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
• 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/