gate voltagegy g g

• Each n-state is parabolaic

• Dashed curves are for the 1st term in Hamiltonian only

• at ng, states have different energy

• ng=1/2 …, degenerate states All th t t h h hi hAll other states have much higher energy

Single Cooper Pair BoxSingle Cooper Pair Box

2. 2nd term: Josephson energy

g pg p

Transition between adjacent charge states

Explain states at ng– see notes

Subspace of 0 and 1

Energy separation is EJEJ

With states |+>, |->

Diabatic states – charge statesEigenstates – evolving with ng

Single Cooper Pair BoxSingle Cooper Pair Box

3. General Hamiltonian – two charge states involved see notes

g pg p

z basis for charge states

4. Degeneracy point – see notes – protect qubits from 1/f noise

Related to decoherenceNoise spectral densityS d d t b ti d fi llSecond order perturbation, and finallyOur recent work on universal quantum degeneracy point

Single Cooper Pair BoxSingle Cooper Pair Boxg pg p

Universal quantum degeneracy point – coupled qubits to protect coherence

Single Cooper Pair BoxSingle Cooper Pair Boxg pg p

New basis states – 2 qubits have four states

eigenstates

All noise operators have off-diagonal elements – immune to low frequency noise

Single Cooper Pair BoxSingle Cooper Pair Boxg pg p

X Deng Y Hu L Tian preprintX. Deng, Y. Hu, L. Tian, preprint

Single Cooper Pair BoxSingle Cooper Pair Boxg pg p

5. Gate operations

Two control parameters for single qubits gate

Flux control x, phase gate at degeneracy pointx, p g g y pVoltage controls z, flip gate at degeneracy pointUniversal single qubit gates possible

See next page for figure and see notes for operation

Single Cooper Pair BoxSingle Cooper Pair Boxg pg p

Single Cooper Pair BoxSingle Cooper Pair Boxg pg p

Persistent Current QubitPersistent Current QubitQQ

Flux quantization relation:

Gauge transformation of EM fields, and no field inside superconductors

Persistent Current QubitPersistent Current QubitQQ

1. Potential energy: two independent phase variable1. Potential energy: two independent phase variable

Depends on flux bias f

f=1/2, we have

Persistent Current QubitPersistent Current QubitQQ

2. Kinetic energy2. Kinetic energy

P: charge on the islandsQ: induced charge on islands

3. Effect of gate voltages

Replace with

Persistent Current QubitPersistent Current QubitQQ

But we have

Boundary condition is same Periodic for 1 2Boundary condition is same. Periodic for 1, 2

Hence, boundary condition for reflects the prefactors – solution differentWhen voltages changedWhen voltages changed

In our system, as very small effect from the voltages

4. System and energy

E

0

D

1 mm 1

I

+Ip

flux bias Fo/2currents ± Ip

-1

0

Icirc

-Ip

0

F~Fo/2

0.5 F/Fo

d t t

( )z xH ½( / 0 5)2 I ground state

excited state( / 0.5)2o o pI

Mooij et al. Science 285 1036 (1999), Orlando et al. PRB 60 15398 (1999)

Persistent Current QubitPersistent Current QubitQQ

Eigenstates at f=1/2g

Persistent Current QubitPersistent Current QubitQQ

5. Effective two level system and tunneling

Double well potential – flux bias controls the energyTunneling controls crossing

Tunneling can be controlled by middle junctionTunneling can be controlled by middle junction

Persistent Current QubitPersistent Current QubitQQ

6. Qubit manipulation –p

Single qubit – again two parametersFluxMiddle junction (also by flux)

Two qubit gates

We have seen effects from environmental noise – decoherenceWe have seen effects from environmental noise decoherence

How to model the microscopic physics and study their effects?

Eventually, how to reduce them by designing clever circuits?

Superposition of alive and deadEither alive or dead

Decoherence is everywhereDecoherence is everywhere

Gaussian free particle

Because of decoherence, localization

• spin-boson model (see notes)Coupling to oscillator bathSpectral densitySpectral densityMaster equation – derivationLow-frequency noise

• Bloch equation (see notes)derivation from the master equationT1, T2 timesstatus que

• circuit model (see notes) Circuit coupling to coherent elementResistance and circuit

Oth t ff• Other stuffleakage problemcircuit imperfectionTLS fluctuatorsTLS fluctuators

Other StuffOther Stuff

Many other factors to cause quantum error/decoherence-- leakage problem

-- circuit imperfection – calibration etc

-- TLS fluctuators – echo experiments, calibration

Other StuffOther Stuff

Outline of LecturesOutline of Lectures

1. Quantum circuit and Hamiltonian2 S d ti t bit ( bit )2. Superconducting quantum bits (qubits)3. Decoherence (spin‐boson model and circuit model) 4. Recent progress (may come back)5. Circuit quantum electrodynamics (QED) and Transmission line resonator6. Two‐level system fluctuators and Superconducting Josephson junction resonator7. Nanomechanical systems, resonator, and laser cooling8. Coherent frequency conversion

• Cavity and Circuit QEDAtomic systems – classic resultsCPW - Transmission line resonatorCPW Transmission line resonatorStrong coupling

• Bose-Hubbard modelcoupling between resonators by capacitorSelf-interactionBHM: second order phase transitionMean field theory approach

• quantum engineering via BHMfour resonator modelEnergy structureAllowed and forbidden transitionsEPR t tEPR states

Cavity QEDCavity QED

2

Node A Node B

2

Quantum

Laserfiber

Laser

network

Atoms in cavityQuantum dot photonicsSolid-state circuit

driven by microwave source1 - freq of qubitc - freq of cavity

Solid state circuit

g1 - coupling, g1 = gc , gd - cavity damping- TLS decoherence

Cavity QEDCavity QED

Cavity QEDCavity QED

Wallraff et al Nature 2004

Superconducting system has uniqueadvantages

Cavity QEDCavity QED

Jaynes-Cummings model

1 pair-wise coupling between state

2. Can be solved exactlySee notes

Vacuum Rabi splittingVacuum Rabi splitting

Cavity QEDCavity QED

Off-Resonant case 3. Dispersive regime

Stark shifts

Can be used to QND detect qubit statesQ q

Transmission Line ResonatorTransmission Line Resonator

• Basics of Circuit QED

• Resonator quantization• Resonator quantization

• Voltage inside resonator

• Coupling with charge qubits

Superconducting transmission line coupling to charge qubits

Quantization of resonator modes – see notes

Bose-Hubbard ModelBose-Hubbard Model

Bose-Hubbard ModelBose-Hubbard Model

Solid-state system provides

1. control of individual cavity2. Control of dynamics3 Detection via various techniques3. Detection via various techniques

We will study the construction and applications

Bose-Hubbard ModelBose-Hubbard Model

• Tunneling between nearest neighbors• On-site interaction of bosons –self interacting photon modes• Classic system for 2nd order quantum phase transitiony q p• See notes for property

Quantum Engineering with BHMQuantum Engineering with BHM

Coupled resonator modes

Can be explored to generateEntangled photon pairsEntangled photon pairs

Y. Hu and L. Tian, 1004.2240

Interaction see notes

coupling

Quantum Engineering with BHMQuantum Engineering with BHM

Without pumping, photon number conservedWith pumping, states can be divided by photon numberp p g y pPumping connects states differ by one photon

Mott insulator regime –Energy levels with 0, 1, 2 photons

When weak driving, transitionsF bid/ ll bForbid/allow by resonance Conditions and symmetry of States and pulses

23 is always eignestate

Quantum Engineering with BHMQuantum Engineering with BHM

Photon out-coupling for real applications

• TLS in amorphous layer Energy structure and distribution – see notesCoupling to junctionsCoupling to junctionsExperimental observations

• circuit QED for TLS – detection (skip this part)( p p )magnetic field effectbad cavity limittransmissionspectrum

• circuit QED for TLS – quantum gateseffective couplingbarriersolutionfid litfidelity

The Presence of Two Level System Defects• Previous phase qubit measurements show spectroscopic

splittings due to amorphous two level system (TLS)

The Presence of Two-Level System Defects

splittings due to amorphous two-level system (TLS) fluctuators inside Josephson junctions (a strong source of decoherence).

• Can we find a way to distinguish the coupling mechanismbetween the two-level systems (TLS) and the junction?, e.g. coupling to critical current or coupling to dielectric field.

Two-level System FluctuatorsTwo-level System Fluctuators

Strong decoherence in superconducting qubits• ubiquitous in solid state systems

o e e Syste uctuato so e e Syste uctuato s

• ubiquitous in solid-state systems • defects in amorphous materials – oxide, glass, …• induces charge/flux/current noise with 1/f spectrum for large # of TLS’sfor large # of TLS s

Experiments showed coherent coupling with phase qubits:previous phase qubit measurements show spectroscopic splittingsprevious phase qubit measurements show spectroscopic splittingsdue to two-level system (TLS) fluctuators inside amorphous junctions (a strong source of decoherence). (Martinis et al. 2005, Neeley et al 2008, ( , y ,Y. Yu et al, 2008, Han group, 2009, …)

Quantum manipulation of TLS – lack of direct controlling

TLS in Amorphous MaterialsTLS in Amorphous Materials

• Model for TLS in solids

distribution• distribution

Coupling to Josephson JunctionCoupling to Josephson Junctionp g pp g p

Critical Current Coupling Dielectric coupling

si

si i

sis

d di l h b i thi k1: total flux in junction d0: dipole, h0: barrier thickness

Circuit QED in JJ ResonatorCircuit QED in JJ Resonator

• atoms ions in cavity

QQ

• atoms, ions in cavity • quantum dot photonic devices • superconducting quantum circuit

cavity - Josesphon junction resonator

coupling microwavedriving

TLS noise

atom – qubit (TLS) cavity damping

c - detuning of microwave modea - detuning of qubit (TLS)g1 - coupling, g1 = gc , gd

Cavity QED in solid-state devices • qubit

TLSg1 coupling, g1 gc , gd • TLS

Idea for Quantum Logic OperationIdea for Quantum Logic OperationQ g pQ g pLong coherence time demonstrated in recent experiments• (de)coherece time longer than that of qubit( ) g q• can we test logic operations with TLS’s inside insulating layer?

A circuit QED idea to achieve universal quantum logic

si

ex

TLS’s inside adriven

J ti t

Challenges in the idea – not trivialTLS’ ll d i ll ff

sJunction resonator

• TLS’s are well spaced in energy – usually off-resonance• lack of control handle on individual TLS

L. Tian & K. Jacobs, PRB 79, 114503 (2009)

Effective Hamiltonian of TLS’sEffective Hamiltonian of TLS’s

• dispersive regime: resonator off-resonance with TLS• driving on resonator• driving on resonator • applying unitary transformation:

• effective Hamiltonian:

CavityHamiltonian

Effective qubitHamiltonian

ResidueCouplingHamiltonian Hamiltonian Coupling

Effective Hamiltonian of TLS’sEffective Hamiltonian of TLS’sEffective Hamiltonian of TLS sEffective Hamiltonian of TLS s

• resonator and TLS are decoupled in • TLS parameters are controllable via resonator• extra noise induced in• TLS’s are off resonance

• Residue coupling is “small” (numerical simulation)

ScalabilityScalabilityyy

exsis

sis

TLS’s in different junctions

• Different junctions corresponds to same cavity mode• TLS’s in different junction coupling with same cavity mode

C ll d l i b f d l b f• Controlled logic gates can be performed exactly as before• Resonator frequency is affected by number of junctions

L. Tian & K. Jacobs, PRB 79, 114503 (2009)

• nanomechanical systemssystemQuantum engineering by coupling to other systemsQuantum engineering by coupling to other systemsLinear couplingQuantum protocol

• cooling in the side-band regimecoupling to superconducting resonatoroptomechanical vs linear couplingcooling

Nanomechanical ResonatorNanomechanical Resonator1. Sometime ago• Vibration of strings• Dynamics – Euler-Bernoulli Eq.

2. Now, the decrease of size provides:, phigh frequency -- GHzhigh Q – 103 –5 & =0/Q

E: Young ModulusE: Young ModulusI: moment of inertiaa : linear density

3. quantum mechanics – (?) => cooling a doubly clamped beam,flexural modes

u(z,t)L

Can It Be Quantum Mechanical?Can It Be Quantum Mechanical?

High quality factor over 10,000,000 (f=20 MHz) very coherent once it becomes coherent(calculation of Q-factor?)

Macroscopic quantum effects?Macroscopic quantum effects? superconducting quantum tunnelingnanomechanical system - cat state, entanglement - test QM (?)

Barrier, thermal fluctuations T=24 mK=500 MHz resonator frequency 100 kHz - 100 MHz

a0

Why interesting? a0

• fundamental physics: quantum/classical b d i S h di tboundary, using e.g. Schroedinger cat state• metrology/calibration with resonators• quantum data bus - ion trap• continuous variable quantum information

Quantum Engineering by Coupling to Other SystemsQuantum Engineering by Coupling to Other Systems

contacts

x

yzresonator

u

Cooper pair box

xu1 u2beam

support Flexual mode

Atomic system - Coulomb interaction with trapped

Solid-state system - Capacitive interaction with- Coulomb interaction with trapped

ion(nanotrap, Tian&Zoller, PRL 2004)

- Capacitive interaction with superconducting devices• diversity in coupling• various quantum devicesvarious quantum devices• flexibility in parameters

Nanomechanical System vs Solid-State QubitNanomechanical System vs Solid-State Qubit

…

|1>

|e>

E

Ion trap

oscillator

|0>|1>

two-level system|g>

EJ r

Cooper pair box

beam

Flexual mode|g>|e>

Cirac, Zoller, PRL

Superconducting charge qubits - Cooper pair box

(1995)

• charge states |0>, |1>, with 2e difference, controlled by flux x(t)• capacitive coupling with nanomechanical vibration• microwave control on couplingg

Armour, Blencowe, Schwab, PRL (2002)

Nanomechanical System vs Solid-State QubitNanomechanical System vs Solid-State Qubit

1. Ground state cooling is possible via microwave driving at the red side band gof qubit and via damping of qubitnf=0.03 vibration quanta

I Martin Shnirman Tian Zoller PRB (2004)

2. Quantum engineering and entanglement by pulses at weak coupling

I. Martin, Shnirman, Tian, Zoller, PRB (2004)

by pulses at weak coupling

T=50 nsec,L. Tian, PRB (2005)

Hz

T 50 nsec, F=0.9952

3. Arbitrary state can be engineered by optimal pulse control, =1.6

50

MH

tJacobs, Tian, Finns, PRL (2009)

Quantum teleportation of nanomechanical modesQuantum teleportation of nanomechanical modes

VVxxCC (x)(x)

VVxxCC (x)(x)

transmission lineresonator resonator

Q pQ p

EEJJ EEJJ

VVgg

CCx x (x)(x)CCgg CCmm

EEJJEEJJ

VVgg

CCx x (x)(x) CCggCCmm

CC CC

LLrr RRrr

JJ JJ

exex

JJJJ

exex

CCrr CCrr

bUnknown state in a ain,a1 b2,a2crstate in ain

Local entangled pair

Remote entangled pairBeam splitterBeam splitter

1. entanglement by parametric amplification: ac driving

After detectionxu, pvUnknown state in a2

2. beam splitter: ac driving3. fidelity calculated by Wigner function approach L. Tian and S. Carr, PRB (2006)

Side Band Cooling RegimeSide Band Cooling Regime

Side band limit provides promise for ground state coolingRecent experiments reach side band limit for NEMS - optical cavity

d NEMS d ti t i t h i l ff tand NEMS - superconducting resonator using optomechanical effectsand NEMS - superconducting qubit (Lehnert, Kippenberg, Wang, Schwab, Mavalvala, Chen ……), Quantum regime is in visible future!

Schliesser et al, Nature Phys (2008)

Regal et al, Nature Phys (2008)

Park & Wang, Nature Phys (2009)

Two Circuits for Mechanical CouplingTwo Circuits for Mechanical Coupling• Resonator capacitively coupling with mode - LC oscillator• Cooling from dynamic backaction capacitance

radiation pressure-like parametrically modulated linear coupling

Previousschemescheme

-

At typical parameters:Cooling:Cooling:

Derivation of coupling Hamiltonian, see notes

Parametric DrivingParametric Driving

• parametric driving provides “up-conversion” of low-energy mechanicalquanta to high energy microwave photons, which then dissipate in circuitq g gy p p

• in rotating frame, effective energy for LC oscillator is -

• thermal bath has temp. T0 with1

effective temp in rotating frame:

nb0

1ehb / kBT0 1

• effective temp. in rotating frame:

• “equilibrium” between thermal bath of mechanical mode and LC mode

Quantum Backaction NoiseQuantum Backaction NoiseCooling of trapped ion at

a

r/2en+1

|e> e

n+1

|e> r /2 b

nn+1

n-1|g>n

n+1n-1|g>

A-=( r)2/e A+=( r / 4a)2 e

O h i l d l li h

( r) e ( r a) e

cooling circle heating circle

Our scheme is related to laser cooling schemecooling transitioncooling rate

heating transitionheating rate

• comparison• also applies to n0

g

Backaction noise comes from counter rotating terms in the coupling

Cooling by Quantum TheoryCooling by Quantum Theory

Quantum explanation - input-output theoryoperator equations can be solved in Heisenberg picture

li t b d i d i l di lfcooling rate can be derived including self-energyequation similar to linearized equations for radiation pressure

red sideband

4gl2

h20(102 /16a

2)=

02 /16a

2 0.0025

• solid - quantum theory• dashed - semiclassical theorymaximal cooling atmaximal cooling at (nearly) red sideband

Derivation of laser cooling, see notes

Occupation NumberOccupation Number

solid-lines: full quantum theoryWe calculated the nf

a with no counter rotating terms: - P-representation q y

dashed-lines: no counter rotating termsQm=a/0: quality factor of resonator

• low Q : 2nd term dominateslow Qm: 2nd term dominates • high Qm: backaction noisedominates, dashed curve can reach 0 solid curve reach nreach 0, solid curve reach n0• Qm=105, na

f =0.01<<1

Tian, PRB (2009)

Parameters in Superconducting CircuitsParameters in Superconducting Circuits

Comparing with parameters in a few experiments, we choose the following:

104 - 106

<

Advantage – no need to pump the LC oscillator to high occupation

ltian@ucmercedhtt //f lt d d /lti /http://faculty.ucmerced.edu/ltian/