Theoretical design of a readout system for the Flux Qubit-Resonator Rabi Model in the ultrastrong coupling regime

Theoretical design of a readout system for the Flux Qubit-Resonator Rabi Model in the ultrastrong coupling regime Ceren Burçak Dağ Supervisors: Dr. Pol Forn-Díaz and Assoc. Prof. Christopher Wilson Institute for Quantum Computing August 2014

Outline 1 Introduction 2 Constructing the Rabi Hamiltonian 3 Results of the Rabi Model of Flux Qubit-LC Resonator system 4 The theoretical design of a QND measurement system for Rabi Model of Flux Qubit-Resonator 5 Conclusions

Introduction Rabi Model qubit coupling resonator Figure : A schematic for the Rabi Model, represents an interaction between a two-level system and a quantum harmonic oscillator, (schematic copyright S. Ashhab and F. Nori).

Introduction Coupling Regimes The main parameters of the system: ω qb, ω r and g coupling strength. When g ω qb,r and the losses dominate the coupling strength: weak coupling regime. When g < ω qb,r but the interaction dominates the losses: strong coupling regime. When g ω qb,r : ultrastrong coupling regime. When g > ω qb,r : deep strong coupling regime.

Introduction Jaynes-Cummings Model The energy level structure of Rabi Model is only solvable in terms of special functions, recently [Braak2011]. So we numerically solve the Rabi Model by truncating its infinite dimensional Hilbert space. An analytical solution is possible for the Rabi Model in the weak coupling regime under Rotating Wave Approximation Jaynes-Cummings Model.

Introduction Comparison of circuit-qed with the atomic QED systems System Type cqed atomic QED Parameter variability: changeable fixed and identical Scalability: high not at all Decoherency: greater lesser Table : The comparison of the cqed and atomic QED systems.

Introduction QND Measurement Quantum Nondemolition Measurement is an indirect method to read-out the information stored in the qubit in a quantum mechanical system. The dispersive regime of the Jaynes-Cummings Model is utilized for QND measurement in the cqed experiments. This research proposes a possible QND measurement method for the Rabi Model in the ultrastrong coupling regime.

Constructing the Rabi Hamiltonian The Quantization of the LC Resonator Figure : Schematic of an LC resonator, shown with its quantized canonical variables.

Classically: H = Q2 2C + φ2 2L By using Hamilton s equations one can obtain the well-known harmonic oscillator differential equation whose solution is, (1) Q(t) = C 1 sin(ωt) + C 2 cos(ωt) (2) where ω = 1/ LC and C 1, C 2 are some arbitrary real constants to be determined by the initial conditions. In order to quantize the LC oscillator, let us rewrite it ( ) 2 H = 1 L 2L C Q + φ 2 ( ) ( ) (3) = 1 L L 2L C Q iφ C Q + iφ

The canonical quantization dictates the exchange of the dynamical variables by the operators with a commutation relation. Then the Hamiltonian will be, Ĥ = ˆQ 2 2C + ˆφ 2 2L. (4) [ where the commutation relation is ˆφ, ˆQ] = i. By applying the same procedure to the factorized parts, â = â = 1 2L ( L C ˆQ + i ˆφ ), (5) ( ) 1 L 2L C ˆQ i ˆφ. (6)

Then finding â â, â â = ˆQ 2 2C + ˆφ 2 2L i 1 [ 2 ˆQ, ˆφ], (7) LC and by exploiting the commutation relation and substituting ω LC = 1/ LC, the Hamiltonian will be, ( Ĥ = ω â â + 1 ) ( = ω ˆn + 1 ). (8) 2 2

Then the field operators in terms of annihilation and creation operators, ( ) 1/2 ( ) C 1/4 ˆQ = (â + â ), (9) 2 L ( ) 1/2 ( ) L 1/4 ˆφ = i (â â ). (10) 2 C just like the electric and magnetic fields in the quantization of the EM field. We define L/C = Z 0, which is introduced as the impedance also used in classical microwave theory.

Constructing the Rabi Hamiltonian Josephson junction SC I SC I = I o sin δ, (11) dδ dt = 2e V. (12) where δ is the phase change between the two ends of the element and I 0 is the critical current, which is the maximum amount of current that can flow through a superconducting junction.

Constructing the Rabi Hamiltonian Flux qubit

The energy of a single Josephson junction, E = t 0 IVdτ = t 0 I o 2e dδ sin δ dτ dτ = I 2e δ o sin δ dδ = E J (1 cos δ). 0 (13) The wave functions must be single valued along a closed path, which motivates the flux quantization condition: φ 1 φ 2 + φ α + 2πf = 0. (14) where f is the magnetic frustration and is defined as f = Φ Φ 0, Φ is the externally applied magnetic flux and Φ 0 = h/2e is the flux quantum. Then the energy of the flux qubit, U = E J (1 cos δ 1 + 1 cos δ 2 + α α cos δ α ) = E J (2 + α cos δ 1 cos δ 2 α cos (2πf + δ 1 δ 2 )), (15)

δ2 2π -2-1 0 1 2 2 2 1 1 0 0-1 -1-2 -2-2 -1 0 1 2 δ1 2π Figure : The contour plot for the energy of the flux qubit with respect to its two different phases when α = 0.8 at symmetry point, f = 0.5.

3.0 2.5 U EJ 2.0 1.5-0.4-0.3-0.2-0.1 0.0 0.1 0.2 Figure : Two dips seen in the energy plot of the flux qubit which corresponds to two stable solutions. δ 1 2π These stable configurations represent the circulating DC currents of opposite direction through the circuit. These currents are called persistent currents, [Mooij1999].

Constructing the Rabi Hamiltonian The Hamiltonian of the Flux Qubit H = 2 σ x + ɛ 2 σ z, (16) where ɛ = 2I p Φ o ( f 1 2) is the magnetic energy of the flux qubit and is the tunnelling rate of the wave functions in the potential energy created by three Josephson junctions in a flux qubit and I p is the persistent current in the flux qubit.

Diagonalized Hamiltonian: ( 1 Λ = 2 ɛ 2 + 2 2 ) 0 ɛ 2 + 2 2, (17) 0 1 2 when the eigenvectors are chosen as, φ 1 = (cos θ/2, sin θ/2), (18) φ 2 = ( sin θ/2, cos θ/2), (19) where tan θ = /ɛ, so that the transformation matrix is T = ( cos θ/2 sin θ/2 sin θ/2 cos θ/2 ). (20)

Constructing the Rabi Hamiltonian Rabi Hamiltonian Rabi Hamiltonian will be in general composed of three parts: Ĥ = Ĥ qb + Ĥ r + Ĥ I. (21) The Hamiltonian for the flux qubit explicitly, Ĥ qb = 1 ( 4 f 1 2 Ip 2 2) 2 Φ 2 0 + 2 2 ω qb 2, (22) The Hamiltonian for the LC resonator, ( Ĥ r = ω r â â + 1 ). (23) 2

Finally the interaction term through electric dipole coupling in general, Ĥ I = ˆd Ê, (24) where ˆd is a dipole operator which consists of atomic transition operators. Let us introduce atomic transition operators in a two-level system, which enable the transitions from ground to excited state (or vice versa): ˆσ + e g, ˆσ g e. (25)

By recalling the form of the quantized charge in the Quantum LC Resonator, the interaction term can be expressed as, Ĥ I = g(ˆσ + + ˆσ )(â + â ) = gσ x (â + â ). (26) By taking the fact that the coupling is through the interaction of the magnetic field of the resonator and the magnetic moment of the qubit into account, the Rabi Hamiltonian for the flux qubit-lc resonator system, H = 2 σ x + I p Φ o ( f 1 2 ) ( σ z + gσ z (â + â ) + ω r â â + 1 ). 2 (27) In order to bring it to the known form of cqed Hamiltonian, we will use the diagonalized form of the qubit Hamiltonian and express the other terms in the eigenbasis of the qubit Hamiltonian.

So we obtain the total Hamiltonian as Ĥ = ω qb 2 σ z g sin θσ x (â + â) + ω r where ω qb ω qb 2 = 1 2 sin θ and (recalling) 4 ( f 2) 1 2 I 2 p Φ 2 0 + 2 2. ( â â + 1 ), (28) 2

Interaction terms: Constructing the Rabi Hamiltonian Rotating Wave Approximation ˆσ + â e i(ω qb ω r )t ˆσ â e i(ω qb+ω r )t ˆσ + â e i(ω qb+ω r )t ˆσ â e i(ω qb ω r )t (29) (30) (31) (32) Equations (30) and (31) represent fast rotating terms and also non-resonant terms in the rotating frame, so that we can apply the so-called rotating wave approximation (RWA) where we drop the fast rotating terms from the Rabi Hamiltonian. RWA is only possible under the condition: g ω qb,r 1. (33)

Constructing the Rabi Hamiltonian Jaynes-Cummings Hamiltonian of Flux Qubit-LC Resonator Figure : A circuit schematic which shows the JC model of a flux qubit coupled to an LC resonator.

Assuming that g ω r 1, the RWA can be applied to the coupling term. Then Jaynes-Cummings Hamiltonian is: Ĥ = ω ( ) ( qb 2 σ z g sin θ ˆσ â + ˆσ + â + ω r â â + 1 ), (34) 2 Now we need to construct the JC Hamiltonian matrix in order to extract the eigenenergies of the Rabi system when g ω r 1.

Ĥ =... ωqb 2 + Ω r (n) g sin θ n + 1 g sin θ n + 1 ω qb 2 + Ω r (n + 1).... (35) ( where Ω r (n) = ω r n + 1 2). If we diagonalize this matrix, we will obtain different eigenvalues in the general form, E ±,n = (n+1) ω r ± 1 ( ) 2 g 2 (n + 1) + 2 2 (ω qb ω r ) 2, (36) ω qb where n is the excitation number. The eigenstates of JC Model are also called the dressed states of the JC Model.

Figure : A schematic which shows the uncoupled Jaynes-Cummings states when g = 0 and dressed states when g > 0.

2.0 10 10 frequency Hz 1.5 10 10 1.0 10 10 0.48 0.49 0.50 0.51 2 Figure : Energy levels with respect to magnetic field frustration in the qubit when up to states with two photons exist in the resonator with g = 0.05ω r, vertical axis is frequency in Hz. Pink: E 0,, blue: E 0,+, green: E 1, and brown: E 1,+.

2.0 10 10 frequency Hz 1.5 10 10 1.0 10 10 0.48 0.49 0.50 0.51 2 Figure : The energy levels with respect to magnetic frustration when two photons exist in the resonator with g = 0, vertical axis is frequency in Hz. Pink: E 0,, blue: E 0,+, green: E 1, and brown: E 1,+.

Results of the Rabi Model of Flux Qubit-LC Resonator system Parity Chains A JC dublet: Rabi parity chains: e0 g1 (37) g0 e1 g2 e3... (p = 1), (38) e0 g1 e2 g3... (p = 1). (39)

Results of the Rabi Model of Flux Qubit-LC Resonator system Numerical Solution of Rabi Model Ĥ = ( 2 (σ x I n ) + I p Φ o f 1 ) 2 + g(σ z (â + â )) + ω r (I qb (σ z I n ) ( â â + 1 2 )). (40) 1.8 10 10 1.6 10 10 frequency Hz 1.4 10 10 1.2 10 10 1.0 10 10 8.0 10 9 6.0 10 9 0.490 0.495 0.500 0.505 0 Figure : The difference between energy levels of QRM with respect to magnetic field frustration in the qubit with g = ω r.

Energy spectrum: 25 25 Frequency [GHz] 20 15 10 5 0-5 20 15 10 5 0-5 e2 E q,r2 g2 E r2 e1 E q,r1 g1 E r1 e0 E q g0 G 0.0 0.2 0.4 0.6 0.8 1.0 g ω ucr Figure : Energy levels of the quantum Rabi model with respect to g/ω r using up to 11 Fock states in the resonator, at the symmetry point, f = 0.5 and ω qb = 0.5ω r.

The theoretical design of a QND measurement system for Rabi Model of Flux Qubit-Resonator Proposed Model

In order to understand why we proposed this model, let us investigate the dispersive regime of Jaynes-Cummings Model. The detuning is defined as δ = ω qb ω r. When there is a large detuning, 2g n + 1/δ 1, we can expand the eigenenergies of JC Model as ( ) E± d = (n + 1) ω r ± g 2 2 ± δ δ 2. (41) The shift ± g 2 δ 2 ω 2 qb is called Stark shift. ω 2 qb

We can obtain more insight about the energy shifts with a Schrieffer-Wolf transformation U of the generator S = i g δ (aσ + a σ ), H eff = UHU 1 = e is He is = H +i[s, H]+ i 2 [S, [S, H]]+... (42) 2 Apply it to the JC Hamiltonian and expanding to second order in g/δ leads to, H d JC ( ω r + g 2 δ sin2 θσ z ) a a + (ω qb + g 2 ) 2 δ sin2 θ σ z. (43)

Hamiltonian: Ĥ = ( 2 σ x + I p Φ o f 1 2 ( + ω RR ˆb ˆb + 1 2 ) ( σ z + gσ z (â + â ) + ω UCR â â + 1 ) 2 ) ( ) g r â + â) (ˆb + ˆb, â: annihilation operator of UCR, ˆb: annihilation operator of RR, g r = ω 2 c 2 ω UCR ω RR. (44) The first set of parameters is /2π = 5 GHz, ω UCR /2π = 10 GHz, ω RR /2π = 9 GHz, ω c /2π = 1 GHz so that the total coupling between the resonators is g r = 52.7 MHz. The second set of parameters is /2π = 12 GHz, ω UCR /2π = 10 GHz, ω RR /2π = 9 GHz and ω c /2π = 1 GHz. The third set of parameters is /2π = 9.5 GHz, ω UCR /2π = 10 GHz, ω RR /2π = 9 GHz and ω c /2π = 1 GHz.

The theoretical design of a QND measurement system for Rabi Model of Flux Qubit-Resonator Results Detuning (between UCR and qubit): 5 GHz. We will show: The compositions of the lowest four states with respect to the coupling strengths φ = i c i ψ i. The amount of the entanglement between RR and the rest of the system. If RR is sufficiently disentagled from rest of the system, energy level structure and the energy shifts observed in the RR s frequency.

1.0 0.8 c i 2 0.6 0.4 g00 G0 e10 E0 q,r1 g20 E0 r2 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 g ω ucr e30 E0 q,r3 g40 E0 r4 Figure : The decomposition of the state G0 to the other states with respect to g coupling.

1.0 0.8 c i 2 0.6 0.4 e00 E0 q g10 E0 r1 e20 E0 q,r2 0.2 g30 E0 r3 e40 E0 q,r4 0.0 0.0 0.2 0.4 0.6 0.8 1.0 g ω ucr Figure : The decomposition of the state E0 q to the other states with respect to g coupling.

1.0 0.8 c i 2 0.6 0.4 g01 G1 e11 E1 q,r1 g21 E1 r2 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 g ω ucr e31 E1 q,r3 g41 E1 r4 Figure : The decomposition of the state G1 to the other states with respect to g coupling.

1.0 0.8 c i 2 0.6 0.4 0.2 g10 E0 r1 e00 E0 q e20 E0 q,r2 g30 E0 r3 e40 E0 q,r4 0.0 0.0 0.2 0.4 0.6 0.8 1.0 g ω ucr g50 E0 r5 Figure : The decomposition of the state E0 r1 to the other states with respect to g coupling.

The energy level structure: 30 30 25 25 g00 G0 g20 E0 r2 Frequency [GHz] 20 15 10 20 15 10 e00 E0 q g01 G1 g10 E0 r1 e01 E1 q e02 E2 q e11 E1 q,r1 e10 E0 q,r1 5 5 g02 G2 g11 E1 r1 0 0 0.0 0.2 0.4 0.6 0.8 1.0 g ω ucr Figure : The energy levels of qubit-ucr-rr system at symmetry point, f = 0.5 with respect to g/ω UCR.

30 30 Frequency [GHz] 25 20 15 10 5 0 25 20 15 10 5 0 e2 E q,r2 g2 E r2 e1 E q,r1 g1 E r1 e0 E q g0 G 0.0 0.2 0.4 0.6 0.8 1.0 g ω ucr Figure : The energy levels of only qubit-ucr states of qubit-ucr-rr system at symmetry point, f = 0.5 with respect to g/ω UCR.

0-500000 6 10 7 4 10 7 g01 - g00-1.0 10 6-1.5 10 6-2.0 10 6-2.5 10 6 e01 - e00 2 10 7 0-2 10 7-3.0 10 6 0.0 0.2 0.4 0.6 0.8 1.0 g -4 10 7 0.0 0.2 0.4 0.6 0.8 1.0 g ω ucr ω ucr Figure : The energy shift between g01 g00 and e01 e00, respectively. g11 - g10 4 10 7 2 10 7 0-2 10 7-4 10 7 e11 - e10 2 10 7 1 10 7 0-1 10 7-2 10 7-3 10 7 0.0 0.2 0.4 0.6 0.8 1.0 g 0.0 0.2 0.4 0.6 0.8 1.0 g ω ucr ω ucr Figure : The energy shift between g11 g10 and e11 e10, respectively.

The first and second sets of parameters are usable, though in the third set (with a detuning of 0.5 GHz) RR is quite entangled to the rest of the system. The numerical results point out to a possible QND measurement setup for Rabi Model in the ultrastrong coupling regime. An analytic expression for the energy shifts and the possibility of a readout-qubit for further research.

PRL 107, 100401 (2011) PRL 105, 263603 (2010) Science, 285, 1036 (1999) PRL A, 81, 042311 (2010)