Physics

Can "observation force" be formulated as a continuous decoherence channel?

Step-by-step physics solution: Can "observation force" be formulated as a continuous decoherence channel?

As an Amazon Associate, I earn from qualifying purchases. For more practice problems like this, see Schaum’s Outline of College Physics, 12th Edition.

1. What the question is really asking

A student proposes to add an extra “observation‑induced decoherence’’ term to the usual Lindblad master equation

[ \frac{d\rho}{dt}= -\frac{i}{\hbar}[H,\rho] +\sum_k!\Big(L_k\rho L_k^{\dagger} -\frac12{L_k^{\dagger}L_k,\rho}\Big) \;-\;\lambda_{\rm obs}(t)\,[L_{\rm obs},[L_{\rm obs},\rho]] . ]

The idea is that simply watching a system (for instance with a detector that never records a click) should increase the decoherence rate, without any exchange of energy.
The student asks:

  • Is the extra double‑commutator term mathematically admissible?
  • Does it already appear in known theories (perhaps under another name)?
  • If it is sensible, what are the physical consequences (e.g. a faster loss of coherence, a quantum‑Zeno effect)?

Below we answer each point in detail, deriving the term from first principles, checking the Lindblad conditions, and linking it to existing literature.

2. From a physical model to the double‑commutator term

2.1 System + measuring device as an environment

Consider a quantum system S (Hilbert space (\mathcal H_S)) that we continuously monitor with a very weak detector M (the “measurement apparatus’’).
A standard way to model a continuous measurement is to couple S to a bosonic probe (or to a stream of ancillae) with an interaction Hamiltonian

[ H_{\rm int}= g(t)\, A \otimes B , ]

where

  • (A) is a Hermitian system operator that we are “looking at’’ (e.g. a position or a spin component).
  • (B) is an operator of the probe (often taken as a field quadrature).
  • (g(t)) is a real coupling strength that can be switched on and off; the square of this strength will become our observation strength (\lambda_{\rm obs}(t)).

The total Hamiltonian is

[ H_{\rm tot}= H_S\otimes\mathbb 1M + \mathbb 1_S\otimes H_M + H{\rm int}. ]

Assume the probe is initially in a Gaussian stationary state (e.g. thermal equilibrium) with zero mean, (\langle B\rangle=0), and short correlation time (\tau_c).
We then trace out the probe under the usual Born–Markov approximation (weak coupling, memoryless bath). The reduced dynamics for (\rho_S) obeys a master equation of the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) form:

[ \dot\rho_S = -\frac{i}{\hbar}[H_S,\rho_S]

  • \Gamma(t)\big( A\rho_S A - \tfrac12{A^2,\rho_S}\big) . \tag{1} ]

The rate (\Gamma(t)) is proportional to the spectral density of the probe evaluated at zero frequency:

[ \Gamma(t)=\frac{2\,g(t)^2}{\hbar^2}\int_{0}^{\infty} ! ! d\tau\, \langle B(\tau) B(0) \rangle . ]

Because the probe is a measurement device, we interpret (\Gamma(t)) as the strength of the continuous observation.

2.2 Re‑writing (1) as a double commutator

If the measured operator (A) is Hermitian ((A=A^{\dagger})), the Lindblad dissipator in (1) can be written

[ A\rho A - \frac12{A^2,\rho} = -\frac12\,[A,[A,\rho]] . ]

Hence (1) becomes

[ \boxed{\; \dot\rho_S = -\frac{i}{\hbar}[H_S,\rho_S] -\frac{\Gamma(t)}{2}\,[A,[A,\rho_S]]\;} \tag{2} ]

which is exactly the form suggested by the student, with the identifications

[ L_{\rm obs}=A ,\qquad \lambda_{\rm obs}(t)=\frac{\Gamma(t)}{2}\ge 0 . ]

Thus the observation‑induced decoherence term is not an ad‑hoc addition; it emerges naturally when a system is weakly and continuously coupled to a measuring apparatus that is later discarded.

2.3 Positivity and the Lindblad condition

The GKSL theorem tells us that any generator of the form

[ \mathcal L[\rho]=\sum_j\Big( L_j\rho L_j^{\dagger} -\frac12{L_j^{\dagger}L_j,\rho}\Big) ]

produces a completely positive, trace‑preserving (CPTP) map for every time interval.

Our double‑commutator can be cast into this form by defining a single Lindblad operator

[ \boxed{L_{\rm obs}= \sqrt{2\lambda_{\rm obs}(t)}\,A } . ]

Indeed,

[ L_{\rm obs}\rho L_{\rm obs}^{\dagger} -\frac12{L_{\rm obs}^{\dagger}L_{\rm obs},\rho} =2\lambda_{\rm obs}(t)\Big(A\rho A-\frac12{A^{2},\rho}\Big) =-\lambda_{\rm obs}(t)[A,[A,\rho]] . ]

Therefore the extra term respects complete positivity provided

[ \lambda_{\rm obs}(t)\;\ge\;0\quad\text{for all }t . ]

If one allowed negative (\lambda_{\rm obs}) the map would cease to be CPTP and could generate unphysical states (e.g. negative eigenvalues).

2.4 No energy exchange (pure dephasing)

If ([A,H_S]=0) the extra term commutes with the Hamiltonian part, so the system’s average energy (\langle H_S\rangle) is unchanged. The dynamics is then a pure dephasing channel: populations in the eigenbasis of (A) stay constant while off‑diagonal coherences decay as

[ \rho_{mn}(t)=\rho_{mn}(0)\, \exp!\Big[-i\omega_{mn}t-\;2\lambda_{\rm obs}(t)\, (a_m-a_n)^2\Big], ]

where (a_m) are eigenvalues of (A) and (\omega_{mn}=(E_m-E_n)/\hbar).
Thus the “information‑only’’ nature claimed by the student is precisely what standard dephasing (phase‑damping) channels describe.

3. Connection to existing theory

Concept How it appears in the literature Relation to the proposed term
Continuous (weak) measurement Quantum‑trajectory theory, stochastic master equations (e.g. Wiseman & Milburn, Quantum Measurement and Control). The deterministic part of the stochastic master equation is exactly Eq. (2).
Quantum‑Zeno effect Frequent (projective) measurements slow down the unitary evolution; in the continuous‑measurement limit the effective decay rate of coherences is (\propto\lambda_{\rm obs}). Larger (\lambda_{\rm obs}) → faster dephasing → slower coherent dynamics, the hallmark of the Zeno regime.
Phase‑damping (dephasing) channel One‑qubit Lindblad with (L=\sqrt{\gamma}\,\sigma_z); master equation (\dot\rho=-\frac{\gamma}{2}[\sigma_z,[\sigma_z,\rho]]). Exact special case with (A=\sigma_z) and (\lambda_{\rm obs}=\gamma/2).
Measurement‑induced decoherence Often discussed in cavity‑QED, optomechanics, and solid‑state qubits where a detector (e.g. a quantum point contact) continuously monitors charge or spin. Same master‑equation structure; (\lambda_{\rm obs}) is proportional to the detector’s shot‑noise power.
Quantum‑filtering / stochastic master equation The stochastic term (\propto dW(t)) (Wiener increment) adds information gain; the deterministic double‑commutator is the associated measurement back‑action. The deterministic term is the one the student wrote; the stochastic term was omitted because they assumed “no record is kept”.

Hence the proposed formulation already exists; the name most commonly used is continuous‑measurement‑induced dephasing or simply measurement‑back‑action.

4. Physical consequences & testable predictions

  1. Decoherence rate increases with observation strength
    For a two‑level system with (A=\sigma_z) the off‑diagonal element obeys
    [ \rho_{01}(t)=\rho_{01}(0)\,e^{-i\omega t}\,e^{-4\lambda_{\rm obs}(t)} . ] If the detector is turned on at (t=0) and (\lambda_{\rm obs}= \lambda) (constant) the coherence decays with time constant (1/(4\lambda)).

  2. Quantum‑Zeno suppression of transitions
    Suppose (H_S = \frac{\hbar\Omega}{2}\sigma_x) (induces Rabi oscillations).
    Adding the dephasing term yields the Bloch‑equation for the population (z(t)=\langle\sigma_z\rangle): [ \dot z = -\Omega y,\qquad \dot y = \Omega z - 4\lambda_{\rm obs} y . ] In the limit (4\lambda_{\rm obs}\gg\Omega) the transverse component (y) is damped so quickly that (z) barely changes – the Zeno freezing of the dynamics.

  3. Energy‑conserving nature
    If ([A,H_S]=0) then (\mathrm{Tr}(H_S\dot\rho)=0). An interferometer that monitors which‑path information (i.e. measures the path operator) therefore reduces fringe visibility without heating the particle.

These effects have been demonstrated experimentally many times, e.g. with superconducting qubits monitored by a linear resonator (see Siddiqi et al., Phys. Rev. Lett. 2004), with quantum dots measured by a quantum point contact (see Gurvitz, Phys. Rev. B 1997), and with trapped ions using weak fluorescence detection.

5. Final answer

  • The extra term (-\lambda_{\rm obs}(t)[L_{\rm obs},[L_{\rm obs},\rho]]) is mathematically sound provided (\lambda_{\rm obs}(t)\ge 0). It can be rewritten as a standard Lindblad dissipator with a single Hermitian Lindblad operator (L_{\rm obs}^{\prime}= \sqrt{2\lambda_{\rm obs}(t)}\,L_{\rm obs}).

  • This term is already known: it is the deterministic part of the master equation that describes a continuous (weak) measurement of the observable (L_{\rm obs}). In quantum‑optics and solid‑state literature it appears under the names measurement‑induced dephasing, phase‑damping channel, or observation‑back‑action.

*

Original question: Can "observation force" be formulated as a continuous decoherence channel? on Physics Stack Exchange, licensed CC BY-SA.

More Physics Problems