Advanced Quantum Mechanics Rajdeep Sensarma sensarma@theory.tifr.res.in Quantum Dynamics Lecture #3
Recap of Last lass Time Dependent Perturbation Theory Linear Response Function and Spectral Decomposition Real and Imaginary Part of Response Fn. Fermi s Golden Rule and Transition Probabilities
Quantum Adiabatic Theorem onsider a time dependent Hamiltonian H(t). If H was time independent, we have seen that expanding the state of the system in this basis simplifies the dynamics. Is it useful to expand the state in the eigenbasis of the instantaneous Hamiltonian H(t) for each time-point t? This would in general be a time-dependent basis set, since Hamiltonians at different time points do not commute. Quantum Adiabatic Theorem (Max Born and Vladimir Fock, 1928) For a continuously evolving of Hamiltonian, whose changes are slow enough, a quantum system, which is initially in the eigenstate of the instantaneous Hamiltonian, follows this state provided the eigenvalue of this state is nondegenerate and separated from the rest of the (instantaneous) Hamiltonian s spectrum at all times.
Quantum Adiabatic Theorem A spin 1/2 object in a magnetic field. 2D Hilbert space with a Hamiltonian H(t) = µ B ~ B(t) ~ Assume that magnitude of the magnetic field is fixed, while its direction varies with time. Define the basis states ˆn, +i ˆn, i and where ~ ˆn ˆn, ±i = ± 1 ˆn, ±i 2 ˆn(0) ˆn(t) If ~ B(t) =Bˆn(t) ˆn(t), ±i is the basis where the instantaneous Hamiltonian H(t) is diagonalized The eigenvalues of the instantaneous Hamiltonian : 1 2 µ BB gapped spectrum at all times The Quantum Adiabatic theorem, in this case, states that, if we start the system in the state n(0), +>, the system will evolve to n(t), +>. Similarly, if we start with n(0), ->, the system will evolve to n(t),->
Quantum Adiabatic Theorem i@ t (t)i = Ĥ[R(t)] (t)i R(t) Vector space of time dependent Parameters The Hamiltonian at time t is completely specified by the values of (possibly >1) parameters which change with time. H(t) is a point in the parameter space As the parameters change in time, one traces out a curve in the parameter space, R(t) orrespondingly, at each time (or for each R), there is a basis which diagonalizes H[R(t)] Ĥ[R(t)] n[r(t)]i = n [R(t)] n[r(t)]i (t)i = X n c n (t)e i R t 0 dt0 n (t) n(t)i = X n c n (t)e i n(t) n(t)i From X [ n c n (t)+ic n (t)]e i n(t) n(t)i + ic n (t)e i n(t) n(t)i = X n n c n (t) n (t)e i n(t) n(t)i Using hm(t) n(t)i = mn c m (t) = c m (t)hm mi X c n (t)e i[ m(t) n (t)] hm ni n6=m
Quantum Adiabatic Theorem H(t) is a point in the parameter space R(t) Taking time derivative Vector space of time dependent Parameters So for m n Adiabaticity condition is equivalent to neglecting the second term on RHS X c n (t) 2 =1) c n (t) < 1 Adiabaticity condition n For lowest energy gap hm Ĥ ni hm mi
Berry Phase For adiabatic processes, starting from a non-degenerate state c m (t) =c m (0)e R t 0 dt0 hm(t 0 ) m(t 0 )i = c m (0)e i m Berry s Phase m(t) =i Z t 0 dt 0 hm(t 0 ) m(t 0 )i Extra phase due to time-dependent basis Separate from the dynamic phase θ m Starting from an eigenstate of the initial Hamiltonian c m (0) = 1 for m = m 0 c m (0) = 0 for m 6= m 0 n (t f )i = e i( n+ n ) (0)i The system remains in the corresponding instantaneous eigenstate R(t) t n(t) =i Z tf 0 dt 0 hn[r(t 0 )] n[r(t 0 )]i = i Z tf 0 dt 0 Ṙhn[R(t 0 )] r R n[r(t 0 )]i Vector space of Parameters Vector Potential/Berry onnection ~A(R) =ihn(r) r R n(r)i H(t) is a point in the parameter space
Berry Phases and Gauge Invariance At each point in the trajectory, the instantaneous Hamiltonian specifies the eigenbasis upto an overall phase which we can choose.this is a position (in the param space) dependent phase rotation ( a U(1) gauge symmetry). Let us assume we can choose the phase of the state we are tracking. All other phases are fixed by H[R(t)]. R(t) Vector space of Parameters H(t) is a point in the parameter space Berry potential is not a gauge invariant quantity Berry, 1973: onsider the evolution on a closed path, so that H(T)=H(0). Berry phase for this evolution I = ~A(R) dr is a gauge invariant quantity R(t) Vector space of Parameters H(t) is a point in the parameter space
Berry Phase for closed paths What happens to the system if we adiabatically take it along a closed path? t The system remains in the corresponding instantaneous eigenstate R(t) Vector space of Parameters H(t) is a point in the parameter space n (t f )i = e i( n+ n ) (0)i Dynamic Phase depends on how fast or slow the circuit is traversed. Geometric or Berry Phase depends on the circuit, but not on how it is traversed in time
Berry urvature onsider 3-dimensional space of parameters, and use Stokes Theorem Z Z Z Z n() =i d S ~ r hn[r] r R n[r]i = Im d S ~ hr R n[r] r R n[r] = Im X m6=n Z Z d ~ S hr R n[r] mi hm r R n[r]i where S is any surface which has as its boundary R(t) Vector space of Parameters H(t) is a point in the parameter space to write Use n() = Z Z d ~ S. ~ B n (R) ~B n (R) =Im X m6=n hn(r) r R Ĥ(R) m(r)i hm(r) r R Ĥ(R) n(r)i (E n (R) E m (R)) 2 Effective Magnetic Field/ Berry urvature The Berry curvature is a gauge invariant quantity
Degeneracy Points and Simplification Berry phase calculation simplifies for trajectories near a degeneracy point (which avoid this degenerate point). Z Z n() = d S. ~ B ~ n (R) Bn ~ (R) =Im X hn(r) r R Ĥ(R) m(r)i hm(r) r R Ĥ(R) n(r)i (E n (R) E m (R)) 2 m6=n Simplest ase: A degeneracy of 2 states ( +> and ->) at R=R 0. Ignore other states: finite energy denominator -----> Work with 2X2 Hamiltonian in this space. Most General 2X2 Hamiltonian H(R) = 1 2 H z H x ih y = 1 R H x + ih y, H z 2 ~ ~ R ~ =(Hx,H y,h z ) r R H = 1 2 ~ E ± ( ~ R)=±R Use all your expertise with spin-1/2 objects to get ~B ± ( ~ R)=± ~ R R 3 Ω is the solid angle subtended by the circuit at the degeneracy point. ±() = 1 2
Topological Invariants For a 2 state system with time varying Hamiltonian, the Berry phase is given by ±() = 1 2 Ω is the solid angle subtended by the circuit at the degeneracy point. Now consider a 2D parameter space, say H y is 0 If H y is 0, exp[i γ()]= -1 if encloses degeneracy point and exp[i γ()]= 1 if it does not. γ()= π if encloses degeneracy point and γ()= 0 if it does not. The Berry phase is topological in the sense that contours that can be smoothly deformed to each other has same Berry phase. The degeneracy point acts as a puncture in this manifold and loops enclosing this point have a different topology than those not enclosing it.
State Systems and Rabi Oscillation 2-state system with time indep. Hamiltonian H 0 = E 1 1ih1 + E 2 2ih2 Harmonic time-dependent off-diagonal term ω E 2 -E 1 H 1 (t) = (e i!t + e i!t ) 1ih2 + h.c. Example: Two level atom interacting with classical radiation field. Transform to a time dependent basis, which rotates in time with the freq of the pert. 1 0 i = e i!t 1i 2 0 i = 2i This is different from going to interaction representation. There, states rotate with their unperturbed energies as freq. Here the rotation freq. is the perturbation frequency. However, at resonance (ω= E 2 -E 1 ), the two approaches are exactly same.
2 State Systems and Rabi Oscillation H 0 = E 1 1ih1 + E 2 2ih2 ω E 2 -E 1 H 1 (t) = (e i!t + e i!t ) 1ih2 + h.c. Example: Two level atom interacting with classical radiation field. In the rotating basis: 1 0 i = e i!t 1i 2 0 i = 2i H = E 1 1 0 ih1 0 + E 2 2 0 ih2 0 + [ 1 0 ih2 0 + 2 0 ih1 0 ] + [e 2i!t 1 0 ih2 0 + e 2i!t 2 0 ih1 0 ] Rotating Wave Approximation: RWA In the basis rotating with the perturbation, terms with explicit dynamics on the scale of the perturbation freq. are neglected (set to 0 by hand). Implicit assumption: we (the measurement process) averages over timescales much larger than ω -1.
RWA and Dressed States Schrodinger Equation: (Time Dep. Basis) i c1 c 2 = (t)i = c 1 (t) 1 0 i + c 2 (t) 2 0 i E1 +! c1 E 2 c 2 Initial ondition: c 1 (0) = 1, c 2 (0) = 0 Dressed Hamiltonian Rabi Frequency E ± = E r 1 + E 2 +! (E2 E 1!) Eigenvalues: ± 2 + 2 4 2! R = r (! E21 ) 2 4 + 2 +( ) u( v) Eigenstates: i = u 2 =1 v 2 = 1 v(u) 2 " 1+! E 21 2 p (! E 21 ) 2 /4+ 2 # Dressed States: Eigenstates of the Hamiltonian in the rotating basis Transform back to get dynamics c 1 (t) =u 2 e ie+t + v 2 e ie t c 2 (t) =uv[e ie+ t e ie t ] So, c 1 (t) 2 =1 2! R 2 sin 2 (! R t) c 2 (t) 2 = 2! 2 R sin 2 (! R t)
2 State Systems and Rabi Oscillation Rabi Frequency! R = r (! E21 ) 2 4 + 2 c 1 (t) 2 c 1 (t) 2 c 1 (t) 2 c 2 (t) 2 c 2 (t) 2 c 2 (t) 2 /! R =0.7 t /! R =0.9 t /! R =1.0 t c 1 (t) 2 c 2 (t) 2 π Pulse: Population in S z =-1/2 state t c 1 (t) 2 π/2 Pulse: Population in c 2 (t) 2 S y state t
Rabi Oscillations and RWA c 1 (t) 2 c 2 (t) 2 RWA vs. Full Numerical Solution As the frequency of the perturbation nears resonance, i.e. for! E 21! R + (! E 21) 2 8 2 t the transition probability oscillates with time with the Rabi frequency. Quite different from linear response theory (where osc. are at driving frequency ω) or from Fermi golden rule where transition probability scales with time. Notice that we have not talked about weak or perturbative drive. In fact, near resonance, the coupling of the time dependent part dominates the whole action. This type of drive is often called coherent coupling of states/coherent drives, since the drive induces phase correlations between the 2 states. t RWA misses the fast wiggles Near resonance we do have a small parameter /, where Δ is the detuning from the transition. This is a question of having different time-scales in the problem.
RWA with 3 states Δ 1 Δ 2 3 Three state system and 2 optical perturbation H 0 = E 1 1ih1 + E 2 2ih2 + E 3 3ih3 E 31 ω2 ω 1 E32 H 1 (t) = 1 (e i! 1t + e i! 2t ) 1ih3 + 2 (e i! 2t + e i! 2t ) 2ih3 + h.c. 1 2 E 21 1 0 i = e ie 1t 1i 2 0 i = e i(e 1+! 1! 2 )t 2i 3 0 i = e i(e 1+! 1 )t 3i H = 0 @ 0 0 1 0 1 2 2 1 2 1 1 0 A + @ 0 0 1 e 2i! 1t 0 0 2 e 2i! 2t 1 e 2i! 1t 2 e 2i! 2t 0 1 A Rotating Wave Approximation Dressed Hamiltonian 2 photon resonance: = 1 2 ' 0 Using tan( ) = 1 2 tan(2 )= Dressed States: p 2 1 + 2 2 1 a + i =sin sin 1i + cos 3i + cos sin 2i Autler Townes Splitting q! ± = 1 ± 2 1 + 2 1 + 2 2 a i =sin cos 1i sin 3i + cos cos 2i a 0 i = cos 1i sin 2i Dark State: Does not overlap with 3>, eigenvalue 0
RWA with 3 states 3 a 0 i = cos 1i sin 2i Dressed State E 31 Δ 1 Δ 2 ω2 Dark State: No overlap with 3>, eigenvalue 0 ω 1 E32 1 2 E 21 If atom is prepared in this dressed state, it cannot be excited to 3> and cannot decay by spontaneous emission. Hence this state is called dark state. If by some mechanism, the atom is prepared in this state, it will remain in this state for a very long time onsider the case : 1 2! 0 a 0 i! 1i a + i!cos 3i +sin 2i a i! sin 3i + cos 2i annot excite the system even if a resonant field (between 1> and 3>) is applied. This is due to the already applied dressing field between 2> and 3>. This phenomenon is called Electromagnetically Induced Transparency (EIT)